Original article
An LMI-based composite nonlinear feedback terminal sliding-mode controller design for disturbed MIMO systems

https://doi.org/10.1016/j.matcom.2012.09.006Get rights and content

Abstract

This paper proposes a new nonlinear sliding surface for a terminal sliding mode (TSM) controller to achieve robustness and high performance tracking for the disturbed MIMO systems. The proposed method improves the transient performance and steady state accuracy in a finite time simultaneously. The control law is designed to guarantee the existence of the sliding mode around the nonlinear surface, and the damping ratio of the closed-loop system is increased as the output approaches the set-point. The conditions on the state error bound in finite time are expressed in the form of linear matrix inequalities (LMIs). A DC motor position tracking problem is considered as a case study. Simulation results are presented to show the effectiveness of the proposed method as a promising approach for controlling similar nonlinear systems.

Introduction

Stabilization and tracking problems are of the most significant problems that the researchers in the areas of linear and nonlinear control theory have considered [1], [27]. The stabilization problem addresses the convergence of system states to a bounded region containing the origin, and has been extensively studied for both linear and nonlinear systems [9]. The performance, robustness and disturbance rejection objectives have been considered for the tracking control strategies [27].

Sliding mode control (SMC) is a powerful robust method which has been successfully used for the control of linear and nonlinear systems [18]. In this method, a sliding surface is first defined, and then, a controller is designed to derive the system states to the sliding surface [13]. An important feature of SMC is the robustness of the closed-loop system against parameter variations and external disturbances after reaching the sliding mode [10]. Linear SMC has been widely used in the literature for asymptotic stabilization of the closed-loop system, which guarantees the system state approach the equilibrium in infinite time [7], [13]. Hence, the switching control may have unsatisfactory performance in finite time. To tackle this problem, finite-time control methods such as terminal sliding mode (TSM) control have been proposed using a fractional power term in the sliding surface [12], [22]. TSM method allows high precision control as it speeds up the rate of convergence near an equilibrium point. Unlike SMC, TSM is based on a set of recursive nonlinear non-smooth differential equations enabling finite time convergence [16]. However, if the initial states are in some particular areas, the singularity problem may occur in TSM which results in large control signals [19]. Nevertheless, most TSM-based works only concentrated on single-input and single-output (SISO) systems. A TSM structure for MIMO linear systems was proposed in [13] for the first time. However, in the TSM offered so far, convergence performance may be lower than that of SMC when the system states are far away from the equilibrium [13], [24].

In most design methods, there is a tradeoff between overshoot and settling time, and the damping ratio is usually a fixed number. Quick response provides high overshoot which is undesirable in many applications. This problem can be solved by using the composite nonlinear feedback (CNF) technique [8]. As explained in [4], a variable damping ratio improves the performance of the system significantly and can resolve the tradeoff. In [3], [4], [5], to improve the transient performance of the system including the overshoot and settling time, a nonlinear sliding surface is used that changes the damping ratio of the system from an initial low value to a final high value during the reaching phase. However, an algebraic Riccati equation (ARE) needs to be satisfied for stabilization of the system as a matching condition. In [15], the ARE and CNF techniques are used for robust tracking and model following of a class of linear systems with time-varying uncertainties and disturbances. However, such equality conditions result in a very conservative design. All the mentioned works require selecting many predefined variables, and the resulting conditions are not presented in LMI form, which are hard to solve.

LMI has emerged as a powerful computational design tool in systems and control engineering problems because of its computational efficiency and flexibility to treat a large class of system design problems [2], [21], [25]. It can be used to solve minimization convex problems such as guaranteed cost control [26], H control [6] and H2 control [6], [21]. An LMI is a semi-definite inequality that is linear in unknown variables. With the recent advances in convex optimization, efficient algorithms have been developed for solving the LMI's [20].

In this paper, a nonlinear sliding surface is developed to improve the transient and steady state responses for finite time control of disturbed MIMO systems. The conditions on the state error bound in finite time are expressed in the form of LMIs. As the output approaches to the desired set-point, the damping ratio of the closed-loop system is increased by adding a significant term to the reaching control input. Unlike the previous works, the resulting LMI conditions have much less pre-assumed design parameters; and thus, the proposed method yields less conservative conditions.

The paper is organized as follows: Section 2 contains the problem statement and the preliminaries. In Section 3, the stability analysis and design procedure of the tracker are discussed. The simulation results and the related discussions are provided in Section 4. Finally, some concluding remarks are given in Section 5.

Section snippets

The problem statement and preliminaries

Consider an uncertain MIMO system described by the following state equations [7]:z˙1(t)=A11z1(t)+A12z2(t),z˙2(t)=A21z1(t)+A22z2(t)+B2u(t)+d2(z,t),y(t)=Cz(t),where z(t)=[z1(t)z2(t)]T, z1(t)  Rnm and z2(t)  Rm are the state vectors, y(t)  Rp is the output of system, u(t)  Rm is the control input, d2(z, t) is the disturbance and unknown model mismatch, where d¯2 is a vector whose elements are the upper bounds of the corresponding elements of d2(z, t), i.e., d¯2(d2(z,t))max, Aij (i, j = 1, 2) are

Main results

In the first theorem in this section, based on the concepts stated in Section 2, a TSM design approach using CNF is provided that guarantees the state error during the sliding phase reaches to desired bound in finite time.

Theorem 1

Consider error system (6) with the nonlinear function ψ(e1(t)) =  ρ|e1(t)|η−1 where η is a ratio of two odd positive integers satisfying 1 > η > 0. Let r and g¯ be given positive constants, where g(t)g¯ and r is the desired bound of the error. If there exist scalars ρ > 0, κ1 > 0 and κ2

Simulation results

Consider the position tracking problem of a DC motor discussed in [23] with equations:z˙1(t)=0100.694z1(t)+0112.36z2(t),z˙2(t)=0161.8z1(t)1500z2(t)+200u(t)+20sin(10z2(t)t),y(t)=100z(t),where z1(t)=[θθ˙]T denotes the angular position and angular velocity of the motor and z2(t) = i is the armature current. The desired trajectory is zd(t)=[000]T. The constant parameters are selected as: d¯2=20, γ = 30, σ = 100, g¯=0.1, r = 0.05 and η = 13/17. For the simulation purpose, the initial conditions are set

Conclusions

This paper offered an LMI-based composite nonlinear feedback sliding-mode controller to improve the transient and steady state responses for a class of MIMO linear systems with time varying disturbances. The proposed method is able to obtain short settling time and high damping ratio simultaneously. The reaching law guarantees the existence of the sliding mode around the nonlinear sliding surface. The stability of the system was proven and system errors are bounded with a desired bound in a

Acknowledgement

The authors would like to appreciate Mr. Mohammad Hassan Asemani for his useful suggestions.

References (27)

  • B. Bandyopadhyay et al.

    Sliding mode control using novel sliding surfaces

    (2009)
  • B. Bandyopadhyay et al.

    A nonlinear sliding surface to improve performance of a discrete-time input-delay system

    International Journal of Control

    (2010)
  • R. Bellman

    Introduction to Matrix Analysis

    (1960)
  • Cited by (0)

    View full text