Chap 4. State-Space Solutions and
Realizations
Outlines
 1.
1 Introduction
 2. Solution of LTI State Equation
 3.
3 Equivalent State Equations
 4. Realizations
 5.
5 Solution of Linear Time-Varying
Time Varying (LTV) Equations
 6. Equivalent Time-Varying Equations
 7.
7 Time-Varying Realizations
2
1.Introduction
 How to find the solution for a linear system
 Convolution
 Discrete version
 LTI systems
 Transfer function, involves Laplace
p
transform, ppoles and zeros, inverse
Laplace transform
 They are inconvenient and inaccurate
2. Solution of LTI State Equations
 Find the solution excited by x(0) and u(t)
 Solution
 Preliminary
l

2. Solution of LTI State Equations
 Verification
 Initial state
 General solution
applying
2. Solution of LTI State Equations
 How to compute eAt
 Alternatively,
2. Solution of LTI State Equations

2. Solution of LTI State Equations
 Discretization

2. Solution of LTI State Equations
 Discretization

2. Solution of LTI State Equations
 Computation of Bd

 If A is nonsingular
 Matlab command
 c2d
2. Solution of LTI State Equations
 Convergence of zero-input
zero input response Akx[0]

 E.g.,
2. Solution of LTI State Equations
 Convergence of zero-input
zero input response Akx[0]
3. Equivalent State Equations
 Equivalent state

3. Equivalent State Equations
 The same eigenvalues
 The same transfer matrix
 Indeed,
Indeed equivalent state equations have the same
characteristic polynomial
3. Equivalent State Equations
 Zero
Zero-state
state equivalent
 The same transfer function matrix
using
 Theorem
 Algebraic equivalence implies zero-state equivalence; however, zero-state
equivalence does not imply Algebraic equivalence
3. Equivalent State Equations
 Canonical forms
 Matlab command: [ab, bb, cb, db, P] = canon(a, b, c, d, ‘type’)
 Type = companion
 {A, b1} is controllable
 Type
yp = model
 Diagonal with complex eigenvalues
3. Equivalent State Equations
 Canonical forms
4. Realization
 Transfer matrix and state-space equation
 Every LTI system has input-output description
 If it is lumped as well
U
Unique
que transformation
t a s o at o from
o SS to TM,,
 How about the inverse problem
p
 Realization = from TM to SS
4. Realization
 Realizable


 Not every TM is realizable, e.g., a distributed system
 If TM iss realizable,
ea a e, itt has
as infinitely
te y many
a y realizations,
ea at o s, not
ot
necessarily of the same dimension
 Theorem
4. Realization
 Proof:
 →:
4. Realization
 ←:
4. Realization
 ←:
See P102 for more details.
4. Realization
 A special case of p = 1
The controllable-canonical-form can be directlyy read out from
the coefficients of TM
4. Realization
 Example
4. Realization
 Example
5. Solution of LTV Equations
 LTV system
 LTI: scalar
because
vector
because
5. Solution of LTV Equations
 LTV system
 LTV: scalar
because
vector
because
but
?
5. Solution of LTV Equations
 Fundamental matrix
 For every initial state xi(t0), there exists a unique solution xi(t).
Let X(t)
( ) = [[x1((t),
), x2((t),…,
),…, xn((t)].
)]. If X(t
( 0) is nonsingular,
g , X(t)
( ) is
the fundamental matrix, satisfying
 X(t0) can be arbitrarily chosen, as long as it is nonsingular, X(t)
is not unique
 X(t) is nonsingular for all t
5. Solution of LTV Equations
 State transition matrix

 Properties
5. Solution of LTV Equations

 Solution
5. Solution of LTV Equations
 Verification
 Initial condition
 Derivative
5. Solution of LTV Equations
 Output
 Zero-input
Z
i t
 Zero-state
5. Solution of LTV Equations
 Input-Output description
 Special case
 If A(t) has commutative property, such as diagonal or constant
we have
constant A
5. Solution of LTV Equations
 Discrete case
 System
 Transition matrix
 Response
 Input-output description
6. Equivalent Time-Varying
Equations
 System
(4
(4.69)
69)
 Let P(t) is nonsingular and both P(t) and its derivative are
continuous for all t. Let
 Equivalence transformation
(
(4.70)
)
where
6. Equivalent Time-Varying
Equations
 Relationship between fundamental matrices
 Proof
 Theorem
6. Equivalent Time-Varying
Equations

6. Equivalent Time-Varying
Equations

 Input-output
I t t t description
d i ti equivalence
i l
 Block diagram
g
6. Equivalent Time-Varying Equations
 Conclusion
7. Time-Varying Realization
 Input-output description
 State
St t equation
ti
 Impulse response matrix
 If such {A(t), B(t), C(t), D(t)} exists,
is said realizable
7. Time-Varying Realization
 Theorem 4.5
45

7. Time-Varying Realization
