On the Determination of the Feedback Gains of Multivariable PI Control Systems Using the Quadratic Cost Performance Index

Abstract

This paper presents a method of determining the optimal feedback gains of a PI control for multivariable linear systems. The control law considered here is an output feedback, and the cost performance index is a quadratic type one which does not contain the integration of the output error or the time derivative of the input. The necessary condition which minimizes the cost performance index is given by the nonlinear simultaneous matrix equations which can be solved by the gradient technique.
This method is based on the fact that, in the steady state, the input u_s is related to the integration of the output error, z_s, as u_s=G₁z_s, where G₁ is the feedback gain matrix of the integral action. Using this relation, we can obtain the feedback gains of the integral action without modifying the cost performance index as mentioned above.
In this method we need not be concerned with the complex problem of obtaining the relationship between the output overshoot and the weighting matrix of the integrated output error or of the derivative of the input. An illustrative example is given.