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Zero dynamics modeling and boundary feedback design for parabolic systems
Mathematical and Computer Modelling 44 (2006) 857–869 www.elsevier.com/locate/mcm
Zero dynamics modeling and boundary feedback design for parabolic systems C.I. Byrnes a , D.S. Gilliam b,∗ , A. Isidori c , V.I. Shubov d a Electrical and Systems Engineering, Washington University, St. Louis, MO 63130, United States b Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409, United States c Dipart. di Info. e Sistemistica, Universit`a di Roma “La Sapienza”, 00184 Rome, Italy d Mathematical Sciences Department, University of Massachusetts Lowell, Lowell, MA 01854, United States
Received 3 February 2006; accepted 21 February 2006
Abstract In this work the authors introduce a notion of zero dynamics for distributed parameter systems governed by linear parabolic equations on bounded domains with controls implemented through first order linear boundary conditions. The idea of zero dynamics presented here is motivated by classical root-locus constructs from finite dimensional linear systems theory. In particular for a scalar proportional output feedback trajectories of the closed loop system converge to those of the zero dynamics system as the gain tends to infinity. The main contribution of this work is a simple modeling and design mechanism for solving certain tracking and disturbance rejection problems. This mechanism provides an alternative to the geometric approach involving regulator equations. Our approach is based on the fact that for certain boundary control systems it is very easy to model the system’s zero dynamics, which, in turn, provides a simple systematic methodology for solving certain problems of output regulation. As special cases we describe dynamic and static controllers from the associated zero dynamics system for set-point and harmonic tracking. In order to provide a simple roadmap to using this methodology we present numerical examples for a one dimensional heat equation. c 2006 Elsevier Ltd. All rights reserved.
1. Introduction The utility of zero dynamics design for nonlinear lumped systems is well documented. Indeed there is a considerable literature devoted to this topic. In a related area, there is also a vast literature in classical automatic control concerned with system zeros, e.g., root-locus design. There have also been a few research articles devoted to zero dynamics and (transmission and invariant) zeros for linear distributed parameter systems. Most of these papers have been concerned with systems having bounded input and output operators. In this work we are primarily interested in linear distributed parameter systems governed by parabolic partial differential equations.
∗ Corresponding author.
E-mail addresses: [email protected] (C.I. Byrnes), [email protected] (D.S. Gilliam), [email protected] (A. Isidori), victor [email protected] (V.I. Shubov). c 2006 Elsevier Ltd. All rights reserved. 0895-7177/$ - see front matter doi:10.1016/j.mcm.2006.02.020
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For distributed parameter system, in general, there is no well defined classical zero dynamics (see [27]) due to a variety of factors including the nonequivalence of different forms of invariance, nonequivalence of transmission and invariant zeros and, even more serious, difficulties that arise due to the presence of unbounded densely defined operators (see [13,27,28]). As an example, even in the simplest cases it may not make sense to write expressions like AB, C A, etc, for unbounded operators A, B and C. However, for systems governed by partial differential equations with certain types of boundary inputs and outputs there is a very natural concept of zero dynamics which (to our knowledge) was first introduced in the authors’ work [10] and has been exploited and developed by the authors for linear and nonlinear distributed parameter systems in several papers, see, for example, [8,6]. For boundary control systems governed by partial differential equations, this alternative concept of zero dynamics proves useful in developing regulator theory for distributed parameter systems with unbounded controls. The main objective of this work is to exploit this notion of zero dynamics to solve problems of output regulation for boundary control systems governed by parabolic partial differential equations. The specific problems of output regulation considered here are tracking and disturbance rejection. The motivation for our design methodology derives from two important points. The first is the well known utility of classical design methodology using a Proportional Error feedback control for stabilization. This idea has been examined in several works by the authors in [9–11,18]. The second motivation comes from the recent development of a theory of nonequilibrium output regulation by Byrnes and Isidori [12]. The methods used in [12] are particularly appealing in the case in which a well defined zero dynamics is available. In the present paper the boundary control systems we consider always provide a well defined zero dynamics. Actually, this methodology has been exploited in earlier work [6,8] in which we established a nonlinear enhancement of classical root-locus constructs by proving convergence of trajectories for the closed loop system (with proportional error feedback) to trajectories of the associated zero dynamics in the high gain limit. We also note that there have been several other works concerned with various things related to zeros or root locus for various types of distributed parameter systems, see, e.g., [1,23,27,28]. 2. The control system, notation, assumptions This work is concerned with a parabolic boundary control system given by z t (x, t) = Lz(x, t),
x ∈ Ω ⊂ Rn , t ≥ 0,
z(x, 0) = ϕ(x), ϕ ∈ Z = L (Ω ), (B j z)(x, t) = u j (x, t), j = 1, 2, . . . , p, x ∈ S j ,
(2.1)
2
(2.2)
where (B j ϕ)(x) = z(x, t) = 0,
∂ + a j (x) ϕ(x), ∂µ
j = 1, 2, . . . , p,
x ∈ S D , (B0 z)(x, t) ≡
∂ + a0 (x) z(x, t) = 0, x ∈ S0 . ∂µ
(2.3) (2.4)
We consider the above system acting in the state space Z = L 2 (Ω ) where Ω is a bounded domain in Rn . Further we assume that the boundary of Ω , denoted by ∂Ω , is piecewise C ∞ which means that Ω can be covered by a finite collection of open sets {Ω j }`j=1 so that if ∂Ω ∩ Ω j 6= ∅ then there exists a C ∞ -diffeomorphism which transforms Ω j into a unit ball Bn = {x ∈ Rn : |x| < 1}, Ω j ∩ Ω into a half-ball Bn = {x ∈ Bn : xn > 0}, and Ω j ∩ ∂Ω into the (n − 1)-dimensional unit ball Bn−1 = {x ∈ Bn : xn = 0}. We also assume that the boundary of Ω is represented as a union of connected hypersurfaces which are closed subsets of ∂Ω and whose interiors are pairwise disjoint (see Fig. 2.1). The (n − 2)-dimensional boundaries of these hypersurfaces are also piecewise C ∞ in the above sense. Here is the list of these hypersurfaces: 1. A region S D on which there are homogeneous Dirichlet boundary conditions. 2. A region S0 on which there are homogeneous Neumann or First Order Linear boundary conditions. p 3. A region SC = ∪ j=1 S j on which we introduce controls u j through Neumann or First Order Linear boundary conditions.
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Fig. 2.1. Region Ω with 3 control surfaces.
Thus we have p
∂Ω = S D ∪ S0 ∪ SC , SC = ∪ j=1 S j .
(2.5)
The operator L is assumed to be a formally self-adjoint uniformly elliptic differential operator: n X ∂ϕ ∂ ai j (x) − a(x)ϕ, Lϕ = ∂x j ∂ xi i, j=1
(2.6)
where a, ai j ∈ C ∞ (Ω ) (Ω is the closure of Ω ), ai j (x) = a ji (x), and a(x) ≥ 0 for all x ∈ Ω . By uniform ellipticity we mean that there exist constants 0 < c1 < c2 < ∞ so that n X c1 |ξ |2 ≤ ai j (x)ξi ξ j ≤ c2 |ξ |2 ∀ ξ ∈ Rn and
x ∈ Ω.
(2.7)
i, j=1
(Here | · | denotes the Euclidean norm in Rn .) The boundary operators in (2.3) are defined in terms of the co-normal derivative operator as follows: Let ν(x) = (ν1 (x), ν2 (x), . . . , νn (x)) denote the exterior unit normal vector at the point x ∈ ∂Ω and define the vector µ(x) = (µ1 (x), µ2 (x), . . . , µn (x))
where µ j (x) =
n X
ai j (x)νi (x).
i=1
Then we define the co-normal derivative on the boundary of Ω by ∂ ϕ = µ · ∇ϕ, ∂µ
(2.8)
where the right side denotes the dot product of µ with the gradient of ϕ. The functions a j ( j = 0, 1, . . . , p) can be extended to C ∞ functions on an open set containing Ω , moreover, a j (x) > 0 for all x ∈ Int(S j ) (where Int means “interior”) and a j (x) = 0 for all x ∈ ∂Ω \S j . Fact 2.1. It is well known (see [20,21]) that under the above assumptions the initial boundary problem (2.1)–(2.4) has a unique strong solution z. For any T > 0 we have z ∈ C([0, T ], L 2 (Ω )) ∩ L 2 ([0, T ], H 1 (Ω )). Moreover, for t > 0 this solution is instantly classical, which means in particular, that z(·, t) ∈ H s (Ω ) for any s > 0. Remark 2.1. Our assumptions about the coefficients of the operator L and the functions a j from (2.3) and (2.4) are somewhat stronger than necessary for all the constructions below to work. We have imposed the above strong requirements only to make the presentation more transparent and to avoid unnecessary technical complications.
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Let us introduce the operator A0 = L in Z with domain ( ) ∂ 2 D(A0 ) = ϕ ∈ H (Ω ) : ϕ|S D = 0, + a0 (x) ϕ = 0 . ∂µ S0
(2.9)
The operator A0 does not possess a full set of boundary conditions, in particular, there are no conditions imposed on the controlled portion of the boundary, SC . Control inputs: We define a boundary input operator > (Bz)(x, t) = (B1 z)(x, t) (B2 z)(x, t) · · · (B p z)(x, t) , j = 1, . . . , p (2.10) where each (B j z)(x, t) is defined in (2.3) for x ∈ S j and we assume that (B j z)(x, t) = 0 for x ∈ ∂Ω \S j . We use similar notation to introduce the vector of measured output, control input and reference signals which will be defined for all x ∈ ∂Ω . Measured output operator: Denote by γ j the trace operator γ j : H s (Ω ) → H s−1/2 (S j ), s > 1/2, j = 0, . . . , p. We define the measured output operator by > (Cz)(x, t) = (C1 z)(x, t) (C2 z)(x, t) · · · (C p z)(x, t) , (2.11) y j (x, t) = (C j z)(x, t) = γ j z(x, t), for x ∈ S j , y j (x, t) = 0,
for x ∈ ∂Ω \S j .
We then define the corresponding Measured output: y(x, t) = y1 (x, t) y2 (x, t)
···
(2.12)
> y p (x, t) ∈ R p .
(2.13)
In this work we seek control inputs u j (x, t) defined on S j ×(0, ∞) and the corresponding control input vector defined by Control inputs: > u(x, t) = u 1 (x, t) u 2 (x, t) · · · u p (x, t) ∈ R p , x ∈ ∂Ω , (2.14) where it is assumed that u j (x, t) = 0 for x ∈ x ∈ ∂Ω \S j . As we have already mentioned in Fact 2.1 for the parabolic systems considered here and t > 0 the solution z to (2.1)–(2.4) is smooth and y j (x, t) = z(x, t),
for x ∈ S j , j = 1, 2, . . . , p.
(2.15)
In this work we are primarily concerned with output regulation problems involving tracking given reference signals. Reference signals: We assume that there are p reference signals w j (t) where w j ∈ Cb1 ([0, ∞]), i.e., w j are infinitely differentiable functions with w j and dw/dt bounded on [0, ∞). We are also given p shape functions q j (x) defined on ∂Ω satisfying q j ∈ C 1 (S j ),
supp(q j ) ⊂ S j .
The reference output vector is then given by yr (x, t) = y1,r (x, t) y2,r (x, t) · · ·
> y p,r (x, t)
where y j,r (x, t) = q j (x)w j (t).
(2.16)
Tracking error: Next we define the tracking error by e(x, t) = y(x, t) − yr (x, t),
so that e j = y j − y j,r .
(2.17)
We are now in a position to state our main problem, Regulation problem: Find control input u in Eq. (2.14) so that the error defined in (2.17) satisfies t→∞
ke(·, t)k H s−1/2 (SC ) → 0
for some s > 1/2,
(2.18)
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where ke(·, t)k2H s−1/2 (S
C)
=
p X
ke j (·, t)k2H s−1/2 (S ) . j
j=1
With the above notation the system (2.1)–(2.4) with outputs defined in (2.11) and (2.13) can be written in the compact abstract form as z˙ = A0 z
(2.19)
z(0) = ϕ Bz(t) = u(t)
(2.20)
y(t) = Cz(t).
(2.21)
At this point we introduce the uncontrolled open loop system dynamics for (2.1)–(2.4) by defining the operator A to be the operator L with the domain further restricted as D(A) = {ϕ ∈ D(A0 ) : B(ϕ) = 0}. Here the boundary operator B defined in (2.3) and (2.14) acts on functions defined on the controlled portion of the boundary SC given in (2.5). For the sake of clarity we write the domain of A explicitly as ) ( ∂ 2 + a j ϕ = 0, j = 0, 1, . . . , p . (2.22) D(A) = ϕ ∈ H (Ω ) : ϕ|S D = 0, ∂µ Sj We note, once again, that in this work we could have significantly relaxed various conditions but at the expense of extended length. Our main goal is rather to provide a fairly general example as a road map to interested readers who want to apply the basic design methodology to other problems. Fact 2.2. 1. The operator A, with domain defined in (2.22), generates an exponentially stable semigroup in L 2 (Ω ) and H s (Ω ) for all s > 0. A is a self-adjoint, uniformly elliptic operator with C ∞ (Ω ) eigenfunctions {ϕ j } and associated eigenvalues {λ j } satisfying 0 > λ1 > λ2 > · · · ,
lim λ j = −∞.
j→∞
All eigenvalues are negative if either one of the following conditions hold: (a) The (n − 1)-dimensional measure, |S D | > 0 and a j (x) ≥ 0 for all j = 0, 1, . . . , p, a(x) ≥ 0. (b) The |S D | = 0 and a j (x) ≥ 0 for all j = 0, 1, 2, . . . , p and for some j0 , |S j0 | > 0 and a j0 > 0 on S j0 and a(x) ≥ 0. 2. The operator A generates an exponentially stable analytic semigroup, denoted by T (t) in L 2 (Ω ), as well as, in H s (Ω ) for s > 0 (and in particular for s > [n/2] + 1 which guarantees the embedding of H s (Ω ) into C(Ω )). The solution to the uncontrolled problem (2.19)– (2.21) with u ≡ 0 (smooth in x and t for t > 0) is given by z(t) = T (t)ϕ. 3. z(·, t) for t > 0 is contained in C ∞ (Ω ) so that the trace on ∂Ω is a continuous function on ∂Ω . The output operator C is continuous from L 2 (Ω ) to H s−1/2 (∂Ω ) for s > [n/2] + 1. The following estimate holds for some positive constant α and positive continuous function Ms (t0 ) for t0 > 0 so that kT (t)ϕk H s (∂ Ω ) ≤ Ms (t0 )e−αt kϕk, where ϕ ∈ Z =
L 2 (Ω ),
∀ t ≥ t0 > 0,
k · k denotes the norm in
L 2 (Ω )
(2.23)
and lim Ms (t0 ) = ∞. t0 →0
4. There is a positive constant α and a positive continuous function Cs (t0 ) for t0 > 0 so that kC(T (t)ϕ)k H s−1/2 (∂ Ω ) ≤ Cs (t0 )e−αt kϕk
∀ t ≥ t0 > 0,
where ϕ ∈ Z = L 2 (Ω ) and lim Cs (t0 ) = ∞. t0 →0
(Note that α, Ms (t), and Cs (t), above all depend on s > 0.)
(2.24)
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3. The zero dynamics For the system (2.19)–(2.21) defined via (2.1)–(2.4) we now define the “zero dynamics” as the system obtained by constraining the output to be zero, i.e., we require Cz = 0 for all time. The control u = B(z) thus obtained is the b to be the operator L with domain control that maintains zero output for all time. Thus we define the operator A b = {ϕ ∈ (D(A0 )) : C(ϕ) = 0}. D( A)
(3.1)
This translates explicitly into ( ) ∂ 2 b = ϕ ∈ H (Ω ) : ϕ|S ∪S = 0, D( A) + a0 (x) ϕ = 0 . D C ∂µ S0
(3.2)
Thus the zero dynamics is given by b ξ˙ = Aξ
(3.3)
ξ(0) = ψ. Remark 3.1. Note that according to our definition of the zero dynamics the term (2.20) involving B(·) does not occur b see (3.1). in (3.3). It has been replaced by the constraint Cz = 0 which is built into (3.3) through the definition of A, b is a strictly negative, Just as with the uncontrolled problem, described above, our assumptions imply that A 2 b(t) unbounded self-adjoint operator in L (Ω ) that generates an exponentially stable analytic semigroup, denoted by T 2 s in L (Ω ), as well as, in H (Ω ) for all s > 0. The solution to the zero dynamics problem (3.3) (smooth in x and t for t > 0) is given by b(t)ψ ξ(t) = T bs (t0 ) for t0 > 0 so that and there is a positive constant b α and a positive continuous function C αt b(t)ψ)k H s−1/2 (∂ Ω ) ≤ C bs (t0 )e−b kC(T kψk
∀ t > t0 > 0,
(3.4)
bs (t0 ) = ∞. (Note that b bs (t) above depend on s > 0). where ψ ∈ Z = L 2 (Ω ), and lim C α and C t0 →0
At least formally, the system (3.3) provides a very natural definition of zero dynamics within the context of boundary feedback control problems for distributed parameter systems governed by partial differential equations. Indeed, if in (2.19)–(2.21) we employ a proportional error feedback law u = −ky.
(3.5)
then we obtain a closed loop system w˙ = Ak w
(3.6)
w(0) = ϕ where the closed loop spatial operator Ak is given by the operator A with domain D(Ak ) = {φ ∈ D(A0 ) : B(φ) + kC(φ) = 0}. For this closed loop system, we see that the uncontrolled problem is obtained when k = 0 and in the high gain limit, at least at a formal level, we obtain the zero dynamics (3.3) as k tends to infinity. This high gain limit can, in fact, be rigorously justified by techniques similar to those used in [6]. Within this context, we could also define the open loop system (2.19)–(2.21) to be minimum phase provided that (3.3) is asymptotically stable. Of course this definition has little meaning unless we can establish some type of concept of left half plane transmission zeros as in the finite dimensional linear case. The necessary generalization is exactly the notion of “zero-pole dynamics” (discussed above) for the closed loop system (2.19)–(2.21). Namely, for a given example, we would expect to prove that the zero dynamics is asymptotically stable and then prove that as we increase
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the gain parameter k, the trajectories of the closed-loop system (3.6) approach the trajectories of the zero dynamics (3.3). Although for some examples this definition of zero dynamics will be different from the one obtained via the invariance concepts and zero dynamics algorithm, at least in the linear self-adjoint case considered here and under certain other technical assumptions the point spectrum of the resulting zero dynamics should be the transmission zeros of the original system. 4. Dynamic controller for boundary control problems In this section we construct a dynamic controller based on the “zero dynamics.” For the boundary control system (2.19)–(2.21) with zero dynamics given in (3.3) define a dynamic controller by replacing the homogeneous constraint Cξ = 0 with the non-homogeneous constraint Cz = yr . The zero dynamics provides the desired dynamic controller. ξt = A0 ξ,
(4.1)
ξ(0) = ψ, Cξ(t) = yr ,
(4.2) (4.3)
u(t) = Bξ(t).
(4.4)
Now we are prepared to prove the following proposition concerning the regulation problem considered here. Proposition 4.1. For the system (2.19)–(2.21), the problem of output regulation (for all initial data ϕ) is solved by the controller given by (4.1)–(4.3) with u in (4.4). The proof of Proposition 4.1 is very simple but it gives a very elementary practical means to construct controllers for boundary controlled distributed parameter systems. Proof. Let u(t) = Bξ(t) as in (4.4) and define η(t) = z(t) − ξ(t) where z is the solution to (2.19)–(2.21) and ξ the solution to (4.1) and (4.2). Then it is easy to show that η satisfies ηt = Aη,
(4.5)
η(0) = ϕ − ψ ≡ η0 ,
(4.6)
e(t) = Cη(t).
(4.7)
The result now follows from Fact 2.1. Indeed, it is well known (cf., [20,21,14]) that η is a continuous (in x and t for t > 0) globally defined solution which satisfies t→∞
sup |η(x, t)| −→ 0. x∈Ω
More precisely, η(·, t) ∈ H s (Ω ) for s > [n/2] + 1 which implies η(·, t) ∈ C(Ω ) and our result follows from (2.23). Namely, we have kη(·, t)k H s (Ω ) ≤ Ms (t0 )e−αt kη0 k,
for some α > 0, t ≥ t0 > 0
and the result follows from (4.7) the estimate (2.24) and the Sobolev Embedding Theorem. 5. Static feedback control: Two SISO examples It is often possible to simplify the controller by replacing the dynamic controller (4.1)–(4.4) by a simple feedback law derived from the steady state locus (cf. [2,12]). These “static” control laws are similar to those obtained by solving the “regulator equations,” as described in [3]. Note that here the input and output operators are unbounded (densely defined) operators in the Hilbert state space so the results of [3] are not directly applicable. For the parabolic problems considered here the steady state response of the zero dynamics typically consists of a single (generally time dependent) asymptotically stable steady state, ξ 0 (x, t). Furthermore, under smoothness assumptions on the signals to be tracked, the convergence to this steady state takes place in C 1 (Ω ). This can be exploited to obtain controls u j (x, t) = B j ξ 0 (x, t) for x ∈ S j .
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For set-point control the signals to be tracked are time independent and the steady state response is also time independent, ξ 0 (x, t) = ξ 0 (x). In this case the resulting controls are given by u j (x) = B j ξ 0 (x) for x ∈ S j . Rather than develop a lengthy discussion of special cases that arise in carrying out the specific analysis, in this section we will present two simple single input single output (SISO) examples (set point control and tracking a sinusoid) that serve to illustrate this methodology for the one dimensional heat equation on a unit interval. Thus we consider the system z t (x, t) = z x x (x, t),
x ∈ [0, 1], t > 0
(B0 z)(t) = z x (0, t) − k0 z(0, t) = 0,
(5.1)
k0 > 0
(B1 z)(t) = z x (1, t) = u(t) z(x, 0) = ϕ(x). The measured output is y(t) = (Cz)(t) = z(1, t). In this case we have d2 , D(A0 ) = {ϕ ∈ H 2 (0, 1) : B0 (ϕ) = 0}, dx 2 d2 dϕ (1) = 0 , A = 2 , D(A) = ϕ ∈ H 2 (0, 1) : B0 (ϕ) = 0. dx dx A0 =
2 b= d , A dx 2
b = {ϕ ∈ H 2 (0, 1) : B0 (ϕ) = 0. ϕ(1) = 0}. D( A)
Given a reference signal yr (t) we want to find u(t) so that the error, defined by e(t) = y(t) − yr (t), goes to zero as t goes to infinity. The design of control laws solving problems of this type has received considerable attention in the literature. A number of authors have extended the finite dimensional geometric methods of Francis [15], Wonham [26], Francis and Wonham [16] and Hautus [17] to the infinite dimensional case including Pohjolainen [24] and Schumacher [25]. In another recent work [3] the authors adapt the ideas of Byrnes and Isidori [19] to characterize solvability of the regulator problem for distributed parameter problems with bounded input and output operators in terms of the solvability of a pair of regulator equations. More specifically these works were concerned with output regulation for systems in the standard form dz (t) = Az(t) + Bu(t), dt dw (t) = Sw(t) dt z(0) = z 0 , w(0) = w0 y(t) = C z(t), yr = Qw,
(5.2) (5.3) (5.4) (5.5)
e(t) = y(t) − yr (t). where B and C are bounded operators and w is the state of the exosystem in a finite dimensional Hilbert space W . Under suitable hypotheses (e.g. A is exponentially stable and S has all its eigenvalues on the imaginary axis) it can be shown that the feedback law u = Γ w solves the problem of output regulation where Γ is obtained as part of the solution to the regulator equations Π S = AΠ + BΓ
(5.6)
CΠ = Q.
(5.7)
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Here Π : W → D(A) ⊂ Z is a bounded linear operator and in the SISO case Γ : W → R is also a linear map. Thus if dim(W ) = p then Γ is a p × 1 constant vector Γ = γ1 , γ2 , . . . , γ p . These techniques have been extended to the unbounded case in certain situations, see for example [4,5]. Example 5.1 (SISO Set-Point Control). For set point control the objective is to have the output y track a prescribed constant M. Thus for SISO set-point control yr (t) = M and we have S = 0, Q = 1. In this case it is easily verified that the regulator equations lead to the formula u = Γ w = G(0)−1 M
(5.8)
where G(s) is the system’s transfer function, G(s) = C(s I − A)−1 B,
s ∈ ρ(A),
(5.9)
(here ρ(A) is the resolvent set of A). For the problem (5.1) we can, after some straightforward calculations, compute the transfer function to obtain √ √ √ s cosh( s) + k0 sinh( s) G(s) = (5.10) √ √ √ . s sinh( s) + k0 s cosh( s) Knowing that a control can be obtained in the form u = Γ M for some Γ ∈ R we consider finding Γ using our zero dynamics methodology. The zero dynamics for the system (5.1) (see (4.1)–(4.3)) is the system ∂ξ ∂ 2ξ (x, t) = 2 (x, t), ξ(x, 0) = ψ(x), ∂t ∂x ∂ξ (0, t) − k0 ξ(0, t) = 0, ∂x ξ(1, t) = M,
(5.11)
and the steady state locus consists of a single stationary solution ξ 0 ∈ D(A0 ). Namely, d2 ξ 0 = 0, dx 2
dξ 0 (0) − k0 ξ 0 (0) = 0, dx
ξ 0 (1) = M,
(5.12)
implies ξ0 =
k0 x + 1 . k0 + 1
Therefore the static feedback law u 0 = B(ξ 0 )M =
dξ 0 (1)M = M dx
solves the problem of output regulation just as the static feedback (5.8), obtained from the regulator equations, does. We notice also that from (5.10) that G(0) = 1. Indeed, in several examples we have verified that both methods provide the same control, i.e., B(ξ0 ) = G(0)−1 . Example 5.2 (SISO Harmonic Tracking). For harmonic tracking the output y in (2.15) is required to track a sinusoid yr (t) = M sin(αt). In this case we can take the exosystem in (5.3) to be the harmonic oscillator written as the two dimensional system 0 α 0 w˙ = Sw = , w(0) = . −α 0 M
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We have w1 (t) M sin(αt) w(t) = = . w2 (t) M cos(αt)
Therefore yr (t) = yr = [1, 0]w = M sin(αt). Once again we can solve the regulator equations as in [3] in terms of the system’s transfer function, given in (5.9) and we have h i u = Re G(iα)−1 , Im G(iα)−1 w (5.13) where, once again, G(s) is the transfer function (see (5.9)) and w = [M sin(αt), M cos(αt)]> . Just as in Example 5.1 we can compute the steady state locus of the zero dynamics in terms of the special stationary solution ξ0 in (5.12). For harmonic tracking the procedure can be carried out quite generally using semigroup theory (see, for example, [22], page 29, Corollary 7.5) or using Laplace transform type techniques like those found in [7]. Here we will not include these details which are quite straightforward. Motivated by the fact that we know from (5.13) that a control can be obtained in terms of a pair of gain parameters γ1 and γ2 our objective here is to obtain these parameters using the steady state response of the zero dynamics (5.11). It is easy to show that the solution to the zero dynamics consists of a sum of decaying transient and a particular periodic motion. After some straightforward calculations based on an analysis of the solution given by the variation of parameters formula, the desired control law can be written in terms of operators applied to the stationary solution ξ0 as follows: bw, u=Γ
b = [b Γ γ1 , b γ2 ]
(5.14)
b2 + α 2 I )−1 ξ0 γ1 = B ξ0 − α 2 B( A b b −1 ξ0 , = Bξ0 + α Im B(iα I − A) bA b2 + α 2 I )−1 ξ0 γ2 = αB A( b b −1 ξ0 . = −α 2 Re B(iα I − A)
(5.15)
where
(5.16) (5.17) (5.18)
In (5.14) we denote by [b γ1 , b γ2 ] the 1 × 2 matrix with entries b γ1 and b γ2 . From (5.15) and (5.17) we need to find b2 + α 2 )−1 ξ0 f ≡ (A which is equivalent to f 0000 + α 2 f = ξ0 , subject to the end conditions f 0 (0) − k0 f (0) = 0,
f (1) = 0,
f 000 (0) − k0 f 00 (0) = 0,
Writing the equation as a first order system we set f 1 = f,
f2 = f 0,
f 3 = f 00 ,
f 4 = f 000
f 00 (1) = 0.
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Fig. 5.1. Solution surface.
Fig. 5.2. Output y(t) (solid) and reference signal yr (t) (dashed).
and consider f2 f1 f1 f2 d f f 3 2 = f3 f4 dx f 3 2 f4 f4 −α f 1 + ξ0
x ∈ [0, 1]
with boundary conditions f 2 (0) − k0 f 1 (0) = 0,
f 1 (1) = 0,
f 4 (0) − k0 f 3 (0) = 0,
f 3 (1) = 0.
Then we obtain from (5.15) and (5.17) γ1 = ξ00 (1) − α f 2 (1), b
γ2 = α f 3 (1) b
so that the desired control is easily computed numerically or we could use the formula given in (5.13). In Fig. 5.1 we have plotted the controlled solution surface and in Fig. 5.2 we have plotted the reference signal and the controlled output. Finally we have plotted the error e(t) = y(t) − yr (t) in Fig. 5.3. In the numerical example given below we solve for the control parameters b g1 and b g2 using the Matlab built-in Boundary Value Solver bvp4c. In the numerical simulation we have set k0 = 1, M = 2, α = 2, z(x, 0) = ϕ(x) =
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Fig. 5.3. The error e(t) = y(t) − yr (t).
2 cos(π x), ξ(x, 0) = ψ(x) = 0 and solved the problem on the time interval 0 < t < 10. In the case we have γ1 = 0.8356 and γ2 = 1.0102 and the values of b g1 and b g2 computed using Matlab agree with these values to six decimal places. In the first figure we have plotted the solution surface for the controlled system. References [1] S.P. Banks, F. Abbasi-Ghelmansarai, Realization theory and the infinite-dimensional root locus, Internat. J. Control 38 (3) (1983) 589–606. [2] C.I. Byrnes, D.S. Gilliam, A. Isidori, J. Ramsey, On the steady-state behavior of forced nonlinear systems, in: New Trends in Nonlinear Dynamics and Control, and their Applications, in: Lecture Notes in Control and Inform. Sci., vol. 295, Springer, Berlin, 2003, pp. 119–143. [3] C.I. Byrnes, D.S. Gilliam, I.G. Lauk´o, V.I. Shubov, Output regulation for linear distributed parameter systems, IEEE Trans. Automat. Control 45 (12) (2000) 2236–2252. [4] C.I. Byrnes, D.S. Gilliam, V.I. Shubov, G. Weiss, The output regulator problem for regular linear systems, Texas Tech University, 2000 (preprint). [5] C.I. Byrnes, D.S. Gilliam, V.I. Shubov, Example of output regulation for a system with unbounded inputs and outputs, in: Proceedings 38th IEEE Conf. Dec. Control, 1999, pp. 4280–4284. [6] C.I. Byrnes, D.S. Gilliam, V.I. Shubov, Boundary control, stabilization and zero-pole dynamics for a nonlinear distributed parameter system, J. Robust Nonlinear Control 9 (1999) 737–768. [7] C.I. Byrnes, D.S. Gilliam, I.G. Lauk´o, V.I. Shubov, Harmonic forcing for linear distributed parameter systems, J. Math. Systems Estim. Control 8 (2) (1998). [8] C.I. Byrnes, D.S. Gilliam, V.I. Shubov, High gain limits of trajectories and attractors for a boundary controlled viscous Burgers’ equation, J. Math. Systems Estim. Control 6 (4) (1996) 40. [9] C.I. Byrnes, D.S. Gilliam, J. He, Root-locus and boundary feedback design for a class of distributed parameter systems, SIAM J. Control Optim. 32 (5) (1994) 1364–1427. [10] C.I. Byrnes, D.S. Gilliam, Asymptotic properties of root locus for distributed parameter systems, in: Proceedings of the 27th Conference on Decision and Control, Austin, Texas, December 1988, pp. 45–51. [11] C.I. Byrnes, D.S. Gilliam, J. He, A root locus methodology for parabolic distributed parameter feedback systems, in: Computation and Control, II (Bozeman, MT, 1990), in: Progr. Systems Control Theory, vol. 11, Birkh¨auser Boston, Boston, MA, 1991, pp. 63–83. [12] C.I. Byrnes, A. Isidori, Limit sets, zero dynamics, and internal models in the problem of nonlinear output regulation, IEEE Trans. Automat. Control AC-48 (2003) 1712–1723. [13] R.F. Curtain, H.J. Zwart, An Introduction to Infinite-Dimensional Linear Systems, Springer-Verlag, New York, 1995. [14] Y.V. Egorov, M.A. Shubin, Foundations of the classical theory of partial differential equations, Springer, 1991 (Second printing 1998). [15] B.A. Francis, The linear multivariable regulator problem, SIAM J. Control Optim. 14 (1977) 486–505. [16] B.A. Francis, W.M. Wonham, The internal model principle of control theory, Automatica 12 (1976) 457–465. [17] M. Hautus, Linear matrix equations with applications to the regulator problem, in: I.D. Landau (Ed.), Outils and Modeles Mathematique pour l’Automatique, C.N.R.S., Paris, 1983, pp. 399–412. [18] J. He, Root locus for control systems with completely separated boundary conditions, in: Computation and Control, III (Bozeman, MT, 1992), in: Progr. Systems Control Theory, vol. 15, Birkh¨auser Boston, Boston, MA, 1993, pp. 229–239. [19] A. Isidori, C.I. Byrnes, Output regulation of nonlinear systems, IEEE Trans. Automat. Control AC-35 (1990) 131–140. [20] O.A. Ladyzhenskaya, V.A. Solonnikov, N.N. Ural’ceva, Linear and Quasilinear Equations of Parabolic Type, in: Translations of AMS, vol. 23, 1968.
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[21] O.A. Ladyzhenskaya, The Boundary Value Problems of Mathematical Physics, in: Applied Mathematical Sciences, vol. 49, Springer-Verlag, 1985. [22] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. [23] S.A. Pohjolainen, N.H. Koivo, Properties and calculation of transmission zeros for distributed parameter systems, optimization techniques, in: Proc. Ninth IFIP Conf., Warsaw, 1979, Part 1, in: Lecture Notes in Control and Information Sci., vol. 22, Springer, Berlin, New York, 1980, pp. 431–438. [24] S.A. Pohjolainen, On the asymptotic regulation problem for distributed parameter systems, Proc. Third Symposium on Control of Distributed Parameter Systems, Toulouse, France, July 1982. [25] J.M. Schumacher, Finite-dimensional regulators for a class of infinite dimensional systems, Systems Control Lett. 3 (1983) 7–12. [26] W.M. Wonham, Linear Multivariable Control: A Geometric Approach, 2nd edition, Springer Verlag, 1979. [27] H.J. Zwart, Geometric Theory for Infinite Dimensional Systems, Ph.D. Thesis, Rijksuniversiteit Groningen, 1988 (LNCIS, vol. 115, SpringerVerlag, 1989). [28] H.J. Zwart, M.B. Hof, Zeros of Infinite-Dimensional Systems, IMA J. Math. Control. Inform. 14 (1997) 85–94.
Zero dynamics modeling and boundary feedback design for parabolic systems C.I. Byrnes a , D.S. Gilliam b,∗ , A. Isidori c , V.I. Shubov d a Electrical and Systems Engineering, Washington University, St. Louis, MO 63130, United States b Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409, United States c Dipart. di Info. e Sistemistica, Universit`a di Roma “La Sapienza”, 00184 Rome, Italy d Mathematical Sciences Department, University of Massachusetts Lowell, Lowell, MA 01854, United States
Received 3 February 2006; accepted 21 February 2006
Abstract In this work the authors introduce a notion of zero dynamics for distributed parameter systems governed by linear parabolic equations on bounded domains with controls implemented through first order linear boundary conditions. The idea of zero dynamics presented here is motivated by classical root-locus constructs from finite dimensional linear systems theory. In particular for a scalar proportional output feedback trajectories of the closed loop system converge to those of the zero dynamics system as the gain tends to infinity. The main contribution of this work is a simple modeling and design mechanism for solving certain tracking and disturbance rejection problems. This mechanism provides an alternative to the geometric approach involving regulator equations. Our approach is based on the fact that for certain boundary control systems it is very easy to model the system’s zero dynamics, which, in turn, provides a simple systematic methodology for solving certain problems of output regulation. As special cases we describe dynamic and static controllers from the associated zero dynamics system for set-point and harmonic tracking. In order to provide a simple roadmap to using this methodology we present numerical examples for a one dimensional heat equation. c 2006 Elsevier Ltd. All rights reserved.
1. Introduction The utility of zero dynamics design for nonlinear lumped systems is well documented. Indeed there is a considerable literature devoted to this topic. In a related area, there is also a vast literature in classical automatic control concerned with system zeros, e.g., root-locus design. There have also been a few research articles devoted to zero dynamics and (transmission and invariant) zeros for linear distributed parameter systems. Most of these papers have been concerned with systems having bounded input and output operators. In this work we are primarily interested in linear distributed parameter systems governed by parabolic partial differential equations.
∗ Corresponding author.
E-mail addresses: [email protected] (C.I. Byrnes), [email protected] (D.S. Gilliam), [email protected] (A. Isidori), victor [email protected] (V.I. Shubov). c 2006 Elsevier Ltd. All rights reserved. 0895-7177/$ - see front matter doi:10.1016/j.mcm.2006.02.020
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For distributed parameter system, in general, there is no well defined classical zero dynamics (see [27]) due to a variety of factors including the nonequivalence of different forms of invariance, nonequivalence of transmission and invariant zeros and, even more serious, difficulties that arise due to the presence of unbounded densely defined operators (see [13,27,28]). As an example, even in the simplest cases it may not make sense to write expressions like AB, C A, etc, for unbounded operators A, B and C. However, for systems governed by partial differential equations with certain types of boundary inputs and outputs there is a very natural concept of zero dynamics which (to our knowledge) was first introduced in the authors’ work [10] and has been exploited and developed by the authors for linear and nonlinear distributed parameter systems in several papers, see, for example, [8,6]. For boundary control systems governed by partial differential equations, this alternative concept of zero dynamics proves useful in developing regulator theory for distributed parameter systems with unbounded controls. The main objective of this work is to exploit this notion of zero dynamics to solve problems of output regulation for boundary control systems governed by parabolic partial differential equations. The specific problems of output regulation considered here are tracking and disturbance rejection. The motivation for our design methodology derives from two important points. The first is the well known utility of classical design methodology using a Proportional Error feedback control for stabilization. This idea has been examined in several works by the authors in [9–11,18]. The second motivation comes from the recent development of a theory of nonequilibrium output regulation by Byrnes and Isidori [12]. The methods used in [12] are particularly appealing in the case in which a well defined zero dynamics is available. In the present paper the boundary control systems we consider always provide a well defined zero dynamics. Actually, this methodology has been exploited in earlier work [6,8] in which we established a nonlinear enhancement of classical root-locus constructs by proving convergence of trajectories for the closed loop system (with proportional error feedback) to trajectories of the associated zero dynamics in the high gain limit. We also note that there have been several other works concerned with various things related to zeros or root locus for various types of distributed parameter systems, see, e.g., [1,23,27,28]. 2. The control system, notation, assumptions This work is concerned with a parabolic boundary control system given by z t (x, t) = Lz(x, t),
x ∈ Ω ⊂ Rn , t ≥ 0,
z(x, 0) = ϕ(x), ϕ ∈ Z = L (Ω ), (B j z)(x, t) = u j (x, t), j = 1, 2, . . . , p, x ∈ S j ,
(2.1)
2
(2.2)
where (B j ϕ)(x) = z(x, t) = 0,
∂ + a j (x) ϕ(x), ∂µ
j = 1, 2, . . . , p,
x ∈ S D , (B0 z)(x, t) ≡
∂ + a0 (x) z(x, t) = 0, x ∈ S0 . ∂µ
(2.3) (2.4)
We consider the above system acting in the state space Z = L 2 (Ω ) where Ω is a bounded domain in Rn . Further we assume that the boundary of Ω , denoted by ∂Ω , is piecewise C ∞ which means that Ω can be covered by a finite collection of open sets {Ω j }`j=1 so that if ∂Ω ∩ Ω j 6= ∅ then there exists a C ∞ -diffeomorphism which transforms Ω j into a unit ball Bn = {x ∈ Rn : |x| < 1}, Ω j ∩ Ω into a half-ball Bn = {x ∈ Bn : xn > 0}, and Ω j ∩ ∂Ω into the (n − 1)-dimensional unit ball Bn−1 = {x ∈ Bn : xn = 0}. We also assume that the boundary of Ω is represented as a union of connected hypersurfaces which are closed subsets of ∂Ω and whose interiors are pairwise disjoint (see Fig. 2.1). The (n − 2)-dimensional boundaries of these hypersurfaces are also piecewise C ∞ in the above sense. Here is the list of these hypersurfaces: 1. A region S D on which there are homogeneous Dirichlet boundary conditions. 2. A region S0 on which there are homogeneous Neumann or First Order Linear boundary conditions. p 3. A region SC = ∪ j=1 S j on which we introduce controls u j through Neumann or First Order Linear boundary conditions.
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Fig. 2.1. Region Ω with 3 control surfaces.
Thus we have p
∂Ω = S D ∪ S0 ∪ SC , SC = ∪ j=1 S j .
(2.5)
The operator L is assumed to be a formally self-adjoint uniformly elliptic differential operator: n X ∂ϕ ∂ ai j (x) − a(x)ϕ, Lϕ = ∂x j ∂ xi i, j=1
(2.6)
where a, ai j ∈ C ∞ (Ω ) (Ω is the closure of Ω ), ai j (x) = a ji (x), and a(x) ≥ 0 for all x ∈ Ω . By uniform ellipticity we mean that there exist constants 0 < c1 < c2 < ∞ so that n X c1 |ξ |2 ≤ ai j (x)ξi ξ j ≤ c2 |ξ |2 ∀ ξ ∈ Rn and
x ∈ Ω.
(2.7)
i, j=1
(Here | · | denotes the Euclidean norm in Rn .) The boundary operators in (2.3) are defined in terms of the co-normal derivative operator as follows: Let ν(x) = (ν1 (x), ν2 (x), . . . , νn (x)) denote the exterior unit normal vector at the point x ∈ ∂Ω and define the vector µ(x) = (µ1 (x), µ2 (x), . . . , µn (x))
where µ j (x) =
n X
ai j (x)νi (x).
i=1
Then we define the co-normal derivative on the boundary of Ω by ∂ ϕ = µ · ∇ϕ, ∂µ
(2.8)
where the right side denotes the dot product of µ with the gradient of ϕ. The functions a j ( j = 0, 1, . . . , p) can be extended to C ∞ functions on an open set containing Ω , moreover, a j (x) > 0 for all x ∈ Int(S j ) (where Int means “interior”) and a j (x) = 0 for all x ∈ ∂Ω \S j . Fact 2.1. It is well known (see [20,21]) that under the above assumptions the initial boundary problem (2.1)–(2.4) has a unique strong solution z. For any T > 0 we have z ∈ C([0, T ], L 2 (Ω )) ∩ L 2 ([0, T ], H 1 (Ω )). Moreover, for t > 0 this solution is instantly classical, which means in particular, that z(·, t) ∈ H s (Ω ) for any s > 0. Remark 2.1. Our assumptions about the coefficients of the operator L and the functions a j from (2.3) and (2.4) are somewhat stronger than necessary for all the constructions below to work. We have imposed the above strong requirements only to make the presentation more transparent and to avoid unnecessary technical complications.
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Let us introduce the operator A0 = L in Z with domain ( ) ∂ 2 D(A0 ) = ϕ ∈ H (Ω ) : ϕ|S D = 0, + a0 (x) ϕ = 0 . ∂µ S0
(2.9)
The operator A0 does not possess a full set of boundary conditions, in particular, there are no conditions imposed on the controlled portion of the boundary, SC . Control inputs: We define a boundary input operator > (Bz)(x, t) = (B1 z)(x, t) (B2 z)(x, t) · · · (B p z)(x, t) , j = 1, . . . , p (2.10) where each (B j z)(x, t) is defined in (2.3) for x ∈ S j and we assume that (B j z)(x, t) = 0 for x ∈ ∂Ω \S j . We use similar notation to introduce the vector of measured output, control input and reference signals which will be defined for all x ∈ ∂Ω . Measured output operator: Denote by γ j the trace operator γ j : H s (Ω ) → H s−1/2 (S j ), s > 1/2, j = 0, . . . , p. We define the measured output operator by > (Cz)(x, t) = (C1 z)(x, t) (C2 z)(x, t) · · · (C p z)(x, t) , (2.11) y j (x, t) = (C j z)(x, t) = γ j z(x, t), for x ∈ S j , y j (x, t) = 0,
for x ∈ ∂Ω \S j .
We then define the corresponding Measured output: y(x, t) = y1 (x, t) y2 (x, t)
···
(2.12)
> y p (x, t) ∈ R p .
(2.13)
In this work we seek control inputs u j (x, t) defined on S j ×(0, ∞) and the corresponding control input vector defined by Control inputs: > u(x, t) = u 1 (x, t) u 2 (x, t) · · · u p (x, t) ∈ R p , x ∈ ∂Ω , (2.14) where it is assumed that u j (x, t) = 0 for x ∈ x ∈ ∂Ω \S j . As we have already mentioned in Fact 2.1 for the parabolic systems considered here and t > 0 the solution z to (2.1)–(2.4) is smooth and y j (x, t) = z(x, t),
for x ∈ S j , j = 1, 2, . . . , p.
(2.15)
In this work we are primarily concerned with output regulation problems involving tracking given reference signals. Reference signals: We assume that there are p reference signals w j (t) where w j ∈ Cb1 ([0, ∞]), i.e., w j are infinitely differentiable functions with w j and dw/dt bounded on [0, ∞). We are also given p shape functions q j (x) defined on ∂Ω satisfying q j ∈ C 1 (S j ),
supp(q j ) ⊂ S j .
The reference output vector is then given by yr (x, t) = y1,r (x, t) y2,r (x, t) · · ·
> y p,r (x, t)
where y j,r (x, t) = q j (x)w j (t).
(2.16)
Tracking error: Next we define the tracking error by e(x, t) = y(x, t) − yr (x, t),
so that e j = y j − y j,r .
(2.17)
We are now in a position to state our main problem, Regulation problem: Find control input u in Eq. (2.14) so that the error defined in (2.17) satisfies t→∞
ke(·, t)k H s−1/2 (SC ) → 0
for some s > 1/2,
(2.18)
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where ke(·, t)k2H s−1/2 (S
C)
=
p X
ke j (·, t)k2H s−1/2 (S ) . j
j=1
With the above notation the system (2.1)–(2.4) with outputs defined in (2.11) and (2.13) can be written in the compact abstract form as z˙ = A0 z
(2.19)
z(0) = ϕ Bz(t) = u(t)
(2.20)
y(t) = Cz(t).
(2.21)
At this point we introduce the uncontrolled open loop system dynamics for (2.1)–(2.4) by defining the operator A to be the operator L with the domain further restricted as D(A) = {ϕ ∈ D(A0 ) : B(ϕ) = 0}. Here the boundary operator B defined in (2.3) and (2.14) acts on functions defined on the controlled portion of the boundary SC given in (2.5). For the sake of clarity we write the domain of A explicitly as ) ( ∂ 2 + a j ϕ = 0, j = 0, 1, . . . , p . (2.22) D(A) = ϕ ∈ H (Ω ) : ϕ|S D = 0, ∂µ Sj We note, once again, that in this work we could have significantly relaxed various conditions but at the expense of extended length. Our main goal is rather to provide a fairly general example as a road map to interested readers who want to apply the basic design methodology to other problems. Fact 2.2. 1. The operator A, with domain defined in (2.22), generates an exponentially stable semigroup in L 2 (Ω ) and H s (Ω ) for all s > 0. A is a self-adjoint, uniformly elliptic operator with C ∞ (Ω ) eigenfunctions {ϕ j } and associated eigenvalues {λ j } satisfying 0 > λ1 > λ2 > · · · ,
lim λ j = −∞.
j→∞
All eigenvalues are negative if either one of the following conditions hold: (a) The (n − 1)-dimensional measure, |S D | > 0 and a j (x) ≥ 0 for all j = 0, 1, . . . , p, a(x) ≥ 0. (b) The |S D | = 0 and a j (x) ≥ 0 for all j = 0, 1, 2, . . . , p and for some j0 , |S j0 | > 0 and a j0 > 0 on S j0 and a(x) ≥ 0. 2. The operator A generates an exponentially stable analytic semigroup, denoted by T (t) in L 2 (Ω ), as well as, in H s (Ω ) for s > 0 (and in particular for s > [n/2] + 1 which guarantees the embedding of H s (Ω ) into C(Ω )). The solution to the uncontrolled problem (2.19)– (2.21) with u ≡ 0 (smooth in x and t for t > 0) is given by z(t) = T (t)ϕ. 3. z(·, t) for t > 0 is contained in C ∞ (Ω ) so that the trace on ∂Ω is a continuous function on ∂Ω . The output operator C is continuous from L 2 (Ω ) to H s−1/2 (∂Ω ) for s > [n/2] + 1. The following estimate holds for some positive constant α and positive continuous function Ms (t0 ) for t0 > 0 so that kT (t)ϕk H s (∂ Ω ) ≤ Ms (t0 )e−αt kϕk, where ϕ ∈ Z =
L 2 (Ω ),
∀ t ≥ t0 > 0,
k · k denotes the norm in
L 2 (Ω )
(2.23)
and lim Ms (t0 ) = ∞. t0 →0
4. There is a positive constant α and a positive continuous function Cs (t0 ) for t0 > 0 so that kC(T (t)ϕ)k H s−1/2 (∂ Ω ) ≤ Cs (t0 )e−αt kϕk
∀ t ≥ t0 > 0,
where ϕ ∈ Z = L 2 (Ω ) and lim Cs (t0 ) = ∞. t0 →0
(Note that α, Ms (t), and Cs (t), above all depend on s > 0.)
(2.24)
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3. The zero dynamics For the system (2.19)–(2.21) defined via (2.1)–(2.4) we now define the “zero dynamics” as the system obtained by constraining the output to be zero, i.e., we require Cz = 0 for all time. The control u = B(z) thus obtained is the b to be the operator L with domain control that maintains zero output for all time. Thus we define the operator A b = {ϕ ∈ (D(A0 )) : C(ϕ) = 0}. D( A)
(3.1)
This translates explicitly into ( ) ∂ 2 b = ϕ ∈ H (Ω ) : ϕ|S ∪S = 0, D( A) + a0 (x) ϕ = 0 . D C ∂µ S0
(3.2)
Thus the zero dynamics is given by b ξ˙ = Aξ
(3.3)
ξ(0) = ψ. Remark 3.1. Note that according to our definition of the zero dynamics the term (2.20) involving B(·) does not occur b see (3.1). in (3.3). It has been replaced by the constraint Cz = 0 which is built into (3.3) through the definition of A, b is a strictly negative, Just as with the uncontrolled problem, described above, our assumptions imply that A 2 b(t) unbounded self-adjoint operator in L (Ω ) that generates an exponentially stable analytic semigroup, denoted by T 2 s in L (Ω ), as well as, in H (Ω ) for all s > 0. The solution to the zero dynamics problem (3.3) (smooth in x and t for t > 0) is given by b(t)ψ ξ(t) = T bs (t0 ) for t0 > 0 so that and there is a positive constant b α and a positive continuous function C αt b(t)ψ)k H s−1/2 (∂ Ω ) ≤ C bs (t0 )e−b kC(T kψk
∀ t > t0 > 0,
(3.4)
bs (t0 ) = ∞. (Note that b bs (t) above depend on s > 0). where ψ ∈ Z = L 2 (Ω ), and lim C α and C t0 →0
At least formally, the system (3.3) provides a very natural definition of zero dynamics within the context of boundary feedback control problems for distributed parameter systems governed by partial differential equations. Indeed, if in (2.19)–(2.21) we employ a proportional error feedback law u = −ky.
(3.5)
then we obtain a closed loop system w˙ = Ak w
(3.6)
w(0) = ϕ where the closed loop spatial operator Ak is given by the operator A with domain D(Ak ) = {φ ∈ D(A0 ) : B(φ) + kC(φ) = 0}. For this closed loop system, we see that the uncontrolled problem is obtained when k = 0 and in the high gain limit, at least at a formal level, we obtain the zero dynamics (3.3) as k tends to infinity. This high gain limit can, in fact, be rigorously justified by techniques similar to those used in [6]. Within this context, we could also define the open loop system (2.19)–(2.21) to be minimum phase provided that (3.3) is asymptotically stable. Of course this definition has little meaning unless we can establish some type of concept of left half plane transmission zeros as in the finite dimensional linear case. The necessary generalization is exactly the notion of “zero-pole dynamics” (discussed above) for the closed loop system (2.19)–(2.21). Namely, for a given example, we would expect to prove that the zero dynamics is asymptotically stable and then prove that as we increase
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the gain parameter k, the trajectories of the closed-loop system (3.6) approach the trajectories of the zero dynamics (3.3). Although for some examples this definition of zero dynamics will be different from the one obtained via the invariance concepts and zero dynamics algorithm, at least in the linear self-adjoint case considered here and under certain other technical assumptions the point spectrum of the resulting zero dynamics should be the transmission zeros of the original system. 4. Dynamic controller for boundary control problems In this section we construct a dynamic controller based on the “zero dynamics.” For the boundary control system (2.19)–(2.21) with zero dynamics given in (3.3) define a dynamic controller by replacing the homogeneous constraint Cξ = 0 with the non-homogeneous constraint Cz = yr . The zero dynamics provides the desired dynamic controller. ξt = A0 ξ,
(4.1)
ξ(0) = ψ, Cξ(t) = yr ,
(4.2) (4.3)
u(t) = Bξ(t).
(4.4)
Now we are prepared to prove the following proposition concerning the regulation problem considered here. Proposition 4.1. For the system (2.19)–(2.21), the problem of output regulation (for all initial data ϕ) is solved by the controller given by (4.1)–(4.3) with u in (4.4). The proof of Proposition 4.1 is very simple but it gives a very elementary practical means to construct controllers for boundary controlled distributed parameter systems. Proof. Let u(t) = Bξ(t) as in (4.4) and define η(t) = z(t) − ξ(t) where z is the solution to (2.19)–(2.21) and ξ the solution to (4.1) and (4.2). Then it is easy to show that η satisfies ηt = Aη,
(4.5)
η(0) = ϕ − ψ ≡ η0 ,
(4.6)
e(t) = Cη(t).
(4.7)
The result now follows from Fact 2.1. Indeed, it is well known (cf., [20,21,14]) that η is a continuous (in x and t for t > 0) globally defined solution which satisfies t→∞
sup |η(x, t)| −→ 0. x∈Ω
More precisely, η(·, t) ∈ H s (Ω ) for s > [n/2] + 1 which implies η(·, t) ∈ C(Ω ) and our result follows from (2.23). Namely, we have kη(·, t)k H s (Ω ) ≤ Ms (t0 )e−αt kη0 k,
for some α > 0, t ≥ t0 > 0
and the result follows from (4.7) the estimate (2.24) and the Sobolev Embedding Theorem. 5. Static feedback control: Two SISO examples It is often possible to simplify the controller by replacing the dynamic controller (4.1)–(4.4) by a simple feedback law derived from the steady state locus (cf. [2,12]). These “static” control laws are similar to those obtained by solving the “regulator equations,” as described in [3]. Note that here the input and output operators are unbounded (densely defined) operators in the Hilbert state space so the results of [3] are not directly applicable. For the parabolic problems considered here the steady state response of the zero dynamics typically consists of a single (generally time dependent) asymptotically stable steady state, ξ 0 (x, t). Furthermore, under smoothness assumptions on the signals to be tracked, the convergence to this steady state takes place in C 1 (Ω ). This can be exploited to obtain controls u j (x, t) = B j ξ 0 (x, t) for x ∈ S j .
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For set-point control the signals to be tracked are time independent and the steady state response is also time independent, ξ 0 (x, t) = ξ 0 (x). In this case the resulting controls are given by u j (x) = B j ξ 0 (x) for x ∈ S j . Rather than develop a lengthy discussion of special cases that arise in carrying out the specific analysis, in this section we will present two simple single input single output (SISO) examples (set point control and tracking a sinusoid) that serve to illustrate this methodology for the one dimensional heat equation on a unit interval. Thus we consider the system z t (x, t) = z x x (x, t),
x ∈ [0, 1], t > 0
(B0 z)(t) = z x (0, t) − k0 z(0, t) = 0,
(5.1)
k0 > 0
(B1 z)(t) = z x (1, t) = u(t) z(x, 0) = ϕ(x). The measured output is y(t) = (Cz)(t) = z(1, t). In this case we have d2 , D(A0 ) = {ϕ ∈ H 2 (0, 1) : B0 (ϕ) = 0}, dx 2 d2 dϕ (1) = 0 , A = 2 , D(A) = ϕ ∈ H 2 (0, 1) : B0 (ϕ) = 0. dx dx A0 =
2 b= d , A dx 2
b = {ϕ ∈ H 2 (0, 1) : B0 (ϕ) = 0. ϕ(1) = 0}. D( A)
Given a reference signal yr (t) we want to find u(t) so that the error, defined by e(t) = y(t) − yr (t), goes to zero as t goes to infinity. The design of control laws solving problems of this type has received considerable attention in the literature. A number of authors have extended the finite dimensional geometric methods of Francis [15], Wonham [26], Francis and Wonham [16] and Hautus [17] to the infinite dimensional case including Pohjolainen [24] and Schumacher [25]. In another recent work [3] the authors adapt the ideas of Byrnes and Isidori [19] to characterize solvability of the regulator problem for distributed parameter problems with bounded input and output operators in terms of the solvability of a pair of regulator equations. More specifically these works were concerned with output regulation for systems in the standard form dz (t) = Az(t) + Bu(t), dt dw (t) = Sw(t) dt z(0) = z 0 , w(0) = w0 y(t) = C z(t), yr = Qw,
(5.2) (5.3) (5.4) (5.5)
e(t) = y(t) − yr (t). where B and C are bounded operators and w is the state of the exosystem in a finite dimensional Hilbert space W . Under suitable hypotheses (e.g. A is exponentially stable and S has all its eigenvalues on the imaginary axis) it can be shown that the feedback law u = Γ w solves the problem of output regulation where Γ is obtained as part of the solution to the regulator equations Π S = AΠ + BΓ
(5.6)
CΠ = Q.
(5.7)
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Here Π : W → D(A) ⊂ Z is a bounded linear operator and in the SISO case Γ : W → R is also a linear map. Thus if dim(W ) = p then Γ is a p × 1 constant vector Γ = γ1 , γ2 , . . . , γ p . These techniques have been extended to the unbounded case in certain situations, see for example [4,5]. Example 5.1 (SISO Set-Point Control). For set point control the objective is to have the output y track a prescribed constant M. Thus for SISO set-point control yr (t) = M and we have S = 0, Q = 1. In this case it is easily verified that the regulator equations lead to the formula u = Γ w = G(0)−1 M
(5.8)
where G(s) is the system’s transfer function, G(s) = C(s I − A)−1 B,
s ∈ ρ(A),
(5.9)
(here ρ(A) is the resolvent set of A). For the problem (5.1) we can, after some straightforward calculations, compute the transfer function to obtain √ √ √ s cosh( s) + k0 sinh( s) G(s) = (5.10) √ √ √ . s sinh( s) + k0 s cosh( s) Knowing that a control can be obtained in the form u = Γ M for some Γ ∈ R we consider finding Γ using our zero dynamics methodology. The zero dynamics for the system (5.1) (see (4.1)–(4.3)) is the system ∂ξ ∂ 2ξ (x, t) = 2 (x, t), ξ(x, 0) = ψ(x), ∂t ∂x ∂ξ (0, t) − k0 ξ(0, t) = 0, ∂x ξ(1, t) = M,
(5.11)
and the steady state locus consists of a single stationary solution ξ 0 ∈ D(A0 ). Namely, d2 ξ 0 = 0, dx 2
dξ 0 (0) − k0 ξ 0 (0) = 0, dx
ξ 0 (1) = M,
(5.12)
implies ξ0 =
k0 x + 1 . k0 + 1
Therefore the static feedback law u 0 = B(ξ 0 )M =
dξ 0 (1)M = M dx
solves the problem of output regulation just as the static feedback (5.8), obtained from the regulator equations, does. We notice also that from (5.10) that G(0) = 1. Indeed, in several examples we have verified that both methods provide the same control, i.e., B(ξ0 ) = G(0)−1 . Example 5.2 (SISO Harmonic Tracking). For harmonic tracking the output y in (2.15) is required to track a sinusoid yr (t) = M sin(αt). In this case we can take the exosystem in (5.3) to be the harmonic oscillator written as the two dimensional system 0 α 0 w˙ = Sw = , w(0) = . −α 0 M
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We have w1 (t) M sin(αt) w(t) = = . w2 (t) M cos(αt)
Therefore yr (t) = yr = [1, 0]w = M sin(αt). Once again we can solve the regulator equations as in [3] in terms of the system’s transfer function, given in (5.9) and we have h i u = Re G(iα)−1 , Im G(iα)−1 w (5.13) where, once again, G(s) is the transfer function (see (5.9)) and w = [M sin(αt), M cos(αt)]> . Just as in Example 5.1 we can compute the steady state locus of the zero dynamics in terms of the special stationary solution ξ0 in (5.12). For harmonic tracking the procedure can be carried out quite generally using semigroup theory (see, for example, [22], page 29, Corollary 7.5) or using Laplace transform type techniques like those found in [7]. Here we will not include these details which are quite straightforward. Motivated by the fact that we know from (5.13) that a control can be obtained in terms of a pair of gain parameters γ1 and γ2 our objective here is to obtain these parameters using the steady state response of the zero dynamics (5.11). It is easy to show that the solution to the zero dynamics consists of a sum of decaying transient and a particular periodic motion. After some straightforward calculations based on an analysis of the solution given by the variation of parameters formula, the desired control law can be written in terms of operators applied to the stationary solution ξ0 as follows: bw, u=Γ
b = [b Γ γ1 , b γ2 ]
(5.14)
b2 + α 2 I )−1 ξ0 γ1 = B ξ0 − α 2 B( A b b −1 ξ0 , = Bξ0 + α Im B(iα I − A) bA b2 + α 2 I )−1 ξ0 γ2 = αB A( b b −1 ξ0 . = −α 2 Re B(iα I − A)
(5.15)
where
(5.16) (5.17) (5.18)
In (5.14) we denote by [b γ1 , b γ2 ] the 1 × 2 matrix with entries b γ1 and b γ2 . From (5.15) and (5.17) we need to find b2 + α 2 )−1 ξ0 f ≡ (A which is equivalent to f 0000 + α 2 f = ξ0 , subject to the end conditions f 0 (0) − k0 f (0) = 0,
f (1) = 0,
f 000 (0) − k0 f 00 (0) = 0,
Writing the equation as a first order system we set f 1 = f,
f2 = f 0,
f 3 = f 00 ,
f 4 = f 000
f 00 (1) = 0.
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Fig. 5.1. Solution surface.
Fig. 5.2. Output y(t) (solid) and reference signal yr (t) (dashed).
and consider f2 f1 f1 f2 d f f 3 2 = f3 f4 dx f 3 2 f4 f4 −α f 1 + ξ0
x ∈ [0, 1]
with boundary conditions f 2 (0) − k0 f 1 (0) = 0,
f 1 (1) = 0,
f 4 (0) − k0 f 3 (0) = 0,
f 3 (1) = 0.
Then we obtain from (5.15) and (5.17) γ1 = ξ00 (1) − α f 2 (1), b
γ2 = α f 3 (1) b
so that the desired control is easily computed numerically or we could use the formula given in (5.13). In Fig. 5.1 we have plotted the controlled solution surface and in Fig. 5.2 we have plotted the reference signal and the controlled output. Finally we have plotted the error e(t) = y(t) − yr (t) in Fig. 5.3. In the numerical example given below we solve for the control parameters b g1 and b g2 using the Matlab built-in Boundary Value Solver bvp4c. In the numerical simulation we have set k0 = 1, M = 2, α = 2, z(x, 0) = ϕ(x) =
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Fig. 5.3. The error e(t) = y(t) − yr (t).
2 cos(π x), ξ(x, 0) = ψ(x) = 0 and solved the problem on the time interval 0 < t < 10. In the case we have γ1 = 0.8356 and γ2 = 1.0102 and the values of b g1 and b g2 computed using Matlab agree with these values to six decimal places. In the first figure we have plotted the solution surface for the controlled system. References [1] S.P. Banks, F. Abbasi-Ghelmansarai, Realization theory and the infinite-dimensional root locus, Internat. J. Control 38 (3) (1983) 589–606. [2] C.I. Byrnes, D.S. Gilliam, A. Isidori, J. Ramsey, On the steady-state behavior of forced nonlinear systems, in: New Trends in Nonlinear Dynamics and Control, and their Applications, in: Lecture Notes in Control and Inform. Sci., vol. 295, Springer, Berlin, 2003, pp. 119–143. [3] C.I. Byrnes, D.S. Gilliam, I.G. Lauk´o, V.I. Shubov, Output regulation for linear distributed parameter systems, IEEE Trans. Automat. Control 45 (12) (2000) 2236–2252. [4] C.I. Byrnes, D.S. Gilliam, V.I. Shubov, G. Weiss, The output regulator problem for regular linear systems, Texas Tech University, 2000 (preprint). [5] C.I. Byrnes, D.S. Gilliam, V.I. Shubov, Example of output regulation for a system with unbounded inputs and outputs, in: Proceedings 38th IEEE Conf. Dec. Control, 1999, pp. 4280–4284. [6] C.I. Byrnes, D.S. Gilliam, V.I. Shubov, Boundary control, stabilization and zero-pole dynamics for a nonlinear distributed parameter system, J. Robust Nonlinear Control 9 (1999) 737–768. [7] C.I. Byrnes, D.S. Gilliam, I.G. Lauk´o, V.I. Shubov, Harmonic forcing for linear distributed parameter systems, J. Math. Systems Estim. Control 8 (2) (1998). [8] C.I. Byrnes, D.S. Gilliam, V.I. Shubov, High gain limits of trajectories and attractors for a boundary controlled viscous Burgers’ equation, J. Math. Systems Estim. Control 6 (4) (1996) 40. [9] C.I. Byrnes, D.S. Gilliam, J. He, Root-locus and boundary feedback design for a class of distributed parameter systems, SIAM J. Control Optim. 32 (5) (1994) 1364–1427. [10] C.I. Byrnes, D.S. Gilliam, Asymptotic properties of root locus for distributed parameter systems, in: Proceedings of the 27th Conference on Decision and Control, Austin, Texas, December 1988, pp. 45–51. [11] C.I. Byrnes, D.S. Gilliam, J. He, A root locus methodology for parabolic distributed parameter feedback systems, in: Computation and Control, II (Bozeman, MT, 1990), in: Progr. Systems Control Theory, vol. 11, Birkh¨auser Boston, Boston, MA, 1991, pp. 63–83. [12] C.I. Byrnes, A. Isidori, Limit sets, zero dynamics, and internal models in the problem of nonlinear output regulation, IEEE Trans. Automat. Control AC-48 (2003) 1712–1723. [13] R.F. Curtain, H.J. Zwart, An Introduction to Infinite-Dimensional Linear Systems, Springer-Verlag, New York, 1995. [14] Y.V. Egorov, M.A. Shubin, Foundations of the classical theory of partial differential equations, Springer, 1991 (Second printing 1998). [15] B.A. Francis, The linear multivariable regulator problem, SIAM J. Control Optim. 14 (1977) 486–505. [16] B.A. Francis, W.M. Wonham, The internal model principle of control theory, Automatica 12 (1976) 457–465. [17] M. Hautus, Linear matrix equations with applications to the regulator problem, in: I.D. Landau (Ed.), Outils and Modeles Mathematique pour l’Automatique, C.N.R.S., Paris, 1983, pp. 399–412. [18] J. He, Root locus for control systems with completely separated boundary conditions, in: Computation and Control, III (Bozeman, MT, 1992), in: Progr. Systems Control Theory, vol. 15, Birkh¨auser Boston, Boston, MA, 1993, pp. 229–239. [19] A. Isidori, C.I. Byrnes, Output regulation of nonlinear systems, IEEE Trans. Automat. Control AC-35 (1990) 131–140. [20] O.A. Ladyzhenskaya, V.A. Solonnikov, N.N. Ural’ceva, Linear and Quasilinear Equations of Parabolic Type, in: Translations of AMS, vol. 23, 1968.
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