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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-11Feng-Li Lian © 2007Theorem 6.2 – 2

Proof:

“1. ⇔ 2.” “(A, B) controllable ⇔ Wc(t) nonsingular, ∀t > 0”


Proof:


Proof:


Proof:

“2. ⇔ 3.” “Wc(t) nonsingular, ∀t > 0 ⇔ C has rank n”


Proof:


Proof:

“3. ⇔ 4.” “C has rank n ⇔ rank [A−λI B] = n,∀ e-value λ of A”


Proof:


Proof:


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


Mass = 0

NTUEE-LS6-CtrbObsv-22Feng-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


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)


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:


• Observable Example:

NTUEE-LS6-CtrbObsv-37Feng-Li Lian © 2007Observability – 6: Physical Representation


the only unknown

Re-write:

total response − zero-state response

Observability involves only zero-input response,

and is decided by A and C


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.


system (A, B, C, D)

Proof:

“⇐”


“⇒”


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


Differentiate repeatedly and set t = 0 to get

or

• An Alternative Way to Decide x(0)


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.~


• The Example:

Canonical Decomposition (6.4)

NTUEE-LS6-CtrbObsv-49Feng-Li Lian © 2007Canonical Decomposition – 2

• Stability

• Controllability

• Observability

are preserved


⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦

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



(A, B, C, D) with

(A, B, C, D) into

(A, B, C, D).


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


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


Proof:

+

∗ ∗ ∗ ∗⎡ ⎤⎢ ⎥= = ⎢ ⎥⎢ ⎥∗ ∗ ∗ ∗⎣ ⎦

1 11 1 [ ] n n nq qqCC q

The controllability matrix of the new state equations is


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)


Because rank B = 2, use C2 = [B AB] to check controllability:

: uncontrollable

Choose


A two-dimensional controllable realization:


(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


Kalman Decomposition

I/O stability only determinedby the controllable and observable parts


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


(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.


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



L.I.

L.I. (≠ 0)

L.I. L.D. (= 0)

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)_ _


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):


and rank Cd = 2, uncontrollableL.D.

線性系統 - 國立臺灣大學cc.ee.ntu.edu.tw/~fengli/teaching/linearsystems/961ls06... ·...

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