44.2. Reference tracking (SISO)

Setting \(\mathbf{F} = \mathbf{I}\) in Fig. 44.1, and neglecting disturbances, the transfer functions between the output and the reference and between the tracking error and the reference signal are

\[\begin{split}\begin{aligned} E(s) & = \frac{1}{1 + G(s) R(s)} Y_{\text{ref}}(s) \\ Y(s) & = \frac{G(s)R(s)}{1 + G(s) R(s)} Y_{\text{ref}}(s) \\ \end{aligned}\end{split}\]

Polynomial forcing.

Laplace transform of polynomial signals

Laplace transform of (casual) polynomial signals immediately follows from the transform of a Dirac’s delta and the property of the Laplace transform of the integral of a function,

\[\mathscr{L}\left\{ \int_{\tau=0}^{t} f(\tau) \, d \tau \right\} = \frac{F(s)}{s} \ , \]

being \(F(s)\) the Laplace transform of function \(f(t)\). Indeed, a ramp can be interpreted as the integral of a Dirac’s delta, a casual parabolic signal is the integral of the ramp, and so on.

Dirac’s delta.

\[F_{-1} := F_{\delta} = \mathscr{L} \{ \delta(t) \} = \int_{t = 0^-}^{+\infty} \delta(t) e^{-st} dt = 1 \ . \]

Step. Step is the integral of the Dirac’s delta.

\[F_0 := F_{\text{step}} = \mathscr{L} \{ 1 \} = \frac{1}{s} \ .\]

Ramp. Step is the integral of the Dirac’s delta.

\[F_1 := F_{\text{ramp}} = \mathscr{L} \{ t \} = \frac{1}{s^2} \ .\]

Signal \(t^n\), \(n \in \mathbb{N}\).

\[\mathscr{L} \left\{ t^n \right\} = \frac{1}{s^{n+1}} \ .\]

Limit of the tracking error of polynomial forcing, for \(t \rightarrow +\infty\). Assuming the system is stable, the free-response to initial condition vanishes. Using final value property of Laplace transform (see Laplace transform definition and properties),

\[\lim_{t \rightarrow + \infty} e(t) = \lim_{s \rightarrow 0} s E(s) \ .\]

Let the open-loop transfer function \(L(s) = G(s) R(s)\) be \(L(s) = \frac{N(s)}{s^p D(s)}\), having explicitly written the integrators (the factors \(\frac{1}{s}\) in the denominator of the TF), so that \(D(0) \ne 0\), \(N(s) \ne 0\). For casual systems \(|N| \le p + |D|\). The closed-loop function reads

\[\frac{1}{1 + L(s)} = \frac{s^p D(s)}{ s^p D(s) + N(s) } \ .\]

Let \(Y_{\text{ref}}(s) = C_0 + \frac{C_1}{s} + \dots + \frac{C_{m+1}}{s^{m+1}}\) the Laplace transform of a \(m\)-degree polynomial reference, the limit value of the tracking error reads

\[\begin{split}\begin{aligned} \lim_{t \rightarrow + \infty} e(t) & = \lim_{s \rightarrow 0} s E(s) = \\ & = \lim_{s \rightarrow 0} \frac{ s^{p+1} D(s) }{ s^p D(s) + N(s) } \left( C_0 + \frac{C_1}{s} + \dots + \frac{C_{m+1}}{s^{m+1}} \right) = \\ & = \lim_{s \rightarrow 0} \frac{ s^{p-m} D(s) }{ s^p D(s) + N(s) } \left( s^{m+1}C_0 + s^m C_1 + \dots + C_{m+1} \right) = \\ & = \left\{ \begin{aligned} & 0 && \ , \qquad p > m \\ & \frac{D(0)}{N(0)} && \ , \qquad p = m \\ & +\infty && \ , \qquad p < m \\ \end{aligned} \right. \end{aligned}\end{split}\]

Type of the system.