Quantitative synthesis of feedback systems with uncertain non-linear multivariable plants†

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Abstract
There is given an n × n non-linear multiple-input-output plant operator W, known only to belong to a defined set . W is imbedded in a feedback system with n × n compensation operators F, G and provision for command inputs r into the system.  = {r 1 [tdot],r α} is a known finite set of possible command input vectors. For each r α, there is given an n-product set of acceptable plant outputs α = α ×[tdot] × n α, F, G are to be chosen so that in response to r α, the plant output Y α = (y1 α, [tdot], yn αεα, (i.e. y1ε1 α) ∀W ε . Two design techniques, motivated by Schauder's fixed point theorem, are presented. They are best suited for ‘ basically non-interacting’ performance specifications, wherein output y1 is to be primarily due to r1, with relatively small components due to the rk, k ≠ i. A detailed 2 × 2 numerical design example is presented, in which F, G are linear time-invariant matrices of transfer functions. It is shown how the ‘ cost of feedback ’ may be significantly reduced, by means of a ‘ nominal cancellation ’ technique, involving non-linear G.