How to get time-domain simulation of a nonlinear state-space model?

I am simulating a nonlinear system in state-space form using MATLAB ODE solvers (e.g., ode23tb). First I starts with initial condition near by equilibrium points then I am correctly getting steady-state equilibrium points after some time I want to pertub the system and see the dynamic respone.

To study the system dynamics, I apply a perturbation (for example +10 or +15 pu) to one of the states. For a stable operating point, the perturbed response behaves as expected.

However, for an unstable operating point, the ODE solver does not show the expected unstable behaviour. The simulation-

• shows only a single spike at the beginning and then nothing,instead of showing the natural unstable response (growing oscillations).

I would like to clearly observe how the system becomes unstable, but the ODE solver is not producing this behaviour.

How can I correctly apply a state perturbation so that the solver shows the full unstable trajectory?

Any explanation or example of how to simulate unstable nonlinear systems in MATLAB would be greatly appreciated.

[Torsten]

You will have to include the ODE system under consideration and your MATLAB code.

Why do you think that "growing oscillations" is the natural behaviour for unstable operating points ? Consider e.g. y' = y with y = 0 as unstable equilibrium point.

[Manish Kumar]

Thank you for the clarification.

You are right that unstable equilibria do not always produce oscillations. I only mentioned “growing oscillations” as one possible behaviour. What I meant is that I want to see any unstable trajectory (whether it diverges exponentially, grows oscillatory, or becomes unbounded in some other way).

[Sam Chak]

Hi @Manish Kumar

Could you provide an example like this so that users in the forum can understand your issue and provide assistance?

%% The system dynamics
function dydt = odefun(t,y)
    dydt= [y(2) - y(1).^99;
         - y(1) + y(2).^99];
end

%% Case 1: ode45, y0 = 0.5
y0      = 0.5;
[t, y]  = ode45(@odefun, [0 10], [y0 y0]);
figure
plot(t, y), grid on, xlabel('t'), ylabel('y'), title('Case 1: Stable, y_0 = 0.5')

%% Case 2: ode15s, y0 = 0.7
y0      = 0.7;
[t, y]  = ode15s(@odefun, [0 10], [y0 y0]);
figure
plot(t, y), grid on, xlabel('t'), ylabel('y'), title('Case 2: Stable, y_0 = 0.7')

●

%% Case 3: ode45, y0 = 0.7
y0      = 0.7;
[t, y]  = ode45(@odefun, [0 10], [y0 y0]);
figure
plot(t, y), grid on, xlabel('t'), ylabel('y'), title('Case 3: Stable, y_0 = 0.7, but ode45 fails to complete')

size(y)

ans = 1×2

57 2

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y(end-24,1)     % y1

ans = -0.2351

y(end-24,2)     % y2

ans = 0.9698

y(end-23,1)     % y1

ans = NaN

y(end-23,2)     % y2

ans = NaN

%% Case 4: ode45, y0 = 1.0
y0      = 1.0;
[t, y]  = ode45(@odefun, [0 10], [y0 y0]);
figure
plot(t, y), grid on, xlabel('t'), ylabel('y'), title('Case 4: At equilibrium, y_0 = 1')

%% Case 5: ode45, y0 = 1.5
y0      = 1.5;
[t, y]  = ode45(@odefun, [0 10], [y0 y0]);
figure
plot(t, y), grid on, xlabel('t'), ylabel('y'), title('Case 5: Unstable, y_0 = 1.5, ode45 no warning')

y(1,1)

ans = 1.5000

y(2,1)

ans = NaN

%% Case 6: ode15s, y0 = 1.5
y0      = 1.5;
[t, y]  = ode15s(@odefun, [0 10], [y0 y0]);

Warning: Failure at t=4.893638e-20. Unable to meet integration tolerances without reducing the step size below the smallest value allowed (9.629650e-35) at time t.

figure
plot(t, y), grid on, xlabel('t'), ylabel('y'), title('Case 6: Unstable, y_0 = 1.5, ode15s gives warning')

[William Rose]

@Sam Chak, that is a very interesting example.

[Sam Chak]

@William Rose, Thanks you. It is to encourage and guide the OP to describe his issue with the solver.