The problem of the synthesis of multivariable controllers which are robust with respect to model-plant mismatch is addressed. A two-step design procedure based on the Internal Model Control (IMC) structure is used. In the first step the IMC controller is designed assuming no modeling error, and in the second step the IMC filter is designed to preserve the closed-loop characteristics in spite of model-plant mismatch.
Two alternatives are provided for the first step. One of them allows the designer to satisfy structural performance specifications, in terms of the structure of the closed-loop interactions, their magnitude and duration. The closed-loop transfer function matrix is directly designed. The method requires only standard linear algebra operations and includes the construction of the IMC or the feedback controller in state-space. The second approach involves the minimization of the appropriately weighted H2-norm of the sensitivity transfer function matrix, that relates the errors to the external inputs (setpoints or disturbances). A method is given for the meaningful selection of a full matrix weight so that the H2-error is minimized for a set of external input directions and their linear combinations. The procedure is extended to open-loop unstable systems. In both approaches, special care is taken to avoid intersample rippling.
The design of the filter in the second step is formulated as an optimization problem over the filter parameters. The objective function is constructed by using the Structured Singular Value theory so that the maximum singular value of the sensitivity transfer function remains bounded in spite of modeling error. The selection of the frequency bound is based on the properties of the design that was obtained in the first step. Analytic gradient expressions have been developed for the objective function. The optimization problem is an unconstrained one, solved with standard gradient search techniques. An iterative method for the selection of the appropriate sampling time is proposed, which explicitly takes into account model uncertainty information and performance specifications.
