# A Simplified Predictive Control of Constrained Markov Jump System with Mixed Uncertainties

Authors: Yanyan Yin; Yanqing Liu; Hamid R. Karimi
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

A simplified model predictive control algorithm is designed for discrete-time Markov
jump systems with mixed uncertainties. The mixed uncertainties include model polytope uncertainty
and partly unknown transition probability. The simplified algorithm involves finite
steps. Firstly, in the previous steps, a simplified mode-dependent predictive controller is presented
to drive the state to the neighbor area around the origin. Then the trajectory of states
is driven as expected to the origin by the final-step mode-independent predictive controller.
The computational burden is dramatically cut down and thus it costs less time but has the
acceptable dynamic performance. Furthermore, the polyhedron invariant set is utilized to enlarge
the initial feasible area. The numerical example is provided to illustrate the efficiency of
the developed results.


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## Body

## 1. Introduction

Hybrid systems are a class of dynamical systems denoted by an interaction between the continuous and discrete dynamics. In control community, the researchers tend to view hybrid systems as continuous state and discrete switching which focuses on the continuous state of dynamic system. Switched systems are a natural result from this point of view. Since switching systems can be applied to model the systems involving abrupt sudden changes which are widely found in the systems of economics and communications as well as manufacturing, more attention has been paid to them (see robust stabilization [1], finite-time analysis [2] and asynchronous switching [3]). When the system model is linear and the switching is driven by Markov process, it leads to Markov jump linear system (MJS). Specifically, MJS presents a stochastic Markov chain to describe the random changes of system parameters or structures, where the dynamic of MJS is switching among the models governed by a finite Markov chain. Due to this superiority, MJS has been widely investigated during the last twenty years. Attractive pioneer works have been obtained (see controller design [4], 2D MJS control [5], peak-to-peak filtering [6], and finite-time control [7, 8]). However, the cases of completely known transition probability (TP) considered in [4–8] are not always achievable since the TP is not easy to be fully accessible (see the delay or packet loss in networked control systems [9]). Thus it is necessary to investigate the partly unknown case [10–12].On the other hand, the systems in practice are usually subject to input/output constraints. Thus, model predictive control (MPC) is then introduced to solve the problem of MJS with constraints since MPC can explicitly solve the constraints in control action. Successful MPC application in discrete-time MJS can be obtained in [13, 14]. Normally, MPC is reformulated as online quadratic program and results have been reported (see stability [15, 16] and enlarged terminal sets [17]). It should be noted that the online computation in the literature [15–17] leads to heavy computational burden. Thus, the researchers attempted to try a new alternative method to solve the problem. For this reason, explicit MPC [18] is presented. However, when the size of system increases, the time of searching explicit MPC law will also increase sharply.Based on the above analysis, a simplified MPC design framework is introduced to reduce the burden of online computation for the constrained MJS with mixed uncertainties. The basic idea is that (N-1) steps of mode-dependent MPC are designed to steer the state to a final neighbour area which includes the origin. Then the final step of robust mode-independent MPC is devised to force the state towards the origin regardless of model uncertainty and transition probability uncertainty. This simplified MPC dramatically reduces the burden of computation with minor performance loss, which implies good balance between the calculation time and dynamical performance. Furthermore, the polyhedron invariant set is applied to further enlarge the initial feasible area.The construction of the paper is as follows. Section2 gives the basic dynamical of the system. Section 3 gives the finite-step simplified MPC algorithm and it is formulated as LMIs. Section 4 presents a numerical example to show the efficiency of the results. Section 5 concludes the paper.Notations. The notations are as follows: Rn denotes a n-dimensional Euclidean space, AT stands for the transpose of a matrix, E{·} denotes the expectation of the stochastic process or vector, a positive-definite matrix is described as P>0, I means the unit matrix with appropriate dimension, and * means the symmetric term in a symmetric matrix.


## 2. Problem Statement and Preliminaries

The constrained discrete-time MJSs with mixed uncertainties are considered in this paper:(1)xk+1=A(rk)xk+B(rk)uk,yk=C(rk)xk,
where xk∈Rnx, uk∈Rnu, yk∈Rny, respectively, denote the state vector, the input vector, and the controlled output vector. The discrete-time Markov stochastic process {rk,k≥0} takes values in a finite set Γ, where Γ contains σ modes of system (1), Γ={1,2,3,…,σ}, and r0 represents the initial mode. The uncertain system model A(rk) and B(rk) belong to the model sets
(2)Ω(rk)={[A(rk),B(rk)],A(rk)=∑ι=1L‍αιAι(rk),B(rk)=∑ι=1L‍αιBι(rk),∑ι=1L‍αι=1}.
Inputs and outputs constraints are subject to
(3)-ulim⁡≤uk≤ulim⁡,(4)-ylim⁡≤yk≤ylim⁡.
The transition probability (TP) matrix is denoted by Π(k)={πij(k)}, i,j∈Γ, where πij(k)=P(rk+1=j∣rk=i) is the transition probability from mode i at time k to mode j at time k+1. The elements in TP matrix satisfy πij(k)≥0 and ∑j=1σπij(k)=1:
(5)π=[π11π12…π1σπ21π22…π2σ⋮⋮⋱⋮πσ1πσ2…πσσ].
The uncertain transition probability (TP) implies that some elements in π are unknown; a four-mode transition probability (TP) matrix π may be
(6)π=[?π12??π21π22??π31?????π43?],
where “?” represents the inaccessible element in TP matrix. For convenience, we denote π=πrkk+πrkuk, for all mode rk∈Γ at sampling time k, if πrkk≠0, and redescribe it as πrkk=(κrk1,…,κrkτ), for all 1≤l≤τ, where κrkl represents the lth exact element in the ith row of π, Πrkk=∑j∈πrkkπrkrk+1.Some preliminaries are introduced before proceeding.Definition 1 (see [6]).
For any initial moder0 and state x0, discrete-time MJS (1) is said to be stochastically stable if
(7)lim⁡k→∞E{xkTxk∣x0,r0}⟶0.Definition 2.
For MJS (1), an ellipsoid set Θ={x∈Rnx∣xkTPk(rk)xk≤γk} associated with the state is said to be asymptotically mode-dependent stable, if the following holds, whenever xk0∈Θ, then xk∈Θ for k≥k0 and xk→0 when k→∞.Next, we first derive the online optimal MPC algorithm for system (1). The aim is to minimize the function cost related to worst-case performance and then in Section 4 the corresponding simplified MPC algorithm will be derived. Finally the polyhedron invariant set is applied to further improve the initial feasible district.


## 3. Simplified MPC Design

### 3.1. Online Optimal MPC

Theorem 3.
Consider MJS (1) with model uncertainties (2) and partly unknown TP matrix (6), at sampling time k, if there exist a set of matrices Fk(rk), such that the following holds:
(8)min⁡Fk(rk)max⁡Aι(rk),Bι(rk),πrkrk+1,rk,rk+1∈ΓJ∞(k)
s.t.                                    (9)-ulim⁡≤uk≤ulim⁡,(10)-ylim⁡≤yk≤ylim⁡,(11)E{V(xk+1,rk+1∣x0,r0)}-E{V(xk,rk∣x0,r0)}≤-E{xkTQ(rk)xk+ukTR(rk)uk∣x0,r0}.
Then, it decides an upper bound on J∞(k), where uk=Fk(rk)xk, J∞(k)=E{∑k=0∞(xkTQ(rk)xk+ukTR(rk)uk)∣x0,r0}, Q(rk), R(rk) are positive definite weighting matrices.Proof.
It is assumed that at the sampling timek, a state-feedback law u(k+i∣k)=Fk(rk)x(k+i∣k), is applied to minimize the worst cost function of Jk; it is easy to show that V(xk,rk∣x0,r0) is an upper bound on J∞(k). Let V(xk)=xkTPk(rk)xk, Pk(rk)>0, be a quadratic Lyapunov function. For any [Aι(rk),Bι(rk)∈Ω(rk)], the following constraint holds
(12)E{V(xk+1,rk+1∣x0,r0)}-E{V(xk,rk∣x0,r0)}≤-E{xkTQ(rk)xk+ukTR(rk)uk∣x0,r0}.
Summing (12) from i=0 to ∞ on both sides and using the fact xk→∞=0 or V(xk→∞)=0, we obtain
(13)J∞(k)≤V(xk,rk∣x0,r0)=xkTPk(rk)xk
which implies that V(xk,rk∣x0,r0) is an upper bound on J∞(k).Theorem 4.
Consider MJS (1) with polytope model uncertainties (2) and partly unknown TP matrix (4), if there exist a set of positive definite matrices Xk(rk), Yk(rk), such that the following optimization problem (12) has an optimal solution:
(14)min⁡Fk(rk)max⁡Aι(rk),Bι(rk),πrkrk+1,rk,rk+1∈Γγk
s.t.                                    (15)[1*xkXk(rk)]≥0,∀rk∈Γ,rk+1∈πrk+1uk,(16)[ZYk(rk)*Xk(rk)]≥0,Ztt≤(ulim⁡t)2,(17)[Xk(rk)*C(rk)θl(rk)M]≥0,Mhh≤(ylim⁡h)2,(18)[Xk(rk)*θl(rk)Xk(rk+1)]≥0,∀rk+1∈πrkuk,(19)[ΠrkkXk(rk)UT(rk)Xk(rk)Q1/2(rk)YkT(rk)Q1/2(rk)*W(rk+1)00**γkI0***γkI]≥0,∀rk+1∈πrkk
then, the mode-dependent state-feedback which minimizes the upper bound γk on J∞(k) and simultaneously stabilizes the closed-loop system within an ellipsoid ɛ={xkTXk-1(rk)xk≤1} is calculated by u(k+i∣k)=Fk(rk+i)xk+i∣k,  Fk(rk+i)=Yk(rk+i)Xk-1(rk+i), where Xk(rk)=γkPk-1(rk), θl(rk)=Al(rk)Xk(rk)+Bl(rk)Yk(rk),  UT(rk)=[κrk1θlT(rk),…,κrkτθlT(rk)], W(rk+1)=diag⁡{Xk(κrk1),Xk(κrk2),…,Xk(κrkτ)}, Ztt, Mhh, respectively, describe the tth, hth diagonal element of Z, M, ulim⁡t and ylim⁡h, respectively, describe the tth and hth element of input and output constraints, t=1,2,…,nu, h=1,2,…,ny.Proof.
LetXk(rk)=γkPk-1(rk); J∞(k)≤γk in (13) can be solved by the following LMIs:
(20)[1*xkXk(rk)]≥0,∀rk∈Γ,rk+1∈πrk+1uk.
The input/output constraints are guaranteed by (16) and (17); the proof is similar to [19]; here we omit the proof. Equation (11) is equivalent to
(21)Ξ(rk)=Pk(rk)-θlT(rk)(∑rk+1∈π‍πrkrk+1Pk(rk+1))×θl(rk)-Q(rk)-FkT(rk)R(rk)Fk(rk)≥0.
Since ∑rk+1∈ππrkrk+1=1, πrkrk+1≥0, Πkrk=∑rk+1∈πrkkπrkrk+1, it leads to
(22)Ξ(rk)=(∑rk+1∈π‍πrkrk+1)Pk(rk)-θlT(rk)×(∑rk+1∈π‍πrkrk+1Pk(rk+1))θl(rk)-Q(rk)-FkT(rk)R(rk)Fk(rk)=ΠrkkP(rk)-θlT(rk)×(∑rk+1∈πrkk‍πrkrk+1Pk(rk+1))θl(rk)+(∑rk+1∈πrkuk‍πrkrk+1)×(Pk(rk)-θlT(rk)Pk(rk+1)θl(rk))-Q(rk)-FkT(rk)R(rk)Fk(rk)≥0.
One sufficient condition to ensure (22) is
(23)ΠrkkPk(rk)-θlT(rk)(∑rk+1∈πrkk‍πrkrk+1Pk(rk+1))×θl(rk)-Q(rk)-FkT(rk)R(rk)Fk(rk)≥0,Pk(rk)-θlT(rk)Pk(rk+1)θl(rk)≥0.
Considering the Schur theory complement lemma, (16) and (17) can be derived.
Actually the feedback controller can make the closed-loop system stable in the ellipsoidɛ={xkTXk-1(rk)xk≤1}. Assume that the optimal Pk*(rk), Fk*(rk) at the moment k are
(24)Pk*(rk)=γk*(Xk*(rk))-1,Fk*(rk)=Yk*(Xk*(rk))-1,θlk*(rk)=Al(rk)+Bl(rk)Fk*(rk),ϑlk*(rk)=Al(rk)Xk*(rk)+Bl(rk)Yk*(rk).
Equations (18) and (19) lead to
(25)xkTPk*(rk)xk≥xkT(θlk*(rk))T×∑rk+1∈π‍πrkrk+1P(rk+1)θlk*(rk)xk+xkTQ(rk)xk+xkTFkT(rk)R(rk)Fk(rk)xk,E{xkTPk*(rk)xk}≥E{xk+1TPk*(rk+1)xk+1}+xkTQ(rk)xk+xkTFkT(rk)R(rk)Fk(rk)xk.Pk+1*(rk+1) is the optimal value at moment k+1; Pk*(rk+1) is a feasible one at moment k+1. By the optimum definition,
(26)xk+1TPk*(rk+1)xk+1≥xk+1TPk+1*(rk+1)xk+1;
then,
(27)E{xkTPk*(rk)xk}≥E{xk+1TPk+1*(rk+1)xk+1}+xkTQ(rk)xk+xkTFkT(rk)R(rk)Fk(rk)xk.
It is shown that E{xkTPk*(rk)xk} decrease strictly as E{xkTΦk*(rk)xk}→0, k→∞
From Definition1, the system is stochastically stable. From (27), then
(28)E{xkTPk*(rk)xk}≥E{xk+1TPk+1*(rk+1)xk+1}.
This implies that the ellipsoid is an asymptotically stable invariant one, which completes the proof.Corollary 5.
Consider MJS (1) with model uncertainties (2) and TP matrix (4) at current moment k; supposing that there exists a set of positive definite matrices X, Y, such that the following optimization problem has an optimal solution:
(29)min⁡γk,X,Yγk
s.t.                                    (30)[1*xkX]≥0,∀rk∈Γ,rk+1∈πrk+1uk,[ZY*X]≥0,Ztt≤(ulim⁡t)2,[X*C(rk)θl(rk)M]≥0,Mhh≤(ylim⁡h)2,[X*θl(rk)X]≥0,∀rk+1∈πrkuk,[ΠrkkXUT(rk)XQ1/2(rk)YTQ1/2(rk)*W(rk+1)00**γkI0***γkI]≥0,∀rk+1∈πrkkthen the mode-independent state-feedback law can minimize the upper boundγkon the objective functionJ∞(k)and stabilize the closed-loop system in the ellipsoidɛ={xkTX-1xk≤1}and it is obtained byu(k+i∣k)=Fxk+i∣k,F=YX-1, whereX=γkP-1,θl(rk)=Al(rk)X+Bl(rk)Y,UT(rk)=[κrk1θlT(rk),…,κrkτθlT(rk)],W(rk+1)=diag{X,X,…,X},ZttandMhh, respectively, describe thetth andhth diagonal element ofZ,M, andulimtandylimh, respectively, describe thetth andhth element of input and output constraints,t=1,2,…,nu,h=1,2,…,ny.


### 3.2.  Simplified MPC Design

In this section, a simplified MPC for uncertain MJS (1) is developed based on the online algorithm in Theorem 4; Figure 1 shows the simplified MPC schematic diagram. Then the simplified mode-independent feedback controller is designed regardless of model uncertainty and TP uncertainty since much more constraints will be nonactive in the neighboring region of origin and this freedom of feasibility is applied to improve the procedure of controller design.Figure 1
Simplified MPC schematic diagram.Theorem 6.
Consider uncertain MJS (1) associated with an initial state x0 satisfying x0TQ0-1(r0)x0≤1; the simplified MPC Algorithm 7 robustly stabilizes the closed-loop system.Proof.
For theN-step implementation at xj, j=1,…,N, the selection for xj in Algorithm 7 implies Qj-1-1(rk)<Qj-1(rk), which means the constructed ellipsoid ξj={x∣xTQj-1(rk)x≤1} is embedded in ξj-1, that is, ξj⊂ξj-1. For a settled x, ξj={x∣xTQj-1(rk)x≤1} is decreasing monotonically associated with j, which guarantees the unique search in the search table for the largest j for ξj={x∣xTQj-1(rk)x≤1}. If xk belongs to ξj={x∣xTQj-1(rk)x≤1} and ξj+1={x∣xTQj+1-1(rk)x>1}, j=1,…,N-1, by applying Theorem 3, the control law uk=Fj(rk)xk will steer the state in ξj-1 to ξj. Finally, the controller uk=FNxk (applying Corollary 5) make the state to be in ξN and converge to the origin. Furthermore, the LP programming algorithm is utilized to remove redundant constraints [20] and construct a sequence of polyhedral invariant set for MJS and thus enlarge the feasible domain.Algorithm 7 (simplified MPC applying polyhedral invariant set).
Simplified MPC design is as follows.(1)
Selectxj, j=1,…,N, which satisfy ɛj+1⊂ɛj, ɛN=δ(0).(2)
For step 1 toN-1, calculate the corresponding mode-dependent gains γj(rk), Qj(rk), Xj(rk), Yj(rk), Fj(rk) by applying Theorem 4 and store them in a search table.(3)
For eachFj(rk), construct the corresponding polyhedral invariant set by the following algorithm: let Sj(rk)=[CT(rk),-CT(rk),FjT(rk),-FjT(rk)]T,  dj(rk)=[ymax⁡T(rk),ymin⁡T(rk),umax⁡T(rk),umin⁡T(rk)]T. Select row m from (Sj(rk),dj(rk)) and then check ∀j if Sj,m(rk(Aj(rk)+BjFj)(rk)≤dj,m(rk)) is redundant through solving the Linear programming:
(31)max⁡ρj,ms.t.ρj,m=Sj,m(rk)(Aj(rk)+BjFj(rk))x-dj,m(rk)Sj(rk)x≤dj(rk).Ifρj,m>0, it implies that the constraint Sj,m(rk(Aj(rk)+BjFj)(rk)≤dj,m(rk)) is nonredundant; then renew the nonredundant constraints as Sj(rk)=[SjT(rk),(Sj,m(rk)(Aj(rk)+Bj(rk)Fj(rk)))T]T, dj(rk)=[dj(rk)T,dj,m(rk)T]T.(4)
Online implementation: search the state in the search table to fix the needed indexj(j<N), decide the smallest polyhedral invariant set χj(rk)={x∣Sj(rk)x≤dj(rk)}, and finally implement uk=Fj(rk)xk.(5)
Online implementation: continue to check ifx∈χN(rk)={x∣SN(rk)x≤dN(rk)} is satisfied; if it is true, then apply uk=FNxk.Remark 8.
It should be noted that the more approximation of optimality can be obtained asN increases; here N can be chosen according to different prior requirements. Thus, we can adjust the numbers of design step in terms of different requirements.


## 3.1. Online Optimal MPC

Theorem 3.
Consider MJS (1) with model uncertainties (2) and partly unknown TP matrix (6), at sampling time k, if there exist a set of matrices Fk(rk), such that the following holds:
(8)min⁡Fk(rk)max⁡Aι(rk),Bι(rk),πrkrk+1,rk,rk+1∈ΓJ∞(k)
s.t.                                    (9)-ulim⁡≤uk≤ulim⁡,(10)-ylim⁡≤yk≤ylim⁡,(11)E{V(xk+1,rk+1∣x0,r0)}-E{V(xk,rk∣x0,r0)}≤-E{xkTQ(rk)xk+ukTR(rk)uk∣x0,r0}.
Then, it decides an upper bound on J∞(k), where uk=Fk(rk)xk, J∞(k)=E{∑k=0∞(xkTQ(rk)xk+ukTR(rk)uk)∣x0,r0}, Q(rk), R(rk) are positive definite weighting matrices.Proof.
It is assumed that at the sampling timek, a state-feedback law u(k+i∣k)=Fk(rk)x(k+i∣k), is applied to minimize the worst cost function of Jk; it is easy to show that V(xk,rk∣x0,r0) is an upper bound on J∞(k). Let V(xk)=xkTPk(rk)xk, Pk(rk)>0, be a quadratic Lyapunov function. For any [Aι(rk),Bι(rk)∈Ω(rk)], the following constraint holds
(12)E{V(xk+1,rk+1∣x0,r0)}-E{V(xk,rk∣x0,r0)}≤-E{xkTQ(rk)xk+ukTR(rk)uk∣x0,r0}.
Summing (12) from i=0 to ∞ on both sides and using the fact xk→∞=0 or V(xk→∞)=0, we obtain
(13)J∞(k)≤V(xk,rk∣x0,r0)=xkTPk(rk)xk
which implies that V(xk,rk∣x0,r0) is an upper bound on J∞(k).Theorem 4.
Consider MJS (1) with polytope model uncertainties (2) and partly unknown TP matrix (4), if there exist a set of positive definite matrices Xk(rk), Yk(rk), such that the following optimization problem (12) has an optimal solution:
(14)min⁡Fk(rk)max⁡Aι(rk),Bι(rk),πrkrk+1,rk,rk+1∈Γγk
s.t.                                    (15)[1*xkXk(rk)]≥0,∀rk∈Γ,rk+1∈πrk+1uk,(16)[ZYk(rk)*Xk(rk)]≥0,Ztt≤(ulim⁡t)2,(17)[Xk(rk)*C(rk)θl(rk)M]≥0,Mhh≤(ylim⁡h)2,(18)[Xk(rk)*θl(rk)Xk(rk+1)]≥0,∀rk+1∈πrkuk,(19)[ΠrkkXk(rk)UT(rk)Xk(rk)Q1/2(rk)YkT(rk)Q1/2(rk)*W(rk+1)00**γkI0***γkI]≥0,∀rk+1∈πrkk
then, the mode-dependent state-feedback which minimizes the upper bound γk on J∞(k) and simultaneously stabilizes the closed-loop system within an ellipsoid ɛ={xkTXk-1(rk)xk≤1} is calculated by u(k+i∣k)=Fk(rk+i)xk+i∣k,  Fk(rk+i)=Yk(rk+i)Xk-1(rk+i), where Xk(rk)=γkPk-1(rk), θl(rk)=Al(rk)Xk(rk)+Bl(rk)Yk(rk),  UT(rk)=[κrk1θlT(rk),…,κrkτθlT(rk)], W(rk+1)=diag⁡{Xk(κrk1),Xk(κrk2),…,Xk(κrkτ)}, Ztt, Mhh, respectively, describe the tth, hth diagonal element of Z, M, ulim⁡t and ylim⁡h, respectively, describe the tth and hth element of input and output constraints, t=1,2,…,nu, h=1,2,…,ny.Proof.
LetXk(rk)=γkPk-1(rk); J∞(k)≤γk in (13) can be solved by the following LMIs:
(20)[1*xkXk(rk)]≥0,∀rk∈Γ,rk+1∈πrk+1uk.
The input/output constraints are guaranteed by (16) and (17); the proof is similar to [19]; here we omit the proof. Equation (11) is equivalent to
(21)Ξ(rk)=Pk(rk)-θlT(rk)(∑rk+1∈π‍πrkrk+1Pk(rk+1))×θl(rk)-Q(rk)-FkT(rk)R(rk)Fk(rk)≥0.
Since ∑rk+1∈ππrkrk+1=1, πrkrk+1≥0, Πkrk=∑rk+1∈πrkkπrkrk+1, it leads to
(22)Ξ(rk)=(∑rk+1∈π‍πrkrk+1)Pk(rk)-θlT(rk)×(∑rk+1∈π‍πrkrk+1Pk(rk+1))θl(rk)-Q(rk)-FkT(rk)R(rk)Fk(rk)=ΠrkkP(rk)-θlT(rk)×(∑rk+1∈πrkk‍πrkrk+1Pk(rk+1))θl(rk)+(∑rk+1∈πrkuk‍πrkrk+1)×(Pk(rk)-θlT(rk)Pk(rk+1)θl(rk))-Q(rk)-FkT(rk)R(rk)Fk(rk)≥0.
One sufficient condition to ensure (22) is
(23)ΠrkkPk(rk)-θlT(rk)(∑rk+1∈πrkk‍πrkrk+1Pk(rk+1))×θl(rk)-Q(rk)-FkT(rk)R(rk)Fk(rk)≥0,Pk(rk)-θlT(rk)Pk(rk+1)θl(rk)≥0.
Considering the Schur theory complement lemma, (16) and (17) can be derived.
Actually the feedback controller can make the closed-loop system stable in the ellipsoidɛ={xkTXk-1(rk)xk≤1}. Assume that the optimal Pk*(rk), Fk*(rk) at the moment k are
(24)Pk*(rk)=γk*(Xk*(rk))-1,Fk*(rk)=Yk*(Xk*(rk))-1,θlk*(rk)=Al(rk)+Bl(rk)Fk*(rk),ϑlk*(rk)=Al(rk)Xk*(rk)+Bl(rk)Yk*(rk).
Equations (18) and (19) lead to
(25)xkTPk*(rk)xk≥xkT(θlk*(rk))T×∑rk+1∈π‍πrkrk+1P(rk+1)θlk*(rk)xk+xkTQ(rk)xk+xkTFkT(rk)R(rk)Fk(rk)xk,E{xkTPk*(rk)xk}≥E{xk+1TPk*(rk+1)xk+1}+xkTQ(rk)xk+xkTFkT(rk)R(rk)Fk(rk)xk.Pk+1*(rk+1) is the optimal value at moment k+1; Pk*(rk+1) is a feasible one at moment k+1. By the optimum definition,
(26)xk+1TPk*(rk+1)xk+1≥xk+1TPk+1*(rk+1)xk+1;
then,
(27)E{xkTPk*(rk)xk}≥E{xk+1TPk+1*(rk+1)xk+1}+xkTQ(rk)xk+xkTFkT(rk)R(rk)Fk(rk)xk.
It is shown that E{xkTPk*(rk)xk} decrease strictly as E{xkTΦk*(rk)xk}→0, k→∞
From Definition1, the system is stochastically stable. From (27), then
(28)E{xkTPk*(rk)xk}≥E{xk+1TPk+1*(rk+1)xk+1}.
This implies that the ellipsoid is an asymptotically stable invariant one, which completes the proof.Corollary 5.
Consider MJS (1) with model uncertainties (2) and TP matrix (4) at current moment k; supposing that there exists a set of positive definite matrices X, Y, such that the following optimization problem has an optimal solution:
(29)min⁡γk,X,Yγk
s.t.                                    (30)[1*xkX]≥0,∀rk∈Γ,rk+1∈πrk+1uk,[ZY*X]≥0,Ztt≤(ulim⁡t)2,[X*C(rk)θl(rk)M]≥0,Mhh≤(ylim⁡h)2,[X*θl(rk)X]≥0,∀rk+1∈πrkuk,[ΠrkkXUT(rk)XQ1/2(rk)YTQ1/2(rk)*W(rk+1)00**γkI0***γkI]≥0,∀rk+1∈πrkkthen the mode-independent state-feedback law can minimize the upper boundγkon the objective functionJ∞(k)and stabilize the closed-loop system in the ellipsoidɛ={xkTX-1xk≤1}and it is obtained byu(k+i∣k)=Fxk+i∣k,F=YX-1, whereX=γkP-1,θl(rk)=Al(rk)X+Bl(rk)Y,UT(rk)=[κrk1θlT(rk),…,κrkτθlT(rk)],W(rk+1)=diag{X,X,…,X},ZttandMhh, respectively, describe thetth andhth diagonal element ofZ,M, andulimtandylimh, respectively, describe thetth andhth element of input and output constraints,t=1,2,…,nu,h=1,2,…,ny.


## 3.2.  Simplified MPC Design

In this section, a simplified MPC for uncertain MJS (1) is developed based on the online algorithm in Theorem 4; Figure 1 shows the simplified MPC schematic diagram. Then the simplified mode-independent feedback controller is designed regardless of model uncertainty and TP uncertainty since much more constraints will be nonactive in the neighboring region of origin and this freedom of feasibility is applied to improve the procedure of controller design.Figure 1
Simplified MPC schematic diagram.Theorem 6.
Consider uncertain MJS (1) associated with an initial state x0 satisfying x0TQ0-1(r0)x0≤1; the simplified MPC Algorithm 7 robustly stabilizes the closed-loop system.Proof.
For theN-step implementation at xj, j=1,…,N, the selection for xj in Algorithm 7 implies Qj-1-1(rk)<Qj-1(rk), which means the constructed ellipsoid ξj={x∣xTQj-1(rk)x≤1} is embedded in ξj-1, that is, ξj⊂ξj-1. For a settled x, ξj={x∣xTQj-1(rk)x≤1} is decreasing monotonically associated with j, which guarantees the unique search in the search table for the largest j for ξj={x∣xTQj-1(rk)x≤1}. If xk belongs to ξj={x∣xTQj-1(rk)x≤1} and ξj+1={x∣xTQj+1-1(rk)x>1}, j=1,…,N-1, by applying Theorem 3, the control law uk=Fj(rk)xk will steer the state in ξj-1 to ξj. Finally, the controller uk=FNxk (applying Corollary 5) make the state to be in ξN and converge to the origin. Furthermore, the LP programming algorithm is utilized to remove redundant constraints [20] and construct a sequence of polyhedral invariant set for MJS and thus enlarge the feasible domain.Algorithm 7 (simplified MPC applying polyhedral invariant set).
Simplified MPC design is as follows.(1)
Selectxj, j=1,…,N, which satisfy ɛj+1⊂ɛj, ɛN=δ(0).(2)
For step 1 toN-1, calculate the corresponding mode-dependent gains γj(rk), Qj(rk), Xj(rk), Yj(rk), Fj(rk) by applying Theorem 4 and store them in a search table.(3)
For eachFj(rk), construct the corresponding polyhedral invariant set by the following algorithm: let Sj(rk)=[CT(rk),-CT(rk),FjT(rk),-FjT(rk)]T,  dj(rk)=[ymax⁡T(rk),ymin⁡T(rk),umax⁡T(rk),umin⁡T(rk)]T. Select row m from (Sj(rk),dj(rk)) and then check ∀j if Sj,m(rk(Aj(rk)+BjFj)(rk)≤dj,m(rk)) is redundant through solving the Linear programming:
(31)max⁡ρj,ms.t.ρj,m=Sj,m(rk)(Aj(rk)+BjFj(rk))x-dj,m(rk)Sj(rk)x≤dj(rk).Ifρj,m>0, it implies that the constraint Sj,m(rk(Aj(rk)+BjFj)(rk)≤dj,m(rk)) is nonredundant; then renew the nonredundant constraints as Sj(rk)=[SjT(rk),(Sj,m(rk)(Aj(rk)+Bj(rk)Fj(rk)))T]T, dj(rk)=[dj(rk)T,dj,m(rk)T]T.(4)
Online implementation: search the state in the search table to fix the needed indexj(j<N), decide the smallest polyhedral invariant set χj(rk)={x∣Sj(rk)x≤dj(rk)}, and finally implement uk=Fj(rk)xk.(5)
Online implementation: continue to check ifx∈χN(rk)={x∣SN(rk)x≤dN(rk)} is satisfied; if it is true, then apply uk=FNxk.Remark 8.
It should be noted that the more approximation of optimality can be obtained asN increases; here N can be chosen according to different prior requirements. Thus, we can adjust the numbers of design step in terms of different requirements.


## 4. Illustrative Example

Consider the discrete-time MJS with four modes (σ=4):
(32)A11=[10.10.010.99],B11=[0.10.187],A12=[10.100.05],B12=[0.10.187],A21=[10.1-0.10.99],B21=[0.10.187],A22=[10.10.10.05],B22=[0.10.187],A31=[10.10.20.99],B31=[0.10.187],A32=[10.10.150.1],B32=[0.10.187],A41=[10.10.050.5],B41=[0.10.187],A42=[10.10.050.1],B42=[0.10.187].
The detailed constraints are umax⁡=2 and ymax⁡=1.5, initial state x0 is [-0.651]T, and C(rk)=[1001]. The positive definite weighting matrices are Q(rk)=[1001] and R(rk)=0.00002. The partly unknown TP matrix is randomly generated in Table 1.Table 1
The partly unknown TP matrix.
1
2
3
4
0.361
?
0.092
?
?
0.090
?
0.248
0.162
0.489
?
?
?
?
0.251
?Here we will show the 5-step example of the proposed Algorithm7. Firstly, a state set {xj∣xj=(0.5,-0.9),(0.4,-0.8),(0.3,-0.7),(0.2,-0.6),(0.1,-0.5)} is designed to compute the corresponding feedback gains Fj(rk). It is noted that the sequence of states xj guarantees that the constructed polyhedral invariant sets are embedded, that is, Sj⊂Sj-1. In this example, the first four mode-dependent feedback laws Fj(rk), j=1,…,4 are obtained. When the state goes into the smallest polyhedral invariant set, the final-step (the fifth-step) gain F5 is designed to steer the state to the origin regardless of model uncertainty and TP uncertainty.For each chosenxj in Figure 2, the 5-step ellipsoid invariant sets (purple solid lines) and 5-step polyhedral invariant sets (blue and orange alternant dot dash lines) are illustrated using the numbers 1 to 5. The stabilizable region of polyhedral invariant set constructed by Algorithm 7 is dramatically larger than that of ellipsoid invariant set while the dynamic response of simplified algorithm is comparable with online algorithm.Figure 2
Trajectory of system states.The results are computed at the same platform (AMD 2.1 GHz, memory 3.0 GB and MATLAB R2010a); the average time and variances of 30 times’ running of the system are shown in Table2. From the table, the burden of computation is significantly reduced by simplified algorithm.Table 2
20 times’ average of 30 iterations of system state.
Algorithm
20 times’ average
Variance
Online
11.6779 s
0.0069
5-step
0.0055 s
1.3173
e
-
006


## 5. Conclusions

The problem of simplified predictive controller design for MJS with mixed uncertainties is investigated. The simplified algorithm drastically reduces the online computational burden with only a little loss of performance. A numerical example is provided to illustrate the validity of the results.



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*Source: 475808-2014-03-26.xml*