Inverse optimal control for asymptotic trajectory tracking of discrete‐time stochastic nonlinear systems in block controllable form

Abstract

This paper concerns an inverse optimal control-based trajectory tracking of discrete-time stochastic nonlinear systems. It is assumed that the nonlinear system can be transformed to the so called nonlinear block controllable form.Additionally, the synthesized control law minimizes a cost functional, which is posteriori determined. In contrast to the optimal control technique, this scheme avoids to solve the stochastic Hamilton-Jacobi-Bellman equation, which is not an easy task. Based on a discrete-time stochastic control Lyapunov function, the proposed optimal controller is analyzed. The proposed approach is applied successfully to the two degrees-of-freedom helicopter with uncertainties in real time. KEYWORDSinverse optimal control, stochastic nonlinear systems, trajectory tracking INTRODUCTIONOptimal stochastic control is related to determining a control law for a given stochastic system, such that the a priori cost functional is minimized; the major drawback is solving the associated stochastic Hamilton-Jacobi-Bellman equation, 1 which is difficult or even impossible to solve. However, in the work of Zhang et al, 2 the optimal control of discrete-time nonlinear stochastic systems is solved based on discrete martingale theory. To overcome the solution of the stochastic Hamilton-Jacobi-Bellman equation, Crandall and Lions 3 introduced a weak solution named as viscosity solution, whereas Krstić et al 4 used the inverse optimal control approach in continuous-time nonlinear systems with uncertainties for stabilization, regulation, and tracking 3,4 ; consider the continuous-time case. On the other hand, the neural network-based approximation methods have been used to solve the optimal control problem via adaptive dynamic programming, such as in the works of Luy, 5,6 where the cooperative control, for stabilization and tracking respectively, of multiple multiple-input-multiple-output nonlinear systems in strict feedback with no knowledge of internal system dynamics and affected by external disturbances is addressed.This paper proposes an inverse optimal control law for asymptotic stability in probability, along a desired trajectory, of discrete-time stochastic nonlinear systems in nonlinear block controllable (NBC) form. Using this approach, a feedback control law, based on the a priori knowledge of a discrete-time stochastic control Lyapunov function (DSCLF), is synthesized first, and then it is established that this control law optimizes a cost functional.The inverse optimal control approach of deterministic systems is presented for the continuous 7-11 and discrete-time 12-14 case, respectively. Although the inverse optimal control for continuous-time stochastic nonlinear systems has been

Inverse optimal control for positive impulsive systems toward personalized therapeutic regimens

“…While the inverse optimal control approach has been explored for the discrete‐time version, 36‐38 an impulsive approach that provides a sudden change at $$$ j $$$th instants of the $$$ k $$$th time steps of the discrete‐time system is needed. Impulsive systems can be pivotal for scheduling therapies in different diseases such as HIV infection, 29,33,39‐41 influenza infection, 28,42 diabetes 43 as well as vaccination 15 .…”

“…Impulsive control is particularly important when a system cannot be controlled by a continuous input, in other words if the input can only be applied at some jth instants of the kth time steps of the discrete-time system, this is also known in the literature as Markov jump systems. [33][34][35] While the inverse optimal control approach has been explored for the discrete-time version, [36][37][38] an impulsive approach that provides a sudden change at jth instants of the kth time steps of the discrete-time system is needed. Impulsive systems can be pivotal for scheduling therapies in different diseases such as HIV infection, 29,33,[39][40][41] influenza infection, 28,42 diabetes 43 as well as vaccination.…”

Observer-Based Finite-Time Inverse Optimal Output Regulation for Uncertain Nonlinear Systems

“…e applications of the inverse optimal control method on Quanser 2 DOF helicopters and unmanned aerial helicopters can be found in references [36,37]. It can be found that the methods above mainly address the stabilization problem of the nonlinear systems.…”