𝐻_∞ Control of Pairwise Distributable Large-Scale TS Fuzzy Systems

Abstract

The paper presents new conditions suitable in design of the stabilizing state controller for a class of continuous-time nonlinear systems, which are representable by pairwise distributable Takagi-Sugeno models. Taking into account the affine properties of the TS model structure and applying the pairwise subsystems fuzzy control scheme relating to the parallel distributed output compensators, the extended bounded real lemma form and the sufficient design conditions for pairwise decentralized control are outlined in terms of linear matrix inequalities. The proposed procedure decouples the Lyapunov matrix and the system parameter matrices in the LMIs and, using free tuning parameter, provides the way to obtain global stability of such large-scale TS systems and optimizes subsystems interaction 𝐻_∞ norm bounds.

1. Introduction

A number of problems that arise in state control can be reduced to a handful of standard convex and quasiconvex problems that involve matrix inequalities (LMI). It is known that the optimal solution can be computed by using the interior point methods [1], which converge in polynomial time with respect to the problem size. Thus, efficient interior point algorithms have recently been developed and further development of algorithms for these standard problems is an area of active research. For this approach, the stability conditions may be expressed in terms of LMIs, which have a notable practical interest due to the existence of numerical solvers. Some progress review in this field can be found, for example, in [2, 3] and the references therein.

Based on the concept of quadratic stability, the control design problems, in respect of the 𝐻_∞ norm of the closed-loop system transfer matrix, are transferred into a standard LMI optimization task, which includes bounded real lemma (BRL) formulation. The first version of BRL presented simple condition under which a transfer function was contractive on the imaginary axis of the complex variable plain. Using this formulation, it was possible to determine the 𝐻_∞ norm of a transfer function, and the BRL has become a significant element to show and prove that the existence of feedback controllers, resulting in a closed-loop transfer matrix having the 𝐻_∞ norm less than a given upper bound, is equivalent to the existence of a solution of certain LMIs. Motivated by the underlying ideas, the technique for BRL representation was extended to state feedback control design and stayed preferable for systems with time-varying parameters [4–6]. When used in robust analysis of linear systems with polytopic uncertainties, as the number of polytops increases, the solution turned out to be very conservative. To reduce conservatism inherent in such use of quadratic methods, the equivalent LMI representations of BRL for continuous-time as well as discrete-time uncertain systems were introduced [7–10]. Moreover, exploiting the sector nonlinearity approach to obtain Takagi-Sugeno (TS) models from the nonlinear system equations [11], suitable BRLs as well as enhanced BRL representation guaranteeing quadratic performances for the closed-loop nonlinear systems are exploited in 𝐻_∞ fuzzy control [12–14].

In last years, modern control methods have found their way into design of interconnected systems and led to a wide variety of new concepts and results. In particular, paradigms of LMIs and 𝐻_∞ norm have appeared to be very attractive due to their promise of handling systems with relative high dimensions, and design of partly decentralized schemes substantially minimized the information exchange between subsystems of a large scale system. With respect to the existing structure of interconnections in a large-scale system, it is generally impossible to stabilize all subsystems and the whole system simultaneously by using decentralized controllers, since the stability of interconnected systems is not only dependent on the stability degree of subsystems but is also closely dependent on the interconnections [15–17]. Analogous principles were applied in decentralized fuzzy control [18].

Considering the decomposition-based control strategy [19] and including into design step the effects of subsystem pairs interconnections [20], a pairwise decentralized control problem was proposed for linear large-scale systems in [21] and for linear large-scale systems with polytopic uncertainties in [22], respectively. Introducing the results as a pairwise partially decentralized control, in [21] there were formulated design conditions for a linear control, while the design conditions given in [22] reflect the robust linear control design task, both solved in the frames of LMI representations. Since the feedback control for TS models is the so-called parallel distributed compensation (PDC) control, the above results have to be significantly reformulated considering the pairwise distributable large-scale TS fuzzy systems.

Inspired by the enhanced design conditions proposed in [13] in designing the optimal control of TS systems with quadratic optimality criterion, the paper is devoted to studying the partially decentralized control problems from the above given viewpoint for pairwise distributable large-scale TS fuzzy systems. Sufficient stability conditions are stated now as a set of LMIs to encompass the quadratic stability case in respect of the 𝐻_∞ approach. Used structures in the presented forms enable us potentially to design systems with an embedded reconfigurable control structure property [21]. To the best of the authors’ knowledge, the paper presents a new formulation of the control principle of pairwise distributable large-scale TS fuzzy systems, newly defines the conditions of existence of solutions for the fuzzy control scheme relating to PDCs in such distributable structure, and offers the possibility of new ways to solve the problem of control law synthesis for TS fuzzy systems with a large number of membership functions.

The paper is organized as follows. In Section 2, the basis preliminaries, concerning the TS models and the 𝐻_∞ norm problems, are presented with results on BRL and enhanced BRL for TS systems. Formulating the pairwise distributable large-scale TS fuzzy systems structure in Section 3 and continuing with this formalism in Section 4, the equivalent TS BRL design methods are outlined to possess the sufficient conditions for the pairwise decentralized control of given class of TS large-scale systems. Finally, the example is given in Section 5, to illustrate the feasibility and properties of the proposed method and some concluding remarks are stated in Section 6.

Throughout the paper, the following notations are used: 𝐱^𝑇, 𝐗^𝑇 denote the transposes of the vector 𝐱 and matrix 𝐗, respectively, diag[𝐗_𝑖], 𝑖=1,2,…,𝑝 denotes a block diagonal matrix with 𝑝 blocks, blk[𝐗_𝑖𝑙], 𝑖,𝑙=1,2,…,𝑝 entails a row- and column-wise partitioned matrix, broken into blocks by partitioning its rows and columns into 𝑝 collection of row-groups and 𝑝 collection of column-groups, for a square matrix, 𝐗<0 means that 𝐗 is a symmetric negative definite matrix, the symbol 𝐈_𝑛 indicates the 𝑛th order unit matrix, ℝ denotes the set of real numbers, and ℝ^𝑛×𝑟 refers to the set of 𝑛×𝑟 real matrices.

2. Preliminaries

2.1. System Model

The class of TS systems, considered in the paper, is formed as follows: 𝐪̇(𝑡)=^𝑠_𝑗=1ℎ_𝑗(𝜽(𝑡))𝐀_𝑗𝐪(𝑡)+𝐁_𝑗𝐮(𝑡),𝐲(𝑡)=𝐂𝐪(𝑡), (1)(2) where 𝐪(𝑡)∈ℝ^𝑛, 𝐮(𝑡)∈ℝ^𝑟, and 𝐲(𝑡)∈ℝ^𝑚 are vectors of the state, input, and measurable output variables, respectively, matrices 𝐀_𝑗∈ℝ^𝑛×𝑛, 𝐁_𝑗∈ℝ^𝑛×𝑟, and 𝐂∈ℝ^𝑚×𝑛 are real matrices, 𝑡∈ℝ is the time variable, and 𝜇_𝑗(𝜽(𝑡)) is the weight for 𝑗th fuzzy rule. Satisfying the following, by definition, the property 0≤𝜇_𝑗(𝜽(𝑡))≤1,^𝑠_𝑗=1𝜇_𝑗(𝜽(𝑡⟨⟩,𝜽(𝑡)=𝜃))=1∀𝑗∈1,2,…,𝑠₁(𝑡)𝜃₂(𝑡)⋯𝜃_𝑣(𝑡) (3) is the vector of the premise variables, where 𝑠, 𝑣 are the numbers of fuzzy rules and premise variables, respectively. It is supposed next that all premise variables are measurable and independent on 𝐮(𝑡) (more details can be found, e.g., in [11, 13]).

2.2. LMI Formulations for 𝐻_∞ Performance

Let the TS systems model (1), (2) be considered (in this subsection only) in the next extended form 𝐪̇(𝑡)=^𝑠_𝑗=1𝜇_𝑗(𝜽(𝑡))𝐀_𝑗𝐪(𝑡)+𝐁_𝑗𝐮(𝑡)+𝐆_𝑗𝐰(𝑡),𝐲(𝑡)=𝐂𝐪(𝑡)+𝐃𝐰(𝑡), (4)(5) where 𝐆_𝑗∈ℝ^𝑛×𝑟, 𝐃∈ℝ^𝑚×𝑟, and 𝑤(𝑡)∈ℝ^𝑟 is the disturbance input that belongs to 𝐿₂⟨0,+∞) and, using the same set of membership functions, the fuzzy state control law is defined as 𝐮(𝑡)=−^𝑠_𝑔=1𝜇_𝑔(𝜽(𝑡))𝐊_𝑔𝐪(𝑡). (6) Evidently, the closed-loop system state variable dynamics is described by the next equation 𝐪̇(𝑡)=^𝑠𝑠_𝑗=1𝑔=1𝜇_𝑗(𝜽(𝑡))𝜇_𝑔(𝜽(𝑡))𝐇_𝑗𝑔𝐪(𝑡)+𝐆_𝑗𝐰(𝑡) (7) and also, owing to the symmetry in summations, by the symmetric equation 𝐪̇(𝑡)=^𝑠𝑠_𝑗=1𝑔=1𝜇_𝑗(𝜽(𝑡))𝜇_𝑔(𝜽(𝑡))𝐇_𝑔𝑗𝐪(𝑡)+𝐆_𝑔𝐰(𝑡), (8) where 𝐇_𝑗𝑔=𝐀_𝑗−𝐁_𝑗𝐊_𝑔,𝐇_𝑔𝑗=𝐀_𝑔−𝐁_𝑔𝐊_𝑗. (9) Thus, adding (8), (9) gives 2𝐪̇(𝑡)=^𝑠𝑠_𝑗=1𝑔=1𝜇_𝑗(𝜽(𝑡))𝜇_𝑔(𝜽(𝑡×𝐇))_𝑗𝑔+𝐇_𝑔𝑗𝐪(𝑡)+𝐆_𝑗+𝐆_𝑔𝐰(𝑡). (10) Rearranging the computation, (10) can be written as 2𝐪̇(𝑡)=^𝑠_𝑗=1𝜇_𝑗(𝜽(𝑡))𝜇_𝑗(𝜽(𝑡×𝐇))_𝑗𝑗+𝐇_𝑗𝑗𝐪(𝑡)+𝐆_𝑗+𝐆_𝑗𝐰(𝑡)+2^𝑠−1𝑠_{𝑗=1𝑔=𝑗+1}𝜇_𝑗(𝜽(𝑡))𝜇_𝑔(𝜽(𝑡×𝐇))_𝑗𝑔+𝐇_𝑔𝑗𝐪(𝑡)+𝐆_𝑗+𝐆_𝑔𝐰(𝑡),𝐪̇(𝑡)=^𝑠_𝑗=1𝜇_𝑗(𝜽(𝑡))𝜇_𝑗(𝜽(𝑡×𝐇))_𝑗𝑗𝐪(𝑡)+𝐆_𝑗𝐰(𝑡)+2^𝑠−1𝑠_{𝑗=1𝑔=𝑗+1}𝜇_𝑗(𝜽(𝑡))𝜇_𝑔(𝜽(𝑡×𝐇))_𝑗𝑔+𝐇_𝑔𝑗2𝐪(𝑡)+𝐆_𝑗+𝐆_𝑔2𝐰(𝑡), (11)(12) respectively.

Considering this, the next lemmas can be introduced.

Lemma 1 (bounded real lemma). The closed-loop system (7), (5) is stable with the performance ∑^𝑠_𝑗=1∑^𝑠_𝑔=1𝜇_𝑗(𝜽(𝑡))𝜇_𝑔(𝜽(𝑡))‖𝐂(𝑠𝐈−𝐇_𝑗𝑔)⁻¹𝐆_𝑗+𝐃‖²_∞≤𝛾 if there exist a positive definite matrix 𝐗∈ℝ^𝑛×𝑛 and a positive scalar 𝛾∈ℝ such that 𝐗=𝐗^𝑇>0,𝛾>0,𝐇_𝑗𝑗𝐗+𝐗𝐇^𝑇_𝑗𝑗𝐆_𝑗𝐗𝐂^𝑇∗−𝛾𝐈_𝑟𝐃^𝑇∗∗−𝐈_𝑚<0,𝐇_𝑗𝑔+𝐇_𝑔𝑗2𝐇𝐗+𝐗^𝑗𝑔+𝐇_𝑔𝑗^𝑇2𝐆_𝑗+𝐆_𝑔2𝐗𝐂^𝑇∗−𝛾𝐈_𝑟𝐃^𝑇∗∗−𝐈_𝑚<0, (13)(14)(15) for all 𝑗∈⟨1,2,…𝑠⟩, and 𝑗<𝑔≤𝑠, 𝑗,𝑔∈⟨1,2,…𝑠⟩, respectively, where 𝐈_𝑟∈ℝ^𝑟×𝑟, 𝐈_𝑚∈ℝ^𝑚×𝑚 are identity matrices and √𝛾 is an upper bound of 𝐻_∞ norm of the disturbance transfer matrix function.
Here and hereafter, ∗ denotes the symmetric item in a symmetric matrix.

Proof. Defining Lyapunov function as 𝑣(𝐪(𝑡))=𝐪^𝑇(𝑡)𝐏𝐪(𝑡)+3^𝑡₀𝐲^𝑇(𝜏)𝐲(𝜏)−𝛾𝐰^𝑇(𝜏)𝐰(𝜏)𝑑𝜏>0, (16) where 𝐏=𝐏^𝑇>0, 𝐏∈ℝ^𝑛×𝑛, 𝛾>0, 𝛾∈ℝ and evaluating the derivative of 𝑣(𝐪(𝑡)) with respect to 𝑡 along a system trajectory, then it yields 𝑣̇(𝐪(𝑡))=𝐪̇^𝑇(𝑡)𝐏𝐪(𝑡)+𝐪^𝑇(𝑡)𝐏𝐪̇(𝑡)+3𝐲^𝑇(𝑡)𝐲(𝑡)−𝛾𝐰^𝑇(𝑡)𝐰(𝑡)<0. (17) Substituting (5) and (12) in (17), the next inequality is obtained: 𝑣̇(𝐪(𝑡=))_𝑠𝑗=1𝜇_𝑗(𝜽(𝑡))𝜇_𝑗(𝜽(𝑡×𝐇))_𝑗𝑗𝐪(𝑡)+𝐆_𝑗𝐰(𝑡)^𝑇𝐏𝐪(𝑡)+𝐪^𝑇(𝑡)𝐏𝐇_𝑗𝑗𝐪(𝑡)+𝐆_𝑗𝐰(𝑡)+(𝐂𝐪(𝑡)+𝐃𝐰(𝑡))^𝑇(𝐂𝐪(𝑡)+𝐃𝐰(𝑡))−𝛾𝐰^𝑇(𝑡)𝐰(𝑡)+2^𝑠−1𝑠_{𝑗=1𝑔=𝑗+1}𝜇_𝑗(𝜽(𝑡))𝜇_𝑔(𝜽(𝑡×𝐇))_𝑗𝑔+𝐇_𝑔𝑗2𝐪(𝑡)+𝐆_𝑗+𝐆_𝑔2𝐰(𝑡)^𝑇𝐏𝐪(𝑡)+𝐪^𝑇(𝑡)𝐏𝐇_𝑗𝑔+𝐇_𝑔𝑗2𝐪(𝑡)+𝐆_𝑗+𝐆_𝑔2𝐰(𝑡)+(𝐂𝐪(𝑡)+𝐃𝐰(𝑡))^𝑇(𝐂𝐪(𝑡)+𝐃𝐰(𝑡))−𝛾𝐰^𝑇(𝑡)𝐰(𝑡)<0 (18) and with the notation 𝐪^𝑇_𝑤(𝑡)=𝐪^𝑇(𝑡)𝐰^𝑇(𝑡), (19) it yields 𝑣̇(𝐪(𝑡))=^𝑠_𝑗=1𝜇_𝑗(𝜽(𝑡))𝜇_𝑗(𝜽(𝑡))𝐪^𝑇_𝑤(𝑡)×𝐑_𝑗𝑗+𝐐𝐪_𝑤(𝑡)+2^𝑠−1𝑠_{𝑗=1𝑔=𝑗+1}𝜇_𝑗(𝜽(𝑡))𝜇_𝑔(𝜽(𝑡))×𝐪^𝑇_𝑤(𝑡)𝐑_𝑗𝑔+𝐐𝐪_𝑤(𝑡)<0, (20) where 𝐑_𝑗𝑗=𝐏𝐇_𝑗𝑗+𝐇^𝑇_𝑗𝑗𝐏𝐏𝐆_𝑗∗−𝛾𝐈_𝑟𝐂,𝐐=^𝑇𝐃^𝑇,𝐑𝐂𝐃_𝑗𝑔=𝐏𝐇_𝑗𝑔+𝐇_𝑔𝑗2+𝐇_𝑗𝑔+𝐇_𝑔𝑗^𝑇2𝐆𝐏𝐏^𝑗+𝐆_𝑔2∗−𝛾𝐈_𝑟. (21) Since (20) implies 𝐑_𝑗𝑗+𝐐<0,𝐑_𝑗𝑔+𝐐<0, (22) for all 𝑗∈⟨1,2,…,𝑠⟩ and 𝑗<𝑔≤𝑠, 𝑖,𝑗∈⟨1,2,…,𝑠⟩, respectively, then using the Schur complement property, (22) can be written as 𝐏𝐇_𝑗𝑗+𝐇^𝑇_𝑗𝑗𝐏𝐏𝐆_𝑗𝐂^𝑇∗−𝛾𝐈_𝑟𝐃^𝑇∗∗−𝐈_𝑚𝐏𝐇<0,_𝑗𝑔+𝐇_𝑔𝑗2+𝐇_𝑗𝑔+𝐇_𝑔𝑗^𝑇2𝐆𝐏𝐏^𝑗+𝐆_𝑔2𝐂^𝑇∗−𝛾𝐈_𝑟𝐃^𝑇∗∗−𝐈_𝑚<0. (23) Defining the transform matrix 𝐋 as 𝐋=diag𝐏⁻¹𝐈_𝑟𝐈_𝑚, (24) premultiplying the left-hand side and the right-hand side of (23) by 𝐋, gives 𝐇_𝑗𝑗𝐏⁻¹+𝐏⁻¹𝐇^𝑇_𝑗𝑗𝐆_𝑗𝐏⁻¹𝐂^𝑇∗−𝛾𝐈_𝑟𝐃^𝑇∗∗−𝐈_𝑚<0,𝐇_𝑗𝑔+𝐇_𝑔𝑗2𝐏⁻¹+𝐏⁻¹𝐇_𝑗𝑔+𝐇_𝑔𝑗^𝑇2𝐆_𝑗+𝐆_𝑔2𝐏⁻¹𝐂^𝑇∗−𝛾𝐈_𝑟𝐃^𝑇∗∗−𝐈_𝑚<0. (25)(26) Thus, using the substitution 𝐗=𝐏⁻¹ then (25) and (26) imply (14) and (15), respectively. This concludes the proof.

Remark 2. Another form of bounded real lemma can be obtained using Lyapunov function of the form 𝑣(𝐪(𝑡))=𝐪^𝑇(𝑡)𝐏𝐪(𝑡)+3𝛾^−1𝑡₀𝐲^𝑇(𝜏)𝐲(𝜏)−𝛾²𝐰^𝑇(𝜏)𝐰(𝜏)𝑑𝜏>0, (27) (compare, e.g., [6, 12, 13, 18]), but the BRL forms implying from (27) result, in general, the higher 𝐻_∞ norm upper bound 𝛾 then using the presented conditions (14), (15).

Lemma 3 (enhanced bounded real lemma). The closed-loop system (7), (5) is stable with the performance ∑^𝑠_𝑗=1∑^𝑠_𝑔=1𝜇_𝑗(𝜽(𝑡))𝜇_𝑔(𝜽(𝑡))‖𝐂(𝑠𝐈−𝐇_𝑗𝑔)⁻¹𝐆_𝑗+𝐃‖²_∞≤𝛾 if for given 𝛿>0, 𝛿∈ℝ, there exist positive definite matrices 𝐕,𝐓∈ℝ^𝑛×𝑛 and a positive scalar 𝛾∈ℝ such that𝐕=𝐕^𝑇>0,𝛾>0,𝐇_𝑗𝑗𝐕+𝐕𝐇^𝑇_𝑗𝑗𝐆_𝑗𝐓−𝛿𝐕+𝐕𝐇^𝑇_𝑗𝑗𝐕𝐂^𝑇∗−𝛾𝐈_𝑟𝐆^𝑇_𝑗𝐃^𝑇∗∗−2𝛿𝐕𝟎∗∗∗−𝐈_𝑚<0,𝐇_𝑗𝑔+𝐇_𝑔𝑗2𝐇𝐕+𝐕^𝑗𝑔+𝐇_𝑔𝑗^𝑇2𝐆_𝑗+𝐆_𝑔2𝐇𝐓−𝛿𝐕+𝐕^𝑗𝑔+𝐇_𝑔𝑗^𝑇2𝐕𝐂^𝑇∗−𝛾𝐈_𝑟𝐆_𝑗+𝐆_𝑔^𝑇2𝐃^𝑇∗∗−2𝛿𝐕𝟎∗∗∗−𝐈_𝑚<0, (28)(29)(30)for all 𝑗∈⟨1,2,…,𝑠⟩ and 𝑗<𝑔≤𝑠, 𝑗,𝑔∈⟨1,2,…,𝑠⟩, respectively.

Proof. Since (12) implies ^𝑠_𝑗=1𝜇_𝑗(𝜽(𝑡))𝜇_𝑗(𝜽(𝑡))𝐇_𝑗𝑗𝐪(𝑡)+𝐆_𝑗𝐰(𝑡)+2^𝑠−1𝑠_{𝑗=1𝑔=𝑗+1}𝜇_𝑗(𝜽(𝑡))𝜇_𝑔(𝜽(𝑡×𝐇))_𝑗𝑔+𝐇_𝑔𝑗2𝐪(𝑡)+𝐆_𝑗+𝐆_𝑔2𝐰(𝑡)−𝐪̇(𝑡)=𝟎, (31) then, with arbitrary regular symmetric square matrices 𝐒₁,𝐒₂∈ℝ^𝑛×𝑛, it yields 𝐪^𝑇(𝑡)𝐒₁+𝐪̇^𝑇(𝑡)𝐒₂^{𝑠∑𝑗=1𝜇𝑗(𝜽(𝑡𝜇))𝑗(𝜽(𝑡𝐇))𝑗𝑗𝐪(𝑡)+𝐆𝑗𝐰(𝑡)}_{+2𝑠−1∑𝑠𝑗=1∑𝑔=𝑗+1𝜇𝑗(𝜽(𝑡𝜇))𝑔(𝜽(𝑡𝐇))𝑗𝑔+𝐇𝑔𝑗2𝐪(𝑡)+𝐆𝑗+𝐆𝑔2𝐰(𝑡−̇𝐪(𝑡))}=𝟎. (32)Thus, adding (32), as well as its transposition to (17), and substituting (5) into (17) give 𝑣̇(𝐪(𝑡=𝐪̇0>))_𝑇(𝑡)𝐏𝐪(𝑡)+𝐪^𝑇(𝑡)𝐏𝐪̇(𝑡)+3(𝐂𝐪(𝑡)+𝐃𝐰(𝑡))^𝑇(𝐂𝐪(𝑡)+𝐃𝐰(𝑡))−3𝛾𝐰^𝑇(𝑡)𝐰(𝑡)+2𝐪^𝑇(𝑡)𝐒₁+𝐪̇^𝑇(𝑡)𝐒₂×^𝑠_𝑗=1𝜇_𝑗(𝜽(𝑡))𝜇_𝑗(𝜽(𝑡×𝐇))_𝑗𝑗𝐪(𝑡)+𝐆_𝑗𝐰(𝑡)−𝐪̇(𝑡)+4𝐪^𝑇(𝑡)𝐒₁+𝐪̇^𝑇(𝑡)𝐒₂×^𝑠−1𝑠_{𝑗=1𝑔=𝑗+1}𝜇_𝑗(𝜽(𝑡))𝜇_𝑔(𝜽(𝑡×𝐇))_𝑗𝑔+𝐇_𝑔𝑗2𝐪(𝑡)+𝐆_𝑗+𝐆_𝑔2𝐰(𝑡), (33) and using the notation 𝐪_𝑤^⋄𝑇(𝑡)=𝐪^𝑇(𝑡)𝐰^𝑇(𝑡)𝐪̇^𝑇(𝑡) (34) can be obtained the following: 𝑣̇(𝐪(𝑡))=^𝑠_𝑗=1𝜇_𝑗(𝜽(𝑡))𝜇_𝑗(𝜽(𝑡))×𝐪_𝑤^⋄𝑇(𝑡)𝐑^⋄_𝑗𝑗+𝐐^⋄𝐪^⋄_𝑤(𝑡)+2^𝑠−1𝑠_{𝑗=1𝑔=𝑗+1}𝜇_𝑗(𝜽(𝑡))𝜇_𝑔(𝜽(𝑡))×𝐪_𝑤^⋄𝑇(𝑡)𝐑^⋄_𝑗𝑔+𝐐^⋄𝐪^⋄_𝑤(𝑡)<0, (35) where𝐑^⋄_𝑗𝑗=𝐒₁𝐇_𝑗𝑗+𝐇^𝑇_𝑗𝑗𝐒₁𝐒₁𝐆_𝑗𝐏−𝐒₁+𝐇^𝑇_𝑗𝑗𝐒₂∗−𝛾𝐈_𝑟𝐆^𝑇_𝑗𝐒₂∗∗−2𝐒₂,𝐑^⋄_𝑗𝑔=𝐒₁𝐇_𝑗𝑔+𝐇_𝑔𝑗2+𝐇_𝑗𝑔+𝐇_𝑔𝑗^𝑇2𝐒₁𝐒₁𝐆_𝑗+𝐆_𝑔2𝐏−𝐒₁+𝐇_𝑗𝑔+𝐇_𝑔𝑗^𝑇2𝐒₂∗−𝛾𝐈_𝑟𝐆_𝑗+𝐆_𝑔^𝑇2𝐒₂∗∗−2𝐒₂,𝐐^⋄=𝐂^𝑇𝐃^𝑇𝟎𝐂𝐃𝟎. (36)Since (35) implies 𝐑^⋄_𝑗𝑗+𝐐^⋄<0,𝐑^⋄_𝑗𝑔+𝐐^⋄<0, (37) for all 𝑗∈⟨1,2,…,𝑠⟩ and 𝑗<𝑔≤𝑠, 𝑖,𝑗∈⟨1,2,…,𝑠⟩, respectively, using the Schur complement property, (37) can be written as 𝐒₁𝐇_𝑗𝑗+𝐇^𝑇_𝑗𝑗𝐒₁𝐒₁𝐆_𝑗𝐏−𝐒₁+𝐇^𝑇_𝑗𝑗𝐒₂𝐂^𝑇∗−𝛾𝐈_𝑟𝐆^𝑇_𝑗𝐒₂𝐃^𝑇∗∗−2𝐒₂𝟎∗∗∗−𝐈_𝑚,𝐒₁𝐇_𝑗𝑔+𝐇_𝑔𝑗2+𝐇_𝑗𝑔+𝐇_𝑔𝑗^𝑇2𝐒₁𝐒₁𝐆_𝑗+𝐆_𝑔2𝐏−𝐒₁+𝐇_𝑗𝑔+𝐇_𝑔𝑗^𝑇2𝐒₂𝐂^𝑇∗−𝛾𝐈_𝑟𝐆_𝑗+𝐆_𝑔^𝑇2𝐒₂𝐃^𝑇∗∗−2𝐒₂𝟎∗∗∗−𝐈_𝑚. (38)Since 𝐒₁, 𝐒₂ are supposed to be regular and symmetric, the transform matrix 𝐋^⋄ can be defined as 𝐋^⋄=diag𝐒₁⁻¹𝐈_𝑟𝐒₂⁻¹𝐈_𝑚, (39) and pre-multiplying the left-hand and the right-hand side of (38) by 𝐋^⋄ gives 𝐇_𝑗𝑗𝐒₁⁻¹+𝐒₁⁻¹𝐇^𝑇_𝑗𝑗𝐆_𝑗𝐓−𝐒₂⁻¹+𝐒₁⁻¹𝐇^𝑇_𝑗𝑗𝐒₁⁻¹𝐂^𝑇∗−𝛾𝐈_𝑟𝐆^𝑇_𝑗𝐃^𝑇∗∗−2𝐒₂⁻¹𝟎∗∗∗−𝐈_𝑚,𝚷_𝑗𝑔𝐆_𝑗+𝐆_𝑔2𝐓−𝐒₂⁻¹+𝐒₁⁻¹𝐇_𝑗𝑔+𝐇_𝑔𝑗^𝑇2𝐒₁⁻¹𝐂^𝑇∗−𝛾𝐈_𝑟𝐆_𝑗+𝐆_𝑔^𝑇2𝐃^𝑇∗∗−2𝐒₂⁻¹𝟎∗∗∗−𝐈_𝑚,𝐓=𝐒₁⁻¹𝐏𝐒₂⁻¹,𝚷_𝑗𝑔=𝐇_𝑗𝑔+𝐇_𝑔𝑗2𝐒₁⁻¹+𝐒₁⁻¹𝐇_𝑗𝑔+𝐇_𝑔𝑗^𝑇2. (40)(41)(42) Thus, using the notation 𝐒₂⁻¹=𝛿𝐕,𝐒₁⁻¹=𝐕, (43) (40), (41) imply (29), (30). This concludes the proof.

This enhanced form of the bounded real lemma eliminates products of a Lyapunov matrix 𝐓 and the system matrix parameters 𝐀_𝑗, 𝐁_𝑗, 𝐂, and 𝐃 in the LMI stability conditions. When used in the synthesis of controllers or observers for TS systems (that are, evidently, the systems with polytopic uncertainties), the enhanced form of BRL gives solutions that are less conservative than ones given by the standard form of BRL. In that sense, the enhanced form of BRL can be preferred in TS systems analysis and design [13] giving less conservative equivalency to the synthesis based on the parameter dependent Lyapunov functions principle [4].

3. Pairwise Distributed Principle in Control Design

The main property of the pairwise distributable structure of TS large-scale systems is given by the next two lemmas.

Lemma 4. Let the system (1), (2) is structured in 𝑝 subsystems in such way that, for 𝑗=1,2,…,𝑠, 𝑟≥3, 𝑖,𝑙=1,2,…,𝑝, 𝐀_𝑗=𝑏𝑙𝑘𝐀_𝑗𝑖𝑙,𝐂=𝑏𝑙𝑘𝐂_𝑖𝑙,𝐁_𝑗=diag𝐁_𝑗𝑖,𝐃=𝟎,𝐪̇_ℎ(𝑡)=^𝑠_𝑗=1𝜇_𝑗(𝜽(𝑡×𝐀))_𝑗ℎℎ𝐪_ℎ(𝑡)+^𝑝_{𝑙=1,𝑙≠ℎ}𝐀_𝑗ℎ𝑙𝐪_𝑙(𝑡)+𝐁_𝑗ℎ𝐮_ℎ(𝑡),𝐲_ℎ(𝑡)=𝐂_ℎℎ𝐪_ℎ(𝑡)+^𝑝_{𝑙=1,𝑙≠ℎ}𝐂_ℎ𝑙𝐪_𝑙(𝑡), (44)(45)(46) where 𝐪_ℎ(𝑡)∈ℝ^𝑛ℎ, 𝐮_ℎ(𝑡)∈ℝ^𝑟ℎ, 𝐲_ℎ(𝑡)∈ℝ^𝑚ℎ, 𝐀_𝑗ℎ𝑙∈ℝ^{𝑛ℎ×𝑛𝑙}, 𝐁_𝑗ℎ∈ℝ^{𝑟ℎ×𝑛ℎ}, and 𝐂_ℎ𝑙∈ℝ^{𝑚ℎ×𝑛ℎ}, respectively, and 𝑛=∑^𝑝_ℎ=1𝑛_ℎ, 𝑟=∑^𝑝_ℎ=1𝑟_ℎ, and 𝑚=∑^𝑝_ℎ=1𝑚_ℎ.
Then the ℎ𝑘th unforced subsystem pair in unforced system (1), (2), and (44) is stable if there exists a set of symmetric matrices 𝐏^∘_ℎ𝑘=𝐏^𝑘_ℎ𝐏_ℎ𝑘𝐏_𝑘ℎ𝐏^ℎ_𝑘,ℎ=1,2,…,𝑝−1,𝑘=ℎ+1,ℎ+2,…,𝑝, (47) such that 𝐪̇^𝑝−1𝑝_{ℎ=1𝑘=ℎ+1}^𝑇_ℎ𝑘(𝑡)𝐏^𝑘_ℎ𝐏_ℎ𝑘𝐏_𝑘ℎ𝐏^ℎ_𝑘𝐪_ℎ𝑘(𝑡)+𝐪^𝑇_ℎ𝑘(𝑡)𝐏^𝑘_ℎ𝐏_ℎ𝑘𝐏_𝑘ℎ𝐏^ℎ_𝑘𝐪̇_ℎ𝑘(𝑡)<0, (48) where 𝐪^𝑇_ℎ𝑘(𝑡)=𝐪^𝑇_ℎ(𝑡)𝐪^𝑇_𝑘(𝑡),𝐪̇_ℎ𝑘(𝑡)=^𝑠_𝑗=1𝜇_𝑗(𝜽(𝑡×𝐀))_𝑗ℎℎ𝐀_𝑗ℎ𝑘𝐀_𝑗𝑘ℎ𝐀_𝑗𝑘𝑘𝐪_ℎ𝑘(𝑡)+^𝑝_{𝑙=1𝑙≠ℎ,𝑘}𝐀_𝑗ℎ𝑙𝐀_𝑗𝑘𝑙𝐪_𝑙(𝑡). (49)(50)

Proof. Defining Lyapunov function as 𝑣(𝐪(𝑡))=𝐪^𝑇(𝑡)𝐏𝐪(𝑡)>0, (51) where 𝐏=𝐏^𝑇>0, 𝐏∈ℝ^𝑛×𝑛, then the time derivative of 𝑣(𝐪(𝑖)) along a solution of the system (1), (2) is 𝑣̇(𝐪(𝑡))=𝐪̇^𝑇(𝑡)𝐏𝐪(𝑡)+𝐪^𝑇(𝑡)𝐏𝐪̇(𝑡)<0. (52) Considering 𝐏 of the next form 𝐏𝐏=¹¹𝐏₁₂…𝐏_1𝑝𝐏₂₁𝐏₂₂…𝐏_2𝑝𝐏⋮⋮⋱⋮_𝑝1𝐏_𝑝2…𝐏_𝑝𝑝,𝐏_ℎℎ=^𝑝_{𝑙=1𝑙≠ℎ}𝐏^𝑙_ℎ,ℎ=1,2,…𝑝, (53) then the next separation is possible 𝐏𝐏=²¹𝐏₁₂𝐏𝟎…𝟎₂₁𝐏¹₂⋮𝐏𝟎…𝟎𝟎𝟎𝟎…𝟎+⋯+^𝑝1𝟎…𝟎𝐏_1𝑝⋮𝐏𝟎𝟎…𝟎𝟎_𝑝1𝟎…𝟎𝐏¹_𝑝⋮+⋯+𝟎…𝟎𝟎𝟎𝟎…𝟎𝐏^𝑝_𝑝−1𝐏_{𝑝−1,𝑝}𝟎…𝟎𝐏_{𝑝,𝑝−1}𝐏_𝑝^𝑝−1. (54) Using (49) and writing (45) as 𝐪̇_ℎ𝑘(𝑡)=^𝑠_𝑗=1𝜇_𝑗(𝜽(𝑡×𝐀))_𝑗ℎℎ𝐀_𝑗ℎ𝑘𝐀_𝑗𝑘ℎ𝐀_𝑗𝑘𝑘𝐪_ℎ𝑘(𝑡)+∑^𝑝_{𝑙=1𝑙≠ℎ,𝑘}𝐀_𝑗ℎ𝑙𝐀_𝑗𝑘𝑙𝐪_𝑙(𝑡)+𝐁_𝑗ℎ𝟎𝟎𝐁_𝑗𝑘𝐮_ℎ(𝑡)𝐮_𝑘(𝑡) (55) and considering that for an unforced regime 𝐮_𝑙(𝑡)=𝟎, 𝑙=1,2,…𝑝, then (55) implies (50). Subsequently, (1), (52), and (54) give ^𝑠_𝑗=1𝜇_𝑗(𝜽(𝑡))^𝑝−1𝑝_{ℎ=1𝑘=ℎ+1}𝐪̇^𝑇_ℎ𝑘(𝑡)𝐏^𝑘_ℎ𝐏_ℎ𝑘𝐏_𝑘ℎ𝐏^ℎ_𝑘𝐪_ℎ𝑘(𝑡)+𝐪^𝑇_ℎ𝑘(𝑡)𝐏^𝑘_ℎ𝐏_ℎ𝑘𝐏_𝑘ℎ𝐏^ℎ_𝑘𝐪̇_ℎ𝑘(𝑡)<0, (56) and the inequality (56) implies (48). This concludes the proof.

Lemma 5. System (1), (2), (44) with the pairwise distributed control takes the form 𝐪̇_ℎ𝑘(𝑡)=^𝑠𝑠_𝑗=1𝑔=1𝜇_𝑗(𝜽(𝑡))𝜇_𝑔(𝜽(𝑡×𝐀))_{∘𝑗ℎ𝑘}−𝐁^∘_𝑗ℎ𝑘𝐊^∘_𝑔ℎ𝑘𝐪_ℎ𝑘(𝑡)+𝝎^∘_{𝑗𝑔ℎ𝑘}(𝑡),𝐲_ℎ𝑘(𝑡)=𝐂^∘_ℎ𝑘𝐪_ℎ𝑘(𝑡)+^𝑝_{𝑙=1𝑙≠ℎ,𝑘}𝐂^𝑙∘_ℎ𝑘𝐪_𝑙(𝑡)+𝟎𝝎_ℎ𝑘(𝑡), (57)(58) where 𝐀^∘_𝑗ℎ𝑘=𝐀_𝑗ℎℎ𝐀_𝑗ℎ𝑘𝐀_𝑗𝑘ℎ𝐀_𝑗𝑘𝑘,𝐁^∘_𝑗ℎ𝑘=𝐁_𝑗ℎ𝟎𝟎𝐁_𝑗𝑘,𝐊^∘_𝑔ℎ𝑘𝐊^𝑘_𝑔ℎ𝐊_𝑔ℎ𝑘𝐊_𝑔𝑘ℎ𝐊^ℎ_𝑔𝑘,𝝎^∘_{𝑗𝑔ℎ𝑘}(𝑡)=^𝑝_{𝑙=1𝑙≠ℎ,𝑘}𝐀^𝑙∘_𝑗ℎ𝑘𝐪_𝑙(𝑡)+𝐁^∘_𝑗ℎ𝑘𝝎_𝑔ℎ𝑘(𝑡),𝐀^𝑙∘_𝑗ℎ𝑘=𝐀_𝑗ℎ𝑙𝐀_𝑗𝑘𝑙,𝝎_𝑔ℎ𝑘(𝑡)=^𝑝_{𝑙=1𝑙≠ℎ,𝑘}𝐮^𝑙_𝑔ℎ(𝑡)𝐮^𝑙_𝑔𝑘(𝑡),𝐮^𝑙_𝑔ℎ(𝑡)=𝐊^𝑙_𝑔ℎ𝐊_𝑔ℎ𝑙𝐪_ℎ(𝑡)𝐪_𝑙(𝑡),𝐂^∘_ℎ𝑘=𝐂^𝑘_ℎ𝐂_ℎ𝑘𝐂_𝑘ℎ𝐂^ℎ_𝑘,𝐂^𝑙∘_ℎ𝑘=𝐂_ℎ𝑙𝐂_𝑘𝑙,𝐂_ℎℎ=^𝑝_{𝑙=1𝑙≠ℎ}𝐂^𝑙_ℎ. (59)(60)(61)(62)

Proof. Considering in (6) the same structure of 𝐊_𝑔, 𝑔=1,2,…𝑠, as is defined for 𝐏 in (53), that is, 𝐊_𝑔=𝐊_𝑔11𝐊_𝑔12…𝐊_𝑔1𝑝𝐊_𝑔21𝐊_𝑔22…𝐊_𝑔2𝑝⋮𝐊_𝑔𝑝1𝐊_𝑔𝑝2…𝐊_𝑔𝑝𝑝,𝐊_𝑔ℎℎ=^𝑝_{𝑙=1𝑙≠ℎ}𝐊^𝑙_𝑔ℎ, (63) then the control law 𝐮_ℎ(𝑡) takes the form 𝐮_ℎ(𝑡)=−^𝑠_𝑔=1𝜇_𝑔(𝜽(𝑡))𝐊_𝑔ℎℎ𝐪_ℎ(𝑡)+^𝑝_{𝑙=1𝑙≠ℎ}𝐊_𝑔ℎ𝑙𝐪_𝑙(𝑡), (64) where 𝐊_𝑔ℎ𝑙(𝑡), ℎ,𝑙=1,2,…,𝑝, are non-zero gain matrices.
Exploiting the main diagonal block property (63), then 𝐮_ℎ(𝑡)=−^𝑠_𝑔=1𝜇_𝑔(𝜽(𝑡×))_{𝑝𝑙=1𝑙≠ℎ}𝐊^𝑙_𝑔ℎ𝐊_𝑔ℎ𝑙𝐪_ℎ(𝑡)𝐪_𝑙(𝑡)=−^𝑠_𝑔=1𝜇_𝑔(𝜽(𝑡))𝐊^𝑘_𝑔ℎ𝐊_𝑔ℎ𝑘𝐪_ℎ(𝑡)𝐪_𝑘(𝑡)+^𝑝_{𝑙=1𝑙≠ℎ,𝑘}𝐊^𝑙_𝑔ℎ𝐊_𝑔ℎ𝑙𝐪_ℎ(𝑡)𝐪_𝑙(𝑡)=−^𝑠_𝑔=1𝜇_𝑔(𝜽(𝑡))𝐮^𝑘_𝑔ℎ(𝑡)+^𝑝_{𝑙=1𝑙≠ℎ,𝑘}𝐮^𝑙_𝑔ℎ(𝑡), (65) where for 𝑙=1,2,…,𝑝, 𝑙≠ℎ,𝑘, 𝐮^𝑙_𝑔ℎ(𝑡)=𝐊^𝑙_𝑔ℎ𝐊_𝑔ℎ𝑙𝐪_ℎ(𝑡)𝐪_𝑙(𝑡). (66) Defining for ℎ=1,2,…,𝑝−1, 𝑘=ℎ+1,ℎ+2,…,𝑝 and with respect to the notations (59) that 𝐮^𝑘_𝑔ℎ(𝑡)𝐮^ℎ_𝑔𝑘(𝑡)=𝐊^𝑘_𝑔ℎ𝐊_𝑔ℎ𝑘𝐊_𝑔𝑘ℎ𝐊^ℎ_𝑔𝑘𝐪_ℎ(𝑡)𝐪_𝑘(𝑡)=𝐊^∘_𝑔ℎ𝑘𝐪_ℎ(𝑡)𝐪_𝑘(𝑡) (67) and combining (64) for ℎ and 𝑘, is obtained the following: 𝐮_ℎ(𝑡)𝐮_𝑘(𝑡)=−^𝑠_𝑔=1𝜇_𝑔(𝜽(𝑡×𝐊))_𝑘𝑔ℎ𝐊_𝑔𝑘ℎ𝐊_𝑔ℎ𝑘𝐊^ℎ_𝑔𝑘𝐪_ℎ(𝑡)𝐪_𝑘(𝑡)−^{𝑝𝑙=1𝑙≠ℎ,𝑘}𝐊^𝑙_𝑔ℎ𝐊_𝑔ℎ𝑙𝐪_ℎ(𝑡)𝐪_𝑙(𝑡)_{𝑝𝑙=1𝑙≠ℎ,𝑘}𝐊^𝑙_𝑔𝑘𝐊_𝑔𝑘𝑙𝐪_𝑘(𝑡)𝐪_𝑙(𝑡),𝐮_ℎ(𝑡)𝐮_𝑘(𝑡)=−^𝑠_𝑔=1𝜇_𝑔(𝜽(𝑡×𝐮))_𝑘𝑔ℎ(𝑡)𝐮^ℎ_𝑔𝑘(𝑡)+^𝑝_{𝑙=1𝑙≠ℎ,𝑘}𝐮^𝑙_𝑔ℎ(𝑡)𝐮^𝑙_𝑔𝑘(𝑡), (68)(69) respectively. Then, substituting (69) in (55) gives 𝐪̇_ℎ𝑘(𝑡)=^𝑠𝑠_𝑗=1𝑔=1𝜇_𝑗(𝜽(𝑡))𝜇_𝑔(𝜽(𝑡))𝐀^∘_{𝑗𝑔ℎ𝑘𝑐}𝐪_ℎ𝑘(𝑡)+𝝎^∘_{𝑗𝑔ℎ𝑘}(𝑡), (70) where 𝐀^∘_{𝑗𝑔ℎ𝑘𝑐}=𝐀_𝑗ℎℎ𝐀_𝑗ℎ𝑘𝐀_𝑗𝑘ℎ𝐀_𝑗𝑘𝑘−𝐁_𝑗ℎ𝟎𝟎𝐁_𝑗𝑘𝐊^𝑘_𝑔ℎ𝐊_𝑔ℎ𝑘𝐊_𝑔𝑘ℎ𝐊^ℎ_𝑔𝑘,𝝎^∘_{𝑗𝑔ℎ𝑘}(𝑡)=^𝑝_{𝑙=1𝑙≠ℎ,𝑘}𝐁_𝑗ℎ𝐮^𝑙_𝑔ℎ(𝑡)+𝐀_𝑗ℎ𝑙𝐪_𝑙(𝑡)𝐁_𝑗𝑘𝐮^𝑙_𝑔𝑘(𝑡)+𝐀_𝑗𝑘𝑙𝐪_𝑙(𝑡). (71)(72) Using (59) and denoting 𝐀^∘_{𝑗𝑔ℎ𝑘𝑗}=𝐀^∘_𝑗ℎ𝑘−𝐁^∘_𝑗ℎ𝑘𝐊^∘_𝑔ℎ𝑘, (73) then (71) implies (57).
Rewriting (72) in the form 𝝎^∘_{𝑗𝑔ℎ𝑘}(𝑡)=^𝑝_{𝑙=1𝑙≠ℎ,𝑘}𝐁^∘_𝑗ℎ𝑘𝐮^𝑙_𝑔ℎ(𝑡)𝐮^𝑙_𝑔𝑘(𝑡)+𝐀_𝑗ℎ𝑙𝐀_𝑗𝑘𝑙𝐪_𝑙(𝑡) (74) and using (61), then (74) implies (60).
Finally, using (62), it can be written as 𝐲(𝑡)=^𝑝−1𝑝_{ℎ=1𝑘=ℎ+1}𝐂^∘_ℎ𝑘𝐪_ℎ𝑘(𝑡)+^𝑝_{𝑙=1,𝑙≠ℎ}𝐂^𝑙_ℎ𝐪_𝑙(𝑡),𝐲_ℎ𝑘(𝑡)=𝐂^∘_ℎ𝑘𝐪_ℎ𝑘(𝑡)+^𝑝_{𝑙=1,𝑙≠ℎ}𝐂^𝑙∘_ℎ𝑘𝐪_𝑙(𝑡), (75)(76) and (76) implies (58). This concludes the proof.

4. Pairwise Control Law Parameter Design

The design conditions, formulated as the set of LMIs, imply from the next theorems.

Theorem 6. Controlled subsystem pair (57), (58) in the system (1), (3) is stable with performances ∑^𝑠_𝑗=1∑^𝑠_𝑔=1𝜇_𝑗(𝜽(𝑡))𝜇_𝑔(𝜽(𝑡))‖𝐂^∘_ℎ𝑘(𝑠𝐈−𝐇^∘_{𝑗𝑔ℎ𝑘})⁻¹𝐁^∘_𝑗ℎ𝑘+𝐃^∘_𝑗ℎ𝑘‖²_∞≤𝛾_ℎ𝑘, if there exist a symmetric positive definite matrix 𝐗^∘_ℎ𝑘∈ℝ^{(𝑛ℎ+𝑛𝑘)×(𝑛ℎ+𝑛𝑘)}, matrices 𝐘^∘_𝑗ℎ𝑘∈ℝ^{(𝑟ℎ+𝑟𝑘)×(𝑛ℎ+𝑛𝑘)}, and positive scalars 𝛾_ℎ𝑘,𝜀_ℎ𝑘𝑙∈ℝ such that 𝐗^∘_ℎ𝑘=𝐗^∘𝑇_ℎ𝑘>0,𝜀_ℎ𝑘𝑙>0,𝛾_ℎ𝑘>0, (77) for ℎ,𝑙=1,…,𝑝, 𝑙≠ℎ,𝑘, ℎ<𝑘≤𝑝, Φ^∘_{𝑗𝑗ℎ𝑘}𝐀^1∘_𝑗ℎ𝑘⋯𝐀^𝑝∘_𝑗ℎ𝑘𝐁^∘_𝑗ℎ𝑘𝐗^∘_ℎ𝑘𝐂^∘𝑇_ℎ𝑘∗−𝜀_ℎ𝑘1𝐈_𝑛1⋯𝟎𝟎𝐂^1∘𝑇_ℎ𝑘⋮⋮⋱⋮⋮⋮∗∗⋯−𝜀_ℎ𝑘𝑝𝐈_𝑛𝑝𝟎𝐂^𝑝∘𝑇_ℎ𝑘∗∗⋯∗−𝛾_ℎ𝑘𝐈_{(𝑟ℎ+𝑟𝑘)}𝟎∗∗⋯∗∗−𝐈_{(𝑚ℎ+𝑚𝑘)}<0, (78) for 𝑗=1,2,…,𝑠, ℎ=1,2,…,𝑝−1, 𝑘=ℎ+1,ℎ+2,…,𝑝, 𝑙=1,2,…,𝑝, 𝑙≠ℎ,𝑘, Φ^∘_{𝑗𝑔ℎ𝑘}𝐀^1∘_𝑗ℎ𝑘+𝐀^1∘_𝑔ℎ𝑘2⋯𝐀^𝑝∘_𝑗ℎ𝑘+𝐀^𝑝∘_𝑔ℎ𝑘2𝐁^∘_𝑗ℎ𝑘+𝐁^∘_𝑔ℎ𝑘2𝐗^∘_ℎ𝑘𝐂^∘𝑇_ℎ𝑘∗−𝜀_ℎ𝑘1𝐈_𝑛1⋯𝟎𝟎𝐂^1∘𝑇_ℎ𝑘⋮⋮⋱⋮⋮⋮∗∗⋯−𝜀_ℎ𝑘𝑝𝐈_𝑛𝑝𝟎𝐂^𝑝∘𝑇_ℎ𝑘∗∗⋯∗−𝛾_ℎ𝑘𝐈_{(𝑟ℎ+𝑟𝑘)}𝟎∗∗⋯∗∗−𝐈_{(𝑚ℎ+𝑚𝑘)}<0, (79)for 𝑗=1,2,…𝑠−1, 𝑗=2,3,…𝑠, ℎ=1,2,…𝑝−1, 𝑘=ℎ+1,ℎ+2,…𝑝, 𝑙=1,2,…𝑝, 𝑙≠ℎ,𝑘, 𝑗<𝑔≤𝑠, with 𝐀^∘_𝑗ℎ𝑘, 𝐁^∘_𝑗ℎ𝑘, 𝐀^𝑙∘_𝑗ℎ𝑘, 𝐂^∘_ℎ𝑘, and 𝐂^𝑙∘_ℎ𝑘 defined in (59), (61), and (62), respectively, Φ^∘_{𝑗𝑗ℎ𝑘}=𝐀^∘_𝑗ℎ𝑘𝐗^∘_ℎ𝑘+𝐗^∘_ℎ𝑘𝐀^∘𝑇_𝑗ℎ𝑘−𝐁^∘_𝑗ℎ𝑘𝐘^∘𝑇_𝑗ℎ𝑘−𝐘^∘𝑇_𝑗ℎ𝑘𝐁^∘𝑇_𝑗ℎ𝑘,Φ^∘_{𝑗𝑔ℎ𝑘}=𝐀^∘_𝑗ℎ𝑘+𝐀^∘_𝑔ℎ𝑘2𝐗^∘_ℎ𝑘+𝐗^∘_ℎ𝑘𝐀^∘_𝑗ℎ𝑘+𝐀^∘_𝑔ℎ𝑘^𝑇2−𝐁^∘_𝑗ℎ𝑘𝐘^∘_𝑔ℎ𝑘+𝐁^∘_𝑔ℎ𝑘𝐘^∘_𝑗ℎ𝑘2−𝐁^∘_𝑗ℎ𝑘𝐘^∘_𝑔ℎ𝑘+𝐁^∘_𝑔ℎ𝑘𝐘^∘_𝑗ℎ𝑘^𝑇2, (80) and where, for given ℎ,𝑘, 𝐀^ℎ∘_𝑗ℎ𝑘, 𝐀^𝑘∘_𝑗ℎ𝑘, and 𝐂^ℎ∘_ℎ𝑘, 𝐂^𝑘∘_ℎ𝑘 are not included into the structure of (78).
When the above conditions hold, the gain matrices 𝐊^∘_𝑗ℎ𝑘 are given by 𝐊^∘_𝑗ℎ𝑘=𝐘^∘_𝑗ℎ𝑘𝐗^∘−1_ℎ𝑘. (81)

Proof. Considering ∑^𝑠_𝑗=1∑^𝑠_𝑔=1𝜇_𝑗(𝜽(𝑡))𝜇_𝑔(𝜽(𝑡))𝝎^∘_{𝑗𝑔ℎ𝑘}(𝑡), with 𝝎^∘_{𝑗𝑔ℎ𝑘}(𝑡) given in (60), as a generalized disturbance acting on the subsystem pair (57), (58), and introducing the notations 𝐆^∘_𝑗ℎ𝑘=𝐀^𝑙∘_𝑗ℎ𝑘^𝑝_{𝑙=1,𝑙≠ℎ,𝑘}𝐁^∘_𝑗ℎ𝑘,𝝎^∘𝑇_ℎ𝑘=𝐪^𝑇_𝑙^𝑝_{𝑙=1,𝑙≠ℎ,𝑘}𝝎^𝑇_ℎ𝑘(𝑡), (82) (57) takes the form 𝐪̇_ℎ𝑘(𝑡)=^𝑠𝑠_𝑗=1𝑔=1𝜇_𝑗(𝜽(𝑡))𝜇_𝑔(𝜽(𝑡×𝐀))_{∘𝑗ℎ𝑘}−𝐁^∘_𝑗ℎ𝑘𝐊^∘_𝑔ℎ𝑘𝐪_ℎ𝑘(𝑡)+𝐆^∘_𝑗ℎ𝑘𝝎^∘_ℎ𝑘(𝑡). (83) Analogously, (58) can be rewritten as 𝐲_ℎ𝑘(𝑡)=𝐂^∘_ℎ𝑘𝐪_ℎ𝑘(𝑡)+𝐃^∘_ℎ𝑘𝝎^𝑙∘_ℎ𝑘, (84) where 𝐃^∘_ℎ𝑘=𝐂^𝑙∘_ℎ𝑘^𝑝_{𝑙=1,𝑙≠ℎ,𝑘}𝟎. (85) Since (56) also gives ^𝑠𝑠_𝑗=1𝑔=1𝜇_𝑗(𝜽(𝑡))𝜇_𝑔(𝜽(𝑡×))_{𝑝−1𝑝ℎ=1𝑘=ℎ+1}𝐪̇^𝑇_ℎ𝑘(𝑡)𝐏^∘_ℎ𝑘𝐪_ℎ𝑘(𝑡)+𝐪^𝑇_ℎ𝑘(𝑡)𝐏^∘_ℎ𝑘𝐪̇_ℎ𝑘(𝑡)<0, (86) defining the matrices Γ^∘_ℎ𝑘=diag𝜀_ℎ𝑘𝑙𝐈_𝑛𝑙^𝑝_{𝑙=1,𝑙≠ℎ,𝑘}𝛾_ℎ𝑘𝐈_{(𝑟ℎ+𝑟𝑘)}, (87) (18) gives0>𝑣̇𝐪_ℎ𝑘(𝑡)=∑^𝑠∑_𝑗=1^𝑝−1𝑝_ℎ=1∑_𝑘=ℎ+1𝜇_𝑗𝜇_𝑗×𝐂^∘_ℎ𝑘𝐪_ℎ𝑘(𝑡)+𝐃^∘_ℎ𝑘𝝎^∘_ℎ𝑘(𝑡)^𝑇𝐂^∘_ℎ𝑘𝐪_ℎ𝑘(𝑡)+𝐃^∘_ℎ𝑘𝝎^∘_ℎ𝑘(𝑡)−𝝎^∘𝑇_ℎ𝑘(𝑡)Γ^∘_ℎ𝑘𝝎^∘_ℎ𝑘(𝑡)+𝐪^𝑇_ℎ𝑘(𝑡)𝐏^∘_ℎ𝑘𝐇^∘_{𝑗𝑗ℎ𝑘}𝐪_ℎ𝑘(𝑡)+𝐆^∘_𝑗ℎ𝑘𝝎^∘_ℎ𝑘(𝑡)+𝐇^∘_{𝑗𝑗ℎ𝑘}𝐪_ℎ𝑘(𝑡)+𝐆^∘_𝑗ℎ𝑘𝝎^∘_ℎ𝑘(𝑡)^𝑇𝐏^∘_ℎ𝑘𝐪_ℎ𝑘(𝑡)+2∑^𝑠−1𝑠_𝑗=1∑∑_𝑔=𝑗+1^𝑝−1𝑝_ℎ=1∑_𝑘=ℎ+1𝜇_𝑗𝜇_𝑔𝐂^∘_ℎ𝑘𝐪_ℎ𝑘(𝑡)+𝐃^∘_ℎ𝑘𝝎^∘_ℎ𝑘(𝑡)^𝑇𝐂^∘_ℎ𝑘𝐪_ℎ𝑘(𝑡)+𝐃^∘_ℎ𝑘𝝎^∘_ℎ𝑘(𝑡)−𝝎^∘𝑇_ℎ𝑘(𝑡)Γ^∘_ℎ𝑘𝝎^∘_ℎ𝑘(𝑡)+𝐪^𝑇_ℎ𝑘(𝑡)𝐏^∘_ℎ𝑘𝐇^∘_{𝑗𝑔ℎ𝑘}+𝐇^∘_{𝑔𝑗ℎ𝑘}2𝐪_ℎ𝑘(𝑡)+𝐆^∘_𝑗ℎ𝑘+𝐆^∘_𝑔ℎ𝑘2𝝎^∘_ℎ𝑘(𝑡)+𝐇^∘_{𝑗𝑔ℎ𝑘}+𝐇^∘_{𝑔𝑗ℎ𝑘}2𝐪_ℎ𝑘(𝑡)+𝐆^∘_𝑗ℎ𝑘+𝐆^∘_𝑔ℎ𝑘2𝝎^∘_ℎ𝑘(𝑡)^𝑇𝐏^∘_ℎ𝑘𝐪_ℎ𝑘(𝑡), (88)where, for simplicity, the argument of membership functions was omitted.
Therefore, inserting appropriate into (14), (15), it is obtained the following: Ψ^∘_{𝑗𝑗ℎ𝑘}𝐀^1∘_𝑗ℎ𝑘⋯𝐀^𝑝∘_𝑗ℎ𝑘𝐁^∘_𝑗ℎ𝑘𝐗^∘_ℎ𝑘𝐂^∘𝑇_ℎ𝑘∗−𝜀_ℎ𝑘1𝐈_𝑛1⋯𝟎𝟎𝐂^1∘𝑇_ℎ𝑘⋮⋮⋱⋮⋮⋮∗∗⋯−𝜀_ℎ𝑘𝑝𝐈_𝑛𝑝𝟎𝐂^𝑝∘𝑇_ℎ𝑘∗∗⋯∗−𝛾_ℎ𝑘𝐈_{(𝑟ℎ+𝑟𝑘)}𝟎∗∗⋯∗∗−𝐈_{(𝑚ℎ+𝑚𝑘)}Ψ<0,_{∘𝑗𝑔ℎ𝑘}𝐀^1∘_𝑗ℎ𝑘+𝐀^1∘_𝑔ℎ𝑘2⋯𝐀^𝑝∘_𝑗ℎ𝑘+𝐀^𝑝∘_𝑔ℎ𝑘2𝐁^∘_𝑗ℎ𝑘+𝐁^∘_𝑔ℎ𝑘2𝐗^∘_ℎ𝑘𝐂^∘𝑇_ℎ𝑘∗−𝜀_ℎ𝑘1𝐈_𝑛1⋯𝟎𝟎𝐂^1∘𝑇_ℎ𝑘⋮⋮⋱⋮⋮⋮∗∗⋯−𝜀_ℎ𝑘𝑝𝐈_𝑛𝑝𝟎𝐂^𝑝∘𝑇_ℎ𝑘∗∗⋯∗−𝛾_ℎ𝑘𝐈_{(𝑟ℎ+𝑟𝑘)}𝟎∗∗⋯∗∗−𝐈_{(𝑚ℎ+𝑚𝑘)}<0, (89)where Ψ^∘_{𝑗𝑗ℎ𝑘}=𝐀^∘_𝑗ℎ𝑘𝐗^∘_ℎ𝑘+𝐗^∘_ℎ𝑘𝐀^∘𝑇_𝑗ℎ𝑘−𝐁^∘_𝑗ℎ𝑘𝐊^∘𝑇_𝑗ℎ𝑘𝐗^∘_ℎ𝑘−𝐗^∘_ℎ𝑘𝐊^∘𝑇_𝑗ℎ𝑘𝐁^∘𝑇_𝑗ℎ𝑘,Ψ^∘_{𝑗𝑔ℎ𝑘}=𝐀^∘_𝑗ℎ𝑘+𝐀^∘_𝑔ℎ𝑘2𝐗^∘_ℎ𝑘+𝐗^∘_ℎ𝑘𝐀^∘_𝑗ℎ𝑘+𝐀^∘_𝑔ℎ𝑘^𝑇2−𝐁^∘_𝑗ℎ𝑘𝐊^∘_𝑔ℎ𝑘+𝐁^∘_𝑔ℎ𝑘𝐊^∘_𝑗ℎ𝑘2𝐗^∘_ℎ𝑘−𝐗^∘_ℎ𝑘𝐁^∘_𝑗ℎ𝑘𝐊^∘_𝑔ℎ𝑘+𝐁^∘_𝑔ℎ𝑘𝐊^∘_𝑗ℎ𝑘^𝑇2. (90) Thus, with the substitutions 𝐘^∘_𝑗ℎ𝑘=𝐊^∘_𝑗ℎ𝑘𝐗^∘_ℎ𝑘,𝐘^∘_𝑔ℎ𝑘=𝐊^∘_𝑔ℎ𝑘𝐗^∘_ℎ𝑘, (91) (89)-(90) imply (78)-(79). This concludes the proof.

Theorem 7. Controlled subsystem pair (57), (58) in the system (1), (3) is stable with performances ∑^𝑠_𝑗=1∑^𝑠_𝑔=1𝜇_𝑗(𝜽(𝑡))𝜇_𝑔(𝜽(𝑡))‖𝐂^∘_ℎ𝑘(𝑠𝐈−𝐇^∘_{𝑗𝑔ℎ𝑘})⁻¹𝐁^∘_𝑗ℎ𝑘+𝐃^∘_𝑗ℎ𝑘‖²_∞≤𝛾_ℎ𝑘, if for given 𝛿>0, 𝛿∈ℝ there exist symmetric positive definite matrices 𝐕^∘_ℎ𝑘,𝐓^∘_ℎ𝑘∈ℝ^{(𝑛ℎ+𝑛𝑘)×(𝑛ℎ+𝑛𝑘)}, matrices 𝐖^∘_𝑗ℎ𝑘∈ℝ^{(𝑟ℎ+𝑟𝑘)×(𝑛ℎ+𝑛𝑘)}, and positive scalars 𝛾_ℎ𝑘,𝜀_ℎ𝑘𝑙∈ℝ such that 𝐕^∘_ℎ𝑘=𝐕^∘𝑇_ℎ𝑘>0,𝐓^∘_ℎ𝑘=𝐓^∘𝑇_ℎ𝑘𝜀>0,_ℎ𝑘𝑙>0,𝛾_ℎ𝑘>0, (92) for ℎ,𝑙=1,…,𝑝, 𝑙≠ℎ,𝑘, ℎ<𝑘≤𝑝, 𝚺^∘_{𝑗𝑗ℎ𝑘}𝐀^1∘_𝑗ℎ𝑘⋯𝐀^p∘_𝑗ℎ𝑘𝐁^∘_𝑗ℎ𝑘Γ^∘_{𝑗𝑗ℎ𝑘}𝐕^∘_ℎ𝑘𝐂^∘𝑇_ℎ𝑘∗−𝜀_ℎ𝑘1𝐈_𝑛1⋯𝟎𝟎𝐀^1∘𝑇_𝑗ℎ𝑘𝐂^1∘𝑇_ℎ𝑘⋮⋮⋱⋮⋮⋮⋮∗∗⋯−𝜀_ℎ𝑘𝑝𝐈_𝑛𝑝𝟎𝐀^𝑝∘𝑇_𝑗ℎ𝑘𝐂^𝑝∘𝑇_ℎ𝑘∗∗⋯∗−𝛾_ℎ𝑘𝐈_{(𝑟ℎ+𝑟𝑘)}𝐁^∘𝑇_𝑗ℎ𝑘𝟎∗∗⋯∗∗−2𝛿𝐕^∘_ℎ𝑘𝟎∗∗⋯∗∗∗−𝐈_{(𝑚ℎ+𝑚𝑘)}<0, (93)for 𝑗=1,2,…,𝑠, ℎ=1,2,…,𝑝−1, 𝑘=ℎ+1,ℎ+2,…,𝑝, 𝑙=1,2,…,𝑝, 𝑙≠ℎ,𝑘, 𝚺^∘_{𝑗𝑔ℎ𝑘}𝐀^1∘_𝑗ℎ𝑘+𝐀^1∘_𝑔ℎ𝑘2⋯𝐀^𝑝∘_𝑗ℎ𝑘+𝐀^𝑝∘_𝑔ℎ𝑘2𝐁^∘_𝑗ℎ𝑘+𝐁^∘_𝑔ℎ𝑘2Γ^∘_{𝑗𝑔ℎ𝑘}𝐕^∘_ℎ𝑘𝐂^∘𝑇_ℎ𝑘∗−𝜀_ℎ𝑘1𝐈_𝑛1𝐀⋯𝟎𝟎^{1∘𝑗ℎ𝑘}+𝐀^1∘_𝑔ℎ𝑘^𝑇2𝐂^1∘𝑇_ℎ𝑘⋮⋮⋱⋮⋮⋮⋮∗∗⋯−𝜀_ℎ𝑘𝑝𝐈_𝑛𝑝𝟎𝐀^𝑝∘_𝑗ℎ𝑘+𝐀^𝑝∘_𝑔ℎ𝑘^𝑇2𝐂^𝑝∘𝑇_ℎ𝑘∗∗⋯∗−𝛾_ℎ𝑘𝐈_{(𝑟ℎ+𝑟𝑘)}𝐁^∘_𝑗ℎ𝑘+𝐁^∘_𝑗ℎ𝑘^𝑇2𝟎∗∗⋯∗∗−2𝛿𝐕^∘_ℎ𝑘𝟎∗∗⋯∗∗∗−𝐈_{(𝑚ℎ+𝑚𝑘)}<0, (94)for 𝑗=1,2,…,𝑠−1, 𝑗=2,3,…,𝑠, ℎ=1,2,…,𝑝−1, 𝑘=ℎ+1,ℎ+2,…,𝑝, 𝑙=1,2,…,𝑝, 𝑙≠ℎ,𝑘, 𝑗<𝑔≤𝑠, with 𝐀^∘_𝑗ℎ𝑘, 𝐁^∘_𝑗ℎ𝑘, 𝐀^𝑙∘_𝑗ℎ𝑘, 𝐂^∘_ℎ𝑘, and 𝐂^𝑙∘_ℎ𝑘 defined in (59), (61), and (62), respectively, and with 𝚺^∘_{𝑗𝑗ℎ𝑘}=𝐀^∘_𝑗ℎ𝑘𝐕^∘_ℎ𝑘+𝐕^∘_ℎ𝑘𝐀^∘𝑇_𝑗ℎ𝑘−𝐁^∘_𝑗ℎ𝑘𝐖^∘_𝑗ℎ𝑘−𝐖^∘𝑇_𝑗ℎ𝑘𝐁^∘𝑇_𝑗ℎ𝑘,𝚺^∘_{𝑗𝑔ℎ𝑘}=𝐀^∘_𝑗ℎ𝑘+𝐀^∘_𝑔ℎ𝑘2𝐕^∘_ℎ𝑘+𝐕^∘_ℎ𝑘𝐀^∘_𝑗ℎ𝑘+𝐀^∘_𝑔ℎ𝑘^𝑇2−𝐁^∘_𝑗ℎ𝑘𝐖^∘_𝑔ℎ𝑘+𝐁^∘_𝑔ℎ𝑘𝐖^∘_𝑗ℎ𝑘2−𝐁^∘_𝑗ℎ𝑘𝐖^∘_𝑔ℎ𝑘+𝐁^∘_𝑔ℎ𝑘𝐖^∘_𝑗ℎ𝑘^𝑇2,Γ^∘_{𝑗𝑗ℎ𝑘}=𝐓^∘_ℎ𝑘−𝛿𝐕^∘_ℎ𝑘+𝐕^∘_ℎ𝑘𝐀^𝑇_𝑗ℎ𝑘−𝐖^∘𝑇_𝑗ℎ𝑘𝐁^∘𝑇_𝑗ℎ𝑘,Γ^∘_{𝑗𝑔ℎ𝑘}=𝐓^∘_ℎ𝑘−𝛿𝐕^∘_ℎ𝑘+𝐕^∘_ℎ𝑘𝐀^∘𝑇_𝑗ℎ𝑘+𝐀^∘𝑇_𝑔ℎ𝑘2−𝐖^∘𝑇_𝑔ℎ𝑘𝐁^∘𝑇_𝑗ℎ𝑘+𝐖^∘𝑇_𝑗ℎ𝑘𝐁^∘𝑇_𝑔ℎ𝑘2, (95)(96)(97)(98) where, for given ℎ,𝑘, 𝐀^ℎ∘_𝑗ℎ𝑘, 𝐀^𝑘∘_𝑗ℎ𝑘, and 𝐂^ℎ∘_ℎ𝑘, 𝐂^𝑘∘_ℎ𝑘 are not included into the structure of (78).
When the above conditions hold, the gain matrices 𝐊^∘_𝑗ℎ𝑘 are given by 𝐊^∘_𝑗ℎ𝑘=𝐖^∘_𝑗ℎ𝑘𝐕^∘−1_ℎ𝑘. (99)

Proof. Using the above given notations, (33) gives0>𝑣̇𝐪_ℎ𝑘(𝑡)=^𝑠_𝑗=1^𝑝−1𝑝_{ℎ=1𝑘=ℎ+1}𝜇_𝑗𝜇_𝑗𝐪̇^𝑇_ℎ𝑘(𝑡)𝐏_ℎ𝑘𝐪_ℎ𝑘(𝑡)+𝐪^𝑇_ℎ𝑘(𝑡)𝐏_ℎ𝑘𝐪̇_ℎ𝑘(𝑡)+3𝐂^∘_ℎ𝑘𝐪_ℎ𝑘(𝑡)+𝐃^∘_ℎ𝑘𝝎^∘_ℎ𝑘(𝑡)^𝑇𝐂^∘_ℎ𝑘𝐪_ℎ𝑘(𝑡)+𝐃^∘_ℎ𝑘𝝎^∘_ℎ𝑘(𝑡)−3𝛾𝝎^∘𝑇_ℎ𝑘(𝑡)𝝎^∘_ℎ𝑘(𝑡)+2𝐪^𝑇_ℎ𝑘(𝑡)𝐒^∘₁+𝐪̇^𝑇_ℎ𝑘(𝑡)𝐒^∘₂𝐇^∘_{𝑗𝑗ℎ𝑘}𝐪_ℎ𝑘(𝑡)+𝐆^∘_𝑗ℎ𝑘𝝎^∘_ℎ𝑘(𝑡)−𝐪̇_ℎ𝑘(𝑡)+4^𝑠−1𝑠_{𝑗=1𝑔=𝑗+1}^𝑝−1𝑝_{ℎ=1𝑘=ℎ+1}𝜇_𝑗𝜇_𝑔𝐪^𝑇_ℎ𝑘(𝑡)𝐒^∘₁+𝐪̇^𝑇_ℎ𝑘(𝑡)𝐒^∘₂𝐇_{𝑗𝑔ℎ𝑘}+𝐇_{𝑔𝑗ℎ𝑘}2𝐪_ℎ𝑘(𝑡)+𝐆_𝑗ℎ𝑘+𝐆_𝑔ℎ𝑘2𝝎^∘_ℎ𝑘(𝑡), (100)and so (29), (30) take the formsΞ^∘_{𝑗𝑗ℎ𝑘}𝐀^1∘_𝑗ℎ𝑘⋯𝐀^𝑝∘_𝑗ℎ𝑘𝐁^∘_𝑗ℎ𝑘𝚼^∘_{𝑗𝑗ℎ𝑘}𝐗^∘_ℎ𝑘𝐂^∘𝑇_ℎ𝑘∗−𝜀_ℎ𝑘1𝐈_𝑛1⋯𝟎𝟎𝐀^1∘𝑇_𝑗ℎ𝑘𝐂^1∘𝑇_ℎ𝑘⋮⋮⋱⋮⋮⋮⋮∗∗⋯−𝜀_ℎ𝑘𝑝𝐈_𝑛𝑝𝟎𝐀^𝑝∘𝑇_𝑗ℎ𝑘𝐂^𝑝∘𝑇_ℎ𝑘∗∗⋯∗−𝛾_ℎ𝑘𝐈_{(𝑟ℎ+𝑟𝑘)}𝐁^∘𝑇_𝑗ℎ𝑘𝟎∗∗⋯∗∗−2𝛿𝐕^∘_ℎ𝑘𝟎∗∗⋯∗∗∗−𝐈_{(𝑚ℎ+𝑚𝑘)}Ξ<0,_{∘𝑗𝑔ℎ𝑘}𝐀^1∘_𝑗ℎ𝑘+𝐀^1∘_𝑔ℎ𝑘2⋯𝐀^𝑝∘_𝑗ℎ𝑘+𝐀^𝑝∘_𝑔ℎ𝑘2𝐁^∘_𝑗ℎ𝑘+𝐁^∘_𝑔ℎ𝑘2𝚼^∘_{𝑗𝑔ℎ𝑘}𝐕^∘_ℎ𝑘𝐂^∘𝑇_ℎ𝑘∗−𝜀_ℎ𝑘1𝐈_𝑛1𝐀⋯𝟎𝟎^{1∘𝑗ℎ𝑘}+𝐀^1∘_𝑔ℎ𝑘^𝑇2𝐂^1∘𝑇_ℎ𝑘⋮⋮⋱⋮⋮⋮⋮∗∗⋯−𝜀_ℎ𝑘𝑝𝐈_𝑛𝑝𝟎𝐀^𝑝∘_𝑗ℎ𝑘+𝐀^𝑝∘_𝑔ℎ𝑘^𝑇2𝐂^𝑝∘𝑇_ℎ𝑘∗∗⋯∗−𝛾_ℎ𝑘𝐈_{(𝑟ℎ+𝑟𝑘)}𝐁^∘_𝑗ℎ𝑘+𝐁^∘_𝑗ℎ𝑘^𝑇2𝟎∗∗⋯∗∗−2𝛿𝐕^∘_ℎ𝑘𝟎∗∗⋯∗∗∗−𝐈_{(𝑚ℎ+𝑚𝑘)}<0, (101)respectively, where Ξ^∘_{𝑗𝑗ℎ𝑘}=𝐀^∘_𝑗ℎ𝑘𝐕^∘_ℎ𝑘+𝐕^∘_ℎ𝑘𝐀^∘𝑇_𝑗ℎ𝑘−𝐁^∘_𝑗ℎ𝑘𝐊^∘_𝑗ℎ𝑘𝐕^∘_ℎ𝑘−𝐕^∘_ℎ𝑘𝐊^∘𝑇_𝑗ℎ𝑘𝐁^∘𝑇_𝑗ℎ𝑘,Ξ^∘_{𝑗𝑔ℎ𝑘}=𝐀^∘_𝑗ℎ𝑘+𝐀^∘_𝑔ℎ𝑘2𝐕^∘_ℎ𝑘+𝐕^∘_ℎ𝑘𝐀^∘_𝑗ℎ𝑘+𝐀^∘_𝑔ℎ𝑘^𝑇2−𝐁^∘_𝑗ℎ𝑘𝐊^∘_𝑔ℎ𝑘+𝐁^∘_𝑔ℎ𝑘𝐊^∘_𝑗ℎ𝑘2𝐕^∘_ℎ𝑘−𝐕^∘_ℎ𝑘𝐁^∘_𝑗ℎ𝑘𝐊^∘_𝑔ℎ𝑘+𝐁^∘_𝑔ℎ𝑘𝐊^∘_𝑗ℎ𝑘^𝑇2,𝚼^∘_{𝑗𝑗ℎ𝑘}=𝐓^∘_ℎ𝑘−𝛿𝐕^∘_ℎ𝑘+𝐕^∘_ℎ𝑘𝐀^𝑇_𝑗ℎ𝑘−𝐕^∘_ℎ𝑘𝐊^∘𝑇_𝑗ℎ𝑘𝐁^∘𝑇_𝑗ℎ𝑘,𝚼^∘_{𝑗𝑔ℎ𝑘}=𝐓^∘_ℎ𝑘−𝛿𝐕^∘_ℎ𝑘+𝐕^∘_ℎ𝑘𝐀^∘𝑇_𝑗ℎ𝑘+𝐀^∘𝑇_𝑔ℎ𝑘2−𝐕^∘_ℎ𝑘𝐊^∘𝑇_𝑔ℎ𝑘𝐁^∘𝑇_𝑗ℎ𝑘+𝐊^∘𝑇_𝑗ℎ𝑘𝐁^∘𝑇_𝑔ℎ𝑘2. (102)(103)(104)(105) Therefore, with the substitutions 𝐖^∘_𝑗ℎ𝑘=𝐊^∘_𝑗ℎ𝑘𝐕^∘_ℎ𝑘,𝐖^∘_𝑔ℎ𝑘=𝐊^∘_𝑔ℎ𝑘𝐕^∘_ℎ𝑘, (106) (101)–(105) imply (93)–(98). This concludes the proof.

To bet both of these theorems in the context of the control design for systems specified by TS models, it is necessary to make some remarks.

Although both theorems solve the same problem, the role of Theorem 6 is primarily methodological and in particular shows that the design problem of pairwise distributable control of large-scale TS systems can be formulated in the terms of 𝐻_∞ approach. Because it is based on the default structure of BRL, the Lyapunov matrices 𝐗^∘_ℎ𝑘 and the subsystem dynamic matrices 𝐀^∘_𝑗ℎ𝑘 form bound pairs with regard to the operation of multiplication. Consequently, when using the design conditions proposed in Theorem 6 and the LMI task is feasible, the obtained solution is very conservative.

Under the conditions defined by Theorem 7, the set of subsystem matrices 𝐀^∘_𝑗ℎ𝑘 are decoupled from the set of Lyapunov matrices 𝐓^∘_ℎ𝑘 by using the set of slack matrix variables 𝐕^∘_ℎ𝑘. This enables the design conditions in respect of natural affine properties of TS models and, compared with the result by Theorem 6, a less conservative solution. In addition, by tuning parameter 𝛿, the stability conditions setting of the whole system can be modified. Less conservatism in this case also means that the linear control of certain subsystem pair can be used instead of the nonlinear TS fuzzy control algorithm (see Section 5).

These theorems, which could be potentially considered as equivalent, give generally different solutions, except for the special case when for a pair ℎ,𝑘 by chance is 𝛿=1 and 𝐕^∘_ℎ𝑘=𝐓^∘_ℎ𝑘=𝐗^∘_ℎ𝑘.

In terms of computational complexity, if there is, for example, only one nonlinear sector in each block of the matrix 𝐀 and every sector is described only by a pair of sector functions, the number of fuzzy rules is 𝑠=2^𝑝2 to formulate the global control design and 𝑠=2^2𝑝 if pairwise distributed control principle is preferred. This can play a major role due to boundaries of LMI solvers. Moreover, as mentioned above, the proposed method can reduce this number in some subsystem pair structures.

On the other hand, although the pairwise subsystem control principle brings more complex control gain, the use of the state control laws does not substantially modify the computational complexity in dependency on the dimensionality of global state vector parts in the control algorithms.

5. Illustrative Example

To demonstrate properties of this approach, a system with four inputs and four outputs is used in the example. The parameters of (1)–(3) are [22] 𝐀=−312−140−1−211521−11−𝑎_𝑗,𝐂−241−2−22123−1−201500−13−2−4_𝑇=121212,𝑎302016−1016−10213110031003,𝐁=₁=0.8,𝑎₂𝜇=1.2,₁(𝜗(𝑡=𝑎))¹−𝑞₁(𝑡)𝑎₁−𝑎₂,𝜇₂(𝜗(𝑡))=1−𝜇₁(𝜗(𝑡,𝜗(𝑡)))=𝑞₁(𝑡). (107) To solve this problem, the next pairs grupping were done 𝐀^∘_•12=−312−1−211−11,𝐀^3∘₁₁₂=,𝐀−1.01.0−0.8_3∘212=−1.01.0−1.2,𝐀^4∘_•12=,𝐀4052−24_∘•13=−3−112,𝐀^2∘_•13=,𝐀12−2−2_4∘•13=,𝐀4012_∘•14=−34032505−4,𝐀^2∘_•14=,𝐀12−1−20−1_3∘•14=03−1,𝐀^1∘_•23=11,𝐀−1_∘123=−211.0−11−0.8−2−22.0,𝐀^∘₂₂₃=,𝐀−211.0−11−1.2−2−22.0_4∘•23=,𝐀52−2412_∘•24=−2152−11−24−1−2150−1−2−4,𝐀^1∘_•24=130,𝐀−1_3∘124=103−0.8,𝐀^3∘₂₂₄=103,𝐀−1.2_∘•34=2120153−2−4,𝐀^1∘_•34=130,𝐀^2∘_•34=,𝐁−2−2−1−20−1_•12=100201,𝐁_•13=1002,𝐁_•14=,𝐁100102_•23=201002,𝐁_•24=20100102,𝐁_•34=,𝐂200102_∘•12=111022,𝐂^3∘_•12=12,𝐂^4∘_•12=,𝐂1100_∘•13=1221,𝐂^2∘_•13=11−1−1,𝐂^4∘_•13=,𝐂1100_∘•14=111011,𝐂^2∘_•14=1100,𝐂^3∘_•14=21,𝐂^∘_•23=221−1−11,𝐂^1∘_•23=02,𝐂^4∘_•23=,𝐂0000_∘•24=22000011,𝐂^1∘_•24=00,𝐂^3∘_•24=11,𝐂^∘_•34=100111,𝐂^1∘_•34=20,𝐂^2∘_•34=,−1−100 (108) where •ℎ𝑘 denotes that the matrix parameters with this subscript are independent of the fuzzy rules. That means that in the pairwise partially decentralized control structure of this example, the control loop pairs 13, 14, and 34 will be surely fuzzy independent.

Solving for given tuning parameter 𝛿, for example, with respect to 𝐕^∘₂₃, 𝐓^∘₂₃, 𝐖^∘₁₂₃, 𝐖^∘₂₂₃, 𝜀₂₃₁, 𝜀₂₃₄, and 𝛾₂₃, it means writing (92)–(98) as 𝐕^∘₂₃=𝐕^∘𝑇₂₃>0,𝐓^∘₂₃=𝐓^∘𝑇₂₃𝜀>0,₂₃₁>0,𝜀₂₃₄>0,𝛾₂₃𝚺>0,_{∘𝑗𝑔23}𝐀^1∘_•23𝐀^4∘_•23𝐁^∘_•23Γ^∘_𝑗𝑔23𝐕^∘₂₃𝐂^∘𝑇_•23∗−𝜀₂₃₁𝟎𝟎𝐀^1∘𝑇_•23𝐂^1∘𝑇_•23∗∗−𝜀₂₃₄𝟎𝐀^4∘𝑇_•23𝐂^4∘𝑇_•23∗∗∗−𝛾₂₃𝐈₂𝐁^∘𝑇_•23𝟎∗∗∗∗−2𝛿𝐙^∘₂₃𝟎∗∗∗∗∗−𝐈₂𝚺<0,_{∘𝑗𝑗23}=𝐕^∘₂₃𝐀^∘𝑇_𝑗23+𝐀^∘_𝑗23𝐕^∘₂₃−𝐁^∘_•23𝐖^∘_𝑗23−𝐖^∘𝑇_𝑗23𝐁^∘𝑇_•23𝚺,𝑗=1,2,_{∘𝑗𝑔23}=𝐀^∘_𝑗23+𝐀^∘_𝑔232𝐕^∘₂₃+𝐕^∘₂₃𝐀^∘_𝑗23+𝐀^∘_𝑔23^𝑇2−𝐁^∘_•23𝐖^∘_𝑔23+𝐁^∘_•23𝐖^∘_𝑗232−𝐁^∘_•23𝐖^∘_𝑔23+𝐁^∘_•23𝐖^∘_𝑗23^𝑇2,Γ𝑗=1,𝑔=2,_{∘𝑗𝑗23}=𝐓^∘₂₃−𝛿𝐕^∘₂₃+𝐕^∘₂₃𝐀^∘𝑇_𝑗23−𝐖^∘𝑇₂₃𝐁^∘𝑇_•23,Γ𝑗=1,2,_{∘𝑗𝑔23}=𝐓^∘₂₃−𝛿𝐕^∘₂₃+𝐕^∘₂₃𝐀^∘𝑇_𝑗23+𝐀^∘𝑇_𝑔232−𝐖^∘𝑇_𝑔23𝐁^∘𝑇_•23+𝐖^∘𝑇_𝑗23𝐁^∘𝑇_•232,𝑗=1,𝑔=2, (109) and using SeDuMi package for Matlab [23], the feasible task for subsystem 23 and 𝛿=10 yields the parameters 𝜀₂₃₁=58.0063,𝜀₂₃₄=70.1966,𝛾₂₃𝐓=32.4549,_∘23=,𝐖27.947010.1742−4.831510.17428.01863.1112−4.83153.111238.0154_∘𝑗23=,𝐕7.56664.47170.6808−1.9191−0.395513.6818_∘23=,𝐊1.20040.2170−0.57730.21700.37930.2991−0.57730.29912.2580_∘⋄23=,4.94828.62320.42454.3630−10.26958.5351 (5) where 𝐊^∘_⋄23 means that 𝐊^∘_𝑗23 were computed equally to 𝑔=1,2.

By the same way, solving the rest LMIs, the gain matrix set is computed as 𝐊^∘₁₁₂=,𝐊7.9346−2.808012.48510.72585.39636.0160_∘212=,𝐊7.8740−2.849712.38280.72505.42686.0364_∘•14=,𝐊4.01065.92231.48482.02494.19413.8514_∘⋄34=,𝐊5.42941.13561.53932.03204.07843.0034_∘•13=,𝐊6.26032.79472.02985.8502_∘⋄24=.𝛾0.83492.27310.59041.1688−0.0995−0.15422.21001.8754₁₂=22.9891,𝛾₁₃𝛾=9.5395,₁₄=9.4173,𝛾₂₄𝛾=9.5641,₃₄=11.0660. (111) The results imply that in this example only the control of the subsystem pair 1,2 has to be realized in a TS fuzzy structure and all other pairs controls can be realized in constant linear pairwise control structure.

Generalizing, with respect the results presented in [13], the design methods based on the enhanced principle tend to produce for TS models the same control gain matrices which radically reduce the control structure, since such results mean very often stabilizing linear control laws for the nonlinear system or, as here, for its pair parts.

It is possible to verify—routinely—that the resulting closed-loop large-scale system will be stable if the tuning parameter is chosen as 𝛿=10. Moreover, this example implies that with 𝛿≤5 such pairwise distributed TS fuzzy control structure ensures all stable closed-loop pairs but does not ensure the stability of the whole closed-loop system. That means that using this design principle, the stability of all closed-loop subsystem pairs does not mean automatically the stability of the whole TS large-scale closed-loop system. Note that the control laws are given in the partly autonomous structure (68), (69), which ensures now that every closed-loop subsystem pair is stable and designed for 𝛿=10, and the large-scale closed-loop system will be stable too.

The block diagram on Figure 1 describes how the control of the subsystems pair (23) is, in general, implemented. This specific subsystem pair in the example is the only one that is controlled by the fuzzy PDC controller. As stated earlier, 𝐁^∘₁₂₃=𝐁^∘₂₂₃=𝐁^∘₂₃, while, for example, 𝐪₂(𝑡) of the global TS system can be obtained as the sum of the second component of vector 𝐪₁₂(𝑡) and the first components of vectors 𝐪₂₃(𝑡) and 𝐪₂₄(𝑡). It is assumed, of course, that bonds to create 𝐀^∘₁₂₃, 𝐀^∘₂₂₃, and 𝐁^∘₂₃ structures can be realized.

Figure 1 

The general closed-loop structure for control of the subsystems pair (23) in the autonomous regime.

Using autonomous control regime, defined by (68), (69), the illustrative simulations were realized under the initial condition 𝐪^𝑇(0)=0.02−0.100.000.10−0.050.05. (112) Figure 2 gives the global closed-loop TS system state response in the used simulation conditions.

Figure 2 

State response of the global closed-loop TS system in the autonomous regime.

For comparison, an equivalent set of gain matrices for the centralized fuzzy control can be constructed using above obtained results, where, for the control law (6), 𝐊₁=,𝐊𝟏𝟖.𝟐𝟎𝟓𝟓−2.808012.48512.79475.92231.48480.7258𝟏𝟑.𝟒𝟑𝟕𝟎𝟐𝟎.𝟐𝟓𝟏𝟎0.4245−1.19141.76462.02984.3630−10.2695𝟏𝟗.𝟖𝟏𝟒𝟕1.13561.53932.02491.5018−5.83332.0320𝟏𝟗.𝟖𝟑𝟏𝟕𝟏𝟎.𝟖𝟔𝟐𝟕₂=,𝟏𝟖.𝟏𝟒𝟒𝟗−2.849712.38282.79475.92231.48480.7250𝟏𝟑.𝟒𝟔𝟕𝟓𝟐𝟎.𝟐𝟕𝟏𝟒0.4245−1.19141.76462.02984.3630−10.2695𝟏𝟗.𝟖𝟏𝟒𝟕1.13561.53932.02491.5018−5.83332.0320𝟏𝟗.𝟖𝟑𝟏𝟕𝟏𝟎.𝟖𝟔𝟐𝟕 (113)and the resulting closed-loop eigenvalue spectrums are stable, where𝜌𝐀₁−𝐁𝐊₁={},𝜌𝐀−0.3278,−8.1143,−18.2169,−35.0038,−41.2029,−48.6512₂−𝐁𝐊₂={−0.3267,−8.1208,−18.1581,−35.0064,−41.1984,−48.7273}. (114)

Note that the structure of the matrices 𝐊_𝑔, 𝑔=1,2 evidently implies that the control laws are block diagonal dominant.

6. Concluding Remarks

The main difficulty of solving the decentralized control problem comes from the fact that the feedback gain is subject to structural constraints. At the beginning the study of large scale system theory, there was prevailing idea that a large of scale system is decentrally stabilizable under controllability condition by strengthening the stability degree of subsystems. But, because of the existence of decentralized fixed modes, some large scale systems can not be decentrally stabilized at all. In this paper, the idea to stabilize all subsystems and the whole system simultaneously by using decentralized controllers is replaced by another one, to stabilize all subsystems pairs and the whole system simultaneously by using partly pairwise decentralized control. In this sense the final scope of the paper is the design conditions for pairwise control of one class of pairwise-distributable continuous time TS large-scale systems. It is shown, based on the equivalent BRL formulations, how to expand the Lyapunov condition for pairwise TS control by using slack matrix variables in LMIs. As mentioned above, such matrix inequalities are linear with respect to the subsystem pairs variables and do not involve any product of the matrices, obtained by the Lyapunov matrix block separation and the subsystem pairs matrices. This enables us to derive a sufficient design condition for control with quadratic performances, optimizing subsystems interaction 𝐻_∞ norm bounds.

The method generally requires to solve, but separate, a larger number of linear matrix inequalities, and so more computational efforts are needed to provide design control parameters. However, used design conditions are less restrictive and are more close to necessity conditions in the sense of [24]. It is a very useful extension to TS system control performance synthesis problem. Numerical example demonstrates the principle effectiveness, although some computational complexity is increased.

Acknowledgments

The work presented in this paper was supported by VEGA, the Grant Agency of Ministry of Education and the Academy of Science of the Slovak Republic under Grant no. 1/0256/11. This support is very gratefully acknowledged.