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Mathematical Representation of
Linear Systems
Deterministic Systems
Memoryless Systems
Systems with Memory
Y (t )  g[ X (t )]
Time-varying
systems
Time-Invariant
systems
Linear systems
Y (t )  L[ X (t )]
Linear-Time Invariant
(LTI) systems
X (t )
h (t )
LTI system

Y (t )     h (t   ) X ( )d

    h ( ) X (t   )d .
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Linearize the Nonlinear System
根據系統的操作點(Operating Point)或平衡點(Equilibrium
Point)做非線性系統之線性化。亦即假設非線性系統在操
作點附近做微量變化，則可將此微量變化的行為視為線性
系統的動態行為。
Taylor series expansion is usually employed.
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非線性非時變系統之線性化
Equlibrium point : x  0 and u=0 or f(x(t), 0)=0
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Robotics Laboratory, M.E., N.T. U.
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Example
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Robotics Laboratory, M.E., N.T. U.
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Solution (1/4)
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臺大機械系機器人實驗室
2022/9/6
Solution (2/4)
NTU ME Robotics Lab.
臺大機械系機器人實驗室
Robotics Laboratory, M.E., N.T. U.
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Robotics Laboratory, M.E., N.T. U.
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Solution (3/4)
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臺大機械系機器人實驗室
2022/9/6
Solution (4/4)
Robotics Laboratory, M.E., N.T. U.
NTU ME Robotics Lab.
臺大機械系機器人實驗室
State Space Model (Internal Description)
Continuous Time

x ( t )  Ax ( t )  Bu (t )
y (t )  Cx ( t )  Du (t )
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Example: In a network if we are interested in terminal
properties we may use the Impedance or Transfer
Function. However, if we want the currents and
voltages of each branch of the network, then loop or
nodal analysis has to be used to find a set of
differential equations that describes the network.
Transfer Function (External Description)
u
y
H
y(s)=H(s)u(s)
y(z)=H(z)u(z)
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• A system is said to be a single-variable if and only if
it has only one input and one output (SISO)
• A system is said to be a multivariable if and only if it
has more than one input or more than one output
(MIMO)
The Input Output Description
The input-output description of a system gives a mathematical
relation between the input and output of the system. In such a
case a system may be considered as a black box and we try to
get system properties through the input-output pairs.
 u1 

 


 
u p 
 y1 

 


 
 y q 
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H
u
y
For a linear system H.

y (t )   g (t , )u (t )d

The time interval in which the inputs and outputs will be
defined is from - ∞ to ∞. We use u to denote a vector
function defined over (- ∞ , ∞); u(t) is used to denote
the value of u at time t.
If the function u is defined only over [t0, t1) we write
u[t0, t1).
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Memoryless and Memory Systems
• Def:
If the output at time t1 of a system
depends only on the input applied at time t1,
the system is called an instantaneous or zeromemory or memoryless system.
• An example for such a system is the resistor.
Def:
A system said to have memory if the output at time t1
depends not only the input applied at t1, but also on the input
applied before and /or after t1.
Hence, if an input u[t1,) is applied to a system, unless we know
the input applied before t1, we will obtain different output y[t1,) ,
although the same input u[t1,) is applied.
So it is clear that such an input-output pair lacks a unique relation.
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Hence in developing the input-output description before an input
is applied, the system must be assumed to be relaxed or at rest,
and the output is excited solely and uniquely by the input applied
thereafter.
A system is said to be relaxed at time t1 if no energy is stored in
the system at that instant. We assume that every system is
relaxed at time - .
Under relaxedness assumption, we can write
y= Hu
where H is some operator or function that specifies
uniquely the output y in terms of the input u of the
system.
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Definition: A system described by the mapping
H is said to be linear if
H(1 u 1 + 2 u 2) = 1H(u 1) + 2 H(u 2)
For all u 1 , u 2 , 1 and 2 .
The Superposition Principle applies
SISO vs. MIMO
• A system is said to be a single-variable if and only if
it has only one input and one output (SISO)
• A system is said to be a multivariable if and only if it
has more than one input or more than one output
(MIMO)
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What about the output for MIMO linear system?
2 Inputs, 1 Ouput
u1
y1
H
u2
 0 
 u1  


y1  H      H    
 0 
 u2  
 H11u1  H12u 2




  g11 (t , )u1 ( )dt   g12 (t , )u2 ( )dt
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For a general MIMO system
y1
u1
H
yq
up




y1 (t )   g11 (t , )u1 ( )d  ...   g1 p (t , )u p ( )d





yq (t )   g q1 (t , )u1 ( )d  ....   g qp (t , )u p ( )d
In a matrix form
y (t )  


G (t ,  )u ( )dt
“Input-Output Description for the linear system”
where
 g11 (t , )  g1 p (t , )


G (t , )   

 g q1 (t , )  g qp (t , )


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• Causality
• Definition: A system is said to be causal or
nonanticipatory if the output of the system at time t
does not depend on the input applied after time t; it
depends only on the input applied before and at time
t.
For a linear system,

y (t )   g (t , )u ( )d

where
g (t , )
is the response to an impulse applied at time
.
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If the linear system is causal, g (t , )  0
for
t

< .
Thus, for a (1) causal and (2) linear system, the output is related
to the input by
y (t ) 
t
 g (t ,  )u ( ) d 

For a linear system
y(t ) 


 G(t, )u( )d


to
  G(t, )u( )d
 G(t, )u( )d
t0



If the system is releaxed
at t t0 , this term is zero
for t t0
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Therefore, for a linear system which is relaxed at t0

 G(t, )u( )d
t
y (t ) 
t t
o
0
For a linear system which is causal and relaxed at t0
y (t ) 
t
 G(t, )u( )d
t
t t
o
0
Convolution Integral Representation
H
u
y
For a linear system H

y (t )   h(t , )u (t )d  u  h  h  u

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Convolution Integral
Derivation of Convolution Integral.
(a) The operator H denotes the
system in which the u(t) is
applied.
(b) Use the linearity property.
(c) Define impulse response as unit
impulse input.
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The time invariance implies that a time-shifted impulse input result
in a time shift impulse response output as in Figure 2.4 below.
y t  

 u  h  t    d

u t  * h t  

 u  h  t    d .
h  t   H   t 

where
Figure 2.4: (a) Impulse response of an LTI system H. (b) The output of an LTI
system to a time-shifted and amplitude-scaled impulse is a time-shifted and
amplitude-scaled impulse response.
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Step for Convolution Integral Computation
To compute the superposition integral
y t   u t  * h t  

 u  h t    d

Step 1: Plot u and h versus t since the convolution sum is on t.
Step 2: Flip h(t) around the vertical axis to obtain h(-t).
Step 3: Shift h(t) by τ to obtain h(t-τ).
Step 4: Multiply to obtain u(t) h(t-τ).
Step 5: Integrate on  to compute

 u  h  t    d

Step 6: Increase  and repeat Step 3-6.
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