Systems & Control Letters. State and output feedback boundary control for a coupled PDE ODE system

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1 Systems & Contro Letters 6 () Contents ists avaiabe at ScienceDirect Systems & Contro Letters journa homepage: wwweseviercom/ocate/syscone State output feedback boundary contro for a couped PDE ODE system Shuxia Tang, Chengkang Xie Schoo of Mathematics System Science, Southwest University, Chongqing 475, China a r t i c e i n f o a b s t r a c t Artice history: Received June Received in revised form 4 December Accepted Apri Avaiabe onine 4 May Keywords: Couped system PDEs Boundary contro Output feedback This note is devoted to stabiizing a couped PDE ODE system with interaction at the interface First, a state feedback boundary controer is designed, the system is transformed into an exponentiay stabe PDE ODE cascade with an invertibe integra transformation, PDE backstepping is empoyed Moreover, the soution to the resuting cosed-oop system is derived expicity Second, an observer is proposed, which is proved to exhibit good performance in estimating the origina couped system, then an output feedback boundary controer is obtained For both the state output feedback boundary controers, exponentia stabiity anayses in the sense of the corresponding norms for the resuting cosed-oop systems are provided The boundary controer observer for a scaar couped PDE ODE system as we as the soutions to the cosed-oop systems are given expicity Esevier BV A rights reserved Introduction In contro engineering, topics concerning couped systems are popuar, which have rich physica backgrounds such as couped eectromagnetic, couped mechanica, couped chemica reactions Many resuts on controabiity of the couped PDE PDE systems have been achieved such as in [ 3] Appicabe controers of state output feedback for couped PDE PDE systems as we as couped PDE ODE systems, however, are origina areas As a beginning, contro design of cascaded PDE ODE systems were considered in [4 9], through decouping PDE backstepping, boundary controers were successfuy estabished The system considered in this note is a couped PDE ODE system with interaction between the ODE the PDE At the interface, the ODE acts back on the PDE at the same time as the PDE acts on the ODE It modes the soid gas interaction of heat diffusion chemica reaction, the interaction occurs at the interface; see Fig This system is certainy more compex than just a singe ODE or a singe PDE, even more compex than a cascade of PDE ODE, in which ony the PDE acts on the ODE, or ony the ODE acts on the PDE Thus, it is needed to overcome some difficuties in contro design Some specia techniques PDE backstepping are used to deveop controers This note is organized as foows In Section, the probem is formuated In Section 3, a state feedback boundary controer is This work is supported by the Fundamenta Research Funds for the Centra Universities under contract XDJK9C99 Nationa Nature Science Foundation under Grant 6 Corresponding author Te: E-mai addresses: tkongzhi@swueducn (S Tang), cxie@swueducn (C Xie) designed to stabiize the couped PDE ODE system In Section 4, an observer, as we as an output feedback boundary controer, is designed And a scaar couped PDE ODE system is given in Section 5 as an exampe, the controer, the observer soutions to the cosed-oop systems are obtained expicity In Section 6, some comments are made on the couped PDE ODE systems Probem formuation Consider the foowing couped PDE ODE system Ẋ(t) = AX(t) + Bu x (, t) () u t (x, t) = u xx (x, t), x (, ) () u(, t) = CX(t) (3) u(, t) = U(t) (4) X(t) R n is the ODE state, the pair (A, B) is assumed to be stabiizabe, u(x, t) R is the PDE state, C T is a constant vector, U(t) is the scaar input to the entire system The couped system is depicted in Fig The contro objective is to exponentiay stabiize the system signa (X(t), u(x, t)) 3 State feedback controer Admittedy, if an invertibe transformation (X, u) (X, w) can be sought to transform the system () (4) into an exponentiay stabe target system companied with a controer, eg, the foowing system Ẋ(t) = (A + BK)X(t) + Bw x (, t) (5) w t (x, t) = w xx (x, t), x (, ) (6) 67-69/$ see front matter Esevier BV A rights reserved doi:6/jsyscone4

2 S Tang, C Xie / Systems & Contro Letters 6 () Fig A physica background of the couped system Fig Contro configuration of the couped system w(, t) = (7) w(, t) = (8) K is chosen such that A + BK is Hurwitz, then, exponentia stabiization of the origina cosed-oop system can be achieved Here the transformation (X, u) (X, w) is postuated in the foowing form X(t) = X(t) (9) w(x, t) = u(x, t) κ(x, y)u(y, t)dy Φ(x)X(t) () the gain functions κ(x, y) R Φ(x) T R n are to be determined The partia derivatives of w(x, t) with respect to x are given by w x (x, t) = u x (x, t) κ(x, x)u(x, t) κ x (x, y)u(y, t)dy Φ (x)x(t) () w xx (x, t) = u xx (x, t) κ(x, x)u x (x, t) d dx κ(x, x) + κ x(x, x) u(x, t) κ xx (x, y)u(y, t)dy Φ (x)x(t) () the notation d dx κ(x, x) = κ x(x, x) + κ y (x, x) is used The derivative of w(x, t) with respect to t is w t (x, t) = u xx (x, t) κ(x, x)u x (x, t) + κ y (x, x)u(x, t) κ yy (x, y)u(y, t)dy + (κ(x, ) Φ(x)B)u x (, t) κ y (x, )u(, t) Φ(x)AX(t) (3) Setting x = in the Eqs () (), by () (3), the foowing equations hod: w(, t) = (C Φ())X(t) w x (, t) = u x (, t) (Φ () + κ(, )C)X(t) d w t (x, t) w xx (x, t) = κ(x, x) u(x, t) dx + (κ xx (x, y) κ yy (x, y))u(y, t)dy + (κ(x, ) Φ(x)B) u x (, t) + (Φ (x) Φ(x)A κ y (x, )C)X(t) the fact that u(, t) = CX(t) is used To satisfy (5) (7), it is sufficient that κ(x, y) Φ(x) satisfy κ xx (x, y) = κ yy (x, y) (4) d κ(x, x) =, dx κ(x, ) = Φ(x)B (5) Φ (x) Φ(x)A κ y (x, )C = (6) Φ() = C, Φ () = K κ(, )C (7) Athough the PDE (4) (5) the ODE (6) (7) are sti couped, they can be decouped soved expicity through some techniques of agebra anaytica mathematics First, the soution to the PDE (4) (5) is κ(x, y) = Φ(x y)b (8) Second, substituting (8) into (6) (7) respectivey, it is obtained that Φ (x) + Φ (x)bc Φ(x)A = (9) Φ () = K Φ()BC = K CBC Let I be an identity matrix, then the expicit soution to the ODE (6) (7) is obtained: Φ(x) = (C K CBC)e Dx I D = I A BC the expicit soution to the PDE (4) (5) is κ(x, y) = (C K CBC)e D(xy) I B The integra transformation (X, u) (X, w) defined by (9) () is invertibe Suppose the inverse transformation (X, w) (X, u) as the foowing form X(t) = X(t) () u(x, t) = w(x, t) + ι(x, y)w(y, t)dy + Ψ (x)x(t) () the kerne functions ι(x, y) R Ψ (x) T R n are to be determined Foowing the same procedure of determination of the kernes κ(x, y) Φ(x), compute the derivatives u x, u xx u t, a sufficient condition for ι(x, y) Ψ (x) to satisfy () (3) is obtained as ι xx (x, y) = ι yy (x, y) () d ι(x, x) =, dx ι(x, ) = Ψ (x)b (3) Ψ (x) Ψ (x)(a + BK) = (4) Ψ () = C, Ψ () = K (5) This cascade system can aso be soved expicity First, the expicit soution to the ODE (4) (5) is Ψ (x) = (C K)e Ex I E = A + BK I Second, the expicit soution to the PDE () (3) is ι(x, y) = Ψ (x y)b = (C K)e E(xy) I B

3 54 S Tang, C Xie / Systems & Contro Letters 6 () Thus, the direct inverse transformations are written as w(x, t) = u(x, t) u(x, t) = w(x, t) + Φ(x y)u(y, t)dyb Φ(x)X(t) (6) Ψ (x y)w(y, t)dyb + Ψ (x)x(t) (7) Evauating (6) at x =, by the boundary condition (4) (8), a controer is obtained as foows: U(t) = Φ( y)u(y, t)dyb + Φ()X(t) (8) Furthermore, the expicit soution to the cosed-oop system () (4) under the controer (8) can aso be obtained if the initia state (X(), u(x, )) is known First, the soution to the heat equation (6) (8) is w(x, t) = µ m = e m π w (ξ) sin t sin x µ m (9) ξ dξ (3) the initia condition w (x) is cacuated by the initia state u(x, ) through (6) Second, signa X(t) is obtained by X(t) = X()e (A+BK)t + t signa u(x, t) is obtained from () e (A+BK)(tτ) Bw x (, τ)dτ (3) Theorem For any initia data X() R u(, ) H (, ), the cosed-oop system consisting of the pant () (4) the contro aw (8) has a cassica soution, which is exponentiay stabiized in the sense of the norm (X(t), u(, t)) = X(t) + u(, t) H (,) denotes the Eucidean norm Proof Consider the foowing Lyapunov function V(t) = X T PX + a w(, t) L (,) + w x(, t) L (,) the matrix P = P T > is the soution to the Lyapunov equation P(A + BK) + (A + BK) T P = Q for some Q = Q T >, the parameter a > is to be chosen ater For simpicity, in the foowing proof, the symbo denotes the L (, ) norm From the backstepping transformations (6) (7), it can be obtained that w α u + α X (3) u β w + β X (33) w x α 3 u x + α 4 u + α 5 X (34) u x β 3 w x + β 4 w + β 5 X (35) α = 3( + ΦB ), α = 3 Φ β = 3( + Ψ B ), β = 3 Ψ α 3 = 4, α 4 = 4((CB) + Φ x B ), α 5 = 4 Φ β 3 = 4, β 4 = 4((CB) + Ψ x B ), β 5 = 4 Ψ Then, it can be obtained that δ( X + u H (,) ) V δ( X + u H (,) ) δ = max λ max (P) + aα + α 5, aα + α 4, α 3 a min δ =,, λ min(p) max{β 3, β + β 4, β + β 5 + } Cacuate the derivative of the Lyapunov function aong the soutions to the system (5) (8), then V λ min(q ) X + PB λ min (Q ) w x(, t) a w x w xx By Agmon s inequaity, it can be proved that the foowing inequaity hods: w xx + w x w x (, t) Thus V λ min(q ) X Now take w x (, t) a > PB λ min (Q ) + + a PB λ min (Q ) + w x then by Poincaré inequaity, it can be shown that V bv b = min Therefore, V(t) V()e bt Let δ = δ/δ, then λmin (Q ) λ max (P), PB + 4 aλ min (Q ) + a X(t) + u(, t) H (,) δ( X() + u(, ) H (,) )ebt hods for a t, which competes the proof 4 Observer output feedback controer Assume that ony u x (, t) is avaiabe for measurement, or for economic consideration, ony u x (, t) is measured To manipuate a contro for the system () (4), an observer is designed to reconstruct the state in the domain With Dirichet actuation, observer of the foowing form ˆX(t) = A ˆX(t) + Bux (, t) + P (u x (, t) û x (, t)) (36) û t (x, t) = û xx (x, t) + p (x)(u x (, t) û x (, t)), x (, ) (37) û(, t) = C ˆX(t) + p (u x (, t) û x (, t)) (38) û(, t) = U(t) (39) is considered, the constant vector P, the function p (x), the constant p are to be determined Write the observer error as ũ(x, t) = u(x, t) û(x, t), X(t) = X(t) ˆX(t)

4 S Tang, C Xie / Systems & Contro Letters 6 () then, to achieve exponentia stabiity of the observer error system X(t) = A X(t) P ũ x (, t) (4) ũ t (x, t) = ũ xx (x, t) p (x)ũ x (, t), x (, ) (4) ũ(, t) = C X(t) p ũ x (, t), ũ(, t) = (4) the foowing transformation w(x, t) = ũ(x, t) Θ(x) X(t) (43) is introduced to transform (4) (4) into the exponentiay stabe system X(t) = (A P Θ ()) X(t) P w x (, t) (44) w t (x, t) = w xx (x, t), x (, ) (45) w(, t) =, w(, t) = (46) Θ(x) is to be determined, P is chosen such that A P Θ () is a Hurwitz matrix By matching (4) (4) (44) (46), a sufficient condition is obtained: Θ (x) Θ(x)A = (47) Θ() = C, Θ() = (48) p (x) = Θ(x)P (49) p = (5) So, it is ony needed to sove the probem of differentia equation (47) (48) To construct the soution to the ODE (47) (48), a emma is shown first Lemma Write A F =, G = ( I)e F I I then G is a nonsinguar matrix if ony if the matrix A has no eigenvaues of the form k π / for k N Proof First, there exists an invertibe matrix H such that H AH is the Jordan s canonica form, that is H AH = diag(j J p ) each Jordan bock J q, q p, is a square matrix of owertrianguar type, a the eements on its main diagona are the eigenvaues of A, which are denoted by ς j, j =,,, n Second, a simpe cacuation gives that m+ G = (m + )! Am m= Thus L : = H m+ GH = (m + )! diag(j m J m p ) m= ( ς ) m (m + )! m= = ( ς n ) m (m + )! m= sinh(ς ) ς = sinh(ς n ) ς n / Therefore, matrix L is singuar if ony if ς j = kπi for some ς j, j n k N, i sts for the imaginary unit, namey, the square root of Thus, G is a nonsinguar matrix if ony if A has no eigenvaues of the form k π / for k N The soution to the Eqs (47) (48) can be represented by Θ(x) = (C Θ ())e Fx I (5) Especiay, for x =, it hods that (C Θ ())e F I = Θ() = When A has no eigenvaues of the form k π / for k N, it can be obtained that Θ () = C(I )e F I G Thus the expicit soution to the Eqs (47) (48) is Θ(x) = C C(I )e F I G e Fx I (5) Choose P such that AP Θ () is Hurwitz, then p (x) p are determined through (49) (5) Thus, a the quantities needed to impement the observer (36) (39) are determined The system (44) (46) is a cascade of the exponentiay stabe heat equation (45) (46) the exponentiay stabe ODE (44) The entire observer error system is exponentiay stabe Theorem Assume that the matrix A has no eigenvaues of the form k π / for k N, then the observer (36) (39) with gains defined through (49), (5) (5), guarantees that the observer error system is exponentiay stabe in the sense of the norm ( X(t), ũ(, t)) = X(t) + ũ(, t) H (,) that is, ˆX(t) û(t) exponentiay track X(t) u(t) in the sense of above norm Proof From the transformation (43), the foowing reations w ũ + Θ X, w x ũ x + Θ X ũ w + Θ X, ũ x w x + Θ X are obtained With the Lyapunov function Ṽ(t) = X T P X + ã w(, t) + w x(, t) P = P T > is the soution to the Lyapunov equation P(A P Θ ()) + (A P Θ ()) T P = Q for some Q = Q T >, it can be obtained that ϱ( X(t) + ũ(t) H (,) ) Ṽ ϱ( X(t) + ũ(t) H (,) ) min ã,, λ min( P) ϱ = max{, + ( Θ + ã Θ )/λ min ( P)} ϱ = max{ã,, Θ + ã Θ + λ max ( P)} Cacuate the time derivative of the Lyapunov function aong the soutions to the system (44) (46), then Ṽ λ min( Q ) X ã PP + w x λ min ( Q ) w x (, t)

5 544 S Tang, C Xie / Systems & Contro Letters 6 () the ast inequaity is obtained by Agmon s inequaity the foowing inequaity w xx + w x w x (, t) Take ã > PP + + λ min ( Q ) by Poincaré inequaity, then Ṽ bṽ λ min ( Q ) b = min, PP + λ max ( P) + 4 ãλ min ( Q ) ã Hence X(t) + ũ(, t) H (,) ϱ( X() + ũ(, ) H (,) )e bt > for a t with ϱ = ϱ/ϱ, which means that the error system (4) (4) is exponentiay stabe in the sense of the norm ( X(t), ũ(, t)) = X(t) + ũ(, t) H (,) thus competes the proof Repace u(y, t) X(t) by û(y, t) ˆX(t) in (8) respectivey, then an output feedback contro aw is obtained as foows U(t) = Φ( y)û(y, t)dyb + Φ() ˆX(t) (53) Theorem 3 Assume that the matrix A has no eigenvaues of the form k π / for k N, then for any initia data X(), ˆX() R u(, ), û(, ) H (, ), the cosed-oop system consisting of the pant () (4), the controer (53) the observer (36) (39) has a cassica soution which is exponentiay stabiized in the sense of the norm (X(t), u(, t), ˆX(t), û(, t)) = X(t) + u(, t) H (,) Proof The transformation ŵ(x, t) = û(x, t) transforms (36) (39) into the system + ˆX(t) + û(, t) H (,) Φ(x y)û(y, t)dyb Φ(x) ˆX(t) (54) ˆX(t) = (A + BK) ˆX(t) + Bŵx (, t) + (B + P )( w x (, t) + Θ () X(t)) (55) ŵ t (x, t) = ŵ xx (x, t) + (p (x) Φ(x)(B + P ) Φ(x y)p (y)dyb ( w x (, t) + Θ () X(t)), x (, ) (56) ŵ(, t) =, ŵ(, t) = (57) The ( X, w)-system (44) (46) the homogeneous part of the ( ˆX, ŵ)-system (55) (57) (without X(t), w(, t)) are exponentiay stabiized The interconnection of the two systems ( ˆX, ŵ, X, w) is a cascade The combined ( ˆX, ŵ, X, w)-system is exponentiay stabiized In fact, this fact can be proved through the weighted Lyapunov function E(t) = ˆX T ˆP ˆX + â ŵ(, t) + ŵ x(, t) + eṽ(t) (58) the matrix ˆP = T ˆP > is the soution to the Lyapunov equation ˆP(A + BK) + (A + BK) T ˆP = ˆQ for some ˆQ = ˆQ T >, the constant â the weighting constant e are to be chosen ater Cacuate the time derivative of (58), then Ė ˆX T ˆQ ˆX + ˆX T P(Bŵ x (, t) + (B + P )( w x (, t) Let + Θ () X(t))) â ŵx + â p (x) Φ(x)(B + P ) ŵ(x) Φ(x y)p (y)dyb ( w x (, t) + Θ () X(t))dx ŵxx + ŵ x (x) p (x) Φ (x)(b + P ) CBp (x) + e θ = max Φ (x y)p (y)dyb ( w x (, t) + Θ () X(t))dx λ min( Q ) X p (x) Φ(x)(B + P ) ã PP + w x λ min ( Q ) ϑ = max p (x) Φ (x)(b + P ) CBp (x) Φ (x y)p (y)dyb Φ(x y)p (y)dyb then by Poincaré, Agmon s Young inequaities after some compex cacuations, it can be obtained that Ė e ˆX e ŵ x e 3 X e 4 w x e = λ min( ˆQ ) ϵ P(B + P ), e = â 4 PB + λ min ( ˆQ ) e 3 = λ min( Q ) e ϵ + 4âθ 3 + ϑ e 4 = e ã PP + λ min ( Q ) 4âθ 3 ϑ Θ () 4 P(B + P ) λ min ( ˆQ ) Choose positive constants â ϵ such that â > 8 PB + 3 +, ϵ < λ min( ˆQ ) λ min ( ˆQ ) P(B + P ) further choose a positive constant e to satisfy e > λ min ( Q ) ϵ + 4âθ 3 + ϑ Θ () positive constant ã so that ã > PP P(B + P ) + 4âθ 3 + ϑ λ min ( Q ) e λ min ( ˆQ )

6 S Tang, C Xie / Systems & Contro Letters 6 () then through a engthy cacuation, it can be obtained that Ė fe, e e f = min, λ max (ˆP) â( + 4 ), e 3 e 4, eλ max ( P) eã( + 4 ) Hence, the ( ˆX, ŵ, X, w)-system is exponentiay stabiized Since the transformations (43) (54) are invertibe, exponentia stabiization of the ( ˆX, ŵ, X, w)-system ensures exponentia stabiization of the ( ˆX, û, X, ũ)-system This directy impies stabiization of the cosed-oop (X, u, ˆX, û)-system 5 Exampe As an exampe, consider the foowing scaar couped contro system Ẋ(t) = X(t) + u x (, t) (59) u t (x, t) = u xx (x, t) (6) u(, t) = X(t), u(, t) = U(t) (6) The state feedback controer, observer, output feedback controer soutions to the cosed-oop systems are derived 5 State feedback controer soutions The feedback gain is taken as K = such that A + BK is Hurwitz, then D =, Φ(x) = ( 3)e Dx the backstepping controer can be derived expicity through (8), which is U(t) = The target system is Φ( y)u(y, t)dy + Φ()X(t) (6) Ẋ(t) = X(t) + w x (, t) (63) w t (x, t) = w xx (x, t), w(, t) =, w(, t) = (64) Furthermore, the soution to the system (59) (6) (6) is expicity avaiabe Suppose an initia condition is u(x, ) = 5x X() = First, the expicit soution to the heat equation (64) is obtained by (9), µ m = 5 cos () m 4 π 4 + 3m π + ( 3)eD 5(m π 6) m 4 π 4 + 3m π + 5 4m π + 9 m π 5 Then, the soution to the cosed-oop system (59) (6) (6) can be obtained expicity from (3) (), which is X(t) = e t + u(x, t) = e t (cosh(ix) i sinh(ix)) + ν m = m π em π t (e (m π )t )µ m (65) e m π t µ m ν m (66) m π ((m π + ) sin(x) + cos(x) e (m π )t (cosh(ix) i sinh(ix))) From (65) (66), it is evident that X(t) u(x, t) exponentiay converges to zero as t tends to the infinity 5 Observer, output feedback soutions Here Θ () = coth, Θ(x) = coth sinh x cosh x Take P = tanh, then the backstepping observer is ˆX(t) = ˆX(t) + ux (, t) + tanh u x (, t) û x (, t) û t (x, t) = û xx (x, t) + (sinh x tanh cosh x)(u x (, t) û x (, t)) û(, t) = ˆX(t), û(, t) = φ( y)û(y, t)dy + Φ() ˆX(t) Taking the observer initia condition û(x, ) =, ˆX() = foowing the simiar steps as seeking for the soution to the cosedoop system in Section 5, the expicit soution to the resuting error system can aso be obtained as foows X(t) = e t + 4 tanh m π em π t ( e (m π )t ) µ m (67) ũ(x, t) = e t (cosh x coth sinh x) + e m π t µ m ν m (68) µ m = m π cos() ν m = sin(x) + (tanh cosh x sinh x) m π (e (m π )t ) From (67) (68), it is obvious that the error system is exponentiay stabiized 6 Comments Contro design of couped PDE ODE systems is an origina area There are many open probems to be considered This paper is just a beginning for studying the couped PDE ODE systems with interaction between the ODE the PDE Other couped PDE ODE systems with interaction between the ODE the PDE, such as the couped system consisting of an ODE a wave equation, are aso subjects of the ongoing research References [] X Zhang, E Zuazua, Contro, observation poynomia decay for a couped heat-wave system, C R Acad Sci Paris Ser I 336 (3) [] X Zhang, E Zuazua, Poynomia decay contro of a -d mode for fuid structure interaction, C R Acad Sci Paris Ser I 336 (3) [3] X Zhang, E Zuazua, Poynomia decay contro of a -d hyperboic paraboic couped system, J Differentia Equations 4 (4) [4] M Krstic, A Smyshyaev, Backstepping boundary contro for first order hyperboic PDEs appication to systems with actuator sensor deays, Systems & Contro Letters 57 (8) [5] M Krstic, Deay Compensation for Noninear, Adaptive, PDE Systems, Birkhauser, 9 [6] M Krstic, Compensating a string PDE in the actuation or in sensing path of an unstabe ODE, IEEE Transaction on Automatic Contro 54 (9) [7] M Krstic, Compensating actuator sensor dynamics governed by diffusion PDEs, Systems & Contro Letters 58 (9) [8] A Smyshyaev, M Krstic, Backstepping observers for a cass of paraboic PDEs, Systems & Contro Letters 54 (5) [9] GA Susto, M Krstic, Contro of PDE ODE cascades with Neumann interconnections, Journa of the Frankin Institute 347 () 84 34