For the resistor, if i(t) is bounded then so is v(t), but for the capacitance this is not true. Consider i(t) = u(t) then v(t) = tu(t) which is unbounded.
5/ Linear systems
for all
,
, a, and b.
6/ Time-invariant systems
for all x(t) and τ.
7/ Linear and time-invariant (LTI) systems
Many man-made and naturally occurring systems can be modeled as LTI systems.
Powerful techniques have been developed to analyze and to characterize LTI systems.
The analysis of LTI systems is an essential precursor to the analysis of more complex systems.
Problem — Multiplication by a time function
A system is defined by the functional description
Is this system linear?
Is this system time-invariant?
Solution — Multiplication by a time function
Let
By definition the response to
Is
This can be rewritten as
Therefore, the system is linear.
Now suppose that
and
, and the response to these two inputs are
and
, respectively. Note that
And
Therefore, the system is time-varying.
Problem — Addition of a constant
Suppose the relation between the output y(t) and input x(t) is y(t) = x(t)+K, where K is some constant. Is this system linear?
Solution — Addition of a constant
Note, that if the input is
then the output will be
Therefore, this system is not linear.
In general, it can be shown that for a linear system if x(t) = 0 then y(t) = 0. Using the definition of linearity, choose a = b = 1 and
then
and
.
Two-minute miniquiz problem
Problem 2-1
The system
is (choose one):
Linear and time-invariant;
Linear but not time-invariant;
Not linear but time-invariant;
Not linear and not time-invariant.
Solution
Note that if
then
Hence, this system is nonlinear.
Note that if
and
then
And
. Hence, this system is time invariant.
V. LINEAR, ORDINARY DIFFERENTIAL EQUATIONS ARISE FOR A VARIETY OF SYSTEM DESCRIPTIONS
1/ Electric network
Kirchhoff’s current law yields
The constitutive relations for each element yield
Combining KCL and the constitutive relations yields
2/ Mechanical system
The simplest possible model of a muscle (the linearized Hill model) consists of the mechanical network shown below which relates the rate of change of the length of the muscle v(t) to the external force on the muscle fe(t).
is the internal force generated by the muscle, K is its stiffness and B is its damping.
The muscle velocity can be expressed as
Combining the muscle velocity equation with the constitutive laws for the elements yields
3/ First-order chemical kinetics
A reversible, first-order chemical reaction can be represented as follows