consider the follo wing sequences yapuno vlik function Gi en strictly increasing se quence we say is yapuno vlik for if x for 1 is monotone nonincreasing on Switching System Theor em If is yapuno lik for then the switched system is stable Pr oof The ID: 30001 Download Pdf

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consider the follo wing sequences yapuno vlik function Gi en strictly increasing se quence we say is yapuno vlik for if x for 1 is monotone nonincreasing on Switching System Theor em If is yapuno lik for then the switched system is stable Pr oof The

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Nonlinear Contr ol Systems Lectur September 15, 2005 Switched System Stability Switched system equation )) )) with (0 for The function is piece wise continuous function of time that describes the switching of the system' ector eld between and generates sequence of switching instants: where is the alue of just before the th switch and is the time of the th switch. consider the follo wing sequences, yapuno v-lik function: Gi en strictly increasing se- quence we say is yapuno v-lik for if x; for +1 is monotone nonincreasing on Switching System Theor em: If is yapuno lik

for then the switched system is stable. Pr oof: The switching sequence is either (0 (1 (0 (1 (0 (1 Consider single ycle ig htar ow Gi en arbitrary let min and dene Choose such that By assumption for +1 +2 So that if +1 then )) for +1 +2 No let min Choose such that By assumption for +1 So if then )) for +1 But which means that +1 which means that for +2 So we can guarantee the -boundedness of er single ycle (if we start within ball of ). also assumed that )) +2 )) Let max can also nd an such that min So if then By the abo assumption on the monotone decreasing nature of on we kno

that for all This implies that we al ays start ycle within of the equilibrium which is suf cient to ensure that remains within of the equilibrium point for all time within the ycle. So the system is stable. yapuno condition is only suf cient. Here are some x- amples of ho we might nd yapuno functions. Consider the ODE where is locally Lipschitz, (0) and xg Clearly is an equilibrium point. Let' consider the candidate yapuno function dy Note that (0) and for So all we need to do is check the directional deri ati of dx dt )] for all So this type of system is asymptotically

stable. Let' reconsider the pendulum xample without friction, sin reat ener gy as candidate yapuno function. (1 cos Note that (0) and that The directional deri ati (we aluated earlier) as

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So this is yapuno function, ut since we can only conclude that the origin is stable. cannot say the origin is asymptotically stable. As can be inferred from the phase portrait of this system we kno that this is the case. No consider what happens if we add friction with the same candidate yapuno function. In this case So is ne gati semi-denite. But it is not ne gati denite

since for and ANY So all we can conclude is that the equilibrium is yapuno stable. cannot say the origin is asymptotically stable. But again, if we appeal to the phase portrait analysis, we kno the equilibrium point is also asymptotically stable. So this candidate yapuno function is not useful in erifying asymptotic stability for this system. So let' try dif ferent candidate yapuno function. (1 cos where is positi denite matrix. Note that 11 12 21 22 is positi denite if and only if 11 22 and 11 22 12 use these constraints in selecting to ensure is ne gati denite. The

directional deri ati of is (1 22 sin 12 sin 11 12 12 22 need to choose to force The cross terms are sign indenite so choose ij to cancel these terms. In other ords, 22 (1 22 sin 11 12 11 12 The other choices are made to assure ne gati deniteness. In other ords, 12 12 sin 22 12 =m 12 22 So we choose 12 =m 22 11 12 to ensure on So we can conclude the pendulum (with friction) equilibrium point is asymptotically stable. This xample illustrates an important limitation of the yapuno stability theorem. It is only suf cient and our inability to nd yapuno function does

not mean the equilibrium point is not stable. The last xample suggests that the appropriate use of yapuno theory in olv es using parametric amily of candidate yapuno functions and then nding parameters to assure The follo wing illustrates the so-called ariable gradient method ariable Gradient Method: Let be scalar function of and let Then the deri ati is ry to select to ensure is ne gati denite. Note that we' re not free to pick an must guarantee that is the gradient of scalar function. The follo wing theorem pro vides necessary and suf cient conditions for this to

occur Theor em: is the gradient of scalar function if and only if the Jacobian matrix is symmetric. Example: Consider ax where is locally Lipschitz, (0) and for ant to nd such that need to select so that the deri ati of is )( ax and dy consider the follo wing candidate The symmetry conditions require

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Cancel the cross terms since the are indenite by setting a This mak es the deri ati of equal to a )] Mak the simple choice in which constant constant constant and let only be function of The symmetry condition then reduces to and the cross terms are remo ed if a So

we get a Inte grate to get a )] dy dy dy where a The condition can therefore be sho wn to become a which is ne gati denite if a and Remark: This as rather tedious computation, ut one might ask whether or not it is possible to automate the search for yapuno function. This has been xplored in the past. The references on the webpage pro vide tw papers sho wing ho to automate the computation. The rst one simply modes where =1 where are smooth “basis functions. This can often be solv ed using con programming methods. Another approach in olv es polytopic form of Both methods are

amenable to allo wing the construction of computer generated yapuno functions.

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