On the generalized PI sliding mode control of DC-to-DC power converters: A tutorial

A tutorial revisiting of the traditional sliding mode control of DC-to-DC power supplies, or power converters, of the 'buck','boost' and 'buck-boost' types is presented. The limitations of the traditional indirect sliding mode controller schemes are critically evaluated. These refer to the availability of the converter inductor current variable and a lack of robustness with respect to unmodelled load resistance variations. These limitations are overcome thanks to recent developments addressed as 'Generalized PI controllers' which are based on integral reconstructors of the unmeasured observable state variables and utilize only available input and output signals.     

On the stability of discrete-time sliding mode control systems

The stability of discrete-time sliding mode control systems is investigated and a new sliding mode condition is suggested. It is shown that the control must have upper and lower bounds. A numerical example is discussed as an illustration.     

Exact linearization in switched-mode DC-to-DC power converters

Three popular DC-to-DC switched mode power converters—Buck, Boost, and Buck-Boost—are shown to be in the same orbit of structural equivalence as a second-order controllable circuit in Brunovsky canonical form. The equivalence is achievable under non-linear feedback and explicit local difleomorphic state coordinate transformations. The implications of local exact linearizability of this class of systems in the design of Variable Structure Feedback Control laws or Pulse-Width-Modulation strategies is thoroughly discussed.     

Sliding-mode control on slow manifolds of DC-to-DC power converters

In this article, general relationships are established among variable structure control strategies and pulse-width modulated control schemes leading to sliding modes in non-linear systems. The results are applied to the design of a sliding regime for the regulation of DC-to-DC switchmode power converters exhibiting a slow integral manifold due to time-scale separation properties among input and output circuits.     

On stability of periodic motions in control systems with first kind pulse-width modulation

We study the stability of the periodic motions for systems with pulse modulated control of the first kind, that are described by differential equations with discontinuous right-hand sides. As an example we consider a control system with pulse-width modulation of the first kind.     

Slow periodic motions with internal sliding modes in variable structure systems

Singularly perturbed relay systems (SPRS) in which the reduced systems have the stable periodic motions with internal sliding modes are studied. The slow motion integral manifold of such systems consists of the parts which correspond to the different values of relay control and the solutions may contain the jumps from one part of the slow manifold to another. For such systems a theorem about existence and stability of the periodic solutions is proved. An algorithm of asymptotic representation for this periodic solutions using boundary layer method is presented. It is proved that in the neighbourhood of the break away point the asymptotic representation starts with the first order boundary layer function.     

On the estimation of sliding domains and stability regions of variable structure control systems with bounded controllers

This article examines the problem of the control of a class of multivariable linear time-invariant uncertain dynamic systems with bounded controllers using the variable structure control (VSC) approach and the second method of Lyapunov. A special coordinate transformation is utilized to facilitate the analysis. Sliding domains and estimates of the domain of attraction along with the regions of asymptotic stability (RAS) are obtained with the aid of Lyapunov-type arguments. Numerical examples are given to illustrate the approach developed in the article.Dedicated to the memory of Professor Janislaw M. Skowronski     

Equations of perturbed motions and stability of state and semi-state systems

Problems in the construction of equations of perturbed motions for state and semi-state systems are discussed. It is shown that in most cases it is not possible to construct the equations of the perturbed motions, so that the classical approach to the stability analysis of a selected motion is not applicable. The necessary and sufficient stability conditions that provide the testing of stability of an arbitrary system motion, without the utilization of the equations of perturbed motions, are developed. The results obtained concern the original Lyapunov stability definition and contain as special cases some other existing stability results. Two examples are worked out: one of them concerns the analysis of a non-linear feedback system that does not possess a state model.     

Estimating regions of asymptotic stability with sliding for relay-control systems

The object of this article is the estimation of stability boundaries and regions of asymptotic stability with sliding for a class of single-input relay-control systems. The direct method of Lyapunov is used to obtain these estimates. A coordinate transformation that brings the system into a special canonical form is utilized to facilitate the stability analysis. The proposed approach to stability regions estimation is applied to a class of second-order systems, and analytical expressions for stability regions are derived.Recommended by K. Mizukami     

On robust stability of the systems

In this paper, to check robust stability for higher order interval systems (n ⩾ 5), a step-by-step procedure is presented using simple conditions, on the basis of Routh criterion. In this, it is pointed out that there is no need to apply Routh criterion to all the four Kharitonov’s polynomials in some class of control system problems, and hence reduces the computational cost. Numerical examples illustrate the procedure. Recommended by Editorial Board member Somanath Majhi under the direction of Editor Jae Weon Choi. Yogesh V. Hote received the B.E. degree in Electrical Engineering from Govt. college of Engineering, Amravati, in 1998. Then, he received the M.E. degree in Control Systems, from Govt. college of Engineering, Pune, in 2000. Since 2001, he has been associated with the Netaji Subhas Institute of Technology (NSIT), Delhi University, New Delhi. Currently, he is holding the post of Sr. Lecturer, in Instrumentation and Control Department. His field of research includes robust control, robotics, numerical analysis and power electronics. D. Roy Choudhury received the B.Tech. and M.Tech degrees in Radio Physics and Electronics from the Institute of Radio Physics and Electronics, University of Calcutta, Calcutta in 1965 and 1966 respectively. He has been awarded the degree of Doctor of Philosophy from the same university in 1971. From 1971 to 1973, he was associated with the Institute de Reglage Automatique, EPFL, Switzerland. Since 1974 he has been associated with Delhi college of Engineering, Delhi. Currently, he is holding the post of Professor in Computer Science Department, I. P. University, Delhi. His field of research includes control systems, digital communications and biomedical engineering. J. R. P. Gupta received the B.Sc (Engg.) degree in Electrical Engineering from Muzaffarpur Institute of Technology, Muzaffarpur and the Ph.D. degree from University of Bihar in 1972 and 1983 respectively. After serving Post and Telegraph Department, Government of India for nearly three years, he joined Muzaffarpur Institute of Technology (MIT) as Assistant professor in Electrical Engineering Department in 1976. He then switched over to Regional Institute of Technology, Jamshedpur in 1986 and then to Netaji Subhas Institute of Technology (NSIT), New Delhi in 1994 where currently he is holding the post of Professor and Head of Department, Instrumentation and control Engineering, University of Delhi. His research interests include power electronics, electrical drives, control theory. He has been awarded K.S. Krishnan memorial award for the best system oriented paper by Institute of Electronics and Telecommunication Engineers (India), in 2008.     

On the stability of fuzzy systems

Studies the global asymptotic stability of a class of fuzzy systems. It demonstrates the equivalence of stability properties of fuzzy systems and linear time invariant (LTI) switching systems. A necessary and sufficient condition for the stability of such systems are given, and it is shown that under the sufficient condition, a common Lyapunov function exists for the LTI subsystems. A particular case when the system matrices can be simultaneously transformed to normal matrices is shown to correspond to the existence of a common quadratic Lyapunov function. A constructive procedure to check the possibility of simultaneous transformation to normal matrices is provided     

On the stability of nonautonomous systems

In Kalitine (RAIRO Automatique/Systems Anal. Control 16(3) (1982) 275) the use of semi-definite Lyapunov functions for exploring the local stability of autonomous dynamical systems has been introduced. In this paper, we give an extension of the results of Kalitine (1982) that allows to study the local stability of nonautonomous differential systems. We give an application to the algebraic Riccati equation.     

On the stability of strategies in competitive systems

This work discusses in some details the mathematical properties of competitive systems. It is demonstrated how an optimal strategy of any dynamic matrix game can be derived analytically. The number of various pure strategies contributing to the optimal strategy is found by analysing the properties of the gain matrix. A relationship between the stability and fitness of equilibrium states is established. It is shown that the fitness of the system can be expressed in terms of the eigenvalue spectrum of the system's stability matrix. The methods developed are applied to a few examples.     

On the stabilization of nonlinear systems via input-dependent sliding surfaces

This article proposes the use of sliding regimes, defined on a suitable input-dependent sliding surface, as a means of making more robust any model-based smooth feedback control scheme designed for the stabilization of a general nonlinear plant. The approach naturally produces a, robust, ‘outer loop’ redundant discontinuous feedback scheme, with several advantageous properties regarding insensitivity to external perturbation signals, modelling errors and sudden failure of the smooth portion of the feedback loop.     

On the robust design of sliding observers for linear systems

The restrictive character of well-known structural constraints, related to the matching conditions, for the sliding mode feedforward state reconstruction problem in linear, time-invariant, perturbed systems is critically reexamined from a new perspective. It is shown that, in generalized state space coordinates, a matched canonical state space realization exists which always allows discontinuous asymptotic stabilization of the observation error dynamics. The well-known structural conditions thus become largely irrelevant and robust asymptotic state estimation is shown to be feasible, for any perturbed observable system, by means of sliding mode observers.     

On periodic motions in relay systems containing blocks with limiters

A method of studying periodic motions in relay systems having a block with limiters is proposed. The method is based on the phase locus of the relay system. The means for constructing the phase locus, for determining with its help periodic motions arising in the system, and for studying their stability are considered.     

On the stability of delay-differential systems in the behavioralframework

In the behavioral approach, a linear time-invariant (LTI) delay-differential system is naturally introduced by Glusing-Luerssen (1997) as a continuous-time system whose dynamics are governed by a set of delay-differential equations, involving both painted and distributed delay operators. For this class of systems the notion of autonomy is introduced and characterized. Finally, asymptotic stability of autonomous delay-differential systems is investigated     

Buck型变换器非奇异固定时间滑模控制

针对存在参数不确定性的Buck变换器系统,提出一种非奇异固定时间滑模控制方法。首先,设计非奇异固定时间滑模面,并基于该滑模面设计固定时间控制器,保证系统输出电压误差在固定时间内收敛到平衡点的邻域内,且其收敛时间上界与系统初始状态无关。其次,设计自适应律估计系统不确定干扰上界,有效抑制不确定干扰对系统的影响,该方法无需干扰上界的先验知识。最后,仿真结果验证了所提方法的有效性。     

On the stability of linear discrete-time systems

In Anderson a connection of two types of stability: Liapunov stability, reflecting the internal stability of a system, and bounded-input bounded-output stability, reflecting the external stability of a system, has been established for time-varying linear finite-dimensional dynamic systems which may be represented in a special canonical form. This correspondence extends the connection to time-varying linear finite-dimensional discrete-time systems.     

On the componentwise stability of linear systems

The componentwise asymptotic stability (CWAS) and componentwise exponential asymptotic stability (CWEAS) represent stronger types of asymptotic stability, which were first defined for symmetrical bounds constraining the flow of the state‐space trajectories, and then, were generalized for arbitrary bounds, not necessarily symmetrical. Our paper explores the links between the symmetrical and the general case, proving that the former contains all the information requested by the characterization of the CWAS/CWEAS as qualitative properties. Complementary to the previous approaches to CWAS/CWEAS that were based on the construction of special operators, we incorporate the flow‐invariance condition into the classical framework of stability analysis. Consequently, we show that the componentwise stability can be investigated by using the operator defining the system dynamics, as well as the standard ε?δ formalism. Although this paper explicitly refers only to continuous‐time linear systems, the key elements of our work also apply, mutatis mutandis, to discrete‐time linear systems. Copyright © 2004 John Wiley & Sons, Ltd.