1ME 233 Advanced Control II
Continuous-time results 5
Frequency-Shaped
Linear Quadratic Regulator(ME233 Class Notes pp.FSLQ1-FSLQ5)

2Outline• Parseval’s theorem• Frequency shaped LQR cost function• Implementation

3

Infinite Horizon LQR
nth order LTI system:

Find the optimal control:

which minimizes the cost functional:

4

Parseval’s theorem• Let• Its (symmetric) Fourier transform is defined byand5

Parseval’s theorem

where(complex conjugate transpose)

6

Parseval’s theorem
Proof:

7

Frequency Cost Function
By Parseval’s theorem, the cost functional:

withis equivalent to8Frequency-Shaped Cost Function
Key idea: Make matrices Q and R functions offrequency

where

9

Frequency-Shaped Cost Function
Define the state and input filters

state filter

input filter

10Frequency-Shaped Cost Function

can be written

11Realizing the filters using LTI’s

can be realized byso thatis causal or strictly causal.

12

Realizing the filters using LTI’s

can be realized by (withso thatis causal (but not strictly causal).

)

13

Example Hard Disk Drive
Consider a simplified model of a voice coil motor and suspension

Magnitude (dB)

Bode Diagram
0-50-100

uncertainty-150

nominal model

Phase (deg)-2000actual plant-180-360010

110

nominal model

231010Frequency (rad/sec)

410

high-frequency resonancemodes are neglected in thenominal model

14

Example Hard Disk Drive

Phase (deg)Magnitude (dB)

Bode Diagram
0-50-100-150

uncertainty-2000

nominal model
actual plant-180-360010

110

2

31010
Frequency (rad/sec)

output is position

410

nominal model

15

Example: Frequency State Weight Q(jω)
Magnitude (dB)

Bode Diagram
0-50-100-150

Phase (deg)-2000

we want zero steady state (i.e. dc)
position-180-360
010

1102341010
Frequency (rad/sec)

10Example
Set weight on

toSet weight on

toset position cost function weight
to go to ∞ as ω → 0

16

Example: Frequency State Weight Q(jω)
Magnitude (dB)

Bode Diagram
0-50-100-150

Phase (deg)-2000set position weight
to go to ∞ as ω → 0-180-360
010

110

Example231010Frequency (rad/sec)

410

17

Example: Frequency State Weight Q(jω)
state space realization

Examplestate space realization

18

Example Hard Disk Drive

weights:nominal model+-s

B

(s)

X(s)

FS-LQR is a dynamic
state feedback

K(s)

U(s)

sufficient condition for robustness

19

Example Hard Disk Drive-s

B

(s)

X(s)

K(s)

U(s)

weights:

1008060
Gain [dB]+40200-20potential
lack of robustness -40

010

110

2

31010
Frequency [rad/sec]

410

20

Example: Frequency Control Weight R(jω)

increase control penalty
at high-frequencies
Example10035803060
Gain [dB]Gain [dB]2520154020100
50 010-20110231010Frequency [rad/sec]

410-40 010110231010Frequency [rad/sec]

41021

Example: Frequency Control Weight R(jω)

Example3530

Gain [dB]25201510state space realization

50 010110231010Frequency [rad/sec]

41022

Example: Frequency Control Weight R(jω)+

Examples

-35B

(s)

X(s)

K(s)

U(s)

30

20100

1510
50 010110231010Frequency [rad/sec]

41050
Gain [dB]Gain [dB]250sufficient robustness
condition is satisfied-50 010110231010
Frequency [rad/sec]

410

23

Cost Function Realization

is equivalent to24

Cost Function Realization

we know that,state space realization

state space realization

25

Cost Function Realization

Plus:

define extended state

26

Cost Function Realization

We can combine state equations and output as follows:27

Extended System Dynamics

28

Extended System Cost

results in29

Extended System Cost

where

30

Extended System Cost

where

31

Extended System LQR
Given the extended dynamics

Find the optimal control:

which minimizes the cost extended functional:

32

Extended LQR Solution

whereThenis stabilizable
is detectable

There exists astabilizing optimal
control shown in thenext page

33

Extended LQR Solution
Optimal Control:

whereand34

Implementation• Control

35

Block Diagram
++++

36

Equivalent Block Diagram

R(s)
-UB

sXK(s)37

FSLQR with reference input• For simplicity, lets assume a scalar output• Assume that we want to design a FSLQR thatwill achieve asymptotic output convergence toa reference input

38

FSLQR with reference input• Assume that the reference input r(s) satisfies• Where Ar(s) has root in the imaginary axis

39

Reference input examples• Assume that r(t) = ro• Assume that

40

FSLQR with reference input• Define the reference frequency weight• WhereAr(s) is the denominator of r(s)41Frequency-Shaped Cost Function• withused for achieving

42Frequency-Shaped Cost Function

43

Realizing the filters using LTI’s

can be realized bysuch that

denominator of r(s)44Realizing the filters using LTI’s

can be realized bysuch that

45

Realizing the filters using LTI’scan be realized bysuch that

46

Cost Function Realization

where,

47

Extended System Dynamics

48

Extended System Cost

49

Extended LQR Solution

whereThenis stabilizable
is detectable

There exists astabilizing optimal
control shown in thenext page

50

Extended LQR Solution
Optimal Control Gain:

whereand51

Block Diagram++++++++52

FSLQR with reference input – Block Diagram+++---

where

53

Block Diagram+++---

Remember that the poles ofare Ar(s) , and54

SISO+++---

The close loop dynamics from r(s) to E(s) will beof the form

55

SISO+++---

Therefore, since Ac(s) is Hurwitz,

56

Example Hard Disk Drive

Cost weights:

035-5030
25-100Gain [dB]

Magnitude (dB)

Bode Diagram-150-200
020
1510Phase (deg)50 010-180-360
010

1102

reference input
1102

31010
Frequency (rad/sec)

410

3

1010
Frequency [rad/sec]

410

Example Hard Disk Drive – Robustness Analysis

57

58

Example Hard Disk Drive
Magnitude (dB)

Bode Diagram
0-50-100-150

35-18030-360
010

110210Frequency (rad/sec)

310

Gain [dB]2541020151050 010Step Response
1.4110231010
Frequency [rad/sec]

4101001.2
150Gain [dB]

Amplitude

Phase (deg)

-200
0

0.8
0.6

00.4
0.2
000.02

0.04

0.06
0.08
Time (sec)

0.1

0.12

0.14

-50 0
101102

31010
Frequency [rad/sec]

41059FSLQR with reference input – State Estimation+++---Assume that :• state x(t) is not measurable• only output y(t) is available
Use state observer

60FSLQR with reference input – State Estimation

Robustness analysis:

61

Loop Transfer Recovery

Assume thatis square and has no unstable zeros

observer:observergain

62

Loop Transfer Recovery
Assume thatis square and has no unstable zeros

Make it approximate(point-wise in s)

63

Example Hard Disk Drive
Magnitude (dB)

Bode Diagram
0-50-100-150-200
03025
Gain [dB]-180-360
01020
15
10

110210Frequency (rad/sec)

31041050 010Bode Diagram

110231010
Frequency [rad/sec]

Bode plots ofMagnitude (dB)

100Phase (deg)Phase (deg)35target

500-50-100-150
13590450-45-90-135-180 0
10target110210Frequency (rad/sec)

31041041064

Example Hard Disk Drive
Magnitude (dB)

Bode Diagram
0-50-100-150
3530

Gain [dB]25-1802015105-360
0
10

1102

10
Frequency (rad/sec)

3104

100 0
10

1102

310
10
Frequency [rad/sec]

41050
Step Response
1.4

Gain [dB]1.2
1
Amplitude

Phase (deg)

-200
00.8
0.6

00.4
0.2
000.02

0.04

0.06
0.08
Time (sec)

0.1

0.12

0.14

-50 2
10

3

10

410