Nonsmooth Control Research Articles

Seam Tracking for Mobile Welding Robot Based on Terminal Sliding Mode

In this paper, non-smooth controller based on terminal sliding mode is applied to mobile welding robot to track a linear welding path. According to the kinematics and dynamics models of welding mobile robot, the seam tracking error system is established considering the external disturbance. Meanwhile, the terminal sliding mode controller (TSMC) is designed to achieve the convergence of the tracking errors in finite time. The new saturation function is used to eliminate the effect of chattering instead of the sign function. Then the Lyapunov stability theory indicates that the closed-loop system is stable. Compared with the linear sliding mode controller (LSMC), the proposed non-smooth controller can offer not only a faster convergence rate but also a higher tracking accuracy. Finally, the numerical simulation is provided to demonstrate the effectiveness of the proposed controller. And it is of great significance for improving welding accuracy.

Non-smooth state feedback prescribed performance control for interconnected nonlinear systems with unmodelled dynamics

ABSTRACTThis paper investigates the problem of non-smooth state feedback control for a group of high-order p-normal form interconnected nonlinear systems with prescribed performance. Each subsystem is in the form of triangular structure with the unmodelled dynamics. With the help of appropriate Lyapunov function, a state feedback controller is constructed via backstepping method. It is strictly proved that the closed-loop system is asymptotically stable and both transient and steady-state performances of the outputs are preserved. Finally, simulation results are presented to illustrate the effectiveness of proposed method.

Necessary optimality conditions for nonautonomous optimal control problems and its applications to bilevel optimal control

This paper focuses on the development of necessary optimality conditions for nonautonomous optimal control problems with nonsmooth mixed state and control constraints. In most of the existing results, the necessary optimality conditions for nonautonomous optimal control problems with mixed state and control constraints are derived under the Mangasarian-Fromovitz condition or even stronger. In this paper we derive the necessary optimality conditions for nonautonomous optimal control problems under constraint qualifications which are weaker than Mangasarian-Fromovitz condition. Moreover necessary optimality conditions with an Euler inclusion taking a bounded explicit multiplier form are derived for certain cases. Specifying these results to bilevel optimal control problems with finite-dimensional lower level we obtain necessary optimality conditions under weaker qualification conditions.

Integrated stability control of AFS and DYC for electric vehicle based on non-smooth control

ABSTRACTA controller which ensures the driving stability of a four-wheel-independent-drive electric vehicle (4WID-EV) is designed in this paper. The controller is structurally hierarchically designed. In order to keep the 4WID-EV running steadily, an upper-level controller integrating the active front-wheel steering control method (AFS) and direct yaw moment control method (DYC) is designed to keep the sideslip angle and yaw rate tracking the ideal values. A non-smooth control method is used to improve the closed-loop system's convergence and anti-disturbance performance. The additional yaw moment generated by the upper-level controller is distributed to four driving wheels by the lower-level controller. An optimal control algorithm is used in the lower-level controller to achieve the minimum sum of tire utilisation, and ensure the power performance and driving stability of the 4WID-EV. In order to verify the effectiveness of the designed controller, a simulation model of the stability control system is established based on Carsim-Matlab/Simulink. And the simulation is performed under double lane change road considering the disturbances. The results of the simulation show that the 4WID-EV with the designed controller achieves smaller sideslip angle than sliding-mode control and the actuator chatter is slight. Then the stability and safety of the 4WID-EV are greatly improved.

Generalized sampling solutions for discontinuous stabilization of nonlinear systems

Sample-and-hold solution and sample stability are useful tools for analyzing stability of nonlinear control systems with discontinuous state feedbacks; for nonlinear control systems on Euclidean spaces, open-loop asymptotic controllability and sample stability of closed-loop systems are equivalent. Moreover, several sample stabilizing state feedback controllers based on nonsmooth control Lyapunov functions (CLFs) are proposed.However, analyzing sample stability based on the nonsmooth CLFs are too complex tasks in general. This is a serious obstacle to introduce more advanced properties such as invariance principle and its application to nonlinear adaptive control or finite-time stability for the sample stability framework.In this paper, we introduce the concept of generalized sampling solution, which is defined as the uniform limit of a sequence of sample-and-hold solutions. Moreover, we define asymptotic stability for generalized sampling solutions.The main contribution of this paper is to prove the existence and the asymptotic stability under the assumption that a locally semiconcave practical CLF (LS-PFCLF) is given.We also briefly mention an invariance principle based on a generalized sampling solution and an LS-PCLF.

Stochastic Feedback Control With One-Dimensional Degenerate Diffusions and Nonsmooth Value Functions

We study the stochastic control problems with degenerate semimartingales and nonsmooth value functions. We first derive the optimality conditions for finite horizon problems with nonsmooth value functions and singular control of infinite horizon discounted problems. Then, we show that the local time at the degenerate point is zero, so if the value function has nonsmooth first-order derivatives at the degenerate points of all policies, the nonsmoothness can be ignored and the optimality equation for stochastic control, including optimal stopping, is simply the classical Hamilton–Jacobi–Bellman equation at the smooth points. Furthermore, we model the singular control by the reflecting force in the Skorokhod problem and show that there are two classes of degenerate points; in the first class, singular control at a degenerate point can be treated in the same way as regular control; and in the second class, the density of the singular control force may be of the order $(\sqrt{dt})^{1+\gamma }$ , $0 , with $\gamma =0$ for nondegenerate points. We apply the direct-comparison-based approach to derive all the results. In all the analysis, viscosity solution is not needed.

Optimal control of implicit control systems and its applications to differential complementarity problems

This paper investigates the optimal control problem of nonsmooth implicit control systems. We construct a special set-valued mapping to transform the implicit control system into a differential inclusion. Based on advanced tools of optimal differential inclusions, we derive the necessary optimality conditions for the optimal control problem of implicit control systems. The results obtained are applied into optimal control problems of implicit systems with complementarity constraints.

Correction to: Stabilization of a class of fractional-order chaotic systems using a non-smooth control methodology

This note provides a corrigendum to the paper “Stabilization of a class of fractional-order chaotic systems using a non-smooth control methodology” [Nonlinear Dynamics, 89 (2017) 1357–1370]. It is pointed out that we have relied on a wrong formula in [2] to compute the upper bounds of the finite settling times of the proposed control methods in [1]. Fortunately, the minor errors appeared in [1] are readily corrected via the Mean Value Theorem for definite integrals. It is proved that the mentioned minor errors do not affect the main results and designs of the control methods in [1].

Optimal Control via Reinforcement Learning with Symbolic Policy Approximation

Model-based reinforcement learning (RL) algorithms can be used to derive optimal control laws for nonlinear dynamic systems. With continuous-valued state and input variables, RL algorithms have to rely on function approximators to represent the value function and policy mappings. This paper addresses the problem of finding a smooth policy based on the value function represented by means of a basis-function approximator. We first show that policies derived directly from the value function or represented explicitly by the same type of approximator lead to inferior control performance, manifested by non-smooth control signals and steady-state errors. We then propose a novel method to construct a smooth policy represented by an analytic equation, obtained by means of symbolic regression. The proposed method is illustrated on a reference-tracking problem of a 1-DOF robot arm operating under the influence of gravity. The results show that the analytic control law performs at least equally well as the original numerically approximated policy, while it leads to much smoother control signals. In addition, the analytic function is readable (as opposed to black-box approximators) and can be used in further analysis and synthesis of the closed loop.

Stabilization of a class of fractional-order chaotic systems using a non-smooth control methodology

This paper is devoted to demonstrate how a class of fractional-order chaotic systems can be controlled in a given finite time using just a single control input. First a novel fractional switching sliding surface is proposed with desired properties such as fast convergence to zero equilibrium and no steady state errors. At the second phase, a smooth reaching control law is derived to guarantee the occurrence of the sliding motion with a finite settling time. Owing to the integration of the control signal discontinuity, chattering oscillations are hindered from the controller. Rigorous stability analysis is performed to validate the design claims. The effects of high frequency external noises as well as modeling errors and dynamic variations are also taken into account, and the robustness of the closed-loop system is ensured. The proposed robust controller is realized for a class of chaotic fractional-order systems with one control input. In accordance, some remarks regarding the inclusion of mismatched uncertainties in the system dynamics are given. The robust functionality and quick convergence property as well as chatter-free attribute of the introduced non-smooth sliding mode technology are demonstrated using oscillation suppression of fractional-order chaotic Lorenz and financial systems.

RBF Nonsmooth Control Method for Vibration of Building Structure with Actuator Failure

In order to accommodate the actuator failure, the finite-time stable nonsmooth control method with RBF neural network is used to suppress the structural vibration. The traditional designed control methods neglect influence of actuator failure in structural vibration. By Lyapunov stable theory, the designed control method is demonstrated to suppress the building structural vibration with actuator failure. Finally, there are some examples to numerically simulate the three-layer building structure which is affected by El Centro seismic wave. Control effect of nonsmooth control is compared with no control and LQR control. The simulation results demonstrate that the designed control method is great for vibration of building structure with actuator failure and great antiseismic effect.

The Maximum Principle for Optimal Control Problems with Time Delays

This paper provides necessary conditions of optimality for optimal control problems with time delays in both state and control variables. Different versions of the necessary conditions cover fixed end-time problems and, under additional hypotheses, free end-time problems. The conditions improve on previous available conditions in a number of respects. They can be regarded as the first maximum principle for fully nonsmooth optimal control problems, involving delays in state and control variables, only special cases of which have previously been derived. Even when the data is smooth, the conditions advance the existing theory. For example, we provide a new “two-sided” generalized transversality condition, associated with the optimal end-time, which gives more information about the optimal end-time than the “one-sided” condition in the earlier literature. But there are improvements in other respects, relating to the treatment of initial data, specifying past histories of the state and control, the nature of ...

Semiactive Nonsmooth Control for Building Structure with Deep Learning

Aiming at suppressing harmful effect for building structure by surface motion, semiactive nonsmooth control algorithm with Deep Learning is proposed. By finite-time stable theory, the building structure closed-loop system’s stability is discussed under the proposed control algorithm. It is found that the building structure closed-loop system is stable. Then the proposed control algorithm is applied on controlling the building structural vibration. The seismic action is chosen as El Centro seismic wave. Dynamic characteristics have comparative analysis between semiactive nonsmooth control and passive control in two simulation examples. They demonstrate that the designed control algorithm has great robustness and anti-interference. The proposed control algorithm is more effective than passive control in suppressing structural vibration.

Direct yaw-moment control for 4WID electric vehicle via finite-time control technique

This paper is concerned with the yaw-moment stabilization for four-wheel independent-drive electric vehicle. A second-order sliding mode observer is first designed to estimate the required sideslip angle in a finite time. Then, the finite-time control technique and nonlinear disturbance observer are applied to construct the upper controller to drive both yaw rate and sideslip angle to their desired values. Finally, the lower controller is developed to distribute the torques to the independent four wheels according to the desired yaw moment obtained by the given upper controller. Comparisons among linear, discontinuous and nonsmooth controllers under different working conditions are given by using CarSim software.

Overview of linear time‐invariant interval observer design: towards a non‐smooth optimisation‐based approach

Some applications in control require the state vector of a system to be expressed in the appropriate coordinates so as to satisfy to some mathematical properties which constrain the studied system dynamics. This is the case with the theory of linear interval observers which are trivial to implement on cooperative systems, a rather limited class of control systems. The available literature shows how to enforce this limiting cooperativity condition for any considered system through a state-coordinate transformation. This study proposes an overview of the existing numerical techniques to determine such a transformation. It is shown that in spite of being practical, these techniques have some limitations. Consequently, a reformulation of the problem is proposed so as to apply non-smooth control design techniques. A solution is obtained in both the continuous- and discrete-time frameworks. Interestingly, the new method allows to formulate additional control constraints. Simulations are performed on three examples.

Non-smooth predictive control for mechanical transmission systems with backlash-like hysteresis

This paper proposes a non-smooth predictive control approach for mechanical transmission systems described by dynamic models with preceded backlash-like hysteresis. In this type of system, the work platform is driven by a DC motor through a gearbox. The work platform is represented by a linear dynamic sub-model connected in series with a backlash-like hysteresis inherent in gearbox. Here, backlash-like hysteresis is modeled as a non-smooth function with multi-valued mapping. In this case, the conventional model predictive control for such system cannot be implemented directly since the gradients of the control objective function with respect to control variables do not exist at non-smooth points. In order to solve this problem, a non-smooth receding horizon strategy is proposed. Moreover, the stability of predictive control of such non-smooth dynamic systems is analyzed. Finally, a numerical example and a simulation study on a mechanical transmission system are presented for validating the proposed method.

Consensus tracking for second-order nonlinear multi-agent systems with switching topologies and a time-varying reference state

ABSTRACTIn this study, the authors consider the consensus tracking problem for second-order nonlinear multi-agent systems with switching topologies and a time-varying reference state. The dynamics of each agent consists of nonlinear inherent dynamics. In order to reach consensus tracking, a class of nonsmooth control protocols is proposed which only depends on the agent's own information and its neighbours’ information. With the aid of algebraic graph theory, matrix theory, and Lyapunov theory, some corresponding sufficient conditions guaranteeing consensus tracking under the proposed control protocols are derived. Finally, the numerical simulations are given to illustrate the effectiveness of the developed theoretical results.

Chattering-free discrete-time sliding mode control

To avoid the chattering problem in the reaching-law-based discrete-time sliding mode control (DSMC) and the generation of over-large control action in the equivalent-control-based DSMC, a new DSMC method based on non-smooth control is proposed in this paper. Since there is no use of any switching term in the proposed DSMC, it is a chattering-free SMC method. Meanwhile, it is shown that the newly proposed non-smooth control-based DSMC can guarantee the same level of accuracy for the sliding mode motion as that of an equivalent control-based DSMC. To demonstrate the effectiveness of the proposed approach, a simulation example is presented.

Feedback preparation of maximally entangled states of two‐qubit systems

Bell states play a central role in the field of quantum information. For two-qubit systems, this study proposes a new non-smooth control strategy within the theoretical framework of quantum continuous measurement, and achieves global feedback stabilisation in any target Bell state. For a target Bell state, the authors select one measured observable and two control channels, where the control law associated with one channel is kept constant. They design a non-smooth control law for the other control channel, and prove the stability of the whole closed-loop system. Some simulation experiments show that the control strategy in this paper has a good control effect.

Finite-Time Composite Position Control for a Disturbed Pneumatic Servo System

This paper investigates the finite-time position tracking control problem of pneumatic servo systems subject to hard nonlinearities and various disturbances. A finite-time disturbance observer is firstly designed, which guarantees that the disturbances can be accurately estimated in a finite time. Then, by combining disturbances compensation and state feedback controller together, a nonsmooth composite controller is developed based on sliding mode control approach and homogeneous theory. It is proved that the tracking errors under the proposed composite control approach can be stabilized to zero in finite time. Moreover, compared with pure state feedback control, the proposed composite control scheme offers a faster convergence rate and a better disturbance rejection property. Finally, numerical simulations illustrate the effectiveness of the proposed control scheme.