Journal of Applied Mathematics

Volume 2015, Article ID 875850, 12 pages

http://dx.doi.org/10.1155/2015/875850

## Dynamic Output Feedback Robust Model Predictive Control via Zonotopic Set-Membership Estimation for Constrained Quasi-LPV Systems

^{1}School of Electro-Mechanical Engineering, Xidian University, Xi’an 710071, China^{2}School of Electronic Engineering, Xidian University, Xi’an 710071, China

Received 5 July 2014; Revised 11 September 2014; Accepted 7 October 2014

Academic Editor: Dewei Li

Copyright © 2015 Xubin Ping and Ning Sun. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

For the quasi-linear parameter varying (quasi-LPV) system with bounded disturbance, a synthesis approach of dynamic output feedback robust model predictive control (OFRMPC) is investigated. The estimation error set is represented by a zonotope and refreshed by the zonotopic set-membership estimation method. By properly refreshing the estimation error set online, the bounds of true state at the next sampling time can be obtained. Furthermore, the feasibility of the main optimization problem at the next sampling time can be determined at the current time. A numerical example is given to illustrate the effectiveness of the approach.

#### 1. Introduction

Model predictive control (MPC) or receding horizon control is a class of optimization based control methods, which explicitly utilizes process model parameters and measurements to optimize a performance function. At the current time, the optimization problem is carried out and then generates the optimal control input sequence. However, only the first element of control inputs is implemented. At the next sampling time, the output measurement is updated and the prediction horizon is shifted one step forward, and then the optimization problem is repeated. Due to its ability to handle the systems with hard constraints, it has attracted many researchers’ attention, from both academic community and industrial society, for example, [1–4]. However, in real processes, the precise model parameters are seldomly available. Hence, robust MPC (RMPC) is more practical for real applications.

Linear parameter varying (LPV) systems are the systems whose parameters take their values in the prespecified sets and the dynamic characteristics depend on the time-varying parameters. When the time-varying parameters of LPV systems can be exactly known at the current time, the systems are called quasi-LPV (quasi-linear parameter varying) systems. It means that the model parameters are exactly known at the current time, but their future evolutions are uncertain and contained in the prescribed bounded sets. In RMPC, the model parametric uncertainty can be dealt with within the frame of LPV systems. For the online RMPC, at each time, a min-max optimization problem is often utilized to minimize the performance function of LPV systems, which considers all the possible realization of model parametric uncertainty [5]. Reference [6] extends the approaches in [5] to the quasi-LPV system with a quasi-min-max optimization. In [7, 8], the controller designment considers the time-varying parameters of quasi-LPV systems having the bounds on their rate of variation. Reference [9] considers the quasi-LPV systems with a parameter-dependent control law.

However, in [5–9], the true states are assumed to be known and the bounded disturbance is not considered. In real processes, the true states are usually unmeasurable, and only the outputs with disturbance are available. In these situations, output feedback RMPC (OFRMPC) is more practical than the state feedback one for real applications. For OFRMPC with bounded disturbance and without model parametric uncertainty, see [10, 11]. For OFRMPC with model parametric uncertainty and without bounded disturbance, see [12–14]. For OFRMPC with both polytopic uncertainty and bounded disturbance, see [15–20].

For OFRMPC, the bounds of estimation error set represent a kind of uncertainty, which has to be considered in the robust stability and physical constraints. When the model parametric uncertainty, bounds of estimation error set, and disturbance are known and contained in the bounded sets, the evolution of state estimation is a compact set which is consistent with the bounded uncertainties. The set-membership estimation considers nonstatistical description of model parameters, estimation error sets, and disturbance in the form of bounded uncertainties. When the set-membership estimation is utilized to state estimation, an appropriate representation of estimation error sets should consider the tradeoff between the precise representation of estimation error sets and the online computational burden. Polytopes can be used to represent exactly the estimation error set. However, if the vertices of polytopes increase dramatically, the computational burden becomes quickly prohibitive [21]. Ellipsoids can overcome the drawback by efficient computation. However, the Minkowski sum of ellipsoids is not necessary an ellipsoid, and an outer approximation ellipsoid is often used to estimate the Minkowski sum of ellipsoids, which may lead to the conservatism of state estimation. In recent years, zonotopes (a particular class of polytopes) have received more attention because of their advantages of having better precision in comparison with ellipsoidal sets and less complexity compared to polyhedral sets. Hence, several set-membership estimation methods based on zonotopes [22–25] have been proposed.

The present paper considers a synthesis approach of dynamic OFRMPC for the quasi-LPV system with bounded disturbance. The main optimization problem is similar to that in [16, 19], while the estimation error sets are represented by zonotopes. The main contribution is a combination between the main optimization problem that calculates control parameters and the set-membership estimation based on zonotopic computation. By properly refreshing the estimation error sets, it can obtain the precise bounds of true states at the next sampling time. Furthermore, the feasibility of the main optimization problem at the next sampling time can be determined by checking a feasible problem at the current time.

#### 2. Notations, Basic Definitions, and Properties

For any vector and positive-definite matrix,. 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. An element belonging to means that it is a convex combination of the elements in , with the scalar combing coefficients nonnegative and their sum equal to 1. 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-dependenceof the MPC decision variables is often omitted for brevity.

Let an interval be defined as the set . The unitary interval is . A box is an interval vector. A unitary box in , denoted by , is a box composed by unitary intervals. The Minkowski sum of two sets and is defined by .

*Definition 1. *A -zonotope in can be defined as the linear image of a -dimensional hypercube in , where is the order of the zonotope and satisfies . Given a vector and a matrix , a -zonotope is the following set:

This is the Minkowski sum of the -segments defined by columns of matrix in . The center of zonotope is vector ; , with , , are the generators of the zonotope. Some properties for zonotopic computation are given as follows.

*Property 1 (the Minkowski sum of two zonotopes [22]). *Given two zonotopes and , the Minkowski sum of two zonotopes is also a zonotope defined by .

*Property 2 (the image of a zonotope [22]). *The image of a zonotope by a linear mapping can be computed by a standard matrix product .

*Property 3 (zonotope reduction [22, 23]). *Given a zonotope and integer with , denote by the matrix resulting from the reordering of the columns of matrix in decreasing order of Euclidean norm and by a diagonal matrix, satisfying the following conditions:and then , where is obtained from the first columns of matrix .

*Remark 2. *From Property 3, it can obtain that is an outer approximation of with a lower order, which can be utilized to estimate a zonotope with high order. In Section 4.2, by applying Property 3, the order of the estimation error set can be reduced.

Zonotopes are the special case of polytopes and can be represented by half-space representation (H-representation) and vertex representation (V-representation). The definitions are as follows.

*Definition 3 (H-representation of a polytope). *For half-spaces, a convex polytope is the set:

*Definition 4 (V-representation of a polytope). *A polytope is the set constructed by the convex combination of finite vertices , ; that is,where is the total number of vertices related to the V-representation of polytope . From Definition 1, the unitary box in has vertices, and each vertex of is a -dimensional vector with the element being or . Hence, according to (1), at most vertices of zonotope can be calculated. We define the vertex which is the interior point of as redundant vertex. To represent the zonotope by V-representation through vertices, a Quickhull Algorithm in [26] can be employed to remove the redundant vertices. When the dimension of zonotope is fixed, the relationship between and can be obtained; for example, when , ; , , respectively [27]. For high-dimensional systems, although the relationship between and is complex, the upper bound of can be calculated in polynomial time for the fixed dimension of zonotopes [28].

Figure 1 shows a zonotope , where , , ; the three generators are , , , which are the three columns of matrix , respectively [27]. Based on Definition 1, the vertices of zonotope can be calculated for the different vertices of unitary box , where the different combinations of “+” and “−” denote the corresponding vertices of . In order to describe the zonotope by V-representation through 6 () vertices, the redundant vertices denoted by “” and “” are removed.