Lecture 23
•Second order system step response
• Governing equation
• Mathematical expression for step response
• Estimating step response directly from
differential equation coefficients
• Examples
•Related educational modules:
–Section 2.5.5
Second order system step response
• Governing equation in “standard” form:
• Initial conditions:
• We will assume that the system is initially “relaxed”
Second order system step response – continued
• We will concentrate on the underdamped response:


A



  n t
y( t )  2 1  e
sind t  
cosd t  
2
 n 
1

 

• Looks like the natural response superimposed with
a step function
Step response parameters
• We would like to get an approximate, but
quantitative estimate of the step response, without
explicitly determining y(t)
• Several step response parameters are directly related to
the coefficients of the governing differential equation
• These relationships can also be used to estimate the
differential equation from a measured step response
• Model parameter estimation
Second order system step response – plot
yp
yss
0.9yss
0.1yss
tr
Steady-state response
• Input-output equation:
• As t, circuit parameters become constant so:
• Circuit DC gain:
• On previous slide, note that DC gain can be
determined directly from circuit.
Rise time
• Rise time is the time required for the response to
get from 10% to 90% of yss
• Rise time is closely related to the natural frequency:
Maximum overshoot, MP
• MP is a measure of the maximum response value
• MP is often expressed as a percentage of yss and is
related directly to the damping ratio:
Maximum overshoot – continued
• For small values of damping ratio, it is often
convenient to approximate the previous
relationship as:
Example 1
• Determine the maximum value of the current, i(t), in the
circuit below
• In previous slide, outline overall approach:
– Need MP, and steady-state value
– Need damping ratio to get MP
– Need natural frequency to get damping ratio
– Need to determine differential equation
Step 1: Determine differential equation
Step 2: Identify n, , and steady-state current
• Governing equation:
Step 3: Determine maximum current
• Damping ratio,  = 0.54
• Steady-state current,
Example 2
• Determine the differential equation governing iL(t) and the
initial conditions iL(0+) and vc(0+)
Example 2 – differential equation, t>0
Example 2 – initial conditions
Example 3 – model parameter estimation
-3
The differential equation
governing a system is known to
be of the form:
2.5
x 10
2
1.5
1
When a 10V step input is
applied to the system, the
response is as shown. Estimate
the differential equation
governing the system.
0.5
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Example 3 – find tr, MP, yss from plot
-3
2.5
x 10
2
1.5
1
0.5
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Example 3 – find differential equation
• From plot, we determined:
– MP  0.25
– tr  0.05
– yss  0.002
Example 4 – Series RLC circuit
• MP  100%, n = 100,000 rad/sec (16KHz)