analysis and design of linear control system –part2-€¦ · instructor: prof. masayuki fujita....

Analysis and Design of Linear Control System –Part2-

Instructor: Prof. Masayuki Fujita

7th Lecture

11.2 Feedforward DesignFeedforward2 Degree of Freedom

Keyword :

11.4 Feedback Design via Loop Shaping: Example

Lead and Lag CompensationKeyword :

11 Frequency Domain Design

11.3 Performance SpecificationsTime Domain Analysis Step Response

Keyword :

(pp. 326--331)

(pp. 319--322)

(pp. 322--326)

(b) Simplified model

: mass: force moment arm: damping: inertia

(2.27)

θθθ sincossin 21 uuxcmgxm −+−−=

θθθ cossin)1(cos 21 uuycmgym ++−−=

1ruJ =θ

11 Fu =mgFu −= 22

mc

Jr

[Ex. 11.6] Roll control for a vectored thrust aircraft*(§2.4, Exe. 8.10)

(a) Harrier “jump jet”Fig. 2.17

from to

Fig. 11.12(a)

Only P control is not allowed

Less than 10%

Specification1u θ

21)(

JsrGsP u == θ

ksC =)( 200=k

)( ωjP )( ωjL

Error in stationary state: Less than 1%

Tracking error up to around10 [rad/s]:

[Ex. 11.6] Roll control for a vectored thrust aircraft*

0=mϕ

0

-180

020

-180

110 210010110−

110 210010110−

110 210010110−

110 210010110−

Lead compensator

Loop transfer function

• improve phase margin (by )

• improve convergence speed (by )

(11.12)

40[deg]

20[dB]loop gain

bsasksC

++

=)(

50 2 950 === bak

r y)(sC )(sPe∑

1−

)()()( sCsPsL =

°= 40mϕ

a b

[Ex. 11.6] Roll control for a vectored thrust aircraft*

)( ωjL

)( ωjP

502

++

ss

k

Gang of Four

Frequency response

In practice, roll-off filter is necessary

T

PS

S

CS

[Ex. 11.6] Roll control for a vectored thrust aircraft*

58.1=TM84.1=SM

closed loop with lead compensator

Feedforward(2 Degree of

Freedom)

P

[Ex. 11.6] Roll control for a vectored thrust aircraft*Step response

11.6 Design Example

(b) Simplified model

[Ex. 11.12] Lateral control of a vectored thrust aircraft

(a) Harrier “jump jet”Fig. 2.17

( 2.4, Exe. 8.10)

Ex. 11.6: controller for the roll dynamicsEx. 11.12: controller for the position of the aircraft

(stabilization of both the attitude and the position)

inner / outer loop design methodology

Fig. 11.17 Inner / outer loop design

inner loop ( ): the roll dynamics and control outer loop: the lateral position dynamics and controller

Design so that provides fast and accurate control of the roll angle.

l.

2. Design for the lateral position under the approximation that we can directly control the roll angle .

iHiC

oC

iH

[Ex. 11.12] Lateral control of a vectored thrust aircraft

Performance specification (entire system)• zero steady-state error in the lateral position• a bandwidth of 1 rad/s• a phase margin of 45°

Performance specification (inner loop)• the low-frequency error to be no more than 5 %• a bandwidth of 10 rad/s (10 times that of the outer loop)

[Ex. 11.12] Lateral control of a vectored thrust aircraft

[Ex. 11.12]

lead compensator ([Ex. 11.6])

transfer function from to

inner loop Actual roll dynamics

Fig. 11.17

Fig. 11.18 (a)

50)2(200)(

++

=s

ssCi

2)(JsrsPi =

ii

iii PC

mgPCsH+−

=1

)1()(

iH

1u

1u

Performance specification (inner loop) ○

2.39)0( −=−≈ mgHi

approximate

Assume that the inner loop will eventually track our commanded input.

Lead compensatorLateral position dynamics

Fig. 11.17

Fig. 11.18 (a)

o

ooo bs

asksC++

−=)(

2.39)0( −≈iH

)()0()( sPHsP oi=

csmsmg+

−= 2

[Ex. 11.12]

At , magnitude 1

Lead compensator

o

ooo bs

asksC++

−=)(

Phase lead flattens out at approximately 10/obDesired crossover ]rad/s[ 1=gcω

10=obEnsure adequate phase lead

ooo bab <<1010/3.0=oa

]rad/s[ 1=gcω

csmsHsPHsP i

i +== 20

)0()()0()(

98.0=ok10

3.098.0)(++

−=sssCo

[Ex. 11.12] Lateral control of a vectored thrust aircraft

ob ob1010/ob

°0°− 45°−90

+

∠obs

1

oa oa1010/oa°0°45°90

( )oas +∠

)45( °≥mϕ

Combine the inner and outer loop controllers and verify that the system has the desired closed loop performance.

Fig. 11.19 (a) Bode plot

(b) Nyquist plot

• : 1 [rad/s]• phase margin : 68°• gain margin : 10

410− 310− 210− 110− 010 110 210 310

90−

180−

270−

360−

[Ex. 11.12] Lateral control of a vectored thrust aircraft

150

100

50

050−

1−

Performance specification(entire system) ○

gcω

mϕ

mg

Gang of Four

Fig. 11.20 Gang of Four for vectored thrust aircraft system

Not have integral action[Ex. 11.12] Lateral control of a vectored thrust aircraft

T PS

CS S

18.1=TM

11.1=SM

Feedforward Design

Controller with two degrees of freedom (2DOF)• A combination of feedforward and feedback controllers. • Response to reference signals can be designed independently of the design for disturbance attenuation and robustness.

Feedforward• Improve the response to

reference signals

Feedback• Give good robustness • Disturbance attenuation

Fig. 11.1 Simple 2DOF controller

FeedbackFeedforward nd

ur

y

yη)(sC )(sPe ν

∑∑∑)(sF

1−

( )SFPFF

PCFPFF

PCFCFPsG

mum

mum

umyr

−+=

+−

+=

++

=

1

1)()(

From reference signal to process output

desired response

Ideal Feedforward

Feedback(11.4)

(11.5)

smallPC

S+

=1

1

PFF m

u =

yr

)(sC∑ye νfbumy

)(sFm

)(sFu

)(sP

r∑

1−

ffu

: ideal response of the system to reference signals

: feedforward reference controller

mF uF

smallFeedback

Ideal Feedforward(11.4)

(11.5)

for all uncertaintiesRobust Performance

PCFPFFsG mu

myr +−

+=1

)(

PCS

+=

11

PFF m

u =

)(sC

y

my

)()(

sPsFm

)(sFu

)(sPr

1−

∑

∑)(sFm

)(sC∑ye νfbumy

)(sFm

)(sFu

)(sP

r∑

1−

ffu

δ

Generalized controller with two degrees of freedom (2DOF)

Fig. 11.3 2DOF controller for improved response to reference signals and measured disturbances

: ideal response of the system to reference signals : signal which gives the

desired output when applied as input to the process

: feedback controller

: desired output

: load disturbance: reference signal

: feedforward reference controller

: feedforward disturbance controller

: process

: feedback control input

mF

C

uF

dF

21, PP

myffu

fbudr

)(sC

∑

∑

d

yηe νfbu

fdu

ffumy

)(sFm

)(sFu

)(1 sP )(2 sP

)(sFdr∑ ∑

1−

12PPP =

12PPP =

Fig. 11.3

from load disturbance to the process output ideal feedforward

feedback

*

(11.6)(11.7)

small

)(sC

∑

∑

d

yηe νfbu

fdu

ffumy

)(sFm

)(sFu

)(1 sP )(2 sP

)(sFdr∑ ∑

1−

dy

SPFPPC

PFPsG

d

dyd

)1( 1

)1()(

12

12

+=++

=

PCS

+=

11

11−−= PFd

12PPP =

Generalized controller with two degrees of freedom (2DOF)

[Ex. 11.2] Vehicle steering

from steering angle to lateral deviation

Fig. 2.16 Vehicle steering dynamics

*2

)1()(sssP +

=γ ba /=γ

δ y

( stable as long as )

Fig. 11.3

from (11.5)

a nice response without overshoot

must be proper

Step Response of

Time [s]

2

2

)()(

asasFm +

=

2

22

))(1( asssa

PFF m

u ++==

γ0>γ

2

)1()(sssP +

=γ

)(sC∑yηe νfbumy

)(sFm

)(sFu

)(1 sP )(2 sP

r∑

1−

ffu12PPP =

[Ex. 11.2] Vehicle steering

desired response2=a 1=a

5.0=a

Fig. 11.4(a) Overhead view

Fig. 11.4(b) Position and steering

5.0=a

[Ex. 11.2] Vehicle steering

3

3

)15.0()1(+

+==

ss

PFF m

u

[Ex. 11.3] Third-order system

process

feedforward controller

desired response

must be proper

Step Response

)(sC

y

my

)()(

sPsFm

)(sFu

)(sPr

1−

∑

∑)(sFm

PI feedback controller

sssC 5.06.0)( +

=

3)1(1)(+

=s

sP3)15.0(

1)(+

=s

sFm

Fig. 11.5 (a) Step response Fig. 11.5 (b) Frequency response

)( ωjGyr

)( ωjGur

[Ex. 11.3] Third-order system

y

u

bandwidth

and are the most widely used

Error tolerance ∆

∆±

sT

pT

pM

rT %90~%10%2=∆ %5=∆

1

0

Peak timeSettling timeRise time

Overshoot

Performance criteria

rT

pM

sT

pT

Step response11.3 Performance Specifications*

Time Responses (§5.3 )

)(sC

y

my

)()(

sPsFm

)(sFu

)(sPr

1−

∑

∑)(sFm

Ideal feedforward

rFy m=PFF m

u =

Feedforward Design*

How to build ?mF

1)(

+=

TsKsFm

First-order System

=∆=∆≈ %2if4

%5if3TTTs

Settling timeTTTr 2.2)9(ln ≈=

Rise time

The step response has no overshoot 0 1 2 3 4 5

Tt /

2.0

4.0

6.0

8.0

1

0

Step Response∆±

sT

K×TK

0≥ζ : damping ratio22

2

2)(

nn

nm ss

KsFωζω

ω++

=

Second-order System*: natural frequency0>nω

21/ ζζπ −−= KeM p

dpT ω

π=Peak time

Overshoot

∆±

sT

pT

pM

rT %90~%10

1

0

Step ResponseK×

=∆

=∆=

%5if3%2if4

n

nsT

ζω

ζω

Settling timed

rT ωζπ arcsin2/ +≈

Rise time ( )21 ζωω −= nd

Butterworth Filter*Low-pass filter

: cutoff frequencycω: order of filtern

n

c

mFjF

20

1)(

+

=

ωω

ω

: DC gain

1=cω

1=n

2=n

3=n4=n

5=n

0F

10 =F

n20− [dB/dec]slope:

Im

Re

3=n1=cω

cs ω+

22 4.1 ccss ωω ++

3223 0.20.2 ccc sss ωωω +++

432234 6.24.36.2 cccc ssss ωωωω ++++

54233245 24.324.524.524.3 ccccc sssss ωωωωω +++++

Denominator polynomial

1=n

2=n

3=n

4=n

5=n

s s1

Low-pass filter High-pass filter

Butterworth Filter*

7th Lecture

11.2 Feedforward DesignFeedforward2 Degree of Freedom

Keyword :

11.4 Feedback Design via Loop Shaping: Example

Lead and Lag CompensationKeyword :

11 Frequency Domain Design

11.3 Performance SpecificationsTime Domain Analysis Step Response

Keyword :

(pp. 326--331)

(pp. 319--322)

(pp. 322--326)