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A control vector simplex approach to variable structure control
Abstract
An effective way to extend to the multi-input case the variable structure control philosophy is the method based on a set of m+1 control vectors forming a simplex in ℛm, and on the corresponding switching of the controlled system from one to another of m+1 different structures. In this paper, the basic method is briefly recalled and conditions for the possible extension of its validity to the case of uncertain nonlinear systems affine in the control law are sought. The possibility of guaranteeing the convergence of the simplex method also in the case of nonlinear systems non-affine in the control law is investigated. © 1997 by John Wiley & Sons, Ltd.
... 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 this paper we refer to a surface ship prototype (AJO-3, Autonomous Jet-propelled Object with 3 dof) equipped by a monodirectional, hydrojet-based, actuation system. We show that the recently proposed design framework referred to as the simplex-based sliding mode control (SB-SMC) method [10], [9], [11] offers a viable solution to the thruster allocation problem for a surface marine vehicle. ...
... The simplex-vector sliding-mode control methodology [8], [11], [9], [10] is now recalled. Consider a nonlinear uncertain systemẋ ...
... Under mild boundedness and smoothness requirements relevant to the system dynamics (10), it has been proven [11], [9] that a constant ρ * can be computed such that for any ρ ≥ ρ * the control strategy (16) guarantees the finitetime robust convergence of vector s towards the origin. ...
This note concerns the design and implementation of a position/attitude sliding-mode controller for a surface vessel prototype. The prototype is equipped with a special, recently patented (V. Arrichiello, et al., 2005), propulsion system based on hydrojets with variable output section. The sliding mode control design is based on the vector simplex method (G, Bartolini, et al., 1997). First we describe the structure and the working principle of the prototype. Then, we present an approximate dynamic model and describe the detailed derivation of the motion controller. Finally, the major implementation issues are discussed and some experimental results are shown
... One of the main feature of SMC is the exploitation of discontinuous control laws (first order SMC) or instead of continuous control inputs obtained with discontinuous terms in the time derivatives of the control inputs (second order SMC). The simplex-based SMC, [6], [7], [8], [9], is a method of the first order, which is particularly well suited to be adopted when the actuation system consists of mono-directional discontinuous devices, [10], [11]. Guldner and Utkin [12], used a combination of APF and SMC for motion planning in the robotic field. ...
... Let us briefly recall the definition and main features of a simplex of vectors, which can be suitably exploited to design SMC, [7], [8]. ...
The paper considers autonomous rendezvous maneuver and proximity operations of two spacecraft in presence of obstacles. A strategy that combines guidance and control algorithms is analyzed. The proposed closed-loop system is able to guarantee a safe path in a real environment, as well as robustness with respect to external disturbances and dynamic obstacles. The guidance strategy exploits a suitably designed Artificial Potential Field (APF), while the controller relies on Sliding Mode Control (SMC), for both position and attitude tracking of the spacecraft. As for the position control, two different first order SMC methods are considered, namely the component-wise and the simplex-based control techniques. The proposed integrated guidance and control strategy is validated by extensive simulations performed with a six degree-of-freedom (DOF) orbital simulator and appears suitable for real-time control with minimal on-board computational effort. Fuel consumption and control effort are evaluated, including different update frequencies of the closed-loop software.
... One of the main feature of SMC is the exploitation of discontinuous control laws (first order SMC) or instead of continuous control inputs obtained with discontinuous terms in the time derivatives of the control inputs (second order SMC). The simplex-based SMC, [6], [7], [8], [9], is a method of the first order, which is particularly well suited to be adopted when the actuation system consists of mono-directional discontinuous devices, [10], [11]. ...
... Let us briefly recall the definition and main features of a simplex of vectors, which can be suitably exploited to design SMC, [7], [8]. ...
The paper considers autonomous rendezvous maneuver and proximity operations of two spacecraft in presence of obstacles. A strategy that combines guidance and control algorithms is analyzed. The proposed closed-loop system is able to guarantee a safe path in a real environment, as well as robustness with respect to external disturbances and dynamic obstacles. The guidance strategy exploits a suitably designed Artificial Potential Field (APF), while the controller relies on Sliding Mode Control (SMC), for both position and attitude tracking of the spacecraft. As for the position control, two different first order SMC methods are considered, namely the component-wise and the simplex-based control techniques. The proposed integrated guidance and control strategy is validated by extensive simulations performed with a six degree-of-freedom (DOF) orbital simulator and appears suitable for real-time control with minimal on-board computational effort. Fuel consumption and control effort are evaluated, including different update frequencies of the closed-loop software.
... In the literature, it has been shown that the problem of making the 2-sliding manifold associated with (1.95) finite-time attractive can be solved by a Second Order Sliding Mode (SOSM) controller of the type (see for instance [12,[28][29][30][31][32][33]) ...
... where U > 0 is the control gain, α * > 1 is the modulation factor, β is the anticipation factor and σ max is the extremal value of the sliding variable σ . The latter can be found by setting at t 0 σ max = σ (t 0 ) and then ∀t > t 0 by inspection of the past three values or two values of the sliding variable σ (t) [12,28]. Moreover, in order to enforce the sliding mode is necessary to tune the parameters according to the following dominance and convergence conditions, respectively ...
This chapter summarizes the basic concepts used in the design of sliding mode controllers, from the definition of conventional sliding set, and the main concept of sliding motion, to the design of the advanced robust exact high-order sliding modes differentiator. These pages describe the basis on which the methodologies presented along the book are designed, with this aim a short historical background is presented in Section 1.1, the main concepts related to the sliding mode theory are presented in Section 1.2. Section 1.3 describes the design process applied to the manifold, which restricts the movement of the state trajectories during the sliding motion. The basis of the design of standard sliding mode controllers are presented in Section 1.4. The main second-order sliding-mode algorithms are presented in Section 1.5 and the robust-exact high-order sliding-mode differentiator is described in Section 1.6.
... Traditionally, one of the main concern in the direct implementations of SMC is that the control signal has not a fixed switching frequency. This has been solved either by resorting to ''regularised'' versions of the SMC or by using more recent Second Order SMC (SOSMC) (Bartolini, Ferrara, & Usai, 1997;Bartolini, Ferrara, Utkin, & Zolezzi, 1997;Cucuzzella, Rosti, Cavallo, & Ferrara, 2017;Levant, 2003;Sabanovic, Fridman, & Spurgeon, 2004), that produce a continuous control, and then employing a standard PWM (Pulse Width Modulation) device operating at fixed switching frequency. In spite of the above concern, however, some direct implementations of SMC, i.e., with the controller driving directly the power switches, both in continuous and discrete time, have been proposed (Lopez, Garcia De Vicuna, Castilla, Matas, & Lopez, 1999;Sira-Ramírez & Silva-Ortigoza, 2006;Yang, Zhong, Kiratipongvoot, Tan, & Hui, 2018), showing the superiority of the direct SMC over the implementation through PWM. ...
The innovative concept of Electric Aircraft is a challenging topic involving different control objectives. For instance, it becomes possible to reduce the size and the weight of the generator by using the battery as an auxiliary generator in some operation phases. However, control strategies with different objectives can be conflicting and they can produce undesirable effects, even instability. For this reason an integrated control design approach is needed, such that stability can be guaranteed in any configuration. In other words, the design of the supervisory controller must be interlaced with that of low-level controllers. Moreover, uncertainties and noisy signals require robust control techniques and the use of adaptiveness in the control algorithm. In this paper, the use of a new adaptive sliding manifold design is proposed for increase robustness against uncertainties and noisy signals, together with a new supervisor exploiting the estimate of the region of attraction of the control laws. A bidirectional voltage converter aiming at recharging batteries and to use the battery to withstand generator overloads is addressed. Detailed and rigorous stability proofs are given for any control configuration, including the switching phases among different control objectives. Effectiveness of the proposed strategies is shown by using a detailed simulator including switching electronic components.
... Traditionally, one of the main concern in the direct implementations of SMC is that the control signal has not a fixed switching frequency. This has been solved either by resorting to ''regularised'' versions of the SMC or by using more recent Second Order SMC (SOSMC) (Bartolini, Ferrara, & Usai, 1997;Bartolini, Ferrara, Utkin, & Zolezzi, 1997;Cucuzzella, Rosti, Cavallo, & Ferrara, 2017;Levant, 2003;Sabanovic, Fridman, & Spurgeon, 2004), that produce a continuous control, and then employing a standard PWM (Pulse Width Modulation) device operating at fixed switching frequency. In spite of the above concern, however, some direct implementations of SMC, i.e., with the controller driving directly the power switches, both in continuous and discrete time, have been proposed (Lopez, Garcia De Vicuna, Castilla, Matas, & Lopez, 1999;Sira-Ramírez & Silva-Ortigoza, 2006;Yang, Zhong, Kiratipongvoot, Tan, & Hui, 2018), showing the superiority of the direct SMC over the implementation through PWM. ...
The innovative concept of Electric Aircraft is a challenging topic involving different control objectives. For instance, it becomes possible to reduce the size and the weight of the generator by using the battery as an auxiliary generator in some operation phases. However, control strategies with different objectives can be conflicting and they can produce undesirable effects, even instability. For this reason an integrated design approach is needed, where stability can be guaranteed in any configuration. In other words, the design of the supervisory controller must be interlaced with that of low-level controllers. Moreover, uncertainties and noisy signals require robust control techniques and the use of adaptiveness in the control algorithm. In this paper, an aeronautic application aiming at recharging batteries and to use the battery to withstand generator overloads is addressed. Detailed and rigorous stability proofs are given for any control configuration, including the switching phases among different control objectives. Effectiveness of the proposed strategies is shown by using a detailed simulator including switching electronic components.
... In particular, as a first term of comparison, we consider the SOSM with optimal reaching (O-SOSM) [39]. Then, we also use the so-called Switched Time Based Adaptive Suboptimal Second Order Sliding Mode (STBA-SSOSM) control, presented in [49], a particular switched/adaptive SOSM law based on the Suboptimal Second Order Sliding Mode (SSOSM) algorithm [50][51][52][53]. Additionally, we consider the Super Twisting Sliding Mode (STSM) algorithm, which is among the most popular solutions for chattering alleviation among the HOSM control algorithms [54]. ...
This paper presents the design of a new adaptive optimization‐based second‐order sliding mode control algorithm for uncertain nonlinear systems. It is designed on the basis of a second‐order sliding mode control with optimal reaching, with the aim of reducing the control effort while maintaining all the positive aspects in terms of finite‐time convergence and robustness in front of matched uncertainties. These features are beneficial to guarantee good performance in case of vehicle dynamics control, a crucial topic in the light of the increasing demand of semiautonomous and autonomous driving capabilities in commercial vehicles. The new proposal is theoretically analyzed, as well as verified relying on an extensive comparative study, carried out on a realistic simulator of a 4‐wheeled vehicle, in the case of a lateral stability control system.
... 39 Then, we also use the so-called switched-time-based adaptive suboptimal second-order SM (STBA-SSOSM) control, presented in the work of Pisano et al, 49 a particular switched/adaptive SOSM law based on the suboptimal second-order SM (SSOSM) algorithm. [50][51][52][53] Additionally, we consider the super-twisting SM (STSM) algorithm, which is among the most popular solutions for chattering alleviation among the HOSM control algorithms. 54 This paper is organized as follows. ...
The control of the longitudinal, lateral and vertical dynamics of two and four-wheeled vehicles, both of conventional type as well as fully-electric, is important not only for general safety of vehicular traffic in general, but also for future automated driving. Sliding Mode Control of Vehicular Dynamics provides an overview of this important topic. Topics covered include an introduction to sliding mode control; longitudinal vehicle dynamics control via sliding modes generation; sliding mode control of traction and braking in two-wheeled vehicles; lateral vehicle dynamics control via sliding modes generation; stability control of heavy vehicles; sliding mode approach in semi-active suspension control; and observer-based parameter identification for vehicle dynamics assessment. Each chapter introduces the problem formulation and a general overview of its physical aspects, provides a survey of the relevant literature on the topic, and reports on the authors’ contributions to solving the control problem.
A new sliding mode control technique for a class of SISO dynamic systems is presented in this chapter. It is seen that the stability status of the closed-loop system is first checked, based on the approximation of the most recent information of the first-order derivative of the Lyapunov function of the closed-loop system, an intelligent sliding mode controller can then be designed with the following intelligent features: (i) If the closed-loop system is stable, the correction term in the controller will continuously adjust control signal to drive the closed-loop trajectory to reach the sliding mode surface in a finite time and the desired closed-loop dynamics with the zero-error convergence can then be achieved on the sliding mode surface. (ii) If, however, the closed-loop system is unstable, the correction term is capable of modifying the control signal to continuously reduce the value of the derivative of the Lyapunov function from the positive to the negative and then drives the closed-loop trajectory to reach the sliding mode surface and ensures that the desired closed-loop dynamics can be obtained on the sliding mode surface. The main advantages of this new sliding mode control technique over the conventional one are that no chattering occurs in the sliding mode control system because of the recursive learning control structure; the system uncertainties are embedded in the Lipschitz-like condition and thus, no priori information on the upper and/or the lower bounds of the unknown system parameters and uncertain system dynamics is required for the controller design; the zero-error convergence can be achieved after the closed-loop dynamics reaches the sliding mode surface and remains on it. The performance for controlling a third-order linear system is evaluated in the simulation section to show the effectiveness and efficiency of the new sliding mode control technique.
In this paper the multi-input Sliding Mode control of nonlinear uncertain non-affine systems with mono-directional actuators is considered. The geometric properties of the regular simplices of vectors are exploited by the mono-directional actuation system. The proposed full-state feedback control method introduces integrators in the input channel and suitably modifies the sliding output to be steered to zero. The resulting high frequency gain matrix (HFGM), which is the matrix multiplying the control vector, turns out to be affected by time and state dependent uncertainties. An effective use of the integral Sliding Mode method in connection with a non-conventional simplex based switching logic is introduced. The proposed simplex control procedure is adequately designed to avoid any possible unbounded increment of the control actions despite the presence of integrators in the input channel and the exploitation of mono-directional devices.
A new sliding mode-based learning control scheme for a class of SISO dynamic systems is developed in this paper. It is seen that, based on the most recent information on the closed-loop stability, a recursive learning chattering-free sliding mode controller can be designed to drive the closed-loop dynamics to reach the sliding mode surface in a finite time, on which the desired closed-loop dynamics with the zero-error convergence can be achieved.
In this chapter a multi-input sliding mode control technique, called the simplex method, is presented. Multi-input sliding
mode control for nonlinear uncertain systems is far from simply being a generalization of the standard single input variable
structure control (VSC) approach, except for the case in which the uncertainties, though affecting the system model, do not
appear in the control matrix, and this latter is invertible. In the case in which the control matrix is uncertain, some structural
properties need to be guaranteed in order to be able to design a sliding mode control vector [l], [2].
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 note concerns the design and practical implementation of a position/attitude sliding-mode controller for a surface vessel prototype. The prototype is equipped with a special, recently patented (Italian Patent, 2005), propulsion system based on hydro-jets with adjustable output section. The sliding-mode control design is based on the combination between three instances of a second-order sliding-mode velocity observer (Automatica 1998; 34:379–384) and a simplex-based sliding-mode controller (Int. J. Robust Nonlinear Control 1997; 7(4):321–335). We first describe the structure and the working principle of the prototype. Then, we detail the derivation of the motion observer/controller. Finally, we discuss the major implementation issues and show some experimental results. Copyright © 2007 John Wiley & Sons, Ltd.
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.
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.
This paper is concerned with the problem of designing a stabilizing controller for a class of uncertain dynamical systems. The vector of uncertain parameters q(¿) is time-varying, and its values q(t) lie within a prespecified bounding set Q in Rp. Furthermore, no statistical description of q(¿) is assumed, and the controller is shown to render the closed loop system "practically stable" in a so-called guaranteed sense; that is, the desired stability properties are assured no matter what admissible uncertainty q(¿) is realized. Within the perspective of previous research in this area, this paper contains one salient feature: the class of stabilizing controllers which we characterize is shown to include linear controllers when the nominal system happens to be linear and time-invariant. In contrast, in much of the previous literature (see, for example, [1],[2],[7] and [9]), a linear system is stabilized via nonlinear control. Another feature of this paper is the fact that the methods of analysis and design do not rely on transforming the system into a more convenient canonical form; e.g., see [3]. It is also interesting to note that a linear stabilizing controller can sometimes be constructed even when the system dynamics are nonlinear. This is illustrated with an example.
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.
Based on concepts from both the theory of variable structure systems and the theory of Leitmann et al., continuous and discontinuous feedback controls are developed which guarantee uniform ultimate boundedness of all motions of a class of imperfectly known dynamical systems with bounded uncertainty. For a subclass of uncertain systems, the zero state can be (a) rendered 'practically' stable (in the sense that, given any neighbourhood of the zero state, there exists a control, in the proposed class of continuous feedback controls, which guarantees global uniform ultimate boundedness with respect to that neighbourhood), or (b) rendered globally uniformly asymptotically stable (in the sense of Lyapunov) by the proposed discontinuous feedback control. The approach is illustrated by application to a Maglev suspension control system.
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.