CONTROL SYSTEMS | Nyquist Stability Criterion | Concepts & Questions #gate2025 #gatepreparation

Name: CONTROL SYSTEMS | Nyquist Stability Criterion | Concepts & Questions #gate2025 #gatepreparation
Uploaded: 2024-12-17T04:03:11.000Z
Duration: 3 h 28 min 36 s

Introduction to Nyquist Stability Criteria

Overview of the Session

The session begins with a warm welcome to participants, emphasizing the importance of the topic: Nyquist Stability Criteria in control systems.

The instructor highlights that this topic is crucial for students across various engineering branches, particularly for GATE examinations, focusing on conceptual understanding rather than mere sketching of Nyquist plots.

Understanding Stability

An introduction to stability concepts is provided, mentioning Harry Nyquist as a key figure in developing these criteria.

The session aims to explain the formula N = P - Z , which relates to stability and its implications in control systems.

Characteristics of Control Systems

Defining Variables

A parabolic curve y = x^2 is used as an analogy for defining functions where x is independent and y is dependent.

Emphasis on plotting real functions with independent variables on the horizontal axis and dependent variables on the vertical axis.

Key Concepts in Stability

Discussion shifts towards how stability depends on parameters defined by the roots of characteristic equations.

The closed-loop transfer function is introduced as GS/(1 + GSHS) , where poles determine system stability.

Characteristic Equation Analysis

Roots and Poles

Explanation of how the nature of roots from the characteristic equation influences system stability; characterized by 1 + GSHS = 0 .

Clarification that GSHS , or open-loop transfer function, is a function of complex variable s = sigma + jomega .

Complex Functions

Transition into discussing complex functions and their representation in two-dimensional planes.

Introduction of another variable z , illustrating it as a complex variable with both real and imaginary components.

Examples Using Complex Variables

Practical Application

Examples are provided using specific values for complex variables (e.g., calculating squares like (1 - i)^2 ) to demonstrate how these calculations relate back to Nyquist's criteria.

Understanding Complex Functions and Their Representation

Introduction to Complex Numbers

The imaginary part Z is expressed as x + iy , indicating that any complex function value is also a complex number, represented as U + iV .

To plot a complex number, we require a 2D plane known as the Z plane, which accommodates both the real and imaginary parts of the complex number.

Plotting in Complex Planes

For plotting function values, another plane called the FZ plane is necessary since functions have both real and imaginary parameters.

Both Z plane and FZ plane are types of complex planes; they are typically named after the variable being plotted (e.g., Z or F(Z) ).

Characteristics of Complex Planes

Each complex plane consists of a real axis and an imaginary axis to represent their respective components.

Example: The first value Z_1 = 1 - i ; here, the real part is 1 and the imaginary part is -1. This point can be plotted on the Z plane.

Function Values in Complex Planes

Another example given is Z_2 = 2 + i , where it has a real part of 2 and an imaginary part of 1.

Since each complex number requires its own representation in a 2D space, two separate planes (Z plane for input values and FZ plane for output values) are essential for clarity.

Mapping Between Planes

Corresponding function values are discussed: for instance, if F(Z_1) = -2i , this indicates how points map from one plane to another.

Similarly, for F(Z_2)=3+4i, it illustrates how different inputs yield distinct outputs in their respective planes.

Nyquist Stability Criteria Overview

The discussion transitions into Nyquist stability criteria within control systems, emphasizing its mathematical foundation related to stability assessment.

It’s highlighted that plotting any complex function requires two distinct planes: one for independent variables (Z plane), another for dependent variables (FZ plane).

Control System Theory Connection

In control system theory, stability relies on roots from characteristic equations like 1 + G(sH(s)) = 0. Here, G(sH(s)) —a function of S—plays a crucial role.

Further Exploration into S Plane Mapping

Transitioning back to mapping concepts between S-plane and G(SH)-plane emphasizes that these mappings help visualize relationships between different functions involving complex variables.

An example function provided is G(S)=S^2 -6S+8; exploring paths along closed loops helps understand behavior around zeros.

This structured overview captures key insights from discussions about complex numbers, their graphical representations in various planes, and connections to control system theories.

Understanding Zeros in Functions and Their Implications

Introduction to Zeros

The discussion begins with identifying zeros of a function, specifically noting that S = 2 and S = 4 are the zeros since substituting these values into the function results in zero.

The speaker emphasizes the importance of understanding why one zero must be inside a closed path while another is outside, indicating that each step has a specific purpose.

Path Selection in the S Plane

A clockwise path is chosen for evaluating values of S, ensuring one zero falls inside and one outside. This setup is crucial for further calculations.

The speaker illustrates taking various values along this path, starting from S1 (S = 2 + 2i), moving through several points to maintain the clockwise direction.

Calculation of Function Values

For each selected value of S, corresponding GSS (function value) calculations are performed. For instance, when S = 2 + 2i:

GSS calculation yields complex results involving imaginary numbers.

When evaluating at S = 3, the result simplifies to -1, showcasing straightforward computation without complex components.

Further Evaluations and Results

At S = 2 - 2i, calculations yield another complex number as output. Each evaluation demonstrates how different inputs affect outputs significantly.

Finally, at S = 1, the function returns a real number (3), illustrating varied outcomes based on input selection.

Plotting Results in Complex Planes

The speaker transitions to plotting these calculated GSS values on a graph with two planes: the S plane and GSS plane.

First plotted point corresponds to (-4,-4i).

As they plot subsequent points based on earlier evaluations (e.g., GS3 at (-4,+4i)), they illustrate how these points connect back to form a closed loop by returning to initial conditions after traversing through all evaluated points.

Understanding the GSHS Plane and S Plane Relationships

Overview of GSHS and S Planes

The speaker discusses the transition from GS3 to GS4, ultimately reaching GS1, illustrating a process in the GSHS plane.

A visual representation of the S plane is provided, with values S1 through S4 plotted in a clockwise manner for clarity.

The relationship between values in the S plane and their corresponding values in the GSHS plane is emphasized, hinting at future discussions on stability.

Vector Representation

The function under discussion is introduced as gshs = (s - 2)(s - 4), explaining how vectors are represented by position differences.

The vector s - 2 is defined as OB - OA, indicating movement from point A to B within the context of these planes.

Movement Along Closed Paths

As values of s move along a closed path (from S1 to S4), it completes one full revolution, equating to a 360° or 2π radians rotation.

This complete rotation indicates that when moving through these points, there’s a consistent change in angle associated with each segment.

Angle Measurement Conventions

Clockwise movements are noted as negative angles while anticlockwise movements are positive; this convention is crucial for understanding angular changes.

The net change in angle for a complete closed path around s - 2 results in an effective measurement of -2π due to its clockwise nature.

Analyzing Changes Between Points

For s - 4, similar analysis applies: moving from point S1 to S3 involves an anticlockwise rotation followed by a clockwise return from S3 back to S1.

The net effect of these rotations results in zero change overall since both movements cancel each other out (Theta clockwise + Theta anticlockwise = 0).

This structured approach provides clarity on complex relationships between different mathematical representations and their implications for stability analysis.

Understanding the Change in Angle of Complex Functions

Introduction to Complex Functions and Angles

The discussion begins with a reference to a function involving complex numbers, specifically mentioning "S equal 2 is a z."

The angle of the function gshs is introduced, defined as the sum of angles from two complex terms: angle s-2 and angle s-4.

Calculating Changes in Angles

The change in angle for gshs is determined by adding changes from both components: change in angle s-2 and change in angle s-4.

It is noted that the net change expected in the angle gshs equals -2π, indicating a significant rotation.

Impact of Zeros on Angle Changes

A key point discussed is that if a zero lies inside the closed path under consideration, it results in a face change of -2π.

Conversely, if a zero is outside this path, there will be no change (noted as zero).

Analyzing Poles and Their Effects

When discussing poles, it’s explained that if you have 1/(s - 2), then at s = 2 (a pole), the angle becomes negative due to division.

If this term represents a pole located inside the closed path, it leads to an increase in face by +2π.

Summary of Key Points on Zeros and Poles

If both zeros and poles are located outside the closed path, there will be no net change observed (0).

The overall conclusion emphasizes understanding how zeros and poles affect angular changes within complex functions.

Visual Representation of Angular Changes

A plot illustrating these concepts shows how angles are calculated based on their position relative to an origin.

Ultimately, all angles measured result in a total rotation equivalent to 360° or 2π clockwise direction leading to -2π as per convention.

Understanding Change in Angle in GSHS Plots

Introduction to GSHS and Angle Changes

The discussion begins with the concept of change in angle for a GSHS plot, emphasizing that a 2π clockwise rotation is represented as -2π.

A new GSHS plot is introduced, illustrating various points (1 through 8) on this plot to demonstrate movement along the curve.

Analyzing Movement Between Points

The speaker describes moving from point to point on the GSHS plot, highlighting the need to complete a closed path by returning to the starting point.

Observations are made about calculating angle changes by joining lines from the origin to each point, noting how many full rotations occur during this process.

Calculating Total Angle Change

After two complete rotations (720° or -4π), it’s explained that this results in a total angle change of -4π due to clockwise directionality.

The first plot encircles the origin once (-2π), while the second encircles it twice (-4π), demonstrating how multiple encirclements affect angle calculations.

Generalizing Encirclement Effects

If a GSHS plot encircles the origin 'n' times, it will yield an angle change of 2πn; anticlockwise movements are positive while clockwise movements are negative.

This leads to understanding that if there are multiple clockwise encirclements, they contribute negatively to the overall angle change.

Impact of Poles and Zeros Inside Closed Paths

The relationship between poles and zeros within closed paths is discussed: if there are P poles inside, they contribute +2πP; if there are Z zeros inside, they contribute -2πZ.

The total change in face for GSHS can be summarized as 2π(P - Z), indicating how both poles and zeros influence overall behavior within closed paths.

Understanding Phase Change in GSS and Its Implications

Phase Change Calculation

The change in phase for GSS is expressed as 2pi P - Z. This can be visualized through a GSHS plot, where the encirclement of the origin indicates how many times it moves around.

The angle change 2pi n correlates with the direction of movement: anticlockwise (positive n) or clockwise (negative n). Thus, comparing both expressions leads to the conclusion that n = P - Z.

Direction of Closed Path

If the closed path in the S-plane is taken clockwise, then we derive n = P - Z. Conversely, if taken anticlockwise, it results in n = Z - P.

This concept is crucial not only for stability analysis but also appears frequently in examination questions.

Example Question Analysis

A specific question from an exam discusses a pole-zero map of a rational transfer function GS. It asks about how many times a closed contour maps into the GS plane and encircles the origin.

The mapping involves understanding how this closed contour relates to encirclements around -1 + j0, which will be elaborated upon later.

Characteristics of Closed Paths

The discussion emphasizes that when analyzing paths on the S-plane, it's essential to note their direction—specifically whether they are clockwise or anticlockwise.

For clockwise paths, use P - Z; for anticlockwise paths, use Z - P.

Counting Poles and Zeros

In determining values for poles (P) and zeros (Z), only those inside the contour are counted. For example:

Poles: 2 (not counted)

Zeros: 3 (counted)

Therefore, with two poles outside and three zeros inside, we find that n = 2 - 3 = -1, indicating one clockwise encirclement of the origin.

Stability Analysis Using Nyquist Criterion

Importance of Characteristic Equation Roots

Stability in control systems hinges on roots of the characteristic equation; specifically, they must not lie in the right half-plane (RHP).

If any root exists within RHP, it signifies instability within the system.

Searching for Roots

To analyze stability effectively, one must search for roots using a defined Nyquist Contour. This contour may traverse either clockwise or anticlockwise depending on its configuration.

Understanding the Nyquist Path in Control Systems

Introduction to Closed Paths in the S Plane

The discussion begins with the concept of closed paths in the S plane, emphasizing that these paths can be either clockwise or anticlockwise. The focus is on covering the entire right half-plane (RHP) to search for roots of characteristic equations.

Clockwise Nyquist Contour

A clockwise Nyquist contour is introduced as a method to cover the RHP. This path starts from a point just above the origin and aims to encompass all points within this region.

Avoiding Singularities at Origin

The speaker explains why it is crucial to avoid the origin when plotting values in the G(s)H(s) plane, noting that certain values lead to undefined results (e.g., G(s) becomes infinite at s = 0).

It is highlighted that s = 0 often represents a pole, making it necessary to approach this point carefully by starting just above it (limit as s approaches 0 from positive side).

Description of Path Movement

The movement along the Y-axis is described: starting from j0 and moving towards +j∞, then taking a semicircular arc back down to -j∞.

After reaching -j∞, movement continues up along the Y-axis until reaching -j0 before returning back up to j0 while avoiding crossing through zero directly.

Completing the Nyquist Path

The complete Nyquist path involves traveling along specific arcs and lines in order to effectively cover all points in RHP without encountering singularities.

Plotting G(s)H(s)

Once values are plotted based on this closed path, they form what is known as a Nyquist plot. This plot visually represents how many times it encircles the origin.

Stability Criteria and Encirclements

The number of encirclements around the origin provides critical information about system stability, following the formula n = p - z where n represents encirclements, p poles, and z zeros.

Emphasis is placed on understanding analysis techniques for interpreting Nyquist plots rather than merely sketching them; recent trends indicate an increased focus on analytical skills over drawing proficiency.

This structured overview captures key concepts discussed regarding Nyquist paths and their significance in control systems analysis.

Understanding the Nyquist Path and Stability in Control Systems

The Nyquist Path and Its Components

The formula for the Nyquist path changes based on its direction; for a clockwise path, it is Z - P , while for an anticlockwise path, it becomes Z - P . Here, P^+ and Z^+ denote poles and zeros of the transfer function (G(s)) that are inside the closed path.

The closed path specifically refers to those poles and zeros within the right half-plane (RHP), which is crucial for analyzing system stability.

In this context, P^+ represents poles located in the RHP, while Z^+ signifies zeros of G(s) also found in this area. This distinction helps clarify their roles in stability analysis.

The equation relating these components is given by N = P^+ - Z^+ , where N indicates how many times the Nyquist plot encircles the origin.

Drawing and Analyzing the Nyquist Plot

When sketching a Nyquist plot, one must observe how many times it encircles the origin; this count directly relates to determining N's value.

If you know P^+ , you can calculate Z^+ . This relationship emphasizes understanding both poles and zeros' contributions to system behavior.

Stability Considerations

For assessing control system stability, we focus on finding roots or zeros of a specific characteristic equation: 1 + G(s) = 0 .

It’s essential to differentiate between zeros of G(s) and those of 1 + G(s) ; only the latter provides relevant information regarding system stability.

Modifications for Better Analysis

To analyze stability effectively, one should consider plotting 1 + G(s) . This approach simplifies understanding how changes affect system behavior compared to just plotting G(s).

When transitioning from plotting G(s) to plotting 1 + G(s), all values shift along the real axis by one unit.

Visualizing Changes in Plots

Adding one to all values results in a horizontal shift of the entire plot without altering its shape—this means that if you already have a plot for G(s), you can derive that for 1 + G(s).

The green plot represents G(s), while another color shows what happens when shifting by one unit along the real axis. This visual representation aids comprehension of how modifications impact overall system dynamics.

Understanding the Shift in Coordinate Systems

Shifting the Imaginary Axis

The discussion begins with a proposal to shift the imaginary axis left instead of moving the plot right, emphasizing that both actions yield similar results.

By shifting the imaginary axis one unit to the left, the new origin is established at (-1, 0), changing its previous position from (0, 0).

Drawing and Modifying Plots

The focus shifts to drawing the plot for 1 + g(s)h(s), which requires adjusting the g(s)h(s) plot by either shifting it or redefining the origin.

A modified origin at (-1, 0) is introduced for analysis, particularly relevant when studying Nyquist plots and their encirclements.

Understanding Poles and Zeros

The formula relating poles and zeros is reiterated: n = p + - z^+, where n represents encirclements around a modified origin.

An example function illustrates how poles remain unchanged while zeros may vary; specifically, poles of g(s) are consistent across transformations.

Implications of Modified Origin

It’s emphasized that while analyzing 1 + g(s)h(s), we will primarily draw from g(s)'s plot but consider a modified origin for accurate representation.

The relationship between poles (p^+) remains constant regardless of whether we analyze g(s) or 1 + g(s).

Nyquist Path Considerations

A critical point about avoiding the origin in Nyquist paths is discussed; if a pole exists at zero, it complicates plotting.

The method involves taking points just below and above the origin to create an arc without directly touching it.

Arc Movement Analysis

Visualizing movement along an arc close to the origin highlights how complex numbers relate to their distance from this point.

As we approach this arc's endpoints near -j0, considerations about angles and magnitudes become crucial in understanding system behavior.

Understanding the Nyquist Plot and Stability Analysis

Limit Behavior as s Tends to Zero

The discussion begins with the behavior of a function as s approaches zero, emphasizing that both left-hand and right-hand limits converge to the same value.

It is noted that if s tends to zero, only terms like K/s^n remain significant, which can lead to undefined values if s = 0 .

Magnitude and Phase Analysis

As R approaches zero in the limit, it is established that the magnitude of certain functions will tend towards infinity while analyzing their phase components.

The angle associated with these functions is described as being influenced by factors such as -nTheta , indicating how phase shifts occur.

Angle Changes in gshs

When evaluating specific points on the complex plane (e.g., at -j0 ), angles are calculated leading to a clockwise direction of rotation.

The change in angle from one point to another is quantified, revealing a total shift of -npi , indicating significant implications for system stability.

Encirclements and Stability Criteria

A critical observation is made regarding how changes in angle correspond to encirclements around points on the Nyquist plot, specifically noting an infinite radius arc due to large magnitudes.

A fundamental question about Nyquist plots for open-loop transfer functions in unity negative feedback systems sets up further analysis.

Application of Nyquist Stability Criterion

The importance of poles and zeros within the right half-plane (RHP) is discussed concerning system stability using the formula: n = p+ - z+ .

With no poles present in the RHP for GS, it follows that p+ = z+ , leading to conclusions about encirclement counts affecting system stability.

Net Encirclement Calculation

Analyzing encirclements around modified origin (-1, 0), it’s determined there are two loops: one clockwise and one anticlockwise.

This results in a net encirclement count of zero; thus, concluding that both zeros ( z+ ) are also equal to zero.

Understanding Unity Feedback Control Systems

Introduction to the Question

The discussion begins with a reference to a question from GATE 2017, which is described as a very good question worth two marks.

Characteristics of the System

A Unity feedback control system is characterized by its open-loop transfer function: 10k s + 2/s^3 + 3s^2 + 2 .

The Nyquist path for this system passes through the origin because there are no poles at the origin, allowing for direct analysis without needing to consider limits approaching zero.

Nyquist Path Analysis

The Nyquist path is identified as clockwise, which influences how stability is determined using the formula n = P - Z .

For an anticlockwise path, the formula would be n = Z - P , emphasizing the importance of direction in stability analysis.

Stability and Poles in RHP

The focus shifts to determining the number of poles of the closed-loop transfer function that lie in the right half-plane (RHP).

It’s clarified that these poles correspond to roots of the characteristic equation located in RHP.

Encirclements and Characteristic Equation Roots

The encirclement around point (-1, 0) is discussed; since K (gain factor) lies between 0 and 1, it indicates no encirclement occurs.

With zero encirclements established, it follows that n = P - Z , leading to conclusions about pole locations based on this relationship.

Determining Poles from Denominator

To find P_+ , or poles in RHP, one must analyze the denominator of G(s) : s^3 + 3s^2 + 2 = 0 .

Since calculators were not allowed during GATE exams, alternative methods such as Routh-Hurwitz criteria or conceptual Nyquist plot analysis are suggested for determining pole counts without exact values.

Methods for Analyzing Poles

Two methods are proposed:

Method One: Use Routh-Hurwitz criteria.

Method Two: Apply Nyquist plot concepts with zero encirclements considered.

This structured approach provides clarity on analyzing unity feedback control systems while addressing specific questions related to stability and pole locations.

Understanding Nyquist Stability Criteria

Key Concepts in Nyquist Stability

The discussion begins with the concept of encirclement in the Nyquist plot, specifically focusing on anticlockwise encirclements. The value of n is determined to be 2, indicating two anticlockwise encirclements.

The speaker emphasizes the importance of understanding the method through the Nyquist plot and mentions using specific points for analysis, such as (-1, 0) and (0, 0), to illustrate concepts effectively.

A direct approach is introduced where no Nyquist plot is needed if certain conditions are met. The characteristic equation can be formed directly from given information about G(s)H(s) .

Characteristic Equation Analysis

The simplified characteristic equation is presented: s^3 + 3s^2 + (10K + 20K + 2) = 0 . This highlights how terms are combined to form a polynomial that represents system stability.

Corrections are made regarding coefficients in the characteristic equation due to typographical errors. It’s clarified that certain values should be adjusted for accurate representation.

Observations on Root Locations

Analyzing signs in the first row of coefficients reveals important insights into root locations. If K < 1 , then 10K - 10 becomes negative, indicating potential instability.

Two sign changes observed in the first row lead to conclusions about RHP roots in the characteristic equation, confirming that there are indeed two RHP roots present.

Practical Application and Homework Assignment

The speaker suggests that when G(s)H(s) is provided completely, it allows for straightforward application of Routh-Hurwitz criteria without needing a Nyquist plot.

A homework question related to gate exam preparation is proposed. Students are encouraged to attempt solving it and share their answers for feedback.

Summary and Additional Resources

A recap of key topics covered includes mapping from S-plane to GH-plane and understanding various aspects of Nyquist plots.

Requests for additional practice sessions on Nyquist plots are welcomed; students can express interest via comments or messages for further assistance.

Information about accessing supplementary materials like PDFs shared through a Telegram group is provided, along with details on password protection for these resources.

PDF Distribution and Closing Remarks

PDF Access Information

The PDF will be distributed through a Telegram group, ensuring easy access for participants.

It is important to note that the PDF will be password protected, adding a layer of security to the document.

Final Thoughts

The speaker, Rakesa, concludes the session with a friendly farewell, emphasizing the importance of safety and self-care.

There is an invitation for further engagement as Maheshwari is encouraged to share answers in the comment box later on.