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Global Asymptotic Stabilization of a Class of Constant Feedback Stabilizable Bilinear Dynamical Systems with Uncertainty
tion (i .e. a set-valued map) , thereby giving rise to a controlled differential inclusion system. A class of discontinuous feedback controls embedded in a multifunction and , henceforth , known as a generalized state-feedback, is proposed which renders the zero-state globally uniformly asymptotically stable for the class of uncertain bilinear systems. Continuous approximations to these discontinuous feedbacks , dependent on design parameters , can be designed . In this case , it is shown that a neighbourhood of the zero-state is globally uniformly asymptotically stable and , since this neighbourhood can be made as small as desired by tuning the design parameters, practical stabilization of the class of uncertain bilinear systems can be achieved .
Hypothesis (a):
The nominal system (2.1) is stabilizable by a constant feedback . Under Hypothesis (a), there exist ki such that m
(A
+L
kiBd is a stable matrix. And hence , i=l there exists a real symmetric matrix P > 0 such that m
m
P(A+ LkiBd + (A i=l
+ LkiBi)Tp+ Q =
0,
i=l
where Q > 0 is a real symmetric matrix . Introduce a multifunction P : lRn which is defined by
x ........ P(x)
:=
=ker(BT),
{y E ker(B T ) : (y , Px) 2: O} .
2. PROBLEM FORMULATION Hypothesis (b) : The class of uncertain systems considered here is based on a nominal known bilinear system, which is homogeneous-in-the-state,
There exist real constants < 1 and
(i) there exist real constants and, for alii and (t , x,u),
m
x(t) = Ax(t) + L ui(t)BiX(t). i=l
(2.1)
0'2
2:
0 , J.L1
2:
0
O'dUil +0'2·
(ii) there exist real constants /32 2: 0 and J.L2 2: 0 , ~(7t(t,
Defn. 1 The bilinear system (2 .1) is stabilizable by a constant feedback if there exists a constant feedback u = U = [U1 ' ... , um]T , Ui E lR,
x, u)
n P(x»
m
< [6(x)]-1 L I(x, PBix)1 i=l
m
+L
2: 0 such that
Iqi(t , x , u)1 ~J.L11(x , PBix)l+
The nominal system is assumed to be stabilizable by a constant feedback.
such that (A
0'1 , /31
0'1 + /31
UiBi) is a stable matrix.
i=l The uncertain system is assumed to be modelled by a differential inclusion with the following structure .
for all (t , x, u) , where 6 : X -+ (0 , 00) E Cl satisfies 6(x) > IIPxll for all x . Remark: 7t(t , x , u) n P(x) is adopted to economize on control gain by exploiting the possible occurrence of 'stability enhancing' uncertainties .
m
x(t) E Ax(t) + L[Ui(t) + qi(t , x(t) , U(t»]BiX(t) i=l +7t(t , x(t) , u(t» ,
(2.2)
The main objective of this paper is to determine a class of discontinuous and continuous feedback controls to globally uniformly asymptotically stabilize this class of bilinear uncertain systems (2 .2) .
where the unknown functions qi : lR x lRn x lRm -+ lR model uncertainty which are 'matched' with respect to Bi and the known multifunction 7t : lR x lRn x lRm lRn models 'residual ' uncertainty. Hypotheses characterizing the known elements of the nominal system (2 .1) and the uncertainty in system (2 .2) are prescribed below.
=
3. STABILIZATION USING DISCONTINUOUS STATE-FEEDBACK CONTROLS
For a compact subset W oflRn or lRm (as appropriate ), we define ~(W) := max{llwll : w E W} if W is nonempty and ~(W) := 0 when W = 0.
The proposed discontinuous state-feed backs u(t) are embedded in a multifunction F(x(t» . The
932
m
multifunction F is assumed to be a gen eralized
L[u;
state-feedback.
+ q;(t , x , u)]B;x + ?-let , x , u)
).
;=1
Introducing Hypothsis (b) ,
Defn . 2 A multifunction F : ffin :: ffim is a generalized state-feedback (i) if F is upper semi continuous with nonempty, convex and compact values; (ii) if F is singleton-valued except on a set I:.r of Lebesgue measure zero in ffin .
1
maxV(t , x , F(t , x) ) ~ -2( x , Qx ) m
-[J.I- (J.ll
The proposed stabilizing generalized statefeedback F(x) = [F1 (x) , .. . , Fm(x)jT has the form
+ J.l2)] L
I( x , PB;x
W
;=1 m
-[p - (02
+ /32 + k)] L
I(
X,
P B;x )1.
;=1
x~F;(x)
-a- 1 [J.I(x , PB;x) + pS«(x , PB;x))]
where k = max(lk; I) . Therefore, if we choose
(3 .1)
I
design parameters J.I , p to satisfy
where J.I 2:: 0 and p 2:: 0 are design parameters , 1 - 01 - /31 , and S : ffi :: ffi is defined by
a=
z
~
S( z ) :=
{I z l-lz}
if z ::f 0,
{ [-1 , 1]
if z = O.
then (3 .2)
vex)
By definition of F and S , F qualifies as a generalized state-feedback .
Theorem 1 If the hypotheses (a) and (b) hold , J.I 2:: J.ll + J.l2 and p 2:: 02 + /32 + k , the generalized state feedback F given by (3 .1) , renders the zero state of the differential inclusion system (2 .2) , subject to x(to) xo , globally uniformly asymptotically stable.
=
Proposition 1 For each xo E ffi n , where xo = x(to) , and continuous u : ffi -+ ffi m, there exists a local solution of (2 .2) which can be extended into a maximal solution x : [to, T) -+ ffin and if x( ·) is bounded, then T = 00.
If a constant feedback K = [k 1 ,· · " km] for the nominal system (2.1) is incorporated in the generalized state-feedback F , then F has the form ,
x
The conditions specified earlier guarantee that the initial value problem (2 .2) subject to x(to) = xo admits a maximal solution.
~
F;(x) = k;
-a- 1 lJ.i(x , PB;x) + pS«(x , PB;x))] .
(3 .3)
Using similar analysis to that used in the proof of Theorem 1, the following theorem may be obtained .
Under Hypothesis (a) , the Lyapunov function candidate is selected as v : ffin -+ [0 , 00) , ~
1 -2( x , Qx) < 0 for x(t)::f 0 ,
along solutions to (2 .2) and for almost all t. From the above analysis and utilizing Proposition 1, the following theorem may be concluded.
To show that the differential inclusion system (2 .2) has indefinite continuation of solutions, the following proposition is required (for existence of local solution see Aubin and Cellina (1984) and for indefinite continuation of solutions see Ryan (1990)) .
x
~
1
vex) = 2( x , Px ),
Theorem 2 If the hypotheses (a) and (b) hold , f.l 2:: J.l l + f.l2 and p 2:: 0 2 + /32 + (01 + /3dk , the generalized state feedback F given by (3.3) , renders the zero state of the differential inclusion system (2.2) with x(to ) = xo globally uniformly asymptotically stable.
then , along maximal solutions to (2 .2) ,
vex ) = ( x , Pi: ) E Vet , x(t ), u(t)) , where
V(t , x,u) :=( PX , Ax+
933
Case I: l(x,PBix)1 ~ Cia.
4. STABILIZATION USING CONTINUOUS STATE-FEEDBACK CONTROLS
1 maxV(t , x , .1") ~ -2( x , Qx ). Case 11: l(x , PBix)1 < Cia.
In the above section , the synthesis of the feedback controls leads to a discontinuous controller. To overcome any difficulties obtained through practical implementation of a controller of this type , a class of continuous state-feedback controls is proposed in this section, which depends on design parameters Ci and each component of the control is a contin.uous ci-approximation to each component of the discontinuous feedbacks. A control function of this type is used by Corless and Leitmann (1981).
1 maxV(t, x , .1") ~ -2( x , Qx ) + PI
i=I Therefore, then , for both cases, (4.3) m
where C =
x ......... .1'i(X) = -a-I[Jl(X,PBiX) (4.1)
is globally uniformly asymptotically stable.
where Jl ~ 0 and PI ~ 0 are design parameters, a = 1 - 0'1 - f31 > 0, and V : IR IR is defined by
=
Theorem 3 Under the Hypotheses (a)-(b) and with generalized state-feedback .1' (given by (4.1)) , if Jl ~ JlI + Jl2 and PI ~ 0'2 + f32 + k, then any compact set containing the ellipsoid £1 is globally uniformly asymptotically stable for the uncertain system (2 .2), with x(to) = xo .
z . . . . . V(z):=
{
L: Ci·
As a consequence of Proposii=I tion 1 and (4.3), all maximal solutions of (2.2) can be extended into a solution on [0,00) . Moreover , by (4.3) , properties of uniform boundedness of solutions, stability and attractivity hold. Hence , any compact set containing the ellipsoid
The proposed stabilizing generalized statefeedback .1'(x) = [.1'1 (x), .. . , .1'm(x)JT has the form
+PI V(c;I a-I (x, P BiX))]
m
L: ci ·
{Izl-Iz}
if Izl> 1,
{y E IR : z2 ~ yz ~ Izl}
if Izl ~ 1.
(4.2)
If a constant feedback K = [k I ,·· " km] IS Incorporated in the controls, the generalized statefeedback is given by
Once again, the Lyapunov function candidate is chosen to be
1 x ......... v(x) = 2( x,Px), then, along maximal solutions to (2.2),
(4.4)
ti(x) = ( x , Pi: ) E V(t , x(t) , u(t)) ,
The following theorem can be obtained.
where
Theorem 4 Under the Hypotheses (a )-(b) and with generalized state-feedback .1' (given by (4.4)), if J.l ~ J.lI +J.l2 and P2 ~ 0'2+f32+(O'I +f31 )k, then any compact set containing the ellipsoid £2 , where £2 := {x E IRn : (x , Qx) ~ 2p2c}, is globally uniformly asymptotically stable under the dynamics of (2.2), with x(to) = xo.
V(t,x , u):=( PX , Ax+ m
L:[Ui i=I
+ qi(t, x , U)]BiX + 1£(t, x , u)
).
If the design parameters Jl, PI are selected to satisfy
Jl ~ JlI + Jl2
Since (0'1 + f3J)k < k , the 'size' of the set £2 is smaller than £1 . When the control design approach is used for tracking problems, the closedloop systems using feedback control (4.4) has higher tracking precision than that obtained by using the feedback controller (4 .1) .
and PI ~ 0'2 + f32 + k,
where k = max(lkd), there are two cases to conI
sider.
934
Hypothesis (a):
The nominal system (2.1) is stabilizable by a constant feedback . Under Hypothesis (a), there exist ki such that m
(A
+L
kiBd is a stable matrix. And hence , i=l there exists a real symmetric matrix P > 0 such that m
m
P(A+ LkiBd + (A i=l
+ LkiBi)Tp+ Q =
0,
i=l
where Q > 0 is a real symmetric matrix . Introduce a multifunction P : lRn which is defined by
x ........ P(x)
:=
=ker(BT),
{y E ker(B T ) : (y , Px) 2: O} .
2. PROBLEM FORMULATION Hypothesis (b) : The class of uncertain systems considered here is based on a nominal known bilinear system, which is homogeneous-in-the-state,
There exist real constants < 1 and
(i) there exist real constants and, for alii and (t , x,u),
m
x(t) = Ax(t) + L ui(t)BiX(t). i=l
(2.1)
0'2
2:
0 , J.L1
2:
0
O'dUil +0'2·
(ii) there exist real constants /32 2: 0 and J.L2 2: 0 , ~(7t(t,
Defn. 1 The bilinear system (2 .1) is stabilizable by a constant feedback if there exists a constant feedback u = U = [U1 ' ... , um]T , Ui E lR,
x, u)
n P(x»
m
< [6(x)]-1 L I(x, PBix)1 i=l
m
+L
2: 0 such that
Iqi(t , x , u)1 ~J.L11(x , PBix)l+
The nominal system is assumed to be stabilizable by a constant feedback.
such that (A
0'1 , /31
0'1 + /31
UiBi) is a stable matrix.
i=l The uncertain system is assumed to be modelled by a differential inclusion with the following structure .
for all (t , x, u) , where 6 : X -+ (0 , 00) E Cl satisfies 6(x) > IIPxll for all x . Remark: 7t(t , x , u) n P(x) is adopted to economize on control gain by exploiting the possible occurrence of 'stability enhancing' uncertainties .
m
x(t) E Ax(t) + L[Ui(t) + qi(t , x(t) , U(t»]BiX(t) i=l +7t(t , x(t) , u(t» ,
(2.2)
The main objective of this paper is to determine a class of discontinuous and continuous feedback controls to globally uniformly asymptotically stabilize this class of bilinear uncertain systems (2 .2) .
where the unknown functions qi : lR x lRn x lRm -+ lR model uncertainty which are 'matched' with respect to Bi and the known multifunction 7t : lR x lRn x lRm lRn models 'residual ' uncertainty. Hypotheses characterizing the known elements of the nominal system (2 .1) and the uncertainty in system (2 .2) are prescribed below.
=
3. STABILIZATION USING DISCONTINUOUS STATE-FEEDBACK CONTROLS
For a compact subset W oflRn or lRm (as appropriate ), we define ~(W) := max{llwll : w E W} if W is nonempty and ~(W) := 0 when W = 0.
The proposed discontinuous state-feed backs u(t) are embedded in a multifunction F(x(t» . The
932
m
multifunction F is assumed to be a gen eralized
L[u;
state-feedback.
+ q;(t , x , u)]B;x + ?-let , x , u)
).
;=1
Introducing Hypothsis (b) ,
Defn . 2 A multifunction F : ffin :: ffim is a generalized state-feedback (i) if F is upper semi continuous with nonempty, convex and compact values; (ii) if F is singleton-valued except on a set I:.r of Lebesgue measure zero in ffin .
1
maxV(t , x , F(t , x) ) ~ -2( x , Qx ) m
-[J.I- (J.ll
The proposed stabilizing generalized statefeedback F(x) = [F1 (x) , .. . , Fm(x)jT has the form
+ J.l2)] L
I( x , PB;x
W
;=1 m
-[p - (02
+ /32 + k)] L
I(
X,
P B;x )1.
;=1
x~F;(x)
-a- 1 [J.I(x , PB;x) + pS«(x , PB;x))]
where k = max(lk; I) . Therefore, if we choose
(3 .1)
I
design parameters J.I , p to satisfy
where J.I 2:: 0 and p 2:: 0 are design parameters , 1 - 01 - /31 , and S : ffi :: ffi is defined by
a=
z
~
S( z ) :=
{I z l-lz}
if z ::f 0,
{ [-1 , 1]
if z = O.
then (3 .2)
vex)
By definition of F and S , F qualifies as a generalized state-feedback .
Theorem 1 If the hypotheses (a) and (b) hold , J.I 2:: J.ll + J.l2 and p 2:: 02 + /32 + k , the generalized state feedback F given by (3 .1) , renders the zero state of the differential inclusion system (2 .2) , subject to x(to) xo , globally uniformly asymptotically stable.
=
Proposition 1 For each xo E ffi n , where xo = x(to) , and continuous u : ffi -+ ffi m, there exists a local solution of (2 .2) which can be extended into a maximal solution x : [to, T) -+ ffin and if x( ·) is bounded, then T = 00.
If a constant feedback K = [k 1 ,· · " km] for the nominal system (2.1) is incorporated in the generalized state-feedback F , then F has the form ,
x
The conditions specified earlier guarantee that the initial value problem (2 .2) subject to x(to) = xo admits a maximal solution.
~
F;(x) = k;
-a- 1 lJ.i(x , PB;x) + pS«(x , PB;x))] .
(3 .3)
Using similar analysis to that used in the proof of Theorem 1, the following theorem may be obtained .
Under Hypothesis (a) , the Lyapunov function candidate is selected as v : ffin -+ [0 , 00) , ~
1 -2( x , Qx) < 0 for x(t)::f 0 ,
along solutions to (2 .2) and for almost all t. From the above analysis and utilizing Proposition 1, the following theorem may be concluded.
To show that the differential inclusion system (2 .2) has indefinite continuation of solutions, the following proposition is required (for existence of local solution see Aubin and Cellina (1984) and for indefinite continuation of solutions see Ryan (1990)) .
x
~
1
vex) = 2( x , Px ),
Theorem 2 If the hypotheses (a) and (b) hold , f.l 2:: J.l l + f.l2 and p 2:: 0 2 + /32 + (01 + /3dk , the generalized state feedback F given by (3.3) , renders the zero state of the differential inclusion system (2.2) with x(to ) = xo globally uniformly asymptotically stable.
then , along maximal solutions to (2 .2) ,
vex ) = ( x , Pi: ) E Vet , x(t ), u(t)) , where
V(t , x,u) :=( PX , Ax+
933
Case I: l(x,PBix)1 ~ Cia.
4. STABILIZATION USING CONTINUOUS STATE-FEEDBACK CONTROLS
1 maxV(t , x , .1") ~ -2( x , Qx ). Case 11: l(x , PBix)1 < Cia.
In the above section , the synthesis of the feedback controls leads to a discontinuous controller. To overcome any difficulties obtained through practical implementation of a controller of this type , a class of continuous state-feedback controls is proposed in this section, which depends on design parameters Ci and each component of the control is a contin.uous ci-approximation to each component of the discontinuous feedbacks. A control function of this type is used by Corless and Leitmann (1981).
1 maxV(t, x , .1") ~ -2( x , Qx ) + PI
i=I Therefore, then , for both cases, (4.3) m
where C =
x ......... .1'i(X) = -a-I[Jl(X,PBiX) (4.1)
is globally uniformly asymptotically stable.
where Jl ~ 0 and PI ~ 0 are design parameters, a = 1 - 0'1 - f31 > 0, and V : IR IR is defined by
=
Theorem 3 Under the Hypotheses (a)-(b) and with generalized state-feedback .1' (given by (4.1)) , if Jl ~ JlI + Jl2 and PI ~ 0'2 + f32 + k, then any compact set containing the ellipsoid £1 is globally uniformly asymptotically stable for the uncertain system (2 .2), with x(to) = xo .
z . . . . . V(z):=
{
L: Ci·
As a consequence of Proposii=I tion 1 and (4.3), all maximal solutions of (2.2) can be extended into a solution on [0,00) . Moreover , by (4.3) , properties of uniform boundedness of solutions, stability and attractivity hold. Hence , any compact set containing the ellipsoid
The proposed stabilizing generalized statefeedback .1'(x) = [.1'1 (x), .. . , .1'm(x)JT has the form
+PI V(c;I a-I (x, P BiX))]
m
L: ci ·
{Izl-Iz}
if Izl> 1,
{y E IR : z2 ~ yz ~ Izl}
if Izl ~ 1.
(4.2)
If a constant feedback K = [k I ,·· " km] IS Incorporated in the controls, the generalized statefeedback is given by
Once again, the Lyapunov function candidate is chosen to be
1 x ......... v(x) = 2( x,Px), then, along maximal solutions to (2.2),
(4.4)
ti(x) = ( x , Pi: ) E V(t , x(t) , u(t)) ,
The following theorem can be obtained.
where
Theorem 4 Under the Hypotheses (a )-(b) and with generalized state-feedback .1' (given by (4.4)), if J.l ~ J.lI +J.l2 and P2 ~ 0'2+f32+(O'I +f31 )k, then any compact set containing the ellipsoid £2 , where £2 := {x E IRn : (x , Qx) ~ 2p2c}, is globally uniformly asymptotically stable under the dynamics of (2.2), with x(to) = xo.
V(t,x , u):=( PX , Ax+ m
L:[Ui i=I
+ qi(t, x , U)]BiX + 1£(t, x , u)
).
If the design parameters Jl, PI are selected to satisfy
Jl ~ JlI + Jl2
Since (0'1 + f3J)k < k , the 'size' of the set £2 is smaller than £1 . When the control design approach is used for tracking problems, the closedloop systems using feedback control (4.4) has higher tracking precision than that obtained by using the feedback controller (4 .1) .
and PI ~ 0'2 + f32 + k,
where k = max(lkd), there are two cases to conI
sider.
934