Boundary Control for a Kind of Coupled PDE-ODE System

A coupled system of an ordinary differential equation (ODE) and a heat partial differential equation (PDE) with spatially varying coefficients is discussed. By using the PDE backstepping method, the state-feedback stabilizing controller is explicitly constructed with the assumptions λ(x) ∈ P[x] n and λ(x) ∈ C∞ [0, l] , respectively. The closed-loop system is proved to be exponentially stable by this controller. A simulation example is presented to illustrate the effectiveness of the proposed method.

In this paper, we replace the spatially constant coefficient  of the PDE subsystem in (2) by the spatially varying coefficient (); that is,   (, ) =   (, ) + ()(), which implies that the effects from the ODE subsystem to the PDE subsystem are varying with the location .In fact, control of the coupled systems is an important subject in control theory since this type of system arises frequently in control engineering.
The objective of this paper is to convert a PDE-ODE coupled system into a closed-loop target system that is exponentially stable in the sense of the norm ‖  ‖ 2 + ‖‖ 2 + || 2 , with a designed stable state-feedback controller by using the backstepping-based predictor design method.
Under the assumptions () ∈ []  and () ∈  ∞ [0,] , respectively, we further obtain the explicit expressions of the kernel function of the backstepping transformation.
This paper is organized as follows.In Section 2, we propose the interaction of PDE-ODE coupled control system.In Section 3, a state-feedback boundary controller is designed for this system by the backstepping-based method.In Section 4, we prove the exponential stability for the designed closed-loop system, and Section 5 is a simulation example.In Section 6, some comments are made on the coupled PDE-ODE systems.

Problem Formulation
Consider a coupled interaction system of an ODE and a heat PDE with () times (): where () ∈  ×1 and (, ) are the vector state and scalar state of the ODE subsystem, (0) =  0 and (, 0) =  0 () are the initial values to the PDE subsystem, respectively,  ∈  × ,  ∈  ×1 , and the pair (, ) are assumed to be stabilization, and () ∈  ∞ [0,] is a known spatially varying parameter with [0, ] ( > 0) which is the length of the PDE domain.
From formulation (4), it can be seen that the output (0, ) of the PDE subsystem acts as the input of the ODE subsystem; meanwhile, the output () of the ODE subsystem affects the PDE along the domain [0, ] times by (), which implies that the effects of the ODE subsystem on the PDE subsystem are varying with the location.To design the state-feedback controller for system (4), an infinite-dimensional backstepping method is adopted, which provides an invertible integral transformation (, )  → (, ) as follows: (, ) =  (, ) − ∫  0  (, )  (, )  − Φ ()  () . (5) This transformation converts the plant (4) into the following target system: Ẋ () = ( + )  () +  (0, )   (, ) =   (, ) where  ∈  1× satisfying  +  is Hurwitz.It should be pointed that it is nontrivial to obtain the kernel function (, ) by the method in the literature [13] as it is related to ().But if we impose some constraints on (), then the kernel can be obtained explicitly, as shown in the next section.Once transformation ( 5) is obtained (namely, (, ) and Φ() are obtained), then the controller subject to the boundary condition (, ) = 0 in ( 6) is given in the form

The Design of the State-Feedback Controller
In this section, we will design the predictor-feedback controller () for the system (4) using the backstepping method.
The key point to this design is to determine the function (, ) and Φ() of transformation (5).In the following procedure, we find that the kernel function (, ) can be expressed by Φ(), while the function Φ() is related to the spatially varying parameter ().
Proof.It follows easily from (8) that By taking the derivative on both sides about  and applying variable substitution, we have Furthermore, take the derivative repeatedly, and then By this equation, we have and with the initial values of (8), we have following initial values: Then the solution to the ODEs (12) with the above initial values can be expressed as follows: with where the initial values of higher-order derivatives of Φ in zero are denoted by (13) and ) , and () ∈  ∞ [0,] , and then (8) has the following unique solution: Proof.By integrating into (8) over [0, ], we have Then, integrating again on both sides of (18) over [0, ] with initial value Φ(0) =  of (8), we conclude where Denoting the following iterative relationship: it suffices to show that if series {Φ  ()} was convergence, then (16) is the unique solution of (19).Considering the difference now, we will estimate ΔΦ  () by induction.First, for ΔΦ 0 (), we have where  = max ∈[0,] |()| is denoted as above and  is the length of the PDE domain.Then, suppose that with  = || +  2 ||.Then |Ψ +1 ()| can be estimated as follows: Noting  ∈ [0, ], we have The series on the right-hand side of (25) converges.Hence by Weierstrass's Discriminance, the series defined by ( 16) converges absolutely and uniformly on 0 ≤  ≤ .Then the existence of the solution to ( 8) is concluded.To show the uniqueness of the solution ( 16) to (8), we assume that Φ and Φ are two different solutions of (8).Substituting these two solutions and after some direct calculation, we have From (25), we know that |Φ()| ≤  √ , which means |Φ()| ≤ 2 √ .Next, we will estimate Φ by induction.

Design of the State-Feedback Controller with Backstepping
Method.Next, we will obtain the backstepping transformation (5).Let  = 0 in (5), and we have and Φ(0) =  by comparing ( 4) and ( 6).The partial derivatives of (, ) in ( 5) with respect to  are given by This equation should be valid for all  and , so we have two conditions that (0, 0) = 0 and Φ  (0) = 0.In order to satisfy the conditions of the target system (6), the (, ) and Φ() in ( 5) should satisfy Note that (37) is a second-order hyperbolic PDE about (, ) and the boundary condition is related to Φ(), and ( 38) is a second-order integral-differential equation about Φ() associated with () and (, ).Next, we will obtain (, ) from (37) and Φ() from (38).

Exponential Stability of the Coupled PDE-ODE System
Now we will prove the exponential stability of the proposed coupled PDE-ODE system.

A Simulation Example
In this section, an example is given to verify the effectiveness of theoretical results for the following simple system: where () ∈ ,0 ≤  ≤ 0.1, and  ≥ 0. In order to show the transient performance of the closed-loop system, a numerical simulation is executed in Matlab.By using the explicit forward Euler method with 1-step discretization in space, simulation Figures 1 and 2 show that both the states () and (, ) converge to zero, which indicates that the closed-loop system is exponentially stable.The convergence rate to zero for the closed-loop system is determined by the eigenvalues of the PDE-ODE system (6).These eigenvalues are the union of the eigenvalues of  + , which are placed at desirable locations by the control vector  and of the eigenvalues of the heat equation with a Neumann boundary condition on one end and a Dirichlet boundary condition on the other end.While exponentially stable, the heat equation PDE need not necessarily have fast decay.Fortunately, the compensated actuator dynamics, that is, the w-dynamics in [6], can be sped up arbitrarily by a modified controller [11,14].

Conclusions
In this paper we have developed an explicit controller for a coupled PDE-ODE system with Dirichlet interconnection (0, ), extending the results in [2,3].Many open problems  (66)