EE611
Deterministic Systems
System Descriptions, State, Convolution
Kevin D. Donohue
Electrical and Computer Engineering
University of Kentucky
Input-Output Systems
SISO- systems with a single input and single output.
MIMO- systems with multiple inputs and multiple
outputs.
Continuous-time systems – inputs and output are
defined over all time t (SISO, input u(t), output y(t),
MIMO, input u(t), output y(t))
Discrete-time systems – inputs and output are defined
at discrete points in time {t=kT∣k ∈ I }, with sampling
interval T (u(kT) = u[k], y(kT) = y[k] ).
System Classes
A system is memoryless (instantaneous) iff (if and
only if) output y( t o) depends only on input at u(t o).
A system is causal (nonanticipatory) iff output y(t o )
depends only on input u(t) for t o ≥ t.
State
The state of a system at t o , x( t o ), is the information
required along with the input u(t) for t ≥ t o that uniquely
determines the output y(t ) for t ≥ t o .
Example: Find a state descriptions for the following
system at t o =0
iL(t)
u(t)
Result:
+
22 ΩΩ vc(t)
_
0.1H
0.5F
200 u t = ÿ t 101 ẏ t 120 y t 
x=
[
][ ]
10iL 0
y 0
=
ẏ 0
100v c 0−10i L 0
+
y(t)
-
10Ω
System Classes: Linear
A system is linear iff for every t o and input-output pair (i =1,2)
x i t o 
 yi t  , t ≥t o
u i t  , t ≥t o
}
then additivity holds:
x 1 t o x 2 t o 
 y1 t  y 2 t  ,t ≥t o
u1 t u2 t  , t≥t o
and homogeneity holds:
}
}
 x i t o 
  yi t  , t ≥t o
 u i t  , t ≥t o
where α∈R
Zero-State, Zero-Input Response
If input is zero, the response that results is due to the system
state, known as the zero-input response:
}
x t o 
 y zi t ,t≥t o
ut≡0 , t ≥t o
If state is zero, the response that results is due to the system
input, known as the zero-state response:
xt o =0
 y zs t  ,t≥t o
ut ,
t ≥t o
}
In general for a linear system superposition holds between the
contributions of the state and input to the response. Therefore,
}
x t o 
 y zi t  y zs ,t≥t o
ut , t ≥t o
Response Classes
Zero-input response - system output due only to system
state (or initial conditions).
N
0=∑n=0 n
n
d y zi
dt
n
Zero-state response - system output due only to the input of
the system.
N
u=∑n=0 n
n
d y zs
dt
n
, x=0
In general: Total response = Zero-input response + Zerostate response
y= y zi y zs
Examples of Linearity Determination
Determine whether or not each system described below is
linear. Assume inputs and outputs are functions of time
denoted by u and y, and constants are denoted k.
•
y=ku
•
u= ÿ ẏ y
•
y=ku10
Input-Output Description
Convolution
For a linear lumped or distributed system, the input-output
relationship for a zero-state response can be expressed in terms
of the convolution integral and the system's impulse response:
∞
y t =∫−∞ g t ,  u d 
where g t , is the system's time-varying impulse response at
time τ. If system is zero-state (relaxed) at t o then integral can
∞
be written:
y t=∫t g t , u d 
o
If system also is causal, impulse response must be zero for τ>t:
t
y t=∫t g t ,u d 
o
MIMO Input-Output
Description
For a p input and q output linear, causal, relaxed at t o , lumped
or distributed system, the input-output relationship for a zerostate response can be expressed in terms of the convolution
integral and the system's impulse response matrix:
t
y t =∫t G t ,ud 
o
where G t ,  is the system's time-varying impulse response
matrix describing the contribution of inputs at all p terminals to
the q outputs:
[
g 11 t ,  g 12 t , 
g t , g 22 t , 
Gt ,= 21
⋮
⋮
g q1 t , g q2 t , 
... g 1p t , 
... g 2p t , 
...
⋮
... g qp t , 
]
State-Space Description
For a lumped system represented by an order N differential
equation governing the state can be written as (p inputs):
ẋ t=A t x tB t ut 
where A is an NxN matrix, x is a Nx1 vector, B is a Nxp matrix
and u is a 1xp vector.
The output (q outputs) is a linear combination of the states and
inputs and can be written as:
y t =Ct x tD t ut 
where C is an qxN matrix, x is an Nx1 vector, D is a qxp matrix
and u is a 1xp vector.
Time Invariance
If system is not changing over time, it is referred to as time
invariant and results in significant simplifications. More
formally stated:
A system is time invariant iff for every t o and input-output pair
}
x t o 
 y t ,t ≥t o
ut , t≥t o
and any time shift T, the following also holds
}
x t oT 
 y t T  , t≥t oT
ut−T  , t≥t oT
Time Invariance
Linear systems that are time invariant are referred to as linear
time invariant (LTI) systems. Their representations simplify to:
t
t
y t =∫t G t ,ud   y t=∫t G t−u d 
o
o
ẋ t=A t  x tB t ut  ẋ t=A x tB ut
y t =Ct x tD t ut  y t =C x tD ut
Transfer Functions
The transfer function TF of an LTI system can be derived from
the Laplace Transform of its input-output description.
Show for a relaxed system, the Laplace Transform of the
impulse response is its transfer function.
y  s 
LT { g  t  }=
= g  s 
u  s 
Transfer Functions and State Space
For a SISO system, derive the relationship between TF and the
zero-state and zero-input responses by taking the LT of a statespace representation to obtain:
y  s =c  s I−A  x  t o   d c  s I−A  b  u  s
−1
−1
Find the formula to convert a state-space representation to a TF
for the zero-state case.
Is it possible for a TF to represent the case when the state is
not zero (not a relaxed system)?
➢
➢
What is the significance of the d parameter?