9.3 Closed-Loop System Analysis
4 min read•july 30, 2024
analysis is crucial for understanding systems. It helps engineers evaluate stability, performance, and robustness of systems like PID controllers. This analysis involves techniques like Routh-Hurwitz and Nyquist criteria for stability assessment.
Performance metrics like and are key in closed-loop analysis. Gain and phase margins provide insights into system robustness. and Bode plots are valuable tools for visualizing system behavior and designing compensators to improve performance.
Closed-loop system stability
Routh-Hurwitz criterion
Top images from around the web for Routh-Hurwitz criterion
Analyze stability of a closed-loop system with Bode - Signal Processing Stack Exchange View original
Is this image relevant?
HurwitzMatrix | Wolfram Function Repository View original
Is this image relevant?
Model Order Reduction Using Routh Approximation and Cuckoo Search Algorithm View original
Is this image relevant?
Analyze stability of a closed-loop system with Bode - Signal Processing Stack Exchange View original
Is this image relevant?
HurwitzMatrix | Wolfram Function Repository View original
Is this image relevant?
1 of 3
Top images from around the web for Routh-Hurwitz criterion
Analyze stability of a closed-loop system with Bode - Signal Processing Stack Exchange View original
Is this image relevant?
HurwitzMatrix | Wolfram Function Repository View original
Is this image relevant?
Model Order Reduction Using Routh Approximation and Cuckoo Search Algorithm View original
Is this image relevant?
Analyze stability of a closed-loop system with Bode - Signal Processing Stack Exchange View original
Is this image relevant?
HurwitzMatrix | Wolfram Function Repository View original
Is this image relevant?
1 of 3
- Algebraic method for determining the stability of a closed-loop system based on the coefficients of its characteristic equation
- Provides necessary and sufficient conditions for the stability of a linear time-invariant (LTI) system
- Constructs a tabular array called the Routh table using the coefficients of the characteristic equation
- The number of sign changes in the first column of the Routh table indicates the number of unstable poles in the closed-loop system
Nyquist stability criterion
- Graphical method for determining the stability of a closed-loop system based on the open-loop
- Relates the encirclements of the -1 point on the complex plane by the open-loop frequency response to the number of unstable poles in the closed-loop system
- Can be applied to systems with time delays and non-minimum phase characteristics (right-half plane zeros)
- The Nyquist plot provides insights into the stability margins and robustness of the closed-loop system
- The Nyquist contour is a closed path that encircles the right-half of the complex plane, including the imaginary axis and an infinite semicircle
Closed-loop system performance
Steady-state error and system type
- Represents the difference between the desired output and the actual output of a closed-loop system in the steady-state condition
- The type of the system (Type 0, Type 1, or Type 2) determines the ability of the system to track different types of inputs (step, ramp, or parabolic) with zero steady-state error
- Type 0 systems have a finite steady-state error for step inputs and an infinite steady-state error for ramp and parabolic inputs
- Type 1 systems have zero steady-state error for step inputs, a finite steady-state error for ramp inputs, and an infinite steady-state error for parabolic inputs
- Type 2 systems have zero steady-state error for step and ramp inputs, and a finite steady-state error for parabolic inputs
Transient response and frequency response
- The transient response describes the behavior of a closed-loop system during the transition from the initial state to the steady-state condition
- Characterized by parameters such as rise time, , , and damping ratio
- Rise time is the time required for the output to rise from 10% to 90% of its final value
- Settling time is the time required for the output to settle within a specified tolerance band (usually ±2% or ±5%) of its final value
- Overshoot is the maximum deviation of the output from its final value, expressed as a percentage
- Damping ratio determines the oscillatory behavior of the system (underdamped, critically damped, or overdamped)
- The frequency response describes the system's behavior in terms of gain and phase shift as a function of input frequency
- Analyzed using Bode plots, which display the magnitude and phase of the system's transfer function in logarithmic scales
- The bandwidth is the range of frequencies over which the system can effectively track the input signal
Gain and phase margins
Gain margin
- The amount of additional gain that can be introduced into the system before it becomes unstable
- Determined by the distance between the 0 dB line and the magnitude plot at the frequency where the phase plot crosses -180 degrees in the
- A positive indicates a stable system, while a negative gain margin indicates an unstable system
- A higher gain margin provides more robustness to variations in system gain
Phase margin
- The amount of additional phase lag that can be introduced into the system before it becomes unstable
- Determined by the distance between the -180 degree line and the phase plot at the frequency where the magnitude plot crosses the 0 dB line in the Bode plot
- A positive indicates a stable system, while a negative phase margin indicates an unstable system
- A higher phase margin provides more robustness to variations in system phase and time delays
- Adequate gain and phase margins are essential for ensuring the reliable operation of the closed-loop system in practical applications
Root locus and Bode plots
Root locus technique
- Graphical technique for analyzing the effect of varying the gain of a closed-loop system on its pole locations in the complex plane
- Shows the trajectories of the closed-loop poles as the system gain is varied from zero to infinity
- Determines the range of gains for which the closed-loop system remains stable
- Helps select an appropriate gain value for desired performance characteristics (damping ratio, settling time, overshoot)
- The root locus plot is symmetric about the real axis and starts at the open-loop poles and ends at the open-loop zeros
Bode plot and compensator design
- Used to analyze the frequency response of a closed-loop system and to design compensators for improving system performance
- Lag compensators improve the steady-state error and low-frequency performance by increasing the low-frequency gain
- Lead compensators improve the transient response and stability margins by increasing the phase margin at the desired crossover frequency
- Lag-lead compensators combine the benefits of both lag and lead compensation to achieve desired performance characteristics over a wide range of frequencies
- Compensators are designed by adjusting the pole and zero locations of the compensator transfer function
- The compensated system's Bode plot is obtained by adding the magnitude and phase plots of the original system and the compensator
Key Terms to Review (18)
Bode Plot: A Bode plot is a graphical representation of a linear time-invariant system's frequency response, displaying both the magnitude and phase of the system's transfer function over a range of frequencies. It helps in understanding how the system reacts to different input frequencies and is essential for analyzing stability, designing controllers, and tuning system parameters.
Closed-loop system: A closed-loop system is a control mechanism that uses feedback to compare the actual output with the desired output in order to minimize the difference between them. This type of system continuously monitors its own output and adjusts its input to achieve the desired performance, making it effective for maintaining stability and accuracy. Feedback is crucial, as it allows the system to respond dynamically to changes and disturbances in its environment.
Feedback control: Feedback control is a process in which a system uses its output to adjust its input in order to maintain a desired performance or behavior. This technique helps systems adapt to changes or disturbances, ensuring stability and accuracy. By continuously monitoring the output and making adjustments based on that information, feedback control is crucial in the analysis and design of dynamic systems.
Frequency Response: Frequency response refers to the measure of a system's output spectrum in response to a sinusoidal input signal. It illustrates how different frequency components of the input signal are amplified or attenuated by the system, giving insight into the system's behavior across various frequencies.
Gain Margin: Gain margin is a measure of stability in control systems that indicates how much gain can be increased before the system becomes unstable. It is derived from frequency response analysis and provides insight into the robustness of a system's control, reflecting how close the system is to instability when subjected to changes in gain.
Lag Compensator: A lag compensator is a control system component that introduces a phase lag to improve the stability and performance of a closed-loop system. By adding a pole closer to the origin in the s-plane, it effectively reduces the bandwidth and enhances the system's ability to track slower changes while improving steady-state error. This compensator is particularly useful for systems that require improved stability margins and reduced sensitivity to disturbances.
Lag-Lead Compensator: A lag-lead compensator is a type of controller used in control systems that combines the properties of both lag and lead compensators to improve system performance, particularly in closed-loop systems. It helps to adjust the phase and gain of the system, allowing for better stability and response characteristics by providing both phase lag (to improve stability) and phase lead (to enhance transient response). This makes it a versatile tool in closed-loop control design, especially when aiming for specific performance requirements.
Lead Compensator: A lead compensator is a control system component designed to improve the transient response and stability of a system by adding phase lead at specific frequencies. By increasing the system's phase margin, it can enhance performance metrics like rise time, settling time, and overshoot, making it a valuable tool in control design.
Nyquist Stability Criterion: The Nyquist Stability Criterion is a graphical method used to determine the stability of a control system based on its open-loop frequency response. It relates the number of clockwise encirclements of the point -1 in the complex plane to the number of poles of the closed-loop transfer function that lie in the right half-plane, providing a powerful tool for assessing system stability without requiring specific numerical values.
Overshoot: Overshoot refers to the phenomenon where a system exceeds its desired output level or target before settling down to the steady-state value. This behavior is crucial in dynamic systems, as it often indicates how well a system responds to changes and how quickly it stabilizes after a disturbance.
Phase margin: Phase margin is a measure of the stability of a control system, specifically indicating how close the system is to the verge of instability. It represents the difference in degrees between the phase angle of the open-loop transfer function and -180 degrees at the gain crossover frequency. A positive phase margin implies a stable system, while a negative value indicates instability, making it a crucial parameter in assessing system performance.
Pid controller: A PID controller is a control loop feedback mechanism widely used in industrial control systems to maintain a desired setpoint by adjusting the process inputs based on proportional, integral, and derivative terms. By tuning these three parameters, the controller can minimize error over time and achieve stable system behavior, making it essential in feedback control, steady-state response analysis, performance metrics, and closed-loop system dynamics.
Root Locus: Root locus is a graphical method used in control systems to analyze the behavior of the roots of a system's characteristic equation as system parameters, typically gain, are varied. This technique helps to visualize how the poles of a transfer function move in the complex plane, aiding in stability analysis and controller design.
Routh-Hurwitz Criterion: The Routh-Hurwitz Criterion is a mathematical test used to determine the stability of a linear time-invariant system by analyzing the characteristic polynomial's coefficients. It establishes conditions under which all roots of the polynomial lie in the left half of the complex plane, ensuring that the system is stable. This criterion is closely related to characteristic equations, transfer functions, and various forms of system analysis.
Settling Time: Settling time is the time taken for a dynamic system's response to reach and stay within a specified tolerance band around the desired final value after a disturbance or input change. This concept is crucial in understanding how quickly a system can stabilize after experiencing a change, which relates to the overall efficiency and performance of control systems and their responses to inputs.
Stability Analysis: Stability analysis is the process of determining whether a dynamic system will return to equilibrium after a disturbance. It involves assessing how system parameters affect system behavior over time, particularly in response to changes or inputs. This concept is essential for designing systems that behave predictably and remain functional under various conditions, connecting deeply with modeling, nonlinear dynamics, feedback systems, and discrete-time analysis.
Steady-State Error: Steady-state error is the difference between the desired output of a system and the actual output as time approaches infinity, indicating how accurately a control system can achieve its target value. This concept is crucial in understanding system performance, particularly how well systems maintain their desired outputs despite disturbances or changes in input.
Transient Response: Transient response refers to the behavior of a dynamic system as it transitions from an initial state to a final steady state after a change in input or initial conditions. This response is characterized by a temporary period where the system reacts to external stimuli, and understanding this behavior is crucial in analyzing the overall performance and stability of systems.