Shuxia Tang

Sliding mode control to the stabilization of a linear 2×2 hyperbolic system with boundary input disturbance

In this paper, sliding mode control approach is used to stabilize a 2×2 system of first-order linear hyperbolic PDEs subject to boundary input disturbance. Disturbance rejection is achieved, and with the designed first-order sliding mode controller, the resulting closed-loop system admits a unique (mild) solution without chattering. Convergence to the chosen infinite-dimensional sliding surface of state trajectories takes place in a finite time. Then on the sliding surface, the system is exponentially stable with a decay rate depending on the spatially varying system coefficients. A simulation example is presented to illustrate the effectiveness and performance of sliding mode control method.

Backstepping stabilization of the linearized Saint-Venant–Exner model

Using backstepping design, exponential stabilization of the linearized Saint-Venant–Exner (SVE) model of water dynamics in a sediment-filled canal with arbitrary values of canal bottom slope, friction, porosity, and water–sediment interaction, is achieved. The linearized SVE model consists of two rightward convecting transport Partial Differential Equations (PDEs) and one leftward convecting transport PDE. A single boundary input control strategy with actuation located only at the downstream gate is employed. A full state feedback controller is designed which guarantees exponential stability of the desired setpoint of the resulting closed-loop system. Using the reconstruction of the distributed state through a backstepping observer, an output feedback controller is established, resulting in the exponential stability of the closed-loop system at the desired setpoint. The proposed state and output feedback controllers can deal with both subcritical and supercritical flow regimes without any restrictive conditions.

State-of-charge estimation for lithium-ion batteries via a coupled thermal-electrochemical model

Accurate online state-of-charge (SoC) estimation is a basic need and also a fundamental challenge for battery applications. In order to achieve accurate SoC estimation for the lithium-ion batteries, we employ a coupled thermal-electrochemical model. This coupled system of an ordinary differential equation (ODE) and a partial differential equation (PDE) is simpler than the Doyle-Fuller-Newman (DFN) model, and is more accurate than the single particle model (SPM) alone. Thus, it could serve as a better fit of model for a full state observer design and accurate SoC estimation. PDE backstepping approach is utilized to develop a Luenberger observer for the electrode concentration, and estimation effectiveness of the proposed method is verified by simulation results.

Stabilization of linearized Korteweg-de Vries systems with anti-diffusion

In this paper, backstepping boundary controllers are designed for a class of linearized Korteweg-de Vries systems with possible anti-diffusion, and the resulting closed-loop systems can achieve arbitrary exponential decay rate. Semigroup of linear operators is constructed in analyzing well-posedness and stability of the target systems, and mathematical induction is used in proving existence of kernel functions. An example is also presented, which illustrates performance of the controller. The decay rate estimate derived in this paper is not necessarily equal to decay rate, which can be seen from the appendix.

Backstepping control of the one-phase stefan problem

In this paper, a backstepping control of the one-phase Stefan Problem, which is a 1-D diffusion Partial Differential Equation (PDE) defined on a time varying spatial domain described by an ordinary differential equation (ODE), is studied. A new nonlinear backstepping transformation for moving boundary problem is utilized to transform the original coupled PDE-ODE system into a target system whose exponential stability is proved. The full-state boundary feedback controller ensures the exponential stability of the moving interface to a reference setpoint and the ℋ1-norm of the distributed temperature by a choice of the setpint satisfying given explicit inequality between initial states that guarantees the physical constraints imposed by the melting process.

State-of-Charge estimation from a thermal–electrochemical model of lithium-ion batteries

Abstract A thermal–electrochemical model of lithium-ion batteries is presented and a Luenberger observer is derived for State-of-Charge (SoC) estimation by recovering the lithium concentration in the electrodes. This first-principles based model is a coupled system of partial and ordinary differential equations, which is a reduced version of the Doyle–Fuller–Newman model. More precisely, the subsystem of Partial Differential Equations (PDEs) is the Single Particle Model (SPM) while the Ordinary Differential Equation (ODE) is a model for the average temperature in the battery. The observer is designed following the PDE backstepping method. Since some coefficients in the coupled ODE–PDE system are time-varying, this results in the time dependency of some coefficients in the kernel function system of the backstepping transformation and it is non-trivial to show well-posedness of the latter system. Adding thermal dynamics to the SPM serves a two-fold purpose: improving the accuracy of SoC estimation and keeping track of the average temperature which is a critical variable for safety management in lithium-ion batteries. Effectiveness of the estimation scheme is validated via numerical simulations.

Active Disturbance Rejection Control for a 2×2 Hyperbolic System with an Input Disturbance

Abstract In this paper, active disturbance rejection control (ADRC) approach is used to stabilize a 2×2 system of first-order linear hyperbolic partial differential equations (PDEs) subject to a boundary input disturbance. Disturbance attenuation is achieved with the designed controller, and the resulting closed-loop control system admits a unique solution, which could tend to any arbitrary vicinity of zero.

Backstepping stabilization of the linearized Saint-Venant-Exner Model: Part II- output feedback

Using the backstepping design, we achieve exponential stabilization of the coupled Saint-Venant-Exner (SVE) PDE model of water dynamics in a sediment-filled canal with arbitrary values of canal bottom slope, friction, porosity, and water-sediment interaction under subcritical or supercritical flow regime. This model consists of two rightward and one leftward convecting transport Partial Differential Equations (PDEs). A single boundary input control (with actuation located only at downstream) strategy is employed and the backstepping approach developed for the first order linear hyperbolic PDEs is used. A full state feedback controller is designed, which guarantees the exponential stability of the closed-loop control system.

Stabilization of linearized Korteweg-de Vries systems with anti-diffusion by boundary feedback with non-collocated observation

This paper addresses the problem of stabilizing a class of one-dimensional linearized Korteweg-de Vries systems with possible anti-diffusion (LKdVA for short), through control at one end and non-collocated observation at the other end. An exponentially convergent observer is designed, and then a dynamical stabilizing output feedback boundary controller is constructed based on the observer. The resulting closed-loop systems can achieve arbitrary exponential decay rate. In order to derive invertibility of the kernel function in the backstepping transformation between the observer error systems and its corresponding target systems, stabilizing of a critical case of LKdVA is considered in the Appendix, which can also be treated as a preliminary problem for the main part of this paper.

Exponential regulation of the anti-collocatedly disturbed cage in a wave PDE-modeled ascending cable elevator

Abstract In cable elevators, large axial vibrations appear when a cage subject to disturbance is lifted up via a compliant cable. The axial vibration dynamics can be described by a wave partial differential equation (PDE) on a time-varying spatial interval with an unknown boundary disturbance. In this paper, we design an output feedback controller actuating at the boundary anti-collocated with the disturbance to regulate the state on the uncontrolled boundary of the wave PDE based on the backstepping idea and the active disturbance rejection control (ADRC) approach. The control law uses the state and disturbance information recovered from the state observer and the disturbance estimator, respectively, which are constructed via limited boundary measurements. The exponential convergence of the state on the uncontrolled boundary and uniform boundedness of all states in the closed-loop system are proved by Lyapunov analysis. Effective vibration suppression in the cable elevator with the designed controller is verified via numerical simulation.