Nonlinear H ∞ -Control, Hamiltonian Systems and Hamilton-Jacobi Equations

Aliyu, M. D. S.

doi:10.1201/b10734

Cited by 62 publications

(43 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…2) The tracking error dynamics (4) with d = 0 is locally asymptotically stable. The main difference between Definition 2 and the standard definition of H ∞ tracking control problem (see [6,Definition 5.2.1]) is that a more general disturbance attenuation condition is defined here. Since the whole control input and the tracking error are penalized in the disturbance attenuation condition (7), the problem formulated in Definition 1 gives an optimal solution, in contrast to the standard definition that results in a suboptimal solution, as stated in [6].…”

Section: H ∞ Tracking Problemmentioning

confidence: 99%

See 1 more Smart Citation

<inline-formula> <tex-math notation="LaTeX">$ {H}_{ {\infty }}$ </tex-math></inline-formula> Tracking Control of Completely Unknown Continuous-Time Systems via Off-Policy Reinforcement Learning

Modares¹,

Lewis²,

Jiang

2015

IEEE Trans. Neural Netw. Learning Syst.

426

View full text Add to dashboard Cite

This paper deals with the design of an H ∞ tracking controller for nonlinear continuous-time systems with completely unknown dynamics. A general bounded L2 -gain tracking problem with a discounted performance function is introduced for the H ∞ tracking. A tracking Hamilton-Jacobi-Isaac (HJI) equation is then developed that gives a Nash equilibrium solution to the associated min-max optimization problem. A rigorous analysis of bounded L2 -gain and stability of the control solution obtained by solving the tracking HJI equation is provided. An upper-bound is found for the discount factor to assure local asymptotic stability of the tracking error dynamics. An off-policy reinforcement learning algorithm is used to learn the solution to the tracking HJI equation online without requiring any knowledge of the system dynamics. Convergence of the proposed algorithm to the solution to the tracking HJI equation is shown. Simulation examples are provided to verify the effectiveness of the proposed method.

show abstract

Section: H ∞ Tracking Problemmentioning

confidence: 99%

“…Remark 1: The disturbance attenuation condition (6) implies that the effect of the disturbance input to the desired performance output is attenuated by a degree at least equal to γ. The minimum value of γ for which the disturbance attenuation condition (6) is satisfied gives the so-called optimal robust control solution.…”

Section: H ∞ Tracking Problemmentioning

confidence: 99%

<inline-formula> <tex-math notation="LaTeX">$ {H}_{ {\infty }}$ </tex-math></inline-formula> Tracking Control of Completely Unknown Continuous-Time Systems via Off-Policy Reinforcement Learning

Modares¹,

Lewis²,

Jiang

2015

IEEE Trans. Neural Netw. Learning Syst.

426

View full text Add to dashboard Cite

show abstract

“…Theorem 1: Let the cooperative dynamics of the multi-NMR systems be defined in (18), which gives the ith dynamics in (22), for all i, i = 1, ..., m. Let the cooperative value function of NMR i be chosen as (28) and the coupled HJI equations as (32). Let the NN weight-tuning law be defined in (46), the control law in (42) and the worst case disturbance law in (43).…”

Section: Stability and Convergence Analysismentioning

confidence: 99%

Intelligent Distributed Cooperative Control for Multiple Nonholonomic Mobile Robots Subject to Unknown Dynamics and External Disturbances

Luy¹

2018

JST

View full text Add to dashboard Cite

This paper addresses an optimal cooperative tracking control method with disturbance rejection in the presence of no knowledge of internal dynamics for multi-nonholonomic mobile robot (NMR) systems. Unlike most existing methods, our method integrates kinematic and dynamic controllers into one using adaptive dynamic programming techniques based on the concept of differential game theory and neural networks, and is therefore entirely optimal. First, with the aim to reduce the computational complexity, the number of neural networks for each agent in the method is chosen to be less than one-third. Second, novel weight-tuning laws of the neural networks and online algorithms are proposed to approximate solutions of the HamiltonJacobi-Isaacs equations. By using Lyapunov theory, value functions and both cooperative control and disturbance laws are proved convergence to the approximately optimal values while the cooperative tracking errors and function approximation errors are uniformly ultimately bounded. Finally, the effectiveness of the proposed method is demonstrated by the results of the compared simulation and experiment on a multi-NMR system equipped with omnidirectional vision.

show abstract

“…The optimal static output-feedback control formulation proceeds along very similar steps as the optimal state-feedback control formulation, as can be seen for example in [25,Sect. 6.5], with the additional condition that the optimal cost-to-go function must satisfy a structural constraint.…”

Section: Adaptive Optimization Algorithmmentioning

confidence: 99%