SOSTOOLS for differential-algebraic systems?

I am trying to find a Lyapunov function for a differential-algebraic system (DAE) of the form:

,

where () represents the dynamic states and () represents the algebraic variables.

Due to certain constraints, I cannot eliminate the algebraic variables, i.e., I cannot explicitly write ().

Has anyone approached this type of problem using SOSTOOLS while keeping the algebraic variables in the formulation? Any suggestions on how to proceed, or any mini examples and references on SOS-based Lyapunov analysis for DAE systems, would be highly appreciated.

[Torsten]

The MATLAB tools to solve such systems are ode15s or ode15i (without Lyapunov function derivations).

[Sam Chak]

Hi @Husni Rois Ali

How do the algebraic variables () influence the search for a Lyapunov function, , which is primarily a function of the state variables ()?

By the way, can the in the expression be treated as coefficients so that you can continue with the regular procedure of searching an abstract, non-physical SOS Lyapunov function that yields for ?

[Husni Rois Ali]

Hi @Sam Chak

Thank you for your answer.

Regarding how (y) influences (x), this is actually the part that is still unclear to me. I am not sure whether the Lyapunov candidate should be formulated as (V(x)) only, or whether it should explicitly include the algebraic variables, i.e., (V(x,y)).

If you have any experience dealing with this type of DAE problem or SOS-based Lyapunov construction, please let me know :-)

[Sam Chak]

Hi @Husni Rois Ali

Your system is defined as dx/dt = f(x, y) subject to the constraint g(x, y) = 0.

If your Lyapunov function is a function of x only, then

dV(x)/dt = dV/dx*dx/dt.

But if your Lyapunov function is a function of x and y, then

dV(x,y)/dt = ∂V/∂x*dx/dt + ∂V/∂y*dy/dt.

However, the derivative dy/dt is unavailable. Moreover, are you trying to prove that the non-system states as via the Lyapunov function?

You can find some examples in Getting started with Sum of Squares and the SOSTOOLS user's guide.

[Torsten]

However, the derivative dy/dt is unavailable.

dg/dx*dx/dt + dg/dy*dy/dt = dg/dx*f(x,y) + dg/dy*dy/dt = 0

thus

dy/dt = -(dg/dy)^(-1) * dg/dx*f(x,y)

[Sam Chak]

Hi @Husni Rois Ali

Please follow @Torsten's insightful approach to derive dy/dt from the equality constraint, . That way, you can formulate the SOS Lyapunov function as , if you wish to also prove that as .

Just to be sure, your purpose in finding the Lyapunov function is to prove that the system eventually comes to rest at the equilibrium, , correct? Suppose you manage to find an SOS Lyapunov function and show that , which implies that and as . When this occurs, does the constraint still hold?