Time-Scaling of SISO and MIMO Discrete-Time Systems

Finally the paper deals with the solution of the following problem: Let signals y1(iT), ..., yq(iT) (elements of output vector y(iT)) be responses of MIMO discretetime system to the input signals u1(iT),..,uq(iT) (elements of input vector u(iT)). Determine discrete-time excitations uA1,......,uAq (elements of vector uA) scaling the system responses to the required forms y1(iA-1T),...,yq(iA-1T), where A is positive number called “time scale coefficient”, T–reference sample time, i–sample number. If A>1, then system responses are speeded up comparing them to reference responses y1(iT),..., yq(iT). Putting 0<A <1 one slows the responses down. The perfect scaling of system makes, that moments of appearances of consecutive amplitudes belonging to the reference responses y1(iT),..., yq(iT) are “controlled” according to value of A. The scaling of system responses yields transparent representations in time and frequency domains. Thus, the timescaling seems to be useful tool for speeding the system responses up, or slowing them down, if forms of reference responses satisfy technical needs. Let us note, that perfect scaling of stable system conserves its stability. The methods presented in current Chapter can be applied for time-scaling of responses of continuous-time systems on the basis of respective discrete–time models. Thus, the considered methods can be treated as alternative solutions to those for time-scaling of continuous–time systems (Durnas & Grzywacz, 2001; Durnas & Grzywacz, 2002; Grzywacz, 2006). Of course, there are other ways of application of time-scaling techniques, like those, for example, presented in (Moya at al., 2002; Respondek & Pogromsky, 2004), where problem of synthesis of controllers and observers for certain classes of non-linear plants has been solved. In current Chapter we do not deal with time-scaling in this meaning.