Output feedback regulation is a classical topic of control theory and engineering practice. A feedback regulator is designed for the controlled system, so that the signal to be regulated can track the target reference signal and the system keeps stable. Many practical problems such as aircraft landing, missile tracking and robot control all depend on output regulation. The wave equation with anti-damping term can simulate many engineering problems like pipeline combustion, acoustic instability or stick slip instability during drilling, which is of great significance to study the output regulation of anti-stable wave model.
In the last few years, a quiet great progress has been made both in the output feedback stabilization  -  and the output feedback regulation  - .  realizes the output tracking and disturbance rejection of a 1-D anti-stable wave system with general boundary disturbance collocated with control by proportional control, in which the external disturbance of the controlled system is at the same end as the control input. Output regulation of 1-D wave equation with both internal and external uncertainties is considered in  and finally achieves exponential tracking. In , the tracking problem of coupled wave equations with external disturbance is solved through the backstepping method. F.F. Jin and B.Z. Guo studies the output tracking problem of 1-D anti-stable wave equation with disturbance generated by external system in  via the reversible backstepping transformation. In  , a transport equation is introduced to deal with the anti-damping term on the boundary. Inspired by this, we further study the output feedback regulation problem of anti-stable wave equation by this method.
In this paper, we focus on the output tracking for 1-D anti-stable wave system with in-domain disturbance generated by an exosystem described by
where is the output to be regulated, the displacement and its derivative with time-delay are the measured output. represents the input (control), an unknown constant parameter ( is to avoid the real part of the plant eigenvalues tending to positive infinity). is initial condition, is a given reference signal, represents the unknown intensity of the distributed external disturbance . are generated by following exosystem
where S is a diagonalizable matrix with all eigenvalues on the imaginary axis. For design purpose, we suppose that the initial value is unknown and so do the state . Rewrite S as and can be written as
Here is known but is unknown due to the uncertainty of . We have and , and assume that the eigenvalues of matrix are distinct and is observable. The objective of this paper is to design an observer-based output feedback regulator for system (1.1) to regulate the tracking error to zero and simultaneously keep all the states bounded. The advanced nature of our result lies in that the measured output is at the left end and may admit time-delay, which makes the regulation problem of (1.1) challenging.
The rest of the paper is organized as follows. In Section 2, an auxiliary stable system is constructed by introducing a transport equation and a regulator equation, and its observer is derived. We propose the output feedback control law for the auxiliary system and obtain the closed-loop system in Section 3. The main results are presented in Section 4 and Section 5 concludes this paper.
2. State Observer Design
In order to deal with the anti-damping in system (1.1), we introduce the following transport equation in  
where is the initial value, a tuning parameter.
In the rest of this paper, we omit the obvious domain and when there is no confusion.
then is governed by
Notice that the parameter in system (1.1) becomes in system (2.3). Moreover, we have
To recover the state of (2.3), we now design an observer for (2.3) using the known signal and as
where is a constant, and , are row vectors where is designed to make the matrix Hurwitz and to be determined later.
Let , then we have
Construct the following transformation
then the observer error system is found to be
where satisfies the boundary value problem (regulator equation) as
We make .
Lemma 2.1: Assume that is a diagonalizable matrix. Then the regulator equation (2.8) admits a unique solution.
Considering our previous assumptions that the matrix is Hurwitz and , the PDE-part of system (2.7) is exponentially stable. (2.7) will be exponentially stable if we can show that is also Hurwitz.
Lemma 2.2: Define a function and be the eigenvector of matrix corresponding to the eigenvalue of . Then is observable if and only if .
Lemma 2.1 and 2.2 are similar to Lemma 3.1 and 3.2 in  respectively, and we omit the details of proof here. Suppose that satisfies the conditions of lemma 2.2, then is observable and consequently can be identified to make the matrix Hurwitz.
Theorem 2.1: Define a function satisfies for all eigen-pairs of . are constants, is the unique solution to (2.8). We make and , matrix is Hurwitz. Then system (2.7) is well-posed and exponentially stable.
We divide system (2.7) into PDE-part and ODE-part, and consider the stability of the solution respectively.
The PDE-part of (2.7) is
Define to be an usual Hilbert space with the following norm induced by the inner product
And . We can directly come to the conclusion that (2.9) is exponentially stable from . In other words, there exist two constants such that
which implies that
The ODE-part of (2.7) is governed by
There are some positive constants such that the solution of (2.11) has the estimation as
Thus we have
for any constants .
Define an invertible bounded operator as
Then there exists which is independent of initial value such that
3. Output Regulator Design
We construct a new transformation as
Then is governed by
where satisfies the BVP as follows
Moreover, we have
according to the second boundary condition in (3.3).
Lemma 3.1: Assume that S is a diagonalizable matrix, then the regulator equation (3.3) admits a unique solution.
Since S is diagonalizable, there exists an invertible matrix such that , where is the eigenvector of S corresponding to the eigenvalue of S. Postmultiplying (3.3) by , n ODEs are found to be
Case 1: When , the BVP (3.5) becomes
A formal simple computation shows that the solution of (3.6) is
Case 2: When , the BVP (3.5) has a general solution as
The last two boundary equations in (3.5) conclude that
Then are determined by solving (3.8) as
Thus the unique solution to the regulator Equation (3.3) is obtained and
Now we design the output feedback controller for (3.2) as
where is a constant. Under (3.9), the closed-loop becomes
Theorem 3.1: For any initial data , are positive constants, there exists a unique (weak) solution to the PDE-part of (3.10) such that . Besides, this solution is exponentially stable in the sense that
Define an operator for the PDE-part of system (3.10) by
Then the PDE-part of (3.10) can be written as an abstract evolutionary equation in as follows
decays to zero exponentially from Theorem 2.1 and the transformation (2.6). It’s well known that can generate an exponentially stable -semigroup by . In other words, there exist two constants , such that . It is a routine exercise that the operator B is admissible for by .
It concludes that for any initial value , there exists a unique solution to the PDE-part of system (3.10), which has the form of
The first term on the right side of (3.15) can be estimated as
The second term on the right side of (3.15) tends to zero exponentially because of the admissibility of B to and the exponential stability of . From Poincare’s inequality we have
Hence holds. Next we show that .
First define the Lyapunov function as
Notice the fact that decays exponentially and .
Then differentiate along the solution to system (3.2) to give
Finally integrating (3.17) from to t with respect to t and obtain
which decays exponentially from the exponential stability of .
Hence holds. £
The closed-loop of system (1.1) corresponding to (1.3), (2.1), (2.4) and (3.9) in the state space yields to
4. Main Results
Considering the closed-loop (3.19) in the state space X and define an operator for (3.19) which satisfies
Then (3.19) can be written as an abstract evolutionary equation in X as
Now we discuss the closed-loop system (4.1) in X.
Theorem 4.1: Define a function satisfies for all eigen-pairs of . are constants, is the unique solution to BVP (2.8). We make and , matrix is Hurwitz. Then for any , system (4.1) admits a unique bounded solution . Moreover, the tracking error exponentially when t tends to infinity.
Define an invertible bounded operator by
An equivalent system of (3.19) is found to be
We can obtain the solution to the transport system (2.1) explicitly as
Let being a designed parameter, then we have the estimation as
Conclusion (3.12) together with exponentially, imply that the solution to (2.1) is bounded as .
According to Theorem 2.1 and the well-posedness and exponential stability of system (3.10), (4.3) admits a unique bounded solution in X and so does the closed-loop (3.19) by the invertible transformation (4.2). As a result, the tracking error
exponentially in the light of (3.12). £
5. Concluding Remarks
In this paper, the output regulation problem for 1-D anti-stable wave equation is solved. The original system has the anti-damping at the position which is anti-collocated with the control, and also subjects to the distributed disturbance with unknown intensity generated by an external system. By proposing an observer-based feedback controller for (1.1), the following objectives are achieved: 1) keep all the states of internal-loop bounded; 2) recover the system state from input and output; 3) regulate the output tracks the given reference signal exponentially.
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