A Kalman Decomposition for Possibly Controllable Uncertain Linear Systems

A Kalman D ecomp osition for P ossibly Controllable Uncertain Linear Systems ⋆ Ian R. Petersen ∗ ∗ Scho ol of Engine ering and Information T e chnolo gy, U niversity of New South Wales at the Austr alian Defenc e F or c e A c ademy, Canb err a ACT 2600, Austr alia (e-mail: i.r. p etersen@gmail.c om) Abstract This pap er considers th e structure of uncertain linear systems bu ilding on concepts of robust unobserv abilit y and p ossible contro llabilit y which w ere introdu ced in prev ious pap ers. The p ap er presents a new geometric characteri zation of th e p ossibly contro llable states. When com bined with previous geometric results on robust unobserv ability , the results of this paper lead to a general Kalman type d ecomposition for uncertain linear systems whic h can b e applied to the p roblem of obtaining redu ced order uncertain sy stem mo dels. 1 In tro duction Controllabilit y and observ ability a re fundamental prop- erties of a linear sys tem; e.g., see [1]. This pap er is co n- cerned with extending these notions to the case of un- certain linear sys tems with the aim of gaining gr eater understanding of the s tructure of uncertain linear sys- tems when applied to problems of reduced order mo d- elling and minimal realizatio n. One reaso n for consider ing the issue of controllabilit y for uncertain systems mig h t b e to determine if a robust state feedback cont roller can b e constructed for the sys- tem; e.g., see [2]. In this case, one w ould be in terested in the q uestion of whether the system is “controllable” for all p o ssible v alue s of the uncertaint y; e.g., see [3–8]. Sim- ilarly , one reason for consider ing observ ability for uncer- tain systems might b e to determine if a robust state es- timator can b e constructed for the sy stem; e.g ., se e [9]. In this case, one would b e in terested in the question o f whether th e system is “observ able” for all p ossible v alues of the uncertaint y; e.g., see [10]. Ho wev er, these q ues- tions of r obust controllability and robust obser v ability are not the questions b eing a ddressed in this pap er. F or the case o f linear sy stems, the notions of controlla- bilit y and observ ability are cen tral to realiza tion theory ; e.g., see [1]. F or example, it is known that if a linear ⋆ This w ork w as supp orted by the Au stralian Researc h Council. Preliminary ve rsions of some of the results of t his pap er appeared in the 46th IEEE Conference on Decision and Con trol, New O rleans and the 2008 I F AC W orld Congress , Seoul. system contains unobserv able or uncontrollable states , those states can b e r emov ed in order to o btain a reduced dimension rea liz ation of the system’s transfer function. F r om this point of view, a natural extensio n of the no- tion o f con trolla bility to the ca se of uncertain s ystems, would be to consider “p ossibly con trolla ble ” states which are controllable for so me p ossible v alues of the uncer - taint y . This idea was developed in the pa per [11] for the case of uncerta in linear sys tems with structured uncer- taint y s ub ject to av erag ed in tegral qua dratic constra ints (IQCs). Similarly , a natural e x tension of the notion o f observ abilit y to uncer tain systems is to consider robus tly unobserv able states whic h are “unobs erv a ble ” for all po s- sible v alues of the uncertaint y . This idea was developed in the pap ers [12, 13]. This pa pe r builds o n co ncepts of “r obust unobser v abil- it y and “p ossible c ontrollabilit y” develop ed in the pa- per s [11, 12]. The results presented in the pa per aim to provide insight in to the str ucture of uncertain systems as it relates to questions of r ealization theo r y and re- duced dimensio n modelling for uncertain sy s tems; e.g., see [14–16]. W e formally define notions of robus t unobserv ability and po ssible controllabilit y in terms o f certain constrained optimization problems. The notion of ro bust unobs erv- ability used in this pa per inv olves extending the standard linear systems definition of the observ abilit y Gra mian to the case of uncertain systems; se e also [17]. Also, the no- tion of p os sible controllability used in this pa pe r inv olves extending the standard linear systems definition of the controllabilit y Gramian to the case of uncer tain systems; see also [18]. W e then apply the S-pro cedure (e.g., see [2]) Preprint submitted to Automatica 6 August 2018 to obtain conditions for robust unobserv ability a nd p o s- sible controllability in terms of unconstra ined LQ opti- mal cont rol problems dep endent on Lagrang e m ultiplier parameters as in [11, 12]. F r om this, w e develop a g eo- metric c hara c terization for the set of r obustly unobserv- able states (as in [13]) and the s et of p oss ibly controllable states. The s e ch ara cterizations imply that the set of ro- bustly unobserv able states is in fact a linear subspace. Similarly , we show that the set of p oss ibly co n trollable states is a linear s ubspace; see also [3, 6, 7]. Thes e charac- terizations lead to a Kalman t ype decomp os itio n for the uncertain systems under consider ation; see also [19 ], [2 0] and Theorem 4 .3 in Cha pter 3 of [1]. This decomp osition is describ ed in the four p ossible cases for which an uncer- tain system mo del can have robus tly uno bserv a ble states or s ta tes which are not p ossibly controllable. These are the cases in which a reduced dimensio n uncertain sy s - tem model can b e obtained whic h reta ins the same set of input-output behaviours as the or iginal mo del. As compared to the pr evious pap ers [11–13], the results of this paper enable a complete geometrical picture to b e obtained which can be a pplied to problems of r educed dimension modelling of uncertain linear systems. Also, the results of this pa p er are muc h mor e co mputation- ally tractable than the res ults of the pa per s [11, 12]. The main assumption requir ed in this pap er as compared to the previous papers [11, 12] is the assumption that the uncertaint y is uns tr uctured and des crib ed b y a sing le av eraged uncer taint y co nstraint. The remainder of the pap er pro ce e ds as follows. In Sec- tion 2, the class of uncertain systems under considera- tion is in tro duced and definitions of robust unobser v- ability a nd p oss ible controllability are g iven. In Section 3, the existing g e o metrical results on robust o bs erv a bil- it y ar e summarized. In Sections 4, 5 , 6, our main results on p ossible con trollability ar e g iven. In Section 7, the re- sults a re c o mbin ed to obtain complete Ka lman decom- po sition results and in Section 8, an illustr ative example is given. T he pa per is concluded in Section 9. 2 Problem F o rm ulation W e consider the following linear time inv ariant uncer tain system: ˙ x ( t ) = Ax ( t ) + B 1 u ( t ) + B 2 ξ ( t ); z ( t ) = C 1 x ( t ) + D 1 u ( t ); y ( t ) = C 2 x ( t ) + D 2 ξ ( t ) (1) where x ∈ R n is the state , y ∈ R l is the me asur e d output , z ∈ R h is the u nc ert ainty output , u ∈ R m is the c ontr ol input , and ξ ∈ R r is the u nc ert ainty input . F o r the system (1), w e de fine the transfer function G ( s ) to b e the transfer function from the input ξ ( t ) to the output y ( t ); i.e ., G ( s ) = C 2 ( sI − A ) − 1 B 2 + D 2 . Also, we define the transfer function H ( s ) to b e the transfer function fro m the input u ( t ) to the output z ( t ); i.e., H ( s ) = C 1 ( sI − A ) − 1 B 1 + D 1 . System Unc ertainty. The uncertaint y in the uncerta in system (1) is required to satisfy a certain “Averaged Int egra l Quadr atic Constraint”. Aver age d Inte gr al Quadr atic Constr aint. Let the time in- terv a l [0 , T ], T > 0 be given and let d > 0 b e a giv en po sitive cons tant a s so ciated with the sy stem (1); see also [11, 12, 2 1]. W e will c o nsider sequences o f uncertain ty in- puts S = { ξ 1 ( · ) , ξ 2 ( · ) , . . . ξ q ( · ) } . The num ber of elements q in any such seq uence is ar bitrary . A sequence of uncer - taint y functions of the form S = { ξ 1 ( · ) , ξ 2 ( · ) , . . . ξ q ( · ) } is an admissible unc ertainty s e quenc e for the sys tem (1 ) if the following conditions hold: Given a n y ξ i ( · ) ∈ S and any corresp onding solution { x i ( · ) , ξ i ( · ) } to (1) defined on [0 , T ], then ξ i ( · ) ∈ L 2 [0 , T ], a nd 1 q q X i =1 Z T 0  k ξ i ( t ) k 2 − k z i ( t ) k 2  dt ≤ d. (2) The class of all such a dmiss ible uncertaint y se q uences is denoted Ξ . One wa y in which such uncerta in ty could be generated is v ia unstructured feedback uncertaint y as shown in the blo ck diagr am in Fig. 1. The averaged IQC uncertaint y description was intro- duced in [21] as a n approach to uncertaint y modelling which gives tigh t results in the ca s e of structured uncer- taint y . The pap er [11] gives a more detailed e xplanation concerning the use of the av eraged IQC uncertaint y de- scription. This pap er contin ues to use the av eraged IQC uncertaint y desc r iption even though it does not consider structured uncertainties since it builds on the r esults of [11, 12] which w ere derived using the averaged I Q C un- certaint y des c ription. It should b e possible to re-derive the results of [11 , 12] using the sta ndard rather than av- eraged IQC uncertain ty description such as co nsidered in [2 2]. These results could then b e us e d to obtain results corres p onding to the re sults of this pap er in the ca s e of a standard IQ C uncertaint y description ra ther than an av eraged IQ C uncer taint y description. Definition 1 The r obust unobse rv ability function for the unc ertain system (1), (2) define d on the time interval [0 , T ] is define d as L o ( x 0 , T ) ∆ = sup S ∈ Ξ 1 q q X i =1 Z T 0 k y ( t ) k 2 dt (3) 2 ✲ ✛ ✲ ✲ ∆( · ) u y ξ z Nominal System Fig. 1. U ncertain system blo ck d iagram’. wher e x (0) = x 0 in (1). This definition extends the s tandard definition of the observ abilit y Gramian for linear systems. Notation. D ∆ = { d : d > 0 } . Definition 2 A non-zer o state x 0 ∈ R n is said to b e robustly unobserv able for the un c ertain system (1), (2) define d on the time int erval [0 , T ] if inf d ∈D L o ( x 0 , T ) = 0 . The set of al l r obustly unobservable states for the unc er- tain system (1), (2) define d on the time interval [0 , T ] is r eferr e d to as the robustly unobs erv a ble s et U ; i.e., U ∆ =  x ∈ R n : inf d ∈D L o ( x, T ) = 0  . Definition 3 The po s sible co ntrollability function for the unc ertain system (1), (2) define d on the time interval [0 , T ] is define d as L c ( x 0 , T ) ∆ = sup ǫ> 0 inf S ∈ Ξ inf U ∈ L q 2 [0 ,T ] 1 q q X i =1 " k x i ( T ) k 2 ǫ + R T 0 k u i ( t ) k 2 dt # (4) wher e x (0) = x 0 in (1). This definition extends the s tandard definition of the controllabilit y Gramian for linea r sys tems . In par ticular, in the spe cial case of systems without uncertaint y , this quantit y will b e infinite for uncontrollable states x 0 . Definition 4 A non-zer o state x 0 ∈ R n is said to b e po ssibly controllable on [0 , T ] for t he u nc ertain system (1), (2) if sup d ∈D L c ( x 0 , T ) < ∞ . This definition reduces to the definition of co n trollable states for the sp ecial case of systems without uncer - taint y ; e.g ., see [1]. Definition 5 A non-zer o state x 0 ∈ R n is said to b e (different ially) p os sibly controllable for the unc ertain system (1), (2) if it is p ossibly c ontr ol lable on [0 , T ] for al l T > 0 su fficiently smal l. The set of al l differ ential ly p ossibly c ontr ol lable states for the unc ertain system (1), ( 2) is r eferr e d to as the p os s ibly controllable set C . Remark 1 It is emphasize d in [11] that t he notion of p ossibly c ontr ol lability for unc ertain syst ems is an ex ten- sion of the standar d notion of c ont r ol lability in its appli- c ation t o pr oblems of minimal r e alization. In p articular, in t he se quel it wil l b e shown that t he existenc e of st ates which ar e not p ossibly c ont r ol lable in an unc ertain system mo del me ans that a r e duc e d dimension unc ertain system mo del c an b e obtaine d with the same input- output b e- haviour as the original mo del. In this sense, states which ar e n ot p ossibly c ontr ol lable c ontr ol lable c orr esp ond to unc ontr ol lable states in standar d line ar systems the ory; e.g., se e [1]. 3 Existing Results on Robust Unobserv abil it y In this sec tio n, we r ecall some existing r esults from [13 ] giving a g e o metrical characterization of r obust unob- serv abilit y . F o r the uncertain system (1), (2) defined on the time int erv al [0 , T ], we define a function V τ ( x 0 , T ) a s follows: V τ ( x 0 , T ) ∆ = inf ξ ( · ) ∈ L 2 [0 ,T ] Z T 0  −k y k 2 + τ k ξ k 2 − τ k z k 2  dt. (5) Here τ ≥ 0 is a given constant. ¯ Γ( x 0 , T ) ∆ = n τ : τ ≥ 0 and V τ ( x 0 , T ) > − ∞ o . Assumption 1 F or al l x 0 ∈ R n , ther e ex ists a c onstant τ ≥ 0 such t hat V τ ( x 0 , T ) > − ∞ . R emark: The abov e assumption is a technical a ssump- tion r equired to establish the r esults o f [13 ]. It represents an assumption on the size of the uncer taint y in the sys- tem rela tive to the time interv al [0 , T ] under cons idera- tion. In general, this assumption c a n alwa ys be s a tisfied by choosing a sufficient ly s mall T > 0. 3 Theorem 1 (Se e [13] for pr o of ). Consider the un c er- tain system (1), (2) and su pp ose that Assumption 1 is satisfie d. Also, supp ose that G ( s ) ≡ 0 . Then a state x 0 is r obust ly u nobservable if and only if it is an unobservable state for the p air ( C 2 , A ) . R emark: F rom the ab ove theorem a nd the fa c t that G ( s ) ≡ 0, it follows that we can apply the s ta ndard Kalman decomp osition to repre sent the uncertain sys- tem as shown in Fig. 2. u Observable Unobservable + y z z z ξ 1 2 ∆ Fig. 2. Observ able-Unobserv able decomp osition for the u n- certain system when G ( s ) ≡ 0. Note that in this cas e, all of the uncertaint y is in the unobserv able subsystem and the co upling b etw een the t wo subsy s tems. Theorem 2 (Se e [13] for pr o of ). Consider the un c er- tain system (1), (2) and su pp ose that Assumption 1 is satisfie d. Also, supp ose that G ( s ) 6≡ 0 . Then a state x 0 is r obust ly u nobservable if and only if it is an unobservable state for the p air ( " C 1 C 2 # , A ) . R emark: The a bove theo rem implies that whe n G ( s ) 6≡ 0, the r obustly unobserv able set is a linea r spa ce eq ual to the unobserv able subspace of the pair ( " C 1 C 2 # , A ). F rom this theo r em, it follows that we ca n a pply the standard Kalman decomp osition to repre sent the uncertain sys- tem as shown in Fig. 3. In this case , all of the uncertaint y is in the observ able subsystem or in the coupling betw een the t wo subsys- tems. u Unobservable Observable y z ∆ ξ Fig. 3. Observ able-Un observ able d ecomposition for the un - certain system when G ( s ) 6≡ 0. 4 Preliminary Results on P ossibl e Contr olla- bility In this section, we will recall the main results of [11] s pe- cialized to the class of uncertain s ystems with unstruc- tured uncertaint y considered in this pap er. 4.1 A F amily of Unc onstr aine d Optimization Pr oblems. F o r the uncertain system (1), (2) defined on the time interv al [0 , T ], we define functions W ǫ τ ( x 0 , λ, T ), W ǫ τ ( x 0 , T ) and W τ ( x 0 , T ) as follows: W ǫ τ ( x 0 , λ, T ) ∆ = inf [ ξ ( · ) ,u ( · )] ∈ L 2 [ λ,T ] k x ( T ) k 2 ǫ + Z T λ  k u k 2 + τ k ξ k 2 − τ k z k 2  dt (6) sub ject to x ( λ ) = x 0 ; W ǫ τ ( x 0 , T ) ∆ = W ǫ τ ( x 0 , 0 , T ); W τ ( x 0 , T ) ∆ = sup ǫ> 0 W ǫ τ ( x 0 , T ) . Here τ ≥ 0 is a given constant. 4.2 A F ormula for the Possible Contr ol lability F unc- tion. Theorem 3 (Se e [11] for pr o of ). Consider the u nc ertain system (1), (2) define d on the time interval [0 , T ] and c orr esp onding p ossible c ontr ol lability function (4). Then for any x 0 ∈ R n , L c ( x 0 , T ) = sup ǫ> 0 sup τ ≥ 0 { W ǫ τ ( x 0 , T ) − τ d } ; = sup τ ≥ 0 { W τ ( x 0 , T ) − τ d } . (7) 4 Corollary 1 ( Se e [11] for pr o of ). If we define ˜ L c ( x 0 , T ) ∆ = sup d ∈D L c ( x 0 , T ) then ˜ L c ( x 0 , T ) = sup ǫ> 0 sup τ ≥ 0 W ǫ τ ( x 0 , T ) = s up τ ≥ 0 W τ ( x 0 , T ) . Observ ation 1 F r om the ab ove c or ol lary, it fol lows im- me diately that a non-zer o state x 0 ∈ R n is (differ en- tial ly) p ossibly c ontr ol lable for t he un c ertain system (1), (2) if and only if sup ǫ> 0 sup τ ≥ 0 W ǫ τ ( x 0 , T ) = s up τ ≥ 0 W τ ( x 0 , T ) < ∞ (8) for al l T > 0 sufficiently smal l. 5 Riccati Equation Solution to the Uncon- strained Optim ization Proble ms In o r der to calcula te W ǫ τ ( x 0 , λ, T ), w e note that if τ > 0, and the optimization pro blem (6) has a finite solution for all initial conditions, then it c a n b e solv ed in terms of the following Riccati differential equation (RDE): − ˙ P ǫ = A ′ P ǫ + P ǫ A − ( P ǫ B 1 − τ C ′ 1 D 1 ) ( I − τ D ′ 1 D 1 ) − 1 ( P ǫ B 1 − τ C ′ 1 D 1 ) ′ − P ǫ B 2 B ′ 2 P ǫ τ − τ C 1 C ′ 1 ; P ǫ ( T ) = I /ǫ (9) which is so lved backw ards in time. Lemma 1 Le t τ > 0 b e such that I − τ D ′ 1 D 1 > 0 . (10) Consider the system ( 1) define d on [0 , T ] and c ost func- tional (6) with λ ∈ [0 , T ) . Then W ǫ τ ( x 0 , λ, T ) > −∞ ∀ x 0 ∈ R n if and only if the RDE ( 9) has a solution P ǫ τ ( t ) define d on [ λ, T ] . In this c ase, W ǫ τ ( x 0 , λ, T ) = x ′ 0 P ǫ τ ( λ ) x 0 . (11) Pr o of. This lemma follows directly from a standar d LQR optimal control r esult; e.g., see pa ge 55 of [2 3]. ✷ In order to calculate W τ ( x 0 , T ) using the Riccati equa- tion approach of [1 1], w e will consider the following RDEs: ˙ S ǫ = AS ǫ + S ǫ A ′ − ( B 1 − τ S ǫ C ′ 1 D 1 ) ( I − τ D ′ 1 D 1 ) − 1 ( B 1 − τ S ǫ C ′ 1 D 1 ) ′ − B 2 B ′ 2 τ − τ S ǫ C 1 C ′ 1 S ǫ ; S ǫ ( T ) = ǫI ; (12) ˙ S = AS + S A ′ − ( B 1 − τ S C ′ 1 D 1 ) ( I − τ D ′ 1 D 1 ) − 1 ( B 1 − τ S C ′ 1 D 1 ) ′ − B 2 B ′ 2 τ − τ S C 1 C ′ 1 S ; S ( T ) = 0 (13) which are solved backw ards in time. Theorem 4 (se e [11] for pr o of.) L et τ > 0 b e such t hat I − τ D ′ 1 D 1 > 0 . Also supp ose ther e ex ists an ǫ 0 > 0 such that for al l ǫ ∈ (0 , ǫ 0 ) , al l non-zer o x 0 ∈ R n and al l λ ∈ [0 , T ] , then W ǫ τ ( x 0 , λ, T ) > 0 . Then for a ny ǫ ∈ (0 , ǫ 0 ) , the Ric c ati e quations ( 12) and (13) have solut ions S ǫ τ ( t ) > 0 and S τ ( t ) ≥ 0 define d on [0 , T ] and for any x 0 6 = 0 W ǫ τ ( x 0 , T ) = x ′ 0 [ S ǫ τ (0)] − 1 x 0 > 0 . Also , if S τ (0) > 0 t hen W τ ( x 0 , T ) = x ′ 0 [ S τ (0)] − 1 x 0 > 0 . F urthermor e, if t he matrix S τ (0) ≥ 0 is singular and x 0 is not c ontaine d within the ra nge sp ac e of S τ (0) , then W τ ( x 0 , T ) = ∞ . The following lemma shows that the time in terv al [0 , T ] can alwa ys b e chosen shor t enough to guarantee that solutions to the RDEs exist. Lemma 2 L et ǫ ∗ > 0 and τ ∗ > 0 b e given. Then ther e exists a sufficiently smal l ˜ T > 0 such that the RDEs (12) and (13) b oth have solutions on [0 , ˜ T ] and S ǫ ∗ τ ∗ ( t ) > 0 . Pr o of. This res ult follows from standard re sults on dif- ferential eq uations and the fact that S ǫ ∗ ( T ) = ǫ ∗ I > 0. ✷ Lemma 3 Corr esp onding t o the syst em (1), we c onsider the dual system: ˙ x ( t ) = − A ′ x ( t ) + C ′ 1 ξ ( t ); y ( t ) = B ′ 1 x ( t ) − D ′ 1 ξ ( t ); z ( t ) = B ′ 2 x ( t ) (14) 5 define d on the time interval [0 , T ] , with initial c ondition x ( λ ) = ˜ x 0 wher e λ ∈ [0 , T ) . Also, supp ose ǫ > 0 and τ > 0 ar e such that the RDEs (12 ) and (13) b oth have solutions on [0 , T ] . Then, we c an write ˜ x ′ 0 S ǫ τ ( λ ) ˜ x 0 = sup ξ ( · ) ∈ L 2 [ λ,T ] ( ǫ k x ( T ) k 2 + R T λ  k y k 2 + 1 τ k z k 2 − 1 τ k ξ k 2  dt ) (15) and ˜ x ′ 0 S τ ( λ ) ˜ x 0 = sup ξ ( · ) ∈ L 2 [ λ,T ] Z T λ  k y k 2 + 1 τ k z k 2 − 1 τ k ξ k 2  dt. (16) F urthermor e, for any λ ∈ [0 , T ) , we have S ǫ τ ( λ ) ≥ S τ ( λ ) ≥ 0 . (17) Pr o of. It follows via some straig ht forward algebraic ma- nipulations that the RDE (12 ) ca n b e re- written as ˙ S ǫ = AS ǫ + S ǫ A ′ − ( S ǫ C ′ 1 − B 1 D ′ 1 )  I τ − D 1 D ′ 1  − 1 ( S ǫ C ′ 1 − B 1 D ′ 1 ) ′ − B 2 B ′ 2 τ − B 1 B ′ 1 ; S ǫ ( T ) = ǫI . (18) Similarly , the RDE (13) can b e re-written as ˙ S = AS + S A ′ − ( S C ′ 1 − B 1 D ′ 1 )  I τ − D 1 D ′ 1  − 1 ( S C ′ 1 − B 1 D ′ 1 ) ′ − B 2 B ′ 2 τ − B 1 B ′ 1 ; S ( T ) = 0 . (19) Then, the formulas (15), (16) follow direc tly from a stan- dard re s ult on the linear quadratic reg ula tor problem; e.g., s e e page 55 of [23 ]. Also, the firs t inequality in (1 7) follows by compar ing (1 5) a nd (16), and the second in- equality in (17 follows by setting ξ ( · ) ≡ 0 in (16 ). ✷ The following simple linea r algebr a result will also b e useful in the pro of of our ma in results. Lemma 4 Le t N b e a given matrix and let M > 0 and ˜ M > 0 b e given p ositive definite matric es such that ˜ M = N N ′ + M If the ve ctor x 0 c an b e written as x 0 = N y 0 , we have x 0 ˜ M − 1 x 0 ≤ y ′ 0 y 0 . Pr o of. It follows from the Matrix Inv ersio n Lemma that we can write I − N ′ ( M + N N ′ ) − 1 N =  I + N ′ M − 1 N  − 1 . Hence, N ′ ( M + N N ′ ) − 1 N = I −  I + N ′ M − 1 N  − 1 ≤ I . Therefore, y ′ 0 N ′ ( M + N N ′ ) − 1 N y 0 = x ′ 0 ˜ M − 1 x 0 ≤ y ′ 0 y 0 . This completes the pro of of the lemma. ✷ 6 Main Results o n P ossible Controllability In this section, we pres e n t r esults which provide a geo- metric characterization of the differentially po ssibly co n- trollable states of the uncertain system (1), (2). W e first consider the ca se in which H ( s ) ≡ 0. Theorem 5 Consider the unc ertain system (1), (2). Also , s upp ose that H ( s ) ≡ 0 . Then a state x 0 is dif- fer ential ly p ossibly c ontr ol lable if and only if i t is a c ont ro l lable state for the p air ( A, B 1 ) . Pr o of. W e first supp ose x 0 is a differentially po ssibly con- trollable state for the uncertain system (1), (2). Hence, using Observ ation 1 it follows that sup ǫ> 0 sup τ ≥ 0 W ǫ τ ( x 0 , T ) < ∞ (20) for all T > 0 sufficiently small. Now let ǫ ∗ > 0 and τ ∗ > 0 be given and cho ose ˜ T > 0 sufficiently small a s in Lemma 2 . Now s ince H ( s ) ≡ 0 , we m ust have D 1 = 0 and it follows fr o m Lemma 3 that we can write ˜ x ′ 0 S τ ∗ ( λ ) ˜ x 0 = sup ξ ( · ) ∈ L 2 [ λ, ˜ T ] Z ˜ T λ k y k 2 + 1 τ ∗ k z k 2 − 1 τ ∗ k ξ k 2 ! dt = Z ˜ T λ    B ′ 1 e − A ′ t ˜ x 0    2 dt + 1 τ ∗ sup ξ ( · ) ∈ L 2 [ λ, ˜ T ] Z ˜ T λ  k z k 2 − k ξ k 2  dt = ˜ x ′ 0 W c ( λ, ˜ T ) ˜ x 0 + 1 τ ∗ ˜ x ′ 0 Q ( λ, ˜ T ) ˜ x 0 6 where ˜ x ′ 0 Q ( λ, ˜ T ) ˜ x 0 = sup ξ ( · ) ∈ L 2 [ λ, ˜ T ] Z ˜ T λ  k z k 2 − k ξ k 2  dt ≥ 0 (21) and W c ( λ, ˜ T ) = Z ˜ T λ e − At B 1 B ′ 1 e − A ′ t dt is the controllability Gramian for the pair ( A, B 1 ); e.g ., see [1]. F rom this, we can conclude that S τ ( λ ) = W c ( λ, ˜ T ) + 1 τ Q ( λ, ˜ T ) (22) is monotone decreas ing as τ incr e a ses and hence, the RDE (13) do es not hav e a finite escap e time o n [0 , ˜ T ] for all τ ≥ τ ∗ . F urthermore, it follows from the c o nt inuit y of solutions to the RDEs (13) a nd (12 ) that for all τ ≥ τ ∗ , there exists a ǫ ∈ (0 , ǫ ∗ ) sufficiently sma ll such that the RDE (12 ) has a solution S ǫ τ ( t ) on [0 , ˜ T ]. W e now o bserve that S ǫ τ ( λ ) > S τ ( λ ) ≥ 0 for all λ ∈ [0 , ˜ T ]. Indeed, g iven any non- z ero ˜ x 0 ∈ R n , it follows from (15) and (16) that ˜ x ′ 0 S ǫ τ ( λ ) ˜ x 0 ≥ ˜ x ′ 0 S τ ( λ ) ˜ x 0 + ǫ k x ∗ ( T ) k 2 (23) where x ∗ ( t ) is the solution to (14 ) with initial condition x ( λ ) = ˜ x 0 and input ξ ∗ ( · ) whic h achiev es the supremum in (16 ). F urthermore, since S τ ( t ) the solutio n to RDE (13) exists on [0 , ˜ T ], it follows by a sta ndard result on linear quadra tic o ptimal co n trol (e.g., see [23 ]) that ξ ∗ ( · ) is defined b y the follo wing state feedback control law for (14) ξ ∗ ( t ) = − τ C 1 S τ ( t ) x ∗ ( t ) . Then, we can write x ∗ ( T ) = Φ( ˜ T , λ ) ˜ x 0 where Φ( ˜ T , λ ) is the state tr ansition matrix for the close d lo o p system ˙ x = ( − A ′ − τ C ′ 1 C 1 S τ ( t )) x. Hence, it follows from (23) tha t ˜ x ′ 0 S ǫ τ ( λ ) ˜ x 0 ≥ ˜ x ′ 0 S τ ( λ ) ˜ x 0 + ǫ k Φ( ˜ T , λ ) ˜ x 0 k 2 > ˜ x ′ 0 S τ ( λ ) ˜ x 0 . Thu s, w e can conclude that S ǫ τ ( λ ) > S τ ( λ ) ≥ 0 for all λ ∈ [0 , ˜ T ]. Also , it follows from Lemma 3, that for an y λ ∈ [0 , ˜ T ) that S ǫ τ ( λ ) is mo notone decreas ing as ǫ → 0. W e have now established that given any τ ≥ τ ∗ , there exists an ǫ ∈ (0 , ǫ ∗ ) such that S ǫ τ ( t ) the s olution to (12) exists on [0 , ˜ T ] and S ǫ τ ( t ) > 0 for a ll t ∈ [0 , ˜ T ]. F rom this, it follo ws that P ǫ τ ( t ) = [ S ǫ τ ( t )] − 1 > 0 is the solution to (9) o n [0 , ˜ T ]. Therefore, it follows fro m Theor em 4 that given any x 0 ∈ R n , then we can write W ǫ τ ( x 0 , ˜ T ) = x ′ 0 [ S ǫ τ (0)] − 1 x 0 . Now we return to the inequality (29) for our differen tially po ssibly controllable state x 0 and co nc lude that there exists a constant M ≥ 0 such that given any in teger k ≥ k 0 ≥ τ ∗ , there exists an ǫ k ∈ (0 , ǫ ∗ ) such tha t W ǫ k ( x 0 , ˜ T ) = x ′ 0 [ S ǫ k k (0)] − 1 x 0 ≤ M . (24) Also, we can as sume without loss of genera lit y that ǫ k → 0 as k → ∞ . W e now define a sequence { y k 0 } ∞ k = k 0 as y k 0 = [ S ǫ k k (0)] − 1 2 x 0 . Hence, we hav e x 0 = [ S ǫ k k (0)] 1 2 y k 0 ∀ k ≥ k 0 (25) and therefore it follows from (2 4) that k y k 0 k 2 ≤ M ∀ k ≥ k 0 . F r om this, w e can c onclude that the sequence { y k 0 } ∞ k = k 0 has a conv ergence subsequence { ˜ y k 0 } ∞ k = k 0 : ˜ y k 0 → ¯ y 0 . Now, using the fact tha t for any τ ≥ τ ∗ , then S ǫ τ (0) → S τ (0) as ǫ → 0, combined with (22), it follows fro m (25) that we can write x 0 = h W c (0 , ˜ T ) i 1 2 ¯ y 0 . That is , x 0 is in the r ange space of the controllability Gramian and hence, x 0 is a controllable state for the pair ( A, B 1 ). Conv ersely , supp ose x 0 is a controllable state for the pair ( A, B 1 ). L e t ǫ ∗ > 0 and τ ∗ > 0 b e any p ositive constants. Also let ˜ T > 0 b e any sufficiently small time horizon chosen as in Lemma 2. Then a s ab ov e, given a ny τ ≥ τ ∗ , S τ ( t ) the solution to (13 ) exists and is p os itive semidefinite on [0 , ˜ T ] a nd sa tis fie s (22). Also, it follows from (22) that for all τ > 0 , the solution to (13) exists and is p ositive semidefinite on [0 , ˜ T ]. F urthermo re a lso as ab ove, given any τ > 0, there exists a sufficiently small ǫ ∈ (0 , ǫ ∗ ) such that S ǫ τ ( t ) the solution to (12) exists and is p ositive definite o n [0 , ˜ T ]. Moreov er, we hav e S ǫ τ ( λ ) > S τ ( λ ) ≥ 0 and S ǫ τ ( λ ) → S τ ( λ ) as ǫ → 0 for all λ ∈ [0 , ˜ T ]. Hence, using (22), we can write S ǫ τ (0) = W c (0 , ˜ T ) + Φ ǫ (26) where Φ ǫ = S ǫ τ (0) − S τ (0) + 1 τ Q (0 , ˜ T ) > 0 a nd Q (0 , ˜ T ) ≥ 0 is defined as in (21). 7 Now using the fac t that x 0 is a controllable state for the pair ( A, B 1 ), it follows that we can write x 0 = h W c (0 , ˜ T ) i 1 2 y 0 for s ome vector y 0 where W c (0 , ˜ T ) is the controllabil- it y Gramian for the pair ( A, B 1 ). Th us, using (26) and Lemma 4, we co nclude that W ǫ τ ( x 0 , ˜ T ) = x ′ 0 [ S ǫ τ (0)] − 1 x 0 ≤ y ′ 0 y 0 . (27) Now for fixed τ > 0, it follows fro m the definition that W ǫ τ ( x 0 , ˜ T ) is monotonically increasing a s ǫ → 0 . Also , (27) holds for all sufficiently small ǫ > 0 . Thus, we must hav e W τ ( x 0 , ˜ T ) = sup ǫ> 0 W ǫ τ ( x 0 , ˜ T ) ≤ y ′ 0 y 0 (28) for all τ > 0. W e now consider the case of τ = 0. In this cas e, W ǫ 0 ( x 0 , 0 , ˜ T ) = inf [ ξ ( · ) ,u ( · )] ∈ L 2 [0 ,T ] k x ( T ) k 2 ǫ + Z T 0  k u k 2  dt. Now since x 0 is a controllable state for the pair ( A, B 1 ), it follows that there exists a control u ∗ ( · ) defined on [0 , ˜ T ] such that with ξ ( · ) ≡ 0 , then x ( T ) = 0. Hence, W ǫ 0 ( x 0 , 0 , ˜ T ) ≤ Z T 0  k u ∗ k 2  dt for all ǫ > 0 . Ther efore, we hav e W 0 ( x 0 , ˜ T ) = sup ǫ> 0 W ǫ 0 ( x 0 , ˜ T ) ≤ Z T 0  k u ∗ k 2  dt. W e hav e now shown that W τ ( x 0 , ˜ T ) < ∞ for all τ ≥ 0 and for a ll ˜ T > 0 sufficiently small. Thus, using Observ ation 1, we can co nclude that x 0 is a differ- ent ially po ssibly controllable s tate. This completes the pro of. ✷ R emark: The above theor e m implies that when H ( s ) ≡ 0 the p ossibly c o nt rolla ble set is a linear s pace equal to the controllable s ubspace of the pair ( A, B 1 ). F rom the ab ov e theor em and the fact that H ( s ) ≡ 0 , it follows that we can apply the standa rd Ka lman decomp osition to represent the uncertain sy stem a s shown in Fig. 4. + Controllable Uncontrollable y z ξ ∆ u Fig. 4. Control-Uncon trollable decomp osition for the uncer- tain system when H ( s ) ≡ 0. In this case, we only have uncer taint y in the uncontrol- lable subsystem or in the coupling b etw een the t w o sub- systems. W e now consider the case in which H ( s ) 6≡ 0. Theorem 6 Consider the u nc ert ain system (1), (2) and supp ose t hat H ( s ) 6≡ 0 . Then, a st ate x 0 is differ ent ial ly p ossibly c ontr ol lable if and only if x 0 is a c ontr ol lable state for the p air ( A, [ B 1 B 2 ]) . Pr o of. Supp ose x 0 is a differentially p ossibly controllable state for the uncertain system (1), (2). Hence, using Ob- serv ation 1 it follows that sup ǫ> 0 sup τ ≥ 0 W ǫ τ ( x 0 , T ) < ∞ (29) for T > 0 sufficiently small. Setting τ = 0 , it follo ws that there exists a constant M > 0 such that inf [ ξ ( · ) ,u ( · )] ∈ L 2 [0 ,T ] k x ( T ) k 2 ǫ + Z T 0 k u k 2 dt ≤ M ∀ ǫ > 0 where the inf is defined for the system (1) with initial condition x (0) = x 0 . F rom this it follows that inf [ ξ ( · ) ,u ( · )] ∈ L 2 [0 ,T ] k x ( T ) k 2 ≤ ǫM ∀ ǫ > 0 and hence, inf [ ξ ( · ) ,u ( · )] ∈ L 2 [0 ,T ] k x ( T ) k 2 = 0 . Therefore, the state x 0 m ust b e a con trollable state for the pair ( A, [ B 1 B 2 ]). W e now supp ose the state x 0 is a co n trolla ble sta te for the pair ( A, [ B 1 B 2 ]) and show that x 0 is a differentially 8 po ssibly controllable state for the uncertain sy stem (1), (2). In o rder to prov e that the state x 0 is p ossibly c o n- trollable, we must show that for all T > 0 s ufficiently small sup τ ≥ 0 W τ ( x 0 , T ) < ∞ . In or der to show this, w e let T > 0 b e given and establish the following claim: Claim. F o r the system (1 ), there ex ists a n input pair { u ∗ ( · ) , ξ ∗ ( · ) } defined on [0 , T ] such that x (0) = x 0 , x ( T ) = 0 and Z T 0  k ξ ∗ k 2 − k z ∗ k 2  dt ≤ 0 . T o establish this claim, we first supp os e that the sta n- dard Kalman decompos ition is applied to the pair ( A, B 1 ) to de c o mpo se it into controllable and uncon- trollable subsystems. That is, w e can assume without loss of g e ne r ality that the system (1) is such that the matrices A , B 1 , B 2 , C 1 and the vector x ar e of the for m A = " A 11 A 12 0 A 22 # ; B 1 = " B 11 0 # ; B 2 = " B 21 B 22 # ; C 1 = h C 11 C 12 i ; x = " x 1 x 2 # (30) where the pair ( A 11 , B 11 ) is controllable. Now consider an input pair { ¯ u ( · ) , ¯ ξ ( · ) } defined on [0 , T 3 ] such that x (0) = x 0 and x ( T 3 ) = 0. Such an input pair exists due to our assumption that x 0 is a controllable state for the pair ( A, [ B 1 B 2 ]). Then, we can write J 1 = Z T 3 0  k ¯ ξ k 2 − k ¯ z k 2  dt < ∞ . Now for t ∈ ( T 3 , 2 T 3 ], co nsider the input pair { ˆ u ( · ) , ˆ ξ ( · ) } defined so that ˆ ξ ( · ) ≡ 0 and so that ˆ u ( · ) is such that the corr esp onding uncertaint y output ˆ z ( · ) 6≡ 0. Such a n input ˆ u ( · ) exists since w e hav e assumed that H ( s ) 6≡ 0. Then, we let γ = Z 2 T 3 T 3 k ˆ z k 2 dt > 0 . Also, since x ( T 3 ) = 0 and ˆ ξ ( t ) = 0 for t ∈ ( T 3 , 2 T 3 ], it follows from (30 ) that x 2 ( t ) = 0 for t ∈ ( T 3 , 2 T 3 ]. Now for t ∈ ( 2 T 3 , T ], c o nsider the input pair { ˇ u ( · ) , ˇ ξ ( · ) } defined so that ˇ ξ ( · ) ≡ 0 and so that ˆ u ( · ) is such that x 1 ( T ) = 0 . Suc h a n input ˇ u ( · ) exists since w e hav e as- sumed tha t the pair ( A 11 , B 11 ) is co n trolla ble . Also, since x 2 ( 2 T 3 ) = 0 and ˇ ξ ( t ) = 0 fo r t ∈ ( 2 T 3 , T ], it follows from (30) that x 2 ( t ) = 0 for t ∈ ( 2 T 3 , T ]. W e let ˇ z ( t ) denote the corr esp onding uncer taint y output for t ∈ ( 2 T 3 , T ]. W e no w consider an input pair { u ∗ ( · ) , ξ ∗ ( · ) } defined as follows: u ∗ ( t ) =        ¯ u ( t ) for t ∈ [0 , T 3 ]; ˆ u ( t ) for t ∈ ( T 3 , 2 T 3 ]; ˇ u ( t ) for t ∈ ( 2 T 3 , T ]; ξ ∗ ( t ) = ( ¯ ξ ( t ) for t ∈ [0 , T 3 ]; 0 for t ∈ ( T 3 , T ] . It follows fro m this construction that the pair { u ∗ ( · ) , ξ ∗ ( · ) } gives x ( T ) = 0 a nd Z T 0  k ξ ∗ k 2 − k z ∗ k 2  dt = Z T 3 0  k ¯ ξ k 2 − k ¯ z k 2  dt − Z 2 T 3 T 3 k ˆ z k 2 dt − Z T 2 T 3 k ˇ z k 2 dt ≤ J 1 − γ . W e now let µ > 0 b e a scaling parameter and intro duce a mo dified input pair { u ∗ ( · ) , ξ ∗ ( · ) } defined a s follows: u ∗ ( t ) =        ¯ u ( t ) for t ∈ [0 , T 3 ]; µ ˆ u ( t ) for t ∈ ( T 3 , 2 T 3 ]; µ ˇ u ( t ) for t ∈ ( 2 T 3 , T ]; ξ ∗ ( t ) = ( ¯ ξ ( t ) for t ∈ [0 , T 3 ]; 0 for t ∈ ( T 3 , T ] . It is straightforward to verify that this input pair also leads to x ( T ) = 0 and Z T 0  k ξ ∗ k 2 − k z ∗ k 2  dt ≤ J 1 − µ 2 γ . Letting, µ = s J 1 γ it follows that Z T 0  k ξ ∗ k 2 − k z ∗ k 2  dt ≤ 0 9 and henc e , the conditions of the cla im are s atisfied. T his completes the pro of of the c la im. W e now use this cla im to complete the proo f. Indeed, for any τ ≥ 0 and ǫ > 0, we hav e W ǫ τ ( x 0 , T ) = inf [ ξ ( · ) ,u ( · )] ∈ L 2 [0 ,T ] k x ( T ) k 2 ǫ + Z T 0  k u k 2 + τ k ξ k 2 − τ k z k 2  dt ≤ Z T 0  k u ∗ k 2 + τ k ξ ∗ k 2 − τ k z ∗ k 2  dt (31) where the input pair { u ∗ ( · ) , ξ ∗ ( · ) } is cons tructed using the ab ov e cla im such tha t x (0) = x 0 and x ( T ) = 0 and Z T 0  k ξ ∗ k 2 − k z ∗ k 2  dt ≤ 0 . Also, z ∗ ( · ) is the cor resp onding uncertain ty output for the sys tem (1). Since ǫ > 0 was ar bitrary , it follo ws fro m (31) that W τ ( x 0 , T ) = sup ǫ> 0 W ǫ τ ( x 0 , T ) ≤ Z T 0 k u ∗ k 2 dt + τ Z T 0  k ξ ∗ k 2 − k z ∗ k 2  dt ≤ Z T 0 k u ∗ k 2 dt (32) for all τ ≥ 0. Thus, we can conclude that sup τ ≥ 0 W τ ( x 0 , T ) < ∞ . Since, T > 0 was arbitrar y , it follows from Observ ation 1 that x 0 is differentially p ossibly co n trollable . This com- pletes the pr o of of the theor em. ✷ R emark: The ab ov e theo rem implies that when H ( s ) 6≡ 0 the p oss ibly controllable set is a linea r space equa l to the controllable subspace of the pa ir ( A, [ B 1 B 2 ]). F r om the ab ov e theorem, it fo llows that we can apply the standa rd Kalma n decompo sition to r epresent the uncertain system a s shown in Fig . 5. In this case, we only have uncerta in ty in the controllable subsystem or in the coupling betw een the t wo subsys- tems. 7 Kalman Decomp ositions W e can now combine the results of Theorems 1 , 2, 5, and 6 to obtain a complete Ka lman decompositio n for the uncertain sys tem in the following cases: Uncontrollable Controllable ∆ ξ u + y + z Fig. 5. Con trol-Uncontrollable decomp osition for the un cer- tain system when H ( s ) 6≡ 0. Case 1 G ( s ) ≡ 0, H ( s ) ≡ 0 . In this case, w e ap- ply the standard Kalman dec o mpo sition to the triple ( C 2 , A, B 1 ) to obtain the situation as illustra ted in the blo ck diagra m shown in Fig. 6 . z Controllable Observable Uncontrollable Observable Controllable Unobservable Uncontrollable Unobservable u + y ∆ + ξ Fig. 6. K alman decomp osition for th e uncertain sy stem when G ( s ) ≡ 0, H ( s ) ≡ 0. This situatio n corr esp onds to uncerta in ty only in the uncontrollable-unobserv able block. Also there is uncertaint y in the coupling b etw een uncontrollable- observ able blo ck and the unco nt rolla ble-unobserv able blo ck. F urthermor e, there is uncer ta in ty in the coupling betw een the uncontrollable-unobser v able blo ck and the controllable-unobserv able block. Case 2 G ( s ) 6≡ 0, H ( s ) ≡ 0. In this case, we apply the standard Kalman decomp ositio n to the triple ( " C 1 C 2 # , A, B 1 ) to obtain the situation as illustrated in the blo ck diag ram shown in Fig. 7 . 10 Controllable Observable Uncontrollable Observable Controllable Unobservable Uncontrollable Unobservable u + ∆ z ξ y Fig. 7. K alman decomp osition for th e uncertain sy stem when G ( s ) 6≡ 0, H ( s ) ≡ 0. This s ituation corr esp onds to uncertaint y only in the uncontrollable-observ able blo ck. Also ther e is uncer- taint y in the coupling b etw een uncontrollable-observ able blo ck a nd the uncontrollable-unobserv able blo ck. F ur- thermore, there is uncer taint y in the coupling b etw een the uncont rolla ble obser v able block and the c o nt rolla ble- unobserv able blo ck. As well, there is uncertain ty in the coupling betw een the uncontrollable-o bs erv a ble blo ck and the co nt rolla ble-observ able block. Note that in or de r to guarantee tha t the condition H ( s ) ≡ 0 we needed to make a further re striction on the con trollable obser v able blo ck in the a bove dia g ram so that in fact it only has an output y . Case 3 G ( s ) ≡ 0, H ( s ) 6≡ 0 . In this case, we apply the standard Ka lma n decomp osition to the triple ( C 1 , A, [ B 1 B 2 ]) to obtain the situation as illustrated in the blo ck diag ram shown in Fig. 8. This s ituation corr esp onds to uncertaint y only in the controllable-unobserv able blo ck. Also ther e is uncer- taint y in the coupling b etw een controllable-o bs erv a ble blo ck and ea ch of the o ther blo cks. Case 4 G ( s ) 6≡ 0, H ( s ) 6≡ 0 . In this case, we apply the standard Ka lma n decomp osition to the triple ( " C 1 C 2 # , A, [ B 1 B 2 ]) to obtain the situation as illustrated in the blo ck diag ram shown in Fig. 9. This s ituation corr esp onds to uncertaint y only in the controllable-observ able blo ck. Also there is uncer- taint y in the coupling b etw een controllable-o bs erv a ble Controllable Observable Uncontrollable Observable Controllable Unobservable Uncontrollable Unobservable u ∆ z ξ y + + + + Fig. 8. K alman decomp osition for th e uncertain sy stem when G ( s ) ≡ 0, H ( s ) 6≡ 0. Controllable Observable Uncontrollable Observable Controllable Unobservable Uncontrollable Unobservable u ∆ z ξ y + + Fig. 9. K alman decomp osition for th e uncertain sy stem when G ( s ) 6≡ 0, H ( s ) 6≡ 0. blo ck and the uncontrollable-observ able blo ck. F ur- thermore, there is uncertaint y in the coupling b etw e en the controllable-o bserv a ble block and the controllable- unobserv able blo ck. As well, there is uncerta in ty in the coupling b etw een the uncontrollable-obse r v able blo ck and the controllable-unobser v able blo ck. R emark Note that each of the four cases considered above corres p onds to uncertaint y only in one of the four blo cks in the Kalma n decomp osition. It migh t b e conjectured that if s tructured uncerta in ty was allow ed then w e could distribute the uncertaint y blo cks around the fo ur blo cks in the Kalman decomp osition r ather than the current re- quirement that the s ingle unce r taint y blo ck cor resp onds to uncer taint y in one of the four blo cks in the Kalman decomp osition. 11 8 Illustrativ e Exampl e s 8.1 Example 1 In this example, we consider an uncer tain sys tem o f the form (1), (2) de fined by the following matr ic e s: A = " − 1 . 283 8 0 . 300 2 − 0 . 760 3 − 0 . 26 6 2 # ; B 1 = " 0 . 3911 0 . 4348 # ; B 2 = " 0 . 7251 0 . 8062 # ; C 1 = h 0 . 6534 − 0 . 0 9 08 i ; D 1 = 0; C 2 = h − 0 . 619 0 0 . 567 8 i ; D 2 = 0 . This system is a mo dificatio n of the system conside r ed in the e xample of [11] to consider the case of unstructured uncertaint y . W e wis h to deter mine if this uncerta in sys- tem co n tains a ny states which are not po ssibly control- lable in order to see if this uncer tain system mo del can be replaced by an equiv alent r educed dimension uncer - tain system mo del. W e first calculate the tra nsfer func- tion H ( s ) = C 1 ( sI − A ) − 1 B 1 + D 1 = 0 . 216 s +0 . 1296 s 2 +1 . 55 s +0 . 57 6≡ 0 . Hence, we apply Theor em 6 to this s ystem a nd con- sider the uncon trollable s tates of the pair ( A, [ B 1 B 2 ]); e.g., see [1]. Indeed, the eigenv alues a nd corr e spo nding left eigen vectors of the ma trix A ar e λ 1 = − 0 . 9500 , λ 2 = − 0 . 6 0 00, x 1 = " − 0 . 915 6 0 . 4020 # , and x 2 = " 0 . 7435 − 0 . 668 7 # . Also, we hav e B ′ 1 x 2 ≈ B ′ 2 x 2 ≈ 0. Hence (to the av ail- able numerical accuracy), x 2 is an uncontrollable s tate for the pair ( A, [ B 1 B 2 ]). Hence using Theo r em 6, w e ca n conclude that x 2 is not a p oss ibly co nt rolla ble state for this uncertain sys tem. W e show that x 2 is not a p ossibly controllable state using Theorem 4. Indeed, we let τ = 1 and so lve the Riccati differential equation (13) for differ en t v alue s of T ∈ [0 , 1]. A plot of the resulting eigen v alue s of S τ (0) v ersus T is shown in Fig. 1 0. F rom this plot, w e can see that the matrix S τ (0) is singula r for all T ∈ [0 , 1 ]. F urthermor e , we find that S τ (0) x 2 = 0 fo r all T ∈ [0 , 1 ]. Thus, using Theorem 4, it follows that with τ = 1 , W τ ( x 2 , T ) = ∞ for all T ∈ [0 , 1]. Hence, it follows fr om Definition 5 that the state x 2 is not (differen tially) p os sibly controllable. W e now apply the Kalman decomp osition to this un- certain system; e.g., see [1, 19, 20]. Indeed, if w e ap- ply the state space transforma tion ˜ x = T x with T = " − 0 . 743 5 0 . 668 7 0 . 6687 0 . 74 3 5 # to this uncertain sys tem, we obtain 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −1 0 1 2 3 4 5 6 T λ min [S τ (0)], λ max [S τ (0)] Fig. 10. λ min [ S τ (0)] and λ max [ S τ (0)] versus T with τ = 1. an uncertain s y stem of the for m (1), (2) de fined by: ˜ A = " − 0 . 600 0 0 . 000 0 1 . 0605 − 0 . 9 5 00 # ; ˜ B 1 = " 0 . 0000 0 . 5848 # ; ˜ B 2 = " 0 . 0000 1 . 0843 # ; ˜ C 1 = h − 0 . 546 5 0 . 369 4 i ; ˜ D 1 = 0; ˜ C 2 = h 0 . 8399 0 . 00 8 2 i ; ˜ D 2 = 0 . F r om this, the co nt rol input u and the uncertain ty input ξ do not a ffect the first state of this system. Thus, we can remov e this state witho ut changing the input-output b e- havior of the system. This leads to a reduced dimens io n uncertain system de s crib ed by the s tate equations ˙ x = − 0 . 9 5 00 x + 0 . 58 4 8 u + 1 . 08 43 ξ ; z = 0 . 369 4 x ; y = 0 . 0 082 x and the av erag e d IQC (2). 8.2 Example 2 This example considers a n uncertain system co rresp ond- ing to the electrical c ir cuit shown in Figure 11. It is straightforward to der ive the fo llowing state s pace mo del for this circ uit: " dV 1 dt dV 2 dt # =   − 1 C 1  1 R 3 + 1 R 1  − 1 R 3 C 1 − 1 R 3 C 2 − 1 C 2  1 R 3 + 1 R 2    " V 1 V 2 # + " 1 C 1 1 C 2 # u ; y = h 0 1 i " V 1 V 2 # . (33) 12 + − + − + − + − P S f r a g r e p la c e m e n t s u R 1 R 2 R 3 C 1 C 2 V 1 V 2 y Fig. 11. Electrical circuit corresponding to Examp le 2. In this example, we choos e the parameter v alues for the nominal to b e R 1 = 0 . 5Ω, R 2 = 1 . 0Ω, R 3 = 0 . 5Ω, C 1 = 2 . 0 F , and C 2 = 1 . 0 F . F or these par ameter v alues, the nominal system is not controllable. W e no w co nsider tw o cas es of uncertain pa rameters for this system. In the first ca se, we find that all non-zero states of the system ar e p ossibly c o nt rolla ble and no reduced dimension model can b e co nstructed using the Kalman deco mp os ition of Section 7. In the second case, we find that there exist non-zero states o f the system which a r e not p ossibly controllable. Then, w e use the Kalman dec ompo sition of Section 7 to construct a mo del of order one which do e s not change the input-output behavior of the system. Case 1. In this cas e, we supp ose that the conductance of the r esistor R 1 is uncer ta in a nd we wr ite 1 R 1 = 2 + ∆ where | ∆ | ≤ 1. This lea ds to an uncertain sys tem of the form (1) where A = " − 2 − 1 − 2 − 3 # ; B 1 = " 0 . 5 1 # ; B 2 = " − 0 . 5 0 # ; C 1 = h 1 0 i ; D 1 = 0; C 2 = h 0 1 i ; D 2 = 0 ; and ξ = ∆ z . Since | ∆ | ≤ 1 , it follows that the av eraged IQC (2) will b e satis fie d. F or this uncerta in sys tem, we calculate the tra nsfer functions G ( s ) and H ( s ) as G ( s ) = 1 s 2 + 5 s + 4 6≡ 0 ; H ( s ) = 0 . 5 s + 0 . 5 s 2 + 5 s + 4 6≡ 0 . F o r this uncertain system, the pair ( A, B 1 ) is not con- trollable. How ev er, the pair ( A, [ B 1 B 2 ]) is controllable. Thu s, it follows from Theore m 6 that the system has no states which are not differentially p os sibly controllable. Also, the pair ( " C 1 C 2 # , A ) is o bserv a ble and hence, us- ing Case 4 of the Kalman decomp os itio ns considered in Section 7, w e cannot c o nstruct an equiv alent reduce d di- mension uncertain system corr esp onding to this uncer- tain system. Note tha t the e xample considered in this case is such that the nominal s ystem is not c ontrollable, but the uncerta in system becomes co n trolla ble for non-zero v alues of the uncertain parameter ∆. If we change the par a meter C 1 to C 1 = 1 , we obtain an uncertain system for which the nominal system is controllable but for whic h the sy stem bec omes unco n trollable for one v alue of the uncertain parameter (∆ = − 1). Case 2. In this cas e, we supp ose that the conductance of the resis tor R 3 is uncer ta in a nd we write 1 R 3 = 2 + ∆ where | ∆ | ≤ 1. This lea ds to an uncertain sys tem of the form (1) where A = " − 2 − 1 − 2 − 3 # ; B 1 = " 0 . 5 1 # ; B 2 = " − 0 . 5 − 1 # ; C 1 = h 1 1 i ; D 1 = 0 ; C 2 = h 0 1 i ; D 2 = 0 ; and ξ = ∆ z . F or this uncertain system, we calc ula te the transfer functions G ( s ) and H ( s ) as G ( s ) = − s − 1 s 2 + 5 s + 4 6≡ 0 ; H ( s ) = 1 . 5 s + 1 . 5 s 2 + 5 s + 4 6≡ 0 . F o r this uncertain system, the pair ( A, [ B 1 B 2 ]) is not controllable. Thus, it follo ws from Theorem 6 that the system has non-zero states whic h are not differentially po ssibly controllable. Also, the pair ( " C 1 C 2 # , A ) is ob- serv able. W e now constr uct the Kalman decomp ositio n for this s ystem as in Ca se 4 of Sec tio n 7. Indeed, we apply a s ta te space transforma tion ˜ x = T x with T = " − 0 . 894 4 0 . 447 2 − 0 . 447 2 − 0 . 89 4 4 # to this uncertain sys tem to obtain an uncertain s y stem of the for m (1), (2) de fined by: ˜ A = " − 1 . 000 0 0 . 000 0 − 1 . 000 0 − 4 . 0 000 # ; ˜ B 1 = " 0 . 0000 − 1 . 118 0 # ; ˜ B 2 = " − 0 . 000 0 1 . 1180 # ; ˜ C 1 = h − 0 . 447 2 − 1 . 3 416 i ; ˜ D 1 = 0; ˜ C 2 = h 0 . 4472 − 0 . 8 9 44 i ; ˜ D 2 = 0 . F r om this, the co nt rol input u and the uncertain ty input ξ do not a ffect the first state of this system. Thus, we can remov e this state witho ut changing the input-output b e- havior of the system. 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