Файл: Microcontroller based applied digital control (D. Ibrahim, 2006).pdf

174 SYSTEM TIME RESPONSE CHARACTERISTICS

7.2 TIME DOMAIN SPECIFICATIONS

The performance of a control system is usually measured in terms of its response to a step input. The step input is used because it is easy to generate and gives the system a nonzero steady-state condition, which can be measured.

Most commonly used time domain performance measures refer to a second-order system with the transfer function:

where ωn is the undamped natural frequency of the system and ζ is the damping ratio of the system.

When a second-order system is excited with a unit step input, the typical output response is as shown in Figure 7.3. Based on this figure, the following performance parameters are usually defined: maximum overshoot; peak time; rise time; settling time; and steady-state error.

The maximum overshoot, M p , is the peak value of the response curve measured from unity. This parameter is usually quoted as a percentage. The amount of overshoot depends on the damping ratio and directly indicates the relative stability of the system.

The peak time, Tp , is defined as the time required for the response to reach the first peak of the overshoot. The system is more responsive when the peak time is smaller, but this gives rise to a higher overshoot.

The rise time, Tr , is the time required for the response to go from 0 % to 100 % of its final value. It is a measure of the responsiveness of a system, and smaller rise times make the system more responsive.

The settling time, Ts , is the time required for the response curve to reach and stay within a range about the final value. A value of 2–5 % is usually used in performance specifications.

The steady-state error, Ess , is the error between the system response and the reference input value (unity) when the system reaches its steady-state value. A small steady-tate error is a requirement in most control systems. In some control systems, such as position control, it is one of the requirements to have no steady-state error.

y(t)

Mp

1

t

Figure 7.3 Second-order system unit step response

Having introduced the parameters, we are now in a position to give formulae for them (readers who are interested in the derivation of these formulae should refer to books on control

theory). The maximum overshoot occurs at at peak time (t = Tp ) and is given by

√

M p = e−(ζ π/ 1−ζ 2),

i.e. overshoot is directly related to the system damping ratio – the lower the damping ratio, the higher the overshoot. Figure 7.4 shows the variation of the overshoot (expressed as a percentage) with the damping ratio.

The peak time is obtained by differentiating the output response with respect to time, letting this equal zero. It is given by

Tp = π , ωd

where

ωd = ωn2 1 − ζ 2

is the damped natural frequency.

The rise time is obtained by setting the output response to 1 and finding the time. It is given

by

Tr = π − β , ωd

where

β= tan−1 wd .

ζωn

The settling time is usually specified for a 2 % or 5 % tolerance band, and is given by

The steady-state error can be found by using the final value theorem, i.e. if the Laplace transform of the output response is y(s), then the final value (steady-state value) is given by

Damping ratio

Figure 7.4 Variation of overshoot with damping ratio

7.3 MAPPING THE s-PLANE INTO THE z-PLANE

The pole locations of a closed-loop continuous-time system in the s-plane determine the behaviour and stability of the system, and we can shape the response of a system by positioning its poles in the s-plane. It is desirable to do the same for the sampled data systems. This section describes the relationship between the s-plane and the z-plane and analyses the behaviour of a system when the closed-loop poles are placed in the z-plane.

First of all, consider the mapping of the left-hand side of the s-plane into the z-plane. Let s = σ + j ω describe a point in the s-plane. Then, along the j ω axis,

z = es T = eσ T e j ωT .

But σ = 0 so we have

z = e j ωT = cos ωT + j sin ωT = 1 ωT .

Hence, the pole locations on the imaginary axis in the s-plane are mapped onto the unit circle in the z-plane. As ω changes along the imaginary axis in the s-plane, the angle of the poles on the unit circle in the z-plane changes.

If ω is kept constant and σ is increased in the left-hand s-plane, the pole locations in the z-plane move towards the origin, away from the unit circle. Similarly, if σ is decreased in the left-hand s-plane, the pole locations in the z-plane move away from the origin in the z-plane. Hence, the entire left-hand s-plane is mapped into the interior of the unit circle in the z-plane. Similarly, the right-hand s-plane is mapped into the exterior of the unit circle in the z-plane. As far as the system stability is concerned, a sampled data system will be stable if the closed-loop poles (or the zeros of the characteristic equation) lie within the unit circle. Figure 7.5 shows the mapping of the left-hand s-plane into the z-plane.

As shown in Figure 7.6, lines of constant σ in the s-plane are mapped into circles in the z-plane with radius eσ T . If the line is on the left-hand side of the s-plane then the radius of the circle in the z-plane is less than 1. If on the other hand the line is on the right-hand side of the s-plane then the radius of the circle in the z-plane is greater than 1. Figure 7.7 shows the corresponding pole locations between the s-plane and the z-plane.

jω

σ

z-plane

s-plane

Figure 7.5 Mapping the left-hand s-plane into the z-plane