Quantitative cascaded multiple-input multiple-output synthesis by an improved method

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 ���� 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 ℬ ⁱ, i = a, b,…, m. The basic problem is how to divide the feedback burden among the m available loop matrices. The paper presents a synthesis procedure, which is easier to implement, in some respects, and is much more economical in loop gains and bandwidths than the previous one. However, the relations and trade-offs between the mn individual loop transmissions are more subtle and complex. It has the following features: (a) conversion ofthe n × n MIMO cascaded m-section problem into n cascaded insertion SISO synthesis problems (the solutions ofthe latter are guaranteed to solve the former); (b) bandwidth propagation, wherein the loop matrices take turns in dominating the design over specific frequency ranges; (c) great transparency in relating the n × m available loop transmissions to the plant parameters and their uncertainties. This provides design perspective, wherein one can make good a priori estimates of the optimum division of the overall feedback burden between the loop matrices. In addition, there is good understanding ofthe trade-offs in transferring to some extent the feedback burden from one loop to another. It is shown how proper use ofthe internal MIMO plant variables may greatly reduce the effect of sensor noise at the plant input. Frequency response is the key tool in the quantitative synthesis.