An Offline Formulation of MPC for LPV Systems Using Linear Matrix Inequalities

An offline model predictive control (MPC) algorithm for linear parameter varying (LPV) systems is presented. The main contribution is to develop an offline MPC algorithm for LPV systems that can deal with both time-varying scheduling parameter and persistent disturbance. The norm-bounding technique is used to derive an offline MPC algorithm based on the parameterdependent state feedback control law and the parameter-dependent Lyapunov functions.The online computational time is reduced by solving offline the linear matrix inequality (LMI) optimization problems to find the sequences of explicit state feedback control laws. At each sampling instant, a parameter-dependent state feedback control law is computed by linear interpolation between the precomputed state feedback control laws. The algorithm is illustrated with two examples. The results show that robust stability can be ensured in the presence of both time-varying scheduling parameter and persistent disturbance.


Introduction
Model predictive control (MPC), also known as receding horizon control, is an effective multivariable control algorithm in which a dynamic optimization problem is solved online.At each sampling time, MPC solves a finite horizon optimal control problem based on an explicit model of the plant.Although an optimal control sequence is determined, only the first control action is applied to the plant.Due to its ability to guarantee optimality while ensuring the satisfaction of constraints on input and state, MPC has received much interest in both industry and academia [1][2][3].
An explicit linear model is typically used in the MPC formulation because the online optimization can be reduced to either a linear program or a quadratic program.Since an industrial process is inherently nonlinear to a certain extent, the control performance of linear MPC can deteriorate as operating conditions significantly change [4,5].For this reason, MPC for linear parameter varying (LPV) systems has been widely developed.LPV systems are linear systems whose dynamics depend on the scheduling parameter that can be measured online.The analysis and synthesis of LPV systems play an important role in control theory since nonlinear systems can be dealt within the framework of LPV systems [6,7].
In the context of MPC for LPV systems, one of the main approaches is to solve a semidefinite problem under linear matrix inequality (LMI) [8].Quasi-Min-Max MPC algorithm for LPV systems was developed by Lu and Arkun [9].Although the scheduling parameter is included in the controller design, it is assumed that there is no disturbance present in the problem formulation, so the algorithm cannot deal with disturbance.MPC for LPV systems using parameter-dependent Lyapunov functions was developed by [10].It is shown that the proposed MPC algorithm can achieve less conservative results as compared with a robust MPC algorithm derived by using a single Lyapunov function [4].However, this algorithm includes only time-varying scheduling parameter in the problem formulation so it cannot ensure robust stability in the presence of disturbance.The bound on the rate of variation of the scheduling parameter can also be taken into account in the MPC formulation [11,12].However, this technique is not applicable to the case where there is the disturbance acting on the system.
MPC for LPV systems can be designed by using ellipsoidal set prediction [13,14].At each sampling instant, the predicted future states on the finite horizon are bounded by using a sequence of ellipsoids.The terminal ellipsoid is contained in a target set guaranteeing stability.The main drawback of this approach lies in the fact that the computational load increases with the length of prediction horizon.In order to reduce online computational time, an offline MPC algorithm for LPV systems was developed [15].The realtime state feedback gain is calculated by linear interpolation between the precomputed state feedback gains.Although the online computational time is significantly reduced, the disturbance is not taken into account in the offline MPC formulation so robust stability cannot be guaranteed in the presence of disturbance.Explicit MPC for LPV systems was proposed by Besselmann et al. [16].Only time-varying scheduling parameter is included in the problem formulation and it is also assumed that there is no disturbance.In Ding [17,18], both time-varying scheduling parameter and disturbance are included in the problem formulation.However, the optimization problem contains a lot of decision variables and constraints, so the algorithm is computationally prohibitive in practical situations.
In the context of tube-based MPC [19][20][21], the disturbance is explicitly taken into account in the MPC design.The basic concept of robust tube-based MPC is to compute the region around the nominal prediction that contains any possible states of the uncertain system.One of the main advantages is that its online computational complexity increases only linearly with the prediction horizon.However, the timevarying scheduling parameter is not incorporated into the MPC design, so robust stability cannot be ensured in the presence of parametric uncertainty.
In this paper, an offline MPC algorithm for LPV systems is presented.Unlike Wan and Kothare [22] where only timevarying scheduling parameter is considered in the offline MPC formulation, the main contribution of this paper is to develop an offline MPC algorithm for LPV systems that can deal with both persistent disturbance and time-varying scheduling parameter.The norm-bounding technique [23] is used to derive an offline MPC algorithm based on the parameter-dependent state feedback control law and the parameter-dependent Lyapunov functions.Most of the optimization problems are solved offline, so the developed MPC algorithm can be applied to fast processes.This article is organized as follows.Section 2 concerns with problem statement and control objectives.The proposed algorithm is described in Section 3. In Section 4, the effectiveness of the proposed MPC algorithm is illustrated.Finally, Section 5 presents some conclusions.
Notation.For any vector  and positive-definite matrix , ‖‖ 2  =   .() is the state measured at real-time  and ( +  | ) is the state at prediction time  +  predicted at real-time .The symbol * denotes symmetric blocks in matrices.An element belonging to a convex hull Co{⋅} means that it is a convex combination of the elements in {⋅}.The time-dependence () of the MPC decision variables is often dropped for simplicity. is the identity matrix with appropriate dimension.
Any (()) and (()) belong to a convex polytope Ω defined by so they can be written as where [  ,   ] are the vertices of Ω and  is the number of the vertices of Ω.Any (()) belongs to a convex polytope Ω  defined by so it can be written as where   are the vertices of Ω  ,   is the number of the vertices of Ω  , and   () is the time-varying parameter that is not necessary to be measurable.The disturbance V() is unmeasurable and persistent.It is assumed to lie in a convex polytope Ω V defined by where  V is the number of the vertices of Ω V .The objective is to find a parameter-dependent state feedback control law (( + )) = ∑  =1   ( + )  , where   ,  ∈ {1, 2, . . ., } are the state feedback gains corresponding to the vertices of Ω that is able to guarantee both robust stability and constraint satisfaction within a positively invariant set.Definition 1.The set  is said to be positively invariant set if it has the property that whenever the current state is contained in this set () ∈ , all possible predicted states must be contained in this set ( +  | ) ∈  for all admissible realizations of ( + ), ( + ) and V( + ),  ≥ 0.
Remark 2. In this paper, the positively invariant set is ((+)) ≤ }, where (( + )) = ∑  =1   ( + )  is a parameter-dependent Lyapunov matrix,   ,  ∈ {1, 2, . . ., } are the Lyapunov matrices corresponding to the vertices of Ω, and  is an upper bound on the infinite horizon cost.From the convexity of the polytopic description,  is an intersection area of ≤ }.Considering the discrete-time LPV system (1) to (9) at each sampling time , a parameter-dependent state feedback control law ( +  | ) = (( + ))( +  | ) that (i) minimizes an upper bound  on  ∞ () and (ii) guarantees both robust stability and robust constraint satisfaction within a positively invariant set  can be calculated by solving the following optimization problem: ( ( +  + 1))  ( where is the predicted state with disturbance, and  and  are symmetric weighting matrices.The cost monotonicity is guaranteed by (11).A positively invariant set containing the state () at each sampling time is computed by (12).All predicted states ( +  + 1 | ) are restricted to lie in a positively invariant set by (13).The input and output constraints are guaranteed by ( 14) and ( 15), respectively.

Offline MPC for LPV Systems with Persistent Disturbances
First of all, we will begin with the preliminary results of Wada et al. [10] where only time-varying scheduling parameter is considered in the problem formulation.Then, the proposed algorithm that can deal with both time-varying scheduling parameter and persistent disturbance will be developed.By following Wada et al. [10], (11) and ( 12) are satisfied and the cost monotonicity is guaranteed if there exist matrices   ,   , symmetric matrices   and a positive scalar  such that the following LMIs are satisfied: Then, it follows that  is the upper bound on  ∞ ().Moreover, a parameter-dependent state feedback gain is given by (( + )) = ∑  =1   ( + )  ,   =    −1  .Next, we will present the results of this paper that can deal with both time-varying scheduling parameter and disturbance.
Proposition 3 (Robust stability in the presence of both time-varying scheduling parameter and disturbance).( 13) is satisfied if there exist matrices   ,   and symmetric matrices   such that the following LMIs are satisfied: where 0 <  < 1 is a prespecified scalar.Then, all predicted states are restricted to lie in a positively invariant set which is an intersection area of Proof.Equation ( 13) is guaranteed by ( 18) and ( 19).The proof detail can be found in Appendix A.
Remark 4.  is a parameter that bounds disturbance-free state trajectories.The value of  should be chosen such that  → 1 as V  → 0.
where Φ  is a scalar that can be obtained by solving the following optimization problem min where  , is the th row of   .
Proof.The input constraint ( 14) is guaranteed by (20).The proof details can be found in Appendix B. The output constraint ( 15) is guaranteed by (21).The proof details can be found in Appendix C.
By considering Propositions 3 and 5, a parameter-dependent state feedback control law that guarantees both robust stability and robust constraint satisfaction can be calculated.Consider the discrete-time LPV system (1) to (9) It is computationally demanding to solve the optimization problem (24) at each sampling time.Inspired by Bumroongsri and Kheawhom [15], we propose an offline MPC algorithm for LPV systems that transfer most of the computations offline.
Example 1.The first example is a continuous stirred tank reactor (CSTR) adapted from Ding and Huang [26] where an exothermic reaction  →  takes place.The dynamic model based on a component balance and an energy balance can be written as where   denotes the concentration of  in the reactor,  denotes the reactor temperature,   denotes the temperature of the coolant stream, and V denotes the disturbance acting on the system.The operating parameters are shown in Table 1.
Figure 1(a) shows the ellipsoids computed offline by Algorithm 6.In this example, four sequences of ellipsoids are computed because the polytope Ω has four vertices.A sequence of positively invariant sets computed by intersection among four sequences of ellipsoids is shown in Figure 1(b).
Figure 2 shows the closed-loop responses of the system when the disturbances are varied as V() = 0.1 sin(10), 0.05 sin(10), and 0.001 sin(10), respectively.It can be observed that   and  are bounded for all values of disturbances so robust stability is ensured by applying Algorithm 6.
The online computational time of Algorithm 6 is very low as shown in Table 2 so it is applicable to fast systems.
Example 2. The second example is an angular positioning system adapted from Kothare et al. [4].The system consists of an electric motor driving a rotating antenna so that is always points in the direction of a moving object.The motion of the antenna can be described by the following discrete-time LPV model: where  1 () is the angular position of the antenna,  2 () is the angular velocity of the antenna, () is the input voltage to the  motor, and V() is the disturbance acting on the system.The scheduling parameter Δ() is measurable at each sampling time and its value is varied between 0.1 and 10.We have that all the solutions of (38) are also the solutions of the following differential inclusion: where   are given by  1 = [ 1 0.1 0 0.99 ],  2 = [ 1 0.1 0 0 ] and   are given by  1 = (10 − Δ())/0.9, 2 = (Δ() − 0.1)/0.9.
Figure 3(a) shows two sequences of ellipsoids corresponding to   .A sequence of positively invariant sets computed by intersection between two sequences of ellipsoids is shown in Figure 3(b).
Figure 4 shows 100 state trajectories of the closed-loop system when V() and Δ() are arbitrarily time-varying in the ranges of −0.01 ≤ V() ≤ 0.01 and 0.1 ≤ Δ() ≤ 10.Two initial points (0.9, 0) and (−0.8, 0.8) are chosen.It can be observed from the figure that all state trajectories are restricted to lie in a sequence of positively invariant sets.The average time for Algorithm 6 to compute a real-time control law is 0.001 s.

Conclusions
In

A. Proof of Proposition 3
Lemma A.1 will be used in the proof.

Figure 1 :
Figure 1: The ellipsoids computed offline in Example 1 (a) four sequences of ellipsoids corresponding to   (b) a sequence of positively invariant sets.

Figure 2 :
Figure 2: The closed-loop responses in Example 1 (a) regulated output and (b) control input.
a) Two sequences of ellipsoids corresponding to   A sequence of positively invariant sets

Figure 3 :Figure 4 :
Figure 3: The ellipsoids computed offline in Example 2 (a) two sequences of ellipsoids corresponding to   and (b) a sequence of positively invariant sets.
this paper, we have presented an offline MPC algorithm for constrained LPV systems.The main contribution is to develop an offline MPC algorithm for LPV systems that can deal with both persistent disturbance and time-varying scheduling parameter.The norm-bounding technique is used to derive an offline MPC algorithm based on the parameterdependent state feedback control law and the parameterdependent Lyapunov functions.Most of the optimization problems are solved offline so the algorithm is applicable to fast systems.At each sampling time, a parameter-dependent state feedback control law is calculated by linear interpolation between the precomputed state feedback control laws.The controller design is illustrated with two examples.

Table 1 :
The operating parameters of nonlinear CSTR in Example 1.