Design of Nonlinear Optimal Control Systems Using Jordan Controlled Form

The problem of control systems design for nonlinear plants still has no exhaustive solution. This problem is commonly solved on the basis of transformation of the nonlinear plant equations to some simple. Such approach simplifies the solution of the control system design problem and makes the solution analytical. For design of the optimal control systems for the nonlinear plants, their equations are expedient to transform to Jordan controlled form (JCF). The analytical design method of the optimal control systems for nonlinear plants with using the JCF of their equations is proposed in this paper. This problem has a solution if all plant state variables are measured. The JCF exists, if the nonlinear plant is completely controllable. The proposed method includes two steps. At the first step, a linearization control is designed by a nonlinear reversible transformation of the plant state variables. Under the linearization control, the system equations are linear and stationary in the new variables. Theorem about existence of a linearization control is proved. At the second step, the optimal control is designed as optimal in the sense of a minimum of nonlinear quadratic criteria. This control is designed using solution of the Riccati equation. Optimality of the obtained nonlinear control is proved also. Design of the optimal control systems for nonlinear plants using JCF is expedient, because the equations of many real plants have JCF or can be easily transformed to this form. Frequently, the plant equations convert into this form if the state variables are designated in appropriate way. The example of the optimal control system design for nonlinear plant is given. Key-Words: Nonlinear plant, reversible transformation, Jordan controlled form, linearization control, optimal control, nonlinear quadratic criteria.