Page 188 - Servo Motors and Industrial Control Theory -

Appendix B: Exercise Problems on State Variable
Feedback Control Theory (Chap. 3)

You should remember that the general form of state variables feedback
control theory is as follows,

d X : = AX + BU
d
t
y :CX=

In the above equations, X is the vector of n state variables, A is a matrix with dimen-
sion n × n. B is the input matrix with dimension of n × m, U is the input with dimen-
sions of m × l. For a single input system U is just a single variable.
Y is the vector of L output variables. The matrix C is known as the output matrix
and its dimension is L × n. For some systems there may be direct contribution from
the input variables to the output variable. In this case a second term must be added
to the output equation.
In the following problems, you should use mathematical software such as
MathCAD or any other mathematical software you are familiar with. When a prob-
lem must be solved by calculation by hand, they will specify explicitly.

1. A system transfer function is given by second order lag transfer function as,
θ ω 2
o := n
θ s + 2 s + ω⋅ ω 2
ξ
2
i n n
For ω = 1000 and ξ = .5
n
a. Find the roots of the characteristic equation.
b. Write the transfer function in state space form.
c. Determine the eigenvalues and eigenvectors of the system matrix using
MathCAD or other mathematical software, and show that the eigenvalues are
the same as the roots of characteristic equation. Discuss the properties of the
eigenvectors.
d. Determine the dynamic matrix and show by hand that the determinant of this
matrix is the same as the characteristic equation.

R. Firoozian, Servo Motors and Industrial Control Theory, Mechanical Engineering Series, 185
DOI 10.1007/978-3-319-07275-3, © Springer International Publishing Switzerland 2014