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Applications of a sub-optimal discontinuous control algorithm for uncertain second order systems
Abstract
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.
... Indeed, we have considered as low-level control strategy a Sliding Mode Controllers [21], [22], that are proving to be very effective for this kind of application. In particular, we have selected Second Order Sliding Mode (SOSM) [23], [24], [25], both for its robustness and its ability to deal with nonlinear and switching devices. By the way, this direct approach to the nonlinear nature of the system avoids linearisations and small-signal approaches and is more rigorous from the mathematical point of view. ...
... Therefore, SOSM techniques can be applied to the system (1)-(4). Proof of the finite time convergence of the sliding function and its time derivative to the origin of the state plane follows directly from [23]. While on the sliding manifold, the switching control takes the form of the so-called equivalent control, being in [0, 1]. ...
... Remark 1: Theorem 1 assumes perfect knowledge of the extremal points. In practice, an estimate of the extremal points can be obtained by using the peak-detection algorithm proposed in [23]. Theorem 1 proves the stability of the closed-loop system when the control objective is to regulate the system state variables such that the supercapacitor supplies the EMA with the current peaks. ...
... When the system considers the system and measurement errors, it is an optimal control problem of an uncertain system. Although there is no study about the optimal control problem for immune systems with uncertain errors, a paper (Bartolini, Ferrara, & Usai, 1997), proposed an optimal control algorithm for the uncertain system. To verify that EBOC method is more effective than other existing algorithms for the uncertain stochastic system with the same initial setting, we design another simulation for applying the algorithm in paper (Bartolini et al., 1997) into our sepsis system with errors. ...
... Although there is no study about the optimal control problem for immune systems with uncertain errors, a paper (Bartolini, Ferrara, & Usai, 1997), proposed an optimal control algorithm for the uncertain system. To verify that EBOC method is more effective than other existing algorithms for the uncertain stochastic system with the same initial setting, we design another simulation for applying the algorithm in paper (Bartolini et al., 1997) into our sepsis system with errors. Simulation results when we applied this optimal control algorithm to the system with errors are shown in Fig. 3.5. ...
... When the system considers possible errors, the errors part p t ( ) can be either positive or negative, so the lower bound of control strategy in our system is 0. Thus, the optimal control at each time period obtained by the algorithm in paper (Bartolini et al., 1997) is set to be either 0 or u l . While the control value of traditional optimal control presented in Section 3.5.2 ...
Treatment strategy of a realistic health care system must consider both system and measurement errors. The traditional optimal control method is commonly applied to deterministic systems instead of dynamic systems with uncertain errors. Therefore, this paper considers uncertain errors and stochastic characteristics in a dynamic health care system and proposes a new evidence-based optimal control (EBOC) approach that combines the traditional optimal control and machine learning methods. Four machine learning algorithms were tested, and the most suitable algorithm was combined with the traditional optimal control method for the sepsis model. Extensive computational studies proved that, compared to the traditional optimal control method, the EBOC method more efficiently controls disease progression and decreases total cost when uncertainty or measurement errors exist in the model, no matter the machine learning algorithm utilized. Moreover, the total settings are possible when numerous parameter í µí± í µí± combinations could affect control results, meaning determination of the optimal parameter set(s) becomes an NP-hardness problem. This paper also uses the genetic algorithm to find superior parameter settings to improve the performance and effectiveness of the control strategy created by the EBOC method.
... Indeed, we have considered as low-level control strategy a Sliding Mode Controllers [21], [22], that are proving to be very effective for this kind of application. In particular, we have selected Second Order Sliding Mode (SOSM) [23], [24], [25], both for its robustness and its ability to deal with nonlinear and switching devices. By the way, this direct approach to the nonlinear nature of the system avoids linearisations and small-signal approaches and is more rigorous from the mathematical point of view. ...
... Therefore, SOSM techniques can be applied to the system (1)-(4). Proof of the finite time convergence of the sliding function and its time derivative to the origin of the state plane follows directly from [23]. While on the sliding manifold, the switching control takes the form of the so-called equivalent control, being in [0, 1]. ...
... Remark 1: Theorem 1 assumes perfect knowledge of the extremal points. In practice, an estimate of the extremal points can be obtained by using the peak-detection algorithm proposed in [23]. Theorem 1 proves the stability of the closed-loop system when the control objective is to regulate the system state variables such that the supercapacitor supplies the EMA with the current peaks. ...
In this paper the problem of reducing stress on aeronautical electric energy generators is considered. The usage of an active Energy Storage Device (a controlled supercapacitor) is considered as a quick device able to absorb or yield quickly energy peaks caused by sudden intervention of Electro-Mechanical Actuators. The control strategy considered is a second-order sliding mode approach, able to guarantee finite-time achievement of the control goal. This characteristic is then exploited by a supervisory control to manage different objectives (including managing the State of Charge of the supercapacitor, with different levels of priority) with guaranteed stability. Detailed simulations in different situations confirm the effectiveness of the proposed strategy.
... It should be noted that different SOSMC algorithms have been reported such as the "twisting" and "super-twisting" [5], "sub-optimal" [6] etc. In Bartolini et al. [6], an optimal version of second-order sliding mode of the so-called twisting algorithm has been presented. ...
... It should be noted that different SOSMC algorithms have been reported such as the "twisting" and "super-twisting" [5], "sub-optimal" [6] etc. In Bartolini et al. [6], an optimal version of second-order sliding mode of the so-called twisting algorithm has been presented. It has been used for systems with the relative degree of r = 2 [7]. ...
... The output voltage error (x 1 ) and its time derivative (x 2 ) are considered as the state variables, so the final state equations for buck converter with a voltage controller, is presented in (6) where ω 2 o = 1/LC is the resonance frequency [29], [31]. ...
This paper presents a simple and systematic approach to design second order sliding mode controller for buck converters. The second order sliding mode control U+0028 SOSMC U+0029 based on twisting algorithm has been implemented to control buck switch mode converter. The idea behind this strategy is to suppress chattering and maintain robustness and finite time convergence properties of the output voltage error to the equilibrium point under the load variations and parametric uncertainties. In addition, the influence of the twisting algorithm on the performance of closed-loop system is investigated and compared with other algorithms of first order sliding mode control such as adaptive sliding mode control U+0028 ASMC U+0029, nonsingular terminal sliding mode control U+0028 NTSMC U+0029. In comparative evaluation, the transient response of the output voltage with the step change in the load and the start-up response of the output voltage with the step change in the input voltage of buck converter were compared. Experimental results were obtained from a hardware setup constructed in laboratory. Finally, for all of the surveyed control methods, the theoretical considerations, numerical simulations, and experimental measurements from a laboratory prototype are compared for different operating points. It is shown that the proposed twisting method presents an improvement in steady state error and settling time of output voltage during load changes.
... First order sliding mode control (SMC), based on [13] and [42]. Second order sub-optimal sliding mode (SOSM) control [13], [39]- [42]. Proportional integral (PI) control with gain scheduling parameterized with (inspired by [8]). ...
... (26) The control law consists of two terms: i) a term , which would keep the system on the sliding surface, if the system dynamics were completely known; and ii) a switching term that ensures robustness with respect to modeling errors and disturbances [13], [39]- [41]. The control law is defined as follows [13]: ...
... The main advantage of second order SOSM control in its chattering avoidance formulation is the ability to achieve robustness with respect to matched disturbances, typical of sliding mode control, while avoiding control input chattering, which would compromise vehicle comfort and drivability [39]- [42]. In fact, with this SOSM formulation the discontinuity is on the time derivative of the control action, and not on the control action itself (see Fig. 9). ...
Extensive literature discusses traction control system designs for electric vehicles. In general, the proposed control structures do not include consideration of the actuation dynamics, which are especially important for vehicles with on-board drivetrains, usually characterized by significant torsional dynamics of the half-shafts. This paper compares the performance of a selection of traction controllers from the literature, with that of PID and control structures specifically designed for on-board electric drivetrains. The analysis in the frequency domain and the simulation results in the time domain show the significant performance improvement provided by the control system designs considering the actuation dynamics.
... As a direct consequence, at the output of the rst integrator in Fig. 1, an estimate of the rst derivative of the input signal x(t) is available Two di erentiators are proposed in this paper, both based on the so-called sub-optimal second order sliding mode controller, recently proposed in literature 6,8]. ...
... Some 2-sliding control schemes, which can solve problem 1, were presented in literature 5,8], and a digital control algorithm, which is e ective for the simpli ed case of constant control gain function, i.e. y(t); t] = 1 (6) is presented in this paper. Under the assumption that only the sequence of the sampled values of the variable y 1 (t), y 1 k] = y 1 (k ) k = 0; 1; 2 : : :, is available, the following proposition is proved. ...
... provides the ful llment of jy 1 (t)j k 3 T 2 t T reach jy 2 (t)j k 4 T (8) provided that the estimatesŷ 1M k] of the singular values of the sliding quantity y 1 are evaluated according to the following algorithmŷ 1M k] = y 1 k 1] if k] 0 y 1M k 1] otherwise k] = (y 1 k 2] y 1 k 1])(y 1 k 1] y 1 k]) y 1 1] =ŷ 1M 1] = y 1 (0) ; y 1 2] = 0 (9) and that the control e ort U M belongs to a nite set as follows U M 2 (2 + k 1 T; k 2 T 2 ) (10) in which k i , i = 1; : : :; 4 are proper positive constants, and T reach is a nite transient time. ...
... In this paper, the PID control remains unchanged, while two different strategies are proposed to reduce the chattering in the FOSM, namely, the introduction of a low-pass filter and the replacement with a continuous approximation of the sign function are considered. To complete the discussion and comparison, a full-state linear-quadradic regulator (LQR) and two second-order sliding mode (SOSM) algorithms, the twisting algorithm [16,17] and the suboptimal algorithm [18], are also presented. ...
... The second algorithm that satisfies the conditions of the SOSM is the suboptimal algorithm [18]. In this version of the sliding mode, the control law is defined as (12): ...
In today’s automotive industry, electrification is a major trend. In-wheel electric motors are among the most promising technologies yet to be fully developed. Indeed, the presence of multiple in-wheel motors acting as independent actuators allows for the implementation of innovative active systems and control strategies. This paper analyzes different design possibilities for a torque vectoring system applied to an originally compact front-wheel drive hybrid electric vehicle with one internal combustion engine for the front axle and two added electric motors integrated in the wheels of the rear axle. A 14 degrees of freedom vehicle model is present o accurately reproduce the nonlinearities of vehicle dynamic phenomena and exploited to obtain high-fidelity numerical simulation results. Different control methods are compared, a PID, an LQR, and four different sliding mode control strategies. All controllers achieve sufficiently good results in terms of lateral dynamics compared with the basic hybrid version. The various aspects and features of the different strategies are analyzed and discussed. Chattering reduction strategies are developed to improve the performance of sliding mode controllers. For a complete overview, control systems are compared using a performance factor that weighs control accuracy and effort in different driving maneuvers, i.e., ramp and step steering maneuvers performed under quite different conditions ranging up to the limits.
... with x = (x 1 , x 2 ) ∈ R 2 the state, u ∈ R the control input, f and g continuous functions such that f (0) = 0, g(x) = 0 for all x ∈ R 2 and d the external disturbance such that |d(t)| < δ. The second-order systems have been widely used in practice, see for instance [23]. The objective is to use the previous results on robust fixed-time stability for designing sliding mode controllers. ...
... When the sliding surface is reached, one haṡ with β 1 > 0, β 2 > 0, α > 1 and 0 < γ < 1 it leads to a singular controller, see for instance [16], [17]. With the classical sliding surface (20), one can get the global robust fixed-time stabilization of the s−system (23), as explained in Proposition 1, but only the global robust asymptotic stabilization of the x−system (19). However, the controller (21) is easy to implement. ...
This article deals with robust fixed-time stability and stabilization. First, new global robust fixed-time stability results are proposed for scalar systems by using constant and variable exponent coefficients. Then, they are applied to global robust fixed-time stabilization of a class of uncertain nonlinear second order systems by using sliding mode control. All the results are illustrated in simulation.
... For example, Bartolini and Pvdvnowski [12] consider a nonlinear system and propose a new method for the asymptotic linearization by means of continuous control law. Also Bartolini et al. [13,14] consider an uncertain secondorder nonlinear system and propose a new approximate linearization and sliding mode to control such systems. In addition to the nonoscillation for two-dimensional systems of first-order equations, periodic and subharmonic solutions are also investigated in [15][16][17], and significant contributions have been made. ...
... i.e., I 2 , ∞. Also, note that 1 t ; À1 À 1 t 2 ÀÁ is a solution of system (14) in N À such that x tends to zero, while y tends to À1, i.e., N À 0, F 6 ¼ ø. ...
Oscillation and nonoscillation theories have recently gotten too much attention and play a very important role in the theory of time-scale systems to have enough information about the long-time behavior of nonlinear systems. Some applications of such systems in discrete and continuous cases arise in control and stability theories for the unmanned aerial and ground vehicles (UAVs and UGVs). We deal with a two-dimensional nonlinear system to investigate the oscillatory behaviors of solutions. This helps us understand the limiting behavior of such solutions and contributes several theoretical results to the literature.
... Second-order sliding mode controllers (2nd-sliding controllers) are an efficient control tool to deal with strong uncertainties and disturbances acting on the plant while finite-time convergence to the desired control task is ensured. [1][2][3][4] This kind of controllers is usually applied to keep at zero outputs of relative degree 2 as well as to reduce the chattering effect of outputs of relative degree 1, and in both cases, the features of the standard (first-order) sliding mode control (SMC) are preserved 1 if actuators are fast. 5 Although the existing 2nd-sliding controllers have been successfully applied in practice, [6][7][8] there are still some practical and theoretical aspects that deserve further attention. ...
... and lim 3 , whereas the solution of the differential equatioṅ ...
The aim of this paper is to propose a novel family of variable‐gain second‐order sliding mode controllers capable of dealing with uncertainties and disturbances that grow with the state. The proposed family provides exact compensation of perturbations having a known upper bound and generalizes some sliding controllers reported in the literature. A novelty of this work is that the designer is able to choose the desired convergence rate: semiglobal/global finite‐time or fixed‐time. The analysis and control design is completely based on a strict Lyapunov function, leading to an effective procedure to tune the gains.
... However, the application of this optimal controller requires a complete information about the states of the system. The problem of the finite time stabilization of uncertain second order nonlinear systems with incomplete state measurements was addressed in [35] in which the suboptimal algorithm, inspired by the classical time optimal bang-bang control of a double integrator, was derived using a sub-time-optimal feedback to fulfill this objective. Unlike the twisting algorithm which makes the trajectories converges non-monotonically in finite time to the origin of the phase plane, this algorithm yields a monotonic convergence. ...
... Moreover, the driven trajectories show twisting and jumping behaviors while converging to the origin, see Fig. 2 (b). This algorithm can be defined as follows [33], [35]: ...
Rotor mass imbalance is a common problem to rotating machines due to the unavoidable imperfections in manufacturing. These imbalance forces can be viewed as harmonic disturbances which lead to a periodic rotor runout during rotation. Furthermore, the runout length increases with the rotational speed squared. Moreover, for variable rotational speed applications, these harmonic disturbances are also time-varying. Active magnetic
bearings (AMB) provide a mean to actively attenuate these disturbances. Although, various imbalance compensation schemes have been proposed in literature to handle this problem, they are often more suitable for constant rotational speed applications where disturbances can be handled at a predetermined rotational speed. This study proposes the application of second-order sliding mode control (2-SMC) to regulate AMB systems throughout a wide operating speed range. The proposed controllers are composed of two components. The first component is a linear controller for the sake of stabilizing the inherently unstable system, while the second component is a 2-SMCto handle the model uncertainties of the system as well as the exogenous harmonic disturbances. Simulation andexperimental results are provided to demonstrate theeffectiveness and superiority of the proposed techniques compared to the conventional linear controller.
... Several examples of algorithms yielding to the finite time stabilization on the second order sliding set 52 = {x EX: s = s = O} can be found in Perruquetti and Barbot (2002), Chapter 3, or in Bartolini et al. (1997). The use of second order sliding mode algorithm is motivated here by the fact that it is robust with respect to a larger class of perturbations than with first order sliding mode. ...
... Am> Co AM > Am + 2Co Remark 4. Other second order sliding mode algorithms that do not require any information on the derivative of the sliding variable could be used here, as for instance the sub-optimal algorithm (Bartolini et al., 1997). ...
This paper is dedicated to the stabilization of the Heisenberg system with some additional integrators in the input path. This objective is achieved by the use of sliding mode control laws: the first one is based on the classical theory and requires the perturbations to be matching while the second one involves higher order sliding mode theory and is robust to a larger class of disturbances. Some simulations illustrate the given results.
... The SOSM technique generalizes the basic SMC design by integrating second order derivatives of the sliding variable [21]. A few of such controllers have been discussed in the literature [22][23][24][25]. ...
... On the other hand, to retain the main advantages of the usual method, they debate the chattering phenomenon and offer advanced precision in practice. In the last decade many research works have applied this type of control [22,23]. ...
This article present a novel direct torque control (DTC) scheme using high order sliding mode (HOSM) and fuzzy logic of a doubly fed induction generator (DFIG) incorporated in a wind turbine system. Conventional direct torque control strategy (C-DTC) using hysteresis controllers presents considerable flux and torque undulations at steady state period. In order to ensure a robust DTC method for the DFIG-rotor side converter and reduce flux and torque ripples, a second order sliding mode (SOSM) technique based on super twisting algorithm and fuzzy logic is used in this paper. Simulation results show the efficiency of the proposed method of control especially on the quality of the provided power comparatively to a C-DTC. Keywords—DFIG; wind turbine; DTC; SOSM; super twisting; fuzzy logic
... As a special case of HOSMC, second order sliding mode control was presented by G. Bartolini and P. Pydynowski [5][6][7][8][9], in which a continuous first-order estimator is acquired to replace the nonlinear dynamic function in the sliding mode firstly and then a second-order auxiliary system is obtained to design the time derivative of the control signal. Nevertheless, some differential inequalities with several conditions are required to be solved, which introduce difficulties for control design. ...
... Afterwards, the method was mended and the results were improved [6]. Sub-optimal variable structure control law is obtained based on the proposed results [7][8], which conduces to better performance of closed-loop systems. The same research about multi-input systems is discussed in [9]. ...
Chattering is one of the main shortcomings of sliding mode control. In this paper method using observer to designing continuous sliding mode control (CSMC) law for a class of nonlinear uncertain systems with uncertainty in control channel is addressed. By constructing a robust observer based on LMI method, variable structure control (VSC) law can be designed on the derivative of the control signal. Consequently chattering phenomenon is eliminated. Simulation results on a numerical example are reported to illustrate the system performance and the feasibility of the control algorithm.
... This type of motor has been neglected by researchers for several years because of its disadvantages. However, it has come back to the forefront because of the progression of the control techniques and the accessibility to its rotor [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. The operation of a variable speed motor needs some control techniques in order to obtain a high performance system. ...
The sliding mode control (SMC) methodology based on the theory of variable structure systems has been widely used for robust control of nonlinear systems. Nevertheless, this type of control has an essential inconvenience, which is the chattering phenomenon caused by the discontinuous control part. In order to reduce the effects of the chattering phenomenon, second order sliding mode (SOSM) seems to be a very attractive solution. To eliminate the remains of chattering phenomenon, a new control scheme based on fuzzy second order sliding mode control (FSOSMC) is proposed in this paper. This fuzzy second order sliding mode controller is destined to the speed control of doubly fed induction motor (DFIM) which is based on the decoupling control to enhance robustness under different operating conditions such as load torque and in the presence of parameters variation. The simulation results for various scenarios show the performances of the proposed control in terms of, precision, rapidity and stability for the high powers DFIM operating at variable speeds.
... where δ is an arbitrarily small time delay, and to adopt the algorithm presented in [BFU98a] which is represented in the Algorithm 1. ...
This thesis settles in the important and actual problem of generation, conversion, utilization and distribution of power in the electrical network of an aircraft. This application, rarely addressed in the system and control literature, naturally calls for two methodological pillars developed by control theorists, namely the Sliding Mode Control (SMC) and the Switched Systems (SS) theory. This thesis introduces the well-known notions of SMC and SS and provides the necessary theoretical knowledge to appreciate the novel contributions to these two topics.
With regards to the SMC, the preliminaries on First Order and Higher Order SMC are given to pave the way to a novel result, that is the design of a mechanism to encounter the actuators saturation limits based on the combination of generic r-order SMC techniques with the Bounded Integral Control algorithm.
Similarly, the SS framework is introduced by presenting the cases of state- and time-dependent switching, the possible solutions of switched systems and the stability properties characterizing switched systems. These notions are of fundamental importance to present the idea of stability of a switched system under slow switching, that occurs when the switching signal is characterized by a given dwell-time.
The original results in the SS framework are represented by the definition of novel stability notions, i.e. quasi-integral-Input-to-State Stability and integral-Input-to-State practical Stability under slow switching, and by the adoption of these two notions to derive sufficient dwell-time conditions to achieve the integral-Input-to-State Stability under slow switching.
The theoretical concepts from the SMC and SS frameworks are then adopted to address the problem of electric power management on-board the aircraft from the control point of view. The concept of the More Electric Aircraft is introduced together with its innovative ideas and challenges. One of these challenges is the usage of auxiliary power sources (such as batteries or supercapacitors) to be exploited in case of generator overload in order to prevent generator over-sizing and thus allowing for lower weight on-board and, consequently, lower fuel consumption. This problem is addressed through the adoption of bidirectional DC/DC converters, located between the main aeronautic generator and the auxiliary power source, capable of managing the power flow from one side to the other and vice versa. Stability of the considered electrical systems is proved through the SS framework. Several SMC algorithms, of first and higher order, are adopted to control the DC/DC converter to achieve the prescribed goal. Moreover, thorough stability proofs are provided to demonstrate the effectiveness of the proposed control strategies. Finally, the theoretical results are confirmed both in simulation and experimental setup.
... A possible solution to obtain a continuous control is to artificially increase the relative degree of the system and to resort to a Second Order Sliding Mode (SOSM). According [25], we can define the auxiliary system aṡ ...
... Similar control method for stabilization control based on second-order sliding mode control is applied to inverted pendulum system [5]. However, in [5] the twisting controller [6] and sub-optimal controller [7] are employed instead of super-twisting algorithm. ...
... Two-dimensional systems of first-order equations have several reallife applications in engineering. For example, Bartolini et al. [4][5][6] considered a secondorder system in order to control uncertain nonlinear systems by some control techniques, for example, sliding mode and approximate linearization. In addition to nonoscillatory solutions of second-order systems, periodic and subharmonic solutions were also considered in [7][8][9], and important results were obtained. ...
This paper deals with long-time behaviors of nonoscillatory solutions of a system of first-order dynamic equations on time scales. Some well-known fixed point theorems and double improper integrals are used to prove the main results.
... Newly developed, the SOSMC generalize the essential sliding mode scheme that acts on the second order time derivatives of the structure variation since the constraint instead of influencing the initial variation derivative as it occurs in usual sliding modes [14]. Some of such controllers were reported in the literature [15][16][17][18]. ...
Traditional filed oriented control strategy including proportional-integral (PI) regulator for the speed drive of the doubly fed induction motor (DFIM) have some drawbacks such as parameter tuning complications, mediocre dynamic performances and reduced robustness. Therefore, based on the analysis of the mathematical model of a DFIM supplied by two five-level SVPWM inverters, this paper proposes a new robust control scheme based on super twisting sliding mode and fuzzy logic. The conventional sliding mode control (SMC) has vast chattering effect on the electromagnetic torque developed by the DFIM. In order to resolve this problem, a second order sliding mode technique based on super twisting algorithm and fuzzy logic functions is employed. The validity of the employed approach was tested by using Matlab/Simulink software. Interesting simulation results were obtained and remarkable advantages of the proposed control scheme were exposed including simple design of the control system, reduced chattering as well as the other advantages.
The work is centered on developing a constructive method of synthesizing an output-feedback second-order sliding-mode (SOSM) controller. By leveraging a coordinate transformation and revamping the technique of adding a power integrator, a state-feedback SOSM controller is first constructed under the full-state measurement. In order to tackle the challenge posed by the unmeasurable first derivative of the sliding variable, a discontinuous observer with appropriately selected scaling gains is put forward to overcome the measurable lack. Through the interactive cooperation of the state-feedback SOSM controller and the discontinuous observer, an output-feedback SOSM controller is successfully designed without reliance upon the separation principle. The finite-time convergence of the whole system is rigorously validated by dint of the Lyapunov function-based analysis. The efficiency of the theoretical results is ultimately corroborated through a simulation study.
This work addresses the vehicle platooning problem on curved paths. The main contribution of the work is a suitable use of curvilinear coordinates so as to make longitudinal and lateral control able to satisfy practical platooning objectives. Specifically, the longitudinal control is designed to guarantee collision avoidance, which is possible thanks to an appropriate contraction property in the framework of sliding mode control. Notably, a suitable choice of the gains of the sliding surface is also able to satisfy acceleration/deceleration limits and string stability requirements, of practical relevance for platooning. The aforementioned curvilinear coordinates allow to decouple the lateral from the longitudinal dynamics: lateral control is then realized via anothe sliding surface designe dusing the vector field approach. Asymptotic convergence to the desired path is shown via Lyapunov techniques. Numerical experiments are carried out to validate the effectiveness of the proposed controller.
Contribution to the control of the dual open-end winding induction motor
Abstract: The work carried out in this thesis concerns the contribution to the control of the dual open-end winding induction machine. The main idea consists in the use of the dual star induction machine with open stator windings supplied by three-phase matrix converters. Our study began with a state of the art on multiphase machines, open winding topologies as well as matrix converters, by first presenting the different topologies and modulations proposed then by modeling the direct and indirect structure of these converters. In addition, the application of a vector modulation to the three matrix converter topologies, direct, indirect and indirect threelevels has been discussed. The decoupled vector modulation was also introduced, specifically geared towards the developed structure of the matrix converter, which presents two outputs to the inversion stage, making it well suited for open winding topology. The sliding mode control based on space vector modulation has been applied to the dual star induction machine with its two structures (conventional, open windings), accompanied by a comparison of the results obtained. Then two advanced nonlinear controls represented by higher order sliding mode control and backstepping control were introduced. These two control approaches have been developed with constant frequency switching based on the vector modulation technique for the dual open-end winding induction machine. This machine was fed in two different stages due to the use of two structures of the two-output indirect matrix converter, the first with two levels and the second with three levels. Our work ended with direct torque control associated with the vector modulation on which we related to ensure the drive of the dual open-end winding induction machine, this control strategy was improved by the introduction of the input-output feedback linearization technique. The results obtained showed the performance and efficiency of the proposed control approaches.
Keywords: dual open-end winding induction machine, matrix converter, vector modulation, direct torque control, input-output feedback linearization.
المساهمة في التحكم في محرك غير متزامن بلفات مزدوجة مفتوحة للجزء الثابت
ملخص: يتعلق العمل الذي تم تقدميه في هذه الأطروحة بالمساهمة في التحكم في آلة غير متزامنة ذات لفات الجزء الثابت المزدوجة المفتوحة. تتمثل الفكرة الرئيسية في استخدام الآلة غير المتزامنة ذات النجم المزدوج مع لفات الجزء الثابت المفتوحة التي يتم تغذيتها بواسطة محولات المصفوفة ثلاثية الطور. بدأت دراستنا بعرض موجز حول الآلات متعددة الأطوار، والطوبولوجيات ذات اللف المفتوح ، بالإضافة إلى حمولات المصفوفة من خلال البدء بتقديم الهياكل والتعديلات المختلفة ثم القيام بنمذجة الهيكل المباشر وغير المباشر لهذه المحولات بالإضافة إلى ذلك، تمت مناقشة تطبيق تعديل المتجه على البنيات الرئيسية الثالثة لمحول المصفوفة، المباشرة وغير المباشرة وغير المباشرة ذات ثالثة مستويات. تم أيضا التطرق الى تعديل المتجه المنفصل الموجه بشكل خاص حنو الهيكل المطور لمحلول المصفوفة، والذي يتكون من مخرجين في مرحلة الانعكاس، مما يجعله مناسب تماما لطبولوجية اللف المفتوح. تم تطبيق التحكم في الوضع الانزلاقي المستند إلى تعديل متجه الفضاء على الالة غير المتزامنة ذات النجمتين بهيكليها (التقليدي، اللف المفتوح(، مصحوبة بمقارنة مفصلة للنتائج التي تم الحصول عليها. بعد ذلك تم إدخال عنصري تحكم غير خطيين متطورين يمثلهما التحكم في الوضع الانزلاقي ذي الترتيب العالي والتحكم بالرجوع للخلف. تم تطوير هاتين الطريقتين للتحكم مع تردد تبديل ثابت على أساس تقنية تعديل المتجه للآلة غير المتزامنة ذات اللفات الثابتة المزدوجة المفتوحة والتي تمت تغذيتها على مرحلتين منفصلتين بسبب استخدام هيكلين لمحول المصفوفة غير المباشر ثنائي المخرجات، الأول بمستويين والثاني بثالثة مستويات. نهاية هذا العمل خصصت للتحكم المباشر في العزم المرتبط بتعديل المتجه الذي اعتمدنا عليه لضمان قيادة الآلة غير المتزامنة ذات اللفات الثابتة المزدوجة المفتوحة، وقد تم تحسين استراتيجية التحكم هذه من خلال إدخال تقنية خطية المدخلات والمخرجات بواسطة ر جوع الحالة. أظهرت النتائج التي تم الحصول عليها أداء وكفاءة أساليب التحكم المقترحة.
كلمات مفتاحية: آلة غير متزامنة مع لفات الجزء الثابت المزدوجة المفتوحة، محول المصفوفة، تعديل المتجه، التحكم المباشر في العزم، خطية المدخلات والمخرجات من خلال رجوع الحالة .
Contribution à la commande d'un moteur asynchrone à double enroulements statoriques ouverts
Résumé : Le travail effectué dans cette thèse porte sur la contribution à la commande d'une machine asynchrone à double enroulements statoriques ouverts. L'idée principale consiste à l'utilisation de la machine asynchrone double étoile avec des enroulements statoriques ouverts alimentée par des convertisseurs matriciels triphasées. Notre étude a commencé par un état de l’art sur les machines multiphasées, les topologies d'enroulements ouverts ainsi que les convertisseurs matriciels, en présentant d'abord les différentes topologies et modulations proposées puis en modélisant ensuite la structure directe et indirecte de ces convertisseurs. De plus, l'application de la modulation vectorielle aux trois topologies de convertisseurs matriciels, directe, indirecte et indirecte à trois niveaux a été discutée. Une modulation vectorielle découplée a été également introduite, spécifiquement orientée vers la structure développée du convertisseur matriciel, qui présente deux sorties à l'étage d'inversion, ce qui la rend bien adaptée à la topologie d'enroulement ouvert. La commande par mode glissant basées sur la modulation vectorielle d’espace a été appliquée à la machine asynchrone double étoile avec ses deux structures (conventionnelle, enroulements ouverts), accompagnée d'une comparaison des résultats obtenus. Ensuite, deux commandes non linéaires avancées représentées par le contrôle en mode glissant d'ordre supérieur et le contrôle backstepping ont été introduites. Ces deux approches de commande ont été développées avec une commutation à fréquence constante basée sur la technique de modulation vectorielle pour la machine asynchrone à double enroulements statoriques ouverts. Cette machine a été alimentée selon deux étapes différentes du fait de l'utilisation de deux structures du convertisseur matriciel indirect à deux sorties, la première à deux niveaux et la seconde à trois niveaux. Notre travail a fini par la commande directe du couple associée à la modulation vectorielle sur laquelle on s'est appuyé pour assurer l'entraînement de la machine asynchrone à double enroulements statoriques ouverts, cette stratégie de commande a été améliorée par l'introduction de la technique de la linéarisation en entrée-sortie par retour d’état. Les résultats obtenus ont montré la performance et l'efficacité des approches de commande proposées.
Mots clés : machine asynchrone à double enroulements statoriques ouverts, convertisseur matriciel, modulation vectorielle, commande directe du couple, linéarisation en entrée-sortie par retour d’état.
The work carried out in this thesis concerns the contribution to the control of the dual open-end winding induction machine. The main idea consists in the use of the dual star induction machine with open stator windings supplied by three-phase matrix converters. Our study began with a state of the art on multiphase machines, open winding topologies as well as matrix converters, by first presenting the different topologies and modulations proposed then by modeling the direct and indirect structure of these converters. In addition, the application of a vector modulation to the three matrix converter topologies, direct, indirect and indirect three-levels has been discussed. The decoupled vector modulation was also introduced, specifically geared towards the developed structure of the matrix converter, which presents two outputs to the inversion stage, making it well suited for open winding topology. The sliding mode control based on space vector modulation has been applied to the dual star induction machine with its two structures (conventional, open windings), accompanied by a comparison of the results obtained. Then two advanced nonlinear controls represented by higher order sliding mode control and backstepping control were introduced. These two control approaches have been developed with constant frequency switching based on the vector modulation technique for the dual open-end winding induction machine. This machine was fed in two different stages due to the use of two structures of the two-output indirect matrix converter, the first with two levels and the second with three levels. Our work ended with direct torque control associated with the vector modulation on which we related to ensure the drive of the dual open-end winding induction machine, this control strategy was improved by the introduction of the input-output feedback linearization technique. The results obtained showed the performance and efficiency of the proposed control approaches.
Model-based approach is the common technique followed in designing a feedback regulator for a controlled system. However, there are always discrepancies (mismatches) between the developed mathematical model and the actual plant. The sources of these uncertainties include variation in system parameters, unmodeled dynamics, system nonlinearities, external disturbances, and measurement noise. Designing a control law in the presence of system uncertainties is one of the challenges a control engineer has to face. Therefore, developing control methods that ensure the required performance in practice in spite of system uncertainties has been gaining much interest since the late 1970s and early 1980s (Safonov, 2012; Shtessel et al., 2014). Control techniques that address this problem are known as robust control methods such as ℋ∞ control (Zhou and Doyle, 1998; Skogestad and Postlethwaite, 2005), μ-synthesis (Zhou and Doyle, 1998; Skogestad and Postlethwaite, 2005), linear matrix inequalities (LMIs) based control (Boyd et al., 1994), quantitative feedback theory (QFT) (Horowitz, 1982), gain scheduling control (Khalil, 2002), linear parameter-varying (LPV) control (White et al., 2013), sliding mode control (SMC) (Shtessel et al., 2014), (Utkin et al., 2009), robust adaptive control (Slotine and Weiping, 1991; Ioannou and Sun, 2013), and passivity based control (Khalil, 2002), as well as intelligent control techniques such as artificial neural network (ANN) based control (Hagan et al., 2002; Hunt et al., 1992), and fuzzy logic control (FLC) (Lee, 1990).
This chapter introduces some fundamentals of SMC including SMC design methods and main approaches to alleviate or limit the chattering. Moreover, some second-order SMC algorithms and definitions are introduced. It also presents the basics for the theoretical results developed in the subsequent chapters.
Sliding mode control is a major robust control technique. Its improved version, higher order sliding mode control, is also developed to remove drawbacks of sliding mode control. The basic concepts of sliding mode, sliding mode control, higher order sliding mode control and discrete-time sliding mode are given in this chapter.
In order to get better dynamic responses of sliding mode control (SMC) for uncertain linear systems, a sliding mode controller based on state norm was obtained in this paper. It is shown that by constituting a new form of nonlinear controller including the quadratic form item, the system states can be driven to sliding mode surface prescribed beforehand in finite time, and then converges to origin along the sliding mode. It has been proved to satisfy the reachability condition of the sliding mode, and the closed-loop systems were asymptotically stable. A numerical simulation example contrasted the performance of the presented controller and other SMC controllers.
In this Chapter a combined discontinues sliding mode control (SMC) with feedback linearization control (FLC) applied for voltage source inverter-fed induction motor are presented. The FLC guarantees the exactly decoupling of the motor speed and rotor flux control. Thus this control method gives a possibility to get very good behavior in both dynamic and steady states. The SMC approach assures direct control of inverter legs and allows using a simple table instead of performing PWM online calculation. Moreover, the SMC is robust to drive uncertainties. The good behaviour of rotor flux and mechanical speed Sliding Mode Observers (SMO) is the important feature of the system. Therefore, the presented approaches are very useful in a variety of applications and, in particular, in drive systems, robotics and power electronics.
Free-flying robotic spacecraft play a significant role in the space industry today. Unlike ground-based robots, the manipulator motion in a space robot can cause undesirable disturbances to the spacecraft platform, causing its attitude to change, potentially disrupting communication and solar energy collection processes as a result. Thus, coordinated control of both the spacecraft attitude and the manipulator motion become essential for successful space operations. Though past research has developed dynamic models for spacecraft manipulators, the contribution of reaction wheels to the angular momentum of the entire system needs further consideration. This paper reformulates the dynamic equations of a free-flying space robot by taking the aspect of reaction wheels into account. A diagonalization method is introduced to decouple the highly nonlinear multi-input/multi-output system model. A novel adaptive variable structure control method is then applied to implement a robust coordination controller for the space robot subjected to system uncertainties. Simulations are carried out on a spacecraft model with a 3-degree-of-freedom manipulator mounted on it to demonstrate the robustness of the proposed approach. Comparisons with sliding mode control show that the new controller results in faster settling times, which leads to maintaining uninterrupted communication links and efficient solar energy harvesting. © Copyright 2016 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.
In this chapter, we analyze the sliding bifurcations that occur in the two-cell DC/DC buck converter controlled using a dynamic feedback controller, then we apply the sliding mode controller to the converter in order to inhibit bifurcations and chaotic behavior. We use a simplified discrete model to analyze the bifurcations in the two-cell converter, which can be regarded as a piecewise smooth nonlinear system with discontinuous iterated maps. Then, we give theoretical conditions of stability according to the parameters values of the dynamic feedback controller. The presence of discontinuities in the converter leads to several types of non-smooth bifurcations namely border collision bifurcation, degenerate flip bifurcation and sliding bifurcations such as switching-sliding, grazing-sliding and adding-sliding also called multi-sliding. Non-smooth bifurcations, and more particularly, sliding bifurcations are caused by structural changes in the system dynamics, then we apply the sliding mode controller which is a variable structure control system (VSCS) to avoid sliding modes in the DC/DC buck converter. Numerical simulations confirm the analytical results and explain the bifurcations and the strange phenomena encountered in the two-cell converter.
In this work, a dynamic feedback control law based on the well known “Twisting” algorithm is under study. Dynamic compensation is added to the Twisting algorithm, in order to use position feedback only, and keep properties such as finite time stability. Moreover, linear terms are considered to improve the robustness of the algorithm. In some mechanical applications, an observer or a differentiator design is required for control purposes when the whole state space is not available for measurement. An alternative solution for this problem is proposed: a finite time stable algorithm that uses dynamic position feedback. Indeed, this new proposal does not require to measure or estimate another signal but the position of the mechanical system. In the stability analysis, strict nonsmooth Lyapunov functions and homogeneity properties are used in order to show finite time stability and robustness. Based on the proposed algorithm, a control law for a one-link pendulum affected by Coulomb friction and bounded external perturbations is designed.
The main purpose is to present the actual development and experimental validation of SOSM controllers, previously reviewed and analysed in this book. As anticipated, the controllers are designed and implemented in the real FC-based generation workbench introduced and modelled in the previous chapter. Three control set-ups based onSuper-Twisting,Twisting andSub-Optimal algorithms are developed. Power conversion optimisation of the laboratory PEM fuel cell system is sought via oxygen stoichiometry regulation. In the implementation, the effect of practical problems such as saturation and possible wind-up are taken into account.
There is also a second intention for this chapter. Taking advantage of the consecutive steps required to present the development of the aforementioned actual FC controllers, it is aimed to provide a condensed recapitulation of the concepts and design procedures reviewed and elaborated along the previous chapters. The objective is twofold. On the one hand, this endows the chapter with a certain degree of self-containment, improving its readability by sparing the reader from awkward goings and comings. On the other hand, the goal is to summarise the successive stages of the FC-SOSM control design process presented in this book, into one concise format. This would offer a unified abridged design guideline to develop proficient practical SOSM controllers for efficiency optimisation of FC generation systems.
We consider a number of dynamical adaptive backstepping control algorithms for the class of observable non-minimum phase nonlinear continuous uncertain systems (triangular and non-triangular), as well as systems with disturbances which can be converted to the parametric semi-strict feedback form. Nonlinear, sliding and second-order sliding control laws are developed. Adaptive backstepping algorithms with tuning functions, and modular parameter identification approaches are presented. BACK, a Maple symbolic algebra package, has been developed as a tool for the design of dynamical adaptive nonlinear controllers for regulation and tracking tasks.
In this Chapter the theory of nonlinear control systems is shortly described. The nonlinear feedback linearization control (FLC) and discontinues sliding mode control (SMC) are presented. Moreover, several applications of nonlinear control methods for induction motor drive are shown. The FLC guarantees the exactly decoupling of the motor speed and rotor flux control. Thus this control method gives a possibility to get very good behavior in both dynamic and steady states. The SMC approach assures direct control of inverter legs and allows using a simple table instead of performing complicated PWM calculation. Moreover the SMC is robust to drive uncertainties. The good behaviour of rotor flux and mechanical speed Sliding Mode Observers (SMO) is the important feature of the system. Therefore, presented approaches are very useful in a variety of applications and, in particular, in the drive systems and power electronics.
In this article, a second-order sliding mode control (2-SMC) is proposed for second-order uncertain plants using equivalent control approach to improve performance of control systems. A Proportional+Integral (PI) sliding surface is defined for the sliding mode. The sliding mode control law is derived using direct Lyapunov stability approach and asymptotic stability is proved theoretically. The second-order plant parameters are experimentally determined using input-output measured data. Performance of the closed-loop system is analysed through an experimental application to an electromechanical plant to show feasibility and effectiveness of the proposed 2-SMC and factors involved in the design. Results of the experimental application are presented to make a quantitative comparison with the traditional (first-order) sliding mode control and PID control. It is demonstrated that the proposed 2-SMC system improves performance of the closed-loop system with better tracking specifications in the case of external disturbances, better behavior of the output and faster convergence of the sliding surface while maintaining the stability.
This paper presents the design of balance control for an inverted pendulum system. The system consists of a cart, linear motor and a pole. The pole is pivoted on the cart where the cart is actuated by a linear motor. The control objective is to stabilize the pole at its upright position. The second-order sliding mode control is used to design the stabilizing control law for the system. For comparison, we consider LQR (Linear Quadratic Regulator) control and first-order sliding mode control to stabilize the system. The dynamic model of the system is given. The linearized model of the system is used to design the LQR control law, while the nonlinear dynamic model is used to design first-order and second-order sliding mode control. The MATLAB/Simulink is used to conduct the simulation studies and the designed control laws are implemented on a digital signal processor. The effectiveness of the designed control laws is demonstrated in the simulations and experiments.
In this paper, an adaptive control scheme for maximum power point tracking of stand-alone PMSG wind turbine systems (WTS) is presented. A novel procedure to estimate the wind speed is derived. To achieve this, a neural network identifier (NNI) is designed in order to approximate the mechanical torque of the WTS. With this information, the wind speed is calculated based on the optimal mechanical torque point. The NNI approximates in real-time the mechanical torque signal and it does not need off-line training to get its optimal parameter values. In this way, it can really approximates any mechanical torque value with good accuracy. In order to regulate the rotor speed to the optimal speed value, a block-backstepping controller is derived. Uniform asymptotic stability of the tracking error origin is proved using Lyapunov arguments. Numerical simulations and comparisons with a standard passivity based controller are made in order to show the good performance of the proposed adaptive scheme.
In this paper a combined algorithm for the control of non-triangular nonlinear systems with unmatched uncertainties will be presented. It relies on a combination of Dynamical Adaptive Backstepping (DAB) and Sliding Mode Control (SMC) of the second order. The role of the sliding mode control is to replace the last step of the DAB procedure, a generalization of the backstepping technique, to obtain more robustness towards disturbances and unmodelled dynamics. The main advantages of the second order sliding mode algorithms are the prevention of chattering and a significant simplification of the control law.
The rotor speed estimation problem in electrical drives, based on shaft encoder measurements, has been dealt with by means of a digital approximate differentiator. The proposed differentiator is based on a digital VSC which is able to perform real 2-sliding modes, so that the observer discontinuous control is confined to the second total derivative of the available encoder signal estimate, and its first derivative is then continuous. Simulation and experimental results, conducted on a field oriented induction motor drive, show the good behaviour of the proposed approximate differentiator, which has been compared with the Matlab-Simulink differentiator algorithm by using a DSP-based system.
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.
• Higher order sliding mode definitions were formulated.
• It was shown that higher order sliding modes are natural phenomena for control systems with discontinuous controllers if the relative degree of the system is more than 1 or a dynamic actuator is present.
• A natural logic of actuator-like algorithm introduction was presented. Such algorithms also provide for the appearance of higher order sliding modes.
• Stability was studied of second order sliding modes in systems with fast stable dynamic actuators of relative degree 1.
• A number of examples of higher order sliding modes were listed. Among them the first example was presented of a third order sliding algorithm with finite time convergence. The discrete switching modification of this algorithm provides for the third order sliding precision with respect to the minimal switching time interval.
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.
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.
In this paper a new method for the solution of the asymptotic linearization of uncertain nonlinear systems by means of a continuous control law is presented. The main feature of this approach consists of a continuous first-order estimator and control laws with piecewise continuous derivatives. As an important by-product of this theory we can deal easily with the case of some uncertain systems which do not satisfy the matching condition, and with systems presenting first order unmodelled dynamics.
The design of multivariable adaptive model-following control systems is described using the theory of variable-structure systems (VSS). Both the model plant transient and the error response transient can be specified in advance. The discontinuous control enforces sliding motion on switching hyperplanes in the error state space, and the dynamic response is insensitive to certain parameter variations and disturbances. The design is straightforward and requires little computational effort.
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
Adaptive control of constrained robot manipulators', Academy of Science of Russia
- E Panteley
- A Stotsky
Panteley, E. and A. Stotsky, 'Adaptive control of constrained robot manipulators', Academy of Science of Russia, Institute of Problems of Mechanical Engineering, Preprint 117, Saint Petersburg, 1994.