These days there is an ongoing debate between people about.
While it is possible to claim that.
My view is that the disadvantages outweigh the advantages.In this essay i shall explain my point of
view
55
Problems
ub(t)
3.4 A fixed, linear system has the unit-step response a(t) te l l(t).
a) Find the impulse response h(t) = Scfw_8(t).
b) Find the response m2(t) to the unit ramp function /x2(0c) Find
3
Principles of time-domain analysis
3.1 Introduction
As we indicated earlier, in the indirect approach to the analysis of linear systems,
when we seek the response of some linear system S to a genera
3.6 Graphical interpretation of the convolution integral
49
Of course, this is just a graphical interpretation of the process of convolution. To
evaluate, we actually integrate the functions inside th
40
2 Finite dimensional linear systems
2.8 Find the state differential equations and the A, B, C, and D matrices for
each of the mechanical systems in Fig. P 2.8. The input is fit) and the
output is z
52
3 Principles of time-domain analysis
where h(k,i) = the response observed at time k when a discrete delta function is
applied at time i. This is called the discrete delta response function or just
48
3 Principles of time-domain analysis
Suppose
and cfw_/2(A.) are as shown in the figure.
1.5
0
0
Then if we plot cfw_f2(k) against A., this looks like
l
Now, shift by an amount t\, so 1/2(^1 A) is
a
Problems
39
2.5 In the circuit shown below, the inputs are the applied voltages e\cfw_t) and
e2(t). The output is the voltage v(t).
Write the state equations and output equations for this circuit.
2.6
2.7 Systems of multiple dynamic equations
35
Example 2.10
h + y\ + h + yi = u
y\ + y\ - h - h = u - u + u.
Procedure 1
Begin by placing the highest order derivative of a different output variable on t
Problems
41
2.11 Find the simulation diagram and the A, B, C, and D matrices for the analog
systems:
a) y'(t) + 2y(t) - y(t) + 3y(t) = AU(t) + u(t)
b) 5t2y(t) + (t- l)y(t) + t2y(t) = u(t) + tu(t).
2.1
54
3 Principles of time-domain analysis
I with all other inputs zero. Then
y(k)= V H(k,l)u(l)
(3.40)
where H(k, I) is the m x r matrix with elements htj(k, I), and it is called the delta
response matr
44
3 Principles of time-domain analysis
3.3 Impulses and the impulse response
The unit impulse or 8-function can be defined by the two properties
F
/
f0
fiX)8iX
- x)dX = <
for t < x
r/
x
(3.3)
r
7-00
42
2 Finite dimensional linear systems
2.20 Write the state difference equations and the A, B, C, and D matrices for
the system
1
ycfw_k + 2) + k2y(k) = u(k + 2) + ku(k + 1) + 2u(k).
k +l
2.21 Find th
36
2 Finite dimensional linear systems
u(t) o-
Fig. 2.15.
Procedure 2
We can frequently get a clue to the order by trying to write a single equation in
terms of one of the output variables. In Example
46
3 Principles of time-domain analysis
which is the integral over the second variable of mt (t, rj) or
rmit, r) = mi+iV>T\
(3 15)
Now, if a system is causal then
h(t,x)=Q
for^<r
a n d a(t,x) = 0 f o
3.8 Systems with multiple inputs and outputs
53
so the discrete step response can be found from the discrete delta response.
Note: It is not always easy to find a closed form expression from this.
3.8
3.5 Relation between the step and impulse response
45
Thus, for t > t0,
rco
y(t) = u(to)a(t,to)+
iicfw_k)a(t,k)dk
(3.9)
Jto
where a(t, z) = the zero-state response observed at time t to a unit step th
38
2 Finite dimensional linear systems
2.2 Write the state differential equations for the circuit below.
R
e(t)l
)
2
Cx
2.3 For the circuit shown below, e(t) is the input voltage and the outputs are
t
50
3 Principles of time-domain analysis
Example 3.2
Suppose we want to find the zero-state response of the system of Example 3.1 to
the input u(t) = 5 1(0 5 \(t 1). Since the system is linear,
Scfw_5
34
2 Finite dimensional linear systems
Formally we can obtain the lowest order simulation for a fixed system with one
input and one output by canceling common factors of the polynomials on each side
o
26
2 Finite dimensional linear systems
We see therefore that the choice of state variables is fairly wide. For circuits,
it is most convenient to choose inductor currents and capacitor voltages (for f
20
2 Finite dimensional linear systems
algebraic equations for z. When Fn is singular (i.e., it has no inverse) then
the equations are degenerate and cannot be put in normal form. (This occurs
when th
2.2 State differential equations of circuits
17
or
iL(t)
(2.6)
L(t)- = vL(t)
dt
dt
can be used to describe the basic terminal relation for an inductor. When the inductor
value does not change with tim
16
2 Finite dimensional linear systems
2.2 State differential equations of circuits
We start by showing how passive electrical circuits can be described by state differential equations in normal form.
24
2 Finite dimensional linear systems
The equations of motion are
at
=
_
K[Z2(t)
(2.15)
(2.16)
- f(t) - K[Z2(t) - D2v2cfw_t) at
where the velocity vt (t) is the derivative of displacement position zt
2.2 State differential equations of circuits
19
a For resistive branches (Rk) the equation is vt Vj = ikRk.
b For inductive branches (Lk) the equation is vt Vj = Lk(dik/dt).
c For capacitative branche