Time-Domain Representations of LTI Systems

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Time-Domain Representations of LTI Systems
2.11 Characteristics of Systems Described by
Differential and Difference Equations
Complete solution y y (n) y (f)
y (n) natural response,
y (f) forced response
2.11.1 The Natural Response
Example 2.24 RC Circuit (continued) Natural
Response
The system In Example 2.17 is described by the
differential equation
Find the natural response of the this system,
assuming that y(0) 2 V, R 1 ? and C 1 F.
ltSol.gt
1. Homogeneous sol.
2. I.C. y(0) 2 V
y (n) (0) 2 V
c1 2
3. Natural Response
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Time-Domain Representations of LTI Systems
Example 2.25 First-Order Recursive System
(Continued) Natural Response
The system in Example 2.21 is described by the
difference equation
Find the natural response of this system.
ltSol.gt
1. Homogeneous sol.
2. I.C. y? 1 8
c1 2
3. Natural Response
The forced response is valid only for t ? 0 or n
? 0
2.11.2 The Forced Response
The forced response is the system output due to
the input signal assuming zero initial conditions.
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Time-Domain Representations of LTI Systems
? The at-rest conditions for a discrete-time
system, y? N 0, , y? 1 0, must be
translated forward to times n 0, 1, , N ? 1
before solving for the undetermined
coefficients, such as when one is determining the
complete solution.
Example 2.26 First-Order Recursive System
(Continued) Forced Response
The system in Example 2.21 is described by the
difference equation
Find the forced response of this system if the
input is xn (1/2)n un.
ltSol.gt
1. Complete solution
2. I.C. Translate the at-rest condition y? 1
to time n 0
y0 1 (1/4) ? 0 1
3. Finding c1
c1 ? 1
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Time-Domain Representations of LTI Systems
4. Forced response
Example 2.27 RC Circuit (continued) Forced
Response
The system In Example 2.17 is described by the
differential equation
Find the forced response of the this system,
assuming that x(t) cos(t)u(t) V, R 1 ? and C
1 F.
From Example 2.22
ltSol.gt
1. Complete solution
2. I.C.
y(0?) y(0) 0
c ? 1/2
3. Forced response
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Time-Domain Representations of LTI Systems
2.11.3 The Impulse Response
? Relation between step response and impulse
response
1. Continuous-time case
2. Discrete-time case
2.11.4 Linearity and Time Invariance
? Forced response ? Linearity
? Natural response ? Linearity
Input Forced response
x1 y1(f)
x2 y2(f)
? x1 ? x2 ?y1(f) ? y2 (f)
Initial Cond. Natural response
I1 y1(n)
I2 y2(n)
? I1 ? I2 ?y1(n) ? y2 (n)
? The complete response of an LTI system is not
time invariant.
Response due to initial condition will not shift
with a time shift of the input.
2.11.5 Roots of the Characteristic Equation
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Time-Domain Representations of LTI Systems
? Roots of characteristic equation
Forced response, natural response, stability, and
response time.
? BIBO Stable
1. Discrete-time case
2. Continuous-time case
2.12 Block Diagram Representations
? A block diagram is an interconnection of the
elementary operations that act on the input
signal.
? Three elementary operations for block diagram

1. Scalar multiplication y(t) cx(t) or yn
   cxn, where c is a scalar.
2. Addition y(t) x(t) w(t) or yn xn
   wn.
3. Integration for continuous-time LTI system

and a time shift for discrete-time LTI system
yn xn ? 1.
Fig. 2.32.
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Time-Domain Representations of LTI Systems
Figure 2.32 (p. 162)Symbols for elementary
operations in block diagram descriptions of
systems. (a) Scalar multiplication. (b) Addition.
(c) Integration for continuous-time systems and
time shifting for discrete-time systems.
Ex. A discrete-time LTI system Fig. 2.33.
? Direct Form I
1. In dashed box
(2.49)
2. yn in terms of wn
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Time-Domain Representations of LTI Systems
(2.50)
3. System output yn in terms of input xn
Cascade Form (Direct Form I)
(2.51)
Figure 2.33 (p. 162)Block diagram
representation of a discrete-time LTI system
described by a second-order difference equation.
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Time-Domain Representations of LTI Systems
? Direct Form II
1. Interchange the order of Direct Form I.
2. Denote the output of the new first system as
fn.
Input xn
(2.52)
3. The signal is also the input to the second
system. The output of the second system is
(2.53)
Fig. 2.35.
Figure 2.35 (p. 164)Direct form II
representation of an LTI system described by a
second-order difference equation.
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Time-Domain Representations of LTI Systems
? Block diagram representation for
continuous-time LTI system
1. Differential Eq.
(2.54)
2. Let v(0)(t) v(t) be an arbitrary signal, and
set
v(n)(t) is the n-fold integral of v(t) with
respect to time
3. Integrator with initial condition
Integrate N times to eq. (2.54)
(2.55)
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Time-Domain Representations of LTI Systems
Ex. Second-order system
(2.56)
(a)
Figure 2.37 (p. 166)Block diagram
representations of a continuous-time LTI system
described by a second-order integral equation.
(a) Direct form I. (b) Direct form II.
(b)
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Time-Domain Representations of LTI Systems
2.13 State-Variable Description of LTI Systems
? The state of a system may be defined as a
minimal set of signals that represent the
systems entire memory of the past.
Given initial point ni (or ti) and the input for
time n ? ni (or t ? ti), we can determine the
output for all times n ? ni (or t ? ti).
2.13.1 The State-Variable Description
1. Direct form II of a second-order LTI system
Fig. 2.39.
2. Choose state variables q1n and q2n.
3. State equation
(2.57)
(2.58)
4. Output equation
(2.59)
5. Matrix Form of state equation
(2.60)
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Time-Domain Representations of LTI Systems
Figure 2.39 (p. 167)Direct form II
representation of a second-order discrete-time
LTI system depicting state variables q1n and
q2n.
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Time-Domain Representations of LTI Systems
6. Matrix form of output equation
(2.61)
Define state vector as the column vector
We can rewrite Eqs. (2.60) and (2.61) as
(2.62)
(2.63)
where matrix A, vectors b and c, and scalar D are
given by
Example 2.28 State-Variable Description of a
Second-Order System
Find the state-variable description corresponding
to the system depicted in Fig. 2.40 by choosing
the state variable to be the outputs of the unit
delays.
ltSol.gt
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Time-Domain Representations of LTI Systems
Figure 2.40 (p. 169)Block diagram of LTI system
for Example 2.28.
1. State equation
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Time-Domain Representations of LTI Systems
2. Output equation
3. Define state vector as
In standard form of dynamic equation
(2.62)
(2.63)
? State-variable description for continuous-time
systems
(2.64)
(2.65)
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Time-Domain Representations of LTI Systems
Example 2.29 State-Variable Description of an
Electrical Circuit
Consider the electrical circuit depicted in Fig.
2.42. Derive a state-variable description of this
system if the input is the applied voltage x(t)
and the output is the current y(t) through the
resistor.
ltSol.gt
Figure 2.42 (p. 171)Circuit diagram of LTI
system for Example 2.29.
1. State variables The voltage across each
capacitor.
2. KVL Eq. for the loop involving x(t), R1, and
C1
Output equation
(2.66)
3. KVL Eq. for the loop involving C1, R2, and C2
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Time-Domain Representations of LTI Systems
(2.67)
Use Eq. (2.67) to eliminate i2(t)
4. The current i2(t) through R2
(2.68)
5. KCL Eq. between R1 and R2
Current through C1 i1(t)
where
(2.69)
? Eqs. (2.66), (2.68), and (2.69)
State-Variable Description.
(2.64)
(2.65)
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Time-Domain Representations of LTI Systems
and
Example 2.30 State-Variable Description from a
Block Diagram
Determine the state-variable description
corresponding to the block diagram in Fig. 2.44.
The choice of the state variables is indicated on
the diagram.
Figure 2.44 (p. 172)Block diagram of LTI system
for Example 2.30.
ltSol.gt
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Time-Domain Representations of LTI Systems
1. State equation
3. State-variable description
2. Output equation
2.13.2 Transformations of The State
The transformation is accomplished by defining a
new set of state variables that are a weighted
sum of the original ones.
The input-output characteristic of the system is
not changed.
1. Original state-variable description
(2.70)
T state-transformation matrix
(2.71)
2. Transformation
q Tq
q T?1 q
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Time-Domain Representations of LTI Systems
3. New state-variable description
1) State equation
2) Output equation
3) If we set
then
and
Ex. Consider Example 2.30 again. Let us define
new states
Find the state-variable description.
ltSol.gt
1. State equation
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Time-Domain Representations of LTI Systems
3. State-variable description
2. Output equation
Example 2.31 Transforming The State
A discrete-time system has the state-variable
description
and
Find the state-variable description A?, b?, c?,
D? corresponding to the new states
and
ltSol.gt
1. Transformation q? Tq, where
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Time-Domain Representations of LTI Systems
2. New state-variable description
and
? This choice for T results in A? being a
diagonal matrix and thus separates the state
update into the two decoupled first-order
difference equations
and
2.14 Exploring Concepts with MATLAB
? Two limitations
1. MATLAB is not easily used in the
continuous-time case.
2. Finite memory or storage capacity and nonzero
computation times.
? Both the MATLAB Signal Processing Toolbox and
Control System Toolbox are use in this
section.
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Time-Domain Representations of LTI Systems
2.14.1 Convolution
x and h are signal vectors.
1. MATLAB command
y conv(x, h)
2. The number of elements in y is given by the
sum of the number of elements in x and h,
minus one.
ltpf.gt
1) Elements in vector x from time n kx to n
lx
2) Elements in vector h from time n kh to n
lh
3) Elements in vector y from time n ky kx
kh to n ly lx ly
4) The length of xn and hn are Lx lx ? kx
1 and Lh lh ? kh 1
5) The length of yn is Ly Lx Lh ? 1
Ex. Repeat Example 2.1
Impulse and Input From time n kh kx 0 to
n lh 1 and n lx 2
Convolution sum From time n ky kx kh 0
to n ly lx lh 3
The length of convolution sum Ly ly ky 1
4
MATLAB Program
gtgt h 1, 0.5 gtgt x 2, 4, -2 gtgt y
conv(x,h)
y 2 5 0 -1
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Time-Domain Representations of LTI Systems
Impulse response
Input
Repeat Example 2.3
Given
and
Find the convolution sum xn ? hn.
ltSol.gt
1. In this case, kh 0, lh 3, kx 0 and lx 9
2. y starts at time n ky 0, ends at time n
ly 12, and has length Ly 13.
3. Generation for vector h with MATLAB
gtgt h 0.25ones(1, 4) gtgt x ones(1, 10)
4. Output and its plot
gtgt n 012 gtgt y conv(x, h) gtgt stem(n, y)
xlabel('n') ylabel('yn')
Fig. 2.45.
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Time-Domain Representations of LTI Systems
Figure 2.45 (p. 177)Convolution sum computed
using MATLAB.
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Time-Domain Representations of LTI Systems
2.14.2 The Step Response
1. Step response the output of a system in
response to a step input
2. In general, step response is infinite in
duration.
3. We can evaluate the first p values of the
step response using the conv function if
hn 0 for n lt kh by convolving the first p
values of hn with a finite-duration step
of length p.
1) Vector h the first p nonzero values of the
impulse response. 2) Define step u ones(1,
p). 3) convolution s conv(u, h).
Ex. Repeat Problem 2.12
Determine the first 50 values of the step
response of the system with impulse response
given by
with ? ? 0.9, by using MATLAB program.
ltSol.gt
1. MATLAB Commands
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Time-Domain Representations of LTI Systems
gtgt h (-0.9).049 gtgt u ones(1, 50) gtgt s
conv(u, h) gtgt stem(049, s(150))
2. Step response Fig. 2.47.
Figure 2.47 (p. 178)Step response computed
using MATLAB.
2.14.3 Simulating Difference equations
1. Difference equation
Command filter
(2.36)
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Time-Domain Representations of LTI Systems
2. Procedure
1) Define vectors a a0, a1, , aN and b b0,
b1, , bM representing the coefficients of
Eq. (2.36). 2) Input vector x 3) y filter(b,
a, x) results in a vector y representing the
output of the system for zero initial
conditions. 4) y filter(b, a, x, zi) results in
a vector y representing the output of the
system for nonzero initial conditions zi.
? The initial conditions used by filter are not
the past values of the output.
? Command zi filtic(b, a, yi), where yi is a
vector containing the initial conditions in
the order y?1, y?2, , y?N, generates
the initial conditions obtained from the
knowledge of the past outputs.
Ex. Repeat Example 2.16
The system of interest is described by the
difference equation
(2.73)
Determine the output in response to zero input
and initial condition y?1 1 and y?2 2.
ltSol.gt
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Time-Domain Representations of LTI Systems
1. MATLAB Program
gtgt a 1, -1.143, 0.4128 b 0.0675, 0.1349,
0.675 gtgt x zeros(1, 50) gtgt zi filtic(b, a,
1, 2) gtgt y filter(b, a, x, zi) gtgt stem(y)
2. Output Fig. 2.28(b).
3. System response to an input consisting of
the Intel stock price data Intc
gtgt load Intc gtgt filtintc filter(b, a, Intc)

• We have assume that the
• Intel stock price data are
• in the file Intc.mat.

? The command h, t impz(b, a, n) evaluates n
values of the impulse response of a system
described by a different equation.
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Time-Domain Representations of LTI Systems
2.14.4 State-Variable Descriptions
Representing the matices A,b,c, and D.
? MATLAB command ss
1. Input MATLAB arrays a, b, c, d
2. Command sys ss(a, b, c, d, -1) produces an
LTI object sys that represents the
discrete-time system in state-variable form.
? Continuous-time case sys ss(a, b, c, d)
No ? 1
? System manipulation
1. sys sys1 sys2
Parallel combination of sys1 and sys2.
2. sys sys1 ? sys2
Cascade combination of sys1 and sys2.
? MATLAB command lsim
1. Command form y lsim(sys, x)
2. Output y, input x.
? MATLAB command impulse
1. Command form h impulse(sys, N)
2. This command places the first N values of the
impulse response in h.
? MATLAB routine ss2ss
Perform the state transformation
1. Command form sysT ss2ss(sys, T), where T
Transformation matrix
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Time-Domain Representations of LTI Systems
Ex. Repeat Example 2.31.
1. Original state-variable description
and
2. State-transformation matrix
3. MATLAB Program
gtgt a -0.1, 0.4 0.4, -0.1 b 2 4 gtgt c
0.5, 0.5 d 2 gtgt sys ss(a, b, c, d, -1)
define the state-space object sys gtgt T
0.5-1, 1 1, 1 gtgt sysT ss2ss(sys, T)
4. Result
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Time-Domain Representations of LTI Systems
a x1 x2 x1 -0.5 0 x2
0 0.3
b u1 x1 1 x2 3
c x1 x2 y1 0 1
d u1 y1 2
Sampling time unspecified Discrete-time model.
Ex. Verify that the two systems represented by
sys and sysT have identical input-output
characteristic by comparing their impulse
responses .
ltSol.gt
gtgt h impulse(sys, 10) hT impulse(sysT,
10) gtgt subplot(2, 1, 1) gtgt stem(09, h) gtgt
title ('Original System Impulse Response') gtgt
xlabel('Time') ylabel('Amplitude') gtgt subplot(2,
1, 2) gtgt stem(09, hT) gtgt title('Transformed
System Impulse Response') gtgt xlabel('Time')
ylabel('Amplitude')
1. MATLAB Program
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Time-Domain Representations of LTI Systems
2. Simulation results Fig. 2.48.
? We may verify that the original and
transformed systems have the (numerically)
identical impulse response by computing
the error, err h hT.
Figure 2.48 (p. 181)Impulse responses
associated with the original and transformed
state-variable descriptions computer using MATLAB.
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Time-Domain Representations of LTI Systems
Plot for err h ? hT