Differential Flatness Theory in the Background of Other Control Methods

Abstract

The chapter demonstrates that differential flatness theory is in the background of other control methods, such as backstepping control and optimal control. First, the chapter shows that differential flatness theory is in the background of backstepping control. The method assumes that the system is already found or can be transformed to the so-called triangular form. The controller design proceeds by showing that each row of the state-space model of the nonlinear system stands for a differentially flat system, where the flat output is chosen to be the associated state variable. Next, for each subsystem which is linked with a row of the state-space model a virtual control input is computed, that can invert the subsystem’s dynamics and can eliminate the subsystem’s tracking error. From the last row of the state-space description, the control input that is actually applied to the nonlinear system is found. This control input contains recursively all virtual control inputs which were computed for the individual subsystems associated with the previous rows of the state-space equation. Thus, by tracing the rows of the state-space model backwards, at each iteration of the control algorithm, one can finally obtain the control input that should be applied to the nonlinear system so as to assure that all its state vector elements will converge to the desirable setpoints. The proposed flatness-based control method can solve efficiently several nonlinear control problems. Indicative evaluation results are presented in the manuscript in the form of simulation experiments. Next, the chapter shows that differential flatness theory is in the background of optimal control. Finally, the aforementioned implementation of flatness-based control in cascading loops is used to solve the problem of boundary control of distributed parameter systems, that is control of nonlinear PDEs using as control inputs their boundary conditions.