Stability state-space model + bound additional variables

Dear forum members,

Let us consider a linear state-space model of the form:

where $e$ is the closed-loop error and d an external variable of interest linked nonlinearly to e as d=h(e).

The stability of e is ensured as A is stable.

However, I would like to ensure not only the stability of e but also that d remains bounded for all e.

Note that d is not really a perturbation that we want to reject, the closed-loop model is designed such that d follows a given trajectory. But I do not know how to include it into the stability analysis.

I have tried Lyapunov and LMI stability analysis such that which gives:

[e d]'[A'P+PA, PG;
       G'P,     0][e d]<0

but this is not feasible due to the 0 on the diagonal.

Any ideas how to overcome this ? I have looked at ISS property or

The final objective would like to be able to write something like : and .

I hope this makes sense, thanks in advance !

Answer 1

[Ayush]

Hi @CN.,

The problem you’re addressing involves analyzing the stability of a closed-loop system where error “e” is controlled to ensure stability, but also ensuring that the external variable “d” remains bounded for all values of “e”.

Following are the approaches you can try:

1. Since d=h(e) is a nonlinear function of “e”, consider Input-to-State Stability (ISS). This property guarantees that if “e” stays stable, “d” will remain bounded as well, provided “h(e)” behaves in a controlled way.

2. If possible, use nonlinear control techniques like backstepping or passivity-based control. These can ensure both the stability of “e” and the boundedness of “d” by directly managing their relationship.

3. Modify your Lyapunov function to include both “e” and “d”, ensuring that the growth of “d” is controlled as “e” decays. You can read more about it here: https://www.mathworks.com/help/predmaint/ref/lyapunovexponent.html

4. If using LMI (Linear Matrix Inequalities), consider relaxing the conditions to account for the nonlinearity of “d”, or introduce new terms to capture the relationship between “e” and “d”. You can read more about it here: https://www.mathworks.com/help/robust/linear-matrix-inequalities.html

In conclusion, combining ISS analysis, nonlinear control and adjusted Lyapunov or LMI methods can help stabilize both “e” and keep “d” bounded.

Hope it helps!