An off-line approach for output feedback robust model predictive control

Abstract

For constrained linear parameter varying systems subject to bounded disturbances and noises, this article investigates an off-line output feedback robust model predictive control approach. The sub-observer gains with robust positively invariant sets, and sub-controller gains with robust control invariant sets are simultaneously off-line optimized and stored in a look-up table. According to real-time estimation error bounds and estimated states, the time-varying sub-observer gains and sub-controller gains are on-line searched. The proposed off-line output feedback robust model predictive control approach with the guarantee of nested robust positively invariant sets and robust control invariant sets in theory reduces the on-line computational burden.

Introduction

Model predictive control (MPC) has been considered as an effective optimal control approach for its ability to explicitly dispose of physical constraints as well as multivariable control in a systematic way. MPC has been originally developed for chemical process control, where the controlled systems often have slow dynamics and a large number of constraints on control inputs and system states [1], [2], [3], [4]. The advantages of MPC have been widely accepted both in academia [5], [6], [7], [8], [9] and other engineering industries such as automotive industries [10], [11], [12], power systems [13], [14], [15], robotics applications [16], [17], [18], brain-machine interface systems [19], [20], cyber-physical systems [21], [22], and mechatronic applications [23]. However, most of real processes have system uncertainties (i.e., model uncertainties, external disturbances and/or noises), and the models of real systems are difficult to be described. Thus, MPC techniques have to be extended to incorporate system uncertainties. The studies of MPC optimization formulations that ensure recursive feasibility and stability in the presence of system uncertainties are often referred to as robust MPC (RMPC) [7], [8], [9]. In RMPC optimization problems, the predicted system future behaviors are often related to the current system true states. In some applications of RMPC approaches, it is either impossible to directly measure all system states or economically nonviable. Hence, some RMPC schemes rely on state estimation techniques for their deployments. These motivate extensive attentions on output feedback RMPC (OFRMPC) approaches, where the synthesis of robust state estimation and observer-based robust MPC control is often considered [24], [25], [26], [27], [28], [29], [30], [31].

Linear parameter varying (LPV) systems can approximate nonlinear systems and represent uncertain systems [32], [33]. For LPV systems subject to bounded disturbances, the min-max OFRMPC approaches (e.g., state feedback robust MPC approaches in [35], [36] and OFRMPC approaches in [26], [27], [28]) consider that all the possible future closed-loop system states predicted at the current sampling time are bounded within one common robust positively invariant (RPI) set, which incorporate system uncertainties and constraint satisfactions in optimizations via closed-loop system worst case predictions. In [30], [31], the on-line OFRMPC approaches consider quasi min-max optimizations to reduce the conservatism in control input optimizations. Compared with [28], [29], [31], the OFRMPC approach in [30] synthesizes the computations on observer and controller gains, and the refreshment of the estimation error bounds in one optimization problem. The optimization problems in [26], [27], [28], [29], [30], [31] are semi-definite programming (SDP) problems and can be solved via the linear matrix inequality (LMI) tool, where the computational burden increases with system dimensions and the number of sub-models in LPV systems.

In [34], [35], [36], [37], [38], [39], the off-line state feedback RMPC approaches reduce the computational burden by searching controller gains from the off-line look-up table. The time-varying controller gains are searched according to the ellipsoids with the closest containment of system true states. Different from the aforementioned off-line state feedback RMPC approaches, the off-line observer-based OFRMPC approaches in [29], [40], [41] and the off-line dynamic OFRMPC approaches in [42], [43], [44] consider the effects of estimation error bounds or system true state bounds on robust stability and physical constraints, which complicate the procedure of off-line OFRMPC algorithms. For off-line OFRMPC approaches, the main challenges are (1) the method on how to design the off-line look-up table with nested robust invariant sets and a larger feasible region; (2) the method on searching the time-varying observer gains and controller gains; and (3) the reduction on algorithm complexities of the search method for the off-line designed look-up table. The off-line OFRMPC algorithms in [41], [43] consider polyhedral estimation error constraints. The off-line OFRMPC approaches with ellipsoidal estimation error sets are investigated in [29], [40], [42]. The off-line OFRMPC approach with both polyhedral and ellipsoidal system true state bounds is investigated in [44], where polyhedral system true states are updated for tightening the ellipsoidal system true state bounds. The observer gains in [41] and sub-observer gains in [29] are fixed, and time-varying controller gains are searched according to the real-time estimated states and estimation error bounds. The off-line OFRMPC approaches in [29], [41], [42], [43], [44] are the extensions of [40] to LPV systems with bounded disturbances and noises, where time-varying estimation error bounds or system true state bounds are considered in the closed-loop stability and physical constraints. In [29], [41], [42], [43], [44], to off-line compute different sizes of ellipsoidal feasible regions for estimated states, a series of estimated states and estimation error sets should be generated in advance. However, the methods on how to generate these estimated states and estimation error sets are not clearly discussed. Moreover, for the OFRMPC approaches in [29], [41], [42], [43], [44], the ellipsoidal feasible regions for estimated states in the look-up table cannot guarantee to be nested in theory, which are disadvantageous for searching real-time controller gains or controller parameters.

The contribution of this paper is to design an off-line OFRMPC method for LPV systems subject to bounded disturbances and noises by generalizing the OFRMPC optimization in [30]. Firstly, a look-up table is off-line designed to save the optimized sub-observer gains with nested RPI sets and sub-controller gains with nested robust control invariant (RCI) sets. Then, based on time-varying estimation error bounds and estimated states, real-time sub-observer gains and sub-controller gains are on-line searched. Different from the off-line OFRMPC approaches in [29], [41], [42], [43], [44], the approach to design the look-up table is not required to pre-specify a set of estimated states and estimation error bounds. However, it is only required to pre-specify a reference estimation error set. Then, the nested RPI sets with the sub-observer gains for the estimation error system, and nested RCI sets with sub-controller gains for the state observer system can be iteratively computed in off-line OFRMPC optimizations. The on-line searched sub-observer gains ensure that estimation errors are convergent within non-increasing RPI sets, and the sub-controller gains steer the estimated states to the neighborhood of the origin. Different from the off-line OFRMPC approach in [29], both time-varying sub-observer gains and sub-controller gains are real-time searched. Compared with the off-line OFRMPC approach in [29], the feasible region for estimated states is larger and the conservatism in OFRMPC controller is reduced. Compared with the on-line OFRMPC approach in [30], the proposed off-line OFRMPC algorithm significantly reduces the on-line computational burden.

$\mathbf{Notations}$ Denote the sets of real numbers, positive real numbers and integers as $\mathbb{R}$, ${\mathbb{R}}_{+}$ and $\mathbb{I}$, respectively. A vector $v$ in $n$-dimensional Euclidean space is denoted as $v\in {\mathbb{R}}^n$. The notation ${\parallel v\parallel }_M\triangleq \sqrt{v^{\mathrm{T}}Mv}$ represents the weight Euclidean norm of the vector $v$, where $M$ is a symmetric weighting matrix. For two integers $t_1\le t_2$, denote ${\mathbb{I}}_{[t_1,t_2]}\triangleq \{t_1,t_1+1,\dots ,t_2\}$. For two integers $a$ and $b$, the notations $min(a,b)$ and $max(a,b)$ represent the minimal and maximal values of $a$ and $b$, respectively. A matrix $M\in \text{Co}\{·\}$ means that the matrix $M$ is the convex combination of the elements in the convex hull $\{·\}$. The notations $x_{i|k}$ are the predicted values of $x$ at time $k+i$, $i\ge 0$. The notation “$★$” denotes a symmetric structure in LMIs. For two matrices $M$ and $N$, the notations $M\ge 0$ and $N>0$, respectively, represent that the matrix $M$ is a symmetric positive semidefinite matrix and the matrix $N$ is a symmetric positive definite matrix. For a symmetric positive definite matrix $Q\in {\mathbb{R}}^{n\times n}$ and a scalar $\eta >0$, denote the ellipsoidal set $\mathcal{S}(Q,\eta )\triangleq \{\xi \in {\mathbb{R}}^n:{\xi }^{\mathrm{T}}Q\xi \le \eta \}$. The notation $\mathcal{S}(Q,1)$ is denoted as $\mathcal{S}\left(Q\right)$ for brevity. For two symmetric positive definite matrices $Q_1$ and $Q_2$, the sets $\mathcal{S}\left(Q_1\right)\subseteq \mathcal{S}\left(Q_2\right)$ when the matrices $Q_1\ge Q_2$. The Minkowski set addition of two sets is defined as ${\mathcal{S}}_1\oplus {\mathcal{S}}_2\triangleq \{s_1+s_2|s_1\in {\mathcal{S}}_1,s_2\in {\mathcal{S}}_2\}$. The notation $\text{diag}\{P_1,P_2\}$ represents a diagonal matrix composed of matrices $P_1$ and $P_2$. $I$ denotes a proper dimensional identity matrix.