Chapter 6-1

Approaches to compensation
In the following sections, we will assume that the process has been improved as much as possible and that the G(s) representing the process is unalterable. First, we shall consider the addition of a so-called phase-lead compensation network and describe the design of the network by frequency response techniques.
Compensation & Compensator
The alteration or adjustment of a control system in order to provide a suitable performance is called compensation. The compensating device may be electric, mechanical, hydraulic, pneumatic, or some other type of device or network and is often called a compensator. A compensator is an additional component or circuit that is inserted into a control system to compensate for a deficient performance.
If we wish a system to have a percent overshoot less than 20% and ess = 0.1 , we obtain a conflicting requirement on the gain, K.
σ % = 20% → K = 1.18
ess = 0.1 → K = 10
ωnt p = 3.3 → ζ = 0.31
Example
K s (Ts + 1)
2 ωn G(s) = s ( s + 2ζω n )
1 1 ζ = 2 TK
σ % = e πζ /
ess = 1 K
1ζ
2
× 100%
ω n = K/T
(for a step input)
(for a unit ramp input)
R(s )
GC (s )
G (s )
Y (s )
H (s )
Compensation & Compensator
Similarly, the other compensation schemes are called feedback, output, and input compensation.
Phase-lead Compensation Network
Consider the first-order compensator with the transfer function
Gc ( s ) =
αTs + 1
Ts + 1
(α > 1)
The design problem then becomes the selection of parameters, α and T, in order to provide a suitable performance.
Introduction
A feedback control system that provides an optimal performance without any necessary adjustments is rare indeed. It is necessary to compromise among the many conflicting and demanding specifications. We have to adjust the system parameters to provide a suitable and acceptable performance when it is not possible to obtain all the desired optimal specifications.
= ∠Gc ( jω ) = tan 1 αTω tan 1 Tω
The maximum value of the phase lead occurs at frequency
ωm =
1
αT
The maximum phase lead is
If we wish a system to have a percent overshoot less than 20% and ω nTp = 3.3 , we obtain a conflicting requirement on the system damping ratio, ζ .
σ % = 20% → ζ = 0.46
Introduction
In Chapter 4, we have illustrated a method of parameter design by using the root locus method. Furthermore, in Chapter 5, we developed suitable measures of performance in terms of the frequency variable and utilized them to adjust parameters.
tp
0
t
tp
ts
Typical frequency domain indices
Crossover frequency: ω c Phase margin: γ Gain margin: h
0 ∠G
20 lg G
ωc
h
ω
γ
Peak value : Mm The bandwidth: ωb The resonant frequency: ω r
Approaches to Compensation
Quite often, in practice, the best and easiest way to improve the performance of a control system is to alter, if possible, the process itself. However, the process is often unalterable or has been altered as much as possible and still results in unsatisfactory performance. Then the addition of compensation networks become useful for improving the performance of the system.
R (s ) G (s ) Y (s )
R (s)
G (s )
Y (s )
GC (s )
GC (s )
H (s )
H (s )
R (s )
GC (s)
G (s)
Y (s )
H (s )
Compound compensation
R (s ) GC (s )
G (s )
Y (s)
H (s )
Gn (s )
Performance specifications
Typical time domain indices: Overshoot: σ %
y (t )
overshoot
Settling time: t s Peak time: e (or Open loop gain) Steady state error:ss
Mm
0.707 M (0)
ωr
ωb
Typical complex domain indices
These indices are represented by the location of the dominant poles
j
πζ
σ% =e
ts =
or
1ζ 2
× 100%
1 T ζω n
η
ωn
N (s )
R (s)
GC (s ) G (s )
Y (s)
Approaches to Compensation
For frequency response methods, we are concerned with altering the system so that the frequency response of the compensated system will satisfy the system specifications. Alternatively the design of a control system can be accomplished in the s-plane by root locus methods. For the case of the s-plane, the designer wishes to alter and reshape the root locus so that the roots of the system will lie in the desired position in the s-plane.