Robust memory state feedback model predictive control for discrete-time uncertain state delayed systems
Introduction
The model predictive control (MPC) technique is a feedback strategy that has been widely adopted in industry and academia. The reason for its success is its ability to deal with input constraints and multi-variable systems [1], [2], [3], [4], [8], [10], [16], [18]. In order to implement the MPC algorithm, MPC requires the recursive solution of the min–max problem, which can be solved by a quadratic program or semi-definite program [1], [9], [10], [17]. The drawbacks of MPC are the difficulty of incorporating plant model uncertainties and time-delays. In most practical control systems, uncertainties and time-delays inevitably exist and often adversely affect performance and stability [11], [12], [13], [14], [15]. Recently, many researchers have focused on the robust MPC of time delay systems but only a few MPC algorithms have been developed to explicitly handle time-delayed systems [1], [5], [6], [7].
Kothare et al. [1] proposed a way to synthesize an MPC law that robustly stabilizes a closed loop system with model uncertainty, the authors assumed that the delay index was known. Kwon et al. [5] suggested the simple receding horizon control for continuous time systems with state delay, where the reduction technique was adopted so that an optimal problem for state delayed systems could be transformed into an optimal problem for delay free ordinary systems. The suggested control method can not guarantee the stability. Jeong and Park [6] proposed a constrained MPC for an uncertain time varying system with an unknown bounded state delay using memoryless state feedback. In order to implement MPC for an unknown delay index, the authors defined the optimization problem for a known upper bound of delay index and proved the equivalence of the two optimization problems between a known upper bound of the delay index and an unknown delay index. They then, proved feasibilities and closed loop stabilities using the equivalence property. However, the results are rather conservative because feedback consists of only state.
In this paper, we propose a new memory state feedback MPC algorithm for uncertain state delayed systems with unknown bounded delay and input constraint. The proposed method is derived from the minimization problem of infinite horizon cost by adopting memory state feedback with a delay index minimizing state estimation errors. Furthermore, we consider a delayed state dependent quadratic function so that the memory state feedback control can be applied to a discrete time uncertain state delayed system. A memory state feedback controller with input constraints yields less conservative sufficient conditions in terms of LMIs so that it allows wider feasible region of numerical optimization. By finding the feedback gain matrices and , system tracking performance is improved and also the upper bound of the cost is reduced. We consider an equivalent optimization problem to design MPC for discrete time uncertain systems with unknown bounded delay, and then solve the original optimization problem using the equivalence property [6]. Finally, we propose a robust memory state feedback MPC law with updating procedures and show the closed loop stability of the proposed method. Finally, a numerical example shows the effectiveness of the proposed method. Notation denotes the n-dimensional Euclidean space, and is the set of real matrix. represents the absolute value. refers to the Euclidean vector norm and the induced matrix norm. For symmetric matrices X and Y, the notation (respectively, ) means that the matrix is positive definite, (respectively, nonnegative). denotes the block diagonal matrix. represents the elements below the main diagonal of a symmetric matrix. I denotes an identity matrix with appropriate dimension. means the transpose of the matrix A. denotes .
Section snippets
Problem statement
Consider the following discrete-time uncertain state delayed systemssubject to input constraintswhere is the state, is the control input, and is the initial condition. d is an unknown constant integer representing the number of delay index in the state, but being assumed with the bounded value . The system matrix is uncertain but belongs to a polytope at
Main result
In this section, we propose a new MPC algorithm for an uncertain state delayed system (1), in which the gain matrices and of the memory state feedback controller (4) are determined from the new sufficient condition for cost monotonicity. The sufficient condition is derived using the delayed state dependent quadratic function. Comparing with previous results [6], the proposed method allows an improvement in system performance in the sense of minimizing the upper bound of the infinite
Numerical example
In this section, a numerical example is presented to illustrate the effectiveness of the proposed robust memory state feedback MPC algorithm. Let us consider the following discrete time uncertain state delayed system [6]:where and are time varying uncertain parameters andThe initial
Conclusions
In this paper, a robust memory state feedback MPC technique was proposed for uncertain state delayed systems with input constraints. The minimization problem for infinite horizon cost was derived using the delayed state Lyapunov function. The optimization problem is reformulated in the form of a finite number of LMIs. Numerical examples demonstrated the effectiveness of the proposed method.
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