Transcript 디지탈제어 강의자료 Chapter6

Chapter 6
Design Using
State-Space Methods
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x (t )  Fx(t )  Gu(t )
y (t )  Hx (t )  Ju(t )
x (k  1)  Φx(k )  Γu(k )
y (k )  Hx(k )  Ju(k )
T
where Φ  e , Γ   eF d G
FT
0
x ( k )  R n , u( k )  R r , y ( k )  R m
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Solution of LTI Discrete-Time State Equation
Assume that x( k0 ) and u( k0 )
u( k  1) are given.
x (k0  1)  Φx(k0 )  Γu( k0 )
x (k0  2)  Φx( k0  1)  Γu( k0  1)
 Φ 2 x (k0 )  ΦΓu(k0 )  Γu(k0  1)
x(k )  Φ
k  k0
x ( k0 ) 
k 1
k  j 1
Φ
Γu( j )

j  k0
The output of the system is as follows.
y (k )  HΦ
k  k0
k 1
x(k0 )  H  Φ k  j 1Γu( j )  Ju( k )
j  k0
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For x (k  1)  Φx (k )  Γu (k )
zX (z )  zx (0)  ΦX ( z )  ΓU ( z )
(zI  Φ ) X ( z )  zx (0)  ΓU ( z )
X ( z )  (zI  Φ ) 1 zx (0)  ( zI  Φ ) 1 ΓU ( z)
x (k )  z1 (zI  Φ ) 1 z x (0)  z- 1 (zI  Φ ) 1 ΓU ( z)
Since k0  0,
If
Φ k  z1 ( zI  Φ ) 1 z
x (0)  0
X ( z )  (zI  Φ ) 1 ΓU ( z )
Y (z )  H (zI  Φ ) 1 Γ  J  U (z)
transfer function
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Lyapunov Stability Analysis
Def : Equilibrium State
The state xe which is a constant solution of the state equation with
u (k )  0 for all k  k0 is defined as the equilibrium state of
the system.
x (k  1)  Φx (k )
xe  Φxe or (I  Φ ) xe  0
Def : Stability in the sense of Lyapunov
xe is said to be stable at k0 if for given any real number ε  0,
there exists a δ (ε , k0 )  0 such that for all initial states x (k0 )
in the sphere of radius δ
x (k0 )  xe  δ (ε , k0 ),
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then the solution of the system remains with a sphere of
radius ε for all k  k0
i .e,
x (k )  x e  ε
 k  k0
x2
ε
xe
δ
x(k0 )
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Def : uniformly stable , δ(ε )
Def : globally stable
x (k0 )  X (entire state space)
Def : asymptotically stable
if
i) it is stable i.s. L
ii) for any k0 and any x (k0 ) which is sufficientally close to xe ,
x (k ) converges to xe as k approaches infinity
Def : uniformly asymptotically stable
Def : asymptotically stable in the large
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Def : exponentially stable
if 
r , a, b  0
 : x (k )  xe  a x (k0 )  xe e  bt
 k , k0  0
 x (k0 )  Br
Remarks :
i) The LTI discrete-time system is asymptotically stable iff
the eigenvalues of Φ i .e, λ i , i  1, 2,
λ i < 1 , i  1, 2,
n satisfy
n
ii) The roots of zI  Φ  0 must all lie inside the unit circle
z  1.
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Def : BIBO ( Bounded Input Bounded Output) Stability
The system is BIBO stable if any bounded input gives
a bounded output
k 1
x (k )  ψ (k ) x (k0 )   ψ (k  i  1)Γu ( j )
i  k0
k 1
x (k )  ψ (k ) x (k0 )   ψ (k  i  1) Γ u ( j )
i  k0
Since u ( j ) is bounded and x (k0 ) is a constant vector,
x (k ) will bounded if ψ (k ) < p   for all k
k 1
and
 ψ (k  i  1)Γ
θ
k
i  k0
asymptotically
BIBO stable
stable
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Def : Positive definite
A scalar function V ( x ) is said to be positive definite in a region Ω
if V ( x )  0 for all nonzero state x in the region Ω
and if V ( x )  0, x  0.
ex )
x12  x22 (p.d)
(x1  x2 )2 (p.s.d)
2
x
2
x12 
(p.d)
2
1  x2
 x12  ( x1  x2 )2 (n.d)
x1 x2  x22 (indefinite)
Similarly , for a time-varying system
V ( x , k )  W ( x ) for all k  k0
V (0, k )  0
for all k  k0
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Def : Lyapunov function
Any function V ( x (k )) is called a Lyapunov function
for the system x (k  1)  f ( x (k )), f (0)  0
If
i) V ( x ) is continuous in x
ii) V ( x (k ))   as
x (k )
  and V (0)  0
iii) V ( x (k )) is p.d. for x  0
iv) V ( x (k )) V ( x (k  1))  V ( x (k )) is n.d. for x  0
ex) In 2 –dim
stable i.s. L
asymptotically stable
x1
x2
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Thm : Lyapunov Second Theorem ( Lyapunov Direct Theorem)
Consider that a discrete - time system is descrived by
x (k  1)  f ( x (k )) where x (k )  R n
f ( x (k )  0) =0  k
Suppose that there exists a scalar function V ( x (k )) which is
continuous on x (k ) such that
i) V ( x (k ))  0 for x  0
ii)  V ( x (k )) 0 for x  0
iii) V ( x (k ))  
as x
 
iv) V ( x (k )  0)  V (0)  0
Then the equilibrium state x (k )  0,  k is asymptotically
stable in the large
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ex )
x1 (k  1)  0.5 x1(k )
x2 (k  1)  0.5 x2 (k )
Assign the Lyapunov function candidate
V ( x (k ))  x12 (k )  x22 (k ) > 0 , x (k )  0
 V ( x (k ))  V ( x (k  1))  V ( x (k ))
 x12 (k  1)  x22 (k  1)  x12 (k )  x22 (k )
 0.75  x12 (k )  x22 (k ) 
V ( x (k ))  0 for x  0
 The system is asymptotically stable
Remarks :
i) eigenvalues =  0.5,  0.5
0 
 0.5
ii) x (k  1)  
x (k )

0.5 
 0
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V ( x (k ))  xT Px  0
P >0
 V ( x (k ))  x (k  1)T Px (k  1)  x (k )T Px (k )
 (Φx (k ))T P (Φx (k ))  x (k )T Px (k )
 x (k )T [ΦT PΦ  P ]x (k )
 ΦT P Φ  P  Q
Given Q > 0,
where Q > 0
 a P 
ΦT P Φ  P  Q
iii) It is a sufficient cond. However , in the case of linear
system, it is a sufficient and necessary condition.
iv) In an LTI continuous-time system, the Lyapunov equation is
described as follows.
AP  PAT  Q
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0 
 0.5
x (k  1)  
x (k )

0.5 
 0
1 0
Q  
 0
good choice

 0 1
ex )
 p11
P
 p21
Let
using
p12 
p22 
ΦT P Φ  P  Q ,
0 
1.33
P 
>0

1.33 
 0
by Lyapunov second theorem, the system is asymptotically stable
Note: Alexander Mikhailovitch Lyapunov (1857 – 1918)
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Controllability ( Reachability )
Def : The system is controllable (reachable) if it is possible to find
a control sequence such that the origin (an arbitrary state)
can be reached from any initial state in finite time.
Remark: finite time / unbounded input
controllable
x (k0 )  0
reachable
x (k0 )  xd (not necessary equal to 0)
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Remarks :
i) If Φ k x (0)  0, it is controllable, but not necessarily to be reachable.
ii) If Φ is invertible , the controllability and the reachability are equivalent.
n 1
x(n )  Φ x(0) + Φ n  j 1Γu( j )
n
j 1
 Φ n x(0) +Φ n 1Γu(0)  Φ n 2 Γu(1) 
x(n )  Φ n x(0)  [Γ : ΦΓ :
 Γu( n  1)
 u(n  1) 
u(n  2)

: Φ n 1Γ ] 




u
(0)


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Define
Wc  [Γ : ΦΓ :
: Φ n 1Γ ]
U  [uT (n  1) :
: uT (0)]
x (n )  Φ n x (0)  Wc U ( i.e, y  Ax )
U  Wc 1[ x (n )  Φ n x (0)]
Wc is called the controllability matrix
Remarks :
i) If Wc has rank n, then it is possible to find n equations
from which the control signals can be found such that
the initial state is transferred to the desired final state x (n )
ii) If rank Wc < n , it is not possible to have x (k )  xf
for all k for some xf
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Theorem : The system is reachable iff the matrix Wc has rank n.
1
 1
 1 
x (k  1)  
x
(
k
)

u( k )



 0.25 0 
 0.5 
2
x (0)   
2
Determine u(0) and u(1) such that xT (2)   0.5 1
ex )
x (2)  Φ 2 x (0)  ΦΓu(0)  Γu(1)
 0.5  3.5   1 
 1    1    0.5  0.5u(0)  u(1)

 
 

0.5u (0)  u (1)  4

u (0)  2, u(1)  3
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Is it controllable ?
Determine u (0) and u (1) such that xT (2)  0.5 1
 no solution exists

0.5 
 1
Wc  
, rank Wc  2

 0.5 0.25 
Remarks :
i) Any initial state x (0) can be transferred to the desired
state in n sampling periods for a sequence of unbounded
control signal as long as Wc is of rank n.
ii) A necessary and sufficient condition for complete controllability
is that no cancellation occurs in the transfer function.
 minimal realization
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ex )
Y (z)
(z  0.2)

U ( z ) ( z  0.8)( z  0.2)
1
 0
 1 
x(k  1)  
x(k )  
u( k )


 0.16 1
 0.8 
y (k )  1 0  x( k )
0.8 
 1
Wc  

 0.8 0.64 
rank Wc  2
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ex )
 0 ω
0
x (t )  
x
(
t
)

u (t )



 ω 0 
ω 
y (t )  1 0 x (t )
det Wc  ω 3
ω2
 h(s )  2
s  ω2
controllable
Using ZOH
 cos ωT sin ωT 
1  cos ωT 
x
(
k
)

  sin ωT cos ωT 
 sin ωT  u(k )




y (k )  1 0  x(k )
x (k  1) 
det Wc  2sin ωT (1  cos ωT )
if ωT  nπ
(2n  1)
if ωT 
π
2
uncontrollable
controllable
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Output Controllability

Any y (0)
arbitrary y (k )  R m
sequence of unbounded control input
finite time
k 1
x (k )  Φ x (0)   Φ k  j 1Γu ( j )
k
j 0
n 1
y (n )  CΦ x (0)   Φ n  j 1Γu ( j )
n
j 0
n 1
y (n )  CΦ x (0)   Φ n  j 1Γu ( j )
n
j 0
 CΦ n 1Γu (0)  CΦ n 2 Γu (1) 
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 CΓu (n  1)
23
 u (n  1) 
u (n  2) 

: CΦ n 1Γ ] 




u
(
0
)


 [CΓ : CΦΓ :
vectors CΓ : CΦΓ :
: CΦ n 1Γ must span the
m-dimensional output space
rank [ CΓ : CΦΓ :
: CΦ n 1Γ ] = m
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Remarks :
i) Complete state controllability implies complete
output controllability iff the m rows of C are
linearly independent
ii) If y (k )  Cx (k )  Du (k )
rank [D : CΓ : CΦΓ : : CΦ n 1Γ ]  m
The presence of the matrix D always helps to establish
complete output controllability
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Regulation via Pole Placement
x (k  1)  Φx (k )  Γu (k )
Assume that the magnitude of the control input u (k ) is unbounded
u (k )  Kx (k )
x (k  1)  (Φ  ΓK ) x (k )
The eigenvalues of (Φ  ΓK ) are the desired closed-loop poles
arbitrary pole
placement
Nec. & Suff.
completely
state
controllable
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Let us define a transformation matrix Q ( P  Q 1 )
from x to z such that z( k )  Qx( k )
P  Wc M 1 or Q  MWc1
 an 1
a
 n 2
where M -1   .

 a1
 1
Wc 
Γ
an 2
.. a1
an 3
.
..
..
1
.
1
..
0
0
..
0
ΦΓ
1
0 
.

0
0 
. . Φ n 1Γ 
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Φ
 0
 0

z(k  1)  

 0
 an
1
0
0
1
0
0
Γ
0 
0 
0 
0 
 
 z (k )    u (k )

 
1 
0 
 1
a1 
an 1 an 2
Controllable canonical form
Φ ~Φ
i.e., Φ  Q 1Φ Q,
z n  a1z n 1 
 an  0
Γ  Q 1 Γ
open - loop characteristic polynomial
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The desired closed-loop system poles are μ1 , μ2 ,...., μn .
φ (z)
(z  μ1 )(z  μ2 )
 z n  α1z n 1 
( z  μn )
 αn
closed-loop characteristic polynomial
Let u (k )  Kz (k )
φ (z )  z n  (a1  k n )z n 1 
 (an  k1 )
Since u (k )  Kz (k )  KQx (k )  Kx (k )
K  [α n  an , α n 1  an 1 ,
K  KQ  [1 0
 [1 0
 [1 0
, α 2  a2 , α1  a1 ]
0]φ (Φ )Q
0]φ (QΦQ 1 )Q
0]Qφ (Φ )
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0]MWc1φ (Φ )
K  [1 0
1]Wc1φ (Φ )
 [0 0
Ackermann's formula
Remarks:
n
i) φ (Φ )  Φ  α1Φ
n 1
 (α1  a1 )Φ

n 1
 αn I

 (α n  an )I
k
The first row of Φ is zero except a position k  1 which is 1
ii)
0 0
1 0 0 0 1
 1 a1
1 
a1   0 0 1
a2  a12 
M
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Remarks:
i) Nonuniqueness of state-space representation
Different state-space representation for a given transfer function
are possible. However, the state equations are related to each other
by the similarity transformation (matrix).
ii) Canonical forms for discrete-time state-space equations
a) controllable canonical form  direct programming method
b) observable canonical form
c) diagonal canonical form
d) Jordan canonical form
 nested programming method
 partial-fraction expansion method
 partial-fraction expansion method
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ex )
1
 0
0 
x (k  1)  
x (k )    u (k )

 0.16 1
 1
open - loop
z 2  z  0.16  0
desired poles z1  0.5  j 0.5, z2  0.5  j 0.5
φ (z )  z 2  z  0.5
0 1 
Wc  [Γ : ΦΓ ]  
full rank

 1 1
K  0 1Wc1φ (Φ )
1
0 1  0.34 2 
 0 1 
 

 1 1 0.32 2.34 
 0.34 2
  k1 k 2 
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1
 0
0 
x (k  1)  
x
(
k
)

u (k )



 0.16 1
 1
ex )
1
0 1  0.34 2 
K  0 1 
 0.34 2   k1



 1 1 0.32 2.34 
k2 
In case of x (0)  5 10  ,
20
20
15
15
15
10
10
10
5
5
5
0
-5
Input u
20
State variable x2
State variable x1
T
0
0
-5
-5
-10
-10
-10
-15
-15
-15
-20
0
2
4
6
8
10
x1 (k )
12
14
16
18
-20
0
2
4
6
8
10
12
14
16
18
x2 (k )
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-20
0
2
4
6
8
10
12
14
16
18
u (k )
33
Deadbeat Control
(Minimum Settling Time Control)
The deadbeat response is unique to discrete-time system.
φ(z)  z n
 Φcn  0 (by Cayley - Hamilton theorem)
i. e. , It will drive all the states to zero in at most n steps.
i. e. , nT is the settling time (only design parameter).
ex )
1
 0
0 
x (k  1)  
x (k )    u (k )

 0.16 1
 1
z 2  (1 k2 )z  0.16  k1  z 2
k1  0.16
k2  1
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34
 0 1
x (k  1)  Φc x (k )  
 x (k )
0
0


 x1 (0)  a 
For
 x (0)   b 
 2   
 x1 (1)  0 1 a   b 
 x (1)   0 0  b    0 
   
 2  
 x1 (2)  0 1  b  0 
 x (2)   0 0  0   0 
   
 2  
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1
 0
0 
x (k  1)  
x
(
k
)

u (k )



 0.16 1
 1
z 2  (1 k2 )z  0.16  k1  z 2
ex )
k1  0.16
k2  1
In case of x (0)  5 10  ,
20
20
15
15
15
10
10
10
5
5
5
0
Input u
20
State variable x2
State variable x1
T
0
0
-5
-5
-10
-10
-10
-15
-15
-15
-5
-20
-20
0
2
4
6
8
10
x1 (k )
12
14
16
18
0
2
4
6
8
10
12
14
16
18
x2 (k )
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-20
0
2
4
6
8
10
12
14
16
u (k )
36
18
ex)
Astrom's Computer-Controlled Systems (pp. 127 ~ )
T 2 / 2
1 T 
x (k  1)  
 u (k )
 x (k )  
0
1


 T 
u (k )  k1x1 (k )  k2 x2 (k )
1 k1T 2 / 2 T  k 2T 2 / 2 
x (k  1)  
 x (k )
1 k 2T 
 k1T
The characteristic equation of the closed-loop system
k1T 2
k1T 2
z (
 k 2T  2)z  (
 k 2T  1)  0
2
2
2
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The desired characteristic equation
z 2  p1z  p2  0
k1 
1
(1 p1  p2 ),
2
T
k2 
1
(3  p1  p2 )
2T
Other approach:
T 2 / 2 3T 2 / 2
Wc  Γ ΦΓ   

T
T


1
c
W
 1/ T 2

2
1
/
T

1.5 / T 

0.5 / T 
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1  p1  p2 2T  p1T 
φ(Φ )  Φ  p1Φ  p2I  

p

p

1
0
2
1

2
K   k1
3  p1  p2 

2T

1  p1  p2
k 2   0 1W φ(Φ )  
T2

1
c
Assume that the desired characteristic polynomial
is s 2  2ζωn s  ωn 2 .
s1,2  ζωn  jωn 1  ζ 2
Then

z  esT
s1,2
 e ζωnT   ωnT 1  ζ 2
p1  2e ζωnT cos(ωnT 1  ζ 2 )
p2  e 2ζωnT
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39
To discuss the magnitude of the control signal, it is assumed
that the system has an initial position x0 and an initial velocity v 0 .
u( k )  k1 x1( k )  k 2 x2 ( k )
u(0)  k1 x0  k 2v 0
where k  0
If the sampling period is short, then the expressions for p1 and p2
can be approxiamated using series expansion.
u(0)  ωn 2 x0  2ζωn v 0
Remark:
The m agnitide of the control signal increases with increasing ωn .
An increase in the speed of the response of the system will require
an increase in the control signals.
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Remarks: Damping ratio and natural frequency
for given the complex pole location in the z-plane
z  esT
Magnitude
Phase
s1,2
 e  ζωnT   ωnT 1  ζ 2  r   θ

2πζ ωd 
r
e
 exp  
2 ω 

1 ζ
s 

ω
ωnT 1  ζ 2  ωd T  2π d  θ
ωs
 ζωnT
or ζωnT   ln r
Taking the ratio of the last two equations and solving this equatuion for ζ yields
ζ 
 ln r
ln2 r  θ 2
from
ζ
1-ζ 2

 ln r
θ
Then ωn becomes
ωn 
1
ln2 r  θ 2
T
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Remarks: Choice of appropriate sampling rate
i) open-loop system
ii) closed-loop system
Tr
 4 ~ 10
T
1.8 1.8 1.8 ωs ωs
ωn 



Tr
10T 10 2π 35
Nr 
2π
ωs
2π
T
N


ωd ω 1  ζ 2 ω T 1  ζ 2
n
n
N  25 to 75 (reasonable)
i.e., the number of samples per period of dominating mode
of the closed-loop system
iii)
ωnT  0.1 to 0.6 for ζ  0.7 where ωn is the desired natural
frequency of the closed-loop system.
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State Feedback Design
by Emulation (Redesign)
Consider x (t )  Ax (t )  Bu (t )
y (t )  Cx (t )
u (t )  Kx (t )  Mr (t )
x (k  1)  Φc x (k )  Γc Mr (k )
where Φc  e AcT  e ( ABK )T     (1)
T
Γc   e
0
Ac s '
ds ' B
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If the digital control is used,
x (k  1)  Φx (k )  Γu (k )
where
Φ  e AT
Γ 

T
0
e
As '
ds ' B
u (k )  Kx (k )  Mr (k )
x (k  1)  (Φ  ΓK ) x (k )  ΓMr (k )
Φc  Φ  ΓK
    (2)
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Assume K  K 0 
T
K1
2
From (1)
T2
Φc  I  ( A  BK )T  [ A  BKA  ABK  (BK ) ]
2
2
2
(3)
T2
Φ  I  AT  A
2
2
T
ΓK   e ds ' B (K 0  K1
As '
0
T
A
T
)  (T  T 2 )B(K 0  K1 )
2
2
2
From (2)
T2 
T2
T2
T3 
Φ  ΓK  I  AT  A
  BK 0T  BK1
 ABK 0
 ABK1

2 
2
2
4 
2
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T2
Φ  ΓK  I  ( A  BK 0 )T  [ A  ABK 0  BK1 ]
2
2
(4)
Comparing (3) with (4)
A  BK  A  BK 0
A2  BKA  ABK  BKBK  A 2  ABK 0  BK1
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Since K 0  K ,
BK1  BKA  BKBK  B (KA  KBK )
K1  K ( A  BK )
T

K  K I  ( A  BK ) 
2

T

Similarly, M   I  KB  M
2

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 0 1
0 
ex ) x (t )  
x (t )    u (t )

0 0 
 1
y (t )  1 0  x (t )
u (t )  Kx (t )  Mr (t )   1 1 x (t )  r (t )
u (k )   1 1 x (k )  r (k ),
T  0.5

u (k )  Kx (k )  Mr (k )
  1 0.5T
 x (kT ) 
1  1
 (1 0.5T )r (kT )

 x2 (kT ) 
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
54
1.4
1.4
1.2
1.2
position
position
ys
1
0.8
yc
0.6
0.4
0.2
0
0
1
2
3
4
5
6
7
8
9
0.6
0.4
0
10
ys
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
7
8
9
10
0.6
0.6
0.5
0.5
velocity
velocity
yc
0.8
0.2
0.7
0.4
yc
0.3
0.2
0.1
0
0.4
0.3
0.2
0.1
0
-0.1
-0.2
ys
1
0
1
2
3
4
5
6
7
8
9
-0.1
10
1
1
0.8
Input
Input
0.6
0.5
0
0.4
0.2
0
-0.2
-0.5
0
1
2
3
4
5
6
7
8
9
10
Fig 1. Control of the double integrator using
the control law in (*) when T=0.5
-0.4
Fig 2. Control of the double integrator using
the modified control law in (**) when T=0.5
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Observability
known
Def. The system
x (k  1)  Φx (k )  Γu (k )
y (k )  Hx (k )
is said to be observable if for any k0 , any state x (k0 )
can be determined from the knowledge of the inputs
u (k0 ),
.u (k  1) and the outputs y (k0 ).
.y (k  1)
for finite time
Let k0  0
y (0)  Hx (0)
y (1)  Hx (1)  H (Φx (0)  Γu (0))
y (n  1)  Hx (n  1)  H[Φ n 1 x (0)  Φ n 2 Γu (0) 
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 Γu (n  2)]
56
given
known
y (0)
 H 


 HΦ 


y
(
1
)

HΓu
(
0
)

 x (0)  








n 1 
n 2
HΦ
y
(
n

1
)

H
[
Φ
Γu
(
0
)


Γu
(
n

2
)]




Define
W0  H T : ΦT H T :
: (Φ n 1 )T H T 
Theorem : The system is observable iff W0 has rank n.
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State Observer (Estimator)
x (k  1)  Φx (k )  Γu(k )
()
y (k )
u (k )
x (k )
state observer
open-loop vs. closed-loop
full-order vs. reduced-order
prediction
vs. current
x (k  1)  Φx (k )  Γu (k )
e(k  1)
(  )
x (k  1)  x (k  1)
 Φe(k )
Note : i) Φ should be stable.
ii) e(0) should be known.
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Prediction Estimator
x (k  1)  Φx (k )  Γu (k ) +Lp [ y (k )  Hx (k )]
e (k )
(  )
correcting term
or innovation term
x (k )  x (k )
e(k  1)  Φe(k )  Lp He(k )  [Φ  Lp H ]e(k )
zI  Φ  Lp H  0
The desired poles are β1 , β2 ,...., βn
(z  β1 )(z  β2 )
(z  βn )  0
Characteristic polynomial of the closed-loop system
φ (z )  z n  α1z n 1 
 αn
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Remarks :
i) Output information is used.
ii) Φ is not necessarily stable.
iii) All eigenvalues of Φ  LP H are chosen to be zero.
It shows deadbea t response.
iv) Ackermann's formula
0 
 
Lp  φ (Φ )W01  
0 
 
 1
where W0 = H T : ΦT H T :
: (Φ n 1 )T H T 
φ(Φ )=Φ n  α1Φ n 1 
Note:
K  [0 0
 α n 1Φ  α n I
1]Wc1φ (Φ )
where Wc  Γ : ΦΓ :
: Φ n 1Γ 
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Current Estimator
x (k )  z (k )  Lc [ y (k )  Hz(k )]
z(k )  Φx (k  1)  Γu (k  1)
e (k )
x (k )  x (k )
e(k  1)  Φx (k )  Γu (k )  {Φx (k )  Γu (k )
 Lc [H [Φx (k )  Γu (k )]  H[Φx (k )  Γu (k )]]}
 (Φ  Lc HΦ )e(k )
 HΦ 
HΦ 2 

Lc  φ (Φ ) 



n
HΦ


1
0 
0 
 
 
 
 1
where φ(Φ )  Φ n  α1Φ n 1 
 α n 1Φ  α n I
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Reduced-Order Estimator
(Minimum-Order Estimator)
x (k  1)  Φx (k )  Γu (k )
co
 xa ( k ) 
 [I 0] 
y (k )
 i .e y (k )  xa (k )
)
k
(
x
 b 
 xa (k  1)  Φaa Φab   xa (k )  Γa 
  x (k )    Γ  u (k )
 x (k  1)   Φ
Φ
bb   b
  b
  ba
 b
xa (k  1)  Φaa xa (k )  Φab xb (k )  Γa u(k )
xa (k  1)  Φaa xa (k )  Γa u(k )  Φab xb (k )
~ y (k )  Hx (k )
xb (k  1)  Φbb xb (k )  Φba xa (k )  Γb u (k )
()
~ x (k  1)  Φx (k )  Γu(k )
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xb (k  1)  Φbb xb (k )  Φba xa (k )  Γb u(k )
 Lr [ xa (k  1)  Φaa xa (k )  Γa u(k )  Φab xb (k )]
()
very inconvenient because of
same time instant
eb (k  1)  [Φbb  Lr Φab ]eb (k ) where eb (k )
xb (k )  xb (k )
If xa (k ) is scalar,
 Φab 
Φ Φ 
Lr  φ(Φbb )  ab bb 



n 2 
Φ
Φ
 ab bb 
1
0 
0 
 
 
 
 1
n 1
n 2
where φ
（Φbb )  Φbb
 α1Φbb

α n 1I
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η (k )
xb (k )  Lr y (k )  xb (k )  Lr xa (k )
η (k )
xb (k )  Lr y (k )
η (k  1)  xb (k  1)  Lr y (k  1)
 (Φbb  Lr Φab ) xb (k )  (Φba  Lr Φaa ) y (k )  (Γb  Lr Γa )u (k )
 (Φbb  Lr Φab )( xb (k )  Lr y (k ))  (Φbb  Lr Φab )Lr y (k )
 (Φba  Lr Φaa ) y (k )  (Γb  Lr Γa )u (k )
 (Φbb  Lr Φab )η (k )  (Φbb  Lr Φab )Lr y (k )
 (Φba  Lr Φaa ) y (k )  (Γb  Lr Γa )u (k )
xb (k )  η (k )  Lr y (k )
xa (k )  y (k )
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u (k )
Γ
+

x (k )
1
Z I
y (k )  xa (k )
H
+
Φ
Γb  Lr Γa
 0 
 η (k ) 


η (k )
0
In  m
+


+
 xa (k ) 


 Lr xa (k ) 
+
Z 1I
η (k  1) +

Φba  Lr Φaa
Φbb  Lr Φab
+
Lr
Im
Lr
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ex)
 1 0.2
0.02 
Φ
,
Γ

, H  1 0



0 1 
 0.2 
y (k )  x1 (k )
co / cc
if z1 , z2  0.6  j 0.4 (state feedback )  K  [8, 3.2]
 y (k ) 
u ( k )    8 3 .2  

x
(
k
)
 2 
if φ (z)  z  0 (deadbeat estimator )
η (k  1)  5 y (k )  0.1u (k )
u (k )  8 y (k )  3.2 x2 (k )
 8 y (k )  3.2(η (k )  5 y (k ))
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u (k  1)  8 y (k  1)  3.2( (k  1)  5 y (k  1))
 8 y (k  1)  16 y (k  1)  3.2(5 y (k )  0.1u (k ))
u (k  1)  0.32u (k )  24 y (k  1)  16 y (k )
u (k )  0.32u (k  1)  24y (k )  16y (k  1)
Y (z )
1 0.667z 1
GD (z) 
 24
U (z )
1 0.32z 1
Robotics Research Laboratory
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Observed-State Feedback Control
With Estimator
Robotics Research Laboratory
68
State Feedback
u (k )  Kx (k )
x (k  1)  Φx (k )  Γu (k )
 Φx (k )  ΓKx (k ) ; e(k )  x (k )  x (k )
 Φx (k )  ΓKx (k )  ΓLe(k )
x (k  1)  (Φ  ΓK ) x (k )  ΓKe(k )
State Observer
x (k  1)  Φx (k )  Γu (k )  L[Hx (k )  Hx (k )]
e(k  1)  (Φ  LH )e(k )
0   e (k ) 
 e(k  1)  Φ  LH
 x (k  1)    ΓK
Φ  ΓK   x (k ) 

 
zI  Φ  LH zI  Φ  ΓK  c (z)o (z)  0
Separation principle
Robotics Research Laboratory
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Compensator (Estimator and Control Mechanism)
For the prediction estimator
x (k )  [Φ  ΓK  Lp H ]x (k  1)  Lp y (k  1)
u (k )  Kx (k )
U (z)
 Dp (z)  K [zI  Φ  ΓK  Lp H ]1 Lp
Y (z)
For the current estimator
x (k )  [Φ  ΓK  Lc HΦ  Lc HΓK ]x (k  1)  Lc y (k )
u (k )  Kx (k )
U (z)
 Dc (z)  K [zI  Φ  ΓK  Lc HΦ  Lc HΓK ]1Lc z
Y (z)
Remarks:
i) As a rule of thumb, an observer response must be at least
4 to 5 times faster than the system response.
ii) Using the output feedback only, we can not place poles
at arbitrary locations.
Robotics Research Laboratory
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Satellite Attitude Control (Franklin’s )
1
G (s )  2
s

 x1  0 1  x1  0 
 x   0 0   x    1 u , y  1 0  x
 2  
 2 
The discrete model for this system is
T 2 
1 T 
Φ
, Γ   2  , H  1 0

 
0 1 
 T 
y (k )  x1 (k )
co / cc
if z1 , z2  0.8  j 0.25 (state feedback )  K  [10, 3.5]
 x (k ) 
u (k )   10 3.5   1 
 x 2 (k ) 
Robotics Research Laboratory
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if φ (z )  (z  0.4  j 0.4)(z  0.4  j 0.4)
where the desired poles of the estimator are assigned so that
s -plane poles have ζ  0.6 and ωn is about three times faster
than the selected poles.
Lp  1.2 5.2 ,
T
Lc  0.68 5.2
T
if φ (z )  z  0.5
Lr  5
where x2 (k ) is the velocity to be estmated.
Robotics Research Laboratory
72
Time histories with prediction estimator
2
1.5
1
OUTPUTS
0.5
0
-0.5
-1
------o---x---*--
-1.5
U/4
X1
X2
X2 TILDE
-2
0
0.5
1
1.5
Time (sec)
Dp (z)  30.4
2
2.5
3
z  0.825
z  0.2  j 0.557
Robotics Research Laboratory
73
Time histories with current estimator
2
1.5
1
OUTPUTS
0.5
0
-0.5
-1
------o---x---*--
-1.5
-2
0
0.5
1
Dc (z)  25.1
1.5
Time (sec)
2
U/4
X1
X2
X2 TILDE
2.5
3
z(z  0.792)
z  0.265  j 0.394
Robotics Research Laboratory
74
Time histories with reduced-order estimator
2
1.5
1
OUTPUTS
0.5
0
-0.5
-1
------o---x---*--
-1.5
-2
0
0.5
1
1.5
Time (sec)
Dr (z)  27.7
2
U/4
X1
X2
X2 TILDE
2.5
3
z  0.8182
z  0.2375
Robotics Research Laboratory
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Reference Inputs for Full-State Feedback
r (k )
xr (k )
Nx
+

K
u (k )
plant
x (k )
H
y (k )
-
Objective :
To make all the states and the outputs of the system follow
the desired trajectory.
x (k  1)  Φx (k )  Γu (k )
y (k )  Hx (k )
u (k )  K [ x (k )  xr (k )]
x r (k )  N x r (k )
Robotics Research Laboratory
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Goal : x()  xss  xr
For a step reference input,
if the system is Type 0  steady-state error
if Type 1 or higher  no steady-state error
For complex systems, the designer has no sufficient knowledge
of the plant. In these cases, it is useful to solve for the equilibrium
condition that satisfies
y ()  yss  r
We need a steady-state control term in order for the solution
to be valid for all system types.
uss  Nu r
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Nu
feedforward
r (k )
xr (k )
Nx
+

K
+

+
u (k )
plant
x (k )
H
y (k )
-
feedback
u (k )  K ( x (k )  xr (k ))  Nu r (k )
Nx r  xr  xss , Hxss  y ss  r
HNx r  r  HNx  I
()
x (k  1)  Φx (k )  Γu (k )
xss  Φxss  Γuss
 (Φ  I ) xss  Γuss  0
 (Φ  I )Nx r  ΓNu r  0
 (Φ  I )Nx  ΓNu  0
()
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Φ  I Γ  Nx  0
 


 H

0  Nu   I 

1
Nx  Φ  I Γ  0
N    H
0   I 
 u 
Remark:
r (k )
N
+
u (k )

plant
x (k )
H
y (k )
-
K
N  Nu  KN x
u   K ( x  x r )  Nu r
 Kx  (KN x  Nu )r
 Kx  Nr
Robotics Research Laboratory
79
Reference Inputs with Estimator
Nu
r (k )
xr (k )
Nx
+

K
+

+
u (k )
plant
y (k )
-
estimator
u  K ( x  xr )  Nu r
 Kx  Kx x  Nu r
 Kx  (KNx  Nu )r
 Kx  Nr
Robotics Research Laboratory
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Output Error Command
ex) thermostats (output error command)
w (k )
r (k ) -

e (k )
x (k )
estimator
K
u (k )
plant
y (k )
+
Estimator
x (k  1)  Φ  ΓK  LpH  x (k )  Lp [ y (k )  r (k )]
 (Φ  ΓK  LpH ) x (k )  LpHx (k )  Lp r (k )
Plant
x (k  1)  Φx (k )  Γu (k )  Γ1w (k )
 Φx (k )  ΓKx (k )  Γ1w (k )
Remark (Fig. 8.17 vs. Figs. 8.19-20 in Franklin’s)
• state space approach > transfer function (I/O) approach
• state command > output-error command
Robotics Research Laboratory
81
Step response time histories for reference input - State command structure
5
4
3
OUTPUTS
2
1
0
-1
-2
-------o----x----*---
-3
-4
-5
0
0.5
1
1.5
Time (sec)
2
Robotics Research Laboratory
U/5
X1
X2
X2 TILDE
2.5
3
82
Output time histories for prediction estimator with output command
5
4
3
OUTPUTS
2
1
0
-1
-2
-------o----x----*---
-3
-4
-5
0
0.5
1
1.5
Time (sec)
2
Robotics Research Laboratory
U/5
X1
X2
X2 TILDE
2.5
3
83
Output time histories for reduced-order estimator with output command
5
4
3
OUTPUTS
2
1
0
-1
-2
------ U/10
--o--- X1
--x--- X2
-3
-4
-5
0
0.5
1
1.5
Time (sec)
2
Robotics Research Laboratory
2.5
3
84
Disk head reader response
2.5
------------ state command structure
ooooooooo output error command
2
.............. classical design
output
1.5
1
0.5
0
0
5
10
15
20
25
30
Time (msec)
35
Robotics Research Laboratory
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45
50
85
Integral Control by State Augmentation
x (k  1)  Φx (k )  Γu(k )  Γ1w (k )
y (k )  Hx (k )
Integral action is useful in eliminating the steady-state error
due to constant disturbance or reference input commands.
-
+
 e (k )
1
z 1
w (k )
xI ( k )
r (k )
KI
xr (k )
Nx
+

K
+

+
u (k )
+

plant
x (k )
y (k )
H
-
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xI (k  1)  xI (k )  e(k )  xI (k )  Hx (k )  r (k )
(i .e X I (z) 
1
E (z) )
z 1
 x (k  1)  Φ 0   x (k )  0 
Γ 
Γ1 
 x (k  1)   H 1  x (k )    1 r (k )  0  u (k )   0  w (k )
 I   
 
 
 I
 
 x (k ) 
u (k )    K K I  
 KN x r
state

augmentation
 xI ( k ) 
Robotics Research Laboratory
87
• The design of [K KI] requires the augmented system matrices.
• For the full-state feedback, an estimator is used to provide x .
• The addition of the extra pole for the integrator state leads to a
deteriorated command input response.
• Elimination of the excitation of the extra root by command inputs
can be done by using the zero. Refer p. 327(b)
• The system has a zero at z  1 
KI
.
KN x
So the selection of the zero
location corresponds to a particular selection of Nx.
Robotics Research Laboratory
88
Response with no integral control
A unit reference input at t = 0
A step disturbance at t = 2
3
2
OUTPUTS
1
0
-1
...
-2
w/2
------ U/5
-3
--o--- X1
--x--- X2
0
1
2
3
Time (sec)
4
Robotics Research Laboratory
5
6
89
Response with integral control
3
2
OUTPUTS
1
0
-1
...
-2
w/2
------ U/5
-3
--o--- X1
--x--- X2
0
1
2
3
Time (sec)
4
Robotics Research Laboratory
5
6
90
Integral control with added zero
3
2
OUTPUTS
1
0
-1
...
-2
w/2
------ U/5
-3
--o--- X1
--x--- X2
0
1
2
3
Time (sec)
4
Robotics Research Laboratory
5
6
91
Disturbance Estimation-Input Disturbance
w (k )
+
r (k )
N
+
+

-
u (k ) +


plant
y (k )
-
w (k )
K
estimator
x (k )
x (k  1)  Φx (k )  Γu (k )  Γ1w (k )
y (k )  Hx (k )
Robotics Research Laboratory
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Disturbance Modeling
constant disturbance
w 0
sinusodial disturbance
w  ω02w
xd  Fd xd ,
w  Hd xd (continuous-time)
xd (k  1)  Φd xd (k ),
w (k )  H d x d ( k )
Robotics Research Laboratory
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ex) sinusoidal disturbance
 xd 1   0
 x    ω 2
 d2   0
1  x d 1 
 Φd xd ,



0   xd 2 
 xd 1 
w  1 0     Hd xd
 xd 2 
 x (k  1)  Φ Γ1Hd   x (k )  Γ 
 x (k  1)    0 Φ   x (k )    0  u (k )
d  d
 d
 
  
Φw
Γw
augmented state
 x (k ) 
y (k )   H 0  

x
(
k
)
 d 
Hw
Φw
Hw  should be observable.
Robotics Research Laboratory
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Bias estimation - Step disturbance
3
2
OUTPUTS
1
0
-1
... w/2
-------o----x--+ +
-2
-3
0
1
2
3
Time (sec)
4
U/5
X1
X2
+ W bar/2
5
6
A unit reference input at t=0 and a step disturbance at t=2 sec with bias estimation
Robotics Research Laboratory
95
Sinusoidal input disturbance rejection
3
2
OUTPUTS
1
0
-1
-2
-3
0
1
2
3
--o--- X1, output
------ w, disturbance
+ + + w bar, disturbance estimate
4
5
6
7
8
9
Time (sec)
10
A sinusoidal input disturbance with disturbance rejection
Robotics Research Laboratory
96
Disturbance Estimation-Output Disturbance
v (k )
r (k )
N
+
u (k )

plant
+
y (k )
-
+

v (k )
K
x (k )
estimator
x (k  1)  Φx (k )  Γu (k )
xd (k  1)  Φd xd (k ),
y (k )  H Hd  ( x (k )T , xd (k )T )T
Robotics Research Laboratory
97
Sinusoidal output disturbance rejection
3
2
OUTPUTS
1
0
-1
-2
-3
0
1
2
3
. . . . . X1+w, measured output, y`
---o--- X1, output y
-------- w, disturbance
+ + + w bar, disturbance estimate
4
5
6
7
8
Time (sec)
9
10
A sinusoidal sensor disturbance (measured error) with disturbance rejection
Robotics Research Laboratory
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Effect of Delays
Robotics Research Laboratory
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Sensor delays
y1d (k  1)  y (k ) one cycle delay
y 2d (k  1)  y1d (k ) two cycle delay
 x(k  1)  Φ 0 0   x(k )  Γ
 y (k  1)    H 0 0   y (k )   0  u (k )
 1d
 
  1d
  
 y 2d (k  1)   0 1 0   y 2d (k )  0 
 x (k ) 
y d (k )  0 0 1  y1d (k ) 
 y 2d (k ) 
Robotics Research Laboratory
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Actuator delays
x(k  1)  Φx(k )  Γ1u(k  1)  Γ2u(k )
xn 1(k )  u(k  1)
 x(k  1)  Φ Γ   x(k )  Γ2 
 x (k  1)    0 0   x (k )    1  u (k )
  n 1   
 n 1
 
 x (k ) 
y (k )  H 0 

x
(
k
)
 n 1 
Robotics Research Laboratory
101
Robotics Research Laboratory
102
0
-4
OUTPUTS
--------- U/3
---o---- Y
-2
0
4
OUTPUTS
2
(b) classical FB w/delay, K=9.3
2
0
4
4
2
2
0
--------- Ud/3
---o---- Y
-2
-4
0
1
2
Time (sec)
--------- Ud/3
---o---- Y
-2
-4
1
2
3
Time (sec)
(c) estimator w/delay, K=9.3
OUTPUTS
OUTPUTS
Fig. 8.37(a) Ideal case (w/o delay, K=9.3)
4
0
1
2
3
Time (sec)
(d) classical FB w/delay, K=4
0
--------- Ud/3
---o---- Y
-2
3
-4
0
Robotics Research Laboratory
1
2
Time (sec)
3
103
2
0
--------- U/4
---o---- Y
-2
-4
(b) Classical FB w/delay, K=4
4
OUTPUTS
OUTPUTS
Fig. 8.38(a) Ideal case (w/o delay, K=9.3)
4
--------- Ud/4
---o---- Y
-2
-4
0
1
2
Time (sec)
--------- Ud/4
---o---- Y
-2
1
2
3
Time (sec)
(d) Cur. estimator w/delay, K=9.3
4
OUTPUTS
OUTPUTS
1
2
3
Time (sec)
(c) Pred. estimator w/delay, K=9.3
4
0
0
-4
0
2
2
0
2
0
--------- Ud/4
---o---- Y
-2
3
-4
0
Robotics Research Laboratory
1
2
Time (sec)
3
104