chapter 9 design via root locus
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CHAPTER 9
DESIGN VIA ROOT LOCUS
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INTRODUCTION
Objectives:How to use the root locus to design cascade
compensators to improve the steady state errorHow to use the root locus to design cascade
compensators to improve the transient responseHow to use the root locus to design cascade
compensators to improve both the steady state error and the transient response
How to use the root locus to design feedback compensators to improve the transient response
How to realize the designed compensators physically
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IMPROVING TRANSIENT RESPONSE
Rather than replacing the existing system with a system whose root locus intersects the desired design of point B, we can augment or compensate the system with additional poles and zeros – so that the compensated system has a root locus that goes through the desired pole location for some value of gain
Two methods – passive or active networkDisadvantages – system order can increase,
subsequently effect the desired responseOne method of compensating for transient response is to
insert differentiator in the forward path in parallel with the gain
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IMPROVING STEADY STATE ERROR
Compensators are not only used to improve the transient response of a system, they are also used independently to improve steady state error characteristics
In Chapter 7 – steady state error can be improved by adding an open loop pole at the origin in forward path, thus increasing the system type and driving the associated steady state error to zero
This additional pole at the origin requires an integrator for its realization
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Configuration of Compensations
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System Improvement Technique
Feeding methodProportional – feed the error forward to the plant Integral – feed the integral of the error to the plantDerivative – feed the derivative of the error to the plant
Implemented using active networks (PI/PD)Using AmplifiersExpensive
Implemented using passive networks (Lag/lead)Less expensive no additional power required
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IMPROVING STEADY-STATE ERROR VIA CASCADE COMPENSATION
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IMPROVING TRANSIENT RESPONSE VIA CASCADE COMPENSATION
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IMPROVING STEADY-STATE ERROR AND TRANSIENT RESPONSE
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The Characteristics of P, I & D Controller
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Mathematical Representation of Proportional Controller
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Mathematical Representation of Integral Controller
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Mathematical Representation of Derivative Controller
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Mathematical Representation of PI Controller
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More on IDEAL INTEGRAL COMPENSATION (PI Controller)
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-θ1-θ2-θ3- θ pc+θzc ≡ (2k+1)1800
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EXAMPLE:Closed-loopsystema. beforecompensation;b. after ideal integralcompensation
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Root locus foruncompensatedsystem
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Root locus forcompensatedsystem
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Ideal integral compensated system response and theuncompensated systemresponse of previous example
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LAG COMPENSATION
If use passive networks, the pole and zero are moved to the left, close to the origin
This placement usually will not increase the system type, but will yield an improvement in the static error constant over an uncompensated system
Although the ideal compensator drives the steady state error to zero, a lag compensator with a pole that is not at the origin will improve the static error constant by a factor equal to Zc/Pc
There will also minimal effect upon the transient response if the pole-zero pair of the compensator close to the origin
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a. Type 1 uncompensated system;b. Type 1 compensatedsystem;c. compensatorpole-zero plot
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Root locus:a. before lag compensation;b. after lag compensation
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Mathematical Representation of PD Controller
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More on IDEAL DERIVATIVE COMPENSATION (PD Controller)
To speed up the original system, we can add a single zero to the forward path
This zero can be represented by a compensator whose transfer function is
Gc(S) = s + Zc
This function, the sum of a differentiator and a pure gain is called ideal derivative – PD controller
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Using ideal derivativecompensation:a. uncompensated;b. compensatorzero at –2;c. compensatorzero at –3;d. compensatorzero at – 4
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Zero is moved to a different position and for each compensated case, the dominant, second order poles are farther out along the 0.4 damping ratio line than the uncompensated system
However, each of the compensated case has dominant poles with the same damping ratio as the uncompensated case thus the percent overshoot is predicted to be similar for each case!
The compensated dominant, closed loop poles has more negative real parts than the uncompensated dominant, closed loop poles thus shorter settling time, Ts
On top of that smaller peak time, Tp
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Predicted characteristics for the previous shown systems
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Uncompensated system and ideal derivativecompensation solutions from previous table
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Example: Feedback control system for Example 9.3
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Root locus for uncompensatedsystem shown in Previous Example
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Uncompensated and compensated system characteristics for Previous example
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Compensateddominant polesuperimposed over the uncompensatedroot locus forprevious example
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Evaluating the location of the compensatingzero
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Root locus for thecompensated system
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Uncompensated andcompensated system step responses ofprevious example
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LEAD COMPENSATION
Similar to the active ideal integral that can be approximated with passive lag network, an active ideal integral can be approximated with a passive lead compensator
Pole is farther from the imaginary axis then the zero – will result in a positive angular distribution of the compensator and thus approximates an equivalent single zero
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Geometry of leadcompensation
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Three of the infinitepossible leadcompensator solutions
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Mathematical Representation of PID Controller
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PID Controller Design
It has two zeros and a pole at the originOne zero and the pole at the origin can be
designed as the ideal integral compensator
Another zero can be designed as the ideal derivative compensator
Follow steps in Text book – page 532Evaluate – Design PD – Design PI
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LAG-LEAD COMPENSATOR DESIGN
First design the lead compensator to improve the transient response
Next, evaluate the improvement in steady state
Finally, design the lag compensator to meet the steady state error requirement
Follow steps in Text Book – Page 537Evaluate – Design Lead – Design Lag
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FEEDBACK COMPENSATION
Approach 1Approach 2
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PHYSICAL REALIZATION OF COMPENSATION
Active circuit RealizationPassive circuit Realization