Output tracking for a 1-D heat equation with non-collocated configurations

Output Tracking for a 1-D Heat Equation with Non-collocated Configurations

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Output Tracking for a 1-D Heat Equation with Non-collocated Configurations Xiao-Hui Wu, Hongyinping Feng PII: DOI: Reference:

S0016-0032(19)30903-2 https://doi.org/10.1016/j.jfranklin.2019.12.016 FI 4333

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Journal of the Franklin Institute

Received date: Revised date: Accepted date:

8 April 2019 4 December 2019 9 December 2019

Please cite this article as: Xiao-Hui Wu, Hongyinping Feng, Output Tracking for a 1-D Heat Equation with Non-collocated Configurations, Journal of the Franklin Institute (2019), doi: https://doi.org/10.1016/j.jfranklin.2019.12.016

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Output Tracking for a 1-D Heat Equation with Non-collocated Configurations Xiao-Hui Wu and Hongyinping Feng

∗

†

School of Mathematical Sciences Shanxi University, Taiyuan, Shanxi, 030006, P.R. China

December 17, 2019

Abstract In this paper, we are concerned with output tracking for a one-dimensional heat equation where both disturbance and performance output are non-collocated to the controller. The method of trajectory planning is used to overcome the difficulties caused by these non-collocated configurations. An observer based output feedback law is designed by using only the tracking error between performance output and reference signal. Exponential convergence of tracking error and uniform boundedness of state of the whole closed-loop system are obtained. These theoretical results are validated by numerical simulations.

Keywords: Disturbance, error feedback, heat equation, observer, output tracking.

1

Introduction

The output tracking is one of the central problems in control theory. The main objective of output tracking is to find a control such that the performance output of the control plant can track the given reference signal in the presence of disturbances. Generally speaking, both reference signal and disturbance can be assumed to be generated by an exosystem. As a result, the problem of output tracking can also be referred as output regulation. In recent years, the output tracking for distributed parameter systems has attracted more and more attention from many researchers. Some recent works can be found in [4, 5, 22, 25, 26]. Different from the finite-dimensional case, we have to face the so-called non-collocated problems in the infinite-dimensional output tracking. The non-collocated problem, which complicates the tracking design immensely, are usually caused by the difference of relative locations, including the locations of measurement sensor, disturbance, performance output and control actuator. There exist various of non-collocated configurations in the problems of output tracking. For example, there are at least three non-collocated configurations ∗ †

This work is supported by the National Natural Science Foundation of China (No. 61873153). Corresponding author. Email: [email protected].

1

for the boundary control actuation: (i) the non-collocated disturbance/actuation configuration; (ii) the non-collocated performance-output/actuation configuration; (iii) the non-collocated sensing/actuation configuration. There are some classical method to cope with non-collocated problem in infinite-dimensional control system. In [17], the backstepping method is applied to stabilize an unstable wave equation with non-collocated actuation/sensing configuration. Soon after, this method have been extended to an anti-stable wave equation where anti-stable term is non-collocated to control actuation [29]. In [4], the backstepping method is used in output regulation problem for a linear 2 × 2 hyperbolic

system. Recently, output regulation with non-collocated harmonic disturbance has been considered by designing an adaptive servomechanism in [13]. The same problem is also considered in [12], but the only measurement available for controller design is the error between performance output and reference signal. The active disturbance rejection control is another effective tool to cope with non-collocated problems. In [16], output tracking for heat equation with non-collocated actuation/disturbance configuration is considered by active disturbance rejection control. The disturbance treatment technique used in [16] is first proposed in [9] and also can be applied to wave equation [35, 36]. Internal model principle is another widely used tool to deal with robust output regulation for infinite-dimensional systems. Examples can be found in [25, 26] and the references therein. In this paper, we limit ourselves to the external disturbance. For the nonlinear and stochastic uncertainties, we refer the author the references [32], [33], and [34]. In existing results such as [12, 13, 16, 35, 36], the performance output is always collocated to the control actuation. Comparing with collocated problems, the problem with non-collocated performance-output/actuation configuration is more challenging and is still scarcely. Very recently, in [10], the trajectory planning method was proposed to cope with the non-collocated problem through the output tracking for a wave equation. In this paper, we apply the trajectory planning method to the output tracking for the heat equation. By proper trajectory planning, the observer and controller can be designed without using the regulation equation, which is very different from the methods in [22] where only the state feedback is considered. Comparing the existing results such as [16] and [13], the main contributions of this paper lie in that: 1), the performance output is non-collocated to the control; 2), the only measurement available for the controller design is the tracking error. We proceed as follows. In Section 2, we present the problem formulation. By exploiting the trajectory planning method, a state feedback law is designed in Section 3. In Section 4, an error based observer is designed to estimate both the disturbance and the state. In Section 5, an output feedback controller is designed by using the tracking error only. The stability for the closed-loop system is also proved. The obtained result is applied to the harmonic disturbance and reference signal in Section 6. Some numerical simulations are presented in Section 7 to validate the theoretical results, followed by some concluding remarks in Section 8.

2

2

Problem formulation

Consider the output tracking for the following one-dimensional heat equation:   wt (x, t) = wxx (x, t), x ∈ (0, 1), t > 0,     wx (0, t) = d(t), wx (1, t) = u(t), t ≥ 0,      y (t) = w(0, t), t ≥ 0, p

(2.1)

where w is the state, u the control input, yp the performance output, and d the disturbance. The main goal of this paper is to design a control law such that the performance output yp tracks a

given reference signal yref in the presence of the external disturbance d. The only measurement available for the control design is the tracking error between the performance output yp and the reference signal yref ye (t) = yref (t) − yp (t).

(2.2)

Throughout this paper, both the reference yref and the disturbance d come from the following exosystem:

  v(t) ˙ = Gv(t), t ≥ 0,

 y (t) = F v(t), d(t) = Qv(t), t ≥ 0, ref

(2.3)

where G ∈ Cn×n , F ∈ C1×n , and Q ∈ C1×n . All of them are assumed to be known. However, the initial state of the exosystem is unknown. Consequently, both the reference yref and the disturbance

d are unknown in some sense. The disturbance and reference given by the dynamic form (2.3) is quite general and can cover usual harmonic signal and polynomial signal as a special case. Even more complicated signal such as the triangle signal also can be approximated by the system (2.3). These facts have been discussed extensively in [24]. The physical meaning of the system (2.1) is clear. The system (2.1), a general 1-d heat equation with boundary convection, depicts the diffusion of heat in a solid rod that is insulated everywhere except the two ends. The heat flux generated by exosystem (2.3) is injected into the rod through the left end. Since the initial state of exosystem (2.3) is unknown, the flux injection at the left end is unknown and thus is a disturbance. Moreover, the reference signal is also unknown. The controller needs to be designed at the right end of the rod so that the temperature at the left end can be stabilized exponentially at the level of given reference temperature. The system (2.1) can be used to describe a wide variety of thermal/fluid systems, such as heat conduction in solids and chemical tubular reactor [18]. For more details of physical modeling of heat equation, we also refer to [14]. Incorporating the exosystem (2.3) into the control plant (2.1), the problem of output tracking

3

then becomes the problem of output regulation   wt (x, t) = wxx (x, t), x ∈ (0, 1), t > 0,        w (0, t) = Qv(t), w (1, t) = u(t), t ≥ 0, x

x

  v(t) ˙ = Gv(t), t ≥ 0,       ye (t) = yref (t) − yp (t) = F v(t) − w(0, t).

(2.4)

Since the performance output yp is non-collocated to the control actuation, the usual servomechanism design in [12], [13], [35] and [36] can not be used directly. In this paper, we apply the method of trajectory planning that is proposed by [10] to cope with the non-collocated problem. More specifically, we construct a proper “auxiliary trajectory” by power series expansion to convert the corresponding non-collocated problem into the canonical collocated form so that the conventional controller design is available. For the sake of simplicity, we drop the obvious time and spatial domain of the control system in the rest of the paper. The spectrum of the operator A is denoted by σ(A); the complex conjugate transpose of matrix A is denoted by A∗ ; the inner product of Hilbert space H is denoted by h·, ·iH ,

and the corresponding norm is denoted by k · kH . Specially, we write h·, ·i as the inner product of

Hilbert space L2 (0, 1), and k · k as the corresponding norm. k · kCn is Euclidean norm of Cn . The p spectral norm of matrix A is defined by k · k2 := max{σ(A∗ A)}.

3

State feedback

By separation principle, the feedback law and the observer can be designed separately. Hence, we first consider the state feedback design. The main difficulty for problem (2.4) lies in that there are two non-collocated configurations: disturbance/actuation and performance-output/actuation. We overcome this difficulty by “ trajectory planning” which was proposed in [10]. The initial idea of this trajectory planning method comes from the monograph [3, 18, 19] where the method of trajectory generation, or motion planning, is given to generate a reference trajectory for the state of a PDE. In order to cancel the disturbance and, at the same time, keep the control invariant, we suppose that the auxiliary trajectory satisfies the following system:   ψt (x, t) = ψxx (x, t), x ∈ (0, 1), t > 0,

 ψ (0, t) = Qv(t), ψ(0, t) = F v(t), t ≥ 0. x

(3.1)

Inspired by [6, p.221, Chapter 4] and [18, Chapter 12], system (3.1) admits a special solution of power series expansion: ψ(x, t) =

∞ X

n=0

αn (t)

xn , x ∈ [0, 1], t ≥ 0, n!

where, by inserting (3.2) into system (3.1), the coefficient functions αn satisfy 4

(3.2)

  α˙ n (t) = αn+2 (t), n = 0, 1, 2, · · · , Hence,

 α (t) = F v(t), α (t) = Qv(t). 0 1 ∞ X

∞

X x2n x2n+1 + Qv (n) (t) (2n)! (2n + 1)! n=0 n=0 ! ! ∞ ∞ X X Gn x2n+1 Gn x2n v(t) + Q v(t). =F (2n)! (2n + 1)!

ψ(x, t) =

F v (n) (t)

n=0

n=0

If we define

(3.3)

 sinh s   , s 6= 0, s G(s) = = (2n + 1)!   1, n=0 s = 0, ∞ X

s2n

(3.4)

then, the function ψ(x, t) can be written analytically ψ(x, t) = Ψ(x)v(t),

x ∈ [0, 1], t ≥ 0,

where Ψ : [0, 1] → Cn is a vector valued function defined by ! ! ∞ ∞ X X Gn x2n+1 Gn x2n +Q Ψ(x) = F (2n)! (2n + 1)! n=0

n=0

1 2

(3.5)

(3.6)

1 2

= F cosh(xG ) + QxG(xG ). For a function f : C → C, the matrix f (G) is well defined, see, for instance [15, p.3, Definition 1.2]. So the functions of matrix in (3.6) make sense. Let

ε(x, t) = w(x, t) − ψ(x, t), x ∈ [0, 1], t ≥ 0. Then, the error ε is governed by   εt (x, t) = εxx (x, t),

 ε (0, t) = 0, ε (1, t) = u(t) − ψ (1, t). x x x

(3.7)

(3.8)

Since ye (t) = −ε(0, t) = ψ(0, t) − w(0, t), it suffices to stabilize system (3.8) to achieve output tracking. Consequently, the controller can be designed easily u(t) = −c1 ε(1, t) + ψx (1, t) = −c1 w(1, t) + [c1 Ψ(1) + Ψx (1)]v(t) h i 1 1 = −c1 w(1, t) + (Q + c1 F ) cosh G 2 + (F G + c1 Q)G(G 2 ) v(t),

(3.9)

where c1 is a positive tuning parameter. Under the controller, we get the following closed-loop

5

system of system (2.4):   v(t) ˙ = Gv(t),        wt (x, t) = wxx (x, t),     wx (0, t) = Qv(t),   i h  1 1   2 + (F G + c1 Q)G(G 2 ) v(t), w (1, t) = −c w(1, t) + (Q + c F ) cosh G  x 1 1       y (t) = F v(t) − w(0, t). e

(3.10)

We consider the control plant (3.10) in the state space H = L2 (0, 1) × Cn .

Theorem 3.1. Suppose that c1 > 0. For any initial state (w(·, 0), v(0)) ∈ H, system (3.10) admits a unique solution (w, v) ∈ C([0, ∞); H) such that, for any t0 > 0, |ye (t)| ≤ L1 e−ω1 t , ∀ t ≥ t0 ,

(3.11)

where L1 and ω1 are positive constants. Moreover, if supt∈[0,∞) kv(t)kCn < +∞, the state of system (3.10) is uniformly bounded, i.e.

sup k(w(·, t), v(t))kH < +∞.

(3.12)

t∈[0,∞)

Proof. Since the “v-subsystem” of system (3.10) is independent of the “w-subsystem” and is just a finite dimensional system, the solution v ∈ C([0, ∞); Cn ) to “v-subsystem” is well defined. Consider the following system

  εt (x, t) = εxx (x, t),

 ε (0, t) = 0, ε (1, t) = −c ε(1, t) x x 1

(3.13)

with initial state ε(·, 0) = w(·, 0) − ψ(·, 0). It is well known that system (3.13) is associated with

a unique exponentially stable analytic semigroup solution ε(·, t) = eAt ε(·, 0), where the generator A : D(A) ⊂ L2 (0, 1) → L2 (0, 1) is defined by   Af = f 00 , ∀ f ∈ D(A), Define

 D(A) = {f ∈ H 2 (0, 1) | f 0 (0) = 0, f 0 (1) = −c f (1)}. 1 w(x, t) = Ψ(x)v(t) + ε(x, t),

x ∈ [0, 1], t ≥ 0.

(3.14)

(3.15)

Then it is easy to verify that such defined (w, v) ∈ C([0, ∞); H) is the solution of system (3.10).

Since the operator A generates an exponentially stable analytic semigroup on L2 (0, 1), there exist positive constants LA and ωA such that kε(·, t)k ≤ LA e−ωA t kε(·, 0)k, t ≥ 0. 6

(3.16)

Moreover, the analyticity of eAt implies that the solution ε(·, t) = eAt ε(·, 0) is classical on (0, ∞)

even if ε(·, 0) ∈ L2 (0, 1) [23, p.104, Corollary 1.5]. Therefore, for any t ≥ t0 > 0, kεt (·, t)k = kεxx (·, t)k ≤ LA kεxx (·, t0 )k e−ωA t .

(3.17)

By using Poincar´e’s inequality along with boundary conditions, we have 1 1 |εx (1, t)| + kεx (·, t)k ≤ (2 + )kεxx (·, t)k, c1 c1

|ye (t)| = |ε(0, t)| ≤

(3.18)

which, together with (3.17), leads to (3.11) easily. Finally, the uniformly boundedness (3.12) can be obtained by (3.15), (3.16) and the assumption supt∈[0,∞) kv(t)kCn < +∞. Thus the proof is completed. 

4

Observer design

This section is devoted to estimate the state of system (2.4). We use the method of trajectory planning again. Suppose that the trajectory satisfies   φt (x, t) = φxx (x, t), x ∈ (0, 1), t > 0,

 φ(0, t) = P v(t), φ (0, t) = P v(t), t ≥ 0, 1 x 2

(4.1)

where P1 , P2 ∈ C1×n will be determined later. Similar to the process (3.2) to (3.6), system (4.1) has a special solution φ(x, t) = Φ(x)v(t), where 1

1

Φ(x) = P1 cosh(xG 2 ) + P2 xG(xG 2 ), x ∈ [0, 1].

(4.2)

Choose P1 and P2 specially so that 1

1

1

P1 G 2 sinh(G 2 ) + P2 cosh(G 2 ) = 0,

(4.3)

which results in φx (1, t) = 0. Define the transform z(x, t) = w(x, t) − φ(x, t), x ∈ [0, 1], t ≥ 0. Then system (2.4) is transformed into   zt (x, t) = zxx (x, t),     zx (0, t) = Q1 v(t), zx (1, t) = u(t),      y (t) = F v(t) − z(0, t), e 1

(4.4)

(4.5)

where

F1 = F − P1 , Q1 = Q − P2 .

7

(4.6)

It’s easy to see F, Q in (2.4) are fixed whereas F1 , Q1 in (4.5) can be regulated by selecting proper P1 , P2 . The following state observer for systems (2.3) and (4.5) is naturally designed:   zˆt (x, t) = zˆxx (x, t),     zˆx (0, t) = c2 [ye (t) + zˆ(0, t)], zˆx (1, t) = u(t),      vˆ˙ (t) = Gˆ v (t) + K[ye (t) − F1 vˆ(t) + zˆ(0, t)],

(4.7)

where c2 > 0 is a tuning parameter and K ∈ Cn×1 such that the matrix G − KF1 is Hurwitz. Choose P1 and P2 specially such that

Q1 − c2 F1 = Q − P2 − c2 (F − P1 ) = 0.

(4.8)

In this way, the measurement/disturbance non-collocated configuration in original system (2.4) can be overcome. Indeed, we let z˜(x, t) = z(x, t) − zˆ(x, t), v˜(t) = v(t) − vˆ(t), x ∈ [0, 1], t ≥ 0. Then the observation error is governed by     z˜t (x, t) = z˜xx (x, t),   z˜x (0, t) = c2 z˜(0, t), z˜x (1, t) = 0,      v˜˙ (t) = G˜ ˜ v (t) + K z˜(0, t),

(4.9)

(4.10)

where

˜ = G − KF1 . G

(4.11)

˜ is Hurwitz, there exists a Hermitian positive definite matrix P such that Since G ˜ ∗P + P G ˜ = −2In . G We consider system (4.10) in state space H with inner product Z 1 h(f1 , h1 ), (f2 , h2 )iH = δ1 f1 (x)f2 (x)dx + hh1 , h2 iP , ∀ (fi , hi ) ∈ H, i = 1, 2,

(4.12)

(4.13)

0

where hh1 , h2 iP = h∗1 P h2 and δ1 is a positive constant that is large enough and will be determined

later.

Now, we prove that the observation error (4.10) decays to zero exponentially in H. Define the

operator A : D(A) ⊂ H → H by    ˜ + Kf (0) , ∀ (f, h) ∈ D(A),  A(f, h) = f 00 , Gh   D(A) = (f, h) ∈ H | f ∈ H 2 (0, 1), f 0 (1) = 0, f 0 (0) = c2 f (0) .

(4.14)

Then, system (4.10) can be written abstractly as

d (˜ z (·, t), v˜(t)) = A(˜ z (·, t), v˜(t)). dt 8

(4.15)

˜ is Hurwitz. Then, the operator A defined by Lemma 4.1. Suppose that c2 > 0 and the matrix G (4.14) generates an exponentially stable analytic semigroup eAt on H.

Proof. We first claim that, there exists constant ϑ1 ∈ (0, π2 ) such that for any ϑ ∈ [0, ϑ1 ), the

operator e±iϑ A generates a C0 -semigroup of contractions on H. In fact, for any (f, h) ∈ D(A), by

(4.12), we have D E e±iϑ A(f, h), (f, h)

H

˜ + Kf (0)), (f, h)iH = he±iϑ (f 00 , Gh h i ˜ hiP + hf (0)K, hiP . = e∓iϑ − c2 δ1 |f (0)|2 − δ1 kf 0 k2 + hGh,

Consequently, D E Re e±iϑ A(f, h), (f, h)

H

  ˜ hiP | + |hf (0)K, hiP |. ≤ cos ϑ −c2 δ1 |f (0)|2 − δ1 kf 0 k2 − khk2Cn + | sin ϑImhGh,

(4.16)

(4.17)

By a simple computation, we can choose proper constant ϑ1 ∈ (0, π2 ) such that cos ϑ − sin ϑkP k2 kGk2 > 0, ∀ 0 ≤ ϑ < ϑ1 .

(4.18)

Using Young’s inequality, we get hf (0)K, hiP ≤

|f (0)| + ζ1 kKk2Cn kP k22 khk2Cn , ∀ ζ1 > 0. 4ζ1

(4.19)

Let 0 < ζ1 < (cos ϑ − sin ϑkP k2 kGk2 )(kKk2Cn kP k22 )−1 and choose δ1 properly such that δ1 c2 cos ϑ −

1 > 0. 4ζ1

There exists a positive constant M1 such that D E Re e±iϑ A(f, h), (f, h) ≤ −κ1 khk2Cn − κ2 |f (0)|2 − δ1 cos ϑkf 0 k2 ≤ −M1 k(f, h)k2H , H

where


 κ1 = cos ϑ − sin ϑkP k2 kGk2 − ζ1 kKk2Cn kP k22 ) > 0,    κ = δ c cos ϑ − 1 > 0, 2 1 2 4ζ1

(4.20)

(4.21)

(4.22)

and the Poincar´e’s inequality is used in the last step of (4.21). The inequality (4.21) implies that ˆ ∈ H, the operator e±iϑ A is dissipative in H for any 0 ≤ ϑ < ϑ1 . On the other hand, for any (fˆ, h) ˆ to get we solve the equation A(f, h) = (fˆ, h) Z xZ τ    f (x) = f (0) + fˆ(s)dsdτ,    0 1   Z 1 1 ˆ f (0) = − f (s)ds,   c2 0      ˆ − Kf (0)], ˜ −1 [h h=G

(4.23)

which yields that A−1 ∈ L(H) is compact on H. Hence, so is for (e±iϑ A)−1 . Hence, D(A) is densely defined in H owing to [30, p.71, Proposition 3.1.6]. By Lumer-Phillips theorem ([23, p.14, 9

Theorem 1.4.3]) and (4.21), the operator e±iϑ A generates a C0 -semigroup of contractions on H. By

[7, p.101, Theorem 4.6], A generates an analytic semigroup on H. Moreover, eAt is exponentially

stable owing to (4.21). The proof is completed.

By a straightforward computation, the solution of equations (4.3) and (4.8) is found to be  1  P1 = (c2 F − Q) cosh G 2 [f0 (G)]−1 ,     1 1 (4.24) P2 = −(c2 F − Q)G 2 sinh G 2 [f0 (G)]−1 ,      f (G) = G 21 sinh(G 12 ) + c cosh(G 21 ), 0 2

provided matrix f0 (G) is invertible. It follows from (4.24) and (4.6) that 1

1

1

F1 = [F G 2 sinh G 2 + Q cosh G 2 ][f0 (G)]−1 .

(4.25)

The following Theorem shows that (4.7) can serve as an observer of systems (2.3) and (4.5). Theorem 4.1. Let P1 , P2 , F1 ∈ C1×n are defined by (4.24) and (4.25). Suppose that c2 > 0, 1

1

1

(G, F G 2 sinh G 2 + Q cosh G 2 ) is observable and f0 (G) is invertible. Then there exists a vec-

tor K ∈ Cn×1 such that the matrix G − KF1 is Hurwitz. As a result, for any initial state (z(·, 0), v(0), zˆ(·, 0), vˆ(0)) ∈ H2 and u ∈ L2loc (0, ∞), system (2.3), (4.5) and (4.7) admits a unique

solution (z, v, zˆ, vˆ) ∈ C([0, ∞); H2 ). Moreover, there exist two positive constants L2 and ω2 , independent of t, such that

k(z(·, t) − zˆ(·, t), v(t) − vˆ(t))kH ≤ L2 e−ω2 t , t ≥ 0.

(4.26)

Proof. It is well known that, for any (z(·, 0), v(0)) ∈ H and u ∈ L2loc (0, ∞), system (2.3) and (4.5) admits a unique solution (z, v) ∈ C([0, ∞); H).

1

1

1

Since the matrix f0 (G) is invertible, (4.25) implies that (G, F G 2 sinh G 2 + Q cosh G 2 ) is ob-

servable if and only if (G, F1 ) is observable. Hence, there exists a vector K ∈ Cn×1 such that the

matrix G − KF1 is Hurwitz. It follows from Lemma 4.1 that error system (4.10) with initial state

(˜ z (·, 0), v˜(0)) = (z(·, 0) − zˆ(·, 0), v(0) − vˆ(0)) admits a unique solution (˜ z , v˜) ∈ C([0, ∞); H) such that

k(˜ z (·, t), v˜(t))kH ≤ LA e−ωA t k(˜ z (·, 0), v˜(0))kH .

(4.27)

With (z, v, z˜, v˜) at hand, we can define zˆ(x, t) = z(x, t) − z˜(x, t), vˆ(t) = v(t) − v˜(t), x ∈ [0, 1], t ≥ 0.

(4.28)

It is easy to check that such defined (z, v, zˆ, vˆ) ∈ C([0, ∞); H2 ) solves system (4.5), (2.3) and (4.7).

Finally, (4.26) can be obtained easily by (4.28) and Lemma 4.1. The proof is completed.

10

yref (t)

Exosystem (1) d(t) Control plant (2.1)

yp (t) = w(0, t)

u(t)

+

− ye (t)

Observer (4.7)

Figure 1: Block diagram of closed-loop system.

5

Closed-loop system

Replacing (v(t), w(1, t)) in (3.9) by (ˆ v (t), zˆ(1, t) + Φ(1)ˆ v (t)), the controller becomes   z (1, t) + (Φ(1) − Ψ(1))ˆ v (t)] + Ψx (1)ˆ v (t)   u(t) = −c1 [ˆ      = −c zˆ(1, t) + Υˆ v (t), 1

1 1   Υ = −c1 (Φ(1) − Ψ(1)) + Ψx (1) = F1 [f0 (G) + c1 (cosh G 2 + c2 G(G 2 ))],      1 1 1  f0 (G) = G 2 sinh(G 2 ) + c2 cosh(G 2 ),

(5.1)

under which we obtain the closed-loop system of (2.4):                                             

v(t) ˙ = Gv(t), ye (t) = F v(t) − w(0, t), d(t) = Qv(t), wt (x, t) = wxx (x, t), wx (0, t) = d(t), wx (1, t) = u(t),

(5.2)

zˆt (x, t) = zˆxx (x, t), zˆx (0, t) = c2 [ye (t) + zˆ(0, t)], zˆx (1, t) = u(t), vˆ˙ (t) = Gˆ v (t) + K[ye (t) − F1 vˆ(t) + zˆ(0, t)], u(t) = −c1 zˆ(1, t) + Υˆ v (t),

where K is a column vector such that the matrix G − KF1 is Hurwitz. For the sake of easy reading,

the corresponding tuning vectors are clustered as follows:  1 1 1   F1 = [F G 2 sinh G 2 + Q cosh G 2 ][f0 (G)]−1 ,    1 1 1 f0 (G) = G 2 sinh(G 2 ) + c2 cosh(G 2 ),      Υ = F [f (G) + c (cosh G 12 + c G(G 21 ))]. 1 0 1 2 1

1

(5.3)

1

Theorem 5.1. Suppose that c1 , c2 > 0, (G, F G 2 sinh G 2 + Q cosh G 2 ) is observable and f0 (G) is invertible. For any initial state (w(·, 0), v(0), zˆ(·, 0), vˆ(0)) ∈ H2 , the closed-loop system (5.2) with 11

setting (5.3) admits a unique solution (w, v, zˆ, vˆ) ∈ C([0, ∞); H2 ) such that, for any t0 > 0, |d(t) − Qˆ v (t)| + |ye (t)| ≤ L3 e−ω3 t , ∀ t ≥ t0 ,

(5.4)

where L3 and ω3 are positive constants independent of t. If supt∈[0,∞) kv(t)kCn < +∞, the state of closed-loop system is uniformly bounded

sup k(w(·, t), v(t), zˆ(·, t), vˆ(t))kH2 < +∞.

(5.5)

t∈[0,∞)

1

1

1

Remark 5.1. The assumption (G, F G 2 sinh G 2 + Q cosh G 2 ) is observable implies that the whole information of the exosystem is injected into the control plant, and the observer can therefore be possibly designed. When the assumption does not hold, there exist some dynamics of exosystem which can not be reflected by the measured output. We point out that the output regulation of heat system (2.1) can also be solved by internal model principle [26] and [28] where the regulation problem of an abstract linear system is considered. In the internal model principle, the forms of the controllers are already given. Different from this, we pay more attention on the process of controller design. In the spirit of estimation and cancelation strategy, the whole process of controller design (or observer design ) is demonstrated by the proper trajectory planning. Before proving Theorem 5.1, we first consider the following transformed system   εt (x, t) = εxx (x, t),        ε (0, t) = 0, εx (1, t) = −c1 ε(1, t) + c1 z˜(1, t) − Υ˜ v (t),    x  z˜t (x, t) = z˜xx (x, t),       z˜x (0, t) = c2 z˜(0, t), z˜x (1, t) = 0,       v˜˙ (t) = G˜ ˜ v (t) + K z˜(0, t)

(5.6)

in the space X = L2 (0, 1) × H with inner product

h(g1 , f1 , h1 ), (g2 , f2 , h2 )iX = hg1 , g2 i + δ2 h(f1 , h1 ), (f2 , h2 )iH , ∀ (gi , fi , hi ) ∈ X , i = 1, 2,

(5.7)

where the inner product h·, ·iH is defined by (4.13) and δ2 > 0, large enough, will be determined by (5.13) below. System (5.6) can be written abstractly

d (ε(·, t), z˜(·, t), v˜(t)) = A (ε(·, t), z˜(·, t), v˜(t)), dt where the operator A : D(A ) ⊂ X → X is defined by 
00  A (g, f, h) = g , A(f, h) , ∀(g, f, h) ∈ D(A ),    n  D(A ) = (g, f, h) ∈ X | g ∈ H 2 (0, 1), g 0 (0) = 0,   o    g 0 (1) = −c1 g(1) + c1 f (1) − Υh, (f, h) ∈ D(A) . 12

(5.8)

(5.9)

Lemma 5.1. Under the assumptions in Lemma 4.1 and c2 > 0. The operator A defined by (5.9) generates an exponentially stable analytic semigroup eA t on X . Proof. For any (g, f, h) ∈ D(A ) and 0 ≤ ϑ < ϑ1 where ϑ1 is the same as (4.18) and, we have D E Re e±iϑ A (g, f, h), (g, f, h) X h i (5.10) = Re he±iϑ g 00 , gi + δ2 he±iϑ A(f, h), (f, h)iH D E   ≤ cos ϑ −c1 |g(1)|2 − kg 0 k2 + |g(1)(c1 f (1) − Υh)| + δ2 Re e±iϑ A(f, h), (f, h) . H

By Young’s inequality, we get

Let

  |g(1)(c1 f (1) − Υh)| ≤ ζ2 |g(1)|2 + 2(4ζ2 )−1 c21 |f (1)|2 + kΥk2Cn khk2Cn , ∀ζ2 > 0. 0 < ζ2 < c1 cos ϑ.

Noting that f (1) = f (0) +

Z

1

(5.11)

(5.12)

f 0 (s)ds, (4.21) and (4.22), there exists δ2 > 0 such that

0

  δ2 κ2 − ζ2−1 c21 > 0,     δ2 κ1 − (2ζ2 )−1 kΥk2Cn > 0,      δ δ cos ϑ − ζ −1 c2 > 0, 2 1 1 2

which, together with (5.10), (4.21) and (5.11) shows that D E Re e±iϑ A (g, f, h), (g, f, h) ≤ 0. X

(5.13)

(5.14)

This inequality implies that the operator e±iϑ A is dissipative in X for any 0 ≤ ϑ < ϑ1 . On the ˆ ∈ X , we solve A (g, f, h) = (ˆ ˆ to obtain other hand, for any (ˆ g , fˆ, h) g , fˆ, h) Z 1Z τ    g(x) = g(1) − gˆ(s)dsdτ,    x 0   Z 1  1 g(1) = − gˆ(s)ds − c1 f (1) + Υh ,   c1 0      ˆ (f, h) = A−1 (fˆ, h).

(5.15)

Hence, A −1 ∈ L(X ) is compact on X . Therefore, σ(A ) consists of isolated eigenvalues of finite algebraic multiplicity only. Similar to Lemma 4.1, the operator A generates an analytic semigroup

eA t on X . Now, we claim that

σ(A ) ⊂ σ(A) ∪ σ(A),

(5.16)

where A and A are defined by (3.14) and (4.14), respectively. Indeed, suppose that λ ∈ σ(A )

and A (g, f, h) = λ(g, f, h) with 0 6= (g, f, h) ∈ D(A ). Comparing (3.14) with (4.14), we get (f, h) ∈ D(A). Hence, λ ∈ σ(A), provided (f, h) 6= 0. When we come across the case that 13

(f, h) = 0 and g 6= 0, it follows from (3.14) and (4.14) that g ∈ D(A) and λ ∈ σ(A). So (5.16) holds. Owing to (5.16) and the exponential stabilities of eAt and eAt , it follows from [23, p.118,

Theorem 4.3] that the operator A generates an exponentially stable analytic semigroup eA t on X . The proof is completed.

Proof of Theorem 5.1. By Lemma 5.1, system (5.6) with initial states   ε(·, 0) = w(·, 0) − Ψ(·)v(0),     z˜(·, 0) = w(·, 0) − Φ(·)v(0) − zˆ(·, 0),      v˜(0) = v(0) − vˆ(0)

(5.17)

admits a unique solution (ε, z˜, v˜) ∈ C([0, ∞); X ) such that for t > 0,

k(ε(·, t), z˜(·, t), v˜(t))kX ≤ LA e−ωA t k(ε(·, 0), z˜(·, 0), v˜(0))kX ,

(5.18)

where LA , ωA are positive constants. Moreover, owing to the analytic semigroup eA t and [23, p.104, Corollary 1.5], the solution of system (5.6) satisfies, for any given t0 > 0 k(εxx (·, t), z˜xx (·, t), v˜˙ (t))kX ≤ LA e−ωA t k(εxx (·, t0 ), z˜xx (·, t0 ), v˜(t0 ))kX , ∀ t ≥ t0 . In terms of (ε(·, t), z˜(·, t), v˜(t)), we define   w(·, t) = ε(·, t) + Ψ(·)v(t),     zˆ(·, t) = ε(·, t) + [Ψ(·) − Φ(·)]v(t) − z˜(·, t),      vˆ(t) = v(t) − v˜(t).

(5.19)

(5.20)

It is easy to check that such defined (w, v, zˆ, vˆ) ∈ C([0, ∞); H2 ) solves the closed-loop system (5.2).

Moreover, the boundedness (5.5) follows from (5.18) and (5.20). By using Poincar´e’s inequality along with (5.19) and boundary conditions of (5.6), for t ≥ t0 ,   1 −ωA t 1 n e k(εxx (·, t0 ), z˜xx (·, t0 ), v˜(t0 ))kX . |ε(0, t)| ≤ LA 2 + (1 + kΥkC ) + c1 c2 Owing to ye (t) = −ε(0, t), (5.4) can be obtained easily by (5.18) and (5.21).

(5.21) 

Remark 5.2. When we consider system with nonlinear uncertainties, the tracking problem of this paper becomes very complicated. [8] provide an alternative idea which comes from the active disturbance rejection control (ADRC). The fuzzy-model-based method (e.g. [31], [20], [32]) is also an alternative idea.

6

Application to harmonic signals

By the Fourier expansion ([1], [2]), the finite sum of harmonic signals can be considered as the approximation of periodic signal which is widely used in practice application. The result in Theorem 14

5.1 can cover the harmonic disturbance and harmonic reference signal, which have been considered in [12] and [13]. Indeed, suppose that the harmonic signal is y(t) = a0 +

m X

(aj sin ωj t + bj cos ωj t) ,

(6.1)

j=1

where the amplitudes aj , bj are unknown. Without loss generality, we suppose that 0 = ω0 < ω1 < ω2 < · · · < ωm , m ∈ N.

(6.2)

Then, we can define the matrices of the exosystem (2.3) as 





0

G = diag 0, G1 , G2 , ..., Gm , Gj =  −ωj

ωj 0

In this case, the harmonic signal can be generated by exosystem (2.3)



.

(6.3)

y(t) = F v(t), F = (F0 , F1 , F2 , · · · , Fm ) with F0 = 1, Fj = (1, 0), j = 1, 2, · · · , m,

(6.4)

where initial value is selected as v(0) = (a0 , b1 , a1 , · · · , bm , am )> . Lemma 6.1. Suppose that C = (c1 , c2 ) is a nonzero vector and Gj is given by (6.3). Then, the pair (Gj , C) is observable, j = 1, 2, · · · , m. Proof. A simple computation shows that    C c1  = det  det  CGj −c2 ωj

c2 c1 ωj




 = c21 + c22 ωj .

Noting that c21 + c22 6= 0, the Kalman rank condition for observability implies that (Gj , C) is

observable. The proof is completed.

Proposition 6.1. Suppose that the matrix G is defined by (6.3). (i). For any c2 > 0, the matrix 1

1

1

1

1

1

f0 (G) = G 2 sinh(G 2 ) + c2 cosh(G 2 ) is invertible; (ii). (G, F G 2 sinh G 2 + Q cosh G 2 ) is observable provided that 1

1

1

he1 , F G 2 sinh G 2 + Q cosh G 2 i2C2m+1 6= 0

(6.5)

and 1

1

1

1

1

1

he2j , F G 2 sinh G 2 + Q cosh G 2 i2C2m+1 + he2j+1 , F G 2 sinh G 2 + Q cosh G 2 i2C2m+1 6= 0

(6.6)

for any j = 1, 2, · · · , m, where ej = (0, · · · , 0, 1, 0 · · · , 0)> , j = 1, 2, · · · , 2m + 1. 1

1

(6.7)

1

Proof. Let f0 (λ) = λ 2 sinh(λ 2 ) + c2 cosh(λ 2 ) for any λ ∈ σ(G) where we keep fixed the choice of 1

branches of square root function λ 2 . By a simple computation, f0 (λ) 6= 0 for any λ ∈ σ(G) ⊂ iR.

This implies that the matrix f0 (G) is invertible according to [15, p.3, Definition 1.2]. The first part of this proposition is proved. The second result can be obtained directly by Lemma 6.1. 15

7

Example and numerical simulation

In this section, we make numerical simulation for system (5.2) validate the theoretical results. The reference signal yref and the disturbance d are chosen as yref (t) =

20 X

Aj sin ωj t, d(t) = 1,

(7.1)

j=1

where

Z 2 π 2 arcsin (sin x) sin jxdx, j = 1, 2, · · · , 20. ωj = j, Aj = π 0 π In this way, the reference yref can be considered as an approximation of a triangle wave with period 2π and amplitude 1. Let   G = diag(G0 , G1 , · · · , G20 ) ∈ R41×41 ,     F = (0, J1 , J2 , · · · , J20 ) ∈ R1×41 ,      Q = (1, 0, 0, · · · , 0, 0) ∈ R1×41 ,

(7.2)

where Jj = (1, 0) and Gj is defined by (6.3), j = 1, 2, · · · , 20. The reference and disturbance can be generated by exosystem (2.3) with initial state    v(0) = (1, v1 (0), v2 (0), · · · , v40 (0)),      k−1   8 (−1) 2 , if k is odd,   π2 k2 v2k−1 (0) = 0, v2k (0) =       0, if k is even,

k = 1, · · · , 20.

The initial states are chosen as w(x, 0) = x(x − 1), zˆ(x, 0) = 0 and vˆ(0) = 0. The corresponding tuning vectors F1 , Υ can be calculated by (5.3). and the parameters are chosen as c1 = c2 = 1, and K = F1∗ . By Proposition (6.1)-(ii), we obtain that G − KF1 is Hurwitz.

With the above setting, it is easy to check that all the assumptions in Theorem 5.1 are satisfied.

The finite difference scheme is adopted in discretization. The numerical results are programmed in Matlab. The time step and the space step are taken as 0.0001 and 0.02 respectively. The solution of closed-loop system (5.2) is plotted in Figure 2. It is seen that all states are bounded. The output tracking is plotted in Figure 3(a) and the disturbance estimation is plotted in Figure 3(b). Both of them show that the convergence is very effective and smooth. Remark 7.1. Different from the methods in [11] and [21] where only the state feedback is considered, the trajectory planning method in this paper is used to design the output feedback based controller. Comparing the recent result [16], the performance output considered in this paper is non-collocated to the control actuation. Moreover, we only need to measure the tracking error for controller design, whereas the output w(0, t) and reference signal are required to be known in [16]. Out results are also different from [13] where the error converges to zero asymptotically. Thanks to the trajectory planning method, we achieve exponential convergence of the tracking error.

16

(a) w(x, t).

(b) zˆ(x, t).

Figure 2: Simulations for system (5.2). 1.5

1.2

1 1 0.5 0

0.8

-0.5 0.6 -1 -1.5

0.4

-2 0.2 -2.5 -3

0

5

10

15

20

25

30

35

40

45

0

50

0

5

10

t/sec

15

20

25

30

35

40

45

50

t/sec

(a) Output tracking.

(b) Disturbance estimation.

Figure 3: Output tracking and disturbance estimation.

8

Concluding remarks

In this paper, a method of trajectory planning is used to deal with the non-collocated problem through the output tracking for a heat equation. Different from the existing results, both the disturbance and the performance output are non-collocated to the control actuation in heat system. The output feedback law is designed in terms of the tracking error only. Exponential stability of the tracking error and uniform boundedness of the state of the closed-loop system are proved mathematically and are validated by numerical simulations. 17

References [1] J. Chauvin, G. Corde, N. Petit, and P. Rouchon, Period input estimation for linear periodic systems: Automotive engine applications, Automatica, 43(2007), 971-980. [2] J. Chauvin, Observer design for a class of wave equation driven by an unknown periodic input, in Prof. 18th IFAC World Congress, 2011, 1332-1337. [3] W.B. Dunbar, N. Petit, P. Rouchon, and P. Martin, Motion planning for a nonlinear Stefan problem, ESAIM Control, Optimisation and Calculus of Variations, 9(2003), 275-296. [4] J. Deutscher, Finite-time output regulation for linear 2 × 2 hyperbolic systems using backstepping, Automatica, 75(2017), 54-62.

[5] J. Deutscher, Backstepping approach to the output regulation of boundary controlled parabolic PDEs, Automatica, 57(2015), 56-64. [6] L. Evans, Partial differential equations, Vol. 19 of Graduate studies in mathematics, American mathematical society, 1997. [7] K.J. Engel and R. Nagel, One-parameter semigroups for linear evolution equations, Vol. 194 of Graduate Texts in Mathematics, Springer, 2000. [8] H. Feng, B.Z. Guo, New unknown input observer and output feedback stabilization for uncertain heat equation, Automatica, 86(2017), 1-10. [9] H. Feng and B.Z. Guo, A new active disturbance rejection control to output feedback stabilization for a one-dimensional anti-stable wave equation with disturbance, IEEE Trans. Automat. Control, 62(2017), 3774-3787. [10] H. Feng, B.Z. Guo and X. H. Wu, Trajectory planning approach to output tracking for a 1-D wave equation, IEEE Trans. Automat. Control, to appear. [11] Y. Gao, P. Shi, H. Li, S. K. Nguang, Output tracking control for fuzzy delta operator systems with time-varying delays, J. Frankl. Inst., 352 (2015), 2951-2970. [12] W. Guo and M. Krstic, Finite-dimensional adaptive error feedback output regulation for 1D wave equation, IFAC PapersOnLine, 50(2017), 9192-9197. [13] W. Guo and B.Z. Guo, Performance output tracking for a wave equation subject to unmatched general boundary harmonic disturbance, Automatica, 68(2016), 194-202. ¨ [14] D.M. Hahn and M.N. Ozisik, Heat conduction. New jersey: John Wiley & Sons,inc. 2012. [15] N.J. Higham, Functions of Matrices Theory and Computation, SIAM, Philadelphia, 2008.

18

[16] F.F. Jin and B.Z. Guo, Performance boundary output tracking for one-dimensional heat equation with boundary unmatched disturbance, Automatica, 96(2018),1-10. [17] M. Krstic, B.Z. Guo, A. Balogh, and A. Smyshlyaev, Output-feedback stabilization of an unstable wave equation, Automatica, 44(2008), 63-74. [18] M. Krstic and A. Smyshlyaev, Boundary Control of PDEs: A Course on Backstepping Designs, Philadelphia, PA, USA: SIAM, 2008. [19] B. Laroche, P. Martin, and P. Rouchon, Motion planning for the heat equation, International Journal of Robust and Nonlinear Control, 10(2000), 629-643. [20] H. Li, Y. Gao, L. Wu, and H. K. Lam, Fault Detection for T-S Fuzzy Time-Delay Systems: Delta Operator and Input-Output Methods, IEEE Trans. Automat. Control, 45(2015), 229-240. [21] H. Li, Y. Wang, L. Xie, Output tracking control of Boolean control networks via state feedback: Constant reference signal case, Automatica, 59(2015), 54-59. [22] V. Natarajan, D.S. Gilliam, and G. Weiss. The state feedback regulator problem for regular linear systems, IEEE Trans. Automat. Control, 59(2014), 2708-2723. [23] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. [24] L. Paunonen, Output regulation theory for linear systems with infinite dimensional and periodic exosystems, PhD thesis, Tampere University of Technology, 2011. [25] L. Paunonen, Controller design for robust output regulation of regular linear systems, IEEE Trans. Automat. Control, 61(2016), 2974-2986. [26] L. Paunonen and S. Pohjolainen, The internal model principle for systems with unbounded control and observations, SIAM J. Control Optim., 52(2014), 3967-4000. [27] P. Rouchon, Motion planning, equivalence, infinite dimensional systems, International Journal of Applied Mathematics and Computer Science, 11(2001), 165-188. [28] R. Rebarber and G. Weiss, Internal model based tracking and disturbance rejection for stable well-posed systems, Automatica, 39(2003), 1555-1569. [29] A. Smyshlyaev and M. Krstic, Boundary control of an anti-stable wave equation with antidamping on the uncontrolled boundary, Systems Control Lett., 58(2009), 617-623. [30] M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkh¨auser, Basel, 2009.

19

[31] Y. Wang, Y. Gao, H. R. Karimi, H. Shen, and Z. Fang, Sliding mode control of fuzzy singularly perturbed systems with application to electric circuit, IEEE Trans. Syst., Man, Cybern., Syst., 48(2018), 1667-1675. [32] X.J. Wei, N. Chen, Composite hierarchical anti-disturbance control for nonlinear systems with DOBC and fuzzy control, Int. J. Robust Nonlinear Control, 24(2014), 362-373. [33] X.J. Wei, Z.J. Wu and H. R. Karimi, Disturbance observer-based disturbance attenuation control for a class of stochastic systems, Automatica, 63(2016), 21-25. [34] H. Zhang, X. J. Wei, L. Zhang and M. Tang, Disturbance rejection for nonlinear systems with mismatched disturbances based on disturbance observer, Journal of the Franklin Institute, 354(2017), 4404-4424. [35] X. Zhang, H. Feng and S. Chai, Performance output exponential tracking for a wave equation with a general boundary disturbance, Systems Control Lett., 98(2016), 79-85. [36] H.C. Zhou and B.Z. Guo, Unknown input observer design and output feedback stabilization for multi-dimensional wave equation with boundary control matched uncertainty, J. Differential Equations, 263(2017), 2213-2246.

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