Analysis of linear systems in the presence of actuator saturation and L2-disturbances

doi:10.1016/j.automatica.2004.02.009

Automatica

Volume 40, Issue 7, July 2004, Pages 1229-1238

https://doi.org/10.1016/j.automatica.2004.02.009 Get rights and content

Abstract

This paper presents a method for the analysis and control design of linear systems in the presence of actuator saturation and $L_{2}$ -disturbances. A simple condition is derived under which trajectories starting from an ellipsoid will remain inside an outer ellipsoid. The stability and disturbance tolerance/rejection ability of the closed-loop system under a given feedback law is measured by the size of these two ellipsoids and the difference between them. Based on the above mentioned condition, the problem of estimating the largest inner ellipsoid and/or the smallest difference between the two ellipsoid is then formulated as a constrained optimization problem. All the constraints are shown to be equivalent to LMIs. In addition, disturbance rejection ability in terms of $L_{2}$ gain is also determined by the solution of an LMI optimization problem. By viewing the feedback gain as an additional free parameter, the optimization problem can easily be adapted for controller design. Numerical examples show that the proposed analysis and design methods significantly improve recent results on the same problems.

Section snippets

Introduction and problem statement

As a natural research topic beyond stabilization, the problem of disturbance rejection for linear systems subject to actuator saturation has been addressed by many authors. The results on this topic can be divided into two categories according to the way the disturbances enter the system. Examples of works on systems with input additive disturbances include Chitour, Liu, and Sontag (1995), Hu and Lin (2001b), Lin (1997), Lin, Saberi, and Teel (1996), and Liu, Chitour, and Sontag (1996). Because

Stability and disturbance tolerance/rejection

Consider the closed-loop system of (1) under the state feedback u=Fx. In the presence of disturbance, the basic requirement for the closed-loop system is the boundedness of state trajectories. We usually use an ellipsoid to bound the state trajectories. In the case where the disturbance is bounded by the $L_{∞}$ norm, we may use an invariant ellipsoid to bound the state trajectory (see, e.g., Hu et al., 2002). However, with disturbances bounded by energy rather than by magnitude, there exists no

Controller synthesis

By viewing F as an additional free parameter, all the optimization problems in the previous section (Problems 1–4) can be adapted for the design of feedback gain F. In particular, by setting Z=FQ, all those LMI optimization problems remain as LMI optimization problems. Once these new LMI problems are solved, the feedback gain can then be computed as F=ZQ⁻¹.

Numerical examples

In this section, we will demonstrate the effectiveness of our methods by some numerical examples.

Example 1

Consider system (1) with $A= 0.6 −0.8 0.8 0.6, B= 24,E= 0.1 0.1, F= 1.2231 −2.2486 .$ By specifying an allowable magnitude of the input to the actuator, r, the algorithm proposed in Hindi and Boyd (1998) (see (27)) determines the largest tolerable disturbance with zero initial conditions, called the r-level disturbance rejection bound α_max,r. Shown in Fig. 1 is the r-level disturbance rejection bound as a function

Conclusions

In this paper, we present a simple condition under which any trajectories starting from an ellipsoid will remain inside an outer ellipsoid for linear systems subject to actuator saturation and $L_{2}$ -disturbances. Based on this condition, the assessment of system stability and disturbance tolerance/rejection can be formulated as optimization problems with LMI constraints. Meanwhile, these optimization problems can be easily adapted for the controller design. Furthermore, it was proved and/or shown

Haijun Fang was born in Xi'an, Shaanxi, China on May 4, 1976. He recieved his B.S. and M.S. degrees of Automation from Tsinghua University, Beijing, China in 1999 and 2001, repectively. He is now pursuing his Ph.D. degree in the department of electrical and computer engineering at University of Virginia.

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Zongli Lin received his B.S. degree in mathematics and computer science from Xiamen University, Xiamen, China, in 1983, his Master of Engineering degree in automatic control from Chinese Academy of Space Technology, Beijing, China, in 1989, and his Ph.D. degree in electrical and computer engineering from Washington State University, Pullman, Washington, in 1994.

Dr. Lin is currently an associate professor with the Department of Electrical and Computer Engineering at University of Virginia. Previously, he has worked as a control engineer at Chinese Academy of Space Technology and as an assistant professor with the Department of Applied Mathematics and Statistics at State University of New York at Stony Brook.

His current research interests include nonlinear control, robust control, and modeling and control of magnetic bearing systems. In these areas he has published several papers. He is also the author or co-author of three books, Low Gain Feedback (Springer-Verlag, London, 1998), Control Systems with Actuator Saturation: Analysis and Design (with Tingshu Hu, Birkhauser, Boston, 2001), and Linear Systems Theory: A Structural Decomposition Approach (with Ben M. Chen and Yacov Shamash, Birkhauser, Boston, 2004).

A senior member of IEEE, Dr. Lin served as an Associate Editor of IEEE Transactions on Automatic Control from 2001 to 2003. He is currently a member of the IEEE Control Systems Society's Technical Committee on Nonlinear Systems and Control and heads its Working Group on Control with Constraints. He is the recipient of a US Office of Naval Research Young Investigator Award.

Tingshu Hu received her B.S. and M.S. degrees in electrical engineering from Shanghai Jiao Tong University, and the Ph.D. degree in electrical engineering from University of Virginia, in 2001. She is currently a research associate at the Department of Electrical and Computer Engineering, University of Virginia. Her research interests include nonlinear systems theory, optimization, robust control theory and control application in mechatronic systems and biomechanical systems. She is a co-author (with Zongli Lin) of the book Control Systems with Actuator Saturation: Analysis and Design (Birkhauser, Boston, 2001). She is currently an associate editor on the Conference Editorial Board of the IEEE Control Systems Society.

^☆: This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associated Editor Faryar Jabbari under the direction of Editor Roberto Tempo. This work was supported in part by the US National Science Foundation under Grant cms-0324329.

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Brief paperAnalysis of linear systems in the presence of actuator saturation and L2-disturbances☆

Abstract

Section snippets

Introduction and problem statement

Stability and disturbance tolerance/rejection

Controller synthesis

Numerical examples

Conclusions

Automatica

Automatica

Regulator problem for linear continuous-time systems with nonsymmetrical constrained control

IEEE Transactions on Automatic Control

On the continuity and incremental-gain properties of certain saturated linear feedback loops

International Journal of Robust and Nonlinear Control

Practical stabilization on the null controllable region of exponentially unstable linear systems subject to actuator saturation nonlinearities and disturbance

International Journal of Robust and Nonlinear Control

Brief paper
Analysis of linear systems in the presence of actuator saturation and $L_{2}$ -disturbances☆