Instability of a parameter varying system whose parameters belong to a compact set:Suppose, there is a systemx˙=f(t,γp(t),x)with x∈R2. For my specific case, parameter vector γp is a scalar and known monotonic function with a compact image set (for example γp(t)=e−t, thus γp∈[1,0)). I would like to show instability of this system

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bobthemightyafm · Accepted Answer

Step 1I don&#039;t think it will be the case in general (I mean without taking advantage of some particular structure of your problem).Just looking at the question &quot;Can instability of all frozen parameter cases let me conclude anything about the instability of the original system?&quot; the answer is: not always.For example, consider this system:x˙=[01γ1(t)γ2(t)]xwhere γ1, γ2 are periodic functions with period 2, and complyγ1(t)=−1,γ2(t)=0.01 for o≤t≤1 and γ1(t)=0.01,γ2(t)=−1. Hence, this system switches betweenx˙=[01−10.01]x, t∈[0,1)andx˙=[010.01−1]x, t∈[1,2)and so on, periodically.Note that in each case, by analyzing the frozen parameters, the matricesA1=[01−10.01]andA2=[010.01−1]are not Hurwitz, so each system is unstable on its own.Despite this, the switching system is asymptotically stable. Look at what happens in the switching instants. For example, at t=1:x(1)=exp⁡(A1)x(0)and at t=2:x(2)=exp⁡(A2)x(1)=exp⁡(A2)exp⁡(A1)x(0)and so on, so that we can look at every two jumps to obtain:x(k+2)=exp⁡(A2)exp⁡(A1)x(k)=A~x(k)for k even. Interestingly, the matrixA~=exp⁡(A2)exp⁡(A1) has eigenvalues in side the unit disk, namely0.1067+0.6002i,0.1067−0.6002i.Hence, the sequence x(0), x(2), x(4),⋯ converge to the origin asymptotically. A similar thing happens to x(1), x(3), x(5),⋯ and its not hard to see that x(t) in the intervals between converge to the origin as well. The reason is that the unstable behavior of A1 is corrected by A2 and viceversa. Through an average argument you might also see that the average of the two systems is stable.In summary, in general you won&#039;t be able to conclude instability of the parameter-varying system by looking at the frozen parameter systems. However, you might take advantage of some particular structure of your setting, namely that γ(t) is scalar or monotone. Hence, without more context/details it is hard to know.

Instability of a parameter varying system whose parameters belong to a compact set: Suppose, the

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