An anti-windup method for a class of uncertain MIMO systems subject to actuator saturation with LADRC

doi:10.1016/j.ins.2018.06.039

Information Sciences

Volume 462, September 2018, Pages 417-429

https://doi.org/10.1016/j.ins.2018.06.039 Get rights and content

Abstract

In this paper, an LMI-based anti-windup approach is proposed for a class of uncertain multiple input multiple output (MIMO) systems subject to actuator saturation with linear active disturbance rejection controller (LADRC). A modified linear extended state observer (MLESO) is presented to deal with the actuator saturation and the observation error is proved to be bounded. Meanwhile, local asymptotic stability of the closed-loop system is achieved and in the meantime estimate of the basin of attraction is obtained through formulating anti-windup gain matrix K_c. Furthermore, estimate of basin of attraction of the closed-loop system is maximized through solving a convex optimization problem. Finally, numerical examples are provided to illustrate the effectiveness of this method.

Introduction

In recent years, actuator saturation as a strong nonlinearity has attracted considerable attention because it may degrade the performance and even induce instability, and exists extensively in practical systems [15], [16]. The most common method to deal with actuator saturation is called anti-windup. Anti-windup is first to design a control law to satisfy performance requirements without actuator saturation, then propose a method to stabilize the closed-loop system with actuator saturation being taken into account. Since global stability can not be obtained for a linear unstable open-loop system with actuator saturation, local stability results have been developed [2], [12], [18]. As a consequence, the anti-windup algorithms that can maximize the basins of attraction of the closed-loop systems have attracted significant interests. In [2], a technique based on anti-windup method to enlarge the basin of attraction was proposed. The anti-windup gain was formulated through solving an iterative optimization problem. A saturation-based switching [23] anti-windup design was proposed to obtain a lager basin of attraction in [12]. Silva and Tarbouriech [18] gave conditions in terms of LMI for the enlargement of domain of attraction. Many achievements have also been made for nonlinear systems with actuator saturation [4], [10], [13], [14], [21]. Due to the difficulty of analysis of nonlinear systems with actuator saturation, most results focused on particular nonlinear classes. However, these methods may be inapplicable because the control laws, designed for the case that actuator saturations are not activated, are based on accurate mathematical models which cannot be obtained in some cases.

On the other hand, recent years have witnessed a rapidly growing interest in Active Disturbance Rejection Controller (ADRC) which is a model-independent controller and was proposed originally in [6], [8]. ADRC introduces a nontrivial innovation in philosophy that parameter uncertainties and external disturbances are lumped as a total disturbance which can be observed and compensated actively by using an extended state observer (ESO). Controlled plants, no matter linear or nonlinear, can be transformed to linear forms provided that the ESOs work properly. Then a control law is designed based on this generated model to achieve the performance requirement. Therefore, this controller has the benefit of generalizability and is model-independent. Much significant progress in the theoretical analysis of ADRC has been made in recent years [3], [7], [11], [27], [28]. More uplifting is that ADRC has been used in industrial products [19], [22], [26].

LADRC is a special class of ADRC and have been extensively studied. However, less attention has been paid to actuator saturation in LADRC. In this paper, we carry out an MLESO to deal with actuator saturation and the boundedness of observation error is obtained. Because the closed-loop system may be unstable when actuator saturation happens, it is significant to obtain and maximize the estimate of basin of attraction of closed-loop system. By using a quadratic candidate Lyapunov function, an LMI-based algorithm is established to formulate the anti-windup gain matrix K_c. This algorithm guarantees that the closed-loop system is local asymptotical stable and provides the estimate of basin of attraction. Furthermore, an optimization criterion is utilized to maximize this estimate. The structure of the closed-loop system is shown in Fig. 1.

This paper is organized as follows. The problem and MLESO are presented in Section 2. The boundedness of the observation error of MLESO is determined in Section 3. Meanwhile, the stabilization of closed-loop system is studied in this section. In Section 4, an algorithm to formulate the anti-windup gain matrix to maximize the estimate of basin of attraction of closed-loop system is given. Numerical examples are given to illustrate the effectiveness of this method in Section 5. Finally, Section 6 offers concluding remarks.

Notation. λ_min(M) denotes the minimum eigenvalue of matrix M. R is the set of real numbers. I and 0 denote the identity and zero matrix of appropriate dimension, respectively. For symmetric matrix A and B, A > 0(A ≥ 0) or A > B(A ≥ B) means A or $A - B$ is positive (semi-) definite.

Section snippets

Problem statement and controller design

Consider the following uncertain MIMO system with actuator saturation, $y^{(n)} = f (y, \dot{y}, \dots, y^{(n - 1)}) + b sat (u),$ where y ∈ R^m is the system output, u ∈ R^q is the controller output, b ∈ R^m × q, f represents the unknown nonlinear dynamics of system, and sat(u) is defined as $s a t (u) = {\begin{matrix} - u_{m} & u \leq - u_{m}, \\ u & - u_{m} < u < u_{m}, \\ u_{m} & u \geq u_{m} . \end{matrix}$ The differential Eq. (1) can be rewritten as $y^{(n)} = f (y, \dot{y}, \dots, y^{(n - 1)}) + b sat (u) - b_{0} sat (u) + b_{0} sat (u)$ where b₀ ∈ R^m × q is a constant matrix to be determined.

Let $F (y, \dot{y}, \dots, y^{(n - 1)}, u) = f (y, \dot{y}, \dots, y^{(n - 1)}) + b sat (u) - b_{0} sat (u)$

Convergence analysis and stabilization of the closed-loop system

The MLESO can estimate the states of system accurately under the condition that the estimation error of MLESO is bounded. Meanwhile, formulating K_c to stabilize the closed-loop system is an important premise for the system working properly. The following assumptions are made for these two problems.

Assumption

(1)
$∥ u - s a t (u) ∥$ and ‖h‖ are bounded.
(2)
$x_{n + 1} \in C^{1} (R^{2 m n}, R^{m}),$ $\tilde{Z_{2}} \in C^{1} (R^{2 m n}, R^{m})$ and $∥ (\begin{matrix} \frac{\partial x_{n + 1}}{\partial \tilde{x}} & \frac{\partial x_{n + 1}}{\partial {\tilde{Z}}_{1}} \\ \frac{\partial {\tilde{Z}}_{2}}{\partial \tilde{x}} & \frac{\partial {\tilde{Z}}_{2}}{\partial {\tilde{Z}}_{1}} \end{matrix}) ∥ \leq ϕ$ for η in the basin of attraction containing the origin.
(3)
If $\tilde{x} = 0$ and ${\tilde{Z}}_{1} = 0,$ $x_{n + 1} = 0$ and ${\tilde{Z}}_{2} = 0$ .

Maximization of the estimate of basin of attraction

Based on the method given in Theorem 3, an algorithm to maximize the estimate of basin of attraction is given in this section. Motivated by Silva and Tarbouriech [18], the key idea is first to define an initial polyhedral set Θ which is a subset of the local region of asymptotic stability η(P). Then, on the premise of αΘ ∈ η(P), obtain W_P and K_c corresponding to the maximum estimate of basin of attraction through finding the maximum positive scalar α.

Let Θ be defined as the initial polyhedral

Numerical studies

Example 1

Consider the following linear open-loop unstable system, ${\begin{matrix} \dot{x} = A_{n} x + B_{n} s a t (u), \\ y = C_{n} x \end{matrix}$ where $A_{n} = [\begin{matrix} 0.1 & - 0.1 \\ 0.1 & - 0.3 \end{matrix}],$ $B_{n} = [\begin{matrix} 5 & 0 \\ 0 & 5 \end{matrix}],$ $C_{n} = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]$ . The system can be rewritten in the following form under the scenario that A_n and B_n can’t be obtained in practice, $\dot{x} = f + b_{0 n} sat (u) .$ Then, the open-loop system and MLESO can be written in the form of (5) and (8), correspondingly. The task of this experiment is to make output y converge to ${[\begin{matrix} 0; & 0 \end{matrix}]}^{T}$ with the initial condition ${[\begin{matrix} - 5; & 5 \end{matrix}]}^{T}$ . The system parameters are given as

Conclusion

In this paper, the MLESO has been presented to deal with actuator saturation and the boundedness of observation error is obtained. An algorithm has been proposed to obtain and to maximize the estimate of basin of attraction through formulating the anti-windup matrix K_c. The local asymptotic stability of closed-loop system has been established through this algorithm. Moreover, the algorithm can be extended to deal with general systems, not limited to the systems incorporating LADRC. It should be

Acknowledgments

The authors would like to thank Prof. Ziqiang Lang of the University of Sheffield, Sheffield, S1 3JD, U.K. for his great help. This work was supported by National Natural Science Foundation of China under Grants 61374072, 61325014 and 61773086.

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An anti-windup method for a class of uncertain MIMO systems subject to actuator saturation with LADRC

Abstract

Introduction

Section snippets

Problem statement and controller design

Convergence analysis and stabilization of the closed-loop system

Maximization of the estimate of basin of attraction

Numerical studies

Conclusion

Acknowledgments

Syst. Control Lett.

Syst. Control Lett.

Inf. Sci. (Ny)

Inf. Sci. (Ny)

Inf. Sci. (Ny)

ISA Trans.

Linear Matrix Inequalities in System and Control Theory

An antiwindup approach to enlarging domain of attraction for linear systems subject to actuator saturation

IEEE Trans. Autom. Control

Advances in disturbance/uncertainty estimation and attenuation

IEEE Trans. Ind. Electron.

Auto disturbances rejection control technique

Front. Sci.

A simulative study on active disturbance rejection control (ADRC) as a control tool for practitioners

Electr. (Basel)

Anti-windup synthesis for nonlinear dynamic inversion control schemes

Int. J. Robust Nonlinear Control