Unified Eigenvalue-Eigenspace Criteria for Functional Properties of Linear Systems and the Generalized Separation Principle

PREPRINT 1 Uniﬁed Eigen v alue–Eigenspace Criter ia f or Functional Proper ties of Linear Systems and the Generaliz ed Separ ation Pr inciple T yrone F er nando Abstract — Classical controllability and observ ability characterise reachability and reconstructibility of the full system state and admit equivalent geometric and eigen value-based P opov–Belevitc h–Hautus (PBH) tests. Motivated by large-scale and networked systems where only selected linear combinations of the state are of inter- est, this paper studies functional generalisations of these properties. A PBH-style framew ork for functional system properties is developed, pr oviding necessary and sufﬁcient spectral characterisations. The results apply uniforml y to diagonalizable and non-diagonalizable systems and re- cover the c lassical PBH tests as special cases. T wo new intrinsic notions are introduced: intrinsic func- tional controllability , and intrinsic functional stabilizabil- ity . These intrinsic properties are formulated directly in terms of in variant subspaces associated with the functional and pro vide veriﬁab le conditions for the existence of ad- missible augmentations required for functional contr oller design and observer-based functional controller design. The intrinsic framew ork enables the generalized separation principle at the functional level, establishing that functional controller s and functional observers can be designed in- dependently . Illustrative examples demonstrate the theory and highlight situations where functional control and esti- mation are possible despite lac k of full-state controllability or observability . Index T erms — Intrinsic Functional controllability , Intrin- sic Functional stabilizability , functional contr ollability , func- tional observability , functional detectability , functional sta- bilizability , tar g et output contr ollability , Jordan chains, gen- eralized eigenvector s. I . I N T R O D U C T I O N Classical controllability and observability are central con- cepts in linear systems theory , characterising state reachability and reconstructibility . They admit geometric interpretations via in variant subspaces and are commonly veriﬁed using controllability and observability matrices. Equiv alent Popo v– Belevitch–Hautus (PBH) tests provide eigenv alue-based rank conditions that expose modal structure and enable veriﬁcation without constructing large matrix powers. Motiv ated by modern applications in large-scale and net- worked systems, increasing attention has been dev oted to functional generalisations of classical system properties [1]- [9]. Rather than controlling or estimating the full state vector , T . Fernando is with the Department of Electrical, Electronic and Computer Engineering, University of Western Australia (UW A), 35 Stir ling Highw ay , Crawley , W A 6009, Australia. (email: ty- rone.f er nando@uwa.edu.au) the objectiv e is to control or reconstruct a prescribed linear functional z ( t ) = F x ( t ) , representing quantities of practical relev ance such as aggre- gate power , population totals, or reduced-order performance variables. This shift leads naturally to functional analogues of controllability , stabilizability , observability , and detectability . Existing work introduces these functional notions using subspace-based deﬁnitions that generalise Kalman’ s geometric framew ork [10], [11]. Subspace-based characterisations are av ailable for functional observability [12], [13], functional detectability [14], functional controllability and functional stabilizability [9], and target output controllability [15]. These formulations preserve classical dualities, provide conceptual clarity , and reduce to standard notions when F is the identity matrix. Extensions of subspace-based methods for functional observability hav e been explored for sample-based linear sys- tems [16], nonlinear systems [17]- [19], networked systems [2]- [6], and di verse applications including power systems [20] and ev ent-triggered estimation [21], [22]. Howe ver , subspace characterisations do not directly e x- pose the role of indi vidual eigenmodes and are often incon- venient for large-scale or structured systems, where local, eigen value-based tests are preferable. Corresponding PBH- style eigen value-eigenspace based tests for functional prop- erties remain limited in scope and have largely been con- ﬁned to restricted system classes [1], [4], [7] while other frequency-domain strategies for functional state estimation appear in [23]. This paper dev elops a uniﬁed PBH-style framew ork [24] for established functional system properties, including functional controllability , functional stabilizability , functional observability , functional detectability , and target output controllability . By w orking directly with eigen vectors and generalized eigen vectors arranged along Jordan chains, we obtain necessary and suf ﬁcient spectral characterisations that av oid the construction of full controllability or observability matrices. The framew ork recovers the classical PBH test as a special case and applies uniformly to both diagonalizable and non-diagonalizable systems. W ithin this framew ork, these functional properties are not separate phenomena, b ut different manifestations of the same underlying eigenspace geometry , from which a generalised separation principle follows natu- rally . Moreov er, for the design of functional controllers and 2 PREPRINT functional observers, augmentation-based formulations [9], [25] remain practically useful. Once suitable augmentation matrices are av ailable, they provide e xplicit realizations and constructiv e synthesis procedures for both functional controller design and functional observer design. Howe ver , unlike in the case of functional observer design, for a gi ven quadruple ( A, B , C , F ) it is not known a priori whether augmentation matrices exist that satisfy the required rank conditions for functional controller design. This paper addresses precisely this issue by introducing two intrinsic functional properties: intrinsic functional contr ollability and intrinsic functional sta- bilizability . For these intrinsic notions, both subspace-based and eigen value–eigenspace-based criteria are developed. In this sense, the intrinsic notions elev ate augmentation- based designs from realization-dependent constructions to veriﬁable structural properties of the quadruple ( A, B , C , F ) , and play a key role in establishing the Generalized Separation Principle by certifying the existence of augmentation matrices required to realise observer-based functional controllers. Contributions. This paper de velops a uniﬁed PBH-style framew ork yielding necessary and suf ﬁcient spectral charac- terisations of seven functional system properties: functional controllability (FC), intrinsic functional controllability (IFC), functional stabilizability (FS), intrinsic functional stabilizabil- ity (IFS), functional observability (FO), functional detectabil- ity (FD), and target output controllability (TOC). Among these, IFC and IFS are introduced as new notions. All condi- tions are expressed in terms of eigen vectors and generalized eigen vectors organised along Jordan chains, av oiding explicit construction of controllability or observ ability matrices. The intrinsic notions provide veriﬁable existence conditions for augmentation-based designs and lead to a generalized sepa- ration principle deriv ed from intrinsic functional properties. Organisation. Section II introduces notation and re views preliminary results used throughout the paper . Section III dev elops PBH-style conditions for functional controllability and introduces Intrinsic Functional Controllability . Section IV dev elops PBH-style conditions for functional stabilizability and introduces Intrinsic Functional Stabilizability . Section V addresses functional observability and functional detectability . Section VI presents PBH-style conditions for target output controllability . Section VII presents a generalized separation principle for functional observer -based control, derived en- tirely from intrinsic functional properties. Conclusions are drawn in Section VIII. I I . N O T A T I O N A N D P R E L I M I N A R I E S System Model. Consider the linear time-in variant system ˙ x ( t )= Ax ( t ) + B u ( t ) , y ( t )= C x ( t ) , z ( t )= F x ( t ) , where F is full row rank, z ( t ) ∈ R r the functional of interest, x ( t ) ∈ R n is the state, u ( t ) ∈ R m the input, and y ( t ) ∈ R p the measured output. Matrices A, B , C and F hav e appropriate dimensions. Notation, Stability , and Inv ariant Subspaces. For a matrix G , G T denotes its transpose and G − a generalized in verse satisfying GG − G = G ; rank( G ) , Im( G ) , and ker( G ) denote its rank, image, and k ernel, respecti vely . The identity matrix of dimension n is denoted I n . For i ∈ { 1 , . . . , n } , let e i ∈ R n denote the i -th standard basis vector . For vectors v 1 , . . . , v k ∈ R n , we write span { v 1 , . . . , v k } := ( k X i =1 α i v i      α i ∈ R ) . The direct sum of vector subspaces is denoted by ⊕ . A vector subspace V ⊆ R n is said to be A -in variant if Av ∈ V ∀ v ∈ V , equiv alently , A V ⊆ V . For a subspace S ⊆ R n , its orthogonal complement is denoted by S ⊥ := { x ∈ R n : x T s = 0 ∀ s ∈ S } . Accordingly , for a vector v ∈ R n and a subspace S , the notation v ⊥ S means v ∈ S ⊥ . Let M ∈ R p × n and v ∈ R n . W e say that M detects the vector v if M v  = 0 , and that M annihilates v if M v = 0 . Equiv alently , v is detected by M if v / ∈ ker( M ) . Controllability and Observability . For A ∈ R n × n , B ∈ R n × m , and C ∈ R p × n , deﬁne C ( A,B ) =  B AB · · · A n − 1 B  , O ( A,C ) =      C C A . . . C A n − 1      . The controllable and uncontrollable subspaces of ( A, B ) are Im( C ( A,B ) ) and k er( C T ( A,B ) ) , respectively , while Im( O T ( A,C ) ) and ker( O ( A,C ) ) are the observable and unobservable sub- spaces of ( A, C ) . Hence, R n = Im( C ( A,B ) ) ⊕ ker( C T ( A,B ) ) = Im( O T ( A,C ) ) ⊕ ker( O ( A,C ) ) . Generalized Eigen vectors and Jordan Chains. Let λ be an eigen value of A . A vector v ∈ C n is a generalized eigen vector of order k if ( λI − A ) k v = 0 , ( λI − A ) k − 1 v  = 0 . A sequence v 1 , . . . , v q satisfying ( λI − A ) v 1 = 0 , ( λI − A ) v i +1 = v i , i = 1 , . . . , q − 1 , is a Jordan chain of length q for ( A, λ ) . All matrices are real, but eigenv ectors and generalized eigen vectors are considered in C n when required for PBH and Jordan-chain arguments. For λ ∈ C , ℜ ( λ ) denotes its real part. Eigenv alues with ℜ ( λ ) ≥ 0 are termed unstable , and the associated eigen vectors and generalized eigen vectors are referred to as unstable modes . All in variant subspaces are understood ov er R . If A has a complex eigenv alue λ ∈ C \ R , the associated r eal generalized eigenspace is deﬁned as the real inv ariant subspace generated by the real and imaginary parts of the corresponding complex PREPRINT 3 Jordan chains. All references to eigenspaces and Jordan chains are interpreted in this real sense unless stated otherwise. Throughout the paper , we adopt the subspace-based deﬁni- tions of functional system properties introduced in [9], [12]. Our objectiv e is not to redeﬁne these notions, but to deriv e equiv alent PBH-style eigen value–eigenspace characterisations and intrinsic conditions. All PBH-style eigenv alue–eigenspace conditions derived are shown to be necessary and sufﬁcient with respect to the adopted subspace deﬁnitions. The follo wing basic lemmas from linear algebra will be used in the sequel. Lemma 1: Let X ∈ R n × m X and Y ∈ R n × m Y be matrices with the same number of rows. The following statements are equiv alent: i. Im( X ) ⊆ Im( Y ) ii. k er( Y T ) ⊆ ker( X T ) iii. rank  Y X  = rank( Y ) iv . rank  Y T X T  = rank( Y T ) v . ∀ v ∈ R n , v T Y = 0 ⇒ v T X = 0 . Pr oof: Using Im( X ) ⊥ = ker( X T ) , we ha ve Im( X ) ⊆ Im( Y ) ⇔ ker( Y T ) ⊆ ker( X T ) . The rank tests follow from Im  Y X  = Im( Y ) + Im( X ) and rank( M ) = rank( M T ) . Finally , k er( Y T ) ⊆ k er( X T ) is equiv alent to ( v T Y = 0 ⇒ v T X = 0 ) for all v . Lemma 2: Let P ∈ R r × n be full row rank and let Q ∈ R n × m be arbitrary . The following statements are equiv alent: i. rank( P Q ) = rank( P ) ; ii. k er( Q T P T ) = { 0 } ; Pr oof: Since rank( P ) = r , rank( P Q ) = rank( P ) if and only if P Q has full row rank, equiv alently ker( Q T P T ) = { 0 } . I I I . E I G E N VA L U E - B A S E D ( P B H - S T Y L E ) C O N D I T I O N S F O R F U N C T I O N A L C O N T R O L L A B I L I T Y A N D I N T R I N S I C F U N C T I O N A L C O N T R O L L A B I L I T Y This section develops eigenv alue–eigenspace (PBH-style) characterisations of functional contr ollability (FC) and intro- duces a stronger notion termed Intrinsic Functional Contr ol- lability (IFC). Throughout this section, vectors v ∈ R n (or C n when considering eigen value-based arguments) denote generic state-space directions, and Jordan chains are tak en with respect to A T . A. Functional Controllability W e ﬁrst recall the subspace-based characterization of func- tional controllability and establish sev eral structural properties of the uncontrollable subspace that underpin the subsequent PBH-style and Jordan-chain characterizations. Lemma 3 ( [9]): The functional z ( t ) = F x ( t ) is con- trollable or the triple ( A, B , F T ) is functional control- lable if and only if ker  C T ( A,B )  ⊆ k er( F ) , equiv alently , rank(  C ( A,B ) F T  ) = rank( C ( A,B ) ) . Lemma 4: The subspace ker( C T ( A,B ) ) is in variant under A T . Pr oof: Let v ∈ k er( C T ( A,B ) ) . Then v T A k B = 0 , k = 0 , 1 , . . . , n − 1 . For k = 0 , 1 , . . . , n − 2 , we have ( A T v ) T A k B = v T A k +1 B = 0 . For k = n − 1 , the Cayley–Hamilton theorem implies that there exist scalars α 0 , . . . , α n − 1 such that A n = n − 1 X i =0 α i A i . Hence ( A T v ) T A n − 1 B = v T A n B = n − 1 X i =0 α i v T A i B = 0 . Therefore A T v ∈ k er( C T ( A,B ) ) . Remark 1: Lemma 4 implies that ker( C T ( A,B ) ) is inv ariant under ( λI − A T ) for ev ery λ ∈ C . Consequently , if v 1 , . . . , v q is a Jordan chain of ( A T , λ ) and v k ∈ ker( C T ( A,B ) ) for some k , then v k − 1 = ( λI − A T ) v k ∈ ker( C T ( A,B ) ) , and iterating yields v 1 , . . . , v k − 1 ∈ ker( C T ( A,B ) ) . This preﬁx- closed structure underlies the Jordan-chain implications used below . Lemma 5: The uncontrollable subspace of the pair ( A, B ) , i.e. ker( C T ( A,B ) ) , admits a basis consisting of generalized eigen vectors of A T . Moreov er, for any eigenv alue λ ∈ σ ( A ) and any Jordan chain v 1 , . . . , v q of ( A T , λ ) , if v k ∈ ker( C T ( A,B ) ) for some k ≤ q , then v 1 , . . . , v k − 1 ∈ ker( C T ( A,B ) ) . Pr oof: By Lemma 4, ker( C T ( A,B ) ) is in variant under A T . Hence the restriction of A T to this subspace deﬁnes a linear operator on a ﬁnite-dimensional space, and therefore this subspace admits a basis consisting of generalized eigen vectors of A T . The second statement follows from in variance under A T together with the Jordan-chain relation ( λI − A T ) v i +1 = v i . Lemma 6: The functional z ( t ) = F x ( t ) is controllable or ( A, B , F T ) is functional controllable if and only if v T C ( A,B ) = 0 = ⇒ F v = 0 . Moreov er, it is sufﬁcient (and necessary) to verify this impli- cation on a basis of generalized eigen vectors of A T spanning k er( C T ( A,B ) ) . Pr oof: By Lemma 3, functional controllability is equiv- alent to ker( C T ( A,B ) ) ⊆ ker( F ) . Equiv alently , for all v ∈ R n , v ∈ k er( C T ( A,B ) ) = ⇒ v ∈ ker( F ) , i.e. v T C ( A,B ) = 0 = ⇒ F v = 0 , which prov es the ﬁrst statement. For the second statement, let { w 1 , . . . , w p } be any ba- sis of ker( C T ( A,B ) ) . Since F is linear , the implication v ∈ 4 PREPRINT k er( C T ( A,B ) ) ⇒ F v = 0 holds for all v in this subspace if and only if it holds for all basis vectors w i . By Lemma 5, k er( C T ( A,B ) ) admits such a basis consisting of generalized eigen vectors of A T , and the claim follows. Using the structural properties of the uncontrollable sub- space established above, we now state a PBH-style Jordan- chain characterization of functional controllability . Theor em 1: The functional z ( t ) = F x ( t ) is functional controllable if and only if, for ev ery eigen value λ of A and ev ery Jordan chain v 1 , . . . , v q of ( A T , λ ) , the implication B T v 1 = · · · = B T v k = 0 for some k ∈ { 1 , . . . , q } implies F v 1 = · · · = F v k = 0 . Pr oof: ( Only if ) Let v 1 , . . . , v q be a Jordan chain of ( A T , λ ) such that B T v 1 = · · · = B T v k = 0 for some k ∈ { 1 , . . . , q } . For each i ≤ k , the Jordan-chain relations together with B T v 1 = · · · = B T v k = 0 imply v T i A ℓ B = 0 , ℓ = 0 , . . . , n − 1 , and hence v i ∈ ker( C T ( A,B ) ) . Functional controllability there- fore yields F v i = 0 for all i = 1 , . . . , k . ( If ) Con versely , assume the stated Jordan-chain implication holds for ev ery eigen value λ and e very Jordan chain of ( A T , λ ) . Let v ∈ ker( C T ( A,B ) ) . By Lemma 5, ker( C T ( A,B ) ) is spanned by vectors belonging to preﬁxes of Jordan chains v 1 , . . . , v q of A T for which B T v 1 = · · · = B T v k = 0 for some k ∈ { 1 , . . . , q } . For each such chain, the hypothesis yields F v 1 = · · · = F v k = 0 . Therefore F v = 0 for e very v ∈ ker( C T ( A,B ) ) , and hence k er( C T ( A,B ) ) ⊆ ker( F ) , which is exactly functional controllability . T o translate the Jordan-chain implication of Theorem 1 into computable algebraic conditions, we introduce the notion of the ﬁrst B –visible index along each Jordan chain. Fix an eigen value λ ∈ σ ( A ) and a Jordan chain v 1 , . . . , v q of ( A T , λ ) . Deﬁne the ﬁrst B –visible index j := min  { i ∈ { 1 , . . . , q } : B T v i  = 0 } ∪ { q + 1 }  . (1) By construction, 1 ≤ j ≤ q + 1 . Moreov er, B T v 1 = · · · = B T v j − 1 = 0 , and either j = q + 1 (in which case B T v i = 0 for all i = 1 , . . . , q ) or else j ≤ q and B T v j  = 0 . Hence span { v 1 , . . . , v j − 1 } is the maximal initial se gment of the chain annihilated by B T . The following lemma reformulates the Jordan-chain condi- tion in terms of explicit nullspaces of PBH-type matrices. Lemma 7: Fix λ , a Jordan chain v 1 , . . . , v q of ( A T , λ ) , and let j be as in (1). For each k = 1 , . . . , j − 1 , deﬁne V k := span { v 1 , . . . , v k } , N k :=  ( λI − A T ) k B T  . Let X λ := k er  ( λI − A T ) n  , the generalized eigenspace of A T associated with λ . Then for all k = 1 , . . . , j − 1 , k er( N k ) ∩ span { v 1 , . . . , v q } = V k . Pr oof: Fix k ∈ { 1 , . . . , j − 1 } . ( ⊆ ) Show V k ⊆ ker( N k ) ∩ span { v 1 , . . . , v q } . For any i ≤ k , the Jordan relations for ( A T , λ ) imply ( λI − A T ) k v i = 0 . Since k < j , the deﬁnition of j gives B T v i = 0 for all i ≤ k . Hence v i ∈ ker( N k ) for i = 1 , . . . , k , and therefore V k ⊆ k er( N k ) . Since V k = span { v 1 , . . . , v k } ⊆ span { v 1 , . . . , v q } , we obtain V k ⊆ ker( N k ) ∩ span { v 1 , . . . , v q } . ( ⊇ ) Show k er( N k ) ∩ span { v 1 , . . . , v q } ⊆ V k . Let x ∈ k er( N k ) ∩ span { v 1 , . . . , v q } . Write x = P q i =1 α i v i . By repeated application of the Jordan-chain relation ( λI − A T ) v i +1 = v i , we obtain ( λI − A T ) k q X i =1 α i v i ! = q − k X i =1 α i + k v i . Because x ∈ k er( N k ) we have ( λI − A T ) k x = 0 , hence α k +1 = · · · = α q = 0 . Thus x = P k i =1 α i v i ∈ V k , pro ving the rev erse inclusion. Combining both inclusions gi ves ker( N k ) ∩ span { v 1 , . . . , v q } = V k . The subspace V k represents the uncontrollable directions associated with the Jordan chain v 1 , . . . , v q up to lev el k , rather than the full uncontrollable subspace ker( C T ( A,B ) ) . The following theorem pro vides a PBH-style characterisa- tion of functional controllability that resolves the contribution of each Jordan chain lev el by lev el. Theor em 2: Let j be deﬁned by (1) for each eigenv alue λ ∈ σ ( A ) and each Jordan chain of ( A T , λ ) . Then ( A, B , F T ) is functional controllable if and only if for each eigenv alue λ ∈ σ ( A ) and for every Jordan chain of ( A T , λ ) , rank    ( λI − A T ) k B T F    = rank  ( λI − A T ) k B T  , k = 1 , . . . , j − 1 . (2) When j = 1 (i.e., B T v 1  = 0 ), the condition is vacuous and no rank test is required on that chain. Pr oof: If j = 1 , the claim follows trivially . Assume henceforth that j > 1 . For each k < j , set N k =  ( λI − A T ) k B T  . By Lemma 1, the rank identity rank  N k F  = rank( N k ) is equiv alent to k er( N k ) ⊆ ker( F ) . Let v 1 , . . . , v q be any Jordan chain of ( A T , λ ) . By Lemma 7 , k er( N k ) ∩ span { v 1 , . . . , v q } = V k . Since (2) is required for every Jordan chain, the inclusion k er( N k ) ⊆ ker( F ) must hold for every chain; equiv alently , for ev ery Jordan chain v 1 , . . . , v q of ( A T , λ ) , k er( N k ) ∩ span { v 1 , . . . , v q } ⊆ k er( F ) . PREPRINT 5 By Lemma 7 this is equiv alent to V k ⊆ ker( F ) . Therefore, (2) holding for all k = 1 , . . . , j − 1 is equiv alent to F v 1 = · · · = F v j − 1 = 0 for ev ery Jordan chain of ( A T , λ ) . By Theorem 1, this is equiv alent to functional controllability . Because the uncontrollable subspaces along each Jordan chain grow monotonically , the sequential PBH-style test re- duces to the following single rank condition ev aluated at the terminal uncontrollable lev el. Theor em 3: Let j be deﬁned by (1) for each eigenv alue λ ∈ σ ( A ) and each Jordan chain of ( A T , λ ) . Then ( A, B , F T ) is functional controllable if and only if, for every such λ and Jordan chain, rank    ( λI − A T ) j − 1 B T F    = rank  ( λI − A T ) j − 1 B T  . (3) When j = 1 , the condition is vacuous and no rank test is required on that chain. Pr oof: If j = 1 , the claim follows trivially . Assume henceforth that j > 1 . For each k = 1 , . . . , j − 1 , deﬁne N k :=  ( λI − A T ) k B T  . By Lemma 1, for each k , rank  N k F  = rank( N k ) ⇐ ⇒ k er( N k ) ⊆ ker( F ) . Fix λ and a Jordan chain. By Lemma 1 with k = j − 1 , (3) is equiv alent to ker( N j − 1 ) ⊆ ker( F ) . W e verify this inclusion chain-by-chain. Let v 1 , . . . , v q be any Jordan chain of ( A T , λ ) . By Lemma 7, for k = 1 , . . . , j − 1 , k er( N k ) ∩ span { v 1 , . . . , v q } = V k := span { v 1 , . . . , v k } . Hence, for this chain, k er( N k ) ∩ span { v 1 , . . . , v q } ⊆ k er( F ) ⇐ ⇒ V k ⊆ ker( F ) , ⇐ ⇒ F v 1 = · · · = F v k = 0 . Because V 1 ⊂ · · · ⊂ V j − 1 , the conditions V k ⊆ k er( F ) for all k = 1 , . . . , j − 1 are equiv alent to V j − 1 ⊆ ker( F ) . Equiv alently , along this chain, F v 1 = · · · = F v j − 1 = 0 . By Theorem 1, this is equiv alent to functional controllability , and the rank condition at k = j − 1 is exactly (3). Remark 2: The sequential test in Theorem 2 exposes how functional controllability may fail at intermediate lev els of a Jordan chain, whereas Theorem 3 provides a minimal single- step condition that is typically preferable for implementation. When A is diagonalizable, ev ery Jordan chain has length q = 1 , and the test reduces to the classical PBH rank condition augmented by F . Cor ollary 1 (Diagonalizable case): Assume A is diagonal- izable. Then ( A, B , F T ) is functional controllable if and only if, for ev ery eigenv alue λ ∈ σ ( A ) , rank    λI − A T B T F    = rank  λI − A T B T  . (4) Finally , setting F = I n recov ers classical controllability . Cor ollary 2 (Reduction to classical controllability): Let F = I n . Then functional controllability of ( A, B , I n ) is equiv alent to classical controllability of ( A, B ) . Equiv alently , for each eigen value λ ∈ σ ( A ) and each Jordan chain v 1 , . . . , v q of ( A T , λ ) , the chain must be B T –visible, i.e., B T v 1  = 0 . This condition is equi valent to the classical PBH controllability test rank  λI − A B  = n, ∀ λ ∈ σ ( A ) . Thus, functional controllability reduces to classical controlla- bility when the functional coincides with the full state. Example 1: Functional Contr ollability T est Using Eigen values and Jor dan Chains Consider A =     4 0 − 2 7 1 2 0 2 − 1 0 4 − 5 − 1 1 1 − 1     , B =     2 1 − 1 − 1     , F =  1 2 1 2  . The eigen values of A are λ = 2 with algebraic multiplicity is 3, and λ = 3 with algebraic multiplicity is 1. Jor dan chain for λ = 2 of A T . A Jordan chain v 1 , v 2 , v 3 of ( A T , 2) satisﬁes (2 I − A T ) v 1 = 0 , (2 I − A T ) v 2 = v 1 , (2 I − A T ) v 3 = v 2 . One such chain is obtained by taking v 1 =  1 − 1 1 0  T , v 2 =  0 − 1 − 1 1  T , v 3 =  0 1 0 1  T , for which the above chain relations hold. Compute the B –visibility along the chain: B T v 1 = 0 , B T v 2 = − 1  = 0 . Hence the ﬁrst B –visible index is j = 2 , so by Theorem 3 it sufﬁces to v erify F v 1 = 0 . Indeed, F v 1 = 0 , so the λ = 2 Jordan structure satisﬁes the FC condition. Eigen vector for λ = 3 of A T . A left eigenv ector of ( A, 3) (i.e., an eigen vector of A T ) is v =  1 1 1 1  T , (3 I − A T ) v = 0 . Moreo ver , B T v = 1  = 0 , so j = 1 on this chain and no additional condition on F is required. Hence all Jordan chains of A T satisfy the PBH functional controllability condition, and ( A, B , F T ) is functional controllable by Theorem 3. 6 PREPRINT B. Intrinsic Functional Controllability Functional controllability (FC) is a static reachability prop- erty: it ensures that the functional z ( t ) = F x ( t ) can be steered to an arbitrary v alue, b ut does not, in general, constrain the internal dynamics through which z ( t ) e volves. In particular , ˙ z ( t ) may depend on uncontrollable state components that cannot be inﬂuenced or stabilised by feedback. This motiv ates a strengthened notion, termed Intrinsic Functional Control- lability (IFC), which excludes such hidden uncontrollable inﬂuences. Example 2: Consider the system ˙ x 1 ( t ) = x 2 ( t ) + u ( t ) , ˙ x 2 ( t ) = 0 , z ( t ) = x 1 ( t ) . The controllability matrix of ( A, B ) is C ( A,B ) =  B AB  =  1 0 0 0  , rank  C ( A,B )  = 1 , so Im( C ( A,B ) ) = span { e 1 } . Since Im( F T ) = span { e 1 } , the functional z ( t ) = x 1 ( t ) is controllable or the triple ( A, B , F T ) is functional controllable. Interpr etation. Although the functional z ( t ) = x 1 ( t ) is directly inﬂuenced by the input, its dynamics satisfy ˙ z ( t ) = x 2 ( t ) + u ( t ) , and therefore depend on the uncontrollable state x 2 ( t ) , which ev olves according to ˙ x 2 ( t ) = 0 . As a result, x 2 ( t ) ≡ x 2 (0) acts as an uncontrollable constant disturbance on the functional dynamics, preventing stabilization of z ( t ) → 0 for all initial conditions. Such hidden uncontrollable inﬂuences are precisely what Intrinsic Functional Contr ollability excludes. W e now formalise this requirement by strengthening func- tional controllability so that, beyond Im( F T ) , the entire A T – in variant subspace generated by the functional directions is required to be reachable. Deﬁnition 1: Let z ( t ) = F x ( t ) with F ∈ R r × n . The triple ( A, B , F T ) is said to be intrinsically functionally contr ollable if and only if there exist an integer d ∈ { r, . . . , n } and a matrix R 1 ∈ R ( d − r ) × n (empty when d = r ) such that ¯ F :=  F R 1  satisﬁes rank( ¯ F ) = d, and rank  ¯ F A ¯ F  =rank( ¯ F ) , (5a) rank  λ ¯ F − ¯ F A ¯ F B  =rank( ¯ F ) , ∀ λ ∈ C . (5b) Lemma 8 ( [26]): Let Z ∈ R m × d be a feedback gain matrix. The control law u ( t ) = − Z ¯ F x ( t ) driv es ¯ F x ( t ) → 0 as t → ∞ (and consequently F x ( t ) → 0 ) with arbitrary rate of con ver gence from any initial condition ¯ F x ( t 0 ) , by assigning d eigen values to a subsystem of order d , if and only if (5a)–(5b) hold. Remark 3: Deﬁnition 1 and Lemma 8 clarify the qualitative difference between FC and IFC. Functional controllability guarantees reachability of F x ( t ) , whereas IFC guarantees that the augmented functional ¯ F x ( t ) ev olves as a control- lable subsystem. Consequently , static feedback of the form u ( t ) = − Z ¯ F x ( t ) allo ws arbitrary eigen v alue assignment for that subsystem and hence arbitrary conv ergence rates for z ( t ) , independently of the remaining (uncontrollable) state components. The discussion above suggests that Intrinsic Functional con- trollability is gov erned not merely by the functional directions themselves, but by the smallest A T –in variant subspace gener- ated by those directions. W e therefore introduce the following subspace, which plays a central role in the characterisation of IFC. Now deﬁne V F := Im  C ( A T ,F T )  the smallest A T –in variant subspace containing Im( F T ) . Lemma 9 (Minimality of V F ): Let F ∈ R r × n and deﬁne V F := Im  C ( A T ,F T )  = Im  F T A T F T . . . ( A T ) n − 1 F T  . Then V F is the smallest A T –in variant subspace containing Im( F T ) . Pr oof: By construction, V F is spanned by ( A T ) k F T , k = 0 , . . . , n − 1 , and therefore contains Im( F T ) . By the Cayley– Hamilton theorem, V F is inv ariant under A T . If W is any A T – in variant subspace containing Im( F T ) , then ( A T ) k Im( F T ) ⊆ W for all k = 0 , . . . , n − 1 , and hence V F ⊆ W . Therefore, V F is the smallest A T –in variant subspace containing Im( F T ) . Lemma 10: Let ¯ F ∈ R d × n hav e full row rank and satisfy rank  ¯ F A ¯ F  = rank( ¯ F ) . Then the following statements are equiv alent: (i). Im( ¯ F T ) ⊆ Im  C ( A,B )  . (ii). k er  C T ( A,B )  ⊆ ker( ¯ F ) . (iii). The triple ( A, B , ¯ F T ) is functionally controllable. (iv). rank  λ ¯ F − ¯ F A ¯ F B  = rank( ¯ F ) , ∀ λ ∈ C . In particular , (i) holds if and only if (iv) holds. Pr oof: The equi valence (i) ⇔ (ii) follows from standard image–kernel duality (Lemma 1). The equi valence (ii) ⇔ (iii) is exactly the subspace charac- terization of functional controllability (Lemma 3). It remains to show (iii) ⇔ (iv). Since ¯ F has full ro w rank, let ¯ F − denote a right in verse satisfying ¯ F ¯ F − = I d . Pre- multiplying ˙ x ( t ) = Ax ( t ) + B u ( t ) by ¯ F yields ¯ F ˙ x ( t ) = ¯ F A ¯ F − ¯ F x ( t ) + ¯ F A ( I − ¯ F − ¯ F ) x ( t ) + ¯ F B u ( t ) . The functional dynamics z ( t ) = ¯ F x ( t ) are decoupled if and only if ¯ F A ( I − ¯ F − ¯ F ) = 0 , which is equiv alent to rank  ¯ F A ¯ F  = rank( ¯ F ) . PREPRINT 7 In this case, the induced subsystem ˙ z ( t ) = ¯ F A ¯ F − z ( t ) + ¯ F B u ( t ) is well deﬁned, in the sense that its dynamics de- pend only on z ( t ) and u ( t ) . Functional controllability of ( A, B , ¯ F T ) is therefore equi valent to controllability of the pair ( ¯ F A ¯ F − , ¯ F B ) . Under the decoupling condi- tion rank  ¯ F A ¯ F  = rank( ¯ F ) , functional controllability of ( A, B , ¯ F T ) reduces to classical controllability of the induced pair ( ¯ F A ¯ F − , ¯ F B ) . By the PBH test for this reduced pair , equiv alently (see Lemma 5 of [26]), rank  λ ¯ F − ¯ F A ¯ F B  = rank( ¯ F ) , ∀ λ ∈ C . W e now giv e a global subspace characterization of in- trinsic functional controllability , expressed in terms of the A T –in variant subspace generated by the functional directions. Theor em 4: Let z ( t ) = F x ( t ) with F ∈ R r × n . Then the triple ( A, B , F T ) is intrinsically functional contr ollable if and only if V F ⊆ Im  C ( A,B )  . (6) Equiv alently , rank  C ( A,B ) C ( A T ,F T )  = rank  C ( A,B )  . (7) Pr oof: The equiv alence between (6) and (7) follows directly from Lemma 1. (Only if) Assume that intrinsic functional controllability holds in the augmentation sense (Deﬁnition 1). Then there exist d and a matrix R such that ¯ F :=  F R  has full row rank d and satisﬁes (5a)–(5b). Condition (5a) is equiv alent to Im( ¯ F A ) ⊆ Im( ¯ F ) , and hence, by taking transposes, Im( A T ¯ F T ) ⊆ Im( ¯ F T ) , so Im( ¯ F T ) is A T –in variant. By minimality of V F (Lemma 9), V F ⊆ Im( ¯ F T ) . Moreov er, by Lemma 10, condition (5b) is equi valent to Im( ¯ F T ) ⊆ Im  C ( A,B )  . Combining the two inclusions yields V F ⊆ Im  C ( A,B )  , which prov es (6). (If) Assume (6) holds and let d := dim( V F ) . Choose a full–row–rank matrix ¯ F ∈ R d × n such that Im( ¯ F T ) = V F . Since Im( ¯ F T ) = V F is A T –in variant, Im( A T ¯ F T ) ⊆ Im( ¯ F T ) , which is equiv alent (by transposing) to Im( ¯ F A ) ⊆ Im( ¯ F ) , and hence to (5a). Furthermore, the assumed inclusion V F ⊆ Im( C ( A,B ) ) implies Im( ¯ F T ) ⊆ Im( C ( A,B ) ) , which, by Lemma 10, is equiv alent to the PBH condition (5b). Finally , since Im( F T ) ⊆ Im( ¯ F T ) , the rows of F are linear combinations of the rows of ¯ F . Hence, after a suitable row basis transformation, ¯ F can be written as ¯ F =  F R  . Therefore ( A, B , F T ) is intrinsically functionally controllable. Theor em 5: Let z ( t ) = F x ( t ) with F ∈ R r × n . The triple ( A, B , F T ) is intrinsically functional contr ollable if and only if ev ery functional z k ( t ) = F A k x ( t ) , k = 0 , 1 , . . . , n − 1 is controllable. Pr oof: By Theorem 4, ( A, B , F T ) is intrinsically func- tional controllable if and only if V F = Im  C ( A T ,F T )  ⊆ Im  C ( A,B )  . By deﬁnition, C ( A T ,F T ) =  F T A T F T · · · ( A T ) n − 1 F T  , and hence V F = span  ( F A k ) T : k = 0 , 1 , . . . , n − 1  . Since Im( C ( A,B ) ) is a subspace, the inclusion V F ⊆ Im( C ( A,B ) ) holds if and only if Im  ( F A k ) T  ⊆ Im  C ( A,B )  , k = 0 , 1 , . . . , n − 1 . By Lemma 1, this is equiv alent to k er  C T ( A,B )  ⊆ ker( F A k ) , k = 0 , 1 , . . . , n − 1 . Finally , by the subspace characterization of functional control- lability (see Lemma 3), the latter condition is equiv alent to the functional controllability of each triple ( A, B , ( F A k ) T ) , i.e., each functional z k ( t ) = F A k x ( t ) is controllable. W e now apply Theorem 4 to Example 2. Here V F = span { e 1 , e 2 } , while Im  C ( A,B )  = span { e 1 } , so V F ⊆ Im  C ( A,B )  . Hence the system is not intrinsically functional controllable, despite being functional controllable. The next result giv es an equiv alent PBH-style characterisa- tion of IFC. In contrast to FC, once the leading eigen vector of a Jordan chain is uncontrollable, IFC requires that the entire chain be invisible to the functional. Theor em 6: Let z ( t ) = F x ( t ) with F ∈ R r × n . Then ( A, B , F T ) is intrinsically functional contr ollable if and only if, for e very eigen value λ ∈ σ ( A ) and ev ery Jordan chain v 1 , . . . , v q of ( A T , λ ) , B T v 1 = 0 = ⇒ F v 1 = · · · = F v q = 0 . (8) In contrapositi ve form, if F v i  = 0 for some i along the chain, then necessarily B T v 1  = 0 . Pr oof: By Theorem 4, intrinsic functional controllability is equiv alent to V F ⊆ Im  C ( A,B )  , V F := Im  C ( A T ,F T )  . (9) 8 PREPRINT Thus it sufﬁces to prove that (9) holds if and only if (8) holds for ev ery eigenv alue and Jordan chain. (Only if) Assume (9). Fix λ ∈ σ ( A ) and a Jordan chain v 1 , . . . , v q of ( A T , λ ) . Suppose B T v 1 = 0 . Since ( A T − λI ) v 1 = 0 , we hav e ( A T ) k v 1 = λ k v 1 , k = 0 , 1 , . . . , n − 1 , and hence v T 1 A k B = λ k v T 1 B = 0 , k = 0 , 1 , . . . , n − 1 , so v 1 ∈ ker  C T ( A,B )  . Since ker  C T ( A,B )  is A T –in variant and ( A T − λI ) v i +1 = v i , it follows that v 1 , . . . , v q ∈ ker  C T ( A,B )  . Moreov er, (9) implies k er  C T ( A,B )  = Im  C ( A,B )  ⊥ ⊆ V ⊥ F , and therefore v i ∈ V ⊥ F for i = 1 , . . . , q . Since Im( F T ) ⊆ V F , we hav e V ⊥ F ⊆ k er( F ) , hence F v i = 0 , for i = 1 , . . . , q , which prov es (8). (If) Assume (8) holds for all eigen values and Jordan chains. W e prove (9), i.e. V F ⊆ Im( C ( A,B ) ) . Recall that V F is the smallest A T –in variant subspace con- taining Im( F T ) , and hence it is generated by Jordan chains of A T that contain at least one vector v i with F v i  = 0 . Fix an eigen value λ ∈ σ ( A ) and a Jordan chain v 1 , . . . , v q of ( A T , λ ) with span { v 1 , . . . , v q } ∩ V F  = { 0 } . Then F v i  = 0 for some i , and by the contrapositiv e of (8) we ha ve B T v 1  = 0 , i.e. the ﬁrst B –visible index of this chain is j = 1 . By the Jordan-chain characterization of the control- lable/uncontrollable decomposition (e.g. Lemma 14 special- ized to the pair ( A, B ) ), a chain with ﬁrst B –visible index j = 1 contributes no uncontrollable directions, and hence span { v 1 , . . . , v q } ⊆ Im  C ( A,B )  . Therefore ev ery Jordan chain that contributes to V F lies in Im( C ( A,B ) ) , and consequently V F ⊆ Im  C ( A,B )  , which is (9). Cor ollary 3: Intrinsic Functional controllability of the triple ( A, B , F T ) implies functional controllability of the triple ( A, B , F T ) . Pr oof: Since Im( F T ) ⊆ V F , the inclusion V F ⊆ Im( C ( A,B ) ) implies Im( F T ) ⊆ Im( C ( A,B ) ) , which is equiv- alent to functional controllability . Example 3: Intrinsic Functional Contr ollability (IFC) Consider the L TI system with A =     0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 1     , B =     0 1 0 0     , F =  1 0 0 0  . Thus the functional of interest is z = x 1 . Step 1: ( A, B ) is not controllable. The controllability matrix is C ( A,B ) =  B AB A 2 B A 3 B  . Since AB =  1 0 0 0  T = e 1 , A 2 B =  0 0 0 0  T , it follo ws that Im( C ( A,B ) ) = span { e 1 , e 2 } , rank( C ( A,B ) ) = 2 < 4 . Hence ( A, B ) is not controllable; in particular , the last two states form an uncontrollable subsystem. Step 2: Compute the A T –in variant closure generated by F T . Note A T e 1 = e 2 , ( A T ) 2 e 1 = A T e 2 = 0 . Therefore, the smallest A T –in variant subspace containing Im( F T ) is V F = Im  C ( A T ,F T )  = span { e 1 , e 2 } . Step 3: V erify IFC via the intrinsic criterion. Since V F = span { e 1 , e 2 } ⊆ Im( C ( A,B ) ) , the intrinsic Func- tional Controllability condition holds. Hence ( A, B , F T ) is intrinsically functional controllable. V eriﬁcation via row augmentation. For this example, the A –in variant row space generated by F is Ro w { F , F A } = span { e T 1 , e T 2 } . Choosing ¯ F =  F R  =  1 0 0 0 0 1 0 0  , R =  0 1 0 0  . This choice of ¯ F spans the A –in variant row space generated by F . W e can verify that conditions (5a) and (5b) are satisﬁed conﬁrming Intrinsic Functional Controllability in the sense of Deﬁnition 1. I V . E I G E N VA L U E - B A S E D ( P B H - S T Y L E ) C O N D I T I O N S F O R F U N C T I O N A L S T A B I L I Z A B I L I T Y A N D I N T R I N S I C F U N C T I O N A L S T A B I L I Z A B I L I T Y This section dev elops PBH-style eigen value and Jordan- chain characterisations of functional stabilizability (FS) and introduces the strengthened notion of intrinsic functional sta- bilizability (IFS). A. Functional Stabilizability Functional stabilizability is the asymptotic analogue of functional controllability: it requires that the functional z ( t ) = F x ( t ) be insensitiv e to uncontrollable modes associated with eigen values in the closed right–half plane. Geometrically , FS excludes any unstable uncontrollable direction detected by the functional. Restricting attention to eigen values with non-negati ve real part leads to the following Jordan-chain characterisation of functional stabilizability . Deﬁnition 2: The functional z ( t ) = F x ( t ) is stabilizable, or the triple ( A, B , F T ) is functional stabilizable , if and only if for e very λ ∈ C with ℜ ( λ ) ≥ 0 and every v ∈ C n satisfying v T ( λI − A ) k = 0 for some k ∈ { 1 , . . . , n } , v T C ( A,B ) = 0 , it follows that F v = 0 . Theor em 7: Let z ( t ) = F x ( t ) with F ∈ R r × n . Then the triple ( A, B , F T ) is functional stabilizable if and only if, for PREPRINT 9 ev ery eigen value λ ∈ σ ( A ) with ℜ ( λ ) ≥ 0 and ev ery Jordan chain v 1 , . . . , v q of ( A T , λ ) , the implication B T v 1 = · · · = B T v k = 0 for some k ∈ { 1 , . . . , q } implies F v 1 = · · · = F v k = 0 . Pr oof: The result follows from Theorem 1 by restricting the Jordan-chain argument to eigen values λ satisfying ℜ ( λ ) ≥ 0 , as required by Deﬁnition 2. The Jordan-chain condition for functional stabilizability admits the following single-step PBH-style rank formulation. Theor em 8: Let z ( t ) = F x ( t ) with F ∈ R r × n . Then ( A, B , F T ) is functional stabilizable if and only if, for ev ery eigen value λ ∈ σ ( A ) with ℜ ( λ ) ≥ 0 and ev ery Jordan chain v 1 , . . . , v q of ( A T , λ ) , letting j := min  { i ∈ { 1 , . . . , q } : B T v i  = 0 } ∪ { q + 1 }  , the rank condition rank    ( λI − A T ) j − 1 B T F    = rank  ( λI − A T ) j − 1 B T  (10) holds. When j = 1 , the condition is vacuous and no rank test is required on that chain. Pr oof: The proof is identical to that of Theorem 3, with all arguments restricted to eigen values λ satisfying ℜ ( λ ) ≥ 0 , as required by Deﬁnition 2. When j = 1 , the condition is vacuous. Cor ollary 4: Assume A is diagonalizable. Then ( A, B , F T ) is functional stabilizable if and only if, for every eigen value λ ∈ σ ( A ) with ℜ ( λ ) ≥ 0 , rank    λI − A T B T F    = rank  λI − A T B T  . (11) Pr oof: If A is diagonalizable then every Jordan chain has length 1 . Thus Theorem 7 reduces to the eigen vector-le vel implication ( λI − A T ) v = 0 , B T v = 0 = ⇒ F v = 0 , ∀ λ ∈ σ ( A ) , ℜ ( λ ) ≥ 0 , which is equiv alent to (11). Cor ollary 5 (Reduction to classical stabilizability): Let F = I n . Then functional stabilizability of ( A, B , I n ) is equiv alent to classical stabilizability of ( A, B ) . Pr oof: If F = I n , the FS chain implication in Theorem 7 requires that no unstable Jordan chain of ( A T , λ ) can have B T v 1 = · · · = B T v k = 0 for any k ≥ 1 , which is exactly the classical stabilizability condition (all uncontrollable modes are stable). Since functional stabilizability differs from functional con- trollability only by restriction to eigen values with nonnegati ve real part, no separate example is included. B. Intrinsic Functional Stabilizability Intrinsic functional stabilizability (IFS) strengthens func- tional stabilizability by imposing the stabilizability require- ment on the A T –in variant subspace gener ated by the func- tional directions . Equiv alently , IFS excludes unstable un- controllable generalized eigenv ectors that contribute to the functional, not merely unstable uncontrollable eigen vectors. As a result, IFS may fail ev en when all unstable uncontrollable eigen vectors are invisible to F , if an unstable uncontrollable generalized eigen vector appears later along a Jordan chain detected by the functional. This mirrors the distinction between functional controllability and functional stabilizability , leading to the following deﬁnition. Deﬁnition 3: The triple ( A, B , F T ) is said to be intrinsi- cally functional stabilisable if it satisﬁes Deﬁnition 1 with the rank condition (5b) required only for eigen v alues λ with ℜ ( λ ) ≥ 0 . W e now state an intrinsic, subspace-lev el characterisation of functional stabilizability , which isolates the unstable functional dynamics via the in variant closure V F . Theor em 9 (Intrinsic functional stabilizability via V F ): Let z ( t ) = F x ( t ) with F ∈ R r × n and recall V F := Im  C ( A T ,F T )  the smallest A T –in variant subspace containing Im( F T ) . Let X + denote the generalized eigenspace of A T associated with eigen values λ satisfying ℜ ( λ ) ≥ 0 . Then ( A, B , F T ) is intrinsically functional stabilizable if and only if V F ∩ X + ⊆ Im  C ( A,B )  . (12) Pr oof: By Deﬁnition 3 and Lemma 10, intrinsic func- tional stabilizability is equiv alent to requiring that for ev ery λ with ℜ ( λ ) ≥ 0 and ev ery Jordan chain of ( A T , λ ) , any uncon- trollable preﬁx of the chain is annihilated by F . Equiv alently , among the generalized eigenv ectors associated with ℜ ( λ ) ≥ 0 , the A T –in variant subspace generated by Im( F T ) is contained in Im( C ( A,B ) ) . Since V F is the smallest A T –in variant subspace containing Im( F T ) , this is exactly V F ∩ X + ⊆ Im( C ( A,B ) ) . Cor ollary 6: If ( A, B , F T ) is intrinsically functional stabi- lizable , then for ev ery k ∈ { 0 , 1 , . . . , n − 1 } the functional z k ( t ) = F A k x ( t ) is stabilizable . Pr oof: Fix k ∈ { 0 , 1 , . . . , n − 1 } and deﬁne F k := F A k . By deﬁnition, V F k = Im  C ( A T ,F T k )  = Im  C ( A T , ( A T ) k F T )  ⊆ V F . Consequently , V F k ∩ X + ⊆ V F ∩ X + . Since ( A, B , F T ) is intrinsically functionally stabilizable, The- orem 9 yields V F ∩ X + ⊆ Im  C ( A,B )  , and hence V F k ∩ X + ⊆ Im  C ( A,B )  . 10 PREPRINT Applying Theorem 9 to ( A, B , F T k ) shows that z k ( t ) = F k x ( t ) = F A k x ( t ) is stabilizable. W e now giv e an equiv alent Jordan–chain characterisation of IFS. Theor em 10: Let z ( t ) = F x ( t ) with F ∈ R r × n . The triple ( A, B , F T ) is intrinsically functional stabilizable if and only if, for e very eigen value λ ∈ σ ( A ) with ℜ ( λ ) ≥ 0 and e very Jordan chain v 1 , . . . , v q of ( A T , λ ) , B T v 1 = 0 = ⇒ F v 1 = · · · = F v q = 0 . (13) Pr oof: By Theorem 9, intrinsic functional stabilizability is equiv alent to V F ∩ X + ⊆ Im  C ( A,B )  , V F := Im  C ( A T ,F T )  , (14) where X + is the generalized eigenspace of A T associated with { λ : ℜ ( λ ) ≥ 0 } . The proof is identical to that of Theorem 6, with the sole difference that we restrict attention to Jordan chains of ( A T , λ ) with ℜ ( λ ) ≥ 0 and replace (9) by (14). Indeed, suppose (14) holds and let v 1 , . . . , v q be a Jordan chain of ( A T , λ ) with ℜ ( λ ) ≥ 0 such that B T v 1 = 0 . Then v 1 ∈ ker( C T ( A,B ) ) , and since k er( C T ( A,B ) ) is A T –in variant we hav e v 1 , . . . , v q ∈ k er( C T ( A,B ) ) ⊆ X + . If F v i  = 0 for some i , then this Jordan chain contributes to V F = Im( C ( A T ,F T ) ) ; equiv alently , span { v 1 , . . . , v q } ∩ V F  = { 0 } , by the Jordan-chain characterization of V F for the pair ( A T , F T ) . Since ℜ ( λ ) ≥ 0 , this implies span { v 1 , . . . , v q } ∩ ( V F ∩ X + )  = { 0 } , contradicting (14). Therefore F v 1 = · · · = F v q = 0 . Con versely , assume (13) holds for every unstable Jordan chain. Arguing as in the con verse direction of Theorem 6, ev ery unstable Jordan chain that contributes to V F must satisfy B T v 1  = 0 , and thus, by the PBH-style test, lies in Im( C ( A,B ) ) . Hence V F ∩ X + ⊆ Im( C ( A,B ) ) , i.e. (14), and Theorem 9 giv es intrinsic functional stabilizability . The follo wing corollaries parallel those for Intrinsic Func- tional controllability . Cor ollary 7: Intrinsic Functional stabilizability implies functional stabilizability . Pr oof: If all unstable generalized eigen vectors con- tributing to the functional are stabilizable, then in particular no unstable uncontrollable eigen vector detected by F exists, which is precisely functional stabilizability . Cor ollary 8 (Diagonalizable case): If A is diagonalizable, then Intrinsic Functional stabilizability coincides with func- tional stabilizability and is characterised by ( λI − A T ) v = 0 , ℜ ( λ ) ≥ 0 , B T v = 0 = ⇒ F v = 0 . Example 4: Intrinsic Functional Stabilizability (IFS) Consider A =     5 − 50 59 64 2 − 29 32 36 − 4 54 − 66 − 72 5 − 67 79 87     , B =     − 4 − 2 4 − 5     , F =  0 1 1 0  . The spectrum is σ ( A ) = { 1 , 1 , − 2 , − 3 } , so the only unstable eigen value is λ = 1 . Unstable Jordan chain of ( A T , 1) . Let N := I − A T . A Jordan chain v 1 , v 2 of ( A T , 1) is v 1 =  − 2 − 2 2 4  T , v 2 =  5 − 3 3 0  T , satisfying N v 1 = 0 and N v 2 = v 1 . Moreover , B T v 1 = 0 , B T v 2 = − 2  = 0 . Thus the unstable Jordan chain contains an uncontrollable direction. V eriﬁcation via the IFS J ordan-chain condition. Direct computation giv es F v 1 = 0 , F v 2 = 0 . Hence the functional annihilates the entire unstable Jordan chain. Since this is the only Jordan chain associated with an eigen value λ satisfying ℜ ( λ ) ≥ 0 , the IFS Jordan-chain condition (13) is satisﬁed. Therefore the triple ( A, B , F T ) is intrinsically functional stabilizable . Interpr etation. Although an unstable uncontrollable direction exists, it is not detected by the functional. Consequently , it does not contribute to the unstable functional subspace, which is precisely the situation permitted by IFS. V . E I G E N VA L U E - B A S E D ( P B H - S T Y L E ) C O N D I T I O N S F O R F U N C T I O N A L O B S E RVA B I L I T Y A N D F U N C T I O N A L D E T E C T A B I L I T Y W e now turn to the dual estimation problem and dev elop PBH-style characterisations of functional observability and functional detectability . In contrast to the controllability re- sults, which are naturally expressed using left eigenv ectors of A , the observability results are formulated in terms of right eigen vectors and Jordan chains of A , reﬂecting the state-space nature of observability . A. Functional Obser vability Consider the pair ( A, C ) and a linear functional z ( t ) = F x ( t ) with F ∈ R r × n . Functional observability concerns the ability to reconstruct z ( t ) uniquely from the measured output y ( t ) = C x ( t ) and the known input, without reconstructing the full state x ( t ) . W e ﬁrst recall a standard rank-based characterization of functional observability . Lemma 11 (see [12]): The triple ( A, C , F ) is functional observable if and only if rank  O ( A,C ) O ( A,F )  = rank  O ( A,C )  . (15) W e now express functional observability in terms of eigen- values and Jordan chains of A . Theor em 11: Let z ( t ) = F x ( t ) . The functional z ( t ) is functional observable if and only if, for ev ery eigen value λ ∈ σ ( A ) and ev ery Jordan chain v 1 , . . . , v q of ( A, λ ) , the implication C v 1 = · · · = C v k = 0 for some k ∈ { 1 , . . . , q } implies F v 1 = · · · = F v k = 0 . PREPRINT 11 Pr oof: By Lemma 1 and Lemma 11, functional observ- ability is equiv alent to the kernel inclusion k er  O ( A,C )  ⊆ ker  O ( A,F )  . (16) Let λ ∈ σ ( A ) and let v 1 , . . . , v q be a Jordan chain of ( A, λ ) . If C v 1 = · · · = C v k = 0 for some k , then the Jordan relations imply v i ∈ ker  O ( A,C )  , i = 1 , . . . , k . By (16), this yields F v 1 = · · · = F v k = 0 , which prov es the stated Jordan–chain implication. Con versely , assume the Jordan–chain implication holds for ev ery eigenv alue and Jordan chain. Any vector in k er( O ( A,C ) ) can be written as a linear combination of generalized eigen- vectors belonging to preﬁxes of Jordan chains annihilated by C . The assumed implication therefore giv es k er( O ( A,C ) ) ⊆ k er( O ( A,F ) ) , proving functional observability . Functional observability admits an equiv alent characteriza- tion in terms of Jordan chains of A . The following lemma is the observability dual of Lemma 7. Deﬁne the ﬁrst C –visible index j := min  { i ∈ { 1 , . . . , q } : C v i  = 0 } ∪ { q + 1 }  . (17) Thus span { v 1 , . . . , v j − 1 } is the C –annihilated initial segment of the chain. Thus C v 1 = · · · = C v j − 1 = 0 , and either j = q + 1 or j ≤ q and C v j  = 0 . Functional observability admits an equiv alent characteriza- tion in terms of Jordan chains of A . The following lemma is the observability dual of Lemma 7. Lemma 12: Fix λ , a Jordan chain v 1 , . . . , v q of ( A, λ ) , and let j be as in (17). For each k = 1 , . . . , j − 1 , deﬁne V k := span { v 1 , . . . , v k } , N k :=  ( λI − A ) k C  . Let X λ := k er  ( λI − A ) n  , the generalized eigenspace of A associated with λ . Then for all k = 1 , . . . , j − 1 , k er( N k ) ∩ span { v 1 , . . . , v q } = V k . Pr oof: The proof is identical to that of Lemma 7, with A T replaced by A and B T replaced by C . Using the monotonicity of the C –annihilated subspaces along a Jordan chain, the PBH rank conditions reduce to a single check per chain. Theor em 12 (PBH-style single-step test): Let j be deﬁned by (17) for each eigen value λ and each Jordan chain. Then ( A, C, F ) is functional observable if and only if for ev ery such λ and chain, rank    ( λI − A ) j − 1 C F    = rank  ( λI − A ) j − 1 C  . (18) When j = 1 , the condition is vacuous and no rank test is required on that chain. Pr oof: The proof is identical to that of Theorem 3 with ( A T , B T ) replaced by ( A, C ) , and Lemma 7 replaced by Lemma 12. This yields (18). Example 5: Functional Observability T est (single-step PBH) Consider A =     5 0 1 4 1 2 0 2 − 2 1 2 − 3 − 1 0 0 0     , C =  1 2 2 0  , F =  1 2 1 2  . The eigenv alues of A are λ = 2 (algebraic multiplicity 3 ) and λ = 3 (multiplicity 1 ). A Jordan form computation yields one Jordan chain of length 3 for λ = 2 and one chain of length 1 for λ = 3 . One con venient Jordan basis is given by the following chains. (i) Eigen value λ = 2 (chain length q = 3 ). Choose v 1 , v 2 , v 3 such that ( A − 2 I ) v 1 = 0 , ( A − 2 I ) v 2 = v 1 , ( A − 2 I ) v 3 = v 2 , with v 1 =  − 2 − 1 2 1  T , v 2 =  − 3 − 1 3 1  T , v 3 =  − 1 1 0 0  T . Compute the C -visibility along the chain: C v 1 = 0 , C v 2 = 1  = 0 . Hence the ﬁrst C –visible index is j = 2 . By the single-step PBH test (Theorem 12), we need only check the rank condition at k = j − 1 = 1 , i.e. at λ = 2 : A direct rank computation giv es rank  2 I − A C  = 3 , rank   2 I − A C F   = 3 , so the required equality holds for the λ = 2 chain. (ii) Eigen value λ = 3 (chain length q = 1 ). Let w 1 satisfy ( A − 3 I ) w 1 = 0 , for example w 1 =  − 3 − 1 2 1  T . Then C w 1 = − 1  = 0 , so j = 1 and, by Theorem 12, no chec k is requir ed on this chain. Since the single-step PBH condition holds for the only chain requiring a check (the λ = 2 chain), the triple ( A, C, F ) is functional observable . B. Functional Detectability Functional detectability is the asymptotic counterpart of functional observability: it requires that every C –unobservable mode associated with an eigen v alue λ with ℜ ( λ ) ≥ 0 be in visi- ble to the functional. As in the observability case, this property admits a Jordan–chain characterisation. The following Jordan- chain characterization is therefore an immediate restriction of Theorem 11 to eigen values with nonnegati ve real parts. Theor em 13: The functional z ( t ) = F x ( t ) is functional detectable if and only if, for ev ery eigenv alue λ ∈ σ ( A ) with 12 PREPRINT ℜ ( λ ) ≥ 0 and ev ery Jordan chain v 1 , . . . , v q of ( A, λ ) , the implication C v 1 = · · · = C v k = 0 for some k ∈ { 1 , . . . , q } implies F v 1 = · · · = F v k = 0 . Pr oof: The proof follo ws verbatim from that of Theo- rem 11, with all arguments restricted to Jordan chains associ- ated with eigen values satisfying ℜ ( λ ) ≥ 0 . As in the functional observ ability case, this condition admits a PBH-style single-step rank characterization along Jordan chains. Theor em 14: Let λ ∈ σ ( A ) satisfy ℜ ( λ ) ≥ 0 , and let v 1 , . . . , v q be a Jordan chain of ( A, λ ) . Deﬁne j := min  { i ∈ { 1 , . . . , q } : C v i  = 0 } ∪ { q + 1 }  . Then the functional z ( t ) = Lx ( t ) is functional detectable if and only if rank    ( λI − A ) j − 1 C F    = rank  ( λI − A ) j − 1 C  (19) for every such eigenv alue and Jordan chain. When j = 1 , the condition is v acuous and no rank test is required on that chain. Pr oof: The result follows immediately from Theorem 12 by restricting attention to eigen values with ℜ ( λ ) ≥ 0 . Remark 4: The PBH-style test for functional detectability is obtained directly from that for functional observability by restricting attention to eigen values with ℜ ( λ ) ≥ 0 . If F = I n , functional detectability reduces exactly to the classical detectability condition for the pair ( A, C ) . V I . E I G E N VA L U E - B A S E D ( P B H - S T Y L E ) C O N D I T I O N S F O R T A R G E T O U T P U T C O N T R O L L A B I L I T Y Classical controllability concerns steering the full state of a system from any initial condition to any desired ﬁnal state in ﬁnite time. In contrast, tar get output contr ollability (TOC) focuses on steering a prescribed linear functional of the state, z ( t ) = F x ( t ) , without requiring controllability of the entire state vector . Even when the pair ( A, B ) is not controllable, it may still be possible to assign the value of z ( t ) at a prescribed time. T arget output controllability was ﬁrst formalised by Bertram and Sarachik [15], who established an algebraic rank condition that is strictly weaker than full-state controllability . In this section, we derive PBH-style conditions that are equiv alent to this classical rank criterion. A. Algebraic Character isation W e begin by recalling the classical algebraic character- isation of target output controllability due to Bertram and Sarachik [15]. Lemma 13 ( [15]): The functional z ( t ) = F x ( t ) is tar get output contr ollable , equiv alently the triple ( A, B , F ) is target output controllable, if and only if rank  F C ( A,B )  = rank( F ) , (20) where C ( A,B ) denotes the controllability matrix of ( A, B ) . B. Structure of the Uncontrollab le Subspace W e now recall a structural PBH fact that underpins all eigen value-based tests. Lemma 14 (PBH pr eparation): A vector v ∈ C n belongs to the uncontrollable subspace k er( C T ( A,B ) ) if and only if there exist an eigenv alue λ ∈ σ ( A ) and an integer k ≥ 1 such that ( λI − A T ) k v = 0 , B T v = 0 . Equiv alently , ker( C T ( A,B ) ) is spanned by initial segments of Jordan chains of A T , truncated at the ﬁrst B T –visible vector . Pr oof: This is a standard consequence of the PBH controllability theorem and the Jordan decomposition of A T . Along any Jordan chain v 1 , . . . , v q of ( A T , λ ) , the vectors v 1 , . . . , v j − 1 with B T v i = 0 span precisely the uncontrollable directions on that chain, where j := min { i : B T v i  = 0 } . C . Geometric Inter pretation The follo wing theorem provides an eigen v alue–based (PBH- style) characterisation of target output controllability . Theor em 15: The triple ( A, B , F ) is targ et output control- lable if and only if, for every eigenv alue λ ∈ σ ( A ) and ev ery Jordan chain v 1 , . . . , v q of ( A T , λ ) , letting j := min  { k ∈ { 1 , . . . , q } : B T v k  = 0 } ∪ { q + 1 }  , one has k er  V T j − 1 F T  = { 0 } , V j − 1 :=  v 1 · · · v j − 1  . (21) If j = 1 , no condition is required on that chain. Pr oof: By Lemma 13, ( A, B , F ) is target output control- lable if and only if rank  F C ( A,B )  = rank( F ) . By Lemma 2 with P = F and Q = C ( A,B ) , this holds if and only if k er  C T ( A,B ) F T  = { 0 } . By Lemma 14, the uncontrollable subspace ker( C T ( A,B ) ) is spanned by the Jordan-chain segments span { v 1 , . . . , v j − 1 } , taken over all eigen values λ ∈ σ ( A ) and all Jordan chains of ( A T , λ ) , where j is the ﬁrst B –visible index. Therefore k er( C T ( A,B ) F T ) = { 0 } holds if and only if, for each such chain, there is no nonzero y such that F T y lies in the segment span { v 1 , . . . , v j − 1 } . Equiv alently , for each eigenv alue λ and each Jordan chain v 1 , . . . , v q , letting V j − 1 := [ v 1 · · · v j − 1 ] , one must have k er  V T j − 1 F T  = { 0 } . If j = 1 , the chain contributes no uncontrollable directions and no condition is required. The following theorem reformulates the Jordan-chain condi- tion as a single-step PBH-style rank test ev aluated along each eigen value. Theor em 16: The triple ( A, B , F ) is targ et output control- lable if and only if, for every eigenv alue λ ∈ σ ( A ) and ev ery Jordan chain v 1 , . . . , v q of ( A T , λ ) , letting j := min  { k ∈ { 1 , . . . , q } : B T v k  = 0 } ∪ { q + 1 }  , the matrix V j − 1 := [ v 1 · · · v j − 1 ] satisﬁes rank   F T V j − 1   = r + ( j − 1) . (22) PREPRINT 13 If j = 1 , no condition is required on that chain. Pr oof: By Lemma 13, ( A, B , F ) is target output control- lable if and only if rank  F C ( A,B )  = rank( F ) . By Lemma 2 with P = F and Q = C ( A,B ) , this is equiv alent to k er  C T ( A,B ) F T  = { 0 } . Fix an eigen value λ ∈ σ ( A ) and a Jordan chain v 1 , . . . , v q of ( A T , λ ) , and let j be the ﬁrst B –visible index. By Lemma 14, the uncontrollable directions contributed by this chain are exactly span { v 1 , . . . , v j − 1 } = Im( V j − 1 ) . Hence k er( C T ( A,B ) F T ) = { 0 } holds if and only if Im( F T ) ∩ Im( V j − 1 ) = { 0 } . Since rank( V j − 1 ) = j − 1 , this is equiv alent to the direct- sum condition Im( F T ) ⊕ Im( V j − 1 ) = Im   F T V j − 1   , and therefore to rank   F T V j − 1   =rank( F T ) + rank( V j − 1 ) =rank( F ) + ( j − 1) . Since rank( F ) = r , this is exactly (22). If j = 1 , then V j − 1 is empty and no condition is required. Example 6: T ar get Output Contr ollability (PBH/Jor dan veriﬁcation) Consider the L TI system A =     4 − 4 2 − 5 1 1 1 − 1 − 1 3 1 4 − 1 3 − 1 5     , B =     1 0 0 0     , F =  1 3 2 1  . The eigen values of A are λ = 2 (algebraic multiplicity 2 ), λ = 3 and λ = 4 . W e v erify TOC using Theorem 16 in its rank form rank   F T V j − 1   = r + ( j − 1) , r = rank( F ) = 1 . Eigen value λ = 2 . A Jordan chain of ( A T , 2) is v 1 =  1 − 1 1 0  T , v 2 =  0 1 1 − 1  T , ( A T − 2 I ) v 1 = 0 , ( A T − 2 I ) v 2 = v 1 . Since B T v 1 = 1  = 0 , we hav e j = 1 and thus V j − 1 = V 0 is empty . Hence no rank test is required on this chain. Eigen value λ = 3 . An eigenv ector of ( A T , 3) is v 3 =  1 1 1 1  T , ( A T − 3 I ) v 3 = 0 . Since B T v 3 = 1  = 0 , again j = 1 and no condition is required. Eigen value λ = 4 . An eigenv ector of ( A T , 4) is v 4 =  0 1 0 1  T , ( A T − 4 I ) v 4 = 0 . Here B T v 4 = 0 , so j = q + 1 = 2 and V j − 1 = [ v 4 ] . Theorem 16 requires rank   F T v 4   = r + ( j − 1) = 2 . In f act rank   F T v 4   = 2 , so the rank condition holds for λ = 4 . All Jordan chains of ( A T , λ ) satisfy the condition in Theo- rem 16 for e very λ ∈ σ ( A ) . Therefore, the triple ( A, B , F ) is tar get output contr ollable . Howe ver , ( A, B , F T ) is not functional contr ollable , since the implication in Theorem 1 fails for at least one Jordan chain of ( A T , λ ) . V I I . G E N E R A L I Z E D S E P A R A T I O N P R I N C I P L E In classical linear systems theory , controllability and observ- ability together imply that state–feedback and observer design problems decouple, leading to the well-known separation principle. In the functional setting, the objectiv e is no longer full-state regulation or estimation, but rather the stabilization and reconstruction of a prescribed functional z ( t ) = F x ( t ) . In this section we establish a generalized separation principle (GSP), showing that functional controller and observer syn- thesis decouple whenev er the system is intrinsically functional controllable (IFC) and functional observable (FO). W e ﬁrst recall a classical augmentation-based characteri- zation of functional observability , which provides necessary and sufﬁcient rank conditions for the existence of a functional observer . Lemma 15 ( [25] ): The triple ( A, C , F ) is functional ob- servable if and only if there exists ¯ q ∈ N such that r ≤ ¯ q ≤ n , and a matrix R ∈ R ( ¯ q − r ) × n (which is empty when ¯ q = r ), such that the following conditions are satisﬁed rank         F A RA C A C F R         = rank     C A C F R     (23a) rank     λF − F A λR − R A C A C     = rank     C A C F R     , ∀ λ ∈ C (23b) Remark 5: The conditions (23a)–(23b) with R = ∅ corre- spond precisely to the existence conditions for a functional observer of order r . In this case, the observer estimates the functional z ( t ) = F x ( t ) directly without augmentation, and coincides with the classical Darouach functional observer [27] existence conditions. The following lemma shows that functional observability is preserved under completion of the functional by rows belonging to the intrinsic functional subspace. Lemma 16 (FO pr eserved under functional completion): Assume ( A, C , F ) is functional observable and let R 1 ∈ R p × n be any matrix whose rows r 1 , . . . , r p satisfy r T i ∈ V F := Im  C ( A T ,F T )  = Im  O T ( A,F )  , i = 1 , . . . , p. Deﬁne ¯ F :=  F R 1  . Then ( A, C, ¯ F ) is functional observable. In particular , if Im( ¯ F T ) = V F , then ( A, C , ¯ F ) is functional observable. 14 PREPRINT Pr oof: Since ( A, C , F ) is functional observ able, we have the kernel inclusion k er  O ( A,C )  ⊆ ker  O ( A,F )  , i.e. for any x ∈ ker( O ( A,C ) ) one has F A k x = 0 for all k ≥ 0 . Now ﬁx any row vector h T ∈ V F . By deﬁnition of V F , there exist vectors α 0 , . . . , α n − 1 such that h T = n − 1 X k =0 α T k F A k . Hence, for x ∈ ker( O ( A,C ) ) and any ℓ ≥ 0 , h T A ℓ x = n − 1 X k =0 α T k F A k + ℓ x = 0 , because F A m x = 0 for all m ≥ 0 . Therefore k er  O ( A,C )  ⊆ ker  O ( A,h T )  . Applying this row-by-ro w to R 1 (whose rows lie in V F ) giv es k er  O ( A,C )  ⊆ ker  O ( A,R 1 )  , and stacking with F yields k er  O ( A,C )  ⊆ ker  O ( A, ¯ F )  , which is exactly functional observability of ( A, C, ¯ F ) . Lemma 17 ( [26]): Let R =  R 1 R 2  and also let R 1 and R 2 be matrices that are either empty or nonempty . If there exists matrices R 1 and R 2 such that conditions (5) and (23) are satisﬁed, then a functional observer and a functional controller can be independently designed such that z ( t ) → 0 at an arbitrarily prescribed rate of con vergence as t → ∞ . W e are now in a position to state a generalized separation principle at the functional lev el. Theor em 17 (Generalized Separation Principle (GSP)): Suppose that 1) ( A, B , F T ) is intrinsically functional contr ollable and 2) ( A, C , F ) is functional observable . Then there exist • a static functional state–feedback law u ( t ) = − K ˆ z ( t ) , where ˆ z ( t ) denotes the estimated functional produced by the observer , • and a functional observer producing ˆ z ( t ) , such that z ( t ) = F x ( t ) → 0 as t → ∞ for all initial conditions. Moreov er, the controller and observer gains can be designed independently . Pr oof: Deﬁne the intrinsic functional subspace V F := Im  C ( A T ,F T )  , the smallest A T –in variant subspace containing Im( F T ) . Contr oller-side existence (IFC). By intrinsic functional con- trollability of the triple ( A, B , F T ) , it follows from the deﬁ- nition of IFC, i.e., Deﬁnition 1, that there exists a matrix R 1 such that the rank conditions (5a)–(5b) are satisﬁed. A minimal choice of R 1 yields Im  F R 1  T = V F . From Lemma 8, it then follows that there exists a static functional state–feedback law u ( t ) = − K ˆ z ( t ) that stabilizes all modes contributing to z ( t ) = F x ( t ) on the intrinsic functional subspace V F . Observer-side existence (FO). Since ( A, C, F ) is functional observable, the functional z ( t ) = F x ( t ) can be reconstructed from the measured output y ( t ) = C x ( t ) and a vailable input u ( t ) . Let R 1 be chosen such that Im  F R 1  T = V F , where V F := Im  C ( A T ,F T )  is the intrinsic functional sub- space. By Lemma 16, functional observability is preserved under A T –in variant completion. Hence ( A, C, F ) functional observable implies that  A, C,  F R 1  is also functional observable. By Lemma 15, there therefore exists a matrix R =  R 1 R 2  satisfying the observer rank conditions (23a)–(23b). Hence a functional observer exists that reconstructs z ( t ) = F x ( t ) with arbitrarily prescribed error con vergence rate. Separation. The controller synthesis depends only on ( A, B , F ) through IFC and the existence of R 1 , while the observer synthesis depends only on ( A, C , F ) through FO and the existence of R 2 . The two synthesis problems are therefore independent. This establishes the generalized separation principle at the functional lev el. Remark 6: Lemma 17 establishes an augmentation-based separation principle: if augmentation matrices R 1 and R 2 exist that satisfy the rank conditions (5) and (23), then functional controller and observer designs decouple. Theorem 17 strengthens this result by providing intrinsic system-theoretic conditions—intrinsic functional controllabil- ity and functional observability—that guarantee the existence of such augmentation matrices. In this sense, Theorem 17 elev ates augmentation-based separation from a realization- dependent statement to a structural property of the quadruple ( A, B , C , F ) . Augmentation-based designs may therefore be viewed as concrete realizations of the generalized separation principle once existence is certiﬁed by the intrinsic conditions. Moreov er, the generalized separation principle admits a con- structiv e formulation, yielding explicit augmentation matrices for independent controller and observer synthesis. PREPRINT 15 Theor em 18 (Constructive GSP): Suppose that 1) ( A, B , F T ) is intrinsically functional controllable (IFC), and 2) ( A, C , F ) is functional observable (FO). Then there exist matrices R 1 and R 2 such that, with ¯ F =  F R 1  , R =  R 1 R 2  , the controller rank conditions (5a)–(5b) and the observer rank conditions (23a)–(23b) are simultaneously satisﬁed. Conse- quently , a functional controller and a functional observer can be designed independently so that z ( t ) = F x ( t ) → 0 as t → ∞ . Closed-form construction of R 1 . Recall that V F = Im  C ( A T ,F T )  . Form the observability matrix of the pair ( A, F ) ¯ F 0 := O ( A,F ) =      F F A . . . F A n − 1      . Let ¯ F be any full–row–rank matrix obtained by selecting a maximal set of linearly independent rows from ¯ F 0 that contains the rows of F . Then rank( ¯ F ) = dim( V F ) =: d, Im( ¯ F T ) = V F . Write ¯ F =  F R 1  , where R 1 consists of the additional ro ws that extend F to a basis of the row space of ¯ F 0 , so that Im( ¯ F T ) = V F . A closed-form solution for R 2 . Since ( A, C , F ) is functional observable, it follows from Lemma 16 that  A, C,  F R 1  is also functional observable. Hence, there exists a similarity transformation T such that T − 1 AT =  A o 0 A 21 A u  , C T =  C o 0  , where A o ∈ R h × h , h ≤ n governs the observable subspace. Functional observability further implies  F R 1  T =  ¯ F o 0  , where ¯ F o ∈ R d × h has full row rank d . Let ¯ F ⊥ o be any matrix whose rows complete those of ¯ F o to a basis of R h , and deﬁne R 2 =  ¯ F ⊥ o 0  T − 1 . Pr oof: Intrinsic functional controllability guarantees the existence of R 1 such that ¯ F =  F R 1  has full row rank, spans V F , and satisﬁes the controller rank conditions (5a)–(5b). Hence all modes contributing to z ( t ) = F x ( t ) are stabilizable. Since ( A, C , F ) is functional observ able and im( F T ) ⊆ V F with V F A T –in variant, functional observ ability extends from ( A, C, F ) to ( A, C , ¯ F ) . By the Kalman observability decomposition, the above choice of R 2 makes   F R 1 R 2   a basis of the observ able subspace and R =  R 1 R 2  satisﬁes the observer rank conditions (23a)–(23b). Thus matrices R 1 and R 2 satisfy all four rank conditions (i.e., (5a)–(5b) and (23a)–(23b)), and by Lemma 17, functional controller and observer designs decouple. Remark 7: The closed-form solution for R 1 yields a matrix with the minimum possible number of rows, namely dim( V F ) , since its rows form a basis of the intrinsic functional subspace. Thus, R 1 is minimal among all matrices satisfying the con- troller rank conditions (5a)–(5b). In contrast, the closed-form solution for R 2 provides a valid completion of the observable subspace and satisﬁes (23a)–(23b), but need not be row-minimal. This lack of mini- mality has no effect on the Generalized Separation Principle, whose aim is only to ensure the existence of matrices R 1 and R 2 satisfying the required rank conditions. Remark 8: When F = I n , intrinsic functional controllabil- ity and functional observ ability reduce to classical controlla- bility and observability , and Theorem 17 recov ers the standard separation principle. Remark 9 (Asymptotic Generalized Separation Principle): The generalized separation principle e xtends naturally to the asymptotic setting. If ( A, B , F T ) is intrinsically functional stabilizable (IFS) and ( A, C, F ) is functional detectable (FD), then there exist a functional state–feedback law and a functional observer such that z ( t ) = F x ( t ) → 0 as t → ∞ . Stabilization and estimation are required only for the unstable modes contributing to V F . Example 7: IFC and FO V eriﬁcation Consider the uncontrollable and unobserv able systems in Examples 2 and 3 of [9]. By Theorem 4, intrinsic functional controllability (IFC) is satisﬁed for the functional z ( t ) = F x ( t ) , and by the functional observability tests de veloped in Section V , the triple ( A, C , F ) is functional observable. Therefore, the conditions of Theorem 18 are satisﬁed. It follows that matrices R 1 and R 2 can be constructed explicitly using the closed form solutions for R 1 and R 2 such that the augmented functional matrix ¯ F =  F R 1  satisﬁes the con- troller rank conditions (5a)-(5b), while the full augmentation R =  R 1 R 2  satisﬁes the observer rank conditions (23a)-(23b). Consequently , a functional controller and a functional observer can be designed independently . V I I I . C O N C L U S I O N This paper de veloped a uniﬁed eigen v alue-based framew ork, inspired by the Popo v–Belevitch–Hautus (PBH) test, for the 16 PREPRINT analysis of functional properties of linear systems. Necessary and suf ﬁcient PBH-style conditions were established for func- tional controllability , stabilizability , observ ability , detectability , target output controllability , and their intrinsic counterparts gov erning the existence of admissible functional completions. By working directly with eigen vectors and generalized eigen- vectors organised along Jordan chains, the resulting condi- tions provide local, mode-by-mode characterisations that apply uniformly to diagonalizable and non-diagonalizable systems, without requiring explicit construction of controllability or observability matrices. A key contribution is the introduction of intrinsic func- tional contr ollability and intrinsic functional stabilizability , which yield veriﬁable structural conditions for the existence of augmentation matrices required in functional controller and observer synthesis. These intrinsic notions underpin a Generalized Separ ation Principle at the functional lev el, estab- lishing that functional controllers and functional observers can be designed independently whenever intrinsic functional con- trollability and functional observability hold. 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J. Hautus, “Controllability and observability conditions of lin- ear autonomous systems, ” Indagationes Mathematicae (Proceedings) , vol. 72, no. 5, pp. 443–448, 1969. [25] T . Fernando, H. T rinh, L. Jennings, “Functional observability and the design of minimum order linear functional observers”, IEEE T rans. Autom. Contr . , 55 (5), pp. 1268-1273, 2010. [26] T . Fernando and M. Darouach, “Existence and design of target output controllers”, IEEE T rans. Autom. Contr . , 70 (9), pp. 6104-6110, 2025. [27] M. Darouach, “Existence and design of functional observ ers”, IEEE T rans. Autom. Contr . , 45(5), pp. 940-943, 2000. B I O G R A P H Y T yrone Fernando receiv ed his B.E. (Hons.) and Ph.D. degrees in Electrical Engineering from the University of Mel- bourne, V ictoria, Australia, in 1990 and 1996, respecti vely . In 1996, he joined the Department of Electrical, Electronic and Computer Engineering, Uni versity of W estern Australia (UW A), Crawle y , W A, Australia, where he currently holds the position of Professor of Electrical Engi- neering. He previously served as Associate Head and Deputy Head of the department from 2008 to 2010. Prof. Fernando is currently the Head of the Power and Clean Energy Research Group at UW A. He has provided professional consultancy to W estern Power on the integration and management of distrib uted energy resources in the electric grid. In recognition of his professional contrib utions, he w as named the Outstanding W A IEEE PES/PELS Engineer in 2018. His research interests include theoretical control, ob- server design, and power system stability and control. He has receiv ed multiple teaching a wards from UW A in recognition of his contributions to control systems and power systems education.

Unified Eigenvalue-Eigenspace Criteria for Functional Properties of Linear Systems and the Generalized Separation Principle

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