Analysis of sliding modes in the frequency domain

A frequency-domain approach to the analysis of real and ideal sliding modes (SM) in single input single output (SISO) systems with linear plants is proposed. The real SM is analysed as a superposition of fast oscillations and slow motions in a relay system. The approach is based on the notion of the locus of a perturbed relay system (LPRS). The proposed LPRS is defined as a characteristic of the response of a linear part to an unequally spaced pulse control of variable frequency in a closed-loop. Formulas for computing the LPRS are derived. It is demonstrated that beside the well know chattering phenomenon, real SM also exhibits a non-ideal disturbance rejection, which can be quantitatively assessed with the use of the proposed approach. An example of the LPRS analysis of a SM system is given.     

An averaging approach to chattering

The singularly perturbed relay control systems (SPRCS) as mathematical models of chattering in the small neighborhood of the switching surface in sliding mode systems are examined. Sufficient conditions for existence and stability of fast periodic solutions to the SPRCS are found. It is shown that the slow motions in such SPRCS are approximately described by equations derived from equations for the slow variables of SPRCS by averaging along fast periodic motions. It is shown that In the general case, when the equations of a plant contain relay control nonlinearly, the averaged equations do not coincide with the equivalent control equations or with the Filippov's definition (1988) for the sliding motions in the reduced system; however, in the linear case, they coincide     

Robust stabilization of a class of uncertain system via block decomposition and VSC

In this paper, a block decomposition procedure for sliding mode control of a class of nonlinear systems with matched and unmatched uncertainties, is proposed. Based on the nonlinear block control principle, a sliding manifold design problem is divided into a number of sub‐problems of lower dimension which can be solved independently. As a result, the nominal parts of the sliding mode dynamics is linearized. A discontinuous feedback is then used to compensate the matched uncertainty. Finally, a step‐by‐step Lyapunov technique and a high gain approach is applied to obtain hierarchical fast motions on the sliding manifolds and to achieve the robustness property of the closed‐loop system motion with respect to unmatched uncertainty. Copyright © 2002 John Wiley & Sons, Ltd.     

Boundary Control for a Flexible Inverted Pendulum System Based on a PDE Model with Input Saturation

This research considers the control problem of a flexible inverted pendulum system (FIPS) in the presence of input saturation. The model for a flexible inverted pendulum system (FIPS) is derived via the Hamilton principle. The FIPS model is divided into a fast subsystem and a slow subsystem via the singular perturbation method. We introduce an auxiliary system to deal with the input saturation of a fast subsystem. To stabilize the fast subsystem, a boundary anti‐windup control force is applied at the free end of the beam. It is proven that the closed‐loop subsystem is stable. For the slow subsystem, a sliding mode control method is employed to design a controller and a new decoupling method to design the sliding surface. Then it is shown that the slow subsystem is stable. Finally, simulation results are provided to confirm the efficacy of the proposed controller.     

Dynamic modelling and adaptive robust tracking control of a space robot with two-link flexible manipulators under unknown disturbances

In this paper, both the closed-form dynamics and adaptive robust tracking control of a space robot with two-link flexible manipulators under unknown disturbances are developed. The dynamic model of the system is described with assumed modes approach and Lagrangian method. The flexible manipulators are represented as Euler–Bernoulli beams. Based on singular perturbation technique, the displacements/joint angles and flexible modes are modelled as slow and fast variables, respectively. A sliding mode control is designed for trajectories tracking of the slow subsystem under unknown but bounded disturbances, and an adaptive sliding mode control is derived for slow subsystem under unknown slowly time-varying disturbances. An optimal linear quadratic regulator method is proposed for the fast subsystem to damp out the vibrations of the flexible manipulators. Theoretical analysis validates the stability of the proposed composite controller. Numerical simulation results demonstrate the performance of the closed-loop flexible space robot system.     

Adjustment of high‐order sliding‐mode controllers

Long‐lasting problems of high‐order sliding‐mode (HOSM) design are solved. Only local uncertainty suppression was previously obtained in the case when the dynamic system uncertainties are unbounded. This restriction is removed in this paper. A universal method is proposed for the proper controller parameter adjustment based on the homogeneity approach. The method allows making the finite‐time convergence arbitrarily fast or slow. In addition, a HOSM regularization procedure is proposed diminishing chattering. Computer simulation confirms the theoretical results. Copyright © 2008 John Wiley & Sons, Ltd.     

Analysis of Closed-Loop Performance and Frequency-Domain Design of Compensating Filters for Sliding Mode Control Systems

For a sliding mode (SM) system containing parasitic dynamics, a model of chattering and a model of the averaged motions of the order equal to the original order of the system are obtained via the use of the locus of a perturbed relay system (LPRS) method. The latter model proves that the averaged motions in an SM system differ from those in the reduced-order model. In particular, such phenomena as the nonideal disturbance rejection and the nonideal input tracking are observed. It is shown that the non-reduced-order model transforms into the reduced one when the parasitic dynamics are removed. Yet, the chattering and the averaged motion - being the two components of the motion in the SM system - belong to two different frequency ranges (spectra). As a result, a frequency characteristic shaping aimed at achieving the desired frequency of chattering and enhancing the closed-loop performance becomes possible. This opportunity is explored in this paper. Examples that illustrate the proposed approach are provided.     

High‐order sliding‐mode observer–based input‐output linearization

The problem of output control in multiple‐input–multiple‐output nonlinear systems is addressed. A high‐order sliding‐mode observer is used to estimate the states of the system and identify the discrepancy between the nominal model and the real plant. The exact and finite‐time estimation may be tackled as long as the system presents the algebraic strong observability property. Thus, a continuous robust input‐output linearization strategy can be obtained with respect to a prescribed output. As a consequence, the closed‐loop dynamics performs robustly to uncertainties/perturbations. To illustrate the advantages of the proposed method, we introduce a study case that demands a robust linear system behavior: the self‐oscillations induced in an underactuated mechanical system through a two‐relay controller. Experiments with an inertial wheel pendulum illustrate the feasibility of the proposed approach.     

Generating self-excited oscillations for underactuated mechanical systems via two-relay controller

A tool for the design of a periodic motion in an underactuated mechanical system via generating a self-excited oscillation of a desired amplitude and frequency by means of the variable structure control is proposed. First, an approximate approach based on the describing function method is given, which requires that the mechanical plant should be a linear low-pass filter–the hypothesis that usually holds when the oscillations are relatively fast. The method based on the locus of a perturbed relay systems provides an exact model of the oscillations when the plant is linear. Finally, the Poincaré map's design provides the value of the controller parameters ensuring the locally orbitally stable periodic motions for an arbitrary mechanical plant. The proposed approach is shown by the controller design and experiments on the Furuta pendulum.     

Adaptive composite suboptimal control for linear singularly perturbed systems with unknown slow dynamics

In this article, using singular perturbation theory and adaptive dynamic programming (ADP) approach, an adaptive composite suboptimal control method is proposed for linear singularly perturbed systems (SPSs) with unknown slow dynamics. First, the system is decomposed into fast‐ and slow‐subsystems and the original optimal control problem is reduced to two subproblems in different time‐scales. Afterward, the fast subproblem is solved based on the known model of the fast‐subsystem and a fast optimal control law is designed by solving the algebraic Riccati equation corresponding to the fast‐subsystem. Then, the slow subproblem is reformulated by introducing a system transformation for the slow‐subsystem. An online learning algorithm is proposed to design a slow optimal control law by using the information of the original system state in the framework of ADP. As a result, the obtained fast and slow optimal control laws constitute the adaptive composite suboptimal control law for the original SPSs. Furthermore, convergence of the learning algorithm, suboptimality of the adaptive composite suboptimal control law and stability of the whole closed‐loop system are analyzed by singular perturbation theory. Finally, a numerical example is given to show the feasibility and effectiveness of the proposed methods.     

Boundary Control for A Flexible Inverted Pendulum System Based on A Pde Model

A partial differential equation (PDE) model for a flexible inverted pendulum system (FIPS) is derived by the use of the Hamilton principle. To solve the coupling system model, a singular perturbation method was adopted. The PDE model was divided into a fast subsystem and a slow subsystem using the singular perturbation method. To stabilize the fast subsystem, a boundary control force was applied at the free end of the beam. It then was proven that the closed‐loop subsystem is appropriate and exponentially stable. For the slow subsystem, a sliding mode control method was employed to design a controller and the Linear Matrix Inequality (LMI) method was used to design the sliding surface. It then was shown that the slow subsystem is exponentially stable.     

On a new adaptive sliding mode control for MIMO nonlinear systems with uncertainties of unknown bounds

This paper proposes a new approach of adaptive sliding mode controller designs for multiple‐input multiple‐output nonlinear systems with uncertainties of unknown bounds and limited available inputs. The goal is to obtain robust, smooth, and fast transient performance for real sliding mode control so that the phenomena of the slow response and the gain overestimation in most adaptive sliding mode controller designs can be greatly improved. We introduce an Integral/Exponential adaptation law with boundary‐layer targeting the reduction of the chatter levels of the sliding mode by significantly reducing the gain overestimation while simultaneously speeding up the system response to the uncertainties. The gain is further reduced when the system state is in the boundary layer. The simulation and experimental results demonstrate the proposed design. Copyright © 2016 John Wiley & Sons, Ltd.     

Coordination and Control of Multi-fingered Robot Hands with Rolling and Sliding Contacts

In this paper, the problem of controlling multi-fingered robot hands with rolling and sliding contacts is addressed. Several issues are explored. These issues involve the kinematic analysis and modeling, the dynamic analysis and control, and the coordination of a multi-fingered robot hand system. Based on a hand-object system in which the contacts are allowed to both roll and slide, a kinematic model is derived and analyzed. Also, the dynamic model of the hand-object system with relative motion contacts is studied. A control law is proposed to guarantee the asymptotic tracking of the object trajectory together with the desired rolling and/or sliding motions along the surface of the object. A planning approach is then introduced to minimize the contact forces so that the desired motion of the object and the relative motions between the fingers and the object can be achieved. Simulation results which support the theoretical development are presented.     

Decomposition of problems of optimal control and estimation for discrete systems with fast and slow variables

Consideration was given to the linear-quadratic problem of optimal control for the discrete linear system with fast and slow variables under incomplete information about system state. Decomposition of the discrete matrix Riccati equations was carried out. The proposed decomposition algorithm relies on a geometrical approach using the properties of the invariant manifolds of slow and fast motions of the nonlinear multirate discrete systems as basis. The splitting transformation was constructed in the form of asymptotic decomposition in the degrees of a small parameter.     

Sliding surface design for singularly perturbed systems

The equilibrium manifold of a singularly perturbed system has a close relationship with the sliding surface of a variable structure system (VSS). The fast time and slow timeresponses has a similar behaviour to the 'reaching mode' and 'sliding mode', respectively. This paper aims to equip the powerful composite control method with robustness through variable structure control design. The major bridge in between is a Lyapunov function. It is found that a singularly perturbed system in sliding mode may preserve two-time-scale attribute, in which a new equilibrium manifold exists on the sliding surface. Sliding motions that are attracted to the manifold can therefore be referred to as 'sliding mode in sliding mode'.     

Robot control for robust stability with finite reachability time in the whole

A robot model incorporates possible discontinuous nonlinearities with unknown forms and values, unknown payload and unknown predictable external disturbance variations, all in known bounds. A control algorithm is synthesized to guarantee the following: 1.Robust global both stability and attraction with finite reachability time of an appropriately chosen sliding set. 2.The robot motions reach, on the sliding set, a desired motion in a prespecified finite time. 3. Robust both stability and global attraction with finite reachability time of the given robot desired motion. 4. A prespecified convergence quality of real motions to the desired motion, independently of the internal dynamics of the system and without oscillations, hence without chattering in the sliding mode. Robot control robustness means that the controller realizes the control without using information about the real robot internal dynamics. All this is achieved by using the Lyapunov method in a new way combined with a sliding mode approach, but without a variation of the controller structure. The theoretical results are applied to a rotational 3‐degree‐of‐freedom robot. The simulations well verify the robustness of the control algorithm and high quality of robot motions with a prespecified reachability time. ©1999 John Wiley & Sons, Inc.     

Decomposition of a linear-quadratic optimal control problem with fast and slow variables

A linear-quadratic optimal control problem for a discrete different time-scale system is studied. The decomposition of the boundary value problem for the maximum principle is based on the geometric approach using the properties of invariant manifolds of slow and fast motions. This approach aids in constructing a transformation for reducing the initial problem to a boundary-value problem for slow variables and two initial-value problems for fast variables. The transformation is expressed as an asymptotic siries in powers of a small parameter.     

Fault detection and isolation for nonlinear systems via high‐order‐sliding‐mode multiple‐observer

In this paper, fault detection and isolation problems are studied for a certain class of nonlinear systems. Under some structural conditions, multiple high‐order sliding‐mode observers are proposed. The value of the equivalent output injection is used for detecting faults and the multiple‐model approach for isolating particular faults in the system. The proposed method provides fast detection and isolation of actuator and plant faults. Simulation results support the proposed approach. Copyright © 2014 John Wiley & Sons, Ltd.     

Proxy-based sliding mode control of a robotic ankle-foot system for post-stroke rehabilitation

Robotic platform-based ankle–foot rehabilitation systems have been proved effective in treating joint spasticity and/or contracture of stroke survivors. However, simple force or velocity limiters are not adequate, since they cannot explicitly guarantee slow and overdamped motions without overshoot. In this paper, we propose a proxy-based sliding mode control (PSMC)-based approach, to avoid unsafe behaviors of a robotic ankle–foot rehabilitation system. The proposed method has three advantages: (1) without deteriorating tracking performance during normal operation, it guarantees overdamped, slow, and safe recoveries after abnormal events; (2) it provides a simple and accurate way to confine the output torque exerted on the subject’s ankle; (3) though effective, the control law avoids the necessity to identify the specific system model or build state observer, which is usually difficult for human–robot interaction system. A 71-year-old stroke patient and 10 able-bodied subjects were recruited for the experiments. Preliminary studies comparing PSMC and PID are performed on trajectory tracking, controlled torque output, slow and safe response under disturbance. Additionally, by fulfilling the rehabilitation method and obtaining biomechanical indicators, the proposed controller is proved to be feasible for the system.     

基于积分流形的柔性机械手组合控制

为解决柔性机械手非最小相位的控制问题以及克服运动中的抖振,采用积分流形和奇异摄动理论,将柔性机械手系统分解为快慢两个子系统.对于慢变子系统,设计一种基于一阶鲁棒微分估计器的二阶滑模控制策略,使其轨迹跟踪期望值;对于快变子系统,采用频率成形滤波器设计动态补偿器来抑制弹性振动,并基于线性二次型最优控制方法给出相应的最优控制规律,使系统的输出快速趋于稳定.仿真结果表明了该控制策略的有效性.