This 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 to make the system stable for each mode of the nominal system. Then, 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. The 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.
Switched time-delay systems are an important kind of hybrid systems, which have attracted extensive attention in recent years. There are some results for switched time-delay 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. Therefore, its study has received considerable attention [
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. Therefore, more and more scholars study distributed optimal control. The method of combining distributed optimal control with the control Lyapunov function-based bounded control also has made further development. Heidarinejad et al. [
In this paper, a distributed optimal control method is applied to switched nonlinear systems with the state time delay. The main ideas are as follows. A bounded controller to make system stable for each mode of the nominal system is designed. Then, 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.
Consider a class of switched nonlinear systems with the state time delay as follows:
In this paper,
The 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 (
The system structure.
The following assumption is satisfied at mode
For some
Assumption
The 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.
For each mode
If the nominal system of system (
Then, system (
Based on controller (
The maximum estimation of
Next, for all initial conditions in
For system (
Because system (
If
Because
For each mode
Firstly, we design controller 2 based on the bounded controller and the received measurement
The optimal solution is denoted by
Once
The optimal solution is expressed as
We emphasize two points: (i) (
In this section, the stability of distributed optimal control scheme is demonstrated.
Considering that switched nonlinear system ( When there is no switch, let if initial state starts from the set When there is switch, the following constraints need to be added to distributed optimal control of (
The proof is divided into two parts. When there is no switch, if From ( The derivative of It follows from ( Using ( For all According to the smoothness of functions Because of the continuity of By ( For Integrating this bound on By recursively applying ( When there is switch, there are two cases.
In case of
In case of
In conclusion, Theorem
The manipulation inputs of the distributed optimal control scheme are defined:
The executive strategy of optimal control in this paper is given as follows: Step 1: controller 2 receives the measurement Step 2: the optimal trajectory of Step 3: once the optimal trajectory of Step 4: controller 1 sends the optimal trajectory of Step 5: go to the next moment.
We consider the following switched system to verify the effectiveness of the proposed distributed optimal control scheme:
We choose the Lyapunov function
Set the system switching mode 2 from mode 1 at time
The trajectories of
The trajectories of
The trajectories of
The trajectories of
The 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. The comparison results are shown in Table
Total performance cost along the closed-loop system trajectories.
Number | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|
Initial value | [−1.1, 0.9] | [1.2, −0.9] | [−0.7, 1.0] | [0.8, 1.05] | [−0.97, −1.3] | [0.86, −1] | [−0.5, 1.4] | [1.5, −1.7] | [−1.4, 0.97] | [1.6, −1.8] |
|
||||||||||
Distributed optimal control | 1778 | 1693 | 1860 | 1792 | 2120 | 1673 | 1732 | 3170 | 1973 | 2970 |
|
||||||||||
Centralized optimal control | 1687 | 1801 | 1778 | 1933 | 1946 | 1854 | 1973 | 3457 | 1669 | 2532 |
It can be seen from Table
This 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. Then, 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.
No data were used to support this study.
The authors declare that they have no conflicts of interest.
This research was supported by the Natural Science Foundation of China (Grant nos. 61374004, 61773237, and 61473170) and Key Research and Development Programs of Shandong Province (Grant no. 2017GSF18116).