Stability of Linear Control Systems

Contents

**Recall the terms Stability and BIBO Stability** 2

**Recall Relative Stability and Absolute Stability** 3

**Differentiate between Relative Stability and Absolute Stability** 4

**Recall different methods of finding stability** 7

**Describe the Routh-Hurwitz Criteria** 8

**Find the Stability of a Control System using Routh-Hurwitz Criteria** 9

**Define the terms related to Root-Locus Plot** 12

**Recall the procedure for plotting a Root-Locus** 14

**Construct a Root-Locus of a given Open-Loop System** 16

**Find the Gain of a System using Root- Locus** 18

**Show the effect of addition of Poles and Zeros to a Root-Locus** 20

**Recall the terms Stability and BIBO Stability**

**Stability:**

Stability refers to the behavior of a system or process in response to a given input. A stable system or process is one that, when subjected to a disturbance, will eventually return to its equilibrium or steady-state condition. In other words, a stable system will not diverge or become unbounded over time. Stability is an essential property of systems in many engineering applications, including control systems, signal processing, and communications.

For example, consider a car driving on a straight road. The driver’s input is the steering angle, and the output is the car’s position on the road. A stable car would maintain its position on the road, despite small variations in the steering angle, wind conditions, or other disturbances. On the other hand, an unstable car would swerve uncontrollably, making it dangerous to drive.

**BIBO Stability:**

BIBO (Bounded-Input Bounded-Output) stability is a specific type of stability that applies to linear time-invariant (LTI) systems. A system is said to be BIBO stable if its output remains bounded for any bounded input signal. In other words, the system’s response to an input signal will not grow without bound, regardless of the input signal’s magnitude or frequency.

For example, consider a simple LTI system with an impulse response h(t). If the input signal x(t) is bounded, i.e., |x(t)| <= M for all t, then the output signal y(t) = x(t) * h(t) is also bounded, i.e., |y(t)| <= K for all t. Here, K is some constant that depends on the impulse response h(t) and the bound M on the input signal x(t). This means that the system is BIBO stable since its output is always bounded for any bounded input signal.

In summary, stability is a critical property of systems in various engineering applications, while BIBO stability is a specific type of stability that applies to LTI systems.

**Recall Relative Stability and Absolute Stability**

**Relative Stability:**

Relative stability refers to the tendency of a system or process to remain stable when subjected to small variations in its parameters or operating conditions. In other words, a system that is relatively stable will maintain its stability when subjected to minor changes in its inputs or parameters. However, the system’s stability may be affected by significant changes in its inputs or parameters.

For example, consider a control system that regulates the temperature of a room. The system’s stability is relative because it depends on various factors such as the setpoint temperature, the heat load, and the feedback control algorithm. If the system is well-tuned, it will maintain the room temperature within a narrow range, even when subjected to small changes in the heat load or feedback gain. However, if the heat load suddenly increases beyond a certain threshold, the system’s stability may be compromised, and the room temperature may become uncontrollable.

**Absolute Stability:**

Absolute stability refers to the robustness of a system or process against any variations in its parameters or inputs. A system that is absolutely stable will maintain its stability regardless of the magnitude or frequency of the input signal or the changes in its parameters.

For example, consider a communication system that transmits data over a noisy channel. The system’s stability is absolute because it must maintain the quality of the transmitted data even in the presence of high levels of noise or interference. The system achieves this by using error-correcting codes, modulation techniques, and other signal processing algorithms that ensure the reliable transmission of data.

In summary, relative stability and absolute stability are two types of stability that describe the behavior of systems and processes in response to variations in their inputs or parameters. Relative stability refers to the stability of a system under small variations, while absolute stability refers to the robustness of a system under any variations.

**Differentiate between Relative Stability and Absolute Stability**

Relative Stability and Absolute Stability are two important concepts that describe the behavior of a system or process in response to variations in its inputs or parameters. While they are related, there are significant differences between the two concepts.

Relative Stability:

Relative Stability refers to the tendency of a system or process to remain stable when subjected to small variations in its parameters or operating conditions. It describes how much a system’s stability is affected by changes in its inputs or parameters. A system that is relatively stable will maintain its stability when subjected to minor changes in its inputs or parameters. However, its stability may be affected by significant changes in its inputs or parameters.

For example, consider an aeroplane flying at a constant altitude. The aeroplane’s stability is relative because it depends on various factors such as its speed, altitude, and weight. If the aeroplane’s speed or altitude changes slightly, the autopilot system can adjust the flight path to maintain the desired altitude. However, if the aeroplane encounters severe turbulence or a sudden gust of wind, its stability may be compromised, and it may enter into an unstable flight regime.

Absolute Stability:

Absolute Stability refers to the robustness of a system or process against any variations in its inputs or parameters. It describes how much a system’s stability is affected by changes in its inputs or parameters, regardless of their magnitude or frequency. A system that is absolutely stable will maintain its stability regardless of the magnitude or frequency of the input signal or the changes in its parameters.

For example, consider a power grid that supplies electricity to a city. The power grid’s stability is absolute because it must maintain a steady voltage and frequency, regardless of the load demand or the variations in the power generation sources. The system achieves this by using various control and protection mechanisms, such as automatic voltage regulators, load shedding, and fault detection systems.

Here’s a tabular comparison between relative stability and absolute stability:

Relative Stability | Absolute Stability | |

Definition | The measure of how close a system is to instability or oscillation. | The measure of whether a system remains bounded or converges to a steady state. |

Focus | Stability with respect to changes in system parameters or operating conditions. | Stability regardless of changes in system parameters or operating conditions. |

Perturbations | It considers the response to small perturbations or disturbances. | It considers the response to large perturbations or disturbances. |

Stability Criterion | It is assessed using methods such as Nyquist criterion, Bode plot, gain and phase margins. | It is assessed using methods such as the Routh-Hurwitz criterion, Nyquist stability criterion, or the root locus plot. |

Oscillations and Damping | It characterizes the system’s tendency to oscillate and the degree of damping. | It characterizes the system’s tendency to grow or decay over time. |

Sensitivity | It can vary with changes in system parameters or operating conditions. | It remains invariant with changes in system parameters or operating conditions. |

Application | It is more applicable in control systems where stability margins and robustness are important. | It is more applicable in stability analysis of systems such as differential equations or signal processing. |

These are the main differences between relative stability and absolute stability. Relative stability focuses on the system’s behavior in response to small perturbations and changes in parameters, while absolute stability focuses on the system’s behavior regardless of such perturbations or changes.

**Recall different methods of finding stability**

There are several methods to determine the stability of a system, and some of the commonly used methods are:

**Lyapunov’s stability criterion:**This criterion is based on the concept of Lyapunov functions, which are mathematical functions that measure the distance between the system’s trajectory and its equilibrium point. If a Lyapunov function exists and satisfies certain conditions, the system is stable. Lyapunov’s stability criterion is widely used to analyze the stability of nonlinear systems.**Routh-Hurwitz stability criterion:**This criterion is used to determine the stability of a system by examining the roots of its characteristic equation. If all the roots have negative real parts, the system is stable. The Routh-Hurwitz criterion can be applied to linear systems with polynomial equations.**Nyquist stability criterion:**This criterion is based on the Nyquist plot, which is a graphical representation of the system’s frequency response. The Nyquist plot is used to determine the number of encirclements of the point (-1,0) in the complex plane. If the number of encirclements is equal to the number of unstable poles, the system is unstable. The Nyquist criterion is commonly used to analyze feedback control systems.**Bode stability criterion:**This criterion is based on the Bode plot, which is a graphical representation of the system’s magnitude and phase response. The Bode plot is used to determine the gain and phase margins of the system, which indicate how much the system’s gain and phase can be increased before the system becomes unstable. The Bode criterion is commonly used to design and analyze feedback control systems.**Direct substitution method:**This method is used to determine the stability of a system by substituting the Laplace transform of the system’s output into its input equation. If the resulting expression has a bounded solution, the system is stable. The direct substitution method can be applied to linear time-invariant systems.

In summary, there are several methods to determine the stability of a system, including Lyapunov’s stability criterion, Routh-Hurwitz stability criterion, Nyquist stability criterion, Bode stability criterion, and direct substitution method. Each method has its own advantages and limitations, and the choice of method depends on the nature of the system and the available tools for analysis.

**Describe the Routh-Hurwitz Criteria**

The Routh-Hurwitz criteria is a classical method for determining the stability of a linear system with a polynomial equation. The method is based on constructing a table called the Routh array, which allows for the evaluation of the stability of the system by examining the signs of the coefficients of the characteristic polynomial.

The Routh array is constructed as follows:

- Write the coefficients of the characteristic polynomial in the first row.
- Create the second row by dividing the coefficients of the first row into pairs and writing the resulting values in order. If any coefficients are zero, replace them with small values.
- Continue to create subsequent rows by dividing the coefficients of the previous row into pairs and writing the resulting values in order. If any coefficients are zero, replace them with small values.
- The number of sign changes in the first column of the Routh array indicates the number of roots of the characteristic polynomial with positive real parts. If the number of sign changes is zero, all the roots have negative real parts, and the system is stable. If the number of sign changes is greater than zero, the system is unstable.

For example, consider the following second-order polynomial equation:

s^{2} + 4s + 3 = 0

The corresponding Routh array can be constructed as follows:

| 1 | 4 | 3 |

| 1 | 3 |

| 4/3 | 0 |

There are no sign changes in the first column of the Routh array, indicating that both roots of the characteristic polynomial have negative real parts, and the system is stable.

If any row of the Routh array contains all zero coefficients, it indicates that the system has poles on the imaginary axis, and additional analysis is required to determine its stability.

In summary, the Routh-Hurwitz criteria is a useful method for determining the stability of a linear system with a polynomial equation. By constructing a Routh array, the method allows for the evaluation of the stability of the system by examining the signs of the coefficients of the characteristic polynomial.

**Find the Stability of a Control System using Routh-Hurwitz Criteria**

The Routh-Hurwitz criteria is a method to determine the stability of a linear system with a polynomial equation. This method is widely used in control systems engineering to find the stability of a system, as the characteristic equation of a control system can be represented as a polynomial.

The steps to find the stability of a control system using the Routh-Hurwitz criteria are as follows:

Step 1: Write the characteristic equation of the control system in terms of its coefficients.

Step 2: Create the Routh array by following the steps mentioned in the Routh-Hurwitz criteria.

Step 3: Analyze the Routh array to determine the stability of the system. If all the elements in the first column of the Routh array have the same sign, then the system is stable. If there is at least one sign change, then the number of sign changes indicates the number of poles with positive real parts. If there are any poles with positive real parts, then the system is unstable.

Let’s consider an example to understand the process of finding the stability of a control system using the Routh-Hurwitz criteria.

Example: Consider the transfer function of a control system given as G(s) = (s^{2} + 3s + 2)/(s^{3} + 5s^{2} + 7s + 2).

The characteristic equation of the system can be obtained by equating the denominator of the transfer function to zero:

s^{3} + 5s^{2} + 7s + 2 = 0

The Routh array for this equation can be constructed as follows:

| 1 | 5 | 2 |

| 3 | 7 |

| 1.4 | 0 |

Since there are no sign changes in the first column of the Routh array, all the poles of the system have negative real parts, and the system is stable.

Therefore, the control system represented by the transfer function G(s) = (s^{2} + 3s + 2)/(s^{3} + 5s^{2} + 7s + 2) is stable.

In summary, the Routh-Hurwitz criteria is a useful method to find the stability of a control system represented by a polynomial equation. By constructing the Routh array and analyzing it, the method allows for the evaluation of the stability of the system.

**Define Root-Locus**

The Root-Locus is a graphical representation of the pole trajectories of a control system in the s-plane as the gain of the system is varied. It is a powerful tool for analyzing the stability and performance of a control system.

The Root-Locus plot displays the locations of the closed-loop poles of a control system as the gain of the system is varied over a specified range. The gain is typically varied from zero to infinity, and the pole locations are traced in the complex plane. The Root-Locus plot is used to analyze the stability and transient response of a control system.

The Root-Locus is based on the concept of the characteristic equation of the control system. The characteristic equation is obtained by equating the denominator of the transfer function to zero. The poles of the closed-loop system are the roots of the characteristic equation. The Root-Locus plot shows the movement of the poles of the closed-loop system as the gain is varied.

The Root-Locus plot has several features that are important in analyzing the stability and performance of a control system. The Root-Locus plot can show the stability of the system, the transient response of the system, and the steady-state error of the system.

The Root-Locus plot is also useful in designing a control system. By analyzing the Root-Locus plot, the gain of the system can be chosen to achieve the desired performance specifications of the system.

Let’s consider an example to understand the concept of the Root-Locus plot.

Example: Consider the transfer function of a control system given as G(s) = K/(s+2)(s+3).

The characteristic equation of the system can be obtained by equating the denominator of the transfer function to zero:

s^{2} + 5s + 6 + K = 0

The Root-Locus plot for this system can be constructed by varying the gain K from zero to infinity and plotting the roots of the characteristic equation. The Root-Locus plot for this system is shown below.

The Root-Locus plot shows the movement of the poles of the closed-loop system as the gain K is varied. The Root-Locus plot also shows the asymptotes, breakaway points, and intersection points of the Root-Locus branches.

In this example, as the gain K is increased, the poles of the closed-loop system move along the Root-Locus branches. The Root-Locus plot shows that the system is stable for all values of K, and the transient response of the system is relatively fast. By analyzing the Root-Locus plot, the gain K can be chosen to achieve the desired performance specifications of the system.

In summary, the Root-Locus plot is a graphical representation of the pole trajectories of a control system in the s-plane as the gain of the system is varied. The Root-Locus plot is used to analyze the stability and performance of a control system and is a powerful tool in designing a control system.

**Define the terms related to Root-Locus Plot**

The Root-Locus plot is a graphical representation of the pole trajectories of a control system in the s-plane as the gain of the system is varied. It is a powerful tool for analyzing the stability and performance of a control system. Here are some terms related to Root-Locus plot that are commonly used:

**Poles and Zeros:**The poles and zeros of a transfer function are the points in the complex plane where the transfer function becomes infinite and zero, respectively. The poles and zeros of the transfer function determine the stability and performance of the control system.**Characteristic Equation:**The characteristic equation of a control system is obtained by equating the denominator of the transfer function to zero. The roots of the characteristic equation are the poles of the closed-loop system.**Root-Locus Branches:**The Root-Locus plot shows the movement of the poles of the closed-loop system as the gain of the system is varied. The Root-Locus branches are the paths followed by the poles of the closed-loop system as the gain is varied.**Asymptotes:**The asymptotes are the straight lines in the Root-Locus plot that show the general direction of the Root-Locus branches as the gain approaches infinity. The number of asymptotes is equal to the number of poles of the open-loop transfer function.**Breakaway Points:**The breakaway points are the points on the Root-Locus branches where the poles move apart from each other and move towards the imaginary axis.**Intersection Points:**The intersection points are the points on the Root-Locus branches where the poles of the closed-loop system cross the imaginary axis. The intersection points can be used to calculate the damping ratio and the natural frequency of the closed-loop system.**Gain Margin and Phase Margin:**The gain margin is the amount of gain that can be added to the system before the system becomes unstable. The phase margin is the amount of phase lag that can be introduced into the system before the system becomes unstable.

Let’s consider an example to understand these terms in the context of Root-Locus plot.

Example: Consider the transfer function of a control system given as G(s) = K/(s+1)(s+2)(s+3).

The characteristic equation of the system can be obtained by equating the denominator of the transfer function to zero:

s^{3} + 6s^{2} + 11s + K = 0

The Root-Locus plot for this system can be constructed by varying the gain K from zero to infinity and plotting the roots of the characteristic equation. The Root-Locus plot for this system is shown below.

In this example, the system has three poles, and hence, there are three Root-Locus branches. The Root-Locus branches start from the poles of the open-loop transfer function and move towards the zeros of the transfer function as the gain K is increased. The asymptotes of the Root-Locus branches are shown in the figure. The breakaway points are the points where the Root-Locus branches move away from each other, and the intersection points are the points where the Root-Locus branches cross the imaginary axis.

The Root-Locus plot shows that the system is stable for all values of K. The gain margin and the phase margin can be calculated from the Root-Locus plot. The gain margin is the amount of gain that can be added to the system before the system becomes unstable. The phase margin is the amount of phase lag that can be introduced into the system before the system becomes unstable.

**Recall the procedure for plotting a Root-Locus**

Root-Locus is a graphical representation of the location of the roots of the characteristic equation of a feedback control system as the system gain varies. The procedure for plotting a Root-Locus involves a series of steps that are essential in determining the stability and performance of a feedback control system.

The procedure for plotting a Root-Locus involves the following steps:

Step 1: Write the open-loop transfer function

The open-loop transfer function relates the output of the system to the input. It is expressed as G(s)H(s), where G(s) is the plant transfer function and H(s) is the controller transfer function. In some cases, the transfer function may have to be simplified to enable easy plotting of the Root-Locus.

Step 2: Determine the poles and zeros of the open-loop transfer function

The poles and zeros of the open-loop transfer function are the roots of the characteristic equation. They determine the stability and performance of the system. To determine the poles and zeros, the transfer function is factored into its individual components.

Step 3: Determine the location of the poles and zeros on the real and imaginary axis

The location of the poles and zeros of the transfer function on the real and imaginary axis is determined. The poles and zeros on the imaginary axis are referred to as breakaway points or break-in points.

Step 4: Sketch the Root-Locus

The Root-Locus is then plotted using the locations of the poles and zeros. The Root-Locus is a set of points that satisfy the characteristic equation. It starts at the poles of the open-loop transfer function and ends at the zeros. It is drawn in such a way that the branches approach the zeros of the transfer function asymptotically.

Step 5: Determine the stability and performance of the system

The stability and performance of the system can be determined by analyzing the Root-Locus. The Root-Locus shows the behavior of the system as the gain of the system varies. The system is stable if all the poles of the transfer function are located in the left half of the s-plane. The performance of the system can be evaluated by analyzing the settling time, overshoot, and damping ratio.

Example:

Consider a feedback control system with the following open-loop transfer function:

G(s)H(s) = (s+3)/(s+2)(s+4)

Step 1: Write the open-loop transfer function

G(s)H(s) = (s+3)/(s+2)(s+4)

Step 2: Determine the poles and zeros of the open-loop transfer function

The poles of the transfer function are -2 and -4. The zero of the transfer function is -3.

Step 3: Determine the location of the poles and zeros on the real and imaginary axis

The poles are located on the real axis at -2 and -4. There are no zeros on the real axis. The breakaway point is located at -3 on the imaginary axis.

Step 4: Sketch the Root-Locus

The Root-Locus is plotted as shown below. The Root-Locus starts at the poles and ends at the zero. The branches of the Root-Locus approach the zero asymptotically.

Step 5: Determine the stability and performance of the system

The system is stable since all the poles of the transfer function are located in the left half of the s-plane. The performance of the system can be evaluated by analyzing the settling time, overshoot, and damping ratio.

**Construct a Root-Locus of a given Open-Loop System**

Root-Locus is a graphical representation of the location of the roots of the characteristic equation of a feedback control system as the system gain varies. Constructing a Root-Locus involves plotting a set of points that satisfy the characteristic equation of the system. The Root-Locus is a useful tool in determining the stability and performance of a feedback control system.

The procedure for constructing a Root-Locus of a given open-loop system involves the following steps:

Step 1: Write the open-loop transfer function

The open-loop transfer function relates the output of the system to the input. It is expressed as G(s)H(s), where G(s) is the plant transfer function and H(s) is the controller transfer function. In some cases, the transfer function may have to be simplified to enable easy plotting of the Root-Locus.

Step 2: Determine the poles and zeros of the open-loop transfer function

The poles and zeros of the open-loop transfer function are the roots of the characteristic equation. They determine the stability and performance of the system. To determine the poles and zeros, the transfer function is factored into its individual components.

Step 3: Determine the location of the poles and zeros on the real and imaginary axis

The location of the poles and zeros of the transfer function on the real and imaginary axis is determined. The poles and zeros on the imaginary axis are referred to as breakaway points or break-in points.

Step 4: Sketch the Root-Locus

The Root-Locus is then plotted using the locations of the poles and zeros. The Root-Locus is a set of points that satisfy the characteristic equation. It starts at the poles of the open-loop transfer function and ends at the zeros. It is drawn in such a way that the branches approach the zeros of the transfer function asymptotically.

Step 5: Determine the stability and performance of the system

The stability and performance of the system can be determined by analyzing the Root-Locus. The Root-Locus shows the behavior of the system as the gain of the system varies. The system is stable if all the poles of the transfer function are located in the left half of the s-plane. The performance of the system can be evaluated by analyzing the settling time, overshoot, and damping ratio.

Example:

Consider a feedback control system with the following open-loop transfer function:

G(s)H(s) = (s+3)/(s+2)(s+4)

Step 1: Write the open-loop transfer function

G(s)H(s) = (s+3)/(s+2)(s+4)

Step 2: Determine the poles and zeros of the open-loop transfer function

The poles of the transfer function are -2 and -4. The zero of the transfer function is -3.

Step 3: Determine the location of the poles and zeros on the real and imaginary axis

The poles are located on the real axis at -2 and -4. There are no zeros on the real axis. The breakaway point is located at -3 on the imaginary axis.

Step 4: Sketch the Root-Locus

The Root-Locus is plotted as shown below. The Root-Locus starts at the poles and ends at the zero. The branches of the Root-Locus approach the zero asymptotically.

Step 5: Determine the stability and performance of the system

The system is stable since all the poles of the transfer function are located in the left half of the s-plane. The performance of the system can be evaluated by analyzing the settling time, overshoot, and damping ratio.

**Find the Gain of a System using Root- Locus**

Root Locus is a graphical method used to study the location and movement of the closed-loop poles of a feedback system, as the system’s gain is varied. The gain of a system is an important parameter that determines the system’s overall performance, stability, and response. By analyzing the Root Locus plot, we can determine the gain required to achieve a desired closed-loop pole location or stability margin.

To find the gain of a system using Root-Locus, the following steps are taken:

- Identify the open-loop transfer function of the system and the desired closed-loop pole location or stability margin.
- Construct the Root Locus plot of the system by varying the system’s gain from zero to infinity while keeping all other parameters fixed.
- Analyze the Root Locus plot to determine the gain required to achieve the desired closed-loop pole location or stability margin.

Example:

Consider a feedback control system with the following open-loop transfer function:

G(s) = (s+2)/(s+3)(s+4)

We want to find the gain required to achieve a closed-loop pole at s = -5.

Step 1: Identify the open-loop transfer function and desired closed-loop pole location

Open-loop transfer function: G(s) = (s+2)/(s+3)(s+4)

Desired closed-loop pole location: s = -5

Step 2: Construct the Root Locus plot

By varying the gain K from zero to infinity, we obtain the following Root Locus plot:

Step 3: Analyze the Root Locus plot to determine the gain required

From the Root Locus plot, we can see that the closed-loop pole moves from -2.68 to -5 as the gain is increased. The gain required to achieve a closed-loop pole at s = -5 is approximately K = 2.6. Therefore, we need to set the gain of the system to K = 2.6 to achieve the desired closed-loop pole location.

In summary, by using the Root Locus method, we can find the gain required to achieve a desired closed-loop pole location or stability margin. This method is particularly useful in the design and analysis of feedback control systems, where the gain plays a crucial role in determining the system’s performance and stability.

**Show the effect of addition of Poles and Zeros to a Root-Locus**

The Root Locus is a graphical representation that shows the locations of closed-loop poles for a given feedback system as a function of a particular parameter. It is an important tool used in control system design, analysis, and stability analysis. By adding poles and zeros to a root locus, we can see how these changes affect the behavior of the system.

Effect of adding Poles to a Root Locus:

When we add a pole to a system, we can observe the following effects on the root locus:

- The Root Locus moves towards the new pole.
- The angle of departure from the previous branch of the root locus to the new pole changes by 180 degrees.
- The number of branches of the root locus increases by one.

For example, let’s consider a transfer function G(s) = (s+1)/(s-2). Here, the pole is located at s = 2. Let’s add a new pole at s = -1. Now, the new transfer function becomes G(s) = (s+1)/(s-2)(s+1).

We can see that the root locus moves towards the new pole at s = -1. Also, the angle of departure from the previous branch of the root locus to the new pole changes by 180 degrees. Additionally, the number of branches of the root locus increases by one.

Effect of adding Zeros to a Root Locus:

When we add a zero to a system, we can observe the following effects on the root locus:

- The Root Locus moves away from the zero.
- The angle of arrival at the previous branch of the root locus to the new zero changes by 180 degrees.
- The number of branches of the root locus decreases by one.

For example, let’s consider the same transfer function G(s) = (s+1)/(s-2). Here, the zero is located at s = -1. Let’s add a new zero at s = 1. Now, the new transfer function becomes G(s) = (s+1)(s-1)/(s-2).

We can see that the root locus moves away from the new zero at s = 1. Also, the angle of arrival at the previous branch of the root locus to the new zero changes by 180 degrees. Additionally, the number of branches of the root locus decreases by one.

In conclusion, by adding poles and zeros to a root locus, we can see how these changes affect the behavior of the system. When we add a pole, the root locus moves towards the new pole, and the number of branches of the root locus increases by one. When we add a zero, the root locus moves away from the new zero, and the number of branches of the root locus decreases by one. Understanding the effect of adding poles and zeros to a root locus is essential for control system design and analysis.