# Model Predictive Control of Uncertain Constrained Linear System Based on Mixedℋ2/ℋ∞ Control Approach

Authors: Patience E. Orukpe
Journal: Journal of Control Science and Engineering
 (2012)

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## Abstract

Uncertain constrained discrete-time linear system is addressed using linear matrix inequality based optimization techniques. The constraints on the inputs and states are specified as quadratic constraints but are formulated to capture hyperplane constraints as well. The control action is of state feedback and satisfies the constraints. Uncertainty in the system is represented by unknown bounded disturbances and system perturbations in a linear fractional transform (LFT) representation. Mixedℋ2/ℋ∞ method is applied in a model predictive control strategy. The control law takes account of disturbances and uncertainty naturally. The validity of this approach is illustrated with two examples.


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## Body

## 1. Introduction

Model predictive control (MPC) is a class of model-based control theories that use linear or nonlinear process models to forecast system behaviour. MPC is one of the control techniques that is able to cope with model uncertainties in an explicit way [1]. MPC has been used widely in practical applications to industrial process systems [2] and active vibration control of railway vehicles [3]. One of the methods used in MPC when uncertainties are present is to minimise the objective function for the worst possible case. This strategy is known as minimax and was originally proposed [4] in the context of robust receding control, [5] in the context of feedback and feedforward control and [6] in the context of ℋ∞ MPC. MPC has been applied to ℋ∞ problems in order to combine the practical advantage of MPC with the robustness of the ℋ∞ control, since robustness of MPC is still being investigated for it to be applied practically.This work is motivated by the work in [7, 8] where uncertainty in the system was modeled by perturbations in a linear fractional representation. In [9], model predictive control based on a mixed ℋ2/ℋ∞ control approach was considered. The designed controller has the form of state feedback and was constructed from the solution of a set of feasibility linear matrix inequalities. However, the issue of handling both uncertainty and disturbances simultaneously was not considered. In this paper, we extend the result of [9] to constrained uncertain linear discrete-time invariant systems using a mixed ℋ2/ℋ∞ design approach and the uncertainty considered is norm-bounded additive. This is more suitable as both performance and robustness issues are handled within a unified framework.The method presented in this paper develops an LMI design procedure for the state feedback gain matrixF, allowing input and state constraints to be included in a less conservative manner. A main contribution is the accomplishment of a prescribed disturbance attenuation in a systematic way by incorporating the well-known robustness guarantees through ℋ∞ constraints into the MPC scheme. In addition, norm-bounded additive uncertainty is also incorporated. A preliminary version of some of the work presented in this paper was presented in [10].The structure of the work is as follows. After defining the notation, we describe the system and give a statement of the mixedℋ2/ℋ∞ problem in Section 2. In Section 3, we derive sufficient conditions, in the form of LMIs, for the existence of a state feedback control law that achieves the design specifications. In Section 4, we consider two examples that illustrate our algorithm. Finally, we conclude in Section 5.The notation we use is fairly standard.ℛ denotes the set of real numbers, ℛn denotes the space of n-dimensional (column) vectors whose entries are in ℛ and ℛn×m denoting the space of all n×m matrices whose entries are in ℛ. For A∈ℛn×m, we use the notation AT to denote transpose. For x,y∈ℛn, x<y (and similarly ≤, >, and ≥) is interpreted element wise. The identity matrix is denoted as I and the null matrix by 0 with the dimension inferred from the context.


## 2. Problem Formulation

We consider the following discrete-time linear time invariant system:(1)xk+1=Axk+Bwwk+Buuk+Bppk,qk=Cqxk+Dquuk+Dqwwk,pk=Δkqk,zk=[CzxkDzuuk],x0given,
where x0 is the initial state, xk∈ℛn is the state, wk∈ℛnw is the disturbance, uk∈ℛnu is the control, zk∈ℛnz is the controlled output, A∈ℛn×n, Bw∈ℛn×nw, Bu∈ℛn×nu, Cz∈ℛnz1×n, and Dzu∈ℛnz2×nu, and where nz=nz1+nz2. The signals qk and pk model uncertainties or perturbations appearing in the feedback loop.The operator,Δk, is block diagonal:(2)Δk∈Δk={Δk=[Δ1k0⋱0Δtk]:‖Δik‖≤1∀i},
and is norm bounded by one. Scalings can be included in Cq and Bp, thus generalizing the bound. Δk can represent either a memoryless time-varying matrix with σ¯(Δik)≤1, for i=1,…,t, k≥0, or the constraints:(3)pikTpik≤qikTqik,i=1,…,t,
where pk=[p1k,…,ptk]T, qk=[q1k,…,qtk]T, and the partitioning is induced by Δk. Each Δk is assumed to be either a full block or a repeated scalar block, and models a number of factors, such as dynamics or parameters, nonlinearities, that are unknown, unmodeled or neglected. In this work, we only consider full blocks for simplicity.In terms of the state space matrices, this formulation can be viewed as replacing a fixed(A,Bu,Bw) by (A,Bu,Bw)∈(𝒜,ℬu,ℬw), where(4)(A,Bu,Bw)={[A+BpΔkCq,Bu+BpΔkDqu,Bw+BpΔkDqw]∣Δk∈Δk}.In robust model predictive control, we consider norm-bounded uncertainty and define stability in terms of quadratic stability [11] which requires the existence of a fixed quadratic Lyapunov function (V(ζ)=ζTPζ, P>0) for all possible choices of the uncertainty parameters.In the case of norm-bounded uncertainties:(5)[ABwBu]∈{[AoBwoBuo]+FAΔH[EAEwEu]:Δ∈Δ},
where [AoBwoBuo] represents the nominal model, ΔH=Δ(I-HΔ)-1, with(6)Δ∈Δ∶={Δ=diag⁡(δ1Iq1,…,δlIql,Δl+1,…,Δl+f):‖Δ‖≤1,δi∈R,Δi∈Rqi×qiδ1Iq1,…,δlIql,Δl+1,…,Δl+f}Δ
and where FA, EA, Ew, Eu, and H are known and constant matrices with appropriate dimensions. This linear fractional representation of uncertainty, which is assumed to be well posed over Δ (i.e., det(I-HΔ)≠0 for all Δ∈Δ), has great generality and is used widely in robust control theory [12].We use the following lemma, which is a slight modification of a result in [13] and which uses the fact that Δ∈Δ to remove explicit dependence on Δ for the solution with norm bounded uncertainties.Lemma 1.
LetΔ be as described in (6) and define the subspaces
(7)𝚺={diag⁡(S1,…,Sl,λ1Iql+1,…,λsIql+f):Si=SiT∈Rqi×qi,λj∈R},Γ={diag⁡(G1,…,Gl,0ql+1,…,0ql+f):Gi=-GiT∈Rqi×qi}.
Let T1=T1T, T2, T3, T4 be matrices with appropriate dimensions. We have det(I-T4Δ)≠0 and T1+T2Δ(I-T4Δ)-1T3+T3T(I-ΔTT4T)-1ΔTT2T<0 for every Δ∈Δ if there exist S∈Σ and G∈Γ such that S>0 and
(8)[T1+T2ST2TT3T+T2ST4T+T2GT3+T4ST2T+GTT2TT4ST4T+T4G+GTT4T-S]<0.
If Δ is unstructured, then (8) becomes
(9)[T1+λT2T2TT3T+λT2T4TT3+λT4T2Tλ(T4T4T-I)]<0,
for some scalar λ>0. In this case, condition (9) is both necessary and sufficient.We also use the following Schur complement result [14].Lemma 2.
LetX11=X11T and X22=X22T. Then
(10)[X11X12X12TX22]≥0⟺X22≥0,X22-X12TX11+X12≥0,X12(I-X11X11+)=0,
where X11+ denotes the Moore-Penrose pseudo-inverse of X11.We assume that the pair(A,Bu) is stabilizable and that the disturbance is bounded as(11)‖w‖2∶=∑k=0∞wkTwk≤w¯,
where w¯≥0is know.The aim is to find a state feedback control law{uk=Fxk} in ℒ2, where F∈ℛnu×n, such that the following constraints are satisfied for all Δk∈Δk.(1)Closed-loop stability: the matrix A+BuF is stable.(2)Disturbance rejection: for given γ>0, the transfer matrix from w to z, denoted as Tzw, is quadratically stable and satisfies the ℋ∞ constraint(12)‖z‖2<γ‖w‖2,
for x0=0.(3)Regulation: for given α>0, the controlled output satisfies the ℋ2 constraint:(13)‖z‖2:=∑k=0∞zkTzk<α.(4)Input constraints: for given H1,…,Hmu∈ℛnu×nu, Hj=HjT≥0, h1,…,hmu∈ℛnu×1, and u¯1,…,u¯mu∈ℛ, the inputs satisfy the quadratic constraints:(14)ukTHjuk+2hjTuk≤u¯j,∀k;forj=1,…,mu.(5)State/output constraints: for given G1,…,Gmx∈ℛn×n, Gj=GjT≥0, g1,…,gmx∈ℛn×1, and x¯1,…,x¯mx∈ℛ the states/outputs satisfy the quadratic constraints:(15)xk+1TGjxk+1+2gjTxk+1≤x¯j,∀k;forj=1,…,mx.
An F∈ℛnu×n satisfying these requirements will be called an admissible state feedback gain.


## 3. LMI Formulation of Sufficiency Conditions

The next theorem, which is the main result of this paper, derives sufficient conditions, in the form of LMIs, for the existence of an admissibleF.Theorem 3.
Let all variables, definitions, and assumptions be as above. Then there exists an admissible state feedback gain matrixF if there exists solutions Q=QT∈ℛn×n, Y∈ℛnu×n, δj≥0, μj≥0, νj≥0, Λ¯=diag(λ¯1I,…,λ¯tI)>0, and Ψ¯j=diag(ψ¯1I,…,ψ¯tI)>0 to the LMIs shown in (16)–(19).(16)[-Q⋆⋆⋆⋆⋆⋆0-α2γ2I⋆⋆⋆⋆⋆00-Λ¯⋆⋆⋆⋆AQ+BuYα2BwBpΛ¯-Q⋆⋆⋆CqQ+DquYα2Dqw00-Λ¯⋆⋆CzQ0000-α2I⋆DzuY00000-α2I]<0,(17)[1⋆⋆γ2w¯2α2γ2w¯2⋆x00Q]≥0,(18)[Q⋆⋆⋆Hj1/2YμjI⋆⋆-hjTY0μju¯j⋆00μj1]≥0,j=1,…,mu,
(19)[Q⋆⋆⋆⋆⋆⋆0δjI⋆⋆⋆⋆⋆00Ψ¯j⋆⋆⋆⋆Gj1/2(AQ+BuY)νjGj1/2BwGj1/2BpΨ¯jνjI⋆⋆⋆CqQ+DquYνjDqw00Ψ¯j⋆⋆-gjT(AQ+BuY)-νjgjTBw-gjTBpΨ¯j00νjx¯j-δjw¯2⋆00000νj1]≥0,j=1,…,mx.Here,⋆ represents terms readily inferred from symmetry and the partitioning of Λ¯ and Ψ¯j is induced by the partitioning of Δk. If such solutions exist, then F=YQ-1.Remark 4.
The variables in the LMI minimization of Theorem3 are computed online at time k, the subscript k is omitted for convenience.Proof.
Usinguk=Fxk, the dynamics in (1) become
(20)xk+1=(A+BuF)︷Aclxk+Bwwk+Bppk,zk=[CzDzuF]︷Cclxk.
Consider a quadratic functionV(x)=xTPx, P>0 of the state xk. It follows from (20) that
(21)V(xk+1)-V(xk)=xkT[AclTPAcl-P]xk+xkTAclTPBwwk+xkTAclTPBppk+wkTBwTPAclxk+wkTBwTPBwwk+wkTBwTPBppk+pkTBpTPAclxk+pkTBpTPBwwk+pkTBpTPBppk=[xkTwkTpkT]K[xkwkpk]-xkTCclTCclxk+γ2wkTwk,
where
(22)K=[AclTPAcl-P+CclTCclAclTPBwAclTPBpBwTPAclBwTPBw-γ2IBwTPBpBpTPAclBpTPBwBpTPBp].
Using qk=(Cq+DquF)xk+Dqwwk,
(23)qkTΛqk=xkT(Cq+DquF)TΛ(Cq+DquF)xk+xkT(Cq+DquF)TΛDqwwk+wkTDqwTΛ(Cq+DquF)xk+wkTDqwTΛDqwwk,
where Λ=diag(λ1I,…λtI).
Substituting (23) into (21), it can be verified that we can write(24)V(xk+1)-V(xk)=[xkTwkTpkT]K¯[xkwkpk]+pkTΛpk-qkTΛqk-xkTCclTCclxk+γ2wkTwk,
where K¯ is defined in (25) and Cpw:=Cq+DquF.(25)K¯=[AclTPAcl-P+CclTCcl+CpwTΛCpwAclTPBw+CpwTΛDqwAclTPBpBwTPAcl+DqwTΛCpwBwTPBw-γ2I+DqwTΛDqwBwTPBpBpTPAclBpTPBwBpTPBp-Λ].
Assuming thatlimk→∞xk=0 we have(26)∑k=0∞[xk+1TPxk+1-xkTPxk]=-x0TPx0.
We write theℋ2 cost function as(27)‖z‖22=∑k=0∞(xkTCclTCclxk-γ2wkTwk)+γ2∑k=0∞wkTwk.
Adding (26) and (27) and carrying out a simple manipulation gives(28)‖z‖22=x0TPx0+γ2‖w‖22+∑k=0∞[xkTwkTpkT]K¯[xkwkpk]+∑k=0∞(pkTΛpk-qkTΛqk),
where K¯ is defined in (25).
Settingx0=0, it follows from (3), (12), and (28) that ∥z∥2<γ∥w∥2 if K¯<0 and Λ≥0. In this work, we will take Λ>0 to simplify our solution [8]. Using (2) and Lemma 1 it can be shown that
(29)K¯<0,
is also sufficient for quadratic stability of Tzw.
Next, we linearize the matrix inequality (29) by applying a Schur complement, to give
(30)[-P⋆⋆⋆⋆⋆⋆0-γ2I⋆⋆⋆⋆⋆00-Λ⋆⋆⋆⋆AclBwBp-P-1⋆⋆⋆CpwDqw00-Λ-1⋆⋆Cz0000-I⋆DzuF00000-I]<0.
Pre- and post-multiplying the equation above by diag(P-1,I,I,I,I,I,I) gives
(31)[-P-1⋆⋆⋆⋆⋆⋆0-γ2I⋆⋆⋆⋆⋆00-Λ⋆⋆⋆⋆AclP-1BwBp-P-1⋆⋆⋆CpwP-1Dqw00-Λ-1⋆⋆CzP-10000-I⋆DzuFP-100000-I]<0,
setting Q=α2P-1, F=YPα-2=YQ-1, Cpw=Cq+DquF and multiplying through by α2, the equation above becomes
(32)[-Q⋆⋆⋆⋆⋆⋆0-α2γ2I⋆⋆⋆⋆⋆00-α2Λ⋆⋆⋆⋆AQ+BuYα2Bwα2Bp-Q⋆⋆⋆CqQ+DquYα2Dqw00-α2Λ-1⋆⋆CzQ0000-α2I⋆DzuY00000-α2I]<0.
Pre- and post-multiplying the equation above by diag(I,I,Λ-1,I,I,I,I) gives
(33)[-Q⋆⋆⋆⋆⋆⋆0-α2γ2I⋆⋆⋆⋆⋆00-α2Λ-1⋆⋆⋆⋆AQ+BuYα2Bwα2BpΛ-1-Q⋆⋆⋆CqQ+DquYα2Dqw00-α2Λ-1⋆⋆CzQ0000-α2I⋆DzuY00000-α2I]<0.
The equation above is a bilinear matrix inequality, thus by defining α2Λ-1 as a variable Λ¯, we get the LMI in (16).
Now, it follows from (3), (11), (28), and (29) that,
(34)‖z‖22≤x0TPx0+γ2‖w‖22≤x0TPx0+γ2w¯2.
Thus the ℋ2 constraint in (13) is satisfied if
(35)x0TPx0+γ2w¯2<α2.
Dividing by α2, rearranging and using a Schur complement give (17) as an LMI sufficient condition for (13).
To turn (14) and (15) into LMIs, we first show that xkTPxk≤α2∀k>0. Since K¯<0, it follows from (3) and (24) that
(36)xk+1TPxk+1-xkTPxk≤γ2wkTwk.
Applying this inequality recursively, we get
(37)xkTPxk≤x0TPx0+γ2∑j=0k-1wjTwj≤x0TPx0+γ2w¯2<α2.
It follows that
(38)‖P1/2xk‖2<α2,
or equivalently,
(39)xkTQ-1xk<1,∀k>0.
Next, we transform the constraints in (14) to a set of LMIs. Setting F=YQ-1=YPα-2 and uk=Fxk,
(40)ej(uk):=ukTHjuk+2hjTuk-u¯j=xkTQ-1YTHjYQ-1xk+2hjTYQ-1xk-u¯j.
Now for anyμj∈ℛ, we can write
(41)ej(uk)=-μj(1-xkTQ-1xk)-[xk1]T×([μjQ-1-Q-1YTHjYQ-1-Q-1YThj-hjTYQ-1-μj+u¯j])×[xk1].
Therefore a sufficient condition forej(uk)≤0 is μj≥0 and
(42)[μjQ-1-Q-1YTHjYQ-1-Q-1YThj-hjTYQ-1u¯j-μj]≥0.
Pre- and post-multiplying bydiag(Q,I) gives a bilinear matrix inequality and applying a Schur complement, we get
(43)[μjQYTHj1/2-YThjHj1/2YI0-hjTY0u¯j-μj]≥0.
Pre- and post-multiplying the above bilinear matrix inequality by diag(μj-1/2,μj1/2,μj1/2) and applying a Schur complement, this is equivalent to the LMI in (18).
Finally, to obtain an LMI formulation of the state constraints (15), the following analogous steps are carried out:
(44)fj(xk+1):=[xkwkpk]T[AclTBwTBpT]Gj[AclBwBp][xkwkpk]+2gjT[AclBwBp][xkwkpk]-x¯j.
Now for any νj,ρj∈ℛ, we can write(45)fj(xk+1)=-νj(1-xkTQ-1xk)-ρj(w¯2-wkTwk)-[xkwk1]T([νjQ-1-(Acl+BpΔkCpw)TGj(Acl+BpΔkCpw)⋆⋆-(Bw+BpΔkDqw)TGj(Acl+BpΔkCpw)ρjI⋆-gjT(Acl+BpΔkCpw)-gjT(Bw+BpΔkDqw)-νj-ρjw¯2+x¯j])[xkwk1].Therefore a sufficient condition for fj(xk+1)≥0 is νj≥0, ρj≥0 and(46)[νjQ-1⋆⋆0ρjI⋆-gjT(Acl+BpΔkCpw)-gjT(Bw+BpΔkDqw)x¯j-νj-ρjw¯2]-[(Acl+BpΔkCpw)T(Bw+BpΔkDqw)T0]Gj[(Acl+BpΔkCpw)(Bw+BpΔkDqw)0]≥0.Applying Schur complement to the above equation, we get(47)[νjQ-1⋆⋆⋆0ρjI⋆⋆-gjT(Acl+BpΔkCpw)-gjT(Bw+BpΔkDqw)x¯j-νj-ρjw¯2⋆Gj1/2(Acl+BpΔkCpw)Gj1/2(Bw+BpΔkDqw)0I]≥0.
WhenΔ is structured we proceed as follows. For norm-bounded uncertainty, we first separate the terms involving modeling uncertainties from the other terms as
(48)[νjQ-1⋆⋆⋆0ρjI⋆⋆-gjTAcl-gjTBwx¯j-νj-ρjw¯2⋆-Gj1/2Acl-Gj1/2Bw0I]︸-T1+[00-gjTBpGj1/2Bp]︸-T2Δk[CpwDqw00]︸T3+[CpwTDqwT00]︸T3TΔkT[00-BpTgjBpTG1/2]︸-T2T≥0.
Equation (48) is equivalent to -T1-T2ΔT3-T3TΔTT2T>0, where T4=0. By using (8) from Lemma 1, we have
(49)[-T1-T3TST3-T2-T2TS]>0.
Applying Schur complement to (49), we get
(50)[-T1T3T-T2T3S-10-T2T0S]>0.
Substituting the variables from (48) into (50) and swapping the third and sixth diagonal elements, we get
(51)[νjQ-1⋆⋆⋆⋆⋆0ρjI⋆⋆⋆⋆00S⋆⋆⋆Gj1/2AclGj1/2BwGj1/2BpI⋆⋆CpwDqw00S-1⋆-gjTAcl-gjTBw-gjTBp00x¯j-νj-ρjw¯2]≥0.
Pre- and post-multiplying by diag(Q,I,I,I,I,I) gives
(52)[νjQ⋆⋆⋆⋆⋆0ρjI⋆⋆⋆⋆00S⋆⋆⋆Gj1/2AclGj1/2BwGj1/2BpI⋆⋆CpwDqw00S-1⋆-gjTAcl-gjTBw-gjTBp00x¯j-νj-ρjw¯2]≥0.
The above equation is bilinear and thus we pre- and post-multiply it by diag(νj-1/2,νj1/2,νj1/2,νj1/2,νj1/2,νj1/2) to obtain(53)[Q⋆⋆⋆⋆⋆0νjρjI⋆⋆⋆⋆00νjS⋆⋆⋆Gj1/2AclνjGj1/2BwνjGj1/2BpνjI⋆⋆CpwνjDqw00νjS-1⋆-gjTAcl-νjgjTBw-νjgjTBp00νjx¯j-νj2-νjρjw¯2]≥0.From the above equation, we can see that the variable S and its inverse appear in the matrix inequality, thus to make it uniform, we pre- and post-multiply by diag(I,I,S-1,I,I,I) to get(54)[Q⋆⋆⋆⋆⋆0νjρjI⋆⋆⋆⋆00νjS-1⋆⋆⋆Gj1/2AclνjGj1/2BwνjGj1/2BpS-1νjI⋆⋆CpwνjDqw00νjS-1⋆-gjTAcl-νjgjTBw-νjgjTBpS-100νjx¯j-νj2-νjρjw¯2]≥0.Thus the above equation is a nonlinear matrix inequality in νj2 and bilinear in νjρj and νjS-1, hence we define new variables Ψ¯j=νjS-1 and δj=νjρj and finally applying a Schur complement, we obtain the LMI of (19).Remark 5.
The input and state constraints used in this paper are more general than those used in [9] in that we allow linear terms and so this makes it possible to include asymmetric or hyperplane constraints.Remark 6.
When there is no uncertainty, the problem reduces to disturbance rejection technique considered in [9].Remark 7.
When there is no disturbance, the results reduce to those of [7].Remark 8.
The method used in this paper guarantees recursive feasibility (see [15, Chapter  4]). Also see [16] for a different approach.


## 4. Numerical Examples

In this section, we present two examples that illustrate the implementation of the proposed scheme. In the first example we consider a solenoid system, and in the second example we consider the coupled spring-mass system. The solution to the linear objective minimization was computed using LMI Control Toolbox in the MATLAB® environment and α2 was set as a variable.


### 4.1. Example 1

We consider a modified version of the solenoid system adapted from [17]. The system (see Figure 1) consists of a central object wrapped with coil and is attached to a rigid surface via a spring and damper, which forms a passive vibration isolator. The solenoid is one of the common actuator components. The basic principle of operation involves a moving ferrous core (a piston) that moves inside a wire coil. Normally, the piston is held outside the core by a spring and damper. When a voltage is applied to the coil and current flows, the coil builds up a magnetic field that attracts the piston and pulls it into the center of the coil. The piston can be used to supply a linear force. Application of this includes pneumatic valves and car door openers.Figure 1
Solenoid system.The system is modeled by(55)[xk+11xk+12]=[0.61480.0315-0.3155-0.0162][xk1xk2]+[0.03850.0315]uk+[0.003850.00315]wk+[010]pk,qk=Cqxk+Dquuk+Dqwwk,pk=Δqk,zk=[CzxkDzuuk],
where
(56)Cq=[10],Dqu=1,Dqw=0,
where x1 and x2 are the position and the velocity of the plate. The cost function is specified using Cz=diag(1,1) and Dzu=10. The magnetic force u is the control variable, and w is the external disturbance to the system, which is bounded in the range [-1,1]. The initial state is given as x0=[10]T.We chooseγ2=0.01 and γ2=1. Figures 2 and 3 compare the closed-loop response for the high and low disturbance rejection levels, respectively, for randomly generated Δ’s. The optimization is feasible, the response is stable, and the performance is good. A control constraint of |uk|≤0.5 is imposed, which is satisfied. The computation time for 100 samples was about 10 s, making 0.1 s per sample.Closed-loop response of the solenoid system withγ2=0.01.
(a)(b)Closed-loop response of the solenoid system withγ2=1.
(a)(b)


### 4.2. Example 2

We revisit a modified version of Example 2 reported in [7]. The system consists of a two-mass-spring model whose discrete-time equivalent is obtained using Euler first-order approximation with a sampling time of 0.1 s. The model in terms of disturbance and perturbation variables is(57)xk+1=[100.100100.1-0.1Km10.1Km1100.1Km2-0.1Km210]xk+[000.1m10]uk+Bwwk+Bppk,qk=Cqxk+Dquuk+Dqwwk,pk=Δqk,yk=[0100]xk,
where
(58)Bw=[00.0100],Bp=[00-0.10.1],Cq=[0.475-0.47500],Dqw=0,Dqu=0,
where x1 and x2 are the positions of body 1 and 2, and x3 and x4 are their velocities, respectively. m1 and m2 are the masses of the two bodies and K is the spring constant. The initial state is given as x0=[1100]T. The cost function is specified using CZ=diag(1,1,1,1), and Dzu=1. We consider the system with m1=m2=1 and K∈[0.5,10].A persistent disturbance of the formwi=0.1 for all sample time was considered. Here we set γ2=0.45 and γ2=4. Figures 4 and 5 compare the closed-loop response for the high and low disturbance rejection levels, respectively, for randomly generated Δ’s. The value of γ2 for high disturbance rejection was the lowest value for which a feasible solution exists. An input constraint of |uk|≤1 is imposed, which is satisfied. The computation time for 300 samples was about 47 s, making 0.16 s per sample.Closed-loop response of the coupled spring-mass system withγ2=0.45.
(a)(b)Closed-loop response of the coupled spring-mass system withγ2=4.
(a)(b)


### 4.3. Discussion

Note that the performance and response of the systems based on the high disturbance rejection level were better than those obtained using low disturbance rejection level, since the states and control are regulated to smaller steady state values. Constraints on the input were satisfied in both cases; however, the constraints were more conservative with respect to the control signal for the low disturbance rejection level. For example in the solenoid system, the control signal for high disturbance rejection level was 0.4826 and that for low disturbance rejection level was 0.4144. The issue of conservativeness in the mixedℋ2/ℋ∞ setting has been considered in [18]. For the systems considered, the upper bound on the ℋ2 cost function α2 is depicted in Figures 6 and 7 for the high and low disturbance rejection levels. It can been seen that the performance coefficient obtained for the high and low disturbance rejection levels is small. However, the offset level on the low disturbance level is higher than that for the high disturbance level. This is due to the higher value of γ2 in x0TPx0+γ2w¯2. In Example 2, we have considered uncertainty in the model by using variable spring constant K.Upper bound onℋ2 cost function for solenoid system.
(a)
System withγ2=0.01(b)
System withγ2=1Upper bound onℋ2 cost function for coupled spring-mass system.
(a)
System withγ2=0.45(b)
System withγ2=4


## 4.1. Example 1

We consider a modified version of the solenoid system adapted from [17]. The system (see Figure 1) consists of a central object wrapped with coil and is attached to a rigid surface via a spring and damper, which forms a passive vibration isolator. The solenoid is one of the common actuator components. The basic principle of operation involves a moving ferrous core (a piston) that moves inside a wire coil. Normally, the piston is held outside the core by a spring and damper. When a voltage is applied to the coil and current flows, the coil builds up a magnetic field that attracts the piston and pulls it into the center of the coil. The piston can be used to supply a linear force. Application of this includes pneumatic valves and car door openers.Figure 1
Solenoid system.The system is modeled by(55)[xk+11xk+12]=[0.61480.0315-0.3155-0.0162][xk1xk2]+[0.03850.0315]uk+[0.003850.00315]wk+[010]pk,qk=Cqxk+Dquuk+Dqwwk,pk=Δqk,zk=[CzxkDzuuk],
where
(56)Cq=[10],Dqu=1,Dqw=0,
where x1 and x2 are the position and the velocity of the plate. The cost function is specified using Cz=diag(1,1) and Dzu=10. The magnetic force u is the control variable, and w is the external disturbance to the system, which is bounded in the range [-1,1]. The initial state is given as x0=[10]T.We chooseγ2=0.01 and γ2=1. Figures 2 and 3 compare the closed-loop response for the high and low disturbance rejection levels, respectively, for randomly generated Δ’s. The optimization is feasible, the response is stable, and the performance is good. A control constraint of |uk|≤0.5 is imposed, which is satisfied. The computation time for 100 samples was about 10 s, making 0.1 s per sample.Closed-loop response of the solenoid system withγ2=0.01.
(a)(b)Closed-loop response of the solenoid system withγ2=1.
(a)(b)


## 4.2. Example 2

We revisit a modified version of Example 2 reported in [7]. The system consists of a two-mass-spring model whose discrete-time equivalent is obtained using Euler first-order approximation with a sampling time of 0.1 s. The model in terms of disturbance and perturbation variables is(57)xk+1=[100.100100.1-0.1Km10.1Km1100.1Km2-0.1Km210]xk+[000.1m10]uk+Bwwk+Bppk,qk=Cqxk+Dquuk+Dqwwk,pk=Δqk,yk=[0100]xk,
where
(58)Bw=[00.0100],Bp=[00-0.10.1],Cq=[0.475-0.47500],Dqw=0,Dqu=0,
where x1 and x2 are the positions of body 1 and 2, and x3 and x4 are their velocities, respectively. m1 and m2 are the masses of the two bodies and K is the spring constant. The initial state is given as x0=[1100]T. The cost function is specified using CZ=diag(1,1,1,1), and Dzu=1. We consider the system with m1=m2=1 and K∈[0.5,10].A persistent disturbance of the formwi=0.1 for all sample time was considered. Here we set γ2=0.45 and γ2=4. Figures 4 and 5 compare the closed-loop response for the high and low disturbance rejection levels, respectively, for randomly generated Δ’s. The value of γ2 for high disturbance rejection was the lowest value for which a feasible solution exists. An input constraint of |uk|≤1 is imposed, which is satisfied. The computation time for 300 samples was about 47 s, making 0.16 s per sample.Closed-loop response of the coupled spring-mass system withγ2=0.45.
(a)(b)Closed-loop response of the coupled spring-mass system withγ2=4.
(a)(b)


## 4.3. Discussion

Note that the performance and response of the systems based on the high disturbance rejection level were better than those obtained using low disturbance rejection level, since the states and control are regulated to smaller steady state values. Constraints on the input were satisfied in both cases; however, the constraints were more conservative with respect to the control signal for the low disturbance rejection level. For example in the solenoid system, the control signal for high disturbance rejection level was 0.4826 and that for low disturbance rejection level was 0.4144. The issue of conservativeness in the mixedℋ2/ℋ∞ setting has been considered in [18]. For the systems considered, the upper bound on the ℋ2 cost function α2 is depicted in Figures 6 and 7 for the high and low disturbance rejection levels. It can been seen that the performance coefficient obtained for the high and low disturbance rejection levels is small. However, the offset level on the low disturbance level is higher than that for the high disturbance level. This is due to the higher value of γ2 in x0TPx0+γ2w¯2. In Example 2, we have considered uncertainty in the model by using variable spring constant K.Upper bound onℋ2 cost function for solenoid system.
(a)
System withγ2=0.01(b)
System withγ2=1Upper bound onℋ2 cost function for coupled spring-mass system.
(a)
System withγ2=0.45(b)
System withγ2=4


## 5. Conclusion

In this paper, we proposed a robust model predictive control design technique using mixedℋ2/ℋ∞ for time invariant discrete-time linear systems subject to constraints on the inputs and states. This method takes account of disturbances naturally by imposing the ℋ∞-norm constraint in (14) and thus extends the work in [9] by the introduction of structured, norm-bounded uncertainty. The uncertain system was represented by LFTs. The development is based on full state feedback assumption and the on-line optimization involves the solution of an LMI-based linear objective minimization (convex optimization). Hence, the resulting state-feedback control law minimizes an upper bound on the robust objective function. The new approach reduces to that of [9] when there are no perturbations present in the system and to [7] when there are no disturbances. Thus, we have been able to show that it is possible to handle uncertainty and disturbance in the mixed ℋ2/ℋ∞ model predictive control design approach. The two examples illustrate the application of the proposed method.



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*Source: 402948-2012-04-04.xml*