STATE-SPACE REPRESENTATION

•How to find mathematical model,

called a state-space representation, for
a linear, time-invariant system

•How to convert between transfer

function and state space models
 State-Space Modeling
• Alternative method of modeling a system than
• Differential / difference equations
• Transfer functions
• Uses matrices and vectors to represent the system
parameters and variables
• In control engineering, a state space representation is
a mathematical model of a physical system as a set of
input, output and state variables related by first-
order differential equations. To abstract from the
number of inputs, outputs and states, the variables are
expressed as vectors.
Motivation for State-Space Modeling
Easier for computers to perform matrix algebra
◦e.g. MATLAB does all computations as
matrix math
Handles multiple inputs and outputs
Provides more information about the system
◦Provides knowledge of internal variables
(states)

Primarily used in complicated, large-scale systems

 Drawbacks of Transfer Function Model Analysis:

• Transfer function is defined under zero initial conditions

• Transfer function is applicable to linear time invariant systems
• Transfer function analysis is restricted to single input and single output
systems
• Does not provide information regarding the internal state of the system

Advantages of State Variable Analysis:

• It can be applied to linear system
• It can be applied to non-linear system
• It can be applied to time varying system
• It can be applied to time invariant system
• It can be applied to multiple input multiple output system
• Its gives idea about the internal state of the system
 Multivariable linear system (MIMO)
•
Representation of x = Ax + Bu

•
+ x • x
Bu 
+
 x dt

A
+
 State representation of a linear system

•
x + y


+
u
B
+

x
C 
+

•
x = Ax + Bu A y = Cx + Du

A= System Matrix(n,n) C= Output Matrix (p,n)

B= Input Matrix (n,r) D= Direct Transmission Matrix (p,r)
x= State Vector (n,1) y= Output Vector (p,1)
u= Input Vector (r,1)
 Electrical Systems
Steps for developing state equations of RLC networks as follows:

➢ Choose all inductor currents and capacitor voltages as state

variables
➢ Choose a set of loop currents and write the relationship between
state variables and derivatives in terms of these loop currents
➢ Write loop equations and eliminate all variables other than state
variables (and their first derivatives) from the equations
Example
State Transition Matrix
Time-Domain Solutions to the State Equations
• Recall our state equations:
x ( t ) = Ax( t ) + Bu ( t )
y( t ) = Cx ( t ) + Du ( t )
• To solve these equations, we will need a few mathematical tools. First:
A 2 t 2 A3t 3
e = I + At +
At
+ +
2! 3!
where I is an NxN Identity Matrix. Ak is simply AxAx…A.
• For any real numbers t and :
e A ( t +  ) = e At e A
• Further, setting  = -t:
e A ( t +  ) = e At e − At = I
• Next:
d At d  A 2 t 2 A3t 3  A3t 2 Homogenous Form of State Equation
e = I + At + + +  = A + A t +
2
+
dt dt  2! 3!  2!
 A 2 t 2 A3t 3 
= A I + At + + +  = Ae At
 2! 3! 
x ( t ) = Ax( t )
• We can use these results to show that the solution to
is: x ( t ) = e At x (0), t  0
• If: x ( t ) = e At x (0), t  0
d
dt dt
 
dt
 
x (t ) = d e At x (0) = d e At x (0) = Ae At x (0) = Ax(t )
• e is referred to as the state-transition matrix and denoted by 𝛟 𝐭 and 𝛟 𝐭 =𝐞𝐀𝐭 .
At

• We can apply these results to the state equations:

x ( t ) = Ax( t ) + Bu ( t )
x ( t ) − Ax( t ) = Bu ( t )
e − At x ( t ) − Ax( t ) = e − At Bu ( t )
• Note that:
  d 
e x ( t ) = e − At x ( t ) +  e − At  x ( t ) = e − At x ( t ) − Ae − At x ( t ) = e − At x ( t ) − Ax( t )
d − At
dt  dt 

d − At
dt

e x ( t ) = e − At Bu ( t )
• Integrating both sides, within limits
t
0 to t, :
Generalization of our
e − At x ( t ) = x (0) +  e − A Bu ()d convolution integral
0
t
x ( t ) = e At x (0) +  e − A ( t −  )Bu ()d, t  0
0
 t
e − At
x ( t ) = x (0) +  e − A
Bu ()d
0
t
−A (t − )
x ( t ) = e x (0) +  e
At
Bu ()d, t  0
0
t
x ( t ) = ( t ) x (0) +  ( t − )Bu ()d, t  0 (A)
0

Zero Input Zero State Component

Component
where 𝛟 𝐭 = 𝐞𝐀𝐭
• The first term on the right hand side of the equation is response due to
input u(t)=0. Hence, it is called zero input response

• The second term depends on input only and not on the initial state vector.
Hence, it is known as zero state component

If the initial state is known at t=t0 rather than at t=0, then =n(A) can be
written as:
t
A (t − )
x(t ) = e A ( t −t 0 )
x(t0 ) +  e Bu ( )d
t0
 Properties of State Transition Matrix

1. Ф(0) is an identity matrix

We Know
x(t ) = e x(0)
At

at t=0
x(t ) =  (t ) x(0)
x (0) = (0) x (0)
(0) = I
2. 𝛟−𝟏 (𝐭)=𝛟(-t)
𝛟 𝐭 = 𝐞𝐀𝐭 = (𝐞−𝐀𝐭 )−𝟏 = [𝛟 −𝐭 ] −𝟏
𝛟 𝐭 =[𝛟 −𝐭 ] −𝟏
Taking inverse on both sides
𝛟−𝟏 (𝐭)=𝛟(-t)
3. Ф(t1+t2)=Ф(t1)Ф(t2)

𝛟(𝐭 𝟏 +𝐭 𝟐 ) = 𝐞𝐀 𝐭 𝟏 +𝐭 𝟐 =𝐞𝐀𝐭𝟏 𝐞𝐀𝐭𝟐 =Ф(t1) Ф(t2)

4. 𝛟(𝐭) 𝐧 = 𝛟(𝐧𝐭)
𝐧
𝛟(𝐭) 𝐧 = 𝐞 𝐀𝐭 =𝐞𝐀𝐧𝐭 =𝛟(𝐧𝐭)