辅导AUTO-1029、讲解C++/python程序语言、讲解LQR留学生
- 首页 >> Python编程Issued: 16‐July‐2018 (Wk 1) Due: 24‐Oct‐2018 (Wk 7)
AUTO1029 PROJECT‐2018 (PART‐1) Page 1
AUTO-1029 “AUTOMOTIVE SYSTEMS & CONTROL”
PROJECT (Part-1)
Consider the mechanical system shown below.
This is a version of a constrained double pendulum and it is modelled as a continuous
system with THREE degrees of freedom: translation x of the trolley and two angles T1
and T 2 for the links.
The differential Equations of motion for this system can be derived using a few different
methods. If the Lagrange’s Equations method is used, the non-linear differential
equations of motion of the system can be derived and then re-written in the matrixvector
form as follows:
1 2 3 2 3 1 1 3 2 2
2
2 3 1 1 2 3 1 3 1 2 1 2 1 2
2
3 2 2 3 1 2 1 2 1 2 3 2
cos cos
cos cos cos sin sin
cos cos cos sin sin
m m m m m l m l
M m m l m m l m l l
m l m l l m l
T T
T T T T T
T T T T T
2 3 1 1 1 3 2 2 2 1
2
3 1 2 2 1 2 2 1 2 1 1 3 1 1
2
3 1 2 1 1 2 1 2 3 2 2
sin sin ( )
sin cos sin cos sin sin
cos sin sin cos sin
m m l m l k x u t
m l l m gl m gl
m l l m gl
T T T T
T T T T T T T
T T T T T T
This system is to be controlled using linear control techniques. However, before this
task can begin, it is necessary to perform a linear analysis of the uncontrolled system.
Therefore, you have been asked to perform a logical analysis of the system using
numerical methods. You are required to do the following tasks:
(1) Linearise the equations of motion about the vertical equilibrium position. Write
these in matrix-vector format.
(2) Assume that the system is characterized with the following parameters:
1 2 3 m k 5 g; m 1.5kg; m 1.2kg; 1 2 l m 1.8 ; l 2.5m; k 100 N / m.
Determine the system natural frequencies and corresponding mode shapes.
Plot the mode shapes for each mode.
(3) Plot the frequency response function for the position of the cart due to u(t) over
the range of frequencies [0.1, 10] rad/sec. Use the definition of the FRF
(4) Using the result from the previous step, calculate the magnitude of the force
transmitted to the wall via the spring constraint for a forcing frequency of 8.25
rad/sec if the amplitude of the forcing function is 500 N.
(5) Rewrite the linearized Equations of Motion of the System in the state space form:
x A x Bu
Issued: 16‐July‐2018 (Wk 1)Due: 24‐Oct‐2018 (Wk 14)
AUTO1029 PROJECT‐2018 (PART‐1) Page 3
Then calculate the forced response of the linear system using MATLAB and/or
SIMULINK to the force input shown in the Figure below with the following initial
conditions: ^x x ,T T1 2 , , ,T T1 2 , ` ^0.05,0.05,0.05,0,0,0` .
0 10 20 30 40 50 60 70 80 90 100
0
50
100
150
200
Time [s]
u(t) [N]
(6) Use the LQR (linear-quadratic regulator) method to control the system to
minimise the quadratic cost function:
0
( 2 ) T T T J x Qx u Ru x Nu dt
f
3
For this purpose:
(a) adopt for your Group a particular control strategy, proposing your Q , R and N
matrices [please explain in the Project your choice of the matrices Q , R and N];
(b) calculate the optimal control gain matrix K using the MATLAB command
[K,S,e] = lqr(SYS,Q,R,N)
(c) simulate the forced response of the optimally controlled system using
SIMULINK. Compare your results with the calculated in the step 5, comment on
your results. Note, that with the LQR method the feedback control law that will
minimise the value of the cost J is u K x .