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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:
NTUEE-LS4-Solution-5Feng-Li Lian © 2007Solution of LTI State Equations – 3
• LTI State Equations:
NTUEE-LS4-Solution-6Feng-Li Lian © 2007Solution of LTI State Equations – 4
• Useful formulae:
• Verification:
NTUEE-LS4-Solution-7Feng-Li Lian © 2007Solution of LTI State Equations – 5
• 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-11Feng-Li Lian © 2007Computing eAt – 2
NTUEE-LS4-Solution-12Feng-Li Lian © 2007Computing eAt – 3
NTUEE-LS4-Solution-13Feng-Li Lian © 2007Computing eAt – 4
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 → ∞
NTUEE-LS4-Solution-20Feng-Li Lian © 2007Solution Characteristics
• 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:
NTUEE-LS4-Solution-23Feng-Li Lian © 2007Discretization
• C.T. systems with piecewise constant inputs
(may be generated by computers)
NTUEE-LS4-Solution-24Feng-Li Lian © 2007Discretization
NTUEE-LS4-Solution-25Feng-Li Lian © 2007Discretization
NTUEE-LS4-Solution-26Feng-Li Lian © 2007Discretization
NTUEE-LS4-Solution-27Feng-Li Lian © 2007Discretization
• If A is nonsingular, then
NTUEE-LS4-Solution-28Feng-Li Lian © 2007Discretization
NTUEE-LS4-Solution-29Feng-Li Lian © 2007Solution of D.T. Equations
•••
NTUEE-LS4-Solution-30Feng-Li Lian © 2007Solution of D.T. Equations
NTUEE-LS4-Solution-31Feng-Li Lian © 2007Solution Characteristics
• λ1 : multiplicity = 4, index = 3
• λ2 : multiplicity = 1, index = 1
NTUEE-LS4-Solution-32Feng-Li Lian © 2007Solution Characteristics
• 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
NTUEE-LS4-Solution-40Feng-Li Lian © 2007State Variable Change or Coordinate Change – 2
are said to be (algebraically) equivalent, and
is called an equivalence transformation.
NTUEE-LS4-Solution-41Feng-Li Lian © 2007State Variable Change or Coordinate Change – 3
NTUEE-LS4-Solution-42Feng-Li Lian © 2007State Variable Change or Coordinate Change – 4
From Sec 3.4
NTUEE-LS4-Solution-43Feng-Li Lian © 2007Equivalent State Equations – 1
• Equivalent state equations
have the same eigenvalues and transfer matrix:
NTUEE-LS4-Solution-44Feng-Li Lian © 2007Equivalent State Equations – 2
• Two state equations may have the same transfer matrix
(and are called zero-state equivalent),
but are NOT algebraically equivalent.
NTUEE-LS4-Solution-45Feng-Li Lian © 2007Example 4.4
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
NTUEE-LS4-Solution-49Feng-Li Lian © 2007Diagonal/Jordan Canonical Form – 3
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
NTUEE-LS4-Solution-50Feng-Li Lian © 2007Diagonal/Jordan Canonical Form – 4
• 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
NTUEE-LS4-Solution-53Feng-Li Lian © 2007Example 4.5
unit-step
t
|x2| too small
|x1| too large
NTUEE-LS4-Solution-54Feng-Li Lian © 2007Example 4.5
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
NTUEE-LS4-Solution-60Feng-Li Lian © 2007
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
NTUEE-LS4-Solution-63Feng-Li Lian © 2007Realization: Controllable Canonical Form
• Let us check the transfer matrix of the following state equation
in controllable canonical form:
rp×rp rp×p
q×rp
A
CB
NTUEE-LS4-Solution-64Feng-Li Lian © 2007Realization: Controllable Canonical Form
• Consider
i.e.,
i.e.,
NTUEE-LS4-Solution-65Feng-Li Lian © 2007Realization: Controllable Canonical Form
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
NTUEE-LS4-Solution-68Feng-Li Lian © 2007Example 4.7
NTUEE-LS4-Solution-69Feng-Li Lian © 2007
For this case, a four-dimensional realization with this method
Overall Realization:
Example 4.7
NTUEE-LS4-Solution-70Feng-Li Lian © 2007Example 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)”.