- Email: [email protected]
State Space Extension Method in the Theory of Absolute Stability
Copyright ~ IFAC Nonlinear Control SystmlS, SI. Petersburg, Russia, 200 I
~
IFAC
0
C>
Publications www.elsevier.comllocatelifac
STATE SPACE EXTENSION METHOD IN THE THEORY OF ABSOLUTE STABILITY Nikita Barabanov
Dept. Software Eng. Electrotechnical University St. Petersburg RUSSIA nikita0freya.etu.ru
Abstract: State space extension method is applied to construct new quadratic constraints and derive new frequency domain criteria for absolute stability. Copyright @2001 IFAC Keywords: absolute stability, quadratic constraints, Lyapunov functions
1. INTRODUCTION
of type "quadratic form of coordinates, nonlinear function, and convolution of coordinates and nonlinear function with given sums of decreasing exponents" .
We consider frequency domain criteria for absolute stability of systems in different classes of nonlinear functions . To apply a standard technique based on Kalman-Yakubovich lemma (Gelig et al. , 1978) or interior point algorithm, and Lyapunov function method it 's necessary to construct particular (local or integral) quadratic constraints, which use information about nonlinear functions.
In this paper there are presented three examples of application of state space extension method.
2. SYSTEMS WITH CONSTRAINTS ON DERIVATIVE OF GAIN FUNCTION
There is no general methods to construct quadratic constraints for nonlinear functions having arbitrary given properties. For example, even for monotone functions there does not exist corresponding quadratic constraint. To overcome this difficulty one can use state space extension method. The idea of this method consists of introducing of additional differential equations to the original system and then to construct new quadratic constraints for extended system. Such constraints may exploit different properties of non linear functions. Each new nontrivial quadratic constraint results in new frequency domain criteria for absolute stability in corresponding class of non linear functions . In terms of Lyapunov functions, state space extension method allow to obtain frequency domain criteria for existence of Lyapunov function
1. Consider system dx/dt
= Ax + b~,
(1)
~(t) =
kl(t)C*X(t) ,
(2)
where A, b, c - are matrices of dimension n x n , n x 1, n x 1 respectively, J-L is positive number and k(-) is differentiable function such, that 0 $ k(t) < J-L for all t ~ o. We transform this system in a following way: dx/dt
= Ax + b~ , ~(t)
1233
a
= c*x -
= k(t)a(t) ,
J-L-l~ ,
(3)
(4)
Assume, that function k satisfies following inequalities:
k(t)
~
~
Cl for all t
Proof According to Kalman-Yakubovich lemma (Gelig et al., 1978) for any f ~ 0 there exists a
Hermitian matrix H. and vector g. such that 0,
(5)
2y* H.((P - ({3 - f)I)y + q(1) + r*y(1 ( ) t2
J
{[(dk(t)/dt - 2a)/k(t)]+ - 2{3}dt
tl
Here Cl and C 2 are positive constants, which depend on function ~. Introduce transfer function of system (3), (4):
W(s)
= cOCA -
=-
~ C2 (6)
for all vectors y E R m and all (1 E R . Moreover, in this case go(P + (31 - iwI)-lq = "IjJ(iw) for all w ~ o. Define 1l".(iw) = Re(Y(iw - (3 + f)). Function 1l". depend on f continuously. Therefore, H. -+ Ho, g. -+ go as f -+ o. Denote y.(w) = (P({3 + f - iw)I)-lq. We have
1l".(iw) =1 g.y.(w) 12, 2fYO(W)* H.yo(w) + 1l"0( iw) = 1 g.yo(w)
sI)-lb,
Now we choose rational function
y.(w)
Yes) = r*(P - sI)-lq, where P + {31 is Hurwitz m x m-matrix, r,q are m-vectors, such that pair (P, r) is controllable, pair (q, P) is observable, and following inequality holds:
1l"(iw) = Re{Y(iw - (3)} for all w ~ o.
~ 0
= Py + q(1.
TP
+ P*T + (3T
~
Tq+r = o.
(9) (10)
yo(w) + f((P + (3I) - iwI)-2 q + 0(f2), 1s
= iw - (3+0(f2) .
2yo(w)* Hoyo(w) = X(iw). According to Kalman-Yaku bovich lemma (Gelig et al., 1978) there exists a Hermitian matrix H such, that 2 (x) * H ( (A + (3I)x +
Inequality (7) means, that there exists a factorization of function 1l", that is, there exists a rational function "IjJ with Hurwitz denominator, such that 1l"(iw) = "IjJ(-iw)"IjJ(iw) for all w ~ o.
,
2fYO(W)* Hoyo(w) = -1l"o(iw) + 1l".(iw) - 2fRe[(goYo(w))"go(P + (31 - iwI)-lq] + 0(f2) dyes) = fRe[d;- 1 s = iw - (3 . d"IjJ(s) . + 2"IjJ( -tw)d;- 1 s = tw] + 0(f2) = fX(iw) + 0(f2),
(8)
0,
2
1
Therefore,
(7)
Inequality (7) may be also presented in form of LMI: there exists Hurwitz m x m-matrix T such that
=
1l".(iw) = 1l"o(iw) + fRe d~~S)
Consider additional equations:
dy/dt
12
2 1 9.y 1
y
b~ )
(P + (3I)y + q(1
+ 6 1c·x 12 +2ay· Hoy for all x ERn, yE Rm ,
~
+ r.
y
~
(13)
~ 0
E R. Define functions
Define function
Vo(x, Y)=(;)*H(;), V1(y)=y·HoY, .) R ([dY(s-{3) X(tW = e ds
2 ()d"IjJ(s)]1 .) "IjJ -s d;- s = tW .
Vex, y, t) = Vo(x, y)
+ k(t)Vl (y) .
Inequality (13) implies: Theorem 1. Assume that there exist positive number 6 such, that for all w ~ 0 following inequality holds:
Re[(Y(iw - (3)(W(iw - (3) + J.L-l)] ~ ax(iw) 1W(iw - (3) + J.L- 1 12 + 61 W(iw - (31 2 .
dV(x(t), yet), t)/dt ~ -6(c·x(t))2 - 2aV1(y(t)) - 2{3Vo(x(t), yet)) + (dk(t)/dt - 2{3k(t))Vt(y(t)) ~ -6(c*x(t))2 + ([(dk(t)/dt - 2a)/k(t)]+ - 2(3)V(x(t) , yet) , t)
(ll)
Then system (3), (4) is absolute stable in class of functions ~ satisfying inequalities (5), (6).
for all solutions of system (3), (4), (8). 1234
(14)
From (12) it follows Ho > O. Similarly, inequality (13) implies H > o. Hence, Vo > 0, VI > 0, and V> o.
limt_oo{ln(V(x(t) , y(t) , t)) ::; -In(2)/T, and system (3),(4) is stable exponentially and uniformly over the class of functions ~, satisfying conditions (5) , (6) . 0
We integrate inequality (14):
This theorem is a modifocation of "average logarithmic" criterion published in (Venkatesh, 1977) . The difference consists of new parameter 0 and less restrictive condition (6) .
for all t2 ~t 1 ~ o. Hence, zero solution of system (3) is Lyapunov stable. Besides, for any T > 0 and p(t) = [(dk(t) / dt - 2o)/k(t)J+ - 2/3 following inequality holds: T
V(x(T), y(T) , T)
+0
T
j exp(j p(T)dT)(C'X(t))2dt o
T
Example 1. Consider system (1) , (2) with transfer function 1
W (8 ) =
T
82
+ 8 + 1.
According to circle criterion this system is absolute stable in class of functions kl (-) such, that o ::; kl (t) ::; 3 for all t 2 o. The maximal number /1max such, that system (1) , (2) is absolute stable in class of functions k l , satisfying inequality 0 ::; kl (t) < /1max for all t 2 0, is equal approximately to 4,63.
::; exp(j p(T)dT)V(X(O), y(O), 0) . o Now we show, that there exists T > 0 such, that for any number t > 0 there exists a number h E [t , t + T ] such, that
Now apply theorem 1. Set /3 = 0, 0 = 1, Y(s) = (s + 1) / (8 + 2/1)2. Then all conditions of theorem 1 are fulfilled if sup{ kl (t)} < 00, inf{ dkl (t)/ dt} > -2. Thus, system (1) , (2) is absolute stable in class of functions kl (-) satisfying these conditions.
V(x(h) , y(h), h) ::; V(x(t), y(t), t) / 2. (15) Assume that property (15) doesn 't hold for all + T] . Then exp(Jtt+ T p(T) dT) > 1/ 2 for all T E [0, T] . Therefore,
h E [t , t T
oj
(c*x(t)fdt ::; 2exp(C2)V(x(t), y(t) , t)
o
3. SYSTEMS WITH INTEGRAL CONSTRAINTS ON GAIN FUNCTION
::; 2 exp( C2)V(x(0), y(O) , 0).
Denote C 3 = 20- 1(HCl ) exp(2C2)V(x(0) , y(O) , 0) . Then
Consider system
j (a(t)2 + ~(t)2)dt ::; C
~(t)
3,
o and for some number C4 , which depend only on coefficients of system (3), (8), following inequality holds:
= Ax + b~,
(16)
= k(t)c'x(t) ,
(17)
dx/dt
T
A is Hurwitz n x n-matrix, b, care n-vectors, and function k(·) satisfies inequalities:
1k(t) I::; /1 ,
(18)
t+T
j
(11 X(T) 112
t
j k(T)dT ::;
+ 11 y(T) 112)dT
::; C 4 (11 x(t) 112
+ 11 y(t) 11
2
).
From Vo > 0, and (15) it follows, that for some number 01 > 0 and all h E [t , t + T] following inequality holds: 11 x(h) 112 + 11 y(h) 112~ 01 (11 x(t) 112 + 11 y(t) 11 2). Then
for all t
X
(11 x(t) 112
~
0 and some positive numbers a , /1 .
To obtain stability criterion we make a coordinate transformation: (1 = kx . Introduce additional system:
j (11 X(T) 112 + 11 y(T) 112) dT
t
(19)
Inequalities (18) , (19) hold, for example, for periodic functions k with zero mean values.
t+T
01T::;
a
o
+ 11 y(t) 112)-1 ::; C4 ·
Hence, T ::; C4 / 01 . Thus , if T > C4 / 01 , then for any number t > 0 there exists a number h E [t , t+ T] such, that V(x(h) , y(h), h) ::; V(x(t) , y(t), t) / 2. Therefore,
Notice, that if x(-) is a solution to system (16), (17) , then y(t) = J~k(T)dTX(t) is a solution to equation (20). Therefore, a quadratic Lyapunov 1235
function for the system (16), (17), (20) is a Lyapunov function for system (16), (17), which is quadratic with coefficients of the form Cl + C2 J~ k(T) dT .
Example 2. Consider differential equations (Kotchenko and Bodunov, 1988)
dX2/ dt =
Theorem 2. Assume that there exists a quadratic form F with 3n + 1 arguments such that 1) For all U1 E en, U2 E e, w 2: 0 following frequency domain inequality holds: Re{F[(iwI A)-lbc*U1 , (iwI - A)-1(U1 + bu 2), U1, U2)]} :-::: 0;
€I (t) = Xl (t) cos(nt),
dyt/dt
k(T) dTZ , k(t)z , k(t)
o
2:
f
11 Z 112 .
dY2/ dt = -mY1 + m6 + 6 , Y2(0) = 0, 6(t) = X2(t) cos(Dt), 6(t) = -Y1(t)(sin(nt))/D.
J
k(T) dTC* z)
°
Proof According to Kalman-Yakubovich lemma (Gelig et al., 1978) there exists a real Hermitian matrix H such, that inequality
rH
(A~:~~c~(l )
+ F(x , y, (1, (2)
(23)
F1 = Xl - ~1 2: 0; F2 = X2 - 6 2: 0; F3 = (Xl - ~1)(X1 - DY1) 2: 0;
Then system (16) , (17) is globally asymptotically stable.
2( ;
= -mY1 + Y2 + ~3 + 6 ,
System (22), (23) has following quadratic constraints:
t
J
m> 0, D > 0.
Y1 (0) = 0,
3) there exist positive numbers f, J and unbounded sequence {tj} of positive numbers such, that for any t E [tj , tj + J), j = 1, 2, ... , Z E Rn following inequality holds:
F(z ,
(22)
According to equation (20) we choose additional system:
2) for all t 2: 0 and all solutions of system (16), (17), (20) following inequality holds: F(x(t), yet) , (1 (t) , (2(t)) 2: 0;
t
+ X2 + ~1, -mx1 + ~1'
dxt/dt = -mx1
(21)
:-::: 0
holds for all x, y , (1 ERn , (2 E R. Define V(x , y) = (x* , y*)Hcol(x, y). First, we show, that V(x,vx) > 0 for all v E [-a , a], x =J o. Fix a E [-a , a], Xo =J o. Define function k : k(t) = J-I-sign(v) if t E [0 , J-I-- 1 1 v I), and k(t) = 0 if t 2: J-I-- 1 1 v I· Solution to system (16) , (17), (20) with initial data x(O) = exp(-(A + bc*sign(v))J-I-- 1 1 v I)xo , y(O) = 0, is such, that
F4 = X2(X2
+ DY2) 2: 0;
Fs = x16 - X2€I == 0; F6 = x16 - Y1~1 == o. For quadratic form F = T1 F1 + ... + T6F6 all conditions of theorem 2 hold if T1 = m(1 - 4/D) , T2 = 0, T3 = 2m/D, T4 = 4/D, TS = 1, T6 = 1, D 2: 4. Hence, for D 2: 4 system (16), (17) is globally asymptotically stable. In this case we may use one more quadratic form: F7 = x2«D 2 + 4)1 /2x2 - D6 + 2DY2) 2: o. If we apply theorem 2 with F=nA+~~+~A+~~+~A
where T1 = m( _(D2 + 4)1 / 2 + 2D - 2), T3 = 2m, TS = D, T6 = mD, T7 = 1, then all conditions of theorem 2 hold if D 2: 8/3. Hence, for D 2: 8/3 system (16), (17) is globally asymptotically stable.
t
yet)
=
J
k(T) dTX(t) , x(J-I--1 1 v I)
= xo ,
o Y(J-I--1 1 v I) = vXo, x(t) -+ 0, yet) -+ Oast ---> 00 . We use condition 2 and (21) to get dV(x(t),y(t))/dt :-::: 0 for all t 2: o. Hence, V(x(t), yet)) 2: o. If V(xo , vxo) = 0, then V(x(t) , yet)) = 0, F(x(t) , yet), (1 (t) , (2(t)) = o for all t 2: 11-- 1 1 v I. Contradiction with condition 3 proves, that V(xo, vxo) > o. From (21) and condition 3 it follows : V(x(t), yet)) :-::: V(x(O) , y(O)) for all t 2: 0, and zero is an wlimit point of any solution of system (16), 17), (20) . Hence zero solution is Lyapunov stable, and all solutions of system (16), (17) tend to zero as t
---> 00 .
REFERENCES Gelig, A.Kh ., G.A. Leonov and V.A. Yakubovich (1978) . Stability of Systems with Nonunique Equilibrium. Nauka, Moscow. Venkatesh, YV. (1977). Energy Methods in TimeVarying Stability and Instability Analysis. Springer, Berlin. Kotchenko, F .F. and N.A .Bodunov (1988). On the dependence of stability of linear periodic systems with respect to period. Differential Equations, 24(2), 338-341.
0
1236
~
IFAC
0
C>
Publications www.elsevier.comllocatelifac
STATE SPACE EXTENSION METHOD IN THE THEORY OF ABSOLUTE STABILITY Nikita Barabanov
Dept. Software Eng. Electrotechnical University St. Petersburg RUSSIA nikita0freya.etu.ru
Abstract: State space extension method is applied to construct new quadratic constraints and derive new frequency domain criteria for absolute stability. Copyright @2001 IFAC Keywords: absolute stability, quadratic constraints, Lyapunov functions
1. INTRODUCTION
of type "quadratic form of coordinates, nonlinear function, and convolution of coordinates and nonlinear function with given sums of decreasing exponents" .
We consider frequency domain criteria for absolute stability of systems in different classes of nonlinear functions . To apply a standard technique based on Kalman-Yakubovich lemma (Gelig et al. , 1978) or interior point algorithm, and Lyapunov function method it 's necessary to construct particular (local or integral) quadratic constraints, which use information about nonlinear functions.
In this paper there are presented three examples of application of state space extension method.
2. SYSTEMS WITH CONSTRAINTS ON DERIVATIVE OF GAIN FUNCTION
There is no general methods to construct quadratic constraints for nonlinear functions having arbitrary given properties. For example, even for monotone functions there does not exist corresponding quadratic constraint. To overcome this difficulty one can use state space extension method. The idea of this method consists of introducing of additional differential equations to the original system and then to construct new quadratic constraints for extended system. Such constraints may exploit different properties of non linear functions. Each new nontrivial quadratic constraint results in new frequency domain criteria for absolute stability in corresponding class of non linear functions . In terms of Lyapunov functions, state space extension method allow to obtain frequency domain criteria for existence of Lyapunov function
1. Consider system dx/dt
= Ax + b~,
(1)
~(t) =
kl(t)C*X(t) ,
(2)
where A, b, c - are matrices of dimension n x n , n x 1, n x 1 respectively, J-L is positive number and k(-) is differentiable function such, that 0 $ k(t) < J-L for all t ~ o. We transform this system in a following way: dx/dt
= Ax + b~ , ~(t)
1233
a
= c*x -
= k(t)a(t) ,
J-L-l~ ,
(3)
(4)
Assume, that function k satisfies following inequalities:
k(t)
~
~
Cl for all t
Proof According to Kalman-Yakubovich lemma (Gelig et al., 1978) for any f ~ 0 there exists a
Hermitian matrix H. and vector g. such that 0,
(5)
2y* H.((P - ({3 - f)I)y + q(1) + r*y(1 ( ) t2
J
{[(dk(t)/dt - 2a)/k(t)]+ - 2{3}dt
tl
Here Cl and C 2 are positive constants, which depend on function ~. Introduce transfer function of system (3), (4):
W(s)
= cOCA -
=-
~ C2 (6)
for all vectors y E R m and all (1 E R . Moreover, in this case go(P + (31 - iwI)-lq = "IjJ(iw) for all w ~ o. Define 1l".(iw) = Re(Y(iw - (3 + f)). Function 1l". depend on f continuously. Therefore, H. -+ Ho, g. -+ go as f -+ o. Denote y.(w) = (P({3 + f - iw)I)-lq. We have
1l".(iw) =1 g.y.(w) 12, 2fYO(W)* H.yo(w) + 1l"0( iw) = 1 g.yo(w)
sI)-lb,
Now we choose rational function
y.(w)
Yes) = r*(P - sI)-lq, where P + {31 is Hurwitz m x m-matrix, r,q are m-vectors, such that pair (P, r) is controllable, pair (q, P) is observable, and following inequality holds:
1l"(iw) = Re{Y(iw - (3)} for all w ~ o.
~ 0
= Py + q(1.
TP
+ P*T + (3T
~
Tq+r = o.
(9) (10)
yo(w) + f((P + (3I) - iwI)-2 q + 0(f2), 1s
= iw - (3+0(f2) .
2yo(w)* Hoyo(w) = X(iw). According to Kalman-Yaku bovich lemma (Gelig et al., 1978) there exists a Hermitian matrix H such, that 2 (x) * H ( (A + (3I)x +
Inequality (7) means, that there exists a factorization of function 1l", that is, there exists a rational function "IjJ with Hurwitz denominator, such that 1l"(iw) = "IjJ(-iw)"IjJ(iw) for all w ~ o.
,
2fYO(W)* Hoyo(w) = -1l"o(iw) + 1l".(iw) - 2fRe[(goYo(w))"go(P + (31 - iwI)-lq] + 0(f2) dyes) = fRe[d;- 1 s = iw - (3 . d"IjJ(s) . + 2"IjJ( -tw)d;- 1 s = tw] + 0(f2) = fX(iw) + 0(f2),
(8)
0,
2
1
Therefore,
(7)
Inequality (7) may be also presented in form of LMI: there exists Hurwitz m x m-matrix T such that
=
1l".(iw) = 1l"o(iw) + fRe d~~S)
Consider additional equations:
dy/dt
12
2 1 9.y 1
y
b~ )
(P + (3I)y + q(1
+ 6 1c·x 12 +2ay· Hoy for all x ERn, yE Rm ,
~
+ r.
y
~
(13)
~ 0
E R. Define functions
Define function
Vo(x, Y)=(;)*H(;), V1(y)=y·HoY, .) R ([dY(s-{3) X(tW = e ds
2 ()d"IjJ(s)]1 .) "IjJ -s d;- s = tW .
Vex, y, t) = Vo(x, y)
+ k(t)Vl (y) .
Inequality (13) implies: Theorem 1. Assume that there exist positive number 6 such, that for all w ~ 0 following inequality holds:
Re[(Y(iw - (3)(W(iw - (3) + J.L-l)] ~ ax(iw) 1W(iw - (3) + J.L- 1 12 + 61 W(iw - (31 2 .
dV(x(t), yet), t)/dt ~ -6(c·x(t))2 - 2aV1(y(t)) - 2{3Vo(x(t), yet)) + (dk(t)/dt - 2{3k(t))Vt(y(t)) ~ -6(c*x(t))2 + ([(dk(t)/dt - 2a)/k(t)]+ - 2(3)V(x(t) , yet) , t)
(ll)
Then system (3), (4) is absolute stable in class of functions ~ satisfying inequalities (5), (6).
for all solutions of system (3), (4), (8). 1234
(14)
From (12) it follows Ho > O. Similarly, inequality (13) implies H > o. Hence, Vo > 0, VI > 0, and V> o.
limt_oo{ln(V(x(t) , y(t) , t)) ::; -In(2)/T, and system (3),(4) is stable exponentially and uniformly over the class of functions ~, satisfying conditions (5) , (6) . 0
We integrate inequality (14):
This theorem is a modifocation of "average logarithmic" criterion published in (Venkatesh, 1977) . The difference consists of new parameter 0 and less restrictive condition (6) .
for all t2 ~t 1 ~ o. Hence, zero solution of system (3) is Lyapunov stable. Besides, for any T > 0 and p(t) = [(dk(t) / dt - 2o)/k(t)J+ - 2/3 following inequality holds: T
V(x(T), y(T) , T)
+0
T
j exp(j p(T)dT)(C'X(t))2dt o
T
Example 1. Consider system (1) , (2) with transfer function 1
W (8 ) =
T
82
+ 8 + 1.
According to circle criterion this system is absolute stable in class of functions kl (-) such, that o ::; kl (t) ::; 3 for all t 2 o. The maximal number /1max such, that system (1) , (2) is absolute stable in class of functions k l , satisfying inequality 0 ::; kl (t) < /1max for all t 2 0, is equal approximately to 4,63.
::; exp(j p(T)dT)V(X(O), y(O), 0) . o Now we show, that there exists T > 0 such, that for any number t > 0 there exists a number h E [t , t + T ] such, that
Now apply theorem 1. Set /3 = 0, 0 = 1, Y(s) = (s + 1) / (8 + 2/1)2. Then all conditions of theorem 1 are fulfilled if sup{ kl (t)} < 00, inf{ dkl (t)/ dt} > -2. Thus, system (1) , (2) is absolute stable in class of functions kl (-) satisfying these conditions.
V(x(h) , y(h), h) ::; V(x(t), y(t), t) / 2. (15) Assume that property (15) doesn 't hold for all + T] . Then exp(Jtt+ T p(T) dT) > 1/ 2 for all T E [0, T] . Therefore,
h E [t , t T
oj
(c*x(t)fdt ::; 2exp(C2)V(x(t), y(t) , t)
o
3. SYSTEMS WITH INTEGRAL CONSTRAINTS ON GAIN FUNCTION
::; 2 exp( C2)V(x(0), y(O) , 0).
Denote C 3 = 20- 1(HCl ) exp(2C2)V(x(0) , y(O) , 0) . Then
Consider system
j (a(t)2 + ~(t)2)dt ::; C
~(t)
3,
o and for some number C4 , which depend only on coefficients of system (3), (8), following inequality holds:
= Ax + b~,
(16)
= k(t)c'x(t) ,
(17)
dx/dt
T
A is Hurwitz n x n-matrix, b, care n-vectors, and function k(·) satisfies inequalities:
1k(t) I::; /1 ,
(18)
t+T
j
(11 X(T) 112
t
j k(T)dT ::;
+ 11 y(T) 112)dT
::; C 4 (11 x(t) 112
+ 11 y(t) 11
2
).
From Vo > 0, and (15) it follows, that for some number 01 > 0 and all h E [t , t + T] following inequality holds: 11 x(h) 112 + 11 y(h) 112~ 01 (11 x(t) 112 + 11 y(t) 11 2). Then
for all t
X
(11 x(t) 112
~
0 and some positive numbers a , /1 .
To obtain stability criterion we make a coordinate transformation: (1 = kx . Introduce additional system:
j (11 X(T) 112 + 11 y(T) 112) dT
t
(19)
Inequalities (18) , (19) hold, for example, for periodic functions k with zero mean values.
t+T
01T::;
a
o
+ 11 y(t) 112)-1 ::; C4 ·
Hence, T ::; C4 / 01 . Thus , if T > C4 / 01 , then for any number t > 0 there exists a number h E [t , t+ T] such, that V(x(h) , y(h), h) ::; V(x(t) , y(t), t) / 2. Therefore,
Notice, that if x(-) is a solution to system (16), (17) , then y(t) = J~k(T)dTX(t) is a solution to equation (20). Therefore, a quadratic Lyapunov 1235
function for the system (16), (17), (20) is a Lyapunov function for system (16), (17), which is quadratic with coefficients of the form Cl + C2 J~ k(T) dT .
Example 2. Consider differential equations (Kotchenko and Bodunov, 1988)
dX2/ dt =
Theorem 2. Assume that there exists a quadratic form F with 3n + 1 arguments such that 1) For all U1 E en, U2 E e, w 2: 0 following frequency domain inequality holds: Re{F[(iwI A)-lbc*U1 , (iwI - A)-1(U1 + bu 2), U1, U2)]} :-::: 0;
€I (t) = Xl (t) cos(nt),
dyt/dt
k(T) dTZ , k(t)z , k(t)
o
2:
f
11 Z 112 .
dY2/ dt = -mY1 + m6 + 6 , Y2(0) = 0, 6(t) = X2(t) cos(Dt), 6(t) = -Y1(t)(sin(nt))/D.
J
k(T) dTC* z)
°
Proof According to Kalman-Yakubovich lemma (Gelig et al., 1978) there exists a real Hermitian matrix H such, that inequality
rH
(A~:~~c~(l )
+ F(x , y, (1, (2)
(23)
F1 = Xl - ~1 2: 0; F2 = X2 - 6 2: 0; F3 = (Xl - ~1)(X1 - DY1) 2: 0;
Then system (16) , (17) is globally asymptotically stable.
2( ;
= -mY1 + Y2 + ~3 + 6 ,
System (22), (23) has following quadratic constraints:
t
J
m> 0, D > 0.
Y1 (0) = 0,
3) there exist positive numbers f, J and unbounded sequence {tj} of positive numbers such, that for any t E [tj , tj + J), j = 1, 2, ... , Z E Rn following inequality holds:
F(z ,
(22)
According to equation (20) we choose additional system:
2) for all t 2: 0 and all solutions of system (16), (17), (20) following inequality holds: F(x(t), yet) , (1 (t) , (2(t)) 2: 0;
t
+ X2 + ~1, -mx1 + ~1'
dxt/dt = -mx1
(21)
:-::: 0
holds for all x, y , (1 ERn , (2 E R. Define V(x , y) = (x* , y*)Hcol(x, y). First, we show, that V(x,vx) > 0 for all v E [-a , a], x =J o. Fix a E [-a , a], Xo =J o. Define function k : k(t) = J-I-sign(v) if t E [0 , J-I-- 1 1 v I), and k(t) = 0 if t 2: J-I-- 1 1 v I· Solution to system (16) , (17), (20) with initial data x(O) = exp(-(A + bc*sign(v))J-I-- 1 1 v I)xo , y(O) = 0, is such, that
F4 = X2(X2
+ DY2) 2: 0;
Fs = x16 - X2€I == 0; F6 = x16 - Y1~1 == o. For quadratic form F = T1 F1 + ... + T6F6 all conditions of theorem 2 hold if T1 = m(1 - 4/D) , T2 = 0, T3 = 2m/D, T4 = 4/D, TS = 1, T6 = 1, D 2: 4. Hence, for D 2: 4 system (16), (17) is globally asymptotically stable. In this case we may use one more quadratic form: F7 = x2«D 2 + 4)1 /2x2 - D6 + 2DY2) 2: o. If we apply theorem 2 with F=nA+~~+~A+~~+~A
where T1 = m( _(D2 + 4)1 / 2 + 2D - 2), T3 = 2m, TS = D, T6 = mD, T7 = 1, then all conditions of theorem 2 hold if D 2: 8/3. Hence, for D 2: 8/3 system (16), (17) is globally asymptotically stable.
t
yet)
=
J
k(T) dTX(t) , x(J-I--1 1 v I)
= xo ,
o Y(J-I--1 1 v I) = vXo, x(t) -+ 0, yet) -+ Oast ---> 00 . We use condition 2 and (21) to get dV(x(t),y(t))/dt :-::: 0 for all t 2: o. Hence, V(x(t), yet)) 2: o. If V(xo , vxo) = 0, then V(x(t) , yet)) = 0, F(x(t) , yet), (1 (t) , (2(t)) = o for all t 2: 11-- 1 1 v I. Contradiction with condition 3 proves, that V(xo, vxo) > o. From (21) and condition 3 it follows : V(x(t), yet)) :-::: V(x(O) , y(O)) for all t 2: 0, and zero is an wlimit point of any solution of system (16), 17), (20) . Hence zero solution is Lyapunov stable, and all solutions of system (16), (17) tend to zero as t
---> 00 .
REFERENCES Gelig, A.Kh ., G.A. Leonov and V.A. Yakubovich (1978) . Stability of Systems with Nonunique Equilibrium. Nauka, Moscow. Venkatesh, YV. (1977). Energy Methods in TimeVarying Stability and Instability Analysis. Springer, Berlin. Kotchenko, F .F. and N.A .Bodunov (1988). On the dependence of stability of linear periodic systems with respect to period. Differential Equations, 24(2), 338-341.
0
1236