Robust optimal control of uncertain nonaffine MIMO nonlinear discrete‐time systems with application to HCCI engines

Abstract

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...

Extremum-seeking for nonlinear discrete-time systems with application to HCCI engines

“…In this section, a NN-based optimal controller proposed in [15] is employed as the inner-loop stabilizing controller, while the plant is a four-stroke HCCI engine whose performance should be maximized. Controlling such engines is difficult because the combustion event depends on chemical kinetics rather than an external trigger.…”

“…In [15] the authors proposed an approach that provides a robust optimal controller that makes the closed loop system converge to an arbitrarily small bound in an optimal manner; this bound, ( ) x b  , is a function of the reconstruction error of the NN and is used here. However, we also need a proper a for the purpose of extremum-seeking.…”

A novel redesign framework to extend the application scope of a class of disturbance‐rejection algorithms

“…The nonaffine-in-control nonlinear systems are very common in practical engineering, such as manipulator, 14 flight vehicle, 15 hypersonic vehicle, 16,17 and electromechanical system. 18,19 Some of the studies for nonaffine nonlinear systems have utilized the neural network and fuzzy control scheme-based methods, such as adaptive neural control, 20,21 adaptive fuzzy control, 17,22 and adaptive neural dynamic surface control. 23 Nevertheless, the heavy computational burdens resulted from adaptive fuzzy or neural weights are unacceptable in practical implementation.…”

“…Remark 4. Note that the time derivative of virtual control input in the ith step is calculated by (18). Thus, the problem of "explosion of complexity" in backstepping design is avoided.…”

“…Then, under the controller designed by (12), (18), and (24) with the existing disturbance control law (x, z) satisfying Assumption 3, there exists a positive constant * > 0 such that, for all 0 < ≤ * , the output y will track the reference output signal y r , which satisfies the Assumption 1, in any finite time interval, and the error statesē n = [e 1 , … , e n ] T will remain bounded despite the presence of the matched and mismatched disturbances.…”