Nonfragile H ∞ tracking control strategies for classes of linear and bilinear uncertain Takagi-Sugeno fuzzy systems

Abstract

This paper presents fuzzy tracking control strategies for classes of linear and bilinear Takagi-Sugeno Fuzzy System with Uncertainties and Disturbances (T-S FSUD). For both classes, the Lyapunov direct approach and the parallel distributed compensation concept are used to design nonfragile H∞ tracking controllers, which ensure the robust asymptotic stability of augmented systems despite the presence of parameter uncertainties, external disturbances, and controllers fragility. Besides that, based on the Schur complement, the separation lemma, and through some variable transformations, sufficient stability conditions of each augmented T-S FSUD are established and formulated in terms of linear matrix inequalities. Finally, numerical simulations are carried out to demonstrate the validity and the applicability of designed control schemes on tracking control of an inverted pendulum and a nonlinear mechanical system.

Keywords

Takagi-Sugeno fuzzy model tracking nonfragile control linear matrix inequality

Introduction

Over the past years, the study of the Fuzzy Logic Control (FLC) has received considerable attention in numerous fields (Chang and Yang, 2014; Huang et al., 2011; Sun et al., 2017), such as robotics, power systems, chemical engineering, image processing, vehicular technology, sensor technology, industrial automation, communications ad networking, consumer electronics, civil engineering, and automobiles. Particularly, in the area of nonlinear control systems design, an important approach is to represent nonlinear dynamic systems as Takagi-Sugeno (T-S) models by a set of If-then rules. Notably, based on the sector nonlinearity approach Takagi and Sugeno (1985), the overall T-S model is achieved by interpolating local linear models through nonlinear fuzzy membership functions. In the controller design procedure, a parallel distributed compensation (PDC) concept has been employed to overcome different stability and stabilization problems of T-S fuzzy systems (Ghorbel et al., 2020; Tlili, 2019; Wei et al., 2016). In the PDC concept, each control rule is designed from the corresponding rule of a T-S fuzzy model. The purposeful fuzzy controller shares the same fuzzy sets with the fuzzy model in the premise parts. The resulting overall controller, which is in general nonlinear, is a fuzzy blending of these linear controllers. However, the obtained stability conditions of the closed-loop system are in general established in terms of Linear Matrix Inequalities (LMI) and the desired controller gains can be solved efficiently using convex programming techniques.

It is known, however, that many nonlinear dynamic systems cannot be adequately approximated by linear models (Penny et al., 2005; Tang et al., 2006). A good example of a bilinear process is the population of biological species, which is described by

\overset{\cdot}{x} = x u

(1)

where $x$ is the population and $u$ denotes the birth rate minus death rate. Moreover, since bilinear systems are linear in input and linear in state but not jointly linear in state and input, it is possible to generalize to them many analysis methods originally developed in the linear systems theory (Gao et al., 2007; Li and Tsai, 2007). However, there are many research works related to the modeling and controller design problems for bilinear T-S fuzzy systems. For example, Li and Tsai (2007) proposed a fuzzy controller for the stabilization of such systems. In Tsai and Li (2008) and Tsai et al. (2012), the authors presented robust $H_{\infty}$ fuzzy controllers for a class of T-S fuzzy uncertain bilinear systems. Saoudi et al. (2015) proposed for this class of systems a fault tolerant control methodology. Lee et al. (2017) studied the problem of robust synchronization of chaotic systems described by bilinear T-S fuzzy systems.

However, in real control design problems, uncertainties, inaccuracies, disturbances, modeling errors, measurement errors, exterior conditions, and parameters variations are frequently a source of system’s instability. As well, most control systems necessitate accurate controllers, in which their coefficients are required to be implemented with exact values as those to be designed. Thus, it is not always possible in practical applications since actuators may be of malfunction and/or round-off errors in numerical computations (Nguang and Shi, 2006; Sun et al., 2017). Furthermore, there are considerable studies on the nonfragile $H_{\infty}$ control and filtering problems of T-S fuzzy systems. For example, Yang et al., 2009 explored the nonfragile $H_{\infty}$ control problem for a class of uncertain T-S fuzzy neutral systems. Chang and Yang, 2011, 2014 studied the problem of nonfragile $H_{\infty}$ filtering for continuous-time T-S fuzzy systems. Kchaou et al. (2014) investigated the problem of nonfragile $H_{\infty}$ control design for T-S fuzzy systems with uncertainties, disturbances, and unmeasurable state variables. In Mardani et al. (2019), the authors proposed a nonfragile polynomial fuzzy controller for a class of polynomial fuzzy systems. Tang and Ma (2021) studied the problem of exponential dissipation stability of such systems using nonfragile control. In spite of this, there is no study explored on the nonfragile $H_{\infty}$ tracking control problem for linear and bilinear fuzzy systems subject to control objectives, such as speed of response, attenuation of the disturbances effect, and measure of controller gain variations.

Hence, the main goal of this work is to purpose fuzzy tracking control strategies for classes of linear and bilinear T-S Fuzzy System with Uncertainties and Disturbances (T-S FSUD). In these strategies, a nonfragile $H_{\infty}$ controller is designed for each class to ensure a good tracking control between the state variables of the closed-loop system and the desired ones despite the presence of uncertainties, inaccuracies, and disturbances.

The main contributions can be summarized as follows:

Compared with the previous works, the problem of tracking T-S fuzzy systems with decay rate, attenuation of the disturbances effect, and measure of controller nonfragility is considered;

New sufficient stability conditions in terms of LMI for the existence of $H_{\infty}$ nonfragile controllers are proposed.

Following the “Introduction,” this paper is structured as follows: In section “Problem formulation and preliminaries,” the problem formulation and preliminaries are exposed. Thereafter, section “Robust $H_{\infty}$ and nonfragile tracking control design” presents robust $H_{\infty}$ and nonfragile tracking control schemes for linear and bilinear T-S FSUD. Subsequently, in section “Numerical examples,” simulation studies are provided to illustrate the merit of our purpose. The paper finalizes with “Conclusion.”

Throughout this paper, $M^{T}$ denotes the transpose of $M$ , $M + (*)$ stands for $M + M^{T}$ , $(\begin{matrix} A & * \\ B & C \end{matrix})$ represents $(\begin{matrix} A & B^{T} \\ B & C \end{matrix})$ , and $χ (z) = χ (z (t))$ .

Problem formulation and preliminaries

Consider a nonlinear continuous-time system described by the following state-space representation

{\begin{matrix} \overset{\cdot}{x} = f (x) + g (x, u) \\ y = h (x) \end{matrix}

(2)

where $f (x)$ , $g (x, u)$ , and $h (x)$ are nonlinear functions of the vector state $x \in R^{n \times n}$ , $u \in R^{m}$ is the input vector, and $y \in R^{p}$ is the output vector.

Using the fuzzy inference method Takagi and Sugeno (1985), the nonlinear system (2) can be represented by a set of linear or bilinear plant rules as below:

Linear rule i

\begin{matrix} If (z_{1} is M_{1 i}) and (z_{2} is M_{2 i}) and \dots and (z_{k} is M_{ki}) \\ then \overset{\cdot}{x} = {\bar{A}}_{i} x + B_{i} u + w, y = C_{i} x + D_{i} u \end{matrix}

(3)

Bilinear rule i

\begin{matrix} If (z_{1} is M_{1 i}) and (z_{2} is M_{2 i}) and \dots and (z_{k} is M_{ki}) \\ then \overset{\cdot}{x} = {\bar{A}}_{i} x + B_{i} u + {\bar{N}}_{i} x u + w, y = C_{i} x + D_{i} u \end{matrix}

(4)

for $i = 1, 2, \dots, r$ and $k \leq n$ where $w \in R^{n}$ is the disturbances vector, $M_{ji}$ is the fuzzy set, $z_{1}, z_{2}, \dots, z_{k}$ are the known premise variables, $M_{ji} (z_{j})$ is the degree of membership of $z_{j}$ , $i = 1, 2, \dots, r$ and $j = 1, 2, \dots, k$ , and $r$ is the number of If-then rules. $A_{i} \in R^{n \times n}$ , $B_{i} \in R^{n \times m}$ , $C_{i} \in R^{p \times n}$ , $D_{i} \in R^{p \times m}$ , and ${\bar{N}}_{i} \in R^{n \times n}$ are known. The matrices ${\bar{A}}_{i}$ and ${\bar{N}}_{i}$ are assumed to be of the following form

\begin{matrix} {\bar{A}}_{i} = A_{i} + Δ A_{i} (t) = A_{i} + H_{i} ψ (t) E_{Ai} \\ {\bar{N}}_{i} = N_{i} + Δ N_{i} (t) = N_{i} + H_{i} ψ (t) E_{Ni} \end{matrix}

(5)

where $H_{i}$ , $E_{Ai}$ , and $E_{Ni}$ are known real matrices of appropriate dimensions, and $ψ (t)$ is a time-varying matrix satisfying $ψ^{T} (t) ψ (t) \leq I$ .

Then, the final dynamic equations of the linear and bilinear fuzzy systems can be described as follows:

Linear T-S FSUD

{\begin{matrix} \overset{\cdot}{x} = \sum_{i = 1}^{r} h_{i} (z) ({\bar{A}}_{i} x + B_{i} u + w) \\ y = \sum_{i = 1}^{r} h_{i} (z) (C_{i} x + D_{i} u) \end{matrix}

(6)

Bilinear T-S FSUD

{\begin{matrix} \overset{\cdot}{x} = \sum_{i = 1}^{r} h_{i} (z) ({\bar{A}}_{i} x + B_{i} u + {\bar{N}}_{i} xu + w) \\ y = \sum_{i = 1}^{r} h_{i} (z) (C_{i} x + D_{i} u) \end{matrix}

(7)

with

h_{i} (z) = \frac{w_{i} (z)}{\sum_{i = 1}^{r} w_{i} (z)}

(8)

where $w_{i} (z) = Π_{j = 1}^{k} M_{ji} (z_{j})$ and $z = {(\begin{matrix} z_{1} & z_{2} & \dots & z_{k} \end{matrix})}^{T}$ .

It should be noted that the membership functions verify the convex conditions

{\begin{matrix} \sum_{i = 1}^{r} h_{i} (z) = 1 \\ h_{i} (z) \geq 0 \end{matrix}

(9)

Furthermore, the objective is to determine a $H_{\infty}$ nonfragile controller for each class of studied fuzzy systems. This controller ensures a good tracking between the state variables of the closed-loop system and the desired ones of the following reference model

{\overset{\cdot}{x}}_{e} = A_{e} x_{e} + B_{e} e

(10)

where $A_{e} \in R^{n \times n}$ is a specific asymptotically stable matrix, $B_{e} \in R^{n}$ is an input matrix, $x_{e} \in R^{n}$ is a reference state, and $e \in R$ is a reference input.

First of all, the following technical lemmas that will be needed throughout the proof of Theorems 1 and 2 are recalled.

Lemma 1. (the Young relation (Tlili, 2019; Wei et al., 2016)): For matrices $A$ and $B$ with appropriate dimensions and a positive scalar $τ$ , the following inequality holds

A^{T} B + B^{T} A \leq τ A^{T} A + τ^{- 1} B^{T} B

(11)

Lemma 2. (the Schur Complement Rtibi et al. (2019)): Given matrices $M = M^{T}$ , $Q = Q^{T}$ , and $L$ with appropriate dimensions, the following statements are equivalent

\begin{matrix} (i) (\begin{matrix} M & * \\ L^{T} & Q \end{matrix}) < 0 \\ (ii) Q < 0, M - L Q^{- 1} L^{T} < 0 \\ (iii) M < 0, Q - L^{T} M^{- 1} L < 0 \end{matrix}

(12)

Robust $H_{\infty}$ and nonfragile tracking control design

This section presents fuzzy tracking control strategies for the linear and bilinear T-S FSUD classes, as are defined in (6) and (7). Thus, for each one, sufficient stability conditions are established to ensure the asymptotic stability of an augmented system with guaranteed control objectives.

The linear T-S FSUD case

In this study, the states of system (6) are assumed to be measurable and the $r -$ pairs $(A_{i}, B_{i})$ are locally controllable. Hence, the PDC concept is used for designing a robust $H_{\infty}$ and nonfragile controller. Then ith local control law is designed for each fuzzy plant rule (3) as follows

Control rule i

\begin{matrix} If (z_{1} is M_{1 i}) and (z_{2} is M_{2 i}) and \dots and (z_{k} is M_{ki}) \\ then u = - {\bar{K}}_{i} ε \end{matrix}

(13)

such that

{\bar{K}}_{i} = K_{i} + Δ K_{i} (t) = (I_{m} + β_{i} ϕ (t)) K_{i}

(14)

where $ε = x_{e} - x$ is the tracking error, $K_{i} \in R^{m \times n}$ is the nominal controller gain, $β_{i} ϕ (t) K_{i}$ represents controller gain variations, $β_{i}$ is a scalar to be designed, and $ϕ (t) \in R^{m \times m}$ is defined as $ϕ^{T} (t) ϕ (t) \leq I_{m}$ .

The final fuzzy controller is

u = - \sum_{i = 1}^{r} h_{i} (z) {\bar{K}}_{i} ε

(15)

By substituting control law equation (15) into state-space equations (6) and (10), the dynamic of the tracking error is described as follows

\overset{\cdot}{ε} = \sum_{i = 1}^{r} \sum_{j = 1}^{r} h_{i} (z) h_{j} (z) (({\bar{A}}_{i} + B_{i} {\bar{K}}_{j}) ε + (A_{e} - {\bar{A}}_{i}) x_{e}) - w + B_{e} e

(16)

Furthermore, equations (10) and (16) lead to the following augmented system

{\overset{\cdot}{x}}_{a} = \sum_{i = 1}^{r} \sum_{j = 1}^{r} h_{i} (z) h_{j} (z) ({\bar{ϒ}}_{ij} x_{a} + Φ φ)

(17)

where $x_{a} = (\begin{matrix} ε \\ x_{e} \end{matrix})$ , $φ = (\begin{matrix} w \\ e \end{matrix})$ , ${\bar{ϒ}}_{ij} = (\begin{matrix} {\bar{A}}_{i} + B_{i} {\bar{K}}_{j} & A_{e} - {\bar{A}}_{i} \\ 0 & A_{e} \end{matrix})$ , and $Φ = (\begin{matrix} - I_{n} & B_{e} \\ 0 & B_{e} \end{matrix})$ .

Mainly, the objective is to determine the gains ${\bar{K}}_{i}$ , $i = 1, 2, \dots, r$ that guarantee the asymptotic stability of the augmented system (equation (17)) and ensure the $H_{\infty}$ tracking control performance

‖ T_{x_{a} φ} ‖_{\infty} = sup_{{‖ φ ‖}_{2} \neq 0} \frac{{‖ x_{a} ‖}_{2}}{{‖ φ ‖}_{2}} < γ

(18)

where $γ > 0$ is a prescribed disturbances attenuation level.

In the following, sufficient stability conditions for the existence of nonfragile $H_{\infty}$ controllers are provided.

Theorem 1. The augmented system (equation (17)) is globally asymptotically stable with a decay rate $α > 0$ and a guaranteed tracking control performance (equation (18)) if there exist common symmetric positive definite matrices $X_{1} \in R^{n \times n}$ and $P_{2} \in R^{n \times n}$ , matrices $M_{i} \in R^{m \times n}$ , and positive scalars $τ_{1}$ , $τ_{2}$ , $τ_{3}$ , $δ_{1} = τ_{1}^{- 1} + τ_{2}^{- 1}$ , and $β_{i}$ , $i = 1, 2, \dots, r$ , verifying the LMI formulation

\begin{matrix} \underset{X_{1}, P_{2}, M_{i}, γ}{maximize} α \\ subject to : \\ (\begin{matrix} L M_{1} & * \\ L M_{2} & L M_{3} \end{matrix}) < 0, i, j = 1, 2, \dots, r \end{matrix}

(19)

where

$L M_{1} = (\begin{matrix} θ_{ij}^{11} & * & * & * \\ X_{1} (A_{e}^{T} - A_{i}^{T}) & θ_{i}^{22} & * & * \\ - X_{1} & 0 & - γ^{2} I & * \\ X_{1} B_{e}^{T} & B_{e}^{T} P_{2} & 0 & - γ^{2} I \end{matrix})$

$θ_{ij}^{11} = A_{i} X_{1} + B_{i} M_{j} + (*) + 2 α X_{1} + δ_{1} H_{i} H_{i}^{T}$

$θ_{i}^{22} = P_{2} A_{e} + (*) + 2 α P_{2} + τ_{2} E_{Ai}^{T} E_{Ai}$

$L M_{2} = (\begin{matrix} E_{Ai} X_{1} & 0 & 0 & 0 \\ M_{j} & 0 & 0 & 0 \\ β_{j} B_{i}^{T} & 0 & 0 & 0 \\ X_{1} & 0 & 0 & 0 \end{matrix})$

$L M_{3} = (\begin{matrix} - τ_{1}^{- 1} I & 0 & 0 & 0 \\ 0 & - τ_{3}^{- 1} I & 0 & 0 \\ 0 & 0 & - τ_{3} I & 0 \\ 0 & 0 & 0 & - Q^{- 1} \end{matrix})$

Hence, if LMI constraints (equation (19)) are satisfied for $i, j = 1, 2, \dots, r$ , then the tracking gains are calulated as follows

K_{i} = M_{i} X_{1}^{- 1}

(20)

Proof. To prove Theorem 1, the following Lyapunov function is considered for augmented system (equation (17))

V (x_{a}) = x_{a}^{T} P x_{a}

(21)

subject to the rapidity performance

\overset{\cdot}{V} (x_{a}) \leq - 2 α V (x_{a})

(22)

where the derivative of the Lyapunov function is $\overset{\cdot}{V} (x_{a}) = {\overset{\cdot}{x}}_{a}^{T} P x_{a} + x_{a}^{T} P {\overset{\cdot}{x}}_{a}$ , $P = P^{T} > 0$ , and $α > 0$ . Thus, the closed-loop system (17) is globally asymptotically stable if

\overset{\cdot}{V} (x_{a}) + 2 α V (x_{a}) + x_{a}^{T} Q^{+} x_{a} - γ^{2} φ^{T} φ < 0

(23)

where $Q^{+} = (\begin{matrix} Q & 0 \\ 0 & 0 \end{matrix})$ , $Q \in R^{n \times n}$ is a positive definite weighting matrix verifying, for a minimum-time $t_{f}$ , the following optimal control problem

\int_{0}^{t_{f}} ε^{T} Q ε dt \leq γ^{2} \int_{0}^{t_{f}} (w^{T} w + e^{T} e) dt

(24)

From equations (17), (21), and (22), equation (23) is equivalent to

\sum_{i = 1}^{r} \sum_{j = 1}^{r} h_{i} (z) h_{j} (z) x_{a}^{T} {\bar{Ω}}_{ij} x_{a} < 0

(25)

where ${\bar{Ω}}_{ij} = (\begin{matrix} P {\bar{ϒ}}_{ij} + (*) + 2 α P + Q^{+} & * \\ Φ^{T} P & - γ^{2} I \end{matrix})$ .

As all nonlinearities $h_{i} (z) \in [0, 1]$ , equation (25) allows writing ${\bar{Ω}}_{ij} < 0$ , $i, j = 1, 2, \dots, r$ . Hereafter, by assuming that $P = diag (P_{1} P_{2})$ , it becomes

(\begin{matrix} Λ_{ij}^{11} + Δ Λ_{ij}^{11} (t) & * & * & * \\ Λ_{i}^{21} + Δ Λ_{i}^{21} (t) & Λ^{22} & * & * \\ - P_{1} & 0 & - γ^{2} I & * \\ B_{e}^{T} P_{1} & B_{e}^{T} P_{2} & 0 & - γ^{2} I \end{matrix}) < 0

(26)

where $Λ_{ij}^{11} = P_{1} (A_{i} + B_{i} K_{j}) + (*) + 2 α P_{1} + Q$ , $Λ_{i}^{21} = (A_{e}^{T} - A_{i}^{T}) P_{1}$ , $Λ^{22} = P_{2} A_{e} + (*) + 2 α P_{2}$ , $Δ Λ_{ij}^{11} (t) = P_{1} (Δ A_{i} (t) + B_{i} Δ K_{j} (t)) + (*)$ , and $Δ Λ_{i}^{21} (t) = - Δ A_{i}^{T} (t) P_{1}$ .

However, it is clear that equation (26) contains certain parts $ϒ_{ij}$ and uncertain ones $Δ ϒ_{ij} (t)$ . This leads to transform it into the form

ϒ_{ij} + Δ ϒ_{ij} (t) < 0

(27)

where $ϒ_{ij} = (\begin{matrix} Λ_{ij}^{11} & * & * & * \\ Λ_{i}^{21} & Λ^{22} & * & * \\ - P_{1} & 0 & - γ^{2} I & * \\ B_{e}^{T} P_{1} & B_{e}^{T} P_{2} & 0 & - γ^{2} I \end{matrix})$ and $Δ ϒ_{ij} (t) = (\begin{matrix} Δ Λ_{ij}^{11} (t) & * & 0 & 0 \\ Δ Λ_{i}^{21} (t) & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix})$ .

After replacing $Δ A_{i} (t)$ and $Δ K_{j} (t)$ by their respective expressions $H_{i} ψ (t) E_{Ai}$ and $β_{j} ϕ (t) K_{j}$ , $Δ ϒ_{ij} (t)$ is equal to

Δ ϒ_{ij} (t) = (\begin{matrix} P_{1} H_{i} ψ (t) E_{Ai} + β_{j} P_{1} B_{i} ϕ (t) K_{j} + (*) & * & 0 & 0 \\ - E_{Ai}^{T} ψ^{T} (t) H_{i}^{T} P_{1} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix})

(28)

As equation (28) contains anti-diagonal terms, the Lemma 1 is used to transform them into diagonal ones. Besides, for positive scalars $τ_{1}$ , $τ_{2}$ , $τ_{3}$ , and $δ_{1} = τ_{1}^{- 1} + τ_{2}^{- 1}$ , it can be transformed as below

Δ ϒ_{ij} (t) \leq (\begin{matrix} ξ_{ij}^{11} & 0 & 0 & 0 \\ 0 & τ_{2} E_{Ai}^{T} E_{Ai} & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix})

(29)

where $ξ_{ij}^{11} = τ_{1} E_{Ai}^{T} E_{Ai} + δ_{1} P_{1} H_{i} H_{i}^{T} P_{1} + τ_{3} β_{j}^{2} P_{1} B_{i} B_{i}^{T} P_{1} + τ_{3}^{- 1} K_{j}^{T} K_{j}$ .

As a consequence, equation (27) is rewritten as

(\begin{matrix} Λ_{ij}^{11} + ξ_{ij}^{11} & * & * & * \\ Λ_{i}^{21} & ρ_{i}^{22} & * & * \\ - P_{1} & 0 & - γ^{2} I & * \\ B_{e}^{T} P_{1} & B_{e}^{T} P_{2} & 0 & - γ^{2} I \end{matrix}) < 0

(30)

where $ρ_{i}^{22} = P_{2} A_{2} + (*) + 2 α P_{2} + τ_{2} E_{Ai}^{T} E_{Ai}$ .

Assuming that $X_{1} = {P_{1}}^{- 1}$ and $M_{i} = K_{i} X_{1}$ , and left and right multiplying equation (30) by positive definite matrix $diag (X_{1}, I, I, I)$ , the condition above becomes

(\begin{matrix} ℓ_{ij}^{11} & * & * & * \\ X_{1} (A_{e}^{T} - A_{i}^{T}) & ρ_{i}^{22} & * & * \\ - X_{1} & 0 & - γ^{2} I & * \\ X_{1} B_{e}^{T} & B_{e}^{T} P_{2} & 0 & - γ^{2} I \end{matrix}) < 0

(31)

where $ρ_{ij}^{11} = ρ_{ij}^{11, a} + ρ_{ij}^{11, b}$ , $ρ_{ij}^{11, a} = A_{i} X_{1} + B_{i} M_{j} + (*) + 2 α X_{1} + δ_{1} H_{i} H_{i}^{T}$ , and $ρ_{ij}^{11, b} = τ_{1} X_{1} E_{Ai}^{T} E_{Ai} X_{1} + τ_{3} M_{j}^{T} M_{j} + τ_{3}^{- 1} β_{j}^{2} B_{i} B_{i}^{T} + X_{1} Q X_{1}$ .

By applying the Schur Complement, as is illustrated in Lemma 2, nonlinear matrix equation (31) is transformed as follows

(\begin{matrix} ρ_{ij}^{11, a} & * & * & * & * & * & * & * \\ X_{1} (A_{e}^{T} - A_{i}^{T}) & ρ_{i}^{22} & * & * & * & * & * & * \\ - X_{1} & 0 & - γ^{2} I & * & * & * & * & * \\ X_{1} B_{e}^{T} & B_{e}^{T} P_{2} & 0 & - γ^{2} I & * & * & * & * \\ E_{Ai} X_{1} & 0 & 0 & 0 & - γ_{1}^{- 1} I & * & * & * \\ M_{j} & 0 & 0 & 0 & 0 & - γ_{3}^{- 1} I & * & * \\ β_{j} B_{i}^{T} & 0 & 0 & 0 & 0 & 0 & - γ_{3} I & * \\ X_{1} & 0 & 0 & 0 & 0 & 0 & 0 & - Q^{- 1} \end{matrix}) < 0

(32)

for $i, j = 1, 2, \dots, r$ . Therefore, the largest lower bound on the decay rate $α > 0$ can be found by solving the optimization problem involving the LMI constraints (equation (19)).

The bilinear T-S FSUD case

In this study, the states of system (7) are assumed to be measurable and the $r -$ pairs $(A_{i}, B_{i})$ are locally controllable. Extending the design of bilinear fuzzy T-S systems control in Tsai and Li (2008) and Tsai et al. (2012), the ith control law for each fuzzy plant rule (4) is transformed as follows

Control rule i

\begin{matrix} If (z_{1} is M_{1 i}) and (z_{2} is M_{2 i}) and \dots and (z_{k} is M_{ki}) \\ then u = ρ \frac{{\bar{L}}_{i} ε}{\sqrt{1 + ε^{T} {\bar{L}}_{i}^{T} {\bar{L}}_{i} ε}} \end{matrix}

(33)

such that

{\bar{L}}_{i} = L_{i} + Δ L_{i} (t) = (I_{m} + η_{i} ζ (t)) L_{i}

(34)

where $ε = x_{e} - x$ is the tracking error, $L_{i} \in R^{m \times n}$ is the nominal controller gain, $η_{i} ζ (t) L_{i}$ represents controller gain variations, $η_{i}$ is a scalar to be designed, and $ζ (t) \in R^{m \times m}$ is defined as $ζ^{T} (t) ζ (t) \leq I_{m}$ .

The final fuzzy controller is

\begin{matrix} u = \sum_{i = 1}^{r} h_{i} (z) ρ \frac{{\bar{L}}_{i} ε}{\sqrt{1 + ε^{T} {\bar{L}}_{i}^{T} {\bar{L}}_{i} ε}} \\ = \sum_{i = 1}^{r} h_{i} (z) ρ \sin (θ_{i}) \\ = \sum_{i = 1}^{r} h_{i} (z) ρ {\bar{L}}_{i} ε \cos (θ_{i}) \end{matrix}

(35)

where $\sin (θ_{i}) = \frac{{\bar{L}}_{i} ε}{\sqrt{1 + ε^{T} {\bar{L}}_{i}^{T} {\bar{L}}_{i} ε}}$ , $\cos (θ_{i}) = \frac{1}{\sqrt{1 + ε^{T} {\bar{L}}_{i}^{T} {\bar{L}}_{i} ε}}$ , and $θ_{i} \in [- \frac{π}{2}, \frac{π}{2}]$ .

By substituting control law (equation (35)) into state-space equations (7) and (10), the dynamic of the tracking error is described as follows

\overset{\cdot}{ε} = \sum_{i = 1}^{r} \sum_{j = 1}^{r} h_{i} (z) h_{j} (z) ({\bar{F}}_{ij} ε + (A_{e} - {\bar{G}}_{ij}) x_{e}) - w + B_{e} e

(36)

where ${\bar{F}}_{ij} = {\bar{A}}_{i} - ρ B_{i} {\bar{L}}_{j} \cos (θ_{j}) + ρ {\bar{N}}_{i} \sin (θ_{j})$ and ${\bar{G}}_{ij} = {\bar{A}}_{i} + ρ {\bar{N}}_{i} \sin (θ_{j})$ .

Furthermore, equations (10) and (36) lead to the following augmented system

{\overset{\cdot}{x}}_{b} = \sum_{i = 1}^{r} \sum_{j = 1}^{r} h_{i} (z) h_{j} (z) ({\bar{ϒ}}_{ij} x_{b} + Φ φ)

(37)

where $x_{b} = (\begin{matrix} ε \\ x_{e} \end{matrix})$ , $φ = (\begin{matrix} w \\ e \end{matrix})$ , ${\bar{ϒ}}_{ij} = (\begin{matrix} {\bar{F}}_{ij} & A_{e} - {\bar{G}}_{ij} \\ 0 & A_{e} \end{matrix})$ , and $Φ = (\begin{matrix} - I_{n} & B_{e} \\ 0 & B_{e} \end{matrix})$ .

Mainly, the control objective is to determine the gains ${\bar{L}}_{i}$ , $i = 1, 2, \dots, r$ that guarantee the asymptotic stability of the augmented system (equation (37)) and ensure the $H_{\infty}$ tracking control performance

‖ H_{x_{b} φ} ‖_{\infty} = sup_{{‖ φ ‖}_{2} \neq 0} \frac{{‖ x_{b} ‖}_{2}}{{‖ φ ‖}_{2}} < γ

(38)

where $γ > 0$ is a prescribed disturbances attenuation level. Consequently, Theorem 2 presents the result.

Theorem 2. The augmented system (37) is globally asymptotically stable with a decay rate $α > 0$ and a guaranteed $H_{\infty}$ tracking control performance (38) if there exist common symmetric positive definite matrices $P_{1} \in R^{n \times n}$ and $P_{2} \in R^{n \times n}$ , vectors $L_{i} \in R^{m \times n}$ , and positive scalars $κ_{1}$ , $κ_{2}$ , $κ_{3}$ , $κ_{4}$ , $δ_{2} = κ_{1}^{- 1} + κ_{2}^{- 1}$ , $δ_{3} = κ_{3}^{- 1} + κ_{4}^{- 1}$ , $ρ$ , and $η_{i}$ , $i = 1, 2, \dots, r$ , verifying the LMI formulation

\begin{matrix} \underset{P_{1}, P_{2}, L_{i}, γ}{maximize} α \\ subject to : \\ (\begin{matrix} L M_{4} & * \\ L M_{5} & L M_{6} \end{matrix}) < 0, i, j = 1, 2, \dots, r \end{matrix}

(39)

where

$L M_{4} = (\begin{matrix} ℓ_{4 ij}^{11} & * & * & * \\ {(A_{e} - A_{i})}^{T} P_{1} & ℓ_{4 i}^{22} & * & * \\ - P_{1} & 0 & - γ^{2} I & * \\ B_{e}^{T} P_{1} & B_{e}^{T} P_{2} & 0 & - γ^{2} I \end{matrix})$

$ℓ_{4 ij}^{11} = P_{1} A_{i} + (*) + Q + 2 α P_{1} + κ_{1} N_{i}^{T} N_{i} + κ_{3} (E_{Ai}^{T} E_{Ai} + E_{Ni}^{T} E_{Ni})$

$ℓ_{4 i}^{22} = P_{2} A_{e} + (*) + 2 α P_{2} + κ_{2} N_{i}^{T} N_{i} + κ_{4} (E_{Ai}^{T} E_{Ai} + E_{Ni}^{T} E_{Ni})$

$L M_{5} = (\begin{matrix} B_{i} L_{j} & 0 & 0 & 0 \\ ρ P_{1} & 0 & 0 & 0 \\ L_{j} & 0 & 0 & 0 \\ \sqrt{1 + ρ^{2}} H_{i}^{T} P_{1} & 0 & 0 & 0 \\ ρ η_{j} B_{i}^{T} P_{1} & 0 & 0 & 0 \end{matrix})$

$L M_{6} = (\begin{matrix} - κ_{1}^{- 1} I & 0 & 0 & 0 & 0 \\ 0 & - δ_{2}^{- 1} I & 0 & 0 & 0 \\ 0 & 0 & - κ_{3}^{- 1} I & 0 & 0 \\ 0 & 0 & 0 & - δ_{3}^{- 1} I & 0 \\ 0 & 0 & 0 & 0 & - κ_{3} I \end{matrix})$

Proof. To prove Theorem 2, the following Lyapunov function is considered for augmented system (37)

V (x_{b}) = x_{b}^{T} P x_{b}

(40)

subject to the rapidity performance

\overset{\cdot}{V} (x_{b}) \leq - 2 α V (x_{b})

(41)

where $\overset{\cdot}{V} (x_{b}) = {\overset{\cdot}{x}}_{b}^{T} P x_{b} + x_{b}^{T} P {\overset{\cdot}{x}}_{b}$ , $P = P^{T} > 0$ , and $α > 0$ . However, the controlled system (37) is globally asymptotically stable with decay rate $α > 0$ if

\overset{\cdot}{V} (x_{b}) + 2 α V (x_{b}) + x_{b}^{T} Q^{+} x_{b} - γ^{2} φ^{T} φ < 0

(42)

From equations (37), (40), and (41), the above condition is equivalent to

\sum_{i = 1}^{r} \sum_{j = 1}^{r} h_{i} (z) h_{j} (z) x_{b}^{T} {\bar{Δ}}_{ij} x_{b} < 0

(43)

where ${\bar{Δ}}_{ij} = (\begin{matrix} P {\bar{ϒ}}_{ij} + (*) + 2 α P + Q^{+} & * \\ Φ^{T} P & - γ^{2} I \end{matrix})$ .

As all nonlinearities $h_{i} (z) \in [0, 1]$ , equation (43) allows writing ${\bar{Δ}}_{ij} < 0$ , $i, j = 1, 2, \dots, r$ . Besides, by assuming that $P = diag (P_{1} P_{2})$ , it becomes

(\begin{matrix} χ_{ij}^{11} + Δ χ_{ij}^{11} (t) & * & * & * \\ χ_{ij}^{21} + Δ χ_{ij}^{21} (t) & χ^{22} & * & * \\ - P_{1} & 0 & - γ^{2} I & * \\ B_{e}^{T} P_{1} & B_{e}^{T} P_{2} & 0 & - γ^{2} I \end{matrix}) < 0

(44)

where

$χ_{ij}^{11} = (P_{1} A_{i} - ρ P_{1} B_{i} L_{j} \cos (θ_{j}) + ρ P_{1} N_{i} \sin (θ_{j})) + (*) + 2 α P_{1} + Q$ ;

$χ_{ij}^{21} = (A_{e}^{T} - A_{i}^{T}) P_{1} - ρ N_{i}^{T} P_{1} \sin (θ_{j})$ ;

$χ^{22} = P_{2} A_{e} + (*) + 2 α P_{2}$ ;

$Δ χ_{ij}^{11} (t) = (P_{1} Δ A_{i} (t) - ρ P_{1} B_{i} Δ L_{j} (t) \cos (θ_{j}) + ρ P_{1} Δ N_{i} (t) \sin (θ_{j})) + (*)$ ;

$Δ χ_{ij}^{21} (t) = - Δ A_{i}^{T} (t) P_{1} - ρ Δ N_{i}^{T} (t) P_{1} \sin (θ_{j})$ .

Furthermore, it is pointing out that equation (44) contains certain parts $Θ_{ij}$ and uncertain ones $Δ Θ_{ij} (t)$ . This leads to transform it into the following form

Θ_{ij} + Δ Θ_{ij} (t) < 0

(45)

where $Θ_{ij} = (\begin{matrix} χ_{ij}^{11} & * & * & * \\ χ_{i}^{21} & χ^{22} & * & * \\ - P_{1} & 0 & - γ^{2} I & * \\ B_{e}^{T} P_{1} & B_{e}^{T} P_{2} & 0 & - γ^{2} I \end{matrix})$ and $Δ Θ_{ij} (t) = (\begin{matrix} Δ χ_{ij}^{11} (t) & * & 0 & 0 \\ Δ χ_{ij}^{21} (t) & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix})$ .

Next, by replacing $Δ A_{i} (t)$ , $Δ L_{j} (t)$ , and $Δ N_{i} (t)$ by their respective expressions $H_{i} ψ (t) E_{Ai}$ , $η_{j} ζ (t) L_{j}$ , and $H_{i} ψ (t) E_{Ni}$ , and using Lemma 1 for positive scalars $κ_{1}$ , $κ_{2}$ , $κ_{3}$ , $κ_{4}$ , $δ_{2} = κ_{1}^{- 1} + κ_{2}^{- 1}$ , and $δ_{3} = κ_{3}^{- 1} + κ_{4}^{- 1}$ , it becomes

Θ_{ij} \leq (\begin{matrix} c_{ij}^{11} & * & * & * \\ {(A_{e} - A_{i})}^{T} P_{1} & c_{i}^{22} & 0 & * \\ - P_{1} & 0 & - γ^{2} I & 0 \\ B_{e}^{T} P_{1} & B_{e}^{T} P_{2} & 0 & - γ^{2} I \end{matrix})

(46)

Δ Θ_{ij} (t) \leq (\begin{matrix} d_{ij}^{11} & 0 & 0 & 0 \\ 0 & d_{i}^{22} & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix})

(47)

where

$c_{ij}^{11} = c_{ij}^{11, a} + c_{ij}^{11, b}$ ;

$c_{ij}^{11, a} = P_{1} A_{i} + (*) + 2 α P_{1} + Q$ ;

$c_{ij}^{11, b} = κ_{1} N_{i}^{T} N_{i} + κ_{1} L_{j}^{T} B_{i}^{T} B_{i} L_{j} + δ_{2} ρ^{2} P_{1}^{2}$ ;

$c_{i}^{22} = P_{2} A_{e} + (*) + 2 α P_{2} + κ_{2} N_{i}^{T} N_{i}$ ;

$d_{ij}^{11} = d_{ij}^{11, a} + d_{ij}^{11, b}$ ;

$d_{ij}^{11, a} = κ_{3} E_{Ai}^{T} E_{Ai} + κ_{3} E_{Ni}^{T} E_{Ni}$ ;

$d_{ij}^{11, b} = κ_{3} L_{j}^{T} L_{j} + δ_{3} (1 + ρ^{2}) P_{1} H_{i} H_{i}^{T} P_{1} + κ_{3}^{- 1} ρ^{2} η_{j}^{2} P_{1} B_{i} B_{i}^{T} P_{1}$ ;

$d_{i}^{22} = κ_{4} E_{Ai}^{T} E_{Ai} + κ_{4} E_{Ni}^{T} E_{Ni}$ .

From equations (46) and (47), equation (45) is equivalent to

(\begin{matrix} χ_{ij}^{11} & * & * & * \\ (A_{e}^{T} - A_{i}^{T}) P_{1} & χ_{i}^{22} & * & * \\ - P_{1} & 0 & - γ^{2} I & * \\ B_{e}^{T} P_{1} & B_{e}^{T} P_{2} & 0 & - γ^{2} I \end{matrix}) < 0

(48)

where

$χ_{ij}^{11} = χ_{ij}^{11, a} + χ_{ij}^{11, b} + χ_{ij}^{11, c}$ ;

$χ_{ij}^{11, a} = P_{1} A_{i} + (*) + 2 α P_{1} + Q + κ_{1} N_{i}^{T} N_{i} + κ_{3} (E_{Ai}^{T} E_{Ai} + E_{Ni}^{T} E_{Ni})$ ;

$χ_{ij}^{11, b} = κ_{1} L_{j}^{T} B_{i}^{T} B_{i} L_{j} + δ_{2} ρ^{2} P_{1}^{2} + κ_{3} L_{j}^{T} L_{j}$ ;

$χ_{ij}^{11, c} = δ_{3} (1 + ρ^{2}) P_{1} H_{i} H_{i}^{T} P_{1} + κ_{3}^{- 1} ρ^{2} η_{j}^{2} P_{1} B_{i} B_{i}^{T} P_{1}$ ;

$χ_{i}^{22} = P_{2} A_{e} + (*) + 2 α P_{2} + κ_{2} N_{i}^{T} N_{i} + κ_{4} (E_{Ai}^{T} E_{Ai} + E_{Ni}^{T} E_{Ni})$ .

By applying the Schur Complement, the above nonlinear matrix inequality is transformed as follows

(\begin{matrix} χ_{ij}^{11, a} & * & * & * & * & * & * & * & * \\ (A_{e}^{T} - A_{i}^{T}) P_{1} & χ_{i}^{22} & * & * & * & * & * & * & * \\ - P_{1} & 0 & - γ^{2} I & * & * & * & * & * & * \\ B_{e}^{T} P_{1} & B_{e}^{T} P_{2} & 0 & - γ^{2} I & * & * & * & * & * \\ B_{i} L_{j} & 0 & 0 & 0 & - κ_{1}^{- 1} I & * & * & * & * \\ ρ P_{1} & 0 & 0 & 0 & 0 & - δ_{2}^{- 1} I & * & * & * \\ L_{j} & 0 & 0 & 0 & 0 & 0 & - κ_{3}^{- 1} I & * & * \\ \sqrt{1 + ρ^{2}} H_{i}^{T} P_{1} & 0 & 0 & 0 & 0 & 0 & 0 & - δ_{3}^{- 1} I & * \\ ρ η_{j} B_{i}^{T} P_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - κ_{3} I \end{matrix}) < 0

(49)

for $i, j = 1, 2, \dots, r$ . Therefore, the largest lower bound on the decay rate $α > 0$ can be found by solving the optimization problem involving the LMI constraints (39).

Numerical examples

In this section, the objective is to evaluate the applicability and effectiveness of the designed nonfragile $H_{\infty}$ controllers.

Tracking control of an inverted pendulum

The studied inverted pendulum is described by the nonlinear equations Srinivasan et al. (2009)

{\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} \\ {\overset{\cdot}{x}}_{2} = \frac{g \sin (x_{1}) - a m l x_{2}^{2} \sin (x_{1}) \cos (x_{1}) - a u}{\frac{4}{3} l - a m l} \\ y = x_{1} \end{matrix}

(50)

where $x_{1} = θ$ is the angle in $rad$ of the pendulum from the vertical, $x_{2} = \overset{\cdot}{θ}$ is the angular velocity in $rad / s$ , $g = 9.81 m / s^{2}$ is the constant gravity, $M = 1.6 kg$ is the mass of the cart, $m = 0.3 kg$ is the mass of the pendulum, $2 l = 0.8 m$ is the length of the pendulum, and $a = 1 / (m + M)$ .

For $x_{1} \in [- x_{10}, x_{10}]$ , $v_{1} = max (x_{2}^{2})$ , and $| ψ (t) | \leq 1$ , the nonlinearities $x_{1} \mapsto \sin (x_{1})$ , $x_{1} \mapsto \cos (x_{1})$ , and $x_{2} \mapsto x_{2}^{2}$ are decomposed as follows

\sin (x_{1}) = M_{11} (x_{1}) x_{1} + M_{12} (x_{1}) \frac{\sin (x_{10})}{x_{10}} x_{1}

(51)

\cos (x_{1}) = M_{21} (x_{1}) 1 + M_{22} (x_{1}) \cos (x_{10})

(52)

x_{2}^{2} = 0 + v_{1} ψ (t)

(53)

where $M_{11} (x_{1}) = \frac{x_{10} \sin (x_{1}) - x_{1} \sin (x_{10})}{x_{1} (x_{10} - \sin (x_{10}))}$ , $M_{12} (x_{1}) = \frac{x_{10} x_{1} - x_{10} \sin (x_{1})}{x_{1} (x_{10} - \sin (x_{10}))}$ , $M_{21} (x_{1}) = \frac{\cos (x_{1}) - \cos (x_{10})}{1 - \cos (x_{10})}$ , and $M_{22} (x_{1}) = \frac{1 - \cos (x_{1})}{1 - \cos (x_{10})}$ .

Figure 1 depicts the evolutions of the membership functions $M_{11} (x_{1})$ , $M_{12} (x_{1})$ , $M_{21} (x_{1})$ , and $M_{22} (x_{1})$ .

As membership functions $M_{11} (x_{1})$ and $M_{12} (x_{1})$ are, respectively, superimposed on $M_{21} (x_{1})$ and $M_{22} (x_{1})$ , the considered system can be described by the following two fuzzy rules:

If (x_{1} is M_{11} (x_{1})) then \overset{\cdot}{x} = {\bar{A}}_{1} x + B_{1} u + w

(54)

If (x_{1} is M_{12} (x_{1})) then \overset{\cdot}{x} = {\bar{A}}_{2} x + B_{2} u + w

(55)

where $x \in R^{2}$ , $u \in R$ , $w \in R^{2}$ ,

\begin{matrix} A_{1} = (\begin{matrix} 0 & 1 \\ \frac{g}{\frac{4}{3} l - aml} & 0 \end{matrix}), A_{2} = (\begin{matrix} 0 & 1 \\ \frac{g \sin (x_{10})}{x_{10} (\frac{4}{3} l - aml)} & 0 \end{matrix}), \\ B_{1} = B_{2} = (\begin{matrix} 0 \\ - \frac{a}{\frac{4}{3} l - aml} \end{matrix}), \end{matrix}

Δ A_{1} (t) = (\begin{matrix} 0 \\ - \frac{aml}{\frac{4}{3} l - aml} \end{matrix}) ψ (t) (\begin{matrix} v_{1} & 0 \end{matrix}),

Δ A_{2} (t) = (\begin{matrix} 0 \\ - \frac{aml \sin (2 x_{10})}{2 x_{10} (\frac{4}{3} l - aml)} \end{matrix}) ψ (t) (\begin{matrix} v_{1} & 0 \end{matrix}),

ψ (t) = \sin (t), v_{1} = 0.5, w (t) = (\begin{matrix} 0.04 \sin (t) \\ 0.04 \cos (t) \end{matrix}) .

Hence, the final linear T-S fuzzy model is described by

\overset{\cdot}{x} = \sum_{i = 1}^{2} h_{i} (x_{1}) ({\bar{A}}_{i} x + B_{i} u + w)

(56)

where the normalized weighting functions are $h_{1} (x_{1}) = M_{11} (x_{1})$ and $h_{2} (x_{1}) = M_{12} (x_{1})$ .

Using Theorem 1 with $γ = 0.73$ , $τ_{1} = 0.9$ , $τ_{2} = 1.6$ , $τ_{3} = 1.4$ , $β_{1} = 0.85$ , $β_{2} = 0.65$ , $ϕ (t) = \sin (0.4 t)$ , $A_{e} = (\begin{matrix} 0 & 1 \\ - 8 & - 6 \end{matrix})$ , $B_{e} = (\begin{matrix} 0 \\ 1 \end{matrix})$ , and $Q = (\begin{matrix} 0.25 & 0 \\ 0 & 0.01 \end{matrix})$ , the obtained solutions are $α = 0.727$ , $P_{1} = (\begin{matrix} 24.338 & - 3.481 \\ - 3.481 & 12.048 \end{matrix})$ , $P_{2} = (\begin{matrix} 40.630 & - 7.366 \\ - 7.366 & 21.627 \end{matrix})$ , $K_{1} = (\begin{matrix} - 107.601 & 324.881 \end{matrix})$ , and $K_{2} = (\begin{matrix} - 64.037 & 382.167 \end{matrix})$ .

Taking into account these considerations, the designed nonfragile $H_{\infty}$ controller can be expressed as

u = - \sum_{i = 1}^{2} h_{i} (x_{1}) {\bar{K}}_{i} ε

(57)

where $ε = x_{e} - x$ is the tracking error, ${\bar{K}}_{1} = (1 + 0.85 \sin (0.4 t)) K_{1}$ , and ${\bar{K}}_{2} = (1 + 0.65 \sin (0.4 t)) K_{2}$ .

Figures 2 and 3 depict, respectively, the tracking trajectories of the angle and the angular velocity under initial conditions $x (0) = {(\begin{matrix} 0.027 & - 0.12 \end{matrix})}^{T}$ and $x_{e} (0) = {(\begin{matrix} 0.015 & - 0.04 \end{matrix})}^{T}$ . Figure 4 shows the quadratic error signal, which is calculated by $ε_{quad} (t) = (1 / 2) \sum_{i = 1}^{2} {(x_{ei} (t) - x_{i} (t))}^{2}$ , whereas Figure 5 exhibits the control signal.

From these simulation results, the state variables of the controlled system, which is described as a linear T-S FSUD, follow perfectly the desired ones of the reference model. Also, the magnitude of the quadratic error tracking is very small in the entire operating region and its trajectory is close to zero. In light of this, the designed nonfragile $H_{\infty}$ controller (57) is effective and ensures a good tracking control despite the presence of uncertainties, controller nonfragility variations, and disturbances.

Tracking control of a mass-spring-damper mechanical system

In this section, a bilinear T-S FSUD is designed for a nonlinear mass-spring-damper mechanical system, which is described by Ghorbel and Benhadj Braiek (2021)

\overset{\cdot\cdot}{y} + g (y, \overset{\cdot}{y}) = h (y, \overset{\cdot}{y}) u

(58)

where

$g (y, \overset{\cdot}{y}) = 3.5 y + 2.8 \sin (\overset{\cdot}{y}) - 0.65 {\overset{\cdot}{y}}^{2} \sin (\overset{\cdot}{y})$ ;

$h (y, \overset{\cdot}{y}) = 1 + 3 \overset{\cdot}{y} - 0.85 {\overset{\cdot}{y}}^{2} \sin (\overset{\cdot}{y})$ .

For $x_{1} = y$ , $x_{2} = \overset{\cdot}{y}$ , $x_{2} \in [- {x_{2}}_{0}, {x_{2}}_{0}]$ , $v_{2} = max (x_{2}^{2})$ , and $| ψ (t) | < 1$ , the nonlinearities $x_{2} \to \sin (x_{2})$ and $x_{2} \to {x_{2}}^{2}$ can be decomposed as follows

\sin (x_{2}) = M_{11} (x_{2}) x_{2} + M_{12} (x_{2}) \frac{\sin ({x_{2}}_{0})}{{x_{2}}_{0}} x_{2}

(59)

{x_{2}}^{2} = 0 + v_{2} ψ (t)

(60)

where $M_{11} (x_{2}) = \frac{{x_{2}}_{0} \sin (x_{2}) - x_{2} \sin ({x_{2}}_{0})}{x_{2} ({x_{2}}_{0} - \sin ({x_{2}}_{0}))}$ and $M_{12} (x_{2}) = \frac{{x_{2}}_{0} x_{2} - {x_{2}}_{0} \sin (x_{2})}{x_{2} ({x_{2}}_{0} - \sin ({x_{2}}_{0}))}$ .

Figure 6 depicts the evolution of the membership functions $M_{11} (x_{2})$ and $M_{12} (x_{2})$ for $x_{2} \in [- \frac{π}{4}, \frac{π}{4}]$ .

The mechanical system can be described by the following two rules

If (x_{1} is M_{11} (x_{2})) then \overset{\cdot}{x} = {\bar{A}}_{1} x + B_{1} u + {\bar{N}}_{1} x u + w

(61)

If (x_{1} is M_{12} (x_{2})) then \overset{\cdot}{x} = {\bar{A}}_{2} x + B_{2} u + {\bar{N}}_{2} x u + w

(62)

where $A_{1} = (\begin{matrix} 0 & 1 \\ - 3.5 & - 2.8 \end{matrix})$ , $A_{2} = (\begin{matrix} 0 & 1 \\ - 3.5 & - 2.52 \end{matrix})$ ,

B_{1} = B_{2} = (\begin{matrix} 0 \\ 1 \end{matrix}), N_{1} = N_{2} = (\begin{matrix} 0 & 0 \\ 0 & 3 \end{matrix}),

Δ A_{1} (t) = (\begin{matrix} 0 \\ 0.65 \end{matrix}) ψ (t) (\begin{matrix} 0 & v_{2} \end{matrix}),

Δ A_{2} (t) = (\begin{matrix} 0 \\ 0.01 \end{matrix}) ψ (t) (\begin{matrix} 0 & v_{2} \end{matrix}),

Δ N_{1} (t) = (\begin{matrix} 0 \\ 0.65 \end{matrix}) ψ (t) (\begin{matrix} 0 & - 0.85 v_{2} \end{matrix}),

Δ N_{2} (t) = (\begin{matrix} 0 \\ 0.01 \end{matrix}) ψ (t) (\begin{matrix} 0 & - 0.015 v_{2} \end{matrix}),

ψ (t) = \sin (0.1), v_{2} = 0.62, w (t) = (\begin{matrix} 0.01 \sin (t) \\ 0.01 \cos (t) \end{matrix}) .

Then, the final bilinear T-S fuzzy model is described by

\overset{\cdot}{x} = \sum_{i = 1}^{2} h_{i} (x_{1}) ({\bar{A}}_{i} x + B_{i} u + {\bar{N}}_{i} x u + w)

(63)

where the normalized weighting functions are $h_{1} (x_{2}) = M_{11} (x_{2})$ and $h_{2} (x_{2}) = M_{12} (x_{2})$ .

Using Theorem 2 with $γ = 0.65$ , $κ_{1} = 0.72$ , $κ_{2} = 0.5$ , $κ_{3} = 0.8$ , $κ_{4} = 0.62$ , $ρ = 0.4$ , $η_{1} = 0.9$ , $η_{2} = 0.85$ , $ζ (t) = \sin (0.05 t)$ , $A_{e} = (\begin{matrix} 0 & 1 \\ - 8 & - 6 \end{matrix})$ , $B_{e} = (\begin{matrix} 0 \\ 0 \end{matrix})$ , and $Q = (\begin{matrix} 0.6 & 0 \\ 0 & 0.05 \end{matrix})$ , the obtained solutions are $α = 0.727$ ,

P_{1} = (\begin{matrix} 18.614 & - 2.144 \\ - 2.144 & 13.705 \end{matrix}), P_{2} = (\begin{matrix} 43.713 & - 10.304 \\ - 10.304 & 24.907 \end{matrix}),

L_{1} = (\begin{matrix} - 72.106 & - 16.480 \end{matrix}), L_{2} = (\begin{matrix} - 42.063 & - 28.145 \end{matrix}) .

Taking into account these considerations, the designed nonfragile $H_{\infty}$ controller can be expressed as

u = - \sum_{i = 1}^{2} h_{i} (x_{2}) {\bar{L}}_{i} ε

(64)

where $ε = x_{e} - x$ is the tracking error, ${\bar{L}}_{1} = (1 + 0.9 \sin (0.05 t)) L_{1}$ , and ${\bar{L}}_{2} = (1 + 0.85 \sin (0.05 t)) L_{2}$ .

Figures 7 and 8 depict, respectively, the tracking trajectories of the position and the velocity under initial conditions $x (0) = {(\begin{matrix} - 1.6 & 0.1 \end{matrix})}^{T}$ and $x_{e} (0) = {(\begin{matrix} - 0.83 & 0.08 \end{matrix})}^{T}$ . Figure 9 shows the quadratic error signal, whereas the control signal is given in Figure 10.

As a result, the state variables of the controlled mechanical system, which is described as bilinear T-S FSUD, follow perfectly the desired ones of the reference model. Also, the magnitude of the quadratic error tracking is very small in the entire operating region and its trajectory is close to zero. Consequently, the designed nonfragile $H_{\infty}$ controller (64) is effective and ensures a good tracking control despite the presence of uncertainties, controller nonfragility variations, and disturbances.

Conclusion

This paper has presented nonfragile $H_{\infty}$ controllers for nonlinear systems as described by linear or bilinear T-S fuzzy models with parameter uncertainties and external disturbances. Based on the parallel distributed compensation concept and the Lyapunov direct approach with decay rate, sufficient conditions in terms of linear matrix inequalities that ensure good tracking control objectives have been provided for both classes. However, each designed controller guaranteed the robust asymptotic stability of the augmented system despite the presence of uncertainties, disturbances, and controller gain variations. Finally, the simulation results show the applicability and effectiveness of the proposed control strategies.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Chekib Ghorbel

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Nonfragile H ∞ tracking control strategies for classes of linear and bilinear uncertain Takagi-Sugeno fuzzy systems

Abstract

Keywords

Introduction

Problem formulation and preliminaries

Robust H ∞ and nonfragile tracking control design

The linear T-S FSUD case

The bilinear T-S FSUD case

Numerical examples

Tracking control of an inverted pendulum

Tracking control of a mass-spring-damper mechanical system

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References

Robust $H_{\infty}$ and nonfragile tracking control design