A Synthesis Approach to Output Feedback MPC for LPV Model With Bounded Disturbance

The model with polytopic parametric uncertainty and bounded disturbance is controlled by the approach named dynamic OFRMPC (Output Feedback Robust Model Predictive Control). A key knob for control performance and region of attraction for this approach is the selection of Lyapunov matrix. A Lyapunov matrix, which does not have structural restriction, is proposed. In the ICCA (Iterative Cone Complementary Approach), which is invoked in optimizing the control law parameters, the starting up steps are designed as a variant CCA (Cone Complementary Approach). ICCA designs an outer loop, over CCA, for searching the minimum cost bound, while the variant CCA omits the outer loop by adding the cost bound in CCA objective function. This starting up can reduce the computational burden. The suboptimal dynamic OFRMPC (where CCA is avoided) is discussed, and a previous approach is re-formulated. A numerical example is given to show the advantages of the proposed approach.


I. INTRODUCTION
In the industrial circle, MPC (Model Predictive Control) has been proven as the most successful among advanced control techniques (see e.g., [1]- [6]). Other advanced control techniques such as sliding mode control (see [7]) and adaptive control (see e.g., [8]- [10]) have also attracted lots of attentions. Some works have combined adaptive control with MPC and applied the approach to highly nonlinear systems such as Brain-Machine Interface (BMI) system (see e.g., [11], [12]). Based on the mathematical model of the system, MPC solves an optimization problem at each control interval. By carefully designing a penalty and a constraint on the terminal state, stability of the closed-loop system has been ensured [13], [14]. LPV (Linear Parameter Varying) model, where a set of linear sub-models are utilized to cover a sufficiently large region around equilibrium, has been widely utilized in MPC literature (see e.g., [15]- [19] theoretically and [20], [21] practically). In this paper, as usual, LPV model refers to the model with polytopic parametric uncertainty, which takes a linear form but can represent a wide variety of nonlinear and/or uncertain systems. The work in [22] has provided an LMI (Linear Matrix Inequality) based min-max The associate editor coordinating the review of this manuscript and approving it for publication was Jun Shen .
MPC approach to handle system with model parametric uncertainty, which is extended in several works with improvements on computational efficiency (see e.g., [21]) or control performance (see e.g., [23]). With an addition of unknown disturbance on linear model, a variety of MPC works are available, where QB (Quadratic Boundedness, see [24], [25]) technique and the constraint tightening approach (see e.g., [26]- [29]) seem to be mostly promising.
When the system state is not measured, the output feedback control may utilize the measured output of the system to estimate the system state (see e.g., [9], [10], [30], [31]), and this scheme applies to MPC (see e.g., [32]- [35]). The existence of state estimation complicates the design in several aspects, e.g., increase of the computational burden (see e.g., [33]), ruin of the recursive feasibility (see e.g., [32]). Regarding the output feedback MPC for LPV model with bounded disturbance, there are a few works available (see e.g., [25], [36]); the notion of QB, which is concise for specifying the invariance and stability properties for the system with additive bounded disturbance, is introduced into dynamic OFRMPC. In [25], a dynamic OFRMPC approach is proposed with guaranteed recursive feasibility and closed-loop stability, and an ICCA (Iterative Cone Complementary Approach) is proposed to find the controller solution. The work [36] improves [25] in several aspects, including the utilization of full VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ Lyapunov matrix, the treatment of physical constraints with norm-bounding technique, and the introduction of multi-step approach.
In [25], a non-diagonal block of the Lyapunov matrix is a fixed transformation of the diagonal block. In [36], it requires a non-diagonal block of the Lyapunov matrix to be invertible. In order to remove these limitations and improve the control performance, in the present paper we develop a dynamic OFRMPC approach without restrictions on the structure of Lyapunov matrix, for LPV model with bounded disturbance. The proposed approach fuses the merits in [36] and previous works, with further improvements. It is shown that the newly proposed optimization problem can cover the one in [36] via appropriate congruence transformations. Then, for handling the nontrivial optimization systematically with less computational burden, we propose an ICCA which includes the previous one and has a potential to reduce the computational burden without sacrifice of the control performance. Finally, by an appropriate congruence transformation and the dilation technique, the suboptimal solution to dynamic OFRMPC is given, which can have considerably lighter computational burden. It is shown that the previous approach in [37] can be re-formulated according to this suboptimal solution.
The main contribution of the paper can be further detailed as follows.
• In [36], the dimension of the controller state x c is required to be equal to the dimension of the true state x. This limitation will be removed by utilizing the general structured Lyapunov matrix and applying some proper matrix transformations. Moreover, this paper will also include the approach in [25].
• The work [36] uses CCA (Cone Complementarity Approach) to handle the mutual inverse positive-definite matrices in the optimization problem, and invoke an outer loop over CCA in order to decrease the cost bound. We call this CCA with outer loop as ICCA. In the starting up steps of ICCA, this paper will use a variant CCA. This variant CCA modifies the objective function of CCA, i.e., adds the cost bound to the objective function of CCA. The completion of variant CCA usually yields solution close to ICCA. Then, the major role of ICCA is to guarantee the recursive feasibility. Therefore, introducing the variant CCA will reduce the computational burden.
• By utilizing a dilation technique, the optimization problem will be solved in a suboptimal manner. This mainly concerns pre-specifying (once off-line, or inherited from the previous control interval) a part of the control law parameters. The approach in [37] is re-formulated with relaxation.
The structure of this paper is organized as follows. In Section II, the problem of dynamic OFMPC with general structured Lyapunov matrix is formulated. In Section III, both near-optimal and sub-optimal dynamic OFRMPC approaches are proposed with guaranteed recursive feasibility and closed-loop stability. In Section IV, the proposed approaches are applied to a fuel cell model. Section V gives the conclusion.
Notations: For any vector x and matrix W , x 2 W := x T Wx. x(i|k) is the value of x at time k + i, predicted at time k. I is the identity matrix with appropriate dimension. ε M := {ξ |ξ T M ξ ≤ 1} denotes the ellipsoid associated with the positive-definite matrix M . An element belonging to a polytope CoS means that it is a convex combination of the elements in set S, with all scalar nonnegative combing coefficients summing as 1. The symbol induces a symmetric structure in a square matrix. A value with superscript * means that it is the solution of the optimization problem. The time-dependence of the MPC decision variables is often omitted for brevity.
As in [36], invoke the following dynamic output feedback control law: where x c ∈ R n xc is the controller state; {A c , B c , C c , D c }(k) are the control law parameters. The closed-loop model is obtained based on (1) and (3), i.e., wherex = x x c , The stability/convergence property of (4) can be specified by the notion of QB, extending the application range of [24].
Definition 1: In the sense for all allowable λ l (i|k), λ j (i|k) and w(i|k), the system (4) is quadratically bounded with a common Lyapunov matrix Q −1 , if Lemma 1: Consider the system (4). In the sense for all allowable λ l (i|k), λ j (i|k) and w(i|k), the following facts are equivalent: (a) (4) is quadratically bounded with a common Lyapunov Define the following disturbance-free model: are the disturbance-free output and input, respectively.
Taking the block-matrix inverse of Q, it is easy to show that In this paper, without loss of generality, we set which naturally satisfy M = Q −1 . Define x 0 = Ux c . Assumption 3: There exist x 0 (0) and M e (0) > 0, such that For all k > 0, it will find x 0 (k) and M e (k), such that Remark 1: In (6), the parameter U has to be nonsingular. This restriction has been removed in (7).
Theorem 1: Consider the system (1) satisfying Assumptions 1-2. The dynamic output feedback control law (3), which drives the closed-loop model (4) quadratically bounded, minimizes the worst-case upper bound γ (k) of performance cost ∞ i=0 y u (i|k) 2 Q + u u (i|k) 2 R , and satisfies |u(i|k)| ≤ū and | z(i + 1|k)| ≤ψ for all i ≥ 0, can be approximated by solving (8)- (14), as shown at the bottom of the next page, where Proof: Refer to [38]. Note that ( (14) and the approach in [38] is the replacement of E T 0 with U . The conditions (9)- (14) are, respectively, on • current estimation error set, current controller state; • QB of the closed-loop model; • optimality of the disturbance-free model; • control input from k onwards; • constrained signal for all future time; • mutual inverse for Lypunov matrices. Remark 2: In this paper, x c is not required to have the same dimension with x, which means that the controller state can have lower dimension than the true state. This property can be invoked to reduce the computational burden. Moreover, note that, in [38] the general form (7) is not given.
Utilize ''Proposition 2'' of [39] to handle the double convex combinations in {(10), (11), (13)}. For example, (10) is guaranteed by either Set 1 or Set 2 in the following: Here, n is the complexity parameter. By increasing n, less conservative and asymptotically necessary conditions can be obtained.

VOLUME 8, 2020
The controller parameters The details can be found in [36]. Both optimization problems (8)-(14) and (15)-(21) are nontrivial (unable to be solved via single convex optimization). Reference [36] has utilized CCA to achieve (21). In order to reduce γ , an iterative procedure is invoked in [36] as an outer loop of CCA. Recently in [40], it shows that the outer loop can be reduced without much influence on the control performance. A linearization method, instead of CCA, is adopted in [40] to handle Q = M −1 . In this paper, we propose a new iterative method based on CCA, which is a close improvement on [36].

III. NEW SYNTHESIS APPROACHES OF DYNAMIC OFRMPC
When {A c , B c , C c , D c }(k) are simultaneously optimized, it is referred to as the near-optimal dynamic controller. In the sequel, we first present a near-optimal dynamic OFRMPC, then give the suboptimal dynamic OFRMPC by taking appropriate transformations.

A. NEAR-OPTIMAL DYNAMIC OFRMPC
When all the four parameters {A c , B c , C c , D c }(k) are optimized simultaneously, one cannot find the solution via single convex optimization; Q = M −1 has to be satisfied, where both Q and M are the decision variables of optimization problem. CCA has been proven as an appropriate approach for handling Q = M −1 . When Q and M are pre-specified, it is easy to solve (8)- (14).
• The problems (23)-(28) are LMI optimizations. Increasing n makes it more likely to yield the feasible solutions, and tends to improve the control performance. However, the larger n, the larger the number of LMIs, and the heavier the computational burden. We restrict n ≤ n 0 for computational reasons.
• The maximum iteration times t 0 is a regular parameter in CCA. • Step a) is a variant CCA. The objective function of CCA (e.g. in (28)) is σ . In the variant CCA (see (25)), the objective function equally include the cost bound γ .
Steps c)-e) composed of ICCA. As compared with ICCA, in the variant CCA there is no gradual reduction of γ (by adding gradual constraint γ ≤ κ o γ o ). Since the variant CCA does not include an outer iteration, it has much lighter computational burden than ICCA. For ICCA, the optimization problem with complexity O(R 3 L), where R is the number of scalar LMI variables and L is the number of scalar LMI rows (see [43]), will be solved for several to thousands times more than variant CCA.
• If the algorithm only takes step a) (i.e., omitting b)-f)), then the whole algorithm becomes variant CCA. However, in order to guarantee the recursive feasibility, we add step b) which can adjust the solution of the variant CCA. Usually, the solution of variant CCA in step a) is close to ICCA. ICCA not only withholds the recursive feasibility, but also may further reduce γ (k). According to our previous studies (see [25], [36] then the problem (8)- (14) can be recursively feasible.
ii) This is similar to [25] and [36], so the details are omitted.

B. SUBOPTIMAL DYNAMIC OFRMPC
Let us take a dilation matrixT = T 1 T 2 T 3 T 3 , and defineÂ c = Theorem 3: When {B c , D c } are pre-specified, the optimization problem (8)- (14) can be transformed into (41)- (46), as shown at the bottom of the next page, where Proof: By taking congruence a transformation via diag{T , I }, and applying the dilation techniqueT T +T − M ≤T T M −1T , it is shown that (10)-(13) are equivalent to (43)- (46). The condition (9) is transformed into (42) by applying the Schur complement.
Proof: This is analogous to Theorem 3. Proposition 2: Adding U as a simultaneous decision variable, the problem (53)-(58) is less conservative than (26) in [37], i.e., the former includes the latter as a special case.
Proof: This is similar to ''Proposition 1'' of [44], only that in [44] Note that all the technical tools/knobs of [37] are improved and incorporated in this paper. Hence, it is deserved that the conclusion holds. Therefore, adding U as a simultaneous decision variable, this paper improves the optimization problem in [37]. However, notice that [37] does not have U , and not pre-specify any element in Q. Compared with (41)- (46), the problem (53)-(58) have too weaknesses: i) the recursive feasibility is not guaranteed; ii) the optimization problem becomes conservative due to the existence of constraint on estimation error, (57).
Hence, this paper retrieves recursive feasibility of the optimization problem in [37].

Remark 5:
The proposed approach provides a systematic way for controlling uncertain systems by utilizing MPC, and it can remove some weaknesses of the previous approaches. Moreover, a sub-optimal algorithm is provided in case the computational efficiency is a crucial issue in real applications. Even through the proposed approach is applicable to moderate dimensional systems, it may not be applicable for high dimensional systems. Other techniques for improving the computational efficiency can be studied and applied.
(58) VOLUME 8, 2020   where any choice of c(0) satisfies Assumption 3. Choose c(0) = 2.0. For the input/state/output responses, see Figures 1-5, which verify the closed-loop QB and constraint satisfactions. The total amounts of computational time for k ≤ 300 for applying Alg2, Alg4, CCC2011 and ASCC2019 are 72.5, 0.81, 161.7 and 68.1 hours, respectively. This is not a real-time simulation. It is observed that the proposed approach in this paper can have a much smaller cost bound and a more aggressive control input.

V. CONCLUSION
Researches on OFRMPC for LPV model with bounded disturbance have been undergone for over 10 years. The dynamic output feedback, where the controller and estimator parameters are not separately designed, has been proven as successful. The controller has been improved over the past ten years, where one of the concentration is on the selection of appropriate Lyapunov matrix. In the previous studies, the Lyapunov matrix either has some fixed structure, or some of its non-diagonal blocks are required to be invertible. In this paper, these restrictions are removed. In this scenario, this paper has found appropriate transformations in order to transform OFRMPC into a solvable form. Not only the general near-optimal solution, but also the suboptimal solution with much lighter computational burden, are proposed in this paper.
In the future work, we plan to introduce free control moves to OFRMPC, which is a subject of great value so that the control performance can be improved and the region of attraction can be enlarged. However, due to the difficulty of guaranteeing the recursive feasibility, this subject needs further exploration.