Brief PaperStabilization of compact sets for passive affine nonlinear systems☆
Introduction
This paper concerns determination of state feedback regulators for a nonlinear controlled system rendering a goal invariant set of the unforced nonlinear system as a local or global attractor of the closed-loop system. More precisely, we will consider an affine in control nonlinear systemwhere , are a state and a control input, respectively. The maps , are smooth vector functions of an appropriate dimensions.
It is assumed that the goal attractor of the closed-loop system coincides with a set of zeros of a known smooth scalar nonnegative function . Introduce dummy outputs of system (1)where and is a smooth vector function. The problem under consideration is to determine a state feedback regulator providing the propertywhere is a solution of the closed-loop system with initial conditions belonging to some prescribed set .
Throughout this paper it will be assumed that
For a positive-definite function the stabilization problem – is reduced to stabilization of the equilibrium point for a passive affine nonlinear system. This special stabilization problem has been widely investigated during the last two decades. The main contributions to its solution were made by Hill and Moylan (1976), Jurdjevic and Quinn (1978), Fradkov (1979), Kalouptsidis and Tsinias (1984), Fradkov (1986), Lee and Araposthatis (1988), Saberi, Kokotovic and Sussmann (1990), Ortega (1991), Byrnes, Isidori and Willems (1991) and others.
The paper (Byrnes et al., 1991) summarized and essentially improved the previous investigations. Let us briefly recall the main scheme of the global (local) stabilization solution presented in this paper: It was assumed that system , is passive with a -smooth storage function , , which is positive definite and proper (positive definite), and that , is (locally) zero-state detectable. Then by Theorem (Byrnes et al., 1991) for any smooth function , such that and for each nonzero , the control lawglobally (locally) asymptotically stabilize the equilibrium of system (1).
Moreover, Proposition (Byrnes et al., 1991) provides a criterion guaranteeing that system , is zero-state detectable. This criterion has the form: Introduce the setswhere is a distribution which is defined by the relationand , are the vector components of the smooth map . If system , is passive with -smooth positive-definite storage function and then system , is zero-state detectable.
This paper owes its origin from the results of Andrievskii, Guzenko and Fradkov (1996), Fradkov (1996), and Åström and Furuta (1996), where successful attempts to take advantage of a passification approach to the stabilization of sets were made. Such problems naturally appear, for example, for Hamiltonian-controlled nonlinear systems when the objective is to stabilize a set corresponding to a desired value of the Hamiltonian function. Compared to the results of Byrnes et al. (1991), for such examples appropriate smooth storage functions are only nonnegative.
The main contribution of this paper is twofold. First, an extension of the stabilization conditions summarized in Byrnes et al. (1991) to the case of a nonnegative storage function is developed; second, it is shown that the zero-state detectability criterion of system , given in Byrnes et al. (1991) under the additional assumption (A3) has a natural extension, and it can be reformulated in a simple equivalent form. Moreover, a new test for detectability of the zero set of the nonnegative smooth storage function, which generalizes the criterion developed in Byrnes et al. (1991), is proposed. This paper also continues the investigations (Fradkov, Makarov, Shiriaev & Tomchina, 1997; Shiriaev 1997, Shiriaev 1998), where some special cases of the stabilization problem – were considered.
Section snippets
Example: energy level stabilization of pendulum
To illustrate the problem stated above consider as an example a pendulum with a controlled suspension point which may perform motions along a straight line having the given angle with the horizontal (see Fig. 1). The motions of that pendulum are described by the equationsHere and are the generalized coordinate and moment, and are the mass and length of the pendulum, is the gravity acceleration and the control law is the
Main results
To solve stabilization problem , , , we seek sufficient conditions guaranteeing asymptotic stabilization of the goal set by a state feedback control of the formwhere is the full derivative of along the solution of (1), and is an arbitrary smooth vector function forming sharp angle with , i.e. for all and . In terms of the coefficients of system , , the control law (12) can be rewritten as follows:
A new notion of detectability
Definition 7 Let be a nonnegative function on . The system , is said to be locally -detectable if there exists a constant such that for any the solutionof the unforced systemsatisfies the implicationIf (21) is valid for then , is -detectable. If the function is positive definite and , is (locally) -detectable then , is (locally) zero-state detectable.
It is worth mentioning that Theorem 2 implicitly contains the test of
Conclusions
In this paper, the stabilization problem of an invariant compact set of an affine nonlinear system is considered. It is assumed that a smooth nonnegative storage function is given in the state space of the system. The problem under consideration is to define the state feedback regulator rendering the set of zeros of the given storage function to be locally or globally attractive. In particular, these settings include the problems of the local or global equilibrium point stabilization of passive
Acknowledgements
The author would like to thank Prof. A.L. Fradkov for helpful discussions, H. Ludvigsen and B. Vik for their significant help during preparation of the paper and the referees for their helpful suggestions and comments.
Anton S. Shiriaev was born in Kirov, Russia, in 1970. He received M.S. and Ph.D. degrees from the St. Petersburg State University, Russia, in 1993 and 1997, respectively, both in Applied Mathematics. He was a research fellow at the Institute for Problems of Mechanical Engineering, Russian Academy of Sciences, St. Petersburg in 1997 and a postdoctoral fellow at the Norwegian Institute of Science and Technology from February 1998 to December 1999. Since January 2000 he is appointed as an
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Anton S. Shiriaev was born in Kirov, Russia, in 1970. He received M.S. and Ph.D. degrees from the St. Petersburg State University, Russia, in 1993 and 1997, respectively, both in Applied Mathematics. He was a research fellow at the Institute for Problems of Mechanical Engineering, Russian Academy of Sciences, St. Petersburg in 1997 and a postdoctoral fellow at the Norwegian Institute of Science and Technology from February 1998 to December 1999. Since January 2000 he is appointed as an assistance professor at the The Maersk Mc-Kinney Moller Institute for Production Technology, University of Southern Denmark. His research interests in systems and control theory are related to stability and stabilization of nonlinear systems with emphasis to frequency-domain theory and to systems with conserved quantities. He has also been interested in special linear quadratic optimization problems such as existence of linear optimal ‘universal’ regulators when a system is subjected either external harmonic disturbances with known frequencies and unknown amplitudes or singular stochastic disturbances with known upper bound for spectral densities.
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This paper was not presemted at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor H. Nijmeijer under the direction of Editor T. Basar. This paper was written while the author was on the Institute for Problems of Mechanical Engineering, RAS, St. Petersburg, Russia, and on the Norwegian University of Science & Technology, and it was supported by the Russian Federal Programme “Integration” (project A0151) and the Norwegian Research Council.