Non-fragile controller design of uncertain saturated polynomial fuzzy systems subjected to persistent bounded disturbance

Abstract

This paper investigates the problem of designing a non-fragile polynomial fuzzy controller for a class of polynomial fuzzy models subjected to persistent bounded disturbances and system uncertainty. The proposed controller is considered to satisfy the input saturation constraint. Furthermore, the resilient controller is affected by linear fractional uncertainties. The overall structure of the controller is obtained by the polynomial parallel distributed compensation concept. The sufficient conditions guarantee an L₁ performance of the perturbed system and exponential stability of the non-perturbed system is achieved in terms of sum-of-squares decomposition conditions. Subsequently, the sum-of-squares-based conditions lead to minimization of the L₁ performance such that the above-mentioned hard constraints on the control input and robustness of the closed-loop system against the system and controller uncertainties will be guaranteed. The controller gain matrices will be achieved by numerically solving the sum-of-squares-based conditions with the third-party MATLAB toolbox ‘SOSTOOLS’. Finally, extensive studies and hardware-in-the-loop simulations on a ball-and-beam system are presented, which illustrate the proposed controller can accurately and smoothly track the set point frequency. Furthermore, the proposed control approach is more robust over prior-art controllers for all the simulation scenarios.

Keywords

Sum-of-squares uncertain polynomial fuzzy models non-fragile controller robust controller input saturation

Introduction

Most of the real-world processes are inherently non-linear. One of the ways to deal with the non-linearity problem of these systems is to utilize the well-known Mamdani or Takagi–Sugeno (TS) fuzzy models (Abadi and Khooban, 2015; Ben Yazid et al., 2017; Dehghani et al., 2016; Khooban et al., 2017). The TS fuzzy models are described by fuzzy blending of some local subsystems (Song et al., 2016). Depending on the consequent part of the fuzzy IF-THEN rules, the TS fuzzy model is called a linear (Zhang et al., 2016), ane (Beyhan, 2017; Wang et al., 2016), bilinear (Chang and Hsu, 2016; Hamdy and Hamdan, 2015) and polynomial (Mardani et al., 2017b; Vafamand et al., 2017b; Zhou and Zeng, 2017) fuzzy model. The simplicity of the TS-based fuzzy control makes this method a very intresting research field (Khooban et al., 2016). Originally, the parallel distributed compensation (PDC) and non-PDC schemes were widely employed for the linear TS fuzzy systems through linear matrix (LMI) techniques (Chang et al., 2012; Dong and Yang, 2017; Mardani et al., 2017a). Recently, polynomial (Mardani et al., 2017b; Vafamand et al., 2017b; Zhou and Zeng, 2017) and polynomial fuzzy (Pitarch et al., 2017) controllers have attracted lots of attention to deal with the polynomial and polynomial fuzzy models. Based on the polynomial fuzzy model approach, sufficient stability conditions are derived in terms of sum-ofsquares (SOS) decomposition conditions (Tanaka et al., 2016). Then, in order to solve SOS-based conditions, the third-party MATLAB toolbox ‘SOSTOOLS’ is used (Sturm, 1999).

From a practical viewpoint, the designed controller must struggle with real world system requirements such as:

System uncertainty (Liu et al., 2016b; Yin et al., 2016, 2017). In most of the practical complex plants, deriving an exact model is a hard task or maybe impossible. Also, some of the system parameters may vary due to physical depreciation. So, considering the system uncertainty in designing a practical controller is an important task.

Controller uncertainty (Tandon and Dhawan, 2016; Wu et al., 2017; Zhang et al., 2007). The designed controller is assumed to be exactly implemented to real-world systems. However, because of several uncertainties and inaccuracies that may happen during the implementation procedure, it is difficult to exactly apply the controller to practical system. These unmodelled behaviours may be caused by implementing digital controller, ageing of the controller components, and numerical computation approximation. Hence, it is necessary to design a controller, which is insensitive to the gain variations. Such a controller is well-known as a non-fragile controller.

Actuator saturation (Vafamand et al., 2016; Zhai and Xia, 2016). Actuator saturation is one of the most usual non-linearity sources. Due to the physical restrictions, applying high amplitude control signal is not applicable in the real world processes. Actuator saturation can severely decrease the closed-loop system performance and sometimes may lead to closed-loop instability. So, it should be considered in the controller design procedure to avoid any performance degradation.

Persistent bounded external disturbance. In several applications, the external disturbance belongs to norm- $\infty$ space, including valve control systems, aircraft flight control systems, shipping navigation control systems, robotic control systems and power motor systems. For such disturbance inputs, the conventional bounded real lemma (Guerra et al., 2017; Liu et al., 2016b) approaches are not applicable and one needs to employ the $L_{\infty}$ performance index.

In recent years, several LMI- and SOS-based approaches have been developed to handle the above-mentioned issues. In the work by Vafamand et al. (2016), the problem of designing a robust $L_{1}$ controller was investigated for the TS fuzzy models. The disturbance and the control saturation were considered and sufficient conditions were derived in terms of LMIs. In the work by Zheng and Wu (2010), a non-fragile controller for polynomial systems was defined. However, the proposed approach suffered from the non-convexity problem and the controller must be designed through an iterative SOS-based numerical technique. In the work by Pitarch et al. (2017), the problem of saturated controller design is formulated in terms of SOS decomposition conditions for partial differential equation systems. This approach was not applicable to ordinary differential equation systems. Tanaka et al. (2016) presented a novel guaranteed controller for uncertain polynomial TS fuzzy models. Furthermore, the amplitude constraint on the control input was guaranteed. However, the work by Tanaka et al. (2016) has some drawbacks. First, the SOS-based conditions were achieved in a non-convex structure. Subsequently, the conditions could not be solved by semi-definite programming. Second, the formulations of the input constraints were derived such that control input signal saturation does not reach the limits. Therefore, the low gain controller designed by Tanaka et al. (2016) degrades the closed-loop performance.

In the work by Vafamand et al. (2017b), a non-iterative input saturation constrained polynomial controller was proposed. Although, the obtained conditions were convex, the problems of external disturbance and a resilient controller were not addressed. Cao et al. (2014) studied a robust controller synthesis of uncertain polynomial fuzzy model. However, in the work by Cao et al. (2014), the sensitivity of the controller against the gain uncertainty and the actuator saturation were not discussed. Unfortunately, the problem of designing a robust polynomial fuzzy controller against the persistent bounded disturbance inputs has not been studied yet. Furthermore, the existing results are confined to LMI-based conditions (Vafamand et al., 2017a,b). According to the best knowledge of the authors, designing a non-fragile $L_{1}$ performance controller with hard constraints for uncertain polynomial fuzzy models in the presence of a persistent bounded disturbance has not been addressed, which is the main idea of this paper.

This paper seeks to provide a framework to design a saturated polynomial $L_{1}$ non-fragile controller for uncertain polynomial fuzzy models subjected to a persistent bounded disturbance. The goal is to improve the performance of the controller in the presence of system and control uncertainty and an exogenous persistent bounded disturbance. Moreover, the $L_{1}$ performance criterion is also investigated in this paper. The sufficient conditions which satisfy the above problems are formulated in terms of an SOS decomposition. The main novelty of the proposed approach is addressing several performances and practical issues (i.e. system uncertainty, non-fragile controller, persistent bounded disturbance and input saturation) in designing the fuzzy controller. In addition, by reviewing the literature it has been revealed that considering each of the mentioned problems in the controller design procedure complicates deriving the SOS conditions so that in the work by Vafamand et al. (2017b) and Zheng and Wu (2010), the problems of non-fragility and input saturation for polynomial systems were derived in non-convex SOS conditions. Also, the $L_{1}$ performance criteria is not investigated for polynomial systems in the literature. However, in the proposed approach such issues are derived in terms of SOS decompositions and also a novel framework is proposed so that the simultaneous consideration of these problems as well as the system uncertainty leads to convex constraints. To perform this, elimination techniques are employed to handle the system uncertainty and non-fragile uncertainty. Also, an S-procedure is used to facilitate formulating the input saturation constraint. Furthermore, a bounded-input-bounded-output (BIBO) scheme is developed to tackle the persistent bounded disturbance inputs. Finally, in order to illustrate the effectiveness and merits of the proposed methods the ball-and-beam system is studied and the experimental results, obtained by the hardware-in-the-loop (HiL) simulation, are compared with the existing methods.

The rest of this paper will be organized as follows: the ‘Preliminaries’ section briefly investigates polynomial fuzzy modelling and the non-fragile polynomial fuzzy controller with hard constraints. Stability analysis of the closed-loop polynomial fuzzy model based (FMB) system will be studied in the ‘Stability analysis’ section. For demonstration of the performance and robustness of the proposed control system, experimental validation using the HiL simulation will be given in the ‘Numerical example’ section. Finally, the conclusions will close the paper in the final section and the ‘Notation’ section can be found in the Appendix.

Preliminaries

Polynomial fuzzy model

Consider the following uncertain polynomial fuzzy model in the presence of persistent bounded external disturbances

{\begin{matrix} \begin{matrix} \begin{matrix} \overset{\cdot}{x} = \sum_{i = 1}^{r} h_{i} (z) {(A_{i} (x) + Δ A_{i} (x)) x \\ + (B_{i} (x) + Δ B_{i} (x)) \bar{u} + (E_{i} (x) + Δ E_{i} (x)) ν} \\ = A_{Δ z} (x) x + B_{Δ z} (x) \bar{u} + E_{Δ z} (x) ν \end{matrix} \end{matrix} \\ \begin{matrix} y = \sum_{i = 1}^{r} h_{i} (z) {(C_{i} (x) + Δ C_{i} (x)) x \\ + (G_{i} (x) + Δ G_{i} (x)) ν} = C_{Δ z} (x) x + G_{Δ z} (x) ν \end{matrix} \end{matrix}

(1)

where $x \in R^{n}$ , $y \in R^{q}$ , $ν \in R^{p}$ and $\bar{u} \in R^{m}$ are the state, output, exogenous disturbance and control input vectors, respectively. Also, $A_{i} (x)$ , $B_{i} (x)$ , $E_{i} (x)$ , $C_{i} (x)$ and $G_{i} (x)$ are the polynomial matrices with appropriate dimensions; $Δ A_{i} (x)$ , $Δ B_{i} (x)$ , $Δ E_{i} (x)$ , $Δ C_{i} (x)$ and $Δ G_{i} (x)$ are the system uncertainties matrices assumed to satisfy the following conditions

\begin{matrix} [\begin{matrix} Δ A_{i} (x) & Δ B_{i} (x) & Δ E_{i} (x) \end{matrix}] \\ = H_{1 i} (x) Δ (x) [\begin{matrix} N_{1 i} (x) & N_{2 i} (x) & N_{3 i} (x) \end{matrix}] \end{matrix}

(2)

[\begin{matrix} Δ C_{i} (x) & Δ G_{i} (x) \end{matrix}] = H_{2 i} (x) Δ (x) [\begin{matrix} N_{4 i} (x) & N_{5 i} (x) \end{matrix}]

(3)

where $H_{1 i} (x)$ , $H_{2 i} (x)$ , $N_{1 i} (x)$ , $N_{2 i} (x)$ , $N_{3 i} (x)$ , $N_{4 i} (x)$ and $N_{5 i} (x)$ are polynomial matrices with appropriate dimensions. Furthermore, $h_{i} (z (t))$ is the normalized membership functions satisfying the convex property $\sum_{i = 1}^{r} h_{i} (z (t)) = 1$ .

Saturated non-fragile polynomial fuzzy controller

Since actuators typically have saturation limits, it is more logical to define the controller with a saturated function as in the work by Vafamand et al. (2016)

\bar{u} = sat (u) = {\begin{matrix} - u_{sat} & if u < - u_{sat} \\ u & if u_{sat} < u < u_{sat} \\ u_{sat} & if u > u_{sat} \end{matrix}

(4)

where $u_{sat}$ is the control input saturation bound. Moreover, based on the resilient (non-fragile) and the PDC concept, $u$ is defined as follows

u = \sum_{j = 1}^{r} h_{j} (z) {F_{j} (x) + Δ F_{j} (x)} x = F_{Δ z} (x) x

(5)

where $F_{j} (x)$ is a polynomial control gain matrix, which will be determined during the design procedure. The fractional uncertainty satisfies the conditions (Zhang et al., 2007)

\begin{matrix} Δ F_{j} (x) = H_{cj} (x) Δ_{c} (x) N_{cj} (x) \\ Δ_{c} (x) = [I - E_{c} J_{c}]^{- 1} F_{c}, 0 < I - J_{c} J_{c}^{T} \end{matrix}

(6)

where $H_{cj} (x)$ and $N_{cj} (x)$ are known polynomial functions with appropriate dimensions. Moreover, $J_{c}$ is a real known matrix with appropriate dimensions, and also the uncertain system is introduced by $E_{c}$ satisfying $E_{c} E_{c}^{T} \leq I$ .

Remark 1. (Zhang et al., 2007). The linear fractional uncertainty considered in this paper satisfies equation (6). For all $E_{c} E_{c}^{T} \leq I$ and $I - J_{c} J_{c}^{T} > 0$ , the expression $[I - E_{c} J_{c}]$ is invertible. Subsequently, $Δ_{c}$ is well-defined. Furthermore, for the special case where $J_{c} = 0$ , the uncertainty $Δ_{c}$ will be reduced to a parametric uncertainty.

Remark 2. If the polynomial matrices in equations (1) and (5) are considered to be constant, then the polynomial fuzzy model and controller will be converted to the well-known TS fuzzy model and PDC controller, respectively. Utilizing the polynomial or polynomial fuzzy approach results in reducing the number of fuzzy rules and also, the number of stability conditions (Tabarisaaadi et al., 2017).

Consider the null term

\frac{1 + ϵ}{2} B_{Δ z} (x) (u - u) = 0

Assume that $0 < ϵ < 1$ . Based on equations (1) and (5), one has

\begin{matrix} \overset{\cdot}{x} = A_{Δ z} (x) x + \frac{1 + ϵ}{2} B_{Δ z} (x) u + B_{Δ z} (x) (\bar{u} - \frac{1 + ϵ}{2} u) \\ + E_{Δ z} (x) ν \\ = A_{Δ z} (x) x + \frac{1 + ϵ}{2} B_{Δ z} (x) u + B_{Δ z} (x) r + E_{Δ z} (x) ν \\ = A_{Δ z} (x) x + \frac{1 + ϵ}{2} B_{Δ z} (x) F_{Δ z} (x) x + B_{Δ z} (x) r + E_{Δ z} (x) ν \end{matrix}

(7)

where $r$ is defined as

r = \bar{u} - \frac{1 + ϵ}{2} u

Lemma 1. (Vafamand et al., 2016). As long as

| u (t) | < \frac{u_{sat}}{ϵ}

the inequality

r^{T} (t) r (t) \leq (\frac{1 - ϵ}{2})^{2} u^{T} (t) u (t)

is held for the saturation constraint.

Lemma 2. (Li et al., 2017). Assume that the matrices $E$ and $G$ and the symmetric matrix $Y$ are given. For any arbitrary diagonal matrix $F$ satisfying $F^{T} F \leq I$ , the inequality $Y + EF G^{T} + G F^{T} E^{T} < 0$ holds if and only if there exists a positive definite diagonal matrix $U > 0$ such that $Y + EU E^{T} + G U^{- 1} G^{T} < 0$ .

Lemma 3. (Zhang et al., 2007). Assume that the matrices $E$ and $G$ and the symmetric matrix $Y$ are given. Moreover, the matrix $F$ is diagonal and satisfies $F^{T} F \leq I$ , $F = [I - EJ]^{- 1} F$ , $I - J J^{T} > 0$ and $E E^{T} > 0$ . Then, the following inequalities are equivalent

the inequality $Y + EFG + G^{T} F^{T} E^{T} < 0$

for some scalar values $α > 0$

[\begin{matrix} α Y & E & α G^{T} \\ * & - I & J^{T} \\ * & * & - I \end{matrix}] < 0

for some scalar values $β > 0$

[\begin{matrix} Y & β E & G^{T} \\ * & - β I & β J^{T} \\ * & * & - β I \end{matrix}] < 0

Lemma 4. (Dong et al., 2011). Assume that the matrices $E$ and $G$ and the symmetric matrix $Y$ are given. For all diagonal matrices $F$ satisfying $F^{T} F \leq I$ and any scalar $α > 0$ , the inequality

EFG + G^{T} F^{T} E^{T} \leq α^{- 1} E E^{T} + α G^{T} G

holds.

Lemma 5. (Guelton et al., 2013). The inequality $\sum_{i = 1}^{r} \sum_{j = 1}^{r} h_{i} h_{j} Ξ_{ij} < 0$ is satisfied if

{\begin{matrix} Ξ_{ii} < 0 & i = 1, 2, \dots, r \\ \frac{1}{r - 1} Ξ_{ii} + \frac{1}{2} (Ξ_{ij} + Ξ_{ji}) < 0 & 1 \leq i \neq j \leq r \end{matrix}

Lemma 6. (Du and Zhang, 2009). Assume that $α$ and $β$ are positive scalar values. For any positive scalar function $S (t)$ with the property $\overset{\cdot}{S} (t) \leq - α S (t) + β ν^{T} ν$ , the following inequality holds

S (t) \leq e^{- α t} S (0) + β \int_{0}^{t} e^{- α t} ν^{T} (t - τ) ν (t - τ) d τ

Stability analysis

Due to the existence of persistent bounded disturbance, we seek to provide a robust framework to guarantee the BIBO stability based on the proposed non-fragile polynomial fuzzy controller given by equation (5). Subsequently, the boundedness of the state vector will be guaranteed as $t \to \infty$ . Moreover, the $L_{1}$ performance criterion is used to minimize the effect of disturbance on the system output

min_{u} sup_{ν \in L_{\infty}} \frac{∥ y ∥_{\infty}}{∥ ν ∥_{\infty}}

(8)

As we have in equation (8), the supremum ratio of the infinity norm of the output to the infinity norm of the input will be minimized. Next, consider the following candidate polynomial Lyapunov function (LF)

V (x) = x^{T} P (\tilde{x}) x

(9)

where $\tilde{x}$ is a vector comprising those states where their corresponding rows of $B_{Δ z}$ and $E_{Δ z}$ are zero, i.e. $\tilde{x} = {x_{k_{i}} | B_{Δ z}^{k_{i}} = 0 p E_{Δ z}^{k_{i}} = 0}$ and $K = {k_{1}, k_{2}, \dots, k_{e}}$ . By considering equation (9), sufficient SOS-based $L_{1}$ performance conditions will be derived.

Theorem 1. The $L_{1}$ performance of the polynomial fuzzy model given by equation (1) enclosed with the saturated non-fragile polynomial fuzzy controller given by equation (4) will be guaranteed, if for a given decay rate $α$ , there exist positive scalars $ρ, θ, ϵ, ζ, \hat{τ}, δ, η$ , a positive diagonal matrix $U$ and polynomial matrices $K_{j} (x)$ and $X (\tilde{x})$ such that the following SOS-based conditions are satisfied:

minimize $ζ$ , subject to

X (\tilde{x}) - λ^{1} (\tilde{x}) is SOS

(10)

Ξ_{ii} - λ_{ii}^{2} (x) is SOS i = 1, 2, \dots, r

(11)

\frac{1}{r - 1} Ξ_{ii} + \frac{1}{2} (Ξ_{ij} + Ξ_{ji}) - λ_{ij}^{3} (x) is SOS 1 \leq i \neq j \leq r

(12)

Θ_{ii} is SOS i = 1, 2, \dots, r

(13)

\frac{1}{r - 1} Θ_{ii} + \frac{1}{2} (Θ_{ij} + Θ_{ji}) is SOS 1 \leq i \neq j \leq r

(14)

Λ_{j} (x) is SOS i = 1, 2, \dots, r

(15)

where $Θ_{ij}$ and $Ξ_{ij}$ are defined in equations (16) and (17), respectively

Θ_{ij} (x) = - [\begin{matrix} - α X (\tilde{x}) & 0 & X (\tilde{x}) C_{i}^{T} (x) & X (\tilde{x}) N_{4 i}^{T} (x) \\ * & - (ζ - γ) I & G_{i}^{T} (x) & N_{5 i}^{T} \\ * & * & - ζ I + H_{2 i} (x) {QH}_{2 j}^{T} (x) & 0 \\ * & * & * & - Q \end{matrix}]

(16)

Ξ_{ij} = - [\begin{matrix} Ξ_{ij}^{11} & \hat{τ} B_{i} (x) & E_{i} (x) & Ξ_{ij}^{14} & Ξ_{ij}^{15} & Ξ_{ij}^{16} & Ξ_{ij}^{17} & Ξ_{ij}^{18} & η Ξ_{ij}^{19} \\ * & - \hat{τ} I & 0 & 0 & \hat{τ} N_{2 i}^{T} (x) & 0 & 0 & 0 & 0 \\ * & * & - γ I & 0 & N_{3 i}^{T} (x) & 0 & 0 & 0 & 0 \\ * & * & * & - \hat{τ} I & 0 & Ξ_{ij}^{46} & 0 & 0 & 0 \\ * & * & * & * & - U & Ξ_{ij}^{56} & 0 & 0 & 0 \\ * & * & * & * & * & - δ I & δ J_{c}^{T} & 0 & 0 \\ * & * & * & * & * & * & - δ I & 0 & 0 \\ * & * & * & * & * & * & * & - η I & 0 \\ * & * & * & * & * & * & * & * & - η I \end{matrix}]

(17)

The matrix $Λ_{j}$ and the elements of $Ξ_{ij}$ are defined as follows

\begin{matrix} Λ_{j} (x) = \\ - [\begin{matrix} - {(\frac{u_{sat}}{ϵ})}^{2} & K_{j} (x) & θ H_{cj} (x) & 0 \\ * & - (\frac{X (\tilde{x})}{ρ}) & 0 & X (\tilde{x}) N_{cj}^{T} (x) \\ * & * & - θ I & θ J^{T} \\ * & * & * & - θ I \end{matrix}] \end{matrix}

\begin{matrix} Ξ_{ij}^{11} = [(A_{i} (x) X (\tilde{x}) + \frac{1 + ϵ}{2} B_{i} (x) K_{j} (x)) + *] + α X (\tilde{x}) \\ + H_{1 i} (x) {UH}_{1 j}^{T} (x) - \sum_{k = 1}^{n} \frac{\partial X (\tilde{x})}{\partial x_{k}} A_{i}^{k} (x) x \\ Ξ_{ij}^{15} = X (\tilde{x}) N_{1 i}^{T} (x) + \frac{1 + ϵ}{2} K_{j}^{T} (x) N_{2 i}^{T} (x), Ξ_{ij}^{19} = x^{T} N_{1 i}^{T} (x) \\ Ξ_{ij}^{16} = δ (\frac{1 + ϵ}{2}) B_{i} (x) H_{cj} (x), Ξ_{ij}^{14} = (\frac{1 - ϵ}{2}) K_{j}^{T} (x) \\ Ξ_{ij}^{17} = X (\tilde{x}) N_{cj}^{T} (x), Ξ_{ij}^{18} = - 0.5 \sum_{k = 1}^{n} \frac{\partial X (\tilde{x})}{\partial x_{k}} H_{1 i}^{k} (x) \\ Ξ_{ij}^{46} = (\frac{1 - ϵ}{2}) δ H_{cj} (x), Ξ_{ij}^{56} = (\frac{1 + ϵ}{2}) δ N_{2 i} (x) H_{cj} (x) \end{matrix}

Furthermore, the $L_{1}$ performance attenuated level $ζ$ will be obtained for the local region ${x | x^{T} P^{- 1} x < ρ}$ . In addition, the controller gain will be computed as follows

F_{j} (x) = K_{j} (x) X^{- 1} (\tilde{x})

(18)

Proof: Time derivative of the polynomial LF given by equation (9) is calculated as

\begin{matrix} \overset{\cdot}{V} = [x^{T} P (\tilde{x}) \overset{\cdot}{x} + *] + x^{T} \frac{\partial P (\tilde{x})}{\partial t} x \\ = [x^{T} P (\tilde{x}) (A_{Δ z} (x) x + \frac{1 + ϵ}{2} B_{Δ z} (x) F_{Δ z} (x) x \\ + B_{Δ z} (x) r + E_{Δ z} (x) ν) + *] + x^{T} \frac{\partial P (\tilde{x})}{\partial t} x \end{matrix}

(19)

Assume $α$ and $γ$ are positive scalars. By adding and subscribing the expression $α x^{T} P (\tilde{x}) x + γ ν^{T} ν$ , one has

\overset{\cdot}{V} = {[\begin{matrix} x \\ r \\ ν \end{matrix}]}^{T} Γ_{zz} [\begin{matrix} x \\ r \\ ν \end{matrix}] - α x^{T} P (\tilde{x}) x + γ ν^{T} ν

(20)

where

Γ_{zz} = [\begin{matrix} Γ_{zz}^{11} & P (\tilde{x}) B_{Δ z} (x) & P (\tilde{x}) E_{Δ z} (x) \\ * & 0 & 0 \\ * & * & - γ I \end{matrix}] < 0

and

\begin{matrix} Γ_{zz}^{11} = [P (\tilde{x}) (A_{Δ z} (x) + \frac{1 + ϵ}{2} B_{Δ z} (x) F_{Δ z} (x)) + *] \\ + \frac{\partial P (\tilde{x})}{\partial t} + α P (\tilde{x}) \end{matrix}

One of the diagonal elements of matrix $Γ_{zz}$ is zero. Thus, we cannot guarantee that matrix $Γ_{zz}$ is negative definite. Subsequently, we use the S-procedure (Vafamand et al., 2016) to deal with this problem. Based on Lemma 1 and the PDC controller given by equation (5), one obtains

\begin{matrix} r^{T} r = [\bar{u} - \frac{1 + ϵ}{2} u]^{T} [\bar{u} - \frac{1 + ϵ}{2} u] \leq {(\frac{1 - ϵ}{2})}^{2} u^{T} u \\ \leq {(\frac{1 - ϵ}{2})}^{2} (x^{T} F_{Δ z}^{T} (x) F_{Δ z} (x) x) \end{matrix}

(21)

Then, equation (21) results in

{[\begin{matrix} x \\ r \\ ν \end{matrix}]}^{T} [\begin{matrix} {(\frac{1 - ϵ}{2})}^{2} F_{Δ z}^{T} (x) F_{Δ z} (x) & 0 & 0 \\ 0 & - I & 0 \\ 0 & 0 & 0 \end{matrix}] [\begin{matrix} x \\ r \\ ν \end{matrix}] \geq 0

(22)

By applying the S-procedure on equation (20) subject to equation (22), one concludes

\begin{matrix} [\begin{matrix} Γ_{zz}^{11} & P (\tilde{x}) B_{Δ z} (x) & P (\tilde{x}) E_{Δ z} (x) \\ * & - τ I & 0 \\ * & * & - γ I \end{matrix}] + \\ [\begin{matrix} (\frac{1 - ϵ}{2}) F_{Δ z}^{T} (x) \\ 0 \\ 0 \end{matrix}] τ [\begin{matrix} (\frac{1 - ϵ}{2}) F_{Δ z} (x) & 0 & 0 \end{matrix}] < 0 \end{matrix}

(23)

By employing the Schur complement (Vafamand et al., 2016), one has

[\begin{matrix} Γ_{zz}^{11} & P (\tilde{x}) B_{Δ z} (x) & P (\tilde{x}) E_{Δ z} (x) & (\frac{1 - ϵ}{2}) F_{Δ z}^{T} (x) \\ * & - τ I & 0 & 0 \\ * & * & - γ I & 0 \\ * & * & * & - τ^{- 1} I \end{matrix}] < 0

(24)

In the following, our goal is to remove the system uncertainties $A_{Δ z}$ , $B_{Δ z}$ and $C_{Δ z}$ from equation (24). The definition given by equation (2) indicates $A_{Δ z} = Δ A_{z} + A_{z}$ , $B_{Δ z} = Δ B_{z} + B_{z}$ and $E_{Δ z} = Δ E_{z} + E_{z}$ . Therefore, the inequality in equation (24) is equivalent to equation (25)

\begin{matrix} [\begin{matrix} \begin{matrix} [P (\tilde{x}) (A_{z} (x) + \frac{1 + ϵ}{2} B_{z} (x) F_{Δ z} (x)) + *] + \frac{\partial P (\tilde{x})}{\partial t} + α P (\tilde{x}) \end{matrix} & P (\tilde{x}) B_{z} (x) & P (\tilde{x}) E_{z} (x) & (\frac{1 - ϵ}{2}) F_{Δ z}^{T} (x) \\ * & - τ I & 0 & 0 \\ * & * & - γ I & 0 \\ * & * & * & - τ^{- 1} I \end{matrix}] \\ + (\begin{matrix} [\begin{matrix} P (\tilde{x}) H_{1 z} (x) \\ 0 \\ 0 \\ 0 \end{matrix}] Δ (x) {[\begin{matrix} {(N_{1 z} (x) + \frac{1 + ϵ}{2} N_{2 z} (x) F_{Δ z} (x))}^{T} \\ N_{2 z}^{T} (x) \\ N_{3 z}^{T} (x) \\ 0 \end{matrix}]}^{T} + * \end{matrix}) < 0 \end{matrix}

(25)

Based on Lemma 2, one has

\begin{matrix} [\begin{matrix} \begin{matrix} [P (\tilde{x}) (A_{z} (x) + \frac{1 + ϵ}{2} B_{z} (x) F_{Δ z} (x)) + *] + \frac{\partial P (\tilde{x})}{\partial t} \\ + α P (\tilde{x}) + P (\tilde{x}) H_{1 z} (x) {UH}_{1 z}^{T} (x) P (\tilde{x}) \end{matrix} & P (\tilde{x}) B_{z} (x) & P (\tilde{x}) E_{z} (x) & (\frac{1 - ϵ}{2}) F_{Δ z}^{T} (x) \\ * & - τ I & 0 & 0 \\ * & * & - γ I & 0 \\ * & * & * & - τ^{- 1} I \end{matrix}] \\ + [\begin{matrix} {(N_{1 z} (x) + \frac{1 + ϵ}{2} N_{2 z} (x) F_{Δ z} (x))}^{T} \\ N_{2 z}^{T} (x) \\ N_{3 z}^{T} (x) \\ 0 \end{matrix}] U^{- 1} {[\begin{matrix} {(N_{1 z} (x) + \frac{1 + ϵ}{2} N_{2 z} (x) F_{Δ z} (x))}^{T} \\ N_{2 z}^{T} (x) \\ N_{3 z}^{T} (x) \\ 0 \end{matrix}]}^{T} < 0 \end{matrix}

(26)

where $U$ is a positive definite diagonal matrix. Utilizing the Schur complement on equation (26), yields

[\begin{matrix} \begin{matrix} [P (\tilde{x}) (A_{z} (x) + \frac{1 + ϵ}{2} B_{z} (x) F_{Δ z} (x)) + *] \\ + \frac{\partial P (\tilde{x})}{\partial t} + α P (\tilde{x}) \\ + P (\tilde{x}) H_{1 z} (x) {UH}_{1 z}^{T} (x) P (\tilde{x}) \end{matrix} & P (\tilde{x}) B_{z} (x) & P (\tilde{x}) E_{z} (x) & (\frac{1 - ϵ}{2}) F_{Δ z}^{T} (x) & \begin{matrix} N_{1 z}^{T} (x) + \\ {(\frac{1 + ϵ}{2} N_{2 z} (x) F_{Δ z} (x))}^{T} \end{matrix} \\ * & - τ I & 0 & 0 & N_{2 z}^{T} (x) \\ * & * & - γ I & 0 & N_{3 z}^{T} (x) \\ * & * & * & - τ^{- 1} I & 0 \\ * & * & * & * & - U \end{matrix}] < 0

(27)

Now, the goal is to remove the controller uncertainty $F_{Δ z}$ . Since $F_{Δ z} = Δ_{Fz} + F_{z}$ , the constraint given by equation (27) is equivalent to

\begin{matrix} (\begin{matrix} [\begin{matrix} (\frac{1 + ϵ}{2}) P (\tilde{x}) B_{z} (x) H_{cz} (x) \\ 0 \\ 0 \\ (\frac{1 - ϵ}{2}) H_{cz} (x) \\ (\frac{1 + ϵ}{2}) N_{2 z} (x) H_{cz} (x) \end{matrix}] Δ_{c} (x) {[\begin{matrix} N_{cz}^{T} (x) \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix}]}^{T} + * \end{matrix}) \\ + {\bar{Γ}}_{zz} < 0 \end{matrix}

(28)

where

{\bar{Γ}}_{zz} = [\begin{matrix} \begin{matrix} [P (\tilde{x}) (A_{z} (x) + \frac{1 + ϵ}{2} B_{z} (x) F_{z} (x)) + *] \\ + \frac{\partial P (\tilde{x})}{\partial t} + α P (\tilde{x}) \\ + P (\tilde{x}) H_{1 z} (x) {UH}_{1 z}^{T} (x) P (\tilde{x}) \end{matrix} & P (\tilde{x}) B_{z} (x) & P (\tilde{x}) E_{z} (x) & (\frac{1 - ϵ}{2}) F_{z}^{T} (x) & \begin{matrix} N_{1 z}^{T} (x) + \\ {(\frac{1 + ϵ}{2} N_{2 z} (x) F_{z} (x))}^{T} \end{matrix} \\ * & - τ I & 0 & 0 & N_{2 z}^{T} (x) \\ * & * & - γ I & 0 & N_{3 z}^{T} (x) \\ * & * & * & - τ^{- 1} I & 0 \\ * & * & * & * & - U \end{matrix}]

From equation (28) and Lemma 3, one has

[\begin{matrix} {\bar{Γ}}_{zz} & δ [\begin{matrix} (\frac{1 + ϵ}{2}) P (\tilde{x}) B_{z} (x) H_{cz} (x) \\ 0 \\ 0 \\ (\frac{1 - ϵ}{2}) H_{cz} (x) \\ (\frac{1 + ϵ}{2}) N_{2 z} (x) H_{cz} (x) \end{matrix}] & [\begin{matrix} N_{cz}^{T} (x) \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix}] \\ * & - δ I & δ J_{c}^{T} \\ * & * & - δ I \end{matrix}] < 0

(29)

where $δ > 0$ . On the other hand, consider the following equality

\begin{matrix} P^{- 1} (\tilde{x}) \frac{\partial P (\tilde{x})}{\partial t} P^{- 1} (\tilde{x}) = - \frac{\partial P^{- 1} (\tilde{x})}{\partial t} \\ = - \sum_{k \in K} \frac{\partial P^{- 1} (\tilde{x})}{\partial x_{k}} A_{z}^{k} (x) x - \sum_{k \in K} \frac{\partial P^{- 1} (\tilde{x})}{\partial x_{k}} Δ A_{z}^{k} (x) x \end{matrix}

(30)

Pre- and post-multiplying equation (29) by the matrix $diag {P^{- 1} (\tilde{x}), I, I, I, I, I, I}$ and considering equation (30), one obtains

\begin{matrix} [\begin{matrix} {\tilde{Γ}}_{zz} & δ [\begin{matrix} (\frac{1 + ϵ}{2}) B_{z} (x) H_{cz} (x) \\ 0 \\ 0 \\ (\frac{1 - ϵ}{2}) H_{cz} (x) \\ (\frac{1 + ϵ}{2}) N_{2 z} (x) H_{cz} (x) \end{matrix}] & [\begin{matrix} P^{- 1} (\tilde{x}) N_{cz}^{T} (x) \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix}] \\ * & - δ I & δ J_{c}^{T} \\ * & * & - δ I \end{matrix}] \\ + diag {- \sum_{k \in K} \frac{\partial P^{- 1} (\tilde{x})}{\partial x_{k}} Δ A_{z}^{k} (x) x, 0, 0, 0, 0, 0, 0} \end{matrix}

(31)

where

{\tilde{Γ}}_{zz} = [\begin{matrix} {\tilde{Γ}}_{zz}^{11} & B_{z} (x) & E_{z} (x) & {\tilde{Γ}}_{zz}^{14} & {\tilde{Γ}}_{zz}^{15} \\ * & - τ I & 0 & 0 & N_{2 z}^{T} (x) \\ * & * & - γ I & 0 & N_{3 z}^{T} (x) \\ * & * & * & - τ^{- 1} I & 0 \\ * & * & * & * & - U \end{matrix}]

and

\begin{matrix} {\tilde{Γ}}_{zz}^{11} = [(A_{z} (x) + \frac{1 + ϵ}{2} B_{z} (x) F_{z} (x)) P^{- 1} (\tilde{x}) + *] \\ + α P^{- 1} (\tilde{x}) + H_{1 z} (x) {UH}_{1 z}^{T} (x) \\ - \sum_{k \in K} \frac{\partial P^{- 1} (\tilde{x})}{\partial x_{k}} A_{z}^{k} (x) x \\ {\tilde{Γ}}_{zz}^{15} = {P^{- 1} (\tilde{x}) N_{1 z}^{T} (x) + P^{- 1} (\tilde{x}) (\frac{1 + ϵ}{2} N_{2 z} (x) F_{z} (x))}^{T} \\ {\tilde{Γ}}_{zz}^{14} = (\frac{1 - ϵ}{2}) P^{- 1} (\tilde{x}) F_{z}^{T} (x) \end{matrix}

The solution to substituting the definition of $Δ A_{z} (x)$ (i.e. equation (2)) into equation (31) can be re-written as follows

\begin{matrix} (\begin{matrix} 0.5 [\begin{matrix} - \sum_{k \in K} \frac{\partial P^{- 1} (\tilde{x})}{\partial x_{k}} H_{1 z}^{k} (x) \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix}] Δ (x) {[\begin{matrix} x^{T} N_{1 z}^{T} (x) \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix}]}^{T} + * \end{matrix}) \\ + {\hat{Γ}}_{zz} < 0 \end{matrix}

(32)

where

\begin{matrix} {\hat{Γ}}_{zz} = \\ [\begin{matrix} {\tilde{Γ}}_{zz} & δ [\begin{matrix} (\frac{1 + ϵ}{2}) B_{z} (x) H_{cz} (x) \\ 0 \\ 0 \\ (\frac{1 - ϵ}{2}) H_{cz} (x) \\ (\frac{1 + ϵ}{2}) N_{2 z} (x) H_{cz} (x) \end{matrix}] & [\begin{matrix} P^{- 1} (\tilde{x}) N_{cz}^{T} (x) \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix}] \\ * & - δ I & δ J_{c}^{T} \\ * & * & - δ I \end{matrix}] \end{matrix}

Subsequently, by utilizing the Schur complement and Lemma 4, one has

[\begin{matrix} {\hat{Γ}}_{zz} & [\begin{matrix} - 0.5 \sum_{k \in K} \frac{\partial P^{- 1} (\tilde{x})}{\partial x_{k}} H_{1 z}^{k} (x) \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix}] & [\begin{matrix} x^{T} N_{1 z}^{T} (x) \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix}] \\ * & - η I & 0 \\ * & * & - η^{- 1} I \end{matrix}] < 0

(33)

The inequality given in equation (33) is equivalent to equation (34) with the following elements

\overset{ˇ}{Γ} = [\begin{matrix} {\overset{ˇ}{Γ}}_{z z}^{11} & B_{z} (x) & E_{z} (x) & {\overset{ˇ}{Γ}}_{z z}^{14} & {\overset{ˇ}{Γ}}_{z z}^{15} & {\overset{ˇ}{Γ}}_{z z}^{16} & {\overset{ˇ}{Γ}}_{z z}^{17} & {\overset{ˇ}{Γ}}_{z z}^{18} & {\overset{ˇ}{Γ}}_{z z}^{19} \\ * & - τ I & 0 & 0 & N_{2 z}^{T} (x) & 0 & 0 & 0 & 0 \\ * & * & - γ I & 0 & N_{3 z}^{T} (x) & 0 & 0 & 0 & 0 \\ * & * & * & - τ^{- 1} I & 0 & {\overset{ˇ}{Γ}}_{z z}^{46} & 0 & 0 & 0 \\ * & * & * & * & - U & {\overset{ˇ}{Γ}}_{z z}^{56} & 0 & 0 & 0 \\ * & * & * & * & * & - δ I & δ J_{c}^{T} & 0 & 0 \\ * & * & * & * & * & * & - δ I & 0 & 0 \\ * & * & * & * & * & * & * & - η I & 0 \\ * & * & * & * & * & * & * & * & - η^{- 1} I \end{matrix}] < 0

(34)

\begin{array}{l} {\overset{ˇ}{Γ}}_{z z}^{11} = [(A_{z} (x) + \frac{1 + ϵ}{2} B_{z} (x) F_{z} (x)) P^{- 1} (\tilde{x}) + *] \\ + α P^{- 1} (\tilde{x}) + H_{1 z} (x) U H_{1 z}^{T} (x) - \sum_{k \in K} \frac{\partial P^{- 1} (\tilde{x})}{\partial x_{k}} A_{z}^{k} (x) x \\ {\overset{ˇ}{Γ}}_{z z}^{15} = P^{- 1} (\tilde{x}) N_{1 z}^{T} (x) + P^{- 1} (\tilde{x}) {(\frac{1 + ϵ}{2} N_{2 z} (x) F_{z} (x))}^{T} \\ {\overset{ˇ}{Γ}}_{z z}^{14} = (\frac{1 - ϵ}{2}) P^{- 1} (\tilde{x}) F_{z}^{T} (x), {\overset{ˇ}{Γ}}_{z z}^{19} = x^{T} N_{1 z}^{T} (x) \\ {\overset{ˇ}{Γ}}_{z z}^{16} = δ (\frac{1 + ϵ}{2}) B_{z} (x) H_{c z} (x) {\overset{ˇ}{Γ}}_{z z}^{17} = P^{- 1} (\tilde{x}) N_{c z}^{T} (x), \\ {\overset{ˇ}{Γ}}_{z z}^{18} = - 0.5 \sum_{k \in K} \frac{\partial P^{- 1} (\tilde{x})}{\partial x_{k}} H_{1 z}^{k} (x) {\overset{ˇ}{Γ}}_{z z}^{46} = (\frac{1 - ϵ}{2}) δ H_{c z} (x), \\ {\overset{ˇ}{Γ}}_{z z}^{56} = (\frac{1 + ϵ}{2}) δ N_{2 z} (x) H_{c z} (x) \end{array}

Pre- and post-multiplying equation (34) by $diag {I, τ^{- 1} I, I, I, I, I, I, I, η I}$ , results in equation (35)

[\begin{matrix} {\overset{ˇ}{Γ}}_{z z}^{11} & τ^{- 1} B_{z} (x) & E_{z} (x) & {\overset{ˇ}{Γ}}_{z z}^{14} & {\overset{ˇ}{Γ}}_{z z}^{15} & {\overset{ˇ}{Γ}}_{z z}^{16} & {\overset{ˇ}{Γ}}_{z z}^{17} & {\overset{ˇ}{Γ}}_{z z}^{18} & η {\overset{ˇ}{Γ}}_{z z}^{19} \\ * & - τ^{- 1} I & 0 & 0 & τ^{- 1} N_{2 z}^{T} (x) & 0 & 0 & 0 & 0 \\ * & * & - γ I & 0 & N_{3 z}^{T} (x) & 0 & 0 & 0 & 0 \\ * & * & * & - τ^{- 1} I & 0 & {\overset{ˇ}{Γ}}_{z z}^{46} & 0 & 0 & 0 \\ * & * & * & * & - U & {\overset{ˇ}{Γ}}_{z z}^{56} & 0 & 0 & 0 \\ * & * & * & * & * & - δ I & δ J_{c}^{T} & 0 & 0 \\ * & * & * & * & * & * & - δ I & 0 & 0 \\ * & * & * & * & * & * & * & - η I & 0 \\ * & * & * & * & * & * & * & * & - η I \end{matrix}] < 0

(35)

Based on Lemma 5, if equations (11) and (12) are satisfied, then $Γ_{zz}$ defined in equation (20) will be negative definite. Therefore, equation (20) leads to

\overset{\cdot}{V} \leq - α V + γ ν^{T} ν

(36)

Recall Lemma 6. It follows from equation (36) that

\begin{matrix} V (t) \leq e^{- α t} V (0) + γ \int_{0}^{t} e^{- α τ} ν^{T} (t - τ) ν (t - τ) d τ \\ \leq sup_{τ \in [0, t]} {e^{- α t} V (0) + γ \int_{0}^{t} e^{- α τ} ν^{T} (t - τ) ν (t - τ) d τ} \\ = sup_{τ \in [0, t]} {e^{- α t} V (0) + \frac{γ}{α} ν^{T} (t - τ) ν (t - τ) (1 - e^{- α t})} \\ \leq V (0) + \frac{γ}{α} ∥ ν (t) ∥_{\infty}^{2} \end{matrix}

(37)

Assume that the initial condition is zero, the inequality given by equation (37) yields

V (t) \leq \frac{γ}{α} ∥ ν (t) ∥_{\infty}^{2}

(38)

In order to guarantee the $L_{1}$ performance given by equation (8), consider the following inequality

y^{T} y \leq ζ {[\begin{matrix} x \\ ν \end{matrix}]}^{T} [\begin{matrix} α P (\tilde{x}) & 0 \\ * & (ζ - γ) I \end{matrix}] [\begin{matrix} x \\ ν \end{matrix}]

(39)

From equations (1) and (39), one has

\begin{matrix} {[\begin{matrix} x \\ ν \end{matrix}]}^{T} [\begin{matrix} C_{Δ z}^{T} (x) \\ G_{Δ z}^{T} (x) \end{matrix}] [\begin{matrix} C_{Δ z} (x) & G_{Δ z} (x) \end{matrix}] [\begin{matrix} x \\ ν \end{matrix}] \leq \\ ζ {[\begin{matrix} x \\ ν \end{matrix}]}^{T} [\begin{matrix} α P (\tilde{x}) & 0 \\ * & (ζ - γ) I \end{matrix}] [\begin{matrix} x \\ ν \end{matrix}] \end{matrix}

(40)

Using the Schure complement, equation (40) leads to

[\begin{matrix} - α P (\tilde{x}) & 0 & C_{Δ z}^{T} (x) \\ * & - (ζ - γ) I & G_{Δ z}^{T} (x) \\ * & * & - ζ I \end{matrix}] \leq 0

(41)

Pre and post multiplying equation (41) by $diag {P^{- 1} (\tilde{x}), I, I}$ gives

[\begin{matrix} - α P^{- 1} (\tilde{x}) & 0 & P^{- 1} (\tilde{x}) C_{Δ z}^{T} (x) \\ * & - (ζ - γ) I & G_{Δ z}^{T} (x) \\ * & * & - ζ I \end{matrix}] \leq 0

(42)

The inequality given by equation (42) is equivalent to

\begin{matrix} [\begin{matrix} - α P^{- 1} (\tilde{x}) & 0 & P^{- 1} (\tilde{x}) C_{z}^{T} (x) \\ * & - (ζ - γ) I & G_{z}^{T} (x) \\ * & * & - ζ I \end{matrix}] \\ + (\begin{matrix} [\begin{matrix} P^{- 1} (\tilde{x}) N_{4 z}^{T} (x) \\ N_{5 z}^{T} (x) \\ 0 \end{matrix}] Δ (x) {[\begin{matrix} 0 \\ 0 \\ H_{2 z} (x) \end{matrix}]}^{T} + * \end{matrix}) \leq 0 \end{matrix}

(43)

Using Lemma 2, one has

\begin{matrix} [\begin{matrix} - α P^{- 1} (\tilde{x}) & 0 & P^{- 1} (\tilde{x}) C_{z}^{T} (x) \\ * & - (ζ - γ) I & G_{z}^{T} (x) \\ * & * & - ζ I \end{matrix}] \\ + [\begin{matrix} 0 \\ 0 \\ H_{2 z} (x) \end{matrix}] Q {[\begin{matrix} 0 \\ 0 \\ H_{2 z} (x) \end{matrix}]}^{T} \\ + [\begin{matrix} P^{- 1} (\tilde{x}) N_{4 z}^{T} (x) \\ N_{5 z}^{T} (x) \\ 0 \end{matrix}] Q^{- 1} {[\begin{matrix} P^{- 1} (\tilde{x}) N_{4 z}^{T} (x) \\ N_{5 z}^{T} \\ 0 \end{matrix}]}^{T} \leq 0 \end{matrix}

(44)

Employing the Schure complement, leads to equation (45).

[\begin{matrix} - α P^{- 1} (\tilde{x}) & 0 & P^{- 1} (\tilde{x}) C_{z}^{T} (x) & P^{- 1} (\tilde{x}) N_{4 z}^{T} (x) \\ * & - (ζ - γ) I & G_{z}^{T} (x) & N_{5 z}^{T} \\ * & * & - ζ I + H_{2 z} (x) {QH}_{2 z}^{T} (x) & 0 \\ * & * & * & - Q \end{matrix}] \leq 0

(45)

By employing Lemma 5, equation (45) will be guaranteed, if equations (13) and (14) are satisfied. By substituting equation (38) into equation (39), one has

\begin{matrix} y^{T} y < ζ α V (t) + (ζ^{2} - ζ γ) ν^{T} ν \\ \leq ζ α (\frac{γ}{α} ∥ ν ∥_{\infty}^{2}) + (ζ^{2} - ζ γ) ∥ ν ∥_{\infty}^{2} = ζ^{2} ∥ ν ∥_{\infty}^{2} \end{matrix}

(46)

Subsequently, the $L_{1}$ performance gain $ζ$ for an uncertain polynomial fuzzy model will be completed. The remaining problem is to assure

| u (t) | < \frac{u_{sat}}{ϵ}

Based on the discussion presented by Du and Zhang (2009), if

F_{Δ z} {(\frac{P (\tilde{x})}{ρ})}^{- 1} F_{Δ z}^{T} \leq {(\frac{u_{sat}}{ϵ})}^{2}

(47)

holds, then $R_{2} = {x^{T} P (\tilde{x}) x < ρ} \in R_{1}$ . Since $R_{2}$ is an invariant set (Du and Zhang, 2009), for any initial conditions $x$ within the $R_{2}$ , the state trajectories will not leave $R_{2} \in R_{1}$ . Employing the Schure complement on equation (47), one obtains

[\begin{matrix} - {(\frac{u_{sat}}{ϵ})}^{2} & F_{Δ z} \\ * & - (\frac{P (\tilde{x})}{ρ}) \end{matrix}] \leq 0

(48)

Pre and post multiplying both sides of equation (48) by $diag {I, P^{- 1} (\tilde{x})}$ , one has

\begin{matrix} [\begin{matrix} - {(\frac{u_{sat}}{ϵ})}^{2} & F_{Δ z} P^{- 1} (\tilde{x}) \\ * & - (\frac{P^{- 1} (\tilde{x})}{ρ}) \end{matrix}] \\ \begin{matrix} = [\begin{matrix} - {(\frac{u_{sat}}{ϵ})}^{2} & F_{z} P^{- 1} (\tilde{x}) \\ * & - (\frac{P^{- 1} (\tilde{x})}{ρ}) \end{matrix}] \\ + (\begin{matrix} [\begin{matrix} H_{cz} (x) \\ 0 \end{matrix}] Δ_{c} (x) {[\begin{matrix} 0 \\ P^{- 1} (\tilde{x}) N_{cz}^{T} (x) \end{matrix}]}^{T} + * \end{matrix}) \leq 0 \end{matrix} \end{matrix}

(49)

Based on Lemma 3, equation (49) is satisfied if

[\begin{matrix} - {(\frac{u_{sat}}{ϵ})}^{2} & F_{z} P^{- 1} (\tilde{x}) & θ H_{cz} (x) & 0 \\ * & - (\frac{P^{- 1} (\tilde{x})}{ρ}) & 0 & P^{- 1} (\tilde{x}) N_{cz}^{T} (x) \\ * & * & - θ I & θ J^{T} \\ * & * & * & - θ I \end{matrix}] \leq 0

(50)

holds. The constraint given by equation (50) leads to equation (15). Thus, the proof is completed.

Remark 3. In this paper, the problems of designing a robust $L_{1}$ performance controller for uncertain polynomial fuzzy models is studied. The designed controller satisfies both the saturation and non-fragile constraints. The existence of several uncertainty terms complicates the derivation of the controller design conditions and prevents the researchers from deriving the SOS conditions for such practical situations. However, in this paper, by considering the assumptions given by equations (2), (3) and (6) and applying Lemmas 2, 3 and 4, the uncertainties are removed. Regrading to each uncertainty, a proper Lemma is used to effectively eliminate the uncertainty and decrease the conservativeness of the SOS conditions. Also, the consideration of actuator saturation increases the complexity of the controller design. To overcome such issue, the input constrained system is re-formulated by a non-constrained system with a vanishing disturbance introduced in equation (7). So, by applying the S-procedure and defining the invariant set $R_{2}$ , the actuator saturation is handled in this paper. Although, providing the $H_{\infty}$ performance is widely used in literature to alleviate the disturbance inputs on the response, the $L_{1}$ performance index is employed for polynomial systems. By considering the BIBO analysis criteria and taking inspiration from Vafamand et al. (2016), a novel $L_{1}$ performance index for fuzzy polynomial systems is proposed.

Remark 4. In Theorem 1, some parameters, including $ϵ$ , $α$ and $ρ$ , are employed to increase the degrees of freedom of the controller design. Although there is no systematic and analytic approach to select these scalars, such parameters may affect the closed-loop system performance and one encounters some tradeoffs for choosing them.

From Lemma 1, one knows that in the controller design procedure, the controller input is artificially bounded by $u_{\max} / ϵ$ By choosing the smaller $ϵ$ , the value of $u_{\max} / ϵ$ is increased which results in a larger amplitude of $u (t)$ and an enhanced transient performance. On the other hand, from equation (17) and $Ξ_{ij}^{11}$ one infers that when $ϵ$ is small, the controller gain $F_{j} (x)$ must be calculated as high to compensate for the small value of $ϵ$ . This fact increases the control input effort.

The parameter $α$ denotes the decay rate of the response and the exponential stability of the closed-loop system. Choosing a higher value for $α$ provides a faster system response. But, this parameter cannot be selected too high so that the SOS conditions must be feasible.

The parameter $ρ$ effects the local region ${x | x^{T} X (\tilde{x}) x < ρ}$ . So, by selecting a higher $ρ$ , a larger local region is obtained. On the other hand, a higher $ρ$ may lead to infeasible results especially for the condition ( $Λ_{j}$ is SOS). Because, the diagonal array $X (\tilde{x}) / ρ$ is small.

The above discussion regarding the effect of parameters on the closed-loop performance can represent a way to select the parameters of Theorem 1.

Remark 5 The overall procedure of the proposed novel saturated non-fragile polynomial fuzzy controller for uncertain polynomial fuzzy models in the presence of persistent bounded disturbance is indicated in Figure 1.

Figure 1.

General scheme of the proposed design approach of the controller given by equation (5).

Numerical example

In this section, a practical ball-and-beam system is given to show the merits and effectiveness of the proposed design method. In order to give a comparison and challenge the proposed approach point by point, some scenarios are proposed. The overall detail of the scenarios is presented in Table 1. All of the simulations have been carried out via the third-party MATLAB toolbox SOSTOOLS 3.3. In order to evaluate the performance of the proposed control method, the HIL simulation approach is utilized. The real time HIL method is used to emulate errors and delays that do not exist in classical off-line simulations. Figure 2 gives a schematic diagram of the HIL setup which consists of:

Table 1.

The overall detail of the scenarios.

Scenario	Performance issue
Scenario	Input saturation	Non-fragile controller	System uncertainty	Persistent disturbance
A	✓	×	×	×
B	✓	✓	×	×
C	✓	✓	✓	×
D	✓	✓	✓	✓

OPAL-RT as a real-time simulator (RTS);

a PC as the command station (programming host) in which the Matlab/Simulink based code, which will be executed on the OPAL-RT, is generated;

a router that is used as a connector of all the setup devices in the same sub-network.

The OPAL-RT, is also connected to the DK60 board through Ethernet ports.

Figure 2.

The real-time experimental setup.

Ball-and-beam system

The ball-and-beam system is described via the following state space model

{\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} \\ {\overset{\cdot}{x}}_{2} = B (x_{1} x_{4}^{2} - g \sin (x_{3})) \\ {\overset{\cdot}{x}}_{3} = x_{4} \\ {\overset{\cdot}{x}}_{4} = u \\ y = [x_{1}, x_{2}, x_{4}]^{T} \end{matrix}

(51)

where the states $x_{1}$ , $x_{2}$ , $x_{3}$ and $x_{4}$ are the position and velocity of the ball and angular position and velocity of the beam, respectively. $B = 0.6$ and $g = 10 m / s^{2}$ . The polynomial fuzzy model of the ball-and-beam system can be described via a two-rule polynomial fuzzy model, as shown in the work by Liu et al. (2016a)

{\begin{matrix} \overset{\cdot}{x} = \sum_{i = 1}^{2} h_{i} (x_{3}) (A_{i} (x) x + B_{i} (x) u) \\ y = \sum_{i = 1}^{2} h_{i} (x_{3}) C_{i} x \end{matrix}

(52)

where $x = [x_{1}, x_{2}, x_{3}, x_{4}]^{T}, B_{1} = B_{2} = [0, 0, 0, 1]^{T}$ and

\begin{matrix} A_{1} (x_{4}) = & [\begin{matrix} 0 & 1 & 0 & 0 \\ {Bx}_{4}^{2} & 0 & - Bg f_{1_{\min}} & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{matrix}] \\ A_{2} (x_{4}) = & [\begin{matrix} 0 & 1 & 0 & 0 \\ {Bx}_{4}^{2} & 0 & - Bg f_{1_{\max}} & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{matrix}] \end{matrix}

C_{1} = C_{2} = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}]

h_{1} (x_{3}) = \frac{f_{1_{\max}} - f_{1} (x_{3})}{f_{1_{\max}} - f_{1_{\min}}}, h_{2} (x_{3}) = 1 - h_{1} (x_{3})

and also $f_{1_{\min}} = 0$ , $f_{1_{\max}} = 1$ and

f_{1} (x_{3}) = \frac{\sin (x_{3})}{x_{3}}

Scenario A

This scenario is presented to compare the results of this paper with the newly published work by Liu et al. (2016a); Vafamand et al. (2016). Since the approach of Liu et al. (2016a) is derived for disturbance-free nominal polynomial fuzzy models, we consider the same assumption and use the system and controller without any uncertainties. And, only the saturation constraint affects the controller. Moreover, the degree of polynomial gain matrices and Lyapunov matrix are chosen to be two and zero, respectively. The saturation bound is selected to be equal to the maximum amplitude of the control signal designed in the work by Liu et al. (2016a) (i.e. $u_{sat} = 4.6999$ ). The other parameters of Theorem 1 are chosen as: $ϵ = 0.01$ and $α = ρ = 0.1$ . By utilizing the SOS conditions of Theorem 1, the feasible solutions are obtained in equations (53) and (54)

P = [\begin{matrix} 6.1954 & 2.2104 & - 3.8098 & - 0.5139 \\ 2.2104 & 0.9797 & - 1.7521 & - 0.2499 \\ - 3.8098 & - 1.7521 & 3.4583 & 0.5517 \\ - 0.5139 & - 0.2499 & 0.5517 & 0.1167 \end{matrix}]

(53)

{\begin{matrix} \begin{matrix} F_{1} (x) = & {[\begin{matrix} 0.051733 x_{1}^{2} + 0.051733 x_{2}^{2} + 0.051733 x_{3}^{2} + 0.12435 x_{4}^{2} + 2149.9383 \\ 0.025157 x_{1}^{2} + 0.025157 x_{2}^{2} + 0.025157 x_{3}^{2} + 0.053176 x_{4}^{2} + 1092.694 \\ - 0.055541 x_{1}^{2} - 0.055541 x_{2}^{2} - 0.055541 x_{3}^{2} - 0.10555 x_{4}^{2} - 2416.7412 \\ - 0.011751 x_{1}^{2} - 0.011751 x_{2}^{2} - 0.011751 x_{3}^{2} - 0.01885 x_{4}^{2} - 522.0697 \end{matrix}]}^{T} \\ F_{2} (x) = & {[\begin{matrix} 0.051733 x_{1}^{2} + 0.051733 x_{2}^{2} + 0.051733 x_{3}^{2} + 0.12281 x_{4}^{2} + 2144.8848 \\ 0.025157 x_{1}^{2} + 0.025157 x_{2}^{2} + 0.025157 x_{3}^{2} + 0.052661 x_{4}^{2} + 1092.0548 \\ - 0.055541 x_{1}^{2} - 0.055541 x_{2}^{2} - 0.055541 x_{3}^{2} - 0.1047 x_{4}^{2} - 2417.4309 \\ - 0.011751 x_{1}^{2} - 0.011751 x_{2}^{2} - 0.011751 x_{3}^{2} - 0.018743 x_{4}^{2} - 521.9315 \end{matrix}]}^{T} \end{matrix} \end{matrix}

(54)

Let $x (0) = [0.25, 0, 0.5, 0]^{T}$ . Figures 3 and 4 illustrate the closed-loop ball-and-beam system states evolutions and control efforts obtained by the proposed approach (solid line), the approach presented by Liu et al. (2016a) (dashed line) and the approach presented by Vafamand et al. (2016) (dot-dash line). The results indicate that the settling times of each closed-loop system states derived by the proposed approach are less than the ones obtained based on the work by Liu et al. (2016a) and Vafamand et al. (2016). Moreover, Table 2 indicates the exact values of the states settling times of the mentioned approaches. As it can be seen in Table 2, the proposed approach significantly improves the closed-loop transient performance in the presence of the input saturation constraint. Furthermore, the Integral of absolute magnitude of the error (IAE) and the integral of the square of the error (ISE) performances of the proposed approach, the approach proposed by Liu et al. (2016a) and the approach proposed by Vafamand et al. (2016) are provided in Table 3. The IAE and ISE performances are computed for the error of the state vector. Also, since the error is persistent bounded, the truncated of the error signal is used.

Figure 3.

Evolution of the control signal of the ball-and-beam system for Scenario A. (Proposed approach: ‘—’, approach by Liu et al. (2016a): ‘- - -’ and approach by Vafamand et al. (2016): ‘- · -’).

Figure 4.

Evolutions of the state variables of the ball-and-beam system for Scenario A: (a) the $x_{1} (t)$ ; (b) the $x_{2} (t)$ ; (c) the $x_{3} (t)$ ; (d) the $x_{4} (t)$ . (Proposed approach: ‘—’, approach by Liu et al. (2016a): `- - -' and approach by Vafamand et al. (2016): ‘- · -’).

Table 2.

Settling time of the state variables for the ball-and-beam system for Scenario A.

	$x_{1}$	$x_{2}$	$x_{3}$	$x_{4}$
Proposed approach (s)	6.00	5.46	5.74	5.13
The approach by Liu et al. (2016a) (s)	12.73	13.50	12.48	11.59
Transient improvement (%)	112.36	147.26	117.48	125.82
The approach by Vafamand et al. (2016) (s)	16.84	14.46	9.64	7.04
Transient improvement (%)	181.23	165.15	68.19	37.40

Table 3.

IAE and ISE for the proposed approach, the approach by Liu et al. (2016a) and the approach by Vafamand et al. (2016).

	IAE	ISE
Proposed approach	0.2931	0.4401
The approach by Liu et al. (2016a)	0.8090	2.8212
The approach by Vafamand et al. (2016)	1.4287	4.5800

Scenario B

In this scenario, the effect of the uncertainty on the resilient controller is investigated. Assume that, the non-fragile controller is affected by the uncertainty $Δ F_{i} = H_{ci} (x) Δ_{c} (x) N_{ci}$ with the following parameters

\begin{matrix} H_{c 1} = & H_{c 2} = 1, Δ_{c} (x) = \sin (0.1 t) \\ N_{c 1} = & N_{c 2} = [\begin{matrix} 1 & 1 & 1 & 1 \end{matrix}] \end{matrix}

(55)

By selecting the same parameters as those given in Scenario A, together with the controller uncertainties given by equation (55) in Theorem 1, the controller gain matrices are achieved in equations (56) and (57)

P = [\begin{matrix} 5.2526, 1.9346, - 3.5175, - 0.5018 \\ 1.9346, 0.8695, - 1.6396, - 0.2483 \\ - 3.5175, - 1.6396, 3.3957, 0.5759 \\ - 0.5018, - 0.2483, 0.5759, 0.1299 \end{matrix}]

(56)

{\begin{matrix} \begin{matrix} F_{1} (x) = & {[\begin{matrix} 0.064037 x_{1}^{2} + 0.064043 x_{2}^{2} + 0.064037 x_{3}^{2} + 0.15172 x_{4}^{2} + 1747.551 \\ 0.03169 x_{1}^{2} + 0.031693 x_{2}^{2} + 0.03169 x_{3}^{2} + 0.066507 x_{4}^{2} + 885.0311 \\ - 0.073494 x_{1}^{2} - 0.0735 x_{2}^{2} - 0.073494 x_{3}^{2} - 0.13907 x_{4}^{2} - 2054.7885 \\ - 0.016576 x_{1}^{2} - 0.016577 x_{2}^{2} - 0.016576 x_{3}^{2} - 0.026462 x_{4}^{2} - 468.9189 \end{matrix}]}^{T} \\ F_{2} (x) = & {[\begin{matrix} 0.064032 x_{1}^{2} + 0.064037 x_{2}^{2} + 0.064037 x_{3}^{2} + 0.15168 x_{4}^{2} + 1744.8647 \\ 0.031688 x_{1}^{2} + 0.03169 x_{2}^{2} + 0.03169 x_{3}^{2} + 0.066491 x_{4}^{2} + 883.8822 \\ - 0.073488 x_{1}^{2} - 0.073494 x_{2}^{2} - 0.073494 x_{3}^{2} - 0.13903 x_{4}^{2} - 2052.9072 \\ - 0.016574 x_{1}^{2} - 0.016576 x_{2}^{2} - 0.016576 x_{3}^{2} - 0.026456 x_{4}^{2} - 468.4379 \end{matrix}]}^{T} \end{matrix} \end{matrix}

(57)

Figures 5 and 6 show the evolution of the control signal and state variables, respectively. It is obvious that the proposed technique is robust against the parameter uncertainties of the control gain matrices. Furthermore, Figures 5 and 6 illustrate the superiority of the proposed non-fragile controller compared to the work by Liu et al. (2016a) (dashed line) and Vafamand et al. (2016) (dot-dash line). Moreover, to clear the differences between the proposed approach and the state-of-the-art methods given by Liu et al. (2016a) and Vafamand et al. (2016), Table 4 is presented.

Figure 5.

Evolution of the control signal of the ball-and-beam system for Scenario B. (Proposed approach: “—”, Liu et al. (2016a): “- - -”, and Vafamand et al. (2016): “- · -”.)

Figure 6.

Evolutions of the state variables of the ball-and-beam system for Scenario B: (a). The $x_{1} (t)$ . (b). The $x_{2} (t)$ . (c). The $x_{3} (t)$ . (d). The $x_{4} (t)$ . (Proposed approach: “—”, Liu et al. (2016a): “- - -”, and Vafamand et al. (2016): “- · -”.)

Table 4.

Settling time of the state variables for the ball-and-beam system for Scenario B.

	$x_{1}$	$x_{2}$	$x_{3}$	$x_{4}$
Proposed approach (s)	6.29	6.66	6.05	5.40
The approach by Liu et al. (2016a) (s)	31.99	31.46	30.52	31.06
Transient improvement (%)	408.47	372.23	404.36	475.57
The approach by Vafamand et al. (2016) (s)	13.14	13.73	9.67	7.73
Transient improvement (%)	108.9	106.3	59.9	43.2

Scenario C

The aim of Scenario C is to analyse the effect of the system uncertainty on the proposed resilient controller. To achieve this goal, the uncertainty is considered to be

\begin{matrix} H_{11} = H_{12} {= [1, 1, 1, 1]}^{T}, N_{11} = N_{12} = [1, 1, 1, 1] \\ N_{21} = N_{22} = N_{31} = N_{32} = 1, Δ (x) = \sin (0.1 t) \end{matrix}

(58)

In addition, the output uncertainties are considered to be zero. Let $ϵ = 0.01$ , $α = 0.05$ and $ρ = 0.1$ in Theorem 1. The matrices $P (\tilde{x})$ and $F_{j} (x), \forall j \in {1, 2}$ are obtained in equations (59) and (60)

P (\tilde{x}) = [\begin{matrix} 23.7300 & 20.3017 & - 36.8505 & - 4.0111 \\ 20.3017 & 18.1138 & - 33.6976 & - 3.8329 \\ - 36.8505 & - 33.6976 & 67.7643 & 9.2994 \\ - 4.0111 & - 3.8329 & 9.2994 & 2.6826 \end{matrix}]

(59)

(\begin{matrix} \begin{matrix} F_{1} (x) = {[\begin{matrix} 0.80541 x_{1}^{2} + 0.80548 x_{2}^{2} + 0.80539 x_{3}^{2} + 0.52125 x_{4}^{2} + 1.3927 \\ 0.76945 x_{1}^{2} + 0.76952 x_{2}^{2} + 0.76943 x_{3}^{2} + 0.51823 x_{4}^{2} + 1.2652 \\ (\begin{matrix} - 1.867 x_{1}^{2} - 0.0163 x_{1} x_{2} - 0.016 x_{1} x_{3} + 0.0127 x_{1} x_{4} - 1.867 x_{2}^{2} \\ - 0.016 x_{2} x_{3} + 0.0125 x_{2} x_{4} - 1.867 x_{3}^{2} + 0.0126 x_{3} x_{4} - 1.4 x_{4}^{2} - 7.953 \end{matrix}) \\ - 0.53878 x_{1}^{2} - 0.53881 x_{2}^{2} - 0.53879 x_{3}^{2} - 0.48513 x_{4}^{2} - 3.3206 \end{matrix}]}^{T} \\ F_{2} (x) = {[\begin{matrix} 0.804 x_{1}^{2} - 0.0121 x_{1} x_{4} + 0.80 x_{2}^{2} - 0.0121 x_{2} x_{4} + 0.804 x_{3}^{2} - 0.0121 x_{3} x_{4} + 0.521 x_{4}^{2} + 1.339 \\ 0.768 x_{1}^{2} - 0.0107 x_{1} x_{4} + 0.768 x_{2}^{2} - 0.0107 x_{2} x_{4} + 0.768 x_{3}^{2} - 0.0107 x_{3} x_{4} + 0.518 x_{4}^{2} + 1.218 \\ (\begin{matrix} - 1.864 x_{1}^{2} - 0.010636 x_{1} x_{2} - 0.010585 x_{1} x_{3} + 0.017466 x_{1} x_{4} - 1.864 x_{2}^{2} - \\ 0.010572 x_{2} x_{3} + 0.017427 x_{2} x_{4} - 1.8638 x_{3}^{2} + 0.01746 x_{3} x_{4} - 1.3993 x_{4}^{2} - 7.8648 \end{matrix}) \\ - 0.53845 x_{1}^{2} - 0.53846 x_{2}^{2} - 0.53842 x_{3}^{2} - 0.48506 x_{4}^{2} - 3.3101 \end{matrix}]}^{T} \end{matrix} \end{matrix}

(60)

The simulations are done based on the calculated gain matrices given by equation (60), the controller uncertainty given by equation (55) and the system uncertainty given by equation (58). Figures 7 and 8 illustrate the control signal and the state variables of the non-linear ball-and-beam system, respectively. The proposed approach and the approach by Liu et al. (2016a) are illustrated by the solid blue and dashed red colors, respectively. Note that the approach by Vafamand et al. (2016) cannot stabilize the non-linear system with the given uncertainties and actuator saturation limit.

Figure 7.

Evolution of the control signal of the ball-and-beam system for Scenario C. (Proposed approach: ‘—’ and the approach by Liu et al. (2016a): ‘- - -’.)

Figure 8.

Evolutions of the state variables of the ball-and-beam system for Scenario C: (a) the $x_{1} (t)$ ; (b) the $x_{2} (t)$ ; (c) the $x_{3} (t)$ ; (d) the $x_{4} (t)$ . (Proposed approach: ‘—’ and the approach by Liu et al. (2016a): ‘- - -’.)

Figure 9.

Evolution of the control signal of the ball-and-beam system for Scenario D. (Proposed approach: ‘—’ and the approach by Liu et al. (2016a): ‘- - -’.)

Figure 10.

Evolutions of the state variables of the ball-and-beam system for Scenario D: (a) the $x_{1} (t)$ ; (b) the $x_{2} (t)$ ; (c) the $x_{3} (t)$ ; (d) the $x_{4} (t)$ . (Proposed approach: ‘—’ and the approach by Liu et al. (2016a): ‘- - -’.)

Figure 11.

Evolution of the control signal of the ball-and-beam system for Scenario D (special case) with white noise into the system. (Proposed approach: ‘—’ and the approach by Liu et al. (2016a): ‘- - -’.)

Figure 12.

Evolutions of the state variables of the ball-and-beam system for Scenario D (special case) with white noise into the system. (Proposed approach: ‘—’ and the approach by Liu et al. (2016a): ‘- - -’.)

Scenario D

In this scenario, all of the destructive effects of all uncertainties and input saturation constraints such as: control saturation, non-fragile controller, system uncertainty and persistent bounded disturbances have been considered in the numerical simulation. In respect to the persistent bounded disturbance, the system matrices $E_{1} = E_{2} = [1, 1, 1, 1]^{T}$ are added to the polynomial fuzzy model given by equation (52). The disturbance is considered to be $ν = 0.1$ . Also, the numerical value of the parameters are $ϵ = 0.01$ and $α = ρ = 0.1$ . The other parameters of Theorem 1 are chosen to be the same as Scenario C. Employing Theorem 1 and the third-party MATLAB toolbox SOS, the controller gain matrices under the above hard constraints are obtained in equations (61) and (62)

P (\tilde{x}) = [\begin{matrix} 14.4927 & 12.6950 & - 22.4170 & - 2.7603 \\ 12.6950 & 12.2726 & - 22.6683 & - 3.0370 \\ - 22.4170 & - 22.6683 & 47.6007 & 7.9396 \\ - 2.7603 & - 3.0370 & 7.9396 & 2.8649 \end{matrix}]

(61)

(\begin{matrix} \begin{matrix} F_{1} (x) = & {[\begin{matrix} 0.54892 x_{1}^{2} + 0.54888 x_{2}^{2} + 0.54894 x_{3}^{2} + 0.53366 x_{4}^{2} + 0.44978 \\ 0.60377 x_{1}^{2} + 0.60372 x_{2}^{2} + 0.60378 x_{3}^{2} + 0.58742 x_{4}^{2} + 0.61117 \\ - 1.578 x_{1}^{2} - 0.0109 x_{1} x_{2} - 0.011 x_{1} x_{3} - 1.578 x_{2}^{2} - 0.0111 x_{2} x_{3} - 1.578 x_{3}^{2} - 1.543 x_{4}^{2} - 6.831 \\ - 0.56973 x_{1}^{2} - 0.56971 x_{2}^{2} - 0.56973 x_{3}^{2} - 0.56439 x_{4}^{2} - 3.3927 \end{matrix}]}^{T} \\ F_{2} (x) = & {[\begin{matrix} 0.54785 x_{1}^{2} + 0.54784 x_{2}^{2} + 0.54786 x_{3}^{2} + 0.53297 x_{4}^{2} + 0.41967 \\ 0.60272 x_{1}^{2} + 0.60271 x_{2}^{2} + 0.60273 x_{3}^{2} + 0.58676 x_{4}^{2} + 0.58416 \\ - 1.576 x_{1}^{2} + 0.012 x_{1} x_{4} - 1.576 x_{2}^{2} + 0.01252 x_{2} x_{4} - 1.576 x_{3}^{2} + 0.01248 x_{3} x_{4} - 1.542 x_{4}^{2} - 6.784 \\ - 0.5694 x_{1}^{2} - 0.56939 x_{2}^{2} - 0.5694 x_{3}^{2} - 0.5642 x_{4}^{2} - 3.3864 \end{matrix}]}^{T} \end{matrix} \end{matrix}

(62)

The designed matrices given by equation (62) are used to construct a saturated non-fragile controller for an uncertain polynomial fuzzy model subject to the persistent bounded disturbance. The evolution of the controller input signal and state variables are shown in Figures 9 and 10, respectively. The accuracy and performance of the controller in the presence of uncertainty, disturbance and input saturation are observed in Figures 9 and 10. In comparison with the work by Liu et al. (2016a), the proposed approach has less settling time. Based on the proposed approach, the norm- $\infty$ of the output and disturbance are calculated as $0.04642$ and $0.1$ , respectively. Hence, the $\infty$ norm of the output to the $\infty$ norm of the disturbance computed in the simulation is $0.4642$ which is less than the computed $γ = 0.5$ in Theorem 1. In order to provide a more challenging situation, it will be considered that the white noise is affecting the proposed controller and also the state variables. White noise can be regarded as the un-modelled dynamics and sensor faults (Vafamand et al., 2016). In this case, white noise with a standard deviation 0.1 is added to the system dynamic and the controller. Figures 10 and 11 demonstrate the evolution of signal control and state variable in the presence of white noise.

Conclusions

In this paper, we have studied the problem of designing a saturated non-fragile polynomial fuzzy controller which is robust against the uncertainty of the persistent bounded disturbance non-linear system. The non-fragile controller was described by fractional uncertainties. The proposed controller guaranteed the $L_{1}$ performance of the closed-loop system. Sufficient SOS-based conditions were obtained and numerically solved by the third-party MATLAB toolbox SOSTOOLS. Finally, we considered a ball-and-beam application. The simulation results performed on an RTS-system clearly illustrated the effectiveness and merits of the proposed approach.

Footnotes

Appendix

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.

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