CLOSED-LOOP PERFORMANCE INFERENCE FOR CONTROLLER SELECTION IN SWITCHING SUPERVISORY CONTROL Edoardo Mosca and Tommaso Agnoloni Dipartimento di Sistemi e Informatica Universit` a di Firenze V. di S. Marta, 3, 50139 Firenze, Italy Tel.: +39.55.4796258 Fax : +39.55.4796363 email: mosca@dsi.unifi.it Keywords : Switching supervisory control; Hybrid control systems; Control of unknown plants. Abstract It is shown that an effective way for inferring the perfor- mance of a linear feedback loop made up by an uncer- tain plant and a given candidate controller is to resort to the concept of a virtual reference. This allows one to compute a performance-related distance measure, called divergence, between the loop made up by the unknown plant in feedback with the candidate controller and the nominal “tuned” loop related to the candidate controller. It is proved that divergence can be obtained by suitably filtering an output prediction error. 1 Introduction The basic problem we wish to address consists of choos- ing, amongst a finite number of candidate controllers K h , h ∈ N := {1, 2,...,N }, the one which yields the best per- formance to the loop made up by the feedback intercon- nection of an uncertain and possibly time-varying plant P with any of the N available controllers K h . Specifically, it is assumed that the plant to be controlled can be rep- resented as a discrete-time SISO linear dynamic system P P : A(d)y(t)= B(d)δu(t)+ A(d)n(t) (1) where: t ∈ Z := {..., −1, 0, 1,...}; y(t) is the plant out- put; δu(t) := u(t) − u(t − 1) the input increment; and n(t) the plant output disturbance. In particular, {n(t)} is assumed to be a w.s. (wide-sense) stationary random sequence with bounded second moment. Further, A(d)= a(d)Δ(d), Δ(d)=1−d, a(d) = 1+a 1 d+a 2 d 2 +...+a na d na , and B(d)= b 1 d + b 2 d 2 + ...b n b d n b are polynomials in the unit backward-shift operator d, with the greatest common divisor between A(d) and B(d) strictly Schur. We shall assume that the plant (1), though unknown, for control purposes can be adequately described by one amongst N different deterministic linear models M h = (A h (d),B h (d)) A h (d)= a h (d)Δ(d), viz. M h : A h (d)y(t)= B h (d)δu(t) h ∈ N (2) We shall also stipulate that the one-degree of freedom lin- ear controller K h =(R h (d),S h (d)) K h : R h (d)δu(t)= −S h (d)[y(t) − r(t)] h ∈ N (3) is tuned on to the model M h via some well defined con- trol design method. In (3), r(t) denotes the ouput refer- ence which is supposed to be a w.s. stationary random sequence with bounded second moment and uncorrelated with {n(t)}. For subsequent discussion, it is convenient to introduce some special notations. Specifically, (P /K h ) will indicate the feedback system wherein the plant (1) is controlled by K h as in (3). Likewise, (M k /K h ) will stand for the feed- back system wherein the model M k is controlled by K h , h, k ∈ N . The I/O variables of P related to (P /K h ) will be denoted δu */h (t) and y */h (t), and similarly, δu k/h (t) and y k/h (t) the I/O pairs related to (M k /K h ). In terms of the foregoing notations, the problem that we wish to address can be formulated in quite ideal terms as follows. Assumed that the current operating loop is (P /K h ), we are first asked to possibly infer the performance of all other possible loops (P /K k ), k = h, k ∈ N . Next, pro- vided that such an inference is obtained, we substitute the formerly acting controller K h by switching on in feedback to the plant, the controller K ˆ k , ˆ k ∈ N , if (P /K ˆ k ) per- forms at least as well as any other candidate loop (P /K k ), k ∈ N . Some seminal papers [1]-[4] on supervisory switching