1 Introduction

Output feedback exponential stabilization for 1-D

unstable wave equations with boundary control

[0.6ex] matched disturbance

[4ex] Hua-Cheng Zhou,    George Weiss  \@footnotetextThis work was supported by the Israel Science Foundation under grant 800/14. \@footnotetextH.-C. Zhou (hczhou@amss.ac.cn) and G. Weiss (gweiss@eng.tau.ac.il) are with the School of Electrical Engineering, Tel Aviv University, Ramat Aviv, Israel, 69978.

Abstract: We study the output feedback exponential stabilization of a one-dimensional unstable wave equation, where the boundary input, given by the Neumann trace at one end of the domain, is the sum of the control input and the total disturbance. The latter is composed of a nonlinear uncertain feedback term and an external bounded disturbance. Using the two boundary displacements as output signals, we design a disturbance estimator that does not use high gain. It is shown that the disturbance estimator can estimate the total disturbance in the sense that the estimation error signal is in . Using the estimated total disturbance, we design an observer whose state is exponentially convergent to the state of original system. Finally, we design an observer-based output feedback stabilizing controller. The total disturbance is approximately canceled in the feedback loop by its estimate. The closed-loop system is shown to be exponentially stable while guaranteeing that all the internal signals are uniformly bounded.

Keywords: Disturbance rejection, output feedback controller, unstable wave equation, exponential stabilization

AMS subject classifications: 37L15, 93D15, 93B51, 93B52.

## 1 Introduction

In this paper, we are concerned with the following one-dimensional wave equation:

 ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩wtt(x,t)= wxx(x,t),\omit\span\@@LTX@noalign\vskip5.690551pt\omitwx(0,t)= −qw(0,t),\omit\span\@@LTX@noalign\vskip5.690551pt\omitwx(1,t)= u(t)+f(w(⋅,t),wt(⋅,t))+d(t),\omit\span\@@LTX@noalign\vskip5.690551pt\omitw(x,0)= w0(x),   wt(x,0) = w1(x),\omit\span\@@LTX@noalign\vskip5.690551pt\omitym(t)= (w(0,t) ,w(1,t)),

where , , is the state, is the control input signal, and is the output signal, that is, the boundary traces and are measured. The equation containing the constant creates a destabilizing boundary feedback at that acts like a spring with negative spring constant. is an unknown possibly nonlinear mapping that represents the internal uncertainty in the model, and represents the unknown external disturbance, which is only supposed to satisfy . For the sake of simplicity, we denote

 F(t):= f(w(⋅,t),wt(⋅,t))+d(t) (1.1)

and we call this signal the total disturbance. We often write instead of .

We consider system (1) in the state Hilbert space with the inner product given by

 ⟨(ϕ1,ψ1),(ϕ2,ψ2)⟩H =∫10[ϕ′1(x)¯¯¯¯¯¯¯¯¯¯¯¯¯ϕ′2(x)+ψ1(x)¯¯¯¯¯¯¯¯¯¯¯¯¯ψ2(x)]d x+ϕ1(0)¯¯¯¯¯¯¯¯¯¯¯¯ϕ2(0) . (1.2)

The objective of this paper is to design a feedback controller which generates the control signal , using only the measurements , such that the state of the closed-loop system (that includes the state of the system (1)) converges to zero, exponentially. Later in the paper we shall also discuss a related problem, where the negative spring is replaced by a negative damper. More precisely, on the right hand-side of the equation in (1)) containing , we have (instead of ). We shall solve the exponential stabilization problem also for this alternative nonlinear wave system (5). These results have been announced (without proof) in the IFAC conference paper [37].

For simplicity of implementation, it is desirable to use a small number of input and output signals for output feedback stabilization. For the disturbance free situation (that is, and ), the stabilization of the system (1) was first investigated in [22], who used two measurement signals to obtain an exponentially stable closed-loop system. Using only one displacement signal as measurement, strong stability of the closed loop system was achieved in [15], using Lyapunov functionals. In the recent paper [12], the output signal is only one displacement signal and an exponentially stabilizing controller is designed by using a new “backstepping” method. However, when the total disturbance acts at the control end, the stabilization problem for (1) becomes much more difficult. Here we present a dynamic compensator which employs a disturbance estimator described by partial differential equations (PDEs) and full state feedback based on the observer state. Our compensator consists of two parts: the first part is to cancel the total disturbance by applying the active disturbance rejection control (ADRC) strategy, which is an unconventional design strategy first proposed by Han in 1998 [19]; the second part is to stabilize the system by using the classic backstepping approach. The stabilization problem of system (1) has been considered first in [14], where the vector of output measurement was taken to be and the disturbance has the following form:

 d(t) = m∑j=1[θjsinαjt+ϑjcosαjt],   t≥0 ,

with known frequencies and unknown amplitudes , and the resulting closed-loop system is asymptotically stable. Obviously, the disturbance signal in this paper is more general than the one described above. Recently, the stabilization problem of system (1) with , has been investigated in [10], where the output measurements are , their result is that the closed-loop system is asymptotically stable. The output feedback of [10] uses one more measurement than [14]. Apart from the more general external disturbance, another point that is different here from [14, 10] is that the closed-loop system in this paper is exponentially stable and we do not require to measure the velocity (or ) which is hard to measure [9]. In this paper, we only use two scalar signals (the components of ) and this is a minimal set of measurement signals. As shown in Figure 1, we apply the control force to deal with both the internal uncertainty and the unknown external disturbance .

Many control methods have been applied to deal with uncertainties in PDE systems. The internal model principle, a classical method to cope with uncertainty, has been generalized to infinite-dimensional systems [3, 29, 27, 24]. In [29], the tracking and disturbance rejection problems for infinite-dimensional linear systems, with reference and disturbance signals that are finite superpositions of sinusoids, are considered. The results are applied to some PDEs including the noise reduction in a structural acoustics model described by a two-dimensional PDE. An interesting PDE example in [29] is disturbance rejection in a coupled beam where the disturbance and control are not matched. Very recently, the backstepping approach has been used to achieve output regulation for the one-dimensional heat equation in [7, 8], and the one-dimensional Schrödinger equation in [36]. For a stochastic PDE, an optimal control problem constrained by uncertainties in system and control is addressed in [30]. An adaptive design is exploited in [1, 21] for dealing with the anti-stable wave equation with unknown anti-damping coefficient. In [13], a boundary control based on the Lyapunov method is designed for the one-dimensional Euler-Bernoulli beam equation with spatial and boundary disturbances. However, there are not so many works, to the best of our knowledge, on exponential stabilization (instead of reference tracking) of PDEs with disturbance by using output feedback. Sliding mode control that is inherently robust is the most popular approach that can achieve exponential stability for infinite-dimensional systems but most often, the literature considers state feedback controllers [28, 5, 16, 34], while here we aim for output feedback.

Output feedback stabilization for one-dimensional anti-stable wave equation has been considered in [17], where a new type of observer has been constructed by using three output signals to estimate the state first and then estimate the disturbance via the state of the observer through an extended state observer (ESO). However, the initial state is required to be smooth in [17] and they obtain asymptotic stability (not exponential, like here). In the recent paper [11] the authors continue to investigate this question and introduce a new disturbance estimator which is different from the traditional one, the smoothness requirement on the initial state being removed. In [11], still three output signals are used as inputs to the controller and the controller achieves asymptotic stability of the closed-loop system. In this paper we consider the output feedback stabilization for a one-dimensional unstable (or anti-stable) wave equation by using two signals only, which is an improvement, and in addition we achieve exponential stability of the state of the controlled original systems, which is stronger than asymptotic stability.

Define the operators ,   by

 ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩A(ϕ,ψ) = (ψ,ϕ′′)∀(ϕ,ψ)∈D(A),\@@LTX@noalign\vskip5.690551pt\omitD(A)={(ϕ,ψ)∈H2(0,1)×H1(0,1) | ϕ′(0)=ϕ(0),ϕ′(1)=0},\@@LTX@noalign\vskip5.690551pt\omitB1=(0,−δ0),    B2=(0,δ1),

where is the Dirac pulse at , with a suitable interpretation. It can be shown (see [25, Example 5.2] for details) that , and

 B∗1(ϕ,ψ) = −ψ(0),   B∗2(ϕ,ψ) = ψ(1)∀(ϕ,ψ)∈D(A∗). (1.3)

We often write a pair as a column vector . The system (1) can be rewritten as

 d d t[w(⋅,t)wt(⋅,t)]= A[w(⋅,t)wt(⋅,t)]−B1((q+1)w(0,t))+B2[f([w(⋅,t)wt(⋅,t)])+u(t)+d(t)]. (1.4)

The equivalence is meant in the algebraic sense, without any reference to existence or uniqueness of solutions, see Remark 10.1.4 in [33]. The proof of the equivalence between (1) and (1.4) uses the theory of boundary control systems in [33, Section 10.1], and the details (for a slightly different system) are in [25, Example 5.2], where the notation and is used in place of and (in this order). About existence and uniqueness of solutions we have the following proposition, whose proof is given in the Appendix.

###### Proposition 1.1.

The above operator generates a unitary group on and are admissible control operators for it. Suppose that satisfies a global Lipschitz condition on and . Then for any and , there exists a unique global solution to (1) such that .

The paper is organized as follows: We consider the exponential stabilization of the unstable wave equation (1) in Sections 2 to 4. More precisely, in Section 2 we desgin an infinite-dimensional total disturbance estimator that does not use high gain, for the system (1). We propose a state observer based on this estimator and develop an output feedback stabilizing controller by compensating the total disturbance in Section 3. The exponential stability of the resulting closed-loop system for (1) is proved in Section 4. Section 5 is devoted to the output feedback exponential stabilization of the alternative anti-stable wave equation mentioned earlier (with the negative damper).

## 2 Disturbance estimator design

In this section, our objective is to design a total disturbance estimator using the input and output signals of the system (1).

###### Remark 2.1.

We explain the need for a disturbance estimator on a simple finite dimensional example. Let , . Consider the system

 ˙x(t) = Ax(t)+Bd(t) (2.1)

where is the state trajectory at time and is the disturbance signal at time . Suppose that is stable (Hurwitz). The solution is given by

 x(t)−eAtx(0) = ∫t0eA(t−s)d(s)d s = eAt2∫t20eA(t2−s)d(s)d s+∫tt2eA(t−s)d(s)d s .

From here, it is easy to verify that as if . Therefore, to design a stabilizing control law for , it suffices to find a control law that generates such that .

For many boundary control systems, the control operator is unbounded but admissible for the underlying operator semigroup. For more on the admissibility concept we refer for instance to [33]. When takes values in a Hilbert space , generates an exponentially stable operator semigroup on and is admissible, we still have a stability result similar to Remark 2.1, see the following lemma. For related results see [23, 20]. As is customary, we denote by the dual of with respect to the pivot space , see [33].

###### Lemma 2.1.

Let be the generator of an exponentially stable operator semigroup on the Hilbert space . Assume that , are admissible control operators for ( are Hilbert spaces). Then the initial value problem

 ˙x(t) = Ax(t)+n∑i=1Biui(t),   x(0)=x0,   ui∈L2loc([0,∞),Ui),\vspace−1mm

admits a unique solution , and if , , then is bounded. If for each index , either or holds, then as . Moreover, if there exist two constants such that , , then   for some .

###### Proof.

Due to the admissibility, by [33, Proposition 4.2.5.], the solution is a continuous -valued function of given by

 x(t) = eAtx0+n∑i=1∫t0eA(t−s)Biui(s)d s .\vspace−1mm

By assumption, there exist constants such that for all . Thus, by superposition, we only have to prove the statements in the lemma for one of the integral terms in the above sum, (with fixed).

Suppose that . Since is -admissible for by virtue of [35, Remark 4.7], it follows from [35, Remark 2.6] that there exists a constant independent of and of such that is bounded:  .

Now suppose that or . For any , there exists such that

 ∥ui∥L2([tσ,∞),Ui) ≤ σ,  or    ∥ui∥L∞([tσ,∞),Ui) ≤ σ .

If then it follows from [35, Remark 2.6] that for any ,

 ∥∥∥∫ttσeA(t−s)Biui(s)d s∥∥∥X ≤ L2∥ui∥L2([tσ,∞),Ui) ≤ L2σ,\vspace−1mm (2.2)

where is a constant that is independent of and of . If , then by [35, Remark 2.6], the -admissibility of implies that for any ,

 ∥∥∥∫ttσeA(t−s)Biui(s)d s∥∥∥X ≤ L1∥ui∥L∞([tσ,∞),Ui) ≤ L1σ .\vspace−1mm (2.3)

Using the exponential stability of again, we have that for any ,

 ∥eA(t−tσ)xi(tσ)∥X ≤ M1e−μ1(t−tσ)∥xi(tσ)∥X . (2.4)

Since  ,  it follows from (2.2) or (2.3), and (2.4) that for ,

 ∥xi(t)∥X ≤ M1e−μ1(t−tσ)∥xi(tσ)∥X+max{L1,L2}σ .

This shows that  . Since was arbitrary, we conclude that the last limsup is 0, whence as .

For the last part of the lemma, suppose that there exist such that . Choose a number , then still generates an exponentially stable operator semigroup. Define the functions and by

 xμi(t) = eμtxi(t) ,uμi(t) = eμtui(t) ,

then it is easy to see that the differential equation holds. Since is bounded, by an argument used at the beginning of this proof (with and in place of and ), there exists such that  . Clearly this implies that tends to zero at the exponential rate . ∎

Now we design a total disturbance estimator for the system (1). This is an infinite dimensional system whose state consists of the functions defined on :

 ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩vtt(x,t)= vxx(x,t),\omit\span\@@LTX@noalign\vskip5.690551pt\omitvx(0,t)= −qw(0,t)+c1[v(0,t)−w(0,t)],    vx(1,t)= u(t)−Wx(1,t),\omit\span\@@LTX@noalign\vskip5.690551pt\omitv(x,0)= v0(x),    vt(x,0) = v1(x),\omit\span\@@LTX@noalign\vskip5.690551pt\omitztt(x,t)= zxx(x,t),\omit\span\@@LTX@noalign\vskip5.690551pt\omitzx(0,t)= c11−c0z(0,t)+c01−c0zt(0,t),    z(1,t)= −v(1,t)+w(1,t)−W(1,t),\omit\span\@@LTX@noalign\vskip5.690551pt\omitz(x,0)= z0(x),    zt(x,0)= z1(x),\omit\span\@@LTX@noalign\vskip5.690551pt\omitWt(x,t)= −Wx(x,t),\omit\span\@@LTX@noalign\vskip5.690551pt\omitW(0,t)= −c0[v(0,t)−w(0,t)],W(x,0) = W0(x),

where and are two positive design parameters, , is the initial state of the disturbance estimator and its input signals are , and . The output of this estimator is .

###### Remark 2.2.

Before going into the tedious technical details, we give an informal overview of how the total disturbance estimator (2) works. The “-part” of (2) is used to channel the total disturbance from the original system to an exponentially stable wave equation with state , where , described in (2). (The equations (2) contain also a -part, but from an input-output point of view, this -part is irrelevant.) The effect of is cancelled in the estimator, so that has no influence on . The wave equation system with state has input and output and it represents from an input-output view the linear part of the plant and the “-part” of (2), taken together, see Figure 2. This is a well-posed boundary control system (in the sense of [33, Definition 10.1.7]), with a bounded observation operator, so that for large , its transfer function satisfies , see for instance [33, Proposition 4.4.6].

The -part of (2) is in fact the same boundary control system as the one just described, but with the roles of input and output reversed. This would be flow inversion in the sense of [32], except that the -part is ill-posed. Indeed, its transfer function is , and from our estimate on it follows that is not proper. Overall, the transfer function from to is the constant 1. The difference depends linearly on the deviation between the initial state of the -part of (2) and the initial state of the -part of (2). Since the -part, in the absence of any input (i.e., when ) is exponentially stable, and its observation operator giving is admissible (as we shall see in Lemma 2.3), it follows that . The overall linear system shown in Figure 2 (with input and output ) is well-posed. If is globally Lipschitz, then also the overall nonlinear system (with input and output ) is well-posed (due to Proposition 1.1).

Figure 2. The total disturbance estimator connected to the plant. The -part of the disturbance estimator (2) is the (ill-posed) flow inverse of the wave system (2) (which has input and output ). The system with input and output is linear and its transfer function is .

Now we start providing the technical details for the operation of the total disturbance estimator. Consider the plant (1) coupled with the estimator (2) and denote

 ˆv(x,t) = v(x,t)−w(x,t). (2.5)

Then it is easy to verify that the subsystem with state satisfies

 ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩ˆvtt(x,t) = ˆvxx(x,t),\@@LTX@noalign\vskip5.690551pt\omitˆvx(0,t)=c1ˆv(0,t),   ˆvx(1,t)+Wx(1,t) = −F(t),\@@LTX@noalign\vskip5.690551pt\omitWt(x,t) = −Wx(x,t),   W(0,t) = −c0ˆv(0,t) ,

where is the total disturbance from (1.1). It will be convenient to change variables once more, by introducing the notation

 p(x,t) = −ˆv(x,t)−W(x,t) ,˜c0 = c01−c0 ,˜c1 = c11−c0 , (2.6)

then from the last part of (2) we see that and hence (using that ) the subsystem with state is governed by

 ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩ptt(x,t) = pxx(x,t),\@@LTX@noalign\vskip5.690551pt\omitpx(0,t) = ˜c1p(0,t)+˜c0pt(0,t),   px(1,t) = F(t),\@@LTX@noalign\vskip5.690551pt\omitWt(x,t) = −Wx(x,t),    W(0,t)% = ˜c0p(0,t),

with the initial state , . The following lemma states some stability properties of the system (2).

###### Lemma 2.2.

Suppose that (or ), is continuous and that (1) admits a unique solution which is bounded. For any initial state with the compatibility condition , there exists a unique solution to (2) and

 supt≥0∥(ˆv(⋅,t),ˆvt(⋅,t),W(⋅,t))∥H×H1(0,1) < ∞ . (2.7)

If we assume further that and , then

 limt→∞∥(ˆv(⋅,t),ˆvt(⋅,t),W(⋅,t))∥H×H1(0,1) = 0 . (2.8)

If we assume that and , then there exist two constants such that

 ∥(ˆv(⋅,t),ˆvt(⋅,t),W(⋅,t))∥H×H1(0,1) ≤ Me−μt   ∀ t≥0 . (2.9)
###### Proof.

We shall use the equivalent system (2). We define the operators and (that resemble and from (1.4)) by

 ⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩A(ϕ,ψ) = (ψ,ϕ′′)∀(ϕ,ψ)∈D(A) ,B = (0,δ1),\@@LTX@noalign\vskip5.690551pt\omitD(A)={(ϕ,ψ)∈H2(0,1)×H1(0,1) | ϕ′(0)=˜c1ϕ(0)+˜c0ψ(0) , ϕ′(1)=0}.

Then the “-part” of (2) can be written in abstract form as

It is well-known [18, Theorem 2.1] that generates an exponentially stable operator semigroup on and is admissible for . Since is continuous and is bounded, we have . Thus, by or by , it follows from Lemma 2.1 that the “-part” of (2) admits a unique bounded solution, so that there exists a constant such that

 supt≥0∥(p(⋅,t),pt(⋅,t))∥H≤M1 . (2.10)

We claim that is uniformly bounded for all . To prove this, first we show that for all ,

 ∫10p2t(0,t−x)d x ≤ 3maxs∈[t−1,t]∥(p(⋅,t),pt(⋅,t))∥2H . (2.11)

Indeed, define

 ρ(t) = 2∫10(x−1)pt(x,t)px(x,t)d%x .

Then . Computing along the solution of the “-part” of (2), using that , yields

 ˙ρ(t)=p2x(0,t)+p2t(0,t)−∫10[p2x(x,t)+p2t(x,t)]d x≥p2t(0,t)−∫10[p2x(x,t)+p2t(x,t)]d x,

which implies that, for ,

 ∫tt−1p2s(0,s)d s ≤∫tt−1∥(p(⋅,s),ps(⋅,s))∥2Hd x+ρ(t)−ρ(t−1)≤ 3maxs∈[t−1,t]∥(p(⋅,t),pt(⋅,t))∥2H . (2.12)

On the other hand, since for any ,  , we obtain (2.11). Define the function

 W(x,t) = ⎧⎪⎨⎪⎩˜c0p(0,t−x),t≥x,\omit\span\@@LTX@noalign\vskip5.690551pt\omitW0(x−t),x>t.\vspace−1mm

Then a simple computation shows that solves the “-part” of (2). It follows from the Sobolev embedding theorem, the last part of (2) and (2.10) that

 |W(0,t)|= ˜c0|p(0,t)|≤˜c0∥p(⋅,t)∥H1(0,1)≤ ˜c0∥(p(⋅,t),pt(⋅,t))∥H≤˜c0M1 . (2.13)

From (2) we derive that for ,  . Then the boundedness of follows from here, using (2.10), (2.13) and (2.11).

Since and , we have

 supt≥0∥(ˆv(⋅,t),ˆvt(⋅,t))∥H≤supt≥0[∥(p(⋅,t),pt(⋅,t))∥H+∥(W(⋅,t),Wx(⋅,t))∥H] .

This with (2.10) and the boundedness of implies that (2.7) holds.

Next, suppose that and . It follows from Lemma 2.1 that the “-part” of (2) admits a unique solution satisfying

 limt→∞∥(p(⋅,t),pt(⋅,t))∥H= 0 .\vspace−2mm (2.14)

By (2.11) and (2.14), we get