Revision: Statement of Lyapunov matrix lemma? Statement of LaSalle’s invariance principle?
Lecture 7 Class K and KL functions Comparison lemma
Recommended reading • Khalil Chapter 3 (2nd edition)
Outline: • Class K, K and KL functions • Properties of the above functions • Comparison lemma • Summary
Class K, K and KL functions • Ubiquitous in stability analysis. • K and K classes of functions are used to upper and lower bound arbitrary nonlinear functions (e.g. positive definite V). • KL functions are used to bound the solutions of UGAS systems. • There are subtle and non-trivial relations between K and KL functions (e.g.comparison principle).
Class K and K functions • Class K satisfy (we write K): • : [0,a) R 0 • is continuous • (0)=0 • is strictly increasing • Class K satisfy (we write K): • All of the above with a= • lims (s)=
Examples Class K: Class K:
Class K? A B D C
A relationship • Note that the following holds: K K • The opposite does not hold in general ! • Example: 1(s)=arctan(s) and 2(s)=tan(s) satisfy 1,2 K and 1,2 K.
Class K ? A C D B
Notation • A composition of two functions 1,2 is denoted: • -1 denotes the inverse function of , that is: • Example: 1(s)=s2, 2(s)=arctan(s)
Properties of K functions
Property 1: • K functions are one-to-one and, hence, they are globally invertible! K-1 • Moreover, we also have that K-1 K
Property 2: • For arbitrary 1,2 K we have that 1,2 K12 K • We can take an arbitrary number of compositions of K functions and we still obtain a K function
Property 3: • The following “weak triangle inequality” holds (s1+s2) (2s1)+(2s2) s1,s2 0 • Actually, given arbitrary , K, we have (s1+s2) (s1+(s1))+(s2+-1(s2)) s1,s2 0
Property 4: • Suppose that a function V is: • positive definite; • radially unbounded Then, there exist 1,2 K
A consequence of Property 4 • We can fit the set c:={x: V(x) c} in the ball of radius 1-1(c) V(x) c |x| 1-1(c) • We can fit the ball of radius r in the set c, c=2(r): |x| r V(x) 2(r)
Graphical interpretation 2(|x|) 1(|x|) V(x) c c 1-1(c) r |x| r V(x) c |x| 1-1(c) V(x) 2(r)=c
Properties of K functions • If 1,2 K with domain [0,a), then • 1-1 K with domain [0,1(a)) • 12 K • (s1+s2) 1(2s1)+1(2s2), s1,s2 a/2 • If V>0 then there exists r>0 such that
Class KL • Class KL satisfy (we write KL): • : [0,a) X R 0 R 0 • is continuous. • s (0,a), (s,.) is decreasing to zero • t 0, (.,t) K NOTE: Sometimes we also require • strictly decreases in the second argument
Examples • Class KL functions:
A property of KL, K functions • Suppose KL, 1,2 K, then we have that 1( (2(s),t) ) KL • A similar statement holds for K functions but the domains need to be appropriately restricted. • The above property is used in the next lecture.
Another property of K functions • Given an arbitrary K, there exists K that is locally Lipschitz and such that (s) (s) s [0,a) (s) (s)
An application to differential equations • Let K be locally Lipschitz. Consider the following differential equation Then, its solution u(s,t) satisfies u(s,t) KL
A comparison principle • Let K be locally Lipschitz and consider the differential inequality: Let the solution of the following equation Be denoted as (u0,t) KL. Then, we have V(t) (V0,t) t 0.
Summary • Class K, K and KL functions are extremely important tools in stability analysis of nonlinear systems. • K functions are globally invertible. • The properties of these functions that we discussed are used in various proofs. • The comparison lemma gives us a bound on solutions of a differential inequality via an auxiliary differential equation.
Next lecture: • Stability of time-varying systems. Homework: read Chapter 3 in Khalil
Thank you for your attention!
