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Multi-Input Second-Order Sliding-Mode Hybrid Control of Constrained Manipulators
Abstract
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.
... In spite of the above cited drawbacks, VSS with Sliding Modes (SM) have been considered often in the technical literature for the robust control of mechanical systems (Slotine and Sastry 1983, Cho and Hedrick 1991, Man Zhihong et al. 1994, Rundell et al. 1995b, 2000b, d, 2002b, Allen et al. 2000, Levant et al. 2000, Shu et al. 2000, Satoshi et al. 2001, Xu and Cao 2001. Most of them used a straightforward approach to avoid chattering: the sign function of the discontinuous control is approximated by the saturation function. ...
... As for the first topic, some differentiation algorithms based on second, and higher, order sliding modes have been recently presented (Levant 1998b, Bartolini et al. 2000, showing an interesting trade-off between precision and noise-immunity. ...
... This control approach suffers from the so-called chattering phenomenon, that is, the high frequency oscillation caused by the discontinuity of the control signals. Various techniques were proposed to eliminate the chattering effect (Corless and Leitmann 1981, Bartolini and Pydynowski 1996, the 2-SMC was shown to be effective to reduce chattering while maintaining robustness, accuracy and simplicity (Bartolini et al. b, 2000. ...
The effective application of sliding mode control to mechanical systems is not straightforward because of the sensitivity of these systems to chattering. Higher-order sliding modes can counteract this phenomenon by confining the switching control to the higher derivatives of the mechanical control variable, so that the latter results are continuous. Generally, this approach requires the availability of a number of time derivatives of the sliding variable, and, in the presence of noise, this requirement could be a practical limitation. A class of second-order sliding mode controllers, guaranteeing finite-time convergence for systems with relative degree two between the sliding variable and the switching control, could be helpful both in reducing the number of differentiator stages in the controller and in dealing with unmodelled actuator dynamics. In this paper different second-order sliding mode controllers, previously presented in the literature, are shown to belong to the above cited class, and some challenging control problems involving mechanical systems are addressed and solved. Simulations and experimental results are provided throughout the paper.
... However, this condition is quite restrictive (for example it does not apply to general Lagrangian systems). For the class of Lagrangian systems solutions are given in [14] and [11]; they require the exploitation of observers, rely on the hierarchical sliding mode concept and ensure asymptotic convergence. The solution in [12] is based on regular form, while the one in [13] is based on quite restrictive conditions. ...
... The discontinuous control law is given by (6), except in a possible initialization phase [3], [21]. The control problem is solved by (6) even when the bounds on the system are not constants and depend on the modulus of s [14], [22]. ...
... Remark 2: The MIMO second order sliding mode control strategy expressed by (24)-(26) is robust with respect to the uncertainties, the norm of which can be upper-bounded. Even in the MIMO case, the knowledge of the bounds (13) and (14) on the uncertain terms results sufficient to compute the control parameters according to (27). ...
The paper proposes an approach to second order sliding mode control for multi-input multi-output (MIMO) non-linear uncertain systems. With respect to standard sliding mode control, the second order sliding mode techniques for single-input single-output (SISO) systems show the same properties of robustness and precision, feature a higher order accuracy and can be exploited to eliminate the chattering effect. The extension of these results to the MIMO nonlinear systems is a challenging matter. In the present paper, the validity is extended to a quite large class of nonlinear processes affected by uncertainties of general nature; the control design is simple; the conditions of existence on the controllers are weak. The proposed procedure represents a general approach to the second order sliding mode control of MIMO systems.
... example it does not apply to general Lagrangian systems). For the class of Lagrangian systems solutions are given in [14] and [11]; they require the exploitation of observers, rely on the hierarchical sliding mode concept and ensure asymptotic convergence. The solution in [12] is based on regular form, while the one in [13] is based on quite restrictive conditions. ...
... The discontinuous control law is given by (6), except in a possible initialization phase [3], [22]. The control problem is solved by (6) even when the bounds on the system are not constants and depend on the modulus of s [14], [23]. ...
... 45th IEEE CDC, San Diego, USA, Dec. [13][14][15]2006 FrA10.1 ...
The paper proposes an approach to second order sliding mode control for multi-input multi-output (MIMO) nonlinear uncertain systems. With respect to standard sliding mode control, the second order sliding mode techniques for single-input single-output (SISO) systems show the same properties of robustness and precision, feature a higher order accuracy and can be exploited to eliminate the chattering effect. The extension of these results to the MIMO nonlinear systems is a challenging matter. In the present paper, the validity is extended to a quite large class of nonlinear processes affected by uncertainties of general nature; the control design is simple; the conditions of existence on the controllers are weak. The proposed procedure, which represents a general approach to the second order sliding mode control of MIMO systems, is applied to the control of mechanical systems
... As a result, the robustness of the system is ensured without manually assigning the controller parameters, which greatly improves the conventional sliding mode controllers. For the safety of systems, sliding mode controllers with hard state-dependent constraints are also investigated ( Bartolini et al., 2000;Incremona et al., 2017). However, robust controllers with both adaptive parameters and hardconstraint compliance are still lacking. ...
... Although the determination of the parameters α i and γ i are theoretically provided by existing framework (Bartolini et al., 2000;Jeong et al., 2018), selecting a proper integral coefficient γ i , is a challenging work in practical applications. In this paper, we propose a adaptive tuning law for γ i which is given by the following theorem. ...
In this paper, a novel adaptive second-order sliding mode controller is designed for Euler-Lagrangian systems with hard safety constraints. Different from the conventional sliding mode controllers, the proposed method provides adaptive controller parameters, such that the robustness of the controller is ensured without bringing up chattering. The controller also guarantees strict compliance to hard state-dependent inequality constraints. The asymptotic convergence of the tracking errors of the proposed controller is proven by a direct Lyapunov method. Finally, the proposed controller is validated by numerical simulation on a three-degree-of-freedom robot platform. The results confirm that the controller ensures strict constraint compliance and precise trajectory tracking, which reveals its potential applicability to the safe control of mechatronic systems.
... The conventional SMC is restricted to systems with output relative-degree of 1. Fortunately, it has been extended to the high-order sliding mode (HOSM), which removes the relative-degree restriction and, in practice, can almost eliminate the chattering [13]. A special class of HOSM controllers that can also avoid chattering and has already been successfully implemented in real problems [14][15][16] is the so-called second-order sliding mode control . ...
... where C σ ∈ R 6×6 is a diagonal parameter matrix. The set σ = 0 defines the sliding manifold corresponding to the sliding variable (14). It turns out that the dynamics of system (12)-(13) is collapsed to the reduced-order dynamicsẋ 1 = −C σ x 1 . ...
This work is concerned with the robust attitude and position tracking control of a multirotor airship subject to a smooth model uncertainty representing unknown aerodynamics coefficients and added mass. The vehicle is assumed to be full-actuated. To tackle this problem, we present a multi-input formulation of a smooth second-order sliding mode control strategy with stability guaranteed on the basis of vector field homogeneity. The method is evaluated using numerical simulation and shows effective.
... While finite-time-convergent arbitrary-order sliding-mode controllers are still theoretically studied (Levant, 2003Levant, , 2005 Floquet et al., 2003}, 2-sliding controllers are already successfully implemented for the solution of practical problems (Bartolini, Ferrara et al., 2000, 1998 Bartolini, Pisano et al., 2000 Levant et al., 2000; Massey et al., 2005, Sira-Ramírez, 2002 Orlov et al., 2003, Krupp et al., 2002 Spurgeon et al., 2002, Shtessel et al., 2003). Almost all known r-sliding controllers possess specific homogeneity called the r-sliding homogeneity (Levant, 2005). ...
... Trajectories of (2) are assumed infinitely extendible in time for any Lebesguemeasurable bounded control u(t, x). Finite-time stabilization of smooth systems at an equilibrium point by means of continuous control is considered in (Bacciotti et al., 2005; Bhat et al., 2000)). In our case any continuous control u = j(s, s & , ..., s (r-1) ) (5) providing for s º 0, would satisfy the equality j(0,0, ..., 0) = -h ( t , x )/g(t,x), whenever (1) holds. ...
Homogeneity features of dynamic systems are known to provide for a number of general practically important features. In particular, the finite-time convergence is easily proved, and the asymptotic accuracy is readily calculated in the presence of input noises, delays and discrete sampling. General uncertain single-input-single-output regulation problems are only solvable by means of discontinuous control via the so-called high-order sliding modes (HOSM). The homogeneity approach facilitates the design and investigation of new HOSM controllers, featuring such attractive properties as practical continuity of the control in the presence of noises. Robust output-feedback controllers are produced, using robust exact homogeneous differentiators. The asymptotic accuracy of the obtained controllers is the best possible under given circumstances. The dangerous chattering effect is removed by means of a standard procedure. The resulting systems are robust with respect to the presence of unaccounted-for fast stable dynamics of actuators and sensors. Simulation results and applications are presented in the fields of control, signal and image processing.
... Among them the step-by-step sliding mode observer with constant and adaptive gains. 4.4.1 Sliding mode observer (SMO) for the rotor and speed estimation [7,8] The control and observation strategy based on sliding mode control is known for its strong robustness against parametric uncertainties and external disturbances [190,191,192,193,194,195,196]. The SMO methodology is first proposed by Slotine and Walcott in the middle of 1980s [191]. ...
This thesis is part of the Renault / Centrale Nantes Chair on improving the performance of electric vehicles (EV / HEV). It is dedicated to the problem of estimating the position / speed of self-sensing permanent magnet synchronous motors (PMSM) without mechanical sensors, using high frequency (HF) signal injection techniques over the full speed range of PMSM. In this context, several contributions have been proposed in the demodulation / signal processing and tracking algorithms parts of HF injection techniques, in order to improve the estimation of the position / speed of the MSAP compared to the existing methods. In the demodulation / signal processing part of the HF injection techniques, the contributions consisted of proposing original solutions making it possible to reduce the filtering effects in the estimation chain and to make the latter independent of the electrical machine parameters. In the tracking part, the contributions mainly concern the use of the function sign of the position error (instead of the position error) as measurement information, to estimate the position, the speed and the acceleration of self-sensing PMSM with firstorder sliding mode observers (conventional, step-by-step and adaptive). The contributions proposed in both parts have the advantages of robustifying the estimation chain by making it independent of electrical and mechanical parameters on the one hand. On the other hand, they allow improving the accuracy and performance of the estimation chain, and therefore the control of self-sensing PMSM, in transient and steady-state phases with an easy tuning method. The estimation methods developed were tested in simulation and experimentation on a test bench of electrical machines. The results obtained made it possible to highlight the performances of these methods in terms of trajectory tracking and robustness over the entire operating range of PMSM self-sensing control.
... The results of Prof. Bartolini's group are based on another second order sliding mode controller that does not (explicitly) need derivatives, the Sub-Optimal Algorithm. It is important to remark that this second order sliding mode algorithm has been used to solve numerous relevant control problems, see for instance [11] or [12] and the references therein. ...
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.
... In this context, the twisting algorithm and super twisting algorithm are cited based on second-order sliding approaches. These algorithms improve the quality of the control by reducing the chattering phenomenon [24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42]. However, these approaches are too complicated to implement. ...
Direct torque control is a powerful approach widely used for the control of electrical machines. However, this approach is sensitive to several problems, such as the stator flux demagnetization phenomenon and the appearance of undesirable torque ripples, especially at low-speed operation. To overcome these problems, a variable structure control approach for a direct torque control–space vector modulation system, with a constant switching frequency, is proposed in this article. The proposed sliding-mode direct torque control–space vector modulation method can (i) enhance the system robustness, (ii) reduce the torque ripples, and (iii) prevent the demagnetization problem, which penalizes the stator flux regulation. To reduce the chattering phenomenon that appeared on the control law, a second-order sliding-mode approach, which is a new second-order procedure, has been proposed. Simulation results dealing with low-speed operation of the induction motor under variable structure control direct torque control–space vector modulation strategy is presented and compared to those yielded by the direct torque control–space vector modulation strategy using hysteresis controllers.
... The control laws known from the literature (see e.g., Bartolini et al.[46]; Bartolini et al.[47];Bartolini et al. ...
This work deals with the problem of the accurate task space trajectory tracking subject to finite-time convergence. Kinematic and dynamic equations of a redundant manipulator are assumed to be uncertain. Moreover, globally unbounded disturbances are allowed to act on the manipulator when tracking the trajectory by the end-effector. Furthermore, the movement is to be accomplished in such a way as to reduce both the manipulator torques and their oscillations thus eliminating the potential robot vibrations. Based on suitably defined task space non-singular terminal sliding vector variable and the Lyapunov stability theory, we propose a class of chattering-free robust controllers, based on the estimation of transpose Jacobian, which seem to be effective in counteracting both uncertain kinematics and dynamics, unbounded disturbances and (possible) kinematic and/or algorithmic singularities met on the robot trajectory. The numerical simulations carried out for a redundant manipulator of a SCARA type consisting of the three revolute kinematic pairs and operating in a two-dimensional task space, illustrate performance of the proposed controllers as well as comparisons with other well known control schemes.
... The robust control scheme designed in the next section is applicable to holonomic mechanical systems comprising both non-redundant and redundant robotic manipulators considered here which are described, in general, by the following dynamic equations, expressed in generalized (joint) coordinates q = (q 1 , . . . , q n ) T [8]: (1) whereq andq represent the velocity and acceleration, respectively. The n × n inertia matrix M(q) is positive definite and symmetric. ...
This work deals with the problem of the accurate task space control subject to finite-time convergence. Kinematic and dynamic equations of a rigid robotic manipulator are assumed to be uncertain. Moreover, unbounded disturbances, i.e., such structures of the modelling functions that are generally not bounded by construction, are allowed to act on the manipulator when tracking the trajectory by the end-effector. Based on suitably defined task space non-singular terminal sliding vector variable and the Lyapunov stability theory, we derive a class of absolutely continuous (chattering-free) robust controllers based on the estimation of a Jacobian transpose matrix, which seem to be effective in counteracting uncertain both kinematics and dynamics, unbounded disturbances and (possible) kinematic and/or algorithmic singularities met on the robot trajectory. The numerical simulations carried out for a 2DOF robotic manipulator with two revolute kinematic pairs and operating in a two-dimensional task space, illustrate performance of the proposed controllers.
... Applications of sliding mode observer for fault diagnosis are found in literature (Iqbal et al., 2011;Bhatti et al., 1999;Goh et al., 2002). The technique is however applied mostly to the continuous time linear or nonlinear models and very few references of its application are available for hybrid model applications (Bartolini et al., 2000;Juloski et al., 2007). In present work we have applied sliding mode on engine hybrid model in which healthy and faulty states of system are considered as two different modes and state variables of system are estimated for identifying system modes. ...
Identification of operating mode is a key step in the fault diagnosis of the hybrid systems. The novelty of this paper is the definition of a hybrid observer where discrete event is identified and then continuous model of a sub-system is selected for the design of observer using sliding mode technique. The observer output is finally used for mode identification and fault diagnosis. The proposed method is applied on Spark Ignition (SI) engine for misfire fault detection. The first order sliding mode observer is used to accomplish the task. The mode identification scheme is validated on data generated through hybrid model simulations as well as for data obtained from a 1.3L production vehicle.
... Cette technique est une généralisation des modes glissants d'ordre un [79,80] et permet que la commande n'agisse plus sur la variable de glissement, mais sur ses dérivées. De nombreuses applications montrent la faisabilité de la commande dans un contexte monovariable [74,112,58,61,7,8,132,12,50] ou multi-variable [112,115,11,71]. ...
Thèse soutenue le 4 décembre 2007 à l'Ecole Centrale de Nantes
... In the sliding mode literature various solutions have been proposed to the observer/differentiator problems Levant [1998], Levant [2003], Bartolini et al. [2000a], Bartolini et al. [2000b]; among these proposals, which guarantee the desired estimates, we choose, for example, the one based on the so called second order sliding mode suboptimal algorithm Bartolini et al. [1998]. ...
In this paper a method to simultaneously achieving the two main objective of smoothing the control while maintaining ideally infinite frequency regimes is particularized to the simplex sliding mode method. The present paper proposes a simplex sliding mode control logic based on the time derivative of the control vector. This practice, beside the standard chattering reduction effect, has the important consequence of making possible the control of a rather wide class of uncertain systems nonlinear in the control. In the uncertain case the increment of the relative degree implicit in the proposed approach requires the introduction of second order sliding mode observer. A separation theorem is proven and sufficient conditions for the finite time convergence of both the estimation and tracking errors are found in terms of further requirements on the control amplitude.
... An interesting sliding mode controller for MIMO systems in is based on the finite time stability of a chain of integrators from Bhat & Bernstein [2005]. We mention that sliding mode controllers had led to a huge number of exciting applications involving real systems as in Bartolini et al. [2000Bartolini et al. [ , 2003, Ghanes et al. [2010], Khan et al. [2003], Levant et al. [2000], ], Orlov et al. [2003, Sira-Ramírez [2002], Spurgeon et al. [2003], Shkolnikov et al. [2000], Shkolnikov & Shtessel [2002], Shtessel & Shkolnikov [2003]. ...
This communication is devoted to a practical comparison between high-order sliding modes and the recently introduced model-free control. The perfect knowledge of the relative degree of the output variable, which is a standard assumption for sliding modes, is assumed here. Our comparisons are based on two concrete case-studies and on numerous computer simulations. The smoothness of the input variables,the robustness with respect to noises and the straightforward extendibility of the model-free controllers to MIMO systems are highlighted.
... Previous research and experimental activities have shown that different approaches may be adopted in order to solve these problems, such as, for instance, decentralized control, see Asada and Slotine [1986], Koivo [1989], Chiacchio et al. [1993], Sciavicco and Siciliano [2000]; feedback linearization, see Kreutz [1989], Kuo and Wang [1989], Abdallah et al. [1991], Spong et al. [1993], , Calanca et al. [2007]; model predictive control, see Richalet et al. [1997], Juang and Eure [1998], Poignet and Gautier [2000]; sliding mode control, see Guldner et al. [1995], Shyu et al. [1996], Chen and Chang [1999], Utkin et al. [1999], Jafarov et al. [2000], Bartolini et al. [2000], , , Davila et al. [2005], and Capisani et al. [2009]; discrete-time sliding mode control, see Capisani et al. [2010]; adaptive control, see, for instance, Balestrino et al. [1983], Craig [1988], Ortega and Spong [1989], Liu [1999], Colbaugh et al. [2000], Perk et al. [2001], and Cheah et al. [2006]. ...
In this work, the joint position tracking control problem of industrial robots is tackled. To cope with the model uncertainties and external disturbances affecting the robot, the Inverse Dynamic Controller (IDC) is combined with an approach based on higher order Sliding Mode Control (SMC) technique. We make use, in particular, of the so-called Twisting Second Order Sliding Mode Controller. Higher order SMC techniques transfer the inherent discontinuities to the time derivative of the input torque and this allows to obtain a continuous profile for the input torque, which is computed through integration of an appropriate discontinuous switching signal. Despite the chattering phenomenon is strongly attenuated, some residual problems (vibration and acustical noise) are still observed during the experimental implementation of such an approach in its standard formulation. To improve the system performance we suggest in this work an adaptation mechanism to adjust on-line the authority of the SMC. The logic is driven by a $lqlq$sliding-mode indicator" that detects, on line, the occurrence of a sliding mode behaviour and uses this information for adaptation purposes. When large and fast control activity is demanded (e.g. to track fast reference trajectories) the adaptation unit reacts by automatically increasing the control authority of the SMC. On the other hand when small control authority is sufficient the control magnitude is lowered. Such a bidirectional adaptation logic significantly reduces the chattering. The proposed technique is theoretically analyzed and experimentally tested, and the results of comparative experiments are discussed in the paper.
... Previous research and experimental activities have shown that different approaches may be adopted in order to solve these problems, such as, for instance, decentralized control, see Asada and Slotine [1986], Koivo [1989], Chiacchio et al. [1993], Sciavicco and Siciliano [2000]; feedback linearization, see Kreutz [1989], Kuo and Wang [1989], Abdallah et al. [1991], Spong et al. [1993], , Calanca et al. [2007]; model predictive control, see Richalet et al. [1997], Juang and Eure [1998], Poignet and Gautier [2000]; sliding mode control, see Guldner et al. [1995], Shyu et al. [1996], Chen and Chang [1999], Utkin et al. [1999], Jafarov et al. [2000], Bartolini et al. [2000], , , Davila et al. [2005], and Capisani et al. [2009]; discrete-time sliding mode control, see Capisani et al. [2010]; adaptive control, see, for instance, Balestrino et al. [1983], Craig [1988], Ortega and Spong [1989], Liu [1999], Colbaugh et al. [2000], Perk et al. [2001], and Cheah et al. [2006]. ...
In this work, the joint position tracking control problem of industrial robots is tackled. To cope with the model uncertainties and external disturbances affecting the robot, the Inverse Dynamic Controller (IDC) is combined with an approach based on higher order Sliding Mode Control (SMC) technique. We make use, in particular, of the so-called "Twisting" Second Order Sliding Mode Controller. Higher order SMC techniques transfer the inherent discontinuities to the time derivative of the input torque and this allows to obtain a continuous profile for the input torque, which is computed through integration of an appropriate discontinuous switching signal. Despite the chattering phenomenon is strongly attenuated, some residual problems (vibration and acustical noise) are still observed during the experimental implementation of such an approach in its standard formulation. To improve the system performance we suggest in this work an adaptation mechanism to adjust on-line the authority of the SMC. The logic is driven by a "sliding-mode indicator" that detects, on line, the occurrence of a sliding mode behaviour and uses this information for adaptation purposes. When large and fast control activity is demanded (e.g. to track fast reference trajectories) the adaptation unit reacts by automatically increasing the control authority of the SMC. On the other hand when small control authority is sufficient the control magnitude is lowered. Such a bidirectional adaptation logic significantly reduces the chattering. The proposed technique is theoretically analyzed and experimentally tested, and the results of comparative experiments are discussed in the paper.
... When the system is uncertain, σ is not available and it is necessary to introduce a different simplex sliding mode control strategy, which is based on an estimated sliding outputσ . In order to provide an estimate ofṡ in finite time, let us introduce a second order sliding mode observer, Bartolini, Ferrara, and Punta (2000); Levant (1998), based on the so-called second order sliding mode suboptimal algorithm Bartolini et al. (2001), Bartolini, Ferrara, Levant, and Usai (1999), ...
In the extension to multi-input nonlinear uncertain systems of the sliding mode methodology, a crucial role is played by the matrix pre-multiplying the control in the dynamic equation of the sliding output. If this matrix is perfectly known and invertible, it is possible to transform a multi-input sliding mode control problem in an almost decoupled set of single-input problems. If this matrix is uncertain then nothing can be done in general, and the investigation is oriented to find conditions ensuring the feasibility of control strategies in a progressively more general set of uncertain matrices. In the case of uncertain and constant matrices, it is possible, in principle, to manage the case in which the matrix in question is invertible. The corresponding adaptive or switching strategy suffers from the curse of dimensionality of the so-called unmixing set. In this article the case of time- and state-varying uncertain matrix is dealt with. A more general class of such a matrices for which there is, at least locally, a solution of the problem is found. The introduction of artificial integrators in the output channel (the integral sliding mode control methodology) allows the practical implementation of the control law without requiring the a priori knowledge of parameters featured by the solution of a relevant nonlinear Lyapunov equation.
... In [21,22], the Lyapunov method is employed to design second-order sliding mode controllers and observers. In addition, the application of two-sliding mode to some specific systems is also discussed in the literature, such as manipulators [23], missile guidance law design [24], diesel engine control [25], stabilisation of hovercraft vessels [26], wheel slip control [27], mobile robot tracking [28], etc. ...
The second-order sliding mode control generates important properties for closed-loop systems, such as robustness, disturbance rejection and finite-time convergence. In this study, it is shown that the adding a power technique plus the nested saturation method will bring in a new second-order sliding mode control scheme for non-linear systems with relative degree two. Based on this, a second-order sliding mode controller is constructed by imposing a natural assumption on the sliding mode dynamics, that is, the uncertainty of the sliding mode dynamics can be bounded by a known function instead of a constant. Under the proposed sliding mode controller, it is proved that the closed-loop system is not only globally convergent, but also locally finite-time stable, which implies the global finite-time stability. Finally, the effectiveness of the proposed method is verified by a numerical example.
... Motion control of rigid robot manipulators is a difficult problem, mainly because of the non-linearities and the coupling effects typical of robotic systems. In past years, different approaches have been proposed in order to solve this problem, such as, for instance, decentralised control (Asada and Slotine 1986;Koivo 1989;Chiacchio, Pierrot, Sciavicco, and Siciliano 1993;Sciavicco and Siciliano 2000), feedback linearisation (Kreutz 1989;Kuo and Wang 1989;Abdallah, Dawson, Dorato, and Jamshidi 1991;Spong, Lewis, and Abdallah 1993;Calanca, Capisani, Ferrara, and Magnani 2007;Ferrara and Magnani 2007), model predictive control (Richalet et al. 1997;Juang and Eure 1998;Poignet and Gautier 2000), as well as sliding mode control (Guldner and Utkin 1995;Shyu, Chu, and Shang 1996;Chen and Chang 1999;Utkin, Guldner, and Shi 1999;Jafarov, Parlakci, and Istefanopulos 2000;Bartolini, Ferrara, and Punta 2000a;Bartolini, Pisano, Punta, and Usai 2003;Davila, Fridman, and Levant 2005). Another manipulator control technique is adaptive control, (Ballestrino and De Maria and Sciavicco 1983;Craig 1988;Ortega and Spong 1989;Liu 1999;Colbaugh, Bassi, Benzi, and Trabatti 2000;Perk, Han, Ahn, and Kim 2001;Cheah, Liu, and Slotine 2006). ...
This article presents an original motion control strategy for robot manipulators based on the coupling of the inverse dynamics method with the so-called second-order sliding mode control approach. Using this method, in principle, all the coupling non-linearities in the dynamical model of the manipulator are compensated, transforming the multi-input non-linear system into a linear and decoupled one. Actually, since the inverse dynamics relies on an identified model, some residual uncertain terms remain and perturb the linear and decoupled system. This motivates the use of a robust control design approach to complete the control scheme. In this article the sliding mode control methodology is adopted. Sliding mode control has many appreciable features, such as design simplicity and robustness versus a wide class of uncertainties and disturbances. Yet conventional sliding mode control seems inappropriate to be applied in robotics since it can generate the so-called chattering effect, which can be destructive for the controlled robot. In this article, this problem is suitably circumvented by designing a second-order sliding mode controller capable of generating a continuous control law making the proposed sliding mode controller actually applicable to industrial robots. To build the inverse dynamics part of the proposed controller, a suitable dynamical model of the system has been formulated, and its parameters have been accurately identified relying on a practical MIMO identification procedure recently devised. The proposed inverse dynamics-based second-order sliding mode controller has been experimentally tested on a COMAU SMART3-S2 industrial manipulator, demonstrating the tracking properties and the good performances of the controlled system.
... The SOA algorithm is also able to guarantee the finite time exact convergence x(t) =ẋ(t) = 0, ∀t ≥ T once the parameter U m has been tuned large enough. This SOA algorithm has been used to solve numerous relevant control problems, see for instance [11] or [16] and the references therein. To smooth out the control law for relative degree one systems, the SOA can be used in the same manner as the TA (1.2.1). ...
The objective of this chapter is to try to analyze the main stages in the development of sliding mode enforcing control algorithms,
starting from the first Variable Structure Systems workshop (VSS90). I would like to underline that this is my personal opinion,
I am just trying to understand the steps we have made as a community during the last twenty years after VSS90 as well as which
problems still remain open. Of course, generally I will concentrate the chapter on results in open problems I have discovered
working with my group and coauthors.
... Asymptotically stable higher-order sliding modes (HOSM) appear in many systems with traditional sliding-mode control (Fridman, 2001(Fridman, , 2003 and are deliberately introduced in systems with dynamical sliding modes (Sira-Ramírez, 1993). While finite-time-convergent arbitrary-order sliding-mode controllers are still studied theoretically (Levant, 2001(Levant, , 2003aFloquet, Barbot, & Perruquetti, 2003), 2-sliding controllers are already successfully implemented for the solution of real problems (Bartolini, Ferrara, & Punta, 2000;Bartolini, Pisano, Punta, & Usai, 2003;Ferrara & Giacomini, 2000;Levant, Pridor, Gitizadeh, Yaesh, & Ben-Asher, 2000;Sira-Ramírez, 2002;Orlov, Aguilar, & Cadiou, 2003;Spurgeon, Goh, & Jones, 2002;Shkolnikov & Shtessel, 2002;Shkolnikov, Sht essel,Lianos, & Thies, 2000;Shtessel & Shkolnikov, 2003). ...
It is shown that a general uncertain single-input–single-output regulation problem is solvable only by means of discontinuous control laws, giving rise to the so-called high-order sliding modes. The homogeneity properties of the corresponding controllers yield a number of practically important features. In particular, finite-time convergence is proved, and asymptotic accuracy is calculated in a very general way in the presence of input noises, discrete measurements and switching delays. A robust homogeneous differentiator is included in the control structure thus yielding robust output-feedback controllers with finite-time convergence. It is demonstrated that homogeneity features significantly simplify the design and investigation of a new family of high-order sliding-mode controllers. Finally, simulation results are presented.
... While finite-time-convergent arbitrary-order slidingmode controllers are mostly still theoretically studied (Levant, 2001(Levant, , 2003Floquet, Barbot and Perruquetti, 2003), 2-sliding controllers with finite-time convergence have already been successfully implemented for solution of real problems (Bartolini, Ferrara & Punta, 2000;Bartolini, Pisano, Punta & Usai, 2003;Levant, Pridor, Gitizadeh, Yaesh & Ben-Asher, 2000;Sira-Ramírez, 2002;Orlov, Aguilar & Cadiou, 2003;Khan, Goh & Spurgeon, 2003;Shkolnikov & Shtessel, 2002;Shkolnikov, Shtessel, Lianos, & Thies, 2000;Shtessel & Shkolnikov, 2003;Shtessel, Shkolnikov & Brown, 2004). There are only few widely used 2-sliding controllers: the sub-optimal controller (Bartolini et al., 1998), the terminal sliding mode controllers (Man, Paplinski & Wu, 1994), and the twisting controller ). ...
Second-order sliding modes are used to keep exactly a constraint of the second relative degree or just to avoid chattering, i.e. in the cases when the standard (first order) sliding mode implementation might be involved or impossible. Design of a number of new 2-sliding controllers is demonstrated by means of the proposed homogeneity-based approach. A recently developed robust exact differentiator being applied, robust output-feedback controllers with finite-time convergence are produced, capable to control any general uncertain single-input–single-output process with relative degree 2. An effective simple procedure is developed to attenuate the 1-sliding mode chattering. Simulation of new controllers is presented.
... For example, Bartolini and others propose a second order sliding mode control applied the sub-optimal algorithm ( Bartolini et al., 1997Bartolini et al., , 1999). After the concept of high order sliding mode control was applied to bound operator in ( Bartolini et al., 2000). Levant used high order sliding mode control in aircraft pitch control (Levant, 2000) as well as the exact robust differentiator (Levant, 1998). ...
Aiming at the chattering problem to sliding mode control systems for permanent magnet synchronous motor (PMSM), a control strategy named as high order sliding control (HOSM) with chattering avoidance is presented. A feedback linearization technology was adopted for the nonlinear model of PMSM. After the decouple analysis, the high order sliding mode controller was designed for PMSM. The HOSM strategy is applied to PMSM experiments based on dSPACE platform. Compared with traditional sliding mode control, HOSM control can remove system chattering and enhance the dynamic performance of the control systems. The experiment results verify the feasibility and effectiveness of the proposed strategy.
The chattering effect, robustness and stability are the major issues in a typical Sliding Mode Control (SMC). The selection of sliding manifold parameters are complicated, time-consuming and tedious. The parameter convergence is also a challenging work. Hence, the paper accords a global optimization of Second-Order Sliding Mode Controller (SOSMC) parameters with new sliding surface using the Jaya algorithm for non-linear uncertain process tank systems. The controller (SOSMC) role is to direct the sliding surface and it’s first order time derivative to zero in a finite time.
The Jaya algorithm uses’optimal features’ by updating the solutions in populations. The effectiveness of proposed strategy is evaluated for five objective functions and compared with Artificial Bee Colony (ABC) based SOSMC, classical SOSMC, typical Sliding Mode Control (SMC), Non-dominated Sorting Genetic Algorithm based First-order SMC (NSGA-II FOSMC) and Proportional+Integral+Derivative (PID) controller which is validated through real-time experimentation. The asymptotic stability is guaranteed through direct Lyapunov candidate function.
The statistical significance of algorithms has been verified by using Friedman, Dunn, critical time efficiency and p-value tests. The Jaya algorithm shows a better performance over reported methods concerning process speed, settling time, rise time, reaching time, overshoot, response time and parameter convergence as explored from simulation, statistical, and experimental results. Furthermore, the elevated dead-time and oscillatory processes have been presented to show the efficacy of proposed strategy.
This work deals with the problem of the robust optimal task space trajectory tracking subject to finite-time convergence. Kinematic and dynamic equations of a redundant manipulator are assumed to be uncertain. Moreover, globally unbounded disturbances are allowed to act on the manipulator when tracking the trajectory by the endeffector. Furthermore, the movement is to be accomplished in such a way as to minimize both the manipulator torques and their oscillations thus eliminating the potential robot vibrations. Based on suitably defined task space non-singular terminal sliding vector variable and the Lyapunov stability theory, we derive a class of chattering-free robust kinematically optimal controllers, based on the estimation of transpose Jacobian, which seem to be effective in counteracting both uncertain kinematics and dynamics, unbounded disturbances and (possible) kinematic and/or algorithmic singularities met on the robot trajectory. The numerical simulations carried out for a redundant manipulator of a SCARA type consisting of the three revolute kinematic pairs and operating in a two-dimensional task space, illustrate performance of the proposed controllers as well as comparisons with other well known control schemes.
This work addresses the problem of the accurate task-space control subject to finite-time convergence. Dynamic equations of a redundant manipulator are assumed to be uncertain. Moreover, globally unbounded disturbances are allowed to act on the manipulator when tracking the trajectory by the end effector. Furthermore, the movement is to be accomplished in such a way as to optimize some performance index. Based on suitably defined task-space non-singular terminal sliding vector variable and the Lyapunov stability theory, we derive a class of inverse-free robust controllers consisting of a Jacobian transpose component plus a compensating term, which seem to be effective in counteracting uncertain dynamics, unbounded disturbances and (possible) kinematic singularities met on the robot trajectory. The numerical simulations carried out for a redundant manipulator of a Selective Compliant Articulated Robot for Assembly (SCARA) type consisting of three revolute kinematic pairs and operating in a two-dimensional task space illustrate performance of the proposed controllers.
Aiming at a kind of piezoelectric actuator (PEA) with hysteresis nonlinearity, a new high order sliding mode controller (HOSMC) is presented. Firstly, the two-order nonlinear dynamic model of the PEA is introduced and the hysteresis nonlinearity part is analyzed as a kind of bounded disturbance acting on the system. Then, the high order sliding mode control law is designed. Further, the finite-time astringencies of the sliding mode variables and the asymptotic astringency of tracking error variable are proved theoretically. The simulation results show the validity of the HOSMC for this kind of piezoelectric actuators with hysteresis nonlinearity.
In this paper a method to simultaneously achieving the two main objective of smoothing the control while maintaining ideally infinite frequency regimes is particularized to the simplex sliding mode method. The present paper proposes a simplex sliding mode control logic based on the time derivative of the control vector. This practice, beside the standard chattering reduction effect, has the important consequence of making possible the control of a rather wide class of uncertain systems nonlinear in the control. In the uncertain case the increment of the relative degree implicit in the proposed approach requires the introduction of second order sliding mode observer. A separation theorem is proven and sufficient conditions for the finite time convergence of both the estimation and tracking errors are found in terms of further requirements on the control amplitude. Copyright © 2007 International Federation of Automatic Control All Rights Reserved.
On the basis of classical studies in robotics, it seems that the conventional sliding mode approach is not a suitable technique to design robotic controllers, due to the presence of the so-called chattering effect. However, studies have shown that a good reduction of the chattering effect can be achieved by relying on higher order sliding modes. This chapter presents the application of the Second Order Sliding Mode (SOSM) design methodology to the control and supervision of industrial manipulators, by proposing a robust control scheme and a diagnostic scheme to detect and, possibly, isolate and identify faults acting on the components of the system. The proposed SOSM motion controller and the SOSM observers designed to construct the diagnostic scheme are theoretically developed, and their practical application is suitably described. Indeed, the proposed approaches are experimentally verified on a COMAU SMART3-S2 industrial robot manipulator, obtaining satisfactory results.
This paper deals with the problem of Sliding Mode Control (SMC) design for a class of non linear system. Two different control designs are developed for a dynamic system Maxpid. The first type is the define of traditional Sliding Mode (SM) control. The second is the Higher Order Sliding Mode (HOSM) control. These two kinds have the same properties of robustness and precision. Note that the implementation drawback of the first type has a chattering phenomenon. The HOSM control is proposed to reduce the phenomenon. The simulation results obtained from the two types are finally compared in terms of good performances.
Nonlinear system control has been widely concern of the research. At present, the nonlinear system decoupling control and static feedback linearization that based on the theory of differential geometry brought the research getting rid of limitation for local linearization and small scale motion. However, differential geometry control must depend on precise mathematical model. As a matter of fact, the control system usually is with parameters uncertainties and output disturbance. In this thesis, nonlinear system of control theory has been studied deeply. Considering sliding mode variable structure control with good robust, which was not sensitive for parameters perturbation and external disturbance, the combination idea of nonlinear system and sliding mode controls was obtained by reference to the large number of documents. Thus, it not only can improve system robustness but solve the difficulties problem of nonlinear sliding mode surface structure. As known to all, traditional sliding mode had a defect that is famous chattering phenomenon. A plenty of research papers focus on elimination/avoidance chattering by using different methods. By comparing, the document is concerned with novel design method for high order sliding mode control, which can eliminate chattering fundamentally. Especially, the approach and realization of nonlinear system high order sliding mode control is presented in this paper. High order sliding mode technique is the latest study. This thesis from the theory analysis to the simulation and experiment deeply study high order sliding mode control principle and its applications. By comparison, the second order sliding mode control law (also known as dynamic sliding mode control, DSM) may be effective to eliminate the chattering phenomenon. But it is still unable to shake off the requirement of system relative degree. Therefore, arbitrary order sliding mode controller is employed, whose relative degree can equal any values instead of one. The robot car model adopted high order sliding mode is taken as an example. The simulation results show that the tracking control is effective. In the control systems design, it is very often to differentiate the variables. Through the derivation of sliding mode, the expression of sliding mode differential value is obtained. The simulation results certificate sliding mode differentiator with robustness and precision. At the same time, the differentiator for arbitrary sliding mode is given to avoiding conventional complex numerical calculation. It not only remains the precision of variables differential value, but also obtains the robustness. A direct application is simplification for high order sliding mode controller. Due to its inherent advantages, the permanent magnet synchronous motor (PMSM) deserves attention and is the most used drive in machine tool servos and modern speed control applications. For improving performance, this paper will applied nonlinear high order sliding mode research achievement to MIMO permanent magnet synchronous motor. It changes the coupling nonlinear PMSM to single input single output (SISO) linear subsystem control problem instead of near equilibrium point linearization. Thereby, the problem of nonlinear and coupling for PMSM has been solved. In addition, Uncertainty nonlinear robust control system has been well-received study of attention. Because the robust control theory is essentially at the expense of certain performance. This kind of robust control strategy often limits bandwidth of closed loop, so that system tracking performance and robustness will be decreased. So, sliding mode control is an effective approach for improving system robust. This thesis first proposed a robust high order sliding mode controller for PMSM. The system has good position servo tracking precision in spite of parameters uncertainties and external torque disturbance. On this basis, According to the principle of high order sliding mode, as well as differentiator, the state variables of PMSM are identified online firstly and successfully. The results of simulation indicate observe value has high precision when sliding mode variable and its differentials are convergent into zero. The same theory is used in external unknown torque disturbance estimation online for PMSM. As if, load torque will no longer be unknown disturbance. System performance can be improved greatly. It establishes theoretical foundation for the future applications. At the end of paper, using advanced half-physical platform controller dSPACE to drive a PMSM, hardware experiment implement is structured completely. The experiment results illustrate that PMSM adopting precious feedback linearization decoupling and high order sliding mode controller can realize system servo tracking control with good dynamic and steady character.
A continuous global robust finite time feedback stabilization control law is proposed for a class of uncertain nonlinear systems. Firstly, a state feedback control law is designed for the nominal system. By using Lyapunov direct method and LaSalle invariance principle, we prove that the resulting closed-loop nominal system is globally asymptotically stable with the negative homogeneity in degrees. Secondly, introducing an auxiliary variable, we design a compensated control law for compensating the uncertainties of the system by using the finite-time convergent second-order sliding mode Super-twisting algorithm. The range of the parameters in the compensation control law is determined by using the finite time Lyapunov function. Finally, a continuous feedback control law is developed for the closed-loop system to converge to its equilibrium point in a finite period of time. Simulation results show the effectiveness of the proposed control law.
The paper documents a new continuous adaptive fast terminal sliding mode control approach for position tracking of robotic manipulators. Combining linear sliding mode and terminal sliding mode, a fast nonsingular terminal sliding mode manifold is presented. Considering the discontinuous property of the sign function, which is often used in traditional sliding mode controller and will result in high-freqsency chattering in the control channel, the proposed controller adopts the continuous saturation function for chattering elimination. Besides the continuous property, convergence to the origin asymptotically and in finite time can be guaranteed in theory with the proposed controller, which is quite different from traditional boundary layer technique, where only bounded motion around the sliding manifold can be ensured. For asymptotic stability, it is only required that the lumped uncertainty is bounded, but the upper bound may be unknown by virtue of the designed adaptive methodology. The obtained results are applied to the problem of position tracking for robotic manipulators. Detailed simulations with some comparisons under various conditions demonstrate the effectiveness of the proposed method.
The pitch channel output tracking of a kind of ummanned aerial vehicle over-load control system was designed based on second order sliding mode control theory. Output-redefinition approach was adopted to resolve the non-minimum phase problem between rudder angle and overload output of ummanned aerial vehicle system, second order sliding mode control was used to eliminate chatter, and guaranteed the tracing of original system output. The observer was designed to deal with the not complete state measurable system. The stability and robustness were verified by the simulations results.
This work offers the solution at the control feed-back level of the accurate trajectory tracking subject to finite-time convergence. Dynamic equations of a rigid robotic manipulator are assumed to be uncertain. Moreover, globally unbounded disturbances are allowed to act on the manipulator when tracking the trajectory. Based on the suitably defined non-singular terminal sliding vector variable and the Lyapunov stability theory, we propose a class of absolutely continuous robust controllers which seem to be effective in counteracting both uncertain dynamics and unbounded disturbances. The numerical simulation results carried out for a robotic manipulator consisting of two revolute kinematic pairs operating in a two-dimensional joint space illustrate performance of the proposed controllers.
The finite-time dynamic feedback stabilization problem is considered for a class of nonlinear systems under the input saturation, and a switch control strategy is proposed. It can divide the complex nonlinear systems into different simple subsystems by a finite number of switching in order to make the states in every step converge to a prespecified point or the equilibrium point in a finite time, and the saturation property is satisfied in every step by the controllers as required as design. Finally, a specific example of the system and a finite-time saturated controller are given; and the effectiveness of the proposed control approach is validated by the simulation results.
This paper considers the finite-time stabilization problem for a class of nonlinear systems with dynamic feedback subject to input saturation. A method is presented by using switching control strategy and state-scaling technique, which can divide a complex nonlinear systems into different simple subsystems in order to make that the states in every step converge to a given point or the equilibrium point in a finite time, and the saturation property of control inputs keeps for the design of controllers proposed in every step. Finally, a simulation example and the finite-time saturated controllers are given to illustrate the effectiveness of theorem.
The novelty of work is the application of first order sliding mode techniques for hybrid model to estimate the system states which are finally analyzed to detect of Misfire fault. The output of hybrid model of SI engine (engine speed) is compared with the observed engine speed to generate the output error. The sliding mode observer is designed to track the engine speed and estimate the other system state i.e. acceleration. The crankshaft acceleration is analyzed to detect the engine misfire fault condition. The stability analysis of application of first order sliding mode for hybrid model is provided. The results of tracking of speed and estimation of acceleration using sliding mode observer are presented. The application of Sliding Mode Technique for hybrid model is first established using data generated by simulating engine model and then using experimental data taken from SI engine.
In this paper an adaptive second order terminal sliding mode (SOTSM) controller is proposed for controlling robotic manipulators. Instead of the normal control input, its time derivative is used in the proposed controller. The discontinuous sign function is contained in the derivative control and the actual control obtained after integration is continuous and hence chatterless. An adaptive tuning method is utilized to deal with the system uncertainties whose upper bounds are not required to be known in advance. The performance of the proposed control strategy is evaluated through the control of a two-link rigid robotic manipulator. Simulation results demonstrate the effectiveness of the proposed control method.
This paper presents a nonlinear disturbance observer-based finite time convergent second order sliding mode control scheme for a tailless aircraft with uncertain parameters. Command angles and angular rate commands are robustly tracked in outer and inner loops correspondingly via finite time convergent second order sliding mode controllers. Nonlinear disturbance observer is employed to retain overall high accuracy of the closed-loop tracking. An optimal control allocation algorithm is employed using nominal mathematical model of an aircraft. The control approach is applied to the innovative control effectors (ICE) fighter aircraft. Computer simulations demonstrate high accuracy tracking performance.
Control under heavy uncertainty conditions is one of the central topics of the modern control theory. The sliding-mode control
approach [51, 53, 9] to the problem is based on the exact keeping of a properly chosen constraint by means of high-frequency
control switching. Although very robust and accurate, the approach also features certain drawbacks. The standard sliding mode
may be implemented only if the relative degree of the constraint is 1, i.e., control has to appear explicitly already in the
first total time derivative of the constraint function. Another problem is that the high-frequency control switching may cause
dangerous vibrations (the so-called chattering effect) [15, 16, 17].
In this paper the tracking problem for rigid robot manipulators with uncertain parameters and unmodelled torque actuators (DC motors) is addressed and solved by combined first/second-order sliding mode control methodology. The robot is electrically-driven through a pulse-width modulated actuator supply voltage. The proposed controller does not require the availability of the joint velocity, is insensitive to smooth load disturbances and guarantees a global domain of attraction. Computer simulations confirm the feasibility of the proposed approach.
This paper deals with the interaction control of robot manipulators. The ideal goal is to control the manipulator so that its end-effector reaches a desired position in the operative space. The attainment of this goal can be prevented by the fact that during its motion the end-effector can interact with the environment. So, a realistic goal is regarded as control objective in the paper, that of asymptotically reaching an equilibrium position jointly determined by the choice of the ideal desired position, and by the presence of the environment. The model of the manipulator is supposed to be affected by uncertainties, so that a classical impedance control scheme cannot be adopted. The approach proposed in this paper relies on the theory of sliding mode control. In general, this methodology is deemed nonappropriate to control mechanical systems like robot manipulators, since it introduces the so-called chattering phenomenon, which produces undesired wear and vibrations. Yet, the final control solution proposed in this paper is based on the generation of sliding modes of the second order, which practically reduces the degradation of performances, and the mechanical stress due to the chattering effect. Indeed, the proposed second-order sliding mode control scheme relies on the use of a continuous control law, in contrast to conventional sliding mode control schemes. Copyright © 2007 John Wiley & Sons, Ltd.
Control under heavy uncertainty conditions remains among the main topics of the control theory. The sliding-mode control [50,
53, 10] is one of the main tools in the field. This approach is based on exactly keeping a properly chosen constraint by means
of control switching of high (theoretically infinite) frequency. Although very robust and accurate, the approach has two basic
restrictions. The direct implementation of standard sliding modes requires the relative degree of the constraint to be 1,
i.e. control has to appear explicitly already in the first total time derivative of the constraint function. Also, high-frequency
control switching may cause the so-called chattering effect [14, 15, 16, 33].
The pitch channel output tracking of a kind of supersonic cruise missile over load control system was designed based on second order sliding mode control theory. Output-redefinition approach was adopted to resolve the non-minimum phase problem between rudder angle and overload output of STT missile system, second order sliding mode control was used to eliminate chatter, and guaranteed the tracing of original system output. The observer was designed to deal with the not complete state measurable system. The stability and robustness were verified by the simulations results.
In this paper the simplex sliding mode control method is endowed with additional properties of practical as well as theoretical relevance. The method is extended to eliminate the chattering phenomenon. To this end an estimation procedure based on second order sliding mode algorithms is coupled with a simplex sliding mode control of an augmented order system driven by the time derivative of the actual control vector.
The simplex sliding mode control method is further developed by considering uncertain control systems non-affine in the control law. In order to reduce chattering effects, a set of integrators is added in the input channels. The augmented system is then controlled by a switching logic based on the simplex control method. As a result, the original control vector turns out to be continuous. A second order sliding mode observer is used when the sliding output is not available. Explicit conditions are identified about systems uncertainties and the simplex geometry in order to guarantee the convergence of the proposed methodology.
This communication is devoted to a "practical" comparison between high-order sliding modes and the recently introduced model-free control. The perfect knowledge of the relative degree of the output variable, which is a standard assumption for sliding modes, is assumed here. Our comparisons are based on two concrete case-studies and on numerous computer simulations. The smoothness of the input variables, the robustness with respect to noises and the straightforward extendibility of the model-free controllers to MIMO systems are highlighted.
The synthesis of a control algorithm that stirs a nonlinear system to a given manifold and keeps it within this constraint is considered. Usually, what is called sliding mode is employed in such synthesis. This sliding mode is characterized, in practice, by a high-frequency switching of the control. It turns out that the deviation of the system from its prescribed constraints (sliding accuracy) is proportional to the switching time delay. A new class of sliding modes and algorithms is presented and the concept of sliding mode order is introduced. These algorithms feature a bounded control continuously depending on time, with discontinuities only in the control derivative. It is also shown that the sliding accuracy is proportional to the square of the switching time delay.
In this paper we study the numerical solution of the differential/algebraic systems F(t, y, y') = 0. Many of these systems can be solved conveniently and economically using a range of ODE methods. Others can be solved only by a small subset of ODE methods, and still others present insurmountable difficulty for all current ODE methods. We examine the first two groups of problems and indicate which methods we believe to be best for them. Then we explore the properties of the third group which cause the methods to fail. We describe a reduction technique which allows systems to be reduced to ones that can be solved. It also provides a tool for the analytical study of the structure of systems.
In this article, the problem of asymptotic output stabilization in nonlinear controlled systems is approached from the perspective of dynamical sliding-mode control. The proposed controller is based on Fliess's Generalized Observability Canonical Form, recently derived from the differential algebraic approach to system dynamics.
We consider a dynamic system containing uncertain elements. Only the set of possible values of these uncertainties is known. Based on this information a class of state feedback controls is proposed in order to guarantee uniform ultimate boundedness of every system response within an arbitrarily small neighborhood of the zero state. These feedback controls are continuous functions of the state.
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).
First order conditions: First order conditions Theory of a weak minimum for the problem on a fixed time interval Theory of the maximum principle Extremals and the Hamiltonian of a control system Hamilton-Jacobi equation and field theory Transformations of problems and invariance of extremals Quadratic conditions: Quadratic conditions and conjugate points for broken extremals Quadratic conditions for a Pontryagin minimum and sufficient conditions for a strong minimum: Proofs Quadratic conditions in the general problem of the calculus of variations and related optimal control problems Investigation of extremals by quadratic conditions: Examples Bibliography.
This paper addresses the problem of robust hybrid position and force control of robot manipulators. Variable structure control with sliding mode is used to implement the hybrid control strategy. Two variable structure control algorithms are developed in task space. One of the algorithms is based on hierarchical control method, and the other is developed for control of robot manipulators used to carry out both unconstrained and constrained taks.
In this paper the authors present a couple of algorithms, one for the case of analogue devices and one for the case of digital devices, that realize a nonlinear controller performing a second order sliding mode. These algorithms allow the finite time stabilization of uncertain second order nonlinear systems with incomplete state measurements. Two different applications of this technique are presented. In the first we deal with the problem of chattering avoidance in sliding mode control of uncertain nonlinear systems without using any observer. The second application shows that the developed algorithms allow the hybrid force/position continuous control of a two arms constrained planar manipulator even when the accelerations are not available. © 1997 by John Wiley & Sons, Ltd.
An adaptive control scheme for the trajectory/force tracking of robot manipulators is presented. Asymptotic stability of state variables and convergence of constraint forces to any prespecified set are proven. The design procedure avoids the restrictive solvability condition of the constraint equation in the whole space of robot co-ordinates. A fairly accurate bound of the tracking error is derived by means of a Lyapunov analysis.
We consider deterministic uncertain control systems, modelled as ordinary differential inclusions. We present computable criteria for possibly discontinuous feedback control laws, guaranteeing prescribed tracking conditions, via a differential inequality approach. Moreover we show that the right concept of the corresponding (Filippov or classical) solutions is related to the choice of the (Lyapounov-like) function defining the differential inequality.
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.
A framework for the analysis and control of manipulator systems with respect to the dynamic behavior of their end-effectors is developed. First, issues related to the description of end-effector tasks that involve constrained motion and active force control are discussed. The fundamentals of the operational space formulation are then presented, and the unified approach for motion and force control is developed. The extension of this formulation to redundant manipulator systems is also presented, constructing the end-effector equations of motion and describing their behavior with respect to joint forces. These results are used in the development of a new and systematic approach for dealing with the problems arising at kinematic singularities. At a singular configuration, the manipulator is treated as a mechanism that is redundant with respect to the motion of the end-effector in the subspace of operational space orthogonal to the singular direction.
The solution of a tracking problem for a secondorder nonlinear system with uncertain dynamics and incomplete state measurement is obtained by means of a procedure directly inspired by the solution of the classical minimum-time optimal control problem. Two different types of uncertainty are considered in the paper: in the first case a constant bound on the uncertain dynamics is assumed to be known; in the second case, the bound is a function of both the measurable and the unmeasurable state variable of the system. In both cases, the possibility of applying the proposed control algorithms is proved to be determined by a proper choice of the control signal features. The resulting system is characterized by a suitable feedback switching logic and the convergence of the system trajectory to the desired one (or to a δ-vicinity of this latter) is proved also in the uncertain case.
We consider the design of a feedback control law for control systems described by a class of nonlinear differential-algebraic equations so that certain desired outputs track given reference inputs. The nonlinear differential-algebraic control system being considered is not in state variable form. Assumptions are introduced and a procedure is developed such that an equivalent state realization of the control system described by nonlinear differential-algebraic equations is expressed in a familiar normal form. A nonlinear feedback control law is then proposed which ensures, under appropriate assumptions, that the tracking error in the closed loop differential-algebraic system approaches zero exponentially. Applications to simultaneous contact force and position tracking in constrained robot systems with rigid joints, constrained robot systems with joint flexibility, and constrained robot systems with significant actuator dynamics are discussed.
This paper deals with the hybrid position/force control of
constrained manipulators subjected to uncertainties and disturbances of
various nature, including Coulomb friction. The proposed solution is
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 drawbacks presented by
the classical sliding mode techniques are the chattering phenomenon and
the algebraic coupling between constraint forces and possibly
discontinuous control signals. Both these problems are addressed in the
paper by exploiting the robustness properties of second-order
sliding-mode control algorithms. A specific algorithm of this kind,
recently developed by the authors, is proven to be effective also when
the dynamic equation of the system includes discontinuous disturbances
A dynamic hybrid position/force control method, which takes into consideration the manipulator dynamics and the constraints on the end-effector specified by the given task, is obtained by extending the method of M.H. Raibert and J.J. Craig (1981). One difficulty in implementing the method is that precise information on the size and position of the object with which the end-effector contacts is not usually available. To cope with this difficulty, a problem of dynamic hybrid control with unknown constraint is studied. An online algorithm is developed that estimates the local shape of the constraint surface by using measured data on the position and force of the end-effector. It is shown by experiments, using a SCARA-type robot, that the combination of this algorithm with the dynamic hybrid control method works fairly well, making the dynamic hybrid control approach more practical
Relying on the possibility of generating a second-order sliding
motion by using, as control, the first derivative of the control signal
instead of the actual control, a new solution to the problem of
chattering elimination in variable structure control (VSC) is presented.
Such a solution, inspired by the classical bang-bang optimal control
strategy, is first depicted and expressed in terms of a control
algorithm by introducing a suitable auxiliary problem involving a
second-order uncertain system with unavailable velocity. Then, the
applicability of the algorithm is extended, via suitable modifications,
to the case of nonlinear systems with uncertainties of more general
types. The proposed algorithm does not require the use of observers and
differential inequalities and can be applied in practice by exploiting
such commercial components as peak detectors or other approximated
methods to evaluate the change of the sign of the derivative of the
quantity accounting for the distance to the sliding manifold
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
A control strategy proposed by the authors in previous works to
reduce the effect of chattering led to the solution of a differential
inequality involving the first and second derivatives of the control
signal and a parameter for a first-order observer. The analysis was
carried out assuming this parameter to be constant. In the present
paper, this assumption is removed, thus allowing the solution of the
differential inequality to be a function of a piecewise continuous first
derivative of the control signal, without need for assuming a priori
knowledge of bounds to the plant trajectories
A simple robust nonlinear control law for n -link robot
manipulators is derived using the Lyapunov-based theory of guaranteed
stability of uncertain systems. The novelty of this result lies in the
fact that the uncertainty bounds needed to derive the control law and to
prove uniform ultimate boundedness of the tracking error depend only on
the inertial parameters of the robot. In previous results of this type,
the uncertainty bounds have depended not only on the inertia parameters
but also on the reference trajectory and on the manipulator state
vector. The presented result also removes previous assumptions regarding
closeness in norm of the computed inertia matrix to the actual inertial
matrix. The design used thus provides the simplest such robust design
available to date
Mathematical models for constrained robot dynamics, incorporating
the effects of constraint force required to maintain satisfaction of the
constraints, are used to develop explicit conditions for stabilization
and tracking using feedback. The control structure allows feedback of
generalized robot displacements, velocities, and the constraint forces.
Global conditions for tracking, based on a modified computed-torque
controller and local conditions for feedback stabilization, using a
linear controller, are presented. The framework is also used to
investigate the closed-loop properties if there are force disturbances,
dynamics in the force feedback loops, or uncertainty in the constraint
functions
Higher order sliding modes as a natural phenomenon in control theory,” Robust Control Via Variable Structure and Lyapunov Techniques
- A Levant
- L Fridmann