a continuo usly differentiable func tion V in ding lyapunov funct io ns R-R is c alled positive definite in a reg ion rans fo rming o ne diffic ult pro blem to a U CRcont aining the origin if ot he 1.V(0)=0 Let the linear system (a)>0,c∈Uand≠ A func tion is c alled positive semidefinite if dt Condit io n 2 is replaced by V(a)20 be stable. Pie k Q positive defi nit e. T he THEORE quat ion If there exists a func tion v: rh r that ATP+PA=-Q positive defi nite suc h that has always a uniq ue solutio n wit h P positive a2 definite and the funt io n f(m)=-W(x) V(x) is neg at ive semidefinite, then the solut ion Lw t=0 is stable. If dv/ dt is negative defi yapunov tunc tion t hen the solut io n is also asym ptotic ally stable m p e rying Syst ems System mat rix a1 a2 亚=( A a ay definition 3 T he solution a(t)=0 is uniformly sta ble if fore>0 the 6(e) dependent of to, suc h that where g1>0 and 92>0 are posit ive. Assume t hat the mat rix p has the form (to<6→‖(t川‖<εVt≥to≥0 T he solut io n is uni formly asymptotically sta ble ifo rmly stable and there independent of to, s ue h that a(t)0 as T he Lya punov equat ion bec omes t→∞, uniformly in to, for all a(to)‖<c.口 a2 a1+ a4 a dEFINITION 4 a continuous fun c tion a:[0,a)→0,∞) to belong to class K Co ndit io ns for existen e e. When is P>0? inc reasing and a( 0)=0. It is said to belong Proofs are given in the mat hematics course on to class Koo if a=∞anda(r)→∞as mat ric es! C K.J. Astrom and BWittenmarkFormalities Definition 2 A continuously dierentiable function V : Rn ! R is called positive denite in a region U  Rn containing the origin if  1. V (0) = 0  2. V (x) > 0; x 2 U and x 6= 0 A function is called positive semidenite if Condition 2 is replaced by V (x)  0. Theorem 1 If there exists a function V : Rn ! R that is positive denite such that dV dt = @V T @x dx dt = @V T @x f (x) = ￾W(x) is negative semidenite, then the solution x(t)=0 is stable. If dV =dt is negative denite, then the solution is also asymptotically stable. Finding Lyapunov Functions "Transforming one dicult problem to an￾other" Let the linear system dx dt = Ax be stable. Pick Q positive denite. The equation AT P + P A = ￾Q has always a unique solution with P positive denite and the funtion V (x) = xT P x is a Lyapunov function Example System matrix A =  a1 a2 a3 a4  Q =  q1 0 0 q2  where q1 > 0 and q2 > 0 are positive. Assume that the matrix P has the form P =  p1 p2 p2 p3  The Lyapunov equation becomes 0 @ 2a1 2a3 0 a2 a1 + a4 a3 0 2a2 2a4 1 A 0 @ p1 p2 p3 1 A = 0 @ ￾q1 0￾q2 1 A Conditions for existence. When is P > 0? Proofs are given in the mathematics course on matrices! Time-Varying Systems dx dt = f (x; t) Definition 3 The solution x(t)=0 is uniformly stable if for " > 0 there exists a number (") > 0, independent of t0, such that kx(t0)k <  ) kx(t)k < " 8t  t0  0 The solution is uniformly asymptotically stable if it is uniformly stable and there is c > 0, independent of t0, such that x(t) ! 0 as t ! 1, uniformly in t0, for all kx(t0)k < c. Definition 4 A continuous function : [0; a) ! [0;1) is said to belong to class K if it is strictly increasing and (0) = 0. It is said to belong to class K1 if a = 1 and (r) ! 1 as r ! 1. c K. J. Åström and B. Wittenmark 3