Brief paperOutput feedback robust MPC for linear systems with norm-bounded model uncertainty and disturbance☆
Introduction
Model predictive control (MPC), also known as receding horizon control, has gained notably attentions in both academia and industry due to its ability to optimally control nonlinear systems subject to physical constraints (Mayne et al., 2000, Morari and Lee, 1999). It is a model-based control technique that solves an optimal control problem at each sampling instant, based on an explicit model of the system. An explicit linear model is typically utilized in MPC because the on-line optimization problem would reduce to a linear or quadratic programming problem (Kouvaritakis et al., 2002, Shead et al., 2010) (sometimes even linear operations (Ghaemi, Sun, & Kolmanovsky, 2012)). However, in real industrial processes, there always exist various uncertainties and nonlinearities, so the theoretical analysis based on the nominal linear model may be impractical. For this reason, different approaches to model system uncertainties have been proposed, e.g., polytopic (represented by Linear Parameter-Varying (LPV) model (Bumroongsri and Kheawhom, 2012, Calafiore and Fagiano, 2013, Gautam et al., 2012, He et al., 2014, Zheng et al., 2013)) and norm-bounded (Casavola et al., 2004, Famularo and Franzè, 2011, Løvaas et al., 2008). Since the seminal work (Kothare, Balakrishnan, & Morari, 1996), Linear Matrix Inequality (LMI) based MPC has become one of the most efficient approaches to handle system with uncertainties and achieve satisfactory control performance (Cuzzola et al., 2002, Garone and Casavola, 2012, Gautam et al., 2012, Imsland et al., 2005, Kouvaritakis et al., 2000). In Imsland et al., 2005, Kouvaritakis et al., 2000, the control move is defined as state feedback plus perturbation. The state feedback gain is designed off-line, and the performance cost is a quadratic term on the perturbations. In Cuzzola et al. (2002), unlike in Kothare et al. (1996) which utilizes a single quadratic Lyapunov function, several quadratic Lyapunov functions, each corresponding to a vertex of the polytope, are utilized to extend the result. In Garone and Casavola (2012), the nonlinear parameter-dependent Lyapunov functions are adopted to reduce the conservativeness. In Gautam et al. (2012), a bounded disturbance is considered, and a so-called uncertainty-based dynamic control policy is used to reduce computational burden.
The assumption that the state is measurable is not generally practical. Indeed, in many applications, all states are not measurable. For this reason, output feedback MPC studies have gained many attentions due to its ability to tolerate state estimation errors (Ding, 2011a, Ding, 2011b, Hu and Ding, 2019, Li et al., 2013, Løvaas et al., 2008, Mayne et al., 2006, Park et al., 2011, Sato and Peaucelle, 2013). When the state is unmeasurable, the system output is utilized to estimate the state. Since the estimation error is unknown, it is advisable to replace it by its outer bound. Hence, the refreshment of this estimation error bound is crucial for guaranteeing the recursive feasibility of the optimization problem (Ding, 2011a). In Ding, 2011a, Ding, 2011b, the dynamic output feedback robust MPC approach is proposed, where the ellipsoid estimation error bound is properly refreshed at each sampling instant, so that the closed-loop system is proven as quadratically bounded. In Løvaas et al. (2008), by pre-specifying the feedback gain, an output feedback MPC approach for systems with unstructured model uncertainty is developed. In Famularo and Franzè (2011), the system with norm-bounded model parametric uncertainty and disturbance is considered, where the estimation error bound is pre-fixed as a constraint of the optimization problem which may bring some conservativeness. The resulting optimization problem is bilinear, so it has to be solved by iterations.
In this paper, we consider linear systems with norm-bounded model parametric uncertainty and disturbance. Usually for this type of problem, when the output feedback MPC approach is adopted, the overall optimization problem adopts both the state estimator gain and state feedback gain as the on-line decision variables (Famularo & Franzè, 2011). The resulting optimization problem needs to be solved by iterative method (e.g., the iterative cone complementary approach similar to Ding (2011a)) which is computationally demanding. In order to overcome this difficulty, we propose a new approach where the estimator state matrix replaces the state estimator gain as the decision variable in this paper. As this new approach does not utilize the bilinear formulation or iterative cone complementary approach, the computational burden can be greatly reduced. Furthermore, since the estimator state matrix has higher dimension than the state estimator gain, it is potential to improve the control performance. The invariance ellipsoid technique (firstly appearing in Kothare et al. (1996)) is applied to handle the state and input constraints. By properly refreshing the ellipsoid estimation error set, we show that the proposed approach is recursively feasible, and the augmented state is guaranteed to converge to the neighborhood of equilibrium with input and state physical constraints being consistently satisfied.
Notations: For any vector and positive-definite matrix , where is omitted when . is the value of at time , predicted at time . is the identity matrix with appropriate dimension. denotes the ellipsoid associated with the symmetric positive-definite matrix . All vector inequalities are interpreted in an element-wise sense. The symbol induces a symmetric structure in the matrix inequalities. A value with superscript means that it is the optimal solution of the optimization problem. The time-dependence of the MPC decision variables is often omitted for brevity.
Section snippets
Problem statement
Consider a class of discrete-time, uncertain, linear systems (Famularo & Franzè, 2011) described by where , , , and are, respectively, the state, input, output, unknown state disturbance and unknown output disturbance vectors. represents the uncertainties in system parameters.
Assumption 1 The pairs and , for all , are, respectively,Keerthi & Gilbert, 1988
Off-line state estimator design
In (5), by choosing (see Remark 4), we obtain the following augmented state dynamics: where . By utilizing (13), we can pre-specify an such that the estimation error is convergent. The disturbance-free system of (13) is Let the following condition be satisfied:
Numerical example
In this section, we present an example that illustrates the implementation of the proposed approach. The LMI Toolbox of Matlab 2017b (Intel Core i5 CPU 2.5 GHz, 8G Memory) is utilized for simulations.
Consider the two-mass–spring system introduced in Kothare et al. (1996). We add disturbance signals and that are randomly generated in the interval , and a multiplier . The parameters for system (1) are
Conclusions
This paper has considered the output feedback robust MPC problem for linear systems with both model parametric uncertainty and norm-bounded disturbance. We have proposed a new approach to solve this problem. The optimization problem is shown to be recursively feasible, and the closed-loop system is convergent. Notably, although the proposed approach is computationally efficient for the moderate dimensional models, it may be impractical for the high dimensional systems. We can apply the off-line
Acknowledgments
The authors would like to thank the Associate Editor and the anonymous reviewers for their constructive comments to improve this manuscript. This work was supported by the National Key R&D Program of China (Grant No. 2017YFA0700300), and by the National Natural Science Foundation of China (Grant No. 61573269).
Jianchen Hu was born in Xi’an, Shaanxi Province, China. He received the B.S. degree from Northwest University, Xi’an, China, in 2011 and the M.S. degree from Arizona State University, AZ, United States, in 2013. He is currently working towards the Ph.D. degree with Xi’an Jiaotong University. His research interests include predictive control, economic optimization and robust control.
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Jianchen Hu was born in Xi’an, Shaanxi Province, China. He received the B.S. degree from Northwest University, Xi’an, China, in 2011 and the M.S. degree from Arizona State University, AZ, United States, in 2013. He is currently working towards the Ph.D. degree with Xi’an Jiaotong University. His research interests include predictive control, economic optimization and robust control.
Baocang Ding was born in Hebei Province, China. He received the M.S. degree from the China University of Petroleum, Beijing, China, in 2000 and the Ph.D. degree from Shanghai Jiaotong University, Shanghai, China, in 2003. From September 2005 to September 2006, he was a Postdoctoral Research Fellow in Department of Chemical and Materials Engineering, University of Alberta, Canada. From November 2006 to August 2007, he was a Research Fellow in the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore. From September 2003 to August 2007, he was an Associate Professor in Hebei University of Technology, Tianjin, China. From September 2007 to December 2008, he was a Professor in Chongqing University, Chongqing, China. He is currently a Professor with Xi’an Jiaotong University, Xi’an, China. His research interests include predictive control, fuzzy control, networked control, and distributed control systems.
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The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Yoshio Ebihara under the direction of Editor Richard Middleton