Optimal Fuzzy Control for a Class of Nonlinear Systems

Abstract

The paper presents conditions suitable in design giving quadratic performances to stabilizing controllers for given class of continuous-time nonlinear systems, represented by Takagi-Sugeno models. Based on extended Lyapunov function and slack matrices, the design conditions are outlined in the terms of linear matrix inequalities to possess a stable structure closest to LQ performance, if premise variables are measurable. Simulation results illustrate the design procedure and demonstrate the performances of the proposed control design method.

1. Introduction

Since a generic method for design of a controller valid for all types of nonlinear systems has not been developed yet, an alternative seems to be fuzzy approach which benefits from the advantages of the approximation techniques approximating nonlinear system model equations. Using the Takagi-Sugeno (TS) fuzzy model [1], the nonlinear system is represented as a collection of fuzzy rules, where each rule utilizes the local dynamics by a linear system model. Since TS fuzzy models can well approximate a large class of nonlinear systems, and the TS-model-based approach can apprehend the nonlinear behavior of a system while keeping the simplicity of the linear models, by employing TS fuzzy models a control design methodology exploits fully advantage of the modern control theory, especially in the state space optimal and robust control.

The main idea of the TS-model-based controller design is to derive control rules so as to compensate each rule of a fuzzy system, determining the local feedback gains. It is known that the separate stabilization of these local modes does not ensure the stability of the overall system, and global design conditions have to be used to guarantee the global stability and control performance. Therefore, a range of stability conditions have been developed for TS fuzzy systems (see e.g., [2–5]), most of them relying on the feasibility of an associated system of linear matrix inequalities (LMIs) [6]. Therefore, the state control based on fuzzy TS system model gives control structures which can be designed using technique derived from equivalent LMIs (some principles and results are reported e.g., in [7–10]).

The problem of controlling a system in such way to optimize a performance index that represents the actual operating performance of a system has been an area of study for several decades [11–14]. In particular, if the attention is restricted to linear quadratic (LQ) control, several works following this approach over the years have been reported in the literature, where some new ones being for example, in [15]. Specifically, this approach has been often made in diverse practical problems for finite-time interval with time-varying feedback gains and full state measurable variables, to bring dynamical systems to a desired final states, as special interest in aircraft, spacecraft, and robots control and diagnosis [16, 17].

Following the given ideas in LQ control [18], the main contribution of the paper is to present new conditions for designing the stabilizing fuzzy state control with LQ performance for nonlinear MIMO systems approximated by a TS model and exploiting measurable premise variables. The proposed design method prefers methodology given in [7, 10] but is constructed on the extended form of quadratic Lyapunov function and enhanced evaluation of its time derivative [19, 20]. Because the Lyapunov synthesis approach is exploited to express global stability conditions in the form of a set of LMIs, resulting conservativeness of stability conditions is reduced, since while a common symmetric positive definite Lyapunov matrix verifying all inequalities is required, this approach eliminates products of this matrix with system model matrix parameters and extensive exploits affine properties of TS models.

The remainder of this paper is organized as follows. In Section 2 the general structure of TS models and LQ control is briefly described, and in Section 3 basic preliminaries are presented. The control design problem for systems with measurable premise variables is given in Sections 4 and 5, where especially new design conditions are derived and proven. Section 6 gives a numerical example to illustrate the effectiveness of the proposed approach and to confirm the validity of the control scheme. The last section draws conclusion remarks.

Throughout the paper, the following notations are used: 𝐱^𝑇, 𝐗^𝑇 denotes the transpose of the vector 𝐱 and matrix 𝐗, respectively, diag[⋅] denotes a block diagonal matrix, for a square matrix 𝐗>0 (resp. 𝐗<0) means that 𝐗 is a symmetric positive definite matrix (resp., negative definite matrix), the symbol 𝐈_𝑛 represents the 𝑛th order unit matrix, ℝ denotes the set of real numbers and ℝ^𝑛×𝑟 the set of all 𝑛×𝑟 real matrices.

2. General Methodologies

2.1. Takagi-Sugeno Fuzzy Models

The systems under consideration is one class of multiinput and multioutput nonlinear (MIMO) dynamic systems, which in the state-space form is represented as 𝐪̇(𝑡)(𝐪(𝑡(𝑡),𝐲(𝑡)(𝑡),=𝐚))+𝐁𝐮=𝐂𝐪 (2.1)(2.2) where 𝐪(𝑡)∈ℝ^𝑛, 𝐮(𝑡)∈ℝ^𝑟, 𝐲(𝑡)∈ℝ^𝑚 are vectors of the state, input, and output variables, respectively, and 𝐁∈ℝ^𝑛×𝑟 and 𝐂∈ℝ^𝑚×𝑛 are real finite values matrices. It is assumed that 𝐚(0)=𝟎, and that 𝐚(𝐪(𝑡)) is bounded in associated sectors, that is, in the regions within the system will operate.

It is considered that the number of the nonlinear terms in the nonlinear part of model 𝐚(𝐪(𝑡)) is 𝑝, and that there exists a set of nonlinear sector functions of this properties 𝑤_𝑙𝑗(𝜽(𝑡𝑤)),𝑗=1,2,…,𝑘,𝑙=1,2,…,𝑝,_𝑙𝑗(𝜽(𝑡))=𝑤_𝑙𝑗𝜃_𝑗(𝑡),𝑤_𝑙1(𝜽(𝑡))=1−^𝑘_𝑗=2𝑤_𝑙𝑗(𝜽(𝑡)), (2.3) where 𝑘 is the number of sector functions, and 𝜽(𝑡)=𝜃₁(𝑡)𝜃₂(𝑡)⋯𝜃_𝑞(𝑡) (2.4) is the vector of premise variables. A premise variable represents any measurable variable and can be in a simple case a state variable.

Using a TS model, the conclusion part of a single rule consists no longer of a fuzzy set but determines a function with state variables as arguments, and the corresponding function is a local function for the fuzzy region that is described by the premise part of the rule [21]. Thus, using linear functions, a system state is described locally (in fuzzy regions) by linear models, and at the boundaries between regions a suitable interpolation is used between the corresponding local models. Thus, given a pair of (𝐪(𝑡),𝐮(𝑡)), the final state of the systems is inferred as follows 𝐪̇(𝑡)=∑^𝑘_ℎ=1⋯∑^𝑘_𝑗=1𝑤_1ℎ𝜃_𝑖(𝑡)⋯𝑤_𝑝𝑗𝜃_𝑗(𝑡)Ω_𝑖⋯𝑗∑^𝑘_ℎ=1⋯∑^𝑘_𝑗=1𝑤_1ℎ𝜃_𝑖(𝑡)⋯𝑤_𝑝𝑗𝜃_𝑗(𝑡),Ω_ℎ⋯𝑗=𝐀_ℎ⋯𝑗𝐪(𝑡)+𝐁𝐮(𝑡), (2.5) where Ω_ℎ⋯𝑗 is the linear model associated with the (ℎ⋯𝑗) combination of sector function indexes. Constructing the aggregated function set {𝑤_𝑖(𝜽(𝑡)),𝑖=1,2,…,𝑠,𝑠=2^𝑘} from all combinations of the sector functions, for example, ordered as follows, 𝑤₁(𝜽(𝑡))=𝑤₁₁𝜃₁(𝑡)⋯𝑤_𝑝1𝜃₁(𝑡)⋮𝑤_𝑠(𝜽(𝑡))=𝑤_1𝑘𝜃_𝑘(𝑡)⋯𝑤_𝑝𝑘𝜃_𝑘(𝑡) (2.6) implies that 𝐪̇(𝑡)=∑^𝑠_𝑖=1𝑤_𝑖(𝜽(𝑡))Ω_𝑖(𝑡)∑^𝑠_𝑖=1𝑤_𝑖(𝜽(𝑡=))^𝑠_𝑖=1ℎ_𝑖(𝜽(𝑡))Ω_𝑖(𝑡),Ω_𝑖=𝐀_𝑖𝐪(𝑡)+𝐁𝐮(𝑡), (2.7)(2.8) where ℎ_𝑖(𝜽(𝑡=𝑤))^𝑖(𝜽(𝑡))∑^𝑠_𝑖=1𝑤_𝑖(𝜽(𝑡)) (2.9) is the 𝑖th aggregated normalized membership function satisfying conditions: 0≤ℎ_𝑖(𝜽(𝑡))≤1,^𝑠_𝑖=1ℎ_𝑖(𝜽(𝑡))=1,∀𝑖∈{1,…,𝑠}, (2.10) and linear consequent equation represented by (2.8) is called a linear subsystem.

Therefore, the TS fuzzy approximation of (2.1) leads to (2.7), (2.8) where the matrix 𝐀_𝑖∈ℝ^𝑛×𝑛 is Jacobian matrix of 𝐚(𝐪(𝑡)) with respect to 𝐪(𝑡)=𝐪_𝑖, and 𝐪_𝑖 is the center of the 𝑖th sector (fuzzy region).

Now, the TS fuzzy model for (2.1), (2.2) takes the form 𝐪̇(𝑡)=^𝑠_𝑖=1ℎ_𝑖(𝜽(𝑡))𝐀_𝑖𝐪(𝑡)(𝑡,𝐲(𝑡)(𝑡).+𝐁𝐮)=𝐂𝐪 (2.11)(2.12)

Assumption 2.1. The matrices 𝐁, 𝐂 are the same for all local models.

Assumption 2.2. The pair (𝐚(𝐪(𝑡)),𝐁) is locally controllable where 𝐚(𝐪(𝑡))=^𝑠_𝑖=1ℎ_𝑖(𝜽(𝑡))𝐀_𝑖, (2.13) and 𝐁 is of full column rank.

2.2. Linear Quadratic Control Background

In order to build up the background of the proposed method, some basics on the continuous-time LQ control are recalled. Considering the linear model (2.1), (2.2), that is, 𝐚(𝐪(𝑡))=𝐀𝐪(𝑡), the control design is possed as an optimal problem with certain combined quadratic performance on 𝐪(𝑡) and 𝐮(𝑡), and the control task is formulated as follows: find the nonzero control 𝐮(𝑡) defined on ⟨0,𝑇⟩ such that the state 𝐪(𝑡) is driven to the state coordinate origin at 𝑡=𝑇, and the following performance index is minimized 𝐽_𝑇=𝐪^𝑇(𝑇)𝐐^•𝐪(𝑇)+^𝑇₀𝑟(𝐪(𝑡)(𝑡,𝑟(𝐪(𝑡)(𝑡,𝐮)),𝐮))=𝐪^𝑇(𝑡)𝐐𝐪(𝑡)+𝐮^𝑇(𝑡)𝐑𝐮(𝑡)=𝐪^𝑇(𝑡)𝐮^𝑇(𝑡)𝐉_𝐽𝐪(𝑡)𝐮(𝑡), (2.14)(2.15) where 𝐉_𝐽∈ℝ^{(𝑛+𝑟)×(𝑛+𝑟)} takes the form: 𝐉_𝐽=diag𝐐𝐑>0, (2.16)𝑇>0 is finite, 𝐐>0, 𝐐∈ℝ^𝑛×𝑛, 𝐑>0, 𝐑∈ℝ^𝑚×𝑚, and 𝐐^•>0, 𝐐^•∈ℝ^𝑛×𝑛.

Proposition 2.3 (equivalent performance index). If the linear system from (2.1), (2.2) is controllable, then the LQ control design task is optimized with respect to the equivalent quadratic cost function (performance index): 𝐽_𝑇=𝐪^𝑇(0)𝐏(0)𝐪(0)+^𝑇₀𝑝(𝐪(𝑡)(𝑡,𝑝(𝐪(𝑡)(𝑡=𝐪,𝐮)),𝐮))^𝑇(𝑡)𝐮^𝑇(𝑡)𝐉(𝑡)𝐪(𝑡)𝐮(𝑡), (2.17)(2.18) where 𝐉(𝑡)=𝐏(𝑡)𝐀+𝐀^𝑇𝐏(𝑡)+𝐏̇(𝑡)(𝑡)𝐁𝐁+𝐐𝐏_𝑇𝐏(𝑡)𝐑, (2.19)𝐏(𝑡)>0, 𝐏(𝑡)∈ℝ^𝑛×𝑛, 𝐉(𝑡)>0, 𝐉(𝑡)∈ℝ^{(𝑛+𝑟)×(𝑛+𝑟)}, respectively.

Proof (compare e.g. [18]). Since now the system (2.1), (2.2) is linear in 𝐪(𝑡), the quadratic Lyapunov function candidate can be chosen as 𝑣(𝐪(𝑡))=𝐪^𝑇(𝑡)𝐏(𝑡)𝐪(𝑡), (2.20) and the derivative of the Lyapunov function candidate takes the form: 𝑣̇(𝐪(𝑡),𝐮(𝑡))=𝐪̇^𝑇(𝑡)𝐏(𝑡)𝐪(𝑡)+𝐪^𝑇(𝑡)𝐏(𝑡)𝐪̇(𝑡)+𝐪^𝑇(𝑡)𝐏̇(𝑡)𝐪(𝑡),𝑣̇(𝐪(𝑡),𝐮(𝑡))=𝐪^𝑇(𝑡)𝐮^𝑇(𝑡)𝐉_𝑉(𝑡)𝐪(𝑡)𝐮(𝑡), (2.21) respectively, where 𝐉_𝑉(𝑡)=𝐏(𝑡)𝐀+𝐀^𝑇𝐏(𝑡)+𝐏̇(𝑡)𝐏(𝑡)𝐁𝐁^𝑇𝐏(𝑡)𝟎. (2.22) Defining, at the time instant 𝑇, the cumulative function 𝑉_𝑇 as 𝑉_𝑇=^𝑇₀𝑣̇(𝐪(𝑡),𝐮(𝑡))𝑑𝑡, (2.23) which, in turn, is equivalent to 𝑉_𝑇=𝐪^𝑇(𝑇)𝐏(𝑇)𝐪(𝑇)−𝐪^𝑇(0)𝐏(0)𝐪(0), (2.24) then adding (2.23) to (2.14), subtracting (2.24) from (2.14), and setting 𝐏(𝑇)=𝐐^•, the performance index (2.14) is brought to the form (2.17), where 𝑝(𝐪(𝑡),𝐮(𝑖))=𝑟(𝐪(𝑖),𝐮(𝑖))+𝑣̇(𝐪(𝑡),𝐮(𝑡)). (2.25) It is evident that with 𝐉(𝑡)=𝐉_𝐽+𝐉_𝑉(𝑡) then (2.16), (2.22) imply (2.19).

Proposition 2.4 (infinite horizon LQ control). LQ control that the control law gain has become constant value is given by 𝐮(𝑡)=−𝐊𝐪(𝑡),𝐊=𝐑⁻¹𝐁^𝑇𝐏, (2.26)(2.27) where 𝐏>0 is a solution of the algebraic Riccati equation (ARE) 𝟎=𝐏𝐀+𝐀^𝑇𝐏+𝐐−𝐏𝐁𝐑⁻¹𝐁^𝑇𝐏. (2.28)

Proof (see e.g., [18]). Considering 𝐏(𝑡)=𝐏,  𝐉(𝑡)=𝐉, 𝐏̇(𝑡)=𝟎, then (2.18), (2.19) imply 𝜕𝑝(𝐪(𝑡),𝐮(𝑡))𝜕𝐮^𝑇(𝑡)=𝟎𝐈_𝑚𝐉𝐪(𝑡)𝐮(𝑡)=𝐁^𝑇𝐪(𝑡)𝐮(𝑡)(𝐪(𝑡)(𝑡𝐏𝐑=𝟎,𝜕𝑝,𝐮))𝜕𝐪^𝑇(𝑡)=𝐈_𝑛𝟎𝐉𝐪(𝑡)𝐮(𝑡)=𝐏𝐀+𝐀^𝑇𝐏+𝐐𝐪(𝑡)+𝐏𝐁𝐮(𝑡)=𝟎, (2.29)(2.30) respectively. It is obvious that (2.29) implies (2.27), and by substituting (2.26), (2.27) into (2.30), (2.28) is obtained.
Note, it makes no practical sense to have a terminal cost term with terminal time being infinite in the performance index.

Summarizing, (2.25) specifies the form of derivative of generalized Lyapunov function to formulate the infinite horizon LQ control design conditions using LMI.

3. Basic Preliminaries

The main concern of this section is to present basic concepts of nonlinear fuzzy control design for systems represented by TS model. Presented structure is partly motivated by minimizing the number of LMIs with respect to LMI solvers limitations.

Definition 3.1. Considering the general form of (2.11): 𝐪̇(𝑡)=^𝑠_𝑖=1ℎ_𝑖(𝜽(𝑡))𝐀_𝑖𝐪(𝑡)+𝐁_𝑖𝐮(𝑡) (3.1) and using the same set of membership function, the nonlinear fuzzy state controller is defined as 𝐮(𝑡)=−^𝑠_𝑗=1ℎ_𝑗(𝜽(𝑡))𝐊_𝑗𝐪(𝑡). (3.2)

Proposition 3.2. If the set of aggregated normalized membership functions (2.9) satisfies (2.10) then ^𝑠_𝑗=1ℎ_𝑗(𝜽(𝑡))𝐊^𝑇_𝑗𝐑^𝑠_𝑘=1ℎ_𝑘(𝜽(𝑡))𝐊_𝑘<^𝑠_𝑙=1𝐊^𝑇_𝑙𝐑𝐊_𝑙. (3.3)

Proof. Considering 𝑠=1 the conditions (2.10) imply ℎ₁(𝜽(𝑡))𝐊^𝑇₁𝐑ℎ₁(𝜽(𝑡))𝐊₁=ℎ²₁(𝜽(𝑡))𝐊^𝑇₁𝐑𝐊₁<𝐊^𝑇₁𝐑𝐊₁. (3.4) Providing the base of mathematical induction principle the number of functions is chosen as 𝑠=2. Thus, left-hand side of (3.3) implies ℎ₁(𝜽(𝑡))𝐊^𝑇₁+ℎ₂(𝜽(𝑡))𝐊^𝑇₂𝐑ℎ₁(𝜽(𝑡))𝐊₁+ℎ₂(𝜽(𝑡))𝐊₂=𝐊^𝑇₁𝐑^1/2𝐊^𝑇₂𝐑^1/2𝐇₂(𝜽(𝑡𝐑))^1/2𝐊₁𝐑^1/2𝐊₂,𝐇₂(𝜽(𝑡=ℎ))²¹(𝜽(𝑡))ℎ₁(𝜽(𝑡))ℎ₂(𝜽(𝑡ℎ))₂(𝜽(𝑡))ℎ₁(𝜽(𝑡))ℎ²₂(𝜽(𝑡,)) (3.5) and right-hand side of (3.3) specifies 𝐊^𝑇₁𝐑𝐊₁+𝐊^𝑇₂𝐑𝐊₂=𝐊^𝑇₁𝐑^1/2𝐊^𝑇₂𝐑^1/2𝐈₂𝐑^1/2𝐊₁𝐑^1/2𝐊₂. (3.6) Applying the Schur complement property to 𝐇₂(𝜽(𝑡)) and 𝐈₂ it is obvious that 1>ℎ²₂(𝜽(𝑡)),1>0=ℎ²₁(𝜽(𝑡))−ℎ₂(𝜽(𝑡))ℎ₁(𝜽(𝑡))ℎ₂⁻²(𝜽(𝑡))ℎ₁(𝜽(𝑡))ℎ₂(𝜽(𝑡)), (3.7) and the condition is satisfied in the sense of the proposition.
Since the statement holds true for at least one value, it is assumed that it holds true for an arbitrary fixed value 𝑠−1𝐈_𝑠−1>𝐇_𝑠−1(𝜽(𝑡=ℎ))¹(𝜽(𝑡⋮ℎ))_𝑠−1(𝜽(𝑡ℎ))¹(𝜽(𝑡))⋯ℎ_𝑠−1(𝜽(𝑡)). (3.8) To prove that the induction hypothesis holds true for all 𝑠 let the 𝑠th membership function is included in prescribed way, that is, 𝐇_𝑠(𝜽(𝑡=𝐇))^𝑠−1(𝜽(𝑡ℎ))¹(𝜽(𝑡⋮ℎ))_𝑠−1(𝜽(𝑡ℎ))^𝑠(𝜽(𝑡))∗ℎ²_𝑠(𝜽(𝑡)), (3.9) where here, and hereafter, ∗ denotes the symmetric item in a symmetric matrix.
Now, comparing the Schur complements of 𝐈_𝑠 and 𝐇_𝑠(𝜽(𝑡)), the first complement is satisfied since 1>ℎ²_𝑠(𝜽(𝑡)), (3.10) and the second gives 𝐈_𝑠−1>𝐇_𝑠−1(𝜽(𝑡))−ℎ²_𝑠(𝜽(𝑡))𝐇_𝑠−1(𝜽(𝑡))ℎ_𝑠⁻²(𝜽(𝑡))=𝐇_𝑠−1(𝜽(𝑡))−𝐇_𝑠−1(𝜽(𝑡))=𝟎. (3.11) Thus, (3.10), (3.11) imply (3.3).

Proposition 3.3. The equilibrium of the system (3.1) under control (3.2) is globally quadratic stable if there exists a positive definite symmetric matrix 𝐏∈ℝ^𝑛×𝑛 such that 𝐇^𝑇_𝑖𝑖𝐏+𝐏𝐇_𝑖𝑖𝐇<0,_𝑇𝑖𝑗+𝐇^𝑇_𝑗𝑖2𝐇𝐏+𝐏^𝑖𝑗+𝐇_𝑗𝑖2<0, (3.12)(3.13) for for all 𝑖∈⟨1,2,…𝑠⟩, 𝑖<𝑗≤𝑠,  𝑖,𝑗∈⟨1,2,…,𝑠⟩ and ℎ_𝑖(𝜽(𝑡))ℎ_𝑗(𝜽(𝑡))≠0, respectively, where 𝐇_𝑖𝑗=𝐀_𝑖−𝐁_𝑖𝐊_𝑗. (3.14)

Proof. Substituting (3.2) into (3.1) results in 𝐪̇(𝑡)=^𝑠_𝑖=1ℎ_𝑖(𝜽(𝑡))𝐀_𝑖𝐪(𝑡)−^𝑠_𝑗=1ℎ_𝑗(𝜽(𝑡))𝐁_𝑖𝐊_𝑗𝐪(𝑡). (3.15) Since ∑^𝑠_𝑖=1ℎ_𝑖(𝜽(𝑡))=1 for all 𝑖∈{1,…,𝑠}, it yields 𝐪̇(𝑡)=^𝑠𝑠_𝑖=1𝑗=1ℎ_𝑖(𝜽(𝑡))ℎ_𝑗(𝜽(𝑡))𝐀_𝑖−𝐁_𝑖𝐊_𝑗𝐪(𝑡) (3.16) and also, owing to the symmetry in summations: 𝐪̇(𝑡)=^𝑠𝑠_𝑖=1𝑗=1ℎ_𝑖(𝜽(𝑡))ℎ_𝑗(𝜽(𝑡))𝐀_𝑗−𝐁_𝑗𝐊_𝑖𝐪(𝑡). (3.17) Thus, adding (3.16), (3.17) gives 2𝐪̇(𝑡)=^𝑠𝑠_𝑖=1𝑗=1ℎ_𝑖(𝜽(𝑡))ℎ_𝑗(𝜽(𝑡))𝐇_𝑖𝑗+𝐇_𝑗𝑖𝐪(𝑡). (3.18) Rearranging the computation, (3.18) can be written as 2𝐪̇(𝑡)=^𝑠_𝑖=1ℎ_𝑖(𝜽(𝑡))ℎ_𝑖(𝜽(𝑡))𝐇_𝑖𝑖+𝐇_𝑖𝑖𝐪(𝑡)+2^𝑠−1𝑠_{𝑖=1𝑗=𝑖+1}ℎ_𝑖(𝜽(𝑡))ℎ_𝑗(𝜽(𝑡))𝐇_𝑖𝑗+𝐇_𝑗𝑖𝐪(𝑡),𝐪̇(𝑡)=^𝑠_𝑖=1ℎ_𝑖(𝜽(𝑡))ℎ_𝑖(𝜽(𝑡))𝐇_𝑖𝑖𝐪(𝑡)+2^𝑠−1𝑠_{𝑖=1𝑗=𝑖+1}ℎ_𝑖(𝜽(𝑡))ℎ_𝑗(𝜽(𝑡𝐇))^𝑖𝑗+𝐇_𝑗𝑖2𝐪(𝑡), (3.19) respectively. Defining Lyapunov function candidate of the form: 𝑣(𝐪(𝑡))=𝐪^𝑇(𝑡)𝐏𝐪(𝑡)>0, (3.20) where 𝐏∈ℝ^𝑛×𝑛 is a positive definite symmetric matrix, then after evaluation the derivative of (3.20) with respect to 𝑡 on a system trajectory it yields 𝑣̇(𝐪(𝑡))=𝐪̇^𝑇(𝑡)𝐏𝐪(𝑡)+𝐪^𝑇(𝑡)𝐏𝐪̇(𝑡)(𝐪(𝑡<0𝑣̇))=𝐪^𝑇(𝑡)^𝑠_𝑖=1ℎ_𝑖(𝜽(𝑡))ℎ_𝑖(𝜽(𝑡))𝐇^𝑇_𝑖𝑖𝐏𝐪(𝑡)+𝐪^𝑇(𝑡)𝐏^𝑠_𝑖=1ℎ_𝑖(𝜽(𝑡))ℎ_𝑖(𝜽(𝑡))𝐇_𝑖𝑖𝐪(𝑡)+2𝐪^𝑇(𝑡)^𝑠−1𝑠_{𝑖=1𝑗=𝑖+1}ℎ_𝑖(𝜽(𝑡))ℎ_𝑗(𝜽(𝑡𝐇))^𝑇𝑖𝑗+𝐇^𝑇_𝑗𝑖2𝐏𝐪(𝑡)+2𝐪^𝑇(𝑡)𝐏^𝑠−1𝑠_{𝑖=1𝑗=𝑖+1}ℎ_𝑖(𝜽(𝑡))ℎ_𝑗(𝜽(𝑡𝐇))^𝑖𝑗+𝐇_𝑗𝑖2𝐪(𝑡)<0, (3.21)(3.22) respectively. Then (3.21) can be compactly written as 𝐪^𝑇(𝑡)^𝑠_𝑖=1ℎ_𝑖(𝜽(𝑡))ℎ_𝑖(𝜽(𝑡))𝐏^∗_𝑖𝑖𝐪(𝑡)+2𝐪^𝑇(𝑡)^𝑠−1𝑠_{𝑖=1𝑗=𝑖+1}ℎ_𝑖(𝜽(𝑡))ℎ_𝑗(𝜽(𝑡))𝐏^∗_𝑖𝑗𝐪(𝑡)<0, (3.23) where 𝐏^∗_𝑖𝑖=𝐇^𝑇_𝑖𝑖𝐏+𝐏𝐇_𝑖𝑖𝐏<0,_∗𝑖𝑗=𝐇^𝑇_𝑖𝑗+𝐇^𝑇_𝑗𝑖2𝐇𝐏+𝐏^𝑖𝑗+𝐇_𝑗𝑖2<0. (3.24) Thus, (3.24) implies (3.12), (3.13). This concludes the proof.

Remark 3.4. If 𝐁_𝑖=𝐁  for all 𝑖∈⟨1,2,…,𝑠⟩ then 𝐇_𝑖𝑗+𝐇_𝑗𝑖=𝐀_𝑖−𝐁𝐊_𝑗+𝐀_𝑗−𝐁𝐊_𝑖=𝐇_𝑖𝑖+𝐇_𝑗𝑗, (3.25) and (3.13) for 𝑖<𝑗≤𝑠,  𝑖,𝑗∈⟨1,2,…,𝑠⟩ takes the form 𝐇^𝑇_𝑖𝑖+𝐇^𝑇_𝑗𝑗2𝐇𝐏+𝐏^𝑖𝑖+𝐇_𝑗𝑗2<0, (3.26) which implies 𝐇^𝑇_𝑖𝑖𝐏+𝐏𝐇_𝑖𝑖<0,𝐇^𝑇_𝑗𝑗𝐏+𝐏𝐇_𝑗𝑗<0. (3.27) It is evident that with 𝐇_𝑖𝑖,𝐇_𝑗𝑗 satisfying (3.12) also (3.27) is satisfied.

Proposition 3.5. The equilibrium of the system (3.1) under control (3.2) is globally asymptotically stable if there exist a positive definite symmetric matrix 𝐗∈ℝ^𝑛×𝑛 and matrices 𝐘_𝑗∈ℝ^𝑟×𝑛 such that 𝐗=𝐗^𝑇>0,𝐓_𝑖𝑖=𝐓^𝑇_𝑖𝑖<0,𝐓_𝑖𝑗=𝐓^𝑇_𝑖𝑗<0, (3.28) for all 𝑖∈⟨1,2,…,𝑠⟩ and 𝑖<𝑗≤𝑠,   𝑖,𝑗∈⟨1,2,…,𝑠⟩ and ℎ_𝑖(𝜽(𝑡))ℎ_𝑗(𝜽(𝑡))≠0, respectively, where 𝐓_𝑖𝑖=𝐗𝐀^𝑇_𝑖+𝐀_𝑖𝐗−𝐘^𝑇_𝑖𝐁^𝑇_𝑖−𝐁_𝑖𝐘_𝑖,𝐓_𝑖𝑗=𝐗𝐀^𝑇_𝑖−𝐘^𝑇_𝑗𝐁^𝑇_𝑖+𝐗𝐀^𝑇_𝑗−𝐘^𝑇_𝑖𝐁^𝑇_𝑗2+𝐀_𝑖𝐗−𝐁_𝑖𝐘_𝑗+𝐀_𝑗𝐗−𝐁_𝑗𝐘_𝑖2. (3.29) Then, the set of control law gain matrices are given as follows: 𝐊_𝑗=𝐘_𝑗𝐗⁻¹,𝑗=1,2,…,𝑠. (3.30)

Proof. Since 𝐏 is considered to be a positive definite matrix, it is obvious that 𝐏⁻¹ is also positive definite, and premultiplying left-hand and right-hand side of (3.12), as well as (3.13) by 𝐏⁻¹ leads to the inequalities: 𝐏⁻¹𝐀_𝑖−𝐁_𝑖𝐊_𝑖^𝑇+𝐀_𝑖−𝐁_𝑖𝐊_𝑖𝐏⁻¹𝐏<0,₋₁𝐀_𝑖−𝐁_𝑖𝐊_𝑗^𝑇+𝐀_𝑗−𝐁_𝑗𝐊_𝑖^𝑇2+𝐀_𝑖−𝐁_𝑖𝐊_𝑗+𝐀_𝑗−𝐁_𝑗𝐊_𝑖2𝐏⁻¹<0. (3.31)
Thus, with the notation 𝐗=𝐏⁻¹,𝐘_𝑗=𝐊_𝑗𝐗, (3.32) (3.31) implies (3.29), respectively. This concludes the proof.

Proposition 3.6. The equilibrium of the fuzzy system (2.11) controlled by the fuzzy controller (3.2) is globally asymptotically stable if there exists a positive definite matrix 𝐗∈ℝ^𝑛×𝑛 and matrices 𝐘_𝑗∈ℝ^𝑟×𝑛 such that 𝐗=𝐗^𝑇>0,𝐓^⋄_𝑖𝑗=𝐓^⋄𝑇_𝑖𝑗<0, (3.33) for ℎ_𝑖(𝜽(𝑡))ℎ_𝑗(𝜽(𝑡))≠0,  𝑖,𝑗=1,2,…,𝑠, where 𝐓^⋄_𝑖𝑗=𝐗𝐀^𝑇_𝑖+𝐀_𝑖𝐗−𝐘^𝑇_𝑗𝐁^𝑇−𝐁𝐘_𝑗. (3.34) The set of control law gain matrices is given by (3.30).

Proof. It Implies directly from Remark 3.4 and Proposition 3.5.

4. Fuzzy Controller with Quadratic Performances

The controller design is accomplished using the concept of asymptotic stability by analyzing the existence of an extended Lyapunov function. The fuzzy static output controller is designed using the concept of parallel distributed compensation, in which the fuzzy controller shares the same sets of normalized membership functions like the TS fuzzy system model. The goal is to achieve a certain level of performance using a guaranteed-cost approach results known from LQ control theory.

Theorem 4.1. The equilibrium of the system (3.1), controlled by the fuzzy controller (3.2), is globally asymptotically stable if there exist positive definite symmetric matrices 𝐗∈ℝ^𝑛×𝑛, 𝐐^⋄∈ℝ^𝑛×𝑛, 𝐑^⋄∈ℝ^𝑟×𝑟, and matrices 𝐘_𝑗∈ℝ^𝑟×𝑛, such that with (3.29) 𝐗=𝐗^𝑇>0,𝐐^⋄=𝐐^⋄𝑇>0,𝐑^⋄=𝐑^⋄𝑇𝐓>0,_𝑖𝑖𝐗𝐘^𝑇₁⋯𝐘^𝑇_𝑠∗−𝐐^⋄𝟎⋯𝟎∗∗−𝐑^⋄⋯𝟎⋮⋮⋮⋱⋮∗∗∗⋯−𝐑^⋄𝐓<0,^𝑖𝑗𝐗𝐘^𝑇₁⋯𝐘^𝑇_𝑠∗−𝐐^⋄𝟎⋯𝟎∗∗−𝐑^⋄⋯𝟎⋮⋮⋮⋱⋮∗∗∗⋯−𝐑^⋄<0, (4.1) for all 𝑖∈⟨1,2,…,𝑠⟩, 𝑖<𝑗≤𝑠,   𝑖,𝑗∈⟨1,2,…,𝑠⟩, and ℎ_𝑖(𝜽(𝑡))ℎ_𝑗(𝜽(𝑡))≠0, respectively. The set of control law gain matrices can be found directly as 𝐊_𝑗=𝐘_𝑗𝐗⁻¹,𝑗=1,2,…,𝑠. (4.2)

Proof. Considering (3.16) and defining with respect to (2.15), (2.25), then the quadratic positive Lyapunov function is as follows 𝑣(𝐪(𝑡))=𝐪^𝑇(𝑡)𝐏𝐪(𝑡)+^𝑡₀𝑟(𝐪(𝑟),𝐮(𝑟))𝑑𝑟>0, (4.3) where 𝐏 is a positive definite symmetric matrix, then it yields 𝑣̇(𝐪(𝑡))=𝐪̇^𝑇(𝑡)𝐏𝐪(𝑡)+𝐪^𝑇(𝑡)𝐏𝐪̇(𝑡)+𝑟(𝐪(𝑟),𝐮(𝑟))<0. (4.4) Substituting (2.15), (3.1) and (3.2) into (4.4) gives 𝑣̇(𝐪(𝑡))=^𝑠_𝑖=1ℎ_𝑖(𝜽(𝑡))𝐪_𝑗^•𝑇(𝑡)𝚷_𝑖𝐪^•_𝑘(𝑡)<0, (4.5) where 𝚷_𝑖=𝐐+𝐀^𝑇_𝑖𝐏+𝐏𝐀_𝑖𝐏𝐁_𝑖𝐁^𝑇_𝑖𝐪𝐏𝐑<0,_𝑗•𝑇(𝑡)=𝐪^𝑇(𝑡)−𝐪^𝑇(𝑡)∑^𝑠_𝑗=1ℎ_𝑗(𝜽(𝑡))𝐊^𝑇_𝑗, (4.6) and after straightforward computation it can be obtained 𝑣̇(𝐪(𝑡))=^𝑠_𝑖=1ℎ_𝑖(𝜽(𝑡))𝐪^𝑇(𝑡)𝚼_𝑖𝑗𝑘(𝜽(𝑡))𝐪(𝑡)<0, (4.7) with 𝚼_𝑖𝑗𝑘(𝜽(𝑡))=𝐐+𝐀^𝑇_𝑖𝐏+𝐏𝐀_𝑖+^𝑠_𝑗=1ℎ_𝑗(𝜽(𝑡))𝐊^𝑇_𝑗𝐑^𝑠_𝑘=1ℎ_𝑘(𝜽(𝑡))𝐊_𝑘−𝐏𝐁_𝑖^𝑠_𝑘=1ℎ_𝑘(𝜽(𝑡))𝐊_𝑘−^𝑠_𝑗=1ℎ_𝑗(𝜽(𝑡))𝐊^𝑇_𝑗𝐁^𝑇_𝑖𝐏. (4.8) Now, exploiting (3.3), then (4.7), (4.8) can be rewritten as 𝑣̇(𝐪(𝑡))<𝐪^𝑇(𝑡)^𝑠𝑠_𝑖=1𝑗=1ℎ_𝑖(𝜽(𝑡))ℎ_𝑗(𝜽(𝑡))Φ_𝑖𝑗𝐪(𝑡)<0, (4.9) where Φ_𝑖𝑗=𝐀_𝑖−𝐁_𝑖𝐊_𝑗^𝑇𝐏+𝐏𝐀_𝑖−𝐁_𝑖𝐊_𝑗+𝐐+^𝑠_𝑙=1𝐊^𝑇_𝑙𝐑𝐊_𝑙<0. (4.10) Analogously to (3.23) then (4.10) can be written as 𝑣̇(𝐪(𝑡))<𝐪^𝑇(𝑡)^𝑠_𝑖=1ℎ_𝑖(𝜽(𝑡))ℎ_𝑖(𝜽(𝑡))Φ^⋄_𝑖𝑖𝐪(𝑡)+2𝐪^𝑇(𝑡)^𝑠−1𝑠_{𝑖=1𝑗=𝑖+1}ℎ_𝑖(𝜽(𝑡))ℎ_𝑗(𝜽(𝑡))Φ^⋄_𝑖𝑗𝐪(𝑡)<0, (4.11) where, with 𝐇_𝑖𝑗 defined in (3.14), it is Φ^⋄_𝑖𝑖=𝐇^𝑇_𝑖𝑖𝐏+𝐏𝐇_𝑖𝑖+𝐐+^𝑠_𝑙=1𝐊^𝑇_𝑙𝐑𝐊_𝑙Φ<0,_⋄𝑖𝑗=𝐇^𝑇_𝑖𝑗+𝐇^𝑇_𝑗𝑖2𝐇𝐏+𝐏^𝑖𝑗+𝐇_𝑗𝑖2+𝐐+^𝑠_𝑙=1𝐊^𝑇_𝑙𝐑𝐊_𝑙<0, (4.12) for all 𝑖∈⟨1,2,…,𝑠⟩, 𝑖<𝑗≤𝑠,  𝑖,𝑗∈⟨1,2,…,𝑠⟩ and ℎ_𝑖(𝜽(𝑡))ℎ_𝑗(𝜽(𝑡))≠0, respectively.
Since 𝐏 is a regular positive definite square matrix, then premultiplying left-hand side and right-hand side of (4.12) by 𝐏⁻¹ give 𝐏⁻¹𝐇^𝑇_𝑖𝑖+𝐇_𝑖𝑖𝐏⁻¹+𝐏⁻¹𝐐𝐏⁻¹+^𝑠_𝑙=1𝐏⁻¹𝐊^𝑇_𝑙𝐑𝐊_𝑙𝐏⁻¹𝐏<0,₋₁𝐇^𝑇_𝑖𝑗+𝐇^𝑇_𝑗𝑖2+𝐇_𝑖𝑗+𝐇_𝑗𝑖2𝐏⁻¹+𝐏⁻¹𝐐𝐏⁻¹+^𝑠_𝑙=1𝐏⁻¹𝐊^𝑇_𝑙𝐑𝐊_𝑙𝐏⁻¹<0. (4.13) Thus, using (3.29) and the notations 𝐗=𝐏⁻¹,𝐘_𝑗=𝐊_𝑗𝐗,𝐐^⋄=𝐐⁻¹,𝐑^⋄=𝐑⁻¹, (4.14) it yields 𝐓_𝑖𝑖+𝐗𝐐^⋄−1𝐗+^𝑠_𝑙=1𝐘^𝑇_𝑙𝐑^⋄−1𝐘_𝑙𝐓<0,_𝑖𝑗+𝐗𝐐^⋄−1𝐗+^𝑠_𝑙=1𝐘^𝑇_𝑙𝐑^⋄−1𝐘_𝑙<0, (4.15)(4.16) for all 𝑖∈⟨1,2,…,𝑠⟩, 𝑖<𝑗≤𝑠,   𝑖,𝑗∈⟨1,2,…,𝑠⟩,ℎ_𝑖(𝜽(𝑡))ℎ_𝑗(𝜽(𝑡))≠0, respectively. It is evident that (4.15) implies (4.2), and (4.16) implies (4.2).

Theorem 4.2. The equilibrium of the fuzzy system (2.11) controlled by the fuzzy controller (3.2) is globally asymptotically stable if there exist positive definite matrices 𝐗∈ℝ^𝑛×𝑛, 𝐐^⋄∈ℝ^𝑛×𝑛, 𝐑^⋄∈ℝ^𝑟×𝑟, and matrices 𝐘_𝑗∈ℝ^𝑟×𝑛 such that 𝐗=𝐗^𝑇>0,𝐐^⋄=𝐐^⋄𝑇>0,𝐑^⋄=𝐑^⋄𝑇>0,𝐗𝐀^𝑇_𝑖+𝐀_𝑖𝐗−𝐘^𝑇_𝑗𝐁^𝑇−𝐁𝐘_𝑗𝐗𝐘^𝑇₁⋯𝐘^𝑇_𝑠∗−𝐐^⋄𝟎⋯𝟎∗∗−𝐑^⋄⋯𝟎⋮⋮⋮⋱⋮∗∗∗⋯−𝐑^⋄<0, (4.17)(4.18) for all 𝑖,𝑗∈⟨1,2,…,𝑠⟩, ℎ_𝑖(𝜽(𝑡))ℎ_𝑗(𝜽(𝑡))≠0. The set of control law gain matrices is given by (4.7).

Proof. Since 𝐁_𝑖=𝐁 for all 𝑖∈⟨1,2,…,𝑠⟩, then (4.10) implies Φ_𝑖𝑗=𝐀_𝑖−𝐁𝐊_𝑗^𝑇𝐏+𝐏𝐀_𝑖−𝐁𝐊_𝑗+𝐐+^𝑠_𝑙=1𝐊^𝑇_𝑙𝐑𝐊_𝑙<0. (4.19) Thus, premultiplying the both side of (4.19) by 𝐏⁻¹ gives 𝐏⁻¹𝐀_𝑖−𝐁𝐊_𝑗^𝑇+𝐀_𝑖−𝐁𝐊_𝑗𝐏⁻¹+𝐏⁻¹𝐐𝐏⁻¹+^𝑠_𝑙=1𝐏⁻¹𝐊^𝑇_𝑙𝐑𝐊_𝑙𝐏⁻¹<0. (4.20) and with (3.34) and the notations (4.14) then (4.20) implies (4.18).

5. Enhanced Controller with Quadratic Performance

The previous section was detailed how to find the fuzzy controller with quadratic performance ensuring the global asymptotic stability of the system. To extend the affine TS model principle by introducing the slack matrix variables into the LMIs, the system matrices are now decoupled from the equivalent Lyapunov matrix.

5.1. Stability Conditions

Theorem 5.1. The equilibrium of the system (3.1) under control (3.2) is globally asymptotically stable if there exist positive definite symmetric matrices 𝐑∈ℝ^𝑟×𝑟, 𝐏, 𝐐, 𝐒₁, 𝐒₂∈ℝ^𝑛×𝑛, such that 𝐏=𝐏^𝑇>0,𝐐=𝐐^𝑇>0,𝐑=𝐑^𝑇>0,𝐒₁=𝐒^𝑇₁>0,𝐒₂=𝐒^𝑇₂𝐖>0,_𝑖𝑖𝐏+𝐒₁−𝐇^𝑇_𝑖𝑖𝐒₂∗2𝐒₂𝐖<0,^𝑖𝑗𝐏+𝐒₁−𝐇^𝑇_𝑖𝑗+𝐇^𝑇_𝑗𝑖2𝐒₂∗2𝐒₂<0, (5.1)(5.2) for all 𝑖∈⟨1,2,…,𝑠⟩, 𝑖<𝑗≤𝑠,   𝑖,𝑗∈⟨1,2,…,𝑠⟩ and ℎ_𝑖(𝜽(𝑡))ℎ_𝑗(𝜽(𝑡))≠0, respectively, where with 𝐇_𝑖𝑗 defined in (3.14) it is 𝐖_𝑖𝑖=𝐐−𝐇^𝑇_𝑖𝑖𝐒₁−𝐒₁𝐇_𝑖𝑖+^𝑠_𝑙=1𝐊^𝑇_𝑙𝐑𝐊_𝑙,𝐖_𝑖𝑗𝐇=𝐐−^𝑇𝑖𝑗+𝐇^𝑇_𝑗𝑖2𝐒₁−𝐒₁𝐇_𝑖𝑗+𝐇_𝑗𝑖2+^𝑠_𝑙=1𝐊^𝑇_𝑙𝐑𝐊_𝑙. (5.3)

Proof. Since (3.1) implies 𝐪̇(𝑡)−^𝑠_𝑖=1ℎ_𝑖(𝜽(𝑡))𝐀_𝑖𝐪(𝑡)+𝐁_𝑖𝐮(𝑡)=𝟎, (5.4) using arbitrary regular symmetric square matrices 𝐒₁,𝐒₂∈ℝ^𝑛×𝑛 it yields 𝐪^𝑇(𝑡)𝐒₁+𝐪̇^𝑇(𝑡)𝐒₂𝐪̇(𝑡)−^𝑠_𝑖=1ℎ_𝑖(𝜽(𝑡))𝐀_𝑖𝐪(𝑡)+𝐁_𝑖𝐮(𝑡)=0. (5.5) Adding (5.5), and transposition of (5.5) to (4.4), and then inserting (3.2) give 𝑣̇(𝐪(𝑡))=𝐪^𝑇(𝑡)𝐐𝐪(𝑡)+𝐪^𝑇(𝑡)^𝑠_𝑗=1ℎ_𝑗(𝜽(𝑡))𝐊^𝑇_𝑗𝐑^𝑠_𝑘=1ℎ_𝑘(𝜽(𝑡))𝐊_𝑘𝐪(𝑡)+𝐪^𝑇(𝑡)𝐒₁𝐪̇(𝑡)+𝐪̇^𝑇(𝑡)𝐒₁𝐪(𝑡)+2𝐪̇^𝑇(𝑡)𝐒₂𝐪̇(𝑡)+𝐪^𝑇(𝑡)𝐏𝐪̇(𝑡)+𝐪̇^𝑇(𝑡)𝐏𝐪(𝑡)−𝐪^𝑇(𝑡)𝐒₁^𝑠𝑠_𝑖=1𝑗=1ℎ_𝑖(𝜽(𝑡))ℎ_𝑗(𝜽(𝑡))𝐀_𝑖−𝐁_𝑖𝐊_𝑗𝐪(𝑡)−𝐪̇^𝑇(𝑡)𝐒₂^𝑠𝑠_𝑖=1𝑗=1ℎ_𝑖(𝜽(𝑡))ℎ_𝑗(𝜽(𝑡))𝐀_𝑖−𝐁_𝑖𝐊_𝑗𝐪(𝑡)−𝐪^𝑇(𝑡)^𝑠𝑠_𝑖=1𝑗=1ℎ_𝑖(𝜽(𝑡))ℎ_𝑖(𝜽(𝑡))𝐀_𝑖−𝐁_𝑖𝐊_𝑗^𝑇𝐒₁𝐪(𝑡)−𝐪^𝑇(𝑡)^𝑠𝑠_𝑖=1𝑗=1ℎ_𝑖(𝜽(𝑡))ℎ_𝑗(𝜽(𝑡))𝐀_𝑖−𝐁_𝑖𝐊_𝑗^𝑇𝐒₂𝐪̇(𝑡)<0. (5.6) Then, using the notation: 𝐪^∘𝑇(𝑡)=𝐪^𝑇(𝑡)𝐪̇^𝑇(𝑡) (5.7) after straightforward computation it can be obtained 𝑣̇(𝐪(𝑡))=^𝑠_𝑖=1ℎ_𝑖(𝜽(𝑡))𝐪^∘𝑇(𝑡)Φ^∘_𝑖(𝜽(𝑡))𝐪^∘(𝑡)<0, (5.8) where Φ^∘_𝑖(𝜽(𝑡=Λ))^∘𝑖(𝜽(𝑡))Γ_𝑖(𝜽(𝑡))∗2𝐒₂Λ_∘𝑖,(𝜽(𝑡))=𝐐−𝐀^𝑇_𝑖𝐒₁−𝐒₁𝐀_𝑖+^𝑠_𝑗=1ℎ_𝑗(𝜽(𝑡))𝐊^𝑇_𝑗𝐑^𝑠_𝑘=1ℎ_𝑘(𝜽(𝑡))𝐊_𝑘+𝐒₁𝐁_𝑖^𝑠_𝑗=1ℎ_𝑗(𝜽(𝑡))𝐊_𝑗+^𝑠_𝑗=1ℎ_𝑗(𝜽(𝑡))𝐊^𝑇_𝑗𝐁^𝑇_𝑖𝐒₁,Γ^∘_𝑖(𝜽(𝑡))=𝐏+𝐒₁−𝐀^𝑇_𝑖𝐒₂+^𝑠_𝑗=1ℎ_𝑗(𝜽(𝑡))𝐊^𝑇_𝑗𝐁^𝑇_𝑖𝐒₂. (5.9)(5.10)(5.11) Exploiting (3.3), then (5.10) can be rewritten as Λ^∘_𝑖(𝜽(𝑡))<Λ_𝑖(𝜽(𝑡))=𝐐−𝐀^𝑇_𝑖𝐒₁−𝐒₁𝐀_𝑖+^𝑠_𝑙=1𝐊^𝑇_𝑙𝐑𝐊_𝑙+𝐒₁𝐁_𝑖^𝑠_𝑗=1ℎ_𝑗(𝜽(𝑡))𝐊_𝑗+^𝑠_𝑗=1ℎ_𝑗(𝜽(𝑡))𝐊^𝑇_𝑗𝐁^𝑇_𝑖𝐒₁, (5.12) and using (5.12) it yields 𝑣̇(𝐪(𝑡))<𝐪^∘𝑇(𝑡)^𝑠𝑠_𝑖=1𝑗=1ℎ_𝑖(𝜽(𝑡))ℎ_𝑗(𝜽(𝑡))𝐏^∘_𝑖𝑗𝐪^∘(𝑡)<0, (5.13) where 𝐏^∘_𝑖𝑗=𝐖_𝑖𝑗𝐏+𝐒₁−𝐀_𝑖−𝐁_𝑖𝐊_𝑗^𝑇𝐒₂∗2𝐒₂<0. (5.14) Analogously to (3.23) and (4.11) now (5.13) can be written as 𝑣̇(𝐪(𝑡))<𝐪^∘𝑇(𝑡)^𝑠_𝑖=1ℎ_𝑖(𝜽(𝑡))ℎ_𝑖(𝜽(𝑡))𝐏^∘_𝑖𝑖𝐪^∘(𝑡)+2𝐪^∘𝑇(𝑡)^𝑠−1𝑠_{𝑖=1𝑗=𝑖+1}ℎ_𝑖(𝜽(𝑡))ℎ_𝑗(𝜽(𝑡))𝐏^∘_𝑖𝑗𝐪^∘(𝑡)<0, (5.15) with 𝐖_𝑖𝑖, 𝐖_𝑖𝑗 defined in (5.3), respectively, and 𝐏^∘_𝑖𝑖=𝐖_𝑖𝑖𝐏+𝐒₁−𝐇^𝑇_𝑖𝑖𝐒₂∗2𝐒₂<0,𝐏^∘_𝑖𝑗=𝐖_𝑖𝑗𝐏+𝐒₁−𝐇^𝑇_𝑖𝑗+𝐇^𝑇_𝑗𝑖2𝐒₂∗2𝐒₂<0. (5.16) Since (5.16) implies (5.1), this concludes the proof.

Corollary 5.2. If 𝐁_𝑖=𝐁 for all 𝑖∈⟨1,2,…,𝑠⟩ then (5.14) implies that the equilibrium of the system (2.11) under control (3.2) is globally asymptotically stable if there exist positive definite symmetric matrices 𝐑∈ℝ^𝑟×𝑟, 𝐏, 𝐐, 𝐒₁, 𝐒₂∈ℝ^𝑛×𝑛, such that 𝐏=𝐏^𝑇>0,𝐐=𝐐^𝑇>0,𝐑=𝐑^𝑇>0,𝐒₁=𝐒^𝑇₁>0,𝐒₂=𝐒^𝑇₂>0,𝐐−𝐇^∘𝑇_𝑖𝑗𝐒₁−𝐒₁𝐇^∘_𝑖𝑗+∑^𝑠_𝑙=1𝐊^𝑇_𝑙𝐑𝐊_𝑙𝐏+𝐒₁−𝐇^∘𝑇_𝑖𝑗𝐒₂∗2𝐒₂<0, (5.17)(5.18) for all 𝑖,𝑗∈⟨1,2,…,𝑠⟩, ℎ_𝑖(𝜽(𝑡))ℎ_𝑗(𝜽(𝑡))≠0, where 𝐇^∘_𝑖𝑗=𝐀_𝑖−𝐁𝐊_𝑗,∀𝑖,𝑗∈⟨1,2,…,𝑠⟩. (5.19)

The importance of Theorem 5.1 is that it separates 𝐏 from system matrices 𝐀_𝑖, 𝐁_𝑖, that is, there are no terms containing product of 𝐏 and any of them. This enables to derive design conditions with respect to natural affine properties of TS models.

5.2. Control Parameter Design

In the next theorems, a scalar 𝛿>0, 𝛿∈ℝ is involved in the set of LMIs. The tuning parameter 𝛿 was added in the LMIs in an attempt to obtain less conservative stability conditions than Theorems 4.1 and 4.2, respectively. This procedure of adding scalar in LMIs has been widely explored in literature (see e.g., [19]).

Theorem 5.3. The equilibrium of the system (3.1) controlled by the fuzzy controller (3.2) is globally asymptotically stable if for given 𝛿>0, 𝛿∈ℝ there exist positive definite symmetric matrices 𝐗, 𝐙, 𝐐^•∈ℝ^𝑛×𝑛, 𝐑^•∈ℝ^𝑟×𝑟, and matrices 𝐘_𝑗∈ℝ^𝑟×𝑛, such that 𝐗=𝐗^𝑇>0,𝐙=𝐙^𝑇>0,𝐐^•=𝐐^•𝑇>0,𝐑^•=𝐑^•𝑇𝐕>0,_𝑖𝑖𝐔_𝑖𝑖𝐘^𝑇₁⋯𝐘^𝑇_𝑠∗−2𝛿𝐗𝟎⋯𝟎∗∗−𝐑^•⋯𝟎⋮⋮⋮⋱⋮∗∗∗⋯−𝐑^•𝐕<0,^𝑖𝑗𝐔_𝑖𝑗𝐘^𝑇₁⋯𝐘^𝑇_𝑠∗−2𝛿𝐗𝟎⋯𝟎∗∗−𝐑^•⋯𝟎⋮⋮⋮⋱⋮∗∗∗⋯−𝐑^•<0, (5.20)(5.21) where 𝐕_𝑖𝑖=𝐓_𝑖𝑖+𝐐^•,𝐕_𝑖𝑗=𝐓_𝑖𝑗+𝐐^•,𝐔_𝑖𝑖=𝐙−𝛿𝐗+𝐗𝐀^𝑇_𝑖−𝐘^𝑇_𝑖𝐁^𝑇_𝑖,𝐔_𝑖𝑗1=𝐙−𝛿𝐗+2𝐗𝐀^𝑇_𝑖+𝐀^𝑇_𝑗−12𝐘^𝑇_𝑗𝐁^𝑇_𝑖+𝐘^𝑇_𝑖𝐁^𝑇_𝑗, (5.22)(5.23)(5.24) for all 𝑖∈⟨1,2,…,𝑠⟩, 𝑖<𝑗≤𝑠,   𝑖,𝑗∈⟨1,2,…,𝑠⟩, ℎ_𝑖(𝜽(𝑡))ℎ_𝑗(𝜽(𝑡))≠0, respectively. The set of control law gain matrices is given as in (3.30).

Proof. Since 𝐒₁, 𝐒₂ are considered to be symmetric positive definite, introducing the congruence transform matrix: 𝐓=diag𝐒₁⁻¹𝐒₂⁻¹ (5.25) and premultiplying left-hand as well as right-hand sides of (5.16) by (5.25) gives 𝐓𝐏^∘_𝑖𝑖𝐖𝐓=^∘𝑖𝑖𝐒₁⁻¹𝐏𝐒₂⁻¹+𝐒₂⁻¹−𝐒₁⁻¹𝐇^𝑇_𝑖𝑖∗2𝐒₂⁻¹<0,𝐓𝐏^∘_𝑖𝑗𝐖𝐓=^∘𝑖𝑗𝐒₁⁻¹𝐏𝐒₂⁻¹+𝐒₂⁻¹−𝐒₁⁻¹𝐇^𝑇_𝑖𝑗+𝐇^𝑇_𝑗𝑖2∗2𝐒₂⁻¹𝐖<0,_∘𝑖𝑖=𝐒₁⁻¹𝐖_𝑖𝑖𝐒₁⁻¹=−𝐒₁⁻¹𝐇^𝑇_𝑖𝑖−𝐇_𝑖𝑖𝐒₁⁻¹+𝐒₁⁻¹𝐐𝐒₁⁻¹+^𝑠_𝑙=1𝐒₁⁻¹𝐊^𝑇_𝑙𝐑𝐊_𝑙𝐒₁⁻¹,𝐖^∘_𝑖𝑗=𝐒₁⁻¹𝐖_𝑖𝑗𝐒₁⁻¹=^𝑠_𝑙=1𝐒₁⁻¹𝐊^𝑇_𝑙𝐑𝐊_𝑙𝐒₁⁻¹+𝐒₁⁻¹𝐐𝐒₁⁻¹−𝐒₁⁻¹𝐇^𝑇_𝑖𝑗+𝐇^𝑇_𝑗𝑖2−𝐇_𝑖𝑗+𝐇_𝑗𝑖2𝐒₁⁻¹. (5.26) Thus, with 𝛿>0, 𝛿∈ℝ and with the notations 𝐗=−𝐒₁⁻¹,𝛿𝐗=−𝐒₂⁻¹,𝐘_𝑗=𝐊_𝑗𝐗,𝐙=𝐒₁⁻¹𝐏𝐒₂⁻¹,𝐐^•=𝐒₁⁻¹𝐐𝐒₁⁻¹,𝐑^•=𝐑⁻¹, (5.27) it yields 𝐖^∘_𝑖𝑖=𝐓_𝑖𝑖+𝐐^•+^𝑠_𝑙=1𝐘^𝑇_𝑙𝐑^•−1𝐘_𝑙,𝐖^∘_𝑖𝑗=𝐓_𝑖𝑗+𝐐^•+^𝑠_𝑙=1𝐘^𝑇_𝑙𝐑^•−1𝐘_𝑙, (5.28) and considering (5.22), we have 𝐕_𝑖𝑖+∑^𝑠_𝑙=1𝐘^𝑇_𝑙𝐑^•−1𝐘_𝑙𝐙−𝛿𝐗+𝐗𝐀^𝑇_𝑖−𝐘^𝑇_𝑖𝐁^𝑇_𝑖𝐕∗−2𝛿𝐗<0,_𝑖𝑗+∑^𝑠_𝑙=1𝐘^𝑇_𝑙𝐑^•−1𝐘_𝑙1𝐙−𝛿𝐗+2𝐗𝐀^𝑇_𝑖+𝐀^𝑇_𝑗−12𝐘^𝑇_𝑗𝐁^𝑇_𝑖+𝐘^𝑇_𝑖𝐁^𝑇_𝑗∗−2𝛿𝐗<0. (5.29) Using Schur complement property, then (5.29) implies (5.21)–(5.24).

Theorem 5.4. The equilibrium of the system (2.11) controlled by the fuzzy controller (3.2) is globally asymptotically stable if for given 𝛿>0, 𝛿∈ℝ there exist positive definite symmetric matrices 𝐗, 𝐙, 𝐐^•∈ℝ^𝑛×𝑛, 𝐑^•∈ℝ^𝑟×𝑟, and matrices 𝐘_𝑗∈ℝ^𝑟×𝑛, such that 𝐗=𝐗^𝑇>0,𝐙=𝐙^𝑇>0,𝐐^•=𝐐^•𝑇>0,𝐑^•=𝐑^•𝑇𝐓>0,_⋄𝑖𝑗+𝐐^•𝐙−𝛿𝐗+𝐗𝐀^𝑇_𝑖−𝐘^𝑇_𝑗𝐁^𝑇𝐘^𝑇₁⋯𝐘^𝑇_𝑠∗−2𝛿𝐗𝟎⋯𝟎∗∗−𝐑^•⋯𝟎⋮⋮⋮⋱⋮∗∗∗⋯−𝐑^•<0, (5.30)(5.31) for all 𝑖,𝑗∈⟨1,2,…,𝑠⟩. Then, the set of control law gain matrices is given as in (3.30).

Proof. If 𝐁_𝑖=𝐁 for all 𝑖∈⟨1,2,…,𝑠⟩ then (5.18) implies 𝐖^⋄_𝑖𝑗𝐏+𝐒₁−𝐀_𝑖−𝐁𝐊_𝑗^𝑇𝐒₂∗2𝐒₂𝐖<0,_⋄𝑖𝑗=𝐐−𝐀_𝑖−𝐁𝐊_𝑗^𝑇𝐒₁−𝐒₁𝐀_𝑖−𝐁𝐊_𝑗+^𝑠_𝑙=1𝐊^𝑇_𝑙𝐑𝐊_𝑙. (5.32)(5.33) Premultiplying left-hand side and right-hand side of (5.32) by (5.25) gives 𝐖^•_𝑖𝑗𝐒₁⁻¹𝐏𝐒₂⁻¹+𝐒₂⁻¹−𝐒₁⁻¹𝐀^𝑇_𝑖+𝐒₁⁻¹𝐊^𝑇_𝑗𝐁^𝑇∗2𝐒₂⁻¹𝐖<0,_•𝑖𝑗=−𝐒₁⁻¹𝐀_𝑖−𝐁𝐊_𝑗^𝑇−𝐀_𝑖−𝐁𝐊_𝑗𝐒₁⁻¹+𝐒₁⁻¹𝐐𝐒₁⁻¹+^𝑠_𝑙=1𝐒₁⁻¹𝐊^𝑇_𝑙𝐑𝐊_𝑙𝐒₁⁻¹, (5.34) and with the notations (5.27), (5.32) then (5.34) implies (5.31).

Note, the forms (5.2), (5.18) are suitable to optimize a solution with respect to LMI variables in an LMI structure. Conversely, the forms (5.21), (5.31) behave LMI structure only if 𝛿 is a prescribed constant design parameter. In the opposite case, the design task has to be formulated as BMI problem.

6. Illustrative Examples

The nonlinear dynamics of the hydrostatic transmission were taken from [22], and this MIMO model was used at first in control design and simulation.

The hydrostatic transmission dynamics is represented by a nonlinear fourth-order state-space model: 𝑞₁̇(𝑡)=−𝑎₁₁𝑞₁(𝑡)+𝑏₁₁𝑢₁(𝑡),𝑞₂̇(𝑡)=−𝑎₂₂𝑞₂(𝑡)+𝑏₂₂𝑢₂(𝑡),𝑞₃̇(𝑡)=𝑎₃₁𝑞₁(𝑡)𝑝(𝑡)−𝑎₃₃𝑞₃(𝑡)−𝑎₃₄𝑞₂(𝑡)𝑞₄(𝑡),𝑞₄̇(𝑡)=𝑎₄₃𝑞₂(𝑡)𝑞₃(𝑡)−𝑎₄₄𝑞₄(𝑡), (6.1) where 𝑞₁(𝑡) is the normalized hydraulic pump angle, 𝑞₂(𝑡) is the normalized hydraulic motor angle, 𝑞₃(𝑡) is the pressure difference [bar], 𝑞₄(𝑡) is the hydraulic motor speed [rad/s], 𝑝(𝑡) is the speed of hydraulic pump [rad/s], 𝑢₁(𝑡) is the normalized control signal of the hydraulic pump, and 𝑢₂(𝑡) is the normalized control signal of the hydraulic motor. It is supposed that the external variable 𝑝(𝑡) as well as the second state variable 𝑞₂(𝑡) are measurable. In given working points the parameters are 𝑎₁₁=7.6923𝑎₂₂=4.5455𝑎₃₃=7.6054.10⁻⁴,𝑎₃₁=0.7877𝑎₃₄=0.9235𝑏₁₁=1.8590.10³,𝑎₄₃=12.1967𝑎₄₄=0.4143𝑏₂₂=1.2879.10³. (6.2) Since the variables 𝑝(𝑡)∈⟨105,300⟩ and 𝑞₂(𝑡)∈⟨0.001,1⟩ are bounded on the prescribed sectors then vector of the premise variables can be chosen as follows: 𝜽(𝑡)=𝜃₁(𝑡)𝜃₂(𝑡)=𝑞₂(𝑡)𝑝(𝑡). (6.3) Thus, the set of nonlinear sector functions: 𝑤₁₁𝑞₂(𝑡)=𝑏₁−𝑞₂(𝑡)𝑏₁−𝑏₂,𝑤₁₂𝑞₂(𝑡)=𝑞₂(𝑡)−𝑏₂𝑏₁−𝑏₂=1−𝑤₁₁𝑞₂(𝑡),𝑏₁=0.001,𝑏₂𝑤=1,₂₁𝑝(𝑡)=𝑐₁−𝑝(𝑡)𝑐₁−𝑐₂,𝑤₂₂𝑝(𝑡)=𝑝(𝑡)−𝑐₂𝑐₁−𝑐₂=1−𝑤₂₁𝑝(𝑡),𝑐₁=105,𝑐₂=300, (6.4) implies the next set of normalized membership functions: ℎ₁𝑞₂(𝑡),𝑝(𝑡)=𝑤₁₁𝑞₂(𝑡)𝑤₂₁𝑝(𝑡),ℎ₂𝑞₂(𝑡),𝑝(𝑡)=𝑤₁₂𝑞₂(𝑡)𝑤₂₁𝑝(𝑡),ℎ₃𝑞₂(𝑡),𝑝(𝑡)=𝑤₁₁𝑞₂(𝑡)𝑤₂₂𝑝(𝑡),ℎ₄𝑞₂(𝑡),𝑝(𝑡)=𝑤₁₂𝑞₂(𝑡)𝑤₂₂𝑝(𝑡). (6.5) The transformation of nonlinear differential equations of the system into a TS fuzzy system in standard form gives 𝐀_𝑖=−𝑎₁₁0000−𝑎₂₂𝑎00₃₁𝑐_𝑘0−𝑎₃₃−𝑎₃₄𝑏_𝑙00𝑎₄₃𝑏_𝑙−𝑎₄₄𝑏,𝐁=¹¹00𝑏₂₂0000,𝐂^𝑇=00100100, (6.6) with the associations ()(),()().𝑖=1⟵𝑙=1,𝑘=1𝑖=2⟵𝑙=2,𝑘=1𝑖=3⟵𝑙=1,𝑘=2𝑖=4⟵𝑙=2,𝑘=2 (6.7) Thus, solving (5.30)-(5.31) for given 𝛿=20 with respect to the LMI matrix variables 𝐗, 𝐙, 𝐐^•, 𝐑^•, and 𝐘_𝑗,𝑗=1, 2, 3, 4 using Self-Dual-Minimization (SeDuMi) package for Matlab [23], the feedback gain matrix design problem was feasible with the results: ,,𝐐𝐗=0.00420.0000−0.00420.00020.00000.02590.00000.0000−0.00420.00000.0269−0.0104−0.00020.0000−0.01040.0537𝐙=1.63600.0000−0.3994−0.03490.00001.29770.00000.0000−0.39940.00000.9726−0.1850−0.03490.0000−0.18501.0982_•=1.86120.0000−0.5123−0.01260.00000.64090.00000.0000−0.51230.00000.21970.0121−0.01260.00000.01210.0314,𝐑^•=,𝐊0.99200.00000.00000.9920_𝑗=0.23860.00000.03500.00750.00000.02070.00000.0000,𝑗=1,2,3,4, (6.8) which rise up a stable set of closed-loop subsystems.

Comparing with the standard approach, presented method tends to produce the same control gain matrices if 𝐁_𝑖=𝐁 for all 𝑖, which radically reduce the control structure, since the result is stabilizing linear control law with quadratic performance for the nonlinear system. Moreover, such control is robust with respect to a premise variable sensor fault.

Specifying simulation conditions for unforced (autonomous) regime and 𝑡≥0 as follows: 𝐩(𝑡)=𝑐₁,𝐪^𝑇(0)=0.10.500, (6.9) then Figure 1(a) shows the solution for constant external system signal 𝑝(𝑡) and nonzero initial condition.

(a)

(b)

Figure 1 

(a) Fault-free system state response, (b) faulty system state response.

If the second variable (premise) sensor fault was modeled as the step function, and the system in unforced regime was controlled by the nominal state control before, and after the sensor fault occurrence time instant 𝑡_𝑓=0.07s, then the state responses of the system are shown in Figure 1(b). Evidently, closed-loop system stayed stable.

Using the decoupling control principe [24], also the forced regime was simulated with the control policy: 𝐮(𝑡)=^𝑠_𝑗=1ℎ_𝑗(𝜽(𝑡))−𝐊_𝑗𝐪(𝑡)+𝐖_𝑤𝑗𝐰(𝑡), (6.10) where 𝐰(𝑡)∈ℝ^𝑟 was the desired output vector, and 𝐖_𝑤𝑗∈ℝ^𝑟×𝑟,𝑗=1,…,𝑠 was the set of signal gain matrices. Using method given in [25], the signal gain matrices were computed as 𝐖_𝑤𝑗1=̇4⁴_𝑖=1𝐂−𝐀_𝑗−𝐁𝐊_𝑖⁻¹𝐁⁻¹,𝐖_𝑤1=0.00000.14010.09700.0000,𝐖_𝑤2=,𝐖0.00000.14010.09700.0000_𝑤3=0.00001.34340.09700.0000,𝐖_𝑤4=,𝐰0.00001.13600.09700.0000_𝑇(𝑡)=0.20000.9000. (6) The simulation results for forced regime are shown in Figures 2(a) and 2(b), reflecting the closed-loop system state, as well as system output responses.

(a)

(b)

Figure 2 

(a) System state response, (b) system output response.

Example 6.1. Consider and give later the problem of balancing an inverted pendulum on a cart adopted from [10, 26], where the objective is to control its state trajectories to the state origin, the state equation of motion of this nonlinear SISO system is, 𝑞₁̇(𝑡)=𝑞₂(𝑡),𝑞₂̇(𝑡)=𝑔sin𝑞₁(𝑡)−𝑐𝑚𝑙𝑞²₂(𝑡)sin2𝑞₁(𝑡)−𝑐cos𝑞₁(𝑡)𝑢(𝑡)(4/3)𝑙−𝑐𝑚𝑙cos²𝑞₁(𝑡,) (6.12) where 𝑞₁(𝑡) denotes the angle of the pendulum from the vertical axis [rad], 𝑞₂(𝑡) is the angular velocity [rad/s], 𝑢(𝑡) is the force applied to the cart [N], 𝑚 is the mass of the pendulum [kg], 𝑀 is the mass of the cart [kg], 𝑙 is the length from the center of mass of the pendulum to the shaft axis [m], and 𝑔 is the gravity constant [9.81 m/s²]. In the design and simulation, the pendulum parameters were set as 𝑙=0.5,𝑚=2,𝑀=8,𝑐=(𝑚+𝑀)⁻¹=0.1. (6.13)

Since the premise variable 𝑞₁(𝑡)∈⟨−𝜋/3,𝜋/3⟩ is bounded on this interval, the next sectors −𝜋3𝜋,−9,−2𝜋9,−𝜋,09,𝜋9,0,2𝜋9,𝜋9,𝜋3 (6.14) were chosen to approximate the system dynamics. The transformation of nonlinear differential equations of the system into a TS fuzzy model gives 𝐀_𝑖=𝑎01_21𝑖0,𝐛_𝑖=0𝑏_2𝑖,𝐜^𝑇=𝑎10,𝑖=1,2,3,4,₂₁₁=𝑔(4/3)𝑙−𝑐𝑚𝑙,𝑎₂₁₂=(9/𝜋)sin(𝜋/9)𝑔(4/3)𝑙−𝑐𝑚𝑙cos²(),𝑎𝜋/9₂₁₃=(9/2𝜋)sin(2𝜋/9)𝑔(4/3)𝑙−𝑐𝑚𝑙cos²(2𝜋/9),𝑎₂₁₄=(3/𝜋)sin(𝜋/3)𝑔(4/3)𝑙−𝑐𝑚𝑙cos²(),𝑏𝜋/3₂₁=−𝑐(4/3)𝑙−𝑐𝑚𝑙,𝑏₂₂=−cos(𝜋/9)𝑐(4/3)𝑙−𝑐𝑚𝑙cos²(),𝑏𝜋/9₂₃=−cos(2𝜋/9)𝑐(4/3)𝑙−𝑐𝑚𝑙cos²(2𝜋/9),𝑏₂₄=−cos(𝜋/3)𝑐(4/3)𝑙−𝑐𝑚𝑙cos²().𝜋/3 (6.15) Now, (3.1), (2.12) take the forms: 𝐪̇(𝑡)=⁴_𝑖=1ℎ_𝑖(𝜽(𝑡))𝐀_𝑖𝐪(𝑡)+𝐛_𝑖𝑢(𝑡),𝑦(𝑡)=𝐜^𝑇𝐪(𝑡), (6.16) where 𝐪^𝑇(𝑡)=𝑞₁(𝑡)𝑞₂(𝑡)^𝑇,𝜽(𝑡)=𝜃₁(𝑡)𝜃₂(𝑡)𝜃₃(𝑡)𝜃₄(𝑡)^𝑇,𝜃_𝑖(𝑡)=𝜃₁(𝑡)if𝑞₁(𝑡)𝜃isabout0,₂(𝑡)if𝑞₁(𝑡)𝜋isabout±9,𝜃₃(𝑡)if𝑞₁(𝑡)isabout±2𝜋9,𝜃₄(𝑡)if𝑞₁(𝑡)𝜋isabout±3,ℎ₁𝜃₁(𝑡9)=1−𝜋𝜃₁(𝑡),ℎ₂𝜃₂(𝑡9)=1−𝜋𝜃₂(𝑡)−𝜋9sign𝜃₂(𝑡,ℎ₃)𝜃₃(𝑡)9=1−𝜃2𝜋³(𝑡)−2𝜋9sign𝜃₃(𝑡),ℎ₄𝜃₄(𝑡)=3𝜋𝜃₄(𝑡)−2𝜋9sign𝜃₄(𝑡). (6.17)

Thus, solving (5.20)–(5.24) for given 𝛿=20 with respect to the LMI matrix variables 𝐗, 𝐙, 𝐐^•, 𝐑^•, and 𝐘_𝑗,𝑗=1,2,3,4 using SeDuMi, the feedback gain matrix design problem was feasible with the results: ,𝐐𝐗=0.0018−0.0066−0.00660.0244,𝐙=0.0446−0.1647−0.16470.6136_•=0.0008−0.0028−0.00280.0101,𝑅^•𝐊=1.1301,_𝑗=−725.0702−195.4523,𝑗=1,2,3,4, (6.18) which rise up a stable set of closed-loop subsystems.

Comparing with the standard approach, the method tends in this case also to produce the same control law gain matrices although 𝐁_𝑖≠𝐁 for all 𝑖. Note that the nonlinear controller (3.2) does not apply for 𝜋/3<|𝑥₁(𝑡)|<𝜋.

Define the simulation conditions for unforced regime as follows: 𝐪^𝑇(0)=𝜋40. (6.19) Figure 3 shows the solution for control of the system with nonzero initial condition.

Figure 3 

Responses of the pendulum angle 𝑞1(𝑡) and angular velocity 𝑞2(𝑡) of the inverted pendulum on a cart with fuzzy control.

The second example was included into the paper to demonstrate more complexity of design. Moreover, functionality properties of proposed method can be verified for example, on the flexible-joint robot arm model [27], Lorenz chaotic system model [28], converter model [29], and so forth.

7. Concluding Remarks

New approach to design of the state control with quadratic performance for a class of continuous-time TS fuzzy systems is presented in this paper. This is achieved by application of TS fuzzy model relating to multimodel approximation structure, the extended Lyapunov function, and its enhanced derivative. Presented version is derived in terms of optimization over LMI constraints using standard LMI numerical optimization procedures to manipulate the global stability of the system. The limitation of this approach is that some state variables must be measurable to construct the fuzzy controller. This is a common limitation for control system design on TS fuzzy approach.

The global quadratic stability of the closed-loop system, solved in the sense of enhanced Lyapunov function derivative, was formulated considering measurable premise variables. Since the stability conditions based on the standard form of the quadratic Lyapunov function are very conservative as a common symmetric positive definite matrix verifying all Lyapunov inequalities is required, the presented principle, naturally exploiting the affine properties of TS fuzzy models and incorporating linear quadratic performance, strictly decouples Lyapunov matrix and the system parameter matrices in the resulting LMIs and significantly reduces the conservativeness in the fuzzy control design. Such LMI-based fuzzy controller is to minimize an upper bound of the performance index.

Used technique denotes that the controller shares the same fuzzy sets with the fuzzy system in the premise parts. As the presented sufficient stability condition did not consider the membership functions of both the TS fuzzy model and the fuzzy controller, the design conditions are valid for any arbitrary membership functions set. Since there is generally no restriction on the design of the membership functions, only the structural complexity of the fuzzy controller may be increased when the membership functions of the TS fuzzy model are more complex. Thus, if it is possible to minimize the design effort and complexity with respect to given system nonlinear sectors, a compromise between complexity and error approximation criterion can be used. In opposite cases, for example, if affine fuzzy system is highly nonlinear, shift operations or diffeomorphic transformations have to be used to transform it to linear types, or staircase membership functions can be employed to approximate the continuous membership functions of the TS fuzzy model and fuzzy controller and to include the membership functions into the stability conditions [30]. However, since this kind of approach yields another structure of matrix inequality conditions, it was not employed in the paper.

Acknowledgment

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