One reason for the use of closed-loop control systems is to automatically control the output of a system to align with a reference input or a set-point as closely as possible. If the refer- ence is fixed, i.e. set to a constant value, the difference between the set-point and the con- trolled output in the steady-state is called the alignment error. However, the reference input may be time varying; in this case it is necessary for the controlled output to track or follow the reference as closely as possible. In order to assess how well a closed-loop control system aligns with - or follows - a reference input a set of test reference signals is devised that pro- vides a measure of the quality of system performance. These tests signals are analysed in the following sections.
The steady-state value of a time-varying signal , i.e. its value after all oscillations associ- ated with any transients have died away, , can be derived from the final-value theorem (FVT), i.e.
. (2.27)
From this result the steady-state value of the output of a system, the alignment and following errors for a given test reference-input can be determined.
The closed-loop control system under study is shown in Figure 2.9, where is the con- trolled output signal, is the reference input signal and the error signal is
. With forward-loop and feedback-path transfer functions and , respectively, the transfer function of the closed-loop system of Figure 2.9 has been shown in (2.21) to be
. x t
xss
xss x t
tlim sX s
slim0
= =
C s R s
E s = R s –H s C s G s
H s
W s C s R s
--- G s 1+G s H s ---
= =
46 Control systems techniques Ch. 2
Figure 2.9 Block diagram of the classical closed-loop system 2.10.1 Steady-state alignment error.
The response of the closed-loop system to a step change in the reference input is shown in Figure 2.10. Under steady-state conditions following the change in the reference input , the steady-state output may not be equal to the constant value of the reference, . Based on the final value theorem the steady-state alignment error is defined as:
, where . (2.28)
Figure 2.10 The steady-state alignment error following a step input.
Thus from (2.28) a general result for the alignment error follows, i.e.
. (2.29)
In the following we will derive the alignment errors for a unity-feedback closed-loop system, i.e. when . In the case of a closed-loop system in which the feedback back is not unity gain, the alignment errors can be derived by using the general result (2.29) or by direct application of the final value theorem. Let us consider two special cases for the unity-feed- back system, namely without and with integration in the forward path.
G s R s
-
E s C s
H s
r t
css rss
eass eass rss–css s R0
---s W s R0 ---s
slim0 –
= = C s = W s R s
eass c t r t
R0
t rss
css c t
tlim
=
eass
eass R01–W s
slim0
=
H s = 1
Sec. 2.10 Steady-state alignment and following errors 47 2.10.1.1 (a) No integration in the forward path, H(s)=1.
Let us assume takes the form .
In the steady-state following a transient, as . Note the form of the factors - not - in the transfer function is important; with no integration in the for- ward path the gain K is often called the static gain. According to (2.29) the associated align- ment error is
,
i.e. , (2.30)
, (2.31)
since .
The result in (2.31) provides a very useful insight, namely, the higher the static gain the smaller is the alignment error. However, there is a downside to high gain settings without suitable compensation, i.e. the dynamic performance of the closed-loop system may become more oscillatory and even unstable. The latter effect will revealed through the analysis of the stability and performance of the closed-loop system using the Bode plot in Section 2.12.2.
2.10.1.2 (b) Single integration in the forward path, H(s)=1.
In this case takes the form . Let ,
then . Substitute this limit for in (2.30) above; the steady-state error is thus .
This is a very useful result; it shows that a single integration in the forward path “integrates out” to zero any error that develops between the reference input and the controlled output. If there is no integration in the forward path the introduction of proportional plus integral (PI) compensation [1], [2] into that path ensures zero alignment error in the steady state. This is highly desirable in some types of closed-loop control systems, however, like the case above, there is a disadvantage. For example, introduction of pure integration also introduces a phase-lag of in the open-loop transfer function. In turn, the Phase Margin is reduced, and consequently the closed-loop system may become unstable. This is discussed in Section 2.12.2.
G s G s K1+sTb11+sTb2
1+sTa1
1+Ca1s C+ a2s2
---
=
G s K s0 1+sT
s a+
eass R01–W s
slim0 R0
slim0 1 G s 1+G s 1 --- –
= =
eass = R0
slim0 1 1+G s --- eass R0 1
1+K ---
= G s K as s0
K
G s G s K
----s 1+sTb11+sTb2
1+sTa1
1+sCa1+Ca2s2
---
= s0
G s K s G s
eass R0
slim0 1
1 K
----s +
--- R0
slim0 s s K+ --- 0
= = =
e t
1s 90
48 Control systems techniques Ch. 2 2.10.2 The steady-state following error
A test reference input of a suitable form for defining the following error is the ramp signal, . Based on the final-value theorem the steady-state following error is given by:
, or
. (2.32)
This is a general result for the following error. However, let us again consider a unity-feed- back system and two special cases, without and with integration in the forward path.
2.10.2.1 (a) No integration in the forward path, H(s)=1.
Assume again takes the form . In the steady-
state following the decay of the transient we find as . By substitution of this limit in (2.32), the following error becomes
, (2.33)
, i.e. . (2.34)
The result reveals that the following error becomes increasingly large with time; the latter is illustrated in Fig. 2.11. The controlled output thus cannot follow a reference input that changes linearly with time. Such a closed-loop system cannot track, for example, a satellite passing overhead.
2.10.2.2 (b) Single integration in the forward path, H(s)=1.
As shown in Section 2.10.1.2 above, as . Substitution for in (2.33) above yields a following error given by:
. (2.35) Thus after any oscillations have died away following the application of the ramp, there is a constant difference, or error, between the ramp input and the controlled output given by (2.35), as illustrated in Figure 2.11.
r t = R0t efss
efss r t –c t
tlim s R0
s2
--- W s R0 s2 ---
slim0 –
= =
efss R0 ---s
slim0 1–W s
=
G s G s K1+sTb11+sTb2
1+sTa1
1+sCa1+Ca2s2
---
=
G s K s0
efss R0 ---s
slim0 1 G s 1+G s 1 ---
– R0
---s 1 1+G s ---
slim0
= =
efss R0 s sK+ ---
slim0
= efss
G s K ----s
s0 G s
efss R0 ---s 1
1 K
----s + ---
slim0 R0
s K+ ---
slim0 R0 ---K
= = =
Sec. 2.11 Frequency response methods 49
Figure 2.11 The following error resulting from a ramp input.
Exercise. Show that if the forward-loop transfer function in a unity feedback system contains two integrations, the following error for a ramp reference input is zero, but is finite for a parabolic input .
Example 8
A unity feedback system has a forward-loop transfer function:
.
Find the value of the gain K’ such that the steady-state following error units for a ramp input of 2 unit/second.
Change the factors in into the form , i.e
, where the ‘static’ gain is .
In order to satisfy the specification, the error , i.e. , or .
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