Hamiltonian‐driven adaptive dynamic programming for mixedH₂/H_∞performance using sum‐of‐squares

Abstract

In this article, the mixed H2/H∞ performance optimization is first formulated as a nonzero‐sum game, of which the sufficient condition guaranteeing the existence of the Nash equilibrium is derived using the Hamilton–Jacobi (HJ) theory. Then, Hamiltonian‐driven inequalities are presented to evaluate the H2 and H∞ performances. Using this Hamiltonian‐inequality driven approach, the coupled HJ equations arising from finding the Nash equilibrium are relaxed to the HJ inequality constraints. A novel mixed policy iteration (PI) algorithm is developed that uses sum‐of‐squares (SOS) program in policy evaluation step, and consists of an H2 performance improvement step and an H∞ performance guarantee step. This constrained‐driven approach allows us to present a PI algorithm that takes into account both robustness and performance objectives. Finally, a numerical simulation is carried out to highlight the efficacy of the proposed framework.

Policy iteration forH∞control of polynomial time‐varying systems

“…In [25], it has been shown that one can efficiently relax the Bellman equation to an SOS optimization problem. Many of other problems in control theory, such as optimal control of discrete‐time systems [26], H ∞ control of polynomial [27], polynomial fuzzy [28], and uncertain polynomial systems [29], mixed H 2 / H ∞ control design [30], H ∞ control of polynomial systems with adjustable parameters ( H ∞ codesign) [31], multi‐objective control design [32], and constrained states control [33], have been solved with the idea of SOS‐based ADP.…”

Observer‐based event triggered ADP approach for input‐constrained nonlinear systems with disturbances

“…Adaptive dynamic programming (ADP), as a powerful tool to approximate the optimal solution to an HJB equation based on function approximators like neural networks (NNs), was proposed and received lots of attention from researchers during the past decades. [3][4][5] The critic-actor network is the classical implementation architecture for the ADP technique, where the critic network is utilized to approximate the optimal performance index and the actor network is employed to obtain the near-optimal control policy. Hence, ADP is essentially a learning-based artificial intelligence algorithm and provide an effective way to solve the optimal control problem for nonlinear systems.…”

“…Therefore, investigating the optimal control for complex nonlinear systems is a hot research topic in the control field recently. Adaptive dynamic programming (ADP), as a powerful tool to approximate the optimal solution to an HJB equation based on function approximators like neural networks (NNs), was proposed and received lots of attention from researchers during the past decades 3‐5 . The critic‐actor network is the classical implementation architecture for the ADP technique, where the critic network is utilized to approximate the optimal performance index and the actor network is employed to obtain the near‐optimal control policy.…”

Learning‐based collision avoidance and robust H∞ optimal formation control for uncertain quadrotor UAV systems

“…28 In general, H ∞ controllers require the solution of Hamilton-Jacobi-Isaacs (HJI) equations for nonlinear systems. 29 However, the HJI equation derived from a nonlinear system is a nonlinear partial differential equation. When dealing with large-scale systems, the issue of "dimension disaster" arises and solving becomes challenging.…”

“…Therefore, the saddle point solution of this game theory represents a solution for $$$ {H}_{\infty } $$$ control problem 28 . In general, $$$ {H}_{\infty } $$$ controllers require the solution of Hamilton‐Jacobi‐Isaacs (HJI) equations for nonlinear systems 29 . However, the HJI equation derived from a nonlinear system is a nonlinear partial differential equation.…”