Quantitative synthesis of uncertain cascaded multiple-input multiple-output feedback systems

There is given an n × n multiple-input multiple-output (MIMO) linear time-invariant plant of m cascaded sections P = P^a P^b… P^m, each an n × n matrix of transfer functions. The m vector outputs y, y^a…, y^m−1;y = P^a y^a= P^a P^b y^b =. = P^a P^b P^mu =δ Pu) can be measured and the data used for feedback purposes. Each Pⁱ is known only to belong to a given set P ⁱ due to uncertainty in the plant parameters. P is embedded in a feedback structure with an n × 1 command input vector r = (r₁,…, r_n)’, and m free n × n loop compensation matrices to be chosen by the designer. There are assigned specifications on the n² system transfer functions f_uv(s) = ŷ(s)/[rcirc]_v(s), to be satisfied for all Pⁱ; in ⁱ, i = a, b,…, m. The basic problem is how to divide the feedback burden among the m available loop matrices. The technique presented in this paper is based upon two synthesis techniques previously devised for the following two problems: (1) the same as above but for single-input single-output (SISO) systems with each plant section a scalar, (2) the single-section (only one loop matrix) n × n system, with no internal plant variables available for feedback. The synthesis procedure has the following fealures: (a) conversion of the n × n MIMO cascaded m-section problem into n cascaded m-section SISO synthesis problems, the solutions of the latter being guaranteed to solve the former; (b) bandwidth propagation, wherein the loop matrices take turns dominating the design over specific frequency ranges; (c) design perspective wherein one can a priori make good estimates of the optimum division of the overall feedback burden between the loop matrices. It is shown how proper use of the internal MIMO plant variables may enormously reduce the effect of sensor noise at the plant input. Frequency response is the key tool in making quantitative synthesis possible.