State-Variable Description
Motivation
Consider a system with the transfer function
H F ( s) 
1
s 1
Clearly the system is unstable
To stabilize it, we can precede HF(s) with a compensator
s 1
H c ( s) 
s 1
v
s 1
s 1
1
s 1
y
The overall transfer function:
1
H f ( s) H c 
s 1
s 1
1

s 1 s 1
This is nice outcome, but unfortunately this technique will not work: After a while the
system will burn or saturate.
To see why, let us first set up an analog computer simulation of the cascade system
    1 0  x   2
1
 x1   
  v



 x   1 1  x2   1 
 2
 x1 (0)   x10 
 x (0)   x 
 2   20 
 x1 
y  0 1 
 x2 
There are general methods of solving such so-called state-space
equation but it will suffice to proceed as follows:
The first equation is

x1   x1  2v , x1 (0)  x10
Which yields
x1(t) = e-t x10 – 2e-t *v
*denotes convolution
The second equation

x2  x1  x2  v
has a solution
y(t) = x2(t) = et x20 +
1
2
(et-e-t) x10 + e-t *v
x20
x10
v( s)
 y ( s )  x2 ( s ) 


s  1 ( s  1)( s  1) s  1
Therefore the overall transfer function, which has to be calculated with zero
initial condition is 1/(s+1) as expected.
Note: However, that unless the initial conditions can always be kept zero,
y(.) will grow without bond.
So the input output description of a system is applicable only when the system
is initially relaxed
State-Variable
Definition: The state of a system at time to is the amount of
information at to that, together with u[to, ), determine
uniquely the behavior of the system for all t  to
Usually x denotes state, u input, y output
Example
1
y (t ) 
C
1 to
1 t
 u( )d  C  u ( )d  C to u ( )d
t
1 t
 y (t o )  to u ( )d
C
where
y (to ) 
1 to
  u( )d
C
So if y(to) is known, the output after t  to can be uniquely determined.
Hence, y(to) regarded on the state at time to
Linearity
Definition: A system is said to be linear if for every to and any
two state-input-output pairs

  y i (t ), t  t o
ui (t ), t  t o 
xi ( to )
for i = 1, 2, we have
1 x1 (to )   2 x2 (to )

  1 y1 (t )   2 y2 (t ), t  t o
1u1 (t )   2u2 (t ) t  t o 
for any real constants α1 and α2. Otherwise the system is said to be
nonlinear.
 Linearity must hold not only at the output but also at all state
variables and must hold for zero initial state and nonzero initial state.
•
This definition is different from
H(α1u1 + α2u2) = α1 H(u1)+ α2 H(u2)
Example
C and L are nonlinear
Because L – C loop is in series connection with the current source, its behavior
will not transmit to the output. Hence the above circuit is linear according the
input-output definition while it is nonlinear according to the above definition
of linearity.
A very important property of any linear system is that the responses of the
system can be decomposed into two parts
Output due to
 x(to )

u(t ),
= output due to
t  to
 x (to )

u(t )  0,
t  to
 x(to )  0
+ output due to 
t  to
u(t ),
Or
Response = zero-input response + zero-state response
A very broad class of systems can be modeled by

x1  f1 ( x1 ,..., xn u1 ,..., u p , t )
.
.
.

x n  f n ( x1 ,..., xn u1 ,..., u p , t )
together with










x  f ( x , u, t )


 
y1  g1 ( x1 ,..., xn u1 ,..., u p , t )
.
.
.
y q  g q ( x1 ,..., xn u1 ,..., u p , t )








y  g ( x , u, t )

where
y 
 
u 
.
.
.
x :  .  , u   .  , and y   . 

   
 
 
 y. 
 x. 
.
u 
 
 
x1
1
1
n
p
q
  
For the special cases:

x  A(t ) x  B(t )u
y  C (t ) x  D(t )u