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Memristor‐based disturbance rejection control for port‐Hamiltonian systems with locally fixed‐time convergence

Memristor‐based disturbance rejection control for port‐Hamiltonian systems with locally... INTRODUCTIONBased on the mechanics formulation introduced by Sir W. R. Hamilton in the 19th century, the port‐Hamiltonian (PH) frameworks derive from network modeling of physical systems in various of domains such as electrics, mechanics, fluids, and electromechanical systems or thermal systems [1–3]. The Hamiltonian function, which is usually the sum of potential energy and kinetic energy in physical systems, is a good Lyapunov function candidate for many physical systems. Due to the above reasons and its nice structural properties with clear physical meaning, the PH systems have been extensively used in practical control and stability analysis [4–6]. The interconnection and damping assignment passivity‐based control (IDA‐PBC) technology, which uses the passivity properties of the PH systems to ensure the convergence of the systems to the desired equilibrium point, has become a simple method to design passive‐based controller and solve the stability problem of PH systems [7–9].Practical PH engineering systems inevitably confront with various disturbances, which bring undesirable effects on the closed‐loop system, always lead to much difficulty for the controller design. Therefore, studies on the disturbance rejection for PH systems have attracted considerable attention. In recent years, several elegant methods have been proposed, such as adaptive control (AC) [10], integral control (IC) [11, 12], sliding mode control (SMC) [13], disturbance observer‐based control (DOBC) [14–16], and active disturbance rejection control (ADRC) [17], etc. However, these control methods have their own shortcomings or limitations. For example, the IC methods cannot effectively remove the effects caused by fast time‐varying disturbances, and the undesirable transient control performances are always unavoidable. SMC, DOBC, and ADRC approaches are usually not available for disturbed PH systems under the so‐called mismatched disturbance, and can not preserve the PH structure of closed‐loop resulting system. In addition, these methods, which are characterized by observer‐based, disturbances estimating and compensating, increase the difficulty of stability analysis and physical explanation.Memristor, which refers to a resistor with memory, was fist postulated the existence of a new basic electrical circuit element by Chua in the early 1970s [18]. A physical passive two‐terminal memristive prototype was realized by scientists of Hewlett‐Packard (HP) Laboratories [19] until May 1, 2008. The authors in [20] pointed out that the memristor and the memristive systems can be described by using the PH form. The inclusion of memristive systems in the PH framework turns out to be almost as straightforward as the inclusion of resistive elements [21]. Adding memristive systems to the existing PH formalism, together with the ‘learning ability’ of the memristors, possibly provides a new method for controller synthesis and design [22] owing to the special characteristic of memristors. The authors in [23] presented the use of the memristor as a new element for designing passivity‐based controllers, and have found some possible benefits (e.g., ensuring better transient response) against the external disturbances. In [24], the authors observed similar numerical results. The memristor has also been used as an analog gain control element for robust‐adaptive control of miniaturized systems [25]. The memristor‐based control can be a promising method againsts disturbances and preserves the PH structure of systems. However, in the references [23–25], theoretical analysis is not sufficient to provide a reasonable interpretation that why the memristor‐based controller can work disturbance rejection.For nonlinear systems in practical applications, the convergence speed is an important performance index. Fixed‐time stability [26] is usually used to evaluating the convergence speed of dynamic systems. Unlike the finite‐time stability [27–30], where the upper bound of the corresponding settling time relies on the initial conditions, fixed‐time stability means that the system states can achieve desired equilibrium in finite‐time and the settling time is bounded by a constant independently of initial values. Fixed‐time control have been proved to be a powerful tool in dealing with disturbances and uncertainties. Because a fixed‐time controller possesses not only faster convergence speed but also better robustness and disturbance rejection performance. Fixed‐time control for dynamical systems have received more and more attention and some elegant studies have been done in a series of works [31–37]. For PH framework, the problems of fixed‐time stability analysis and fixed‐time control have been solved in references [38] and [39]. In [38], the locally fixed‐time stability of dynamical systems is first distinguished from the globally fixed‐time stability. Neither the reference [38] nor the reference [39] has involved using memristor to suppress external disturbance.Motivated by the previous studies, a framework of memristor‐based locally fixed‐time disturbance rejection control for PH systems is developed via IDA‐PBC technique. Two novel memristor‐based controllers are designed to accomplish this task. Fist, a memristor‐based locally fixed‐time controller is designed to suppress the external disturbance of PH systems. The settling‐times related to the memristor‐based controller in locally fixed‐time stabilization for the same PH system are shorter, thus the convergence speed are faster than the classical IDA‐PBC technique without memristive control action. Second, in view of the H∞$H_{\infty }$ performance is also significant in terms of system disturbance rejection performance analysis, the memristor‐based locally fixed‐time H∞$H_{\infty }$ controller is also designed. Compared with the existing results, the present paper has the following new highlights:(i)Compared with the existing disturbance rejection schemes [10–17], the memritor‐based method does not sacrifice its nominal control performance. The disturbance rejection is supplied by a state‐dependent memristor, which is almost as straightforward as the inclusion of resistive elements to the original PH system. The memritor‐based control has nothing to do with object disturbances, no need for system on‐line identification and adjustment, simple physical implementation and so on. The method proposed can not only preserve the PH structure and clear physical meaning of the closed‐loop resulting system, but also can obtain a better control performance, that is, the settling‐time related to the memristor‐based controllers in stabilizing the same PH systems are shorter, and the memristor‐based controllers can make the oscillation in the case of the strong periodic disturbance be better suppressed. In addition, the memritor‐based control can compatible with other disturnace rejection control methods (e.g., IC, DOBC and ADRC) via the IDA‐PBC technique, to further improve the disturbance suppression performance of PH systems.(ii)Although the fixed‐time stability analysis and fixed‐time control of PH systems have been addressed in references [38, 39], all of them have not involved using memristor to suppress external disturbance. The present study focuses on the locally fixed‐time stability of PH systems due to the disturbance rejection memristor only works in a neighborhood of the desired equilibrium point.(iii)Although the authors in the references [23] and [24] have observed some possible benefits against the external disturbances via memristive IDA‐PBC method, their theoretical analysis is not sufficient to provide a reasonable interpretation that why the memristor‐based controller can work disturbance rejection. They have not considered the locally fixed‐time stability of systems. In this work, the locally fixed‐time stability of PH system is used to quantitatively describe the convergence speed. The shorter is the settling time, the faster is the convergence speed, and thus the better is the disturbance rejection. The memristor‐based controller possesses good disturbance rejection owing to it can make the system states in a neighbourhood accelerate convergence to the desired equilibrium point.Therefore, the proposed approach has application potential in situation where robust and adaptive control is needed in the presence of external disturbance. Two numerical examples are presented to show the effectiveness of the memristor‐based controllers designed and the theoretical analysis.The remainder of the article is organized as follows. In Section 2, we briefly review some preliminary results on locally fixed‐time stability which will be used in the following discussion, incorporate memristor into PH systems and present the problem statement of the memristor‐based locally fixed‐time H∞$H_{\infty }$ control. In Section 3, we analyze theoretically the memristor‐based locally fixed‐time stabilization and H∞$H_{\infty }$ control problems of PH systems via IDA‐PBC method, and discuss the disturbance rejection performance of the memristor‐based controllers proposed. Two illustrative examples are presented in Section 4. Conclusions and discussions are given in Section 5.Notations: The set of real numbers is denoted by R$\mathbb {R}$. The set of positive real numbers is denoted by R+$\mathbb {R}_{+}$. For any matrix A∈Rn×n$A\in \mathbb {R}^{n\times n}$, we denote the positive definiteness (respectively, positive semi‐definiteness) of A by A≻0$A\succ 0$ (respectively, A⪰0$A\succeq 0$ ). Let AT$A^{T}$ denote the transpose of a matrix A∈Rn×n$A\in \mathbb {R}^{n\times n}$.PRELIMINARIES AND PROBLEM STATEMENTLocally fixed‐time stabilityIn this subsection, we briefly review preliminaries on locally fixed‐time stability which will be used in the following discussion.Consider a dynamical system1ẋ=f(x),f(0)=0,x(t0)=x0,x∈U⊆Rn,\begin{equation} \dot{x}=f(x), f(0)=0, x(t_0)=x_0, x\in U\subseteq \mathbb {R}^n, \end{equation}where f:U↦Rn$f:U\mapsto \mathbb {R}^n$ is continuous on an open neighborhood U of the origin such that the solution to system (1) is unique in the forward time.1Definition[39] The null solution O of system (1) is said to be locally fixed‐time stable if it is locally finite‐time stable in a round neighborhood with the centrality O and the radius δ, Uδ=:U(O,δ),δ>0$U_\delta =:U(O,\delta ),\delta &gt;0$, and the settling‐time function T(x0)$T(x_0)$ is bounded by a positive number, that is, ∃Tmax>0,s.t.,T(x0)≤Tmax,∀x0∈Uδ$\exists T_{max}&gt;0, s.t., T(x_0)\le T_{max}, \forall x_0\in U_\delta$.2DefinitionA nonlinear control system2ẋ=f(x)+g(x)u,x∈Rn,u∈Rm,\begin{equation} \dot{x}=f(x)+g(x)u, \quad x\in \mathbb {R}^n, u\in \mathbb {R}^m, \end{equation}is called locally fixed‐time stabilizable via continuous state feedback, if there exists a continuous feedback law u=u(x)$u=u(x)$ such that the equilibrium of the closed‐loop system is locally fixed‐time stable, where g(x):Rn→Rn×m$g(x):\mathbb {R}^{n}\rightarrow \mathbb {R}^{n\times m}$ is the control gain matrix and is assumed full column rank.1Lemma(Jensen's inequality [29]).3∑i=1n|xi|a21a2≤∑i=1n|xi|a11a1,0<a1<a2,\begin{equation} {\left(\sum \limits _{i=1}^{n} |x_i|^{a_2}\right)}^{\frac{1}{a_2}}\le {\left(\sum \limits _{i=1}^{n} |x_i|^{a_1}\right)}^{\frac{1}{a_1}}, \quad 0&lt;a_1&lt;a_2, \end{equation}where a1,a2$a_1,a_2$ and xi,i=1,2,…,n$x_i,i=1,2,\ldots,n$ are all real numbers.2Lemma[27] Consider a dynamical system (1) If there exist a real number β>1$\beta &gt; 1$ and a C1 radially unbounded Lyapunov function V(x)$V (x)$ (i.e., V(x)$V(x)$ have continuous first‐order partial derivative and V(x)→∞$V(x)\rightarrow \infty$ as ∥x∥→∞$\Vert x\Vert \rightarrow \infty$, where ∥·∥$\Vert \cdot \Vert$ is the Euclidean norm) of the system such that4V̇≤−kV1β(x(t)),k>0,\begin{eqnarray} \dot{V}\le -kV^{\frac{1}{\beta }}(x(t)), \quad k&gt;0, \end{eqnarray}holds along the trajectories of the system (1) starting from any x0∈Rn$x_0 \in \mathbb {R}^n$, then the origin is a global finite‐time stable equilibrium of system (1). Furthermore, the settling time of system (1) with respect to x0 satisfies5T(x0)≤t0+βk(β−1)Vβ−1β(x0),∀x0∈Rn.\begin{equation} T(x_0) \le t_0+\frac{\beta }{k(\beta -1)}V^{\frac{\beta -1}{\beta }}(x_0),\quad \forall x_0\in \mathbb {R}^n. \end{equation}Incorporating memristor into PH systemsThe input‐state‐output port controlled‐Hamiltonian system is6ΣP:ẋ=J(x)−R(x)∂H∂x(x)+Gu,y=GT∂H∂x(x),\begin{eqnarray} \Sigma _{P}: {\left\lbrace \def\eqcellsep{&}\begin{array}{ll} \dot{x}={\left(J(x)-R(x)\right)}\dfrac{\partial H}{\partial x}(x)+Gu,\\[11pt] y=G^{T}\dfrac{\partial H}{\partial x}(x), \end{array} \right.} \end{eqnarray}where x∈Rn$x\in \mathbb {R}^{n}$ is the state vector, u∈Rm,(m≤n)$u\in \mathbb {R}^{m}, (m\le n)$ is the control input. H(x):Rn→R$H(x):\mathbb {R}^{n}\rightarrow \mathbb {R}$ is the Hamiltonian , J(x):Rn→Rn×n,R(x):Rn→Rn×n$J(x) : \mathbb {R}^{n}\rightarrow \mathbb {R}^{n\times n},R(x):\mathbb {R}^{n}\rightarrow \mathbb {R}^{n\times n}$, with J(x)=−JT(x)$J(x)=-J^{T}(x)$ and R(x)=RT(x)⪰0$R(x)=R^{T}(x)\succeq 0$, are the natural interconnection and damping matrices, respectively, and G(x):Rn→Rn×m$G(x):\mathbb {R}^{n}\rightarrow \mathbb {R}^{n\times m}$ is the gain matrix and is full column rank.Before the effect of memristive systems can be studied in the PH framework, we first need to extend the PH formalism by adding a memristive port, with port variables (fM,eM)∈FM×EM$(f_M,e_M)\in \mathcal {F}_M \times \mathcal {E}_M$, to the Dirac structure [20, 21]. Assuming that the memristive port can be described by an xf$x_f$‐controlled constitutive relationship (i.e., the memristance of memristor depending on state variable xf$x_f$), we can define the memristive Dirac structure U$\mathcal {U}$ as7U={(fM,eM)∈FM×EM|ẋf−fM=0,eM−Mf(xf)fM=0},\begin{equation} \def\eqcellsep{&}\begin{array}{ll} \mathcal {U}= \bigg \lbrace (f_M,&e_M)\in \mathcal {F}_{M}\times \mathcal {E}_{M}\bigg |\\[3pt] &\dot{x}_f-f_M=0,e_M-M_f (x_f)f_M=0 \bigg \rbrace , \end{array} \end{equation}where Mf(xf)$M_f(x_f)$ is an incremental memristance matrix (or memductance matrix if the admittance form is adopted) with appropriate dimension. Then the dynamics on U$\mathcal {U}$ derive the form8ẋf=fM,eM=∂HM∂xf(xf)+Mf(xf)fM,\begin{eqnarray} {\left\lbrace \def\eqcellsep{&}\begin{array}{ll} \dot{x}_f= f_M,\\[11pt] e_M=\dfrac{\partial H_M}{\partial x_f}(x_f)+M_f(x_f)f_M, \end{array} \right.} \end{eqnarray}where the memristive port variables fM$f_M$ and eM$e_M$ can be considered as the inputs uM$u_M$ and outputs yM$y_M$, respectively. It gives9ΣM:ẋf=uM,yM=∂HM∂xf(xf)+Mf(xf)uM,\begin{eqnarray} \Sigma _{M}: {\left\lbrace \def\eqcellsep{&}\begin{array}{ll} \dot{x}_f= u_M,\\[15pt] y_M= \dfrac{\partial H_M}{\partial x_f}(x_f)+M_f(x_f)u_M, \end{array} \right.} \end{eqnarray}where HM$H_M$ represents the stored energy in the memristor. Notice that, because of the ‘no energy discharge property’, the energy stored in the memristor is defined by a null‐Hamiltonian, HM(xf)≡0$H_M(x_f) \equiv 0$.Differentiating HM$H_M$ with respect to time and making use of (10) give10ḢM=∂HM∂xfẋf=yMuM−Mf(xf)uM2≡0.\begin{eqnarray} \dot{H}_M=\frac{\partial H_M}{\partial x_f}\dot{x}_f =y_Mu_M-M_f(x_f)u^{2}_M\equiv 0. \end{eqnarray}Hence, a memristor is passive if and only if its memristance is positive semi‐definite, that is, Mf(xf)⪰0$M_f(x_f)\succeq 0$.Using the modularity property of the PH systems (i.e., two different PH systems can be interconnected to constitute a new PH system), the memristor system (10) can be easily interconnected with a PH system in the form (6). The feedback interconnection described in Figure 1, y=uM,u=−yM$y = u_M, u=-y_M$, yields another memristive PH system11ẋẋf=J−R−GMfGT−GGT0∂H∂x∂HM∂xf+G∼0u∼.\begin{equation} \def\eqcellsep{&}\begin{array}{ll}{\left( \def\eqcellsep{&}\begin{array}{c} \dot{x}\\[3pt] \dot{x}_f \end{array} \right)}= {\left( \def\eqcellsep{&}\begin{array}{c c}J-R-GM_fG^{T} & -G \\[5pt] G^{T} & 0 \end{array} \right)} {\left( \def\eqcellsep{&}\begin{array}{c} \dfrac{\partial H}{\partial x}\\[15pt] \dfrac{\partial H_M}{\partial x_f} \end{array} \right)} \\[3pt] \qquad\qquad +\,{\left( \def\eqcellsep{&}\begin{array}{c}\tilde{G}\\[3pt] 0 \end{array} \right)}\tilde{u}. \end{array} \end{equation}1FIGUREAdding memristor to PH systemsThe new output for the system is naturally defined by12y∼=G∼T∂H∂x(x).\begin{equation} \tilde{y}=\tilde{G}^{T}\frac{\partial H}{\partial x}(x). \end{equation}The power‐balance inequality associated to Equations (11) and (12) takes the form13Ḣ(x)+ḢM(xf)=y∼Tu∼(x)−∂H∂x(x)T(RM(x)+R(x))∂H∂x(x)≤y∼Tu∼(x),\begin{equation} \def\eqcellsep{&}\begin{array}{lll} \dot{H}(x)+\dot{H}_M(x_f)=\tilde{y}^{T}\tilde{u}(x)-\left(\frac{\partial H}{\partial x}(x)\right)^{T}(R_M(x)\\[11pt] \qquad\qquad\qquad\qquad +\,R(x))\dfrac{\partial H}{\partial x}(x)\le \tilde{y}^{T}\tilde{u}(x), \end{array} \end{equation}where u∼,y∼$\tilde{u}, \tilde{y}$ and G∼$\tilde{G}$ are respectively new input, output and gain matrix of the resulting PH system memristor added, RM≜GMf(xf)GT≻0$R_M\triangleq GM_f(x_f)G^{T}\succ 0$, since Mf(xf)≻0$M_f(x_f)\succ 0$, for all xf$x_f$. Hence, if the Hamiltonian function H(x)$H(x)$ is bounded from below, then the system is passive with respect to the supply rate y∼Tu∼$\tilde{y}^{T}\tilde{u}$.Memristor‐based IDA‐PBC techniqueThe IDA‐PBC technique, introduced in [7], has emerged as an easy methodology for designing passivity‐based controllers [8] and exploits the stability properties of PH system. It uses the passivity properties of the PH system to ensure the convergence of the system to the desired equilibrium point.The memristor system can be included to improve the classical IDA‐PBC method, the key idea is to define a desired target dynamic with the form14ẋẋf=Jd−Rd−GMfGT−GGT0∂Hd∂x∂HM∂xf,\begin{equation} {\left( \def\eqcellsep{&}\begin{array}{c}\dot{x}\\[3pt] \dot{x}_f \end{array} \right)}= {\left( \def\eqcellsep{&}\begin{array}{cc} J_d-R_d-GM_fG^{T} & -G \\[5pt] G^{T} & 0 \end{array} \right)} {\left( \def\eqcellsep{&}\begin{array}{c} \dfrac{\partial H_d}{\partial x}\\[15pt] \dfrac{\partial H_M}{\partial x_f} \end{array} \right)}, \end{equation}where Jd,Rd$J_d, R_d$ and Hd$H_d$ can be obtained with the classical IDA‐PBC approach, Md$M_d$ is the incremental memristance matrix of the added memristor and ∂HM/∂xf≡0$\partial H_M/\partial x_f\equiv 0$.Designing a method to stabilize the system (6) consists in finding a control law u such that (6) behaves as (14). In that case, the design procedure reduces to find matrices Jd(x),Rd(x)$J_d(x),R_d(x)$ and Mf(x)$M_f(x)$ and a desired Hamiltonian function, Hd(x)$H_d(x)$, such that the so‐called matching equation15(J−R)∂H∂x+Gu=(Jd−Rd−GMfGT)∂Hd∂x,\begin{equation} (J-R)\frac{\partial H}{\partial x}+Gu=(J_d-R_d-GM_fG^{T})\frac{\partial H_d}{\partial x}, \end{equation}is solved with the result that the controller becomes16ucm=uc+um=(GTG)−1GT(Jd−Rd)∂Hd∂x−(J−R)∂H∂x+(−MfGT)∂Hd∂x,\begin{eqnarray} u_{cm} &=&u_c+u_m\nonumber\\ &=& (G^{T}G)^{-1}G^{T}{\left((J_d-R_d)\frac{\partial H_d}{\partial x} -(J-R)\frac{\partial H}{\partial x}\right)}\\ &&+\,(-M_fG^{T})\frac{\partial H_d}{\partial x}, \nonumber\qquad \end{eqnarray}where uc$u_c$ corresponds to the controller design via the classical IDA‐PBC, and um$u_m$ includes the memristor dynamic. Let RM=GMf(x)GT≻0$R_M= GM_f (x)G^{T}\succ 0$, we can obtain the control laws17ucm=G+(Jd−Rd−RM)∂Hd∂x−(J−R)∂H∂x,\begin{equation} u_{cm}=G^{+}{\left((J_d-R_d-R_M)\frac{\partial H_d}{\partial x}-(J-R)\frac{\partial H}{\partial x}\right)}, \end{equation}and18uc=G+(Jd−Rd)∂Hd∂x−(J−R)∂H∂x,\begin{equation} u_c=G^{+}{\left((J_d-R_d)\frac{\partial H_d}{\partial x}-(J-R)\frac{\partial H}{\partial x}\right)}, \end{equation}19um=−MfGT∂Hd∂x,\begin{equation} u_m=-M_fG^{T}\frac{\partial H_d}{\partial x}, \end{equation}where G+≜(GTG)−1GT$G^{+}\triangleq (G^{T}G)^{-1}G^{T}$ is the Moore‐Penrose pseudo‐inverse of the matrix G(x)$G(x)$.1RemarkNote that, the matching equation corresponding to the classical IDA‐PBC is(J−R)∂H∂x+Gu=(Jd−Rd)∂Hd∂x,\begin{equation*} (J-R)\frac{\partial H}{\partial x}+Gu=(J_d-R_d)\frac{\partial H_d}{\partial x}, \end{equation*}which is just missing the third item on the right hand compared with the equation (15).Locally fixed‐time H∞$H_{\infty }$ controlConsider the following general nonlinear system20ẋ=f(x,u,ω)z=h(x,u,ω),\begin{equation} {\left\lbrace \def\eqcellsep{&}\begin{array}{ll} \dot{x }= f (x, u, \omega )\\[3pt] z = h(x,u,\omega ) \end{array} \right.}, \end{equation}where z∈Rq$z \in \mathbb {R}^q$ and ω∈Rs$\omega \in \mathbb {R}^s$ are the penalty signal and disturbance of the system, respectively; x,u$x, u$ are the same as those in the system (1), by considering the pre‐HamiltonianKγ(x,H,ω,u):=∂HT(x)∂xf(x,u,ω)−12γ2∥ω∥2+12∥z∥2.\begin{equation*} K_{\gamma }(x, H, \omega , u) := \frac{\partial H^T(x)}{\partial x}f(x, u, \omega )-\frac{1}{2}\gamma ^2\Vert \omega \Vert ^2 +\frac{1}{2}\Vert z\Vert ^2. \end{equation*}Suppose that Kγ$K_{\gamma }$ has a saddle‐point (u∗(x,H),ω∗(x,H))$(u^{*}(x, H),\omega ^{*}(x, H))$, that is, for all u,ω$u,\omega$,Kγ(x,H,ω(x,H),u∗(x,H))≤Kγ(x,H,ω∗(x,H),u∗(x,H))≤Kγ(x,H,ω∗(x,H),u(x,H)),\begin{equation*} \def\eqcellsep{&}\begin{array}{ll} &K_{\gamma }(x, H, \omega (x, H), u^{*}(x, H)) \\[5pt] &\quad\le\, K_{\gamma }(x, H, \omega ^{*}(x, H), u^{*}(x, H)) \\[5pt] &\quad \le\, K_{\gamma }(x, H, \omega ^{*}(x, H), u(x, H)), \end{array} \end{equation*}then we consider the Hamilton‐Jacobi inequality21(HJ1)Kγ(x,H,ω∗(x,H),u∗(x,H))≤0.\begin{equation} \mathbf {(H J1)}K_{\gamma }(x, H, \omega ^{*}(x, H), u^{*}(x, H))\le 0. \end{equation}3Definition[40] Consider the input signal space Rm$\mathbb {R}^m$ and the output signal space Rp$\mathbb {R}^p$, together with an input‐output mapping G:Rm↦Rp,u→y=G(u)$\mathcal {G}:\mathbb {R}^m\mapsto \mathbb {R}^p, u\rightarrow y=\mathcal {G}(u)$. The map G(u)$\mathcal {G}(u)$ is said to have finite L2‐gain if there exist finite constants γ and b such that∥G(u)∥≤γ∥u∥+b,\begin{equation*} \Vert \mathcal {G}(u)\Vert \le \gamma \Vert u\Vert +b, \end{equation*}G$\mathcal {G}$ is said to have finite L2‐gain with zero bias if b can be taken equal to zero, here norm ∥·∥$\Vert \cdot \Vert$ is the L2 norm. Let G$\mathcal {G}$ have finite L2$\mathcal {L}_2$‐gain. Then the L2‐gain of G$\mathcal {G}$ is defined asγ(G):=inf{γ|∃b,satisfies∥G(u)∥≤γ∥u∥+b}.\begin{equation*} \gamma (\mathcal {G}):=\inf {\lbrace \gamma |\exists b, satisfies\quad \Vert \mathcal {G}(u)\Vert \le \gamma \Vert u\Vert +b \rbrace }. \end{equation*}3Lemma(Proposition 6.1.3 of [40]) Let γ>0$\gamma &gt; 0$. Assume there exists saddle point (u∗(x,H),ω∗(x,H))$(u^{*}(x, H), \omega ^{*}(x, H))$ of Kγ(x,H,ω,u)$K_{\gamma }(x, H, \omega , u)$. Suppose there exists a Cr(k>r>1)$C^r (k &gt; r &gt; 1)$ solution H>0$H&gt; 0$ to the Hamilton‐Jacobin inequality (HJ1)$\mathbf {(H J1)}$ given by (21). Then the Cr−1$C^{r-1}$ state feedbacku=u∗x,∂HT(x)∂x,\begin{equation*} u = u^{*}{\left(x, \frac{\partial H^T(x)}{\partial x}\right)}, \end{equation*}is such that the closed‐loop systemẋ=fx,u∗x,∂HT∂x(x),ωz=hx,u∗x,∂HT(x)∂x,ω,\begin{equation*} {\left\lbrace \def\eqcellsep{&}\begin{array}{ll} \dot{x }= f{\left(x, u^{*}{\left(x, \dfrac{\partial H^T}{\partial x}(x)\right)}, \omega \right)}\\[11pt] z = h{\left(x,u^{*}{\left(x, \dfrac{\partial H^T(x)}{\partial x}\right)},\omega \right)} \end{array} , \right.} \end{equation*}has L2‐gain <γ$&lt; \gamma$.Consider the following PH system22ẋ=(J(x)−R(x))∂H∂x(x)+G(x)u+G1(x)ωz=M(x)GT(x)∂Hd∂x(x),\begin{equation} {\left\lbrace \def\eqcellsep{&}\begin{array}{ll} \dot{x}= (J(x)-R(x))\dfrac{\partial H}{\partial x}(x)+G(x)u+G_1(x)\omega \\[11pt] z= M(x)G^{T}(x)\dfrac{\partial H_d}{\partial x}(x) \end{array} \right.}, \end{equation}where x∈Uδ⊂Rn,H∈Rn,u∈Rm,G∈Rn×m,J$x\in U_{\delta }\subset \mathbb {R}^{n}, H\in \mathbb {R}^{n}, u\in \mathbb {R}^{ m},G\in \mathbb {R}^{n\times m},J$ and R are the same as those in system (6), ω∈Rq$\omega \in \mathbb {R}^{q}$ is the disturbance in L2, G1∈Rn×q$G_1\in \mathbb {R}^{n\times q}$ is the disturbance gain, z is the system's penalty function and M(x)∈Rm×m$M(x)\in \mathbb {R}^{m\times m}$ is a weighting matrix.4DefinitionThe locally fixed‐time H∞$H_{\infty }$ control problem of system (22) can be described as follows. For a given disturbance attenuation level γ>0$\gamma &gt; 0$, design a feedback controller23ucr(x)=uc(x)+ur(x)=uc−12MT(x)M(x)+12γ2ImGT(x)∂Hd∂x(x),\begin{eqnarray} \def\eqcellsep{&}\begin{array}{ll} u_{cr}(x)=u_c (x)+u_r(x)\\[11pt] \qquad \ \ =\,u_c-\left[\dfrac{1}{2}M^{T}(x)M(x)+\dfrac{1}{2 \gamma ^{2}} I_m\right]G^T(x) \dfrac{\partial H_{d}}{\partial x}(x), \end{array} \nonumber\\ \end{eqnarray}where uc$u_c$ is given as (18), such that the L2‐gain of the closed‐loop system (from ω to z) is not larger than γ , and the closed‐loop system is locally fixed‐time stable when ω=0$\omega = 0$.How to use the memristor improving the convergence speed and rejection disturbance properties of the PH system (6)? In the following section, the details are presented.MEMRISTOR‐BASED LOCALLY FIXED‐TIME STABILIZATION CONTROL FOR PH SYSTEMSIn order to investigate the locally fixed‐time stability of the closed‐loop system (6) with controllers (17) and (18), the desired Hamiltonian function of the PH system (6) is taken as24Hd(x)=∑i=1n(xi2)α2α−1,α>1.\begin{equation} H_d(x)=\sum \limits _{i=1}^{n}(x_{i}^{2})^\frac{\alpha }{2\alpha -1},\quad \alpha &gt;1. \end{equation}4Lemma(Theorem 1 of [39]) Consider the system (6), if the desired Hamiltonian function Hd$H_d$ is given as (24), Uδ$U_{\delta }$ is a neighborhood of the origin, and25k0≜min1≤i≤ninfx∈Uδ{σiRd(x)}>0,\begin{equation} k_0\triangleq \min \limits _{1\le i \le n}\inf \limits _ {x\in U_{\delta }} \lbrace \sigma _{i}^{R_d(x)}\rbrace &gt;0, \end{equation}where σiRd(x)$\sigma _{i}^{R_d(x)}$ is the eigenvalue of Rd(x)$R_d(x)$.Furthermore, if there exists a scalar C≜supx∈UδHd(x)=supx∈Uδ∑i=1n(xi2(t))α2α−1$C\triangleq \sup \nolimits _{x\in U_{\delta }}H_d(x)=\sup \nolimits _{x\in U_{\delta }}\sum \nolimits _{i=1}^{n}(x^{2}_{i}(t))^\frac{\alpha }{2\alpha -1}$ and 0<C<+∞$0&lt; C&lt;+\infty$, then the origin of system (6) is locally fixed‐time stabilized by classical IDA‐PBC controller26uc=G+(Jd−Rd)∂Hd∂x−(J−R)∂H∂x,\begin{equation} u_c=G^{+}{\left((J_d-R_d)\frac{\partial H_d}{\partial x}-(J-R)\frac{\partial H}{\partial x}\right)}, \end{equation}and the upper bound of the settling times can be estimated as27T0≜(2α−1)24k0α(α−1)cα−1α,\begin{equation} T_{0} \triangleq \frac{(2\alpha -1)^2}{4k_0 \alpha (\alpha -1)}c^{\frac{\alpha -1}{\alpha }}, \end{equation}which is independently of initial conditions.1TheoremIf Jd,Rd$J_d, R_d$ can be obtained with the classical IDA‐PBC approach, the desired Hamiltonian function Hd$H_d$ is given as (24), Uδ,k0$U_{\delta }, k_0$ and C are the same as those in Lemma 4, then the origin of the closed‐loop system (6) can be locally fixed‐time stabilized by memristor‐based IDA‐PBC controller28ucm=G+(Jd−Rd−RM)∂Hd∂x−(J−R)∂H∂x,\begin{equation} u_{cm}=G^{+}{\left((J_d-R_d-R_M)\frac{\partial H_d}{\partial x}-(J-R)\frac{\partial H}{\partial x}\right)}, \end{equation}and the settling time satisfies29T(x0)≤(2α−1)24k(x)α(α−1)Hdα−1α≤Tm≜(2α−1)24kmα(α−1)Cα−1α,∀x0∈Uδ,\begin{equation} \def\eqcellsep{&}\begin{array}{ll} T(x_0)\le \dfrac{(2\alpha -1)^2}{4k(x) \alpha (\alpha -1)}H_d^{\frac{\alpha -1}{\alpha }} \\[15pt] \qquad \ \ \le T_{m}\triangleq \dfrac{(2\alpha -1)^2}{4k_m \alpha (\alpha -1)}C^{\frac{\alpha -1}{\alpha }},\forall x_0\in U_{\delta }, \end{array} \end{equation}where k(x)≜k0+min1≤i≤n{σiRM(x)}$k(x)\triangleq k_0+\min \nolimits _{1\le i \le n}\lbrace \sigma _{i}^{R_M(x)}\rbrace$, σiRM(x)$\sigma _{i}^{R_M(x)}$ denotes the eigenvalue of RM(x)$R_M(x)$, RM=GMf(x)GT≻0$R_M= GM_f (x)G^{T}\succ 0$, Mf(x)$M_f (x)$ is the memristance, and km=infx∈Uδk(x)$k_m=\inf \nolimits _ {x\in U_{\delta }}k(x)$.ProofSince Hd(x)$H_d(x)$ can be a Lyapunov function candidate. Consider the following desired systems:30ẋ=Jd(x)−Rd(x)−RM(x)∂Hd(x)∂x,\begin{equation} \dot{x}={\left(J_d(x)-R_d(x)-R_M(x)\right)}\frac{\partial H_d(x)}{\partial x}, \end{equation}which is the equivalent equations for the state equations of PH system (6) corresponding to controller (28). For any x0∈Uδ$x_0\in U_{\delta }$, let x(t)≜x(t;0,x0)$x(t)\triangleq x(t;0,x_0)$ be the trajectory of system (30) starting from x0.Notice the skew‐symmetry of the matrix Jd(x)$J_d(x)$, by computing the derivative of Hd(x)$H_d(x)$ along the trajectory of (30), the following inequality can be obtainedḢd(x)=−∂HdT(x)∂xRd(x)+RM(x)∂Hd(x)∂x≤−k(x)∂HdT(x)∂x∂Hd(x)∂x=−k(x)2α2α−1(x12)1−α2α−1x1,…,2α2α−1(xn2)1−α2α−1xn×2α2α−1(x12)1−α2α−1x1,…,2α2α−1(xn2)1−α2α−1xnT=−k(x)(2α2α−1)2∑i=1n(xi2)12α−1.≤−k(x)(2α2α−1)2∑i=1n[(xi2)α2α−1]1α≤−k(x)(2α2α−1)2∑i=1n(xi2)α2α−11α=−k(x)(2α2α−1)2Hd1α,\begin{eqnarray*} \dot{H}_d(x) &=&-\frac{\partial H^{T}_d(x)}{\partial x}{\left(R_d(x)+R_M(x)\right)}\frac{\partial H_d(x)}{\partial x} \\ &\le& -k(x)\frac{\partial H^{T}_d(x)}{\partial x}\frac{\partial H_d(x)}{\partial x} \\ &=& -k(x){\left(\frac{2\alpha }{2\alpha -1}(x_1^2)^{\frac{1-\alpha }{2\alpha -1}}x_1,\ldots,\frac{2\alpha }{2\alpha -1}(x_n^2)^{\frac{1-\alpha }{2\alpha -1}}x_n\right)}\\ &&\times\, {\left(\frac{2\alpha }{2\alpha -1}(x_1^2)^{\frac{1-\alpha }{2\alpha -1}}x_1,\ldots,\frac{2\alpha }{2\alpha -1}(x_n^2)^{\frac{1-\alpha }{2\alpha -1}}x_n\right)}^T\\ &=& -k(x)(\frac{2\alpha }{2\alpha -1})^2 \sum \limits _{i=1}^{n}(x_i^2)^{\frac{1}{2\alpha -1}}. \\ &\le& -k(x)(\frac{2\alpha }{2\alpha -1})^2 \sum \limits _{i=1}^{n}[(x_i^2)^{\frac{\alpha }{2\alpha -1}}]^{\frac{1}{\alpha }}\\ &\le& -k(x)(\frac{2\alpha }{2\alpha -1})^2 {\left[\sum \limits _{i=1}^{n}(x_i^2)^{\frac{\alpha }{2\alpha -1}}\right]}^{\frac{1}{\alpha }}\\ &= &-k(x)(\frac{2\alpha }{2\alpha -1})^2 H_d^{\frac{1}{\alpha }}, \end{eqnarray*}notice that31k(x)=k0+min1≤i≤n{σiRM(x)},RM(x)≻0,\begin{equation} k(x)=k_0+\min \limits _{1\le i \le n}\lbrace \sigma _{i}^{R_M(x)}\rbrace , R_M(x)\succ 0, \end{equation}which implies32k(x)>k0>0.\begin{equation} k(x) &gt; k_0&gt;0. \end{equation}From Lemma 2 and Definition 1, it can be concluded that the origin is locally fixed‐time stable, and the settling time satisfies33T(x0)≤(2α−1)24k(x)α(α−1)Hdα−1α<Tm=(2α−1)24kmα(α−1)Cα−1α,∀x0∈Uδ.\begin{equation} \def\eqcellsep{&}\begin{array}{ll} T(x_0) \le \dfrac{(2\alpha -1)^2}{4k(x) \alpha (\alpha -1)}H_d^{\frac{\alpha -1}{\alpha }}\\[15pt] \qquad \ \ &lt; T_{m}= \dfrac{(2\alpha -1)^2}{4k_m \alpha (\alpha -1)}C^{\frac{\alpha -1}{\alpha }},\forall x_0\in U_{\delta }. \end{array} \end{equation}Therefore, the proof of Theorem 1 is completed.□$\Box$2RemarkNote that, in the proof of Theorem 1,k(x)=k0+min1≤i≤n{σiRM(x)},\begin{equation*} k(x)=k_0+\min \limits _{1\le i \le n}\lbrace \sigma _{i}^{R_M(x)}\rbrace , \end{equation*}andk(x)>k0>0,\begin{equation*} k(x) &gt;k_0&gt;0, \end{equation*}it can be obtained34T(x0)≤Tm=(2α−1)24kmα(α−1)Cα−1α<T0=(2α−1)24k0α(α−1)Cα−1α,∀x0∈Uδ.\begin{equation} \def\eqcellsep{&}\begin{array}{ll} T(x_0) \le T_m= \dfrac{(2\alpha -1)^2}{4k_m \alpha (\alpha -1)}C^{\frac{\alpha -1}{\alpha }} \\[15pt] \qquad \ \ &lt; T_{0}= \dfrac{(2\alpha -1)^2}{4k_0 \alpha (\alpha -1)}C^{\frac{\alpha -1}{\alpha }},\forall x_0\in U_{\delta }. \end{array} \end{equation}Because the settling time is larger (smaller), the energy decay speed of free moving system is slower (faster). Hence, the convergence speed of the memristor‐based controller ucm$u_{cm}$ in locally fixed‐time stabilizing PH system (6) is faster than that of the classical IDA‐PBC controller uc)$u_{c})$ in fixed‐time stabilizing PH system (6). Thus, the memristor‐based controller ucm$u_{cm}$ possesses better disturbance rejection performance than the classical IDA‐PBC controller uc)$u_{c})$.In (34), the settling time Tm$T_m$ is dependent on the value of k(x)=k0+min1≤i≤n{σiRM(x)}$k(x)=k_0+\min \nolimits _{1\le i \le n}\lbrace \sigma _{i}^{R_M(x)}\rbrace$, so it also depends on the memristance RM(x)=G(x)M(x)GT(x)$R_M(x)=G(x)M(x)G^T (x)$. In order to further enhance the disturbance rejection performance of the memristor‐based controller ucm$u_{cm}$, we impose on the following assumptions for the memristance function, see Figure 2.1Assumption(H1)$(\mathbf {H}_1)$The memristance function r(t)=m(ei(t))$r(t)=m(e_i(t))$ is even and continuous when ei(t)∈(−Q,Q)$e_i(t)\in (-Q,Q)$, Q∈R+$Q\in \mathbb {R}^{+}$, Roff>Ron>0$R_{off}&gt;R_{on}&gt;0$ andm(ei(t))=Ron,ei∼(t)≥Q,Roff,ei∼(t)=0.\begin{equation*} m(e_i(t))={\left\lbrace \def\eqcellsep{&}\begin{array}{ll} R_{on},\widetilde{e_i}(t)\ge Q,\\[7pt] R_{off},\widetilde{e_i}(t)= 0. \end{array} \right.} \end{equation*}(H2)$(\mathbf {H}_2)$The memristance function r(t)=m(ei∼(t))$r(t)=m(\widetilde{e_i}(t))$ is monotonically decreasing and its inverse function ei∼(t)=m−1(r(t))$\widetilde{e_i}(t)=m^{-1}(r(t))$ exists, as ei∼(t)∈(0,Q),i=1,…,n$\widetilde{e_i}(t)\in (0,Q),i=1,\ldots,n$,where e(t)=(e1(t),…,en(t))=((x1(t)−x1∗),…,(xn(t)−xn∗))=x(t)−x∗$e(t)=(e_1(t),\ldots,e_n(t))=((x_1(t)-{x}_{1}^{*}),\ldots,(x_n(t)-{x}_{n}^{*}))=x (t)-x^{*}$ is the state error, x is the state and x∗$x^{*}$ is the desired equilibrium state of system (6), and ei∼(t)=|ei(t)|,i=1,…,n$\widetilde{e_i}(t)=|e_i(t)|,i=1,\ldots,n$.3RemarkIn [23] and [24], the memristance functions are respectively designed asm1(ei(t))=ρe−σei(t),\begin{equation*} m_1(e_i(t))=\rho e^{-\sigma e_i(t)}, \end{equation*}andm2(ei(t))=Ron,|e(t)|≥QRon+(Roff−Ron)Q−|ei(t)|Q,|ei(t)|<Q,\begin{equation*} m_2(e_i(t))={\left\lbrace \def\eqcellsep{&}\begin{array}{ll} R_{on}, |e(t)|\ge Q\\[15pt] R_{on}+(R_{off}-R_{on})\dfrac{Q-|e_i(t)|}{Q}, |e_i(t)|&lt;Q, \end{array} \right.} \end{equation*}which obviously meet the Assumption 1, where ei=xi−xi∗$e_i=x_i-x_i^{*}$ is the error of partial state. We will adopt appropriate memristance function of memristor, such that the state‐dependent memristor should be designed as that the value of damping is low when the error of state is high, and then it drastically increases when the error is low.2TheoremConsider the PH system (6), x∗$x^{*}$ is an equilibrium point, and the desired Hamiltonian function Hd(x)=∑i=1n[(xi−xi∗)2]α2α−1,α>1$H_d(x)=\sum \nolimits _{i=1}^{n}[(x_{i}-x_{i}^{*})^{2}]^\frac{\alpha }{2\alpha -1}, \alpha &gt;1$, Uδ$U_{\delta }$ is a neighborhood of x∗$x^{*}$, and35k01=min1≤i≤ninfx∈Uδ{σiR(x)}>0,\begin{equation} k_{01}=\min \limits _{1\le i \le n}\inf \limits _ {x\in U_{\delta }} \lbrace \sigma _{i}^{R(x)}\rbrace &gt;0, \end{equation}36C1=supx∈Uδ∑i=1n[(xi−xi∗)2]α2α−1>0,\begin{equation} C_{1}=\sup \limits _{x\in U_{\delta }} \left\lbrace \sum \limits _{i=1}^{n}[(x_{i}-x_{i}^{*})^{2}]^\frac{\alpha }{2\alpha -1}\right\rbrace &gt;0, \end{equation}where σiR(x),i=1,2,…,n$\sigma _{i}^{R(x)}, i=1,2,\ldots,n$, denote the eigenvalues of R(x)$R(x)$.(a)Then, the equilibrium point x∗$x^{*}$ of the closed‐loop system (6) is locally fixed‐time stabilized in the neighborhood Uδ$U_{\delta }$ by the memristor‐based feedback controller37ucM≜uc+uM,\begin{equation} u_{cM}\triangleq u_c+u_M, \end{equation}and the settling time satisfies38T(x0)≤T̂=(2α−1)24kM(x)α(α−1)C1α−1α≤TM≜(2α−1)24kMα(α−1)C1α−1α,∀x0∈Uδ,\begin{equation} \def\eqcellsep{&}\begin{array}{ll} T(x_0) \le \hat{T}=\dfrac{(2\alpha -1)^2}{4k_M(x) \alpha (\alpha -1)}C_{1}^{\frac{\alpha -1}{\alpha }}\\[19pt] \qquad \ \ \ \le T_{M}\triangleq \dfrac{(2\alpha -1)^2}{4k_M \alpha (\alpha -1)}C_{1}^{\frac{\alpha -1}{\alpha }}, \forall x_0\in U_{\delta }, \end{array} \end{equation}where uc$u_c$ is given as (18), uM=−M(e(t))GT∂H∂x$u_M=-M(e(t))G^{T}\frac{\partial H}{\partial x}$, M(e(t))=diag{m(e1(t)),m(e2(t)),…,m(em(t))}$M(e(t)) = diag\lbrace m(e_1(t)),m(e_2(t)), \ldots,m(e_m(t))\rbrace$, andkM(x)≜k01+min1≤i≤n{σiG(x)M(x−x∗)GT(x)},kM=infx∈UδkM(x);{\fontsize{9.8}{11.8}{\selectfont{ \begin{equation*} k_M(x)\triangleq k_{01}+\min \limits _{1\le i \le n}\lbrace \sigma _{i}^{G(x)M(x-x^{*})G^{T}(x)}\rbrace , k_M=\inf \limits _ {x\in U_{\delta }}k_M(x); \end{equation*}}}}(b)In addition, if the memristance function M(ei(t)),i=1,2,…,m$M(e_i(t)), i=1,2,\ldots, m$, meet the Assumption 1, then the state error e(t)$e(t)$ accelerate convergence to the origin of the closed‐loop system (6).2FIGUREThe dependence of memristance functions on state errorProof(a)The Proof is similar to the Proof of the Theorem 1, so it is omitted here.(b)The minimum positive eigenvalue of RM=G(x)M(x)GT(x)$R_M=G(x)M(x)G^T(x)$ is larger, the value of kM(x)$k_M(x)$ is larger, the settling time T̂$\hat{T}$ is shorter. Because assumptions (H1)$(\mathbf {H}_1)$ and (H2)$(\mathbf {H}_2)$ are satisfied, whenever ei∼(t)∈(0,Q),i=1,…,n$\widetilde{e_i}(t)\in (0,Q), i=1,\ldots,n$, it will lead to the values of m(ei(t)),i=1,…,m$m(e_i(t)), i=1,\ldots,m$, continuously increase to Roff$R_{off}$. As the minimum positive eigenvalue of RM=G(x)M(x)GT(x)$R_M=G(x)M(x)G^T(x)$ and kM(x)$k_M(x)$ are increase, the settling time T̂$\hat{T}$ is decrease, thus the convergence speed of controller (37) in locally fixed‐time stabilization PH system (6) is accelerated. Hence, the value of m(ei(t)),i=1,…,m$m(e_i(t)),i=1,\ldots,m$, continuously increases to Roff$R_{off}$, and can make error state e(t)$e(t)$ accelerate convergence to the origin. The proof of Theorem 2 is complete.□$\Box$MEMRISTOR‐BASED LOCALLY FIXED‐TIME H∞$H_{\infty }$ CONTROL FOR PH SYSTEMSIn the following, we discuss memristor‐based locally fixed‐time H∞$H_{\infty }$ control for PH systems in the presence of external disturbances.Consider the following PH systems39ẋ=(J(x)−R(x))∂H∂x(x)+G(x)u+G1(x)ωz=M(x)GT(x)∂Hd∂x(x),\begin{equation} {\left\lbrace \def\eqcellsep{&}\begin{array}{ll} \dot{x}=(J(x)-R(x))\dfrac{\partial H}{\partial x}(x)+G(x)u+G_1(x)\omega \\[15pt] z= M(x)G^{T}(x)\dfrac{\partial H_d}{\partial x}(x) \end{array} , \right.} \end{equation}where x,H∈Rn,u∈Rm,G∈Rn×m,J$x, H\in \mathbb {R}^{n}, u\in \mathbb {R}^{ m},G\in \mathbb {R}^{n\times m},J$ and R are the same as those in system (6), ω∈Rq$\omega \in \mathbb {R}^{q}$ is the disturbance in L2, G1∈Rn×q$G_1\in \mathbb {R}^{n\times q}$ is the disturbance gain, z is the system's penalty function, M(x)∈Rm×m$M(x)\in \mathbb {R}^{m\times m}$ is an incremental memristance matrix and Hd(x)$H_d (x)$ is given as (24).3TheoremIf let rankG(x)=m<n$rankG(x)=m&lt;n$ and the disturbance attenuation level γ>0$\gamma &gt;0$ be given, Rd,Jd,Uδ,k0,C$R_d, J_d, U_{\delta }, k_0, C$ and uc$u_c$ are the same as those in the Lemma 4, and40Rd(x)+12γ2[G(x)GT(x)−G1(x)G1T(x)]⪰0,\begin{equation} R_d(x)+\frac{1}{2\gamma ^{2}}[G(x)G^{T}(x)-G_{1}(x)G_{1}^{T} (x)]\succeq 0, \end{equation}then the locally fixed‐time H∞$H_{\infty }$ control problem (39) can be solved by the following memristor‐based H∞$H_{\infty }$ controller41ucr(x)=uc(x)+ur(x)=uc−12MT(x)M(x)+12γ2ImGT(x)∂Hd∂x(x).\begin{eqnarray} u_{cr}(x) &=& u_c (x)+u_r(x)\nonumber\\ &=& u_c-\left[\frac{1}{2}M^{T}(x)M(x)+\frac{1}{2 \gamma ^{2}} I_m\right]G^T(x) \frac{\partial H_{d}}{\partial x}(x).\nonumber\\ \end{eqnarray}Furthermore, if the Assumption 1 also holds, then state x(t)=(x1(t),x2(t),…,xn(t)$x(t)=(x_1(t),x_2(t),\ldots,x_n(t)$ of system (39) accelerate converge to the origin, when x(t)∈(−Q,Q)$x(t)\in (-Q,Q)$, and ω=0$ \omega =0$.ProofSubstituting (41) into system (39), the following closed‐loop PH system can be obtained42ẋ=(J(x)−R(x))∂H∂x(x)+G(x)uc−G(x)12MT(x)M(x)+12γ2ImGT(x)∂Hd∂x(x)+G1(x)ω=[Jd(x)−(Rd(x)+Rr(x))]∂Hd∂x(x)+G1(x)ω,{\fontsize{9.5}{11.5}{\selectfont{ \begin{eqnarray} \dot{x}&=&(J(x)-R(x))\frac{\partial H}{\partial x}(x)+G(x)u_c\nonumber\\ &&-\,G(x) \left[\frac{1}{2}M^{T}(x)M(x)+\frac{1}{2 \gamma ^{2}} I_m\right]G^T(x) \frac{\partial H_{d}}{\partial x}(x)+G_1(x)\omega\nonumber\\ &=&[J_d(x)-(R_d(x)+R_r(x))]\frac{\partial H_d}{\partial x}(x)+G_1(x)\omega , \end{eqnarray}}}}where Rd$R_d$ and Jd$J_d$ are obtained from classical IDA‐PBC technique, Rr(x)=G(x)[12MT(x)M(x)+12γ2Im]GT(x)$R_r(x)=G(x)[\frac{1}{2}M^{T}(x)M(x)+\frac{1}{2 \gamma ^{2}} I_m]G^T(x)$. Notice the positivity properties of memristance M(x)MT(x)$M(x)M^T(x)$ and Rr(x)$R_r(x)$, let43kr(x)≜k0+min1≤i≤n{σiRr(x)},\begin{equation} k_r(x)\triangleq k_0+\min \limits _{1\le i \le n}\lbrace \sigma _{i}^{R_r(x)}\rbrace , \end{equation}where σiRr(x)$\sigma _{i}^{R_r(x)}$ denotes the eigenvalues of Rr(x)$R_r(x)$, which implies44kr(x)>k0>0.\begin{equation} k_r(x) &gt; k_0&gt;0. \end{equation}When ω=0$\omega = 0$, taking advantage of the method similar to the proof of Theorem 1, it is not difficult to obtain that the closed‐loop system (42) can be locally fixed‐time stabilized at the origin. Computing the derivative of Hd(x)$H_d(x)$ along the trajectory of (42) we can obtain the following inequalityḢd(x)=−∂HdT(x)∂x(Rd(x)+Rr(x))∂Hd(x)∂x≤−kr(x)∂HdT(x)∂x∂Hd(x)∂x=−kr(x)2α2α−1(x12)1−α2α−1x1,…,2α2α−1(xn2)1−α2α−1xn×2α2α−1(x12)1−α2α−1x1,…,2α2α−1(xn2)1−α2α−1xnT=−kr(x)2α2α−12∑i=1n(xi2)12α−1.\begin{eqnarray*} \dot{H}_d(x) &=&-\frac{\partial H^{T}_d(x)}{\partial x}(R_d(x)+R_r(x))\frac{\partial H_d(x)}{\partial x}\\ &\le& -k_r(x)\frac{\partial H^{T}_d(x)}{\partial x}\frac{\partial H_d(x)}{\partial x}\\ &=& -k_r(x){\left(\frac{2\alpha }{2\alpha -1}(x_1^2)^{\frac{1-\alpha }{2\alpha -1}}x_1,\ldots,\frac{2\alpha }{2\alpha -1}(x_n^2)^{\frac{1-\alpha }{2\alpha -1}}x_n\right)}\\ &&\times\, {\left(\frac{2\alpha }{2\alpha -1}(x_1^2)^{\frac{1-\alpha }{2\alpha -1}}x_1,\ldots,\frac{2\alpha }{2\alpha -1}(x_n^2)^{\frac{1-\alpha }{2\alpha -1}}x_n\right)}^T\\ &=& -k_r(x)\left(\frac{2\alpha }{2\alpha -1}\right)^2 \sum \limits _{i=1}^{n}(x_i^2)^{\frac{1}{2\alpha -1}}. \end{eqnarray*}Because of α>1$\alpha &gt;1$, by using the Jenson's inequality (3) we obtainḢd(x)≤−kr(x)2α2α−12∑i=1n(xi2)α2α−11α≤−kr(x)2α2α−12∑i=1n(xi2)α2α−11α=−kr(x)2α2α−12Hd1α.\begin{eqnarray*} \dot{H}_d(x) &\le& -k_r(x)\left(\frac{2\alpha }{2\alpha -1}\right)^2 \sum \limits _{i=1}^{n} \left[(x_i^2)^{\frac{\alpha }{2\alpha -1}}\right]^{\frac{1}{\alpha }}\\ &\le& -k_r(x) \left(\frac{2\alpha }{2\alpha -1}\right)^2 {\left[\sum \limits _{i=1}^{n}(x_i^2)^{\frac{\alpha }{2\alpha -1}}\right]}^{\frac{1}{\alpha }}\\ &=& -k_r(x)\left(\frac{2\alpha }{2\alpha -1}\right)^2 H_d^{\frac{1}{\alpha }}. \end{eqnarray*}From Lemma 2, we obtain that the origin is a locally fixed‐time stable equilibrium of PH system (39), and that the settling time satisfies45T(x0)≤T̂r=(2α−1)24kr(x)α(α−1)Cα−1α≤Tr≜(2α−1)24krα(α−1)Cα−1α,∀x0∈Uδ,\begin{eqnarray} T(x_0) &\le& \hat{T}_r=\frac{(2\alpha -1)^2}{4k_r(x) \alpha (\alpha -1)}C^{\frac{\alpha -1}{\alpha }}\nonumber\\ &\le& T_{r}\triangleq \frac{(2\alpha -1)^2}{4k_r \alpha (\alpha -1)}C^{\frac{\alpha -1}{\alpha }}, \forall x_0\in U_{\delta }, \end{eqnarray}where kr=infx∈Uδkr(x)$k_r=\inf \nolimits _ {x\in U_{\delta }}k_r(x)$.Next, we show that the L2 gain (from ω to z) of the closed‐loop system consisting of system (39) is bounded by γ.When ω≠0$\omega \ne 0$, computing the derivative of the Hamiltonian function Hd(x)$H_d(x)$ along the trajectories of system (42) and using the penalty function yield46Hḋ(x)=−∂HdT∂x(x)[Jd(x)−(Rd(x)+Rr(x))]∂Hd∂x(x)+∂HdT∂x(x)G1(x)ω=−∂HdT∂x(x)[Jd(x)−(Rd(x)+Rr(x))]∂Hd∂x(x)−12∥γω−1γG1T(x)∂Hd∂x(x)∥2+12{γ2∥ω∥2−∥z∥2}+12∂HdT∂x(x)G(x)MT(x)M(x)GT(x)∂Hd∂x(x)+12γ2∂HdT∂x(x)G1(x)G1T(x)∂Hd∂x(x)=−∂HdT∂x(x){Rd(x)+12γ2[G(x)GT(x)−G1(x)G1T(x)]}∂Hd∂x(x)+12{γ2∥ω∥2−∥z∥2}−12∥γω−1γG1T(x)∂Hd∂x(x)∥2≤−∂HdT∂x(x){Rd(x)+12γ2[G(x)GT(x)−G1(x)G1T(x)]}∂Hd∂x(x)+12{γ2∥ω∥2−∥z∥2},\begin{eqnarray} \dot{H_d}(x)&=& -\frac{\partial H^T_d}{\partial x}(x)[J_d(x)-(R_d(x)+R_r(x))] \frac{\partial H_d}{\partial x}(x)\nonumber\\ &&+\frac{\partial H^T_d}{\partial x}(x)G_1(x)\omega \nonumber\\ &=&-\frac{\partial H^T_d}{\partial x}(x)[J_d(x)-(R_d(x)+R_r(x))]\frac{\partial H_d}{\partial x}(x)\nonumber\\ &&-\frac{1}{2}\Vert \gamma \omega -\frac{1}{\gamma }G^T_1 (x)\frac{\partial H_d}{\partial x}(x)\Vert ^2+\frac{1}{2}\lbrace \gamma ^2 \Vert \omega \Vert ^2-\Vert z\Vert ^2\rbrace \nonumber\\ &&+\frac{1}{2}\frac{\partial H^T_d}{\partial x}(x)G(x)M^T(x)M(x)G^T(x)\frac{\partial H_d}{\partial x}(x)\nonumber\\ &&+\frac{1}{2\gamma ^2}\frac{\partial H^T_d}{\partial x}(x)G_1(x)G^T_1(x)\frac{\partial H_d}{\partial x}(x)\nonumber\\ &=&-\frac{\partial H^T_d}{\partial x}(x)\lbrace R_d(x)+\frac{1}{2\gamma ^{2}}[G(x)G^{T}(x)\nonumber\\ &&-G_{1}(x)G_{1}^{T} (x)]\rbrace \frac{\partial H_d}{\partial x}(x)+\frac{1}{2}\lbrace \gamma ^2 \Vert \omega \Vert ^2-\Vert z\Vert ^2\rbrace \nonumber\\ &&-\frac{1}{2}\Vert \gamma \omega -\frac{1}{\gamma }G^T_1 (x)\frac{\partial H_d}{\partial x}(x)\Vert ^2\nonumber\\ &\le& -\frac{\partial H^T_d}{\partial x}(x)\lbrace R_d(x)+\frac{1}{2\gamma ^{2}}[G(x)G^{T}(x)\nonumber\\ &&-G_{1}(x)G_{1}^{T} (x)]\rbrace \frac{\partial H_d}{\partial x}(x)+\frac{1}{2}\lbrace \gamma ^2 \Vert \omega \Vert ^2-\Vert z\Vert ^2\rbrace , \end{eqnarray}where ∥·∥$\Vert \cdot \Vert$ is the L2 norm. From (46), the γ‐dissipation inequality is obtained47Hḋ(x)+P(x)≤12{γ2∥ω∥2−∥z∥2},\begin{equation} \dot{H_d}(x)+P(x)\le \frac{1}{2}\lbrace \gamma ^2 \Vert \omega \Vert ^2-\Vert z\Vert ^2\rbrace , \end{equation}the condition (40) implies48P(x)=∂HdT∂x(x){Rd(x)+12γ2[G(x)GT(x)−G1(x)G1T(x)]}∂Hd∂x(x)≥0.\begin{equation} \def\eqcellsep{&}\begin{array}{ll} P(x)=\dfrac{\partial H^T_d}{\partial x}(x)\lbrace R_d(x)+\dfrac{1}{2\gamma ^{2}}[G(x)G^{T}(x)\\[15pt] \qquad\quad -\,G_{1}(x)G_{1}^{T} (x)]\rbrace \dfrac{\partial H_d}{\partial x}(x)\ge 0. \end{array} \end{equation}So, it is easy to obtainHḋ(x)≤12{γ2∥ω∥2−∥z∥2},\begin{eqnarray*} \dot{H_d}(x)\le \frac{1}{2}\lbrace \gamma ^2 \Vert \omega \Vert ^2-\Vert z\Vert ^2\rbrace , \end{eqnarray*}this means the (HJ1)$\mathbf {(HJ1)}$ given by (21) holds.Thus, it follows from the Lemma 3, the L2‐gain of the closed‐loop system (39) is bounded by the given disturbance attenuation level γ. Therefore, the locally fixed‐time H∞$H_{\infty }$ control problem (39) is solved by control law (41).At last, if the Assumption 1 holds, whenever x∼(t)=|x(t)|∈(0,Q)$\widetilde{x}(t)=|x(t)|\in (0,Q)$ will lead to the values of M(xi(t)),i=1,…,m$M(x_i(t)), i=1,\ldots,m$ continuously increase to Roff$R_{off}$. From (45), when the minimum eigenvalue ofRr(x)=G(x)[12MT(x)M(x)+12γ2Im]GT(x)\begin{eqnarray*} R_r(x)=G(x)[\frac{1}{2}M^{T}(x)M(x)+\frac{1}{2 \gamma ^{2}} I_m]G^T(x) \end{eqnarray*}and kr(x)$k_r(x)$ increase, the settling time T̂r$\hat{T}_r$ decreases, thus the convergence speed of fixed‐time stabilization for PH system (39) with controller (41) is accelerated. Hence, the value of Rr(xi(t)),i=1,…,m$R_r (x_i(t)),i=1,\ldots,m$, which continuously increases to Roff$R_{off}$, can lead to system states x(t)$x(t)$ accelerate converge to the origin.□$\Box$4RemarkIn Theorem 3, when the disturbance ω≠0$\omega \ne 0$, and its amplitude is η, η∈(0,Q)$\eta \in (0,Q)$, if the Assumption 1 holds, then system's state x(t)$x(t)$ in fixed time accelerately converges to a neighborhood (U(o,ε),0<ε≪η$U(o,\epsilon ), 0&lt;\epsilon \ll \eta$) of the origin. Besides, the incremental memristance matrix M(x)$M(x)$, emerges not only in penalty function but also in the memristor‐based locally fixed‐time H∞$H_{\infty }$ feedback controller ucr=uc−[12MT(x)M(x)+12γ2Im]GT(x)∂Hd∂x(x)$u_{cr}=u_c-[\frac{1}{2}M^{T}(x)M(x)+\frac{1}{2 \gamma ^{2}} I_m]G^T(x)\frac{\partial H_{d}}{\partial x}(x)$.5RemarkFrom the Theorem 1–Theorem 3, the energy level C≜supx∈Uδ∑i=1n(xi2(t))α2α−1$C\triangleq \sup \nolimits _{x\in U_{\delta }}\sum \nolimits _{i=1}^{n}(x^{2}_{i}(t))^\frac{\alpha }{2\alpha -1}$ and the upper bounds of the settling time Tξ≜(2α−1)24kξα(α−1)Cα−1α$T_{\xi }\triangleq \frac{(2\alpha -1)^2}{4k_{\xi } \alpha (\alpha -1)}C^{\frac{\alpha -1}{\alpha }}$, where ξ denote the subscripts m,M$m, M$ and r, respectively. Therefore, Uδ,C$U_{\delta }, C$ and Tξ$T_{\xi }$ change in the same direction, that is, the bigger is Uδ$U_{\delta }$, the bigger is C, thus the bigger is Tξ$T_{\xi }$. Furthermore, in order to obtain faster convergence speed, the key parameter α should better be 2>α>1$2&gt;\alpha &gt;1$, and kξ$k_{\xi }$ should be as large as possible.ILLUSTRATIVE EXAMPLEExample 1:In this subsection, numerical simulations are used to demonstrate the correctness of our theoretical results. In order to compare our memristor‐based control laws with the classical IDA‐PBC method in convergence speed and the effect in rejection periodic disturbance, we consider the locally fixed‐time stabilization problem of the following PH system:49ẋ=(J(x)−R(x))∂H∂x(x)+G(x)u+G1(x)ω,\begin{equation} \dot{x}=(J(x)-R(x))\frac{\partial H}{\partial x}(x)+G(x)u+G_1(x)\omega , \end{equation}where x=(x1,x2,x3)T∈R3,x0=x(t0)=x(0)=(1,−2,2)T$x {=}(x_1,x_2,x_3)^T {\in} \mathbb {R}^{3}, x_0=x(t_0)=x(0)=(1,-2,2)^T$, H(x)=12(x12+x22)+x3127,u=(u1,u2)T∈R2$H(x)=\frac{1}{2}(x_1^2+x_2^2)+x_3^{\frac{12}{7}}, u=(u_1,u_2)^T\in \mathbb {R}^{ 2}$, ω=5sign(sin6t),t∈[0.5,4]$ \omega = 5sign(sin6t),t\in [0.5,4]$ is a periodic disturbance in L2, andJ(x)=010−10−1010,R(x)=000010002,\begin{eqnarray*} J(x)={\left( \def\eqcellsep{&}\begin{array}{ccc}0 & 1 & 0 \\[3pt] -1 & 0 & -1 \\[3pt] 0 & 1 & 0 \end{array} \right)},\quad R(x)={\left( \def\eqcellsep{&}\begin{array}{ccc}0 & 0 & 0 \\[3pt] 0 & 1 & 0 \\[3pt] 0 & 0 & 2 \end{array} \right)}, \end{eqnarray*}G(x)=100110,G1(x)=100001.\begin{eqnarray*} G(x)={\left( \def\eqcellsep{&}\begin{array}{cc}1 & 0 \\[3pt] 0& 1 \\[3pt] 1 & 0 \end{array} \right)},\quad G_1(x)={\left( \def\eqcellsep{&}\begin{array}{cc}1 & 0 \\[3pt] 0& 0 \\[3pt] 0 & 1 \end{array} \right)}. \end{eqnarray*}The desired Hamiltonian function Hd$H_d$ is given asHd(x)=(x12)67+(x22)67+(x32)67,\begin{eqnarray*} H_d(x)=(x_1^2)^\frac{6}{7}+(x_2^2)^\frac{6}{7}+(x_3^2)^\frac{6}{7}, \end{eqnarray*}Jd,Rd$J_d, R_d$ can be obtained with the IDA‐PBC approachJd(x)=001000−100,Rd(x)=100010001,\begin{eqnarray*} J_d(x)={\left( \def\eqcellsep{&}\begin{array}{ccc}0 & 0 & 1 \\[3pt] 0 & 0 & 0 \\[3pt] -1 & 0 & 0 \end{array} \right)},\quad \quad R_d(x)={\left( \def\eqcellsep{&}\begin{array}{ccc}1 & 0 & 0 \\[3pt] 0 & 1 & 0 \\[3pt] 0 & 0 & 1 \end{array} \right)}, \end{eqnarray*}From (18), it can be obtained50uc=G+(Jd−Rd)∂Hd∂x−(J−R)∂H∂x=−x2−127x157+127x357x1+x2−127x257+127x357,\begin{eqnarray} u_c &=& G^{+}{\left((J_d-R_d)\frac{\partial H_d}{\partial x}-(J-R) \frac{\partial H}{\partial x}\right)}\nonumber\\ &=& \left( \def\eqcellsep{&}\begin{array}{c} -x_2-\dfrac{12}{7}x_1^{\frac{5}{7}}+ \dfrac{12}{7}x_3^{\frac{5}{7}}\\[11pt] x_1+x_2-\dfrac{12}{7}x_2^{\frac{5}{7}}+\dfrac{12}{7}x_3^{\frac{5}{7}} \end{array} \right), \end{eqnarray}where G+≜(GTG)−1GT$G^{+}\triangleq (G^{T}G)^{-1}G^{T}$ is the Moore–Penrose pseudo‐inverse of the matrix G(x)$G(x)$. Let the incremental memrisance matrix51M(x)=m(x1)00m(x2),\begin{equation} M(x)={\left( \def\eqcellsep{&}\begin{array}{cc}m(x_1) & 0 \\[3pt] 0 & m(x_2) \end{array} \right)}, \end{equation}where m(xi(t)),i=1,2$m(x_i(t)), i=1,2$, meet with the relation52m(xi(t))=ρe−σxi2(t)2,\begin{equation} m(x_i(t))=\rho e^{-\sigma \frac{x^2_i(t)}{2}}, \end{equation}xi(t),i=1,2$x_i(t),i=1,2$, are partial state of system (49), ρ=6,σ=0.1$\rho =6, \sigma =0.1$, see Figure 3.3FIGUREMemristance function depends on the stateThus the memristive control law becomes53um=−M(x)GT∂Hd∂x=−127m(x1)x157+m(x1)x357m(x2)x257,\begin{eqnarray} u_m &=& -M(x)G^{T}\frac{\partial H_d}{\partial x}\nonumber\\ &=&-\frac{12}{7} \left( \def\eqcellsep{&}\begin{array}{c} m(x_1)x_1^{\frac{5}{7}}+m(x_1)x_3^{\frac{5}{7}} \\[9pt] m(x_2)x_2^{\frac{5}{7}} \end{array} \right), \end{eqnarray}and54ucm=uc+um=−x2−12+12m(x1)7x157+12−12m(x1)7x357x1+x2−12+12m(x2)7x257+127x357.\begin{eqnarray} u_{cm} &=& u_c+u_m\nonumber\\ &=& \left[ \def\eqcellsep{&}\begin{array}{c} -x_2-\dfrac{12+12m(x_1)}{7}x_1^{\frac{5}{7}}+ \dfrac{12-12m(x_1)}{7}x_3^{\frac{5}{7}}\\[15pt] x_1+x_2-\dfrac{12+12m(x_2)}{7}x_2^{\frac{5}{7}}+\dfrac{12}{7}x_3^{\frac{5}{7}} \end{array} \right]. \qquad \end{eqnarray}Similarly, from (42) we get55ucr(x)=uc(x)+ur(x)=uc−12MT(x)M(x)+12γ2ImGT(x)∂Hd∂x(x)=−x2−12+6m2(x1)+1γ27x157+12−6m2(x1)+1γ27x357x1+x2−12+6m2(x2)+1γ27x257+127x357.{\fontsize{8.3}{10.3}{\selectfont{ \begin{eqnarray} u_{cr}(x) &=& u_c (x)+u_r(x)\nonumber\\ &=&u_c-\left[\frac{1}{2}M^{T}(x)M(x)+\frac{1}{2 \gamma ^{2}} I_m\right]G^T(x) \dfrac{\partial H_{d}}{\partial x}(x)\nonumber\\ &=& \left[ \def\eqcellsep{&}\begin{array}{c} -x_2-\dfrac{12+6\left(m^2(x_1)+\frac{1}{\gamma ^2}\right)}{7}x_1^{\frac{5}{7}} + \dfrac{12-6 \left(m^2(x_1)+\frac{1}{\gamma ^2}\right)}{7}x_3^{\frac{5}{7}}\\[15pt] x_1+x_2-\dfrac{12+6 \left(m^2(x_2)+\frac{1}{\gamma ^2}\right)} {7}x_2^{\frac{5}{7}}+\dfrac{12}{7}x_3^{\frac{5}{7}} \end{array} \right].\nonumber\\ \end{eqnarray}}}}Noticek0=min1≤i≤3infx∈Rn{σiRd(x)}=1>0,\begin{eqnarray*} k_0=\min \limits _{1\le i \le 3}\inf \limits _ {x\in \mathbb {R}^{n}} \lbrace \sigma _{i}^{R_d(x)}\rbrace =1&gt;0 , \end{eqnarray*}from Theorem 1–Theorem 3, we can draw a conclusion that the locally fixed‐time stabilizing problem (49) can be solved by control laws uc,ucm$u_c,u_{cm}$ and ucr$u_{cr}$. Furthermore, when ω=0$\omega =0$, the settling times satisfyTc(x0)≤4924k0C16,Tcm(x0)≤4924km(x)C16≤4924kmC16,\begin{eqnarray*} T_c(x_0) \le \frac{49}{24k_0}C^{\frac{1}{6}}, \quad T_{cm}(x_0) \le \frac{49}{24k_m(x)}C^{\frac{1}{6}}\le \frac{49}{24k_m}C^{\frac{1}{6}}, \end{eqnarray*}Tcr(x0)≤4924kr(x)C16≤4924krC16,\begin{eqnarray*} T_{cr}(x_0) \le \frac{49}{24k_r(x)}C^{\frac{1}{6}}\le \frac{49}{24k_r}C^{\frac{1}{6}}, \end{eqnarray*}respectively, wherekm=infx∈Uδkm(x),kr=infx∈Uδkr(x),\begin{eqnarray*} k_m=\inf \limits _ {x\in U_{\delta }}k_m(x), \quad \quad k_r=\inf \limits _ {x\in U_{\delta }}k_r(x), \end{eqnarray*}andkm(x)=min1≤i≤3{σiRd(x)+RM(x)}=min1≤i≤2{1+m(xi)},\begin{eqnarray*} \quad k_m(x)=\min \limits _{1\le i \le 3}\lbrace \sigma _{i}^{R_d(x)+R_M(x)}\rbrace =\min \limits _{1\le i \le 2}\lbrace 1+m(x_i)\rbrace , \end{eqnarray*}kr(x)=min1≤i≤3{σiRd(x)+Rr(x)}=min1≤i≤21+m2(xi)2+12γ2,\begin{eqnarray*} \quad k_r(x)=\min \limits _{1\le i \le 3}\lbrace \sigma _{i}^{R_d(x)+R_r(x)}\rbrace =\min \limits _{1\le i \le 2} \left\lbrace 1+\frac{m^2(x_i)}{2}+\frac{1}{2\gamma ^2}\right\rbrace , \end{eqnarray*}C=supx∈Uδ∑i=1n(xi2(t))α2α−1=supx∈Uδ∑i=13(xi2(t))1.22×1.2−1,\begin{eqnarray*} C=\sup \limits _{x\in U_{\delta }}\left[\sum \limits _{i=1}^{n}(x^{2}_{i}(t))^\frac{\alpha }{2\alpha -1}\right] =\sup \limits _{x\in U_{\delta }} \left[\sum \limits _{i=1}^{3}(x^{2}_{i}(t))^\frac{1.2}{2\times 1.2-1}\right], \end{eqnarray*}RM=GM(x)GT,Rr(x)=G(x)12MT(x)M(x)+12γ2ImGT(x),\begin{eqnarray*} R_M &=& GM(x)G^{T},\\ R_r(x) &=& G(x)\left[\frac{1}{2}M^{T}(x)M(x)+\frac{1}{2 \gamma ^{2}} I_m\right]G^T(x), \end{eqnarray*}Uδ$U_{\delta }$ is a neighborhood of the origin.It is not difficult to find that the convergence speed of the memristor‐based controllers ucr$u_{cr}$ and ucm$u_{cm}$ stabilizing the system (49) are faster than that of the classical IDA‐PBC controller uc$u_{c}$ when the disturbance ω=0$\omega =0$, see Figures 4–6.4FIGUREω=0$\omega =0$5FIGUREω=0$\omega =0$6FIGUREω=0$\omega =0$If ω≠0$\omega \ne 0$, owing to the dependence on memristance M(x(t))$M(x(t))$, control laws ucm$u_{cm}$ and ucr$u_{cr}$ have better disturbance rejection performance than that of the controller uc$u_c$ related to the classical IDA‐PBC method without memristive control action.Furthermore, from the condition56Rd(x)+12γ2[G(x)GT(x)−G1(x)G1T(x)]=1012γ201+12γ2012γ201⪰0,\begin{equation} \def\eqcellsep{&}\begin{array}{ll} R_d(x)+\dfrac{1}{2\gamma ^{2}}[G(x)G^{T}(x)-G_{1}(x)G_{1}^{T} (x)]\\[15pt] \qquad =\, \left( \def\eqcellsep{&}\begin{array}{ccc} 1 & 0 & \dfrac{1}{2\gamma ^2}\\[11pt] 0 & 1+\dfrac{1}{2\gamma ^2} & 0 \\[11pt] \dfrac{1}{2\gamma ^2} & 0 & 1 \end{array} \right) \succeq 0, \end{array} \end{equation}it can be obtained 14γ2≤1$\frac{1}{4\gamma ^2}\le 1$, or γ≥0.5$\gamma \ge 0.5$. Following Theorem 3, let γ=0.6>0.5$\gamma =0.6&gt;0.5$, andz=M(x)GT(x)∂Hd∂x(x),\begin{eqnarray*} z=M(x)G^{T}(x)\frac{\partial H_d}{\partial x}(x), \end{eqnarray*}is the penalty function, then the control law ucr$u_{cr}$ is locally fixed‐time stabilizing system (49) when input disturbance ω=0$\omega =0$. Moreover, the the memristor‐based H∞$H_{\infty }$ controller ucr$u_{cr}$ has the best performance in inhibiting the periodic disturbance, and the fastest speed converge to origin after disturbance vanishes, among uc,ucr$u_c,u_{cr}$ and ucm$u_{cm}$, see Figures 7–9.7FIGUREω=5sign(sin(6t)),t∈[0.5,4]$\omega =5sign(sin(6t)), t\in [0.5,4]$8FIGUREω=5sign(sin(6t)),t∈[0.5,4]$\omega =5sign(sin(6t)), t\in [0.5,4]$9FIGUREω=5sign(sin(6t)),t∈[0.5,4]$\omega =5sign(sin(6t)), t\in [0.5,4]$In summary, from Figures 4–12, it can be found that the memristor‐based controller ucm$u_{cm}$ has the fastest speed in stabilizing the system (49) when disturbance ω=0$\omega =0$. However, the memristor‐based H∞$H_{\infty }$ controller ucr$u_{cr}$ has less control costs than controller ucm$u_{cm}$ while it has the best performance in offsetting the periodic disturbance, and fastest converge speed to origin after disturbance vanishes. So, the memristor‐based H∞$H_{\infty }$ controller ucr$u_{cr}$ is the first choice in those situations coping with continuously periodic disturbances.6RemarkIn the simulation example 1, the memristors are mainly used to inhibiting the external periodic disturbance. However, in the case of the other kinds of bounded disturbances, the memristor‐based controllers ucm$u_{cm}$ and ucr$u_{cr}$ will produce a marked effect also.10FIGUREω=5sign(sin(6t)),t∈[0.5,4]$\omega =5sign(sin(6t)), t\in [0.5,4]$11FIGUREControl action corresponding to the controllers uc$u_{c}$, ucm$u_{cm}$ and ucr$u_{cr}$, respectively, and ω=0$\omega =0$12FIGUREControl action corresponding to the controllers uc$u_{c}$, ucm$u_{cm}$ and ucr$u_{cr}$, respectively, and ω=5sign(sin(6t)),t∈[0.5,4]$\omega =5sign(sin(6t)), t\in [0.5,4]$Example 2:In this subsection, a more realistic example is presented to illustrate the memristor‐based locally fixed‐time controller improving the performance of a nonlinear circuit system against the strongly periodic disturbance.Consider the nonlinear circuit system in Figure 13, where the capacitance and the inductance are controlled by electric charge q and magnetic flux ψ, respectively; and their characteristic are represented by u1=f1(q1),i3=f2(ψ2)$u_1 = f_1(q_1), i_3 = f_2(\psi _2)$; the voltĺCampere characteristic of the nonlinear resistance R4 is represented by u4=f4(i4)$u_4 = f_4(i_4)$, and iω$i_{\omega }$ is disturbance signal.13FIGURENonlinear circuitAccording to Kirchhoffa̧ŕs Voltage Law and Kirchhoffa̧ŕs Current Law, the system can be expressed as570=f1(q1)−f4(i4)+Us,q1̇=−Is−i4−f2(ψ2)−iω,y=Us+f1(q1)−R3f2(ψ2).\begin{eqnarray} {\left\lbrace \def\eqcellsep{&}\begin{array}{lll} 0=f_1(q_1)-f_4(i_4)+U_s,\\[9pt] \dot{q_1}= -I_s-i_4-f_2(\psi _2)-i_{\omega },\\[9pt] y= U_s+f_1(q_1)-R_3f_2(\psi _2). \end{array} \right.} \end{eqnarray}Let f1(q1)=85q135,f2(ψ2)=85ψ235,f4(i4)=(58)4i45+i4,R3=3Ω$f_1(q_1)=\frac{8}{5}q_1^{\frac{3}{5}},f_2(\psi _2)=\frac{8}{5}\psi _2^{\frac{3}{5}},f_4(i_4)=(\frac{5}{8})^4i_4^5+i_4,R_3=3\Omega$. Denote x=(x1,x2,x3)T=[(58i4)5,q1,ψ2]T,ω=iω,u=[Us,Is]T$x =(x_1,x_2,x_3)^T= [(\frac{5}{8} i_4)^5, q_1, \psi _2]^T , \omega = i_\omega , u = [U_s, I_s]^T$. Then the system (57) can be transformed into58000010001ẋ=−1−x12510−10−101−3∂Hd(x)∂x+100−110u+0−10ω,\begin{equation} \def\eqcellsep{&}\begin{array}{ll}{\left( \def\eqcellsep{&}\begin{array}{ccc} 0 & 0 & 0 \\[3pt] 0 & 1 & 0 \\[3pt] 0 & 0 & 1 \end{array} \right)}\dot{x}&={\left( \def\eqcellsep{&}\begin{array}{ccc}-1-x_1^{\frac{2}{5}} & 1 & 0 \\[3pt] -1 & 0 & -1 \\[3pt] 0 & 1 & -3 \end{array} \right)}\frac{\partial H_d(x)}{\partial x} \\[35pt] &\quad +\, {\left( \def\eqcellsep{&}\begin{array}{cc}1 & 0 \\[3pt] 0 & -1\\[3pt] 1 & 0 \end{array} \right)}u+{\left( \def\eqcellsep{&}\begin{array}{c}0 \\[3pt] -1 \\[3pt] 0 \end{array} \right)}\omega , \end{array} {} \end{equation}where Hd(x)=∑i=13(xi2(t))45=∑i=13(xi2(t))α2α−1,α=43$H_d(x)=\sum \nolimits _{i=1}^{3}(x^{2}_{i}(t))^{\frac{4}{5}}=\sum \nolimits _{i=1}^{3}(x^{2}_{i}(t))^{\frac{\alpha }{2\alpha -1}},\alpha =\frac{4}{3}$. For u=(0,0)T,ω=0$u=(0,0)^T,\omega =0$, the origin of the system (58) is locally fixed‐time stable, see Figure 14.14FIGURETime history of the system (58) without disturbance and control inputThe memristor‐based controller is designed as59um=−M(x)GT∂Hd∂x=−85m(x1)x135+m(x1)x335−m(x2)x235,\begin{eqnarray} u_m &=& -M(x)G^{T}\dfrac{\partial H_d}{\partial x}\nonumber\\ &=& -\frac{8}{5} \left( \def\eqcellsep{&}\begin{array}{c} m(x_1)x_1^{\frac{3}{5}}+m(x_1)x_3^{\frac{3}{5}} \\[9pt] -m(x_2)x_2^{\frac{3}{5}} \end{array} \right), \end{eqnarray}where M(x)$M(x)$ is given as (51) and (52), ρ=0.44,σ=0.00001$\rho =0.44, \sigma =0.00001$.When ω=−6sin12t$\omega =-6 sin 12t$, the controller (59) improves the performance of the system (58) against the strongly periodic disturbance, and the system states converge to the origin in a shorter convergence time after the disturbance vanishes, see Figures 15 and 16.15FIGURETime history of the system (58) with disturbance ω=−6sin12t,t∈[0.5,1.5]$\omega =-6sin12t,t\in [0.5,1.5]$, and control input u=(0,0)T$u=(0,0)^T$16FIGURETime history of the system (58) with disturbance ω=−6sin12t,t∈[0.5,1.5]$\omega =-6sin12t,t\in [0.5,1.5]$ and controller (59): u=um$u=u_m$CONCLUSIONSIn this paper, two novel memristor‐based locally fixed‐time controllers are designed to improve the disturbance rejection performance of PH systems via the IDA‐PBC method. The memristor‐based controllers proposed can make the states in the neighbourhood of equilibrium point accelerate convergence to desired equilibrium point, and lead to the oscillation be quashed considerably in the case of periodic disturbances. One of our future work is to find some applications in those situations where robust and adaptive control is needed, due to system parameter uncertainty and external disturbance.ACKNOWLEDGEMENTSThis work is supported in part by the Doctoral Research Start‐Up Fund of Yunnan Agricultural University under Grant A2032002548, and in part by the Philosophy and Social Science Planning Project of Yunnan Province under Grant YB2021016.CONFLICT OF INTERESTThe authors have declared no conflict of interest.DATA AVAILABILITY STATEMENTAll data needed to evaluate the conclusions in the paper are present in the paper itself. Additional data related to this paper may be requested to the corresponding author.REFERENCESCouenne, F., Jallut, C., Maschke, B.M., Tayakout, M., Breedveld, P.: Bond graph for dynamic modelling in chemical engineering. Chem. Eng. Process. 47, 1994–2003 (2008)Jeltsema, D., Scherpen, J.: Multi‐domain modeling of nonlinear networks and systems: energy‐and power‐based perspectives. IEEE Control Syst. Mag. 29(4), 28–59 (2009)Macchelli, A.: Port Hamiltonian systems: A unified approach for modeling and control finite and infinite dimensional physical systems. Ph.D. thesis, University of Bologna‐DEIS, Bologna, Italy (2003)Fujimoto, K., Sakurama, K., Sugie, T.: Trajectory tracking control of portcontrolled Hamiltonian systems via generalized canonical transformations. Automatica 39(12), 2059–2069 (2003)Ortega, R., Galaz, M., Astolfi, A., Sun, Y., Shen, T.: Transient stabilization of multi‐machine power systems with nontrivial transfer conductances. IEEE Trans. Autom. Control 50(1), 60–75 (2005)Ortega, R., Spong, M.W., Gómez‐Estern, F., Blankenstein, G.: Stabilization of a class of under actuated mechanical systems via interconnection and damping. IEEE Trans. Autom. Control 47(8), 1218–1233 (2002)Ortega, R., van der Schaft, A.J., Maschke, B., Escobar, G.: Interconnection and damping assignment passivity‐based control of port‐controlled Hamiltonian systems. Automatica 38(4), 585–596 (2002)Dörfler, F., Johnsen, J., Allgöwer, F.: An introduction to interconnection and damping assignment passivity‐based control in process engineering. J. Process Control 19(9), 1413–1426 (2009)Romero, J.G., Gandarilla, I., Santibáñez, V.: Stabilization of a class of nonlinear underactuated mechanical systems with 2‐DOF via immersion and invariance. European J. Control 63, 196–205 (2022)Ryalat, M., Laila, D.S., ElMoaqet, H.: Adaptive Interconnection and Damping Assignment Passivity Based Control for Underactuated Mechanical Systems. Int. J. Control Autom. Syst. 19(X), 1–14 (2021)Guerrero‐Sánchez, M.E., Hernández‐González, O., Valencia‐Palomo, G., Mercado‐Ravell, D.A., López‐Estrada, F.R., Hoyo‐Montaño, J.A.: Robust IDA‐PBC for under‐actuated systems with inertia matrix dependent of the unactuated coordinates: application to a UAV carrying a load. Nonlinear Dyn. 105, 3225–3238 (2021)Ferguson, J., Donaire, A., Ortega, R., Middleton, R.H.: Matched disturbance rejection for a class of nonlinear systems. IEEE Trans. Autom. Control 65(4), 1710–1715 (2020)Farid, Y., Ruggiero, F.: Finite‐time extended state observer and fractional‐order sliding mode controller for impulsive hybrid port‐Hamiltonian systems with input delay and actuators saturation: application to ball‐juggler robots. Mech. Mach. Theory 167(104577), 1–23 (2022)Yaghmaei, A., Yazdanpanah, M.J.: Structure preserving observer design for port‐Hamiltonian systems. IEEE Trans. Autom. Control 64(3), 1214–1220 (2019)Lv, C., Yu, H., Zhao, N., Chi, J., Liu, H., Li, L.: Robust state‐error port‐controlled Hamiltonian trajectory tracking control for unmanned surface vehicle with disturbance uncertainties. Asian J. Control 2467, 1–13 (2020) https://doi.org/10.1002/asjc.2467Fu, B., Li, S., Yang, J., Guo, L.: Global output regulation for a class of single input port‐controlled Hamiltonian disturbed systems. Appl. Math. Comput. 325, 322–331 (2018)Romero, J.G., Donaire, A., Navarro‐Alarcon, D., Ramirez, V.: Passivity‐based tracking controllers for mechanical systems with active disturbance rejection. IFAC‐PapersOnLine 48(13), 129–134 (2015)Chua, L.O.: Memristor‐the missing circuit element. IEEE Trans. Circ. Theory 18(2), 507–519 (1971)Strukov, D.B., Snider, G.S., Stewart, D.R., Williams, R.S.: The missing memristor found. Nature 453, 80–83 (2008)Jeltsema, D., vander Schaft, A.J.: Memristive port‐Hamiltonian systems. Math. Comput. Modell. Dyn. Syst. 16(2), 75–93 (2010)Jeltsema, D., Dòria‐Cerezo, A.: Port‐Hamiltonian formulation of systems with memory. Proc. IEEE 100(6), 1928–1937 (2012)Pershin, Y.V., Lafontaine, S., DiVentra, M.: Memristive model of amoeba's learning. Phys. Rev. E 80, 021926 (2009)Dòria‐Cerezo, A., van der Heijden, L., Scherpen, J.M.A.: Memristive port‐Hamiltonian control: Path‐dependent damping injection in control of mechanical systems. European J. Control 19, 454–460 (2013)Pasumarthy, R., Saha, G., Kazi, F., Singh, N.: Energy and power based perspective of memristive controllers. In: 52nd IEEE Conference on Decision and Control. IEEE, Piscataway (2013)Sala, G., Pasumarthy, R., Khatavkar, P.: Towards analog memristive controllers. IEEE Trans. Circuit Syst. 62, 205–214 (2015)Polyakov, A.: Nonlinear feedback design for fixed‐time stabilization of linear control systems. IEEE Trans. Autom. Control 57, 2106–2110 (2012)Bhat, S.P., Bernstein, D.S.: Finite‐time stability of continuous autonomous systems. SIAM J. Control Optim. 38, 751–766 (2000)Moulay, E., Perruquetti, W.: Finite time stability and stabilization of a class of continuous systems. J. Math. Anal. Appl. 323, 1430–1443 (2006)Sun, L., Feng, G., Wang, Y.: Finite‐time stabilization and H∞$H_\infty$ control for a class of nonlinear Hamiltonian descriptor systems with application to affine nonlinear descriptor systems. Automatica 50, 2090–2097 (2014)Lu, X., Li, H.: Prescribed finite‐time H∞$H_{\infty }$ control for nonlinear descriptor systems. IEEE Trans. Circu. Syst. II 28, 2917–2921 (2021)Yang, X., Lam, J., Ho, D.W.C., Feng, Z.: Fixed‐time synchronization of complex networks with impulsive effects via non‐chattering control. IEEE Trans. Autom. Control 62(11), 5511–5521 (2018)Sun, J., Yi, J., Pu, Z.: Augmented fixed‐time observer‐based continuous robust control for hypersonic vehicles with measurement noises. IET Control Theory Appl. 13(3), 422–433 (2019)Langueh, K.A.A., Zheng, G., Floquet, T.: Fixed‐time sliding mode‐based observer for non‐linear systems with unknown parameters and unknown inputs. IET Control Theory Appl. 14(14), 1920–1927 (2020)Hu, C.: Special functions‐based fixed‐time estimation and stabilization for dynamic systems. IEEE Trans. Systems, Man, and Cybernetics: Systems (2021). https://doi.org/10.1109/TSMC.2021.3062206Zhang, W., Yang, X., Yang, S., Alsaedi, A.: Finite‐time and fixed‐time bipartite synchronization of complex networks with signed graphs. Math. Comput. Simul 188, 319–329 (2021)Kong, F., Zhu, Q., Huang, T.: Fixed‐time stability for discontinuous uncertain inertial neural networks with time‐varying delays. IEEE Trans. Syst. Man Cybern.: Syst. (2021). https://doi.org/10.1109/TSMC.2021.3096261Wang, Y., Li, C., Cheng, D.: Generalized Hamiltonian realization of time‐invariant nonlinear systems. Automatica 39(8), 1437–1443 (2003)Liu, X., Liao, X.: Fixed‐time H∞$\mathcal {H}_{\infty }$ control for port‐controlled Hamiltonian systems. IEEE Trans. Autom. Control 64(7), 2753–2765 (2019)Liu, X., Liao, X.: Fixed‐time stabilization control for port‐Hamiltonian systems. Nonlinear Dyn. 96(2), 1497–1509 (2019)van der Schaft, A.J.: L2L2‐gain and passivity techniques in nonlinear control, 2nd ed. Springer, London (2000) http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png IET Control Theory & Applications Wiley

Memristor‐based disturbance rejection control for port‐Hamiltonian systems with locally fixed‐time convergence

Liu, Xinggui; Zhao, Mei
IET Control Theory & Applications , Volume 16 (13) – Sep 1, 2022
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Wiley
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© 2022 The Institution of Engineering and Technology.
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1751-8652
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10.1049/cth2.12307
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Abstract

INTRODUCTIONBased on the mechanics formulation introduced by Sir W. R. Hamilton in the 19th century, the port‐Hamiltonian (PH) frameworks derive from network modeling of physical systems in various of domains such as electrics, mechanics, fluids, and electromechanical systems or thermal systems [1–3]. The Hamiltonian function, which is usually the sum of potential energy and kinetic energy in physical systems, is a good Lyapunov function candidate for many physical systems. Due to the above reasons and its nice structural properties with clear physical meaning, the PH systems have been extensively used in practical control and stability analysis [4–6]. The interconnection and damping assignment passivity‐based control (IDA‐PBC) technology, which uses the passivity properties of the PH systems to ensure the convergence of the systems to the desired equilibrium point, has become a simple method to design passive‐based controller and solve the stability problem of PH systems [7–9].Practical PH engineering systems inevitably confront with various disturbances, which bring undesirable effects on the closed‐loop system, always lead to much difficulty for the controller design. Therefore, studies on the disturbance rejection for PH systems have attracted considerable attention. In recent years, several elegant methods have been proposed, such as adaptive control (AC) [10], integral control (IC) [11, 12], sliding mode control (SMC) [13], disturbance observer‐based control (DOBC) [14–16], and active disturbance rejection control (ADRC) [17], etc. However, these control methods have their own shortcomings or limitations. For example, the IC methods cannot effectively remove the effects caused by fast time‐varying disturbances, and the undesirable transient control performances are always unavoidable. SMC, DOBC, and ADRC approaches are usually not available for disturbed PH systems under the so‐called mismatched disturbance, and can not preserve the PH structure of closed‐loop resulting system. In addition, these methods, which are characterized by observer‐based, disturbances estimating and compensating, increase the difficulty of stability analysis and physical explanation.Memristor, which refers to a resistor with memory, was fist postulated the existence of a new basic electrical circuit element by Chua in the early 1970s [18]. A physical passive two‐terminal memristive prototype was realized by scientists of Hewlett‐Packard (HP) Laboratories [19] until May 1, 2008. The authors in [20] pointed out that the memristor and the memristive systems can be described by using the PH form. The inclusion of memristive systems in the PH framework turns out to be almost as straightforward as the inclusion of resistive elements [21]. Adding memristive systems to the existing PH formalism, together with the ‘learning ability’ of the memristors, possibly provides a new method for controller synthesis and design [22] owing to the special characteristic of memristors. The authors in [23] presented the use of the memristor as a new element for designing passivity‐based controllers, and have found some possible benefits (e.g., ensuring better transient response) against the external disturbances. In [24], the authors observed similar numerical results. The memristor has also been used as an analog gain control element for robust‐adaptive control of miniaturized systems [25]. The memristor‐based control can be a promising method againsts disturbances and preserves the PH structure of systems. However, in the references [23–25], theoretical analysis is not sufficient to provide a reasonable interpretation that why the memristor‐based controller can work disturbance rejection.For nonlinear systems in practical applications, the convergence speed is an important performance index. Fixed‐time stability [26] is usually used to evaluating the convergence speed of dynamic systems. Unlike the finite‐time stability [27–30], where the upper bound of the corresponding settling time relies on the initial conditions, fixed‐time stability means that the system states can achieve desired equilibrium in finite‐time and the settling time is bounded by a constant independently of initial values. Fixed‐time control have been proved to be a powerful tool in dealing with disturbances and uncertainties. Because a fixed‐time controller possesses not only faster convergence speed but also better robustness and disturbance rejection performance. Fixed‐time control for dynamical systems have received more and more attention and some elegant studies have been done in a series of works [31–37]. For PH framework, the problems of fixed‐time stability analysis and fixed‐time control have been solved in references [38] and [39]. In [38], the locally fixed‐time stability of dynamical systems is first distinguished from the globally fixed‐time stability. Neither the reference [38] nor the reference [39] has involved using memristor to suppress external disturbance.Motivated by the previous studies, a framework of memristor‐based locally fixed‐time disturbance rejection control for PH systems is developed via IDA‐PBC technique. Two novel memristor‐based controllers are designed to accomplish this task. Fist, a memristor‐based locally fixed‐time controller is designed to suppress the external disturbance of PH systems. The settling‐times related to the memristor‐based controller in locally fixed‐time stabilization for the same PH system are shorter, thus the convergence speed are faster than the classical IDA‐PBC technique without memristive control action. Second, in view of the H∞$H_{\infty }$ performance is also significant in terms of system disturbance rejection performance analysis, the memristor‐based locally fixed‐time H∞$H_{\infty }$ controller is also designed. Compared with the existing results, the present paper has the following new highlights:(i)Compared with the existing disturbance rejection schemes [10–17], the memritor‐based method does not sacrifice its nominal control performance. The disturbance rejection is supplied by a state‐dependent memristor, which is almost as straightforward as the inclusion of resistive elements to the original PH system. The memritor‐based control has nothing to do with object disturbances, no need for system on‐line identification and adjustment, simple physical implementation and so on. The method proposed can not only preserve the PH structure and clear physical meaning of the closed‐loop resulting system, but also can obtain a better control performance, that is, the settling‐time related to the memristor‐based controllers in stabilizing the same PH systems are shorter, and the memristor‐based controllers can make the oscillation in the case of the strong periodic disturbance be better suppressed. In addition, the memritor‐based control can compatible with other disturnace rejection control methods (e.g., IC, DOBC and ADRC) via the IDA‐PBC technique, to further improve the disturbance suppression performance of PH systems.(ii)Although the fixed‐time stability analysis and fixed‐time control of PH systems have been addressed in references [38, 39], all of them have not involved using memristor to suppress external disturbance. The present study focuses on the locally fixed‐time stability of PH systems due to the disturbance rejection memristor only works in a neighborhood of the desired equilibrium point.(iii)Although the authors in the references [23] and [24] have observed some possible benefits against the external disturbances via memristive IDA‐PBC method, their theoretical analysis is not sufficient to provide a reasonable interpretation that why the memristor‐based controller can work disturbance rejection. They have not considered the locally fixed‐time stability of systems. In this work, the locally fixed‐time stability of PH system is used to quantitatively describe the convergence speed. The shorter is the settling time, the faster is the convergence speed, and thus the better is the disturbance rejection. The memristor‐based controller possesses good disturbance rejection owing to it can make the system states in a neighbourhood accelerate convergence to the desired equilibrium point.Therefore, the proposed approach has application potential in situation where robust and adaptive control is needed in the presence of external disturbance. Two numerical examples are presented to show the effectiveness of the memristor‐based controllers designed and the theoretical analysis.The remainder of the article is organized as follows. In Section 2, we briefly review some preliminary results on locally fixed‐time stability which will be used in the following discussion, incorporate memristor into PH systems and present the problem statement of the memristor‐based locally fixed‐time H∞$H_{\infty }$ control. In Section 3, we analyze theoretically the memristor‐based locally fixed‐time stabilization and H∞$H_{\infty }$ control problems of PH systems via IDA‐PBC method, and discuss the disturbance rejection performance of the memristor‐based controllers proposed. Two illustrative examples are presented in Section 4. Conclusions and discussions are given in Section 5.Notations: The set of real numbers is denoted by R$\mathbb {R}$. The set of positive real numbers is denoted by R+$\mathbb {R}_{+}$. For any matrix A∈Rn×n$A\in \mathbb {R}^{n\times n}$, we denote the positive definiteness (respectively, positive semi‐definiteness) of A by A≻0$A\succ 0$ (respectively, A⪰0$A\succeq 0$ ). Let AT$A^{T}$ denote the transpose of a matrix A∈Rn×n$A\in \mathbb {R}^{n\times n}$.PRELIMINARIES AND PROBLEM STATEMENTLocally fixed‐time stabilityIn this subsection, we briefly review preliminaries on locally fixed‐time stability which will be used in the following discussion.Consider a dynamical system1ẋ=f(x),f(0)=0,x(t0)=x0,x∈U⊆Rn,\begin{equation} \dot{x}=f(x), f(0)=0, x(t_0)=x_0, x\in U\subseteq \mathbb {R}^n, \end{equation}where f:U↦Rn$f:U\mapsto \mathbb {R}^n$ is continuous on an open neighborhood U of the origin such that the solution to system (1) is unique in the forward time.1Definition[39] The null solution O of system (1) is said to be locally fixed‐time stable if it is locally finite‐time stable in a round neighborhood with the centrality O and the radius δ, Uδ=:U(O,δ),δ>0$U_\delta =:U(O,\delta ),\delta &gt;0$, and the settling‐time function T(x0)$T(x_0)$ is bounded by a positive number, that is, ∃Tmax>0,s.t.,T(x0)≤Tmax,∀x0∈Uδ$\exists T_{max}&gt;0, s.t., T(x_0)\le T_{max}, \forall x_0\in U_\delta$.2DefinitionA nonlinear control system2ẋ=f(x)+g(x)u,x∈Rn,u∈Rm,\begin{equation} \dot{x}=f(x)+g(x)u, \quad x\in \mathbb {R}^n, u\in \mathbb {R}^m, \end{equation}is called locally fixed‐time stabilizable via continuous state feedback, if there exists a continuous feedback law u=u(x)$u=u(x)$ such that the equilibrium of the closed‐loop system is locally fixed‐time stable, where g(x):Rn→Rn×m$g(x):\mathbb {R}^{n}\rightarrow \mathbb {R}^{n\times m}$ is the control gain matrix and is assumed full column rank.1Lemma(Jensen's inequality [29]).3∑i=1n|xi|a21a2≤∑i=1n|xi|a11a1,0<a1<a2,\begin{equation} {\left(\sum \limits _{i=1}^{n} |x_i|^{a_2}\right)}^{\frac{1}{a_2}}\le {\left(\sum \limits _{i=1}^{n} |x_i|^{a_1}\right)}^{\frac{1}{a_1}}, \quad 0&lt;a_1&lt;a_2, \end{equation}where a1,a2$a_1,a_2$ and xi,i=1,2,…,n$x_i,i=1,2,\ldots,n$ are all real numbers.2Lemma[27] Consider a dynamical system (1) If there exist a real number β>1$\beta &gt; 1$ and a C1 radially unbounded Lyapunov function V(x)$V (x)$ (i.e., V(x)$V(x)$ have continuous first‐order partial derivative and V(x)→∞$V(x)\rightarrow \infty$ as ∥x∥→∞$\Vert x\Vert \rightarrow \infty$, where ∥·∥$\Vert \cdot \Vert$ is the Euclidean norm) of the system such that4V̇≤−kV1β(x(t)),k>0,\begin{eqnarray} \dot{V}\le -kV^{\frac{1}{\beta }}(x(t)), \quad k&gt;0, \end{eqnarray}holds along the trajectories of the system (1) starting from any x0∈Rn$x_0 \in \mathbb {R}^n$, then the origin is a global finite‐time stable equilibrium of system (1). Furthermore, the settling time of system (1) with respect to x0 satisfies5T(x0)≤t0+βk(β−1)Vβ−1β(x0),∀x0∈Rn.\begin{equation} T(x_0) \le t_0+\frac{\beta }{k(\beta -1)}V^{\frac{\beta -1}{\beta }}(x_0),\quad \forall x_0\in \mathbb {R}^n. \end{equation}Incorporating memristor into PH systemsThe input‐state‐output port controlled‐Hamiltonian system is6ΣP:ẋ=J(x)−R(x)∂H∂x(x)+Gu,y=GT∂H∂x(x),\begin{eqnarray} \Sigma _{P}: {\left\lbrace \def\eqcellsep{&}\begin{array}{ll} \dot{x}={\left(J(x)-R(x)\right)}\dfrac{\partial H}{\partial x}(x)+Gu,\\[11pt] y=G^{T}\dfrac{\partial H}{\partial x}(x), \end{array} \right.} \end{eqnarray}where x∈Rn$x\in \mathbb {R}^{n}$ is the state vector, u∈Rm,(m≤n)$u\in \mathbb {R}^{m}, (m\le n)$ is the control input. H(x):Rn→R$H(x):\mathbb {R}^{n}\rightarrow \mathbb {R}$ is the Hamiltonian , J(x):Rn→Rn×n,R(x):Rn→Rn×n$J(x) : \mathbb {R}^{n}\rightarrow \mathbb {R}^{n\times n},R(x):\mathbb {R}^{n}\rightarrow \mathbb {R}^{n\times n}$, with J(x)=−JT(x)$J(x)=-J^{T}(x)$ and R(x)=RT(x)⪰0$R(x)=R^{T}(x)\succeq 0$, are the natural interconnection and damping matrices, respectively, and G(x):Rn→Rn×m$G(x):\mathbb {R}^{n}\rightarrow \mathbb {R}^{n\times m}$ is the gain matrix and is full column rank.Before the effect of memristive systems can be studied in the PH framework, we first need to extend the PH formalism by adding a memristive port, with port variables (fM,eM)∈FM×EM$(f_M,e_M)\in \mathcal {F}_M \times \mathcal {E}_M$, to the Dirac structure [20, 21]. Assuming that the memristive port can be described by an xf$x_f$‐controlled constitutive relationship (i.e., the memristance of memristor depending on state variable xf$x_f$), we can define the memristive Dirac structure U$\mathcal {U}$ as7U={(fM,eM)∈FM×EM|ẋf−fM=0,eM−Mf(xf)fM=0},\begin{equation} \def\eqcellsep{&}\begin{array}{ll} \mathcal {U}= \bigg \lbrace (f_M,&e_M)\in \mathcal {F}_{M}\times \mathcal {E}_{M}\bigg |\\[3pt] &\dot{x}_f-f_M=0,e_M-M_f (x_f)f_M=0 \bigg \rbrace , \end{array} \end{equation}where Mf(xf)$M_f(x_f)$ is an incremental memristance matrix (or memductance matrix if the admittance form is adopted) with appropriate dimension. Then the dynamics on U$\mathcal {U}$ derive the form8ẋf=fM,eM=∂HM∂xf(xf)+Mf(xf)fM,\begin{eqnarray} {\left\lbrace \def\eqcellsep{&}\begin{array}{ll} \dot{x}_f= f_M,\\[11pt] e_M=\dfrac{\partial H_M}{\partial x_f}(x_f)+M_f(x_f)f_M, \end{array} \right.} \end{eqnarray}where the memristive port variables fM$f_M$ and eM$e_M$ can be considered as the inputs uM$u_M$ and outputs yM$y_M$, respectively. It gives9ΣM:ẋf=uM,yM=∂HM∂xf(xf)+Mf(xf)uM,\begin{eqnarray} \Sigma _{M}: {\left\lbrace \def\eqcellsep{&}\begin{array}{ll} \dot{x}_f= u_M,\\[15pt] y_M= \dfrac{\partial H_M}{\partial x_f}(x_f)+M_f(x_f)u_M, \end{array} \right.} \end{eqnarray}where HM$H_M$ represents the stored energy in the memristor. Notice that, because of the ‘no energy discharge property’, the energy stored in the memristor is defined by a null‐Hamiltonian, HM(xf)≡0$H_M(x_f) \equiv 0$.Differentiating HM$H_M$ with respect to time and making use of (10) give10ḢM=∂HM∂xfẋf=yMuM−Mf(xf)uM2≡0.\begin{eqnarray} \dot{H}_M=\frac{\partial H_M}{\partial x_f}\dot{x}_f =y_Mu_M-M_f(x_f)u^{2}_M\equiv 0. \end{eqnarray}Hence, a memristor is passive if and only if its memristance is positive semi‐definite, that is, Mf(xf)⪰0$M_f(x_f)\succeq 0$.Using the modularity property of the PH systems (i.e., two different PH systems can be interconnected to constitute a new PH system), the memristor system (10) can be easily interconnected with a PH system in the form (6). The feedback interconnection described in Figure 1, y=uM,u=−yM$y = u_M, u=-y_M$, yields another memristive PH system11ẋẋf=J−R−GMfGT−GGT0∂H∂x∂HM∂xf+G∼0u∼.\begin{equation} \def\eqcellsep{&}\begin{array}{ll}{\left( \def\eqcellsep{&}\begin{array}{c} \dot{x}\\[3pt] \dot{x}_f \end{array} \right)}= {\left( \def\eqcellsep{&}\begin{array}{c c}J-R-GM_fG^{T} & -G \\[5pt] G^{T} & 0 \end{array} \right)} {\left( \def\eqcellsep{&}\begin{array}{c} \dfrac{\partial H}{\partial x}\\[15pt] \dfrac{\partial H_M}{\partial x_f} \end{array} \right)} \\[3pt] \qquad\qquad +\,{\left( \def\eqcellsep{&}\begin{array}{c}\tilde{G}\\[3pt] 0 \end{array} \right)}\tilde{u}. \end{array} \end{equation}1FIGUREAdding memristor to PH systemsThe new output for the system is naturally defined by12y∼=G∼T∂H∂x(x).\begin{equation} \tilde{y}=\tilde{G}^{T}\frac{\partial H}{\partial x}(x). \end{equation}The power‐balance inequality associated to Equations (11) and (12) takes the form13Ḣ(x)+ḢM(xf)=y∼Tu∼(x)−∂H∂x(x)T(RM(x)+R(x))∂H∂x(x)≤y∼Tu∼(x),\begin{equation} \def\eqcellsep{&}\begin{array}{lll} \dot{H}(x)+\dot{H}_M(x_f)=\tilde{y}^{T}\tilde{u}(x)-\left(\frac{\partial H}{\partial x}(x)\right)^{T}(R_M(x)\\[11pt] \qquad\qquad\qquad\qquad +\,R(x))\dfrac{\partial H}{\partial x}(x)\le \tilde{y}^{T}\tilde{u}(x), \end{array} \end{equation}where u∼,y∼$\tilde{u}, \tilde{y}$ and G∼$\tilde{G}$ are respectively new input, output and gain matrix of the resulting PH system memristor added, RM≜GMf(xf)GT≻0$R_M\triangleq GM_f(x_f)G^{T}\succ 0$, since Mf(xf)≻0$M_f(x_f)\succ 0$, for all xf$x_f$. Hence, if the Hamiltonian function H(x)$H(x)$ is bounded from below, then the system is passive with respect to the supply rate y∼Tu∼$\tilde{y}^{T}\tilde{u}$.Memristor‐based IDA‐PBC techniqueThe IDA‐PBC technique, introduced in [7], has emerged as an easy methodology for designing passivity‐based controllers [8] and exploits the stability properties of PH system. It uses the passivity properties of the PH system to ensure the convergence of the system to the desired equilibrium point.The memristor system can be included to improve the classical IDA‐PBC method, the key idea is to define a desired target dynamic with the form14ẋẋf=Jd−Rd−GMfGT−GGT0∂Hd∂x∂HM∂xf,\begin{equation} {\left( \def\eqcellsep{&}\begin{array}{c}\dot{x}\\[3pt] \dot{x}_f \end{array} \right)}= {\left( \def\eqcellsep{&}\begin{array}{cc} J_d-R_d-GM_fG^{T} & -G \\[5pt] G^{T} & 0 \end{array} \right)} {\left( \def\eqcellsep{&}\begin{array}{c} \dfrac{\partial H_d}{\partial x}\\[15pt] \dfrac{\partial H_M}{\partial x_f} \end{array} \right)}, \end{equation}where Jd,Rd$J_d, R_d$ and Hd$H_d$ can be obtained with the classical IDA‐PBC approach, Md$M_d$ is the incremental memristance matrix of the added memristor and ∂HM/∂xf≡0$\partial H_M/\partial x_f\equiv 0$.Designing a method to stabilize the system (6) consists in finding a control law u such that (6) behaves as (14). In that case, the design procedure reduces to find matrices Jd(x),Rd(x)$J_d(x),R_d(x)$ and Mf(x)$M_f(x)$ and a desired Hamiltonian function, Hd(x)$H_d(x)$, such that the so‐called matching equation15(J−R)∂H∂x+Gu=(Jd−Rd−GMfGT)∂Hd∂x,\begin{equation} (J-R)\frac{\partial H}{\partial x}+Gu=(J_d-R_d-GM_fG^{T})\frac{\partial H_d}{\partial x}, \end{equation}is solved with the result that the controller becomes16ucm=uc+um=(GTG)−1GT(Jd−Rd)∂Hd∂x−(J−R)∂H∂x+(−MfGT)∂Hd∂x,\begin{eqnarray} u_{cm} &=&u_c+u_m\nonumber\\ &=& (G^{T}G)^{-1}G^{T}{\left((J_d-R_d)\frac{\partial H_d}{\partial x} -(J-R)\frac{\partial H}{\partial x}\right)}\\ &&+\,(-M_fG^{T})\frac{\partial H_d}{\partial x}, \nonumber\qquad \end{eqnarray}where uc$u_c$ corresponds to the controller design via the classical IDA‐PBC, and um$u_m$ includes the memristor dynamic. Let RM=GMf(x)GT≻0$R_M= GM_f (x)G^{T}\succ 0$, we can obtain the control laws17ucm=G+(Jd−Rd−RM)∂Hd∂x−(J−R)∂H∂x,\begin{equation} u_{cm}=G^{+}{\left((J_d-R_d-R_M)\frac{\partial H_d}{\partial x}-(J-R)\frac{\partial H}{\partial x}\right)}, \end{equation}and18uc=G+(Jd−Rd)∂Hd∂x−(J−R)∂H∂x,\begin{equation} u_c=G^{+}{\left((J_d-R_d)\frac{\partial H_d}{\partial x}-(J-R)\frac{\partial H}{\partial x}\right)}, \end{equation}19um=−MfGT∂Hd∂x,\begin{equation} u_m=-M_fG^{T}\frac{\partial H_d}{\partial x}, \end{equation}where G+≜(GTG)−1GT$G^{+}\triangleq (G^{T}G)^{-1}G^{T}$ is the Moore‐Penrose pseudo‐inverse of the matrix G(x)$G(x)$.1RemarkNote that, the matching equation corresponding to the classical IDA‐PBC is(J−R)∂H∂x+Gu=(Jd−Rd)∂Hd∂x,\begin{equation*} (J-R)\frac{\partial H}{\partial x}+Gu=(J_d-R_d)\frac{\partial H_d}{\partial x}, \end{equation*}which is just missing the third item on the right hand compared with the equation (15).Locally fixed‐time H∞$H_{\infty }$ controlConsider the following general nonlinear system20ẋ=f(x,u,ω)z=h(x,u,ω),\begin{equation} {\left\lbrace \def\eqcellsep{&}\begin{array}{ll} \dot{x }= f (x, u, \omega )\\[3pt] z = h(x,u,\omega ) \end{array} \right.}, \end{equation}where z∈Rq$z \in \mathbb {R}^q$ and ω∈Rs$\omega \in \mathbb {R}^s$ are the penalty signal and disturbance of the system, respectively; x,u$x, u$ are the same as those in the system (1), by considering the pre‐HamiltonianKγ(x,H,ω,u):=∂HT(x)∂xf(x,u,ω)−12γ2∥ω∥2+12∥z∥2.\begin{equation*} K_{\gamma }(x, H, \omega , u) := \frac{\partial H^T(x)}{\partial x}f(x, u, \omega )-\frac{1}{2}\gamma ^2\Vert \omega \Vert ^2 +\frac{1}{2}\Vert z\Vert ^2. \end{equation*}Suppose that Kγ$K_{\gamma }$ has a saddle‐point (u∗(x,H),ω∗(x,H))$(u^{*}(x, H),\omega ^{*}(x, H))$, that is, for all u,ω$u,\omega$,Kγ(x,H,ω(x,H),u∗(x,H))≤Kγ(x,H,ω∗(x,H),u∗(x,H))≤Kγ(x,H,ω∗(x,H),u(x,H)),\begin{equation*} \def\eqcellsep{&}\begin{array}{ll} &K_{\gamma }(x, H, \omega (x, H), u^{*}(x, H)) \\[5pt] &\quad\le\, K_{\gamma }(x, H, \omega ^{*}(x, H), u^{*}(x, H)) \\[5pt] &\quad \le\, K_{\gamma }(x, H, \omega ^{*}(x, H), u(x, H)), \end{array} \end{equation*}then we consider the Hamilton‐Jacobi inequality21(HJ1)Kγ(x,H,ω∗(x,H),u∗(x,H))≤0.\begin{equation} \mathbf {(H J1)}K_{\gamma }(x, H, \omega ^{*}(x, H), u^{*}(x, H))\le 0. \end{equation}3Definition[40] Consider the input signal space Rm$\mathbb {R}^m$ and the output signal space Rp$\mathbb {R}^p$, together with an input‐output mapping G:Rm↦Rp,u→y=G(u)$\mathcal {G}:\mathbb {R}^m\mapsto \mathbb {R}^p, u\rightarrow y=\mathcal {G}(u)$. The map G(u)$\mathcal {G}(u)$ is said to have finite L2‐gain if there exist finite constants γ and b such that∥G(u)∥≤γ∥u∥+b,\begin{equation*} \Vert \mathcal {G}(u)\Vert \le \gamma \Vert u\Vert +b, \end{equation*}G$\mathcal {G}$ is said to have finite L2‐gain with zero bias if b can be taken equal to zero, here norm ∥·∥$\Vert \cdot \Vert$ is the L2 norm. Let G$\mathcal {G}$ have finite L2$\mathcal {L}_2$‐gain. Then the L2‐gain of G$\mathcal {G}$ is defined asγ(G):=inf{γ|∃b,satisfies∥G(u)∥≤γ∥u∥+b}.\begin{equation*} \gamma (\mathcal {G}):=\inf {\lbrace \gamma |\exists b, satisfies\quad \Vert \mathcal {G}(u)\Vert \le \gamma \Vert u\Vert +b \rbrace }. \end{equation*}3Lemma(Proposition 6.1.3 of [40]) Let γ>0$\gamma &gt; 0$. Assume there exists saddle point (u∗(x,H),ω∗(x,H))$(u^{*}(x, H), \omega ^{*}(x, H))$ of Kγ(x,H,ω,u)$K_{\gamma }(x, H, \omega , u)$. Suppose there exists a Cr(k>r>1)$C^r (k &gt; r &gt; 1)$ solution H>0$H&gt; 0$ to the Hamilton‐Jacobin inequality (HJ1)$\mathbf {(H J1)}$ given by (21). Then the Cr−1$C^{r-1}$ state feedbacku=u∗x,∂HT(x)∂x,\begin{equation*} u = u^{*}{\left(x, \frac{\partial H^T(x)}{\partial x}\right)}, \end{equation*}is such that the closed‐loop systemẋ=fx,u∗x,∂HT∂x(x),ωz=hx,u∗x,∂HT(x)∂x,ω,\begin{equation*} {\left\lbrace \def\eqcellsep{&}\begin{array}{ll} \dot{x }= f{\left(x, u^{*}{\left(x, \dfrac{\partial H^T}{\partial x}(x)\right)}, \omega \right)}\\[11pt] z = h{\left(x,u^{*}{\left(x, \dfrac{\partial H^T(x)}{\partial x}\right)},\omega \right)} \end{array} , \right.} \end{equation*}has L2‐gain <γ$&lt; \gamma$.Consider the following PH system22ẋ=(J(x)−R(x))∂H∂x(x)+G(x)u+G1(x)ωz=M(x)GT(x)∂Hd∂x(x),\begin{equation} {\left\lbrace \def\eqcellsep{&}\begin{array}{ll} \dot{x}= (J(x)-R(x))\dfrac{\partial H}{\partial x}(x)+G(x)u+G_1(x)\omega \\[11pt] z= M(x)G^{T}(x)\dfrac{\partial H_d}{\partial x}(x) \end{array} \right.}, \end{equation}where x∈Uδ⊂Rn,H∈Rn,u∈Rm,G∈Rn×m,J$x\in U_{\delta }\subset \mathbb {R}^{n}, H\in \mathbb {R}^{n}, u\in \mathbb {R}^{ m},G\in \mathbb {R}^{n\times m},J$ and R are the same as those in system (6), ω∈Rq$\omega \in \mathbb {R}^{q}$ is the disturbance in L2, G1∈Rn×q$G_1\in \mathbb {R}^{n\times q}$ is the disturbance gain, z is the system's penalty function and M(x)∈Rm×m$M(x)\in \mathbb {R}^{m\times m}$ is a weighting matrix.4DefinitionThe locally fixed‐time H∞$H_{\infty }$ control problem of system (22) can be described as follows. For a given disturbance attenuation level γ>0$\gamma &gt; 0$, design a feedback controller23ucr(x)=uc(x)+ur(x)=uc−12MT(x)M(x)+12γ2ImGT(x)∂Hd∂x(x),\begin{eqnarray} \def\eqcellsep{&}\begin{array}{ll} u_{cr}(x)=u_c (x)+u_r(x)\\[11pt] \qquad \ \ =\,u_c-\left[\dfrac{1}{2}M^{T}(x)M(x)+\dfrac{1}{2 \gamma ^{2}} I_m\right]G^T(x) \dfrac{\partial H_{d}}{\partial x}(x), \end{array} \nonumber\\ \end{eqnarray}where uc$u_c$ is given as (18), such that the L2‐gain of the closed‐loop system (from ω to z) is not larger than γ , and the closed‐loop system is locally fixed‐time stable when ω=0$\omega = 0$.How to use the memristor improving the convergence speed and rejection disturbance properties of the PH system (6)? In the following section, the details are presented.MEMRISTOR‐BASED LOCALLY FIXED‐TIME STABILIZATION CONTROL FOR PH SYSTEMSIn order to investigate the locally fixed‐time stability of the closed‐loop system (6) with controllers (17) and (18), the desired Hamiltonian function of the PH system (6) is taken as24Hd(x)=∑i=1n(xi2)α2α−1,α>1.\begin{equation} H_d(x)=\sum \limits _{i=1}^{n}(x_{i}^{2})^\frac{\alpha }{2\alpha -1},\quad \alpha &gt;1. \end{equation}4Lemma(Theorem 1 of [39]) Consider the system (6), if the desired Hamiltonian function Hd$H_d$ is given as (24), Uδ$U_{\delta }$ is a neighborhood of the origin, and25k0≜min1≤i≤ninfx∈Uδ{σiRd(x)}>0,\begin{equation} k_0\triangleq \min \limits _{1\le i \le n}\inf \limits _ {x\in U_{\delta }} \lbrace \sigma _{i}^{R_d(x)}\rbrace &gt;0, \end{equation}where σiRd(x)$\sigma _{i}^{R_d(x)}$ is the eigenvalue of Rd(x)$R_d(x)$.Furthermore, if there exists a scalar C≜supx∈UδHd(x)=supx∈Uδ∑i=1n(xi2(t))α2α−1$C\triangleq \sup \nolimits _{x\in U_{\delta }}H_d(x)=\sup \nolimits _{x\in U_{\delta }}\sum \nolimits _{i=1}^{n}(x^{2}_{i}(t))^\frac{\alpha }{2\alpha -1}$ and 0<C<+∞$0&lt; C&lt;+\infty$, then the origin of system (6) is locally fixed‐time stabilized by classical IDA‐PBC controller26uc=G+(Jd−Rd)∂Hd∂x−(J−R)∂H∂x,\begin{equation} u_c=G^{+}{\left((J_d-R_d)\frac{\partial H_d}{\partial x}-(J-R)\frac{\partial H}{\partial x}\right)}, \end{equation}and the upper bound of the settling times can be estimated as27T0≜(2α−1)24k0α(α−1)cα−1α,\begin{equation} T_{0} \triangleq \frac{(2\alpha -1)^2}{4k_0 \alpha (\alpha -1)}c^{\frac{\alpha -1}{\alpha }}, \end{equation}which is independently of initial conditions.1TheoremIf Jd,Rd$J_d, R_d$ can be obtained with the classical IDA‐PBC approach, the desired Hamiltonian function Hd$H_d$ is given as (24), Uδ,k0$U_{\delta }, k_0$ and C are the same as those in Lemma 4, then the origin of the closed‐loop system (6) can be locally fixed‐time stabilized by memristor‐based IDA‐PBC controller28ucm=G+(Jd−Rd−RM)∂Hd∂x−(J−R)∂H∂x,\begin{equation} u_{cm}=G^{+}{\left((J_d-R_d-R_M)\frac{\partial H_d}{\partial x}-(J-R)\frac{\partial H}{\partial x}\right)}, \end{equation}and the settling time satisfies29T(x0)≤(2α−1)24k(x)α(α−1)Hdα−1α≤Tm≜(2α−1)24kmα(α−1)Cα−1α,∀x0∈Uδ,\begin{equation} \def\eqcellsep{&}\begin{array}{ll} T(x_0)\le \dfrac{(2\alpha -1)^2}{4k(x) \alpha (\alpha -1)}H_d^{\frac{\alpha -1}{\alpha }} \\[15pt] \qquad \ \ \le T_{m}\triangleq \dfrac{(2\alpha -1)^2}{4k_m \alpha (\alpha -1)}C^{\frac{\alpha -1}{\alpha }},\forall x_0\in U_{\delta }, \end{array} \end{equation}where k(x)≜k0+min1≤i≤n{σiRM(x)}$k(x)\triangleq k_0+\min \nolimits _{1\le i \le n}\lbrace \sigma _{i}^{R_M(x)}\rbrace$, σiRM(x)$\sigma _{i}^{R_M(x)}$ denotes the eigenvalue of RM(x)$R_M(x)$, RM=GMf(x)GT≻0$R_M= GM_f (x)G^{T}\succ 0$, Mf(x)$M_f (x)$ is the memristance, and km=infx∈Uδk(x)$k_m=\inf \nolimits _ {x\in U_{\delta }}k(x)$.ProofSince Hd(x)$H_d(x)$ can be a Lyapunov function candidate. Consider the following desired systems:30ẋ=Jd(x)−Rd(x)−RM(x)∂Hd(x)∂x,\begin{equation} \dot{x}={\left(J_d(x)-R_d(x)-R_M(x)\right)}\frac{\partial H_d(x)}{\partial x}, \end{equation}which is the equivalent equations for the state equations of PH system (6) corresponding to controller (28). For any x0∈Uδ$x_0\in U_{\delta }$, let x(t)≜x(t;0,x0)$x(t)\triangleq x(t;0,x_0)$ be the trajectory of system (30) starting from x0.Notice the skew‐symmetry of the matrix Jd(x)$J_d(x)$, by computing the derivative of Hd(x)$H_d(x)$ along the trajectory of (30), the following inequality can be obtainedḢd(x)=−∂HdT(x)∂xRd(x)+RM(x)∂Hd(x)∂x≤−k(x)∂HdT(x)∂x∂Hd(x)∂x=−k(x)2α2α−1(x12)1−α2α−1x1,…,2α2α−1(xn2)1−α2α−1xn×2α2α−1(x12)1−α2α−1x1,…,2α2α−1(xn2)1−α2α−1xnT=−k(x)(2α2α−1)2∑i=1n(xi2)12α−1.≤−k(x)(2α2α−1)2∑i=1n[(xi2)α2α−1]1α≤−k(x)(2α2α−1)2∑i=1n(xi2)α2α−11α=−k(x)(2α2α−1)2Hd1α,\begin{eqnarray*} \dot{H}_d(x) &=&-\frac{\partial H^{T}_d(x)}{\partial x}{\left(R_d(x)+R_M(x)\right)}\frac{\partial H_d(x)}{\partial x} \\ &\le& -k(x)\frac{\partial H^{T}_d(x)}{\partial x}\frac{\partial H_d(x)}{\partial x} \\ &=& -k(x){\left(\frac{2\alpha }{2\alpha -1}(x_1^2)^{\frac{1-\alpha }{2\alpha -1}}x_1,\ldots,\frac{2\alpha }{2\alpha -1}(x_n^2)^{\frac{1-\alpha }{2\alpha -1}}x_n\right)}\\ &&\times\, {\left(\frac{2\alpha }{2\alpha -1}(x_1^2)^{\frac{1-\alpha }{2\alpha -1}}x_1,\ldots,\frac{2\alpha }{2\alpha -1}(x_n^2)^{\frac{1-\alpha }{2\alpha -1}}x_n\right)}^T\\ &=& -k(x)(\frac{2\alpha }{2\alpha -1})^2 \sum \limits _{i=1}^{n}(x_i^2)^{\frac{1}{2\alpha -1}}. \\ &\le& -k(x)(\frac{2\alpha }{2\alpha -1})^2 \sum \limits _{i=1}^{n}[(x_i^2)^{\frac{\alpha }{2\alpha -1}}]^{\frac{1}{\alpha }}\\ &\le& -k(x)(\frac{2\alpha }{2\alpha -1})^2 {\left[\sum \limits _{i=1}^{n}(x_i^2)^{\frac{\alpha }{2\alpha -1}}\right]}^{\frac{1}{\alpha }}\\ &= &-k(x)(\frac{2\alpha }{2\alpha -1})^2 H_d^{\frac{1}{\alpha }}, \end{eqnarray*}notice that31k(x)=k0+min1≤i≤n{σiRM(x)},RM(x)≻0,\begin{equation} k(x)=k_0+\min \limits _{1\le i \le n}\lbrace \sigma _{i}^{R_M(x)}\rbrace , R_M(x)\succ 0, \end{equation}which implies32k(x)>k0>0.\begin{equation} k(x) &gt; k_0&gt;0. \end{equation}From Lemma 2 and Definition 1, it can be concluded that the origin is locally fixed‐time stable, and the settling time satisfies33T(x0)≤(2α−1)24k(x)α(α−1)Hdα−1α<Tm=(2α−1)24kmα(α−1)Cα−1α,∀x0∈Uδ.\begin{equation} \def\eqcellsep{&}\begin{array}{ll} T(x_0) \le \dfrac{(2\alpha -1)^2}{4k(x) \alpha (\alpha -1)}H_d^{\frac{\alpha -1}{\alpha }}\\[15pt] \qquad \ \ &lt; T_{m}= \dfrac{(2\alpha -1)^2}{4k_m \alpha (\alpha -1)}C^{\frac{\alpha -1}{\alpha }},\forall x_0\in U_{\delta }. \end{array} \end{equation}Therefore, the proof of Theorem 1 is completed.□$\Box$2RemarkNote that, in the proof of Theorem 1,k(x)=k0+min1≤i≤n{σiRM(x)},\begin{equation*} k(x)=k_0+\min \limits _{1\le i \le n}\lbrace \sigma _{i}^{R_M(x)}\rbrace , \end{equation*}andk(x)>k0>0,\begin{equation*} k(x) &gt;k_0&gt;0, \end{equation*}it can be obtained34T(x0)≤Tm=(2α−1)24kmα(α−1)Cα−1α<T0=(2α−1)24k0α(α−1)Cα−1α,∀x0∈Uδ.\begin{equation} \def\eqcellsep{&}\begin{array}{ll} T(x_0) \le T_m= \dfrac{(2\alpha -1)^2}{4k_m \alpha (\alpha -1)}C^{\frac{\alpha -1}{\alpha }} \\[15pt] \qquad \ \ &lt; T_{0}= \dfrac{(2\alpha -1)^2}{4k_0 \alpha (\alpha -1)}C^{\frac{\alpha -1}{\alpha }},\forall x_0\in U_{\delta }. \end{array} \end{equation}Because the settling time is larger (smaller), the energy decay speed of free moving system is slower (faster). Hence, the convergence speed of the memristor‐based controller ucm$u_{cm}$ in locally fixed‐time stabilizing PH system (6) is faster than that of the classical IDA‐PBC controller uc)$u_{c})$ in fixed‐time stabilizing PH system (6). Thus, the memristor‐based controller ucm$u_{cm}$ possesses better disturbance rejection performance than the classical IDA‐PBC controller uc)$u_{c})$.In (34), the settling time Tm$T_m$ is dependent on the value of k(x)=k0+min1≤i≤n{σiRM(x)}$k(x)=k_0+\min \nolimits _{1\le i \le n}\lbrace \sigma _{i}^{R_M(x)}\rbrace$, so it also depends on the memristance RM(x)=G(x)M(x)GT(x)$R_M(x)=G(x)M(x)G^T (x)$. In order to further enhance the disturbance rejection performance of the memristor‐based controller ucm$u_{cm}$, we impose on the following assumptions for the memristance function, see Figure 2.1Assumption(H1)$(\mathbf {H}_1)$The memristance function r(t)=m(ei(t))$r(t)=m(e_i(t))$ is even and continuous when ei(t)∈(−Q,Q)$e_i(t)\in (-Q,Q)$, Q∈R+$Q\in \mathbb {R}^{+}$, Roff>Ron>0$R_{off}&gt;R_{on}&gt;0$ andm(ei(t))=Ron,ei∼(t)≥Q,Roff,ei∼(t)=0.\begin{equation*} m(e_i(t))={\left\lbrace \def\eqcellsep{&}\begin{array}{ll} R_{on},\widetilde{e_i}(t)\ge Q,\\[7pt] R_{off},\widetilde{e_i}(t)= 0. \end{array} \right.} \end{equation*}(H2)$(\mathbf {H}_2)$The memristance function r(t)=m(ei∼(t))$r(t)=m(\widetilde{e_i}(t))$ is monotonically decreasing and its inverse function ei∼(t)=m−1(r(t))$\widetilde{e_i}(t)=m^{-1}(r(t))$ exists, as ei∼(t)∈(0,Q),i=1,…,n$\widetilde{e_i}(t)\in (0,Q),i=1,\ldots,n$,where e(t)=(e1(t),…,en(t))=((x1(t)−x1∗),…,(xn(t)−xn∗))=x(t)−x∗$e(t)=(e_1(t),\ldots,e_n(t))=((x_1(t)-{x}_{1}^{*}),\ldots,(x_n(t)-{x}_{n}^{*}))=x (t)-x^{*}$ is the state error, x is the state and x∗$x^{*}$ is the desired equilibrium state of system (6), and ei∼(t)=|ei(t)|,i=1,…,n$\widetilde{e_i}(t)=|e_i(t)|,i=1,\ldots,n$.3RemarkIn [23] and [24], the memristance functions are respectively designed asm1(ei(t))=ρe−σei(t),\begin{equation*} m_1(e_i(t))=\rho e^{-\sigma e_i(t)}, \end{equation*}andm2(ei(t))=Ron,|e(t)|≥QRon+(Roff−Ron)Q−|ei(t)|Q,|ei(t)|<Q,\begin{equation*} m_2(e_i(t))={\left\lbrace \def\eqcellsep{&}\begin{array}{ll} R_{on}, |e(t)|\ge Q\\[15pt] R_{on}+(R_{off}-R_{on})\dfrac{Q-|e_i(t)|}{Q}, |e_i(t)|&lt;Q, \end{array} \right.} \end{equation*}which obviously meet the Assumption 1, where ei=xi−xi∗$e_i=x_i-x_i^{*}$ is the error of partial state. We will adopt appropriate memristance function of memristor, such that the state‐dependent memristor should be designed as that the value of damping is low when the error of state is high, and then it drastically increases when the error is low.2TheoremConsider the PH system (6), x∗$x^{*}$ is an equilibrium point, and the desired Hamiltonian function Hd(x)=∑i=1n[(xi−xi∗)2]α2α−1,α>1$H_d(x)=\sum \nolimits _{i=1}^{n}[(x_{i}-x_{i}^{*})^{2}]^\frac{\alpha }{2\alpha -1}, \alpha &gt;1$, Uδ$U_{\delta }$ is a neighborhood of x∗$x^{*}$, and35k01=min1≤i≤ninfx∈Uδ{σiR(x)}>0,\begin{equation} k_{01}=\min \limits _{1\le i \le n}\inf \limits _ {x\in U_{\delta }} \lbrace \sigma _{i}^{R(x)}\rbrace &gt;0, \end{equation}36C1=supx∈Uδ∑i=1n[(xi−xi∗)2]α2α−1>0,\begin{equation} C_{1}=\sup \limits _{x\in U_{\delta }} \left\lbrace \sum \limits _{i=1}^{n}[(x_{i}-x_{i}^{*})^{2}]^\frac{\alpha }{2\alpha -1}\right\rbrace &gt;0, \end{equation}where σiR(x),i=1,2,…,n$\sigma _{i}^{R(x)}, i=1,2,\ldots,n$, denote the eigenvalues of R(x)$R(x)$.(a)Then, the equilibrium point x∗$x^{*}$ of the closed‐loop system (6) is locally fixed‐time stabilized in the neighborhood Uδ$U_{\delta }$ by the memristor‐based feedback controller37ucM≜uc+uM,\begin{equation} u_{cM}\triangleq u_c+u_M, \end{equation}and the settling time satisfies38T(x0)≤T̂=(2α−1)24kM(x)α(α−1)C1α−1α≤TM≜(2α−1)24kMα(α−1)C1α−1α,∀x0∈Uδ,\begin{equation} \def\eqcellsep{&}\begin{array}{ll} T(x_0) \le \hat{T}=\dfrac{(2\alpha -1)^2}{4k_M(x) \alpha (\alpha -1)}C_{1}^{\frac{\alpha -1}{\alpha }}\\[19pt] \qquad \ \ \ \le T_{M}\triangleq \dfrac{(2\alpha -1)^2}{4k_M \alpha (\alpha -1)}C_{1}^{\frac{\alpha -1}{\alpha }}, \forall x_0\in U_{\delta }, \end{array} \end{equation}where uc$u_c$ is given as (18), uM=−M(e(t))GT∂H∂x$u_M=-M(e(t))G^{T}\frac{\partial H}{\partial x}$, M(e(t))=diag{m(e1(t)),m(e2(t)),…,m(em(t))}$M(e(t)) = diag\lbrace m(e_1(t)),m(e_2(t)), \ldots,m(e_m(t))\rbrace$, andkM(x)≜k01+min1≤i≤n{σiG(x)M(x−x∗)GT(x)},kM=infx∈UδkM(x);{\fontsize{9.8}{11.8}{\selectfont{ \begin{equation*} k_M(x)\triangleq k_{01}+\min \limits _{1\le i \le n}\lbrace \sigma _{i}^{G(x)M(x-x^{*})G^{T}(x)}\rbrace , k_M=\inf \limits _ {x\in U_{\delta }}k_M(x); \end{equation*}}}}(b)In addition, if the memristance function M(ei(t)),i=1,2,…,m$M(e_i(t)), i=1,2,\ldots, m$, meet the Assumption 1, then the state error e(t)$e(t)$ accelerate convergence to the origin of the closed‐loop system (6).2FIGUREThe dependence of memristance functions on state errorProof(a)The Proof is similar to the Proof of the Theorem 1, so it is omitted here.(b)The minimum positive eigenvalue of RM=G(x)M(x)GT(x)$R_M=G(x)M(x)G^T(x)$ is larger, the value of kM(x)$k_M(x)$ is larger, the settling time T̂$\hat{T}$ is shorter. Because assumptions (H1)$(\mathbf {H}_1)$ and (H2)$(\mathbf {H}_2)$ are satisfied, whenever ei∼(t)∈(0,Q),i=1,…,n$\widetilde{e_i}(t)\in (0,Q), i=1,\ldots,n$, it will lead to the values of m(ei(t)),i=1,…,m$m(e_i(t)), i=1,\ldots,m$, continuously increase to Roff$R_{off}$. As the minimum positive eigenvalue of RM=G(x)M(x)GT(x)$R_M=G(x)M(x)G^T(x)$ and kM(x)$k_M(x)$ are increase, the settling time T̂$\hat{T}$ is decrease, thus the convergence speed of controller (37) in locally fixed‐time stabilization PH system (6) is accelerated. Hence, the value of m(ei(t)),i=1,…,m$m(e_i(t)),i=1,\ldots,m$, continuously increases to Roff$R_{off}$, and can make error state e(t)$e(t)$ accelerate convergence to the origin. The proof of Theorem 2 is complete.□$\Box$MEMRISTOR‐BASED LOCALLY FIXED‐TIME H∞$H_{\infty }$ CONTROL FOR PH SYSTEMSIn the following, we discuss memristor‐based locally fixed‐time H∞$H_{\infty }$ control for PH systems in the presence of external disturbances.Consider the following PH systems39ẋ=(J(x)−R(x))∂H∂x(x)+G(x)u+G1(x)ωz=M(x)GT(x)∂Hd∂x(x),\begin{equation} {\left\lbrace \def\eqcellsep{&}\begin{array}{ll} \dot{x}=(J(x)-R(x))\dfrac{\partial H}{\partial x}(x)+G(x)u+G_1(x)\omega \\[15pt] z= M(x)G^{T}(x)\dfrac{\partial H_d}{\partial x}(x) \end{array} , \right.} \end{equation}where x,H∈Rn,u∈Rm,G∈Rn×m,J$x, H\in \mathbb {R}^{n}, u\in \mathbb {R}^{ m},G\in \mathbb {R}^{n\times m},J$ and R are the same as those in system (6), ω∈Rq$\omega \in \mathbb {R}^{q}$ is the disturbance in L2, G1∈Rn×q$G_1\in \mathbb {R}^{n\times q}$ is the disturbance gain, z is the system's penalty function, M(x)∈Rm×m$M(x)\in \mathbb {R}^{m\times m}$ is an incremental memristance matrix and Hd(x)$H_d (x)$ is given as (24).3TheoremIf let rankG(x)=m<n$rankG(x)=m&lt;n$ and the disturbance attenuation level γ>0$\gamma &gt;0$ be given, Rd,Jd,Uδ,k0,C$R_d, J_d, U_{\delta }, k_0, C$ and uc$u_c$ are the same as those in the Lemma 4, and40Rd(x)+12γ2[G(x)GT(x)−G1(x)G1T(x)]⪰0,\begin{equation} R_d(x)+\frac{1}{2\gamma ^{2}}[G(x)G^{T}(x)-G_{1}(x)G_{1}^{T} (x)]\succeq 0, \end{equation}then the locally fixed‐time H∞$H_{\infty }$ control problem (39) can be solved by the following memristor‐based H∞$H_{\infty }$ controller41ucr(x)=uc(x)+ur(x)=uc−12MT(x)M(x)+12γ2ImGT(x)∂Hd∂x(x).\begin{eqnarray} u_{cr}(x) &=& u_c (x)+u_r(x)\nonumber\\ &=& u_c-\left[\frac{1}{2}M^{T}(x)M(x)+\frac{1}{2 \gamma ^{2}} I_m\right]G^T(x) \frac{\partial H_{d}}{\partial x}(x).\nonumber\\ \end{eqnarray}Furthermore, if the Assumption 1 also holds, then state x(t)=(x1(t),x2(t),…,xn(t)$x(t)=(x_1(t),x_2(t),\ldots,x_n(t)$ of system (39) accelerate converge to the origin, when x(t)∈(−Q,Q)$x(t)\in (-Q,Q)$, and ω=0$ \omega =0$.ProofSubstituting (41) into system (39), the following closed‐loop PH system can be obtained42ẋ=(J(x)−R(x))∂H∂x(x)+G(x)uc−G(x)12MT(x)M(x)+12γ2ImGT(x)∂Hd∂x(x)+G1(x)ω=[Jd(x)−(Rd(x)+Rr(x))]∂Hd∂x(x)+G1(x)ω,{\fontsize{9.5}{11.5}{\selectfont{ \begin{eqnarray} \dot{x}&=&(J(x)-R(x))\frac{\partial H}{\partial x}(x)+G(x)u_c\nonumber\\ &&-\,G(x) \left[\frac{1}{2}M^{T}(x)M(x)+\frac{1}{2 \gamma ^{2}} I_m\right]G^T(x) \frac{\partial H_{d}}{\partial x}(x)+G_1(x)\omega\nonumber\\ &=&[J_d(x)-(R_d(x)+R_r(x))]\frac{\partial H_d}{\partial x}(x)+G_1(x)\omega , \end{eqnarray}}}}where Rd$R_d$ and Jd$J_d$ are obtained from classical IDA‐PBC technique, Rr(x)=G(x)[12MT(x)M(x)+12γ2Im]GT(x)$R_r(x)=G(x)[\frac{1}{2}M^{T}(x)M(x)+\frac{1}{2 \gamma ^{2}} I_m]G^T(x)$. Notice the positivity properties of memristance M(x)MT(x)$M(x)M^T(x)$ and Rr(x)$R_r(x)$, let43kr(x)≜k0+min1≤i≤n{σiRr(x)},\begin{equation} k_r(x)\triangleq k_0+\min \limits _{1\le i \le n}\lbrace \sigma _{i}^{R_r(x)}\rbrace , \end{equation}where σiRr(x)$\sigma _{i}^{R_r(x)}$ denotes the eigenvalues of Rr(x)$R_r(x)$, which implies44kr(x)>k0>0.\begin{equation} k_r(x) &gt; k_0&gt;0. \end{equation}When ω=0$\omega = 0$, taking advantage of the method similar to the proof of Theorem 1, it is not difficult to obtain that the closed‐loop system (42) can be locally fixed‐time stabilized at the origin. Computing the derivative of Hd(x)$H_d(x)$ along the trajectory of (42) we can obtain the following inequalityḢd(x)=−∂HdT(x)∂x(Rd(x)+Rr(x))∂Hd(x)∂x≤−kr(x)∂HdT(x)∂x∂Hd(x)∂x=−kr(x)2α2α−1(x12)1−α2α−1x1,…,2α2α−1(xn2)1−α2α−1xn×2α2α−1(x12)1−α2α−1x1,…,2α2α−1(xn2)1−α2α−1xnT=−kr(x)2α2α−12∑i=1n(xi2)12α−1.\begin{eqnarray*} \dot{H}_d(x) &=&-\frac{\partial H^{T}_d(x)}{\partial x}(R_d(x)+R_r(x))\frac{\partial H_d(x)}{\partial x}\\ &\le& -k_r(x)\frac{\partial H^{T}_d(x)}{\partial x}\frac{\partial H_d(x)}{\partial x}\\ &=& -k_r(x){\left(\frac{2\alpha }{2\alpha -1}(x_1^2)^{\frac{1-\alpha }{2\alpha -1}}x_1,\ldots,\frac{2\alpha }{2\alpha -1}(x_n^2)^{\frac{1-\alpha }{2\alpha -1}}x_n\right)}\\ &&\times\, {\left(\frac{2\alpha }{2\alpha -1}(x_1^2)^{\frac{1-\alpha }{2\alpha -1}}x_1,\ldots,\frac{2\alpha }{2\alpha -1}(x_n^2)^{\frac{1-\alpha }{2\alpha -1}}x_n\right)}^T\\ &=& -k_r(x)\left(\frac{2\alpha }{2\alpha -1}\right)^2 \sum \limits _{i=1}^{n}(x_i^2)^{\frac{1}{2\alpha -1}}. \end{eqnarray*}Because of α>1$\alpha &gt;1$, by using the Jenson's inequality (3) we obtainḢd(x)≤−kr(x)2α2α−12∑i=1n(xi2)α2α−11α≤−kr(x)2α2α−12∑i=1n(xi2)α2α−11α=−kr(x)2α2α−12Hd1α.\begin{eqnarray*} \dot{H}_d(x) &\le& -k_r(x)\left(\frac{2\alpha }{2\alpha -1}\right)^2 \sum \limits _{i=1}^{n} \left[(x_i^2)^{\frac{\alpha }{2\alpha -1}}\right]^{\frac{1}{\alpha }}\\ &\le& -k_r(x) \left(\frac{2\alpha }{2\alpha -1}\right)^2 {\left[\sum \limits _{i=1}^{n}(x_i^2)^{\frac{\alpha }{2\alpha -1}}\right]}^{\frac{1}{\alpha }}\\ &=& -k_r(x)\left(\frac{2\alpha }{2\alpha -1}\right)^2 H_d^{\frac{1}{\alpha }}. \end{eqnarray*}From Lemma 2, we obtain that the origin is a locally fixed‐time stable equilibrium of PH system (39), and that the settling time satisfies45T(x0)≤T̂r=(2α−1)24kr(x)α(α−1)Cα−1α≤Tr≜(2α−1)24krα(α−1)Cα−1α,∀x0∈Uδ,\begin{eqnarray} T(x_0) &\le& \hat{T}_r=\frac{(2\alpha -1)^2}{4k_r(x) \alpha (\alpha -1)}C^{\frac{\alpha -1}{\alpha }}\nonumber\\ &\le& T_{r}\triangleq \frac{(2\alpha -1)^2}{4k_r \alpha (\alpha -1)}C^{\frac{\alpha -1}{\alpha }}, \forall x_0\in U_{\delta }, \end{eqnarray}where kr=infx∈Uδkr(x)$k_r=\inf \nolimits _ {x\in U_{\delta }}k_r(x)$.Next, we show that the L2 gain (from ω to z) of the closed‐loop system consisting of system (39) is bounded by γ.When ω≠0$\omega \ne 0$, computing the derivative of the Hamiltonian function Hd(x)$H_d(x)$ along the trajectories of system (42) and using the penalty function yield46Hḋ(x)=−∂HdT∂x(x)[Jd(x)−(Rd(x)+Rr(x))]∂Hd∂x(x)+∂HdT∂x(x)G1(x)ω=−∂HdT∂x(x)[Jd(x)−(Rd(x)+Rr(x))]∂Hd∂x(x)−12∥γω−1γG1T(x)∂Hd∂x(x)∥2+12{γ2∥ω∥2−∥z∥2}+12∂HdT∂x(x)G(x)MT(x)M(x)GT(x)∂Hd∂x(x)+12γ2∂HdT∂x(x)G1(x)G1T(x)∂Hd∂x(x)=−∂HdT∂x(x){Rd(x)+12γ2[G(x)GT(x)−G1(x)G1T(x)]}∂Hd∂x(x)+12{γ2∥ω∥2−∥z∥2}−12∥γω−1γG1T(x)∂Hd∂x(x)∥2≤−∂HdT∂x(x){Rd(x)+12γ2[G(x)GT(x)−G1(x)G1T(x)]}∂Hd∂x(x)+12{γ2∥ω∥2−∥z∥2},\begin{eqnarray} \dot{H_d}(x)&=& -\frac{\partial H^T_d}{\partial x}(x)[J_d(x)-(R_d(x)+R_r(x))] \frac{\partial H_d}{\partial x}(x)\nonumber\\ &&+\frac{\partial H^T_d}{\partial x}(x)G_1(x)\omega \nonumber\\ &=&-\frac{\partial H^T_d}{\partial x}(x)[J_d(x)-(R_d(x)+R_r(x))]\frac{\partial H_d}{\partial x}(x)\nonumber\\ &&-\frac{1}{2}\Vert \gamma \omega -\frac{1}{\gamma }G^T_1 (x)\frac{\partial H_d}{\partial x}(x)\Vert ^2+\frac{1}{2}\lbrace \gamma ^2 \Vert \omega \Vert ^2-\Vert z\Vert ^2\rbrace \nonumber\\ &&+\frac{1}{2}\frac{\partial H^T_d}{\partial x}(x)G(x)M^T(x)M(x)G^T(x)\frac{\partial H_d}{\partial x}(x)\nonumber\\ &&+\frac{1}{2\gamma ^2}\frac{\partial H^T_d}{\partial x}(x)G_1(x)G^T_1(x)\frac{\partial H_d}{\partial x}(x)\nonumber\\ &=&-\frac{\partial H^T_d}{\partial x}(x)\lbrace R_d(x)+\frac{1}{2\gamma ^{2}}[G(x)G^{T}(x)\nonumber\\ &&-G_{1}(x)G_{1}^{T} (x)]\rbrace \frac{\partial H_d}{\partial x}(x)+\frac{1}{2}\lbrace \gamma ^2 \Vert \omega \Vert ^2-\Vert z\Vert ^2\rbrace \nonumber\\ &&-\frac{1}{2}\Vert \gamma \omega -\frac{1}{\gamma }G^T_1 (x)\frac{\partial H_d}{\partial x}(x)\Vert ^2\nonumber\\ &\le& -\frac{\partial H^T_d}{\partial x}(x)\lbrace R_d(x)+\frac{1}{2\gamma ^{2}}[G(x)G^{T}(x)\nonumber\\ &&-G_{1}(x)G_{1}^{T} (x)]\rbrace \frac{\partial H_d}{\partial x}(x)+\frac{1}{2}\lbrace \gamma ^2 \Vert \omega \Vert ^2-\Vert z\Vert ^2\rbrace , \end{eqnarray}where ∥·∥$\Vert \cdot \Vert$ is the L2 norm. From (46), the γ‐dissipation inequality is obtained47Hḋ(x)+P(x)≤12{γ2∥ω∥2−∥z∥2},\begin{equation} \dot{H_d}(x)+P(x)\le \frac{1}{2}\lbrace \gamma ^2 \Vert \omega \Vert ^2-\Vert z\Vert ^2\rbrace , \end{equation}the condition (40) implies48P(x)=∂HdT∂x(x){Rd(x)+12γ2[G(x)GT(x)−G1(x)G1T(x)]}∂Hd∂x(x)≥0.\begin{equation} \def\eqcellsep{&}\begin{array}{ll} P(x)=\dfrac{\partial H^T_d}{\partial x}(x)\lbrace R_d(x)+\dfrac{1}{2\gamma ^{2}}[G(x)G^{T}(x)\\[15pt] \qquad\quad -\,G_{1}(x)G_{1}^{T} (x)]\rbrace \dfrac{\partial H_d}{\partial x}(x)\ge 0. \end{array} \end{equation}So, it is easy to obtainHḋ(x)≤12{γ2∥ω∥2−∥z∥2},\begin{eqnarray*} \dot{H_d}(x)\le \frac{1}{2}\lbrace \gamma ^2 \Vert \omega \Vert ^2-\Vert z\Vert ^2\rbrace , \end{eqnarray*}this means the (HJ1)$\mathbf {(HJ1)}$ given by (21) holds.Thus, it follows from the Lemma 3, the L2‐gain of the closed‐loop system (39) is bounded by the given disturbance attenuation level γ. Therefore, the locally fixed‐time H∞$H_{\infty }$ control problem (39) is solved by control law (41).At last, if the Assumption 1 holds, whenever x∼(t)=|x(t)|∈(0,Q)$\widetilde{x}(t)=|x(t)|\in (0,Q)$ will lead to the values of M(xi(t)),i=1,…,m$M(x_i(t)), i=1,\ldots,m$ continuously increase to Roff$R_{off}$. From (45), when the minimum eigenvalue ofRr(x)=G(x)[12MT(x)M(x)+12γ2Im]GT(x)\begin{eqnarray*} R_r(x)=G(x)[\frac{1}{2}M^{T}(x)M(x)+\frac{1}{2 \gamma ^{2}} I_m]G^T(x) \end{eqnarray*}and kr(x)$k_r(x)$ increase, the settling time T̂r$\hat{T}_r$ decreases, thus the convergence speed of fixed‐time stabilization for PH system (39) with controller (41) is accelerated. Hence, the value of Rr(xi(t)),i=1,…,m$R_r (x_i(t)),i=1,\ldots,m$, which continuously increases to Roff$R_{off}$, can lead to system states x(t)$x(t)$ accelerate converge to the origin.□$\Box$4RemarkIn Theorem 3, when the disturbance ω≠0$\omega \ne 0$, and its amplitude is η, η∈(0,Q)$\eta \in (0,Q)$, if the Assumption 1 holds, then system's state x(t)$x(t)$ in fixed time accelerately converges to a neighborhood (U(o,ε),0<ε≪η$U(o,\epsilon ), 0&lt;\epsilon \ll \eta$) of the origin. Besides, the incremental memristance matrix M(x)$M(x)$, emerges not only in penalty function but also in the memristor‐based locally fixed‐time H∞$H_{\infty }$ feedback controller ucr=uc−[12MT(x)M(x)+12γ2Im]GT(x)∂Hd∂x(x)$u_{cr}=u_c-[\frac{1}{2}M^{T}(x)M(x)+\frac{1}{2 \gamma ^{2}} I_m]G^T(x)\frac{\partial H_{d}}{\partial x}(x)$.5RemarkFrom the Theorem 1–Theorem 3, the energy level C≜supx∈Uδ∑i=1n(xi2(t))α2α−1$C\triangleq \sup \nolimits _{x\in U_{\delta }}\sum \nolimits _{i=1}^{n}(x^{2}_{i}(t))^\frac{\alpha }{2\alpha -1}$ and the upper bounds of the settling time Tξ≜(2α−1)24kξα(α−1)Cα−1α$T_{\xi }\triangleq \frac{(2\alpha -1)^2}{4k_{\xi } \alpha (\alpha -1)}C^{\frac{\alpha -1}{\alpha }}$, where ξ denote the subscripts m,M$m, M$ and r, respectively. Therefore, Uδ,C$U_{\delta }, C$ and Tξ$T_{\xi }$ change in the same direction, that is, the bigger is Uδ$U_{\delta }$, the bigger is C, thus the bigger is Tξ$T_{\xi }$. Furthermore, in order to obtain faster convergence speed, the key parameter α should better be 2>α>1$2&gt;\alpha &gt;1$, and kξ$k_{\xi }$ should be as large as possible.ILLUSTRATIVE EXAMPLEExample 1:In this subsection, numerical simulations are used to demonstrate the correctness of our theoretical results. In order to compare our memristor‐based control laws with the classical IDA‐PBC method in convergence speed and the effect in rejection periodic disturbance, we consider the locally fixed‐time stabilization problem of the following PH system:49ẋ=(J(x)−R(x))∂H∂x(x)+G(x)u+G1(x)ω,\begin{equation} \dot{x}=(J(x)-R(x))\frac{\partial H}{\partial x}(x)+G(x)u+G_1(x)\omega , \end{equation}where x=(x1,x2,x3)T∈R3,x0=x(t0)=x(0)=(1,−2,2)T$x {=}(x_1,x_2,x_3)^T {\in} \mathbb {R}^{3}, x_0=x(t_0)=x(0)=(1,-2,2)^T$, H(x)=12(x12+x22)+x3127,u=(u1,u2)T∈R2$H(x)=\frac{1}{2}(x_1^2+x_2^2)+x_3^{\frac{12}{7}}, u=(u_1,u_2)^T\in \mathbb {R}^{ 2}$, ω=5sign(sin6t),t∈[0.5,4]$ \omega = 5sign(sin6t),t\in [0.5,4]$ is a periodic disturbance in L2, andJ(x)=010−10−1010,R(x)=000010002,\begin{eqnarray*} J(x)={\left( \def\eqcellsep{&}\begin{array}{ccc}0 & 1 & 0 \\[3pt] -1 & 0 & -1 \\[3pt] 0 & 1 & 0 \end{array} \right)},\quad R(x)={\left( \def\eqcellsep{&}\begin{array}{ccc}0 & 0 & 0 \\[3pt] 0 & 1 & 0 \\[3pt] 0 & 0 & 2 \end{array} \right)}, \end{eqnarray*}G(x)=100110,G1(x)=100001.\begin{eqnarray*} G(x)={\left( \def\eqcellsep{&}\begin{array}{cc}1 & 0 \\[3pt] 0& 1 \\[3pt] 1 & 0 \end{array} \right)},\quad G_1(x)={\left( \def\eqcellsep{&}\begin{array}{cc}1 & 0 \\[3pt] 0& 0 \\[3pt] 0 & 1 \end{array} \right)}. \end{eqnarray*}The desired Hamiltonian function Hd$H_d$ is given asHd(x)=(x12)67+(x22)67+(x32)67,\begin{eqnarray*} H_d(x)=(x_1^2)^\frac{6}{7}+(x_2^2)^\frac{6}{7}+(x_3^2)^\frac{6}{7}, \end{eqnarray*}Jd,Rd$J_d, R_d$ can be obtained with the IDA‐PBC approachJd(x)=001000−100,Rd(x)=100010001,\begin{eqnarray*} J_d(x)={\left( \def\eqcellsep{&}\begin{array}{ccc}0 & 0 & 1 \\[3pt] 0 & 0 & 0 \\[3pt] -1 & 0 & 0 \end{array} \right)},\quad \quad R_d(x)={\left( \def\eqcellsep{&}\begin{array}{ccc}1 & 0 & 0 \\[3pt] 0 & 1 & 0 \\[3pt] 0 & 0 & 1 \end{array} \right)}, \end{eqnarray*}From (18), it can be obtained50uc=G+(Jd−Rd)∂Hd∂x−(J−R)∂H∂x=−x2−127x157+127x357x1+x2−127x257+127x357,\begin{eqnarray} u_c &=& G^{+}{\left((J_d-R_d)\frac{\partial H_d}{\partial x}-(J-R) \frac{\partial H}{\partial x}\right)}\nonumber\\ &=& \left( \def\eqcellsep{&}\begin{array}{c} -x_2-\dfrac{12}{7}x_1^{\frac{5}{7}}+ \dfrac{12}{7}x_3^{\frac{5}{7}}\\[11pt] x_1+x_2-\dfrac{12}{7}x_2^{\frac{5}{7}}+\dfrac{12}{7}x_3^{\frac{5}{7}} \end{array} \right), \end{eqnarray}where G+≜(GTG)−1GT$G^{+}\triangleq (G^{T}G)^{-1}G^{T}$ is the Moore–Penrose pseudo‐inverse of the matrix G(x)$G(x)$. Let the incremental memrisance matrix51M(x)=m(x1)00m(x2),\begin{equation} M(x)={\left( \def\eqcellsep{&}\begin{array}{cc}m(x_1) & 0 \\[3pt] 0 & m(x_2) \end{array} \right)}, \end{equation}where m(xi(t)),i=1,2$m(x_i(t)), i=1,2$, meet with the relation52m(xi(t))=ρe−σxi2(t)2,\begin{equation} m(x_i(t))=\rho e^{-\sigma \frac{x^2_i(t)}{2}}, \end{equation}xi(t),i=1,2$x_i(t),i=1,2$, are partial state of system (49), ρ=6,σ=0.1$\rho =6, \sigma =0.1$, see Figure 3.3FIGUREMemristance function depends on the stateThus the memristive control law becomes53um=−M(x)GT∂Hd∂x=−127m(x1)x157+m(x1)x357m(x2)x257,\begin{eqnarray} u_m &=& -M(x)G^{T}\frac{\partial H_d}{\partial x}\nonumber\\ &=&-\frac{12}{7} \left( \def\eqcellsep{&}\begin{array}{c} m(x_1)x_1^{\frac{5}{7}}+m(x_1)x_3^{\frac{5}{7}} \\[9pt] m(x_2)x_2^{\frac{5}{7}} \end{array} \right), \end{eqnarray}and54ucm=uc+um=−x2−12+12m(x1)7x157+12−12m(x1)7x357x1+x2−12+12m(x2)7x257+127x357.\begin{eqnarray} u_{cm} &=& u_c+u_m\nonumber\\ &=& \left[ \def\eqcellsep{&}\begin{array}{c} -x_2-\dfrac{12+12m(x_1)}{7}x_1^{\frac{5}{7}}+ \dfrac{12-12m(x_1)}{7}x_3^{\frac{5}{7}}\\[15pt] x_1+x_2-\dfrac{12+12m(x_2)}{7}x_2^{\frac{5}{7}}+\dfrac{12}{7}x_3^{\frac{5}{7}} \end{array} \right]. \qquad \end{eqnarray}Similarly, from (42) we get55ucr(x)=uc(x)+ur(x)=uc−12MT(x)M(x)+12γ2ImGT(x)∂Hd∂x(x)=−x2−12+6m2(x1)+1γ27x157+12−6m2(x1)+1γ27x357x1+x2−12+6m2(x2)+1γ27x257+127x357.{\fontsize{8.3}{10.3}{\selectfont{ \begin{eqnarray} u_{cr}(x) &=& u_c (x)+u_r(x)\nonumber\\ &=&u_c-\left[\frac{1}{2}M^{T}(x)M(x)+\frac{1}{2 \gamma ^{2}} I_m\right]G^T(x) \dfrac{\partial H_{d}}{\partial x}(x)\nonumber\\ &=& \left[ \def\eqcellsep{&}\begin{array}{c} -x_2-\dfrac{12+6\left(m^2(x_1)+\frac{1}{\gamma ^2}\right)}{7}x_1^{\frac{5}{7}} + \dfrac{12-6 \left(m^2(x_1)+\frac{1}{\gamma ^2}\right)}{7}x_3^{\frac{5}{7}}\\[15pt] x_1+x_2-\dfrac{12+6 \left(m^2(x_2)+\frac{1}{\gamma ^2}\right)} {7}x_2^{\frac{5}{7}}+\dfrac{12}{7}x_3^{\frac{5}{7}} \end{array} \right].\nonumber\\ \end{eqnarray}}}}Noticek0=min1≤i≤3infx∈Rn{σiRd(x)}=1>0,\begin{eqnarray*} k_0=\min \limits _{1\le i \le 3}\inf \limits _ {x\in \mathbb {R}^{n}} \lbrace \sigma _{i}^{R_d(x)}\rbrace =1&gt;0 , \end{eqnarray*}from Theorem 1–Theorem 3, we can draw a conclusion that the locally fixed‐time stabilizing problem (49) can be solved by control laws uc,ucm$u_c,u_{cm}$ and ucr$u_{cr}$. Furthermore, when ω=0$\omega =0$, the settling times satisfyTc(x0)≤4924k0C16,Tcm(x0)≤4924km(x)C16≤4924kmC16,\begin{eqnarray*} T_c(x_0) \le \frac{49}{24k_0}C^{\frac{1}{6}}, \quad T_{cm}(x_0) \le \frac{49}{24k_m(x)}C^{\frac{1}{6}}\le \frac{49}{24k_m}C^{\frac{1}{6}}, \end{eqnarray*}Tcr(x0)≤4924kr(x)C16≤4924krC16,\begin{eqnarray*} T_{cr}(x_0) \le \frac{49}{24k_r(x)}C^{\frac{1}{6}}\le \frac{49}{24k_r}C^{\frac{1}{6}}, \end{eqnarray*}respectively, wherekm=infx∈Uδkm(x),kr=infx∈Uδkr(x),\begin{eqnarray*} k_m=\inf \limits _ {x\in U_{\delta }}k_m(x), \quad \quad k_r=\inf \limits _ {x\in U_{\delta }}k_r(x), \end{eqnarray*}andkm(x)=min1≤i≤3{σiRd(x)+RM(x)}=min1≤i≤2{1+m(xi)},\begin{eqnarray*} \quad k_m(x)=\min \limits _{1\le i \le 3}\lbrace \sigma _{i}^{R_d(x)+R_M(x)}\rbrace =\min \limits _{1\le i \le 2}\lbrace 1+m(x_i)\rbrace , \end{eqnarray*}kr(x)=min1≤i≤3{σiRd(x)+Rr(x)}=min1≤i≤21+m2(xi)2+12γ2,\begin{eqnarray*} \quad k_r(x)=\min \limits _{1\le i \le 3}\lbrace \sigma _{i}^{R_d(x)+R_r(x)}\rbrace =\min \limits _{1\le i \le 2} \left\lbrace 1+\frac{m^2(x_i)}{2}+\frac{1}{2\gamma ^2}\right\rbrace , \end{eqnarray*}C=supx∈Uδ∑i=1n(xi2(t))α2α−1=supx∈Uδ∑i=13(xi2(t))1.22×1.2−1,\begin{eqnarray*} C=\sup \limits _{x\in U_{\delta }}\left[\sum \limits _{i=1}^{n}(x^{2}_{i}(t))^\frac{\alpha }{2\alpha -1}\right] =\sup \limits _{x\in U_{\delta }} \left[\sum \limits _{i=1}^{3}(x^{2}_{i}(t))^\frac{1.2}{2\times 1.2-1}\right], \end{eqnarray*}RM=GM(x)GT,Rr(x)=G(x)12MT(x)M(x)+12γ2ImGT(x),\begin{eqnarray*} R_M &=& GM(x)G^{T},\\ R_r(x) &=& G(x)\left[\frac{1}{2}M^{T}(x)M(x)+\frac{1}{2 \gamma ^{2}} I_m\right]G^T(x), \end{eqnarray*}Uδ$U_{\delta }$ is a neighborhood of the origin.It is not difficult to find that the convergence speed of the memristor‐based controllers ucr$u_{cr}$ and ucm$u_{cm}$ stabilizing the system (49) are faster than that of the classical IDA‐PBC controller uc$u_{c}$ when the disturbance ω=0$\omega =0$, see Figures 4–6.4FIGUREω=0$\omega =0$5FIGUREω=0$\omega =0$6FIGUREω=0$\omega =0$If ω≠0$\omega \ne 0$, owing to the dependence on memristance M(x(t))$M(x(t))$, control laws ucm$u_{cm}$ and ucr$u_{cr}$ have better disturbance rejection performance than that of the controller uc$u_c$ related to the classical IDA‐PBC method without memristive control action.Furthermore, from the condition56Rd(x)+12γ2[G(x)GT(x)−G1(x)G1T(x)]=1012γ201+12γ2012γ201⪰0,\begin{equation} \def\eqcellsep{&}\begin{array}{ll} R_d(x)+\dfrac{1}{2\gamma ^{2}}[G(x)G^{T}(x)-G_{1}(x)G_{1}^{T} (x)]\\[15pt] \qquad =\, \left( \def\eqcellsep{&}\begin{array}{ccc} 1 & 0 & \dfrac{1}{2\gamma ^2}\\[11pt] 0 & 1+\dfrac{1}{2\gamma ^2} & 0 \\[11pt] \dfrac{1}{2\gamma ^2} & 0 & 1 \end{array} \right) \succeq 0, \end{array} \end{equation}it can be obtained 14γ2≤1$\frac{1}{4\gamma ^2}\le 1$, or γ≥0.5$\gamma \ge 0.5$. Following Theorem 3, let γ=0.6>0.5$\gamma =0.6&gt;0.5$, andz=M(x)GT(x)∂Hd∂x(x),\begin{eqnarray*} z=M(x)G^{T}(x)\frac{\partial H_d}{\partial x}(x), \end{eqnarray*}is the penalty function, then the control law ucr$u_{cr}$ is locally fixed‐time stabilizing system (49) when input disturbance ω=0$\omega =0$. Moreover, the the memristor‐based H∞$H_{\infty }$ controller ucr$u_{cr}$ has the best performance in inhibiting the periodic disturbance, and the fastest speed converge to origin after disturbance vanishes, among uc,ucr$u_c,u_{cr}$ and ucm$u_{cm}$, see Figures 7–9.7FIGUREω=5sign(sin(6t)),t∈[0.5,4]$\omega =5sign(sin(6t)), t\in [0.5,4]$8FIGUREω=5sign(sin(6t)),t∈[0.5,4]$\omega =5sign(sin(6t)), t\in [0.5,4]$9FIGUREω=5sign(sin(6t)),t∈[0.5,4]$\omega =5sign(sin(6t)), t\in [0.5,4]$In summary, from Figures 4–12, it can be found that the memristor‐based controller ucm$u_{cm}$ has the fastest speed in stabilizing the system (49) when disturbance ω=0$\omega =0$. However, the memristor‐based H∞$H_{\infty }$ controller ucr$u_{cr}$ has less control costs than controller ucm$u_{cm}$ while it has the best performance in offsetting the periodic disturbance, and fastest converge speed to origin after disturbance vanishes. So, the memristor‐based H∞$H_{\infty }$ controller ucr$u_{cr}$ is the first choice in those situations coping with continuously periodic disturbances.6RemarkIn the simulation example 1, the memristors are mainly used to inhibiting the external periodic disturbance. However, in the case of the other kinds of bounded disturbances, the memristor‐based controllers ucm$u_{cm}$ and ucr$u_{cr}$ will produce a marked effect also.10FIGUREω=5sign(sin(6t)),t∈[0.5,4]$\omega =5sign(sin(6t)), t\in [0.5,4]$11FIGUREControl action corresponding to the controllers uc$u_{c}$, ucm$u_{cm}$ and ucr$u_{cr}$, respectively, and ω=0$\omega =0$12FIGUREControl action corresponding to the controllers uc$u_{c}$, ucm$u_{cm}$ and ucr$u_{cr}$, respectively, and ω=5sign(sin(6t)),t∈[0.5,4]$\omega =5sign(sin(6t)), t\in [0.5,4]$Example 2:In this subsection, a more realistic example is presented to illustrate the memristor‐based locally fixed‐time controller improving the performance of a nonlinear circuit system against the strongly periodic disturbance.Consider the nonlinear circuit system in Figure 13, where the capacitance and the inductance are controlled by electric charge q and magnetic flux ψ, respectively; and their characteristic are represented by u1=f1(q1),i3=f2(ψ2)$u_1 = f_1(q_1), i_3 = f_2(\psi _2)$; the voltĺCampere characteristic of the nonlinear resistance R4 is represented by u4=f4(i4)$u_4 = f_4(i_4)$, and iω$i_{\omega }$ is disturbance signal.13FIGURENonlinear circuitAccording to Kirchhoffa̧ŕs Voltage Law and Kirchhoffa̧ŕs Current Law, the system can be expressed as570=f1(q1)−f4(i4)+Us,q1̇=−Is−i4−f2(ψ2)−iω,y=Us+f1(q1)−R3f2(ψ2).\begin{eqnarray} {\left\lbrace \def\eqcellsep{&}\begin{array}{lll} 0=f_1(q_1)-f_4(i_4)+U_s,\\[9pt] \dot{q_1}= -I_s-i_4-f_2(\psi _2)-i_{\omega },\\[9pt] y= U_s+f_1(q_1)-R_3f_2(\psi _2). \end{array} \right.} \end{eqnarray}Let f1(q1)=85q135,f2(ψ2)=85ψ235,f4(i4)=(58)4i45+i4,R3=3Ω$f_1(q_1)=\frac{8}{5}q_1^{\frac{3}{5}},f_2(\psi _2)=\frac{8}{5}\psi _2^{\frac{3}{5}},f_4(i_4)=(\frac{5}{8})^4i_4^5+i_4,R_3=3\Omega$. Denote x=(x1,x2,x3)T=[(58i4)5,q1,ψ2]T,ω=iω,u=[Us,Is]T$x =(x_1,x_2,x_3)^T= [(\frac{5}{8} i_4)^5, q_1, \psi _2]^T , \omega = i_\omega , u = [U_s, I_s]^T$. Then the system (57) can be transformed into58000010001ẋ=−1−x12510−10−101−3∂Hd(x)∂x+100−110u+0−10ω,\begin{equation} \def\eqcellsep{&}\begin{array}{ll}{\left( \def\eqcellsep{&}\begin{array}{ccc} 0 & 0 & 0 \\[3pt] 0 & 1 & 0 \\[3pt] 0 & 0 & 1 \end{array} \right)}\dot{x}&={\left( \def\eqcellsep{&}\begin{array}{ccc}-1-x_1^{\frac{2}{5}} & 1 & 0 \\[3pt] -1 & 0 & -1 \\[3pt] 0 & 1 & -3 \end{array} \right)}\frac{\partial H_d(x)}{\partial x} \\[35pt] &\quad +\, {\left( \def\eqcellsep{&}\begin{array}{cc}1 & 0 \\[3pt] 0 & -1\\[3pt] 1 & 0 \end{array} \right)}u+{\left( \def\eqcellsep{&}\begin{array}{c}0 \\[3pt] -1 \\[3pt] 0 \end{array} \right)}\omega , \end{array} {} \end{equation}where Hd(x)=∑i=13(xi2(t))45=∑i=13(xi2(t))α2α−1,α=43$H_d(x)=\sum \nolimits _{i=1}^{3}(x^{2}_{i}(t))^{\frac{4}{5}}=\sum \nolimits _{i=1}^{3}(x^{2}_{i}(t))^{\frac{\alpha }{2\alpha -1}},\alpha =\frac{4}{3}$. For u=(0,0)T,ω=0$u=(0,0)^T,\omega =0$, the origin of the system (58) is locally fixed‐time stable, see Figure 14.14FIGURETime history of the system (58) without disturbance and control inputThe memristor‐based controller is designed as59um=−M(x)GT∂Hd∂x=−85m(x1)x135+m(x1)x335−m(x2)x235,\begin{eqnarray} u_m &=& -M(x)G^{T}\dfrac{\partial H_d}{\partial x}\nonumber\\ &=& -\frac{8}{5} \left( \def\eqcellsep{&}\begin{array}{c} m(x_1)x_1^{\frac{3}{5}}+m(x_1)x_3^{\frac{3}{5}} \\[9pt] -m(x_2)x_2^{\frac{3}{5}} \end{array} \right), \end{eqnarray}where M(x)$M(x)$ is given as (51) and (52), ρ=0.44,σ=0.00001$\rho =0.44, \sigma =0.00001$.When ω=−6sin12t$\omega =-6 sin 12t$, the controller (59) improves the performance of the system (58) against the strongly periodic disturbance, and the system states converge to the origin in a shorter convergence time after the disturbance vanishes, see Figures 15 and 16.15FIGURETime history of the system (58) with disturbance ω=−6sin12t,t∈[0.5,1.5]$\omega =-6sin12t,t\in [0.5,1.5]$, and control input u=(0,0)T$u=(0,0)^T$16FIGURETime history of the system (58) with disturbance ω=−6sin12t,t∈[0.5,1.5]$\omega =-6sin12t,t\in [0.5,1.5]$ and controller (59): u=um$u=u_m$CONCLUSIONSIn this paper, two novel memristor‐based locally fixed‐time controllers are designed to improve the disturbance rejection performance of PH systems via the IDA‐PBC method. The memristor‐based controllers proposed can make the states in the neighbourhood of equilibrium point accelerate convergence to desired equilibrium point, and lead to the oscillation be quashed considerably in the case of periodic disturbances. One of our future work is to find some applications in those situations where robust and adaptive control is needed, due to system parameter uncertainty and external disturbance.ACKNOWLEDGEMENTSThis work is supported in part by the Doctoral Research Start‐Up Fund of Yunnan Agricultural University under Grant A2032002548, and in part by the Philosophy and Social Science Planning Project of Yunnan Province under Grant YB2021016.CONFLICT OF INTERESTThe authors have declared no conflict of interest.DATA AVAILABILITY STATEMENTAll data needed to evaluate the conclusions in the paper are present in the paper itself. Additional data related to this paper may be requested to the corresponding author.REFERENCESCouenne, F., Jallut, C., Maschke, B.M., Tayakout, M., Breedveld, P.: Bond graph for dynamic modelling in chemical engineering. Chem. Eng. Process. 47, 1994–2003 (2008)Jeltsema, D., Scherpen, J.: Multi‐domain modeling of nonlinear networks and systems: energy‐and power‐based perspectives. IEEE Control Syst. Mag. 29(4), 28–59 (2009)Macchelli, A.: Port Hamiltonian systems: A unified approach for modeling and control finite and infinite dimensional physical systems. Ph.D. thesis, University of Bologna‐DEIS, Bologna, Italy (2003)Fujimoto, K., Sakurama, K., Sugie, T.: Trajectory tracking control of portcontrolled Hamiltonian systems via generalized canonical transformations. Automatica 39(12), 2059–2069 (2003)Ortega, R., Galaz, M., Astolfi, A., Sun, Y., Shen, T.: Transient stabilization of multi‐machine power systems with nontrivial transfer conductances. IEEE Trans. Autom. Control 50(1), 60–75 (2005)Ortega, R., Spong, M.W., Gómez‐Estern, F., Blankenstein, G.: Stabilization of a class of under actuated mechanical systems via interconnection and damping. IEEE Trans. Autom. Control 47(8), 1218–1233 (2002)Ortega, R., van der Schaft, A.J., Maschke, B., Escobar, G.: Interconnection and damping assignment passivity‐based control of port‐controlled Hamiltonian systems. Automatica 38(4), 585–596 (2002)Dörfler, F., Johnsen, J., Allgöwer, F.: An introduction to interconnection and damping assignment passivity‐based control in process engineering. J. Process Control 19(9), 1413–1426 (2009)Romero, J.G., Gandarilla, I., Santibáñez, V.: Stabilization of a class of nonlinear underactuated mechanical systems with 2‐DOF via immersion and invariance. European J. Control 63, 196–205 (2022)Ryalat, M., Laila, D.S., ElMoaqet, H.: Adaptive Interconnection and Damping Assignment Passivity Based Control for Underactuated Mechanical Systems. Int. J. Control Autom. Syst. 19(X), 1–14 (2021)Guerrero‐Sánchez, M.E., Hernández‐González, O., Valencia‐Palomo, G., Mercado‐Ravell, D.A., López‐Estrada, F.R., Hoyo‐Montaño, J.A.: Robust IDA‐PBC for under‐actuated systems with inertia matrix dependent of the unactuated coordinates: application to a UAV carrying a load. Nonlinear Dyn. 105, 3225–3238 (2021)Ferguson, J., Donaire, A., Ortega, R., Middleton, R.H.: Matched disturbance rejection for a class of nonlinear systems. IEEE Trans. Autom. Control 65(4), 1710–1715 (2020)Farid, Y., Ruggiero, F.: Finite‐time extended state observer and fractional‐order sliding mode controller for impulsive hybrid port‐Hamiltonian systems with input delay and actuators saturation: application to ball‐juggler robots. Mech. Mach. Theory 167(104577), 1–23 (2022)Yaghmaei, A., Yazdanpanah, M.J.: Structure preserving observer design for port‐Hamiltonian systems. IEEE Trans. Autom. Control 64(3), 1214–1220 (2019)Lv, C., Yu, H., Zhao, N., Chi, J., Liu, H., Li, L.: Robust state‐error port‐controlled Hamiltonian trajectory tracking control for unmanned surface vehicle with disturbance uncertainties. Asian J. Control 2467, 1–13 (2020) https://doi.org/10.1002/asjc.2467Fu, B., Li, S., Yang, J., Guo, L.: Global output regulation for a class of single input port‐controlled Hamiltonian disturbed systems. Appl. Math. Comput. 325, 322–331 (2018)Romero, J.G., Donaire, A., Navarro‐Alarcon, D., Ramirez, V.: Passivity‐based tracking controllers for mechanical systems with active disturbance rejection. IFAC‐PapersOnLine 48(13), 129–134 (2015)Chua, L.O.: Memristor‐the missing circuit element. IEEE Trans. Circ. Theory 18(2), 507–519 (1971)Strukov, D.B., Snider, G.S., Stewart, D.R., Williams, R.S.: The missing memristor found. Nature 453, 80–83 (2008)Jeltsema, D., vander Schaft, A.J.: Memristive port‐Hamiltonian systems. Math. Comput. Modell. Dyn. Syst. 16(2), 75–93 (2010)Jeltsema, D., Dòria‐Cerezo, A.: Port‐Hamiltonian formulation of systems with memory. Proc. IEEE 100(6), 1928–1937 (2012)Pershin, Y.V., Lafontaine, S., DiVentra, M.: Memristive model of amoeba's learning. Phys. Rev. E 80, 021926 (2009)Dòria‐Cerezo, A., van der Heijden, L., Scherpen, J.M.A.: Memristive port‐Hamiltonian control: Path‐dependent damping injection in control of mechanical systems. European J. Control 19, 454–460 (2013)Pasumarthy, R., Saha, G., Kazi, F., Singh, N.: Energy and power based perspective of memristive controllers. In: 52nd IEEE Conference on Decision and Control. IEEE, Piscataway (2013)Sala, G., Pasumarthy, R., Khatavkar, P.: Towards analog memristive controllers. IEEE Trans. Circuit Syst. 62, 205–214 (2015)Polyakov, A.: Nonlinear feedback design for fixed‐time stabilization of linear control systems. IEEE Trans. Autom. Control 57, 2106–2110 (2012)Bhat, S.P., Bernstein, D.S.: Finite‐time stability of continuous autonomous systems. SIAM J. Control Optim. 38, 751–766 (2000)Moulay, E., Perruquetti, W.: Finite time stability and stabilization of a class of continuous systems. J. Math. Anal. Appl. 323, 1430–1443 (2006)Sun, L., Feng, G., Wang, Y.: Finite‐time stabilization and H∞$H_\infty$ control for a class of nonlinear Hamiltonian descriptor systems with application to affine nonlinear descriptor systems. Automatica 50, 2090–2097 (2014)Lu, X., Li, H.: Prescribed finite‐time H∞$H_{\infty }$ control for nonlinear descriptor systems. IEEE Trans. Circu. Syst. II 28, 2917–2921 (2021)Yang, X., Lam, J., Ho, D.W.C., Feng, Z.: Fixed‐time synchronization of complex networks with impulsive effects via non‐chattering control. IEEE Trans. Autom. Control 62(11), 5511–5521 (2018)Sun, J., Yi, J., Pu, Z.: Augmented fixed‐time observer‐based continuous robust control for hypersonic vehicles with measurement noises. IET Control Theory Appl. 13(3), 422–433 (2019)Langueh, K.A.A., Zheng, G., Floquet, T.: Fixed‐time sliding mode‐based observer for non‐linear systems with unknown parameters and unknown inputs. IET Control Theory Appl. 14(14), 1920–1927 (2020)Hu, C.: Special functions‐based fixed‐time estimation and stabilization for dynamic systems. IEEE Trans. Systems, Man, and Cybernetics: Systems (2021). https://doi.org/10.1109/TSMC.2021.3062206Zhang, W., Yang, X., Yang, S., Alsaedi, A.: Finite‐time and fixed‐time bipartite synchronization of complex networks with signed graphs. Math. Comput. Simul 188, 319–329 (2021)Kong, F., Zhu, Q., Huang, T.: Fixed‐time stability for discontinuous uncertain inertial neural networks with time‐varying delays. IEEE Trans. Syst. Man Cybern.: Syst. (2021). https://doi.org/10.1109/TSMC.2021.3096261Wang, Y., Li, C., Cheng, D.: Generalized Hamiltonian realization of time‐invariant nonlinear systems. Automatica 39(8), 1437–1443 (2003)Liu, X., Liao, X.: Fixed‐time H∞$\mathcal {H}_{\infty }$ control for port‐controlled Hamiltonian systems. IEEE Trans. Autom. Control 64(7), 2753–2765 (2019)Liu, X., Liao, X.: Fixed‐time stabilization control for port‐Hamiltonian systems. Nonlinear Dyn. 96(2), 1497–1509 (2019)van der Schaft, A.J.: L2L2‐gain and passivity techniques in nonlinear control, 2nd ed. Springer, London (2000)

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