Stabilization of a Class of Nonlinear ODE/Wave PDE Cascaded Systems

We investigate stabilization of a class of cascaded systems of nonlinear ordinary differential equation (ODE)/wave partial differential equation (PDE) with time-varying propagation speed based on a two-step PDE backstepping transformation. A time-varying propagation velocity of wave PDE leads to two difficulties. One is how to prove the well-posedness and uniqueness of the time-varying kernel PDEs in the first-step backstepping transformation, the other is how to construct a backstepping transform to map the original system into a suitable target system during the second-step transformation. We prove that there exists a unique continuous $2 \times 2$ matrix-valued solution to the time-varying kernel PDEs, and design a predictor control for the original cascaded system. An example is provided to illustrate the feasibility of the proposed design.


I. INTRODUCTION
Since the pioneering works [1], [2] revealed PDE backstepping method as a new way to stabilize input delay systems, various interesting results [3]- [11] have been achieved. PDE backstepping design has also been utilized to control various PDE-ODE cascaded systems [12]- [15]. Boundary stabilization of one-dimensional linear hyperbolic PDEs with time and space varying parameters is presented, the well-posedness of time and space varying kernel PDEs has been solved in [16]. Global stabilization of a class of switched nonlinear systems is investigated in [17], in which a state feedback sampled-data controller is constructed by backstepping design.
In oil drilling, torsional vibrations of a drill string caused by the friction between drill bit and rock will seriously damage the drilling facilities [18]. The torsional dynamics of a drill string is modeled as a wave PDE, which is coupled with a nonlinear ODE that describes dynamics of angular velocity of the drill bit at the bottom of the drill string [19]. This engineering application inspires researchers to study how to control cascaded systems of nonlinear ODE/wave PDE.
The associate editor coordinating the review of this manuscript and approving it for publication was Nasim Ullah .
A predictor control is presented for nonlinear ODE/wave PDE cascaded system [20]. Stabilization of wave PDE dynamics with a moving controlled/uncontrolled boundary has been solved in [21] and [22], respectively. Boundary control of a nonlinear ODE actuated through a wave PDE with spatially-varying propagation speed is explored in [23].
Note that a variable propagation speed increases the intricacies arising in this class of problems and also causes difficulties in the analysis of control design. Therefore, the study of a time-varying propagation speed is challenging and practical. This motivates us to investigate nonlinear ODE/wave PDE cascaded system with time-varying propagation speed.
In this paper, we develop a stabilization design for cascaded system of nonlinear ODE/wave PDE with time-varying propagation speed. Based on a two-step PDE backstepping transformation and Lyapunov arguments, we prove globally asymptotical stability of the closed-loop system. In addition, the time-varying propagation speed leads to two difficulties. One is how to prove the well-posedness and uniqueness of time-varying kernel PDEs during the first-step backstepping transformation. The other is how to construct predictors during the second-step backstepping transformation. We prove that there exists a unique continuous 2 × 2 VOLUME 10, 2022 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ matrix-valued solution to the time-varying kernel PDEs, and design predictors for the second-step backstepping transformation. This paper is organized as follows: System description and main results are in Section II. Coordinate transformations and backstepping transformations are in Section III. Stability analysis of the closed-loop system is in Section IV, and an example is in V. Concluding remarks are shown in Section VI.

II. SYSTEM DESCRIPTION AND MAIN RESULTS
Consider the nonlinear ODE/wave PDE cascaded systeṁ where X ∈ R n , u ∈ R, U ∈ R are ODE state, PDE state, and control input, respectively, and f : R n × R → R n is locally Lipschitz with f (0, 0) = 0, and v is a propagation speed satisfying the following Assumption.
Assumption 1: Propagation speed v : R → R is continuously differentiable, and there are υ > 0, υ > 0, M 1 > 0 such that and for all t ∈ R. Remark 1: Propagation speed v > 0 is bounded, and its rate of change is also bounded. Denote for all t ≥ 0, and φ −1 (t) is the inverse function of φ(t).

A. CONTROL DESIGN
If a nominal controller κ : R n → R is such thatẊ (t) = f (X (t), κ(X (t))) is globally asymptotically stable, then a predictor control for system (1)-(4) is designed as where p 1 ∈ R n , p 2 ∈ R are given by with for all x ∈ [0, L]. The initial conditions of (9) and (10) are given as dαdy, (14) for all x ∈ [0, L]. The gain c 1 > 0 in (8), and the kernel gains k 11 and k 12 are solutions to the following kernel PDEs: 35654 VOLUME 10, 2022 where (15) Remark 2: For an implementation of control law (8), we have to numerically integrate a finite interval in (9) and (10) by one of the numerical quadratures. In the simulations, we use the composite left-endpoint rectangle rule.

III. COORDINATE TRANSFORMATIONS AND BACKSTEPPING TRANSFORMATIONS
First, introducing the following change of coordinate the reverse is Using change of coordinate (31), (32), system (1)-(4) is expressed asẊ where and A(t) is given by (19).
The inverse transform of (47) is designed as , to map the target system (48)-(52) to system (43)-(46), with the inverse kernel PDEs as follows: and B(t) are given by (19). Based on the successive approximation method, Coron, et al. solved the existence and uniqueness of time-varying kernel PDEs in Theorem 2.6 in [16], but the boundary conditions of the kernel PDE (15)- (18) are different from those of the kernel PDEs in Theorem 2.6 in [16]. So the result of [16] cannot be directly applied to the kernel PDEs (15)- (18). Following the method in [16], we prove the existence and uniqueness of the kernel PDEs (15)- (18).
in order to avoid such a condition.

IV. STABILITY ANALYSIS OF THE CLOSED-LOOP SYSTEM
Lemma 4: Under assumptions 1 and 2, consider system (146)-(150), there exists a class KL function β, such that for all t ≥ 0. VOLUME 10, 2022 Proof: We introduce a new variable z(x, t), x ∈ [−L, L], t ≥ 0, such that Let g,n (t) be the following norm for any g > M 1 2υ and positive integer n, the derivative of g, n (t) satisfieṡ Using Assumption 2, from (165), it is easy to get (162).

V. EXAMPLE
Example 1: For a second-order systeṁ a nominal control law [24] is u(0, t) = −X 3 − (X 1 + 2X 2 + X 3 + 0.25X 2 2 + 0. It is clear that v(t) satisfies Assumption 1. Following [25], a forward finite difference scheme is used for the explicit time  integral with a negative time step to archive a backward in time computation of kernel PDEs (15)- (18). Responses of the states X 1 , X 2 and X 3 of the closed-loop system under the proposed control law, the uncompensated control law are shown in Fig. 1 and Fig.2. Wave dynamics under the compensated control is in Fig.3. One can conclude that the proposed control law ensures asymptotic stability of the closed-loop system while the uncompensated control (178) leads to instability.

VI. CONCLUSION
We consider a class of nonlinear ODE/wave PDE cascaded systems. A predictor control is designed such that the closed-loop system is globally asymptotically stable. One difficulty is how to prove the well-posedness and uniqueness of time-varying kernel PDEs (15)- (18), the other is how to construct predictors p(x, t), q(x, t) in backstepping transforms (143), (144). Stability of the closed-loop system is proved using a two-step backstepping transformation and Lyapunov-like arguments. Generalization of the result to a wider class of propagation speed and robustness analysis with respect to disturbances will be considered in our future work.