Simulation Systems Department.
The solution of problems for multidimensional nonlinear control systems encounters serious difficulties, which are both mathematical and technical in nature. Therefore it is imperative to develop methods of reduction of nonlinear systems to a simpler form, for example, decomposition into systems of lesser dimension. Approaches to reduction are diverse, in particular, techniques based on approximation methods. We elaborate the most natural and obvious (in our opinion) approach, which is essentially inherent in any theory of mathematical entities, for instance, in the theory of linear spaces, theory of groups, etc. Reduction in our interpretation is based on assigning to the initial object an isomorphic object, a quotient object, and a subobject. In the theory of linear spaces, for instance, reduction consists in reducing to an isomorphic linear space, quotient space, and subspace. Strictly speaking, the exposition of any mathematical theory essentially begins with the introduction of these reduced objects and determination of their basic properties in relation to the initial object. Therefore, we can say that the theory of reduction of nonlinear control systems discussed here outlines the elements of the general theory of such systems. This theory is purely differential geometric and theoretical group by nature.
A formal definition of these reduced objects that is suitable for any mathematical theory can be given within the framework of the theory of categories or the Bourbaki theory of structures. At the descriptive level every category (for example, the category of linear spaces or category of groups) is a class of objects, in which every object S is a set M with the same sort of predefined structure. This structure can be interpreted as a collection of a particular type of relationships between the elements of the set M. Along with objects, a category also contains morphisms that implement the object-object interrelations. If two objects S1 and S2 are defined on the sets M1 and M2, respectively, then a morphism f of the object S1 into the object S2 is a mapping f from M1 to M2 that preserves the structure of a given kind (i.e., preserves the corresponding relationships between the elements of the sets). For example, in the category of linear spaces, morphisms are linear mappings, while in the category of groups they are homomorphisms.
For nonlinear control systems dy/dt=f(y,u), it is possible to construct a category along the following lines. Objects in this category, which is denoted by NS, are control systems. Morphisms are defined as follows. Along with some system S1 described by relations dy/dt=f(y,u), let us consider a control system S2 described by the relations dx/dt=g(x,v). A morphism of the system S1 to the system S2 is called a mapping f of the phase space M of the system S1 into the phase space L of the system S2 that carries the solutions (phase trajectories) of the system S1 into the solutions of the system S2. Isomorphism in any category is a morphism f that is a one-one mapping, moreover, the inverse mapping is also a morphism. If for the objects S1 and S2 there exists an isomorphism of S1 into S2, then the objects S1 and S2 are said to be isomorphic. Isomorphic objects have identical properties within the framework of the category. For example, in the category of linear spaces, isomorphisms are linear isomorphisms.
It is appropriate to reduce the control system to an isomorphic or, alternatively, equivalent system if the latter is simpler in form. For example, a complicated nonlinear system can be equivalent to a linear system. In this case, nonlinearity is a `causal trait,' which vanishes in an equivalent system. Vital properties of control systems like controllability, stability, and optimality of solutions are preserved in changing over to an equivalent system. Therefore, it is natural to solve any control problem for a simpler equivalent system and then `carry over' the results thus obtained to the initial system through an isomorphism.
The concept of a subobject results from our desire to construct correctly the restriction of an object S1 defined on a set M to a subset N of M. It is generally not possible to restrict an object S1 to an arbitrary set. An object S2 defined on a subset N is called a subobject if the canonical injection of N to M is a morphism. For example, in the category of linear space, subobjects are linear subspaces.
The need for restricting a control system S1, i.e., changing over to a subsystem S2 on a subset N of the set M, is dictated by practical considerations when the elements of the set M are forced to obey certain constraints (initial conditions, boundary conditions, etc.). In such cases, it is natural to restrict a system S1 to some subset N for which these constraints are satisfied. The subsystem S2 on N defines a part of the solutions of the initial system S1 that lie within N and, in particular, satisfy the given constraints. Therefore, the solution of a control problem formulated for a system S1 can be reduced to the solution of an analogous problem for its subsystem S2 with phase space of diminished dimension.
While simplification in restriction is achieved via passage to a subset N of the set M, in factorization it results from `contraction' of the set M, i.e., passage to the quotient set M/R over some equivalence relations R. In this passage, points belonging to one equivalence class are `glued' to the same point of the quotient set M/R. An object S2 defined on the quotient set M/R is called the factor object of the object S1 on the set M if the canonical projection of Mto M/R is a morphism. For example, in the category of linear spaces, factor objects are quotient spaces.
The significance of factorization for the reduction of control systems lies primarily in the decomposition of the initial system it generates. More exactly, if a system S1 has a factor system dz/dt=g(z,v) on some quotient set L=M/R, then system S1 is equivalent to the system dz/dt=g(z,v), dx/dt=h(z,x,v). This system by its structure suggests that its solution z(t), x(t) can be determined as follows. First we solve of the factor system dz/dt=g(z,v) (for some control v(t)) and then, substituting z(t) into dx/dt=h(z,x,v) , we obtain x(t). This is the underlying principle of decomposition of algorithms designed for solving control problems. Let us note that many properties of a control system (observability, autonomity, etc.) are determined by its factor systems of a special type.
The concepts of an isomorphic object, a factor object, and a subobject may be applied for reducing an object jointly and in any order. A concrete sequence of transitions to a reduced system is called the reduction scheme and the number of transitions is called the reduction depth. To solve a control problem, a reduction scheme and its depth are chosen from the conditions of the problem.
A pivotal topic in reduction is the construction of a mathematical apparatus for determining reduced objects. The mathematical apparatus includes those concepts that are invariant to morphisms. An effective instrument for studying reduction in the category NS and in its different subcategories is presented by the associative differetial geometric and theoretical group objects: transformation group, Lie algebra of vector fields, affine distribution, codistribution, Pfaffian system etc. Objects of such a kind can be linked with every control system; moreover, their structure is preserved under the action of morphisms of the category of control systems. Note for example the following results. The subsets, on which subsystems may exist, must be invariant manifolds of the associated group. Consequently, this group must be intransitive so that (nontrivial) subsystems may exist. On the other hand the associative group must be imprimitive so that (nontrivial) factor systems may exist.
This approach to the problem of reduction of control systems is presented in the following publications.
1. Elkin V.I. Affine control systems: their equivalence, classification, quotient systems and subsystems //Journal of Mathematical Sciences. 1998. Vol. 88. No. 5. P. 675-722.
2. Elkin V.I. Reduction of nonlinear control systems. Dordrecht/Boston/London: Kluwer Academic Publishers, 1999. 249 p.
Copyright 2000, Simulation Systems.