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Questions and Answers
The root locus technique is used to adjust the location of closed loop ______ to achieve desired system performance.
The root locus technique is used to adjust the location of closed loop ______ to achieve desired system performance.
poles
Root Locus is the plot of the roots of the characteristics ______ of the closed loop system.
Root Locus is the plot of the roots of the characteristics ______ of the closed loop system.
equation
The condition of the root locus can be defined using the ______ Condition and Magnitude Condition.
The condition of the root locus can be defined using the ______ Condition and Magnitude Condition.
Angle
The angle condition states that G(S)H(S) = ±(2q + 1)180°, where q = 0, 1, 2, and represents ______ multiples of 180°.
The angle condition states that G(S)H(S) = ±(2q + 1)180°, where q = 0, 1, 2, and represents ______ multiples of 180°.
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The magnitude condition can be stated as |G(S)H(S)| = |-1 + j0| = ______.
The magnitude condition can be stated as |G(S)H(S)| = |-1 + j0| = ______.
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G(S)H(S) cannot be found if it contains unknown variable ______.
G(S)H(S) cannot be found if it contains unknown variable ______.
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If any value of 's' satisfies the angle condition, then that point is considered to be on the root ______.
If any value of 's' satisfies the angle condition, then that point is considered to be on the root ______.
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To determine whether S = -0.5 lies on the root locus, one must use the ______ condition.
To determine whether S = -0.5 lies on the root locus, one must use the ______ condition.
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The study of whether a system is stable or unstable is known as __________ analysis.
The study of whether a system is stable or unstable is known as __________ analysis.
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A necessary condition for a feedback system to be stable is that all the poles of the system transfer function have __________ real parts.
A necessary condition for a feedback system to be stable is that all the poles of the system transfer function have __________ real parts.
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A system is said to be __________ stable if it produces a bounded output for a given bounded input.
A system is said to be __________ stable if it produces a bounded output for a given bounded input.
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In the absence of input, a system is said to have __________ stability if its output tends to zero irrespective of initial conditions.
In the absence of input, a system is said to have __________ stability if its output tends to zero irrespective of initial conditions.
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A system is called __________ stable if it is stable for all values of the parameters.
A system is called __________ stable if it is stable for all values of the parameters.
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A system is said to be __________ stable if it exhibits sustained oscillation with constant magnitude.
A system is said to be __________ stable if it exhibits sustained oscillation with constant magnitude.
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A system is __________ if the output is unbounded for a bounded input.
A system is __________ if the output is unbounded for a bounded input.
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If the system output is stable for a limited range of parameter variations, it is termed __________ stable.
If the system output is stable for a limited range of parameter variations, it is termed __________ stable.
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The characteristic equation related to root locus technique is given by 1 + G(s)H(s) = ______
The characteristic equation related to root locus technique is given by 1 + G(s)H(s) = ______
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Break away points occur where multiple roots of the characteristic equation 1 + G(s)H(s) = ______ occur.
Break away points occur where multiple roots of the characteristic equation 1 + G(s)H(s) = ______ occur.
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The center of gravity or centroid is denoted by ______ and is calculated using poles and zeros in the transfer function.
The center of gravity or centroid is denoted by ______ and is calculated using poles and zeros in the transfer function.
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The angle of asymptotes can be calculated using the formula where p = 0, 1, 2, ..., (N - M - ______).
The angle of asymptotes can be calculated using the formula where p = 0, 1, 2, ..., (N - M - ______).
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Asymptotes provide direction to the root locus when they depart from break away ______.
Asymptotes provide direction to the root locus when they depart from break away ______.
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Routh Hurwitz criterion is used to find out the point of intersection of root locus with the ______ axis.
Routh Hurwitz criterion is used to find out the point of intersection of root locus with the ______ axis.
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Break in points occur when root locus is present between two adjacent ______ on the real axis.
Break in points occur when root locus is present between two adjacent ______ on the real axis.
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K is the open loop system ______ in the transfer function G(s).
K is the open loop system ______ in the transfer function G(s).
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The point of intersection of the asymptotes on the real axis is called the ______.
The point of intersection of the asymptotes on the real axis is called the ______.
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The formula for the centroid σ is given by the sum of real parts of poles minus the sum of real parts of zeros divided by the ______.
The formula for the centroid σ is given by the sum of real parts of poles minus the sum of real parts of zeros divided by the ______.
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For K = -S^3 - 4S^2 - 3S, the characteristic equation is S(S^2 + 4S + 3) + K = ______.
For K = -S^3 - 4S^2 - 3S, the characteristic equation is S(S^2 + 4S + 3) + K = ______.
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When S = -0.451, the value of K is calculated to be ______.
When S = -0.451, the value of K is calculated to be ______.
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The formula for the closed loop transfer function is C(S) / R(S) = K / ______.
The formula for the closed loop transfer function is C(S) / R(S) = K / ______.
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To find the breakaway point, we differentiate K with respect to ______.
To find the breakaway point, we differentiate K with respect to ______.
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The closed loop transfer function is equal to K divided by S(S + 1)(S + 3) + ______.
The closed loop transfer function is equal to K divided by S(S + 1)(S + 3) + ______.
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The formula for the centroid σ is the sum of real parts of poles minus the sum of real parts of ______.
The formula for the centroid σ is the sum of real parts of poles minus the sum of real parts of ______.
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The characteristic equation is represented as S(S+2)(S+4) + ______ = 0.
The characteristic equation is represented as S(S+2)(S+4) + ______ = 0.
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Differentiating K with respect to S gives us the equation dK/dS = ______.
Differentiating K with respect to S gives us the equation dK/dS = ______.
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For K to be positive and real, the corresponding S value is ______.
For K to be positive and real, the corresponding S value is ______.
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When S = -3.154, K is calculated to be ______.
When S = -3.154, K is calculated to be ______.
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The term used for when K is positive and real in relation to S is known as the ______ point.
The term used for when K is positive and real in relation to S is known as the ______ point.
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To find breakaway points, we set dK/dS equal to ______.
To find breakaway points, we set dK/dS equal to ______.
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The characteristic equation is S(S+2)(S+4) + K = 0, which expands to S3 + 6S2 + 8S + ______ = 0.
The characteristic equation is S(S+2)(S+4) + K = 0, which expands to S3 + 6S2 + 8S + ______ = 0.
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To find the crossing points on the root-locus, we substitute s = ______.
To find the crossing points on the root-locus, we substitute s = ______.
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The equation −w^3 + 8w = ______ shows the condition for the imaginary part of the characteristic equation.
The equation −w^3 + 8w = ______ shows the condition for the imaginary part of the characteristic equation.
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The crossing points of the root locus are ______ with the value of K at this point being 48.
The crossing points of the root locus are ______ with the value of K at this point being 48.
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For δ = 0.5, α can be calculated as cos⁻¹______ which equals to 60°.
For δ = 0.5, α can be calculated as cos⁻¹______ which equals to 60°.
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The product of vector length from open loop poles to the point s gives Ksd, which is the ______ of the corresponding point.
The product of vector length from open loop poles to the point s gives Ksd, which is the ______ of the corresponding point.
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The Root Locus Plot technique is particularly useful in investigating the stability ______ of the system.
The Root Locus Plot technique is particularly useful in investigating the stability ______ of the system.
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One application of the Root Locus method is to determine the ______ boundaries of the system.
One application of the Root Locus method is to determine the ______ boundaries of the system.
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Study Notes
Stability Analysis in Control Systems
- Stability is a crucial characteristic of a control system's transient performance
- A stable system's output is predictable, finite, and stable for a given input
- System stability is determined by analyzing the location of its poles in the complex plane.
Concepts of Stability
- BIBO Stability: A system is BIBO stable if it produces a bounded output for any bounded input
- Asymptotic Stability: In the absence of input, the output of the system goes to zero, regardless of initial conditions.
- Stable System: A stable system produces a bounded output for a bounded input. Four cases of system stability are possible
- Absolute Stable: Stable for all parameter values
- Marginally Stable: Output has sustained oscillations at a constant magnitude.
- Unstable: Output is unbounded for bounded input
- Conditionally Stable: Stable only within a limited range of parameter variations
Effect of Pole Location on Stability
- Real and Negative Poles: Exponentially decaying time response; stable
- Complex Conjugate Poles with Negative Real Parts: Damped oscillations; stable
- Real and Positive Poles: Exponentially increasing response; unstable
- Complex Conjugate Poles with Positive Real Parts: Growing oscillations; unstable
Routh-Hurwitz Criterion
- An algebraic method for determining the stability of a linear time-invariant (LTI) system
- Examines the coefficients of the characteristic equation to determine the number of roots with positive real parts (unstable).
- By examining the signs in the first column of the Routh array, the stability of the system is determined.
- The number of sign changes in the first column of the Routh array equals the number of roots with positive real parts in the characteristic equation, indicating the number of unstable poles.
Root Locus Technique
- A graphical method used to analyze the stability and transient response of a closed-loop control system as controller gain (K) changes
- Plots the location of closed-loop poles in the complex s-plane as K varies from zero to infinity.
- Poles and zeros of the open-loop system determine the root locus plot's characteristics.
- Useful for determining stability boundaries and designing suitable system parameters
- Utilizes angle and magnitude conditions to plot the root locus.
Terms in Root Locus
- Characteristic Equation: The equation relating the stability of a closed loop system to its controller gain (K)
- Breakaway Points: Points where multiple roots of the characteristic equation occur. Breakaway points on the real axis are where the root locus branches depart.
- Break-In Points: Points where multiple roots of the characteristic equation occur. Points where root locus branches intersect the real axis and extend back into the complex plane.
- Centroid (σa): The center of gravity of the poles and zeros of the open-loop system
- Asymptotes: Lines originating from the centroid (σa), dictating how the root locus branches approach infinity. The angles of asymptotes are calculated from the number of poles and zeros
- Intersection points with the Imaginary Axis: The auxiliary equation and Routh-Hurwitz stability criterion are used
- Gain Margin: The amount by which the gain can be increased before the system becomes unstable
- Phase Margin: The amount of additional phase shift required before the system becomes unstable
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Description
This quiz covers the crucial concepts of stability in control systems, including BIBO stability and its significance. Understand how pole locations impact system behavior and explore different stability classifications such as absolute, marginally stable, and unstable systems.
