Elsevier

Automatica

Volume 139, May 2022, 110189
Automatica

Brief paper
Orbital decompositions and integrable pseudosymmetries of control systems

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Abstract

Local decompositions of nonlinear dynamical control systems are investigated. We use orbital equivalences to transform systems into decomposed forms. In the language of infinite jets, every control system is interpreted as an infinite dimensional manifold. Each orbital decomposition of the system defines an invariant distribution on this manifold. A local basis of this distribution is an integrable pseudosymmetry of the system. An algorithm for constructing orbital decompositions is given and demonstrated by the example of the Kapitsa pendulum system. It is also shown how to use orbital decompositions to plan trajectories.

Introduction

We study the problem of constructing local decompositions of control systems. In difference from well-known classical works (Fliess, 1985, Grizzle and Marcus, 1985, Hirshorn, 1982, Isidori, 1995, Isidori et al., 1981, Krener, 1977, Nijmeijer, 1983, Nijmeijer and van der Schaft, 1985, Respondek, 1982) we use orbital equivalences (Fliess et al., 1999) for transforming systems to decomposed forms. This is reason why the decompositions examined below are called orbital.

Since we use a wider group of transformations, we get a wider class of decomposable forms for a given system. Therefore, orbital decompositions will have wider application if one learns to construct them. At the moment, we can mention only two control problems that were investigated using decomposition: the construction of flat outputs (Schöberl & Schlacher, 2014) and the point-to-point steering problem (Belinskaya & Chetverikov, 2016).

In this paper, we present a method for constructing orbital decompositions. We use the geometry of infinite jets and characterize local orbital decompositions in terms of integrable pseudosymmetries. The notion of pseudosymmetry of evolution equations with one spatial variable was introduced in Sokolov (1988). We generalize this concept to control systems and introduce the notion of integrable pseudosymmetry. Note that orbital equivalences of systems can be interpreted as diffeomorphisms of infinite dimensional manifolds corresponding to the systems (see Krasil’shchik and Vinogradov (1999) or Fliess et al. (1999)). Any pseudosymmetry determines an involutive distribution on the corresponding infinite dimensional manifold. But involutive distributions on infinite-dimensional manifolds may be nonintegrable. Similarly to Chetverikov (1991), where integrable higher symmetries were described, we introduce an integrability condition for pseudosymmetries such that an integrable pseudosymmetry determines an integrable distribution. If an integrable pseudosymmetry depends on the state only, then its integrable distribution is a distribution invariant under dynamics (Isidori, 1995, Isidori et al., 1981). In this case the corresponding decomposition is classical. Like the invariant distribution in the classical case (see Isidori (1995)), the distribution of an integrable pseudosymmetry can be a fundamental tool in the analysis of the internal structure of nonlinear systems.

We prove that any orbital decomposition of a system determines and is determined by its integrable pseudosymmetry. Any pseudosymmetry is determined by two matrices of functions. The equations for these matrices are deduced. The method for constructing orbital decompositions is given and demonstrated by the example of the Kapitsa pendulum system.

Section snippets

Decompositions of control systems

Consider the system ẋ=f(t,x,u),xRn,uRm,where f=(f1,,fn) is a smooth vector function, x=(x1,,xn) is the state, and u=(u1,,um) is the control input. Here and throughout the following, smoothness is understood as infinite differentiability.

Suppose system (1) is orbitally equivalent (Fliess et al., 1999) to a system of the form z=f1(τ,z,v),zRn1,vRmζ=f2(τ,ζ,z,v),ζRn2, where τ is the independent variable, z=dzdτ. Then we say that system (2)–(3) is an orbital decomposition of system (1).

We

A geometric framework

A detailed exposition of geometry of infinitely prolonged differential equations can be found in Krasil’shchik and Vinogradov (1999).

Let us associate to system (1) an infinite dimensional manifold E with coordinates (t,x,u(0),u(1),,u(l),),where u(l) denotes the vector variable corresponding to the lth order derivative of u with respect to t. Every solution s=(sx(t),su(t)) of system (1) and a point t0R determine a point of E with the coordinates t=t0,x=sx(t0),u(l)=lsu(t0)tl,l0.This point

Pseudosymmetries and decompositions

Let (E,Dt) be the diffiety of system (1). We say that a set of vector fields X1,,Xq on E is an integrable pseudosymmetry of system (1) if the following conditions hold:

(i) the fields X1,,Xq generate an involutive distribution of dimension q on E;

(ii) the column X=(X1,,Xq)T satisfies the equality [X,Dt]=AX+BDt,where A is a square matrix of some functions on E, AX is the product of the matrix A and the column X, and B is a column of some functions on E;

(iii) there exists a ring K of

Techniques of computing integrable pseudosymmetries

When q=1 and A0, Eq. (9) means that X1 defines a higher symmetry of the system. The next theorem generalizes well-known theorems on higher symmetries (see Krasil’shchik and Vinogradov (1999)).

First consider the diffiety E of system (1) and a q-column of the form Эφ,A=φxx+k0(Dt+A)k(φu)u(k),where φ=(φx,φu), and φx,φu, and A are matrices of functions on E of dimensions q×n,q×m, and q×q respectively, (Dt+A)k is the kth degree of the (q×q)-matrix differential operator Dt+A acting on the

Proofs of theorems

First let us prove that the definition of integrable pseudosymmetry is invariant under C-diffeomorphisms. Consider an integrable pseudosymmetry X={X1,,Xq} of a system E and a C-diffeomorphism F:ES. Since a diffeomorphism takes involutive distributions of dimension q into involutive distributions of dimension q, we see that the distribution generated by the fields F(X1),,F(Xq) is involutive and has dimension q. Thus condition (i) is invariant under C-diffeomorphisms.

Let t and τ be the

Conclusion

The presented method allows one to construct both orbital and classical decompositions and consists of 2 stages:

(1) solving equation (16) using the splitting procedure, Theorem 8, and condition (iii);

(2) constructing decompositions for the found integrable pseudosymmetries.

In the case m=1, the ring K (see condition (iii)) coincides with Fl(E), where the integer l is determined by the matrices φ and A (cf. with Example 11). In the case m>1, we suggest first choosing K and then solving equation 

Vladimir Chetverikov graduated from the Moscow State University, Mechanical & Mathematical Faculty in 1983. He obtained the Ph.D. degree in Mathematics in 1988 and Doctor of Science degree in Mathematics in 2007. He is currently a full professor at the Bauman Moscow State Technical University. His research interests include theoretical aspects of nonlinear control, Lie–Backlund transformations of differential equations, invertible differential operators, Backlund transformations,

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Vladimir Chetverikov graduated from the Moscow State University, Mechanical & Mathematical Faculty in 1983. He obtained the Ph.D. degree in Mathematics in 1988 and Doctor of Science degree in Mathematics in 2007. He is currently a full professor at the Bauman Moscow State Technical University. His research interests include theoretical aspects of nonlinear control, Lie–Backlund transformations of differential equations, invertible differential operators, Backlund transformations, transformations of functional–differential equations.

This work was partially supported by the Ministry of Science and Higher Education of the Russian Federation, project no. 0705-2020-0047 and the Russian Foundation for Basic Research , project no. 20-07-00279. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Warren E. Dixon under the direction of Editor Daniel Liberzon.

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