北京化工大学自动控制原理期末复习
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x(ABK)x
then the characteristic equation of the CL system is
detsI(AB K )0
1 ,2 , ,n
(a)
2 Assuc m( s e) th( s e de1 s) ( irs e d 2 l) oca( ts i onn a) r e0 known,
x
1
x 1 a 1 x 1 a 2 x 2 a 3 x 3 u 0
0 1
0
x
0
u
0 0
x2 x 1
x3 x2
y b1 b2 b3
5
• Dynamic response of SS model
Consider the following linear time-invariant system
– Control canonical form – Observer canonical form – Modal canonical form
U
b1
b2
Y
1
s
1 x1c s
x 2c
1 s
b
x3c 3
a1 a2
a3
Namely x1,x2,x3
a1 a2 a3 1
We obtain
• Observability : output ----- state • Their physical meanings?
• Criterion
• rank criterion
rB a A n A 2 B B k A n 1 B n
• PBH rank criterion
r a n k s I AB n , s C
(1) represents the power series of the matrix At
eAt IAt1A2t2 1Aktk
2!
k0k!
(2) using typical matrix, such as Diagonal or Jordan matrix
1
A
Pwk.baidu.com
0
0
P
1
n
e1t
eAt
Course Review
Youqing Wang
College of Information Science & Technology 2019.11.15
Final Exam
• Total 50 scores, 6 questions • 3 questions for linear continuous systems (29) • 2 questions for linear discrete systems (16)
L(eA t)IssA 2A s3 2 (sIA )1 eAt L1[(sIA)1]
example：
L1(s2s 5s56)L1(s32s23)
3e2t 2e3t
8
Controllability and Observability
• Controllability : input ----- state
How to select pole position
• Dominant second-order poles---deals with pole selection without explicit regard for control effort
– Natural frequency + damping factor
then the corresponding desired characteristic equation:
10
F H
G
J
Nx Nu
0 1
NNuxHF
G10 J 1
r
Nu
uNurK(xNxr)
Kx(Nu KNx)r
Nx
-K
u
y
Plant
x
uKxNr
r
N
u
y
Plant
x
-K Full state feedback with reference inpu1t1
• Linear continuous-time system theory • Linear discrete-time system theory
• Overview: transfer function state space model • Classical control vs modern control • “Transfer function” to “SS model” (multi-choice) • Canonical form
P
0
0
P1
ent
1 1
1 1
AQ
1 2
Q 1 1
2
e1t
te1t
t 2 e1t 2
e At
Q
e1t te1t e1t
Q
1
e2t te2t
e2t
7
Calculation of e At -2
（3） taking the inverse Laplace transform of (sIA)1
Observer (duality)
correction term L(yC~ x), close-loop observer：
~ x A ~ x B L ( u y C ~ x ) ( A L ) ~ x C B L uy
• optimal control (or symmetric root locus)--specifically address the issue of achieving a balance between good system response and
control effort1 .G 0sG 0s0
• System destruction according to controllability
and observability
9
Controller design
• Check controllability • Full-state feedback control
1 Substituting the feedback law, yields
xAxBu,
Its dynamic response is:
x(t)eA(tt0)x0
t t0
eA(t)Bu()d
eAtx0
teA(t)Bu()d
0
zero-input response Freedom motion
zero-state response Forced motion
6
Calculation of e At (1)
then the characteristic equation of the CL system is
detsI(AB K )0
1 ,2 , ,n
(a)
2 Assuc m( s e) th( s e de1 s) ( irs e d 2 l) oca( ts i onn a) r e0 known,
x
1
x 1 a 1 x 1 a 2 x 2 a 3 x 3 u 0
0 1
0
x
0
u
0 0
x2 x 1
x3 x2
y b1 b2 b3
5
• Dynamic response of SS model
Consider the following linear time-invariant system
– Control canonical form – Observer canonical form – Modal canonical form
U
b1
b2
Y
1
s
1 x1c s
x 2c
1 s
b
x3c 3
a1 a2
a3
Namely x1,x2,x3
a1 a2 a3 1
We obtain
• Observability : output ----- state • Their physical meanings?
• Criterion
• rank criterion
rB a A n A 2 B B k A n 1 B n
• PBH rank criterion
r a n k s I AB n , s C
(1) represents the power series of the matrix At
eAt IAt1A2t2 1Aktk
2!
k0k!
(2) using typical matrix, such as Diagonal or Jordan matrix
1
A
Pwk.baidu.com
0
0
P
1
n
e1t
eAt
Course Review
Youqing Wang
College of Information Science & Technology 2019.11.15
Final Exam
• Total 50 scores, 6 questions • 3 questions for linear continuous systems (29) • 2 questions for linear discrete systems (16)
L(eA t)IssA 2A s3 2 (sIA )1 eAt L1[(sIA)1]
example：
L1(s2s 5s56)L1(s32s23)
3e2t 2e3t
8
Controllability and Observability
• Controllability : input ----- state
How to select pole position
• Dominant second-order poles---deals with pole selection without explicit regard for control effort
– Natural frequency + damping factor
then the corresponding desired characteristic equation:
10
F H
G
J
Nx Nu
0 1
NNuxHF
G10 J 1
r
Nu
uNurK(xNxr)
Kx(Nu KNx)r
Nx
-K
u
y
Plant
x
uKxNr
r
N
u
y
Plant
x
-K Full state feedback with reference inpu1t1
• Linear continuous-time system theory • Linear discrete-time system theory
• Overview: transfer function state space model • Classical control vs modern control • “Transfer function” to “SS model” (multi-choice) • Canonical form
P
0
0
P1
ent
1 1
1 1
AQ
1 2
Q 1 1
2
e1t
te1t
t 2 e1t 2
e At
Q
e1t te1t e1t
Q
1
e2t te2t
e2t
7
Calculation of e At -2
（3） taking the inverse Laplace transform of (sIA)1
Observer (duality)
correction term L(yC~ x), close-loop observer：
~ x A ~ x B L ( u y C ~ x ) ( A L ) ~ x C B L uy
• optimal control (or symmetric root locus)--specifically address the issue of achieving a balance between good system response and
control effort1 .G 0sG 0s0
• System destruction according to controllability
and observability
9
Controller design
• Check controllability • Full-state feedback control
1 Substituting the feedback law, yields
xAxBu,
Its dynamic response is:
x(t)eA(tt0)x0
t t0
eA(t)Bu()d
eAtx0
teA(t)Bu()d
0
zero-input response Freedom motion
zero-state response Forced motion
6
Calculation of e At (1)