mathematical representation of linear systems state space model ( internal description)
DESCRIPTION
Mathematical Representation of Linear Systems State Space Model ( Internal Description) Continuous Time. Discrete Time. Transfer Function ( External Description) y(s)=H(s)u(s) y(z)=H(z)u(z). U. y. P. - PowerPoint PPT PresentationTRANSCRIPT
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)()()()()()(tDutCxtytButAxtx
Mathematical Representation of Linear Systems
• State Space Model (Internal Description)
Continuous Time
Discrete Time
)()()()()()1(
kDukCxkykBukAxkx
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• Transfer Function (External Description)
y(s)=H(s)u(s)
y(z)=H(z)u(z)
PU y
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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.
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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)
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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.
pu
u1
qy
y1
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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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• 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 zero-
memory or memoryless system.
• An example for such a system is the resistor.
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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 -
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Under relaxedness assumption, we can write
y= Hu
Where H is some operator or function that specifies
uniquely the ouput 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
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The linearity condition:H(1 u 1 + 2 u 2) = 1H(u 1) + 2 H(u 2)
is equivalent to:
1. H(u) = H(u) (Homogeneity)
2. H(u 1 + u 2) = H(u 1) + H(u 2) (Additivity)
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We can take advantage of the linearity property to calculate the output of a given linear system for any input.
• First, define to be the pulse:
A given input can be approximated by a sequence of pulses as follows:
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i
ii tttutu )()()(
)()( i
ii tttuHHuy
By linearity
),()()(
)()(),(
ii
i
ttg
ii
i
ttgtuty
ttHtuyi
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Taking the limit as 0
),()()( dtguty
where )()(:),( ttHtg
i.e. ),( tg the output at time t when the input is function applied at time .
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For a linear system H.
u
H
y
dtutgty )(),()(
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What about the output for MIMO linear system?
2 Input, 1 Ouput
u1
u2H
y1
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dtutgdtutg
uHu
uH
uHy
)(),()(),(
H
00
212111
12111
2
11
2
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up
u1y1
H yq
For a general MIMO system
dutgdutgty
dutgdutgty
pqpqq
pp
)(),(....)(),()(
)(),(...)(),()(
11
11111
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In a matrix form
dtutGty )(),()(
“Input-Output Description for the linear system”Where
),(),(
),(),(),(
1
111
tgtg
tgtgtG
qpq
p
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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.
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For a linear system,
dutgty )(),()(
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Where is the response to an impulse applied at time .
If the linear system is causal, for < .
Thus, for a (1) causal (2) linear system, the output is related to the input by:
),( tg
0),( tg t
tdutgty )(),()(
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o
o
tdutG
tt
tt
tdutG
dutGty
)(),(
0for
zero is term this,0
at
releaxed is system theIf
)(),(
)(),()(
For a linear system
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Therefore, for a linear system which is relaxed at t0
ot
dutGty o t t )(),()(
For a linear system which is causal and relaxed at t0
t
otdutGty o t t )(),()(