# A Summary of Dynamic Output Feedback Robust MPC for Linear Polytopic Uncertainty Model with Bounded Disturbance

Authors: Baocang Ding; Xiaoming Tang; Jianchen Hu
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

This paper is the summary and enhancement of the previous studies on dynamic output feedback robust model predictive control (MPC) for the linear parameter varying model (described in a polytope) with additive bounded disturbance. When the state is measurable and there is no bounded disturbance, the robust MPC has been developed with several paradigms and seems becoming mature. For the output feedback case for the LPV model with bounded disturbance, we have published a series of works. Anyway, it lacks a unification of these publications. This paper summarizes the existing results and exposes the ideas in a unified framework. Indeed there is a long way to go for the output feedback case for the LPV model with bounded disturbance. This paper can pave the way for further research on output feedback MPC.


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## Body

## 1. Introduction

In the control community, it is widely recognized that linear parameter varying (LPV) model, whose system matrices lie in the polytope, is a good tool for representing the nonlinearity and uncertainty. The well-known Takagi–Sugeno (T-S) model (see, e.g., [1, 2]), often when the stability is considered, can be considered as the LPV model. Therefore, it is not surprising that there are a lot of research works on the LPV model-based and T-S model-based controls. Moreover, it is impossible that all the uncertainties can be included in the parametric polytopes. The additive bound disturbance, with its real-time value arbitrarily changing, without useful statistics, is another widely accepted uncertainty description. This paper considers the above LPV model (including T-S model) with additive bound disturbance.The research on robust model predictive control (MPC) for LPV model has begun as early as in 1996 (see [3]). After researching for slightly longer than a decade, the robust MPC for LPV model (excluding T-S model and bounded disturbance), when the state is assumed measurable, seems becoming mature; there are four types in this robust MPC community, i.e., open-loop MPC, partial feedback MPC, feedback MPC, and parameter-dependent open-loop MPC (see the Introduction of [4]). In the partial feedback, the control move u is defined as u=Fx+c (i.e., state feedback Fx plus perturbation c); when c=0, the partial feedback becomes the feedback and when F=0, open-loop. When the switching horizon N=0 or N=1, the four types are equivalent. When N≥2, u can be defined as parameter-dependent as in [4]; in this parameter-dependent case, open-loop is equivalent to partial feedback.From 2006, we have begun research on robust MPC for LPV model (including T-S model and bounded disturbance), where the state can be unmeasurable. We have published several works, emphasizing onN=0, i.e., a close generalization of [3]. For N>1, we have not reached to a technique which is, to us, as satisfactory as that in the case when the state x is measurable. Therefore, this paper concentrates on the output feedback robust MPC with N=0. N=0 here indicates that there is no free control move, i.e., both u and c will not be the immediate decision variables.Although we have published several works on output feedback MPC, there lacks a unified and updated framework. These works are given across more than 10 years. The results are scattered in different works; there are necessary overlaps due to problem statements and recalls; some of the results are improved which are not easy to trace back; some of the details are missed in all published results; the original thoughts may be overlooked. In this paper, we rearrange the results of output feedback MPC for the LPV model during these years, compromising the above demerits in the existing works. We think that this is useful for future research; it is not only a guideline, but also a summary for readers.Notations:I is the unitary matrix with appropriate dimension; xk+ik is the prediction of xk+i at time k. Moreover,(i)
u: in ℜnu, the control input signal(ii)
w: in ℜnw, the disturbance(iii)
x: in ℜnx, the true state(iv)
xc: in ℜnxc, the estimator state or controller state(v)
y: in ℜny, the output(vi)
ξ: the component-wise absolute value of ξ(vii)
εM: the ellipsoid associated with the positive-definite matrix M, i.e., εM=ξξTMξ≤1(viii)
CoS: an element belonging to CoS means that it is a convex combination of the elements in the polytope S, with the scalar combing coefficients being nonnegative and summing as 1(ix)
★: this symbol induces a symmetric structure in any square matrix(x)
∗: a value with superscript ∗ means that it is the solution of the optimization problem


## 2. Dynamic Output Feedback Robust MPC Problem

Consider the following linear parameter varying (LPV) model:(1)xk+1=Akxk+Bkuk+Dkwk,yk=Ckxk+Ekwk,zk=Ckxk+Ekwk,z′k=Fkxk+Gkwk,where zk∈ℜnz (see [5, 6]) and z′k∈ℜnz′ (see [7, 8]) are the constrained signal and penalized signal, respectively, and w is unknown, norm-bounded, and persistent.Assumption 1.
w
k
≤
1 for all k≥0.Assumption 2.
A
B
C
D
E
C
E
F
G
k
∈
Ω
≔
Co
A
l
B
l
C
l
D
l
E
l
C
l
E
l
F
l
G
l
l
=
1
,
…
,
L, i.e., there exist nonnegative coefficients λlk,l=1,…,L such that ∑l=1Lλlk=1 and ABCDECEFGk=∑l=1LλlkAlBlClDlElClElFlGl.
SinceDk,Ek are shaping matrices, Assumption 1 applies to any norm-bounded disturbance. If λlk’s are exactly known at the current time k, but λlk+i for all i>0 are unknown at the current time k, then we specially call (1) the quasi-LPV model.
The hard physical constraints are(2)uk≤u¯,Ψzk+1≤ψ¯,k≥0,where u¯=u¯1,u¯2,…,u¯nuT; ψ¯=ψ¯1,ψ¯2,…,ψ¯qT; u¯j>0, j=1,…,nu; ψ¯j>0, j=1,…,q; Ψ∈ℜq×nz.
Whenx is fully measurable and wk≡0, Kothare et al. [3] have developed a technique which, at each time k, solves a linear matrix inequality (LMI) optimization problem with four constraints (confinement of the current state, invariance/stability/optimality condition, input constraint, and state/output constraint). In the following, we will generalize the procedure of [3] to the cases when x can be unmeasurable and wk≠0.Theorem 1 (see [9]).
Consider system (1), with Assumptions 1 and 2 being satisfied. Adopt the dynamic output feedback controller, i.e.,(3)xck+1=Ackxck+Bckyk,uk=Cckxck+Dckyk,where the controller parameters are defined as parameter-dependent, i.e.,(4)Ack=∑l=1L∑j=1LλlkλjkA¯cljk,Bck=∑l=1LλlkB¯clk,Cck=∑j=1LλjkC¯cjk,Dck=D¯ck.
The controller parametric matricesA¯clj,B¯cl,C¯cj,D¯ck are taken as(5)D¯c=D^c,C¯cj=C^cj−D¯cCjQ1Q2−1,B¯cl=M2−TB^cl−M1BlD¯c,A¯clj=M2−TA^clj−M1AlQ1−M1BlD¯cCjQ1−M2TB¯clCjQ1−M1BlC¯cjQ2Q2−1,where “k” is omitted for brevity. Further, A^clj,B^cl,C^cj,D^ck are obtained by solving(6)minγ,αlj,ϱ,Q1,M1,A^clj,B^cl,C^cj,D^ckγk,(7)s.t.M1k≤ϱkMek,(8)1−ϱk⋆⋆UkxckQ1k⋆0IM1k≥0,(9)∑l=1LClℓd,2ϒllQBk+∑l=1L−1∑j=l+1LCljℓd,1,1ϒljQBk+ϒjlQBk≥0,ℓ=1,…,Kd+2,(10)M1k⋆⋆⋆IQ1k⋆⋆00I⋆11−η1sξsD^ckCj11−η1sξsC^cjk1η1sξsD^ckEju¯s2≥0,j=1,…,L,s=1,…,nu,(11)∑l=1LClℓd,2ϒhllszk+∑l=1L−1∑j=l+1LCljℓd,1,1ϒhljszk+ϒhjlszk≥0,ℓ=1,…,Kd+2,h=1,…,L,s=1,…,q,where Uk is a transformation matrix being given before solving (6)–(11), d is a fixed nonnegative integer, η1s∈0,1 are the fixed scalars, and ξs is the s-th row of nu-ordered identity matrix. In (9) and (11), Kd+2 is the set of L-tuples obtained from all possible combinations of d1,d2,…,dL, dl≥0,l=1,…,L such that d1+d2+⋯+dL=d+2. The number of elements of Kd+2 is given by Kd+2=L+d+1!/d+2!L−1!. The L-tuples of Kd+2 are lexically ordered as ℓ=1,…,Kd+2. Moreover,(12)Clℓd,2=d!d1!⋯dl−1!dl−2!dl+1!⋯dL!,dl≥2,0,otherwise,Cljℓd,1,1=d!d1!⋯dl−1!dl−1!dl+1!⋯dj−1!dj−1!dj+1!⋯dL!,dl≥1,dj≥1,0,otherwise.
In (9),(13)ϒljQB=1−αM1⋆⋆⋆⋆⋆⋆1−αI1−αQ1⋆⋆⋆⋆⋆00αI⋆⋆⋆⋆Al+BlD^cCjAlQ1+BlC^cjBlD^cEj+DlQ1⋆⋆⋆M1Al+B^clCjA^cljB^clEj+M1DlIM1⋆⋆Q11/2FjQ11/2FjQ1Q11/2Gj00γI⋆ℛ1/2D^cCjℛ1/2C^cjℛ1/2D^cEj000γI,where “k” is omitted, and Q1,ℛ are the weighting matrices. In (11),(14)ϒhljsz=M1⋆⋆⋆IQ1⋆⋆00I⋆♠1♠211−η2sη3sΨsEhBlD^cEj+Dlψ¯s2−1η2sΨsEhEhTΨsT,♠1=11−η2s1−η3sΨsChAl+BlD^cCj,♠2=11−η2s1−η3sΨsChAlQ1+BlC^cj,where “k” is omitted, Ψs is the s-th row of Ψ, and η2s,η3s∈0,1 are the fixed scalars.
TakeU0=I and an xc0, and suppose x0−xc0∈εMe0. At each k≥0,(a)
Fork>0, apply (4) and (5) to obtain Ac,Bck−1, then calculate xck=Ack−1xck−1+Bck−1yk−1(b)
Fork>0, take(15)Uk=Uk−1,(16)ϱk=1−xckTM3∗k−1−Uk−1TM1∗k−1Uk−1xck,(17)Mek=M1∗k−1ϱk−1,where(18)M3k−1=M2k−1M1k−1−Q1k−1−1−1M2k−1T,with M2k−1=−Uk−1TM1k−1(c)
Fork>0, find Me′,U′k satisfying(19)xk−1−Uk−1xck−1∈εMek−1,wk−1≤1⟹xk−U′kxck∈εMe′k,(20)Me′k≥Mek,and if (19) and (20) are feasible, then change Mek=Me′k and Uk=U′k(d)
Solve (6)–(11) to find Q1,M1,A^clj,B^cl,C^cj,D^c∗k(e)
TakeQ1,M1k=Q1,M1∗k, Q2k=Uk−1Q1k−M1k−1, and M2k=−UkTM1k(f)
Apply (4) and (5) to obtain Cck and Dck, then implement uk=Cckxck+Dckyk
Suppose (6)–(11) is feasible at time k=0. Then,(i)
(6)–(11) will be feasible at each k>0(ii)
γ,z′,u will converge to a neighborhood of 0, and the constraints in (2) are satisfied for all k≥0In (6)–(11), the four constraints of [3] are generalized (i.e., the confinement of xk being generalized to (7) and (8) which is the confinement of both xk and xck, invariance/stability/optimality condition to (9) which is the combination of quadratic boundedness and optimality conditions, input constraint to (10), and state/output constraint to (11) which is the constraint on z).In the following, let us show the details how the above generalizations happen, taking Theorem1 as one of the examples.


## 3. Model and Controller Descriptions

The predictive form of (1) is(21)xk+i+1k=Ak+ixk+ik+Bk+iuk+ik+Dk+iwk+i,yk+ik=Ck+ixk+ik+Ek+iwk+i,for all i≥0. The predictive form of (2) is(22)uk+ik≤u¯,Ψzk+i+1k≤ψ¯,i≥0,where zk+ik=Ck+ixk+ik+Ek+iwk+i. According to Assumption 2,(23)ABCDECEFGk+i=∑l=1Lλlk+iAlBlClDlElClElFlGl.


### 3.1. Controller for LPV Model

For the LPV model (1), the dynamic output feedback controller is of the following form (see firstly [10, 11]):(24)xck+1=Ackxck+Lckyk,uk=Fxkxck+Fykyk,where Ac,Lc are controller gain matrices and Fx,Fy are feedback gain matrices. It is unnecessary that nx=nxc. The predictive form of (24) is(25)xck+i+1k=Ackxck+ik+Lckyk+ik,uk+ik=Fxkxck+ik+Fykyk+ik.Remark 1.
There are 4 controller parametersAc,Lc,Fx,Fy in (24) and (25). In the literature, often there are only 2 controller parameters Lc,Fx for output feedback. We found that with only 2 controller parameters Lc,Fx, for (1), it is more difficult to find the feasible solution to the optimization problem of output feedback MPC. With 4 parameters Ac,Lc,Fx,Fy, output feedback MPC can be applied to a much larger range of system models.
Define the augmented statex˜=xxc. By applying (1) and (24), the augmented closed-loop system is(26)x˜k+1=Φkx˜k+Γkwk,where(27)Φk=Ak+BkFykCkBkFxkLckCkAck,Γk=BkFykEk+DkLckEk.
The predictive form of (26) is(28)x˜k+i+1k=Φi,kx˜k+ik+Γi,kwk+i,where(29)Φi,k=Ak+i+Bk+iFykCk+iBk+iFxkLckCk+iAck,Γi,k=Bk+iFykEk+i+Dk+iLckEk+i.
By applying (23), it is shown that(30)Φi,k=∑l=1Lλlk+i∑j=1Lλjk+iΦljk,Γi,k=∑l=1Lλlk+i∑j=1Lλjk+iΓljk,Φljk=Al+BlFykCjBlFxkLckCjAck,Γljk=BlFykEj+DlLckEj.


### 3.2. Controller for Quasi-LPV Model

For the quasi-LPV model (1), the dynamic output feedback controller is (3) and (4) (see firstly [12, 13]), where nx=nxc. The predictive form of (3) is(31)xck+i+1k=Ack+ixck+ik+Bck+iyk+ik,uk+ik=Cck+ixck+ik+Dck+iyk+ik,where(32)Ack+i=∑l=1L∑j=1Lλlk+iλjk+iA¯cljk,Bck+i=∑l=1Lλlk+iB¯clk,Cck+i=∑j=1Lλjk+iC¯cjk,Dck+i=D¯ck.Remark 2.
For the quasi-LPV, sinceλlk are known, we can utilize A¯clj,B¯cl,C¯cj,D¯ck to calculate the parameter-dependent Ac,Bc,Cck. Such Ac,Bc,Cck allows to find convex optimization problem to simultaneously give A¯clj,B¯cl,C¯cj,D¯ck. Hence, the parameter-dependent Ac,Bc,Cck is considerably better than the non-parameter-dependent Ac,Lc,Fxk.
Define the augmented statex˜=xxc. By applying (1) and (3), the augmented closed-loop system is(33)x˜k+1=Φkx˜k+Γkwk,where(34)Φk=Ak+BkDckCkBkCckBckCkAck,Γk=BkDckEk+DkBckEk.
The predictive form of (33) is(35)x˜k+i+1k=Φi,kx˜k+ik+Γi,kwk+i,where(36)Φi,k=Ak+i+Bk+iDck+iCk+iBk+iCck+iBck+iCk+iAck+i,Γi,k=Bk+iDck+iEk+i+Dk+iBck+iEk+i.
By applying (32), it is shown that(37)Φi,k=∑l=1Lλlk+i∑j=1Lλjk+iΦljk,Γi,k=∑l=1Lλlk+i∑j=1Lλjk+iΓljk,Φljk=Al+BlD¯ckCjBlC¯cjkB¯clkCjA¯cljk,Γljk=BlD¯ckEj+DlB¯clkEj.In the sequel, we often use the notations for LPV, but the results can be simply transplanted to quasi-LPV.


## 3.1. Controller for LPV Model

For the LPV model (1), the dynamic output feedback controller is of the following form (see firstly [10, 11]):(24)xck+1=Ackxck+Lckyk,uk=Fxkxck+Fykyk,where Ac,Lc are controller gain matrices and Fx,Fy are feedback gain matrices. It is unnecessary that nx=nxc. The predictive form of (24) is(25)xck+i+1k=Ackxck+ik+Lckyk+ik,uk+ik=Fxkxck+ik+Fykyk+ik.Remark 1.
There are 4 controller parametersAc,Lc,Fx,Fy in (24) and (25). In the literature, often there are only 2 controller parameters Lc,Fx for output feedback. We found that with only 2 controller parameters Lc,Fx, for (1), it is more difficult to find the feasible solution to the optimization problem of output feedback MPC. With 4 parameters Ac,Lc,Fx,Fy, output feedback MPC can be applied to a much larger range of system models.
Define the augmented statex˜=xxc. By applying (1) and (24), the augmented closed-loop system is(26)x˜k+1=Φkx˜k+Γkwk,where(27)Φk=Ak+BkFykCkBkFxkLckCkAck,Γk=BkFykEk+DkLckEk.
The predictive form of (26) is(28)x˜k+i+1k=Φi,kx˜k+ik+Γi,kwk+i,where(29)Φi,k=Ak+i+Bk+iFykCk+iBk+iFxkLckCk+iAck,Γi,k=Bk+iFykEk+i+Dk+iLckEk+i.
By applying (23), it is shown that(30)Φi,k=∑l=1Lλlk+i∑j=1Lλjk+iΦljk,Γi,k=∑l=1Lλlk+i∑j=1Lλjk+iΓljk,Φljk=Al+BlFykCjBlFxkLckCjAck,Γljk=BlFykEj+DlLckEj.


## 3.2. Controller for Quasi-LPV Model

For the quasi-LPV model (1), the dynamic output feedback controller is (3) and (4) (see firstly [12, 13]), where nx=nxc. The predictive form of (3) is(31)xck+i+1k=Ack+ixck+ik+Bck+iyk+ik,uk+ik=Cck+ixck+ik+Dck+iyk+ik,where(32)Ack+i=∑l=1L∑j=1Lλlk+iλjk+iA¯cljk,Bck+i=∑l=1Lλlk+iB¯clk,Cck+i=∑j=1Lλjk+iC¯cjk,Dck+i=D¯ck.Remark 2.
For the quasi-LPV, sinceλlk are known, we can utilize A¯clj,B¯cl,C¯cj,D¯ck to calculate the parameter-dependent Ac,Bc,Cck. Such Ac,Bc,Cck allows to find convex optimization problem to simultaneously give A¯clj,B¯cl,C¯cj,D¯ck. Hence, the parameter-dependent Ac,Bc,Cck is considerably better than the non-parameter-dependent Ac,Lc,Fxk.
Define the augmented statex˜=xxc. By applying (1) and (3), the augmented closed-loop system is(33)x˜k+1=Φkx˜k+Γkwk,where(34)Φk=Ak+BkDckCkBkCckBckCkAck,Γk=BkDckEk+DkBckEk.
The predictive form of (33) is(35)x˜k+i+1k=Φi,kx˜k+ik+Γi,kwk+i,where(36)Φi,k=Ak+i+Bk+iDck+iCk+iBk+iCck+iBck+iCk+iAck+i,Γi,k=Bk+iDck+iEk+i+Dk+iBck+iEk+i.
By applying (32), it is shown that(37)Φi,k=∑l=1Lλlk+i∑j=1Lλjk+iΦljk,Γi,k=∑l=1Lλlk+i∑j=1Lλjk+iΓljk,Φljk=Al+BlD¯ckCjBlC¯cjkB¯clkCjA¯cljk,Γljk=BlD¯ckEj+DlB¯clkEj.In the sequel, we often use the notations for LPV, but the results can be simply transplanted to quasi-LPV.


## 4. Characterization of Stability and Optimality

Consider the closed-loop systems (28) and (35). They have the same form. Both (28) and (35) have uncertain system parametric matrices which are composed of double convex combinations (i.e., convex combinations by coefficients λlk+i and λjk+i).We will borrow the notion of quadratic boundedness (QB) in [14, 15] to characterize the closed-loop stability of (28) and (35).


### 4.1. Review of Quadratic Boundedness

In [14], the following model with nominal parametric matrices is considered:(38)xk+1=Axk+Dvk,where A and D are time-invariant (fixed) matrix, v∈ℜnv. In [14], it is firstly assumed that v∈V where V is a compact (bounded and closed) set, and V⊂ℜnv.Definition 1.
(see [14]). System (38) is said to be quadratically bounded with Lyapunov matrix P>0 if(39)xTPx≥1⟹Ax+DvTPAx+Dv≤xTPx,∀v∈V.
System (38) is said to be strictly quadratically bounded with Lyapunov matrix P>0 if(40)xTPx>1⟹Ax+DvTPAx+Dv<xTPx,∀v∈V.Lemma 1.
(see [14]). Suppose there exists a ξ∈V such that Dξ≠0. If (38) is quadratically bounded with the Lyapunov matrix P>0, then it is strictly quadratically bounded with the same Lyapunov matrix.Definition 2.
The setS is a robust positively invariant set for (38), if(41)x∈S⟹Ax+Dv∈S,∀v∈V.Theorem 2.
(see [14]). Suppose v∈εPv with Pv>0. The following facts are equivalent:(i)
(38) is quadratically bounded with Lyapunov matrix P>0(ii)
(38) is strictly quadratically bounded with Lyapunov matrix P>0(iii)
The ellipsoidεP is a robust positively invariant set for (38)(iv)
xTPx≥vTPvv⟹Ax+DvTPAx+Dv≤xTPx(v)
There existsα>0 such that(42)1−αP−ATPA⋆−DTPAαPv−DTPD≥0;(vi)
A is exponentially stable (i.e., there existsP¯>0 such that P¯−ATP¯A>0)
In [15], the following model with uncertain parametric matrices is considered:(43)xk+1=Akxk+Dkvk,where AkDk belongs to a known bounded set, i.e., AkDk∈P for all k≥0, and D≠0 for at least one AD∈P.Definition 3.
(see [15]). Suppose vk∈εPv for all k≥0, in (43). System (43) is said to be strictly quadratically bounded with a common Lyapunov matrix P>0, if(44)xTPx>1⟹Ax+DvTPAx+Dv<xTPx,∀v∈εPv,∀AD∈P.
SinceD≠0 for at least one AD∈P, and v∈εPv, there exists a Dv≠0. Similarly to Lemma 1. if (43) is quadratically bounded with Lyapunov matrix P>0, then it is strictly quadratically bounded with the same Lyapunov matrix. The definition of quadratic boundedness is similar to Definition 1.Definition 4.
Supposevk∈εPv for all k≥0, in (43). The set S is a positively invariant set for (43), if(45)x∈S⟹Ax+Dv∈S,∀v∈εPv,∀AD∈P.Theorem 3.
(see [15]). Suppose vk∈εPv for all k≥0, in (43). The following facts are equivalent:(i)
(43) is strictly quadratically bounded with a common Lyapunov matrix P>0(ii)
The ellipsoidεP is a positively invariant set for (43)(iii)
There existsαk∈0,1 such that(46)1−αkP−AkTPAk⋆−DkTPAkαkPv−DkTPDk≥0.
Note that in the above theorem it is necessary to use a time-varyingαk.


### 4.2. Stability Condition

In the output feedback MPC, QB is equivalent to strict QB (see [16]). For the closed-loop systems (28) and (35), by generalizing the results in Section 4.1, we obtain the following results.Definition 5.
(see firstly [12, 13] for quasi-LPV and [11, 17] for LPV). Suppose (referring to Assumptions 1 and 2), at time k and for all i≥0:(47)wk+i≤1;there exist nonnegative coefficients λlk+i,l=1,…,L such that ∑l=1Lλlk+i=1 and ABCDEk=∑l=1Lλlk+iAlBlClDlEl.
System (28) or (35) is said to be quadratically bounded with a common Lyapunov matrix Mk>0, if(48)x˜k+ikMk2≥1⟹x˜k+i+1kMk2≤x˜k+ikMk2,∀i≥0.Definition 6.
With the assumptions in Definition5 satisfied, the set S is a positively invariant set for (28) or (35), if(49)x˜k+ik∈S⟹x˜k+i+1k∈S,∀i≥0.Theorem 4.
(see firstly [10, 11] for LPV and [12, 18] for quasi-LPV). With the assumptions in Definition 5 satisfied, the following facts are equivalent:(i)
(28) or (35) is quadratically bounded with a common Lyapunov matrix Mk>0(ii)
The ellipsoidεMk is a positively invariant set for (28) or (35)(iii)
There existsαi,k∈0,1 such that(50)1−αi,kMk−Φi,kTMkΦi,k⋆−Γi,kTMkΦi,kαi,kI−Γi,kTMkΓi,k≥0,i≥0,(iv)
Φi,k is exponentially stable for all i>0 (i.e., there exists M¯k>0 such that M¯k−Φi,kTM¯kΦi,k>0).In [19], the single-valued α is firstly replaced by(51)αi,k=∑l=1L∑j=1Lλlk+iλjk+iαlj.


### 4.3. Optimality Condition

The disturbance-free form of (28) or (35) is(52)x˜uk+i+1k=Φi,kx˜uk+ik,∀i≥0,x˜ukk=x˜k.Correspondingly,(53)uuk+ik=Fxkxc,uk+ik+Fykyuk+ik,yuk+ik=Ck+ixuk+ik,zuk+ik=Ck+ixuk+ik,zu′k+ik=Fk+ixuk+ik.Let us introduce the quadratic cost(54)Jk=∑i=0∞Jik,Jik=zu′k+ikQ12+xc,uk+ikQ22+uuk+ikℛ2,where Q1, Q2, and ℛ are positive-definite weighting matrices, and consider the condition(55)x˜uk+i+1kMk2−x˜uk+ikMk2≤−1γkJik,∀i≥0.In Theorem1, it has taken Q2=0. For exponentially stable Φi,k, it will result in limi⟶∞zu′k+ik=0, limi⟶∞xc,uk+ik=0, and limi⟶∞uuk+ik=0. Hence, summing (55) from i=0 to i=∞ yields(56)Jk≤γkx˜ukkMk2=γkx˜kMk2.Further, let(57)x˜k∈εMk.Then, applying (57) to (56) yields(58)Jk≤γk,that is, γk is an upper bound of Jk. We will take γk as the cost function of the optimization problems which finds the controller parametric matrices.The condition (55) can be rewritten as(59)x˜uk+ikTΠi,kx˜uk+ik≥0,where(60)Πi,k=Mk−Φi,kTMkΦi,k−1γkdiagFk+iTQ1Fk+i,Q2−1γkFykCk+iFxkTℛFykCk+iFxk.Hence, (55) is guaranteed by Πi,k≥0. By applying the Schur complement, it is shown that Πi,k≥0 can be transformed into(61)Mk−Φi,kTMkΦi,k⋆⋆Q1/2diagFk+i,IγkI⋆ℛ1/2FykCk+iFxk0γkI≥0,i≥0,where Q=diagQ1,Q2.The condition (55) or (61) is for optimality, not primarily for stability. However, if(62)diagFk+iTQ1Fk+i,Q2+FykCk+iFxkTℛFykCk+iFxk>0,then (61) means that M¯k−Φi,kTM¯kΦi,k>0, i.e., Φi,k is exponentially stable (referring to point (iv) of Theorem 4). We can indeed combine the optimality and stability conditions by imposing (see firstly [11, 17] for LPV and [12, 13] for quasi-LPV)(63)x˜k+ikMk2≥1⟹x˜k+i+1kMk2−x˜k+ikMk2≤−1γkz′k+ikQ12+xck+ikQ22+uk+ikℛ2,∀i≥0.It is easy to show that (63) is equivalent to (in the sense for any x˜k+ik and wk+i)(64)1−αi,kMk−Φi,kTMkΦi,k⋆⋆⋆−Γi,kTMkΦi,kαi,kI−Γi,kTMkΓi,k⋆⋆Q1/2diagFk+i,IQ1/2Gk+i0γkI⋆ℛ1/2FykCk+iFxkℛ1/2FykEk+i0γkI≥0,i≥0.Remark 3.
It is apparent that feasibility of (64) guarantees both (50) and (61). With γk free (i.e., as a decision variable), feasibility of (50) guarantees both (61) and (64). Therefore, on the feasibility aspect, (64) and (50) are equivalent.


### 4.4. A Paradox for State Convergence

Consider the condition group {(57), (64)} or {(57), (50)}. Condition (64) or (50) means that, if the augmented state x˜k lies outside of the ellipsoid εMk, then x˜k+ik will converge to εMk with the increase of i≥0. However, condition (57) requires that the initial augmented state lies within the ellipsoid εMk. With the satisfaction of (57), condition (64) or (50) cannot guarantee the convergence of x˜k+ik; condition (64) or (50) only guarantees the invariance of x˜k+ik within εMk.In the above, although there is no guarantee thatx˜k+ik will converge, the convergence of x˜k+ik will happen when x˜k is not small (see firstly [19] for LPV and [20] for quasi-LPV). The main reason lies in that (64) or (50) is a robust condition.Let us impose that, if the augmented statex˜k lies outside of the ellipsoid εβk−1Mk, then x˜k+ik converges to εβk−1Mk with the increase of i≥0. Here, εβk−1Mk is an ellipsoid not larger than εMk since 0<βk≤1 (see firstly [17] for LPV and [13, 20] for quasi-LPV). By applying such βk, we can change (48) as(65)x˜k+ikMk2≥βk⟹x˜k+i+1kMk2≤x˜k+ikMk2,∀i≥0,which is equivalent to (in the sense for any x˜k+ik and wk+i)(66)♡⋆−Γi,kTMkΦi,kαi,kβkI−Γi,kTMkΓi,k≥0,♡=1−αi,kMk−Φi,kTMkΦi,k,i≥0.We can also change (63) as(67)x˜k+ikMk2≥βk⟹x˜k+i+1kMk2−x˜k+ikMk2≤−1γkz′k+ikQ12+xck+ikQ22+uk+ikℛ2,∀i≥0,which is equivalent to (in the sense for any x˜k+ik and wk+i)(68)1−αi,kMk−Φi,kTMkΦi,k⋆⋆⋆−Γi,kTMkΦi,kαi,kβkI−Γi,kTMkΓi,k⋆⋆Q1/2diagFk+i,IQ1/2Gk+i0γkI⋆ℛ1/2FykCk+iFxkℛ1/2FykEk+i0γkI≥0,i≥0.Addingβk∈0,1 as a free variable, due to the special position of βk in either (66) or (68), does not affect the minimization of γk and feasibility. It is suggested to minimize βk after the minimization of γk (see firstly [19] for LPV and [20] for quasi-LPV). If the controller parametric matrices are not reoptimized in minimizing βk, it is easy to know that we do not need βk, i.e., we can simply remove it.


## 4.1. Review of Quadratic Boundedness

In [14], the following model with nominal parametric matrices is considered:(38)xk+1=Axk+Dvk,where A and D are time-invariant (fixed) matrix, v∈ℜnv. In [14], it is firstly assumed that v∈V where V is a compact (bounded and closed) set, and V⊂ℜnv.Definition 1.
(see [14]). System (38) is said to be quadratically bounded with Lyapunov matrix P>0 if(39)xTPx≥1⟹Ax+DvTPAx+Dv≤xTPx,∀v∈V.
System (38) is said to be strictly quadratically bounded with Lyapunov matrix P>0 if(40)xTPx>1⟹Ax+DvTPAx+Dv<xTPx,∀v∈V.Lemma 1.
(see [14]). Suppose there exists a ξ∈V such that Dξ≠0. If (38) is quadratically bounded with the Lyapunov matrix P>0, then it is strictly quadratically bounded with the same Lyapunov matrix.Definition 2.
The setS is a robust positively invariant set for (38), if(41)x∈S⟹Ax+Dv∈S,∀v∈V.Theorem 2.
(see [14]). Suppose v∈εPv with Pv>0. The following facts are equivalent:(i)
(38) is quadratically bounded with Lyapunov matrix P>0(ii)
(38) is strictly quadratically bounded with Lyapunov matrix P>0(iii)
The ellipsoidεP is a robust positively invariant set for (38)(iv)
xTPx≥vTPvv⟹Ax+DvTPAx+Dv≤xTPx(v)
There existsα>0 such that(42)1−αP−ATPA⋆−DTPAαPv−DTPD≥0;(vi)
A is exponentially stable (i.e., there existsP¯>0 such that P¯−ATP¯A>0)
In [15], the following model with uncertain parametric matrices is considered:(43)xk+1=Akxk+Dkvk,where AkDk belongs to a known bounded set, i.e., AkDk∈P for all k≥0, and D≠0 for at least one AD∈P.Definition 3.
(see [15]). Suppose vk∈εPv for all k≥0, in (43). System (43) is said to be strictly quadratically bounded with a common Lyapunov matrix P>0, if(44)xTPx>1⟹Ax+DvTPAx+Dv<xTPx,∀v∈εPv,∀AD∈P.
SinceD≠0 for at least one AD∈P, and v∈εPv, there exists a Dv≠0. Similarly to Lemma 1. if (43) is quadratically bounded with Lyapunov matrix P>0, then it is strictly quadratically bounded with the same Lyapunov matrix. The definition of quadratic boundedness is similar to Definition 1.Definition 4.
Supposevk∈εPv for all k≥0, in (43). The set S is a positively invariant set for (43), if(45)x∈S⟹Ax+Dv∈S,∀v∈εPv,∀AD∈P.Theorem 3.
(see [15]). Suppose vk∈εPv for all k≥0, in (43). The following facts are equivalent:(i)
(43) is strictly quadratically bounded with a common Lyapunov matrix P>0(ii)
The ellipsoidεP is a positively invariant set for (43)(iii)
There existsαk∈0,1 such that(46)1−αkP−AkTPAk⋆−DkTPAkαkPv−DkTPDk≥0.
Note that in the above theorem it is necessary to use a time-varyingαk.


## 4.2. Stability Condition

In the output feedback MPC, QB is equivalent to strict QB (see [16]). For the closed-loop systems (28) and (35), by generalizing the results in Section 4.1, we obtain the following results.Definition 5.
(see firstly [12, 13] for quasi-LPV and [11, 17] for LPV). Suppose (referring to Assumptions 1 and 2), at time k and for all i≥0:(47)wk+i≤1;there exist nonnegative coefficients λlk+i,l=1,…,L such that ∑l=1Lλlk+i=1 and ABCDEk=∑l=1Lλlk+iAlBlClDlEl.
System (28) or (35) is said to be quadratically bounded with a common Lyapunov matrix Mk>0, if(48)x˜k+ikMk2≥1⟹x˜k+i+1kMk2≤x˜k+ikMk2,∀i≥0.Definition 6.
With the assumptions in Definition5 satisfied, the set S is a positively invariant set for (28) or (35), if(49)x˜k+ik∈S⟹x˜k+i+1k∈S,∀i≥0.Theorem 4.
(see firstly [10, 11] for LPV and [12, 18] for quasi-LPV). With the assumptions in Definition 5 satisfied, the following facts are equivalent:(i)
(28) or (35) is quadratically bounded with a common Lyapunov matrix Mk>0(ii)
The ellipsoidεMk is a positively invariant set for (28) or (35)(iii)
There existsαi,k∈0,1 such that(50)1−αi,kMk−Φi,kTMkΦi,k⋆−Γi,kTMkΦi,kαi,kI−Γi,kTMkΓi,k≥0,i≥0,(iv)
Φi,k is exponentially stable for all i>0 (i.e., there exists M¯k>0 such that M¯k−Φi,kTM¯kΦi,k>0).In [19], the single-valued α is firstly replaced by(51)αi,k=∑l=1L∑j=1Lλlk+iλjk+iαlj.


## 4.3. Optimality Condition

The disturbance-free form of (28) or (35) is(52)x˜uk+i+1k=Φi,kx˜uk+ik,∀i≥0,x˜ukk=x˜k.Correspondingly,(53)uuk+ik=Fxkxc,uk+ik+Fykyuk+ik,yuk+ik=Ck+ixuk+ik,zuk+ik=Ck+ixuk+ik,zu′k+ik=Fk+ixuk+ik.Let us introduce the quadratic cost(54)Jk=∑i=0∞Jik,Jik=zu′k+ikQ12+xc,uk+ikQ22+uuk+ikℛ2,where Q1, Q2, and ℛ are positive-definite weighting matrices, and consider the condition(55)x˜uk+i+1kMk2−x˜uk+ikMk2≤−1γkJik,∀i≥0.In Theorem1, it has taken Q2=0. For exponentially stable Φi,k, it will result in limi⟶∞zu′k+ik=0, limi⟶∞xc,uk+ik=0, and limi⟶∞uuk+ik=0. Hence, summing (55) from i=0 to i=∞ yields(56)Jk≤γkx˜ukkMk2=γkx˜kMk2.Further, let(57)x˜k∈εMk.Then, applying (57) to (56) yields(58)Jk≤γk,that is, γk is an upper bound of Jk. We will take γk as the cost function of the optimization problems which finds the controller parametric matrices.The condition (55) can be rewritten as(59)x˜uk+ikTΠi,kx˜uk+ik≥0,where(60)Πi,k=Mk−Φi,kTMkΦi,k−1γkdiagFk+iTQ1Fk+i,Q2−1γkFykCk+iFxkTℛFykCk+iFxk.Hence, (55) is guaranteed by Πi,k≥0. By applying the Schur complement, it is shown that Πi,k≥0 can be transformed into(61)Mk−Φi,kTMkΦi,k⋆⋆Q1/2diagFk+i,IγkI⋆ℛ1/2FykCk+iFxk0γkI≥0,i≥0,where Q=diagQ1,Q2.The condition (55) or (61) is for optimality, not primarily for stability. However, if(62)diagFk+iTQ1Fk+i,Q2+FykCk+iFxkTℛFykCk+iFxk>0,then (61) means that M¯k−Φi,kTM¯kΦi,k>0, i.e., Φi,k is exponentially stable (referring to point (iv) of Theorem 4). We can indeed combine the optimality and stability conditions by imposing (see firstly [11, 17] for LPV and [12, 13] for quasi-LPV)(63)x˜k+ikMk2≥1⟹x˜k+i+1kMk2−x˜k+ikMk2≤−1γkz′k+ikQ12+xck+ikQ22+uk+ikℛ2,∀i≥0.It is easy to show that (63) is equivalent to (in the sense for any x˜k+ik and wk+i)(64)1−αi,kMk−Φi,kTMkΦi,k⋆⋆⋆−Γi,kTMkΦi,kαi,kI−Γi,kTMkΓi,k⋆⋆Q1/2diagFk+i,IQ1/2Gk+i0γkI⋆ℛ1/2FykCk+iFxkℛ1/2FykEk+i0γkI≥0,i≥0.Remark 3.
It is apparent that feasibility of (64) guarantees both (50) and (61). With γk free (i.e., as a decision variable), feasibility of (50) guarantees both (61) and (64). Therefore, on the feasibility aspect, (64) and (50) are equivalent.


## 4.4. A Paradox for State Convergence

Consider the condition group {(57), (64)} or {(57), (50)}. Condition (64) or (50) means that, if the augmented state x˜k lies outside of the ellipsoid εMk, then x˜k+ik will converge to εMk with the increase of i≥0. However, condition (57) requires that the initial augmented state lies within the ellipsoid εMk. With the satisfaction of (57), condition (64) or (50) cannot guarantee the convergence of x˜k+ik; condition (64) or (50) only guarantees the invariance of x˜k+ik within εMk.In the above, although there is no guarantee thatx˜k+ik will converge, the convergence of x˜k+ik will happen when x˜k is not small (see firstly [19] for LPV and [20] for quasi-LPV). The main reason lies in that (64) or (50) is a robust condition.Let us impose that, if the augmented statex˜k lies outside of the ellipsoid εβk−1Mk, then x˜k+ik converges to εβk−1Mk with the increase of i≥0. Here, εβk−1Mk is an ellipsoid not larger than εMk since 0<βk≤1 (see firstly [17] for LPV and [13, 20] for quasi-LPV). By applying such βk, we can change (48) as(65)x˜k+ikMk2≥βk⟹x˜k+i+1kMk2≤x˜k+ikMk2,∀i≥0,which is equivalent to (in the sense for any x˜k+ik and wk+i)(66)♡⋆−Γi,kTMkΦi,kαi,kβkI−Γi,kTMkΓi,k≥0,♡=1−αi,kMk−Φi,kTMkΦi,k,i≥0.We can also change (63) as(67)x˜k+ikMk2≥βk⟹x˜k+i+1kMk2−x˜k+ikMk2≤−1γkz′k+ikQ12+xck+ikQ22+uk+ikℛ2,∀i≥0,which is equivalent to (in the sense for any x˜k+ik and wk+i)(68)1−αi,kMk−Φi,kTMkΦi,k⋆⋆⋆−Γi,kTMkΦi,kαi,kβkI−Γi,kTMkΓi,k⋆⋆Q1/2diagFk+i,IQ1/2Gk+i0γkI⋆ℛ1/2FykCk+iFxkℛ1/2FykEk+i0γkI≥0,i≥0.Addingβk∈0,1 as a free variable, due to the special position of βk in either (66) or (68), does not affect the minimization of γk and feasibility. It is suggested to minimize βk after the minimization of γk (see firstly [19] for LPV and [20] for quasi-LPV). If the controller parametric matrices are not reoptimized in minimizing βk, it is easy to know that we do not need βk, i.e., we can simply remove it.


## 5. General Optimization Problem

Definepar⟶=Ac,Lc,Fx,Fy for LPV and par⟶=A^clj,B^cl,C^cj,D^c for qusi-LPV. The dynamic OFRMPC aims at solving, at each k,(69)minγ,M,par⟶kmaxABCDECEFGk+i∈Ω,wk+i≤1γk,s.t.22,57,48and55.Lemma 2.
(see firstly [19] for LPV and [18, 20] for quasi-LPV) (recursive feasibility). Assume that the state x is measurable. At each time k≥0, solve (69) and implement uk. Problem (69) is feasible for any k>0 if and only if it is feasible at k=0.Theorem 5.
(see firstly [20] for quasi-LPV and [19] for LPV) (stability). Assume that the state x is measurable. At each time k≥0, solve (69) and implement uk. If (69) is feasible at k=0, then with the evolution of k, γ,z′,xc,u will converge to a neighborhood of the origin, and stay in this neighborhood thereafter, and the constraints in (22) are satisfied for all k≥0.
According to the above section, (69) is transformed into (equivalently in the sense for any x˜k+ik and wk+i)(70)minγ,αlj,M,par⟶kmaxABCDECEFGk+i∈Ωγk,s.t.22,57,50and61,with recursive feasibility and stability properties retained.


### 5.1. Handling Physical Constraints

In [21, 22], the following lemma is utilized to handle the physical constraints (e.g., the magnitude constraints on x, y, and u).Lemma 3.
Suppose a and b are vectors with appropriate dimensions. Then for any scalarη∈0,1, a+b2≤1−ηa2+1/ηb2.
In [5, 7, 23, 24], it is found that applying the above lemma, although simple, can greatly reduce the conservativeness for physical constraint handling. In essence, the physical constraints are handled based on the invariance of x˜k+ik within εMk.Theorem 6 (see firstly [5, 23] for LPV and [9] for quasi-LPV).
Suppose at time k, there exist scalarsαi,k∈0,1 and ηrs, and matrix Mk>0, such that (57) and (50) hold, and(71)Mk⋆⋆0I⋆11−η1sξsFykCk+iFxk1η1sξsFykEk+iu¯s2≥0,s=1,…,nu,i≥0,(72)Mk⋆⋆0I⋆11−η2s1−η3sΨsCk+i+1Φ1i,k11−η2sη3sΨsCk+i+1Γ1i,kψ¯s2−1η2sΨsEk+i+1Ek+i+1TΨsT≥0,s=1,…,q,i≥0,where Φ1i,k (Γ1i,k) is the first of the two rows of Φi,k (Γi,k). Take care of the special cases:(a)
Ifεk+i+1=0, then take 1/η2sΨsEk+i+1Ek+i+1TΨsT=0 and η2s=0(b)
IfEk+i=0, then take 1/η1sξsFykEk+i=0 and η1s=0(c)
IfDk+i=0 and Ek+i=0, then take 1/η3sΨsCk+i+1Γ1i,k=0 and η3s=0
Then, (22) is satisfied.
In the above theorem, one may want to chooseηrs be time-varying. However, we have not found a good method to online optimize ηrs, so we take ηrs as time-invariant.
According to Theorem6, the problem (70) is approximated as (by no means equivalent to)(73)minγ,αlj,M,par⟶kmaxABCDECEFGk+i∈Ωγk,s.t.57,50,61,71and72,with recursive feasibility and stability properties retained. In (73), ηrs is prespecified (see firstly [5, 23] for LPV and [9] for quasi-LPV).


### 5.2. Current Augmented State

The condition (57) (i.e., x˜kMk2≤1 or x˜k∈εMk) is the current condition on the augmented state. At time k, in x˜k=xkT,xckTT, xk can be unmeasurable, while xck is always known. When xk is unmeasurable, we need to remove it from (57) for the sake of solving (73).Let us define an error signal(74)ek=xk−x0k,where(75)x0k=Ukxck,with Uk being a known transformation matrix. When Uk=I, defining ek is usual; when Uk=E0T is fixed, see firstly [7, 25]; when Uk is online refreshed, see firstly [5, 26] for LPV and [9] for quasi-LPV. When xk is unmeasurable, ek is unknown (nondeterministic). If we can obtain the outer bounding set of ek, say Dek, then we can utilize x0k⊕Dek to replace xk. Since Dek is known (deterministic), by replacing xk by x0k⊕Dek, (57) becomes deterministic.Define(76)M=M1M2TM2M3.Usingx=e+Uxc, we obtain(77)x˜TMx˜=e+UxcTM1e+Uxc+2e+UxcTM2Txc+xcTM3xc=eTM1e+2eTM1U+M2Txc+xcTUTM1U+2UTM2T+M3xc.If we can remove the cross item2eTM1U+M2Txc, then the treatment of (57) will become easier, and the treatment of recursive feasibility of the resultant optimization will become simpler.Lemma 4.
In order to remove the cross item2eTM1U+M2Txc in x˜TMx˜, we need to take U=−M1−1M2T.For quasi-LPV, [12, 18] firstly impose M2=−M1 and [9] firstly imposes U=−M1−1M2T, both removing the cross item. For LPV, [19] firstly imposes M2=−M1, [7, 25] firstly impose M2=−E0M1, and [5, 26] firstly impose U=−M1−1M2T, all removing the cross item.By substitutingU=−M1−1M2T into (77), we obtain(78)x˜TMx˜=eTM1e+xcTM3−UTM1Uxc.By introducing a scalarϱk and imposing(79)ekTM1kek≤ϱk,(80)xckTM3k−UkTM1kUkxck≤1−ϱk,it is apparent that (57) is guaranteed. Condition (79) is guaranteed by (7) if we can firstly guarantee that(81)ek∈εMek.The conditionek∈εMek can be guaranteed by appropriately refreshing Mek at each k>0. However, for the initial time k=0, ek∈εMek has to be assumed.Assumption 3.
e
0
=
x
0
−
x
0
0
∈
ε
M
e
0.
Based on Assumption3 and the fact that ek∈εMek, problem (73) is approximated as (by no means equivalent to)(82)minγ,αlj,ϱ,M,par⟶kmaxABCDECEFGk+i∈Ωγk,s.t.80,7,50,61,71and72,with recursive feasibility and stability properties retained in case Mek is appropriately refreshed.Lemma 5.
(for quasi-LPV, see [20] firstly with M2=−M1, and [9] firstly with U=−M1−1M2T; for LPV, see [19] firstly with M2=−M1, [7, 25] firstly with M2=−E0M1, and [5, 26] firstly with U=−M1−1M2T). At each k>0, if we choose (12)–(14), then at time k, (80) and (7) can be satisfied with equalities, i.e.,(83)xckTM3k−UkTM1kUkxck=1−ϱk,M1k=ϱkMek.Remark 4.
In the above, the ellipsoidal bound one or x has been discussed. We have also utilized polyhedral outer bounding sets of x, e.g.,(a)
Polyhedron with plane representation (see firstly [27] for LPV), i.e.,(84)xk∈Pxk≔x−Gke¯≤Hx−xˇk≤Gke¯,where xˇ is a bias item, H=HaHb a prespecified transformation matrix with Ha being nonsingular, Gk a diagonal matrix, and e¯=e¯1,e¯2,…,e¯pT with p>nx and e¯j>0 (for all j=1,…,p) being prespecified(b)
Polyhedron with vertex representation, i.e.,(85)xk∈P¯xk=Coϑjkj=1,2,…,nϑk,(see, e.g., [20] for quasi-LPV and [19] for LPV) which is a general formulation of convex polyhedron.
We will not discuss the details for utilizing polyhedral bounds, but the following points are promising:(i)
For the output feedback MPC,Pxk in (84) is a general formulation of convex polyhedron, which includes the other polyhedral sets (e.g., [11–13, 18, 22]) as special cases, and is equivalent to the expression Pxk=ξℋξ≤Gk1⟶ (with ℋ∈ℜp×nx being prespecified, and 1⟶=1,1,…,1T) of [10].(ii)
Before [28], either ellipsoidal bound or polyhedral bound is solely applied in the optimization problem. The recursive feasibility is guaranteed by a simple refreshment of the ellipsoidal bound but might be lost by applying polyhedral bound. In [28], it utilizes either the ellipsoidal bound or the polyhedral bound, the latter being used if and only if it is contained in the former. Moreover, [28] shows the sufficient conditions under which some approaches based on polyhedral bound preserve the property of recursive feasibility. In [29], the potentiality of applying both ellipsoidal and polyhedral bounds is further explored.


### 5.3. Some Usual Transformations

In order to solve (82), we need to transform (50) and (61) into familiar forms (comparing with, e.g., [3]). Define Q=M−1 and(86)Q=Q1Q2TQ2Q3.By applying the Schur complement, (50) is transformed into(87)1−αi,kMk⋆⋆0αi,kI⋆Φi,kΓi,kQk≥0,i≥0.By applying the Schur complement, (61) is transformed into(88)Mk⋆⋆⋆Φi,kQk⋆⋆Q1/2diagFk+i,I0γkI⋆ℛ1/2FykCk+iFxk00γkI≥0,i≥0.Then, we need to remove or handle the convex combinations in {(87), (88), (71), (72)}. By invoking the double convex combinations (DbCCs), (87) and (88) are equivalent to, respectively,(89)∑l=1L∑j=1Lλlk+iλjk+iϒljQBk≥0,i≥0,∑l=1L∑j=1Lλlk+iλjk+iϒljoptk≥0,i≥0,where(90)ϒljQBk=1−αljMk⋆⋆0αljI⋆ΦljkΓljkQk,ϒljoptk=Mk⋆⋆⋆ΦljkQk⋆⋆Q1/2diagFj,I0γkI⋆ℛ1/2FykCjFxk00γkI.By removing the single convex combination, (71) is guaranteed by(91)ϒjuk≥0,j=1,…,L,s=1,…,nu,where(92)ϒjuk=Mk⋆⋆0I⋆11−η1sξsFykCjFxk1η1sξsFykEju¯s2.By removing the single convex combination, and invoking DbCC, (72) is guaranteed by(93)∑l=1L∑j=1Lλlk+iλjk+iϒhljzk≥0,h=1,…,L,s=1,…,q,i≥0,where(94)ϒhljzk=Mk⋆⋆0I⋆11−η2s1−η3sΨsChΦlj1k11−η2sη3sΨsChΓlj1kψ¯s2−1η2sΨsChChTΨsT.In summary, problem (85) is approximated as (not strictly equivalent to)(95)minγ,αlj,ϱ,M,Q,par⟶kmaxABCDECEFGk+i∈Ωγks.t.80,7,89,91,93andQ=M−1,with recursive feasibility and stability properties retained in case Mek is appropriately refreshed.


### 5.4. Handling Double Convex Combinations

In the literature of fuzzy control based on Takagi–Sugeno model and robust feedback control, the double convex combinations as in {(89), (93)} have been extensively studied. Some well-known examples include [30] (being invoked by MPC in [11, 18]), [31, 32] (being invoked by MPC firstly in [10, 25]), and [1] (being invoked by MPC firstly in [12]).By analogy to “Theorem1” in [31], the following result can be obtained.Lemma 6 (see firstly [10, 25]).
The conditions(96)∑l=1L∑j=1Lλlk+iλjk+iϒljk≥0,i≥0,hold if and only if there exists a sufficiently large d≥0 such that(97)∑l=1LClℓd,2ϒllk+∑l=1L−1∑j=l+1LCljℓd,1,1ϒljk+ϒjlk≥0,ℓ∈1,…,Kd+2.
Moreover, if (97) holds for d=d^, then they hold for any d>d^.This lemma has been utilized in Theorem1. In this lemma, ϒljk∈ϒljQBk,ϒljoptk,ϒhljzk. Equivalently, the techniques for the positivity of DbCC, as in [1], can be exactly utilized to obtain finite dimensional sufficient conditions for the nonnegativity of DbCC in (96). For example, (96) is guaranteed by any one set of the following sets of conditions (see “Proposition 2” of [1]):Set 1: (n=2) (i) ϒllk≥0, l∈1,…,L, (ii) ϒljk+ϒjlk≥0, j>l, l,j∈1,…,LSet 2: (n=3) (i) ϒllk≥0, l∈1,…,L, (ii) ϒllk+ϒljk+ϒjlk≥0, j≠l, l,j∈1,…,L, (iii) ϒljk+ϒjlk+ϒjtk+ϒtjk+ϒtlk+ϒltk≥0, t>j>l, l,j,t∈1,…,LIn Sets 1 and 2,n is the complexity parameter of [1]. With a larger n, the conditions are less conservative but the computational burden is heavier. There exists a finite n such that necessary and sufficient conditions for satisfaction of (96) can be obtained for a concrete model.


## 5.1. Handling Physical Constraints

In [21, 22], the following lemma is utilized to handle the physical constraints (e.g., the magnitude constraints on x, y, and u).Lemma 3.
Suppose a and b are vectors with appropriate dimensions. Then for any scalarη∈0,1, a+b2≤1−ηa2+1/ηb2.
In [5, 7, 23, 24], it is found that applying the above lemma, although simple, can greatly reduce the conservativeness for physical constraint handling. In essence, the physical constraints are handled based on the invariance of x˜k+ik within εMk.Theorem 6 (see firstly [5, 23] for LPV and [9] for quasi-LPV).
Suppose at time k, there exist scalarsαi,k∈0,1 and ηrs, and matrix Mk>0, such that (57) and (50) hold, and(71)Mk⋆⋆0I⋆11−η1sξsFykCk+iFxk1η1sξsFykEk+iu¯s2≥0,s=1,…,nu,i≥0,(72)Mk⋆⋆0I⋆11−η2s1−η3sΨsCk+i+1Φ1i,k11−η2sη3sΨsCk+i+1Γ1i,kψ¯s2−1η2sΨsEk+i+1Ek+i+1TΨsT≥0,s=1,…,q,i≥0,where Φ1i,k (Γ1i,k) is the first of the two rows of Φi,k (Γi,k). Take care of the special cases:(a)
Ifεk+i+1=0, then take 1/η2sΨsEk+i+1Ek+i+1TΨsT=0 and η2s=0(b)
IfEk+i=0, then take 1/η1sξsFykEk+i=0 and η1s=0(c)
IfDk+i=0 and Ek+i=0, then take 1/η3sΨsCk+i+1Γ1i,k=0 and η3s=0
Then, (22) is satisfied.
In the above theorem, one may want to chooseηrs be time-varying. However, we have not found a good method to online optimize ηrs, so we take ηrs as time-invariant.
According to Theorem6, the problem (70) is approximated as (by no means equivalent to)(73)minγ,αlj,M,par⟶kmaxABCDECEFGk+i∈Ωγk,s.t.57,50,61,71and72,with recursive feasibility and stability properties retained. In (73), ηrs is prespecified (see firstly [5, 23] for LPV and [9] for quasi-LPV).


## 5.2. Current Augmented State

The condition (57) (i.e., x˜kMk2≤1 or x˜k∈εMk) is the current condition on the augmented state. At time k, in x˜k=xkT,xckTT, xk can be unmeasurable, while xck is always known. When xk is unmeasurable, we need to remove it from (57) for the sake of solving (73).Let us define an error signal(74)ek=xk−x0k,where(75)x0k=Ukxck,with Uk being a known transformation matrix. When Uk=I, defining ek is usual; when Uk=E0T is fixed, see firstly [7, 25]; when Uk is online refreshed, see firstly [5, 26] for LPV and [9] for quasi-LPV. When xk is unmeasurable, ek is unknown (nondeterministic). If we can obtain the outer bounding set of ek, say Dek, then we can utilize x0k⊕Dek to replace xk. Since Dek is known (deterministic), by replacing xk by x0k⊕Dek, (57) becomes deterministic.Define(76)M=M1M2TM2M3.Usingx=e+Uxc, we obtain(77)x˜TMx˜=e+UxcTM1e+Uxc+2e+UxcTM2Txc+xcTM3xc=eTM1e+2eTM1U+M2Txc+xcTUTM1U+2UTM2T+M3xc.If we can remove the cross item2eTM1U+M2Txc, then the treatment of (57) will become easier, and the treatment of recursive feasibility of the resultant optimization will become simpler.Lemma 4.
In order to remove the cross item2eTM1U+M2Txc in x˜TMx˜, we need to take U=−M1−1M2T.For quasi-LPV, [12, 18] firstly impose M2=−M1 and [9] firstly imposes U=−M1−1M2T, both removing the cross item. For LPV, [19] firstly imposes M2=−M1, [7, 25] firstly impose M2=−E0M1, and [5, 26] firstly impose U=−M1−1M2T, all removing the cross item.By substitutingU=−M1−1M2T into (77), we obtain(78)x˜TMx˜=eTM1e+xcTM3−UTM1Uxc.By introducing a scalarϱk and imposing(79)ekTM1kek≤ϱk,(80)xckTM3k−UkTM1kUkxck≤1−ϱk,it is apparent that (57) is guaranteed. Condition (79) is guaranteed by (7) if we can firstly guarantee that(81)ek∈εMek.The conditionek∈εMek can be guaranteed by appropriately refreshing Mek at each k>0. However, for the initial time k=0, ek∈εMek has to be assumed.Assumption 3.
e
0
=
x
0
−
x
0
0
∈
ε
M
e
0.
Based on Assumption3 and the fact that ek∈εMek, problem (73) is approximated as (by no means equivalent to)(82)minγ,αlj,ϱ,M,par⟶kmaxABCDECEFGk+i∈Ωγk,s.t.80,7,50,61,71and72,with recursive feasibility and stability properties retained in case Mek is appropriately refreshed.Lemma 5.
(for quasi-LPV, see [20] firstly with M2=−M1, and [9] firstly with U=−M1−1M2T; for LPV, see [19] firstly with M2=−M1, [7, 25] firstly with M2=−E0M1, and [5, 26] firstly with U=−M1−1M2T). At each k>0, if we choose (12)–(14), then at time k, (80) and (7) can be satisfied with equalities, i.e.,(83)xckTM3k−UkTM1kUkxck=1−ϱk,M1k=ϱkMek.Remark 4.
In the above, the ellipsoidal bound one or x has been discussed. We have also utilized polyhedral outer bounding sets of x, e.g.,(a)
Polyhedron with plane representation (see firstly [27] for LPV), i.e.,(84)xk∈Pxk≔x−Gke¯≤Hx−xˇk≤Gke¯,where xˇ is a bias item, H=HaHb a prespecified transformation matrix with Ha being nonsingular, Gk a diagonal matrix, and e¯=e¯1,e¯2,…,e¯pT with p>nx and e¯j>0 (for all j=1,…,p) being prespecified(b)
Polyhedron with vertex representation, i.e.,(85)xk∈P¯xk=Coϑjkj=1,2,…,nϑk,(see, e.g., [20] for quasi-LPV and [19] for LPV) which is a general formulation of convex polyhedron.
We will not discuss the details for utilizing polyhedral bounds, but the following points are promising:(i)
For the output feedback MPC,Pxk in (84) is a general formulation of convex polyhedron, which includes the other polyhedral sets (e.g., [11–13, 18, 22]) as special cases, and is equivalent to the expression Pxk=ξℋξ≤Gk1⟶ (with ℋ∈ℜp×nx being prespecified, and 1⟶=1,1,…,1T) of [10].(ii)
Before [28], either ellipsoidal bound or polyhedral bound is solely applied in the optimization problem. The recursive feasibility is guaranteed by a simple refreshment of the ellipsoidal bound but might be lost by applying polyhedral bound. In [28], it utilizes either the ellipsoidal bound or the polyhedral bound, the latter being used if and only if it is contained in the former. Moreover, [28] shows the sufficient conditions under which some approaches based on polyhedral bound preserve the property of recursive feasibility. In [29], the potentiality of applying both ellipsoidal and polyhedral bounds is further explored.


## 5.3. Some Usual Transformations

In order to solve (82), we need to transform (50) and (61) into familiar forms (comparing with, e.g., [3]). Define Q=M−1 and(86)Q=Q1Q2TQ2Q3.By applying the Schur complement, (50) is transformed into(87)1−αi,kMk⋆⋆0αi,kI⋆Φi,kΓi,kQk≥0,i≥0.By applying the Schur complement, (61) is transformed into(88)Mk⋆⋆⋆Φi,kQk⋆⋆Q1/2diagFk+i,I0γkI⋆ℛ1/2FykCk+iFxk00γkI≥0,i≥0.Then, we need to remove or handle the convex combinations in {(87), (88), (71), (72)}. By invoking the double convex combinations (DbCCs), (87) and (88) are equivalent to, respectively,(89)∑l=1L∑j=1Lλlk+iλjk+iϒljQBk≥0,i≥0,∑l=1L∑j=1Lλlk+iλjk+iϒljoptk≥0,i≥0,where(90)ϒljQBk=1−αljMk⋆⋆0αljI⋆ΦljkΓljkQk,ϒljoptk=Mk⋆⋆⋆ΦljkQk⋆⋆Q1/2diagFj,I0γkI⋆ℛ1/2FykCjFxk00γkI.By removing the single convex combination, (71) is guaranteed by(91)ϒjuk≥0,j=1,…,L,s=1,…,nu,where(92)ϒjuk=Mk⋆⋆0I⋆11−η1sξsFykCjFxk1η1sξsFykEju¯s2.By removing the single convex combination, and invoking DbCC, (72) is guaranteed by(93)∑l=1L∑j=1Lλlk+iλjk+iϒhljzk≥0,h=1,…,L,s=1,…,q,i≥0,where(94)ϒhljzk=Mk⋆⋆0I⋆11−η2s1−η3sΨsChΦlj1k11−η2sη3sΨsChΓlj1kψ¯s2−1η2sΨsChChTΨsT.In summary, problem (85) is approximated as (not strictly equivalent to)(95)minγ,αlj,ϱ,M,Q,par⟶kmaxABCDECEFGk+i∈Ωγks.t.80,7,89,91,93andQ=M−1,with recursive feasibility and stability properties retained in case Mek is appropriately refreshed.


## 5.4. Handling Double Convex Combinations

In the literature of fuzzy control based on Takagi–Sugeno model and robust feedback control, the double convex combinations as in {(89), (93)} have been extensively studied. Some well-known examples include [30] (being invoked by MPC in [11, 18]), [31, 32] (being invoked by MPC firstly in [10, 25]), and [1] (being invoked by MPC firstly in [12]).By analogy to “Theorem1” in [31], the following result can be obtained.Lemma 6 (see firstly [10, 25]).
The conditions(96)∑l=1L∑j=1Lλlk+iλjk+iϒljk≥0,i≥0,hold if and only if there exists a sufficiently large d≥0 such that(97)∑l=1LClℓd,2ϒllk+∑l=1L−1∑j=l+1LCljℓd,1,1ϒljk+ϒjlk≥0,ℓ∈1,…,Kd+2.
Moreover, if (97) holds for d=d^, then they hold for any d>d^.This lemma has been utilized in Theorem1. In this lemma, ϒljk∈ϒljQBk,ϒljoptk,ϒhljzk. Equivalently, the techniques for the positivity of DbCC, as in [1], can be exactly utilized to obtain finite dimensional sufficient conditions for the nonnegativity of DbCC in (96). For example, (96) is guaranteed by any one set of the following sets of conditions (see “Proposition 2” of [1]):Set 1: (n=2) (i) ϒllk≥0, l∈1,…,L, (ii) ϒljk+ϒjlk≥0, j>l, l,j∈1,…,LSet 2: (n=3) (i) ϒllk≥0, l∈1,…,L, (ii) ϒllk+ϒljk+ϒjlk≥0, j≠l, l,j∈1,…,L, (iii) ϒljk+ϒjlk+ϒjtk+ϒtjk+ϒtlk+ϒltk≥0, t>j>l, l,j,t∈1,…,LIn Sets 1 and 2,n is the complexity parameter of [1]. With a larger n, the conditions are less conservative but the computational burden is heavier. There exists a finite n such that necessary and sufficient conditions for satisfaction of (96) can be obtained for a concrete model.


## 6. Solutions to Output Feedback MPC

For solving (95), LPV is much more difficult than quasi-LPV. For quasi-LPV, by setting(98)Q=Q1−Q1−M1−1M1M2−1−M2−TM1Q1−M1−1M2−TM1Q1−M1−1M1M2−1,(99)M=M1M2TM2M2M1−Q1−1−1M2T,which naturally satisfies M=Q−1, and using the transformation (equivalent to (5), with “k” being omitted for brevity)(100)D^c=D¯cC^cj=D¯cCjQ1+C¯cjQ2B^cl=M1BlD¯c+M2TB¯clA^clj=M1AlQ1+M1BlD¯cCjQ1+M2TB¯clCjQ1+M1BlC¯cjQ2+M2TA¯cljQ2,a solution to (95) can be obtained through a single optimization problem (6)–(11). For the prespecified α,η1s,η2s,η3s, (6)–(11) is an LMI optimization problem. Before [9], for the quasi-LPV, some special solutions to (95) can be found in [20, 28]. For LPV, even with prespecified αlj,η1s,η2s,η3s, one cannot find all the parameters Ac,Lc,Fx,Fyk in a single LMI optimization problem. In the following, we give two solutions to (95) for LPV.


### 6.1. Full Online Method for LPV

By applying the block-matrix inversion onQ=Q1Q2TQ2Q3, it is easy to show that(101)M=M1−M1Q2TQ3−1−Q3−1Q2M1Q3−1+Q3−1Q2M1Q2TQ3−1.TakeU=−M1−1M2T. Then, it is easy to show that U=−Q2TQ3−1 and(102)x˜kTMkx˜k=xk−x0kTM1kxk−x0k+xckTQ3k−1xck.Lemma 7.
Let Assumption3 hold and at each k>0, find x0,Mek such that xk−x0k∈εMek. Choose U,xc0 such that U0xc0=x00 and at each k>0, Uk such that Ukxck=x0k. Then, condition (80) holds if(103)1−ϱk⋆xckQ3k≥0.Further defineN1=M1−1 and P3=Q3−1. Then,(104)Q=N1+UQ3UTUQ3Q3UTQ3,M=M1−M1U−UTM1P3+UTM1U,which naturally satisfies M=Q−1. By applying (104), problem (95) becomes (equivalently)(105)minγ,αlj,ϱ,N1,M1,P3,Q3,Ac,Lc,Fx,FykmaxABCDECEFGk+i∈Ωγk,s.t.103,7,89,91,93,104,N1k=M1k−1,P3k=Q3k−1.This approach is proposed in [6, 7] where Uk=E0T, and hence,(106)Q=Q1E0TQ3Q3E0Q3,M=M1−M1E0T−E0M1M3.In solving (105), usually αljk=αk can be prespecified. One can line-search αk over the interval 0,1. Indeed, we found that the improvement on control performance is negligible by online optimizing αk. The problem (105) has been solved by the iterative cone-complementary approach (ICCA) (see firstly in [10, 25]). ICCA has two major loops. The inner loop is the cone-complementary approach (CCA) which minimizes TraceM1kN1k+N1kM1k+Q3kP3k+P3kQ3k in order to achieve N1k=M1k−1 and P3k=Q3k−1. The outer loop gradually reduces γk. Note that, even with αk being prespecified, (105) cannot be transformed into LMI optimization problem.In Algorithm1, while first and second equations in step (c) are natural for refreshing the bound of xk, third equation in step (c) is imposed for the recursive feasibility of (105). Finding Me′k satisfying equations in step (c) in Algorithm 1 can be achieved via LMI techniques.Theorem 7 (see [5, 26]).
Adopt Algorithm1. Suppose that Assumption 3 holds, and (105) is feasible at time k=0. Then,(i)
(105) will be feasible at each k>0(ii)
γ,z′,xc,u will converge to a neighborhood of 0, and the constraints in (2) are satisfied for all k≥0Algorithm 1:Full dynamic OFRMPC.
At eachk≥0,
(a)
fork=0, take U0=I;
(b)
fork>0, calculate xck=Ack−1xck−1+Lck−1yk−1, and refresh Me,U,x0k as in (15)–(17);
(c)
fork>0, find Me′k satisfying (the same as (19)–(20))
x
k
−
1
−
x
0
k
−
1
∈
ε
M
e
k
−
1
,
w
k
−
1
≤
1
⟹
x
k
−
x
′
0
k
∈
ε
M
e
′
k
,
x
′
0
k
≥
U
′
k
x
c
k
,
M
e
′
k
≥
M
e
k
,
and, if equations in step (c) in Algorithm1 are feasible, then change Mek=Me′k, Uk=U′k and x0k=U′kxck;
(d)
solve (105) to find Ac,Lc,Fx,Fy,M1,N1,Q3,P3∗k;
(e)
implementuk=Fxkxck+Fykyk.


### 6.2. Partial Online Method for LPV

In order to alleviate the computational burden, we can prespecifyLc,Fy in (105). In this way, M1,P3 are no longer the decision variables. Therefore, {(7), (89), (91), (93)} will be modified accordingly.By applying the Schur complement, (7) is equivalent to(107)ϱkMekIIN1k≥0.Taking congruence transformations viadiagQk,I on (89) yields(108)∑l=1Lλlk+i∑j=1Lλjk+i1−αljkQk⋆⋆0αljkI⋆Φ˘ljkΓljkQk≥0,i≥0,∑l=1Lλlk+i∑j=1Lλjk+iQk⋆⋆Φ˘ljkQk⋆♠0γkI≥0,i≥0,(109)♠=Q11/2Fj♡Q11/2FjUkQ3kQ21/2Q3kUkTQ21/2Q3kℛ1/2FyCj♡+F˘xkUkTℛ1/2FyCjUkQ3k+F˘xk,♡=N1k+UkQ3kUkT,where(110)Φ˘ljk=⋄Q1k+BlF˘xkUkT⋄Q2kT+BlF˘xkLcCjQ1k+A˘ckUkTLcCjQ2kT+A˘ck,Φ˘lj1k=⋄Q1k+BlF˘xkUkT⋄Q2kT+BlF˘xk,⋄=Al+BlFyCj,A˘ck=AckQ3k,F˘xk=FxkQ3k.Taking congruence transformations on (91) and (93) via diagQk,I, and applying the Schur complement, yields(111)Qk⋆11−η1sξs♠FyCjUkQ3k+F˘xku¯s2−1η1sξsFyEjEjTFyTξsT≥0,♠=FyCjN1k+UkQ3kUkT+F˘xkUkT,j=1,…,L,s=1,…,nu,(112)∑l=1L∑j=1Lλlk+iλjk+iQk⋆⋆0I⋆♠1♠2♠3≥0,♠1=11−η2s1−η3sΨsChΦ˘lj1k,♠2=11−η2sη3sΨsChΓlj1k,♠3=ψ¯s2−1η2sΨsChChTΨsT,h=1,…,L,s=1,…,q,i≥0.In summary, problem (105) is simplified as(113)minγ,αlj,ϱ,N1,Q3,A˘c,F˘xkmaxABCDECEFGk+i∈Ωγk,s.t.103,107and108,111,112,with Ack,Fxk calculated by(114)Ack=A˘ckQ3k−1,Fxk=F˘xkQ3k−1.The solution to (113) can be obtained by LMI toolbox. Since CCA is not involved, it is computationally less expensive than (105).Theorem 8 (see [5, 26]).
Adopt Algorithm2. Suppose that Assumption 3 holds, and (113) is feasible at time k=0. Then(i)
(113) will be feasible at each k>0;(ii)
γ,z′,xc,u will converge to a neighborhood of 0, and the constraints in (2) are satisfied for all k≥0.Algorithm 2:Partial dynamic OFRMPC.
At eachk≥0,
(a)
see steps (a)–(c) in Algorithm1;
(b)
solve (113) to find A˘c,F˘x,N1,Q3∗k;
(c)
calculateAc,Fx∗k via (114), and implement uk=Fxkxck+Fyyk.


### 6.3. Prespecifying Relaxation Scalars

The scalarsηrs appear nonaffine and nonlinear in (105) and (113). Although it is suggested that ηrs can be line-searched over the interval 0,1 for online optimizations, in this way, the computational burden will be considerably increased. An alternative is to offline optimize ηrs. In [5, 26], we offline calculated ηrs by applying the norm-bounding technique.The condition (111) is satisfied if(115)Qk⋆ξs♠FyCjUkQ3k+F˘xku˜s2≥0,♠=FyCjN1k+UkQ3kUkT+F˘xkUkT,j=1,…,L,s=1,…,nu,(116)11−η1su˜s2+1η1sζsu2≤u¯s2,where ζsu=maxξsFyEjEjTFyTξsT1/2j=1,…,L. The maximum u˜s satisfying (116) is calculated by(117)u˜s=u¯s−ζsu,by taking η1s=ζsu/u¯s.The condition (112) is satisfied if(118)∑l=1Lλlk+i∑j=1Lλjk+iQk⋆ΨsChΦ˘lj1kψ˜s2≥0,h=1,…,L,s=1,…,q,i≥0,(119)11−η2s1−η3sψ˜s2+11−η2sη3sζ¯sz2+1η2sζsz2≤ψ¯s2,where ζsz=maxΨsEhEhTΨsT1/2h=1,…,L and(120)ζ¯sz=minζ¯szζ¯szs.t.∑l=1Lλlk+i∑j=1Lλjk+iζ¯szI⋆ΨsChΓlj1kζ¯sz≥0,h=1,…,L,i≥0.The maximumψ˜s satisfying (119) is calculated by(121)ψ˜s=ψ¯s−ζsz−ζ¯sz,by taking η2s=ζsz/ψ¯s and η3s=ζ¯sz/ψ¯s−ζsz.In the above, sinceζsu and ζsz,ζ¯sz are the norms of the disturbance-related items, the method for optimizing η1s,η2s,η3s has been called the norm-bounding technique. In this way, we obtain “the second best” values of ηrs (though may not be the best).The constraints (115) and (118) will not be utilized in the optimization problems (though they could be utilized), since they are more conservative than (111) and (112). In (111) the item u¯s2−1/η1sξsFyEjEjTFyTξsT applies for each j, while in (115), the item u˜s2 imposes for all j=1,…,L. Similarly, in (112), the item 1/1−η2sη3sΨsChΓlj1k,ψ¯s2−1/η2sΨsεhεhTΨsT applies for each pair of l,j, while in (118), the item ψ˜s2 imposes for all l,j=1,…,L.


### 6.4. Alternative Transformations for LPV

Takenx=nxc and Q2=0 in this subsection. Based on (98) and (99), let us define(122)N1≔M1−1,P1≔Q1−1,U≔−M1−1M2T,ek≔xk−Uxck,A^c≔−UAcQ2,L^c≔−ULc,F^x=FxQ2,A¯c≔−UAcM2−TM1−P1,F¯x≔FxM2−TM1−P1,T0≔I00M2−TM1−P1,T1≔Q1N1Q20,T2≔I00−UT,ℳP≔M1⋆M1−P1M1−P1,QN≔Q1⋆N1−Q1Q1−N1,NQ≔Q1⋆N1N1,Φ¯lj≔Al+BlFyCjBlF¯xL^cCjA¯c,Φ^lj≔Al+BlFyCjQ1+BlF^xAl+BlFyCjN1L^cCjQ1+A^cL^cCjN1,Γ^lj≔Dl+BlFyEjL^cEj,Φ¯lj1≔Al+BlFyCjBlF¯x,Γ^lj1≔Dl+BlFyEj,Φ^lj1=Al+BlFyCjQ1+BlF^xAl+BlFyCjN1.Based on these notations, we have(123)Ac=−U−1A¯cM1−P1−1M2T,Lc=−U−1L^c,Fx=F¯xM1−P1−1M2T,M2=−UTM1,(124)Ac=−U−1A^cQ2−1,Lc=−U−1L^c,Fx=F^xQ2−1,Q2=U−1Q1−N1.According to (98), we have Q3=U−1Q1−N1U−T. Applying a congruence transformation on (103), via diagI,UkT, yields(125)1−ϱk⋆UkxckQ1k−N1k≥0.Based on (98) and (99), applying congruence transformations on (89), via diagT0,I,T2 and diagT0,T2,I,I, respectively, yields(126)∑l=1Lλlk+i∑j=1Lλjk+iϒljQB≥0,i≥0,(127)∑l=1Lλlk+i∑j=1Lλjk+iϒljopt≥0,i≥0,where ϒljQB≔1−αljℳP⋆⋆0αljI⋆Φ¯ljΓ^ljQN, ϒljopt≔ℳP⋆⋆⋆Φ¯ljQN⋆⋆Q11/2Fj00γI⋆ℛ1/2FyCjF¯x00γI. Applying congruence transformations on (91) and (93), via diagT0,I, yields(128)ℳP⋆⋆0I⋆11−η1sξsFyCjF¯x1η1sξsFyEju¯s2≥0,j=1,…,L,s=1,…,nu,(129)∑l=1L∑j=1Lλlk+iλjk+i×ℳP⋆⋆0I⋆11−η2s1−η3sΨsChΦ¯lj111−η2sη3sΨsChΓ^lj1ψ¯s2−1η2sΨsεhεhTΨsT≥0,h=1,…,L,s=1,…,q,i≥0.Summarizing the above, an equivalent transformation of (105) is (see [5])(130)minγ,αlj,ϱ,M1,N1,Q1,P1,A¯c,L^c,F¯x,FykmaxABCDECEFGk+i∈Ωγk,s.t.125,7,126−129andM1k=N1k−1,Q1k=P1k−1,with Ac,Lc,Fxk calculated by (123). The optimization problem (130) is nonconvex, but its near-optimal solution arbitrarily close to the theoretically optimal one can be found by applying ICCA.Based-on (98) and (99), applying congruence transformations on (89) and (90), via diagT1,I,T2 and diagT1,T2,I,I respectively, yields(131)∑l=1Lλlk+i∑j=1Lλjk+i1−αljNQ⋆⋆0αljI⋆Φ^ljΓ^ljQN≥0,i≥0,(132)∑l=1Lλlk+i∑j=1Lλjk+iNQ⋆⋆Φ^ljQN⋆Q11/2FjQ1Q11/2FjN1ℛ1/2FyCjQ1+F^xℛ1/2FyCjN10γI≥0,i≥0.Applying congruence transformations on (91) and (93), via diagT1,I, yields(133)NQ⋆11−η1sξsFyCjQ1+F^xFyCjN1u¯s2−1η1sξsFyEjEjTFyTξsT≥0,j=1,…,L,s=1,…,nu,(134)∑l=1L∑j=1Lλlk+iλjk+iNQ⋆⋆0I⋆11−η2s1−η3sΨsChΦ^lj111−η2sη3sΨsChΓ^lj1ψ¯s2−1η2sΨsEhεhTΨsT≥0,h=1,…,L,s=1,…,q,i≥0.In summary, an equivalent transformation of (113) is (see [5])(135)minγ,αlj,ϱ,N1,Q1,A^c,F^xkmaxABCDECEFGk+i∈Ωγk,s.t.125,107and131−134,with Ac,Lc,Fxk calculated by (124) and L^c,Fy prespecified. The solution to (135) can be obtained by LMI toolbox.


## 6.1. Full Online Method for LPV

By applying the block-matrix inversion onQ=Q1Q2TQ2Q3, it is easy to show that(101)M=M1−M1Q2TQ3−1−Q3−1Q2M1Q3−1+Q3−1Q2M1Q2TQ3−1.TakeU=−M1−1M2T. Then, it is easy to show that U=−Q2TQ3−1 and(102)x˜kTMkx˜k=xk−x0kTM1kxk−x0k+xckTQ3k−1xck.Lemma 7.
Let Assumption3 hold and at each k>0, find x0,Mek such that xk−x0k∈εMek. Choose U,xc0 such that U0xc0=x00 and at each k>0, Uk such that Ukxck=x0k. Then, condition (80) holds if(103)1−ϱk⋆xckQ3k≥0.Further defineN1=M1−1 and P3=Q3−1. Then,(104)Q=N1+UQ3UTUQ3Q3UTQ3,M=M1−M1U−UTM1P3+UTM1U,which naturally satisfies M=Q−1. By applying (104), problem (95) becomes (equivalently)(105)minγ,αlj,ϱ,N1,M1,P3,Q3,Ac,Lc,Fx,FykmaxABCDECEFGk+i∈Ωγk,s.t.103,7,89,91,93,104,N1k=M1k−1,P3k=Q3k−1.This approach is proposed in [6, 7] where Uk=E0T, and hence,(106)Q=Q1E0TQ3Q3E0Q3,M=M1−M1E0T−E0M1M3.In solving (105), usually αljk=αk can be prespecified. One can line-search αk over the interval 0,1. Indeed, we found that the improvement on control performance is negligible by online optimizing αk. The problem (105) has been solved by the iterative cone-complementary approach (ICCA) (see firstly in [10, 25]). ICCA has two major loops. The inner loop is the cone-complementary approach (CCA) which minimizes TraceM1kN1k+N1kM1k+Q3kP3k+P3kQ3k in order to achieve N1k=M1k−1 and P3k=Q3k−1. The outer loop gradually reduces γk. Note that, even with αk being prespecified, (105) cannot be transformed into LMI optimization problem.In Algorithm1, while first and second equations in step (c) are natural for refreshing the bound of xk, third equation in step (c) is imposed for the recursive feasibility of (105). Finding Me′k satisfying equations in step (c) in Algorithm 1 can be achieved via LMI techniques.Theorem 7 (see [5, 26]).
Adopt Algorithm1. Suppose that Assumption 3 holds, and (105) is feasible at time k=0. Then,(i)
(105) will be feasible at each k>0(ii)
γ,z′,xc,u will converge to a neighborhood of 0, and the constraints in (2) are satisfied for all k≥0Algorithm 1:Full dynamic OFRMPC.
At eachk≥0,
(a)
fork=0, take U0=I;
(b)
fork>0, calculate xck=Ack−1xck−1+Lck−1yk−1, and refresh Me,U,x0k as in (15)–(17);
(c)
fork>0, find Me′k satisfying (the same as (19)–(20))
x
k
−
1
−
x
0
k
−
1
∈
ε
M
e
k
−
1
,
w
k
−
1
≤
1
⟹
x
k
−
x
′
0
k
∈
ε
M
e
′
k
,
x
′
0
k
≥
U
′
k
x
c
k
,
M
e
′
k
≥
M
e
k
,
and, if equations in step (c) in Algorithm1 are feasible, then change Mek=Me′k, Uk=U′k and x0k=U′kxck;
(d)
solve (105) to find Ac,Lc,Fx,Fy,M1,N1,Q3,P3∗k;
(e)
implementuk=Fxkxck+Fykyk.


## 6.2. Partial Online Method for LPV

In order to alleviate the computational burden, we can prespecifyLc,Fy in (105). In this way, M1,P3 are no longer the decision variables. Therefore, {(7), (89), (91), (93)} will be modified accordingly.By applying the Schur complement, (7) is equivalent to(107)ϱkMekIIN1k≥0.Taking congruence transformations viadiagQk,I on (89) yields(108)∑l=1Lλlk+i∑j=1Lλjk+i1−αljkQk⋆⋆0αljkI⋆Φ˘ljkΓljkQk≥0,i≥0,∑l=1Lλlk+i∑j=1Lλjk+iQk⋆⋆Φ˘ljkQk⋆♠0γkI≥0,i≥0,(109)♠=Q11/2Fj♡Q11/2FjUkQ3kQ21/2Q3kUkTQ21/2Q3kℛ1/2FyCj♡+F˘xkUkTℛ1/2FyCjUkQ3k+F˘xk,♡=N1k+UkQ3kUkT,where(110)Φ˘ljk=⋄Q1k+BlF˘xkUkT⋄Q2kT+BlF˘xkLcCjQ1k+A˘ckUkTLcCjQ2kT+A˘ck,Φ˘lj1k=⋄Q1k+BlF˘xkUkT⋄Q2kT+BlF˘xk,⋄=Al+BlFyCj,A˘ck=AckQ3k,F˘xk=FxkQ3k.Taking congruence transformations on (91) and (93) via diagQk,I, and applying the Schur complement, yields(111)Qk⋆11−η1sξs♠FyCjUkQ3k+F˘xku¯s2−1η1sξsFyEjEjTFyTξsT≥0,♠=FyCjN1k+UkQ3kUkT+F˘xkUkT,j=1,…,L,s=1,…,nu,(112)∑l=1L∑j=1Lλlk+iλjk+iQk⋆⋆0I⋆♠1♠2♠3≥0,♠1=11−η2s1−η3sΨsChΦ˘lj1k,♠2=11−η2sη3sΨsChΓlj1k,♠3=ψ¯s2−1η2sΨsChChTΨsT,h=1,…,L,s=1,…,q,i≥0.In summary, problem (105) is simplified as(113)minγ,αlj,ϱ,N1,Q3,A˘c,F˘xkmaxABCDECEFGk+i∈Ωγk,s.t.103,107and108,111,112,with Ack,Fxk calculated by(114)Ack=A˘ckQ3k−1,Fxk=F˘xkQ3k−1.The solution to (113) can be obtained by LMI toolbox. Since CCA is not involved, it is computationally less expensive than (105).Theorem 8 (see [5, 26]).
Adopt Algorithm2. Suppose that Assumption 3 holds, and (113) is feasible at time k=0. Then(i)
(113) will be feasible at each k>0;(ii)
γ,z′,xc,u will converge to a neighborhood of 0, and the constraints in (2) are satisfied for all k≥0.Algorithm 2:Partial dynamic OFRMPC.
At eachk≥0,
(a)
see steps (a)–(c) in Algorithm1;
(b)
solve (113) to find A˘c,F˘x,N1,Q3∗k;
(c)
calculateAc,Fx∗k via (114), and implement uk=Fxkxck+Fyyk.


## 6.3. Prespecifying Relaxation Scalars

The scalarsηrs appear nonaffine and nonlinear in (105) and (113). Although it is suggested that ηrs can be line-searched over the interval 0,1 for online optimizations, in this way, the computational burden will be considerably increased. An alternative is to offline optimize ηrs. In [5, 26], we offline calculated ηrs by applying the norm-bounding technique.The condition (111) is satisfied if(115)Qk⋆ξs♠FyCjUkQ3k+F˘xku˜s2≥0,♠=FyCjN1k+UkQ3kUkT+F˘xkUkT,j=1,…,L,s=1,…,nu,(116)11−η1su˜s2+1η1sζsu2≤u¯s2,where ζsu=maxξsFyEjEjTFyTξsT1/2j=1,…,L. The maximum u˜s satisfying (116) is calculated by(117)u˜s=u¯s−ζsu,by taking η1s=ζsu/u¯s.The condition (112) is satisfied if(118)∑l=1Lλlk+i∑j=1Lλjk+iQk⋆ΨsChΦ˘lj1kψ˜s2≥0,h=1,…,L,s=1,…,q,i≥0,(119)11−η2s1−η3sψ˜s2+11−η2sη3sζ¯sz2+1η2sζsz2≤ψ¯s2,where ζsz=maxΨsEhEhTΨsT1/2h=1,…,L and(120)ζ¯sz=minζ¯szζ¯szs.t.∑l=1Lλlk+i∑j=1Lλjk+iζ¯szI⋆ΨsChΓlj1kζ¯sz≥0,h=1,…,L,i≥0.The maximumψ˜s satisfying (119) is calculated by(121)ψ˜s=ψ¯s−ζsz−ζ¯sz,by taking η2s=ζsz/ψ¯s and η3s=ζ¯sz/ψ¯s−ζsz.In the above, sinceζsu and ζsz,ζ¯sz are the norms of the disturbance-related items, the method for optimizing η1s,η2s,η3s has been called the norm-bounding technique. In this way, we obtain “the second best” values of ηrs (though may not be the best).The constraints (115) and (118) will not be utilized in the optimization problems (though they could be utilized), since they are more conservative than (111) and (112). In (111) the item u¯s2−1/η1sξsFyEjEjTFyTξsT applies for each j, while in (115), the item u˜s2 imposes for all j=1,…,L. Similarly, in (112), the item 1/1−η2sη3sΨsChΓlj1k,ψ¯s2−1/η2sΨsεhεhTΨsT applies for each pair of l,j, while in (118), the item ψ˜s2 imposes for all l,j=1,…,L.


## 6.4. Alternative Transformations for LPV

Takenx=nxc and Q2=0 in this subsection. Based on (98) and (99), let us define(122)N1≔M1−1,P1≔Q1−1,U≔−M1−1M2T,ek≔xk−Uxck,A^c≔−UAcQ2,L^c≔−ULc,F^x=FxQ2,A¯c≔−UAcM2−TM1−P1,F¯x≔FxM2−TM1−P1,T0≔I00M2−TM1−P1,T1≔Q1N1Q20,T2≔I00−UT,ℳP≔M1⋆M1−P1M1−P1,QN≔Q1⋆N1−Q1Q1−N1,NQ≔Q1⋆N1N1,Φ¯lj≔Al+BlFyCjBlF¯xL^cCjA¯c,Φ^lj≔Al+BlFyCjQ1+BlF^xAl+BlFyCjN1L^cCjQ1+A^cL^cCjN1,Γ^lj≔Dl+BlFyEjL^cEj,Φ¯lj1≔Al+BlFyCjBlF¯x,Γ^lj1≔Dl+BlFyEj,Φ^lj1=Al+BlFyCjQ1+BlF^xAl+BlFyCjN1.Based on these notations, we have(123)Ac=−U−1A¯cM1−P1−1M2T,Lc=−U−1L^c,Fx=F¯xM1−P1−1M2T,M2=−UTM1,(124)Ac=−U−1A^cQ2−1,Lc=−U−1L^c,Fx=F^xQ2−1,Q2=U−1Q1−N1.According to (98), we have Q3=U−1Q1−N1U−T. Applying a congruence transformation on (103), via diagI,UkT, yields(125)1−ϱk⋆UkxckQ1k−N1k≥0.Based on (98) and (99), applying congruence transformations on (89), via diagT0,I,T2 and diagT0,T2,I,I, respectively, yields(126)∑l=1Lλlk+i∑j=1Lλjk+iϒljQB≥0,i≥0,(127)∑l=1Lλlk+i∑j=1Lλjk+iϒljopt≥0,i≥0,where ϒljQB≔1−αljℳP⋆⋆0αljI⋆Φ¯ljΓ^ljQN, ϒljopt≔ℳP⋆⋆⋆Φ¯ljQN⋆⋆Q11/2Fj00γI⋆ℛ1/2FyCjF¯x00γI. Applying congruence transformations on (91) and (93), via diagT0,I, yields(128)ℳP⋆⋆0I⋆11−η1sξsFyCjF¯x1η1sξsFyEju¯s2≥0,j=1,…,L,s=1,…,nu,(129)∑l=1L∑j=1Lλlk+iλjk+i×ℳP⋆⋆0I⋆11−η2s1−η3sΨsChΦ¯lj111−η2sη3sΨsChΓ^lj1ψ¯s2−1η2sΨsεhεhTΨsT≥0,h=1,…,L,s=1,…,q,i≥0.Summarizing the above, an equivalent transformation of (105) is (see [5])(130)minγ,αlj,ϱ,M1,N1,Q1,P1,A¯c,L^c,F¯x,FykmaxABCDECEFGk+i∈Ωγk,s.t.125,7,126−129andM1k=N1k−1,Q1k=P1k−1,with Ac,Lc,Fxk calculated by (123). The optimization problem (130) is nonconvex, but its near-optimal solution arbitrarily close to the theoretically optimal one can be found by applying ICCA.Based-on (98) and (99), applying congruence transformations on (89) and (90), via diagT1,I,T2 and diagT1,T2,I,I respectively, yields(131)∑l=1Lλlk+i∑j=1Lλjk+i1−αljNQ⋆⋆0αljI⋆Φ^ljΓ^ljQN≥0,i≥0,(132)∑l=1Lλlk+i∑j=1Lλjk+iNQ⋆⋆Φ^ljQN⋆Q11/2FjQ1Q11/2FjN1ℛ1/2FyCjQ1+F^xℛ1/2FyCjN10γI≥0,i≥0.Applying congruence transformations on (91) and (93), via diagT1,I, yields(133)NQ⋆11−η1sξsFyCjQ1+F^xFyCjN1u¯s2−1η1sξsFyEjEjTFyTξsT≥0,j=1,…,L,s=1,…,nu,(134)∑l=1L∑j=1Lλlk+iλjk+iNQ⋆⋆0I⋆11−η2s1−η3sΨsChΦ^lj111−η2sη3sΨsChΓ^lj1ψ¯s2−1η2sΨsEhεhTΨsT≥0,h=1,…,L,s=1,…,q,i≥0.In summary, an equivalent transformation of (113) is (see [5])(135)minγ,αlj,ϱ,N1,Q1,A^c,F^xkmaxABCDECEFGk+i∈Ωγk,s.t.125,107and131−134,with Ac,Lc,Fxk calculated by (124) and L^c,Fy prespecified. The solution to (135) can be obtained by LMI toolbox.


## 7. Conclusion

We have summarized the existing results for dynamic output feedback robust MPC for the polytopic LPV model with additive bounded disturbance. This kind of research is still undergoing. For example, the free control moves are not included satisfactorily as in the disturbance-free case whenx is measurable (e.g., the partial feedback MPC, feedback MPC, open-loop MPC, and parameter-dependent open-loop MPC). The summary in this paper may pave the way for future research.



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*Source: 3830724-2020-02-01.xml*