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

(is 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 becomingmature. For the output feedback case for the LPVmodel with bounded disturbance, we have published a series of works. Anyway, it lacks a unification of these publications. (is 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 LPVmodel with bounded disturbance.(is paper can pave the way for further research on output feedback MPC.


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. e 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. erefore, it is not surprising that there are a lot of research works on the LPV model-based and T-S modelbased controls. Moreover, it is impossible that all the uncertainties can be included in the parametric polytopes. e additive bound disturbance, with its real-time value arbitrarily changing, without useful statistics, is another widely accepted uncertainty description. is paper considers the above LPV model (including T-S model) with additive bound disturbance.
e 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 parameterdependent as in [4]; in this parameter-dependent case, openloop 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 on N � 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. erefore, 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. ese works are given across more than 10 years. e 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; x(k + i | k) is the prediction of x(k + i) at time k. Moreover, (i) u: in R n u , the control input signal (ii) w: in R n w , the disturbance (iii) x: in R n x , the true state (iv) x c : in R n x c , the estimator state or controller state (v) y: in R n y , the output (vi) |ξ|: the component-wise absolute value of ξ (vii) ε M : the ellipsoid associated with the positive-definite matrix M, i.e., ε M � ξ | ξ T Mξ ≤ 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

Dynamic Output Feedback Robust MPC Problem
Consider the following linear parameter varying (LPV) model: where z(k) ∈ R n z (see [5,6]) and z ′ (k) ∈ R n z ′ (see [7,8]) are the constrained signal and penalized signal, respectively, and w is unknown, norm-bounded, and persistent.
i.e., there exist nonnegative coefficients λ l (k), l � 1, . . . , L such that L l�1 λ l (k) � 1 and Since D(k), E(k) are shaping matrices, Assumption 1 applies to any norm-bounded disturbance. If λ l (k)'s are exactly known at the current time k, but λ l (k + i) for all i > 0 are unknown at the current time k, then we specially call (1) the quasi-LPV model. e hard physical constraints are When x is fully measurable and w(k) ≡ 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 w(k) ≠ 0.
Theorem 1 (see [9]). Consider system (1), with Assumptions 1 and 2 being satisfied. Adopt the dynamic output feedback controller, i.e., where the controller parameters are defined as parameterdependent, i.e., where "(k)" is omitted for brevity. Further, {A lj where U(k) is a transformation matrix being given before solving (6)- (11), d is a fixed nonnegative integer, η 1s ∈ [0, 1) are the fixed scalars, and ξ s is the s-th row of n u -ordered identity matrix. In (9) and (11), K(d + 2) is the set of L-tuples obtained from all possible combinations of e number of elements of K(d + 2) is given by In (9), Mathematical Problems in Engineering where "(k)" is omitted, Ψ s is the s-th row of Ψ, and η 2s , η 3s ∈ [0, 1) are the fixed scalars. Take U(0) � I and an x c (0), and suppose (a) For k > 0, apply (4) and (5) where and if (19) and (20) are feasible, then change M e (k) � M e ′ (k) and U(k) � U ′ (k) (4) and (5) to obtain C c (k) and D c (k), then Suppose (6)-(11) is feasible at time k � 0. en, (i) (6)- (11) will be feasible at each k > 0 (ii) c, z ′ , u will converge to a neighborhood of 0, and the constraints in (2) are satisfied for all k ≥ 0 In (6)- (11), the four constraints of [3] are generalized (i.e., the confinement of x(k) being generalized to (7) and (8) which is the confinement of both x(k) and x c (k), 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 Theorem 1 as one of the examples.

Model and Controller Descriptions
e predictive form of (1) is for all i ≥ 0. e predictive form of (2) is According to Assumption 2,4 Mathematical Problems in Engineering 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]): where A c , L c are controller gain matrices and F x , F y are feedback gain matrices. It is unnecessary that n x � n x c . e predictive form of (24) is Remark 1. ere are 4 controller parameters A c , L c , F x , F y in (24) and (25). In the literature, often there are only 2 controller parameters L c , F x for output feedback. We found that with only 2 controller parameters L c , F x , for (1), it is more difficult to find the feasible solution to the optimization problem of output feedback MPC. With 4 parameters A c , L c , F x , F y , output feedback MPC can be applied to a much larger range of system models.
Define the augmented state x � x x c . By applying (1) and (24), the augmented closed-loop system is where e predictive form of (26) is where By applying (23), it is shown that

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 n x � n x c . e predictive form of (3) is where Remark 2. For the quasi-LPV, since λ l (k) are known, we can Define the augmented state x � x x c . By applying (1) and (3), the augmented closed-loop system is where e predictive form of (33) is Mathematical Problems in Engineering where By applying (32), it is shown that In the sequel, we often use the notations for LPV, but the results can be simply transplanted to quasi-LPV.

Characterization of Stability and Optimality
Consider the closed-loop systems (28) and (35). ey 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 λ l (k + i) and λ j (k + i)). We will borrow the notion of quadratic boundedness (QB) in [14,15] to characterize the closed-loop stability of (28) and (35).

Review of Quadratic Boundedness.
In [14], the following model with nominal parametric matrices is considered: where A and D are time-invariant (fixed) matrix, v ∈ R n v . In [14], it is firstly assumed that v ∈ V where V is a compact (bounded and closed) set, and V ⊂ R n v .
Definition 1 (see [14]). System (38) is said to be quadratically bounded with Lyapunov matrix P > 0 if System (38) is said to be strictly quadratically bounded with Lyapunov matrix P > 0 if (40) 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.
e set S is a robust positively invariant set for (38), if Theorem 2 (see [14]). Suppose v ∈ ε P v with P v > 0. e following facts are equivalent: In [15], the following model with uncertain parametric matrices is considered: where [A(k) | D(k)] belongs to a known bounded set, i.e., [A(k) | D(k)] ∈ P for all k ≥ 0, and D ≠ 0 for at least one Definition 3 (see [15]). Suppose v(k) ∈ ε P v for all k ≥ 0, in (43). System (43) is said to be strictly quadratically bounded with a common Lyapunov matrix P > 0, if 6 Mathematical Problems in Engineering Since D ≠ 0 for at least one [A | D] ∈ P, and v ∈ ε P v , 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. e definition of quadratic boundedness is similar to Definition 1.
Theorem 3 (see [15]). Suppose v(k) ∈ ε P v for all k ≥ 0, in (43). e following facts are equivalent: Note that in the above theorem it is necessary to use a time-varying α(k).

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.

Optimality Condition.
e disturbance-free form of (28) or (35) is Correspondingly, Let us introduce the quadratic cost where Q 1 , Q 2 , and R are positive-definite weighting matrices, and consider the condition Mathematical Problems in Engineering In eorem 1, it has taken Q 2 � 0. For exponentially Further, let en, applying (57) to (56) yields that is, c(k) is an upper bound of J(k). We will take c(k) as the cost function of the optimization problems which finds the controller parametric matrices. e condition (55) can be rewritten as where Hence, (55) is guaranteed by Π(i, k) ≥ 0. By applying the Schur complement, it is shown that Π(i, k) ≥ 0 can be transformed into where Q � diag Q 1 , Q 2 . e condition (55) or (61) is for optimality, not primarily for stability. However, if is exponentially stable (referring to point (iv) of eorem 4). We can indeed combine the optimality and stability conditions by imposing (see firstly [11,17] for LPV and [12,13] for quasi-LPV) It is easy to show that (63) is equivalent to (in the sense for any x(k + i | k) and w(k + i)) Remark 3. It is apparent that feasibility of (64) guarantees both (50) and (61). With c(k) free (i.e., as a decision variable), feasibility of (50) guarantees both (61) and (64). erefore, on the feasibility aspect, (64) and (50) are equivalent. In the above, although there is no guarantee that x(k + i | k) will converge, the convergence of x(k + i | k) will happen when ‖x(k)‖ is not small (see firstly [19] for LPV and [20] for quasi-LPV). e main reason lies in that (64) or (50) is a robust condition.

A Paradox for State
Let us impose that, if the augmented state x(k) lies outside of the ellipsoid ε β(k) − 1 M(k) , then x(k + i | k) converges to ε β(k) − 1 M(k) with the increase of i ≥ 0. Here, ε β(k) − 1 M(k) is an ellipsoid not larger than ε M(k) since 0 < β(k) ≤ 1 (see firstly [17] for LPV and [13,20] for quasi-LPV). By applying such β(k), we can change (48) as which is equivalent to (in the sense for any x(k + i | k) and We can also change (63) as which is equivalent to (in the sense for any x(k + i | k) and w(k + i)) Adding β(k) ∈ (0, 1] as a free variable, due to the special position of β(k) in either (66) or (68), does not affect the minimization of c(k) and feasibility. It is suggested to minimize β(k) after the minimization of c(k) (see firstly [19] for LPV and [20] for quasi-LPV). If the controller parametric matrices are not reoptimized in minimizing β(k), it is easy to know that we do not need β(k), i.e., we can simply remove it.  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 u(k). If (69) is feasible at k � 0, then with the evolution of k, c, z ′ , x c , 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 + i | k) and w(k + i)) min c,α lj ,M,par

General Optimization Problem
with recursive feasibility and stability properties retained.

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. en for any scalar η
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 + i | k) within ε M(k) . Theorem 6 (see firstly [5,23] for LPV and [9] for quasi-LPV). Suppose at time k, there exist scalars α(i, k) ∈ (0, 1) and η rs , and matrix M(k) > 0, such that (57) and (50) hold, and , k)). Take care of the special cases: Ψ s C(k + i + 1)Γ 1 (i, k) � 0 and η 3s � 0 en, (22) is satisfied. In the above theorem, one may want to choose η rs be time-varying. However, we have not found a good method to online optimize η rs , so we take η rs as time-invariant.

Current Augmented State.
e condition (57) (i.e., x(k) can be unmeasurable, while x c (k) is always known. When x(k) is unmeasurable, we need to remove it from (57) for the sake of solving (73).
Let us define an error signal where with U(k) being a known transformation matrix. When U(k) � I, defining e(k) is usual; when U(k) � E T 0 is fixed, see firstly [7,25]; when U(k) is online refreshed, see firstly [5,26] for LPV and [9] for quasi-LPV. When x(k) is unmeasurable, e(k) is unknown (nondeterministic). If we can obtain the outer bounding set of e(k), say D e (k), then we can utilize Using x � e + Ux c , we obtain If we can remove the cross item 2e T (M 1 U + M T 2 )x c , then the treatment of (57) will become easier, and the treatment of recursive feasibility of the resultant optimization will become simpler.
with recursive feasibility and stability properties retained in case M e (k) is appropriately refreshed.
(ii) Before [28], either ellipsoidal bound or polyhedral bound is solely applied in the optimization problem. e 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.

Handling Double Convex
Lemma 6 (see firstly [10,25]). e conditions hold if and only if there exists a sufficiently large d ≥ 0 such that (97) Moreover, if (97) holds for d � d, then they hold for any d > d.
is lemma has been utilized in eorem 1. In this lemma, Υ lj (k) ∈ Υ QB lj (k), Υ opt lj (k), Υ z hlj (k) . 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]): In 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. ere exists a finite n such that necessary and sufficient conditions for satisfaction of (96) can be obtained for a concrete model. min c,α lj , 9,N 1 (103), (7), (89), (91), (93), (104), is approach is proposed in [6,7] where U(k) � E T 0 , and hence, In solving (105), usually α lj (k) � α(k) can be prespecified. One can line-search α(k) over the interval (0, 1). Indeed, we found that the improvement on control performance is negligible by online optimizing α(k). e problem (105) has been solved by the iterative cone-complementary approach (ICCA) (see firstly in [10,25]). ICCA has two major loops. e inner loop is the cone-complementary approach (CCA) which minimizes Trace M 1 (k) e outer loop gradually reduces c(k). Note that, even with α(k) being prespecified, (105) cannot be transformed into LMI optimization problem.
In Algorithm 1, while first and second equations in step (c) are natural for refreshing the bound of x(k), third equation in step (c) is imposed for the recursive feasibility of (105). Finding M e ′ (k) satisfying equations in step (c) in Algorithm 1 can be achieved via LMI techniques.
Theorem 7 (see [5,26] Taking congruence transformations via diag Q(k), ♠ � where Taking congruence transformations on (91) and (93) via diag Q(k), I { }, and applying the Schur complement, yields In summary, problem (105) is simplified as with A c (k), F x (k) calculated by e 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] (ii) c, z ′ , x c , u will converge to a neighborhood of 0, and the constraints in (2) are satisfied for all k ≥ 0.

Prespecifying Relaxation Scalars.
e scalars η rs appear nonaffine and nonlinear in (105) and (113). Although it is suggested that η rs 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 η rs . In [5,26], we offline calculated η rs by applying the norm-bounding technique.

(134)
In summary, an equivalent transformation of (113) is (see [5]) with A c , L c , F x (k) calculated by (124) and L c , F y prespecified. e solution to (135) can be obtained by LMI toolbox.

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

Conflicts of Interest
e authors declare that they have no conflicts of interest.