mor.stability

Main interface for LTI dynamical model stability estimation.

Syntax

stab = mor.stability(sys,W)

Description

mor.stability(sys,W) evalutes if the dynamical model sys is stable, over a frequency space W. The mor.stability function evalutes the stability of any dynamical model sys, defining a meromorphic function. This includes time-delayed models, functions defined by trigonometric operators, etc. The input sys can either be

a state-space ODE (DAE) realisation described as sys = ss(A,B,C,D) (or sys = dss(A,B,C,D,E)) with or without internal delays
a single complex variable function handle of an ny outputs, nu inputs responses described sys = @(s) myFun

stab = mor.stability(sys,W) provides a positive scalar stab. If stab is close to zero, then the model is stable. Otherwise it is unstable.

Input arguments

sys	Dynamical system, or handle function. More specifically `sys` can either be a state-space ODE (DAE) realization described as `sys = ss(A,B,C,D)` (or `sys = dss(A,B,C,D,E)`) with or without internal delays a single complex variable function handle of an nu inputs, ny outputs responses described `sys = @(s) myFun`
W	Vector of frequencies, in rad/s, over which the system is defined.

Output arguments

stab	Returns the real scalar stability evaluation tag (stable if `stab` close to zero precision, unstable otherwise).

Examples

In this example, we consider a dynamical system with an internal delay. The model is then defined as a transfer function, parametrized by s, the Laplace variable, and tau, the delay. It is known that the destabilising tau value is pi/2. This property is recovered by the mor.stability function.

% In this delayed model, the stability is analysed as a function of the
% delay value tau (the destabilizing theoretical value is pi/2)
W       = logspace(-2,1,100)*2*pi;
H       = @(s,p) 1./(s+exp(-p*s));
% Let p lower than pi/2, for this case, model is stable
p       = pi/2-1e-6;
Hp      = @(s) H(s,p);
stabEst = mor.stability(Hp,W) % = 2.9477e-04, close to zero, thus stable
% Now, let p greather than pi/2, then the model is unstable
p       = pi/2+1e-6;
Hp      = @(s) H(s,p);
stabEst = mor.stability(Hp,W) % = 9.3105e+05, far from zero, thus unstable