The Robustness of Controllability and ... - Berkeley Robotics
EEE TRANSACTIONS
ON AUTOMATIC CONTROL, VOL.
AC-27, NO. 4,
AUGUST 933
1982
adaptive techniques.’” IEEE Tram. Auromar. Conrr., voL AC-23, pp. 97-99, Feb. 1978. C. R Johnson. Jr., “A convergence proof for a hyperstable adaptive recursive filter.” IEEE Trum. Inform. Theor), vol. IT-25, pp. 745-749, Nov. 1979. Y.-H. Lin and K. S. Pr‘arendra, “A new error model for adaptive systems,” IEEE Trans. Auromar. Conrr., vol. AC-25, pp. 585-587, June 1980. R. R Bitmead and B. D. 0. Anderson, “Performance of adaptive estimation algorithms in dependent random environments.” IEEE Trum. Auromnr. Conrr.. vol. AC-25. pp. 788-794. Aug. 1980. C. R Johnson. Jr., “Another use of the Lin-Narendra error model: HAW.” IEEE Tram. Auromur. Conrr., vol. AC-25, pp. 985-988, Oct. 1980. M. G. Larimore. J. R. Treicbler. and C. R Johnson, Jr., “SHARF:An algoritbm for adapting IIR digital filters,” IEEE Trum. A c m r . , Speech. S i p 1 Processing, vol. ASSP-28, pp. 428-440, Aug. 1980. C. R Johnson, Jr., “An output error identification interpretation of model reference adaptive control,” Auromaricu. vol. 16. pp. 419-421. July 1980. -, “The common parameter estimation basis of adaptive filtering, identification, and control.” in Proc. I9rh IEEE C o d . Decision Conrr.., Albuauemue. PiM.. Dec. 1980, pp. 447-452. C. R. Johnson, Jr. and B. D. 0. Anderson, “Sufficient excitation and stable reducedorder adaptive IIR filtering,” IEEE Tram. Acourr., SDeech. Sinnul Processmn. vol. ASSP-29,pp. 1212-1215, 6ec. 1981. C. A. Desoer and M. Vidyasagar. Feedback Sysrem: Inpur-Oupur Propemes. Ken. York: Academic, 1975. pp. 133-135. J. L. Willems. Srubrlrry Theor). of &namicul Svsrem. London: Nelson, 1970, pp. 113-114. P. Ioannou and C. R. Johnson. Jr.. “Reduced-order performance of parallel and series-parallel identifiers of mc-time-scale systems.” in Proc. 2nd Yule Workshop Appl. Adupr. Sysr., New Haven, C T , May 1981, pp. 169-174.
. . .
The Robustness of Controllabilityand Observability of Linear Time-Varying Systems
s. s. SASTRY, MEMBER, IEEE, AND c.A. DESOER, FELLOW, IEEE Abstract-Fixedpointmethods from nonlinearanalysis are used to establish conditions under which the uniform complete controllability of linear time-varying systems is preserved under nonlinear perturbations in the state dynamics and the zero-input uniform complete observability of linear timevarying systems is preservedunder nonlinear perturbation in the state dynamics and output read-out map. Robustness of partial controllability, observability, and a specific kind of nonzero input observability are also proven. I. INTRODUCTION Controllability and observability are key issues in system theory. To be specific, consider a class of physical dynamical systems which are adequately modeled by ordinary differential equations with inputs u and a static read-out map h : more precisely, R=/(x,u,t)
(1.1)
y=h(x,t)
(1.2)
put observable, then the original nonlinear system is also uniformly completely controllable and zero-input observable. Further, if the linear system is only partially controllable or observable, then the original nonlinear system at least retains these partial controllability and observability properties (see Section V). These results are termed robustness results for the ,controllability and observability of linear time-vaqing systems because of the perturbational nature of the proofs and estimates about a nominal linear system. The major mathematical tool is a solvability theorem for operator equations with a quasibounded nonlinearity due to Granas [4] which is reminiscent of the familiar small gain theorem (see [2], for example). The heart of the theorem lies in the Rothe fixed point theorem. The use of fixed point theorems in proving global nonlinear controllability results is not nev-the hela-Ascoli theorem was used in Lukes [5], the contraction mapping theorem by Mirza and Womack [8], and the Schauder fixed point theorem by Vidyasagar [14] and by Dauer [16], [17] and Aronsson [ 181. For global nonlinear observability. we have the work of Yamamoto and Sugiura [15]. Our estimates by virtue of our new technique are different from those reported in the literature (with some overlap, Theorem V.l is also proven by Lukes).’ Detailed comments on how the results reported above are special cases of our theorems appear in the text of the paper. The results of Dauer [ 171 and Aronsson [18] start from rather different assumptions from ours on the nonlinear perturbations and are not compared. We have illustrated the use of our results in this paper in the derivation of control laws during a very specific emergency, for interconnected power systems, by posing the alert state control problem as a steering problem in [ 1 11. 11. NOTATION The dynamical systems that we study are differential dvnamical svscems (DDS) with finite dimensional cector spaces as input. output, and state space, respectively R“,, Rn0. and R”, with the representation i=f(x,u.t)
(11.1)
Y =h ( x , t )
(11.2)
where t E R - , f is a C o function from R “ X R ” ~ X R , -.Rn which is globally Lipschitz continuous in its first argument (to guarantee uniqueness of solution to (2.1) when the initial condition is given), and h is a Co function from R “ X R R “ 0 . Finite dimensional litwar dynamical systems (FDLS) with a bourlded realization are defined as differential dynamical systems of the form (11.3)$ (11.4): -f
(11.3)
i=A(t)xtB(r)u
y = C( t ) x withx~R“,u~R“~,~ER”~,t€R+,and/,hareCofunctions;/satisfie~ Lipschitz and growth conditions for existence and uniqueness of solutions withIIA(.)II, IIB(.)Il. IIC(.)ll boundedonR+.
on R + . The definition of uniform complete controllability for (1.1) is as follows: 3 T E R + such that given any t o , initial time, and any two states xOl the initial state, and X , , the final state, there exists a control u E L;l([to*t o TI) whichwill steer the system (1.1) from x0 at t o to x, at to T. Zero-input uniform complete observability is defined as: 3 T € R , such that given any t o and the output of the system with zero input on [ t o . to TI. we can determine (uniquely) the state of the system at to. In this paper, we prove that if a system of the form (1.1), (1.2) is “close” to a linear system which is uniformly completely controllable and zero-in-
+
+
+
Manuscript received November 26,1979: revised September 2, 1980 and September 3, 1981. Paper recommended by A. N. Micbel, Past Chairman of the Stability, Nonlinear. and Distributed Systems Committee. This work was supported by the h:ational Science Foundation under Grant ENG78-09032-AOI and the US. Department of Energy under Contract DE-AC01-79ET29364. S. S. Sastry was with the Department of Uectrical Engineering and Computer Scienca and the Electronics Research Laboratory. University of California, Berkeley, CA 94720. He is now wiib the Laboratory for Information and Decision S c i e n c e s , M.I.T.. Cambridge. MA 02139. C. A. Desoer is with the Department of Electrical Engineering and Computer Sciences and the Elecvonics Research Laboratoq-. University of California, Berkeley, CA 94720.
111.
CHARACTERIZATION OF CONTROLLABILIN FOR
(11.4) (11.5) FINITE
LINEARSYSTEhlS DIMENSIONAL The definitions and propositions of this section are well known. although not standardized. We restate them here to establish the terminology and notation. The definitions are d r a w from Silverman [12], and the proofs maybe found in standard books (see.e.g., [l]). To obtain the desired characterization define, for fixed t o E R + , the linear map C R (called the reachabilig map) from L:L([t o . t o TI) to R ” by
+
(111.1) where @(I, T ) is the state-transition function for the linear system. Then at 7, (111.1) leads to
t = to
+
’Since this paper was written. the work of Dacka [3]. d i c h is similar in spirit, has appeared. His main result is a generalization of our Theorem V. 1. This generalization can be derived from our methods as well (see Section V-A).
OO18-9286/82/0800-0933$O00.75 a1982IEEE
934
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL.
(111.2)
e,.
The adjoint map of denoted E,: L2,([t0,t o + 71)defined by
?:.X=
then is the linear map from R " to
B*( .)Q*( to + T , . ) X .
(111.3)
The burden of this paper consists of demonstrating the robustness of strong uniform complete controllubility of an FDLS in the faceof nonlinear perturbations in the dynamics, both bounded and unbounded. Our methcds seemto indicate thatFDLS with a smaller reachability condition number are more robust than others with a larger reachability condition number.
Since the realization (11.5) is bounded (say by K ) , we have from the Bellman-Gronwall lemma that ilQ(t.~)liSexpK(t-~)
V ~ , T E W + t, a ~ . (111.4)
Also, llB(T)1! S K
b'TER+.
(111.5)
Using (111.4) and (111.5). it is easy to check that and E: are continuous linear maps. The following is well known. Theorem III.1 (Characterization of Uniform Complete Controllabilig for FDLS): The FDLS with bounded realization represented by (11.3) and (11.4) is uniformly completely controllable V t o E R + , the reachability map E:, L;,([t o ,to + T I ) -,R " defined in (111.1) is onto V t o E W + the composition of the reachability map, and its adjoint. namely, R " R". is a bijection. 0 Definition 111.3 (Reachability Grammian): Given t o € W + , the matrix representation of the continuous linear map R " -88" is the reachabilig grammian, denoted WR[to.t o + T I E R n X "
-
AC-27,NO. 4, AUGUST 1982
SOLVABILITY OF AN OPERATOR EQUATION WITH A QUASIBOUNDED NONLINEARITY IN NORMED SPACES
IV.
The main mathematical tool used in the investigation of the robustness of controllability is a solvability theorem for an operatorequation in normed spaces with a quasibounded nonlinearity proved in its present form by Granas [ I 11; see also Mawhin [IS]. The heart of the theorem lies in fixed point methods in nonlinear analysis: specifically. the Rothe fixed point theorem whch we state in the Appendix. For details, the reader is referred to the excellent monograph of Smart [ 131. Definition IV.1 (Quasibounded Maps): Given % and 9 Banach spaces ' to %, F is said with respective norms I . I x and I . I and F a map from % to be quasibow1ded if the number
!2Rf?z:
-
(1V.I)
e,?::
is finite, and this number is called the quasinorm of F. 0 Comments: I ) A continuous linear map is quasibounded and its quasinorm correW R [ t O . t O + T ] = ~ r o + ~ ~ ( t o + ~ , ~ ) B ( ~ ) B * ( ~ ) ~ * ( t sponds o + T . to r ) the d ~ usual . induced norm. 10 2) If, for instance, for some cl. c2, c,E R (111.6) I F ( X ) ~ ~ < C J X ~ ~V+X €C{ X ~: I x I ~ ~ c ~ } W.2)
0 Thus. the FDLS with bounded realization given by (11.3) and (11.4) is uniformly completely controllable iff Wh[to.to TI is nonsingular V t o E IW-. Notice, however. that the FDLS can be uniformly completely controllable with the smallest eigenvalue of WR[ to. to + TI tending to 0 as
+
to
-
33.
To guarantee th~s,we define a slightly stronger form of controllability. Defitzition (Ir1.3) (Strong L'niform Complete Conrro1labiliiy))c An FDLS ~ i t hbounded realization represented by (11.3) and (11.4) is srrong!y uniforml~completell: controllable if 3 T > 0. A , > 0 such that VroE R + W,[r,,t,+T]aA',I.
(that is, (IV.2) holds for all x € X outside a ball of radius c 3 ) , then F is quasibounded and its quasinorm is less than or equal to cI. In particular, if cI = 0, then the quasinorm of F is zero. 3) If F is a compact map on X, then F is quasibounded I F(x ) 19 < c,ix~~+c2forsomecl,c,~W. (Recall that a continuous map F: 'Ji 3 ' is said to be compact if the closure of the image of any bounded set is compact.) Theorem I K I (Soltability Theorem): If F: ?X %. is a continuous, quasibounded, compact map on the Banach space 3 and if
-
+
-
P(F) 0, and let X, > 0 be defined by
i L = sup roE R
sup X - ( W R [ 1 0 . t 0 + t ] ) 1 ~ ' 2
(111.9)
P.3)
Now p ( F )
forxEX
1 implies that
+
where 6 and rl are some constants. Choose E > 0 such that E 6 < 1 and define r: = max( r l , 1 . ~ ~ /E). 1 Now S, = ( x € X : / x I = r } is the (topological) boundaryof the ball B , = ( x E X : I x I G r } . and for XES,, we have
r E [ O . TI
and hence. then the reachability condition number over T seconds x R is defined by
1x1 (111.1 1)
By the definition of
E,
< E + 6 (by the definition of r ) we have
935
lEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. AC-27, NO. 4, AUGUST 1982
P.4)
V X E SI, F . (x)l=e"
= Eo(xo)+N(xo)
where A': R " -,L;o([ t o , t o + T ) ] is a continuous map. We next consider the projection of y2( .) E L;o([ t o , t o + T ] )onto ?,(R "1 denotedj2( ,) = Eo(xo)+ $(xo). Note that eof N is one to one ifeo + N is one to one. From the fact that $( .) is Lipschitz. we have. using the previous estimate, that
e,
Given that the map is injective from R " to L;O([r,. to + T I ) [and, in fact, a bijection from Iw " to e,(R ")I. we demand that E, + A' be one to one. BY the incremental small gain theorem (see. for example. ill). E, + R is one to one if
VtER,,
and for some y( +) < (x), IJ.C.(xz,~)-J.txi,~)l~Y(J.)IX2-XiI
V t € R + , V X I . X ~ E R " . (VII.3)
The system Perturbed in the output channel is represented by i=A(r)x
(VII.4)
y=C(t)x+f(x,t)
(VII.5)
where f is a C0 function: R n X R +
+
118" with
m,,t)=eno
VIER+
(VII.6) r E R , XER"
Further, let y ( C ) : = sup IC(t)l,
ON AUTOMATIC CONTROL, VOL.
AC-27, NO. 4,
AUGUST 933
1982
adaptive techniques.’” IEEE Tram. Auromar. Conrr., voL AC-23, pp. 97-99, Feb. 1978. C. R Johnson. Jr., “A convergence proof for a hyperstable adaptive recursive filter.” IEEE Trum. Inform. Theor), vol. IT-25, pp. 745-749, Nov. 1979. Y.-H. Lin and K. S. Pr‘arendra, “A new error model for adaptive systems,” IEEE Trans. Auromar. Conrr., vol. AC-25, pp. 585-587, June 1980. R. R Bitmead and B. D. 0. Anderson, “Performance of adaptive estimation algorithms in dependent random environments.” IEEE Trum. Auromnr. Conrr.. vol. AC-25. pp. 788-794. Aug. 1980. C. R Johnson. Jr., “Another use of the Lin-Narendra error model: HAW.” IEEE Tram. Auromur. Conrr., vol. AC-25, pp. 985-988, Oct. 1980. M. G. Larimore. J. R. Treicbler. and C. R Johnson, Jr., “SHARF:An algoritbm for adapting IIR digital filters,” IEEE Trum. A c m r . , Speech. S i p 1 Processing, vol. ASSP-28, pp. 428-440, Aug. 1980. C. R Johnson, Jr., “An output error identification interpretation of model reference adaptive control,” Auromaricu. vol. 16. pp. 419-421. July 1980. -, “The common parameter estimation basis of adaptive filtering, identification, and control.” in Proc. I9rh IEEE C o d . Decision Conrr.., Albuauemue. PiM.. Dec. 1980, pp. 447-452. C. R. Johnson, Jr. and B. D. 0. Anderson, “Sufficient excitation and stable reducedorder adaptive IIR filtering,” IEEE Tram. Acourr., SDeech. Sinnul Processmn. vol. ASSP-29,pp. 1212-1215, 6ec. 1981. C. A. Desoer and M. Vidyasagar. Feedback Sysrem: Inpur-Oupur Propemes. Ken. York: Academic, 1975. pp. 133-135. J. L. Willems. Srubrlrry Theor). of &namicul Svsrem. London: Nelson, 1970, pp. 113-114. P. Ioannou and C. R. Johnson. Jr.. “Reduced-order performance of parallel and series-parallel identifiers of mc-time-scale systems.” in Proc. 2nd Yule Workshop Appl. Adupr. Sysr., New Haven, C T , May 1981, pp. 169-174.
. . .
The Robustness of Controllabilityand Observability of Linear Time-Varying Systems
s. s. SASTRY, MEMBER, IEEE, AND c.A. DESOER, FELLOW, IEEE Abstract-Fixedpointmethods from nonlinearanalysis are used to establish conditions under which the uniform complete controllability of linear time-varying systems is preserved under nonlinear perturbations in the state dynamics and the zero-input uniform complete observability of linear timevarying systems is preservedunder nonlinear perturbation in the state dynamics and output read-out map. Robustness of partial controllability, observability, and a specific kind of nonzero input observability are also proven. I. INTRODUCTION Controllability and observability are key issues in system theory. To be specific, consider a class of physical dynamical systems which are adequately modeled by ordinary differential equations with inputs u and a static read-out map h : more precisely, R=/(x,u,t)
(1.1)
y=h(x,t)
(1.2)
put observable, then the original nonlinear system is also uniformly completely controllable and zero-input observable. Further, if the linear system is only partially controllable or observable, then the original nonlinear system at least retains these partial controllability and observability properties (see Section V). These results are termed robustness results for the ,controllability and observability of linear time-vaqing systems because of the perturbational nature of the proofs and estimates about a nominal linear system. The major mathematical tool is a solvability theorem for operator equations with a quasibounded nonlinearity due to Granas [4] which is reminiscent of the familiar small gain theorem (see [2], for example). The heart of the theorem lies in the Rothe fixed point theorem. The use of fixed point theorems in proving global nonlinear controllability results is not nev-the hela-Ascoli theorem was used in Lukes [5], the contraction mapping theorem by Mirza and Womack [8], and the Schauder fixed point theorem by Vidyasagar [14] and by Dauer [16], [17] and Aronsson [ 181. For global nonlinear observability. we have the work of Yamamoto and Sugiura [15]. Our estimates by virtue of our new technique are different from those reported in the literature (with some overlap, Theorem V.l is also proven by Lukes).’ Detailed comments on how the results reported above are special cases of our theorems appear in the text of the paper. The results of Dauer [ 171 and Aronsson [18] start from rather different assumptions from ours on the nonlinear perturbations and are not compared. We have illustrated the use of our results in this paper in the derivation of control laws during a very specific emergency, for interconnected power systems, by posing the alert state control problem as a steering problem in [ 1 11. 11. NOTATION The dynamical systems that we study are differential dvnamical svscems (DDS) with finite dimensional cector spaces as input. output, and state space, respectively R“,, Rn0. and R”, with the representation i=f(x,u.t)
(11.1)
Y =h ( x , t )
(11.2)
where t E R - , f is a C o function from R “ X R ” ~ X R , -.Rn which is globally Lipschitz continuous in its first argument (to guarantee uniqueness of solution to (2.1) when the initial condition is given), and h is a Co function from R “ X R R “ 0 . Finite dimensional litwar dynamical systems (FDLS) with a bourlded realization are defined as differential dynamical systems of the form (11.3)$ (11.4): -f
(11.3)
i=A(t)xtB(r)u
y = C( t ) x withx~R“,u~R“~,~ER”~,t€R+,and/,hareCofunctions;/satisfie~ Lipschitz and growth conditions for existence and uniqueness of solutions withIIA(.)II, IIB(.)Il. IIC(.)ll boundedonR+.
on R + . The definition of uniform complete controllability for (1.1) is as follows: 3 T E R + such that given any t o , initial time, and any two states xOl the initial state, and X , , the final state, there exists a control u E L;l([to*t o TI) whichwill steer the system (1.1) from x0 at t o to x, at to T. Zero-input uniform complete observability is defined as: 3 T € R , such that given any t o and the output of the system with zero input on [ t o . to TI. we can determine (uniquely) the state of the system at to. In this paper, we prove that if a system of the form (1.1), (1.2) is “close” to a linear system which is uniformly completely controllable and zero-in-
+
+
+
Manuscript received November 26,1979: revised September 2, 1980 and September 3, 1981. Paper recommended by A. N. Micbel, Past Chairman of the Stability, Nonlinear. and Distributed Systems Committee. This work was supported by the h:ational Science Foundation under Grant ENG78-09032-AOI and the US. Department of Energy under Contract DE-AC01-79ET29364. S. S. Sastry was with the Department of Uectrical Engineering and Computer Scienca and the Electronics Research Laboratory. University of California, Berkeley, CA 94720. He is now wiib the Laboratory for Information and Decision S c i e n c e s , M.I.T.. Cambridge. MA 02139. C. A. Desoer is with the Department of Electrical Engineering and Computer Sciences and the Elecvonics Research Laboratoq-. University of California, Berkeley, CA 94720.
111.
CHARACTERIZATION OF CONTROLLABILIN FOR
(11.4) (11.5) FINITE
LINEARSYSTEhlS DIMENSIONAL The definitions and propositions of this section are well known. although not standardized. We restate them here to establish the terminology and notation. The definitions are d r a w from Silverman [12], and the proofs maybe found in standard books (see.e.g., [l]). To obtain the desired characterization define, for fixed t o E R + , the linear map C R (called the reachabilig map) from L:L([t o . t o TI) to R ” by
+
(111.1) where @(I, T ) is the state-transition function for the linear system. Then at 7, (111.1) leads to
t = to
+
’Since this paper was written. the work of Dacka [3]. d i c h is similar in spirit, has appeared. His main result is a generalization of our Theorem V. 1. This generalization can be derived from our methods as well (see Section V-A).
OO18-9286/82/0800-0933$O00.75 a1982IEEE
934
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL.
(111.2)
e,.
The adjoint map of denoted E,: L2,([t0,t o + 71)defined by
?:.X=
then is the linear map from R " to
B*( .)Q*( to + T , . ) X .
(111.3)
The burden of this paper consists of demonstrating the robustness of strong uniform complete controllubility of an FDLS in the faceof nonlinear perturbations in the dynamics, both bounded and unbounded. Our methcds seemto indicate thatFDLS with a smaller reachability condition number are more robust than others with a larger reachability condition number.
Since the realization (11.5) is bounded (say by K ) , we have from the Bellman-Gronwall lemma that ilQ(t.~)liSexpK(t-~)
V ~ , T E W + t, a ~ . (111.4)
Also, llB(T)1! S K
b'TER+.
(111.5)
Using (111.4) and (111.5). it is easy to check that and E: are continuous linear maps. The following is well known. Theorem III.1 (Characterization of Uniform Complete Controllabilig for FDLS): The FDLS with bounded realization represented by (11.3) and (11.4) is uniformly completely controllable V t o E R + , the reachability map E:, L;,([t o ,to + T I ) -,R " defined in (111.1) is onto V t o E W + the composition of the reachability map, and its adjoint. namely, R " R". is a bijection. 0 Definition 111.3 (Reachability Grammian): Given t o € W + , the matrix representation of the continuous linear map R " -88" is the reachabilig grammian, denoted WR[to.t o + T I E R n X "
-
AC-27,NO. 4, AUGUST 1982
SOLVABILITY OF AN OPERATOR EQUATION WITH A QUASIBOUNDED NONLINEARITY IN NORMED SPACES
IV.
The main mathematical tool used in the investigation of the robustness of controllability is a solvability theorem for an operatorequation in normed spaces with a quasibounded nonlinearity proved in its present form by Granas [ I 11; see also Mawhin [IS]. The heart of the theorem lies in fixed point methods in nonlinear analysis: specifically. the Rothe fixed point theorem whch we state in the Appendix. For details, the reader is referred to the excellent monograph of Smart [ 131. Definition IV.1 (Quasibounded Maps): Given % and 9 Banach spaces ' to %, F is said with respective norms I . I x and I . I and F a map from % to be quasibow1ded if the number
!2Rf?z:
-
(1V.I)
e,?::
is finite, and this number is called the quasinorm of F. 0 Comments: I ) A continuous linear map is quasibounded and its quasinorm correW R [ t O . t O + T ] = ~ r o + ~ ~ ( t o + ~ , ~ ) B ( ~ ) B * ( ~ ) ~ * ( t sponds o + T . to r ) the d ~ usual . induced norm. 10 2) If, for instance, for some cl. c2, c,E R (111.6) I F ( X ) ~ ~ < C J X ~ ~V+X €C{ X ~: I x I ~ ~ c ~ } W.2)
0 Thus. the FDLS with bounded realization given by (11.3) and (11.4) is uniformly completely controllable iff Wh[to.to TI is nonsingular V t o E IW-. Notice, however. that the FDLS can be uniformly completely controllable with the smallest eigenvalue of WR[ to. to + TI tending to 0 as
+
to
-
33.
To guarantee th~s,we define a slightly stronger form of controllability. Defitzition (Ir1.3) (Strong L'niform Complete Conrro1labiliiy))c An FDLS ~ i t hbounded realization represented by (11.3) and (11.4) is srrong!y uniforml~completell: controllable if 3 T > 0. A , > 0 such that VroE R + W,[r,,t,+T]aA',I.
(that is, (IV.2) holds for all x € X outside a ball of radius c 3 ) , then F is quasibounded and its quasinorm is less than or equal to cI. In particular, if cI = 0, then the quasinorm of F is zero. 3) If F is a compact map on X, then F is quasibounded I F(x ) 19 < c,ix~~+c2forsomecl,c,~W. (Recall that a continuous map F: 'Ji 3 ' is said to be compact if the closure of the image of any bounded set is compact.) Theorem I K I (Soltability Theorem): If F: ?X %. is a continuous, quasibounded, compact map on the Banach space 3 and if
-
+
-
P(F) 0, and let X, > 0 be defined by
i L = sup roE R
sup X - ( W R [ 1 0 . t 0 + t ] ) 1 ~ ' 2
(111.9)
P.3)
Now p ( F )
forxEX
1 implies that
+
where 6 and rl are some constants. Choose E > 0 such that E 6 < 1 and define r: = max( r l , 1 . ~ ~ /E). 1 Now S, = ( x € X : / x I = r } is the (topological) boundaryof the ball B , = ( x E X : I x I G r } . and for XES,, we have
r E [ O . TI
and hence. then the reachability condition number over T seconds x R is defined by
1x1 (111.1 1)
By the definition of
E,
< E + 6 (by the definition of r ) we have
935
lEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. AC-27, NO. 4, AUGUST 1982
P.4)
V X E SI, F . (x)l=e"
= Eo(xo)+N(xo)
where A': R " -,L;o([ t o , t o + T ) ] is a continuous map. We next consider the projection of y2( .) E L;o([ t o , t o + T ] )onto ?,(R "1 denotedj2( ,) = Eo(xo)+ $(xo). Note that eof N is one to one ifeo + N is one to one. From the fact that $( .) is Lipschitz. we have. using the previous estimate, that
e,
Given that the map is injective from R " to L;O([r,. to + T I ) [and, in fact, a bijection from Iw " to e,(R ")I. we demand that E, + A' be one to one. BY the incremental small gain theorem (see. for example. ill). E, + R is one to one if
VtER,,
and for some y( +) < (x), IJ.C.(xz,~)-J.txi,~)l~Y(J.)IX2-XiI
V t € R + , V X I . X ~ E R " . (VII.3)
The system Perturbed in the output channel is represented by i=A(r)x
(VII.4)
y=C(t)x+f(x,t)
(VII.5)
where f is a C0 function: R n X R +
+
118" with
m,,t)=eno
VIER+
(VII.6) r E R , XER"
Further, let y ( C ) : = sup IC(t)l,
