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 \(G(s)\),
For rational transfer functions
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
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.
Example 20.1 (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,
The value of the coefficients \(A\), \(B\), \(C\) is computed comparing the first and the last expression of the numerator of \(G(s)\)
and thus
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,
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
and thus the the diagram of function \(F(s)\) performs
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
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
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…