27. LTI: stability and feedback

Contents

• Stability of a SISO LTI, w.r.t. non-zero i.c. or impulsive input

27.1. Stability of a LTI - SISO

27.1.1. Transfer function

Let the SISO input-output relation of a LTI system be represented in Laplace domain by the transfer function \(G(s)\),

\[y(s) = G(s) u(s)\]

For rational transfer functions

\[G(s) = k \dfrac{\prod_{n=1}^N (z_n - s)}{\prod_{d=1}^{D} (p_d - s)} = \dfrac{\sum_{n=1}^{N} a_n s^n}{\sum_{d=0}^{D} b_d s^d} \ ,\]

with \(z_n\) the zeros, \(p_d\) the poles, and \(k = \frac{a_0}{b_0} = \lim_{s \rightarrow 0} G(s)\) the static gain. If the system is strictly proper, \(N < D\), and the TF can be written as a sum of partial functions. As an example, for a transfer function with simple poles

\[G(s) = \sum_{d=1}^{N} \dfrac{A_d}{(p_d-s)} \ , \]

while for a pole \(p_d\) with multiplicity \(m_d> 1\) all the terms \(\propto \frac{1}{(p_d-s)^{e_d}}\), with \(e_d = 1:m_d\) must be included, see example below.

\[G(s) = \sum_{d=1}^{N} \dfrac{A_d}{(p_d-s)^{e_d}} \ , \]

Example 27.1 (Sum of partial fractions of a TF with poles with multiplicity \(> 1\))

Let’s write the rational function \(G(s) = \frac{s+1}{(s+2)^3}\) as a sum of partial functions,

\[\begin{split}\begin{aligned} G(s) & = \dfrac{s+1}{(s+2)^3} = \\ & = \dfrac{A}{s+2} + \dfrac{B}{(s+2)^2} + \dfrac{C}{(s+2)^3} = \\ & = \dfrac{A(s+2)^2+B(s+2)+C}{(s+2)^3} = \\ & = \dfrac{s^2(A) + s(4A + B) + 4A+2B+C}{(s+2)^3} \ , \end{aligned}\end{split}\]

The value of the coefficients \(A\), \(B\), \(C\) is computed comparing the first and the last expression of the numerator of \(G(s)\)

\[\begin{split}\begin{cases} s^2 : \quad A = 0 \\ s\ \ : \quad 4A + B = 1 \\ 1\ \ : \quad 4A + 2B + C = 1 \\ \end{cases} \qquad \rightarrow \qquad \begin{cases} A = 0 \\ B = 1 \\ C = -1 \end{cases}\end{split}\]

and thus

\[G(s) = \dfrac{s+1}{(s+2)^3} = \dfrac{1}{(s+2)^2} - \dfrac{1}{(s+2)^3} \ . \]

27.1.2. Stability w.r.t. non-zero initial conditions - or w.r.t. implusive input - in time domain

Transfer function \(G(s)\) represents the free the function w.r.t. impulsive input. Response in time domain can be evaluated as the inverse Laplace transform of the transfer function,

\[y(t) = \mathscr{L}^{-1} \left\{ y(s) \right\} = \mathscr{L}^{-1} \left\{ \sum_{n_d=1}^{D} \dfrac{R_d}{(s-p_d)} \right\} = \sum_{d=1}^{D} R_d \, e^{p_d t} \ .\]

If \(\text{re}\{ p_d \} < 0\) for \(\forall d\), then the response is asymptotically stble for \(t \rightarrow +\infty\), as \(|e^{p_d t}| = e^{\text{re}\{p_d\} t} \rightarrow 0\), for \(t \rightarrow + \infty\).