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Performance of second-order systems

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Performance of second-order systems

Modern Control Systems

Lecture 12

- Design specifications & trade-off
- Test input signals
- Performance of second-order systems to step input
- Standard time-domain performance measures for second-order systems
- Effects of a third pole and a zero on the second-order system response

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?

- The actual input signal is unknown.
- Good correlation between the response of the system to a test input signal and the system response under normal operating conditions.
- Using a standard test input signal allows different designs to be compared.
- Many control systems experience input signals similar to test input signals.

Test input signals commonly used are the step input, the ramp input, and the parabolic input.

- Step input

derivative of ramp input

- Ramp input

derivative of parabolic input divided by a scalar

- Parabolic input

Δ→0

1

e(t)

D

t

D

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:

- impulse response = derivative of step response
- step response = derivative of ramp response
- ramp response = derivative of parabolic response

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 characteristic equation of

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

Closed-loop unit step response:

- Rise time
- Peak time
- Percentage overshoot

where fv is the final value of the response. For the 2nd-order system under investigation, fv=1

- Settling time. The time for which the response remains within 2% of the final value.

where τ is the time constant 1/ζωn.

or use

to obtain

- The swiftness of response (rise time, peak time)
- The closeness of the response to the desired response (overshoot, settling time)

conflicting requirements

To obtain an explicit relation for the peak value and peak time in terms of ζ, we can differentiate

impulse response

Then

Therefore,

P.O. is independent ofωn.

- For a given ωn, step response is faster for lower ζ.

- For a given ζ, step response is faster for larger ωn.

ζ=0.2

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.

- third-pole

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

- second-order system with one zero

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