University of Notre Dame
Aerospace and Mechanical Engineering

ME 698: Geometric Nonlinear Control
Homework 3

B. Goodwine
Fall, 1999
Issued: September 9, 1999
Due: September 16, 1999

1.
Consider the system defined by

\begin{displaymath}\dot x = A x + B u.
\end{displaymath}

Let

\begin{displaymath}\Delta = \left[ B \quad AB \quad A^2 B \quad \cdots \quad A^{n-1}B
\right]
\end{displaymath}

and

\begin{displaymath}M = \left[ \begin{array}{ccccc}
a_{n-1} & a_{n-2} & \cdots & ...
...& \cdots & 0 & 0\\
1 & 0 & \cdots & 0 & 0
\end{array} \right]
\end{displaymath}

where the ai come from the characteristic equation for A:

\begin{displaymath}\mbox{det}A = \lambda^n + a_1 \lambda^{n-1} + \cdots + a_{n-1} \lambda
+ a_n.
\end{displaymath}

By considering the case n=3, show that

\begin{displaymath}T^{-1}AT = \left[ \begin{array}{ccccc}
0 & 1 & 0 & \cdots & ...
... -a_{n-1} & -a_{n-2} & \cdots & -a_1 \\
\end{array} \right].
\end{displaymath}

2.
Consider the system defined by

\begin{displaymath}\dot x = A x + B u, \end{displaymath}

where

\begin{displaymath}A = \left[ \begin{array}{ccc} 1 & 2 & 1 \\ 0 & 1 & 3 \\ 1 & 1...
...d
B = \left[ \begin{array}{c} 1 \\ 0 \\ 1 \end{array} \right].
\end{displaymath}

Transform the system into controllable canonical form.

3.
Consider the system defined by

\begin{displaymath}\dot x = A x + B u, \end{displaymath}

where

\begin{displaymath}A = \left[ \begin{array}{cc} 0 & 1 \\ -2 & -3 \end{array} \ri...
...\qquad
B = \left[ \begin{array}{c} 0 \\ 2 \end{array} \right].
\end{displaymath}

The characteristic equation of the system with no controller is

\begin{displaymath}\mbox{det}(\lambda I - A) = \lambda^2 + 3 \lambda + 2,
\end{displaymath}

so the eigenvalues of A are -1 and -2.

It is desired to have eigenvalues at -3 and -5 by using a state feedback control u = -K x. Determine the necessary feedback gain matrix, K and control signal u.

4.
Consider the inverted pendulum shown in the figure where

\begin{eqnarray*}M &=& 2 \mbox{ kg}\\
m &=& 0.1 \mbox{ kg}\\
l &=& 0.5 \mbox{ m}.\\
\end{eqnarray*}

(a)
Determine the equations of motion for the system.
(b)
Linearize the equations of motion.
(c)
Show that the linearized system is controllable.
(d)
Using the method of pole placement, design a controller which stabilizes the system about the $\theta = 0$ position.
i.
Indicate where you chose to place the poles.
ii.
Simulate the controller on both the linearized as well as the actual equations of motion.



The burden of the lecture is just to emphasize the fact that it is impossible to explain honestly the beauty of the laws of nature in a way that people can feel, without their having some deep understanding of mathematics. I am sorry, but this seems to be the case.

--- Richard Feynman, The Character of Physical Law.


Last updated: September 9, 1999.
B. Goodwine (jgoodwin@nd.edu)