線性系統cc.ee.ntu.edu.tw/~fengli/Teaching/LinearSystems/961ls04...Linear Systems Chapter 04 State-Space Solutions & Realizations Feng-Li Lian NTU-EE Sep07 – Jan08 Materials used

Fall 2007

線性系統Linear Systems

Chapter 04State-Space Solutions & Realizations

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-LS4-Solution-2Feng-Li Lian © 2007

Introduction

Solution of LTI State Equations (4.2)

Equivalent State Equations (4.3)

Realizations (4.4)

Outline

NTUEE-LS4-Solution-3Feng-Li Lian © 2007Solution of LTI State Equations (4.2) – 1

• Derivative of Exponential Function:

NTUEE-LS4-Solution-4Feng-Li Lian © 2007Solution of LTI State Equations – 2

• LTI State Equation and its Solution:


• LTI State Equations:


• Useful formulae:

• Verification:


• Output Equation:

NTUEE-LS4-Solution-8Feng-Li Lian © 2007Solution of LTI State Equations (4.2): By Laplace Transform

• Laplace transform:

NTUEE-LS4-Solution-9Feng-Li Lian © 2007Computing eAt & ( sI - A )-1

NTUEE-LS4-Solution-10Feng-Li Lian © 2007Computing eAt – 1




NTUEE-LS4-Solution-14Feng-Li Lian © 2007Computing ( sI - A )-1 – 1

NTUEE-LS4-Solution-15Feng-Li Lian © 2007Computing ( sI - A )-1 – 2

NTUEE-LS4-Solution-16Feng-Li Lian © 2007Computing ( sI - A )^-1 – 3

the Jordan form for A

NTUEE-LS4-Solution-17Feng-Li Lian © 2007Computing ( sI - A )^-1 – 4

( )s s s s1 1 2 3 2I A I A A− − − −− = + + +

(Problem 3.26)

NTUEE-LS4-Solution-18Feng-Li Lian © 2007Example 4.2

NTUEE-LS4-Solution-19Feng-Li Lian © 2007Solution Characteristics

• If Re(λi) < 0 for all i,

then every zero-input response will approach zero as t → ∞

• If Re(λi) > 0 for some i,

then part of zero-input response may grow unbounded as t → ∞


• If Re(λi) ≤ 0 for all i, and λj with Re(λj) = 0 has only index 1,

then zero-input response will be bounded for all t

• If Re(λi) ≤ 0 for all i,

but some λj with Re(λj) = 0 has index 2 or higher,

then part of zero-input response may grow unbounded as t → ∞

NTUEE-LS4-Solution-21Feng-Li Lian © 2007Discretization (4.2.1)

• Finite difference approximation of C.T. systems

NTUEE-LS4-Solution-22Feng-Li Lian © 2007Discretization

• Approximation:


• C.T. systems with piecewise constant inputs

(may be generated by computers)





• If A is nonsingular, then


NTUEE-LS4-Solution-29Feng-Li Lian © 2007Solution of D.T. Equations

•••

NTUEE-LS4-Solution-30Feng-Li Lian © 2007Solution of D.T. Equations


• λ1 : multiplicity = 4, index = 3

• λ2 : multiplicity = 1, index = 1


• If |λi| < 1 for all i, thenevery zero-input response will approach zero as k → ∞

• If |λi| > 1 for some i, thenpart of zero-input response may grow unbounded as k → ∞

• If |λi| ≤ 1 for all i, and λj with |λj| = 1 has only index 1, then zero-input response will be bounded for all k.

• If |λi| ≤ 1 for all i, but some λj with |λj| = 1 has index 2 or higher,

then part of zero-input response may grow unboundedas k → ∞.

NTUEE-LS4-Solution-33Feng-Li Lian © 2007In Summary: CT

NTUEE-LS4-Solution-34Feng-Li Lian © 2007In Summary: DT

NTUEE-LS4-Solution-35Feng-Li Lian © 2007In Summary: A in CT & DT

NTUEE-LS4-Solution-36Feng-Li Lian © 2007In Summary: exp(A)

NTUEE-LS4-Solution-37Feng-Li Lian © 2007Equivalent State Equations (4.3) – 1

• Example 4.3: Equivalent state equations

NTUEE-LS4-Solution-38Feng-Li Lian © 2007Equivalent State Equations – 2

• Example 4.3: Equivalent state equations Two sets of state variables:

NTUEE-LS4-Solution-39Feng-Li Lian © 2007State Variable Change or Coordinate Change – 1


are said to be (algebraically) equivalent, and

is called an equivalence transformation.



From Sec 3.4


• Equivalent state equations

have the same eigenvalues and transfer matrix:


• Two state equations may have the same transfer matrix

(and are called zero-state equivalent),

but are NOT algebraically equivalent.


NTUEE-LS4-Solution-46Feng-Li Lian © 2007Theorem 4.1

⇔

NTUEE-LS4-Solution-47Feng-Li Lian © 2007Diagonal/Jordan Canonical Form (4.3.1) – 1

NTUEE-LS4-Solution-48Feng-Li Lian © 2007Diagonal/Jordan Canonical Form – 2


Q = [ n L.I. eigenvectors/generalized eigenvectors ]

has the form, which may have diagonal/Jordan c eomplex lements.⇒ A

For , the may be obtained

with a further equivalence

modal f

transfo

orm

rma

real

tion:

A


• Combined equivalence transformation for the modal form:

• Please figure out the modal form

and the associated equivalence transformation for systems

with generalized eigenvectors of grades larger than unity

NTUEE-LS4-Solution-51Feng-Li Lian © 2007Diagonal/Jordan Canonical Form (4.3.1)

• Special algebraically equivalent forms –

the canonical form & companion canonical form:

NTUEE-LS4-Solution-52Feng-Li Lian © 2007Magnitude Scaling (4.3.2)

• Magnitude scaling:

in hardware (op-amp circuit)

or software (digital computation) simulations,

usually need to ensure that

magnitude of all signals are not too large and not too small


unit-step

t

|x2| too small

|x1| too large


Use state variable change to adjust the magnitudes:

t

NTUEE-LS4-Solution-55Feng-Li Lian © 2007Realizations (4.4)

• A transfer matrix is realizable

if there exists a finite dimensional state equation {A,B,C,D},

a realization of , such that

ˆ ( )sG

ˆ ( ) ( )s s −= − +1G IC A B Dˆ ( )sG

NTUEE-LS4-Solution-56Feng-Li Lian © 2007Special Case: Single-Input-Single-Output Systems

NTUEE-LS4-Solution-57Feng-Li Lian © 2007Special Case: Single-Input-Single-Output Systems

NTUEE-LS4-Solution-58Feng-Li Lian © 2007Special Case: Single-Input-Multi-Output Systems (p = 1)

a realization

NTUEE-LS4-Solution-59Feng-Li Lian © 2007Example 4.6: Multi-Input-Multi-Output Systems

N(s)

N1 N2 N3


N1 N2 N3

−α1I2 −α2I2 −α3I2

a six-dimensional realization

Example 4.6

NTUEE-LS4-Solution-61Feng-Li Lian © 2007Realizations (4.4)

: strictly proper

Proof:

“⇒”

and is proper)(ˆ sG

NTUEE-LS4-Solution-62Feng-Li Lian © 2007Realization: Controllable Canonical Form

ˆ: monic l.c.d. of all entries of ( )sp sG

“⇐” Let


• Let us check the transfer matrix of the following state equation

in controllable canonical form:

rp×rp rp×p

q×rp

A

CB


• Consider

i.e.,

i.e.,


Also

i.e.,

i.e., { } is realizaˆ ˆ, , , ( ta io of n) ( )s∞A B C G G

In addition to the controllable canonical form, there is also the “observable” canonical form. See Prob. 4.9

NTUEE-LS4-Solution-66Feng-Li Lian © 2007Special Case: Single-Input Systems (p = 1)

a realization

NTUEE-LS4-Solution-67Feng-Li Lian © 2007Special Case: Single-Input Systems (p = 1)

A multi-input LTI system is

the sum of many single-input LTI systems,

so can realize each single-input subsystem and form the sum:

1st and 2nd columns of



For this case, a four-dimensional realization with this method

Overall Realization:

Example 4.7


Overall Realization:

,

q q q q

1 1 1 1A 0 B c 0 d

x x u y x u

0 A B 0 c d

⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥= + = +⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

realize with {Ai, Bi, ci, di}

ˆth row of ( )i sG

• Can also focus on the realizations of single-output systems,

then treat LTI systems with multi-outputs

as combinations of single-outputs subsystems.

NTUEE-LS4-Solution-71Feng-Li Lian © 2007Realizations for D.T. Systems

• Realizations for discrete-time systems:

all discussions apply,

except “s” is changed to “z”, “x(t)” is changed to “x(k)”,

and “dx(t)/dt” is changed to “x(k+1)”.

Documents

線性系統cc.ee.ntu.edu.tw/~fengli/Teaching/LinearSystems/961ls04...Linear Systems Chapter 04 State-Space Solutions & Realizations Feng-Li Lian NTU-EE Sep07 – Jan08 Materials used