A STATE-FEEDBACK CONTROLLER WITH MINIMUM CONTROL EFFORT

The pole placement problem, in the case of multivariable systems, does, in general, result in many solutions as opposed to that of single input systems. Parameterized controllers are commonly used when it is desired to satisfy some performance index beyond the pole placement. While the parameters do not affect the locations of the poles (eigenvalues), they can be chosen in such a manner as to satisfy some desired performance index. Among those desired criteria is the minimization of the control effort necessary for the pole allocation. A suitable measure of control effort is the Frobinious norm of the control feedback gain matrix. A parametric state feedback controller was recently proposed by the authors. This parametric state feedback controller has the advantage of being linear in free parameters, which makes it attractive for mathematical operations such as differentiation. In the following, two methods are proposed to solve the pole placement with the control effort minimization problem. Both methods lead to a one-step explicit solution, as opposed to many existing iterative techniques. Those proposed one-step solutions result in global minimization, as compared to the local minimization characterizing iterative techniques. Moreover, those proposed techniques do not impose particular restrictions (other than the natural complex conjugate pairing) on the destination, nature, or multiplicity of the eigenvalues, as is the case with many existing norm minimization schemes. Numerical examples are used to illustrate the methods and to demonstrate their advantages over currently used ones.