Stabilizing Fuzzy Output Control for a Class of Nonlinear Systems

Abstract

The paper presents new conditions suitable in design of a stabilizing output controller for a class of continuous-time nonlinear systems, represented by Takagi-Sugeno models. Taking into account the affine properties of the TS model structure and applying the fuzzy control scheme relating to the parallel distributed output compensators, the sufficient design conditions are outlined in terms of linear matrix inequalities. The proposed procedure decouples the Lyapunov matrix and the system parameter matrices in the LMIs and guarantees global stability of the system. Simulation result illustrates the design procedure and demonstrates the performances of the proposed design method.

1. Introduction

Contrarily to the linear framework, nonlinear systems are too complex to be represented by unified mathematical resources, and so a generic method has not been developed yet to design a controller valid for all types of nonlinear systems. An alternative to nonlinear system models is Takagi-Sugeno (TS) fuzzy approach [1], which benefits from the advantages of suitable linear approximation. Using the TS fuzzy model, each rule utilizes the local system dynamics by a linear model and the nonlinear system is represented by a collection of fuzzy rules. Recently, TS model based fuzzy control approaches are being fast and successfully used in nonlinear control frameworks. As a result, a range of stability analysis conditions [2–5], as well as control design methods [6–11], have been developed for TS fuzzy systems, relying mostly on the feasibility of an associated set of linear matrix inequalities (LMI) [12]. An important fact is that the design problem is a standard feasibility problem with several LMIs, potentially reformulated such that the feedback gains can be solved numerically. In consequence, the state feedback control based on fuzzy TS systems model is mostly realized in such structures which can be designed using a technique based on LMIs.

The TS fuzzy model-based state control is based on an implicit assumption that all states are available for measurement. If it is impossible, TS fuzzy observers are used to estimate the unmeasurable states variables, and the state controller exploits the system state variable estimate values [13–15]. The nonlinear output feedback design is so formulated as the two LMI set problems, and treated by the two-stage procedure, that is, dealing with a set of LMIs for the observer parameters at first and then solving another set of LMIs for the controller parameters [16]. Since the fuzzy control design problem is preferred to be formulated as a one LMI set problem, such formulation for the output feedback control design is proposed in [17, 18].

The main contribution of the paper is the presentation of the original design condition of the fuzzy output feedback control for the continuous-time nonlinear MIMO systems approximated by a TS model. The central idea of the TS fuzzy model-based control design, that is, to derive control rules so as to compensate each rule of a fuzzy system and construct the control strategy based on the parallel distributed compensators, is reflected in the approach of output control, taking into account the fact that the desired output variables are measurable. Motivated by the above-mentioned observations, the proposed design method combines the principles given in [15, 17], respecting the results presented in [19], and is constructed on an enhanced form of quadratic Lyapunov function [20]. Comparing with the approaches based on a quadratic Lyapunov matrix [15, 17, 21], which are particularly in the case of a large number of rules very conservative as a common symmetric positive definite matrix is used to verify all Lyapunov inequalities, presented principle naturally extends the affine TS model properties using slack matrix variables to decouple Lyapunov matrix and the system matrices in LMIs, does not use iterative algorithms based on LMIs or matrix norm bounds, and gives substantial reducing of conservativeness. Potentially, extra constraints can be imposed to the slack matrices, but such additive constraints can potentially increase the conservativeness of the proposed design conditions.

The remainder of this paper is organized as follows. In Section 2 the structure of TS model for considered class of nonlinear systems is briefly described, and some of its properties are outlined. The output feedback control design problem for systems with measurable premise variables is given in Section 3, where the design conditions that guarantee global quadratic stability are formulated and proven. The method is reformulated in Section 4 in a newly defined enhanced criterion for fuzzy output feedback control design. Section 5 gives the numerical example to illustrate the effectiveness of the proposed approach and to confirm the validity of the control scheme. The last section draws conclusions and some future directions.

Throughout the paper, the following notations are used: 𝐱^𝑇, 𝐗^𝑇 denote the transpose of the vector 𝐱 and matrix 𝐗, respectively, diag[⋅] denotes a block diagonal matrix, for a square matrix 𝐗=𝐗^𝑇>0 (resp., 𝐗=𝐗^𝑇<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 denotes the set of all 𝑛×𝑟 real matrices.

2. Takagi-Sugeno Fuzzy Models

The systems under consideration are from one class of multiinput and multioutput nonlinear (MIMO) dynamic systems, represented in state-space form as 𝐪̇(𝑡)(𝐪(𝑡(𝐪(𝑡𝐮(𝑡),𝐲(𝑡)(𝑡),=𝐚))+𝐛))=𝐂𝐪 (1)(2) where 𝐪(𝑡)∈ℝ^𝑛, 𝐮(𝑡)∈ℝ^𝑟, and 𝐲(𝑡)∈ℝ^𝑚 are vectors of the state, input, and output variables, 𝐂∈ℝ^𝑚×𝑛 is a real finite values matrix, and 𝑚≤𝑛, 𝑟≤𝑛, respectively.

Considering that the number of the nonlinear terms in 𝐚(𝐪(𝑡)) is 𝑝, there exists a set of nonlinear sector functions as follows: 𝑚_𝑙𝑗𝜃_𝑗(𝑡𝑚),𝑗=1,2,…,𝑘,𝑙=1,2,…,𝑝,_𝑙1𝜃_𝑗(𝑡)=1−^𝑘_𝑗=2𝑚_𝑙𝑗𝜃_𝑗(𝑡), (3) where 𝑘 is the number of sectors, and 𝜽(𝑡)=𝜃₁(𝑡)𝜃₂(𝑡)⋯𝜃_𝑞(𝑡) (4) is the vector of premise variables. It is assumed that the premise variable is a system state variable, or a measurable external variable, and none of the premise variables does not depend on the inputs 𝐮(𝑡).

Using a TS model, the conclusion part of a single rule consists no longer of a fuzzy set [3], 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. Thus, using linear functions, a system state is described locally (in fuzzy regions) by linear models, and at the boundaries between regions an interpolation is used between the corresponding local models.

Thus, constructing the set of membership functions {𝑤_𝑖(𝜽(𝑡))=∏^𝑠_𝑗=1𝑚_𝑙𝑗(𝜃_𝑗(𝑡)), 𝑖=1,2,…𝑠,𝑠=2^𝑘} from all combinations of sector functions, the final states of the systems are inferred as follows: 𝐪̇(𝑡)=^𝑠_𝑖=1ℎ_𝑖(𝜽(𝑡))𝐀_𝑖𝐪(𝑡)+𝐁_𝑖𝐮(𝑡) (5) with the output given by the relation 𝐲(𝑡)=𝐂𝐪(𝑡), (6) where ℎ_𝑖(𝜽(𝑡=𝑤))^𝑖(𝜽(𝑡))∑^𝑠_𝑖=1𝑤_𝑖(𝜽(𝑡)) (7) is the average weight for the 𝑖th rule, representing the normalized grade of membership (membership function). By definition, the membership functions satisfy the following convex sum properties:0≤ℎ_𝑖(𝜽(𝑡))≤1,^𝑠_𝑖=1ℎ_𝑖(𝜽(𝑡))=1∀𝑖∈⟨1,…,𝑠⟩. (8)

Assuming that 𝐚(𝐪(𝑡)) and 𝐛(𝐪(𝑡)) are bounded in sectors, that is, in the fuzzy regions within the system will operate, and 𝐚(𝐪(𝑡)) takes the value 𝐚(𝟎)=𝟎, the fuzzy approximation of (1) leads to (6). Thus, 𝐀_𝑖∈ℝ^𝑛×𝑛 is the matrix of 𝐚(𝐩(𝑡)), 𝐁_𝑖∈ℝ^𝑛×𝑟 is the matrix of 𝐛(𝐩(𝑡)), respectively, both for 𝐩(𝑡)=𝐩_𝑖, where 𝐩_𝑖 is the 𝑖th combination of the bounds of the sector functions with respect to the center of the 𝑖th fuzzy region, dedicated by (3). It is evident that a general fuzzy model is achieved by fuzzy amalgamation of the linear systems models.

Note, the model (5) and (6) is mostly considered for analysis, control, and state estimation of nonlinear systems.

Assumption 1. Each triplet (𝐀_𝑖,𝐁_𝑖,𝐂) is locally controllable and observable, the matrix 𝐂 is the same for all local models, and the number of input variables is equal to the number of output variables.

It is supposed in the following considerations that the aforementioned model does not include parameter uncertainties or external disturbances, and the premise variables are measured.

3. Fuzzy Output Control Design

In the next, the fuzzy 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.

Definition 1. Considering (5), (6), and using the same set of normalized membership function (8), the fuzzy static output controller is defined as 𝐮(𝑡)=−^𝑠_𝑗=1ℎ_𝑗(𝜽(𝑡))𝐊_𝑗𝐲(𝑡)=−^𝑠_𝑗=1ℎ_𝑗(𝜽(𝑡))𝐊_𝑗𝐂𝐪(𝑡). (9) Note that the fuzzy controller (9) is in general nonlinear.

Theorem 2. The equilibrium of the fuzzy system (5) and (6), controlled by the fuzzy controller (9), is global asymptotically stable if there exist a positive definite symmetric matrix 𝐖∈ℝ^𝑛×𝑛 and matrices 𝐍_𝑗,𝐘_𝑖𝑗∈ℝ^𝑟×𝑛 such that 𝐖=𝐖^𝑇𝐘>0,¹¹𝐘₁₂⋯𝐘_1𝑠𝐘₁₂𝐘₂₂⋯𝐘_2𝑠𝐘⋮⋮⋱⋮_1𝑠𝐘_2𝑠⋯𝐘_𝑠𝑠𝐀>0,_𝑖𝐖+𝐖𝐀^𝑇_𝑖−𝐁_𝑖𝐍_𝑗𝐂−𝐂^𝑇𝐍^𝑇_𝑗𝐁^𝑇_𝑖+𝐘_𝑖𝑗<0 (10)(11) for ℎ_𝑖(𝜽(𝑡))ℎ_𝑗(𝜽(𝑡))≠0,  𝑖,𝑗=1,2,…,𝑠.
If the above conditions hold, the set of control law gain matrices is given as 𝐊_𝑗=𝐍_𝑗𝐌⁻¹,𝑗=1,2,…,𝑠, (12) where 𝐌=𝐂𝐖𝐂^⊖1,𝐂^⊖1=𝐂^𝑇𝐂𝐂^𝑇−1. (13)𝐂^⊖1 is Moore-Penrose pseudo-inverse of 𝐂.

Proof. Considering the model (6) and (2) and the control law (9), then it yields 𝐪̇(𝑡)=^𝑠𝑠_𝑖=1𝑗=1ℎ_𝑖(𝜽(𝑡))ℎ_𝑗(𝜽(𝑡))𝐀_𝑖−𝐁_𝑖𝐊_𝑗𝐂𝐪(𝑡). (14) In order to analyze the convergence of the system state, the quadratic positive Lyapunov function is considered as follows: 𝑣(𝐪(𝑡))=𝐪^𝑇(𝑡)𝐏𝐪(𝑡)>0, (15) where 𝐏∈ℝ^𝑛×𝑛 is a positive definite symmetric matrix. Then, the time derivative of 𝑣(𝐪(𝑡)) along the system trajectory is 𝑣̇(𝐪(𝑡))=𝐪̇^𝑇(𝑡)𝐏𝐪(𝑡)+𝐪^𝑇(𝑡)𝐏𝐪̇(𝑡)<0. (16) Substituting (14) in (16), and introducing the term as follows: 𝑣_𝑣(𝜽(𝑡))=𝐪^𝑇(𝑡)𝐙(𝜽(𝑡))𝐪(𝑡), (17) where 𝐙(𝜽(𝑡))=^𝑠𝑠_𝑖=1𝑗=1ℎ_𝑖(𝜽(𝑡))ℎ_𝑗(𝜽(𝑡))𝐗_𝑖𝑗>0 (18) and {𝐗_𝑖𝑗=𝐗^𝑇_𝑗𝑖∈ℝ^𝑛×𝑛,𝑖,𝑗=1,2,…,𝑠} is the set of matrices. In the sense of Krasovskii theorem (see, e.g., [22]) it can be set up as follows: 𝑣̇(𝐪(𝑡))=𝐪^𝑇(𝑡)^𝑠𝑠_𝑖=1𝑗=1ℎ_𝑖(𝜽(𝑡))ℎ_𝑗(𝜽(𝑡))𝐏𝐀_𝑖−𝐁_𝑖𝐊_𝑗𝐂𝐪(𝑡)+𝐪^𝑇(𝑡)^𝑠𝑠_𝑖=1𝑗=1ℎ_𝑖(𝜽(𝑡))ℎ_𝑗(𝜽(𝑡×𝐀))_𝑖−𝐁_𝑖𝐊_𝑗𝐂^𝑇𝐏𝐪(𝑡)<−𝐪^𝑇(𝑡)^𝑠𝑠_𝑖=1𝑗=1ℎ_𝑖(𝜽(𝑡))ℎ_𝑗(𝜽(𝑡))𝐗_𝑖𝑗𝐪(𝑡)=−𝐪^𝑇(𝑡)𝐙(𝜽(𝑡𝐪(𝑡)𝑣̇(𝐪(𝑡=))<0,))^𝑠𝑠_𝑖=1𝑗=1ℎ_𝑖(𝜽(𝑡))ℎ_𝑗(𝜽(𝑡))𝐪^𝑇(𝑡)𝐏_𝑖𝑗𝐪(𝑡)<0, (19)(20) respectively, which implies 𝐏_𝑖𝑗=𝐏𝐀_𝑖−𝐁_𝑖𝐊_𝑗𝐂+𝐀_𝑖−𝐁_𝑖𝐊_𝑗𝐂^𝑇𝐏+𝐗_𝑖𝑗<0. (21) It is evident that 𝐏_𝑖𝑗 has to be negative definite.
Since 𝑟=𝑚, it is possible to set 𝐁_𝑖𝐊_𝑗𝐂=𝐁_𝑖𝐊_𝑗𝐌𝐌⁻¹𝐂=𝐁_𝑖𝐍_𝑗𝐂𝐏, (22) where 𝐊_𝑗𝐌=𝐍_𝑗,𝐌⁻¹𝐂=𝐂𝐏 (23) and 𝐌∈ℝ^𝑚×𝑚 is a regular square matrix. Substituting (22) into (21) results in 𝐏𝐀_𝑖−𝐁_𝑖𝐍_𝑗𝐂𝐏+𝐀_𝑖−𝐁_𝑖𝐍_𝑗𝐂𝐏^𝑇𝐏+𝐗_𝑖𝑗<0. (24) Premultiplying the left side and the right side of (24) by 𝐏⁻¹ leads to 𝐀_𝑖𝐏⁻¹−𝐁_𝑖𝐍_𝑗𝐂+𝐏⁻¹𝐀^𝑇_𝑖−𝐂^𝑇𝐍^𝑇_𝑗𝐁^𝑇_𝑖+𝐏⁻¹𝐗_𝑖𝑗𝐏⁻¹<0 (25) and using the notations 𝐖=𝐏⁻¹,𝐘_𝑖𝑗=𝐖𝐗_𝑖𝑗𝐖 (26) then (25) implies (11). Since (25) also implies 𝐏^∼_𝑖𝑗=𝐀_𝑖𝐖+𝐖𝐀^𝑇_𝑖−𝐁_𝑖𝐍_𝑗𝐂−𝐂^𝑇𝐍^𝑇_𝑗𝐁^𝑇_𝑖+𝐘_𝑖𝑗<0. (27) Using the membership functions property (8) and defining 𝐪^⋄(𝑡)=𝐖⁻¹𝐪(𝑡), it can be written as follows: 𝑣̇_𝑣(𝐪(𝑡=))_{𝑠𝑠𝑖=1𝑗=1}ℎ_𝑖(𝜽(𝑡))ℎ_𝑗(𝜽(𝑡))𝐪^𝑇(𝑡)𝐖⁻¹𝐖𝐗_𝑖𝑗𝐖𝐖⁻¹𝐪(𝑡)=^𝑠𝑠_𝑖=1𝑗=1ℎ_𝑖(𝜽(𝑡))ℎ_𝑗(𝜽(𝑡))𝐪^⋄𝑇(𝑡)𝐘_𝑖𝑗𝐪^⋄(𝑡)=𝐪^⋄𝑇(𝑡)𝐖𝐙(𝜽(𝑡))𝐖𝐪^⋄(𝑡)>0. (28) Writing 𝐙(𝜽(𝑡)) as follows:𝐙(𝜽(𝑡=ℎ))¹(𝜽(𝑡))𝐪^⋄(𝑡)ℎ₂(𝜽(𝑡))𝐪^⋄(𝑡)⋮ℎ_𝑠(𝜽(𝑡))𝐪^⋄(𝑡)^𝑇𝐘₁₁𝐘₁₂⋯𝐘_1𝑠𝐘₂₁𝐘₂₂⋯𝐘_2𝑠𝐘⋮⋮⋱⋮_𝑠1𝐘_𝑠2⋯𝐘_𝑠𝑠×ℎ₁(𝜽(𝑡))𝐪^⋄(𝑡)ℎ₂(𝜽(𝑡))𝐪^⋄(𝑡)⋮ℎ_𝑠(𝜽(𝑡))𝐪^⋄(𝑡)>0 (29) then (29) implies (10). In addition, (23) gives𝐌𝐂=𝐂𝐏⁻¹=𝐂𝐖. (30) Premultiplying the right side of (30) by 𝐂^𝑇 gives 𝐂𝐖𝐂^𝑇=𝐌𝐂𝐂^𝑇 (31) and so it is 𝐌=𝐂𝐖𝐂^𝑇𝐂𝐂^𝑇−1=𝐂𝐖𝐂^⊖1. (32) Thus, (23) and (32) imply (12)–(13), respectively. This concludes the proof.

Note, the derived results are linked to some existing finding when the design problem involves additive performance requirements and the relaxed quadratic stability conditions of fuzzy control systems (see, e.g., [13, 23]) are equivalently steered.

Trying to minimize the number of LMIs owing to the limitation of solvers, Theorem 2 can be presented in the equivalent structure in which the number of stabilization conditions, used in fuzzy controller design, is equal to 𝑁=(𝑠²+𝑠)/2+1. Evidently, the number of stabilization conditions is substantially reduced if 𝑠 is large.

Theorem 3. The equilibrium of the fuzzy system (5) and (6), controlled by the fuzzy controller (9), is global asymptotically stable if there exist a positive definite symmetric matrix 𝐖∈ℝ^𝑛×𝑛 and matrices 𝐍_𝑗,𝐘_𝑖𝑗∈ℝ^𝑟×𝑛 such that 𝐖=𝐖^𝑇𝐘>0,¹¹𝐘₁₂⋯𝐘_1𝑠𝐘₂₁𝐘₂₂⋯𝐘_2𝑠𝐘⋮⋮⋱⋮_𝑠1𝐘_𝑠2⋯𝐘_𝑠𝑠𝐇>0_𝑖𝑖+𝐇^𝑇_𝑖𝑖+𝐘_𝑖𝑖𝐇<0,_𝑖𝑗+𝐇_𝑗𝑖2+𝐇^𝑇_𝑖𝑗+𝐇^𝑇_𝑗𝑖2+𝐘_𝑖𝑗+𝐘_𝑗𝑖2<0 (33)(34) for all 𝑖∈⟨1,2,…,𝑠⟩, 𝑖<𝑗≤𝑠,𝑖,𝑗∈⟨1,2,…,𝑠⟩, respectively, and ℎ_𝑖(𝜽(𝑡))ℎ_𝑗(𝜽(𝑡))≠0. Here it is 𝐇_𝑖𝑗=𝐀_𝑖𝐖−𝐁_𝑖𝐍_𝑗𝐂 (35) and if the above conditions hold, the set of control law gain matrices is given as in (12) and (13).

Proof. Now (28) can be written as 𝑣̇(𝐪(𝑡))=^𝑠𝑠_𝑖=1𝑗=1ℎ_𝑖(𝜽(𝑡))ℎ_𝑗(𝜽(𝑡))𝐪^𝑇(𝑡)×𝐇_𝑖𝑗+𝐇^𝑇_𝑖𝑗+𝐘_𝑖𝑗𝐪(𝑡), (36) where 𝐇_𝑖𝑗 is given in (35). Permuting the subscripts 𝑖 and 𝑗 in (36) gives 𝑣̇(𝐪(𝑡))=^𝑠𝑠_𝑖=1𝑗=1ℎ_𝑖(𝜽(𝑡))ℎ_𝑗(𝜽(𝑡))𝐪^𝑇(𝑡)×𝐇_𝑗𝑖+𝐇^𝑇_𝑗𝑖+𝐘_𝑗𝑖𝐪(𝑡) (37) and adding (36) and (37) results in 2𝑣̇(𝐪(𝑡=))_{𝑠𝑠𝑖=1𝑗=1}ℎ_𝑖(𝜽(𝑡))ℎ_𝑗(𝜽(𝑡))𝐪^𝑇(𝑡)×𝐇_𝑖𝑗+𝐇_𝑗𝑖+𝐇_𝑖𝑗+𝐇_𝑗𝑖^𝑇+𝐘_𝑖𝑗+𝐘_𝑗𝑖𝐪(𝑡). (38) Rearranging the computation, (38) takes the following form: 𝑣̇(𝐪(𝑡))=^𝑠_𝑖=1ℎ²_𝑖(𝜽(𝑡))𝐪^𝑇(𝑡)𝐇_𝑖𝑖+𝐇^𝑇_𝑖𝑖−𝐘_𝑖𝑖𝐪(𝑡)+2^𝑠−1𝑠_{𝑖=1𝑗=𝑖+1}ℎ_𝑖(𝜽(𝑡))ℎ_𝑗(𝜽(𝑡))𝐪^𝑇(𝑡)×𝐇_𝑖𝑗+𝐇_𝑗𝑖2+𝐇^𝑇_𝑖𝑗+𝐇^𝑇_𝑗𝑖2+𝐘_𝑖𝑗+𝐘_𝑗𝑖2𝐪(𝑡) (39) and, evidently, (39) implies (34). This concludes the proof.

4. Enhanced Criterion for Output Control Design

Theorem 4. The equilibrium of the fuzzy system (5) and (6), controlled by the fuzzy controller (9) is global asymptotically stable if there exist positive definite symmetric matrices 𝐔,𝐕∈ℝ^𝑛×𝑛, a matrix 𝐓∈ℝ^𝑛×𝑛, and matrices 𝐍_𝑗,𝐘_𝑖𝑗∈ℝ^𝑟×𝑛 such that 𝐔=𝐔^𝑇>0,𝐕=𝐕^𝑇𝐘>0,₁₁𝐘₁₂⋯𝐘_1𝑠𝐘₁₂𝐘₂₂⋯𝐘_2𝑠𝐘⋮⋮⋱⋮_1𝑠𝐘_2𝑠⋯𝐘_𝑠𝑠𝐏>0,_•𝑖𝑗=𝐀_𝑖𝐕+𝐕𝐀^𝑇_𝑖−𝐁_𝑖𝐍_𝑗𝐂−𝐂^𝑇𝐍^𝑇_𝑗𝐁^𝑇_𝑖+𝐘_𝑖𝑗∗𝐓−𝐔+𝐀_𝑖𝐕−𝐁_𝑖𝐍_𝑗𝐂−2𝐔<0 (40)(41) for ℎ_𝑖(𝜽(𝑡))ℎ_𝑗(𝜽(𝑡))≠0, 𝑖,𝑗=1,2,…,𝑠.
If the above conditions hold, the set of control law gain matrices is given as 𝐊_𝑗=𝐍_𝑗𝐌⁻¹,𝑗=1,2,…,𝑠, (42) where 𝐌=𝐂𝐕𝐂^⊖1,𝐂^⊖1=𝐂^𝑇𝐂𝐂^𝑇−1, (43)𝐂^⊖1 is Moore-Penrose pseudo-inverse of 𝐂.
Here, and hereafter, ∗ denotes the symmetric item in a symmetric matrix.

Proof . Writing (14) in the following form: 𝐪̇(𝑡)−^𝑠𝑠_𝑖=1𝑗=1ℎ_𝑖(𝜽(𝑡))ℎ_𝑗(𝜽(𝑡))𝐀_𝑖−𝐁_𝑖𝐊_𝑗𝐂𝐪(𝑡)=0 (44) then with arbitrary symmetric regular matrices 𝐒₁,𝐒₂∈ℝ^𝑛×𝑛 it yields −𝐪^𝑇(𝑡)𝐒₁+𝐪̇^𝑇(𝑡)𝐒₂×𝐪̇(𝑡)−^𝑠𝑠_𝑖=1𝑗=1ℎ_𝑖(𝜽(𝑡))ℎ_𝑗(𝜽(𝑡))𝐀_𝑖−𝐁_𝑖𝐊_𝑗𝐂𝐪(𝑡)=0. (45) Thus, adding (45) as well as the transposition of (45) to (19), it yields 𝑣̇(𝐪(𝑡))=𝐪̇^𝑇(𝑡)𝐏𝐪(𝑡)+𝐪^𝑇(𝑡)𝐏𝐪̇(𝑡)−𝐪^𝑇(𝑡)𝐒₁+𝐪̇^𝑇(𝑡)𝐒₂×𝐪̇(𝑡)−^𝑠𝑠_𝑖=1𝑗=1ℎ_𝑖(𝜽(𝑡))ℎ_𝑗(𝜽(𝑡×𝐀))_𝑖−𝐁_𝑖𝐊_𝑗𝐂𝐪(𝑡)−𝐪̇^𝑇(𝑡)−𝐪^𝑇(𝑡)^𝑠𝑠_𝑖=1𝑗=1ℎ_𝑖(𝜽(𝑡))ℎ_𝑗(𝜽(𝑡×𝐀))_𝑖−𝐁_𝑖𝐊_𝑗𝐂^𝑇𝐒₁𝐪(𝑡)+𝐒₂𝐪̇(𝑡)<−𝐪^𝑇(𝑡)^𝑠𝑠_𝑖=1𝑗=1ℎ_𝑖(𝜽(𝑡))ℎ_𝑗(𝜽(𝑡))𝐗_𝑖𝑗𝐪(𝑡)=−𝐪^𝑇(𝑡)𝐙(𝜽(𝑡))𝐪(𝑡)<0. (46) Using the notation 𝐪^∘𝑇(𝑡)=𝐪^𝑇(𝑡)𝐪̇^𝑇(𝑡) (47) the inequality (46) can be set and written as 𝑣̇(𝐪(𝑡))=^𝑠𝑠_𝑖=1𝑗=1ℎ_𝑖(𝜽(𝑡))ℎ_𝑗(𝜽(𝑡))𝐪^∘𝑇(𝑡)𝐏^∘_𝑖𝑗𝐪^∘(𝑡)<0, (48) where 𝐏^∘_𝑖𝑗=𝐒₁𝐀_𝑖−𝐁_𝑖𝐊_𝑗𝐂+𝐀_𝑖−𝐁_𝑖𝐊_𝑗𝐂^𝑇𝐒₁+𝐗_𝑖𝑗∗𝐏−𝐒₁+𝐒₂𝐀_𝑖−𝐁_𝑖𝐊_𝑗𝐂−2𝐒₂𝐙<0_∘(𝜽(𝑡))=diag𝐙(𝜽(𝑡))𝟎≥0. (49)(50) Since 𝑟=𝑚, it is now possible to set 𝐁_𝑖𝐊_𝑗𝐂=𝐁_𝑖𝐊_𝑗𝐌𝐌⁻¹𝐂=𝐁_𝑖𝐍_𝑗𝐂𝐒₁, (51) where 𝐊_𝑗𝐌=𝐍_𝑗,𝐌⁻¹𝐂=𝐂𝐒₁, (52) and 𝐌∈ℝ^𝑚×𝑚 is a regular square matrix. Substituting (51) into (49) results in 𝐏^∘_𝑖𝑗=Φ_𝑖𝑗∗𝐏−𝐒₁+𝐒₂𝐀_𝑖−𝐁_𝑖𝐍_𝑗𝐂𝐒₁−2𝐒₂, (53) where Φ_𝑖𝑗=𝐗_𝑖𝑗+𝐒₁𝐀_𝑖−𝐁_𝑖𝐍_𝑗𝐂𝐒₁+𝐀_𝑖−𝐁_𝑖𝐍_𝑗𝐂𝐒₁^𝑇𝐒₁. (54) Defining the congruence transform matrix 𝐓_𝑐 as follows: 𝐓_𝑐=𝐒₁⁻¹𝐒₂⁻¹. (55) Premultiplying the left and the right side of (53) and (50) by 𝐓_𝑐 gives 𝚷_𝑖𝑗∗𝐒₂⁻¹𝐏𝐒₁⁻¹−𝐒₂⁻¹+𝐀_𝑖𝐒₁⁻¹−𝐁_𝑖𝐍_𝑗𝐂−2𝐒₂⁻¹𝐒<0diag₁⁻¹𝐙(𝜽(𝑡))𝐒₁⁻¹𝟎≥0, (56)(57) where 𝚷_𝑖𝑗=𝐒₁⁻¹Φ_𝑖𝑗𝐒₁⁻¹=𝐒₁⁻¹𝐗_𝑖𝑗𝐒₁⁻¹+𝐀_𝑖𝐒₁⁻¹−𝐁_𝑖𝐍_𝑗𝐂+𝐀_𝑖𝐒₁⁻¹−𝐁_𝑖𝐍_𝑗𝐂^𝑇. (58) Thus, using the notations 𝐕=𝐒₁⁻¹,𝐔=𝐒₂⁻¹,𝐓=𝐔𝐏𝐕,𝐘_𝑖𝑗=𝐕𝐗_𝑖𝑗𝐕 (59) then (56), (58) implies (41).
Since now, 𝐕𝐙(𝜽(𝑡))𝐕 in (57) and 𝐖𝐙(𝜽(𝑡))𝐖 in (28), both have the same structure if  𝐕=𝐖, (57) implies (40).
In addition, (52) gives 𝐌𝐂=𝐂𝐒₁⁻¹=𝐂𝐕 (60) and since (60) and (30) both have the same structure if 𝐕=𝐖, then (60) implies (42). This concludes the proof.

This principle naturally exploits the affine TS model properties. Introducing the slack matrix variables 𝐔, 𝐕 into the LMIs, the system matrices are decoupled from the equivalent Lyapunov matrix 𝐓. Note, the above-presented inequalities are linear matrix inequalities, but the equivalent Lyapunov matrix 𝐓 is not a symmetric matrix. Introducing a scalar design parameter 𝛿>0, 𝛿∈ℝ, Theorem 4 can be modified in the next form.

Corollary 5. If, instead of the notations (56), there are used in (59) the next substitutions 𝐔=𝛿𝐕,𝐓=𝛿𝐕𝐏𝐕,𝛿>0,𝛿∈ℝ. (61) It is evident that (53) and (54) with (61) imply 𝐓=𝐓^𝑇>0,𝐕=𝐕^𝑇𝐘>0,₁₁𝐘₁₂⋯𝐘_1𝑠𝐘₁₂𝐘₂₂⋯𝐘_2𝑠𝐘⋮⋮⋱⋮_1𝑠𝐘_2𝑠⋯𝐘_𝑠𝑠𝐀>0,_𝑖𝐕+𝐕𝐀^𝑇_𝑖−𝐁_𝑖𝐍_𝑗𝐂−𝐂^𝑇𝐍^𝑇_𝑗𝐁^𝑇_𝑖+𝐘_𝑖𝑗∗𝐓−𝛿𝐕+𝐀_𝑖𝐕−𝐁_𝑖𝐍_𝑗𝐂−2𝛿𝐕<0 (62)(63) for ℎ_𝑖(𝜽(𝑡))ℎ_𝑗(𝜽(𝑡))≠0, 𝑖,𝑗=1,2,…,𝑠.
Thus, the equilibrium of the fuzzy system (5) and (6), controlled by the fuzzy controller (9), is global asymptotically stable if for given positive scalar 𝛿>0, 𝛿∈ℝ there exist positive definite symmetric matrices 𝐓,𝐕∈ℝ^𝑛×𝑛, and matrices 𝐍_𝑗,𝐘_𝑖𝑗∈ℝ^𝑟×𝑛 such that (62)–(63) hold. Subsequently, the set of control law gain matrices is given by (42)–(43).

Note, (63) represents the set of LMIs only if 𝛿 is a prescribed constant (𝛿 can be considered as a tuning parameter). Considering 𝛿 as a LMI variable, (63) represents the set of bilinear matrix inequalities (BMI).

Theorem 6. The equilibrium of the fuzzy system (5) and (6), controlled by the fuzzy controller (9) is global asymptotically stable if there exist positive definite symmetric matrices 𝐔,𝐕∈ℝ^𝑛×𝑛, a matrix 𝐓∈ℝ^𝑛×𝑛, and matrices 𝐍_𝑗,𝐘_𝑖𝑗∈ℝ^𝑟×𝑛 such that 𝐔=𝐔^𝑇>0,𝐕=𝐕^𝑇𝐘>0,₁₁𝐘₁₂⋯𝐘_1𝑠𝐘₁₂𝐘₂₂⋯𝐘_2𝑠𝐘⋮⋮⋱⋮_1𝑠𝐘_2𝑠⋯𝐘_𝑠𝑠𝐇>0,_𝑖𝑖+𝐇^𝑇_𝑖𝑖+𝐘_𝑖𝑖∗𝐓−𝐔+𝐇_𝑖𝑖𝐇−2𝐔<0,_𝑖𝑗+𝐇_𝑗𝑖2+𝐇_𝑖𝑗+𝐇_𝑗𝑖^𝑇2+𝐘_𝑖𝑗+𝐘_𝑗𝑖2∗𝐇𝐓−𝐔+^𝑖𝑗+𝐇_𝑗𝑖2−2𝐔<0 (64)(65)(66) for all 𝑖∈⟨1,2,…𝑠⟩, 𝑖<𝑗≤𝑠,𝑖,𝑗∈⟨1,2,…𝑠⟩, respectively, and ℎ_𝑖(𝜽(𝑡))ℎ_𝑗(𝜽(𝑡))≠0. Here it is 𝐇_𝑖𝑗=𝐀_𝑖𝐕−𝐁_𝑖𝐍_𝑗𝐂 (67) and if the above conditions hold, the set of control law gain matrices is given as in (42)–(43).

Proof. Considering (41) and (67), (48) can be written as 𝑣̇(𝐪(𝑡))=^𝑠𝑠_𝑖=1𝑗=1ℎ_𝑖(𝜽(𝑡))ℎ_𝑗(𝜽(𝑡))×𝐪^∘𝑇(𝑡)𝐓_𝑐⁻¹𝐓_𝑐𝐏^∘_𝑖𝑗𝐓_𝑐𝐓_𝑐⁻¹𝐪^∘(𝑡)=^𝑠𝑠_𝑖=1𝑗=1ℎ_𝑖(𝜽(𝑡))ℎ_𝑗(𝜽(𝑡))𝐪^•𝑇(𝑡)𝐏^•_𝑖𝑗𝐪^•(𝑡)<0, (68) where with 𝐓_𝑐 defined in (55), it is 𝐪^•(𝑡)=𝐓_𝑐⁻¹𝐪^∘(𝑡),𝐏^•_𝑖𝑗=𝐓_𝑐𝐏^∘_𝑖𝑗𝐓_𝑐=𝐇_𝑖𝑗+𝐇^𝑇_𝑖𝑗+𝐘_𝑖𝑗∗𝐓−𝐔+𝐇_𝑖𝑗−2𝐔<0. (69) Permuting the subscripts 𝑖 and 𝑗 in (68) gives 𝑣̇(𝐪(𝑡))=^𝑠𝑠_𝑖=1𝑗=1ℎ_𝑖(𝜽(𝑡))ℎ_𝑗(𝜽(𝑡))𝐪^•𝑇(𝑡)𝐏^•_𝑗𝑖𝐪^•(𝑡) (70) and adding (68) and (70) results in 2𝑣̇(𝐪(𝑡))=^𝑠𝑠_𝑖=1𝑗=1ℎ_𝑖(𝜽(𝑡))ℎ_𝑗(𝜽(𝑡))𝐪^•𝑇(𝑡)𝐏^∼•_𝑖𝑗𝐪^•(𝑡), (71) where 𝐏^∼•_𝑖𝑗=𝐇_𝑖𝑗+𝐇_𝑗𝑖+𝐇^𝑇_𝑖𝑗+𝐇^𝑇_𝑗𝑖+𝐘_𝑖𝑗+𝐘_𝑗𝑖∗2𝐓−2𝐔+𝐇_𝑖𝑗+𝐇_𝑗𝑖−4𝐔<0. (72) Rearranging the computation, (71) takes the form 𝑣̇(𝐪(𝑡))=∑^𝑠_𝑖=1ℎ²_𝑖(𝜽(𝑡))𝐪^𝑇(𝑡)12^∼𝐏^•_𝑖𝑖𝐪^•(𝑡)+2^𝑠−1𝑠_{𝑖=1𝑗=𝑖+1}ℎ_𝑖(𝜽(𝑡))ℎ_𝑗(𝜽(𝑡))𝐪^•𝑇(𝑡)12^∼𝐏^•_𝑖𝑗𝐪^•(𝑡), (73) where 12^∼𝐏^•_𝑖𝑖=𝐇_𝑖𝑖+𝐇^𝑇_𝑖𝑖+𝐘_𝑖𝑖∗𝐓−𝐔+𝐇_𝑖𝑖1−2𝐔<02^∼𝐏^•_𝑖𝑗=𝐇_𝑖𝑗+𝐇_𝑗𝑖2+𝐇_𝑖𝑗+𝐇_𝑗𝑖^𝑇2+𝐘_𝑖𝑗+𝐘_𝑗𝑖2∗𝐇𝐓−𝐔+^𝑖𝑗+𝐇_𝑗𝑖2−2𝐔<0 (74)(75) and, evidently, (74), (75) imply (65),(66) respectively. This concludes the proof.

5. Illustrative Example

The nonlinear dynamics of the hydrostatic transmission was taken from [24] and this model was used in control design and simulation.

The hydrostatic transmission dynamics is represented by a nonlinear fourth-order state-space model as follows. 𝑞̇₁(𝑡)=−𝑎₁₁𝑞₁(𝑡)+𝑏₁₁𝑢₁(𝑡),𝑞̇₂(𝑡)=−𝑎₂₂𝑞₂(𝑡)+𝑏₂₂𝑢₂(𝑡),𝑞̇₃(𝑡)=𝑎₃₁𝑞₁(𝑡)𝑝(𝑡)−𝑎₃₃𝑞₃(𝑡)−𝑎₃₄𝑞₂(𝑡)𝑞₄(𝑡),𝑞̇₄(𝑡)=𝑎₄₃𝑞₂(𝑡)𝑞₃(𝑡)−𝑎₄₄𝑞₄(𝑡), (76) 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 point the parameters are 𝑎₁₁=7.6923𝑎₂₂=4.5455𝑎₃₃=7.6054.10⁻⁴,𝑎₃₁=0.7877𝑎₃₄=0.9235𝑏₁₁=1.8590.10³𝑎,and₄₃=12.1967𝑎₄₄=0.4143𝑏₂₂=1.2879.10³. (77) Since the variables 𝑝(𝑡)∈⟨105,300⟩ and 𝑞₂(𝑡)∈⟨0.0001,1⟩ 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). (78) Thus, the set of nonlinear sector functions as follows. 𝑤₁₁𝑞₂(𝑡)=𝑏₁−𝑞₂(𝑡)𝑏₁−𝑏₂,𝑤₁₂𝑞₂(𝑡)=𝑞₂(𝑡)−𝑏₂𝑏₁−𝑏₂=1−𝑤₁₁𝑞₂(𝑡),𝑏₁=0,𝑏₂𝑤=1,₂₁𝑝(𝑡)=𝑐₁−𝑝(𝑡)𝑐₁−𝑐₂,𝑤₂₂𝑝(𝑡)=𝑝(𝑡)−𝑐₂𝑐₁−𝑐₂=1−𝑤₂₁𝑝(𝑡),𝑐₁=105,𝑐₂=300 (79) implies the next set of normalized membership functions as follows: ℎ₁𝑞₂(𝑡),𝑝(𝑡)=𝑤₁₁𝑞₂(𝑡)𝑤₂₁𝑝(𝑡),ℎ₂𝑞₂(𝑡),𝑝(𝑡)=𝑤₁₂𝑞₂(𝑡)𝑤₂₁𝑝(𝑡),ℎ₃𝑞₂(𝑡),𝑝(𝑡)=𝑤₁₁𝑞₂(𝑡)𝑤₂₂𝑝(𝑡,ℎ₄)𝑞₂(𝑡),𝑝(𝑡)=𝑤₁₂𝑞₂(𝑡)𝑤₂₂𝑝(𝑡). (80) The transformation of nonlinear differential equation systems into a TS fuzzy system in standard form gives 𝐀_𝑖=−𝑎₁₁0000−𝑎₂₂𝑎00₃₁𝑐_𝑘0−𝑎₃₁−𝑎₃₄𝑏_𝑙00𝑎₄₃𝑏_𝑙−𝑎₄₄𝑎,𝐁=¹¹00𝑏₂₂,𝐂0000_𝑇=00100100 (81) with the associations ()()()().𝑖=1⟵𝑙=1,𝑘=1𝑖=2⟵𝑙=2,𝑘=1𝑖=3⟵𝑙=1,𝑘=2𝑖=4⟵𝑙=2,𝑘=2 (82) Thus, solving (62), (63) with respect to the LMI matrix variables 𝐓, 𝐕, 𝐍_𝑗,𝑗=1,2,3,4, and with 𝛿=9 then, using Self-Dual-Minimization (SeDuMi) package for Matlab [25, 26], the feedback gain matrix design problem was feasible with the results 𝐊𝐓=0.00170.0000−0.02230.00060.00000.92580.00000.0000−0.02230.00000.5679−0.14640.00060.0000−0.14641.7684>0𝐕=0.00010.0000−0.0017−0.00010.00000.04810.0000−0.0000−0.00170.00000.0410−0.0253−0.00010.0000−0.02530.2027>0_𝑖=0.00000.00020.00470.0000,𝑖=1,2,3,4 (83) which rise up a stable set of closed-loop subsystems. It can be seen that with an enough precision the used design conditions imply the approximately equal control gain matrices. Comparing with design methods proposed in [17], the fuzzy control is so less conservative.

The conditions in simulations were specified for the system in the forced regime, where 𝐮(𝑡)=^𝑠_𝑗=1ℎ_𝑗(𝜽(𝑡))−𝐊_𝑗(𝑡)(𝑡.(𝑡)=,𝐪(0)(𝑡)𝐂𝐪+𝐖𝐰)𝐖=0.00000.00020.00730.0000,𝐰0.50.3=𝟎,𝑝=105. (84) Figure 1 shows the simulation result for the system with zero initial state.

Figure 1 

System output response.

6. Concluding Remarks

New approach for output feedback control design, taking into account the output matrix of the system model and the affine properties of the TS model structure, is presented in this paper. Applying the fuzzy output control scheme relating to the parallel distributed output compensators and introducing the slack matrices into an enhanced Lyapunov inequality, the method significantly reduces the conservativeness in the control design conditions. By the proposed procedure, strictly decoupling a Lyapunov matrix and the system parameter matrices in the LMIs, the control problem is parameterized in such LMI structure which admit more freedom in guaranteeing the output feedback control performances.

Sufficient conditions of the controller existence, manipulating the global stability of the system, imply the control structure which stabilizes the nonlinear system in the sense of Lyapunov, and the design of controller parameters directly from these conditions is a solved numerical problem. An additional benefit of the method is that controllers use minimum feedback information with respect to desired system output and the approach is flexible enough to allow the inclusion of additional design conditions such as fuzzy Lyapunov functions. The validity and applicability of the approach is demonstrated through a numerical design example.

Acknowledgments

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.