Application of Conjugate Gradient Approach for Nonlinear Optimal Control Problem with Model-Reality Differences

In this paper, an efficient computational algorithm is proposed to solve the nonlinear optimal control problem. In our approach, the linear quadratic optimal control model, which is adding the adjusted parameters into the model used, is employed. The aim of applying this model is to take into account the differences between the real plant and the model used during the calculation procedure. In doing so, an expanded optimal control problem is introduced such that system optimization and parameter estimation are mutually interactive. Accordingly, the optimality conditions are derived after the Hamiltonian function is defined. Specifically, the modified model-based optimal control problem is resulted. Here, the conjugate gradient approach is used to solve the modified model-based optimal control problem, where the optimal solution of the model used is calculated repeatedly, in turn, to update the adjusted parameters on each iteration step. When the convergence is achieved, the iterative solution approaches to the correct solution of the original optimal control problem, in spite of model-reality differences. For illustration, an economic growth problem is solved by using the algorithm proposed. The results obtained demonstrate the efficiency of the algorithm proposed. In conclusion, the applicability of the algorithm proposed is highly recommended.

A Gauss-Newton Approach for Nonlinear Optimal Control Problem with Model-Reality Differences

Output measurement for nonlinear optimal control problems is an interesting issue. Because the structure of the real plant is complex, the output channel could give a significant response corresponding to the real plant. In this paper, a least squares scheme, which is based on the Gauss-Newton algorithm, is proposed. The aim is to approximate the output that is measured from the real plant. In doing so, an appropriate output measurement from the model used is suggested. During the computation procedure, the control trajectory is updated iteratively by using the Gauss-Newton recursion scheme. Consequently, the output residual between the original output and the suggested output is minimized. Here, the linear model-based optimal control model is considered, so as the optimal control law is constructed. By feed backing the updated control trajectory into the dynamic system, the iterative solution of the model used could approximate to the correct optimal solution of the original optimal control problem, in spite of model-reality differences. For illustration, current converted and isothermal reaction rector problems are studied and the results are demonstrated. In conclusion, the efficiency of the approach proposed is highly presented.

An Expanded Optimal Control Policy for a Coupled Tanks System with Random Disturbance

In this paper, an expanded optimal control policy is proposed to study the coupled tanks system, where the random disturbance is added into the system. Since the dynamics of the coupled tanks system can be formulated as a nonlinear system, determination of the optimal water level in the tanks is useful for the operation decision. On this point of view, the coupled tanks system dynamics is usually linearized to give the steady state operating height. In our approach, a model-based optimal control problem, which is adding with adjusted parameters, is considered to obtain the true operating height of the real coupled tanks system. During the computation procedure, the differences between the real plant and the model used are measured repeatedly, where the optimal solution of the model used is updated. On this basis, system optimization and parameter estimation are integrated. At the end of the iteration, the iterative solution approximates to the correct optimal solution of the original optimal control problem, in spite of model-reality differences. In conclusion, the efficiency of the approach proposed is shown.

Discrete-Time Nonlinear Stochastic Optimal Control Problem Based on Stochastic Approximation Approach

In this paper, a computational approach is proposed for solving the discrete-time nonlinear optimal control problem, which is disturbed by a sequence of random noises. Because of the exact solution of such optimal control problem is impossible to be obtained, estimating the state dynamics is currently required. Here, it is assumed that the output can be measured from the real plant process. In our approach, the state mean propagation is applied in order to construct a linear model-based optimal control problem, where the model output is measureable. On this basis, an output error, which takes into account the differences between the real output and the model output, is defined. Then, this output error is minimized by applying the stochastic approximation approach. During the computation procedure, the stochastic gradient is established, so as the optimal solution of the model used can be updated iteratively. Once the convergence is achieved, the iterative solution approximates to the true optimal solution of the original optimal control problem, in spite of model-reality differences. For illustration, an example on a continuous stirred-tank reactor problem is studied, and the result obtained shows the applicability of the approach proposed. Hence, the efficiency of the approach proposed is highly recommended.