How it works

The Virtual Reference Feedback Tuning is a direct method for the design of feedback controllers when the plant is unknown. It is assumed that the plant is a linear SISO discrete-time dynamical system with transfer function P(z). Such transfer function is unknown and a set of I/O data, collected during an experiment on the plant, is available for design purposes.

In the following, details are given on how the method works for the simplest set-up where one wants to shape the reference-to-output transfer function.

The control specifications are assigned via a reference model Mr(z). This describes the desired r(t) to y(t) transfer function of the closed-loop system (see figure 1). Attention is restricted to linearly parameterized controllers:

where βk(z) are known transfer functions and θk are parameters to be selected.

The control objective is the minimization of the following model-reference criterion:

where W(z) is a weighting function chosen by the user.

Figure 1: 1 degree of freedom control scheme.

The idea behind the VRFT method can be summarized in 4 steps:

• STEP 1: generate a set of I/O data {u(t),y(t)}, t = 1,...,N, by a single experiment on the plant.

 

Figure 2: open loop experiment scheme.

 

• STEP 2: given the measured y(t), generate in your computer a reference signal r*(t) such that M(z)r*(t) = y(t), where M(z) is the desired reference model. r*(t) is called "virtual reference" because it was not used to generate y(t) and it only exists as a computer file. Notice that y(t) is by construction of r*(t) the desired output of the closed-loop system when the reference signal is r*(t).

 

• STEP 3: generate the corresponding tracking error e*(t) = r*(t) - y(t).

 

• STEP 4: even though plant P(z) is not known, we know that when P(z) is fed by u(t) (the actually measured input signal), it generates y(t) as an output. Therefore, a good controller is one that generates u(t) when fed by e*(t). The idea is then to search for such a controller. Since both signals u(t) and e*(t) are known, this task boils down to the problem of identifying the dynamical relationship between e*(t) and u(t).

 

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