state space representation,State Space Model Controllability and Observability Derive Transfer Function from State Space Equation Time Response and State Transition Matrix Eigen Value

State Space Representation
1
 State Variables of a Dynamical System
 State Variable Equation
 Why State space approach
 Block DiagramRepresentation Of State Space Model
 Controllabilityand Observability
 Derive Transfer Function from State Space Equation
 Time Response and State Transition Matrix
 Eigen Value

Introduction
2
 The state-space description provide the dynamics as a set of coupled
first-order differential equations in a set of internal variables known as
state variables, together with a set of algebraic equations that combine
the state variables into physical output variables.

Deﬁnition of System State
(𝒙𝟏, 𝒙𝟐, … … , 𝒙𝒏) (called State Variables or State Vector) such that knowledge of
these variables at 𝑡 = 𝑡0, together with knowledge of the input for 𝑡 ≥ 𝑡0 ,
completely determines the behavior of the system for any time t to t0 .
 The number of state variables to completely define the dynamics of the system is
equal to the number of integrators involved in the system (System Order).
 Assume that a multiple-input, multiple-output system involves n integrators (State
Variables).
 Assume also that there are r inputs u1(t), u2(t),……. ur(t) and p outputs y1(t),
y2(t), …….. yp(t). 4
Inner state variables
x1,x2,xn
 
 State: The state of a dynamic system is the smallest set of variables
u1(t)
u2(t)
u (t)
r
y1(t)
y2(t)
yp(t)

General State Representation
 State equation:
 Output equation:
 𝑥 = 𝑆𝑡𝑎𝑡𝑒 𝑉𝑒𝑐𝑡𝑜𝑟
 𝑥 = 𝑑 𝑥(𝑡)
= 𝐷𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑆𝑡𝑎𝑡𝑒 𝑉𝑒𝑐𝑡𝑜𝑟
𝑡
 𝑢 = 𝐼𝑛𝑝𝑢𝑡 𝑉𝑒𝑐𝑡𝑜𝑟
 𝑦 = 𝑂𝑢𝑡𝑝𝑢𝑡 𝑉𝑒𝑐𝑡𝑜𝑟
 𝐴 = 𝑆𝑡𝑎𝑡𝑒 𝑀𝑎𝑡𝑟𝑖𝑥 = 𝑆𝑦𝑠𝑡𝑒𝑚 𝑀𝑎𝑡𝑟𝑖𝑥
 𝐵 = 𝐼𝑛𝑝𝑢𝑡 𝑀𝑎𝑡𝑟𝑖𝑥
 𝐶 = 𝑂𝑢𝑡𝑝𝑢𝑡 𝑀𝑎𝑡𝑟𝑖𝑥
 𝐷 = 𝐹𝑒𝑒𝑑𝑏𝑎𝑐𝑘 𝑚𝑎𝑡𝑟𝑖𝑥/ direct Transmission Matrix
x
(t)  Ax(t)  Bu(t)
y(t)  C x(t)  Du(t)
Dynamic equations

State-Space Equations (Model)
 State equation:
 Output equation:
6
x
(t)  Ax(t)  Bu(t)
y(t)  C x(t)  Du(t)
Dynamic equations
1


 
n1
 n
x (t)
x (t)
x(t)  x2 (t)

1


 
r1
 r
u (t)
u (t)
u(t)  u2 (t)

1
 
 

p1
 p
y (t)
 y (t)
y(t)   y2 (t)

State Vector
State variable
Input Vector Output Vector







A   n n







B   nr







C   pn
1
 
 
n1
 n
x (0)
x (0)
x(0)  x2 (0)








D   pr

Block Diagram Representation Of State Space Model
C
A
D
B
1
s
6
+
+
+
+
u(t) y(t)
x(t)
x
(t)
x
(t)  Ax(t)  Bu(t)
y(t)  C x(t)  Du(t)

sXs AX s BU s
Ys CX s DUs
 Then, the transfer function is
X s sI  A1
BU s
 The state space model
x
  Ax  Bu
y  Cx  Du
 by Laplace transform
Ys 
CsI  A1
B  D
Us
Us
Ts
Ys CsI  A1
B  D
State Space model to Transfer Function

Controllability
 Plant:
x
  Ax  Bu, x  Rn
y  Cx  Du
 Definition of Controllability
A system is said to be (state) controllable at time t0 , if
8
0, 1
t ]
there exists a finite t1  t0 such for any x(t0 ) and any x1 ,
there exist an input u[t that will transfer the state 0
x(t )
x1
to the state at time t1 , otherwise the system is said to
be uncontrollable at time t0 .

Controllability Matrix
 Consider a single-input system (u ∈ R):
 The Controllability Matrix is deﬁned as
 We say that the above system is controllable if its controllability matrix
𝐶(𝐴, 𝐵) is invertible.
Condition for controllability(system is controllable if’f)
det(C )  0  rank(C)  n, full rank
9

Observability
 Plant:
x
  Ax  Bu, x  Rn
y  Cx  Du
 Definition of Observability
and the
10
unobservable at
A system is said to be (completely state) observable at
time t0 , if there exists a finite t1  t0 such that for any x(t0 )
0, 1
t ]
output over the time interval [t0 ,t1] suffices to
determine the state x0 , otherwise the system is said to be
t0 .
at time t0 , the knowledge of the input u[t
0, 1
y[t t ]

Observability Matrix
11
A,CObservable  rank(V)  n  det(V )  0





 

CAn1 

 CA 
C

Observability Matrix V   CA2


if y  R

Controllability and Observability
Theorem I
x
c (t)  Ac xc (t) Bc u(t)
Controllable canonical form Controllable
Theorem II x
o (t)  Ao xo (t) Bo u(t)
y(t)  Co xo (t)
Observable canonical form Observable
 A system in Controller Canonical Form (CCF) is always controllable!!
12
 A system in Observable Canonical Form (OCF) is always controllable!!

Controllable Canonical Form
 We consider the following state-space representation, being called a
controllable canonical form, as
 Note that the controllable canonical form is important in discussing the
pole-placement approach to the control system design.
13

Observable Canonical Form
 We consider the following state-space representation, being called an
observable canonical form, as
14

Diagonal Canonical Form
 Diagonal Canonical Form greatly simpliﬁes the task of computing the
analytical solution to the response to initial conditions.
15

State transition matrix
 State transition matrix
x
(t)  Ax(t)
sX(s)  x(0)  AX (s)
X (s)  (sI  A)1
x(0)
x(t)  L1
[(sI  A)1
]x(0)
 eAt
x(0)

(t)eAt
 L1
[(sI  A)1
]
0
0
0
0
0
At
x(t0 )  (t t0 )x(t0 )
x(t )  e
x(t)  e e
x(0)  e x(t )
x(t0 )  e 0
x(0)
A(tt )
At At
At
16

Eigen values
The eigenvalues of an nxn matrix A are the roots of the characteristic equation.
Consider, for example, the following matrix A:

General State Representation
1. Select a particular subset of all possible system variables, and call
state variables.
2. For nth-order, write n simultaneous, first-order differential equations
in terms of the state variables (state equations).
3. If we know the initial condition of all of the state variables at 𝑡0 as
well as the system input for 𝑡 ≥ 𝑡0, we can solve the equations

State-Space Representation of nth-Order Systems of Linear
Differential Equations
 Consider the following nth-order system:
(𝒏) (𝒏−𝟏)
𝒚 + 𝒂𝟏 𝒚 + … + 𝒂𝒏−𝟏𝒚 + 𝒂𝒏 𝒚 = 𝒖
 where y is the system output and u is the input of the System.
 The system is nth-order, then it has n-integrators (State Variables)
 Let us define n-State variables
20

Differential Equations (Cont.)
 Then the last Equation can be written as
21

Differential Equations (Cont.)
 Then, the stat-space state equation is
 where
22

State Space Model (Example)
 Find the state space model for a system that described by the following
differential equation

c
 9c
 26c
  24c  24r
 Solution:
 The system is 3rd order, then it has three states as follows
x1  c x1  x2
x2  x3
x
3  24x1  26x2 9x3  24r
x2  c

x3  c
 The output equation is
y  c  x1
differentiation

 2 

x3 

0 0x 
x1 
y  
1
x
  
24
 0
1
0

 24  26
 0
  
    
9
 
x3 

1  x    0  r
2
0  x1   0 
2

x
3 


x
1 
.
State Space Model (Example)
x1  x2
x2  x3
x
3  24x1  26x2 9x3  24r
y  c  x1

Transfer Function (Example)
 Find the transfer function from the following transfer function
 Solution:
  
3
 
 0 

 0 1 0  10
 0 1  x   0 u

1  2
x
   0 y  
1 0 0x


1
s 1 0
sI  A 0 s 1 

2 s  3

adj(sI  A)
det(sI  A)
sI  A1

1 

s3
 3s2
 2s 1

s2 

s 
(s2
 3s  2) s  3

 1 s(s  3)

  s  (2s 1)
Ts CsI  A1
B  D

Ts CsI  A1
B  D
s3
 3s2
 2s 1
 3s  2)
10(s2
Ts

 
1  10

1 0 0


s3
 3s2
 2s 1
s2 
 
 0 

s   0 
 3s  2) s  3
1 s(s  3)
 s  (2s 1)
(s2
T(s) 
Transfer Function (Example)

System Poles from State Space model
 poles and check the stability of the following state space Example find the
System model
 Solution:
 Since
 To find the poles 
 Then the poles are {-1, -2 }, the system is stable
  

5
 0 2 
x
  1 3 x  0u y  
1 0x
s  2
sI  A   s(s 3)  2  0
1 s  3


1
 2 
s 3
sI  A
s

controllability and Obervability (Example)
28
 Hence the system is both controllable and observable.

1,
 

0
1
 

0 1
C  0 1
0, B
A
 Plant:
x
  Ax  Bu, x  Rn
y  Cx  Du
   
 
1
Obervability Matrix
0
1
1
CA 1 0
N 
 C 

0
Controllability Matrix V  B AB
0
rank(V )  rank(N)  2

Example
29
c
c
c
y  2 1x
1

0
u
  
1 
x

 2 3
x 
 0
Controllable canonical form
   


 2 1
1 
3
1
1 
CA
V 
 C 

 2
U  B AB
0 rank[U]  2  n
rank[V] 1 n
o
o
o
y  0 1x
2
x  1u

  

1 3
0  2
x 

Observable canonical form

  



3
1 
1 1
2  2
CA 1
V 
 C 

0
U  B AB
rank[U] 1 n
rank[V]  2  n
(s 1)(s  2)
T(s) 
s  2

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