Lead-lag compensator
Encyclopedia
A lead–lag compensator is a component in a control system
that improves an undesirable frequency response
in a feedback and control system
. It is a fundamental building block in classical control theory
.
,
satellite
control, automobile diagnostics, laser
frequency stabilization, and many more.
They are an important building block in analog control systems, and
can also be used in digital control.
Given a control plant, desired specifications can be achieved using compensators. I, D, PI, PD, and PID
, are optimizing controllers which are used to improve system parameters (such as reducing steady state error, reducing resonant peak, improving system response by reducing rise time). All these operations can be done by compensators as well.
pair into the open loop transfer function
. The transfer function can be written in the Laplace domain as
where X is the input to the compensator, Y is the output, s is the complex Laplace transform variable, z is the zero frequency and p is the pole frequency. The pole and zero are both typically negative. In a lead compensator, the pole is left of the zero in the complex plane
,
,
while in a lag compensator
.
A lead-lag compensator consists of a lead compensator cascaded with a lag compensator. The overall transfer function can be written as
Typically
, where z1 and p1 are the zero and pole of the lead compensator and z2 and p2 are the zero and pole of the lag compensator. The lead compensator provides phase lead at high frequencies. This shifts the poles to the left, which enhances the responsiveness and stability of the system. The lag compensator provides phase lag at low frequencies which reduces
the steady state error.
The precise locations of the poles and zeros depend on both the desired characteristics of the closed loop response and the characteristics of the system being controlled. However, the pole and zero of the lag compensator should be close together so as not to cause the poles to shift right, which could cause instability or slow convergence.
Since their purpose is to affect the low frequency behaviour, they should be near the origin.
In analog control systems, where integrators are expensive, it is common to group terms
together to minimize the number of integrators required:
In analog control, the control signal is typically an electrical voltage
or current
(although other signals such as hydraulic pressure can be used).
In this case a lead-lag compensator will consist of
a network of operational amplifier
s ("op-amps") connected as integrators and
weighted adders. In digital control, the operations are performed numerically.
The reason for expressing the transfer function as an integral equation
is that
differentiating signals amplify the noise
on the signal, since even very small
amplitude noise has a high derivative if its frequency is high, while integrating a
signal averages out the noise. This makes implementations in terms of integrators
the most numerically stable.
needing correction can be classified as a lead-network, a lag-network, or a combination
of the two: a lead-lag network (hence the name "lead-lag compensator"). The electrical
response of this network to an input signal is expressed by the network's Laplace-domain
transfer function, a complex
mathematical function which itself can be expressed as one
of two ways: as the current-gain ratio transfer function or as the voltage-gain ratio
transfer function. Remember that a complex function can be in general written as
, where 
is the "Real Part" and 
is the "Imaginary Part" of
the single-variable function,,
.
The "phase angle" of the network is the argument of
; in the left half plane this is 
. If the phase angle
is negative for all signal frequencies in the network then the network is classified
as a "lag network". If the phase angle is positive for all signal frequencies
in the network then the network is classified as a "lead network". If the total network
phase angle has a combination of positive and negative phase as a function of frequency
then it is a "lead-lag network".
Depending upon the nominal operation design parameters of a system under an active
feedback control, a lag or lead network can cause instability
and poor speed and
response times.
Control system
A control system is a device, or set of devices to manage, command, direct or regulate the behavior of other devices or system.There are two common classes of control systems, with many variations and combinations: logic or sequential controls, and feedback or linear controls...
that improves an undesirable frequency response
Frequency response
Frequency response is the quantitative measure of the output spectrum of a system or device in response to a stimulus, and is used to characterize the dynamics of the system. It is a measure of magnitude and phase of the output as a function of frequency, in comparison to the input...
in a feedback and control system
Control system
A control system is a device, or set of devices to manage, command, direct or regulate the behavior of other devices or system.There are two common classes of control systems, with many variations and combinations: logic or sequential controls, and feedback or linear controls...
. It is a fundamental building block in classical control theory
Control theory
Control theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems. The desired output of a system is called the reference...
.
Applications
Lead–lag compensators influence disciplines as varied as roboticsRobotics
Robotics is the branch of technology that deals with the design, construction, operation, structural disposition, manufacture and application of robots...
,
satellite
Satellite
In the context of spaceflight, a satellite is an object which has been placed into orbit by human endeavour. Such objects are sometimes called artificial satellites to distinguish them from natural satellites such as the Moon....
control, automobile diagnostics, laser
Laser
A laser is a device that emits light through a process of optical amplification based on the stimulated emission of photons. The term "laser" originated as an acronym for Light Amplification by Stimulated Emission of Radiation...
frequency stabilization, and many more.
They are an important building block in analog control systems, and
can also be used in digital control.
Given a control plant, desired specifications can be achieved using compensators. I, D, PI, PD, and PID
PID controller
A proportional–integral–derivative controller is a generic control loop feedback mechanism widely used in industrial control systems – a PID is the most commonly used feedback controller. A PID controller calculates an "error" value as the difference between a measured process variable and a...
, are optimizing controllers which are used to improve system parameters (such as reducing steady state error, reducing resonant peak, improving system response by reducing rise time). All these operations can be done by compensators as well.
Theory
Both lead compensators and lag compensators introduce a pole–zeroZero (complex analysis)
In complex analysis, a zero of a holomorphic function f is a complex number a such that f = 0.-Multiplicity of a zero:A complex number a is a simple zero of f, or a zero of multiplicity 1 of f, if f can be written asf=g\,where g is a holomorphic function g such that g is not zero.Generally, the...
pair into the open loop transfer function
Transfer function
A transfer function is a mathematical representation, in terms of spatial or temporal frequency, of the relation between the input and output of a linear time-invariant system. With optical imaging devices, for example, it is the Fourier transform of the point spread function i.e...
. The transfer function can be written in the Laplace domain as
where X is the input to the compensator, Y is the output, s is the complex Laplace transform variable, z is the zero frequency and p is the pole frequency. The pole and zero are both typically negative. In a lead compensator, the pole is left of the zero in the complex plane
Complex plane
In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...
,
,while in a lag compensator
.A lead-lag compensator consists of a lead compensator cascaded with a lag compensator. The overall transfer function can be written as
Typically
, where z1 and p1 are the zero and pole of the lead compensator and z2 and p2 are the zero and pole of the lag compensator. The lead compensator provides phase lead at high frequencies. This shifts the poles to the left, which enhances the responsiveness and stability of the system. The lag compensator provides phase lag at low frequencies which reducesthe steady state error.
The precise locations of the poles and zeros depend on both the desired characteristics of the closed loop response and the characteristics of the system being controlled. However, the pole and zero of the lag compensator should be close together so as not to cause the poles to shift right, which could cause instability or slow convergence.
Since their purpose is to affect the low frequency behaviour, they should be near the origin.
Implementation
Both analog and digital control systems use lead-lag compensators. The technology used for the implementation is different in each case, but the underlying principles are the same. The transfer function is rearranged so that the output is expressed in terms of sums of terms involving the input, and integrals of the input and output. For example,In analog control systems, where integrators are expensive, it is common to group terms
together to minimize the number of integrators required:
In analog control, the control signal is typically an electrical voltage
Voltage
Voltage, otherwise known as electrical potential difference or electric tension is the difference in electric potential between two points — or the difference in electric potential energy per unit charge between two points...
or current
Electric current
Electric current is a flow of electric charge through a medium.This charge is typically carried by moving electrons in a conductor such as wire...
(although other signals such as hydraulic pressure can be used).
In this case a lead-lag compensator will consist of
a network of operational amplifier
Operational amplifier
An operational amplifier is a DC-coupled high-gain electronic voltage amplifier with a differential input and, usually, a single-ended output...
s ("op-amps") connected as integrators and
weighted adders. In digital control, the operations are performed numerically.
The reason for expressing the transfer function as an integral equation
Integral equation
In mathematics, an integral equation is an equation in which an unknown function appears under an integral sign. There is a close connection between differential and integral equations, and some problems may be formulated either way...
is that
differentiating signals amplify the noise
Noise
In common use, the word noise means any unwanted sound. In both analog and digital electronics, noise is random unwanted perturbation to a wanted signal; it is called noise as a generalisation of the acoustic noise heard when listening to a weak radio transmission with significant electrical noise...
on the signal, since even very small
amplitude noise has a high derivative if its frequency is high, while integrating a
signal averages out the noise. This makes implementations in terms of integrators
the most numerically stable.
Intuitive explanation
To begin designing a lead-lag compensator, an engineer must consider whether the systemneeding correction can be classified as a lead-network, a lag-network, or a combination
of the two: a lead-lag network (hence the name "lead-lag compensator"). The electrical
response of this network to an input signal is expressed by the network's Laplace-domain
transfer function, a complex
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
mathematical function which itself can be expressed as one
of two ways: as the current-gain ratio transfer function or as the voltage-gain ratio
transfer function. Remember that a complex function can be in general written as
, where is the "Real Part" and is the "Imaginary Part" ofthe single-variable function,,
.The "phase angle" of the network is the argument of
; in the left half plane this is . If the phase angleis negative for all signal frequencies in the network then the network is classified
as a "lag network". If the phase angle is positive for all signal frequencies
in the network then the network is classified as a "lead network". If the total network
phase angle has a combination of positive and negative phase as a function of frequency
then it is a "lead-lag network".
Depending upon the nominal operation design parameters of a system under an active
feedback control, a lag or lead network can cause instability
Stability theory
In mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions...
and poor speed and
response times.
External links
- http://www.library.cmu.edu/ctms/ctms/extras/leadlag.htm
- lead controller using Matlab
- Lead-Lag Frequency Response at MathPages
- Lead-Lag Algorithms at MathPages