Distributed Optimal Control for a Class of Switched Nonlinear Systems with the State Time Delay

,is paper is focused on a kind of distributed optimal control design for a class of switched nonlinear systems with the state time delay which have a prescribed switching sequence. Firstly, we design a bounded controller tomake the system stable for eachmode of the nominal system.,en, a distributed optimal controller which can satisfy input constraint is designed based on the bounded stabilization controller. A sufficient condition to guarantee ultimate boundedness of the system is given based on appropriate assumption.,e significance of this paper is that distributed optimal control method is applied to switched nonlinear systems with the state time delay. Finally, a simulation example is given to verify the effectiveness of the proposed method.


Introduction
Switched time-delay systems are an important kind of hybrid systems, which have attracted extensive attention in recent years. ere are some results for switched time-delay systems [1][2][3][4]. However, the input of switched time-delay systems is often constrained by objective conditions, and some switched systems need to operate according to a prescribed switching sequence in many practical control systems. So, there are still many challenging control problems need to be solved for this class of nonlinear systems.
Regression optimal control is an optimal control method that can deal with system constraints. It obtains the current control action by solving a finite time open loop optimal control problem at every sampling moment. It has well dynamic control effect and is helpful to improve the stability of the system. erefore, its study has received considerable attention [5][6][7][8]. Moreover, the method of combining regression optimal control with the control Lyapunov function-based bounded control has made further development in the fields of single-objective control, multiobjective control, and so on. For example, the objective function in [9] adopts the nonquadratic cost function and studies a regression optimal control based on the Lyapunov economic model, which can directly solve economic problems. In [10], a fast feedback controller that stabilizes the fast dynamics and a regression optimal controller that stabilizes the slow dynamics and enforces desired performance objectives in the slow subsystem are designed for a kind of nonlinear singularly perturbed systems. He et al. [11] study an alternative utopia-tracking multiobjective economic regression optimal control scheme of constrained nonlinear systems with guaranteed asymptotic stability and convergence of average performance.
In most existing results, centralized optimal control methods are adopted. When this kind of control method is applied to the system, the computational complexity reduces the performance of the system with the increase of variables and the expansion of model size. Distributed optimal control can reduce computational burden and fault tolerance of the system by using communication and cooperation among multiple controllers.
erefore, more and more scholars study distributed optimal control. e method of combining distributed optimal control with the control Lyapunov function-based bounded control also has made further development. Heidarinejad et al. [12] study a distributed optimal control of the switched nonlinear system. It redesigns the local control system (LCS) and the new control system (NCS) by using an optimal control method based on Lyapunov function to coordinate their respective operations. e distributed structure adopted in [12] enables LCS to stabilize the closed-loop system, and NCS can communicate and cooperate with LCS, which can further improve the closed-loop performance. A distributed optimal control scheme with such distributed structure is considered for switched nonlinear time-delay systems with delayed measurements and communication disruption in [13]. Liu et al. [14] propose a distributed optimal control strategy for nonlinear systems with asynchronous and time-delay measurements. Heidarinejad et al. [15] study a distributed optimal control scheme of nonlinear systems subject to communication disruptions-communication channel noise and data losses-between distributed controllers. It is noted that the distributed optimal control method is seldom applied to nonlinear time-delay systems. Hence, it is necessary to study the problem of distributed optimal control for switched nonlinear systems with the state time delay.
In this paper, a distributed optimal control method is applied to switched nonlinear systems with the state time delay. e main ideas are as follows. A bounded controller to make system stable for each mode of the nominal system is designed. en, a kind of distributed optimal control which can satisfy input constraint is designed based on the bounded stabilization controller. A sufficient condition to guarantee ultimate boundedness of the switched system is given. Finally, a simulation example is given to verify the effectiveness of the proposed distributed optimal control strategy.

Problem Statement and Preliminaries
Consider a class of switched nonlinear systems with the state time delay as follows: being a continuous function and ‖φ σ(t) ‖ � max −τ σ(t) ≤ξ≤0 ‖φ σ(t) (ξ)‖, ‖·‖ denotes the Euclidean norm. u j σ(t) (t) is restricted on a nonempty set for all i, which means that a finite number of switches can only be switched in a finite time interval.
In this paper, t i in r and t i out r denote the time of the rth switch in and out for the ith mode respectively, i.e., (2) respectively. e objective of this paper is to propose a distributed optimal control scheme for a class of switched nonlinear systems with the state time delay. System (1) adopts the distributed structure (see Figure 1), where u 1 and u 2 are designed by controller 1 and controller 2, respectively. e following assumption is satisfied at mode i (i � 1, 2, . . . , s).
where Ψ i (·) is a continuous positive definite nondecreasing function.
Remark 1. Assumption 1 illustrates that the decoupling between the delay state and the current state can be realized by the derivative of the Lyapunov function. is assumption is also used to realize the separation of time-delay state and nontime-delay state in [16].

Remark 2.
e distributed optimal control can set up multiple part controllers to make the system stable and improve its performance. For simplicity, only two parts are considered in this paper. Distributed optimal controller designed in this paper is composed of controller 1 and controller 2. Controller 1 can stabilize the closed-loop system. Controller 2 can improve the performance of the closed-loop system by communicating with controller 1. If the distributed optimal controller is composed by multiple controllers, then controller 1 makes the closed-loop system stable, remaining controllers improve the performance of the closed-loop system through cooperation and communication with controller 1.

Design of Bounded Controller.
For each mode i (i � 1, 2, . . . , s), we can design bounded controllers acting on the following proposition.
satisfies Assumption 1, we can choose bounded controllers en, system (4) is asymptotically stable under the control of Proof. Based on controller (6), from [17,18], the following stable set is given: e maximum estimation of Φ(u max 1 i ) can be described by where ρ i > 0 is the largest number for Next, for all initial conditions in Ω ρ , we can prove that the controllers are bounded. From (6), we have For system (4), Razumikhin's original idea is that if Mathematical Problems in Engineering Because system (4) satisfies Assumption 1, we can obtain Because 1, 2, . . . , s), a distributed optimal control scheme is designed.
Firstly, we design controller 2 based on the bounded controller and the received measurement x(t k ) at time t k . e input u 2 i can be calculated by the following optimal problem: min where p(Δ) is a family of piecewise constants function with a sampling time Δ, NΔ is the predicted range, Q i , R 1 i , and R 2 i are the right of positive definite matrix, x(t) is the trajectory of system (4) with u 1 i � h i (x(t)), and u 2 is the input trajectory computed by (15)-(21). e optimal solution is denoted by u * 2 i (t | t k ) (t ∈ [t k , t k + NΔ)).
Once u * 2 i (t | t k ) is calculated, it is sent to controller 1 and the corresponding control actuator. en, we design controller 1 based on the measurement x(t k ) and u * 2 i (t | t k ). e input u 1 i is calculated by the following optimal problem: s.t.
x t k � x t k , e optimal solution is expressed as u * 1 i (t | t k ) (t ∈ [t k , t k + NΔ)).

Stability Analysis.
In this section, the stability of distributed optimal control scheme is demonstrated.

Theorem 1. Considering that switched nonlinear system (1) satisfies Assumption 1, and under distributed optimal control of (15)-(21) and (23)-(28), (i) When there is no switch, let
if initial state starts from the set Ω ρ i , and ρ m i < ρ i , When there is switch, the following constraints need to be added to distributed optimal control of (15)-(21) and (23)-(28): where V j (x(t j in w )) denotes the value of Lyapunov functions of mode j for wth time and V j (x(t j in w−1 )) is the value of

Lyapunov functions of mode j for (w − 1)th time. If (29) is satisfied, then x(t) is ultimately bounded in
Proof. e proof is divided into two parts.
(i) When there is no switch, if x(t k ) ∈ Ω ρ i , from (21) and (28), we have From (14), it can be deduced that It follows from (31) and (33) that is acontinuous differentiable and f i (·), g 1 i (·), g 2 i (·), q i (·), and k i (·) are local Lipschitz. So, there are positive numbers L x which make the following equations true for ξ 1 , ξ 2 ∈ Ω ρ i : Mathematical Problems in Engineering 5 Using (35)-(39), we can rewrite (34) as follows: For all x(t k ) ∈ Ω ρ i /Ω ρ s i , it follows from (32) that According to the smoothness of functions f i (·), g 1 i (·), g 2 i (·), and q i (·), there exists positive constants M j i (j � 1, 2, 3, 4) such that Because of the continuity of x(t), for t ∈ [t k , t k+1 ), there are By (40) and all initial states For Integrating this bound on t ∈ [t k , t k+1 ), we have By recursively applying (46), we can obtain that if the initial state starts from Ω i /Ω ρ s i , the state converges to Ω ρ s i . If the state converges to Ω ρ s i ⊂ Ω ρ m i (or starts at Ω ρ s i ), the state remains in Ω ρ m i , determined by the definition of ρ m i . So, x(t) is ultimately bounded in Ω ρ m i . (ii) When there is switch, there are two cases.
In case of t i in r ≤ t < t i out r and t j in w � t i out r < ∞, we consider that the system has an infinite number of switches. At this point, ith mode is activated. If V i (x) > ρ m i , then the constraints of (21) and (28) guarantee that V i (x(t i out r )) < V i (x(t i in r )), and the constraints of (30) ensure that and t j in w � t i out r � ∞, we consider that the system has an finite number of switches. Similarly, V i (x(t i in r )) < V i (x(t i in r−1 )) < · · · < ρ m i . When switching to the ith mode, there is x(t i in r ) ∈ ρ m i . From this point on, distributed optimal control strategy is not subject to any switching constraints. So, x(t) is ultimately bounded stable.
In conclusion, eorem 1 is proved. e manipulation inputs of the distributed optimal control scheme are defined: (47) e executive strategy of optimal control in this paper is given as follows: Step 1: controller 2 receives the measurement x(t k ) at time t k , and then controller 2 calculates the optimal trajectory of u 2 i .
Step 2: the optimal trajectory of u 2 i is transmitted to the actuator and controller 1.
Step 3: once the optimal trajectory of u 2 i is received by controller 1, the optimal trajectory of u 1 i is calculated.
Step 4: controller 1 sends the optimal trajectory of u 1 i to the actuator.
Step 5: go to the next moment.

Simulation Example
We consider the following switched system to verify the effectiveness of the proposed distributed optimal control scheme: where f 1 (x) � (3x 3 2 (t), 0) T and f 2 (x) � (x 3 2 , 0) T ; g 1 1 (x) � (0, 1) T and g 1 2 (x) � (0, 1) T ; g 2 1 (x) � (1, 0) T and g 2 2 (x) � (1, 0) T ; q 1 ((x(t − 1))) � (2x 2 2 (t − 1), 2x 2 1 (t − 1)) T and q 2 ((x (t − 1))) � (x 2 2 (t − 1), x 2 1 (t − 1)) T ; k 1 (x(t)) � (0.2, 0) T and k 2 (x(t)) � (0.23, 0) T ; and ω 1 (t) � (sin(x 2 (t)), 0) T and ω 2 (t) � (sin(x 2 (t)), 0) T . Our goal is to adjust the initial state x 0 to the stable state x T s � [0, 0] for the system. e control constraints are We choose the Lyapunov function 2 2 , and a 2 � 0.79 to satisfy Assumption 1. We use Proposition 1 to construct the bounded controller u 1 i � h i (x) and u 2 i � 0. We apply the distributed optimal control and the centralized optimal control, respectively. Both control methods use the same parameters. e prediction horizon is N � 6, the sampling interval is Δ � 0.01, and the weighted matrices are chosen as Q i � diag [10,10], R 1 i � 0.50, and R 2 i � 0.60(i � 1, 2). Select the initial state as x T 0 � [−2.6, 1.9]. Set the system switching mode 2 from mode 1 at time t � 3.50 s (the system state enters the stable region of mode 2). e simulation results are shown in  e solid lines and the dotted lines represent the state trajectories under distributed optimal control and centralized optimal control, respectively. Simulation results indicate that optimal control via distributed optimal control method can ensure that the time-delay system state is ultimately bounded and stable. It also shows that both control designs give similar closed-loop performance and drive the state and the state close to the stable state in about 7.5 s and 8.5 s, respectively. To further illustrate that the two control designs have similar closed-loop performance, we illustrate this point from the perspective of the performance index. A series of simulation comparisons are made between the distributed optimal control and the centralized optimal control with the same parameters by using ten sets of different initial values. We calculated the total performance cost along the closed-loop system trajectories in 10s at different initial values. e comparison results are shown in Table 1.
It can be seen from Table 1 that the cost of distributed optimal control method is lower than that of the centralized optimal control method in 6 out of 10 simulations.
is shows that the closed-loop performance of the distributed optimal control method is comparable to that of the centralized optimal control method. Among the 10 simulation results, we also compared the computation    Mathematical Problems in Engineering time of the distributed optimal control method and centralized optimal control method. e average time to calculate the centralized optimal control is 2.64 s. e average time to calculate controller 1 and controller 2 is 0.9 s and 1.05 s, respectively. Obviously, the computation time of the distributed optimal control method is less than that of centralized optimal control. is is because centralized optimization control must optimize two inputs simultaneously in an optimization problem, while distributed optimization control is to solve two optimization problems which only have one decision variable.

Conclusions
is paper solves the problem of distributed optimal control for switched nonlinear systems with the state time delay. For each mode, a bounded controller is designed to stabilize the system. en, a distributed optimal control scheme which can satisfy the input constraint is designed based on the bounded stabilization controller. Finally, it is proved that the system state is ultimately bounded stable.

Data Availability
No data were used to support this study.

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