Memory-based passive control design for complex networked fuzzy delta operator systems

Abstract

This article investigates the memory-based state-feedback control with strict passivity performance for nonlinear systems in delta domain. Takagi–Sugeno fuzzy model of delta operator form is adopted as an approximator of the investigated nonlinear plant with system perturbations/exogenous disturbances. The nonlinear output of the plant, which is accompanied with complex exogenous disturbances, is formulated as the passive output satisfying a pre-designed strict passivity performance. The memory-based state-feedback controller is expressed with a known memory coefficient to significant the influence of the delayed states. The resulting closed-loop system is inferred as a time-varying delay system, of which the stability and passivity criterion is carried out via a general fuzzy Lyapunov–Krasovskii functional approach. Then, parametric gains of the desired strictly passive memory-based controller are designed subject to a set of solvable matrix inequalities. Finally, a numerical example shows the validity of the proposed memory-based passive control method.

Introduction

It is well known that the Takagi–Sugeno (T-S) fuzzy model^1,2 has been recognized as one of the most effective approaches to formulate complex nonlinear systems. In T-S fuzzy model, some local linear systems can be interpolated by the so-called membership functions in a unit framework. In the past few decades, the problem of stability analysis and controller synthesis by the utilization of T-S fuzzy models has been widely investigated (see, for example, Feng, ³ Liu et al., ⁴ Su et al., ⁵ Lam and Lauber, ⁶ Liu et al., ⁷ Wu et al., ⁸ and Wei et al. ⁹ and references therein).

As an impactful control methodology in control systems, passivity control has been successfully applied in engineering applications, such as electrical circuit systems, mechanical systems, and complex network systems.^10–13 Under the passive control forces, the internal stability of the plant can be kept and even the stability can be improved. Therefore, the passive control problems have been investigated for fuzzy systems.^14–16 For instance, considering the time-varying delays in T-S fuzzy systems, Zhang et al. ¹⁵ designed a very-strict passive controller. A robust passive control scheme was presented in Wu et al. ¹⁶ for networked fuzzy systems with randomly occurring uncertainties.

It is remarkable that as a branch of sampling systems, delta operator–based systems^17,18 have been made good use for excellent finite word length performance under fast sampling. For discrete-time control systems, delta operator systems abide the concept that the shorter the sampling period, the better the system performances. ¹⁹ Benefiting the virtue of delta operator systems, numerical ill-conditioning experienced in algorithms of discrete-time system can be avoided when the sample period is sufficiently small. To point a few, Li and Gevers ²⁰ introduced the relationships of optimal realization sets between shift and delta operator. According to the delta operator formulation, Tadjine et al. ²¹ overviewed the design features of a loop transfer recovery controller both at input and output. A strictly positive real control with low-frequency range was studied in Yang and Xia ²² for delta operator systems. For a class of time-delay systems with mismatched parametric uncertainties, Xia et al. ²³ investigated the robust sliding-mode control. An output-tracking control scheme of delta operator time-delay systems was presented in Gao et al. ²⁴ By using the T-S fuzzy model, the fault detection problem was concentered in Li et al. ²⁵ for nonlinear delta operator systems. Very little results have been reported on delta operator–based fuzzy controller in literatures, such as time-delay system. ²⁶ However, to the best of our knowledge, the memory-based controller design, especially with passivity performance for delta operator nonlinear systems, receives little attention now.

Based on the discussion above, in this article, we devote to the memory-based passive control of discrete-time T-S fuzzy systems using delta operator approach. For the investigated fuzzy delta operator systems, a very-strict passivity index is established. We formulate memorized information of the system state in a fuzzy controller including a memory coefficient. Then, the closed-loop system is formulated as a time-varying delay system. We use a fuzzy Lyapunov–Krasovskii functional (LKF) to analyze the passivity and stability of the resulting time-delay system. This work contributes to the following points: (1) a memory-based fuzzy controller is established fully considering memorized information of the system state; (2) passivity criterion of the investigated delta operator system is constructed; and (3) a solvable solution of the memory-based passive control subject to a set of linear matrix inequalities (LMIs) is presented. Finally, the presented memory-based passive controller is validated by using a numerical example.

The subsequent context of this article is organized into four sections: Section “Problem descriptions” describes the formulations of the delta operator system and memory-based fuzzy controller. Section “Main results” presents the passive control design and stability analysis. Section “Simulation example” provides an illustrative example to show the effectiveness of the proposed memory-based passive controller. Section “Conclusion” concludes this article.

Notation

Let $t_k=\mathrm{kT}$ for the convenience and $T$ is a sampling period. For matrix $X\in {\mathbb{R}}^{n\times n}$ , $X>0$ means that the matrix $X$ is real symmetric positive definite. The star “★“ in a matrix $X$ stands for the transposed elements in the symmetric positions. The superscripts “ $T$ “ and “ $-1$ ” denote the matrix transpose and inverse, respectively. Identity matrices of appropriate dimensions will be denoted by “ $I$ ”. The shorthand $\mathrm{diag}\{X_1,X_2,\dots ,X_r\}$ denotes a block diagonal matrix with diagonal blocks being the matrices $X_1$ , $X_2$ , …, $X_r$ . If not explicitly stated, all matrices are assumed to have compatible dimensions for algebraic operations.

Problem descriptions

Based on the delta operator, we present the following T-S fuzzy model with $N$ rules and $v$ fuzzy sets to approximate the nonlinear plant. ²⁴

Fuzzy rule $r$

If ${\mu }_1(t_k)$ is $S_{r1}$ , and …, and ${\mu }_j(t_k)$ is $S_{\mathrm{rj}}$ , and …, and ${\mu }_v(t_k)$ is $S_{\mathrm{rv}}$ , then

$$\{\begin{array}{c}\delta x(t_k)=A_{r}x(t_k)+B_{r}u(t_k)+B_{\mathrm{dr}}d(t_k)\hfill \\ \multicolumn{1}{c}{z(t_k)=C_{r}x(t_k)+D_{r}u(t_k)+D_{\mathrm{dr}}d(t_k)}\end{array}$$

(1)

where $S_{\mathrm{rj}}$ and ${\mu }_j(t_k)$ are the fuzzy sets and the premise variables, respectively ( $r=1,2,\mathrm{\dots },N$ and $j=1,2,\dots ,v$ ). $x(t_k)\in {\mathbb{R}}^n$ and $d(t_k)\in {\mathbb{R}}^l$ are the state variable and the bounded uncertainties such as system perturbations and exogenous disturbances, respectively. $u(t_k)\in {\mathbb{R}}^m$ and $z(t_k)\in {\mathbb{R}}^g$ are the memory-based control input variable to be designed and the control output, respectively. $A_r$ , $B_r$ , $B_{\mathrm{dr}}$ , $C_r$ , $D_r$ , and $D_{\mathrm{dr}}$ are system matrices with appropriate dimensions. $\delta x(t_k)$ is the delta operator of $x(t_k)$ , which is defined by

$$\delta x(t_k)=\{\begin{array}{c}\frac{d}{d{t}_k}x(t_k),T=0\hfill \\ \frac{x(t_k+T)-x(t_k)}{T},T\ne 0\hfill \end{array}$$

(2)

Then, the defuzzified model of system (1) is inferred as follows

$$\{\begin{array}{c}\delta x(t_k)=\sum _{r=1}^N{\lambda }_r[A_{r}x(t_k)+B_{r}u(t_k)+B_{\mathrm{dr}}d(t_k)]\hfill \\ \multicolumn{1}{c}{z(t_k)=\sum _{r=1}^N{\lambda }_r[C_{r}x(t_k)+D_{r}u(t_k)+D_{\mathrm{dr}}d(t_k)]}\end{array}$$

(3)

in which, we simplify ${\lambda }_r(\mu (t_k))$ to ${\lambda }_r$ for saving space and

$$\begin{array}{c}{\lambda }_r(\mu (t_k)){\xi }_r(\mu (t_k))/\sum _{r=1}^N{\xi }_r(\mu (t_k))\hfill \\ {\xi }_r(\mu (t_k))\underset{j=1}{\overset{v}{\Pi }}{S}_{\mathrm{rj}}({\mu }_j(t_k))\ge 0\hfill \end{array}$$

and $S_{\mathrm{rj}}({\mu }_j(t_k))$ is the membership degree of the premise variable ${\mu }_j(t_k)$ in fuzzy set $S_{\mathrm{rj}}$ . According to the parallel distributed compensation, we suppose the fuzzy rules of the desired controller sharing the same fuzzy rules of plant (1). The memory-based fuzzy state-feedback controller can be constructed as follows

$$u(t_k)=\sum _{s=1}^N{\lambda }_s[(1-\rho )K_{1s}x(t_k)+\rho K_{2s}x(t_{k-i_k})]$$

(4)

where $\rho$ is the memory coefficient satisfying $0\le \rho \le 1$ . ${\sigma }_k=i_{k}T$ denotes a bounded delayed time satisfying

$$0<{\sigma }_1\le {\sigma }_k\le {\sigma }_2$$

(5)

with ${\sigma }_1=i_{m}T$ and ${\sigma }_2=i_{M}T$ ( $i_m$ and $i_M$ are known positive integers). When $t_k=-i_{M}T,-i_{M}T+T,\dots ,0$ , $x(t_k)=\chi (t_k)$ , where $\chi (t_k)$ is a continuous vector–valued initial function. The parametric matrices $K_{1s}$ and $K_{2s}$ $(s=1,2,...,N)$ are the local control gain to be determined. Then, substituting equation (4) into equation (3) yields the following closed-loop system

$$\{\begin{array}{c}\delta x(t_k)=A(t_k)x(t_k)+B_K(t_k)x(t_{k-i_k})+B_d(t_k)d(t_k)\hfill \\ \multicolumn{1}{c}{z(t_k)=C(t_k)x(t_k)+D_K(t_k)x(t_{k-i_k})+D_d(t_k)d(t_k)}\end{array}$$

(6)

where

$$\begin{array}{c}A(t_k)-\sum _{r=1}^N\sum _{s=1}^N{\lambda }_r{\lambda }_s(A_r+B_r{K}_s)\hfill \\ B_K(t_k)-\sum _{r=1}^N\sum _{s=1}^N{\lambda }_r{\lambda }_s{B}_r{K}_{2s},B_d(t_k)=\sum _{r=1}^N{\lambda }_r{B}_{\mathrm{dr}}\hfill \\ C(t_k)-\sum _{r=1}^N\sum _{s=1}^N{\lambda }_r{\lambda }_s(C_r+D_r{K}_s)\hfill \\ D_K(t_k)-\sum _{r=1}^N\sum _{s=1}^N{\lambda }_r{D}_r{K}_{2s},D_d(t_k)=\sum _{r=1}^N{\lambda }_r{D}_{\mathrm{dr}}\hfill \end{array}$$

We detail the following definition and a lemma for the basis of the design procedure in the following context.

Definition 1

System (6) is said to be very strictly passive if there exist constants $\alpha >0$ , $\beta >0$ , and $\gamma$ such that ²⁷

$$\begin{array}{l}2{\displaystyle \sum _{l=0}^k{z}^T(t_k)d(t_k)}\ge \gamma \\ +\alpha {\displaystyle \sum _{l=0}^k{z}^T(t_k)z(t_k)}\\ +\beta {\displaystyle \sum _{l=0}^k{d}^T(t_k)d(t_k)}\end{array}$$

Lemma 1

For any matrix $X>0$ and positive integers $k$ and $k_0$ satisfying $1\le k_0\le k$ , the following inequality holds ²⁸

$$\begin{array}{c}{(\sum _{r=k_0}^{k}x(t_r))}^{T}X(\sum _{r=k_0}^{k}x(t_r))\hfill \end{array}\le \tilde{k}\sum _{r=r_0}^k{x}^T(t_r)\mathrm{Xx}(t_r)$$

where $\tilde{k}=(k-k_0+1)$ .

Remark 1

According to Definition 2, this article aims to design the desired memory-based controller in such that the resulting time-delay system (6) is very strictly passive and its asymptotic stability can be guaranteed. The very strictly passive control design procedure for system (6) will be detailed in the following section.

Main results

In terms of the overall system (6), the fuzzy Lyapunov functional approach to derive the passivity criterion with a sufficient condition is used in this section. Then, a set of LMIs are developed to solve the solution of the parametric matrices of the desired memory-based fuzzy controller.

Theorem 1

In terms of the delta operator system (6) with a sampling period $T$ , for some given constants ${\sigma }_1$ and ${\sigma }_2$ satisfying (5), system (6) can be very strictly passive if there exist scalars $\alpha >0$ and $\beta >0$ and appropriately dimensioned matrices $P>0$ , $Z_1>0$ , $Z_2>0$ , $M_l>0$ , $N_{1l}>0$ , and $N_{2l}>0$ ( $l=1,2,\dots ,N$ ) such that for $l,m,n,o,r,s=1,2,\dots ,N$ , the following inequalities hold

$${\mathrm{\Phi }}_{\mathrm{lmnorr}}<0$$

(7)

$${\mathrm{\Phi }}_{\mathrm{lmnors}}+{\mathrm{\Phi }}_{\mathrm{lmnosr}}<0,s<r$$

(8)

where

$$\begin{array}{c}{\mathrm{\Theta }}_1\stackrel{\mathrm{\Delta }}{=}\left[\begin{array}{cc}{\mathrm{\Theta }}_{11}\hfill & {\mathrm{\Theta }}_{12}\hfill \\ \multicolumn{1}{c}{★}& {\mathrm{\Theta }}_{13}\hfill \end{array}\right],{\mathrm{\Phi }}_{\mathrm{lmnors}}\stackrel{\mathrm{\Delta }}{=}\left[\begin{array}{cc}{\mathrm{\Theta }}_1\hfill & {\mathrm{\Theta }}_2^T\hfill \\ \multicolumn{1}{c}{★}& -\epsilon I\hfill \end{array}\right]\hfill \\ {\mathrm{\Theta }}_{11}\stackrel{\mathrm{\Delta }}{=}\left[\begin{array}{ccc}{\mathrm{\Gamma }}_1\hfill & P(A_r+B_r{K}_{1s})\hfill & P{B}_r{K}_{2s}\hfill \\ \multicolumn{1}{c}{★}& {\mathrm{\Gamma }}_{2\mathrm{lrs}}\hfill & P{B}_r{K}_{2s}\hfill \\ \multicolumn{1}{c}{★}& ★\hfill & -M_m\hfill \end{array}\right]\hfill \\ {\mathrm{\Theta }}_{12}\stackrel{\mathrm{\Delta }}{=}\left[\begin{array}{ccc}0\hfill & 0\hfill & P{B}_{\mathrm{dr}}\hfill \\ \multicolumn{1}{c}{\frac{1}{{\sigma }_1}{Z}_1}& \frac{1}{{\sigma }_2}{Z}_2\hfill & {\mathrm{\Gamma }}_{3\mathrm{rs}}\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & -K_{2s}^T{D}_r^T\hfill \end{array}\right]\hfill \\ {\mathrm{\Theta }}_{13}\stackrel{\mathrm{\Delta }}{=}\mathrm{diag}\{-N_{1n}-\frac{1}{{\sigma }_1}{Z}_1,-N_{2o}-\frac{1}{{\sigma }_2}{Z}_2,\beta I-{\left[D_{\mathrm{dr}}\right]}_s\}\hfill \\ {\mathrm{\Theta }}_2\stackrel{\mathrm{\Delta }}{=}\left[\begin{array}{cccccc}0\hfill & C_r+D_r{K}_{1s}\hfill & D_r{K}_{2s}\hfill & 0\hfill & 0\hfill & D_{\mathrm{dr}}\hfill \end{array}\right]\hfill \\ {\mathrm{\Gamma }}_1\stackrel{\mathrm{\Delta }}{=}(T-2)P+{\sigma }_1{Z}_1+{\sigma }_2{Z}_2\hfill \\ {\mathrm{\Gamma }}_2\stackrel{\mathrm{\Delta }}{=}{[P(A_r+B_r{K}_s)]}_s+({\sigma }_2-{\sigma }_1+T+1)M_l\hfill \\ +N_{1l}+N_{2l}-\frac{1}{{\sigma }_1}{Z}_1-\frac{1}{{\sigma }_2}{Z}_2\hfill \\ {\mathrm{\Gamma }}_3=P{B}_{\mathrm{dr}}-(C_r+D_r{K}_s{)}^T\hfill \end{array}$$

Proof

For simplicity, we denote $\stackrel{⌣}{M}(t_k)={\sum }_{l=1}^N{\lambda }_l{M}_l$ , ${\stackrel{⌣}{N}}_1(t_k)={\sum }_{l=1}^N{\lambda }_l{N}_{1l}$ , and ${\stackrel{⌣}{N}}_2(t_k)={\sum }_{l=1}^N{\lambda }_l{N}_{2l}$ . Select the LKF $V(x(t_k))$ for system (6) as follows

$$V(t_k)=V_1(t_k)+V_2(t_k)+V_3(t_k))+V_4(t_k)$$

(9)

where

$$\begin{array}{l}{V}_1(t_k)=x^T(t_k)Px(t_k)+T{\displaystyle \sum _{r=1}^n{x}^T}(t_{k-r})\stackrel{⌣}{M}(t_{k-r})x(t_{k-r})\\ V_2(t_k)=T{\displaystyle \sum _{r=1}^{i_m}{x}^T}(t_{k-r}){\stackrel{⌣}{N}}_1(t_{k-r})x(t_{k-r})\\ +T{\displaystyle \sum _{r=1}^{i_M}{x}^T}(t_{k-r}){\stackrel{⌣}{N}}_2(t_{k-r})x(t_{k-r})\\ V_3(t_k)=T^2{\displaystyle \sum _{r=i_m}^{i_M}{\displaystyle \sum _{s=1}^r{x}^T}}(t_{k-s})\stackrel{⌣}{M}(t_{k-s})x(t_{k-s})\\ V_4(t_k)={\displaystyle \sum _{r=1}^{i_m}{\displaystyle \sum _{s=1}^r{e}^T}}(t_{k-s})Z_{1}e(t_{k-s})\\ +{\displaystyle \sum _{r=1}^{i_M}{\displaystyle \sum _{s=1}^r{e}^T}}(t_{k-s})Z_{2}e(t_{k-s})\end{array}$$

in which $e(t)=x(t)-x(t+T)$ . Thus, we have the following formula: $e(t_{k-r})=x(t_{k-r})-x(t_k-(r-1)T)$ and $\delta x(t_k)=-e(t_k)/T$ .

According to the following formula ²⁹

$$\begin{array}{c}\delta (x(t_k)y(t_k))=\delta (x(t_k))y(t_k)+x(t_k)\delta (y(t_k))\hfill \\ +T\delta (x(t_k))\delta (y(t_k))\hfill \end{array}$$

we calculate $V_1(x(t_k))$ , $V_1(x(t_k))$ , $V_2(x(t_k))$ , and $V_3(x(t_k))$ , respectively, as follows

$$\begin{array}{c}\delta V_1(t_k)=x^T(t_k){\left[PA(t_k)\right]}_{s}x(t_k)+2x^T(t_k)P{B}_K(t_k)x(t_{k-i_k})\\ +2x^T(t_k)P{B}_d(t_k)d(t_k)+T{\delta }^T\left(x(t_k)\right)P\delta \left(x(t_k)\right)\\ +\frac{1}{T}[T{\displaystyle \sum _{r=1}^n{x}^T}(t_{k-r+1})\stackrel{⌣}{M}(t_{k-r+1})x(t_{k-r+1})-T{\displaystyle \sum _{r=1}^n{x}^T}(t_{k-r})\stackrel{⌣}{M}(t_{k-r})x(t_{k-r})]\\ \le x^T(t_k)\stackrel{⌣}{M}(t_k)x(t_k)-x^T(t_{k-i_k})\stackrel{⌣}{M}(t_{k-i_k})x(t_{k-i_k})\\ +T{\displaystyle \sum _{r=i_m}^{i_M}{x}^T}(t_{k-r})\stackrel{⌣}{M}(t_{k-r})x(t_{k-r})\\ \delta V_2(t_k)=x^T(t_k)({\stackrel{⌣}{N}}_1(t_k)+{\stackrel{⌣}{N}}_2(t_k))x(t_k)\\ -x^T(t_{k-i_m}){\stackrel{⌣}{N}}_1(t_{k-i_m})x(t_{k-i_m})\\ -x^T(t_{k-i_M})N_2(t_{k-i_M})x(t_{k-i_M})\\ \delta V_3(t_k)=T[{\displaystyle \sum _{s=1}^r\left(x^T(t_{k-s+1})\stackrel{⌣}{M}(t_{k-s+1})x(t_{k-s+1})\right)}-{\displaystyle \sum _{s=1}^r\left(x^T(t_{k-s})\stackrel{⌣}{M}(t_{k-s})x(t_{k-s})\right)}]\\ =\left({\sigma }_2-{\sigma }_1+T\right)x^T(t_k)\stackrel{⌣}{M}(t_k)x(t_k)\\ -T{\displaystyle \sum _{r=i_m}^{i_M}{x}^T}(t_{k-r})\stackrel{⌣}{M}(t_{k-r})x(t_{k-r})\end{array}$$

Using Lemma 1, we have

$$\begin{array}{c}\multicolumn{1}{c}{\delta V_4(t_k)=\frac{1}{T}[\sum _{r=1}^{i_m}\sum _{s=1}^r{e}^T\left(t_{k-s+1}\right)Z_{1}e\left(t_{k-s+1}\right)}\\ -\sum _{r=1}^{i_m}\sum _{s=1}^r{e}^T(t_{k-s})Z_{1}e(t_{k-s})\hfill \\ +\sum _{r=1}^{i_M}\sum _{s=1}^r{e}^T\left(t_{k-s+1}\right)Z_{2}e\left(t_{k-s+1}\right)\hfill \\ -\sum _{r=1}^{i_M}\sum _{s=1}^r{e}^T(t_{k-s})Z_{2}e(t_{k-s})]\hfill \\ \le e^T(t_k)(\frac{i_m}{T}{Z}_1+\frac{i_M}{T}{Z}_2)e(t_k)\hfill \\ -\frac{1}{i_{m}T}{[\sum _{r=1}^{i_m}e(t_{k-r})]}^T{Z}_1[\sum _{r=1}^{i_m}e(t_{k-r})]\hfill \\ -\frac{1}{i_{M}T}{[\sum _{r=1}^{i_M}e(t_{k-r})]}^T{Z}_2[\sum _{r=1}^{i_M}e(t_{k-r})]\hfill \\ \multicolumn{1}{c}{={\delta }^T(x(t_k))({\sigma }_1{Z}_1+{\sigma }_2{Z}_2)\delta (x(t_k))}\\ -\frac{1}{{\sigma }_1}{[x(t_k)-x(t_{k-i_m})]}^T{Z}_1[x(t_k)-x(t_{k-i_m})]\hfill \\ -\frac{1}{{\sigma }_2}{[x(t_k)-x(t_{k-i_M})]}^T{Z}_2[x(t_k)-x(t_{k-i_M})]\hfill \end{array}$$

Besides, for any matrix $P>0$ , it always holds that

$$\begin{array}{c}\multicolumn{1}{c}{0=-2{\delta }^T(x(t_k))P[\delta (x(t_k))-A(t_k)x(t_k)}\\ -B_K(t_k)x(t_{k-i_k})-B_d(t_k)d(t_k)]\hfill \end{array}$$

Then, according to the statement in Definition 1, it results that

$$\begin{array}{c}\multicolumn{1}{c}{\Im ={\alpha }^{-1}{z}^T(t_k)z(t_k)+\beta d^T(t_k)d(t_k)-2z^T(t_k)d(t_k)+\delta V(t_k)}\\ \le {\phi }^T(t_k)\sum _{l=1}^N\sum _{m=1}^N\sum _{n=1}^N\sum _{o=1}^N\sum _{r=1}^N\sum _{s=1}^N{\lambda }_l{\lambda }_m{\lambda }_n{\lambda }_o{\lambda }_r{\lambda }_s\hfill \\ \times [{\mathrm{\Theta }}_{1\mathrm{lmnors}}+{\alpha }^{-1}{\mathrm{\Theta }}_{2\mathrm{rs}}^T{\mathrm{\Theta }}_{2\mathrm{rs}}]\phi (t_k)\hfill \end{array}$$

(10)

where

$$\begin{array}{c}\phi (t_k)={\left[\begin{array}{cc}{\phi }_1^T(t_k)\hfill & {\phi }_2^T(t_k)\hfill \end{array}\right]}^T\hfill \\ {\phi }_1(t_k)={\left[\begin{array}{ccc}{\delta }^T(x(t_k))\hfill & x^T(t_k)\hfill & x^T(t_{k-i_k})\hfill \end{array}\right]}^T\hfill \\ {\phi }_2(t_k)={\left[\begin{array}{ccc}{x}^T(t_{k-i_m})\hfill & x^T(t_{k-i_M})\hfill & d^T(t_k)\hfill \end{array}\right]}^T\hfill \end{array}$$

Furthermore, from Theorem 1, we can rewrite that

$$\begin{array}{c}\sum _{l=1}^N\sum _{m=1}^N\sum _{n=1}^N\sum _{o=1}^N\sum _{r=1}^N\sum _{s=1}^N{\lambda }_l{\lambda }_m{\lambda }_n{\lambda }_o{\lambda }_r{\lambda }_s{\mathrm{\Phi }}_{\mathrm{lmnors}}\hfill \\ =\sum _{l=1}^N\sum _{m=1}^N\sum _{n=1}^N\sum _{o=1}^N{\lambda }_l{\lambda }_m{\lambda }_n{\lambda }_o[\sum _{r=1}^N{\lambda }_r^2{\mathrm{\Phi }}_{\mathrm{lmnorr}}\hfill \\ +\sum _{r=1}^{N-1}\sum _{s=r+1}^N{\lambda }_r{\lambda }_s({\mathrm{\Phi }}_{\mathrm{lmnors}}+{\mathrm{\Phi }}_{\mathrm{lmnosr}})]<0\hfill \end{array}$$

Using Schur complement, we know that ${\mathrm{\Phi }}_{\mathrm{lmnors}}<0$ is equivalent to ${\mathrm{\Theta }}_{1\mathrm{lmnors}}+{\alpha }^{-1}{\mathrm{\Theta }}_{2\mathrm{rs}}^T{\mathrm{\Theta }}_{2\mathrm{rs}}<0$ . It means

$$\begin{array}{c}\Im =\sum _{l=0}^k{\alpha }^{-1}{z}^T(t_k)z(t_k)+\sum _{l=0}^k\beta d^T(t_k)d(t_k)\hfill \\ -\sum _{l=0}^{k}2z^T(t_k)d(t_k)+\sum _{l=0}^k\delta V(t_k)<0\hfill \end{array}$$

That is to say

$$\begin{array}{l}2{\displaystyle \sum _{l=0}^k{z}^T}(t_l)d(t_l)\ge \alpha {\displaystyle \sum _{l=0}^k{z}^T}(t_l)z(t_l)+\beta {\displaystyle \sum _{l=0}^k{d}^T}(t_l)d(t_l)\\ +\mathrm{V}({\mathrm{t}}_{\mathrm{k}})-\mathrm{V}({\mathrm{t}}_0)\\ \ge \alpha {\displaystyle \sum _{\mathrm{l}=0}^{\mathrm{k}}{\mathrm{z}}^{\mathrm{T}}}({\mathrm{t}}_{\mathrm{l}})\mathrm{z}({\mathrm{t}}_{\mathrm{l}})+\beta {\displaystyle \sum _{\mathrm{l}=0}^{\mathrm{k}}{\mathrm{d}}^{\mathrm{T}}}({\mathrm{t}}_{\mathrm{l}})\mathrm{d}({\mathrm{t}}_{\mathrm{l}})-\mathrm{V}({\mathrm{t}}_0)\\ =\alpha {\displaystyle \sum _{\mathrm{l}=0}^{\mathrm{k}}{\mathrm{z}}^{\mathrm{T}}}({\mathrm{t}}_{\mathrm{l}})\mathrm{z}({\mathrm{t}}_{\mathrm{l}})+\beta {\displaystyle \sum _{\mathrm{l}=0}^{\mathrm{k}}{\mathrm{d}}^{\mathrm{T}}}({\mathrm{t}}_{\mathrm{l}})\mathrm{d}({\mathrm{t}}_{\mathrm{l}})+\gamma \end{array}$$

where $\gamma =-V(t_0)$ . Recalling Definition 1, we know the very-strict passivity can be guaranteed for the time-delay delta operator system (6). This completes the proof.

In the following theorem, we develop the conditions in Theorem 1 to some solvable LMIs for designing the parametric matrices of controller (4).

Theorem 2

In terms of the delta operator system (6) with a sampling period $T$ , for some given constants ${\sigma }_1$ and ${\sigma }_2$ satisfying equation (5), the desired very-strict passivity can be guaranteed by system (6) if there exist scalars $\alpha >0$ and $\beta >0$ and appropriately dimensioned matrices $X>0$ , ${\overline{Z}}_1>0$ , ${\overline{Z}}_2>0$ , ${\overline{M}}_l>0$ , ${\overline{N}}_{1l}>0$ , and ${\overline{N}}_{2l}>0$ ( $l=1,2,\dots ,N$ ), such that for $l,m,n,o,r,s=1,2,\dots ,N$ , the following LMIs hold

$${\overline{\mathrm{\Phi }}}_{\mathrm{lmnors}}<0$$

(11)

$${\overline{\mathrm{\Phi }}}_{\mathrm{lmnors}}+{\overline{\mathrm{\Phi }}}_{\mathrm{lmnosr}}<0,s<r$$

(12)

where

$$\begin{array}{c}{\overline{\mathrm{\Theta }}}_1\stackrel{\mathrm{\Delta }}{=}\left[\begin{array}{cc}{\overline{\mathrm{\Theta }}}_{11}\hfill & {\overline{\mathrm{\Theta }}}_{12}\hfill \\ \multicolumn{1}{c}{★}& {\overline{\mathrm{\Theta }}}_{13}\hfill \end{array}\right],{\overline{\mathrm{\Phi }}}_{\mathrm{lmnors}}\stackrel{\mathrm{\Delta }}{=}\left[\begin{array}{cc}{\overline{\mathrm{\Theta }}}_1\hfill & {\overline{\mathrm{\Theta }}}_2^T\hfill \\ \multicolumn{1}{c}{★}& -\epsilon I\hfill \end{array}\right]\hfill \\ {\overline{\mathrm{\Theta }}}_{11}\stackrel{\mathrm{\Delta }}{=}\left[\begin{array}{ccc}{\overline{\mathrm{\Gamma }}}_1\hfill & A_{r}X+B_r{\overline{K}}_{1s}\hfill & B_r{\overline{K}}_{2s}\hfill \\ \multicolumn{1}{c}{★}& {\overline{\mathrm{\Gamma }}}_{2\mathrm{lrs}}\hfill & D_r{\overline{K}}_{2s}\hfill \\ \multicolumn{1}{c}{★}& ★\hfill & -{\overline{M}}_m\hfill \end{array}\right]\hfill \\ {\overline{\mathrm{\Theta }}}_{12}\stackrel{\mathrm{\Delta }}{=}\left[\begin{array}{ccc}0\hfill & 0\hfill & B_{\mathrm{dr}}\hfill \\ \multicolumn{1}{c}{\frac{1}{{\sigma }_1}{\overline{Z}}_1}& \frac{1}{{\sigma }_2}{\overline{Z}}_2\hfill & {\overline{\mathrm{\Gamma }}}_{3\mathrm{rs}}\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & -{\overline{K}}_{2s}^T{D}_r^T\hfill \end{array}\right]\hfill \\ {\overline{\mathrm{\Theta }}}_{13}\stackrel{\mathrm{\Delta }}{=}\mathrm{diag}\{-{\overline{N}}_{1n}-\frac{1}{{\sigma }_1}{\overline{Z}}_1,-{\overline{N}}_{2o}-\frac{1}{{\sigma }_2}{\overline{Z}}_2,-{\overline{N}}_{2o}-\frac{1}{{\sigma }_2}{\overline{Z}}_2\}\hfill \\ {\overline{\mathrm{\Theta }}}_2\stackrel{\mathrm{\Delta }}{=}\left[\begin{array}{cccccc}0\hfill & C_{r}X+D_r{\overline{K}}_{1s}\hfill & D_r{\overline{K}}_{2s}\hfill & 0\hfill & 0\hfill & D_{\mathrm{dr}}\hfill \end{array}\right]\hfill \\ {\overline{\mathrm{\Gamma }}}_1\stackrel{\mathrm{\Delta }}{=}(T-2)X+{\sigma }_1{\overline{Z}}_1+{\sigma }_2{\overline{Z}}_2\hfill \\ {\overline{\mathrm{\Gamma }}}_2\stackrel{\mathrm{\Delta }}{=}{[A_{r}X+B_r{\overline{K}}_{1s}]}_s+({\sigma }_2-{\sigma }_1+T+1){\overline{M}}_l\hfill \\ +{\overline{N}}_{1l}+{\overline{N}}_{2l}-\frac{1}{{\sigma }_1}{\overline{Z}}_1-\frac{1}{{\sigma }_2}{\overline{Z}}_2\hfill \\ {\overline{\mathrm{\Gamma }}}_3=B_{\mathrm{dr}}-{\mathrm{XC}}_r^T-{\overline{K}}_{1s}^T{D}_r^T\hfill \end{array}$$

The fuzzy memory–based control gain matrices are determined by

$$K_{1s}={\overline{K}}_{1s}{X}^{-1},K_{2s}={\overline{K}}_{2s}{X}^{-1}$$

(13)

Proof

First, we let $P=X^{-1}$ , $K_{1s}={\overline{K}}_{1s}{X}^{-1}$ , $K_{2s}={\overline{K}}_{2s}{X}^{-1}$ , $Z_1=X^{-1}{\overline{Z}}_1{X}^{-1}$ , $Z_2=X^{-1}{\overline{Z}}_2{X}^{-1}$ , $M_r=X^{-1}{\overline{M}}_r{X}^{-1}$ , $N_{1r}=X^{-1}{\overline{N}}_{1r}{X}^{-1}$ , and ${\overline{N}}_{2r}=X^{-1}{\overline{N}}_{2r}{X}^{-1}$ . Then, applying a congruent transformation by $\mathrm{diag}\{X^{-1},X^{-1},X^{-1},X^{-1},X^{-1},I,I\}$ to equations (11) and (12), it derives that the conditions (7) and (8) hold. That is to say system (6) is very strictly passive if equations (11) and (12) hold. This ends the proof.

As a supplement, we give an asymptotic stability criterion of the proposed memory-based control system in $\delta$ domain. Considering the exogenous disturbance $d(t_k)=0$ , we obtain the memory-based control system dynamics as follows

$$\delta x(t_k)=\sum _{r=1}^N{\lambda }_r(\mu (t_k))[A_{r}x(t_k)+B_{r}u(t_k)]$$

(14)

Following the procedure in Theorems 1 and 2, we infer the following two corollaries for memory-based control system (14).

Corollary 1

For some constants ${\sigma }_1$ and ${\sigma }_2$ ( $0<{\sigma }_1\le {\sigma }_2$ ), the delta operator control system (14) with the memory-based controller (4) is asymptotically stable if there exist appropriately dimensioned matrices $X>0$ , ${\overline{Z}}_1>0$ , ${\overline{Z}}_2>0$ , ${\overline{M}}_l>0$ , ${\overline{N}}_{1l}>0$ , and ${\overline{N}}_{2l}>0$ ( $l=1,2,\dots ,N$ ) such that for $l,m,n,o,r,s=1,2,\dots ,N$ , the following LMIs satisfy

$${\hat{\mathrm{\Phi }}}_{\mathrm{lmnorr}}<0$$

(15)

$${\hat{\mathrm{\Phi }}}_{\mathrm{lmnors}}+{\hat{\mathrm{\Phi }}}_{\mathrm{lmnosr}}<0,s<r$$

(16)

where

$$\begin{array}{c}{\hat{\mathrm{\Phi }}}_{\mathrm{lmnors}}\stackrel{\mathrm{\Delta }}{=}\left[\begin{array}{cc}{\hat{\mathrm{\Phi }}}_1\hfill & {\hat{\mathrm{\Phi }}}_2\hfill \\ \multicolumn{1}{c}{★}& {\hat{\mathrm{\Phi }}}_3\hfill \end{array}\right]\hfill \\ {\hat{\mathrm{\Phi }}}_1\stackrel{\mathrm{\Delta }}{=}\left[\begin{array}{cc}{\overline{\mathrm{\Gamma }}}_1\hfill & A_{r}X+B_r{\overline{K}}_{1s}\hfill \\ \multicolumn{1}{c}{★}& {\overline{\mathrm{\Gamma }}}_{2\mathrm{lrs}}\hfill \end{array}\right]\hfill \\ {\hat{\mathrm{\Phi }}}_2\stackrel{\mathrm{\Delta }}{=}\left[\begin{array}{ccc}{B}_r{\overline{K}}_{2s}\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{B_r{\overline{K}}_{2s}}& \frac{1}{{\sigma }_1}{\overline{Z}}_1\hfill & \frac{1}{{\sigma }_2}{\overline{Z}}_2\hfill \end{array}\right]\hfill \\ {\hat{\mathrm{\Phi }}}_3\stackrel{\mathrm{\Delta }}{=}\mathrm{diag}\{-{\overline{M}}_m,-{\overline{N}}_{1n}-\frac{1}{{\sigma }_1}{\overline{Z}}_1,-{\overline{N}}_{2o}-\frac{1}{{\sigma }_2}{\overline{Z}}_2\}\hfill \end{array}$$

Then, the parametric matrices of memory-based controller (4) are determined by $K_{1s}={\overline{K}}_{1s}{X}^{-1}$ and $K_{2s}={\overline{K}}_{2s}{X}^{-1}$ .

Simulation example

In terms of the proposed memory-based control method, we provide a numerical example to illustrate its validity. Consider the following system matrices of the two-rule fuzzy nonlinear plant with exogenous disturbance in equation (1)

$$\begin{array}{c}{A}_1=\left[\begin{array}{ccc}-1.000\hfill & -0.060\hfill & 0.100\hfill \\ \multicolumn{1}{c}{0.450}& 0.001\hfill & -0.100\hfill \\ \multicolumn{1}{c}{0.700}& 0.100\hfill & 0.001\hfill \end{array}\right],B_1=\left[\begin{array}{c}0.080\hfill \\ \multicolumn{1}{c}{0.050}\\ \multicolumn{1}{c}{0.065}\end{array}\right]\hfill \\ A_2=\left[\begin{array}{ccc}-2.000\hfill & 0.010\hfill & 0.100\hfill \\ \multicolumn{1}{c}{0.550}& 0.001\hfill & -0.100\hfill \\ \multicolumn{1}{c}{0.700}& 0.100\hfill & 0.005\hfill \end{array}\right],B_2=\left[\begin{array}{c}0.078\hfill \\ \multicolumn{1}{c}{0.090}\\ \multicolumn{1}{c}{0.065}\end{array}\right]\hfill \\ B_{d1}=\left[\begin{array}{c}0.10\hfill \\ \multicolumn{1}{c}{-0.10}\\ \multicolumn{1}{c}{-0.10}\end{array}\right],B_{d2}=\left[\begin{array}{c}0.20\hfill \\ \multicolumn{1}{c}{-0.10}\\ \multicolumn{1}{c}{-0.10}\end{array}\right]\hfill \\ C_1=\left[\begin{array}{cc}0.010\hfill & 0.010\hfill \end{array}\right],D_1=0.010,D_{d1}=0.012\hfill \\ C_2=\left[\begin{array}{cc}0.020\hfill & 0.010\hfill \end{array}\right],D_2=0.020,D_{d2}=0.021\hfill \end{array}$$

The corresponding sampling period is supposed as $T=0.1\mathrm{s}$ . The corresponding membership functions ${\lambda }_1(\mu (t_k))$ and ${\lambda }_2(\mu (t_k))$ are chosen as the trigonometric functions. Next, we calculate the parameters of the memory-based controller according to the statement in Theorem 2. Suppose the delay coefficient $\rho =0.2$ . When setting the lower bound ${\sigma }_1=0.1s$ of the delay time $\sigma (t_k)$ , we received the maximum upper bound ${\sigma }_{2\max }=6.0676$ s by solving the condition in equations (11) and (12). Then, we assume that the delayed time is randomly in closed interval [0.1,2]. Via solving the condition in equations (11) and (12), the index level $\beta =2.0311$ , $\delta =0.0030$ , and the controller gains

$$\begin{array}{c}{K}_{11}=\left[\begin{array}{ccc}-3.8453\hfill & -1.1850\hfill & -3.6204\hfill \end{array}\right]\hfill \\ K_{12}=\left[\begin{array}{ccc}-3.1987\hfill & -2.3213\hfill & -5.2212\hfill \end{array}\right]\hfill \\ K_{21}=\left[\begin{array}{ccc}3.3493\hfill & 1.0469\hfill & 3.1820\hfill \end{array}\right]\hfill \\ K_{22}=\left[\begin{array}{ccc}5.0861\hfill & 1.5787\hfill & 4.8096\hfill \end{array}\right]\hfill \end{array}$$

Assume that the disturbance is $d(t_k)=0.1(\mathrm{rand}-0.5)$ and the initial condition is $x(t_0)={\left[\begin{array}{ccc}-2\hfill & 4\hfill & 3\hfill \end{array}\right]}^T$ . Then, the simulation results are shown in Figures 1–5. According to Definition 1, let

$$\begin{array}{c}\multicolumn{1}{c}{\upsilon (t_k)=2\sum _{k=0}^{100}{z}^T(t_k)d(t_k)-\alpha \sum _{k=0}^{100}{z}^T(t_k)z(t_k)}\\ +\beta \sum _{k=0}^{100}{d}^T(t_k)d(t_k)\hfill \end{array}$$

Figure 1.

State response of the open-loop system.

Figure 2.

Output response of the open-loop system.

Figure 3.

State response of the closed-loop system.

Figure 4.

Output response of the closed-loop system.

Figure 5.

Control force of the closed-loop system.

On one hand, the open-loop system is divergent from the trajectories as depicted in Figures 1 and 2. On the other hand, when applying the memory-based controller to the considered plant, we obtained the following simulation results. By the following calculations, $\upsilon (t_k)=-2.3067\ge \gamma =-V(t_0)=-7.5229$ , we know that the simulated closed-loop system is very strictly passive. Besides, Figures 3–5 illustrate that the closed-loop system has been effectively stabilized under the designed memory-based control force.

Remark 2

The simulation results show that the simulated plant which is unstable has been effectively stabilized, and the passivity of the control system is satisfied. In engineering applications, the discrete-time control system in terms of the delta operator can be represented as the form of system (1). ²⁶ Then, the system parametric matrices can be obtained and the controller matrices can be solved according to the criterion in Theorem 2. Specific control procedure can be applied referencing the provided simulation example.

Conclusion

This article has coped with the memory-based very strictly passive control problem of nonlinear systems in delta domain. The delta operator nonlinear plant with nonlinear controllable output is approximated by T-S fuzzy models. The output passive index has been formulated and passivity criterion has been detailed. Considering a known memory coefficient, the memory-based state-feedback controller has been expressed, which is subject to a set of solvable LMIs, to memorize the influence of the delayed states. Stability condition has been provided using Lyapunov stability theory. Finally, the numerical example has validated the effectiveness of the proposed memory-based passive control scheme. Additionally, future work will be concerned with some advanced memory-based control problems when the system states are constrained, such as unmeasurable states and states with stochastic process. Furthermore, motivated by the approach to deal with the time delay and filtering,^30–32 some conservativeness can be reduced.