Design of Positive Linear Observers for Positive Systems via Coordinates Transformation and Positive Realization

Design of Positive Linear Observers for Positive Systems via Coordinates Transformation and Positive Realization Juhoon Back 1 Alessandro Astolfi 1 1 Department of Electrical and Electronic Engineering Imperial College, UK ICM 3 May 2006

Outline 1 Background 2 Postive Linear Observer via Coordinates Trasformation Necessary Condition on the Existence of PLO via CT Sufficient Condition: Design of PLO via CT 3 Positive Linear Observer via Positive Realization Positive Realization and PLO via PR Existence of PLO via PR 4 Conclusion and Discussion

Background Positive Linear System Positive System: ẋ = f (x, u), x R n, u R m Positive orthant: R n + := [0, ) n x(0) R n +, u(t) R m +, t 0 x(t) R n +, t 0. Natural if x represents density, population,... Field: Economics, Biology, etc. Positive Linear System: ẋ = Ax + Bu, y = Cx Positive Linear System (wrt x, y) A is Metzler (off-diagonal nonnegative), B 0, C 0. If A is Metzler, the eigenvalue of maximal real part λ max (A) is real.

Background Positive Linear System Example ẋ = [ ] 1 1 x + 1 2 u = 0.5(2 + sin t). [ ] 1 u 0

Background Positive Linear Observer Linear observer for linear systems Luenberger type observer ˆx = Aˆx + K(y Cˆx), A KC : Hurwitz Existence condition : detectability of (A, C) Positive linear observer: A positive linear system with input y, output ˆx Guarantees error convergence ( ˆx x 0) Reasonable for positive systems: ˆx 0. Good candidate: ˆx = Aˆx + K(y Cˆx) A KC : Hurwitz, Metzler, K 0 Existence condition has not been found for general case

Background Previous Results Observer Structure: Luenberger Type Observer ˆx = Aˆx + K(y Cˆx) A KC : Hurwitz, Metzler, K 0 Van Den Hof (1998, SIAM) : Compartmental Systems, MO Compartmental System: Every column sum of A 0, A Metzler, B 0, C 0. Dautrebande, Bastin (1999, ECC) : Positive Systems, MO Leenheer, Aeyels (2001, SCL): Stabilization, SI

Coordinates Change Motivating Example Example ẋ = y = [ 1 [ ] 1 1 x =: Ax, 2 0 0 ] x =: Cx. Identity PLO not possible ˆx = Aˆx + K(y Cˆx) A KC : Hur. and Met., K 0 [ ] 1 k1 1 A KC = 2 k 2 0

Coordinates Change Motivating Example Example ẋ = y = [ 1 [ ] 1 1 x =: Ax, 2 0 0 ] x =: Cx. [ ] 2 Take k =. 3 Luenberger [ ] Observer: [ ] 3 1 2 ˆx = ˆx + y 2 0 4 Not positive System Identity PLO

Coordinates Change Motivating Example Example ẋ = y = [ 1 [ ] 1 1 x =: Ax, 2 0 0 ] x =: Cx. Identity PLO PLO via Transformation ˆx = [ ] 3 1 ˆx + 2 0 Transform: z = Tx = [ ] 2 y (Not PLO) 4 [ 2 1 5 1 1 5 2 2 ] 1 x ẑ = T(A KC)T 1 ẑ + TKy [ ] [ ] 2 0 0 = ẑ + 0 1 2 y, ˆx = T 1 ẑ 5 PLO

PLO via Coordinates Transformation Sylvester Equation Structure: ẑ = Fẑ + Gy ˆx = T 1 z F: Hurwitz and Metzler matrix, G R n q + T: nonsingular matrix with T 1 R n n +. Equivalent to solve the Sylvester Equation: TA FT = GC, F, G : Design parameters Coodinate transformation involved Hard: T 1 R n n +, inverse eigenvalue problem F. Sufficient condition for the existence T: σ(f) σ(a) = φ When σ(f) σ(a) = φ, necessary condition for invertible T: Observability (A, C) and Controllability (F, G)

PLO via Coordinates Transformation Design Procedure Procedure with Nonresonnance Condition 1 Choose a controllable pair (F, G) such that F is Hurwitz and Metzler, σ(f) σ(a) = φ, and G R n k +. 2 Solve the unique T in TA FT = GC. 3 Check if T is nonsingular and T 1 R n n +. 4 If T fails to meet the conditions of step 3 of the procedure, then choose other (F, G) pair and repeat the process. σ(f) σ(a) = φ: restrictive (for uniqueness of T). Procedure without σ(f) σ(a) = φ is an observer design procedure [Chen1984]. Requires iteration (not constructive).

Necessary Condition on the Existence of PLO via CT Restriction on the Spectrum Basic Tool Lemma Let A R n n be Metzler and C R 1 n +. Suppose λ max(a) 0. If there exists K R n 1 + such that A KC is Metzler and λ max (A KC) < λ max (A), then the algebraic multiplicity of λ max (A) (=: m) is at most 1. Proof: by Contradiction. Assume m > 1. Detectable. For a Metzler matrix A, the algebraic multiplicity of λ max (A) equals the geometric multiplicity of λ max (A) (nullity of (A λ max (A)I)). [ ] C rank rank C + rank [A λ A λ max (A)I max (A)I] = 1 + n m < n Condtradiction.

Necessary Condition on the Existence of PLO via CT Restriction on the Spectrum Theorem Let A R n n be a Metzler matrix and C := [c 1,, c n ] R 1 n +. If there exists K R n + and S R n n + with det S 0 such that S 1 AS KCS is Metzler and Hurwitz, then the number of nonnegative real eigenvalue of A counting the multiplicity is at most 1. Note: 1. TA FT = GC, T 1 0 AS SF = SGCS, S 0. 2. F = TAT 1 GCT 1 = S 1 AS GCS.

Necessary Condition on the Existence of PLO via CT Restriction on the Spectrum Proof: F = TAT 1 GCT 1 = S 1 AS GCS, by Contradiction idea: Reduce parameters to one and use Root Locus 1. A = S 1 AS, C = CS. 2. Suppose A KC = S 1 AS KCS is Metzler and Hurwitz. 3. A δkc, δ [0, 1] is Metzler. 4. δ [0, 1) s.t. A δkc is Metzler and Hurwitz δ [δ, 1]. 5. One parameter det(si A + δkc) = det((si A)(I + δ(si A) 1 KC)) Root Locus analysis. = det(si A)(1 + δc(si A) 1 K) = det(si A) + δcadj(si A)K =: D(s) + δn(s)

Necessary Condition on the Existence of PLO via CT Necessary Condition when F is a diagonal Simple structure of PLO ẑ = Fẑ + Gy = λ 1... λ n + g 1. g n y Assumption 1 (Non Resonnance): σ(f) σ(a) = φ Assumption 2 (F diagonal): F = diag{λ 1, λ 2,, λ n }, λ i R, and λ i λ j if i j Theorem Single-output System. Suppose Assumption 1 and 2 hold. If there exists a positive linear observer of dimension n, then Card(σ(A) R ) n 1.

Necessary Condition on the Existence of PLO via CT Necessary Condition when F is a diagonal-proof Assume λ 1 < λ 2 < < λ n. TI IT = 0 TA FT = GC. TA n F n T = GCA n 1 + FGCA n 2 + + F n 2 GCA + F n 1 GC T (A) (F)T = U F Λ α V A α n 1 α n 2 α 1 1 C := [ G FG F n 1 G ] α n 2 α n 3 1 0 CA..... α 1 1 0 0 CA n 2 1 0 0 0 CA n 1 T = ( (F)) 1 U F Λ α V A. (1)

Necessary Condition on the Existence of PLO via CT Necessary Condition when F is a diagonal-proof n PLO, S R n n + s.t. TS = I. U F Λ α V A S = (F). (2) g 1 λ 1 g 1 λ n 1 1 g 1 U F Λ α =... Λ α g n λ n g n λ n 1 n g n g 1 θ 1 (λ 1 ) g 1 θ 2 (λ 1 ) g 1 θ n (λ 1 ) =... g n θ 1 (λ n ) g n θ 2 (λ n ) g n θ n (λ n ) θ i (s) := s n i + α 1 s n i 1 + α 2 s n i 2 + + α n i, i = 1,, n 1 θ n (s) := 1. (3)

Necessary Condition on the Existence of PLO via CT Necessary Condition when F is a diagonal-proof Θ := θ 1 (λ 1 ) θ 2 (λ 1 ) θ n (λ 1 )... θ 1 (λ n ) θ 2 (λ n ) θ n (λ n ) det Θ = (λ i λ j ). (4) 1 i<j n F ( (F) as well) is diagonal and g i 0, (λ 1 ) g 1 0 0 (λ 0 2 ) ΘV A S = g 2 0..... 0 0 ΘV A S i = 0 (i 1) 1 (λ i) g i 0 (n i) 1 (λ n) g n,, S i : ith column of S (5)

Necessary Condition on the Existence of PLO via CT Necessary Condition when F is a diagonal-proof Cramer s rule: CS i = 1 0 (i 1) 1. θ 2 (λ 1 ) θ n (λ 1 ) det Θ det (λ i) g i... 0 (n i) 1. θ 2 (λ n ) θ n (λ n ) = ( 1) i (λ i ) g i det Θ det Θ [i] θ 2 (λ 1 ) θ n (λ 1 ).. Θ [i] := θ 2 (λ i 1 ) θ n (λ i 1 ) θ 2 (λ i+1 ) θ n (λ i+1 ).. θ 2 (λ n ) θ n (λ n ) which should be nonnegative because C 0, S i 0. (6)

Necessary Condition on the Existence of PLO via CT Necessary Condition when F is a diagonal-proof The nonsingularity of the matrices in (6) and Assumption 1 ( (λ i ) 0) implies Using (4), compute det Θ [i] det Θ = CS i = ( 1) i (λ i) det Θ [i] g i det Θ = 1 u<v n,u i,v i (λ u λ v ) 1 u<v n (λ u λ v ) > 0, i = 1,, n. (7) 1 (λ 1 λ i ) (λ i 1 λ i )(λ i λ i+1 ) (λ i λ n), 2 i n 1 1 (λ 1 λ 2 ) (λ 1 λ n), i = 1 1 (λ 1 λ n) (λ n 1 λ n), i = n.

Necessary Condition on the Existence of PLO via CT Necessary Condition when F is a diagonal-proof Since λ i < λ i+1, i = 1,, n 1, CS i = ( 1) i+n 1 det Θ (λ i ) [i] g i det Θ ( 1) i+n 1 (λ i ) > 0, i = 1,, n. ( 1) i+n 1 (λ i ) ( 1) i+1+n 1 (λ i+1 ) > 0, i = 1,, n 1, (λ i ) (λ i+1 ) < 0, i = 1,, n 1. Note: ( ) is a continuous function λ i s.t. (λ i ) = 0, λ i < λ i < λ i+1, i = 1,, n 1. there are at least n 1 negative real eigenvalues of A.

Sufficient Condition: Design of PLO via CT A with Distinct Real Eigenvalues Definition Reducible: if there exists a permutation matrix P R n n s.t. ( ) PAP t A11 0 = A 21 A 22 where A 11, A 22 are square matrices. A is said to be irreducible if A is not reducible. Note: if A is irreducible, it has an eigenvector V 1 whose elements are all positive.

Sufficient Condition: Design of PLO via CT A with Distinct Real Eigenvalues Assume: A has n real distinct eigenvalues λ 1,, λ n s.t. λ 1 > > λ n. Jordan Form: A = VJV 1 (8) J := diag{λ 1,, λ n } (9) V := [ V 1 V n ] (10) Theorem Single-output positive system. Suppose A is irreducible and (A, C) is observable. There exists a positive linear observer of dimension n if (A, C) satisfies 1) σ(a) R and σ(a) contains one (or no) nonnegative real number, 2) all the eigenvalues of A are simple.

Sufficient Condition: Design of PLO via CT A with Distinct Real Eigenvalues: proof Observability assumption CV i 0. Scale V i s.t. CV i = 1, i = 1,, n. Main idea: Construct S and G to solve AS SF = SGCS Let β, γ be real numbers in the open interval (0, 1). We define [ ] 1 β (1 γ)1 S(β, γ) = 1 (n 1). (11) β1 (n 1) 1 γi n 1 Properties of S: det S = γ n 2 [γ nβγ + (n 1)β]. [ S 1 = γn 3 γ 2 ] γ(1 γ)1 1 (n 1) det S β(1 γ)1 n 1 + π β,γ I n 1 βγ1 (n 1) 1 where π β,γ := γ(1 β) + (n 1)β(1 γ). Note: π β,γ > 0, β, γ (0, 1). (12)

Sufficient Condition: Design of PLO via CT A with Distinct Real Eigenvalues: proof Construction of S: Let β (0, 1), γ (0, 1) such that S(β, γ) = VS(β, γ). (13) S(β, γ) 0, β (0, β ), γ (0, γ ). (14) β, γ always exist. In fact, let S i be the ith column of S. Then, S 1 = (1 β)v 1 + β( V 2 V n ) S i = (1 γ)v 1 + γv i, i = 2,, n. (15) Note: S i starts from V 1 (i.e., β = 0, γ = 0) and ends at (V 2 + + V n ) or V i (i.e., β = 1, γ = 1). Thus, by continuity, the existence of β, and γ is proved.

Sufficient Condition: Design of PLO via CT A with Distinct Real Eigenvalues: proof T(β, γ) = S 1 (β, γ). F = TAT 1 GCT 1. (16) Note: first column of S 1 is elementwise positive G = µs 1 e 1 (17) where e 1 := [ 1 chosen later. 0 1 (n 1) ] t and µ is a positive number to be F = S 1 VJV 1 S GCS = S 1 V 1 VJV 1 VS S 1 µe 1 CVS = S 1 [J µe 1 1 1 n ]S.

Sufficient Condition: Design of PLO via CT A with Distinct Real Eigenvalues: proof Fine µ, β, γ rendering F Hurwitz and Metzler. Then, F := det S γ n 3 F T := det S S 1 γn 3 F = T[J µe 1 1 1 n ]S [ ] π β,γ [λ 1 (1 γ) µ]1 = T 1 (n 1) β [ ] t λ 2 λ n γdiag{λ 2,, λ n } where π β,γ := (λ 1 µ)(1 β) + (n 1)µβ.

Sufficient Condition: Design of PLO via CT A with Distinct Real Eigenvalues: proof Thus, F 11 F 21 F n1 γ 2 π β,γ + βγ(1 γ) n i=2 λ i. = βγπ β,γ βπ β,γ λ 2 + β 2 (1 γ) n F nk i=2 λ i. βγπ β,γ βπ β,γ λ n + β 2 (1 γ) n i=2 λ i and for k = 2,, n, F 1k γ 2 [(λ 1 λ k )(1 γ) µ] F 2k. = βγ[(λ 1 λ k )(1 γ) µ] + δ 2,k γπ β,γ λ k. βγ[(λ 1 λ k )(1 γ) µ] + δ n,k γπ β,γ λ k where δ i,j = 1 if i = j and δ i,j = 0 otherwise.

Sufficient Condition: Design of PLO via CT A with Distinct Real Eigenvalues: proof From this computation, we derive some properties of F. P1. For all k = 3,, n, F 21 F k1. P2. For all k = 2,, n, F 1k 0 if and only if F jk 0, j = 2,, n, j k. P3. For all k = 3,, n, F 12 F 1k. P1 can be proved easily since F 21 F k1 = βπ β,γ (λ 2 λ k ) < 0, P2 follows from 1 γ F 1k = 1 β F jk = γ[(λ 1 λ k )(1 γ) µ], and the relation F 12 F 1k = γ 2 (1 γ)( λ 2 + λ k ) < 0, k = 3,, n, ensures P3.

Sufficient Condition: Design of PLO via CT A with Distinct Real Eigenvalues: proof From P1, P2, P3, and the fact that F is similar to J µe 1 1 1 n whose spectrum is {λ 1 µ, λ 2,, λ n }, it follows that F is Hurwitz and Metzler if there exist µ, β (0, β ), and γ (0, γ ) such that the following inequalities hold λ 1 µ < 0 (18) n γπ β,γ π β,γ λ 2 + β(1 γ) λ i 0 (19) i=2 (λ 1 λ 2 )(1 γ) µ 0. (20) We choose µ in (λ 1, λ 1 λ 2 ) and define { γ = min γ, λ } 1 λ 2 µ. λ 1 λ 2 Then, for each γ in (0, γ ), the inequality (20) holds.

Sufficient Condition: Design of PLO via CT A with Distinct Real Eigenvalues: proof With µ and γ chosen above, it remains to choose β for (19). To do this, we expand the left hand side of (19) as follows γπ β,γ π β,γ λ 2 + β(1 γ) n i=2 λ i = γ(λ 1 λ 2 nµ)β + (1 γ)β n i=2 ( λ 2 + λ i ) + (λ 1 λ 2 µ)γ Thus, the inequality (19) is equivalent to (λ 1 λ 2 µ)γ Υ(γ, µ, λ 1,, λ n )β (21) where Υ(γ, µ, λ 1,, λ n ) = γ(λ 1 λ 2 nµ) + (1 γ) n i=2 (λ 2 λ i ). This inequality (hence (19)) holds true for any β (0, β ) where { β, if Υ 0 β = { } min β,, if Υ > 0. Thus, the proof is complete. λ 1 λ 2 γ Υ(γ,µ,λ 1,,λ n)

PLO via Positive Ralization Positive Realization Definition Transfer function matrix T(s) with input u and output y. If there exists ż = Fz + Gu, y = Hx F : Metzler, G 0, H 0 such that T(s) = H(sI F) 1 G, then (F, G, H) is said to be a positive realization of T(s). Finite Dimensional PR: Find a polyhedral cone with certain property (F invariant). SISO: If (A, B, C) is a minimal realization of G(s), G(s) has a PR if and only if a polyhedral convex cone M s.t. (A + λi)m M for some λ 0, R M O R : Reachability Cone, O : Observability Cone.

PLO via Positive Ralization Positive Realization Existence PR - impulse response and dominant pole Theorem (SISO) Let T(s) be a rational transfer function. T(s) has a positive realization if and only if 1. The impulse response function h(t) is positive, i.e., h(t) > 0 for every t > 0 and h(0) 0, and 2. There is a unique (possibly multiple) pole of T(s) with maximal real part. MIMO: T(s) has a PR iff each component of T(s) has a PR.

PLO via Positive Ralization Problem Formulation- Observer Definition Given a positive linear system, a linear system ẑ = Fẑ + Gy, ẑ R N ˆx = Hẑ, ˆx R n (22) is called a positive linear observer if F is Hurwitz and Metzler, G R N q +, H Rn N +, and lim t Hẑ(t) x(t) = 0 for all ẑ(0) R N + and x(0) R n +. Observer can be interpreted as a Transfer Function. Positive Realization is employed (N : additional degree of freedom). Under what condition the existence of positive linear observer is guaranteed? How to construct (F, G, H)?.

PLO via Positive Ralization Existence of PLO via PR Theorem Detectable Positive linear system (A, C). If there exists K R n q such that A KC is a Hurwitz matrix and the transfer function G o (s) = (si A + KC) 1 K admits a positive realization, then there exists a PR (F, G, H) with F being Hurwitz such that dynamic system ẑ = Fẑ + Gy, ˆx = Hẑ, ˆx R n is a positive linear observer for (A, C). ẑ R N N is a design variable N can be larger than n or smaller than n

PLO via Positive Ralization Existence of PLO via PR-proof Let A K = A KC and T be the transformation decomposing (A K, K, I) into controllable and uncontrollable part. [ ] [ ] AK,c A TA K = 12 Kc T, TK =, T 1 = [ ] C 0 A K,c 0 c C c such that (A K,c, K c ) is controllable. Then, G o (s) = C c (si A K,c ) 1 K c. Let (F, G, H) be a positive realization of G o (s). Note that this realization may not be Hurwitz (i.e., there may be an positive real eigenvalue which is uncontrollable or unobservable). Following the same procedure of Anderson et al (Theorem 3.2), one can exclude these modes to obtain a reduced oder positive realization (F, G, H) such that λ max (F) is a pole of G o (s), i.e., F is Hurwitz.

PLO via Positive Ralization Existence of PLO via PR-proof Now, suppose (F, G, H) with F being Hurwitz be a positive realization of G o (s) with a cone M generated by M, i.e., A K,c M = MF, K c = MG, H = C c M. (23) Since A K T 1 [ ] [ ] [ ] A = A K Cc C c = Cc C K,c A 12 c = 0 A [ ] K,c T 1 AK,c A 12, it follows that A 0 A K C c = C c A K,c. Similarly, one K,c has C c K c = K. Hence, AH HGCH = A K H + KCH HGCH Let e = ˆx x. Then, = A K C c M + C c K c CH HGCH = C c A K,c M + C c MGCH HGCH = HF. ė = HFẑ + HGCx Ax = = (A KC)e

PLO via Positive Ralization Existence of PLO via PR Theorem Consider a positive linear observer (F, G, H) for (A, C) ((F, G, H) is a positive realization of G o (s) = (si A + KC) 1 K). Let n c be the dimension of controllable part of (A KC, K). Then, 1. K 0. 2. rank H = n c. 3. AH FH = HGCH. Theorem A positive linear system (A, C), assume (A, C) is an observable pair. If there is K R n q such that K > 0 and G o (s) = (si A + KC) 1 K admits a PR (F, G, H) which is a PLO via PR, then there exists K such that A KC is Hurwitz, (A KC, K) is controllable, and G o (s) = (si A + KC) 1 K admits a PR (F, G, H) with rank H = n. (Proof: Underway)

PLO via Positive Ralization Existence of PLO via PR Theorem For a single output positive system (A, C), suppose A is irreducible and (A, C) is detectable. Let λ 1 = λ max (A). There exists a PLO via PR (F, G, H) if all eigenvalues of A except λ 1 lie in the open left half complex plain. Stable engenvalues can be complex (compare PLO via CT). One dimensional PLO via PR can be constructed (estimate the mode of λ 1 ).

PLO via Positive Ralization Existence of PLO via PR- irreducible case, proof A: irreducible, (A, C) is detectable v max int R n + such that Av max = λ max (A)v max, Cv max = 1. λ := max{re λ λ σ(a), λ λ 1 }. K := (λ 1 + ɛ)v max, λ < ɛ < 0 As observered by Brauer, 1952 (Theorem 2.9), we have σ(a KC) = (σ(a) {λ 1 }) { ɛ} since Cv max = 1. All the eigenvalues of A and A KC coincide except that λ 1 is replaced by ɛ. A KC is Hurwitz and the eigenvalue with the maximum real part is real.

PLO via Positive Ralization Existence of PLO via PR- irreducible case, proof Define G o (s) = (si A + KC) 1 K. g o (t) = e (A KC)t K = e (A (λ 1+ɛ)v maxc)t (λ 1 + ɛ)v max [ ] 1 = (λ 1 + ɛ) i! [A (λ 1 + ɛ)v max C] i t i i=0 [ ] 1 = (λ 1 + ɛ) i! ( ɛt)i v max i=0 = (λ 1 + ɛ)e ɛt v max > 0 which implies that G o (s) has a positive realization. v max

Conclusion and Discussion Done Postive linear observer for positive linear systems. Coordinates Transformation and Positive Realization. Existence Conditions and Constructive Designs. Todo More general case: Forced Systems, Nonlinear Systems. Study on dual problem: Negative Stabilization