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Journal ArticleDOI

Nonlinear control via approximate input-output linearization: the ball and beam example

TL;DR: In this paper, an approximate input-output linearization of nonlinear systems which fail to have a well defined relative degree is studied, and a method for constructing approximate systems that are input output linearizable is provided.
Abstract: Approximate input-output linearization of nonlinear systems which fail to have a well defined relative degree is studied. For such systems, a method for constructing approximate systems that are input-output linearizable is provided. The analysis presented is motivated through its application to a common undergraduate control laboratory experiment-the ball and beam-where it is shown to be more effective for trajectory tracking than the standard Jacobian linearization. >
Citations
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Journal ArticleDOI
TL;DR: Using the Stone-Weierstrass theorem, it is proved that linear combinations of the fuzzy basis functions are capable of uniformly approximating any real continuous function on a compact set to arbitrary accuracy.
Abstract: Fuzzy systems are represented as series expansions of fuzzy basis functions which are algebraic superpositions of fuzzy membership functions. Using the Stone-Weierstrass theorem, it is proved that linear combinations of the fuzzy basis functions are capable of uniformly approximating any real continuous function on a compact set to arbitrary accuracy. Based on the fuzzy basis function representations, an orthogonal least-squares (OLS) learning algorithm is developed for designing fuzzy systems based on given input-output pairs; then, the OLS algorithm is used to select significant fuzzy basis functions which are used to construct the final fuzzy system. The fuzzy basis function expansion is used to approximate a controller for the nonlinear ball and beam system, and the simulation results show that the control performance is improved by incorporating some common-sense fuzzy control rules. >

2,575 citations


Cites background or methods from "Nonlinear control via approximate i..."

  • ...But the controller of [ 5 ] requires the mathematical model of the system, whereas our method does not....

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  • ...Then, from [ 5 ], the system can be represented by the state-space model...

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  • ...We also simulated two extreme cases: (i) using the original controller of [ 5 ] and (ii) using only the pure linguistic controller fL(z) based on Rf-R;, for the same initial V. CONCLUSIONS...

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  • ...where the control U is the acceleration of 0, and the parameters B and G are defined in [ 5 ]....

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  • ...The input-output linearization algorithm of [ 5 ] determines the control law U(.) as follows: for state 2, compute ~(z) =...

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Journal ArticleDOI
01 Mar 1995
TL;DR: After synthesizing a FLS, it is demonstrated that it can be expressed mathematically as a linear combination of fuzzy basis functions, and is a nonlinear universal function approximator, a property that it shares with feedforward neural networks.
Abstract: A fuzzy logic system (FLS) is unique in that it is able to simultaneously handle numerical data and linguistic knowledge. It is a nonlinear mapping of an input data (feature) vector into a scalar output, i.e., it maps numbers into numbers. Fuzzy set theory and fuzzy logic establish the specifics of the nonlinear mapping. This tutorial paper provides a guided tour through those aspects of fuzzy sets and fuzzy logic that are necessary to synthesize an FLS. It does this by starting with crisp set theory and dual logic and demonstrating how both can be extended to their fuzzy counterparts. Because engineering systems are, for the most part, causal, we impose causality as a constraint on the development of the FLS. After synthesizing a FLS, we demonstrate that it can be expressed mathematically as a linear combination of fuzzy basis functions, and is a nonlinear universal function approximator, a property that it shares with feedforward neural networks. The fuzzy basis function expansion is very powerful because its basis functions can be derived from either numerical data or linguistic knowledge, both of which can be cast into the forms of IF-THEN rules. >

2,024 citations

Book
01 Jan 1996
TL;DR: This text is the first to combine the study of neural networks and fuzzy systems, their basics and their use, along with symbolic AI methods to build comprehensive artificial intelligence systems.
Abstract: From the Publisher: "Covering the latest issues and achievements, this well documented, precisely presented text is timely and suitable for graduate and upper undergraduate students in knowledge engineering, intelligent systems, AI, neural networks, fuzzy systems, and related areas. The author's goal is to explain the principles of neural networks and fuzzy systems and to demonstrate how they can be applied to building knowledge-based systems for problem solving. Especially useful are the comparisons between different techniques (AI rule-based methods, fuzzy methods, connectionist methods, hybrid systems) used to solve the same or similar problems." -- Anca Ralescu, Associate Professor of Computer Science, University of Cincinnati Neural networks and fuzzy systems are different approaches to introducing human-like reasoning into expert systems. This text is the first to combine the study of these two subjects, their basics and their use, along with symbolic AI methods to build comprehensive artificial intelligence systems. In a clear and accessible style, Kasabov describes rule- based and connectionist techniques and then their combinations, with fuzzy logic included, showing the application of the different techniques to a set of simple prototype problems, which makes comparisons possible. A particularly strong feature of the text is that it is filled with applications in engineering, business, and finance. AI problems that cover most of the application-oriented research in the field (pattern recognition, speech and image processing, classification, planning, optimization, prediction, control, decision making, and game simulations) are discussed and illustrated with concrete examples. Intended both as a text for advanced undergraduate and postgraduate students as well as a reference for researchers in the field of knowledge engineering, Foundations of Neural Networks, Fuzzy Systems, and Knowledge Engineering has chapters structured for various levels of teaching and includes original work by the author along with the classic material. Data sets for the examples in the book as well as an integrated software environment that can be used to solve the problems and do the exercises at the end of each chapter are available free through anonymous ftp.

977 citations


Cites methods from "Nonlinear control via approximate i..."

  • ...…prediction—Deboeck (1994); Kaufman (1987); Goonatilake and Kheball (1994) Control—Werbos (1992); Wang (1994); for a mathematical solution of the Ball and Beam Problem, see Hanser et al. (1992) Statistical methods—Metcalfe (1994) Generic and specific AI problems: Schwefel and Manner (1990) Page 75 2...

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  • ...Imagine a problem called the Ball and Beam Problem....

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  • ...Symbolic AI, expert systems—Giarratano and Riley (1989); Pratt (1994); Dean et al. (1995) Genetic algorithms—Holland (1992); Goldberg (1989); Davis (1991); Michaliewicz (1992); Fogel (1995) Pattern recognition—Weiss and Kulikowski (1991); Nigrin (1994); Pao (1989); Bezdek and Pal (1992) Speech and language processing—Morgan and Scofield (1992); Owens (1993); Kasabov and Watson (1994) Time-series prediction—Weigend and Gershenfeld (1993) Financial and business prediction—Deboeck (1994); Kaufman (1987); Goonatilake and Kheball (1994) Control—Werbos (1992); Wang (1994); for a mathematical solution of the Ball and Beam Problem, see Hanser et al. (1992) Statistical methods—Metcalfe (1994) Generic and specific AI problems: Schwefel and Manner (1990) Page 75 2...

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Journal ArticleDOI
TL;DR: This work describes a class of systems for which IDA-PBC yields a smooth asymptotically stabilizing controller with a guaranteed domain of attraction, given in terms of solvability of certain partial differential equations.
Abstract: We consider the application of a formulation of passivity-based control (PBC), known as interconnection and damping assignment (IDA) to the problem of stabilization of underactuated mechanical systems, which requires the modification of both the potential and the kinetic energies. Our main contribution is the characterization of a class of systems for which IDA-PBC yields a smooth asymptotically stabilizing controller with a guaranteed domain of attraction. The class is given in terms of solvability of certain partial differential equations. One important feature of IDA-PBC, stemming from its Hamiltonian formulation, is that it provides new degrees of freedom for the solution of these equations. Using this additional freedom, we are able to show that the method of "controlled Lagrangians"-in its original formulation-may be viewed as a special case of our approach. As illustrations we design asymptotically stabilizing IDA-PBCs for the classical ball and beam system and a novel inertia wheel pendulum.

803 citations


Cites background from "Nonlinear control via approximate i..."

  • ...We refer the reader to [14] for further details on the model....

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Journal ArticleDOI
TL;DR: It is shown that the performance of a globally bounded partial state feedback control of an input-output linearizable system can be recovered by a sufficiently fast high-gain observer.
Abstract: It is shown that the performance of a globally bounded partial state feedback control of a certain class of nonlinear systems can be recovered by a sufficiently fast high-gain observer. The performance recovery includes recovery of asymptotic stability of the origin, the region of attraction, and trajectories.

655 citations


Cites background from "Nonlinear control via approximate i..."

  • ...Examples of such models can be found in [3], [ 5 ], [11], [17], [19], and [26]....

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  • ...In [ 5 ] and [26], the models given of the ball and beam system fit in the form of (1)‐(4)....

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References
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Book
01 Jan 1985
TL;DR: In this paper, a systematic feedback design theory for solving the problems of asymptotic tracking and disturbance rejection for linear distributed parameter systems is presented, which is intended to support the development of flight controllers for increasing the high angle of attack or high agility capabilities of existing and future generations of aircraft.
Abstract: : The principal goal of this three years research effort was to enhance the research base which would support efforts to systematically control, or take advantage of, dominant nonlinear or distributed parameter effects in the evolution of complex dynamical systems. Such an enhancement is intended to support the development of flight controllers for increasing the high angle of attack or high agility capabilities of existing and future generations of aircraft and missiles. The principal investigating team has succeeded in the development of a systematic methodology for designing feedback control laws solving the problems of asymptotic tracking and disturbance rejection for nonlinear systems with unknown, or uncertain, real parameters. Another successful research project was the development of a systematic feedback design theory for solving the problems of asymptotic tracking and disturbance rejection for linear distributed parameter systems. The technical details which needed to be overcome are discussed more fully in this final report.

8,525 citations

Book
01 Jan 1967

2,437 citations

Book
24 Apr 1986
TL;DR: This chapter discusses the development of Geometric Theory of State Feedback for Multi-Input Multi-Output Systems and its applications in control systems.
Abstract: Contents: Local Decompositions of Control Systems.- Global Decompositions of Control Systems.- Input-Output Maps and Realization Theory.- Elementary Theory of Nonlinear Feedback for Single-Input Single-Output Systems.- Elementary Theory of Nonlinear Feedback for Multi-Input Multi-Output Systems.- Geometric Theory of State Feedback: Tools.- Geometric Theory of State Feedback: Applications.- Appendix A.- Appendix B.- Bibliographical Notes.- References.- Subject Index.

1,696 citations

Book ChapterDOI
TL;DR: In this paper, the problem of controlling a fixed nonlinear plant in order to have its output track (or reject) a family of reference (or disturbance) signal produced by some external generator is discussed.
Abstract: The problem of controlling a fixed nonlinear plant in order to have its output track (or reject) a family of reference (or disturbance) signal produced by some external generator is discussed. It is shown that, under standard assumptions, this problem is solvable if and only if a certain nonlinear partial differential equation is solvable. Once a solution of this equation is available, a feedback law which solves the problem can easily be constructed. The theory developed incorporates previously published results established for linear systems. >

1,639 citations

Journal ArticleDOI
TL;DR: In this paper, a technique for constructing a transformation under the assumption that {g\ldot[f\dotg],...,(adn-1}f\ldotsg)} span an n -dimensional space and that the set is an involutive set.
Abstract: Recent results have established necessary and sufficient conditions for a nonlinear system of the form \dot{x}(t) = f(x(t))-u(t)g(x(t)) . with f(0) = 0 , to be locally equivalent in a neighborhood of the origin in Rnto a controllable linear system. We combine these results with several versions of the global inverse function theorem to prove sufficient conditions for the transformation of a nonlinear system to a linear system. In doing so we introduce a technique for constructing a transformation under the assumptions that {g\ldot[f\dotg],...,(ad^{n-1}f\ldotg)} span an n -dimensional space and that {g\ldot[f\ldot g],...,(ad^{n-2}f\ldotg)} is an involutive set.

592 citations