[Paper Review] On feedback classification of control-affine systems with one and two-dimensional inputs

Andrei Agrachev, Igor Zelenko|ArXiv.org|Feb 1, 2005

Adaptive Control of Nonlinear Systems1 references4 citations

TL;DR

This paper provides a complete local classification of generic control-affine systems with one or two-dimensional inputs on n-dimensional manifolds (n ≥ 4 for scalar input, n = 4,5 for two inputs) up to state-feedback equivalence. It introduces a canonical frame and feedback invariants via Poincaré series, reducing the classification problem to functional moduli of varying variables, with explicit parameterization and intrinsic moduli counts in the C^∞ and C^ω categories.

ABSTRACT

The paper is devoted to the local classification of generic control-affine systems on an n-dimensional manifold with scalar input for any n>3 or with two inputs for n=4 and n=5, up to state-feedback transformations, preserving the affine structure. First using the Poincare series of moduli numbers we introduce the intrinsic numbers of functional moduli of each prescribed number of variables on which a classification problem depends. In order to classify affine systems with scalar input we associate with such a system the canonical frame by normalizing some structural functions in a commutative relation of the vector fields, which define our control system. Then, using this canonical frame, we introduce the canonical coordinates and find a complete system of state-feedback invariants of the system. It also gives automatically the micro-local (i.e. local in state-input space) classification of the generic non-affine n-dimensional control system with scalar input for n>2. Further we show how the problem of feedback-equivalence of affine systems with two-dimensional input in state space of dimensions 4 and 5 can be reduced to the same problem for affine systems with scalar input. In order to make this reduction we distinguish the subsystem of our control system, consisting of the directions of all extremals in dimension 4 and all abnormal extremals in dimension 5 of the time optimal problem, defined by the original control system. In each classification problem under consideration we find the intrinsic numbers of functional moduli of each prescribed number of variables according to its Poincare series.

Motivation & Objective

• To establish a complete local classification of generic control-affine systems with scalar input for n ≥ 4 and two inputs for n = 4,5 under state-feedback equivalence.
• To introduce intrinsic numbers of functional moduli via Poincaré series, quantifying the dependence of classification on functions of varying numbers of variables.
• To develop a canonical frame and feedback-invariant structure for scalar-input systems using normalization of structural functions in commutator relations.
• To reduce the two-input classification problem in dimensions 4 and 5 to the scalar-input case by identifying extremal subsystems (time-optimal and abnormal extremals).
• To provide a characteristic parameterization of the classification problem, explicitly listing the number and weight of functional invariants for each variable count.

Proposed method

• Construct a canonical frame by normalizing structural functions in the commutator identities of the vector fields defining the control system.
• Define canonical coordinates using the canonical frame to extract a complete system of state-feedback invariants.
• Use the Poincaré series to compute the intrinsic number of functional moduli of each prescribed number of variables in the classification problem.
• Reduce the two-input problem in n=4 and n=5 to scalar-input problems by isolating the subsystem of extremals (time-optimal in n=4, abnormal in n=5).
• Introduce the recovering invariant R, a feedback-invariant function of five variables, to reconstruct the original system uniquely up to feedback equivalence.
• Apply normalization of the parameterization matrix P to derive the characteristic matrix C = Norm(P), yielding the characteristic parameterization of functional invariants by weight and variable count.

Experimental results

Research questions

• RQ1 What is the complete set of state-feedback invariants for generic control-affine systems with scalar input on n-dimensional manifolds for n ≥ 4?
• RQ2 How can the classification problem for two-input systems in dimensions 4 and 5 be reduced to the scalar-input case?
• RQ3 What is the intrinsic number of functional moduli of each prescribed number of variables in the classification of such systems, as encoded by the Poincaré series?
• RQ4 What is the characteristic parameterization of the classification problem, and how are functional invariants distributed by weight and variable count?
• RQ5 How does the recovering invariant R enable unique reconstruction of the system from its reduction and invariants?

Key findings

• For generic real analytic 5D control-affine systems with two inputs, the classification is parameterized by 4 germs of functions of 5 variables, 10 germs of functions of 4 variables, 10 germs of functions of 3 variables, and 5 germs of functions of 2 variables, plus the recovering invariant R.
• The Poincaré series of the classification problem is M(t) = 1/(1−t)^6 + t^3(5/(1−t)^3 + 10/(1−t)^4 + 10/(1−t)^5 + 3/(1−t)^6).
• The characteristic matrix C = Norm(P) is given by [[1,3,6,5],[0,0,0,16],[0,0,0,13],[0,0,0,4]], encoding the number of invariants by weight and variable count.
• The classification of generic scalar-input systems with n ≥ 5 is micro-local and equivalent to the classification of differential 1-forms in the C^ω category.
• The characteristic parameterization includes 4 invariants of 5 variables (weight 0), 13 of 4 variables (weight 3), 16 of 3 variables (weight 3), and 1 invariant of 2 variables (weight 0), plus 3, 6, and 5 invariants of 2 variables at weights 1, 2, and 3 respectively.
• The system is uniquely recoverable from its reduction and the recovering invariant R, which is a weight-0 feedback invariant of five variables.