Adaptive dynamic programming algorithm for uncertain nonlinear switched systems

This paper studies an approximate dynamic programming (ADP) strategy of a group of nonlinear switched systems, where the external disturbances are considered. The neural network (NN) technique is regarded to estimate the unknown part of actor as well as critic to deal with the corresponding nominal system. The training technique is simultaneously carried out based on the solution of minimizing the square error Hamilton function. The closed system’s tracking error is analyzed to converge to an attraction region of origin point with the uniformly ultimately bounded (UUB) description. The simulation results are implemented to determine the effectiveness of the ADP based controller.


INTRODUCTION
It is worth noting that many systems in industry can be described by switched system such as DC-DC converter [1]- [3], H-bridge inverter [4], multilevel inverter [5], photovoltaic inverter [6]. Although many different approaches for switched systems have been proposed, e.g., switching-delay tolerant control [7], classical nonlinear control [8]- [12], the optimization approaches with the advantage of mentioning the input/state constraint has not been mentioned much. The approaches of fuzzy and neural network as well as ANN, particle swarm optimization (PSO) technique were investigated in several different systems such as photovoltaic inverter, transmission line. [13]- [17].
Adaptive dynamic programming has been considered in many situations, such as nonlinear continuous time systems [18], actuator saturation [19], linear systems [20]- [22], output constraint [23]. In the case of nonlinear systems, the algorithm should be implemented based on Neural Networks (NNs). However, Kronecker product was employed in linear systems. Furthermore, the data driven technique should to be mentioned to compute the actor/critic precisely. It should be noted that the robotic systems has been controlled by ADP algorithm [24]- [25].
Our work proposed the solution of adaptive dynamic programming in nonlinear perturbed switching systems based on the neural networks. The consideration of the Halminton function enables us obtaining the learning technique of these neural networks. The UUB stability of closed system is analyzed and simulation results illustrate the high effectiveness of given controller.
where ξ (t) ∈ Ω x ∈ R n denotes the state variables and u (t) ∈ Ω u ∈ R m describes the control variables. The function β : [ 0, +∞) → Ω = {1, 2, ..., l} is a information of switching processing, which is known as a function with many continuous piecewise depending on time, and l is the subsystems number. f i (ξ) are uncertain smooth vector functions with f i (0) = 0. g i (ξ) are mentioned as smooth vector functions with the property G min g i (ξ) G max . The switching index β (t) is unknown. Assumption 1: ∆ (ξ, t) is bounded by a certain function (ξ) as ∆ (ξ, t) (ξ) Consider the cost function connected with the uncertain switched system (1): The main purpose is to achieve the state feedback control design and give the upper bound term to guarantee the closed systems under this controller is robustly stable. Additionally, the performance index (2) Definition: The term K(u) is given by the appropriate performance index. As a result, the control input u * = arg min u∈Ωu K (ξ, u) is mentioned as the optimal appropriate performance index method.

3.
CONTROL DESIGN The obtained nominal system after eliminating the disturbance in switched system (3) is described by: The performance index of system (3) is modified as (4) We prove that Q 1 (ξ, u) with γ R is the one of appropriate performance indexes of dynamical system (1). Define: based on nominal system and cost function (4), it leads to Halminton function as (6) H by using optimality principle, the optimal control input can be obtained as (7).
Obtain the description as (14).
It follows that e N N converges uniformly to zero as N → ∞. For each number N , e N N is bounded on a region as e N N e max . Under the structure of ADP-based controller, a critic NN is computed as (15).
It is able to achieve that: The training law is handled based on a steepest descent method: with E = 1 2 e T HJB e HJB . Remark 1: The weight w is trained to minimize the network error part G = 1 2 e T HJB e HJB . This result is obtained from (18).
Adaptive dynamic programming algorithm for uncertain nonlinear switched systems (Dao Phuong Nam) Ì

ISSN: 2088-8694
Theorem 2: Consider the feedback controller in (15) and the critic weight is updated by (18), the weight estimate errorw = w −ŵ and the closed system's state vector x(t) are uniformly ultimately bounded (UUB).
Proof: Let's choose the Lyapunov function: Using the Assumption 3: f i (ξ) + g i (ξ) u * ρ max and the definition: we imply that:V It leads to the estimation:V 1 (t) −π 1 . For the term V 2 (t) , from (20) we have (21).
. Remark 2: The coefficients ϑ 1 ; ϑ 2 can be chosen by renovating the NN of the optimal performance index. Moreover, for arbitrary switching index, after V (0) min(π1;π2) the variable ξ and w tend to the accurate domains. The ADP controllerû is proposed in (15), which tends to the neighborhood of u * .
Proof: The deviation of control input is estimated as (25).
Thus the proof is completed.

SIMULATION RESULTS
In this section, we consider the simulations to validate the performance of the established control scheme: Let N = 2 and the subsystems of the switched system are (26) and (27).
The initial state vectors can be chosen as (28).
The simulation results shown in Figure 1 and Figure 2 validate the effectiveness of proposed algorithm.

CONCLUSION
This paper has investigated the ADP problem of switched nonlinear systems under the external disturbance. We consider previously for nominal system by eliminating the disturbance, then using classical nonlinear control technique. The neural networks have been designed to estimate the actor and critic NN of iteration. It is possible to develop the learning algorithm with simultaneous tuning. Finally, UUB description of the closed system is guaranteed under this work.

ACKNOWLEDGEMENT
This research was supported by research foundation funded by Thai Nguyen University of Technology.