Welcome to the next phase of our course on Principles of Automatic Control. In this notebook, we will delve into the Root Locus Design.
Let’s begin by revisiting some of the concepts we have previously discussed and then move on to practical design examples.
A compensator is a type of controller, like PI (Proportional-Integral), PD (Proportional-Derivative), or PID (Proportional-Integral-Derivative), designed to enhance a system’s performance.
Performance in control systems is generally categorized into two types:
Transient performance is crucial because it determines how quickly and accurately the system reaches its desired state after a disturbance. Steady-state performance, on the other hand, is important for the long-term stability and accuracy of the system.
Let’s consider a typical control system structure:
|
In this setup, the compensator (\(D(s)\)) is placed in cascade with the plant. This arrangement can also be modified with minor feedback loops, which can be represented in a similar structural format for design purposes.
It’s vital to understand that the choice of compensator affects both the transient and steady-state behaviors of the system. A well-chosen compensator can significantly enhance the system’s response to changes and maintain stability over time.
Definition and Transfer Function
A phase lead compensator is used to improve transient performance by shifting the root locus to the left, which enhances system stability. The transfer function of a phase lead compensator is given by:
\[ D(s) = K_c \frac{(s + z_c)}{(s + p_c)} \]
Where: - $ K_c $ is the gain. - $ z_c $ is the zero of the compensator. - $ p_c $ is the pole of the compensator.
|
Figure: plot illustrating the pole-zero diagram of a phase lead compensator. The zero is at $ -z_c $ and the pole at $ -p_c $, with the pole farther from the origin than the zero.
The zero in the phase lead compensator (at $ s = -z_c $) introduces a derivative action, pulling the root locus plot towards the left half of the plane, thus enhancing stability and transient performance.
An additional pole is included for noise attenuation purposes, especially for high-frequency noise, represented by the pole at $ s = -p_c $.
Placement of the lead compensator is critical in cases where transient performance needs improvement.
The PD controller is a special case of a phase lead compensator that does not include the pole.
Pop-up Question: Why is a phase lead compensator used to improve transient performance? Answer: Because it shifts the root locus to the left, enhancing the system’s stability and transient response.
Let’s apply these concepts to a practical example: the attitude control of a satellite. The system can be modeled as $ $.
System Model:
\[ G(s) = \frac{1}{s^2} \]
Input: Torque $ T(t) $
Output: Attitude angle $ (t) $
System Type: Type-2 (implies good steady-state performance: zero error for step and ramp inputs, and finite error to acceleration inputs)
Problem Statement: Addressing instability due to a double integrator at the origin.
|
We start considering the simple situation where
\[ D(s) = K \]
This corresponds to the situation where \(D(s)=K\) is the constant of an actuator tht converts the elctrical signals into a torque \(T(t)\) for the thrusters.
Open-Loop Transfer Function:
\[ G(s) = \frac{K}{s^2} \]
Closed-Loop Behavior:
We can analyse it using the root locus as \(K\) varies.
|
Remember, the root locus plot is a graphical representation of how the closed-loop poles of a control system vary with changes in a system parameter, typically the gain.
For the system $ $, the root locus plot indicates an oscillatory nature. This is because for all values of ‘s’, the roots lie on the imaginary axis of the s-plane, which is characteristic of undamped oscillations.
We can plot the root locus using Python in this way:
import numpy as np
import matplotlib.pyplot as plt
import control as ctl
numerator = [1]
denominator = [1, 0, 0]
G = ctl.TransferFunction(numerator, denominator)
# Plot the root locus
plt.figure(figsize=(10, 6))
ctl.root_locus(G, plot=True);
plt.title(f'Root locus of {G}');
Here is a Python script that simulates the closed-loop system:
import numpy as np
import matplotlib.pyplot as plt
import control as ctl
# System parameters
K = 1 # You can modify this gain value as needed
# Transfer function of the system
numerator = [K]
denominator = [1, 0, 0] # s^2 term has a coefficient of 1, s term is 0, constant term is 0
system = ctl.tf(numerator, denominator)
# Closed loop transfer function with a unity feedback
# For a unity feedback system, the feedback transfer function is simply 1
closed_loop_system = ctl.feedback(system, 1)
# Time parameters for simulation
t = np.linspace(0, 100, 1000) # Simulation from 0 to 10 seconds, with 1000 points
# Step response
t, y = ctl.step_response(closed_loop_system, t)
# Plotting
plt.plot(t, y)
plt.title('Closed-Loop Step Response of the System K/s^2')
plt.xlabel('Time (seconds)')
plt.ylabel('Response')
plt.grid(True)
plt.show()
In our current scenario, the primary concern is not steady-state accuracy but transient performance. Therefore, a phase lead compensator is the preferred choice.
The phase lead compensator is represented by the transfer function:
\[ D(s) = K_c \frac{(s + z_c)}{(s + p_c)} \]
where $ K_c $, $ z_c $, and $ p_c $ are the parameters we can determine to achieve our requirements.
Note: The concepts behind the design are more important than the specific design. More than one design can be suitable.
There are three parameters under our control. How we implement these parameters depend on the hardware.
We call $ $ as uncompensated system that in this case includes the design parameter \(K\).
Consider an attitude control system for a satellite: - Uncompensated System: \[ \frac{K}{s^2} \] - Compensator: \[ D(s) = \frac{s + z_c}{s + p_c} \]
Suppose the design requirements are a damping ratio $ = 0.707 $ and a settling time $ t_s = 2$ seconds. Our goal is to translate these requirements into desired closed-loop pole locations.
Remember that transient requirements might be given to you in terms of $_n $ (natural frequency), \(M_p\) (Overshoot Peak), etc.
Remember also that we can often go from one specific requirement to another. For example, we can calculate:
\[ \zeta\omega_n = \frac{4}{t_s} = 2 \]
The closed-loop pole locations can be determined from these values. Below the steps to do it.
To calculate the position of the desired closed-loop poles for a control system with a specified damping ratio $ $ and settling time $ t_s $, we use the standard formulas related to the second-order system’s response characteristics.
The desired closed-loop poles are generally complex conjugates for underdamped systems, and their position in the s-plane is determined by the damping ratio $ $ and the natural frequency $ _n $.
Given: - Damping ratio $ = 0.707 $ - Settling time $ t_s = 2 $ seconds
**Calculate the Natural Frequency ($ _n $)**: The settling time for a second-order system is approximated by $ t_s $. We rearrange this to solve for $ _n $:
\[ \omega_n = \frac{4}{\zeta t_s} \]
Substituting the given values:
\[ \omega_n = \frac{4}{0.7 \times 2} = \frac{4}{1.4} \approx 2.857 \, \text{rad/s} \]
Determine the Real and Imaginary Parts of the Poles: The general form of the closed-loop poles for a second-order system is:
\[ s = -\zeta \omega_n \pm j\omega_n\sqrt{1 - \zeta^2} \]
The real part (sigma) is given by $ -_n $ and the imaginary part (omega) by $ _n $.
Calculating these:
Closed-loop Pole Locations: Therefore, the closed-loop poles are located at approximately:
\[ s = -2 \pm j2 \]
These calculations give you the desired locations for the closed-loop poles in the s-plane to achieve the specified damping ratio and settling time in your control system design.
We can plot this on the s-plane:
|
Figure: Plot of the root locus of the uncompensated system. It is possible to identify where the closed-loop poles should be for the desired performance.
We need to modify the behaviour of the root locus to pass through the desired closed-loop poles so that the requirements on the transient accuracy are satisfied.
Start by placing the zero ($ z_c \() and the pole (\) p_c $) of the compensator tentatively on the s-plane. This initial placement doesn’t have to be exact; it’s a starting point for fine-tuning.
To ensure the system’s closed-loop response meets our requirements, the root locus must pass through the desired closed-loop pole locations. This requires satisfying the angle criterion at these points.
We need to find the angles $ {z_c} $ and $ {p_c} $ such that they fulfill the equation:
\[ \theta_{z_c} - \theta_{p_c} - 2\theta_1 = -180^\circ \]
Here, $ {z_c} $ and $ {p_c} $ are the angles from the compensator’s zero and pole to the desired closed-loop pole location, respectively, and $ 2_1 $ is the contribution from the plant’s poles.
|
|
Net Angle Contribution: The compensator must provide a net angle $ {z_c} - {p_c} $, where $ {z_c} $ and $ {p_c} $ are the angles from the compensator’s zero and pole to the desired closed-loop poles.
Formula: Given $ _1 = 135^$, we calculate:
\[ \theta_{z_c} - \theta_{p_c} = -180^\circ + 2 \cdot 135^\circ = -180^\circ + 270^\circ = 90^\circ \]
|
Preferred Location: Ideally, place the zero to the left of the desired complex conjugate closed-loop poles. This ensures that the third pole lies to the left of the desired poles, satisfying the dominance condition. This is because the root locus branch will end at that zero.
Exceptions: Sometimes, this ideal placement isn’t possible. In such cases, as with our 90° net angle contribution, we may need to place the zero to the right.
To solve for the gain $ K $ in our control system example, where the zero of the compensator $ z_c $ is at -1 and the pole $ p_c $ is at -6, we follow the steps laid out earlier. We need to calculate $ K $ such that the magnitude condition is satisfied at the desired closed-loop poles $ -2 j2 $.
Given:
The open-loop transfer function $ G(s) $ is:
\[ D(s)G(s) = \frac{K \cdot (s + z_c)}{s^2 \cdot (s + p_c)} \]
Substituting $ z_c = -1 $ and $ p_c = -6 $ into the transfer function, we get:
\[ D(s)G(s) = \frac{K \cdot (s + 1)}{s^2 \cdot (s + 6)} \]
Now, we apply the magnitude condition at one of the desired closed-loop poles, say $ s = -2 + j2 $:
\[ |G(-2 + j2)| = 1 \]
\[ \frac{K \cdot \left|(-2 + j2) + 1\right|}{\left|(-2 + j2)^2\right| \cdot \left|(-2 + j2) + 6\right|} = 1 \]
Let’s simplify this equation:
\[ \frac{K \cdot \left|-1 + j2\right|}{\left| (-2 + j2)^2\right| \cdot \left|4 + j2\right|} = 1 \]
First, we need to simplify the complex numbers in the equation:
Let’s do these calculations:
Then, calculate the magnitudes:
Finally, solve for $ K $ using:
\[ \frac{K \cdot \text{Magnitude of Numerator}}{\text{Magnitude of Denominator}} = \frac{8(4.47)}{2.23} = 16\]
In the design of a compensator, particularly a phase lead compensator, a crucial aspect to consider is the relationship between the compensator’s pole and zero. This relationship is not just a matter of meeting geometric conditions for angle requirements but also plays an important role in noise attenuation, especially for high-frequency signals.
To calculate the location of the third pole in a control system where a phase lead compensator has been added, and the gain $ K $ has been determined, we can follow these steps:
Open-Loop Transfer Function: Recall the open-loop transfer function with the compensator: \[ D(s)G(s) = \frac{K \cdot (s + z_c)}{s^2 \cdot (s + p_c)} \] where $ z_c $ is the zero and $ p_c $ is the pole of the compensator.
Substitute Known Values: Substitute the known values for $ K $, $ z_c $, and $ p_c $ into the open-loop transfer function.
Root Locus Method: To find the location of the third pole, you can use the root locus method. This involves plotting the root locus of the open-loop transfer function and identifying where the third pole lies based on the value of $ K $.
Analytical Approach: Alternatively, for an analytical approach, you can set up the characteristic equation of the closed-loop system, which is: \[ 1 + G(s)H(s) = 0 \] Solving this equation will give you the poles of the closed-loop system.
Solving the Characteristic Equation: The characteristic equation will be a cubic equation in $ s $ (since the original system is second-order and the compensator adds one order). Solving this cubic equation will give you three solutions, corresponding to the three poles of the closed-loop system.
Identifying the Third Pole: Two of these poles will be the desired closed-loop poles (e.g., $ -2 j2 $ in our example). The third solution will be the additional pole introduced by the compensator.
Computational Tools: Solving the cubic equation analytically can be complex. It’s often more practical to use computational tools like MATLAB or Python, which can efficiently compute the roots of polynomial equations.
Dominance of Poles: Once you find the third pole, you should check its dominance in the system’s response. If it’s far left in the s-plane, its effect on the system’s transient response might be negligible. If it’s closer to the imaginary axis, it might significantly influence the system’s behavior.
import numpy as np
# Coefficients of the cubic equation
coefficients = [1, 6, 16, 16]
# Finding the roots
roots = np.roots(coefficients)
# Display the roots
print("The poles of the system are:", roots)The poles of the system are: [-2.+2.j -2.-2.j -2.+0.j]import numpy as np
import matplotlib.pyplot as plt
import control as ctl
from ipywidgets import interact, FloatSlider
def calculate_settling_time(T, yout, tol=0.02):
# Settling time is the time at which the response remains within a certain tolerance
settled_value = yout[-1]
lower_bound = settled_value * (1 - tol)
upper_bound = settled_value * (1 + tol)
within_tol = np.where((yout >= lower_bound) & (yout <= upper_bound))[0]
if within_tol.size == 0:
return np.nan # Return NaN if the system never settles
return T[within_tol[0]]
def calculate_steady_state_error(system_type, G, K, time_span):
G_closed_loop = ctl.feedback(K * G)
if system_type == 'step':
# Steady-state error for step input (Type 0 system)
T, yout = ctl.step_response(G_closed_loop, T=np.linspace(0, time_span[-1], 500))
steady_state_val = yout[-1]
return 1 - steady_state_val
elif system_type == 'ramp':
# Steady-state error for ramp input (Type 1 system)
Kv = ctl.dcgain(G_closed_loop * ctl.TransferFunction([1, 0], [1]))
return 1 / Kv if Kv != 0 else np.inf
else:
return np.nan
# def plot_step_response(G, K, time_span):
# G_closed_loop = ctl.feedback(K * G)
# T, yout = ctl.step_response(G_closed_loop, T=np.linspace(0, time_span[-1], 500))
# plt.plot(T, yout)
# settling_time = calculate_settling_time(T, yout)
# steady_state_error = calculate_steady_state_error('step', G, K, time_span)
# plt.title(f'Step Response for Gain K={K}\nSettling Time: {settling_time:.2f}, Steady-State Error: {steady_state_error:.2f}')
# plt.xlabel('Time')
# plt.ylabel('Amplitude')
# plt.grid(True)
# def plot_ramp_response(G, K, time_span):
# G_closed_loop = ctl.feedback(K * G)
# T, yout = ctl.forced_response(G_closed_loop, T=time_span, U=time_span)
# plt.plot(T, yout)
# steady_state_error = calculate_steady_state_error('ramp', G, K, time_span)
# plt.title(f'Ramp Response for Gain K={K}\nSteady-State Error: {steady_state_error:.2f}')
# plt.xlabel('Time')
# plt.ylabel('Amplitude')
# plt.grid(True)
def plot_root_locus_with_gain(K, numerator, denominator):
# Define the transfer function G(s)
G = ctl.TransferFunction(numerator, denominator)
# Calculate the closed-loop transfer function for the given gain
G_closed_loop = ctl.feedback(K * G)
# Find the poles for the specific gain
poles = ctl.pole(G_closed_loop)
# Plot the root locus
plt.figure(figsize=(10, 6))
ctl.root_locus(G, plot=True)
# Plot the poles for the specific gain
plt.plot(np.real(poles), np.imag(poles), 'ro', markersize=10, label=f'Poles for K={K}')
# Enhance plot
plt.xlabel('Real Axis')
plt.ylabel('Imaginary Axis')
plt.title(f'Root Locus of G(s) with Poles for Gain K={K}')
plt.grid(True)
plt.legend()
plt.show()
def plot_all(K):
# Define the transfer function D(s)G(s)
numerator = [1, 1 * zc] # 16s+16
denominator = [1, 6, 0, 0] # s^3 + 6s^2
G = ctl.TransferFunction(numerator, denominator)
# Time span for the responses
time_span = np.linspace(0, 10, 1000)
# Plot Root Locus
plot_root_locus_with_gain(K, numerator, denominator)
# # Plot Step Response
# plt.figure(figsize=(10, 4))
# plot_step_response(G, K, time_span)
# plt.show()
# # Plot Ramp Response
# plt.figure(figsize=(10, 4))
# plot_ramp_response(G, K, time_span)
# plt.show()
zc = 1 # Zero of the compensator
interact(plot_all,
K=FloatSlider(value=16, min=0, max=50, step=0.5, description='Gain K:'))<function __main__.plot_all(K)>import numpy as np
import matplotlib.pyplot as plt
import control as ctl
# Define system parameters
K = 1 # This is one to have K to vary in the plot. We need to click on the value of gain 16.
zc = 1 # Zero of the compensator
pc = 6 # Pole of the compensator
# Define the transfer function
numerator = [K, K * zc] # 16s+16
denominator = [1, 6, 0, 0] # s^3 + 6s^2
system = ctl.tf(numerator, denominator)
# Plot the root locus
ctl.rlocus(system)
plt.show()
# Use the plot to find the third pole at the specific gain K
Consider designing a phase lead compensator for a system where transient performance is as crucial as noise reduction. The choice of pole and zero locations would need to:
Interactive Exercise: Students can engage in an exercise using a software tool like MATLAB or Python to experiment with different pole-zero ratios. Observing how these adjustments affect both the transient response and noise attenuation will provide practical insights into the trade-offs involved in compensator design.
Given that we have a type-2 system this is not a concern, certainly not for steps and ramps. If acceleration error is to be calculated this can be done from:
\[ D(s)G(s) = \frac{K \cdot (s + z_c)}{s^2 \cdot (s + p_c)} = \frac{16 \cdot (s + 1)}{s^2 \cdot (s + 6)} \]
and hence, the acceleration constant is:
\[ K_a = \frac{16}{6} \]
and the corresponding acceleration error is:
\[ e_{ss} = \frac{6}{16} radians \]
This is a finite value and most likely acceptable. Following acceleration is difficult and typically we are satisfied with step and ramp inputs.
In this part, we’ll delve into the concept of phase lag compensation. Unlike phase lead compensators that are designed to improve transient performance, phase lag compensators are primarily used to enhance steady-state accuracy.
The transfer function for a phase lag compensator is generally given by:
\[ D(s) = \frac{s + z_c}{s + p_c} \]
Where: - $ z_c $ is the zero of the compensator. - $ p_c $ is the pole of the compensator.
In a phase lag compensator, the pole is usually placed close to the origin but not necessarily at it. Placing a pole at the origin makes it a Proportional-Integral (PI) controller.
The zero ($ z_c \() is positioned close to the pole (\) p_c $), which is essential for maintaining system stability. Otherwise, the presence the pole at the origin might destabilise the system.
|
Let’s consider a type-1 system represented by:
\[ G(s) = \frac{K}{s(s + 2)} \]
This model could represent a motor servo system for position tracking.
In our control system example, let’s begin by assessing the uncompensated system’s performance. A key parameter in this assessment is the velocity error constant, denoted as $ K_v $. For our system, $ K_v $ is calculated as follows:
\[ K_v = \frac{K}{2} \]
For a damping ratio ($ $) of 0.45, the closed-loop poles are located at $ -1 2j $.
This calculation is based on standard second-order system dynamics where the pole locations are determined by the damping ratio and the natural frequency.
The location of these poles in the s-plane directly affects the transient behavior of the system, including aspects like overshoot and settling time.
|
Figure: root locus plot that shows how varying $ K $ affects the pole locations.
When $ K = 5 $, the velocity error constant is:
\[ K_v = \frac{K}{2} = \frac{5}{2} = 2.5 \]
However, this value of $ K_v $ is less than the required 20 for adequate steady-state performance.
To accomplish our desired outcome, we will position both the zero and the pole of the compensator near the origin. The goal here is to ensure that their combined contribution to the phase angle of the system is minimal, ideally within a range of 1 to 5 degrees. By doing so, we can effectively ensure that these elements have a negligible impact on the system’s original root locus plot. This strategic placement is crucial for maintaining the original characteristics of the system while implementing the phase lag compensator.
\[ D(s) G(s) = \frac{K (s+z_c)}{s(s + 2)(s+p_c)} \]
and
\[ K_v = \frac{K z_c}{2p_c} = 20 \]
this means that we can use the ratio \(\frac{z_c}{p_c}\) to improve our velocity constant.
More specifically, for \(K=5\) (note that for the same reason why the zero and pole do not disturb the transient, they also do not change much the value of the root locus gain, the boost that we need is \(\frac{z_c}{p_c}=8\).
This helps us define the relationship between the zero and the pole of the compensator.
The root locus is (this is not too scale to highlight the relationships):
|
It is possible to verify that the third pole must be real, and hence very close to the zero. This means that the dominance condition is satisfied (even though it is close to the imaginary axis). If the third pole is not close enough we can make an adjustment to the zero/pole locations to move it closer to the zero. This is typically done using software.
Finally through simulation we can verify the full performance (transient and steady-state) of the system.
In concluding our discussion on control system compensators, let’s consider how to choose between different types of compensators based on the performance of the uncompensated system. Here, by ‘uncompensated,’ we mean a system where only gain adjustments are permissible.
Conclusion: The design and selection of compensators depend on both the transient and steady-state performance requirements of your system. The decision-making process involves a careful evaluation of the system’s current performance and the objectives you aim to achieve with compensation.
Note for Instructor: Encourage students to analyze different system scenarios and decide on the type of compensator needed. Practical exercises involving Op-Amp circuit design for lead-lag and lag-lead compensators can be very beneficial for understanding these concepts.