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Lecture – 27
Linear Quadratic Regulator (LQR) Design – I
Dr. Radhakant PadhiAsst. Professor
Dept. of Aerospace EngineeringIndian Institute of Science - Bangalore
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
2
LQR Design: Problem Objective
To drive the state of a linear (rather linearized) system to the origin by minimizing the following quadratic performance index (cost function)
( ) ( )0
1 12 2
where, 0 (psdf), 0 (pdf)
ftT T Tf f f
t
f
J X S X X Q X U RU dt
S Q R
= + +
≥ >
∫
X A X BU= +X
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
3
LQR Design: Guideline for Selection of Weighting Matrices
( )( )
2
2
0 (psdf), 0 (psdf), 0 (pdf)
These are usually chosen as diagonal matrices, with
maximum expected/acceptable value of 1/
maximum expected/acceptable value of 1/
maximum expected/a
i f
f
f i
i i
i
S Q R
s x
q x
r
≥ ≥ >
=
=
= ( )2cceptable value of 1/ iu
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
4
LQR Design: Some Facts to Remember
The pair needs to be controllable and the pairneeds to be detectable
(these are usually chosen as diagonal matrices)
By default, it is assumed that
Constrained problems (state and control inequality constraints) are not considered here. Those will be considered later.
{ },A B
0 (psdf), 0 (psdf), 0 (pdf) fS Q R≥ ≥ >
ft →∞
{ },A Q
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
5
LQR Design: Problem Statement
Performance Index (to minimize):
Path Constraint:
Boundary Conditions:
( )( )
( )( )
0,
1 12 2
f
f
tT T Tf f f
tL X UX
J X S X X Q X U RU dt
ϕ
= + +∫
X A X BU= +
( )( )
00 :Specified
: Fixed, : Freef f
X X
t X t
=
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
6
LQR Design:Necessary Conditions of Optimality
Terminal penalty:
Hamiltonian:
State Equation:
Costate Equation:
Optimal Control Eq.:
Boundary Condition:
( ) ( )12
T T TH X Q X U RU AX BUλ= + + +
( ) ( )12
Tf f f fX X S Xϕ =
X AX BU= +
( ) ( )/ TH X QX Aλ λ= − ∂ ∂ = − +
( ) 1/ 0 TH U U R B λ−∂ ∂ = ⇒ = −
( )/f f f fX S Xλ ϕ= ∂ ∂ =
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
7
LQR Design: Derivation of Riccati Equation
Guess:
Justification:
( ) ( ) ( )t P t X tλ =
( )( )
( )
From functional analysis theory of normed linear space,
lies in the "dual space" of , which is the space consisting
of all continuous linear functionals of .
Reference: Optimization by Vecto
t
X t
X t
λ
r Space Methods D. G. Luenberger, John Wiley & Sons, 1969.
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
8
LQR Design: Derivation of Riccati Equation
Guess ( ) ( ) ( )t P t X tλ =
( )( )( )
1
1
T
T
PX PXPX P AX BU
PX P AX BR B
PX P AX BR B PX
λ
λ−
−
= +
= + +
= + −
= + −
( ) ( )( )
1
1 0
T T
T T
QX A PX P PA PBR B P X
P PA A P PBR B P Q X
−
−
− + = + −
+ + − + =
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
9
LQR Design: Derivation of Riccati Equation
Riccati equation
Boundary condition
1 0T TP PA A P PBR B P Q−+ + − + =
( ) ( )is freef f f f fP t X S X X=
( )f fP t S=
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
10
LQR Design: Solution Procedure
Use the boundary condition and integrate the Riccati Equation backwards from to Store the solution history for the Riccati matrixCompute the optimal control online
( )f fP t S=
ft 0t
( )1 TU R B P X K X−= − = −
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
11
LQR Design: Infinite Time Regulator ProblemTheorem (By Kalman)
Algebraic Riccati Equation (ARE)
Note:ARE is still a nonlinear equation for the Riccati matrix. It is not straightforward to solve. However, efficient numerical methods are now available.
A positive definite solution for the Riccati matrix is needed toobtain a stabilizing controller.
1 0T TPA A P PBR B P Q−+ − + =
As , for constant and matrices, 0ft Q R P t→∞ → ∀
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Example – 1:
Stabilization of Inverted Pendulum
Dr. Radhakant PadhiAsst. Professor
Dept. of Aerospace EngineeringIndian Institute of Science - Bangalore
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
13
A Motivating Example:Stabilization of Inverted Pendulum
( )
2 2
System dynamics:
, /Linearized about vertical equilibrium point
n nu g Lθ ω θ ω= − =
1 2
1 12
2 2
System dynamics (state space form):
Define: ,0 1 0
0 1n
BXX A
x xx x
ux x
θ θ
ω
= =
⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤= +⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥−⎣ ⎦⎣ ⎦ ⎣ ⎦⎣ ⎦
2 22
0
2
Performance Index (to minimize):
1 1 2
1 0 1 ,0 0
J u dtc
Q Rc
θ∞ ⎛ ⎞= +⎜ ⎟⎝ ⎠
⎡ ⎤= =⎢ ⎥⎣ ⎦
∫
mgθ Lu
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
14
A Motivating Example:Stabilization of Inverted Pendulum
mgθ Lu1
1 2
2 3
2 2 2 22 22 1 2 2 32 3
2 2 2 23 2 2 3 31 2
2 2 2 22 2 2 2
ARE: 0
Let (a symmetric matrix)
1 0 0 00 0 0 0
Equations:12 1 0
T T
n n n
n
n n
PA A P PBR B P Qp p
Pp p
p p c p c p pp pp p c p p c pp p
p c p pc
ω ω ωω
ω ω ω
−+ − + =
⎡ ⎤= ⎢ ⎥⎣ ⎦
⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤+ − + =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎣ ⎦ ⎣ ⎦⎣ ⎦⎣ ⎦ ⎣ ⎦
− + = ⇒ = ± 4 2
2 21 3 2 3
2 22 3 3 2
0 (repeated)12 0 2
n
n
c
p p c p p
p c p p pc
ω
⎡ ⎤+⎣ ⎦
+ − =
− = ⇒ = ±
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
15
A Motivating Example:Stabilization of Inverted Pendulum
mgθ Lu
3
2
2 4 22 3 22
2 21 3 2 3
2 21 2 3 3
However, is a diagonal term, which needs to be real and positive.Hence, needs to be positive. Therefore
1 1, 2
Moreover,0
(not needed i
n n
n
n
pp
p c p pc c
p p c p p
p c p p p
ω ω
ω
ω
⎡ ⎤= + + =⎣ ⎦
+ − =
= −
( )
1 2 22 3
22 3
n this problem)
Gain Matrix:
Control:
TK R B P c p c p
u K X c p pθ θ
− ⎡ ⎤= = − −⎣ ⎦
= − = +
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
16
A Motivating Example:Stabilization of Inverted Pendulum
mgθ Lu
Analysis
( )
2 22
Open-Loop System:1
0
right half pole: unstable system
nn
n
I Aλ
λ λ ωω λ
λ ω
−− = = − =
−
= ±
( )
( )
2 4 2
2 22 2
1/ 22 23 2 2
Define: 1
1 2 2
n
n
n
c
pc
p pc c
ω ω
ω ω
ω ω
= +
= +
= = +
2 2 22 3
Closed-Loop System:0 1
Closed-Loop Poles: 0
CLn
CL
A A BKc p c p
I A
ω
λ
⎡ ⎤= − = ⎢ ⎥− −⎣ ⎦
− =
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
17
A Motivating Example:Stabilization of Inverted Pendulum
mgθ Lu
Analysis
( )( ) ( )
( )
1/ 22 2 2 2
1/ 2 1/ 22 2 2 21,2
2 4 2 2
Closed-Loop Poles:
2 0
1 1 2 2
Note:
n
n n
n n
j
c
λ ω ω λ ω
λ ω ω ω ω
ω ω ω
+ + + =
= − + ± −
= + >
Both of the closed-loop poles are strictly in the left-half plane.
Hence, the closed-loop is guaranteed to be “asymptotically stable”.
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Example – 2:
Finite-time Temperature Control
Dr. Radhakant PadhiAsst. Professor
Dept. of Aerospace EngineeringIndian Institute of Science - Bangalore
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
19
Example: Finite Time Temperature Control Problem
( )
0
where, : Constants : Temperature
: Ambient temperature (Constant = 20 C) : Heat input
n
n
a bu
a b
u
θ θ θ
θ
θ
= − − +
System dynamics :
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
20
Problem formulations
Case – 1: Case – 2:
( )
2
0
0
Cost Function:
12
30
(Hard constraint)
ft
f f
J u dt
t Cθ θ
=
= =
∫ ( )
( )
2 2
0
0
Cost Function:
1 302
0 : Weightage
i.e. 30
(Soft Constraint)
ft
f f
f
f
J s u dt
s
t C
θ
θ
⎡ ⎤= − +⎢ ⎥
⎢ ⎥⎣ ⎦>
≈
∫
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
21
Solution:
( ) ( )( ) ( )
( )2
Solution:, 0
, 0 012
0
a a
a a
x
x ax bu x
u ax bu
ax
u bu
θ θ θ θ
θ θ
λ
λ λ
λ
− =
= − + = − =
Η = + − +
∂Η⎛ ⎞= − =⎜ ⎟∂⎝ ⎠∂Η
= ⇒ = −∂
Necessary conditions
x ax bua
u bλ λ
λ
= − +
== −
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
22
Solution: Case – 1 (Hard constraint)( ) ( )
( )
( )
( ) ( )
2
Taking laplace transform:
0
f f
f
f
a t t a t tf f
a t tf
a t tf
e e
u be
x ax b e
sX s x
λ λ λ
λ
λ
− − −
− −
− −
= =
= −
= − −
− ( ) 2
0
1fatfaX s b e
s aλ −
⎡ ⎤ ⎛ ⎞⎢ ⎥ = − − ⎜ ⎟−⎢ ⎥ ⎝ ⎠⎣ ⎦
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
23
Solution: Case – 1 (Hard constraint)
( )
( )
( )
22 2
2
2
Unknown
0
1
1 1 12
1Hence ( )2
However, ( ) 10
f
f
f
atf
atf
at at atf
f f a
X s b es a
b ea s a s a
x t b e e ea
x t C
λ
λ
λ
θ θ
−
−
− −
⎛ ⎞= − ⎜ ⎟−⎝ ⎠⎛ ⎞= − −⎜ ⎟− +⎝ ⎠
= − −
= − =
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
24
Solution: Case – 1 (Hard constraint)
( )
( )
( )
2
22
22
2
1( ) 102
10 12
201
( )
f f f
f
f
at at atf f
atf
f at
x t b e e ea
be
a
ab e
x t b
λ
λ
λ
− −
−
−
= = − −
⎛ ⎞= − −⎜ ⎟⎜ ⎟
⎝ ⎠−
=−
= −20 a−
2b ( ) ( ) ( )( )2
10121
f
f f f
at atat at at
at at at
e ee e e
ae e e
−− −
− −
⎛ ⎞ −⎜ ⎟ − =⎜ ⎟− −⎝ ⎠
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
25
Solution: Case – 1 (Hard constraint)
( )10( )
f fat at
f
e ex t
−−=
Note :
( )f fat ate e−−10
(i.e. The boundary condition is "exactly met".)
( )u t b
=
= −
Controller :
( )2
20fa t t aeb
− − −
( ) ( )2
201 f f f
at
at at at
aee b e e− −
⎡ ⎤ ⎡ ⎤−⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥− −⎣ ⎦ ⎣ ⎦
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
26
Solution: Case – 2 (Soft constraint)
( )
( )
( ) ( )
0 0
2 2
0
2
30 10 .
Hence the cost function is
1 102
10 10
However, we have
2
f
f
f f
t
f f
ff f f f
f
at at atf
C x C
J s x u dt
s x xs
bx t e e ea
θ
λλ
λ − −
→ ⇒ →
⎡ ⎤= − +⎢ ⎥
⎢ ⎥⎣ ⎦⎛ ⎞
= − ⇒ = +⎜ ⎟⎜ ⎟⎝ ⎠
= − −
∫
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
27
Solution: Case – 2 (Soft constraint)
( ) ( )
( )
( )( ) ( )
( )
22
22
22
22
At , 1 102
1. . 1 102
20. .
2 1
20Hence
2 1
f
f
f
f f
f
at ff f f
f
atf
f
ff at
f
a t t a t t ff at
f
bt t x t ea s
bi e es a
s ai e
a s b e
s ae e
a s b e
λλ
λ
λ
λ λ
−
−
−
− − − −
−
= = − − = +
⎡ ⎤+ − = −⎢ ⎥
⎢ ⎥⎣ ⎦⎡ ⎤−⎢ ⎥=⎢ ⎥+ −⎣ ⎦
⎡ ⎤−⎢ ⎥= =⎢ ⎥+ −⎣ ⎦
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
28
Solution: Case – 2 (Soft constraint)
( )
( )
( )
( )
( )
22
2
20
2 1
10
2
f
f
f
f f f
a t t fat
f
ata t t f
at at atf
u t b
s abe
a s b e
s abebe
s bae e e
λ
− −
−
− −
−
= −
⎡ ⎤−⎢ ⎥= −⎢ ⎥+ −⎣ ⎦⎡ ⎤⎢ ⎥⎢ ⎥= −⎢ ⎥+ −⎢ ⎥⎣ ⎦
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
29
Correlation between hard and soft constraint results
( )( )
( ) ( )
. . 2
. .
As ,
10lim lim1
2
20
. . The "soft constraint" problem behaves like the "hard constraint" problem when .
f ff f f
f f
f
at
S Cs sat at at
f
at
at at H C
f
s
abeu tbae e e
s
ae u tb e e
i es
→∞ →∞−
−
→ ∞
=⎛ ⎞
+ −⎜ ⎟⎜ ⎟⎝ ⎠
= =−
→∞
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
30