In-domain control of a heat equation: an approach combining zero-dynamics inverse and differential flatness

# In-domain control of a heat equation: an approach combining zero-dynamics inverse and differential flatness

## Abstract

This paper addresses the set-point control problem of a heat equation with in-domain actuation. The proposed scheme is based on the framework of zero dynamics inverse combined with flat system control. Moreover, the set-point control is cast into a motion planing problem of a multiple-input, multiple-out system, which is solved by a Green’s function-based reference trajectory decomposition. The validity of the proposed method is assessed through the convergence and solvability analysis of the control algorithm. The performance of the developed control scheme and the viability of the proposed approach are confirmed by numerical simulation of a representative system.

###### keywords:
Distributed parameter systems; heat equation; zero-dynamics inverse; differential flatness.
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proofProof

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[mycorrespondingauthor]Corresponding author
Tel: +1-514-340-4711 ext. 5868

## 1 Introduction

Control of parabolic partial differential equations (PDEs) is a long-standing problem in PDE control theory and practice. There exists a very rich literature devoted to this topic and it is continuing to draw a great attention for both theoretical studies and practical applications. In the existing literature, the majority of work is dedicated to boundary control, which may be represented as a standard Cauchy problem to which functional analytic setting based on semigroup and other related tools can be applied (see, e.g., (1); (2); (3); (4)). It is interesting to note that in recent years, some methods that were originally developed for the control of finite-dimensional systems have been successfully extended to the control of parabolic PDEs, such as backstepping (see, e.g., (5); (6); (7)), flat systems (see, e.g., (8); (9); (10); (11); (12); (13); (14)), as well as their variations (see, e.g., (15); (16); (17)).

This paper addresses the in-domain (or interior point) control problem of a heat equation, which may arise in application related concerns for, e.g., the enhancement of control efficiency. Integrating a number of control inputs acting in the domain will lead to non-standard inhomogeneous PDEs (1); (18), which should be treated differently than the standard boundary control problem. The control scheme developed in the present work is based-on the framework of zero-dynamics inverse (ZDI), which was introduced by Byrnes and his collaborators in (19) and has been exploited and developed in a series of work (20); (21); (22); (23); (24). It is pointed out in (23) 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.” Indeed, the construction of zero dynamics for output regulation of certain in-domain controlled PDEs is also straightforward (see, e.g., (22)) and hence, the control design can be carried out in a systematic manner. A main issue related to the ZDI design is that the computation of dynamic control laws requires resolving the corresponding zero dynamics, which may be very difficult for generic regulation problems, such as set-point control. To overcome this difficulty, we leverage one of the fundamental properties of flat systems, that is if a lumped or distributed parameter system is differentially flat (or flat for short), then its states and inputs can be explicitly expressed by the so-called flat output and its time-derivatives (25); (13). In the context of ZDI design, the control can be derived from the flat output without explicitly solving the zero dynamics. Moreover, in the framework of flat systems, set-point control can be cast into a problem of motion planning, which can also be carried out in a systematic manner.

The system model used in this work is taken from (22). In order to perform control design, we present the original system in a form of parallel connection. This formulation allows a significant simplification of computation. As the control with multiple actuators located in the domain leads to a multiple-input, multiple-output (MIMO) problem, the design of reference trajectories is not trivial. To overcome this problem, we introduce a Green’s function-based reference trajectory decomposition scheme that enables a simple and computational tractable implementation of the proposed control algorithm.

The remainder of the paper is organized as follows. Section 2 describes the model of the considered system and its equivalent settings. Section 3 presents the detailed control design. Section 4 deals with motion planning and addresses the convergence and the solvability of the proposed control scheme. A simulation study is carried out in Section 5, and, finally, some concluding remarks are presented in Section 6.

## 2 Problem Setting

In the present work, we consider a scaler parabolic equation describing one-dimensional heat transfer with boundary and in-domain control, which is studied in (22). Denote by the heat distribution over the one-dimensional space, , and the time, . The derivatives of with respect to its variables are denoted by and , respectively. For notational simplicity, we may not show all the variables of functions if there is no ambiguity, e.g., . Consider points , , in the interval and assume, without loss of generality, that . Let . The considered heat equation with boundary and in-domain control in a normalized coordinate is of the form:

 zt(x,t)−zxx(x,t)=0,  x∈Ω, t>0, (1a) z(x,0)=ϕ(x), (1b) B0z=zx(0,t)−k0z(0,t)=0, (1c) B1z=zx(1,t)+k1z(1,t)=0, (1d) z(x+j)=z(x−j), j=1,...,m, (1e) Bxjz=[zx]xj=uj(t), j=1,2,...,m, (1f)

where for a function and a point we define

 [ψ]x=ψ(x+)−ψ(x−),

with and denoting, respectively, the usual meaning of left and right hand limits to . The initial condition is specified by (1b) with . In System (1), we assume that we can control the heat flow in and out of the system at the points , i.e.,

 uj(t)=[zx]xj=zx(x+j,t)−zx(x−j,t).

Note that in (1), represents the point-wise control located on the boundary or in the domain.

The space of weak solutions to System (1) is chosen to be . Note that System (1) is exponentially stable in if the boundary controls and are chosen such that , , and (23).

Denote a set of reference signals by . Let

 ei(t)=z(xi,t)−zDi(xi,t)

be the regulation errors. Let and .

###### Problem 1

The considered regulation problem is to find a dynamic control such that the regulation error satisfies as .

The above in-domain control problem can also be formulated in another way by replacing the jump conditions in (1f) by point-wise controls as source terms. The resulting system will be of following the form

 zt(x,t)−zxx(x,t)=m∑j=1δ(x−xj)αj(t), 00, (2a) z(x,0)=ϕ(x), (2b) B0z=zx(0,t)−k0z(0,t)=0, (2c) B1z=zx(1,t)+k1z(1,t)=0, (2d)

where is the Dirac delta function supported at the point , denoting an actuation spot, and , , are the in-domain control signals.

###### Lemma 1

Considering weak solutions in , System (1) and System (2) are equivalent if

 αj(t)=−uj(t)=−[zx]xj,  j=1,...,m.
{proof}

The proof follows the idea presented in (26). Indeed, it suffices to prove “System (1) System (2).” Let be a Hilbert space equipped with the inner product . Let the operator be defined by , with domain . It is easy to see that , the adjoint of , is equal to . Let be an extension of with domain . Let . Using integration by parts we obtain that

 ⟨˜Au,v⟩=⟨u,Av⟩+m∑j=1(ux(x−j)−ux(x+j))v(xj). (3)

Let , the dual space of . We need to define another extension for . Let be defined by

 ⟨^Au,v⟩=⟨u,A∗v⟩  for\ % all v∈D(A∗), (4)

with . Note that is not in , but it is in the large space . It follows from (3), (4), and that

 ˜Au=^Au+m∑j=1(ux(x−j)−ux(x+j))δ(x−xj), (5)

in . If satisfies System (1), then , which yields, considering (5), . Finally, we can see that System (1) becomes System (2) with , where we look for generalized solutions such that (5) is true in .

To establish in-domain control at every actuation point, we will proceed in the way of parallel connection, i.e., for every , consider the following two systems

 zt(x,t)−zxx(x,t)=0,  x∈(0,xj)∪(xj,1), t>0, (6a) z(x,0)=ϕj(x), (6b) B0z=zx(0,t)−k0z(0,t)=0, (6c) B1z=zx(1,t)+k1z(1,t)=0, (6d) z(x+j)=z(x−j), (6e) Bxjz=[zx]xj=vj(t). (6f)

and

 zt(x,t)−zxx(x,t)=δ(x−xj)βj(t), 00, (7a) z(x,0)=ϕj(x), (7b) B0z=zx(0,t)−k0z(0,t)=0, (7c) B1z=zx(1,t)+k1z(1,t)=0, (7d)

with . Similarly, System (6) and (7) are equivalent provided and . Let for any , where denotes the solution to System (7). One may directly check that is a solution to System (2). Moreover,

 [zx]xi=m∑j=1[zjx]xi=[zix]xi=ui,

for any . Hence is a solution to System (1). Therefore, throughout this paper, we assume for any . Due to the equivalences of System (1) and (2), and System (6) and (7), we may consider (2) and System (6) in the following parts.

## 3 Control Design Based on Zero-Dynamics Inverse and Differential Flatness

In the framework of zero-dynamics inverse, the in-domain control is derived from the so-called forced zero-dynamics that are constructed from the original system dynamics by replacing the control constraints by the regulation constraints. To work with the parallel connected system (6), we first split the reference signal as:

 zD(x,t)=m∑j=1γj(x,xj)zdj(xj,t), (8)

in which the function will be determined in Proposition 4 (see Section 4). Denoting by the regulation error corresponding to System (6) and replacing the control constraint by , we obtain the corresponding zero-dynamics for a fixed :

 ξt(x,t)=ξxx(x,t), x∈(0,xj)∪(xj,1), t>0, (9a) ξ(x,0)=0, (9b) ξx(0,t)−k0ξ(0,t)=0, (9c) ξx(1,t)+k1ξ(1,t)=0, (9d) ξ(xj,t)=zdj(xj,t). (9e)

For simplicity, we denote by and the solutions to the systems (6) and (9), respectively. Also, we write henceforth as the reference trajectory in the system (9) if there is no ambiguity. The in-domain control signal of System (6) can then be computed by

 vj=[zjx]xj=[ξjx]xj. (10)
###### Remark 1

Note that for with , arguing as (22), we have

 εj(t)=zj(xj,t)−zdj(xj,t)→0 as t→∞.

Obviously, to find the control signals, we need to solve the corresponding zero-dynamics (9). For this purpose, we leverage the technique of flat systems (27); (11); (13). In particular, we apply a standard procedure of Laplace transform-based method to find the solution to (9). Henceforth, we denote by the Laplace transform of a function with respect to time variable. Then, for fixed , the transformed equations of (9) in the Laplace domain read as

 sˆξ(x,s)=ˆξxx(x,s), x∈(0,xj)∪(xj,1), s∈C, (11a) ˆξ(x,0)=0, (11b) ˆξx(0,s)−k0ˆξ(0,s)=0, (11c) ˆξx(1,s)+k1ˆξ(1,s)=0, (11d) ˆξ(xj,s)=ˆzdj(xj,s). (11e)

We divide (11) into two sub-systems, i.e., for fixed , considering

 sˆξ(x,s)=ˆξxx(x,s), 0

and

 sˆξ(x,s)=ˆξxx(x,s), xj

Let and be the general solutions to (12) and (13), respectively, and denote their inverse Laplace transforms by and . The solution to (9) can be written as

 ξj(x,t)=ξj−(x,t)χ{(0,xj)}+ξj+(x,t)χ{[xj,1)},

where

 χ(x){Ωj}={1,x∈Ωj⊆(0,1);0,otherwise.

Then at each point , by (10) and the argument of “parallel connection” (see Section 2), we have . Hence the in-domain control signals of System (1) can be computed by

 ui=[zx]xi=[ξix]xi, i=1,…,m. (14)

In the following steps, we present the computation of the solution to System (9), . Issues related to the reference trajectory for System (1) will be addressed in Section 4.

Note that and , the general solutions to (12) and (13), are given by

 ˆξj−(x,s) =C1ϕ1(x,s)+C2ϕ2(x,s), ˆξj+(x,s) =C3ϕ1(x,s)+C4ϕ2(x,s),

with

 ϕ1(x,s)=sinh(√sx)√s, ϕ2(x,s)=cosh(√sx).

We obtain by applying (12c) and (12d)

 C1ϕ1(xj,s)+C2ϕ2(xj,s)=ˆzdj(xj,s), C1−k0C2=0,

which can be written as

 (ϕ1(xj,s)ϕ2(xj,s)1−k0)(C1C2)=(ˆzdj(xj,s)0).

Let

 Rj−=(ϕ1(xj,s)ϕ2(xj,s)1−k0)

and

 ˆzdj(xj,s)=−det(Rj−)ˆyj−(xj,s). (15)

We obtain

 Unknown environment '%

Therefore, the solution to (12) can be expressed as

 ˆξj−(x,s)=(k0ϕ1(x)+ϕ2(x))ˆyj−(xj,s). (16)

We may proceed in the same way to deal with (13). Indeed, letting

 Rj+=(ϕ1(xj,s)ϕ2(xj,s)ϕ2(1,s)+k1ϕ1(1,s)sϕ1(1,s)+k1ϕ2(1,s))

and

 ˆzdj(xj,s)=det(Rj+)ˆyj+(xj,s), (17)

we get from (13)

 (C3C4)=((sϕ1(1,s)+k1ϕ2(1,s))ˆyj+(xj,s)−(ϕ2(1,s)+k1ϕ1(1,s))ˆyj+(xj,s)),

and

 ˆξj+(x,s)= ((sϕ1(1,s)+k1ϕ2(1,s))ϕ1(x) +(ϕ2(1,s)+k1ϕ1(1,s))ϕ2(x))ˆyj+(xj,s). (18)

Applying modulus theory (28); (29) to (15) and (17), we may choose as the basic output such that

 ˆyj+(xj,s) =−det(Rj−)ˆyj(xj,s), (19) ˆyj−(xj,s) =det(Rj+)ˆyj(xj,s). (20)

Then, using the property of hyperbolic functions, we obtain from (16) and (3) that

 ˆξj−(x,s)= (k1sinh(√sxj−√s)√s−cosh(√sxj−√s)) ×(k0sinh(√sx)√s+cosh(√sx))ˆyj(xj,s), (21) ˆξj+(x,s)= (k1sinh(√sx−√s)√s−cosh(√sx−√s)) ×(k0sinh(√sxj)√s+cosh(√sxj))ˆyj(xj,s). (22)

Note that

 ˆξj(x,s)=ˆξj−(x,s)χ{(0,xj)}+ˆξj+(x,s)χ{[xj,1)} (23)

is a solution to (11). Using the fact

 sinhx=∞∑n=0x2n+1(2n+1)!, coshx=∞∑n=0x2n(2n)!,

we obtain

 ˆξj(x,s) = [(k0k1∞∑n=0n∑k=0x2k+1(xj−1)2(n−k)+1(2k+1)![2(n−k)+1]!sn−k0∞∑n=0n∑k=0x2k+1(xj−1)2(n−k)(2k+1)![2(n−k)]!sn +k1∞∑n=0n∑k=0x2k(xj−1)2(n−k)+1(2k)![2(n−k)+1]!sn−∞∑n=0n∑k=0x2k(xj−1)2(n−k)(2k)![2(n−k)]!sn)χ{(0,xj)} +(k0k1∞∑n=0n∑k=0x2k+1j(x−1)2(n−k)+1(2k+1)![2(n−k)+1]!sn−k0∞∑n=0n∑k=0x2k+1j(x−1)2(n−k)(2k+1)![2(n−k)]!sn +k1∞∑n=0n∑k=0x2kj(x−1)2(n−k)+1(2k)![2(n−k)+1]!sn−∞∑n=0n∑k=0x2kj(x−1)2(n−k)(2k)![2(n−k)]!sn)χ{[xj,1)}]ˆyj.

It follows that

 ξj(x,t) = [(k0k1∞∑n=0n∑k=0x2k+1(xj−1)2(n−k)+1(2k+1)![2(n−k)+1]!y(n)j−k0∞∑n=0n∑k=0x2k+1(xj−1)2(n−k)(2k+1)![2(n−k)]!y(n)j +k1∞∑n=0n∑k=0x2k(xj−1)2(n−k)+1(2k)![2(n−k)+1]!y(n)j−∞∑n=0n∑k=0x2k(xj−1)2(n−k)(2k)![2(n−k)]!y(n)j)χ{(0,xj)} +(k0k1∞∑n=0n∑k=0x2k+1j(x−1)2(n−k)+1(2k+1)![2(n−k)+1]!y(n)j−k0∞∑n=0n∑k=0x2k+1j(x−1)2(n−k)(2k+1)![2(n−k)]!y(n)j +k1∞∑n=0n∑k=0x2kj(x−1)2(n−k)+1(2k)![2(n−k)+1]!y(n)j−∞∑n=0n∑k=0x2kj(x−1)2(n−k)(2k)![2(n−k)]!y(n)j)χ{[xj,1)}]. (24)

By a direct computation we get

 [ˆξjx]xj= [(k0k1√s+√s)sinh(√s)+(k0+k1)cosh(√s)]×ˆyj(xj,s) = (k0k1∞∑n=0sn(2n+1)!+(k0+k1)∞∑n=0sn(2n)!+∞∑n=0sn+1(2n+1)!)ˆyj(xj,s). (25)

It follows from (14) that

 uj(t)= [ξjx]xj = k0k1∞∑n=0y(n)j(xj,t)(2n+1)!+(k0+k1)∞∑n=0y(n)j(xj,t)(2n)!+∞∑n=0y(n+1)j(xj,t)(2n+1)!. (26)
###### Remark 2

The reference signal can be derived in the same way from the flat output from (15) and (17). However, as the flatness-based control is driven by flat output, there is no need to explicitly compute .

## 4 Motion Planning

For control purpose, we have to choose appreciate reference trajectories, or equivalently the basic outputs. In the present work, we consider the set-point control problem, i.e. to steer the heat distribution to a desired steady-state profile, denoted by . Without loss of generality, we consider a set of basic outputs of the form:

 yj(t)=¯¯¯y(xj)φj(t), j=1,…,m, (27)

where is a smooth function evolving from 0 to 1. Motion planning amounts then to deriving from and to determining appropriate functions , for .

To this aim and due to the equivalence of the systems (1) and (2), we consider the steady-state heat equation corresponding to System (2):

 ¯zxx(x)=m∑j=1δ(x−xj)¯αj, 00, (28a) ¯zx(0)−k0¯z(0)=0, (28b) ¯zx(1)+k1¯z(1)=0. (28c)

Based on the principle of superposition for linear systems, the solution to the steady-state heat equation (28) can be expressed as:

 ¯z(x)=∫10m∑j=1G(x,ζ)δ(ζ−xj)¯αjdζ=m∑j=1G(x,xj)¯αj. (29)

where is the Green’s function corresponding to (28), which is of the form

 G(x,ζ)=⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩(k1ζ−k1−