#### Date of Award

Spring 1-1-2011

#### Document Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematics

#### First Advisor

Jeanne Clelland

#### Second Advisor

George Wilkens

#### Third Advisor

Carla Farsi

#### Abstract

Control systems are underdetermined systems of ** n** ordinary differential equations (ODEs),

ẋ = f (x, u), (1)

that show up in the design of electrical and mechanical systems, among other things. The variables **x** whose time evolution is determined by the ODEs are called state variables, while the "free parameters" **u** are called control variables. A control system can be viewed as a submanifold ∑ of the tangent bundle of the state space in the following way: given a manifold *M* and a curve **x** :* I* → *M*, we say that **x** is a solution to the system ∑ ⊂ *TM* if (x(t), ẋ (t)) lies in ∑ for all *t* ∈ *I*. The map ℝ^{s} → *T*x*M* given by **u** ↦ (x, f(x, u)) is a parametrization of ∑x = ∑∩ *T*x*M* with the parameters **u** seen as local coordinates on ∑x.

A dynamic equivalence takes trajectories of one system, ẋ = f(x, u), to those of another, ẏ = g(y, v), and back again via maps between jet spaces which allow state derivatives to get mixed in:

(x, u, u̇, …, u^{(J)}) ↦ y(x, u, u̇, …, u^{(J)}).

Through the defining equation (1), derivatives of state variables can be expressed in terms of control variables and their derivatives as well. Static (feedback) equivalence, which is a diffeomorphism of the state space, is a special case when y = y(x).

Up to dynamic equivalence at the first jet level (*J* = 0), i.e. x = x(y, v) and y = y(x, u), my results classify all affine linear control systems,

ẋ = f^{0}(x) + uif^{i}(x),

of at most three states and two controls through the use of Cartan's method of equivalence. My main result is that every affine linear control system of three states and two controls falls into one of three classes under dynamic equivalence. The numbered rows represent these three classes. The entries in each row are systems that, while dynamically equivalent, are not statically equivalent.

1 ẋ_{1 }= u_{1 }ẋ_{1 }= u_{1 }ẋ_{1 }= u_{1}

ẋ_{2 }= u_{2 }ẋ_{2 }= u_{2 }ẋ_{2 }= u_{2}

ẋ_{3 }= u_{2 }ẋ_{3 }= x_{2}u_{1 }ẋ_{3 }= 1+x_{2}u_{1}

2 ẋ_{1 }= u_{1}

ẋ_{2 }= u_{2}

ẋ_{3 }= 0

3 ẋ_{1 }= u_{1}

ẋ_{2 }= u_{2}

ẋ_{3 }= 1

#### Recommended Citation

Stackpole, Matthew Ward, "Dynamic Equivalence of Control Systems via Infinite Prolongations" (2011). *Mathematics Graduate Theses & Dissertations*. 5.

https://scholar.colorado.edu/math_gradetds/5