線性系統 - 國立臺灣大學cc.ee.ntu.edu.tw/~fengli/Teaching/LinearSystems/961ls06... ·...
Transcript of 線性系統 - 國立臺灣大學cc.ee.ntu.edu.tw/~fengli/Teaching/LinearSystems/961ls06... ·...
Fall 2007
線性系統Linear Systems
Chapter 06Controllability & Observability
Feng-Li LianNTU-EE
Sep07 – Jan08
Materials used in these lecture notes are adopted from“Linear System Theory & Design,” 3rd. Ed., by C.-T. Chen (1999)
NTUEE-LS6-CtrbObsv-2Feng-Li Lian © 2007Outline
Introduction
Controllability (6.2)
Observability (6.3)
Canonical Decomposition (6.4)
Conditions in Jordan-Form Equations (6.5)
Discrete-Time State Equations (6.6)
Controllability after Sampling (6.7)
NTUEE-LS6-CtrbObsv-3Feng-Li Lian © 2007Introduction (6.1)
Controllability and observability reveal
the internal structure of the system (model)
1 input: u2 states: x1, x2
1 output: y
The state x2 is NOT “controllable” by the input u
The state x1 is NOT “observable” at the output y = −x2 +2u
NTUEE-LS6-CtrbObsv-4Feng-Li Lian © 2007Controllability (6.2)
NTUEE-LS6-CtrbObsv-5Feng-Li Lian © 2007Controllability – 2
if x(0) = 0,
then x(t) = 0, ∀t ≥ 0,no matter what u(t) is
if x1(0) = x2(0),
then x1(t) = x2(t), ∀t ≥ 0,no matter what u(t) is
• Un-Controllable Examples:
NTUEE-LS6-CtrbObsv-6Feng-Li Lian © 2007Controllability – 3
• Controllable Example:
NTUEE-LS6-CtrbObsv-7Feng-Li Lian © 2007Controllability – 4: Diagonal Representation
NTUEE-LS6-CtrbObsv-8Feng-Li Lian © 2007Controllability – 5: Controllable Representation
NTUEE-LS6-CtrbObsv-9Feng-Li Lian © 2007Controllability – 6: Physical Representation
NTUEE-LS6-CtrbObsv-10Feng-Li Lian © 2007Theorem 6.1 (6.2)
NTUEE-LS6-CtrbObsv-11Feng-Li Lian © 2007Theorem 6.2 – 2
Proof:
“1. ⇔ 2.” “(A, B) controllable ⇔ Wc(t) nonsingular, ∀t > 0”
NTUEE-LS6-CtrbObsv-12Feng-Li Lian © 2007Theorem 6.2 – 3
Proof:
NTUEE-LS6-CtrbObsv-13Feng-Li Lian © 2007Theorem 6.2 – 4
Proof:
NTUEE-LS6-CtrbObsv-14Feng-Li Lian © 2007Theorem 6.2 – 5
Proof:
“2. ⇔ 3.” “Wc(t) nonsingular, ∀t > 0 ⇔ C has rank n”
NTUEE-LS6-CtrbObsv-15Feng-Li Lian © 2007Theorem 6.2 – 6
Proof:
NTUEE-LS6-CtrbObsv-16Feng-Li Lian © 2007Theorem 6.2 – 7
Proof:
“3. ⇔ 4.” “C has rank n ⇔ rank [A−λI B] = n,∀ e-value λ of A”
NTUEE-LS6-CtrbObsv-17Feng-Li Lian © 2007Theorem 6.2 – 8
Proof:
NTUEE-LS6-CtrbObsv-18Feng-Li Lian © 2007Theorem 6.2 – 9
Proof:
NTUEE-LS6-CtrbObsv-19Feng-Li Lian © 2007Theorem 6.2 – 10
Proof:
“2. ⇔ 5.”“For stable A, Wc(t) nonsingular, ∀t > 0 ⇔
AWc + WcA′ = −BB′ has a unique P.D. sol. Wc(∞)”
NTUEE-LS6-CtrbObsv-20Feng-Li Lian © 2007Example 6.2 (6.2)
rank = 4
NTUEE-LS6-CtrbObsv-21Feng-Li Lian © 2007Example 6.3 (6.2)
Mass = 0
NTUEE-LS6-CtrbObsv-22Feng-Li Lian © 2007Example 6.3 – 2
NTUEE-LS6-CtrbObsv-23Feng-Li Lian © 2007Example 6.3 – 2
“Larger” u1 transfers x(0) = [10 −1]′ to x(2) = 0 in 2 seconds, &
“Smaller” u2 transfers x(0) = [10 −1]′ to x(4) = 0 in 4 seconds.
Note: Given the same x(0), t1, and x(t1), the formula in Theorem 6.1 for u(·) gives the minimal energy control than other u(·):
NTUEE-LS6-CtrbObsv-24Feng-Li Lian © 2007Controllability Index (6.2.1)
NTUEE-LS6-CtrbObsv-25Feng-Li Lian © 2007Controllability Index – 2
Given a controllable pair (A, B) ∈ Rn×n × Rn×p and rank B = p
search for n L.I. columns from left to right
NTUEE-LS6-CtrbObsv-26Feng-Li Lian © 2007Controllability Index – 3
b1 b2 • • • bp
I
A
A2
A3
•••
An−1
12{ , 1, 2,, , , }, , ii i i i i pμ − =b Ab A b A b ……
is a set of n L.I. columns,
and the set
{μ1, μ2, …, μp} with μ1+μ2+…+μp = n
is the set of controllability indices
μ = max{μ1, μ2, …, μp} is called
the controllability index of (A, B)
NTUEE-LS6-CtrbObsv-27Feng-Li Lian © 2007Controllability Index – 4
If μi = 1 for all i ≠ i0, then μ = μi0 = n − (p − 1)
α
μ
α α
α α α
− −
− −
= + + +
= + + +
≤
1 21 2
1 21 2
If is the degree of the of ,
then
an
minimal polyn
d .
Thus
o
mial
.
n nn
n n
n
nn
n
n
B B
A
A A A I
A A A B B
=
NTUEE-LS6-CtrbObsv-28Feng-Li Lian © 2007Corollary 6.1 (6.2.1)
NTUEE-LS6-CtrbObsv-29Feng-Li Lian © 2007Example 6.5 (6.2.1)
μ1 = μ2 = μ = 2
NTUEE-LS6-CtrbObsv-30Feng-Li Lian © 2007Theorem 6.2 (6.2.1)
Proof:
NTUEE-LS6-CtrbObsv-31Feng-Li Lian © 2007Theorem 6.3 (6.2.1)
Proof:
NTUEE-LS6-CtrbObsv-32Feng-Li Lian © 2007Observability (6.3)
above
NTUEE-LS6-CtrbObsv-33Feng-Li Lian © 2007Observability – 2
if u(t) = 0, ∀t ≥ 0,
then y(t) = 0, ∀t ≥ 0,no matter what x(0) is
if u(t) = 0, ∀t ≥ 0 and x2(0) = 0,
then y(t) = 0, ∀t ≥ 0,no matter what x1(0) is
• Un-Observable Examples:
NTUEE-LS6-CtrbObsv-34Feng-Li Lian © 2007Observability – 3
• Observable Example:
NTUEE-LS6-CtrbObsv-35Feng-Li Lian © 2007Observability – 4: Diagonal Representation
NTUEE-LS6-CtrbObsv-36Feng-Li Lian © 2007Observability – 5: Controllable Representation
NTUEE-LS6-CtrbObsv-37Feng-Li Lian © 2007Observability – 6: Physical Representation
NTUEE-LS6-CtrbObsv-38Feng-Li Lian © 2007Observability – 7
the only unknown
Re-write:
total response − zero-state response
Observability involves only zero-input response,
and is decided by A and C
NTUEE-LS6-CtrbObsv-39Feng-Li Lian © 2007Observability – 8
q×n (known) n×1 (unknown) q×1 (known)
: Linear equations
Because x(0) generates ,
the linear equations always have solutions,
and the problem is to determine x(0) uniquely
For q < n, need at an interval of t
to find the unique solution.
NTUEE-LS6-CtrbObsv-40Feng-Li Lian © 2007Theorem 6.4 (6.3)
system (A, B, C, D)
Proof:
“⇐”
NTUEE-LS6-CtrbObsv-41Feng-Li Lian © 2007Theorem 6.4 – 2
“⇒”
NTUEE-LS6-CtrbObsv-42Feng-Li Lian © 2007Theorem 6.5 (6.3)
Proof:
NTUEE-LS6-CtrbObsv-43Feng-Li Lian © 2007
NTUEE-LS6-CtrbObsv-44Feng-Li Lian © 2007Observability Index (6.3.1)
Given an observable pair (A, C) ∈ Rn×n × Rq×n and rank C = q
Search forn L.I. rows from top to bottom
12, , , { 1, 2, }, , , ii i i i i qν − =c c A c A c A ……
is a set of n L.I. rows, and
the set {ν1, ν2, …, νq} with ν1+ν2+…+νq = n
is the set of observability indices
ν = max{ν1, ν2, …, νq} is called
the observability index of (A, C), and
is the least integer such that
also,
NTUEE-LS6-CtrbObsv-45Feng-Li Lian © 2007Observability Index – 2
NTUEE-LS6-CtrbObsv-46Feng-Li Lian © 2007Observability Index – 3
Differentiate repeatedly and set t = 0 to get
or
• An Alternative Way to Decide x(0)
NTUEE-LS6-CtrbObsv-47Feng-Li Lian © 2007Observability Index – 3
if and only if (A, C) is observable (rank Oν = n)
But the method is not very practical,
because derivatives of y(0) are needed_
The linear equations have solutions
because y(0) is generated by x(0), and have a unique sol.~
NTUEE-LS6-CtrbObsv-48Feng-Li Lian © 2007
• The Example:
Canonical Decomposition (6.4)
NTUEE-LS6-CtrbObsv-49Feng-Li Lian © 2007Canonical Decomposition – 2
• Stability
• Controllability
• Observability
are preserved
NTUEE-LS6-CtrbObsv-50Feng-Li Lian © 2007Canonical Decomposition – 3
⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦
o
c
co
co
c
o
x
xx
x
x
controllable and observable part
controllable and unobservable part
uncontrollable and observable part
uncontrollable and unobservable part
• With appropriate equivalence transformations,
we may obtain new state equations with following property
NTUEE-LS6-CtrbObsv-51Feng-Li Lian © 2007Canonical Decomposition – 4
NTUEE-LS6-CtrbObsv-52Feng-Li Lian © 2007Theorem 6.6 (6.4)
(A, B, C, D) with
(A, B, C, D) into
(A, B, C, D).
NTUEE-LS6-CtrbObsv-53Feng-Li Lian © 2007Theorem 6.6 – 2
Proof:
−⎡ ⎤⎣ ⎦ ⎡ ⎤⎣ ⎦1 1 2 12
112 1, , , , , , ra , , , , nk ,= r ank p pn
np pb b b b b b b bA A A A Aq q
{ } ⎡ ⎤⊂ ⎣ ⎦1 11 1, , span n nq q q qA A
{ } { }−⊂1 1 2
11
2 21 1, , , , , , , , ,, , , , p p
np pn b b b b b b b bA A A A Aq q
−⎡ ⎤⎣ ⎦ ⎡ ⎤⎣ ⎦1 1 2 12
112 1, , , , , , sp , , , , an ,= s pan p pn
np pb b b b b b b bA A A A Aq q
{ }+ ⎡ ⎤∉ ⎣ ⎦1 11 1 span , ,n n nq q q q
+=1 1 11 2 + + + + + + i n nnq q q qq qA
+=111 2 1 + + + + + + i nn nb q q q qq
NTUEE-LS6-CtrbObsv-54Feng-Li Lian © 2007Theorem 6.6 – 2
Proof:
+ +
+ +
=
∗ ∗ ∗ ∗⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥∗ ∗ ∗ ∗
= = ⎢ ⎥∗ ∗⎢ ⎥⎢ ⎥⎢ ⎥
∗ ∗⎣ ⎦
1 11 1
1 11 1
1 1 1
1 11
1
1
[ ] [ ]
[ ] [ ]0 0
0 0
n n
n
n n n n
n n n nn
A
A A A A
q q q q Aq q q q
q qq qq qq q
+ +
∗ ∗⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥∗ ∗
= = ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
1 11 11 11 1 [ ] [ ] 0 0
0 0
n n n nn nq q B q q q qq qB
= 1 [ ]pB b b
NTUEE-LS6-CtrbObsv-55Feng-Li Lian © 2007Theorem 6.6 – 3
Proof:
+
∗ ∗ ∗ ∗⎡ ⎤⎢ ⎥= = ⎢ ⎥⎢ ⎥∗ ∗ ∗ ∗⎣ ⎦
1 11 1 [ ] n n nq qqCC q
The controllability matrix of the new state equations is
NTUEE-LS6-CtrbObsv-56Feng-Li Lian © 2007Theorem 6.6 – 4
Proof:
Thus implies that is controllable
Transfer Matrix:
NTUEE-LS6-CtrbObsv-57Feng-Li Lian © 2007Controllable Realization (6.4)
In the new state equations
The state space is divided into a subspace for (dim. = n1)
And a subspace for (dim. = n − n1);
is controllable by u, while is not controllable
After dropping the uncontrollable subspace,
becomes a controllable realization of smaller dimension which is zero-state equivalent to (A, B, C, D)
NTUEE-LS6-CtrbObsv-58Feng-Li Lian © 2007Example 6.8 (6.4)
Because rank B = 2, use C2 = [B AB] to check controllability:
: uncontrollable
Choose
NTUEE-LS6-CtrbObsv-59Feng-Li Lian © 2007Example 6.8 – 2
A two-dimensional controllable realization:
NTUEE-LS6-CtrbObsv-60Feng-Li Lian © 2007
(A, B, C, D) into
(A, B, C, D).
(A, B, C, D) with
NTUEE-LS6-CtrbObsv-61Feng-Li Lian © 2007Canonical Decomposition
⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦
o
c
co
co
c
o
x
xx
x
x
controllable and observable part
controllable and unobservable part
uncontrollable and observable part
uncontrollable and unobservable part
NTUEE-LS6-CtrbObsv-62Feng-Li Lian © 2007Theorem 6.7 (6.4)
Kalman Decomposition
I/O stability only determinedby the controllable and observable parts
NTUEE-LS6-CtrbObsv-63Feng-Li Lian © 2007Example 6.9 (6.4)
ˆ( ) 1
u
g
y
s
==
co- -
co-co-
co-
NTUEE-LS6-CtrbObsv-64Feng-Li Lian © 2007Conditions in Jordan-Form Equations (6.5)
Without loss of generality, consider only the case
[ ]
11 11
12 12
13 13
21 21
22 22
11 12 13 21 22
,
⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥= =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦
=
J 0 B
J B
J BJ B
J B
0 J B
C C C C C C
Jordan blocks
for λ1
Jordan blocksfor λ2
the last row of Bij
is denoted as blij
the first column ofCij is denoted as cfij
NTUEE-LS6-CtrbObsv-65Feng-Li Lian © 2007Theorem 6.8 (6.5)
(J, B, C)
(J, B, C)
Proof:(for a case where λ1 has only 2 blocks & λ2 has only 1 block)
1. Use the controllability condition
rank[ J−sI B ] = rank[ sI−J B ] = n, for s = λ1, λ2.
NTUEE-LS6-CtrbObsv-66Feng-Li Lian © 2007Theorem 6.8 – 2
Substitute s by λ1 and get
Examination of the rows reveals that bl11 and bl12 should be L.I. for the matrix to have full row rank.
Similarly, substituting s by λ2 requires that bl21 be L.I. (≠ 0 for one vector).
2. Proof is similar for observability
NTUEE-LS6-CtrbObsv-67Feng-Li Lian © 2007Example 6.10 (6.5)
NTUEE-LS6-CtrbObsv-68Feng-Li Lian © 2007Example 6.10 – 2
L.I.
L.I. (≠ 0)
L.I. L.D. (= 0)
NTUEE-LS6-CtrbObsv-69Feng-Li Lian © 2007Corollary 6.8 & 8.O8 (6.5)
NTUEE-LS6-CtrbObsv-70Feng-Li Lian © 2007Example 6.11 (6.5)
NTUEE-LS6-CtrbObsv-71Feng-Li Lian © 2007Discrete-Time State Equations (6.6): Controllability
above
NTUEE-LS6-CtrbObsv-72Feng-Li Lian © 2007Theorem 6.D1 (6.6)
NTUEE-LS6-CtrbObsv-73Feng-Li Lian © 2007Theorem 6.D1 – 2
Proof:
“1. ⇔ 3.” “(A, B) controllable ⇔ Cd has rank n”
i.e.,
arbitraryvector
Cd full row rank⇔ input u[·] canalways be found
“2. ⇔ 3.” “Wdc[n−1] nonsingular (P.D.) ⇔ Cd has rank n”
NTUEE-LS6-CtrbObsv-74Feng-Li Lian © 2007Theorem 6.D1 – 3
“3. ⇔ 4.” “Cd has rank n ⇔ rank [A−λI B] = n, ∀e-value λ of A”
The proof is exactly the same as that for the C.T. systems
“Suppose A has eigenvalues with magnitudes < 1.Wdc[n − 1] nonsingular ⇔ Wdc − AWdcA′ = BB′ has a unique positive definite solution Wdc(∞)”
Theorem 5.D6 says that Wdc − AWdcA′ = BB′ has the unique solution
= Wdc(∞) = Wdc[n − 1] +
> 0≥ 0> 0
“2. ⇔ 5.”
NTUEE-LS6-CtrbObsv-75Feng-Li Lian © 2007Observability (6.6)
above
NTUEE-LS6-CtrbObsv-76Feng-Li Lian © 2007Theorem 6.DO1 (6.6)
(dual to Theorem 6.D1)
NTUEE-LS6-CtrbObsv-77Feng-Li Lian © 2007Observability
Controllability/Observability Indices, Kalman Decomposition,
& Jordan-Form Controllability/Observability Conditions
for discrete-time systems parallels those for C.T. systems
Controllability Index =
Length of the shortest input sequencethat can transfer any state to any other state
Observability Index =
Lengths of the shortest input and output sequencesneeded to determine the initial state uniquely
For discrete-time systems,
NTUEE-LS6-CtrbObsv-78Feng-Li Lian © 2007Controllability to & from the Origin (6.6)
In addition to the regular controllability,
there are two other “weaker” definitions of controllability:
1. Controllability to the origin:
transfer any state to the zero state;
2. Controllability from the origin:
transfer the zero state to any other state,
also called reachability.
It can be shown that for continuous-time systems,
all definitions of controllability are equivalent,
but not for discrete-time systems
NTUEE-LS6-CtrbObsv-79Feng-Li Lian © 2007Controllability to & from the Origin – 2
rank Cd = rank =1:
not controllable, not reachable,
But controllable to the origin:
transfers [0] to [ [ ]2 10]u α βαβ⎡ ⎤⎢+⎣ ⎦
= =⎥= x x 0
NTUEE-LS6-CtrbObsv-80Feng-Li Lian © 2007Controllability after Sampling (6.7)
NTUEE-LS6-CtrbObsv-81Feng-Li Lian © 2007Theorem 6.9 & 6.10 (6.7)
(A, B) (A, B)_ _
NTUEE-LS6-CtrbObsv-82Feng-Li Lian © 2007Example 6.12 (6.7)
Eigenvalues: −1, −1±j2
Discretized systems will be controllableif and only if the sampling period
for m = 1, 2, ….
Let us try T = 0.5π (m = 1):
NTUEE-LS6-CtrbObsv-83Feng-Li Lian © 2007Example 6.12 – 2
and rank Cd = 2, uncontrollableL.D.