A multilayer neural-network (NN) controller is designed to deliver a desired tracking performance for the control of a class of unknown nonlinear systems in discrete time where the system nonlinearities do not satisfy a matching condition. Using the Lyapunov approach, the uniform ultimate boundedness of the tracking error and the NN weight estimates are shown by using a novel weight updates. Further, a rigorous procedure is provided from this analysis to select the NN controller parameters. The resulting structure consists of several NN function approximation inner loops and an outer proportional derivative tracking loop. Simulation results are then carried out to justify the theoretical conclusions. The net result is the design and development of an NN controller for strict-feedback class of nonlinear discrete-time systems.
MIMO optimal control of unknown nonaffine nonlinear discrete-time systems is a challenging problem owing to the presence of control inputs inside the unknown nonlinearity. In this paper, the nonaffine nonlinear discrete-time system is transformed to an affine-like equivalent nonlinear discrete-time system in the inputoutput form. Next, a forward-in-time Hamilton-Jacobi-Bellman equation-based optimal approach, without using value and policy iterations, is developed to control the affine-like nonlinear discrete-time system by using both NN as an online approximator and output measurements alone. To overcome the need to know the control gain matrix in the optimal controller, a new online discrete-time NN identifier is introduced. The robustness of the overall closed-loop system is shown via singular perturbation analysis by using an additional auxiliary term to mitigate the higher-order terms. Lyapunov stability of the overall system, which includes the online identifier and robust control term, demonstrates that the closed-loop signals are bounded and the approximate control input approaches the optimal control signal with a bounded error. The proposed optimal control approach is applied to a cycle-by-cycle discrete-time representation of an experimentally validated homogeneous charge compression ignition fuel-flexible engine whose dynamics are modeled as uncertain nonlinear, nonaffine, and MIMO discrete-time system. Simulation results are included to demonstrate the efficacy of the approach in presence of actuator disturbances.ROBUST OPTIMAL CONTROL WITH APPLICATION TO HCCI ENGINES 593 value and policy iterations. Whereas [1] and [2] present offline-based schemes, others [3] address optimal control in an online manner for affine nonlinear discrete-time systems.In [1] and [3], the input gain matrix ‡ (IGM) of the affine system is considered known whereas the internal system dynamics are considered unknown. The work in [5] introduces an adaptive dynamic programming-based scheme for optimal control of unknown affine systems. The authors in [1] and [6] deal with online optimal control of affine nonlinear system whose IGM is considered known. Here in these works [1] and [6], the cost function is estimated through the HJB equation offline, whereas the work in [3] estimates the cost function with an online NN-based estimator while proving the overall convergence of the NN-based controller. In [6], convergence of the heuristic dynamic programming algorithm via value and policy iterations is demonstrated, and closed-loop stability is not shown. It is found that an insufficient number of iterations in the value and policy iteration-based optimal control schemes [3, 6] will not only cause convergence issues but also instability. Therefore, the optimal controller in [3] is developed without using value and policy iterations, and closed-loop stability analysis is demonstrated. However, all these methods [1-6] assume that the states of the system are measurable. Unfortunately, in many practical applications, such as the propose...
A novel reinforcement-learning-based dual-control methodology adaptive neural network (NN) controller is developed to deliver a desired tracking performance for a class of complex feedback nonlinear discrete-time systems, which consists of a second-order nonlinear discrete-time system in nonstrict feedback form and an affine nonlinear discrete-time system, in the presence of bounded and unknown disturbances. For example, the exhaust gas recirculation (EGR) operation of a spark ignition (SI) engine is modeled by using such a complex nonlinear discrete-time system. A dual-controller approach is undertaken where primary adaptive critic NN controller is designed for the nonstrict feedback nonlinear discrete-time system whereas the secondary one for the affine nonlinear discrete-time system but the controllers together offer the desired performance. The primary adaptive critic NN controller includes an NN observer for estimating the states and output, an NN critic, and two action NNs for generating virtual control and actual control inputs for the nonstrict feedback nonlinear discrete-time system, whereas an additional critic NN and an action NN are included for the affine nonlinear discrete-time system by assuming the state availability. All NN weights adapt online towards minimization of a certain performance index, utilizing gradient-descent-based rule. Using Lyapunov theory, the uniformly ultimate boundedness (UUB) of the closed-loop tracking error, weight estimates, and observer estimates are shown. The adaptive critic NN controller performance is evaluated on an SI engine operating with high EGR levels where the controller objective is to reduce cyclic dispersion in heat release while minimizing fuel intake. Simulation and experimental results indicate that engine out emissions drop significantly at 20% EGR due to reduction in dispersion in heat release thus verifying the dual-control approach.
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