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