Dear Ibrahim,

You are right, the stability analysis of TV systems is more complicated. But, the fundamental approaches are the same as for any other system. I guess I have collected the most widelly used methods in my book Continuous-Time Systems, Springer, 2007. It is available on this site.

Dear Ibrahim,

You can find some stability analysis of TV systems in my paper:

G. Hovhannisyan. “Stability of planar non autonomous dynamic systems.” Advances in Difference Equations 2013 (2013), 144. 38 pp. doi:10.1186/1687-1847-2013-144

It is available on this site.

Dears Yuriy/Gro,

Thanks a lot for your valuable comments. I reviewed your publications and they are quite interesting and personally I am impressed with their depth. More comments on the differences between the TIV and TV systems in the sense of solution existence and stability analysis are more than welcomed.

More specifically, if you know the impulse response of the SISO LTV system and intergate its absolute value over the auxiliary time-variable, then the system is BIBO stable if the integral does not produce infinity for any time variable t. See p. 371 in my book. In state space, do the same for MIMO LTV systems with respect to the norm of the impulse response matrix. See p. 393. It is a very reliable way to ascertain stability of any LTV system.

I do agree with Prof.Shmaily, especially considering stability issues of LTV.

That's true. However, let's consider the more pragmatic situation (that we face in many control applications) in which:

1. The system is an LTV.

2. The parameters of the system (A(t), B(t), C(t), and D(t)) are varying and unknown.

Now, two important questions are raised:

I. What are the properties of the time varying parameters that can achieve the solution existence for the system states?

II. What are the boundaries (or roughly saying assumptions) that should be imposed on the time varying parameters for guaranteeing the system stability?

Personally, I believe the first step is to study the transition matrix properties and from them we can conclude the corresponding properties and assumptions that can be drawn to the parameters.

Your feedback is more than welcomed.

If you want to ascertain stability via the state transition matrix, then specifics follow:

1. The measurement equation has nothing to do with system stability, so forget about C(t) and D(t).

2. Now recall the definition of stability: a system is stable if its output does not diverge as long as its intput does not diverge. That means that you may assume any finite input. Then assume it zero and forget about B(t).

3. You now have the homogenous state equation x´(t) = A(t)x(t). Then write its solution as x(t) = F(t,t_0) x(t_0), where F is the state transition matrix.

4. Again follow the definition and arrive at a conclusion that the system is stable (x(t) does not diverge) if F has finite norm. Any norm can be applied. The infinity norm is most simple to mean that no one component of F goes to infinity with time t.

Hope it will help.

Dear Ibrahim

i would like to add my little experience to the discussion.

Let us assume that the matrix A(t) is a function of a time varying paramter w(t), say A(t) = A(w(t)).

About the existence of solution, it is shown (see, for example, "Robust control of linear parameter varying systems" from Francesco Amato, Springer) that if A is differentiable in w and piecewise continuous in t, then the solution of dx(t)/dt = A(w(t))*x(t) exists and is unique.

About stability, i think it is a little hard to derive general conditions on the parameters w(t). However, there exists a straightforward way to check stability if the system dx(t)/dt = A(w(t))*x(t) is a polytopic one, i.e., if the matrix A(w(t)) belongs to a convex set determined by some (matrix) vertices A1, A2,...An.

In this case, the exponential stability of x=0 holds if a positive definite matrix P exists, such that

transpose(Ai)*P + P*Ai < 0, for i = 1...n

In other words, a classic Lyapunov equation must be checked for each matrix vertex of the politope within which A(w(t)) varies.

The stability condition that i wrote is only sufficient and is proved to be quite conservative. However, there are several extensions to this condition, based on similar ideas.

Hope this was helpful

Hello Leonardo,

That´s quite interesting. However, this would raise the issue of finding the vertices A1, A2,...An.

I guess the comment of Prof. Shmaliy is interesting in the sense that it is simple and generic for checking the stability of a tv system. However, I think it is difficult to apply it in case one of the time varying parameters is unknown (which is unfortunately the case of many control problems). Therefore, I would suggest looking on this interesting case and find the features of those parameters that can ascertain stable behavior of the system.

Hi Ibrahim,

You got some very good responses and I only add a few thoughts of my own.

RG tells me that you showed interest in my paper "Can stability analysis be really simplified?" This paper expresses my interest in stability, which comes from Adaptive Control, which is nonlinear, beyond being nonstationary.

Looking around, I found lots of methodologies, one more complex than the other, all requiring lots of preliminary conditions.

My first question was: if d(t) is impulse function (I can't write here delta(t)) and as we know that the linear equation xdot(t)=-x+d(t) has a solution (impulse response) does the change to  the nonlinear xdot(t)=-x^3+d(t) make the solution just disappear? Definitely not!

But then, many more conditions are related to the proofs of stability, uniform continuity being the most common.  Here I thought: If I use a pulse command to move a robot from A to B and then back from B to A, is the robot going to brake just because the f(x,t) in xdot=f(x,t) is not continuous? OK, here one enters the nonstationary/nonlinear world and finds out that linear thinking and linear theory are far from sufficient.

For starters, assume that just one parameter k in your (otherwise) LTI plant can vary between 1 and 10. You check stability and find out that it remains asymptotically stable for all fixed k values between 1 and 10. Then, you let the parameter vary freely within the admissible region 10 (for matrices, >0 means positive definite) such that P(t)A(t)+A'(t)P(t)=-Q(t). In general, the equation is Pdot(t)+ P(t)A(t)+A'(t)P(t)=-Q(t).

I should not make this message into a dissertation, so I will stop here. Please feel free to come back with any question you may have.

Best regards,

Itzhak

Hi Itzhak,

Thanks for your explanation.

Actually, your comment raises a couple of questions:

Q1: Don't you think linearizing the nonlinear TV system (by employing its Jacobian A(t)) might result in a significant inaccuracy for describing the behavior of the original nonlinear system?

Q2: For simplicity, suppose that we are given a linear TV system. Do you think the nature of parameters variation-stationary or nonstationary- is correlated to the choice of a Lyapunov candidate? 

Kind regards,

Ibrahim

Hi Ibrahim,

1. I DON"T linearize! PERIOD.

Nevertherless, when you deal with a new system and want to get any idea about it, any method is right. For example, in particular if you want to see what happens around equilibrium points, doesn't hurt to linearize locally and to get some idea.

In one of my examples of nonlinear systems, where i analyzed its stabilty without any approximation, I finally got that the system does not diverge and yet, that it has three equilibrium points. So, how do you know which equilibrium is also attractive and which is not? Linearizing aound each one of of them gave the answer pretty easily.

In some instances, instead of working with the general xdot=F(x,t), I found it more convenient to use the form xdot=A(x,t)x(t)+B(x,t)u(t). However, the system matrices contain the nonlinear terms without any approximation.

2. Yes, finding the right Lyapunov function for your specific system is an art.