Performance of second-order systems. Modern Control Systems Lecture 12. Outline. Design specifications & trade-off Test input signals Performance of second-order systems to step input Standard time-domain performance measures for second-order systems
Modern Control Systems
Design specifications for control systems are explicit statements in terms of performance measures, which indicate how well the system should perform the task for which it is designed.
Trade-off is often necessary to provide a sub-optimal solution in control system design.
Test input signals are selected and standard input signals, aiming to test the response of a control system.
Why test input signals are used?
Test input signals commonly used are the step input, the ramp input, and the parabolic input.
derivative of ramp input
derivative of parabolic input divided by a scalar
DUnit impulse function (Dirac Delta)
derivative of unit step
The unit impulse δ(t)
has useful properties.
The response of the system to a unit impulse signal is
TF of a system is the Laplace transform of its impulse response.
Because of the derivative (or integral) relationship between standard test input signals, the response of linear time-invariant (LTI) systems satisfies:
The above property only applies to LTI systems.
the response of the system to a unit step input is
Taking the inverse Laplace transform, we have
The system response contains two parts – steady-state response & transient response
The steady-state response is y(∞)=0.9.
The steady-state error is
The time response of a control system consists of transient response and steady-state response.
Time-domain performance specifications/measures are important indices that represent the performance of the control system.
The ability to adjust the transient and steady-state responses is a distinct advantage of feedback control systems. Based on the performance measures, system parameters may be adjusted to provide the desired response.
Let us consider a typical second-order system to unit step input.
The closed-loop output is
Rewrite the above equation in standard form
The output response is
Step response of the second-order system
The complex conjugate roots
Step response of second-order systems
As ζ varies with ωn constant,
the locus of the closed-loop roots is as shown in the left figure. As ζdecreases, the closed-loop roots approach the imaginary axis, and the output becomes increasingly oscillatory. If ζis increased beyond unity, the response becomes overdamped.
Let us consider the second-order system to unit impulse input.
The closed-loop output written in standard form is
Impulse response of second-order systems
The output response is
where fv is the final value of the response. For the 2nd-order system under investigation, fv=1
where τ is the time constant 1/ζωn.
To obtain an explicit relation for the peak value and peak time in terms of ζ, we can differentiate
P.O. is independent ofωn.
What we have discussed thus far is only for second-order systems in the form of
However, the information is useful because many systems possess a dominant pair of roots and the step response can be estimated by the second-order system counterpart.
For a third-order system with a closed-loop TF
its step response can be approximated by the dominant roots of the second-order system if
The second-order system in the form of
has no finite zeros. When adding a zero to the system so that its TF becomes
if the location of the zero is relatively near the dominant poles, the zero will materially affect the transient response.
without finite zeros
with one zero s+a, a/ζωn=A,B,C, or D andζ=0.45