MODULE-II
COMPENSATOR DESIGN
Every control system which has been designed for a specific application should
meet certain performance specification. There are always some constraints which are
imposed on the control system design in addition to the performance specification. The
choice of a plant is not only dependent on the performance specification but also on the
size , weight & cost. Although the designer of the control system is free to choose a new
plant, it is generally not advised due to the cost & other constraints. Under this
circumstances it is possible to introduce some kind of corrective sub-systems in order to
force the chosen plant to meet the given specification. We refer to these sub-systems as
compensator whose job is to compensate for the deficiency in the performance of the
plant.
REALIZATION OF BASIC COMPENSATORS
Compensation can be accomplished in several ways.
Series or Cascade compensation
Compensator can be inserted in the forward path as shown in fig below. The transfer
function of compensator is denoted as Gc(s), whereas that of the original process of the
plant is denoted by G(s).
feedback compensation
Combined Cascade & feedback compensation
31
,Compensator can be electrical, mechanical, pneumatic or hydrolic type of device. Mostly
electrical networks are used as compensator in most of the control system. The very
simplest of these are Lead, lag & lead-lag networks.
Lead Compensator
Lead compensator are used to improve the transient response of a system.
Fig: Electric Lead Network
Taking i2=0 & applying Laplace Transform, we get
𝑉2(𝑠) 𝑅2(𝑅1𝐶𝑠 + 1)
=
𝑉1(𝑠) 𝑅2 + 𝑅2𝑅1𝐶𝑠 + 𝑅1
Let 𝜏 = 𝑅1 𝐶 , 𝛼= 𝑅2 <1
𝑅1+𝑅2
𝑉2(𝑠) 𝛼(𝜏𝑠+1)
= Transfer function of Lead Compensator
𝑉1 (𝑠) (1+𝜏𝛼𝑠 )
Fig: S-Plane representation of Lead Compensator
Bode plot for Lead Compensator
Maximum phase lead occurs at 𝑤 𝑚 = 1
𝜏 𝛼
Let 𝜙𝑚 = maximum phase lead
1−𝛼
sin 𝜙𝑚 =
1+𝛼
1 − sin 𝜙𝑚
𝛼=
1 + sin 𝜙𝑚
Magnitude at maximum phase lead 𝐺𝑐(𝑗𝑤) = 1
𝛼
32
, Fig: Bode plot of Phase Lead network with amplifier of gain 𝐴 = 1 𝛼
Lag Compensator
Lag compensator are used to improve the steady state response of a system.
Fig: Electric Lag Network
Taking i2=0 & applying Laplace Transform, we get
𝑉2(𝑠) 𝑅2𝐶𝑠 + 1
=
𝑉1(𝑠) (𝑅2 + 𝑅1)𝐶𝑠 + 1
Let 𝜏 = 𝑅2 𝐶 , 𝛽 = 𝑅1+𝑅2 > 1
𝑅2
𝑉2(𝑠) 𝜏𝑠+1
𝑉1 (𝑠) = 1+𝜏𝛽𝑠 Transfer function of Lag Compensator
33
, Fig: S-Plane representation of Lag Compensator
Bode plot for Lag Compensator
1
Maximum phase lag occurs at 𝑤𝑚 =
𝜏 𝛽
Let 𝜙𝑚 = maximum phase lag
1−𝛽
sin 𝜙𝑚 =
1+𝛽
1 − sin 𝜙𝑚
𝛽=
1 + sin 𝜙𝑚
Fig: Bode plot of Phase Lag network
Cascade compensation in Time domain
Cascade compensation in time domain is conveniently carried out by the root locus
technique. In this method of compensation, the original design specification on dynamic
response are converted into 𝜁 & 𝑤𝑛 of a pair of desired complex conjugate closed loop pole
34
COMPENSATOR DESIGN
Every control system which has been designed for a specific application should
meet certain performance specification. There are always some constraints which are
imposed on the control system design in addition to the performance specification. The
choice of a plant is not only dependent on the performance specification but also on the
size , weight & cost. Although the designer of the control system is free to choose a new
plant, it is generally not advised due to the cost & other constraints. Under this
circumstances it is possible to introduce some kind of corrective sub-systems in order to
force the chosen plant to meet the given specification. We refer to these sub-systems as
compensator whose job is to compensate for the deficiency in the performance of the
plant.
REALIZATION OF BASIC COMPENSATORS
Compensation can be accomplished in several ways.
Series or Cascade compensation
Compensator can be inserted in the forward path as shown in fig below. The transfer
function of compensator is denoted as Gc(s), whereas that of the original process of the
plant is denoted by G(s).
feedback compensation
Combined Cascade & feedback compensation
31
,Compensator can be electrical, mechanical, pneumatic or hydrolic type of device. Mostly
electrical networks are used as compensator in most of the control system. The very
simplest of these are Lead, lag & lead-lag networks.
Lead Compensator
Lead compensator are used to improve the transient response of a system.
Fig: Electric Lead Network
Taking i2=0 & applying Laplace Transform, we get
𝑉2(𝑠) 𝑅2(𝑅1𝐶𝑠 + 1)
=
𝑉1(𝑠) 𝑅2 + 𝑅2𝑅1𝐶𝑠 + 𝑅1
Let 𝜏 = 𝑅1 𝐶 , 𝛼= 𝑅2 <1
𝑅1+𝑅2
𝑉2(𝑠) 𝛼(𝜏𝑠+1)
= Transfer function of Lead Compensator
𝑉1 (𝑠) (1+𝜏𝛼𝑠 )
Fig: S-Plane representation of Lead Compensator
Bode plot for Lead Compensator
Maximum phase lead occurs at 𝑤 𝑚 = 1
𝜏 𝛼
Let 𝜙𝑚 = maximum phase lead
1−𝛼
sin 𝜙𝑚 =
1+𝛼
1 − sin 𝜙𝑚
𝛼=
1 + sin 𝜙𝑚
Magnitude at maximum phase lead 𝐺𝑐(𝑗𝑤) = 1
𝛼
32
, Fig: Bode plot of Phase Lead network with amplifier of gain 𝐴 = 1 𝛼
Lag Compensator
Lag compensator are used to improve the steady state response of a system.
Fig: Electric Lag Network
Taking i2=0 & applying Laplace Transform, we get
𝑉2(𝑠) 𝑅2𝐶𝑠 + 1
=
𝑉1(𝑠) (𝑅2 + 𝑅1)𝐶𝑠 + 1
Let 𝜏 = 𝑅2 𝐶 , 𝛽 = 𝑅1+𝑅2 > 1
𝑅2
𝑉2(𝑠) 𝜏𝑠+1
𝑉1 (𝑠) = 1+𝜏𝛽𝑠 Transfer function of Lag Compensator
33
, Fig: S-Plane representation of Lag Compensator
Bode plot for Lag Compensator
1
Maximum phase lag occurs at 𝑤𝑚 =
𝜏 𝛽
Let 𝜙𝑚 = maximum phase lag
1−𝛽
sin 𝜙𝑚 =
1+𝛽
1 − sin 𝜙𝑚
𝛽=
1 + sin 𝜙𝑚
Fig: Bode plot of Phase Lag network
Cascade compensation in Time domain
Cascade compensation in time domain is conveniently carried out by the root locus
technique. In this method of compensation, the original design specification on dynamic
response are converted into 𝜁 & 𝑤𝑛 of a pair of desired complex conjugate closed loop pole
34
