This paper considers the output feedback stabilization for a cascaded-wave PDE-ODE system with velocity recirculation by boundary control. First, we choose a well-known exponentially stable system as its target system and find a backstepping transformation to design a state feedback controller for the original system. Second, we attempt to give an output feedback controller for the original system by introducing the observer. The resulting closed-loop system admits a unique solution which is proved to be exponentially stable. Finally, we give some numerical examples to prove the validity for the theoretical results.

1. Introduction

In control engineering, ordinary differential equations (ODEs) and partial differential equations (PDEs) are widely used to model such problems. Many researchers have done detailed research on this content in the past decades. Systems modeled by ODEs are common in this respect. On the other hand, there are more and more works that contributed to stabilization for systems described by PDEs in recent years ([17]). For instance, aiming at solving the problem of system with variable coefficients and Neumann boundary actuation, Wu et al. used the backstepping approach that converted the wave equation into a same type equation in [8]. Not only that, the coupled PDE-ODE problem was also studied in control systems, where PDEs mainly included heat equations and wave equations. With the development of research on such problems, more results of PDE-ODE emerged. For example, in [916], the authors solved the stabilization for heat PDE-ODE by boundary control systems which used the PDE backstepping method. In [10, 13], Tang and Xie considered the systems where in boundary of “PDE part.” The system of in “PDE part” was considered in [12]. The stabilization for wave PDE-ODE systems by the PDE backstepping method was solved in [6, 7, 17, 18]. But beyond that, in [19], Meglio solved the problem of stabilizing a linear ODE coupled first-order hyperbolic PDE, which was also based on the backstepping approach. This paper proposed a unified framework of iterative learning control for typical flexible structures under spatiotemporally varying disturbances in [20]. In [21], the control problem was addressed for a hybrid PDE-ODE system that described a nonuniform gantry crane system with constrained tension. In addition, the PDE-ODE models also played a significant role in practical application, such as 3D-printing [22], oil drilling [23], cable elevator [24], battery management [25], traffic [26], and so on.

In fact, the backstepping method played an important role in solving many PDEs and PDE-ODE problems, which were introduced by Krstic [2730]. This approach was dedicated to finding an invertible Volterra integral transformation which mapped the considered plant with the boundary feedback law into a known stable target system. About some PDE systems, Jin and Guo also used the backstepping method to solve the PDEs system of nonlocal terms in [31]. In [32], Hasan and Tang used the backstepping method to model boundary stabilization of the Korteweg–De Vries (KdV) equation with sensors and actuators. For some PDE-ODE systems, the authors also used the backstepping approach to design the controller in [6, 7, 918]. In [17], Zhou et al. used directly the known backstepping method (from [30]) to achieve the stabilization of system matched disturbance with boundary control. It can be said that backstepping is a significant method and key move to deal with such PDE and PDE-ODE problems.

In this paper, our main focus is on the output feedback stabilization for a cascaded-wave PDE-ODE system subject to boundary control with velocity recirculation:where is the output (measurement), is the control input of the entire system, is the initial value, is a known constant, , and . We suppose that is stabilizable and is observable. System (1) is considered in the state space .

On some special cases, we consider the following. For system (1) without “ part,” Su et al. gave the backstepping method and the state feedback controller in [33], which was . When , by the backstepping method shown in [30], one can easily design a control law to stabilize system (1). And in [16], Zhou and Xu used backstepping transformation to settle the problem of coupled PDE-ODE where there was no non-local term in heat equation. On such coupling problems, there was few research on PDE-ODE problem with non-local term. Non-local terms, including both boundary terms and strict feedback/Volterra terms, have been the mainstay of the applications of PDE backstepping methods to parabolic PDEs. Equation models the string vibration of an electric guitar with a pickup at the location . The move of leads to the location change of the pickup. Inspired by it, we let in . Compared with [30], we add a non-local term to the right end of the wave equation. This makes it more difficult for us to research this problem. In [30], the form of backstepping transformation is “”; it does not work to solve the problem in their way. Inspired by (4) in [34], we obtain the new form of backstepping transformation. Using the backstepping and introducing the exponentially stable target system to design state feedback controller are also nice methods in this paper.

The paper is organized as follows. We design the state feedback controller for system (1) in Section 2. Section 3 is contributed to designing the output feedback control and proving the exponential stability of the closed-loop system. In Section 4, we give some numerical examples to prove the validity of the proposed controller.

2. State Feedback Controller Design

In this section, we are devoted to designing a state feedback controller for system (1). First, we introduce the backstepping approach:where the kernel functions , , , and will be defined later. Compared with the backstepping in [30], we add an additional term “” in transformation (2). And we hope that turns system (1) into the target systemwhere are tuning parameters and is chosen such that is Hurwitz.

It is well known that there exists a exponentially stable solution for “ part” in the state space , which is equipped with the norm induced by inner product

In order to get these kernel functions, we make the following calculation. Taking derivative of (2) with respect to , we can obtain

Taking derivative of (2) with respect to twice, we find

In the same way, taking derivative of (2) with respect to and taking derivative of (2) with respect to twice and using the partial integral method, we can get the following:

Then, we have

By the boundary condition in (3) and (5), we obtain

Hence, we choose the functions to satisfy

We can obtain by . In [30], we find that the solution of is as follows:

By (2) and (10), we have ; then, the first equation in (3) is obtained.

Next we can let ; by “ part” in (10), we can have

Through , we get

Because , we can acquire . According to the above formulas, we obtain

According to , simplify the above three formulas in (14)(16). Let ; we get , , and

By differentiating (17) with respect to twice, we transform (17) into a second-order linear ODE and get the solution of :

We substitute (20) for (18) and obtain the solution of . In the same way, the solution of can also be obtained.

Because , . , , , so , and .

We suppose that the inverse transformation of (2) exists in the formwhere are the kernel functions.

Derive (21) with respect to , and we obtain

Differentiating (21) with respect to twice, we have

We differentiate (21) with respect to and get

Differentiating (21) twice on both sides with respect to yields

Then, we have

The boundary condition at in (1) implies

Hence, we choose the functions to satisfy

We obtain the solution of in [30]:

Let ; then,

Combining the forth equations in (1) and (3), we can obtain the controller for (1)

For convenience, we describe system (1) under controller (31) as

Define the state space for system (32) with the normwhere .

Theorem 1. Assume thatandis chosen such thatis Hurwitz. For any initial condition such that, closed-loop system (32) admits a unique solution. Moreover, closed-loop system (32) is exponentially stable: there exist two positive constantsdepending onandsuch that

Proof. It is a direct result from the equivalence between system (3) and (32). Obviously, we define the operator for the “ part” of target system (3):Then the “ part” of system (3) can be written as an abstract evolutionary equation in :As we all know, the operator can generate an exponentially stable -semigroup on [35]. In other words, there exist such thatNext we consider the “X part” of target system (3) on . Because , using the constant variational formula, the solution of “X part” is given byBecause is Hurwitz, there exist positive constants such thatFrom Theorem 1 in [17], the solution of ” part” is exponentially stable. So, system (3) admits a unique solution and the solution is exponentially stable. According to transformations (2) and (21), we define a bounded invertible operator in :whereand are defined in (14)(16) and (30), respectively.
By the formula above, we have the following expression:Since the operator is bounded and invertible, there exists such that . Besides, the “ part” admits a unique solution and the solution is exponentially stable, so the solution of closed-loop system (32) is exponentially stable by (42). There exist such that (34) holds.

3. Observer and Output Feedback Controller Design

In this section, we focus on designing output feedback controller for system (1) on the grounds of the observer. Next, we give the observer for system (1).where are positive design parameters.

The error between system (1) and observer (43) is governed by

Define the state space for system (44) with the norm

Theorem 2. Suppose that. For any initial value, system44admits a unique exponentially stable solutionin the sense thatfor some positive constantsand, whereis given by

Moreover,for and , where

Proof. As we all know, the “ part” of system (44) admits a unique exponentially stable solution from [33] and is Hurwitz, so system (44) has a unique exponentially stable solution.
Notice that the energy is equivalent to . We only prove (48) and (49). By Cauchy inequality and Hölder inequality, we havefor some constant .
It is obvious that is equivalent to . is exponentially stable, so (48) holds. Next we prove (49).
Differentiating , we findIntegrating above equation from to , we obtainDefine another functionIt is clear that . Taking derivative of (54), we getIntegrating above equation from to , we havefor some constant . Hence, (48) and (49) hold. Especially, when , .
We give the output feedback controller about system (1):In terms of control law (57), closed-loop system (1) can be described as