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Feedback Principle & Root Locus Technique Study notes For EE/EC
By BYJU'S Exam Prep
Updated on: September 25th, 2023
Welcome to the comprehensive study notes on the Feedback Principle and Root Locus Technique specifically tailored for Electrical Engineering (EE) and Electronics and Communication Engineering (EC) students. Understanding the principles of feedback and the root locus technique is essential for analyzing and designing control systems, which play a crucial role in numerous applications, from industrial automation to communication systems. These study notes are carefully crafted to provide a clear and systematic exploration of these concepts, empowering students in their journey to master the art of control systems engineering.
In the first section of these study notes, we delve into the Feedback Principle, a fundamental concept in control theory. You will learn about the different types of feedback, their advantages, and challenges in various systems. Gain insight into stability analysis and how feedback influences the stability and performance of control systems. The second section focuses on the Root Locus Technique, a powerful graphical method to analyze the closed-loop stability of control systems. Learn to plot root loci, determine system behaviour based on their characteristics, and design controllers to meet specific requirements. Through this comprehensive study material, EE and EC students will build a strong foundation in control systems engineering and be well-prepared to tackle real-world challenges in this dynamic field.
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Table of content
Feedback Principle
In a feedback control system, at least part of the information used to change the output variable is derived from measurements performed on the output variable itself. This type of closed-loop control is often used in preference to open-loop control (where the system does not use output-variable information to influence its output) since feedback can reduce the sensitivity of the system to externally applied disturbances and changes in system parameters.
Familiar examples of feedback control systems include residential heating systems, most high-fidelity audio amplifiers, and the iris-retina combination that regulates light entering the eye.
Closed-loop control system
- Sometimes, we may use the output of the control system to adjust the input signal. This is called feedback. Feedback is a special feature of a closed-loop control system.
- A closed-loop control system compares the output with the expected result or command status, then it takes appropriate control actions to adjust the input signal.
- Therefore, a closed-loop system is always equipped with a sensor, which is used to monitor the output and compare it with the expected result.
The following figure shows a simple closed-loop system.
- The output signal is feedback to the input to produce a new output.
- A well-designed feedback system can often increase the accuracy of the output.
Block diagram of a closed-loop control system
Feedback can be divided into positive feedback and negative feedback.
Positive Feedback:
- Positive feedback causes the new output to deviate from the present command status.
- For example, an amplifier is put next to a microphone, so the input volume will keep on increasing, resulting in a very high output volume.
Negative Feedback:
- Negative feedback directs the new output towards the present command status, so as to allow more sophisticated control.
- For example, a driver has to steer continuously to keep his car on the right track.
Most modern appliances and machinery are equipped with closed loop control systems. Examples include air conditioners, refrigerators, automatic rice cookers, automatic ticketing machines, etc. An air conditioner, for example, uses a thermostat to detect the temperature and control the operation of its electrical parts to keep the room temperature at a preset constant.
Block diagram of the control system of an air conditioner
- One advantage of using the closed-loop control system is that it is able to adjust its output automatically by feeding the output signal back to the input.
- When the load changes, the error signals generated by the system will adjust the output. However, closed-loop control systems are generally more complicated and thus more expensive to make.
- Feedback is a common and powerful tool when designing a control system.
- The feedback loop is the tool which takes the system output into consideration and enables the system to adjust its performance to meet a desired result of the system.
In any control system, output is affected due to changes in environmental conditions or any kind of disturbance. So one signal is taken from the output and is fed back to the input. This signal is compared with reference input and then an error signal is generated. This error signal is applied to the controller and the output is corrected. Such a system is called a feedback system.
- When the feedback signal is positive then the system is called the positive feedback system. For a positive feedback system, the error signal is the addition of a reference input signal and feedback signal.
- When the feedback signal is negative then the system is called a negative feedback system. For a negative feedback system, the error signal is given by the difference between the reference input signal and the feedback signal.
Feedback characteristics: Including feedback into the control of a system results in the following.
Advantages:
- Increased accuracy. The output can be made to reproduce the input.
- Reduced sensitivity to system characteristics.
- Reduction in the effect of non-linearities.
- Increased bandwidth. The system can be made to respond to a larger range of input frequencies.
The major disadvantages resulting from feedback are the increased risk of instability and the additional cost of design and implementation.
Applications of Feedback: Flight control systems, Robotics, Chemical process control, Communications & Networks, Automotive, Biological Systems, Environmental Systems & Quantum Systems.
Rule to Draw the Root Locus Plot
Rule 1-Symmetry: Since the characteristic equation has real coefficients, any zeros must occur in complex conjugate pairs (which are symmetric about the real axis). Since the root locus is just a diagram of the roots of the characteristic equation as K varies, it must also be symmetric about the real axis.
Rule 2-Number of Branches: Since the order of the characteristic equation is the same as that of the denominator of the loop gain, the number of branches is n, the order of the denominator polynomial.
Rule 3- Starting and Ending Points: Start from the magnitude condition: K|N(s)/D(s)| =1
- So the locus starts (when K=0) at the poles of the loop gain, and ends (when K→∞) at the zeros.
Rule 4- Locus On The Real Axis: The locus exists on that portion of the real axis which contains an odd number of poles and zeroes towards its right side.
Rule 5- Number of Branch Terminating to Infinity: If For a given system the no of Open Loop Poles is P & the number of Open Loop Zeros is Z then there will be P-Z branches which are terminating to Infinity.
Rule 6- Asymptotes Angle & Centroid: We know that if we have a characteristic equation P(s) that has more poles (P=N) than zeros (Z=M) then N−M of the root locus branches tend to zeros at infinity.
These asymptotes intercept the real axis at a point, Called it Centroid σ_{A},
Or in other words
- The angles of the asymptotes φ_{k} are given by for K>0
where, q = 0, 1, 2, …, n – m – 1.
- The angles of the asymptotes φ_{k} are given by for K<0
where, q = 0, 1, 2, …, n – m – 1.
Rule 7-Determining the Breakaway Points:
- First of all, we have to identify the portions of the real axis where a breakaway point must exist.
- Assuming we have already marked the segments of the real axis that are on the root locus, we need to find the segments that are the part of the root locus by either two poles or two zeros (either finite zeros or zeros at infinity).
- To estimate the values of s at the breakaway points, the characteristic equation ⇒
- 1 + KP(s) = 0 is rewritten in terms of K as
To find the breakaway points we find the values of s corresponding to the maxima in K(s). i.e. where dK/ds = K'(s) = 0.
- As the last step, we check the roots of K'(s) that lie on the real axis segments of the locus. The roots that lie in these intervals are the breakaway points.
Rule 8- Find the angles of departure/arrival for complex poles/zeros:
- The angle of departure (θ_{d}) is given by θ_{d} = 180° + arg [G(s )H(s ) where arg [G(s )H(s)] is the angle of G(s)H(s) excluding the pole where the angle is to be calculated.
- Similarly, the angle of arrival is given by θ_{a} = 180° – arg [G(s)H(s)], where arg [G(s)H(s)] is the angle of G(s)H(s) excluding the zero, where the angle is to be calculated.
- When there are complex poles or zeros of P(s), the root locus branches will either depart or arrive at an angle θ where, for a complex pole or zero at s = p or s = z,
where
In other words
Therefore, exploiting the rules of complex numbers, we can rewrite ∠P^{∗}(p)and ∠P^{∗}(z) as
Rule 9- Intersection of the Root Locus with the jω Axis: To find the intersection of root locus with the imaginary axis. The following procedures are followed
Step 1: Construct the characteristic equation 1+ G(s) H(s) = 0
Step 2: Develop the Routh array in terms of K.
Step 3: Find K_{mar} that creates one of the roots of the Routh array as a row of zeros.
Step 4: Frame auxiliary equation A(s) = 0 with the help of the coefficient of a row just above the row of zeros.
Step 5: The roots of the auxiliary equation A(s) = 0 for K = K_{mar} give the intersection points of the root locus with the imaginary axis.
Value Gain Margin:
Grain Margin (GM) =
Note: If the root locus does not cross the jω axis, the gain is ∞. GM represents the maximum gain that can be multiplied for the system to be just on the verge of instability.
Phase Margin: Phase margin (PM) can be determined for a given value of K as follows
- Calculate ω for which |G (jω) H (jω)| = 1 for the given value of K.
- Calculate [G (jω) H (jω)]
- Phase margin = 180° + arg [G (jω) H (jω)]
For getting more on Root Locus find here PDF on Root Locus Techniques
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