2 Signals^ system s and m athem atical m odels
Appendix 2.1; Some elementary results on system stability
2.1.1 External or B ounded-Input B ounded-O utput (BIBO) stab ility
It is n o t possible to form ally define BIBO stab ility w ith o u t refe rrin g to Lebesque spaces. H ow ever this is beyond the scope of this appendix an d the interested reader is refered to V idyasagar (1978) for an excellent exposition of in p u t-o u tp u t stability theory. W e shall here restrict ourselves to an inform al definition w hich is given in ord er to clarify certain points that w ere m ad e in the m ain body of chapter 2.
D efin itio n ; A continuous-tim e causal system is ex tern ally sta b le if a bou n d ed input, u(t) < Ml, -oo < T < t < oo, produces a b o u n d ed o u tp u t y(t) < M j, T < t < oo. A w ell-know n necessary and sufficient condition for such BIBO sta b ility is that the im pulse response in (2.7) be such that
•foe
j|h (t)|d t < M <oo A2.1.1
i.e. h(t) is absolutely integrable over [0, «»). For an inform al proof of this result see Kailath (1980) or Chen (1970).
A necessary a n d sufficient co ndition for BIBO stab ility of d iscrete-tim e system s is that the im pulse response in (2.41) is such that
+ 00
^ h ( k ) | < M < oo A2.1.2
2.1.2 Lyapunov stability definitions
The definitions given in this section are entirely based on V idyasagar (1978). Notice that the stability conditions given becom e stricter as w e progress from just stable system s to globally asym ptotically stable systems.
C onsider the vector differential equation
w here x(t) g R" and f:R g x R" —>R ". A ssum e th at the function f is such that
(A2.1.3) h as a u n iq u e solution over [0, ©o) c o rre sp o n d in g to each initial condition for x(0) and that the solution d ep en d s continuously on x(0). The point Xo G R" is said to be an equilibrium point of the system (A2.1.3) at time to if
f ( t , X o ) = 0 , V t > t q A2.1.4
N ow , w ithout loss of generality w e can assum e that
f ( t , 0 ) = 0 , V t > t „ A2.1.5
From equation (A2.1.5) it is easy to see th at 0 is a solution of (A2.1.3) and hence if the system is in the initial state x(to) = 0, the resulting trajectory is x (t)
= 0, V t > to. N ow suppose that x(to) is n o t 0 b u t "close" to it. Then w h at is the
natu re of the resulting trajectory ?
D efin itio n : The equilibrium point 0 at tim e to is said to be sta b le at tim e to if, for each e > 0, there exists a 6 ( t o , e) > 0 such that
|x (t„ )||< 8 (t„ ,e ) = > ||x (t)||< e , V t S t „ A2.1.6
It is said to be u n ifo rm ly stable over [ t o , ©©) if, for each e > 0, there exists a 5(e) > 0 such that
||x (t,)||< 5(e) , t , > t „ = î.||x (t)||< e , V t > t , A2.1.7
(i.e. the sam e 8(e) applies for all t,).
D efinition: The equilibrium point 0 at tim e to is asym ptotically stab le at time to if it is stable at time to, and there exists a num ber 8i(to) > 0 such that
l|x (t)||-^0 as t-^o o A2.1.8
It is u n ifo rm ly asym ptotically stab le over [ t o , ©©) if it is uniform ly stable over
[ t o , ©©) and there exists a num ber 8i > 0 such that
||x (ti)||< 5^ , t^ > t(j => ||x (t)||-> 0 as t-^o© A2.1.9
D efinition: The equilibrium point 0 at time to is g lo b ally asy m p to tically stable if x(t) —> 0 as t —> oo (regardless of w hat x(to) is).
2.1.3 L yapunov's direct m eth o d
H aving given the L yapunov stability definitions in the previous section, it is n o w a p p ro p ria te to give w ith o u t p ro o f a v ery im p o rtan t theorem o n the asym ptotic stability of the class of non-linear system s defined by (A2.1.3). This theorem , together w ith theorem s on stability an d in stab ility of non-linear system s form the basis for L y ap u n o v 's first m ethod. H ow ever no fu rth er details w ill be given here and the interested read er is refered to V idyasagar (1978) for a detailed exposition of system stability.
L em m a A 2.1.1a: A continuous function W :R " is a locally positive
definite function (l.p.d.f.) if and only if
(i) W(0) = 0
(ii) W ( x ) > 0 , V x ? i O belonging to some ball defined by
Br = {x :||x ||< r} , r > 0
L em m a A 2.1.1b: A continuous function V i R ^ x R " -> R is an l.p.d.f. if and only if there exists an l.p.d.f. W :R ” -» R such that
V (t,x) > W (x ) , V t >0 , V X e Br , r > 0
D e fin itio n : A continuous function V :R g x R" —^R is said to be decrescent if there exists a function P(-) such that
V (t,x) < P(||x|l) , V t > 0 , V x e B r , r > 0
w here p(-) has the follow ing properties:
(i) p(-) is nondecreasing
(ii) p(0) = 0
(iii) p(p) > 0 w henever p > 0
T h eo rem A2.1.1: The equilibrium point 0 at time to of the system (A2.1.3) is
continuously differentiable decrescent l.p.d.f. V (t,x) swc/i that - V ( t ,x ) is an l.p.d.f..
2.1.4 Internal stability
Internal stability refers to the stability of a realisation of a system (Kailath, 1980). The realisation (2.14)
x(t) = A x (t) + b u (t) A2.1.10a
y(t) = c^x (t) + d u (t) A2.1.10b
is in te rn ally stable or stable in the sense of L yapunov if the solution of
x(t) = A x ( t ) , x(tg) = Xg , V t > tg A2.1.11
is globally asym ptotically stable. In fact it is well k n o w n (Kailath, 1980) that (A2.1.11) is going to be globally asym ptotically stable if an d only if
R e{X .(A )} < 0 A2.1.12
w here {^i(A)} are the eigenvalues of A. Sim ilarly, for discrete-tim e system s, the realisation (2.48) will be internally stable if an d only if
|%;(A)| < 1 A2.1.13
2.1.5 The L yapunov S tability C riterion
T h e o re m A2.1.2: A matrix A is a stability matrix, i.e. Re(^i(A)} < 0 for all eigenvalues of A , if and only if for any given positive-definite^ sym m etric matrix Q there exists a positive-definite symmetric matrix P that satisfies
A ^ P + P A = - Q A2.1.14
^ Any one of the following is a necessary and sufficient condition for a matrix M to be positive definite (Strang ,1980):
( i ) x^^ix > 0 for all vectors x.
(i i ) All principal submatrices have positive determinants. ( i i i ) All the eigenvalues of M satisfy > 0.
For a proof of theorem A2.1.2 refer to Kailath (1980) or V idyasagar (1978).
C o ro lla ry A 2.1.2a: If A is a stability matrix, then the Lyapunov equation
(A2.1.14) has a unique solution for every Q (proof in K ailath, 1980).
The physical basis of theorem A2.1.2 is the following. The quantity
V (x (t)) = xT (t)P x(t) A2.1.15
can be regarded as a generalised energy associated w ith the realisation. In a stable system the en erg y sh o u ld decay w ith tim e, co n sisten t w ith the calculation
• ^ V ( x ( t) ) = x^(t)P x(t) + x^Px(t)
= x ^ (t)[A ’'P + P A ]x (t)
= - x '^ ( t) Q x ( t) A2.1.16
The first derivative of V(x(t)) in the last equation w ill be negative, pro v id ed th at Q is positive definite. Sim ilar results exist of course for discrete-tim e system s, w ith the only m odification that the L yapunov equation in theorem A2.1.2 is now replaced by
P - A^PA = Q A2.1.17
By finding suitable functions V(x(t)), stability criteria can be established for several classes of non-linear system s, as w ill be d em o n strated in the proof of theorem A2.1.3 below.
2.1.6 A stab ility resu lt for lin earised system s (L yapunov's in d irect m eth o d or first m ethod)
T h eo rem A2.1.3: Consider the system
x(t) = f(x (t)) , t > 0 A2.1.18
Suppose that
and that f(-) is continuously differentiable. Define df ( x) A = 9x x = 0 a nd f^(x) = f(x) - Ax
and assume that
A2.1.20 A2.1.21 lim l l x l U o f^(x = 0 A2.1.22
Under these conditions, if the equilibrium point 0 of the system
q (t) = Aq (t) A2.1.23
is uniformly asymptotically stable over [0, «>), then the equilibrium point 0 of the system (A2.1.18) will also be uniformly asymptotically stable over [0, «»).
N ote that u n d e r condition (A2.1.22), (A2.1.23) is the linearisation of (A2.1.18) about X = 0.
Proof: Let
V ( x ) = x^Px A2.1.24
be a Lyapunov function candidate, w here P is the solution to the equation
A P + PA = - I A2.1.25
(a solution exists because of corollary A2.1.2a an d because A in (A2.1.23) is a stability m atrix by assum ption). Since P is positive definite, it satisfies for som e positive a and p the inequalities
a x ^ x <x^Px < p x ^ x , Vx g R"
From (A2.1.24) and using (A2.1.18) an d (A2.1.25)
V(x) = x^Px + x^Px = f^(x)Px + x^Pf(x)
= [f^(x) + A x] Px + x^P[f^(x) + A x ]
= f’[(x)Px + x'^Pf ^(x) + x'^( A ^P + P A )x
= - x^x + 2x^Pf j(x) A2.1.27
Because (A2.1.22) holds, choose a num ber r > 0, such that
||f^(x)||< -^11x11, V x :||x ||< r A2.1.28
Inequality (A2.1.26) m ay be rew ritten as
|x '^ P x |< p ||x |f A2.1.29
M ultiplying (A2.1.28) w ith (A2.1.29)
I x ’^ P x l • ||f^(x)|| < ^ | | x | | • pllxf A2.1.30
b u t |xT'Px|- ||f^(x)|| = |xT'Pf^(x)|- ||x|| A2.1.31
which m ay be easily show n by m ultiplying o u t the tw o sides of (A2.1.31) and com paring terms. Hence
|x'^Pf^(x)| • ||x|| < ^ ||x || • \\xf
=> |2x^Pf^(x)| < ^ x ^ x , V X :||x|| < r A2.1.32
From (A2.1.27) and (A2.1.32)
V ( x ) < - ^ , V x : ||x ||< r A2.1.33
But (A2.1.33) satisfies all the conditions of theorem A2.1.1. H ence (A2.1.18) is uniform ly asym ptotically stable.