advertisement

## Feedback Control Theory and Applications in Engineering

### Introduction

Feedback control is a fundamental concept in engineering, which involves using the output of a system to adjust its input in order to achieve a desired behavior. It is widely used in various fields, including mechanical, electrical, chemical, and aerospace engineering, among others.

In a feedback control system, a sensor measures the output of the system, and the measured value is compared to a reference or setpoint value. The difference between the two values, called the error, is used to adjust the input of the system in order to reduce the error and bring the output closer to the desired value. This process of comparing the output to the reference value and adjusting the input is repeated continuously in a closed-loop fashion, resulting in the desired behavior of the system.

advertisement

#### A. Definition of feedback control

Feedback control is a process in which the output of a system is measured and used to adjust the input of the system in order to achieve a desired behavior. It involves comparing the actual output of a system to a reference or setpoint value, calculating the difference between the two values (called the error), and using this error to adjust the input of the system in a closed-loop fashion. By continuously adjusting the input based on the measured output, feedback control can compensate for changes in the system or its environment, maintain stability, and improve performance.

In essence, feedback control aims to regulate a dynamic system by using information from the system's output to influence its input, with the goal of achieving a desired output. This is achieved by continuously measuring the output and adjusting the input to minimize the difference between the output and the desired value. Feedback control is used in a wide range of applications, including mechanical, electrical, chemical, and aerospace engineering, among others, where it is crucial for maintaining stability and achieving optimal performance.

#### B. Importance of feedback control in engineering

Feedback control is of significant importance in engineering for various reasons, some of which are outlined below:

**Regulation of dynamic systems:** Many systems in engineering are
dynamic and subject to changes and uncertainties. Feedback control can
regulate these systems by continuously adjusting the input based on the
measured output, compensating for changes and disturbances, and maintaining
the system's stability and performance.

**Improvement of system performance:** Feedback control can improve the
performance of systems by reducing errors and ensuring that the output
follows a desired trajectory. By adjusting the input of the system based on
the measured output, feedback control can optimize the system's response
time, accuracy, and stability.

**Robustness:** Feedback control can ensure that the system is robust to
changes and uncertainties in the environment. By continuously measuring the
output and adjusting the input, feedback control can compensate for changes
in the system's behavior and maintain stability.

**Automation:** Feedback control is essential for automation in
engineering. By using sensors and actuators, feedback control can
automatically adjust the input of the system to achieve a desired output,
reducing the need for human intervention.

**Safety:** Feedback control can improve safety in engineering systems.
By continuously monitoring the output and adjusting the input, feedback
control can prevent the system from exceeding safe operating limits or from
entering dangerous states.

advertisement

#### C. Basic components of a feedback control system

**Sensor: **A sensor is used to measure the output of the system, which
is then fed back to the controller. The sensor can be a physical device that
converts the output into an electrical or mechanical signal, such as a
temperature sensor, pressure sensor, or position sensor.

**Controller:** The controller is the brain of the feedback control
system. It processes the measured output from the sensor and calculates the
required input to achieve the desired behavior of the system. The controller
can be implemented using analog or digital circuits, microcontrollers, or
software algorithms.

**Actuator: **The actuator is the device that adjusts the input of the
system based on the controller output. It can be a motor, valve, heater, or
any other device that can modify the system's behavior. The actuator
converts the electrical or mechanical signal from the controller into a
physical action that affects the system's input.

**Feedback loop: **The feedback loop is the connection between the
output, sensor, controller, and actuator. It enables the continuous
adjustment of the input based on the measured output, allowing the system to
regulate itself and achieve the desired behavior.

**Setpoint:** The setpoint is the desired value of the output that the
system is designed to achieve. It is set by the operator or the system
designer and serves as the reference for the feedback control system.

These components work together in a closed-loop system to continuously monitor and adjust the system's behavior. The sensor measures the output, the controller calculates the required input, the actuator adjusts the input, and the feedback loop ensures that the output follows the desired trajectory. By continuously adjusting the input based on the measured output, the feedback control system can regulate the system's behavior and achieve the desired performance.

### II. Feedback Control Theory

#### A. Feedback loop and closed-loop control

A feedback loop is an essential component of a closed-loop control system. It enables the continuous adjustment of the system's input based on the measured output, allowing the system to regulate itself and achieve the desired behavior. In a closed-loop control system, the feedback loop connects the system's output to the input, forming a closed circuit.

advertisement

Feedback loop and closed-loop control |

**The feedback loop in a closed-loop control system operates as
follows:
**

- The output of the system is measured using a sensor, and the resulting signal is fed back to the controller.
- Te controller compares the measured output to the desired setpoint and calculates the difference (error).
- The controller uses the error to adjust the system's input through an actuator, which modifies the system's behavior.
- The adjusted input affects the output of the system, which is measured again, and the feedback loop repeats the cycle.

By continuously adjusting the input based on the measured output, the closed-loop control system can regulate the system's behavior and achieve the desired performance. The closed-loop control system is designed to maintain the output within a specific range around the setpoint, ensuring that the system operates reliably and efficiently.

Closed-loop control systems are used in a wide range of applications, including process control, robotics, and aerospace engineering. They are preferred over open-loop control systems because they are more robust to disturbances and uncertainties, and they can maintain stability even in the presence of changes in the system or the environment.

#### C. Transfer function and frequency response

The transfer function and frequency response of a feedback control system can be derived by considering the transfer function of each individual component in the system and using the feedback loop equation to relate the system's input and output.

Let's consider a simple feedback control system with a plant G(s), a controller C(s), and a feedback loop with transfer function H(s), as shown below:

where u(t) is the input to the system, y(t) is the output of the system, e(t) is the error signal, and H(s) is the transfer function of the feedback loop.

The transfer function of the system, from the input u(t) to the output y(t), can be obtained by considering the output of the plant G(s) with input e(t), which is given by:

G(s) * e(t)

where * denotes the convolution operation.

The error signal e(t) can be obtained by subtracting the output of the plant G(s) with input C(s)*y(t) (i.e., the feedback signal) from the input u(t), which is given by:

e(t) = u(t) - C(s) * y(t)

Substituting the above expression for e(t) into the equation for G(s)*e(t), we get:

y(t) = G(s) * e(t) = G(s) * [u(t) - C(s) * y(t)]

Solving for Y(s)/U(s), we get the transfer function of the system:

Y(s) G(s)

---- = ------

U(s) 1 + G(s)C(s)H(s)

where H(s) is the transfer function of the feedback loop.

advertisement

The frequency response of the system can be obtained by substituting s with jÏ‰, where Ï‰ is the angular frequency, and plotting the magnitude response and phase response of the transfer function on a logarithmic scale versus the angular frequency. The frequency response analysis can be used to analyze the stability, sensitivity, and performance of the feedback control system, and to design appropriate controllers based on the system's frequency response characteristics.

#### D. Stability analysis

Stability analysis is an important aspect of feedback control theory, as it helps determine whether a feedback control system is stable or unstable, and how the system responds to disturbances or changes in the input.

A feedback control system is said to be stable if the output of the system remains bounded for any bounded input, and unstable if the output of the system grows without bound for certain inputs. A stable system is desirable in control engineering, as it ensures that the system responds predictably to the input and does not exhibit oscillations, overshoot, or instability.

There are several methods for analyzing the stability of a feedback control system, including:

**Bode stability criterion:** This method uses the frequency response of
the system to determine the stability of the system. According to the Bode
stability criterion, a feedback control system is stable if and only if the
phase shift of the system is less than 180 degrees at the frequency where
the magnitude of the transfer function is unity (0 dB).

**Routh-Hurwitz stability criterion:** This method uses the
characteristic equation of the system to determine the stability of the
system. According to the Routh-Hurwitz stability criterion, a feedback
control system is stable if all the coefficients of the characteristic
equation have the same sign, and there are no sign changes in the first
column of the Routh array.

**Nyquist stability criterion: **This method uses the complex plane
representation of the frequency response of the system to determine the
stability of the system. According to the Nyquist stability criterion, a
feedback control system is stable if and only if the Nyquist plot of the
system does not encircle the point (-1,0) in the complex plane.

**Root locus method:** This method uses the root locus plot of the
system to determine the stability and response of the system to changes in
the system's parameters. The root locus plot shows the movement of the
closed-loop poles of the system as a function of a parameter in the system's
transfer function

advertisement

### III. PID Control

Proportional-Integral-Derivative (PID) control is a common feedback control technique used in many industrial processes to control the output of a system. The PID controller continuously calculates an error value as the difference between the desired set point and the actual output of the system. It then applies proportional, integral, and derivative terms to the error value to compute the output of the controller.

The proportional term of the controller produces an output that is proportional to the current error value. The integral term of the controller produces an output that is proportional to the accumulated error over time, while the derivative term produces an output that is proportional to the rate of change of the error over time. By combining these three terms, the PID controller produces an output that compensates for both the steady-state error and the transient response of the system, leading to faster and more accurate control.

The PID control algorithm can be mathematically represented by the following equation:

u(t) = Kpe(t) + KiIntegral[e(t)] + Kd*Derivative[e(t)]

where u(t) is the output of the controller at time t, e(t) is the error between the desired set point and the actual output of the system at time t, Kp, Ki, and Kd are the proportional, integral, and derivative gain coefficients, respectively, and Integral[e(t)] and Derivative[e(t)] are the integrals and derivatives of the error signal with respect to time.

The PID controller is widely used in many applications, such as temperature control, speed control, level control, and pressure control, due to its simplicity, flexibility, and robustness. However, the tuning of the PID parameters can be challenging and requires some knowledge of the system's dynamics and response characteristics.

#### B. Mathematical model of a PID controller

The mathematical model of a PID controller can be expressed as a transfer function that relates the output of the controller to the input error signal. The transfer function of a PID controller can be written as:

G(s) = Kp + Ki/s + Kd*s

where G(s) is the transfer function of the controller, Kp, Ki, and Kd are the proportional, integral, and derivative gains, respectively, and s is the Laplace variable.

The first term, Kp, represents the proportional gain, which is multiplied by the error signal to produce the output of the controller. The proportional gain determines how much the controller responds to the current error value.

The second term, Ki/s, represents the integral gain, which is multiplied by the integral of the error signal over time. The integral gain reduces the steady-state error of the system and helps the controller to reach the desired set point.

The third term, Kd*s, represents the derivative gain, which is multiplied by the derivative of the error signal with respect to time. The derivative gain improves the transient response of the system and helps the controller to respond faster to changes in the input.

The transfer function of the PID controller can be used to analyze the stability and performance of the closed-loop system and to design appropriate controller parameters for a given system. The PID controller is a widely used feedback control technique due to its simplicity, flexibility, and robustness, and it is applied in many industrial and engineering applications, such as temperature control, speed control, and process control.

PID control tuning is the process of adjusting the controller parameters to achieve the desired closed-loop system performance. There are several methods for tuning PID controllers, and the choice of the tuning method depends on the application requirements, the system dynamics, and the available resources.

advertisement

#### C. Tuning methods

**Ziegler-Nichols Method:** This is a popular and widely used tuning
method that involves applying a step input to the system and then measuring
the system response to determine the ultimate gain and period of
oscillation. These values are then used to calculate the PID controller
parameters.

**Cohen-Coon Method:** This is another popular tuning method that
involves estimating the process time constant and the process gain based on
the system response to a step input. These estimates are then used to
calculate the PID controller parameters.

**Trial and Error Method: **This method involves manually adjusting the
PID controller parameters until the desired closed-loop system performance
is achieved. This method is simple but can be time-consuming and may not
guarantee optimal performance.

**Auto-Tuning Method:** This method involves using an algorithm or
software to automatically adjust the PID controller parameters based on the
system response. Auto-tuning methods can save time and improve performance
but require more resources and may not work well for highly nonlinear or
time-varying systems.

**Frequency Response Method:** This method involves analyzing the
frequency response of the closed-loop system and designing the PID
controller parameters based on the desired gain and phase margins. This
method is effective for systems with known transfer functions and is
commonly used in aerospace and control systems engineering.

#### Limitations of PID control

**Nonlinear systems:** PID control assumes that the system dynamics are
linear and time-invariant, which may not be the case for highly nonlinear
systems. In such cases, more advanced control techniques, such as adaptive
control or model predictive control, may be required.

**Dead time:** Dead time is the delay between the input and output of
the system, which can lead to instability and poor control performance in
PID control. Dead time can be compensated for by using a Smith predictor or
a modified PID controller.

**Saturation and nonlinearity:** PID control assumes that the control
signal can vary continuously between zero and the maximum output, which may
not be the case for systems with saturation or nonlinearity. In such cases,
anti-windup schemes or nonlinear control techniques may be required.

**Robustness:** PID control is not inherently robust to disturbances or
model uncertainties, which can affect the closed-loop system performance.
Robust control techniques, such as H-infinity control or sliding mode
control, can be used to improve the robustness of the system.

**Tuning:** PID control requires careful tuning of the controller
parameters to achieve the desired closed-loop system performance. The tuning
process can be time-consuming and may require expert knowledge and
experience.

advertisement

### IV. Advanced Control Techniques

#### A. Nonlinear control

Nonlinear control techniques are used to address the limitations of linear control methods such as PID control and are effective in handling systems with nonlinearities and uncertainties. Some of the commonly used nonlinear control techniques are:

**Feedback linearization:** This technique transforms a nonlinear system
into a linear system by using a change of coordinates and feedback control.
It is particularly useful for systems with known or measurable inputs and
outputs.

**Sliding mode control:** This technique creates a sliding surface where
the system behavior is constrained to follow a desired trajectory. The
control law is designed such that the sliding motion is maintained,
resulting in robustness to uncertainties and disturbances.

**Backstepping control:** This technique is a recursive feedback design
that constructs a control law for the system based on a virtual subsystem.
The control law is designed such that the closed-loop system asymptotically
follows the desired trajectory.

#### B. Optimal control

Optimal control techniques are used to optimize a certain objective function while satisfying the system constraints. These techniques are particularly useful for systems with complex dynamics and multiple inputs and outputs. Some of the commonly used optimal control techniques are:

**Linear quadratic regulator (LQR):** This technique is a state-feedback
control design that minimizes a quadratic cost function of the system state
and control inputs. It is particularly useful for systems with linear
dynamics and Gaussian disturbances.

**Model predictive control (MPC):** This technique uses a dynamic model
of the system to predict the future behavior of the system and optimize a
performance index over a finite time horizon. It is particularly useful for
systems with nonlinear dynamics and constraints on the inputs and
outputs.

**Optimal control of distributed parameter systems:** This technique is
used to design control laws for systems described by partial differential
equations, such as heat transfer or fluid flow systems. The control law is
designed to minimize an objective function over the entire domain.

advertisement

#### C. Robust control

Robust control techniques are used to design control laws that are insensitive to uncertainties and disturbances in the system. These techniques are particularly useful for systems with modeling errors and parameter variations. Some of the commonly used robust control techniques are:

**H-infinity control:** This technique is a frequency-domain control
design that minimizes the worst-case performance of the closed-loop system
subject to a prescribed level of disturbance attenuation. It is particularly
useful for systems with uncertain dynamics and disturbances.

**Mu-synthesis:** This technique is a control design method that
combines the H-infinity and classical control approaches to obtain a robust
control law. It is particularly useful for systems with both parametric and
nonparametric uncertainties.

**Robust control of nonlinear systems:** This technique uses feedback
linearization, sliding mode control, and other nonlinear control techniques
to design control laws that are robust to uncertainties and disturbances in
nonlinear systems.

#### D. Adaptive control

Adaptive control techniques are used to design control laws that adapt to changes in the system dynamics or parameter variations. These techniques are particularly useful for systems with unknown or time-varying parameters. Some of the commonly used adaptive control techniques are:

**Model reference adaptive control: **This technique uses a reference
model of the system to adjust the control law based on the difference
between the actual and desired system behavior. It is particularly useful
for systems with linear dynamics and parameter variations.

**Self-tuning control:** This technique adjusts the control law
parameters based on an estimate of the system parameters. It is particularly
useful for systems with unknown parameters or systems that operate under
varying conditions.

**Adaptive control of nonlinear systems:** This technique uses nonlinear
control techniques, such as feedback linearization and sliding mode control,
to design control laws that adapt to changes in the system dynamics or
parameter variations.

advertisement

### V. Applications of Feedback Control

#### A. Control of mechanical systems

Control of mechanical systems involves the design of control laws for machines and mechanical systems such as robots, vehicles, and industrial equipment. Some of the common applications of control of mechanical systems include:

**Motion control: **This involves the control of position, velocity, and
acceleration of mechanical systems such as vehicles, aircraft, and robotics.
Motion control is critical for achieving accurate and reliable performance
in applications such as autonomous vehicles, unmanned aerial vehicles, and
robotic manufacturing.

**Robotic control:** Robotic control involves the design of control laws
for robots, including manipulators, mobile robots, and humanoid robots.
Robotic control is essential for achieving precise and efficient robotic
movements and interactions with the environment, and has applications in
fields such as manufacturing, healthcare, and search and rescue
operations.

The design of control laws for mechanical systems involves modeling the dynamics of the system, designing a suitable control algorithm, and implementing the control system on the hardware platform. Control of mechanical systems requires a multidisciplinary approach that integrates principles of mechanical engineering, electrical engineering, and computer science. Advances in control theory, sensing and actuation technologies, and artificial intelligence have enabled the development of more sophisticated control systems for mechanical systems, leading to improved performance, efficiency, and safety.

#### B. Control of chemical and process systems

Control of chemical and process systems involves the design of control strategies for chemical processes, which include various unit operations such as reactors, separators, and heat exchangers. Control of chemical and process systems is critical for ensuring safe, efficient, and reliable operation of chemical plants, refineries, and other process industries. Some of the common applications of control of chemical and process systems include:

**Process control:** Process control involves the control of process
variables such as temperature, pressure, flow rate, and chemical
concentrations to achieve desired process performance. Process control is
critical for maintaining product quality, minimizing waste, and maximizing
production efficiency.

**Batch control:** Batch control involves the control of a sequence of
operations that are carried out in batches, such as in pharmaceutical
manufacturing or food processing. Batch control is critical for achieving
consistent product quality and minimizing waste in batch processes.

advertisement

#### C. Control of electrical and electronic systems

Control of electrical and electronic systems involves the design of control strategies for electrical systems, including power electronics, electric machines, and electronic circuits. Control of electrical and electronic systems is critical for ensuring safe, efficient, and reliable operation of electrical systems in various applications, such as electric vehicles, renewable energy systems, and power grids. Some of the common applications of control of electrical and electronic systems include:

**Power electronics control:** Power electronics control involves the
design of control strategies for power converters, such as inverters and
rectifiers, used in various applications such as renewable energy systems,
electric vehicles, and motor drives. Power electronics control is critical
for achieving efficient and reliable power conversion and regulation.

**Motor control:** Motor control involves the design of control
strategies for electric machines such as motors and generators. Motor
control is critical for achieving efficient and reliable operation of
electric machines in various applications such as electric vehicles,
robotics, and industrial automation.

### VI. Conclusion

In summary, feedback control theory is a fundamental discipline in engineering that deals with the design of control systems that can achieve desired performance and stability. Feedback control systems consist of sensors, actuators, and controllers that work together to regulate a process or a system. The basic components of a feedback control system include a plant, a controller, and a feedback loop, which together can achieve desired performance specifications such as stability, accuracy, and robustness. Proportional-Integral-Derivative (PID) control is a widely used control technique that can provide good performance for many applications. However, other advanced control techniques such as nonlinear control, optimal control, robust control, and adaptive control are also important for more complex and challenging applications.

Feedback control theory has wide-ranging applications in various fields of engineering, including mechanical systems, chemical and process systems, and electrical and electronic systems. Control of mechanical systems involves the design of control strategies for motion control and robotic control. Control of chemical and process systems involves the design of control strategies for process control and batch control. Control of electrical and electronic systems involves the design of control strategies for power electronics control and motor control. In all these applications, feedback control is critical for ensuring safe, efficient, and reliable operation of various systems and processes.

advertisement

### Table Of Contents

I. Introduction

A. Definition of feedback control

B. Importance of feedback control in engineering

C. Basic components of a feedback control system

**II. Feedback Control Theory
**

A. Feedback loop and closed-loop control

B. Block diagram representation

C. Transfer function and frequency response

D. Stability analysis

- Routh-Hurwitz criterion
- Bode plot
- Nyquist plot

**III. PID Control
**

A. Definition of proportional-integral-derivative control

B. Mathematical model of a PID controller

C. Tuning methods

- Ziegler-Nichols method
- Cohen-Coon method
- Tyreus-Luyben method

D. Limitations of PID control

**IV. Advanced Control Techniques
**

A. Nonlinear control

- Feedback linearization
- Sliding mode control
- Backstepping control

B. Optimal control

Linear quadratic regulator (LQR)

Model predictive control (MPC)

Optimal control of distributed parameter systems

C. Robust control

- H-infinity control
- Mu-synthesis
- Robust control of nonlinear systems

D. Adaptive control

- Model reference adaptive control
- Self-tuning control
- Adaptive control of nonlinear systems

**V. Applications of Feedback Control
**

A. Control of mechanical systems

- Motion control
- Robotic control

B. Control of chemical and process systems

- Process control
- Batch control

C. Control of electrical and electronic systems

- Power electronics control
- Motor control

**VI. Conclusion
**

A. Summary of feedback control theory and applications

B. Future developments in feedback control.

Download PDF in comment section

advertisement

Feedback Control Theory and Applications in Engineering

ReplyDeleteNot downloading

ReplyDelete