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EE392m - Spring 2005Gorinevsky
Control Engineering 13-1
Lecture 13 – Programmed Control
• Optimal control: programmed, synthesis• Rocket ascend • Grade change in process control
– Example
• QP optimization • Flexible dynamics: input shaping, input trajectory
– example
• Robotics path planning
EE392m - Spring 2005Gorinevsky
Control Engineering 13-2
Open-loop (programmed) control• Control u(t) found by solving an
optimization problem. Constraints on control and state variables.
• Used in space, missiles, aircraft FMS – Mission planning – Complemented by feedback corrections
• Sophisticated mathematical methods were developed in the 60s-70s to overcome computing limitations.
UX ∈∈→
=
uxtuxJ
tuxfx
,min),,(),,(&
)(* tuu =Optimal control:
EE392m - Spring 2005Gorinevsky
Control Engineering 13-3
Optimal Control Synthesis
• Example: Time optimal control – 2nd order system
• Toy problems, building blocks• MPC - will discuss in Lecture 14
1|| ≤=
uux&&
x
x&
1−=u
1=u
• Find optimal control program for any initial conditions• At any point in time apply control that is optimal for the
current state. This is feedback control!– Optimality principle: an optimal trajectory part is an optimal trajectory
• Example: LQ control – Linear system; Quadratic index– Analytically solvable problem
EE392m - Spring 2005Gorinevsky
Control Engineering 13-4
Optimal Control• Formulate performance index and constraints • Programmed control
– compute optimal control as a time function for particular initial (and final) conditions
x
t
0)(0)(
min)(
≤=→
XgXhXJMinimize:
subject to:NLP:
• Modern practical approach– Discretize trajectory and control – Solve using a Nonlinear Programming
(NLP) software package– Use sparse matrix arithmetics
EE392m - Spring 2005Gorinevsky
Control Engineering 13-5
Ascend trajectory optimization • Rocket launch vehicles
– fuel (payload) optimality– orbital insertion constraint– flight envelope constraints– booster drop constraint
Kourou, French Guiana
EE392m - Spring 2005Gorinevsky
Control Engineering 13-6
Trajectory optimization
• Nonlinear constrained optimization problem• Nonlinear Programming (NLP): not QP, not LP
– Iterative optimization methods: Gradient, Newton, Levenberg-Marquardt, SQP, SSQP
– Need supervision by a human
• Recall: QP, LP are guaranteed to always produce a solution if the problem is feasible
- Can be possibly used on-line, inside a control loop
EE392m - Spring 2005Gorinevsky
Control Engineering 13-7
Optimization of Process Transitions
• Process plants manufacture different product varieties (grades)• Need to optimize transitions from grade to grade
EE392m - Spring 2005Gorinevsky
Control Engineering 13-8
Product Grade Change
• The requirement: to change from manufacturing grade A to grade B with minimum off-spec production.
• The implementation: using detailed models of process and operating procedures.
• The results: optimum setpointtrajectories for key process controls during the changeover, resulting in minimum lost revenue.
Polymerization process
EE392m - Spring 2005Gorinevsky
Control Engineering 13-9
Grade change control example• Example process model: chemical reactor
0 20 400
0.02
0.04
0.06
0.08
0 20 40-0.2
-0.1
0
0.1
0.2
uy1
y2
internal temperature
quality variable
• The process is in an initial steady state: u = 0; y1= y2 = 0• Need to transition, as quickly as possible, to other steady state:
u = const; y1= const; y2 = yd
995.0001746.0
1 −=
zP
8187.003625.05
12 −−= −
zzPP
EE392m - Spring 2005Gorinevsky
Control Engineering 13-10
Grade change control example
• Linear system model in the convolution formuhy *=
• Inequality constraints– Control
– Temperature
• Equality constraint (process transitioning to the new grade)fd TTtTytyty +≤≤≡≡ for ,)(,0)( 21&
• Quadratic-optimal controlmin)1()()( 22
2 →−−+−∑t
d tuturyty
*2
*
)(
)(
Tty
utu
≤
≤
T fT0
EE392m - Spring 2005Gorinevsky
Control Engineering 13-11
Grade change control example• Sampled time: t = 1, …, N• Vector-matrix formulation of the problem
HUUHUH
YY
Y =⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡=
2
1
2
1
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
)(
)1(,
)(
)1(,
)(
)1(
2
2
2
1
1
1
Ny
yY
Ny
yY
Nu
uU MMM
uhUH *11 =uhUH *22 =
• H is a block-Toeplitz matrix
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−=
)1()2()(0)1()1(
00)1(
111
11
1
1
hhNhhNh
h
H
L
L
MOOM
L
EE392m - Spring 2005Gorinevsky
Control Engineering 13-12
Grade change control example
• Inequality constraints– Control
– Temperature *1
**
0 TYuUu
≤≤≤≤−
• Quadratic-optimal control
min)()( 22 →+−−= DUDrUYYYYJ TTd
Td
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
1
1Mdd yY
min...2 222 →+−+= YYYYDUDrUJ Td
TTT
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−
−
=
000110
011
L
L
MOOM
L
D
• Dynamics as an equality constraint:
0=− YHU⎥⎦
⎤⎢⎣
⎡=
2
1
YY
Y
Notations
EE392m - Spring 2005Gorinevsky
Control Engineering 13-13
Terminal constraint
⎥⎦
⎤⎢⎣
⎡=
f
T
YY
Y,1
,11
• Equality constraints (new grade steady state)
fdf YY ,,2 =
0,1 =fDY
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−
−
=
000110
011
L
L
MOOM
L
D
⎥⎦
⎤⎢⎣
⎡=
f
T
YY
Y,2
,22
- transition - new steady state
- steady in the end
- at target in the end
t = 0 t = T t = Tf
EE392m - Spring 2005Gorinevsky
Control Engineering 13-14
Quadratic Programming
• QP Problem:
min21 →+=
=≤
xfQxxJ
bxAbAx
TT
eqeq
• Matlab Optimization Toolbox: QUADPROG
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
2
1
YYU
x......
...
=
==
eq
eq
bAA
...
...==
fQ
See previous slides
EE392m - Spring 2005Gorinevsky
Control Engineering 13-15
SimQP Program for
the grade change, no terminal constraint; slow T=40
20205.01.0
75.0
*
*
==
==
=
uTr
yd
τ
0 5 10 15 20 25 30 35 400
5
10
15
20
CO
NTR
OL
INPU
T (M
V)
0 5 10 15 20 25 30 35 400
0.5
1
1.5
2TE
MPE
RAT
UR
E
0 5 10 15 20 25 30 35 40-2
-1
0
1
QU
ALIT
Y PA
RAM
ETE
R
TIME (HRS)
EE392m - Spring 2005Gorinevsky
Control Engineering 13-16
SimQP Program for
the grade change with a terminal constraint at T = 8
20205.01.0
75.0
*
*
==
==
=
uTr
yd
τ
0 1 2 3 4 5 6 7 8 9 10-5
0
5
10
15
20
CO
NTR
OL
INPU
T (M
V)
0 1 2 3 4 5 6 7 8 9 10-0.5
0
0.5
1
1.5
2TE
MPE
RAT
UR
E
0 1 2 3 4 5 6 7 8 9 10-3
-2
-1
0
1
QU
ALI
TY P
ARAM
ETER
TIME (HRS)
EE392m - Spring 2005Gorinevsky
Control Engineering 13-17
Flexible Satellite Slew Control
• Single flexible mode model• Franklin, Section 9.2
)()()()(
212122
212111
xxbxxkxJguxxbxxkxJ
&&&&
&&&&
−+−=+−−−−=
BusFirst
flexiblemode
gu
2x→1x→ k
breaction wheel
EE392m - Spring 2005Gorinevsky
Control Engineering 13-18
Flexible Satellite Slew Control
• Linear system modelCxy
BuAxx=
+=&
BusFirst
flexiblemode
gu
2x→1x→ k
b
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
2
2
1
1
xxxx
x
&
&
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−=
1
12
1
xxx
xy
&
• slew angle
• deformation
• slew rate
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−−
−−=
2222
1111
////1000////0010
JbJkJbJk
JbJkJbJkA
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
00/0
1JgB
EE392m - Spring 2005Gorinevsky
Control Engineering 13-19
Flexible Satellite Slew Control
• Linear system model in the convolution formuhy *=
• Inequality constraints– Control
– Deformation
– Slew rate
• Equality constraint (system coming to the target slew angle)fd TTtTyty +≤≤≡ for ,)(
• Quadratic-optimal controlmin)( 2 →∑
t
tu
*3
*2
)(
)(
1)(
vty
dty
tu
≤
≤
≤
EE392m - Spring 2005Gorinevsky
Control Engineering 13-20
Flexible Satellite Slew Control• Sampled time: t = 1, …, N• Y is a 3N vector; H is a block-Toeplitz matrix
HUY =
• Inequality constraints– Control
– Deformation
– Slew rate
• Equality constraint (system coming to the target slew angle)dYYS =1
• Quadratic-optimal controlmin→UU T
*3*
*2*
11
vYSvdYSd
U
≤≤≤≤
≤≤− This is a QP problem
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−=
1
12
1
xxx
xy
&
EE392m - Spring 2005Gorinevsky
Control Engineering 13-21
SimQP Program
for the flexible satellite slew
0 2 4 6 8 10 12 14 16 18 20-1
-0.5
0
0.5
1
CO
NTR
OL
0 2 4 6 8 10 12 14 16 18 200
0.5
1
1.5
2S
LEW
AN
GLE
0 2 4 6 8 10 12 14 16 18 20-0.05
0
0.05
DE
FOR
MA
TIO
N
0 2 4 6 8 10 12 14 16 18 20-0.05
0
0.05
0.1
0.15
SLE
W R
ATE
2.002.0
0036.0091.01.0
102.0
*
*
2
1
==
====
=
vdbkJJg
EE392m - Spring 2005Gorinevsky
Control Engineering 13-22
Robust design approach
• Replace exact terminal constraint by a given residual error• Consider the system for several different values of
parameters and group the results together• As an optimality index, consider an average performance
index or the worst residual error
EE392m - Spring 2005Gorinevsky
Control Engineering 13-23
Mobile Robot Path Planning→ min
(p,q)
Point mass model:
Constrained optimization problemof finding an optimal path
From V. Lumelsky, Univ. Wisconsin, Madison
Future Combat Systems (FCS)
Follow-OnHeavy Forces(if Necessary)
Enem
y Fo
rce
Build
Up
• Technology-based transformation of military force • Ground and air robotics vehicles• Application of robotics research• Path planning and optimization are important