3.5 Time-domain design specifications

Time-domain design specifications are crucial for evaluating control system performance. They help engineers assess transient and steady-state behavior, guiding the selection of controller parameters to achieve desired system responses.

Key specifications include rise time, settling time, overshoot, and steady-state error. Understanding these metrics allows designers to fine-tune system behavior, balancing speed, accuracy, and stability in control applications.

Time response specifications

• Time response specifications are a set of performance criteria used to evaluate the transient and steady-state behavior of a control system in the time domain
• These specifications help designers determine the required controller parameters to achieve the desired system response
• Common time response specifications include rise time, settling time, overshoot, and steady-state error

Transient response of second-order systems

• Transient response refers to the system's behavior during the initial period after a change in input or disturbance
• Second-order systems are widely used in control theory due to their simplicity and ability to model many physical systems
• Understanding the transient response of second-order systems is crucial for designing controllers that meet the desired performance specifications

Standard second-order transfer function

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• The standard second-order transfer function is given by: $G(s) = \frac{\omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2}$
• $\omega_n$ is the natural frequency, which determines the speed of the system's response
• $\zeta$ is the damping ratio, which characterizes the system's tendency to oscillate or settle

Damping ratio and natural frequency

• The damping ratio ($\zeta$) is a dimensionless quantity that describes the system's ability to dissipate energy and reduce oscillations
• The natural frequency ($\omega_n$) is the frequency at which the system would oscillate if no damping were present
• These two parameters play a crucial role in determining the system's transient response characteristics

Effect of damping ratio on system response

• The damping ratio affects the system's response in the following ways:
  • Underdamped ($0 < \zeta < 1$): The system exhibits oscillatory behavior before settling to the final value
  • Critically damped ($\zeta = 1$): The system reaches the final value in the shortest time without oscillations
  • Overdamped ($\zeta > 1$): The system reaches the final value without oscillations, but more slowly than the critically damped case

Overshoot vs damping ratio

• Overshoot is the percentage by which the system's response exceeds the final value during the transient period
• The overshoot decreases as the damping ratio increases
• For an underdamped system, the overshoot can be calculated using: $\text{Overshoot} = e^{-\frac{\zeta\pi}{\sqrt{1-\zeta^2}}} \times 100\%$

Settling time vs damping ratio

• Settling time is the time required for the system's response to settle within a specified tolerance band (usually ±2% or ±5%) around the final value
• The settling time increases as the damping ratio decreases
• For an underdamped system, the settling time can be approximated using: $t_s \approx \frac{4}{\zeta\omega_n}$

Rise time vs damping ratio

• Rise time is the time required for the system's response to rise from 10% to 90% of its final value
• The rise time increases as the damping ratio increases
• For an underdamped system, the rise time can be approximated using: $t_r \approx \frac{1.8}{\omega_n}$

Peak time vs damping ratio

• Peak time is the time at which the system's response reaches its maximum value (peak)
• The peak time increases as the damping ratio increases
• For an underdamped system, the peak time can be calculated using: $t_p = \frac{\pi}{\omega_n\sqrt{1-\zeta^2}}$

Steady-state error

• Steady-state error is the difference between the desired output and the actual output of a system in the steady-state (as time approaches infinity)
• It is a measure of the system's ability to track a reference input or reject disturbances
• The steady-state error depends on the system type and the input signal

Position error constant

• The position error constant ($K_p$) is used to determine the steady-state error for a step input
• For a unity feedback system with a forward transfer function $G(s)$, $K_p$ is calculated as: $K_p = \lim_{s \to 0} G(s)$
• The steady-state error for a step input is given by: $e_{ss} = \frac{1}{1 + K_p}$

Velocity error constant

• The velocity error constant ($K_v$) is used to determine the steady-state error for a ramp input
• For a unity feedback system with a forward transfer function $G(s)$, $K_v$ is calculated as: $K_v = \lim_{s \to 0} sG(s)$
• The steady-state error for a ramp input is given by: $e_{ss} = \frac{1}{K_v}$

Acceleration error constant

• The acceleration error constant ($K_a$) is used to determine the steady-state error for a parabolic input
• For a unity feedback system with a forward transfer function $G(s)$, $K_a$ is calculated as: $K_a = \lim_{s \to 0} s^2G(s)$
• The steady-state error for a parabolic input is given by: $e_{ss} = \frac{1}{K_a}$

System type and steady-state error

• The system type is determined by the number of pure integrators (poles at the origin) in the forward transfer function $G(s)$
• The system type determines which error constant is non-zero and, consequently, the system's ability to track different types of inputs with zero steady-state error
  • Type 0 systems have a non-zero $K_p$ and can track step inputs with zero steady-state error
  • Type 1 systems have a non-zero $K_v$ and can track ramp inputs with zero steady-state error
  • Type 2 systems have a non-zero $K_a$ and can track parabolic inputs with zero steady-state error

Dominant poles and time-domain specifications

• In systems with multiple poles, the dominant poles are the poles that have the most significant impact on the system's transient response
• By focusing on the dominant poles, designers can simplify the analysis and design of control systems

Dominant vs non-dominant poles

• Dominant poles are the poles closest to the imaginary axis in the complex plane
• Non-dominant poles are the poles further away from the imaginary axis
• The effect of non-dominant poles on the system's response decays much faster than that of dominant poles

Second-order approximation

• When a system has a pair of complex conjugate dominant poles and other non-dominant poles, the system's response can be approximated by considering only the dominant poles
• This approximation is called the second-order approximation because the system is reduced to a second-order transfer function
• The second-order approximation simplifies the analysis and design process while providing a good estimate of the system's transient response

Dominant poles and transient response

• The location of the dominant poles in the complex plane determines the system's transient response characteristics
• The real part of the dominant poles affects the decay rate of the response (settling time)
• The imaginary part of the dominant poles affects the oscillation frequency of the response (peak time)
• By placing the dominant poles at the desired locations, designers can achieve the desired transient response specifications

Time-domain design using root locus

• The root locus is a graphical method used to analyze how the poles of a closed-loop system change as a parameter (usually the controller gain) varies
• It is a powerful tool for designing controllers to meet time-domain specifications

Root locus review

• The root locus plots the locations of the closed-loop poles in the complex plane as a function of the controller gain
• The root locus starts at the open-loop poles and ends at the open-loop zeros and infinity
• The root locus provides information about the stability, damping, and transient response of the closed-loop system

Selecting closed-loop pole locations

• To meet the desired time-domain specifications, designers select the appropriate closed-loop pole locations on the root locus
• The desired pole locations are usually specified in terms of the damping ratio and natural frequency
• The selected pole locations should provide a good balance between the transient response and stability

Designing controllers for time-domain specs

• Once the desired closed-loop pole locations are selected, designers can determine the required controller gain to place the poles at those locations
• The controller gain is found by solving the characteristic equation at the desired pole locations
• Additional controller elements (lead, lag, or lead-lag compensators) may be needed to shape the root locus and achieve the desired pole locations

Time-domain design using frequency response

• Frequency response methods, such as Bode plots and Nyquist plots, can also be used to design controllers for time-domain specifications
• These methods provide insight into the system's stability and performance in the frequency domain

Frequency response review

• The frequency response of a system describes its behavior when subjected to sinusoidal inputs of varying frequencies
• Bode plots display the magnitude and phase of the system's frequency response, while Nyquist plots display the real and imaginary parts
• Frequency response methods help designers analyze the system's stability, bandwidth, and robustness

Bandwidth and rise time

• Bandwidth is the range of frequencies over which the system's gain is within 3 dB of its maximum value
• The bandwidth is inversely related to the rise time of the system's step response
• A higher bandwidth generally results in a faster rise time and a more responsive system

Resonant peak and overshoot

• The resonant peak is the maximum value of the system's frequency response magnitude
• A higher resonant peak indicates a more oscillatory response and a larger overshoot in the time domain
• The resonant peak can be reduced by increasing the system's damping or by using notch filters

Phase margin and stability

• Phase margin is the difference between -180° and the system's phase at the gain crossover frequency (where the magnitude crosses 0 dB)
• A positive phase margin indicates a stable system, while a negative phase margin indicates an unstable system
• A larger phase margin provides more stability robustness and reduces the overshoot in the time domain

Gain margin and stability

• Gain margin is the reciprocal of the system's magnitude at the phase crossover frequency (where the phase crosses -180°)
• A gain margin greater than 1 (or 0 dB) indicates a stable system, while a gain margin less than 1 (or 0 dB) indicates an unstable system
• A larger gain margin provides more stability robustness and allows for more uncertainty in the system's gain

Designing controllers for time-domain specs

• To design controllers using frequency response methods, designers shape the system's frequency response to achieve the desired time-domain specifications
• Lead compensators can be used to increase the phase margin and improve stability
• Lag compensators can be used to increase the low-frequency gain and reduce steady-state error
• Notch filters can be used to reduce the resonant peak and limit overshoot

Time-domain design using state-space methods

• State-space methods provide a powerful framework for designing controllers to meet time-domain specifications
• These methods rely on the state-space representation of the system, which describes the system's dynamics using a set of first-order differential equations

State-space representation review

• The state-space representation consists of two equations:
  • State equation: $\dot{x} = Ax + Bu$
  • Output equation: $y = Cx + Du$
• $x$ is the state vector, $u$ is the input vector, $y$ is the output vector, and $A$, $B$, $C$, and $D$ are the system matrices
• The state-space representation provides a compact and general description of the system's dynamics

Controllability and observability

• Controllability is the ability to steer the system's states from any initial condition to any desired final condition in a finite time using the available inputs
• Observability is the ability to determine the system's initial state based on the measured outputs over a finite time
• Controllability and observability are essential properties for the design of state feedback controllers and observers

Pole placement using state feedback

• State feedback is a control technique that uses the system's state variables to generate the control input
• The state feedback control law is given by: $u = -Kx$, where $K$ is the state feedback gain matrix
• Pole placement involves selecting the desired closed-loop pole locations and determining the required state feedback gain matrix $K$ to achieve those pole locations

Observer design for state estimation

• In practice, not all state variables may be directly measurable
• An observer is a dynamical system that estimates the system's state variables based on the measured inputs and outputs
• The observer's poles are placed at desired locations to ensure fast and accurate state estimation
• The estimated states can then be used in the state feedback control law

Designing controllers for time-domain specs

• To design controllers using state-space methods, designers follow these steps:
  1. Determine the desired closed-loop pole locations based on the time-domain specifications
  2. Check the system's controllability and observability
  3. Design a state feedback controller using pole placement to achieve the desired pole locations
  4. Design an observer to estimate the system's states if necessary
  5. Combine the state feedback controller and the observer to form the overall control system
• State-space methods provide a systematic approach to controller design and can handle systems with multiple inputs and outputs