State Estimation and Optimal Control of Four-Tank System with Stochastic Approximation Approach

This study aims to optimally control the level of a four-tank system at the steady state in the random disturbance environment using the stochastic approximation (SA) approach. Firstly, the stochastic optimal control problem of an equivalent discrete-time is introduced, where the voltages to the pumps are the control inputs. By minimizing the sum of squared errors, the liquid levels are estimated. Then, first-order necessary conditions are derived by defining the Hamiltonian function. Thus, the optimal voltages are calculated based on the estimated liquid levels to update the gradient of the cost function. Finally, for illustration, parameters in the system are considered and a simulation is conducted. The simulation results show that the state estimation and control law design can perform well, and the liquid levels are addressed along the steady state. In conclusion, the applicability of the SA approach for handling a four-tank system with random disturbances is demonstrated.