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Multi-input sliding mode control of a class of uncertain nonlinear systems
Abstract
An effective way to extend to the multi-input case the variable
structure control philosophy is based on a set of m+1 control vectors
forming a simplex in R<sup>m</sup> and on the corresponding switching of
the controlled system from one to another of m+1 different structures.
Yet, some problems arise when uncertainties are present in certain
matrices characterizing the controlled systems. In this paper,
conditions are identified under which, even in the presence of
uncertainty, the convergence to the sliding manifold is ensured via the
application of a multi-input control strategy still based on a simplex
of control vectors
... Besides, SSMC is also an effective way to extend the sliding-mode control methodology to the multi-input case. The design of the so-called simplex control for multivariable systems [2,3] is relatively straightforward in comparison with the conventional SMC, and some successful applications of SSMC in practice have been reported [4]. ...
... An alternative approach to multi-input nonlinear sliding-mode control is the simplex control [2,3]. The advantage of the simplex control approach lies in the fact that the number of structures among which the controlled system switches is reduced with respect to the component-wise case (m þ 1 instead of 2 m ). ...
... Note that given a simplex U , the control switching surfaces for the simplex control law are defined by the boundaries of X i , which is apparently not accordant with the sliding manifold. The following theorem [2,3] investigates the stability and convergence of the simplex nonlinear sliding mode control approach. ...
As an emerging effective approach to nonlinear robust control, simplex sliding mode control demonstrates some attractive features not possessed by the conventional sliding mode control method, from both theoretical and practical points of view. However, no systematic approach is currently available for computing the simplex control vectors in nonlinear sliding mode control. In this paper, chaos-based optimization is exploited so as to develop a systematic approach to seeking the simplex control vectors; particularly, the flexibility of simplex control is enhanced by making the simplex control vectors dependent on the Euclidean norm of the sliding vector rather than being constant, which result in both reduction of the chattering and speedup of the convergence. Computer simulation on a nonlinear uncertain system is given to illustrate the effectiveness of the proposed control method.
... In [8] an adaptive scheme of designing SMC for affine class of multiinput multi-output (MIMO) nonlinear systems with uncertainty in the systems dynamics and control distribution gain is developed. Also in [9] the problem of MIMO SMC of a class of uncertain nonlinear systems is faced. The problem has been also faced from an experimental point of view: in [10], for example, a motion control algorithm for microsatellite mock-up with flexible rods on an air table is investigated, even if focused specifically on the docking with noncooperative target. ...
... When the analog, time varying, control signal from the actuation interface circuit is connected to the PZT actuators, they deform and exert an elastic load on the panel which can be described as a pair of moments with opposite signs acting on the two ends of the rectangular PZT patch: (9) where E p is the PZT Young modulus, d 31 is the PZT electromechanical coupling coefficient, b is the PZT width (see Fig. 4), h is the thickness of the panel, V (t), is the voltage applied to the PZT. ...
The interaction between angular motion and flexible vibrations can heavily affect the stability of the spacecraft. Many control strategies have been developed for solving this issue. Some of them are focused on the attitude dynamics while a different approach consists in facing the problem from the structural point of view, trying to actively damp the vibrations induced by the attitude control, using smart material (like piezoelectric) devices.
In this research, an approach unifying the two aspects is proposed. The satellite is modeled as a flexible multibody system, in which the two sets of actuators (attitude and structural devices) are commanded by means of a common sliding mode control algorithm. In such a way, the two systems are not considered as competitors (each one trying to cancel the disturbing effects caused by the other one), but they are cooperating for the common goal of acquiring a desired attitude in a given time without residual oscillations. This synergetic approach is first developed in a numerical environment, then it is tested by means of a free-floating platform equipped with flexible appendages, designed and built as a multilayer composite material with a net of embedded PZT patches (sensors and actuators). The overall navigation and control loop is based on the information coming from the Inertial Measurement Unit and from the PZT sensors, which are filtered and sent to the Synergetic controller, with the goal of reaching a desired attitude. The output of the Synergetic controller consists in both the thrusters firing sequence and the PZT actuators voltage difference required to reach the goal. The experimental results are compared with the ones obtained by more classic approaches (attitude and structural control computed independently), commenting both advantages and drawbacks of the different approaches.
... For example, Utkin et al. [17] have discussed SMC with an application domain of electro-mechanical systems. SMC has also been demonstrated for multi-input systems with uncertainties [18]. An SMC with an adaptive switching gain based on Neural Network has been proposed in. ...
... Theorem 1: Suppose the system defined in (18) satisfies Assumption 1. Then this system is finite time stable using control law (21), if there exist a gain η such that η > ρ. ...
In this work, we present a Sliding Mode Control (SMC) based approach to address the velocity tracking and head angle control problem of a planar snake robot. The motion characteristics of a snake exhibit the generation of propulsive force as a result of anisotropic friction with respect to the ground. To imitate the motion of a snake, all the joints of the snake robot are tracked to a Serpenoid gait function utilizing Virtual Holonomic Constraints (VHCs). The parameters of the gait function are obtained from the SMC resulting in head-angle control and velocity tracking. SMC has been chosen to ensure robustness and stability of the system in the presence of uncertainties arising from variation in the friction force coefficients between the robot and the ground. Lyapunov’s stability analysis proves the finite-time stability of the system. The control scheme has also been verified and compared with an existing approach through simulation studies.
... One of the most stimulating aspect of the Sliding Mode Control (SMC) [1][2][3][4] is the discontinuous nature of the controller, whose primary function is to switch between two characteristically different system structures such that a new type of system motion, called sliding modes [5], exists in a manifold, known as the sliding manifold. This inspirational system phenomenon results in splendid system performance, which includes parameter invariance and remarkable robustness against disturbances and model uncertainties. ...
... where ξ T = [|x 1 | y sign(x 1 ) x 2 ], y = ρ−1 ρ and P ∈ ℜ 2×2 is symmetric and positive definite matrix, which is a solution of the arithmetic Lyapunov equation (ALE) (4). ...
Novel robustness and performance parameters are established for Smooth Super Twisting Algorithm (SSTA). The stability of SSTA is well established for arbitrary gains using homogeneity approach. The design and tuning of the controller parameters is a major issue and no analytic design method is available so far. A novel Lyapunov function is proposed and by the virtue of stability analysis, the stability bounds for a certain class of uncertainties are determined. In addition, the issue of finite time convergence is also explored, resulting in determination of the settling time as a function of the controller parameters. The proposed settling time formulation suggests a methodical approach to SSTA design in contrast to the available rules of thumb. Unlike the literature available for Higher Order Sliding Mode (HOSM) controllers, the proposed design framework is validated against a challenging problem of the Underground Coal Gasification (UCG) process control. Like the other process control problems the chosen problem is nonlinear and contains significant uncertainties. © 2016 Institute of Control, Robotics and Systems and The Korean Institute of Electrical Engineers and Springer-Verlag Berlin Heidelberg
... The mathematical problem to be faced can be described as follows: differentiate twice the quantity s to obtaiṅ s =ẋ 2 + cẋ 1 = f (x, u) + cx 2 (2) ...
... To conclude this treatment we outline the fact that in the more general situation (m < n − 1), a straightforward extension of the above methodology can be found only for the particular case in which M −1 * (q 1 )J 1 (q) is diagonally dominant. A possible generalization of the simplex control method [1], [2] is under investigation. ...
This paper deals with the hybrid position/force control problem for constrained manipulators subjected to uncertainties and disturbance of various natures. This problem can be formulated in terms of the solution to differential algebraic equations with a structure which allows the separation of the force and position control when the system is perfectly known. A solution for the uncertain case has been carried out based on sliding mode control theory which has been shown to be highly effective in counteracting uncertainties and disturbances for some classes of uncertain, nonlinear systems. Specific problems related to this technique are chattering elimination and the algebraic coupling between constraint forces and possibly discontinuous control signals. This paper presents a solution to the particular case of a manipulator with n degrees of freedom and n-1 holonomic constraints, leaving the solution to the general problem for further investigations. The main contribution is the use of a new second order sliding mode control algorithm which is proved to yield the solution of the problem after a transient of finite duration without requiring the availability of the acceleration vector.
... SMC-based robust control approach has been widely utilized to address uncertainties in systems like electromechanical systems [57], wheeled mobile robot [58], snake robot [59,60], and aerospace systems [61]. A switching control law is designed with a gain chosen based on the known upper bound of the uncertainty to achieve finitetime convergence [62,63]. ...
This paper reports a method for regulating the internal forces during in hand manipulation of an unknown shaped object with soft robotic fingers. The internal forces ensure that the object does not move between the robotic fingers, thus improving the grip. It is shown that if soft fingers show bounded conformity and the finger-object interface have bounded relative slip, then the relative angular velocity between the object and the fingertip coordinate frame in contact is bounded. Detailed derivation of the proof is presented. The proof is used to define a new metric of relative slip. The metric is used to design a sliding mode control algorithm that results in an efficient grip which is robust towards uncertainty in object shape. The robotic fingers are assumed to be under virtual rigidity constraint, that is, the distance between the fingertips do not change. The control algorithm is attractive as it skirts the requirement of information of the shape of the object or to solve optimization problems. The grip with the robust control algorithm is shown to be finite-time stable through Lyapunov's method. The methodology is demonstrated using simulations.
... SMC-based robust control approach has been widely utilized to address uncertainties in systems like electromechanical systems [57], wheeled mobile robot [58], snake robot [59,60], and aerospace systems [61]. A switching control law is designed with a gain chosen based on the known upper bound of the uncertainty to achieve finitetime convergence [62,63]. ...
This paper reports a method for regulating the internal forces during in hand manipulation of an unknown shaped object with soft robotic fingers. It is known that for the case of multifingered manipulation, a part of the forces applied by the fingers result in the motion of the object, whereas the other part is considered to be an internal force. The internal forces do not result in the motion of the object but are used to improve the grip on the object. For an object with unknown shape, the internal forces are regulated to ensure that the object does not slip off during manipulation. It is shown that if soft fingers show bounded conformity and the finger-object interface does not have relative slip (or a bounded slip), then the relative angular velocity between the object and the fingertip frame in contact is bounded. The proof is used to define of a new metric of relative slip. The metric is used to design a sliding mode control algorithm. The robotic fingers are assumed to be under virtual rigidity constraint, that is, the distance between the fingers do not change. The control algorithm is attractive as it skirts requirement of information of the shape of the object or to solve optimization problems. The control algorithm developed controls the internal forces and does not require the knowledge of the shape of the object. The methodology is simulated for the case of one spherical object and one conical object.
... SMC is able to achieve finite time reaching and exact keeping of a suitably chosen sliding manifold in the state space by means of discontinuous control. So, over the past decade, SMC has been one of the most popular control methods that has found wide applications to different fields of science and industry, and many important results have been reported for this kind of control strategy [15,16]. Unfortunately, lack of extension of SMC to accommodate FO nonlinear systems is apparent. ...
This paper concerns with the problem of designing a passivity-based fractional-order (FO) integral sliding mode controller for uncertain FO nonlinear systems. Utilizing the FO calculus, it is showed that the state trajectories of the closed-loop system reach the FO switching manifold in finite time. The control law ensures the asymptotical stability on the sliding surface. A parameter adjustment scheme for FO integral sliding surface is proposed by using the linear matrix inequality (LMI) approach. The proposed controller can be applied to different systems such as chaotic systems. Finally, simulation results are provided to show the effectiveness of the proposed method controlling chaos in FO Chua circuit and FO Van-der-Pol oscillator.
... The presence of uncertain nonlinearity in physical systems has been extensively studied because most of the real systems are rather complex dynamical nonlinear with large uncertainties which can affect inaccuracy and poor robustness of the control system. To do this, numerous control schemes have been developed; for example, sliding mode control [1][2][3], intelligent control [4,5], feedback linearization [6], and adaptive control [7][8][9]. ...
The sliding mode control (SMC) technique with a first-order low-pass filter (LPF) is incorporated with a new adaptive PID controller. It is proposed for tracking control of an uncertain nonlinear system. In the proposed control scheme, the adaptation law is able to update the PID controller online during the control process within a short period. The chattering phenomenon of the SMC can be alleviated by incorporation of a first-order LPF, while the robustness of the control system is similar to that of the sliding mode. In the closed-loop control analysis, the convergence condition in the reaching phase and the existence condition of the sliding mode were analyzed. The stability of the closed-loop control is guaranteed in the sense of Lyapunov’s direct method. The simulations and experimental applications of a speed tracking control of a spark ignition (SI) engine via electronic throttle valve control architecture are provided to verify the effectiveness and the feasibility of the proposed control scheme.
... 2001). Bartolini et.al (1996), identified some conditions under which, even in the presence of uncertainty, the convergence to the sliding manifold is ensured via the application of a multi-input control. Han Ho Choi (2002), proposed a linear matrix inequality (LMI)-based sliding surface design method for integral sliding-mode control of systems which may have mismatched norm bounded uncertainties in the state matrix as well as the input matrix. ...
In this paper, the authors propose a novel dynamic integral sliding mode controller for state dependent matched and unmatched uncertainties. An output feedback methodology based designed dynamic controller attenuate the effects of the both matched and unmatched uncertainties along with considerable reduction in chattering phenomena. An integral manifold is used which provides sliding mode without reaching phase and further enhances the robustness of the controller against uncertainties. A Lyapunov candidate is used for the stability analysis against uncertainties. The feasibility of the control law is illustrated via a kinematic car model.
... During the last years, the increasing interest in higher order sliding modes (14), (15), (16), introduced for differentiation (17), (18), and widely applied to the field of mechanical systems (see, among others, (19), (20), (21), and (22)), has led to many interesting proposals of application of these advanced variable structure control schemes to induction motor control (see, for instance, (13), (23), and (24)). ...
A novel sliding mode observer for current-based sensorless speed control of induction motors is presented in this article. The control objective is to guarantee asymptotic tracking of prespecified references for speed and rotor flux magnitude, without sensors measuring the mechanical speed and the flux, and assuming to have some kind of uncertainties on the value of the rotor resistance. To this end, the proposed observer is designed by coupling a second-order sliding mode observer of the stator current with a nonlinear flux and speed observer, adaptive with respect to the rotor resistance. Estimation of unknown inputs is based on a different and original approach with respect to the widely used equivalent control-based techniques. As for the control aspects, in the present proposal, the problem of chattering, typical of sliding mode controllers, is made less critical since the derivative of the stator currents are used as discontinuous forcing actions, while the actual control signals are continuous, thus limiting the mechanical stress.
The ability of a control scheme for a mobile robot to ensure satisfactory performance in an unstructured or unknown environment is what makes the control law unique. Various sources of uncertainties pose a serious challenge to the tracking performance of the system. The state-of-the-art control law presented in Chap. 1 cannot ensure stability in the presence of uncertainties due to feedback linearization methodology, as it requires exact cancellation of the nonlinearities in the system [1, 2]. Moreover, singular perturbation-based approach is prone to generate high control gains which may jeopardize the integrity of the actuators [1, 2]. This is where robust control techniques step in, to assure stable and acceptable performance in the presence of uncertainties. To imitate variation of ground condition, time-varying uncertainties have been induced through the friction coefficients in the planar snake robot model. These uncertainties have been assumed to be bounded with a known upper bound to implement a Sliding-Mode Control (SMC) law with the aim of achieving efficient head-angle and velocity tracking. Furthermore, to relax the constraint on the uncertainty bound and also to solve the overestimation of switching gain, an adaptive SMC has been proposed to improve the tracking performance.
A backstepping adaptive sliding mode control strategy based on extended state observer (ESO) is proposed for uncertain nonlinear system with unknown system functions and unknown external disturbance. The backstepping adaptive sliding mode control and the ESO theory are combined in the developed controller, and thus the restriction of all the states in the nonlinear system should be completely measurable is abolished. Through a simple coordinate transformation, the general nonlinear model is transformed into a strict-parametric-feedback form, which is more suitable for the design of the sliding mode controller by backstepping method. In the presence of unknown states and total uncertainties, the ESO is employed to estimate the unknown states and the total uncertainties. Thereafter, the backstepping sliding mode control law and adaptive law of switch gain are designed to guarantee that the tracking errors can converge to zeros rapidly and stably. And then, the derivatives of the virtual control are computed by using a filter without having to analytically or numerically differentiate it. Therefore, the implementation of the backstepping design method is significantly simplified. Comparative simulation experiment results all show that the proposed controller can improve the chattering problem of the sliding mode control and improve the dynamic performance of the system.
In this paper, a novel fuzzy simplex sliding-mode controller is proposed for controlling a multivariable nonlinear system. Here, the fuzzy logic control (FLC) technique and simplex sliding-mode control (SSMC) theory are integrated to improve the system response and to reduce system states chattering phenomenon of the controlled system. Hence, from this motivation yields the so-called fuzzy simplex sliding mode control (FSSMC) scheme. Comparing the proposed FSSMC with the conventional fuzzy sliding-mode control (FSMC), some differences between these two control schemes should be addressed. Unlike the design procedure in FSMC, we design the FSSMC by utilizing the switching function σ and its norm value ||σ|| as the inputs of the controller instead of error signal e and error rate signal ė used in FSMC. The output actions of the proposed FSSMC, however, are represented by the SSMC vector form and its entries are determined by FLC. Finally, a numerical example is performed to illustrate the merits of the proposed controllers; the simulation results verify the superiority of the proposed FSSMC over SSMC.
In this paper, a multi-input multi-output(MIMO) integral variable structure system with an integral-augmented sliding surface is designed for the improved robust control of uncertain multivariable system under the matched persistent disturbance. To effectively remove the reaching phase problems, the integral augmented sliding surface is proposed. Then for its design, the eigenstructure assignment technique is introduced to. To guarantee the designed performance against the persistent disturbance, the stabilizing control for multi-input system is also designed to generate the sliding mode on the integral sliding surface. The stability of the global system together with the existence condition of the sliding mode are investigated and proved for the case of multi input system in the presence of uncertainty and disturbance. The reaching phase is completely removed in proposed MIMO VSS by satisfying the two requirements. An example and computer simulations will be present for showing the usefulness of algorithm.
In this paper, a novel fuzzy simplex sliding-mode controller is proposed for controlling a multivariable nonlinear system. Here, the fuzzy logic control (FLC) algorithm and simplex sliding-mode control (SSMC) theory are integrated to improve the system states response and to reduce system states chattering phenomenon of the controlled system for simplex control method. Hence, from this motivation yields the so-called fuzzy simplex sliding mode control (FSSMC) scheme. the fuzzy logic control algorithm and simplex sliding mode control algorithm is integrated to improve the system states response and chattering phenomenon. In this paper, at first, we introduce the principle of simplex method, and then develop the fuzzy controls based on the simplex method. Finally, a numerical example is proposed to illustrate the advantages of the proposed controllers, the simulation results demonstrate that the fuzzy simplex type sliding mode control scheme is a good solution to the chattering problem in the simplex sliding mode control.
In this work the authors present a dynamic integral sliding mode controller which is based on the existing dynamic sliding mode control and integral sliding mode control techniques. The proposed control law makes use of an integral manifold instead of the conventional sliding manifold which provides dynamic sliding mode without reaching phase. The robustness is inherited from dynamic sliding mode control and is enhanced by the elimination of reaching phase. In addition, this new designed control law reduces chattering with the incorporation of the dynamic sliding mode control concept. Furthermore, the performance is improved via the linear control law design. A comprehensive comparative analysis carried out with dynamic sliding mode control demonstrates the superiority of the proposed control law. A chatter free regulation control of a kinematic car model with im,proved performance in the presence of uncertainties certifies the robustness of the proposed dynamic integral sliding mode controller.
The uncertain integral linear system is the integral-augmented uncertain system to improve the resultant performance. In this note, for a MI(Multi Input) uncertain integral linear case, Utkin`s theorem is proved clearly and comparatively. With respect to the two transformations(diagonalizations), the equation of the sliding mode is invariant. By using the results of this note, in the SMC for MIMO uncertain integral linear systems, the existence condition of the sliding mode on the predetermined sliding surface is easily proved. The effectiveness of the main results is verified through an illustrative example and simulation study.
In this paper, the authors propose a novel dynamic integral sliding mode controller for state dependent matched and unmatched uncertainties. An output feedback methodology based designed dynamic controller attenuate the effects of the both matched and unmatched uncertainties along with considerable reduction in chattering phenomena. An integral manifold is used which provides sliding mode without reaching phase and further enhances the robustness of the controller against uncertainties. A Lyapunov candidate is used for the stability analysis against uncertainties. The feasibility of the control law is illustrated via a kinematic car model
In this chapter an adaptive sliding mode controller for the Furuta pendulum is proposed. The Furuta pendulum is a class of underactuated mechanical systems commonly used in many control systems laboratories due to its complex stabilization which allows the analysis and design of different nonlinear and multivariable controllers that are useful in some fields such as aerospace and robotics. Sliding mode control has been extensively used in the control of mechanical systems as an alternative to other nonlinear control strategies such as backstepping, passivity based control etc. The design and implementation of an adaptive sliding mode controller for this kind of system is explained in this chapter, along with other sliding mode controller variations such as second order sliding mode (SOSMC) and PD plus sliding mode control (PD TeX SMC) in order to compare their performance under different system conditions. These control techniques are developed using the Lyapunov stability theorem and the variable structure design procedure to obtain asymptotically stable system trajectories. In this chapter the adaptive sliding mode consist of a sliding mode control law with an adaptive gain that makes the controller more flexible and reliable than other sliding mode control (SMC) algorithms and nonlinear control strategies. The adaptive sliding mode control (ASMC) of the Furuta pendulum, and the other SMC strategies shown in this chapter, are derived according to the Furuta’s pendulum dynamic equations making the sliding variables, position errors and velocity errors reach the zero value in a specified reaching time. The main reason of deriving two well known sliding mode control strategy apart from the proposed control strategy of this chapter (adaptive sliding mode control) is for comparison purposes and to evince the advantages and disadvantages of adaptive sliding mode control over other sliding mode control strategies for the stabilization of the Furuta pendulum. A chattering analysis of the three SMC variations is done, to examine the response of the system, and to test the performance of the ASMC in comparison with the other control strategies explained in this chapter.
In this paper, based on the adaptive control and the simplex sliding-mode control schemes, a novel simplex-type adaptive fuzzy sliding-mode control (STA-FSMC) is presented. An irregular simplex manifold is generated by rotating the original regular simplex one and then is used to infer the control action by fuzzy algorithm. Design procedures of the proposed STA-FSMC are explored in detail and the stabilization of multi-input systems by using adaptive fuzzy control based on simplex-type sliding-mode control philosophy is also examined thoroughly. The adaptive control laws modify the irregular simplex control vectors such that the magnitude of simplex control vectors and chattering phenomena are improved. The computer simulation demonstrates the feasibility and effectiveness of the proposed methodology.
In this paper, a novel simplex-type adaptive sliding-mode control (ST-ASMC) is presented. An irregular simplex manifold is introduced in stead of the conventional regular one for tuning the control performance. The adaptive control laws modify the irregular simplex control vectors such that the selection of the magnitude and direction of the control vectors can be accomplished automatically. The computer simulation is performed to demonstrate the feasibility and validity of the proposed methodology.
In this paper the sliding mode control theory is applied to a particular underwater gripper actuated by linear motors which, acting on a hydraulic circuit, generate monodirectional forces. In practical realizations actuators often show imprecise relationships between electrical input signals and mechanical output; such a situation constitutes a source of uncertainties. A sliding mode control methodology based on the use of a simplex of constant control vectors is presented. The proposed approach is general enough to also work with different applications. [S0022-0434(00)01204-1].
This paper deals withthe hybrid position/force control of constrainedmanipulators subjected to uncertainties. A solution is proposedthat is based on sliding-mode control theory, which proved tobe highly effective in counteracting uncertainties for some classesof nonlinear systems. Specific problems involved in this techniqueare chattering elimination and the algebraic coupling betweenconstraint forces and possibly discontinuous control signals.Both the problems are addressed in this paper by exploiting therobustness properties of a second-order sliding-mode controlalgorithm. This algorithm, recently proposed by the authors forsolving the single-input hybrid control problem, is generalizedin this paper to deal with the class of multi-input differentialalgebraic systems describing the behaviour of constrained mechanicalsystems.
Deals with the application of sliding mode control theory to the
specific case of a manipulator for which mono-directional control
actions only have to be considered. In particular the control of an
underwater gripper is presented. Many even complex robotic structures,
can be actuated by mono-directional control actions, for example the so
called tendon-arms, jet-actuated vehicles, underwater vehicles with
mono-directional thrusters, etc. The robotic system which is considered
in the paper belongs to the above class, since it is actuated by voice
coil motors which, acting on a hydraulic circuit, are able to generate
mono-directional forces. In practical realizations actuators often show
imprecise relationships between the electrical input signals and the
mechanical outputs, that is joint forces or torques. Such a situation
constitutes a source of uncertainties we have to deal with. A sliding
mode control methodology based on the use of a simplex of constant
control vectors is presented, which has been revealed to be general
enough to work with different applications too
General nonlinear control systems described by ordinary differential equations with a prescribed sliding manifold are considered. A method of designing a feedback control law such that the state variable fulfills the sliding condition in finite time is based on the construction of a suitable simplex of vectors in the tangent space of the manifold. The convergence of the method is proved under an obtuse angle condition and a way to build the required simplex is indicated. An example of engineering interest is presented.
In this paper, the design of variable structure control systems is presented with incorporation of the dynamics of an actuator. The VSS controller design is formulated, including the dynamics of the actuator expressed as a state equation. An example is given to show the effectiveness of the developed algorithm
This paper deals with the hybrid position/force control of
constrained manipulators subject to uncertainties and disturbances of
various nature. A solution is proposed based on the sliding mode control
theory, which revealed to be highly effective in counteracting
uncertainties and disturbances for some class of uncertain nonlinear
systems. Specific problems connected to this technique are the
chattering elimination and the algebraic coupling between constraint
forces and possibly discontinuous control signals. The solution proposed
in this paper overcomes these problems by applying a new second order
sliding mode control algorithm
We develop a new analysis of the behavior of simplex control methods applied to multiple-input-multiple-output nonlinear control systems under uncertainties. According to such sliding-mode control methods the control vector is constrained to belong to a finite set of (fixed or varying) vectors, with an appropriate switching logic to guarantee the specified sliding condition. Bounded uncertainties acting on the nominal system are allowed. The proposed sliding control methodology relies on the knowledge of the nominal system only. We prove rigorously the convergence of these methods to the sliding manifold in a finite time under explicit quantitative conditions on the system parameters and the available bounds of the uncertainty. Application to a robotic problem is discussed and a nonlinear example is presented.
One of the major obstacles in the use of efficient tools for designing control systems is the high order of equations that describe their behaviour. In many cases they may be reduced to a lower order model by neglecting small time constants or rejecting fast components of the system overall motion. Fast motions may, for instance, be caused by small impedances in equations of electromechanical energy converters [155], time constants of electric motors in systems controlling slow processes [68], nonrigidity of flying vehicles construction [74] and many other reasons. The design of control systems resting upon the use of low-order models may be carried out both by analytical and by various computational techniques. (Application of computational techniques to the design of control systems may be seriously hindered not only by their high dimension, but also by the fact that the computational problems in such systems are generally ill-posed and require ad-hoc methods to be developed).
An introduction to sliding mode variable structure control.- An algebraic approach to sliding mode control.- Robust tracking with a sliding mode.- Sliding surface design in the frequency domain.- Sliding mode control in discrete-time and difference systems.- Generalized sliding modes for manifold control of distributed parameter systems.- Digital variable structure control with pseudo-sliding modes.- Robust observer-controller design for linear systems.- Robust stability analysis and controller design with quadratic Lyapunov functions.- Universal controllers: Nonlinear feedback and adaptation.- Lyapunov stabilization of a class of uncertain affine control systems.- The role of morse-Lyapunov functions in the design of nonlinear global feedback dynamics.- Polytopic coverings and robust stability analysis via Lyapunov quadratic forms.- Model-following VSC using an input-output approach.- Combined adaptive and Variable Structure Control.- Variable structure control of nonlinear systems: Experimental case studies.- Applications of VSC in motion control systems.- VSC synthesis of industrial robots.
A new definition of well-posedness, called generalized approximability, is introduced for variable-structure control systems described by ordinary differential equations. This definition isolates the following basic property: states fulfilling only approximately the sliding condition converge toward some well-defined sliding state, as the perturbations preventing exact sliding disappear. Approximability rules out ambiguous systems, which defy successful implementation of sliding feedback controls. The existence of the equivalent control is not required. It is shown that approximability is in essence of a form of uniqueness of the Filippov sliding state. Explicit criteria for approximability are obtained.
This paper develops a calculus for computing Filippov's differential inclusion which simplifies the analysis of dynamical systems described by differential equations with discontinuous right-hand-side. In particular, when a slightly generalized Lyapunov theory is used, the rigorous stability analysis of variable structure systems is routine. As an example, a variable structure control law for rigid-link robot manipulators is described and its stability is proved.
In direct contrast to adaptive controllers the deterministic control of uncertain time-varying systems control is achieved using fixed nonlinear feedback control functions, which operate effectively over a specified magnitude range of a class of system parameter variations. There is no requirement for online identification of the values of the system parameters. Furthermore no statistical information of the system variations is required to yield the desired robust dynamic behaviour. If the parameter variations satisfy certain matching conditions, complete insensitivity to system variations can be achieved. The two main approaches, variable structure control (VSC) and Lyapunov control, are described
This paper develops a calculus for computing Filippov's differential inclusion which simplifies the analysis of dynamical systems described by differential equations with a discontinuous right-hand side. In particular, when a slightly generalized Lyapunov theory is used, the rigorous stability analysis of variable structure systems is routine. As an example, a variable structure control law for rigid-link robot manipulators is described and its stability is proved.