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

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], y_n ^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 r_k, 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.