20. LTI: stability and feedback#

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

20.1. Stability of a LTI - SISO#

20.1.1. 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.

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

20.1.2. 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 \(t \rightarrow + \infty\).

20.2. Stability of closed-loop systems#

20.2.1. Cauchy argument principle#

For the Cauchy argument principle, the difference of argument of function \(F(s)\) when \(s\) performs a counter-clockwise loop over th contour \(\Gamma\) enclosing poles \(p_n\), and zeros \(z_d\) of the function \(F(s)\) reads

\[\Delta \text{arg} \{ F(s) \} = \sum_{n} \text{arg}(s - z_n) - \sum_{d} \text{arg}(s-p_d) = Z 2 \pi - P 2 \pi = (Z-P) 2 \pi \ , \]

and thus the the diagram of function \(F(s)\) performs

(20.1)#\[N = Z - P\]

loops around the origin \(0+i 0\) of the complex plane.

20.2.2. Nyquist stability criterion#

Nyquist stability criterion provides some conditions for the stability of a closed loop transfer function

\[G_c(s) := \dfrac{G(s)}{1+G(s)} \ , \]

being \(G(s)\) the open loop transfer function, and the feedback with the opposite of the output (\(y = G(s) \left( u - y \right)\)).

The poles of the closed loop systems are the zeros of the function \(1 + G(s)\). If the closed-loop system is asymptotically stable, it must have no pole with positive real part. As the complex variable \(s\) performs a clockwise (and thus, the signs of the relation change w.r.t. Cauchy argument criterion) loop over Nyquist path (semicircle in the RHS half-plane; then todo discuss the case where poles and zeros are on the imaginary axis…deform the path…). No zero of \(1 + G(s)\) with positive real part means \(Z = 0\). Using the relation between poles, zeros and loops around the origin given by Cauchy argument principle (20.1), and recalling the opposite direction, it follows that

\[N = P \ ,\]

in order to have \(Z = 0\), i.e. the diagram of \(1+G(s)\) must perform \(N\) counter-clockwise loops around \(0+i0\) equal to the number of its poles with positive real parts. As the poles of \(1+ G(s)\) are the same as the poles of \(G(s)\), it’s possible to formulate Nyquist criterion looking at \(G(s)\), as

Theorem 20.1 (Nyquist criterion)

In order for the closed-loop system to be asymptotically stable, the diagram of the open-loop transfer function \(G(s)\) must perform a number \(N\) of counter-clocwise loops around the critical point \(-1 + i \, 0\) equal to the number of its zeros with positve real part.

20.2.3. Bode stability criterion for minimal phase systems#

Definition 20.1 (Minimal phase systems)

Safetry gain margin, safety phase margin…