Robust Receding Horizon Control with Free Control Moves for Polytopic Parametric Uncertainty Systems Subject to Multiple Input Delays

A robust receding horizon control (RHC) with free control moves is applied to polytopic parametric uncertainty systems with multiple input delays and unstable system matrices. A difficulty in the previous robust RHC work is that the free control moves are unsuitable for the system with input time delay, which is overcome in this paper by the design based on the augmented state. As a result, the synthesis of local control based on augmented feedback is given offline to alleviate the online computation burden. +e free control moves before the augmented feedback are the online decision variables, which are solved by minimizing a sequence of nonnegative scalars online. +e recursive feasibility is guaranteed by adopting the augmented state space equation. By adjusting the robust positively invariant set, the stability of the closedloop system is guaranteed. Simulation results demonstrate that the proposed algorithm improves the control performance effectively.


Introduction
Receding horizon control (RHC) is known for the capacity to deal with the multivariable constraint by the receding horizon optimization. It has obtained a lot of attention in academic [1][2][3][4][5][6], and the control policy has been applied to many actual systems such as multiagent systems [7], chemical engineering [8], and hybrid electric vehicles [9].
Since modeling errors cannot be avoided in the practical systems, the uncertainties and disturbance usually exist in the system model. ere are many results focused on the robust control for the system with the uncertainty and disturbance ( [10][11][12]). In particular, the robust RHC is a hot topic in recent years (see, e.g., [13][14][15][16]). A lot of significant results for the state feedback RHC with uncertainty have appeared in the literature ( [17][18][19][20]). Besides, time delays usually exist in dynamic systems ( [10,21]). e presence of time delays often leads to oscillation and instability of the closed-loop system. e controller design for the time delay systems is often crucial for the system operation. A lot of researchers have devoted to designing a robust MPC controller for the system with uncertainties and time delay. In [22], robust RHC for the model with polytopic parametric uncertainty and one state delay is studied. e local control is a memoryless feedback. e recursive feasibility is well guaranteed under the proper assumption of terminal weighting matrices. e stability of the system is proved by choosing the Lyapunov function and feedback gain properly. In [23], for the positive systems with polytopic parametric uncertainty and multiple state delays, a robust RHC scheme is proposed. A linear copositive Lyapunov function and a cone invariant set are used to prove the stability of the system. In [24], a robust RHC for the system with state and random input delays is studied. e robust RHC for the systems with one input delay and more than one state delay is first proposed in [25], but the implementation details are not given. In [26], the implementation details are rectified. If the local feedback is adopted the same as [25], the system matrices are required to be asymptotically stable.
For the robust RHC, many researchers have made effort to obtain a better control performance. In [25], the infinite horizon robust RHC is parameterized into a single feedback gain K. One free control move is added before K in [27]. As a result, the region of attraction is enlarged, and a better control performance is achieved. In [28], the approach with N ≥ 1 free control moves is adopted. First, the local feedback gain K is calculated offline, so the online computation burden is reduced significantly. At each time k, the predictive control input is defined as is the free perturbation being the online decision variable. is approach not only guarantees the closed-loop stability but also can get a bigger region of attraction compared with the approach of one free control move. Hence, a better performance and a wider applicability can be achieved. For the state feedback robust RHC, the successful applications of this approach have been witnessed, e.g., [29], N free control moves are applied to the systems with norm-bounded uncertainty. In [19], by incorporating more uncertain system information, an improved free control move approach is proposed. In [26], the free control move approach is extended to the systems with uncertainty and multiple input delays. By modifying the local feedback, this approach can be applied to the system with unstable matrices. However, this approach make the optimization problem lose the recursive feasibility. is motivates us to design a new RHC synthesis control scheme to restore the recursive feasibility. e contribution of this paper will be highlighted as follows: (1) e local control adopts augmented state feedback, where the augmented state contains past inputs and states, and the heavy computation burden incurred by augmented feedback is removed offline (2) e recursive feasibility and closed-loop stability are guaranteed, even when the free control moves are introduced before the augmented state, since the augmented state space equation is used (3) e unstable system matrices can be handled by adopting the augmented state space model, which is the least conservative treatment of state delays and input delays, whether the free control moves are introduced or not e organization of our work is summarized as follows. In Section 2, the problem statement is given. e offline robust controller synthesis for the multiple time delay systems is presented in Section 3. A general robust positively invariant set is proposed which serves as the terminal invariant set. In Section 4, the receding horizon control policy with free control moves is given. Simulation results demonstrate that receding horizon control with free control moves is effective in Section 5. Finally, a conclusion of the paper is stated.

Problem Statement
Consider a polytopic parametric uncertainty system with multiple time delays: where x ∈ R p is measurable state, u ∈ R q is the control input, μ ≥ 0, τ ≥ 0, and m and n are positive integers. e following assumptions are given.

Assumption 2.
e multiple time delay system (1) satisfies the following physical constraints: A local feedback control is designed: where F μ , μ � 0, 1, . . . , m and H τ , τ � 1, 2, . . . , n are the feedback gains to be designed.  (1) and (5), one obtains the following augmented model: where Γ � Remark 1. It is worth noting that a polytopic parametric uncertainty system with multiple time delay is all inclusive in the mathematical model.
(1) When all values of μ and τ are equal to zero, system (1) is translated to the system without time delay: (2) When the values of μ and τ are 0 or 1, system (1) is translated to the system with state delay or input delay. For example, (3) e mathematical model in this paper is a more general case for the dynamic system.

Offline Robust Controller Synthesis
Given matrices Q > 0 and R > 0, the following cost function is considered: We will design a local feedback (4) by minimizing J, for system (6) satisfying constraints (3) and (4). e statement is summarized into the following optimization problem: where Consider the following Lyapunov function for system (6): where e following inequality is imposed: Mathematical Problems in Engineering where λ is a scalar, then the robust stability of system (6) holds.

Remark 3.
e local feedback gain can also be optimized online, but the computation burden is heavy. In this paper, the local feedback gain is designed offline.

Robust Receding Horizon
e following receding horizon control policy is considered: where T � 0, 1, . . . , N − 1 { } and K is calculated offline. A sequence of control moves v s (k) are the online decision variables.
Based on (36), the prediction model is represented as where Λ � B 0 0

e Online Optimization.
At every time k, the following problem is solved to find the free control moves v s (k): Let J s (k): � ‖x s (k)‖ 2 Q + ‖u s (k)‖ 2 R , s ∈ T, which is represented as the stage cost; J ∞ (k): � ∞ s�N ‖x s (k)‖ 2 Q + ‖u s (k)‖ 2 R , which is represented as the terminal cost.

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For handling the stage cost easily, a sequence of nonnegative scalars λ s (k) are introduced [26,29], which satisfy J s (k) ≤ λ s (k), s ∈ T.
(39) e robust positively invariant set ε Q − 1 is the terminal constraint set and ξ s (k) ∈ ε Q − 1 . e quadratic function By summing (40), from s � N to s � ∞, one has erefore, the online optimization problem is stated as follows: e prediction of the augmented state ξ s (k) is used in the optimization problem. Since the polytopic parametric uncertainty exists in the system, the deterministic value of ξ s (k) cannot be given directly. However, a tightest set can be constructed which contains all possible ξ s (k). e following lemma will be proposed to calculate the tightest set. It is a generalization of "Lemma 1" in [30].
ere exist a sequence of scalars λ s (k) > 0, s ∈ T, such that (a) (43) is guaranteed by

Recursive Feasibility and Stability
Theorem 2. For system (6), at time k, if (59) has feasible solution, then (59) is solvable at time k + 1 and system (37) is exponentially stable.
Proof. Denote the solution of (59) at time k as follows: We will prove, if one takes the following solutions: (59) is solvable at time k + 1. us, one has en, (52)-(55) hold at time k + 1. In summary, (59) is feasible at time k + 1.
□ Remark 4. In [26], the predictive control input u s (k) � Kx τ+s (k) + v s (k) are implemented to the abovementioned system. e x τ+i (k) is uncertain, which is hard to construct u N (k + 1). erefore, the recursive feasibility is lost in theory.
Remark 5. In this paper, the control scheme is divided into two parts: offline robust controller synthesis and online robust receding horizon. Offline, the controller is designed to reduce the computation burden. Because the augmented state space model is used to handle the unstable system matrices, the computation burden is huge. Online, the free control moves are introduced into the online optimization problem to improve the control performance. is is the significance of the proposed control scheme.

Simulation Examples.
In this section, we give two examples to demonstrate the effectiveness of the proposed approach. Example 1 is a numerical example, and Example 2 is an application-oriented example.

Example 1.
e example is adapted from [31]. e system with polytopic uncertainty is where the uncertainty parameter ρ(k) ∈ [0.4, 1.2]. e input constraint ] � 2, u(− 3) � u(− 2) � 0, and u(− 1) � 0.2, and there is no constraint for the system state. Q � I and R � 1. Algorithm 1 in this paper is denoted by Algorithm 1, Algorithm of type-III in [31] without free control moves is denoted by IJC 2010. e comparisons between the Algorithm 1 and IJC 2010 are reported in this section. e online receding horizon N � 2. Figures 1 and 2 depict the state responses of x 1 and x 2 , respectively, for β � 0.37. From Figures 1 and 2, x 1 and x 2 fluctuate as time goes and tend to the zero as k increases; therefore, the proposed controller stabilizes the closed-loop systems with unstable matrices. Besides, the fluctuation range of IJC 2010 is stronger than Algorithm 1. On the contrary, Algorithm 1 with free control moves makes the states of the system smoother. Figure 3 depicts the evolution of c(k). It is shown that c(k) monotonically decrease with the increase of k. Compared with IJC 2010, a smaller upper of cost function and a more quick convergence rate are achieved for Algorithm 1. erefore, the control performance is improved by introducing free control moves for Algorithm 1.

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Hence, Algorithm 1 has a better control performance and stronger robustness.
e state responses of x 1 and x 2 , are depicted in Figures 5 and 6, respectively. Similar to Figures 1 and 2, all states converge to zero as time goes by, which demonstrates that the closed-loop system with unstable matrices is stable. e evolution of c(k) is shown in Figure 7. It is clear that c(k) is monotonically decreasing for both algorithms, and the cost function of Algorithm 1 sharply reduces.
us, Algorithm 1 with free control moves has a better control performance.
e control inputs are depicted in Figure 8. According to Figure 8, for Algorithm 1 and IJC 2010, the input constraints are guaranteed. Moreover, the initial value of the control input for Algorithm 1 is − 1.2, while IJC 2010 is − 0.59. Compared with IJC 2010, the maximum value of the control input for Algorithm 1 is larger. erefore, a wider input range is achieved for Algorithm 1.
Hence, Algorithm 1 with free control moves not only can stabilize the system with unstable system matrices but also improve the control performance.
To further validate the advantage of the proposed algorithm with free control moves, the other two cases are considered.
For β � 0.28, the control inputs are depicted in Figure 9, and the evolution of c(k) are shown in Figure 10. For β � 0.28, the control inputs and the evolution of c(k) are shown in Figures 11 and 12, respectively. Similar to Case 1 and Case 2, simulation results come to the same conclusions for Case 3 and Case 4. ese simulation results demonstrate the advantage of the proposed algorithm.
In summary, the following conclusions are obtained from the figures.
Firstly, controller (5) can stabilize the system with unstable system matrices and more than one input delay, which is consistent with " eorem 2". Secondly, Algorithm 1 has a more aggressive input and a better control performance. erefore, it is very necessary to apply the receding horizon policy with free control moves to the systems with unstable system matrices and more than one input delay. erefore, it is very necessary to apply the receding horizon policy with free control moves to the systems with unstable system matrices and more than one input delay.
Example 2. An application-oriented example is adopted to verify the proposed result. e example is adapted from [23]. e urban water system is established based on the underground water network. e details can refer to [23]. e system parameters are as follows: Initial state x(k) � (2 1) T , u(− 1) � 0.2, input constraint ] � 1.8, state constraint ϕ � 1.5. Q � 0.5, and R � 0.5, and the online receding horizon N � 3.
In [23], the MPC control scheme is without the free control moves in the controller designing. e algorithm of [23] is denoted by IJCAS 2018. e comparisons between Algorithm 1 and IJCAS 2018 are given to show the effectiveness of the proposed approach.
For Algorithm 1 and IJCAS 2018, the control input u is shown in Figure 13, and the evolution of c is depicted in Figure 14.
e analysis of simulation results is the same as Example 1. e simulation results also reflect that Algorithm 1 has a better control performance.

Conclusions
In this paper, the free control move approach is generalized to the systems with unstable system matrices and more than one input delay. By designing a new local feedback controller based on the augmented state, the recursive feasibility and closed-loop stability are guaranteed, where the local feedback gain is designed offline and the free control moves are minimized by online optimization problem. In this paper, the time delay is a constant and disturbance is not considered. erefore, the free control move approach will be further applied to the system with time-varying time delays and disturbance in the future works.

Data Availability
e data used to support the findings of this paper have been deposited in the Taylor and Francis repository (DOI: 10. 1080/00207179/2010.498058).

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper. Mathematical Problems in Engineering 11