Brief paperSynthesis of dynamic output feedback RMPC with saturated inputs☆
Introduction
In the last decade, model predictive control (MPC) and robust MPC (RMPC) have attracted extensive attention from both academic and industrial communities. For RMPC, many previous works assumed that system states were measurable online (see, e.g., Kothare, Balakrishnan, and Morari (1996), Li, Xi, and Zheng (2009), Lu and Arkun (2000), Wan and Kothare (2003)). But in practical applications, system states are not always measurable. Some researchers have paid attention to this issue. In Lee and Kouvaritakis (2001) and Mayne, Rakovic, Findeisen, and Allgower (2006), a state observer was adopted to design the output feedback RMPC (OFRMPC) for a linear time-invariant (LTI) system with bounded disturbance, where the state observer was given in advance (i.e. the parameters of the state observer were fixed) and the control inputs were optimized online by OFRMPC. A similar method was also adopted by Ding, Xi, Cychowski, and O’Mahony (2008) and Kim, Park, and Sugie (2006) for polytopic uncertain systems. On the other hand, Bemporad and Garulli (2000) used the dynamic output feedback control (DOFC) approach to design OFRMPC for LTI systems with bounded disturbance. Recently, Ding (2010) designed OFRMPC based on DOFC for polytopic uncertain systems with bounded disturbance and online known system parameters.
In most literature on OFRMPC for systems with plant uncertainties, an observer or a DOFC is usually adopted and the augmented system constructed by the estimated states and estimated errors is formulated to analyze and design the OFRMPC, such as Ding (2010) and Ding et al. (2008). But the unknown estimated errors and uncertainties make these designs difficult to fully utilize the capability of actuators, which was clearly reflected by the simulation results of Ding (2010) and Ding et al. (2008). This means that the control performance of these designs was subject to more stringent input constraints than they should be. Obviously, compared with other designs (see, e.g., Cao and Lin (2005), Huang, Li, Lin, and Xi (2011), Kothare et al. (1996) and Li et al. (2009)), this is a conservative point for OFRMPC design, and overcoming it can lead to improved performance, which motivates this paper.
In this paper, the synthesis of OFRMPC for polytopic uncertain systems is proposed. After introducing a new freedom of design to reduce the conservativeness, the DOFC design in Bemporad and Garulli (2000) is applied to polytopic uncertain systems. Based on it, an augmented system is constructed to design the OFRMPC. In order to fully utilize the capability of actuators, the saturation function is applied to the output of the DOFC, i.e. the control input of the controlled plant. Thus, the input constraints of the original system are satisfied naturally. This enables us to design the OFRMPC control law to achieve better performance without considering the input constraints directly. The whole design of the OFRMPC is based on the idea of an invariant set and formulated as an optimization problem with LMIs, which was also adopted by Ding (2010), Ding et al. (2008) and Kothare et al. (1996). The recursive feasibility and closed-loop stability of the proposed OFRMPC are proved to be guaranteed. Furthermore, an algorithm to update the estimated error set online is developed to improve the control performance. In addition, a simplified algorithm is also presented to reduce the online computational burden of the proposed OFRMPC, which makes the design more practical.
This paper is organized as follows. Section 2 introduces the problem statement, the adopted DOFC, and some preliminary knowledge. Section 3 details the design of the OFRMPC with an online algorithm to update the estimated error set. The simplified design of the OFRMPC is also introduced in this section. For the proposed OFRMPC, Section 4 illustrates its effectiveness by a numerical example.
Notation Denote as the value of vector at time , predicted at time . , , and is the th element of vector . The symbol induces a symmetric structure of a symmetric matrix. In addition, 0 and are the zero and identity matrices with proper dimensions, respectively. The abbreviation co stands for convex hull. Also, for two integers , .
Section snippets
Problem statement
Consider the following system with plant uncertainties: where is an unmeasurable system state, is the control input, and is the measurable system output. Eq. (3) implies that the system model is time varying and belongs to a convex hull. The constraints on the system inputs and outputs are given as follows. Here, without loss of generality, we
Design of output feedback RMPC
In this section, we first formulate the original system controlled by DOFC (8)–(9) as an augmented system, which belongs to a convex hull according to Lemma 1. Then, the OFRMPC optimization problem is designed. Furthermore, the approach to update the estimated error set is also developed and is embedded into the OFRMPC optimization problem. Based on it, a simplified algorithm is suggested for the proposed OFRMPC at the end of this section to reduce the online computational burden, which makes
Numerical example
Consider the following system: where . This is a system with two vertices, whose system matrices are denoted as and , respectively, and one is unstable. The constraint is .
Choose parameters of DOFC (8)–(9) as , , , . In addition, and are identity matrices. The OFRMPC based on Algorithm 1 with , and
Conclusion
OFRMPC for a polytopic uncertain system was studied. The saturation function was applied to the output of a DOFC to fully utilize the capability of the actuator. This can lead to improved control performance. Meanwhile, an algorithm was also proposed to update the estimated error set online. The proposed OFRMPC was proved to be recursively feasible and stable. A simplified design of OFRMPC was also developed to make the proposed design more practical.
Dewei Li is an associated professor in the Department of Automation at Shanghai Jiao Tong University. He received his B.S. degree and his Ph.D. degree in Automation from Shanghai Jiao Tong University in 1993 and 2009, respectively. He worked as a postdoctoral researcher in Shanghai Jiao Tong University from 2009 to 2010. His research interests include predictive control and robust control.
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Dewei Li is an associated professor in the Department of Automation at Shanghai Jiao Tong University. He received his B.S. degree and his Ph.D. degree in Automation from Shanghai Jiao Tong University in 1993 and 2009, respectively. He worked as a postdoctoral researcher in Shanghai Jiao Tong University from 2009 to 2010. His research interests include predictive control and robust control.
Yugeng Xi received his Dr.-Ing. degree in automatic control from the Technical University Munich, Germany, in 1984. Since then, he has been with the Department of Automation, Shanghai Jiao Tong University, and as a professor since 1988. His research interests include predictive control, large scale and complex systems, and intelligent robotic systems. Currently, he is the Vice Chair of IFAC TC Large Scale Complex Systems, the Vice President of the Chinese Association of Automation and the Editor or Associate Editor of 11 academic journals, including Control Engineering Practice, Int. J. of Humanoid Robotics, and ACTA Automatica Sinica.
Furong Gao is a Professor in the Department of Chemical and Biomolecular Engineering at the Hong Kong University of Science and Technology (HKUST). He obtained his B.Eng. in Automation from East China Institute of Petroleum in 1985, and his M.Eng. and Ph.D. degrees in Chemical Engineering from McGill University, Montreal, Canada, in 1989 and 1993, respectively. He worked as a Senior Research Engineer at Moldflow International, Melbourne, Australia, from 1993 to 1995 before joining HKUST as a professor. His research interests include process monitoring and fault diagnosis, batch process control, polymer processing control, and optimization. He is a recipient of numerous best paper awards, and is currently on the Editorial Board of a number of journals such as Journal of Process Control.
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This work is supported by the National Natural Science Foundation of China (Grant No. 60934007, 61074060, 61104078). The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Mario Sznaier under the direction of Editor Roberto Tempo.
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