The control problem addressed by Virtual Reference Feedback Tuning (VRFT) can be formally stated as follows.Consider a linear time-invariant discrete-time Single Input Single Output (SISO) plant P(z) as in the control scheme of figure 1.

Figure 1: 1 degree of freedom control scheme.

 

P(z) is unknown and only a finite set of Input/Output (I/O) data {u(t),y(t)}t=1,...,N (whatever collected, either open-loop or closed-loop) is available. Given a set of parameterized controllers:

where βk(z) are known transfer functions and θk are parameters to be selected, and a reference model Mr(z), the problem consists in finding the parameter vector which minimizes the following model-reference performance index:

where W(z) is a user-chosen weighting function. More general control set-ups are considered in "what it is used for".

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