Consider an observable system and consider a Lyapunov function which is definite positive in the output space (see image). does the convergence of the output error imply the convergence of the state error ?
If it is true, how can we prove it mathematically ? otherwise, give a counterexample which demonstrate the invalidity of the proposed preposition.
Hi Omar,
For the specific Lyapunov function that you are using, I feel the statement will be conditionally true. The reason is as follows.
Since ey = C.ex,
V = (C.ex)T.P.(C.ex) = exT.(CT.P.C).ex
So only if (CT.P.C) is positive definite regardless of C, does convergence of output error imply convergence of state error - otherwise not !!
Now (courtesy my pathetic math !) I don't immediately recall a theorem or lemma that establishes positive definiteness of (CT.P.C) if P is positive definite, regardless of C ! Maybe you could look around and check if that is universally true !!
If so, then the two convergence are linked to each other; if not, then it depends on the C and how the Ricatti equation plays out thereafter !!
Cheers !
-Sanjay
Hi Omar,
For the specific Lyapunov function that you are using, I feel the statement will be conditionally true. The reason is as follows.
Since ey = C.ex,
V = (C.ex)T.P.(C.ex) = exT.(CT.P.C).ex
So only if (CT.P.C) is positive definite regardless of C, does convergence of output error imply convergence of state error - otherwise not !!
Now (courtesy my pathetic math !) I don't immediately recall a theorem or lemma that establishes positive definiteness of (CT.P.C) if P is positive definite, regardless of C ! Maybe you could look around and check if that is universally true !!
If so, then the two convergence are linked to each other; if not, then it depends on the C and how the Ricatti equation plays out thereafter !!
Cheers !
-Sanjay
Dear Omar,
It is true for linear system - the stability of output (and also of the whole state) depends on function exp(At), A- state matrix, (see the time solution y(t)=...).
For nonlinear system the output can converge stable point and the state can be unstable, e.g. for the system:
x1'=2x1+x2;
x2'=-x2+1/x1;
y=x2
the output will converge 0 whereas x1 infinity (+ or -).
Therefore when you use Lyapunov theory for nonlinear system the Lyapunov function should be the function of every state variable.
Best regards,
Joanna
Hi Omar,
RG tells me that you might have looked at a paper of mine on Adaptive Control and this brought me to your question.
If the system has n state variables and m outputs (n>m), the output matrix is Cm*n, then your P is of order m*m and no combination CTPC, which is of order n*n, can become positive definite.
If you look at other papers on stability, then a new Theorem of Stability claims that the trajectories end with Vdot identical zero, which implies that next derivative of Vdot is also zero and so is next and all next that you may need, and you end with enough relations that may tell you what really happens with all variables of the system.
Hope this helps.
Yes, things are delicate and can be very confusing and I also got a bit confused, or at least, I rushed to conclusions.
A second (or third) reading always helps. In order to analyze stability of the entire system, the Lyapunov function must be Positive Definite (i.e, as a function of ALL variables). My conclusion assumed that V was PD wrt all variables.
If your V is PD only in terms of ey, you can only prove stability with respect to ey.
On the other hand, if a system contains many variables, x,y,z,... and you cannot find a PD Lyapunov that "works," yet you can find that V=x2 has a negative definite or semidefinite derivative (no matter what the others do), you do know how x will behave.
If you can find functions for others, too, then maybe you can reach conclusions for more, or even for all variables.
You remind me the Vectorial Lyapunov function approach, which tries to find more than one function and combine their partial conclusions. Others claim that, if the system is stable, then there must be a PD Lyapunov function that contains them all. This is true, yet in nonlinear systems is not that easy to find the "best" Lyapunov and so, once you have the problem and get a solution, you use the solution, even if, maybe (or maybe not), later you may find a simpler solution.
...and to complete my answer, in LTI systems, the new result on final identity (rather than simple equality) relation, combined with observability, does lead to asymptotic convergence of the entire state.
If your system is edot=Ae and you end with ey=Ce=0, then you get dotey=CAe=0, ,,, CAn-1e=0. and this leads to Oe=0, where O is the NON-SINGULAR Observability Matrix, which then implies e=0.
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