Meng-Bi Cheng

Brief paper: Sliding mode boundary control of a parabolic PDE system with parameter variations and boundary uncertainties

This paper considers the stabilization problem of a one-dimensional unstable heat conduction system (rod) modeled by a parabolic partial differential equation (PDE), powered with a Dirichlet type actuator from one of the boundaries. By applying the Volterra integral transformation, a stabilizing boundary control law is obtained to achieve exponential stability in the ideal situation when there are no system uncertainties. The associated Lyapunov function is used for designing an infinite-dimensional sliding manifold, on which the system exhibits the same type of stability and robustness against certain types of parameter variations and boundary disturbances. It is observed that the relative degree of the chosen sliding function with respect to the boundary control input is zero. A continuous control law satisfying the reaching condition is obtained by passing a discontinuous (signum) signal through an integrator.

A Sampled-Data Singularly Perturbed Boundary Control for a Heat Conduction System With Noncollocated Observation

This note presents a sampled-data strategy for a boundary control problem of a heat conduction system modeled by a parabolic partial differential equation (PDE). Using the zero-order-hold, the control law becomes a piecewise constant signal, in which a step change of value occurs at each sampling instant. Through the dasialiftingpsila technique, the PDE is converted into a sequence of constant input problems, to be solved individually for a sampled-data formulation. The eigenspectrum of the parabolic system can be partitioned into two groups: a finite number of slow modes and an infinite number of fast modes, which is studied via the theory of singular perturbations. Controllability and observability of the sampled-data system are preserved, irrelevant to the sampling period. A noncollocated output-feedback design based upon the state observer is employed for set-point regulation. The state observer serves as an output-feedback compensator with no static feedback directly from the output, satisfying the so-called dasialow-pass propertypsila. The feedback controller is thus robust against the observation error due to the neglected fast modes.

Robust tracking control of a unicycle-type wheeled mobile manipulator using a hybrid sliding mode fuzzy neural network

This article presents a robust tracking controller for an uncertain mobile manipulator system. A rigid robotic arm is mounted on a wheeled mobile platform whose motion is subject to nonholonomic constraints. The sliding mode control (SMC) method is associated with the fuzzy neural network (FNN) to constitute a robust control scheme to cope with three types of system uncertainties; namely, external disturbances, modelling errors, and strong couplings in between the mobile platform and the onboard arm subsystems. All parameter adjustment rules for the proposed controller are derived from the Lyapunov theory such that the tracking error dynamics and the FNN weighting updates are ensured to be stable with uniform ultimate boundedness (UUB).

Sliding mode boundary control of unstable parabolic PDE systems with parameter variations and matched disturbances

This paper considers the stabilization problem of a one-dimensional unstable heat conduction system subject to parametric variations and boundary uncertainties. This system is modeled as a parabolic partial differential equation (PDE) and is only powered from one boundary with a Dirichlet type of actuator. By taking the Volterra integral transformation, we obtain a nominal PDE with asymptotic stability characteristics in the new coordinates when an appropriate boundary control input is applied. The associated Lyapunov function can then be used for designing an infinite-dimensional sliding surface, on which the system exhibits exponential stability, invariant of the bounded matched disturbance, and is robust against certain types of parameter variations. A continuous variable structure boundary control law is employed to attain the sliding mode on the sliding surface. The proposed method can be extended to other parabolic PDE systems such as diffusion-advection system. Simulation results are demonstrated and compared with the other outstanding back-stepping control schemes.

Robust Backstepping Tracking Control Using Hybrid Sliding-Mode Neural Network for a Nonholonomic Mobile Manipulator with Dual Arms

This paper presents a methodology for trajectory tracking control of a wheeled dual-arm mobile manipulator with parameter uncertainties and external load variations. Based on backstepping technique, the proposed control laws comprise two levels: kinematic and dynamic. First, the auxiliary kinematic velocity control laws for the mobile robot and the two onboard arms are separately established. Second, a robust backstepping tracking control based on hybrid sliding-mode neural networks (HSMNN) is presented to ensure the velocity tracking ability in spite of the uncertainties. The proposed robust backstepping tracking controller is actually composed of a neural network controller, a robust controller, and a proportional controller. To achieve the overall trajectory tracking goal, a neural network controller is developed to imitate an equivalent control law in the sliding-mode control, a robust controller is designed to incorporate the system dynamics into the sliding surface for guaranteeing the asymptotical stability, and the proportional controller is designed to improve the transient performance for randomly initializing neural network. All the adaptive learning algorithms for the proposed controller are derived from the Lyapunov stability theory so that the close-loop asymptotical tracking ability can be guaranteed no matter the uncertainties taken place or not. Simulation results demonstrate the feasibility as well as usefulness of the proposed control strategy in comparison with other conventional control methods.

Boundary stabilization and matched disturbance rejection of hyperbolic PDE systems: A sliding-mode approach

This paper deals with the robust boundary stabilization problem of hyperbolic partial differential equations (PDEs) subject to boundary uncertainties. These systems include a second-order undamped wave equation and a first-order delay system. By taking the integral transformation, we obtain a nominal PDE with asymptotic stability in the new coordinates when an appropriate boundary control input is applied. The associated Lyapunov function can then be used for designing an infinite-dimensional sliding surface, on which the system exhibits exponential stability, invariant of the bounded matched disturbance. A continuous sliding-mode boundary control law satisfied with reaching condition is employed to ensure the systems will reach the proposed sliding surface within finite time. Simulation results of a second-order wave equation are provided to demonstrate and compare with the other control schemes.

A proportional-plus-integral boundary control design for an unstable heat conduction system

This paper presents a simple proportional-plus-integral (PI) boundary controller for an unstable parabolic systems. For an infinite-dimensional system, the characteristics of minimum-phase is not easily observed from those governing equation and boundary conditions. In this paper the method of infinite product expansion is firstly employed to identify exact locations of system zeros/poles. Furthermore, if the system parameters are given, the Nyquist plot can be used for minimum-phase check. Once the minimum-phase condition is satisfied, a simple PI output feedback boundary controller can achieve stabilizing even if there are multiple unstable modes.

TRAJECTORY TRACKING CONTROL FOR AN UNCERTAIN MOBILE MANIPULATOR: COMBINING SLIDING MODE AND NEURAL NETWORK

Abstract This paper deals with the trajectory tracking problem of a nonholonomic wheeled robot mounted with n-degree-of-freedom (n-DOF) onboard manipulator under modelling uncertainties and external load changes. Based on backstepping technique, the original problem has converted into kinematics and dynamic issues. First, the auxiliary kinematic velocity control schemes for the mobile robot and the onboard arm are developed. Second, a tracking controller, merged with the benefits of the robustness of sliding mode approach and the function approximation of neural network (NN), is presented to guarantee the velocity tracking ability in dynamic level with uncertainties. The approximated error of NN and the dynamic uncertainties are integrated into matched uncertainties. Once the system’s states are forced on the sliding surface, the robustness of the controller is obtained. All the adaptive learning algorithms of the proposed control law are derived from the Lyapunov stability theory so that the closed-loop system tracking ability can be ensured. Simulations are presented to illustrate the effectiveness of the proposed control.

Sliding mode dirichlet boundary stabilization of uncertain parabolic PDE systems with spatially varying coefficients

We consider the robust boundary stabilization problem of an unstable parabolic partial differential equation (PDE) system with uncertainties entering from both the spatially-dependent parameters and from the boundary conditions. The parabolic PDE is transformed through the Volterra integral into a damped heat equation with uncertainties, which contains the matched part (the boundary disturbance) and the mismatched part (the parameter variations). In this new coordinates, an infinite-dimensional sliding manifold that ensures system stability is constructed. For the sliding mode boundary control law to satisfy the reaching condition, an adaptive switching gain is used to cope with the above uncertainties, whose bound is unknown.