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7/27/2019 AUTO_L7_2012 Control Lecture Note
1/23
AUTO-1029 Lecture 6 Slide 1
AUTO-1029: AUTOMOTIVESYSTEMS & CONTROL
Prof Pavel M. TRIVAILO, PhD(Professor of Aerospace Engineering)
E-mail: [email protected]
SAMME, RMIT University, Melbourne, Australia
Lecturer: Prof P.M.Trivailo c2012 SAMME, RMIT
7/27/2019 AUTO_L7_2012 Control Lecture Note
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AUTO-1029 Lecture 6 Slide 2
Lecture 7:
INTRODUCTION INTO OPTIMAL CONTROL
LINEAR QUADRATIC REGULATOR (LQR) LQR Control of 2-DOF Systems using SIMULINK
(diagram method)
Lecturer: Prof P.M.Trivailo c2012 SAMME, RMIT
7/27/2019 AUTO_L7_2012 Control Lecture Note
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AUTO-1029 Lecture 6 Slide 3
Lecture 7:FEEDBACK? PROBLEMS?
ANNOUNCEMENT: A-2
Assignment is due on Friday, 21-Sept-2012, 18:00
Groups (with up to 4 Members)
Registration of your Groups is due today!
Lecturer: Prof P.M.Trivailo c2012 SAMME, RMIT
7/27/2019 AUTO_L7_2012 Control Lecture Note
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AUTO-1029 Lecture 6 Slide 4
LINEAR QUADRATIC
REGULATOR(LQR)
Lecturer: Prof P.M.Trivailo c2012 SAMME, RMIT
7/27/2019 AUTO_L7_2012 Control Lecture Note
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AUTO-1029 Lecture 6 Slide 5
HINTS: LQR
Let us inspect a cost function J for a linearised n-
DOF system with state vector x = [q q]T
COST: J=
0
(xTQx + uTRu + 2xTNu) dt
Firstly, let us determine the dimensions of the ma-
trices in the cost function matrix expression:
for n-DOF: x2n1; un1; Q2n2n; Rnn; N2nnLecturer: Prof P.M.Trivailo c2012 SAMME, RMIT
7/27/2019 AUTO_L7_2012 Control Lecture Note
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AUTO-1029 Lecture 6 Slide 6
These matrices will be used in the lqr(SYS,Q,R,N)or lqr(A,B,Q,R,...) MATLAB commands.
Note, that in these commands, there are peculiar
requirements to the matrices, for example:the [R] matrix must be symmetric positive definite
with as many columns as [B]. Also, [Q N;N R] must
be positive definite (you can use EIG to check pos-
itivity).
Lecturer: Prof P.M.Trivailo c2012 SAMME, RMIT
7/27/2019 AUTO_L7_2012 Control Lecture Note
7/23
AUTO-1029 Lecture 6 Slide 7
Let us then distinquish a special case (we will call it
Case-V), for which
[Q] =
[k]nn [0]nn[0]nn [0]nn
; [R] [0]nn; [N] = [0]2nn;
Please observe that for this case the cost is approxi-
mately equal to the doubled POTENTIAL ENERGY
of the system V:
J 2qT[k]q = 2V
Lecturer: Prof P.M.Trivailo c2012 SAMME, RMIT
7/27/2019 AUTO_L7_2012 Control Lecture Note
8/23
AUTO-1029 Lecture 6 Slide 8
Therefore, Case-V will correspond to the control
strategy, where the main consideration is given to
the minimisation of the potential energy of the
system.
Similarly, let us distinquish a case (we will call it
Case-T), for which
[Q] =
[0]nn [0]nn
[0]nn [m]nn
; [R] [0]nn; [N] = [0]2nn
Lecturer: Prof P.M.Trivailo c2012 SAMME, RMIT
7/27/2019 AUTO_L7_2012 Control Lecture Note
9/23
AUTO-1029 Lecture 6 Slide 9
Prove that for this case the cost is approximately
equal to the doubled KINETIC ENERGY of the
system T:
J 2qT[m]q = 2T
Therefore, Case-T will correspond to the control
strategy, where the main consideration is given to
the minimisation of the kinetic energy of the sys-
tem.
Lecturer: Prof P.M.Trivailo c2012 SAMME, RMIT
7/27/2019 AUTO_L7_2012 Control Lecture Note
10/23
AUTO-1029 Lecture 6 Slide 10
Combining two previous cases, we can introduce
case (we will call it Case-E), for which
[Q] =
[k]nn [0]nn
[0]nn [m]nn
; [R] [0]nn; [N] = [0]2nn
Please observe that for this case the cost is approx-
imately equal to the doubled TOTAL ENERGY
of the system E = V+ T:
J 2 qT[k]q + 2 qT[m]q = 2V+ 2T = 2 E
Lecturer: Prof P.M.Trivailo c2012 SAMME, RMIT
7/27/2019 AUTO_L7_2012 Control Lecture Note
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AUTO-1029 Lecture 6 Slide 11
Therefore, the last Case-E will correspond to the
control strategy, where the main consideration is
given to the minimisation of the total energy of
the system.
We can also introduce some other control strtate-
gies, including minimisation of the control efforts
(Case-u) : [Q] [0]2n2n; [R] = [diag(1)]nn; [N] =[0]2nn OR a mixture of the abovementioned Cases.
Lecturer: Prof P.M.Trivailo c2012 SAMME, RMIT
7/27/2019 AUTO_L7_2012 Control Lecture Note
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AUTO-1029 Lecture 6 Slide 12
All of these control strategies can be easily imple-mented in MATLAB and/or SIMULINK. You may
employ the following steps:
Enter the system matrices [m], [k], [c].
Calculate the state-space matrices [A], [B], [C]
and [D].
Lecturer: Prof P.M.Trivailo c2012 SAMME, RMIT
7/27/2019 AUTO_L7_2012 Control Lecture Note
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AUTO-1029 Lecture 6 Slide 13
Describe (represent) the system in MATLAB, us-
ing the ss MATLAB comand:
>> SYS = ss(A,B,C,D)
Based on the requirements to your system, in-troduce matrices [Q], [R] and [N], as discussed
before on Slides 5-11.
Lecturer: Prof P.M.Trivailo c2012 SAMME, RMIT
http://-/?-http://-/?-http://-/?-http://-/?-7/27/2019 AUTO_L7_2012 Control Lecture Note
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AUTO-1029 Lecture 6 Slide 14
Calculate the optimal control gain matrix K using
the MATLAB command
>> [K,S,e] = lqr(SYS,Q,R,N)
In the Project, we do not need to use calculated
matrices [S] a nd [e]. However, the matrix [K]
will be very important. The values in this matrix,gains, will be used to calculate the optimal control
feedback as u = Kx.
Lecturer: Prof P.M.Trivailo c2012 SAMME, RMIT
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AUTO-1029 Lecture 6 Slide 15
Create a SIMULINK closed-loop model for yoursystem
Simulate your system in SIMULINK
Plot results of the simulation in MATLAB
Lecturer: Prof P.M.Trivailo c2012 SAMME, RMIT
7/27/2019 AUTO_L7_2012 Control Lecture Note
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AUTO-1029 Lecture 6 Slide 16
LQR EXAMPLE: Test 2-DOF System
Parameters: m1 = 2; m3 = 3 kg
k1 = 600; k2 = 280 N/mADDED DAMPING: c1 = 3.8; c2 = 10 Ns/m.
Lecturer: Prof P.M.Trivailo c2012 SAMME, RMIT
7/27/2019 AUTO_L7_2012 Control Lecture Note
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AUTO-1029 Lecture 6 Slide 17
EXCITATION EXTERNAL FORCES
0 5 10 15
0
1
2
3
4
5
Q1
[N]
0 5 10 151
0.5
0
0.5
1
Q
2[N]
time [s]
Lecturer: Prof P.M.Trivailo c2012 SAMME, RMIT
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AUTO-1029 Lecture 6 Slide 18
LQR: SIMULINK DIAGRAM
Lecturer: Prof P.M.Trivailo c2012 SAMME, RMIT
7/27/2019 AUTO_L7_2012 Control Lecture Note
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AUTO-1029 Lecture 6 Slide 19
TRICK: option for GAIN
Lecturer: Prof P.M.Trivailo c2012 SAMME, RMIT
7/27/2019 AUTO_L7_2012 Control Lecture Note
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AUTO-1029 Lecture 6 Slide 20
LQR: RESULTS for displacements q1 and q2
0 5 10 150
2
4
Q1,2
[N]
Q1
Q2
0 5 10 155
0
5
10
15x 10
3
q1
[m]
0 5 10 150.01
0
0.01
0.02
q2
[m]
time [s]
Lecturer: Prof P.M.Trivailo c2012 SAMME, RMIT
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AUTO-1029 Lecture 6 Slide 21
LQR: RESULTS for velocities q1 and q2
0 5 10 150
2
4
Q1,2
[N]
Q1
Q2
0 5 10 150.1
0.05
0
0.05
0.1
q1
[m/s]
0 5 10 150.1
0.05
0
0.05
0.1
q2
[m/s]
time [s]
Lecturer: Prof P.M.Trivailo c2012 SAMME, RMIT
7/27/2019 AUTO_L7_2012 Control Lecture Note
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AUTO-1029 Lecture 6 Slide 22
COMARISON: NO CONTROL vs LQR
0 5 10 15
5
0
5
10
15x 10
3
q1
[m]
NO CONTROL
0 5 10 15
5
0
5
10
15x 10
3
LQR CONTROL
0 5 10 15
5
0
5
10
15
x 103
q2[m]
time [s]0 5 10 15
5
0
5
10
15
x 103
q2[m]
time [s]
Lecturer: Prof P.M.Trivailo c2012 SAMME, RMIT
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AUTO-1029 Lecture 6 Slide 23
INSPECT MATLAB FILE
My detailed MATLAB/SIMULINK example file is
provided. In order to re-produce simulation you can
run one single file twoDOFprojectALL.m:
>> twoDOFprojectALL
Note that the SIMULINK model twoDOFprojectLQR.m
(shown on Slide 18) is run from inside MATLAB
using the following command:
>> sim(twoDOFprojectLQR);
Lecturer: Prof P.M.Trivailo c2012 SAMME, RMIT
http://-/?-http://-/?-