SYSTEM DYNAMICS (ME 344)
Homework #6
(2 pages)
Due 10/16/2012
1. Consider the system from Homework 5 problem 7, for
which the equations of motion are
I1q1 = -k1q1 - k2 (q1 - q 2 )
I 2q 2 = -k2 (q 2 - q1 ) - k3q 2 + T2
Define the state variables for this system to be the
angle and angular velocity of the two drums (that is,
q1, w 1 , q 2 , and w 2 ) and the (only) input variable to
be the applied torque T2 . Rewrite the two second order differential equations as a set of four
first order differential equations, each one the formula for the derivative of one of the state
variables (in terms of state and input variables). Then, assemble the four differential
equations into standard matrix form, x = Ax + Bu (that is, determine the necessary matrices
A and B to rewrite your four individual equations as a matrix equation).
2. The system shown at right has two inputs: a directly
applied force fa and a motion z imposed at the end of the
z
spring. The equation of motion for the system is
mx = f a - k(x - z) - cx . Use the position and velocity of
the mass as state variables and define two output
fa
variables that are the amount of compression in the spring
and the total force exerted on the mass. Rewrite the equation of motion as a pair of first order
equations, write definition equations for the output variables, and then put all of your
equations into standard matrix form x = Ax + Bu and y = Cx + Du .
3. Consider the system defined by
é
ê
ê
ë
é
ê
ê
ë
x1 ù é 0 1 ù é x1 ù é 1 ù
ú=ê
ú+ê
úê
ú éëu ùû
ú
ê
x2
-3 -4 û x2 ú ë 0 û
ë
û
ë
û
y1 ù é 1 0 ù é x1 ù
ú=ê
ú (D is zero).
úê
ú
ê
y2
0 1 û x2 ú
û ë
ë
û
and
Find the transfer functions Y1(s)/U(s) and Y2(s)/U(s) by hand (you may verify w/MATLAB).
4. Consider the system defined by
é
ê
ê
ë
é
ê
ê
ë
é
ù
x1 ù é 0
1 ù ê x1 ú + é
ú=ê
ú
ê
x2 ú ë -25 -4 û ê x2 ú ë
û
ë
û
y1 ù é 1 0 ù é x1 ù é 1
ú=ê
ú+ê
úê
y2 ú ë 0 1 û ê x2 ú ë 0
û
ë
û
é
ù
1 1 ù ê u1 ú
ú
0 1 û ê u2 ú
ë
û
é
ù
0 ù ê u1 ú .
ú
1 û ê u2 ú
ë
û
and
This system has two inputs (u1 and u2) and two outputs (y1 and y2). There are thus four
transfer functions: Y1/U1, Y2/U1, Y1/U2, and Y2/U2. Using MATLAB (or other software of your
choice) obtain all four transfer functions (the four elements of the 2x2 transfer matrix).
NOTE: The MATLAB ss2tf command doesn’t give the transfer matrix directly – it gives the
coefficients of the various polynomials in the matrix. If you use this command, you must
then write out the answer properly as a function of s. Alternatively, you can define a
symbolic variable s and used the matrix formula given in class to perform the conversion.
5. Consider the system described by the following state and output equations, corresponding to
the quarter-car model with specific values for suspension stiffness and damping and tire
stiffness (the outputs y1 and y2 are the displacements of the car body and wheel, respectively):
é
ê
ê
ê
ê
ë
x1
x2
x3
x4
ù
ú é
ú=ê
ú ê
ú êë
û
0
1
0
0
-10 -2 10
2
0
0
0
1
10 2 -410 -2
é
é y1 ù é 1 0 0 0 ù ê
ê
ê
ú=ê
úû ê
y
0
0
1
0
ë
ë 2 û
ê
ë
x1
x2
x3
x4
ù
ú
ú
ú
ú
û
é
ùê
úê
úê
úû ê
ë
x1
x2
x3
x4
ù
ú é
ú+ê
ú ê
ú êë
û
0
0
0
400
ù
ú
ú éëu ùû
úû
(D is zero)
Obtain (using MATLAB or other software of your choice)
(a) the unit step response (the input u(t) is a unit step function occuring at t = 0), and
(b) the unit impulse response (the input u(t) is a Dirac delta function occuring at t = 0),
For both cases, generate a plot of both outputs (y1 and y2) vs. t for the first 5 seconds.
6. For the same system described in problem #5, obtain
(a) the response to initial conditions x1 = 1, x2 = 0, x3 = 1, x4 = 0 (without any input),
(b) the response to an input u(t) that is 1 for the first second ( 0 £ t £1) and zero after that
(when t > 1), when there are no initial conditions, and
(c) the response to the same input (u(t) = 1 for the first second and zero after that) when the
initial conditions are x1 = 1, x2 = 0, x3 = 1, x4 = 0.
For all three cases, generate a plot of both outputs (y1 and y2) vs. t for the first 5 seconds.