In this paper, we present Lyapunov-based {\color{black}time varying} controllers for {\color{black}fast} stabilization of a perturbed chain of integrators with bounded uncertainties. We refer to such controllers as {\color{black}time varying} higher order sliding mode controllers since they are designed for nonlinear Single-Input-Single-Output (SISO) systems with bounded uncertainties such that the uncertainty bounds are unknown. %{\color{blue} OLD: Our main result states that, given any neighborhood $\varepsilon$ of the origin, we determine a controller insuring, for every uncertainty bounds, that every trajectory of the corresponding closed loop system enters $\varepsilon$ and eventually remains there. Furthermore, based on the homogeneity property, a new asymptotic accuracy, which depends on the size of $\varepsilon$, is presented.} We provide a time varying control feedback law insuring verifying the following: there exists a family $(D(t))_{t\geq 0}$ of time varying open sets decreasing to the origin as $t$ tends to infinity, such that, for any unknown uncertainty bounds and trajectory $z(\cdot)$ of the corresponding system, there exists a positive positve $t_z$ for which $z(t_z)\in D(t_z)$ and $z(t)\in D(t)$ for $t\geq t_z$. %enters convergence in finite time of all the trajectories to a time varying domain $D(t)$ shrinking to the origin and their maintenance there. Hence, since the function $η(t)$ tends to zero, this leads the asymptotic convergence of all the trajectories to zero. The effectiveness of these controllers is illustrated through simulations.
In this note, it is shown by counterexamples, theoretical analysis and simulation tests that the two types of sliding-mode controllers presented in the paper [1] fail to solve the stabilization problem of a class of under-actuated mechanical systems.
Conventional sliding mode controllers are based on the assumption of switching control but a well-known drawback of this approach is the chattering phenomenon. To overcome the undesirable chattering effects, the discontinuity in the control law can be smoothed out in a thin boundary layer neighboring the switching surface. In this work, rigorous proofs of the boundedness and convergence properties of smooth sliding mode controllers are presented. This result corrects flawed conclusions previously reached in the literature.
In this paper, two classes of continuous higher order adaptive sliding mode controllers based on barrier functions are developed for a perturbed chain of integrators with unbounded perturbations. Both classes provide finite-time convergence of system states to a predefined domain using a continuous control signal. The first class of adaptive controllers does not require any assumptions about the perturbation; however, it can provide unbounded control gains. To ensure bounded control gains, and in the case of a Lipschitz perturbation, a second class of adaptive controllers, called the adaptive …
Different time-discretization methods for equivalent-control based sliding mode control (ECB-SMC) are presented. A new discrete-time sliding mode control scheme is proposed for linear time-invariant (LTI) systems. It is error-free in the discretization of the equivalent part of the control input. Results from simulations using the various discretized SMC schemes are shown, with and without perturbations. They illustrate the different behaviours that can be observed. Stability results for the proposed scheme are derived.
In this paper, we present a Lyapunov-based homogeneous controller for the stabilization of a perturbed chain of integrators of arbitrary order $r\geq 1$. The proposed controller is based on homogeneous controller for stabilization of pure integrator chains. The advantages to control the homogeneity degree of the controller are also discussed. A bounded-controller with minimum amplitude of discontinuous control and a controller with fixed-time convergence are synthesized, using control of homogeneity degree, and their performances are shown in simulations. It is demonstrated that the homogeneou…
Sliding mode control (SMC) is a robust and computationally efficient solution for tracking control problems of highly nonlinear systems with a great deal of uncertainty. High frequency oscillations due to chattering phenomena and sensitivity to data sampling imprecisions limit the digital implementation of conventional first order continuous-time SMC. Higher order discrete SMC is an effective solution to reduce the chattering during the controller software implementation, and also overcome imprecisions due to data sampling. In this paper, a new adaptive second order discrete sliding mode contr…
This paper is concerned with model reference adaptive controller design for a class of nonlinear fractional order systems. Recent works on this topic rarely include direct methods and they are mostly based on indirect methods where the frequency distributed model is used to prove the stability of the closed loop system. Since the chain rule cannot be applied in fractional derivations, in order to prove the lyapunov stability here fractional inequalities are used. Finally, by means of a numerical example, the controller performance is demonstrated.
In this paper, we present Lyapunov-based robust and adaptive controllers for the finite time stabilization of a perturbed chain of integrators with bounded uncertainties. The proposed controllers can be designed for integrator chains of any arbitrary length. The uncertainty bounds are known in the robust control problem whereas they are unknown in the adaptive control problem. Both controllers are developed from a class of finite time stabilization controllers for pure integrator chains. Lyapunov-based design permits to calculate upper bound on convergence time.
Analog-to-digital conversion (ADC) and uncertainties in modeling the plant dynamics are the main sources of imprecisions in the design cycle of model-based controllers. These implementation and model uncertainties should be addressed in the early stages of the controller design, otherwise they could lead to failure in the controller performance and consequently increase the time and cost required for completing the controller verification and validation (V&V) with more iterative loops. In this paper, a new control approach is developed based on a nonlinear discrete sliding mode controller (DSM…
We propose a learning-based method for Lyapunov stability analysis of piecewise affine dynamical systems in feedback with piecewise affine neural network controllers. The proposed method consists of an iterative interaction between a learner and a verifier, where in each iteration, the learner uses a collection of samples of the closed-loop system to propose a Lyapunov function candidate as the solution to a convex program. The learner then queries the verifier, which solves a mixed-integer program to either validate the proposed Lyapunov function candidate or reject it with a counterexample, …
This paper studies switching stabilization problems for general switched nonlinear systems. A piecewise smooth control-Lyapunov function (PSCLF) approach is proposed and a constructive way to design a stabilizing switching law is developed. The switching law is constructed via the directional derivatives of the PSCLF with a careful discussion on various technical issues that may occur on the nonsmooth surfaces. Sufficient conditions are derived to ensure stability of the closed-loop Filippov solutions including possible sliding motions. The proposed PSCLF approach contains many existing result…
This paper deals with the design of robust tracking and model following (RTMF) controller for linear time-invariant (LTI) systems with uncertainties. The controller is based on the second order sliding mode (SOSM) algorithm (super twisting) which is the most effective and popular in the family of higher order sliding modes (HOSM). The use of super twisting algorithm (STA) eliminates the chattering problem encountered in traditional sliding mode control while retaining its robustness properties. The proposed robust tracking controller can guarantee the asymptotic stability of tracking error in …
In this work, an intelligent controller is employed to the chaos control problem in a nonlinear pendulum. The adopted approach is based on the sliding mode control strategy and enhanced by an adaptive fuzzy algorithm to cope with modeling inaccuracies. The convergence properties of the closed-loop system are analytically proven using Lyapunov's direct method and Barbalat's lemma. Numerical results are also presented in order to demonstrate the control system performance.
Sliding mode control (SMC) is a robust and computationally efficient model-based controller design technique for highly nonlinear systems, in the presence of model and external uncertainties. However, the implementation of the conventional continuous-time SMC on digital computers is limited, due to the imprecisions caused by data sampling and quantization, and the chattering phenomena, which results in high frequency oscillations. One effective solution to minimize the effects of data sampling and quantization imprecisions is the use of higher order sliding modes. To this end, in this paper, a…
In this paper, the control problem for underactuated systems in the presence of external disturbances and model uncertainties is considered. An adaptive fuzzy sliding mode controller (AFSMC) is proposed to solve the problem, satisfying the robustness properties against uncertainties and disturbances. The designed controller can be applied to a wide class of underactuated systems and have fewer restrictions, compared with many pervious works. To evaluate the performance of proposed algorithm, it is applied to an underactuated inverted pendulum and a Translational Oscillator/Rotational Actuator …
The performance of a conventional model-based controller significantly depends on the accuracy of the modeled dynamics. The model of a plant's dynamics is subjected to errors in estimating the numerical values of the physical parameters, and variations over operating environment conditions and time. These errors and variations in the parameters of a model are the major sources of uncertainty within the controller structure. Digital implementation of controller software on an actual electronic control unit (ECU) introduces another layer of uncertainty at the controller inputs/outputs. The imple…
The dead-zone is one of the most common hard nonlinearities in industrial actuators and its presence may drastically compromise control systems stability and performance. In this work, an adaptive variable structure controller is proposed to deal with a class of uncertain nonlinear systems subject to a non-symmetric dead-zone input. The adopted approach is primarily based on the sliding mode control methodology but enhanced by an adaptive fuzzy algorithm to compensate the dead-zone. Using Lyapunov stability theory and Barbalat's lemma, the convergence properties of the closed-loop system are a…
This paper deals with the design of a model-based rapid fault detection and isolation strategy using sliding mode observers. To address this problem, a new scheme is proposed by adaptively combining the information provided by a bank of observers. In this regard, a new structure for sliding mode observers is considered. Then, the well-known recursive least square algorithm is utilized to merge individual state estimations suitably such that the system fault is detected faster. The required condition for enhancing perfect state estimation is derived, and the stability of the overall system is p…
Incremental stability is a property of dynamical and control systems, requiring the uniform asymptotic stability of every trajectory, rather than that of an equilibrium point or a particular time-varying trajectory. Similarly to stability, Lyapunov functions and contraction metrics play important roles in the study of incremental stability. In this paper, we provide characterizations and descriptions of incremental stability in terms of existence of coordinate-invariant notions of incremental Lyapunov functions and contraction metrics, respectively. Most design techniques providing controllers…
The paper introduces novel families of cruise controllers for autonomous vehicles on lane-free ring-roads. The design of the cruise controllers is based on the appropriate selection of a Control Lyapunov Function expressed on measures of the energy of the system with the kinetic energy expressed in ways similar to Newtonian or relativistic mechanics. The derived feedback laws (cruise controllers) are decentralized (per vehicle), as each vehicle determines its control input based on: (i) its own state; (ii) either only the distance from adjacent vehicles (inviscid cruise controllers) or the sta…
This paper presents three types of sliding mode controllers for a magnetic levitation system. First, a proportional-integral sliding mode controller (PI-SMC) is designed using a new switching surface and a proportional plus power rate reaching law. The PI-SMC is more robust than a feedback linearization controller in the presence of mismatched uncertainties and outperforms the SMC schemes reported recently in the literature in terms of the convergence rate and settling time. Next, to reduce the chattering phenomenon in the PI-SMC, a state feedback-based discrete-time SMC algorithm is developed…
This paper presents an intelligent controller for uncertain underactuated nonlinear systems. The adopted approach is based on sliding mode control and enhanced by an artificial neural network to cope with modeling inaccuracies and external disturbances that can arise. The sliding surfaces are defined as a linear combination of both actuated and unactuated variables. A radial basis function is added to compensate the performance drop when, in order to avoid the chattering phenomenon, the sign function is substituted by a saturation function in the conventional sliding mode controller. An applic…
A constructive tool of nonlinear control systems design, the method of Control Lyapunov Functions (CLF) has found numerous applications in stabilization problems for continuous time, discrete-time and hybrid systems. In this paper, we address the fundamental question: given a CLF, corresponding to the continuous-time controller with some predefined (e.g. exponential) convergence rate, can the same convergence rate be provided by an event-triggered controller? Under certain assumptions, we give an affirmative answer to this question and show that the corresponding event-based controllers provid…
This paper studies the time-varying bearing-based tracking of leader-follower formations. The desired constraints between agents are specified by bearing vectors, and several leaders are moving with a bounded reference velocity. Each followers can measure the relative positions of its neighbors, its own velocities, and receive information from their neighbors. Under the assumptions that the desired formation is infinitesimally bearing rigid and the local reference frames of followers are aligned with each other, two control laws are presented in this paper based on sliding mode control approac…
As more inverter-connected renewable resources are integrated into the grid, frequency stability may degrade because of the reduction in mechanical inertia and damping. A common approach to mitigate this degradation in performance is to use the power electronic interfaces of the renewable resources for primary frequency control. Since inverter-connected resources can realize almost arbitrary responses to frequency changes, they are not limited to reproducing the linear droop behaviors. To fully leverage their capabilities, reinforcement learning (RL) has emerged as a popular method to design n…
In this work, we establish different control design approaches for discrete-time systems, which build upon the notion of finite-step control Lyapunov functions (fs-CLFs). The design approaches are formulated as optimization problems and solved in a model predictive control (MPC) fashion. In particular, we establish contractive multi-step MPC with and without reoptimization and compare it to classic MPC. The idea behind these approaches is to use the fs-CLF as running cost. These new design approaches are particularly relevant in situations where information exchange between plant and controlle…
In this paper the existence of a quadratic control Lyapunov function for bilinear systems is considered. The existence of a control Lyapunov function ensures the existence of a control law which ensures the global asymptotic stability of the closed loop control system. In this paper we will derive conditions for the existence of a control Lyapunov function for bilinear systems. These conditions will be derived for the whole class of two dimensional bilinear systems with one control input. This will lead to a simple flow diagram representation for the controller design for bilinear systems usin…
In this technical note, we generalize the well-known Lyapunov-based stabilizability and detectability tests for linear time-invariant (LTI) systems to the context of discrete-time (DT) polytopic linear parameter-varying (LPV) systems. To do so, we exploit the mathematical structure of the class of poly-quadratic Lyapunov functions, which enables us to formulate conditions in the form of linear matrix inequalities (LMIs). Our results differ from existing conditions in that we adopt weaker assumptions on the parameter dependence of the controllers/observers and our method does not require explic…
Electro-hydraulic servo-systems are widely employed in industrial applications such as robotic manipulators, active suspensions, precision machine tools and aerospace systems. They provide many advantages over electric motors, including high force to weight ratio, fast response time and compact size. However, precise control of electro-hydraulic systems, due to their inherent nonlinear characteristics, cannot be easily obtained with conventional linear controllers. Most flow control valves can also exhibit some hard nonlinearities such as dead-zone due to valve spool overlap. This work describ…
Applied Lyapunov Stability on Output Tracking Problem for a Class of Discrete-Time Linear Systems
Laboratory of Analysis Modeling and Simulation, Department of Mathematics and Computer Science, Faculty of Sciences Ben M’Sik, Hassan II University of Casablanca（卡萨布兰卡哈桑二世大学本米斯克理学院数学与计算机科学系分析、建模与仿真实验室）
