Variable structure systems with sliding modes
TL;DR: Design and analysis forVariable structure systems are surveyed in this paper and it is shown that advantageous properties result from changing structures according to this switching logic.
Abstract: Variable structure systems consist of a set of continuous subsystems together with suitable switching logic. Advantageous properties result from changing structures according to this switching logic. Design and analysis for this class of systems are surveyed in this paper.
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TL;DR: Bendixson's theorem is extended to the case of Lipschitz continuous vector fields, allowing limit cycle analysis of a class of "continuous switched" systems.
Abstract: We introduce some analysis tools for switched and hybrid systems. We first present work on stability analysis. We introduce multiple Lyapunov functions as a tool for analyzing Lyapunov stability and use iterated function systems theory as a tool for Lagrange stability. We also discuss the case where the switched systems are indexed by an arbitrary compact set. Finally, we extend Bendixson's theorem to the case of Lipschitz continuous vector fields, allowing limit cycle analysis of a class of "continuous switched" systems.
3,289 citations
Cites background from "Variable structure systems with sli..."
...” Similar examples showing stable systems constructed from unstable ones appear in [16]; Åström had qualitatively similar ones in [17]....
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TL;DR: A tutorial account of variable structure control with sliding mode is presented, introducing in a concise manner the fundamental theory, main results, and practical applications of this powerful control system design approach.
Abstract: A tutorial account of variable structure control with sliding mode is presented. The purpose is to introduce in a concise manner the fundamental theory, main results, and practical applications of this powerful control system design approach. This approach is particularly attractive for the control of nonlinear systems. Prominent characteristics such as invariance, robustness, order reduction, and control chattering are discussed in detail. Methods for coping with chattering are presented. Both linear and nonlinear systems are considered. Future research areas are suggested and an extensive list of references is included. >
2,884 citations
TL;DR: It turns out that the deviation of the system from its prescribed constraints (sliding accuracy) is proportional to the switching time delay and a new class of sliding modes and algorithms is presented and the concept of sliding mode order is introduced.
Abstract: 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.
2,714 citations
TL;DR: This survey is the first to bring to the attention of the controls community the important contributions from the tribology, lubrication and physics literatures, and provides a set of models and tools for friction compensation which will be of value to both research and application engineers.
2,658 citations
TL;DR: A direct adaptive tracking control architecture is proposed and evaluated for a class of continuous-time nonlinear dynamic systems for which an explicit linear parameterization of the uncertainty in the dynamics is either unknown or impossible.
Abstract: A direct adaptive tracking control architecture is proposed and evaluated for a class of continuous-time nonlinear dynamic systems for which an explicit linear parameterization of the uncertainty in the dynamics is either unknown or impossible. The architecture uses a network of Gaussian radial basis functions to adaptively compensate for the plant nonlinearities. Under mild assumptions about the degree of smoothness exhibit by the nonlinear functions, the algorithm is proven to be globally stable, with tracking errors converging to a neighborhood of zero. A constructive procedure is detailed, which directly translates the assumed smoothness properties of the nonlinearities involved into a specification of the network required to represent the plant to a chosen degree of accuracy. A stable weight adjustment mechanism is determined using Lyapunov theory. The network construction and performance of the resulting controller are illustrated through simulations with example systems. >
2,254 citations
References
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TL;DR: In this paper, the authors discuss the possibility of lowering a system sensitivity by the application of a sliding mode, which can be obtained by means of control functions which on certain hypersurfaces have first order discontinuities in a state space.
832 citations
TL;DR: The algorithm presented in this paper solves the non-linear programming problem by transforming it into a sequence of unconstrained maximization problems and is shown to be a dual feasible method.
Abstract: The non-linear programming problem seeks to maximize a function f(x) where the n component vector x must satisfy certain constraints gi(x) = 0, i = 1, …, m1 and gi(z) ≧ 0, i = m1 + 1, …, m. The algorithm presented in this paper solves the non-linear programming problem by transforming it into a sequence of unconstrained maximization problems. Essentially, a penalty is imposed whenever x does not satisfy the constraints. Although the algorithm appears most useful in the concave case, the convergence proof holds for non-concave functions as well. The algorithm is especially interesting in the concave case because the programming problem reduces to a single unconstrained maximization problem or, at most, to a finite sequence of unconstrained maximization problems. In addition, the paper presents a new class of dual problems, and the algorithm is shown to be a dual feasible method. Another property of the algorithm is that it appears particularly well suited for large-scale problems with a sizable number of c...
454 citations
TL;DR: In this article, necessary and sufficient conditions are derived for a minimal order linear time-invariant differential feedback control system to exist for a linear time invariant multivariable system with unmeasurable arbitrary disturbances of a given class occurring in it.
Abstract: Necessary and sufficient conditions are derived for a minimal order linear time-invariant differential feedback control system to exist for a linear time-invariant multivariable system with unmeasurable arbitrary disturbances of a given class occurring in it, such that the outputs of the system asymptotically become equal to preassigned functions of a given class of outputs, independent of the disturbances occurring in the system, and such that the closed-loop system is controllable. The feedback gains of the control system are obtained so that the dynamic behavior of the closed-loop system is specified by using either an integral quadratic optimal control approach or a pole assignment approach. The result may be interpreted as being a generalization of the single-input, single-output servomechanism problem to multivariable systems or as being a solution to the asymptotic decoupling problem.
241 citations
01 Jan 1975
217 citations
TL;DR: In this paper, the authors generalized the results obtained in [1] to accommodate the case of unmeasurable disturbances, which are known only to satisfy a given ρ th-degree linear differential equation.
Abstract: In a previous paper [1], the conventional optimal linear regulator theory was extended to accommodate the case of external input disturbances \omega(t) which are not directly measurable but which can be assumed to satisfy d^{m+1}\omega(t)/dt^{m+1} = 0 , i.e., represented as m th-degree polynomials in time t with unknown coefficients. In this way, the optimal controller u^{0}(t) was obtained as the sum of: 1) a linear combination of the state variables x_{i}, i = 1,2,...,n , plus 2) a linear combination of the first (m + 1) time integrals of certain other linear combinations of the state variables. In the present paper, the results obtained in [1] are generalized to accommodate the case of unmeasurable disturbances \omega(t) which are known only to satisfy a given \rho th-degree linear differential equation D: d^{\rho}\omega(t)/dt^{\rho} + \beta_{\rho}d^{\rho-1}\omega(t)/dt^{\rho-1}+...+\beta_{2}d\omega/dt + \beta_{1}\omega=0 where the coefficients \beta_{i}, i = 1,...,\rho , are known. By this means, a dynamical feedback controller is derived which will consistently maintain state regulation x(t) \approx 0 in the face of any and every external disturbance function \omega(t) which satisfies the given differential equation D -even steady-state periodic or unstable functions \omega(t) . An essentially different method of deriving this result, based on stabilization theory, is also described, In each cases the results are extended to the case of vector control and vector disturbance.
117 citations