This document provides a lecture summary on modeling digital control systems. It discusses the sampling theorem, analog-to-digital converter (ADC) and digital-to-analog converter (DAC) models, and how to combine these models. The sampling theorem establishes the minimum sampling rate needed to reconstruct a band-limited signal. The ADC and DAC are modeled as an ideal sampler and zero-order hold, respectively. Their combination transfer function is derived as (1 - z-1) times the z-transform of the analog subsystem transfer function divided by s. Examples are provided to illustrate selecting suitable sampling rates and deriving combined digital system transfer functions.
10 Discrete Time Controller Design.pptxSaadAzhar15
Β
This document discusses digital control system design. It begins with an overview of discretization methods and the effect of zero-order hold. Examples are provided to illustrate discretization and digital controller design. Design of PI and PID digital controllers via pole placement is covered. An example designs a cruise control system for a car using a digital PI controller. The controller is designed by deriving specifications from the design problem, discretizing the plant, determining controller parameters, and simulating the closed-loop response. The controller meets specifications when applied to both the discretized and actual continuous-time plant.
This document discusses PID (proportional-integral-derivative) controllers. It explains that PID controllers use three terms: proportional to the error, integral of the error, and derivative of the error. The document provides equations for continuous and discrete PID controllers. It also describes Ziegler-Nichols tuning, which is a common method for adjusting the PID parameters (Kp, Ti, Td) based on open-loop step response testing of the plant. Ziegler-Nichols tuning values are given for proportional, PI, and PID controllers to minimize error. An example problem demonstrates identifying plant parameters from step response data and applying Ziegler-Nichols tuning to design proportional, PI, and PID controllers.
The document discusses techniques for designing discrete-time infinite impulse response (IIR) filters from continuous-time filter specifications. It covers the impulse invariance method, matched z-transform method, and bilinear transformation method. The impulse invariance method samples the continuous-time impulse response to obtain the discrete-time impulse response. The bilinear transformation maps the entire s-plane to the unit circle in the z-plane to avoid aliasing. Examples are provided to illustrate the design process using each method.
DSP_2018_FOEHU - Lec 07 - IIR Filter DesignAmr E. Mohamed
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The document discusses the design of discrete-time IIR filters from continuous-time filter specifications. It covers common IIR filter design techniques including the impulse invariance method, matched z-transform method, and bilinear transformation method. An example applies the bilinear transformation to design a first-order low-pass digital filter from a continuous analog prototype. Filter design procedures and steps are provided.
The document discusses vibration isolation of a LEGO platform using low-cost instrumentation as part of an open source project. It summarizes using inertial mass actuators to apply skyhook damping control to actively isolate the platform from environmental vibrations. Preliminary measurements identify the platform's modal properties. A real implementation is presented using 4 symmetric actuators. Control design applies skyhook damping in modal coordinates to optimally isolate each mode.
This document discusses the damping ratio of unit step responses in control systems. It defines damping ratio as the ratio of the actual damping coefficient to the critical damping coefficient. It describes the different types of damping including underdamped, overdamped, and critically damped systems. It discusses using a unit step function as a common test input and analyzing the step response to identify system properties. MATLAB coding examples are provided to simulate step responses and the document discusses applications in identification from step response testing.
This document provides a lecture summary on modeling digital control systems. It discusses the sampling theorem, analog-to-digital converter (ADC) and digital-to-analog converter (DAC) models, and how to combine these models. The sampling theorem establishes the minimum sampling rate needed to reconstruct a band-limited signal. The ADC and DAC are modeled as an ideal sampler and zero-order hold, respectively. Their combination transfer function is derived as (1 - z-1) times the z-transform of the analog subsystem transfer function divided by s. Examples are provided to illustrate selecting suitable sampling rates and deriving combined digital system transfer functions.
10 Discrete Time Controller Design.pptxSaadAzhar15
Β
This document discusses digital control system design. It begins with an overview of discretization methods and the effect of zero-order hold. Examples are provided to illustrate discretization and digital controller design. Design of PI and PID digital controllers via pole placement is covered. An example designs a cruise control system for a car using a digital PI controller. The controller is designed by deriving specifications from the design problem, discretizing the plant, determining controller parameters, and simulating the closed-loop response. The controller meets specifications when applied to both the discretized and actual continuous-time plant.
This document discusses PID (proportional-integral-derivative) controllers. It explains that PID controllers use three terms: proportional to the error, integral of the error, and derivative of the error. The document provides equations for continuous and discrete PID controllers. It also describes Ziegler-Nichols tuning, which is a common method for adjusting the PID parameters (Kp, Ti, Td) based on open-loop step response testing of the plant. Ziegler-Nichols tuning values are given for proportional, PI, and PID controllers to minimize error. An example problem demonstrates identifying plant parameters from step response data and applying Ziegler-Nichols tuning to design proportional, PI, and PID controllers.
The document discusses techniques for designing discrete-time infinite impulse response (IIR) filters from continuous-time filter specifications. It covers the impulse invariance method, matched z-transform method, and bilinear transformation method. The impulse invariance method samples the continuous-time impulse response to obtain the discrete-time impulse response. The bilinear transformation maps the entire s-plane to the unit circle in the z-plane to avoid aliasing. Examples are provided to illustrate the design process using each method.
DSP_2018_FOEHU - Lec 07 - IIR Filter DesignAmr E. Mohamed
Β
The document discusses the design of discrete-time IIR filters from continuous-time filter specifications. It covers common IIR filter design techniques including the impulse invariance method, matched z-transform method, and bilinear transformation method. An example applies the bilinear transformation to design a first-order low-pass digital filter from a continuous analog prototype. Filter design procedures and steps are provided.
The document discusses vibration isolation of a LEGO platform using low-cost instrumentation as part of an open source project. It summarizes using inertial mass actuators to apply skyhook damping control to actively isolate the platform from environmental vibrations. Preliminary measurements identify the platform's modal properties. A real implementation is presented using 4 symmetric actuators. Control design applies skyhook damping in modal coordinates to optimally isolate each mode.
This document discusses the damping ratio of unit step responses in control systems. It defines damping ratio as the ratio of the actual damping coefficient to the critical damping coefficient. It describes the different types of damping including underdamped, overdamped, and critically damped systems. It discusses using a unit step function as a common test input and analyzing the step response to identify system properties. MATLAB coding examples are provided to simulate step responses and the document discusses applications in identification from step response testing.
This document discusses PID control. It begins by explaining that PID controllers are the most common type used in process industries. It then provides the general feedback control loop model and shows how to derive the closed-loop transfer functions. Next, it defines the characteristic equation and provides an example using a P-only controller. The document continues by presenting the position and velocity forms of the PID algorithm as well as digital equivalents. It concludes by discussing different PID configurations available in distributed control systems.
The document discusses real time implementation of active noise control. It begins with an introduction to noise and different noise reduction techniques. It then discusses active noise control in more detail, including its structure, block diagram, and adaptive algorithm techniques like the steepest descent and least mean square algorithms. The document presents simulation results of noise cancellation using MATLAB. It describes implementing active noise control on a Texas Instruments C6713 DSK board using Code Composer Studio and MATLAB Simulink. Issues faced during the hardware implementation are also discussed.
The document discusses digital filters and their design process. It explains that the design process involves four main steps: approximation, realization, studying arithmetic errors, and implementation.
For approximation, direct and indirect methods are used to generate a transfer function that satisfies the filter specifications. Realization generates a filter network from the transfer function. Studying arithmetic errors examines how quantization affects filter performance. Implementation realizes the filter in either software or hardware.
The document also outlines the basic building blocks of digital filters, including adders, multipliers, and delay elements. It introduces linear time-invariant digital filters and explains their input-output relationship using difference equations and the z-transform.
This chapter introduces analog computing techniques. It discusses the components of an analog computer including operational amplifiers, resistors, capacitors and inductors. It describes how operational amplifiers can be used to simulate linear systems using inverting amplifiers, non-inverting amplifiers, summer amplifiers, integrators and differentiators. The chapter also covers how to apply magnitude and time scaling to model systems within the voltage range of an analog computer. An example shows how to derive the scaled dynamic model of a system and realize it using operational amplifiers.
This document describes an automatic agriculture assistance system that uses autonomous vehicles guided by GPS navigation to help farmers cultivate crops on a large scale. The system uses commercial tractors outfitted with encoders, ultrasonic sensors, an onboard computer and PLC controller. The vehicle's motion is controlled through parameters like driving speed and steering angle. A Kalman filter is used to estimate the vehicle's state based on encoder readings and reduce errors between the actual and planned paths. The system is intended to help save farmer's time and money during crop cultivation.
Advanced Nonlinear PID-Based Antagonistic Control for Pneumatic Muscle Actuatorsmanikuty123
Β
a seminar on "Advanced Nonlinear PID-Based Antagonistic Control for Pneumatic Muscle Actuators" for control enginerring,It involves the control of pneumatic mucle actuators using Advanced nonlinear PID control.It is a model less control.These actuators are mainly used in humanoid robborts.this leads to more easier and robust control.
Development of Digital Controller for DC-DC Buck ConverterIJPEDS-IAES
Β
This paper presents a design & implementation of 3P3Z (3-pole 3-zero)
digital controller based on DSC (Digital Signal Controller) for low voltage
synchronous Buck Converter. The proposed control involves one voltage
control loop. Analog Type-3 controller is designed for Buck Converter using
standard frequency response techniques.Type-3 analog controller transforms
to 3P3Z controller in discrete domain.Matlab/Simulink model of the Buck
Converter with digital controller is developed. Simualtion results for steady
Keyword: state response and load transient response is tested using the model.
- The group aims to filter ECG signals acquired from patients in an MRI environment to improve signal quality which is currently lacking.
- Objectives are to filter 80% of MRI machine interference on ECG signals, convert analog ECG/MRI signals to digital for processing, and convert back to analog for legacy devices.
- Goals are to further adaptive filtering in medicine and improve ECG signal quality to potentially save lives.
This file contains slides that explains the IIR filter design techniques. Especially the time invariance and bilinear transformations. The material found in this presentation was taken from Oppenheim second edition reference book, I hope that anyone who read this presentation to leave a feedback that mention its suitability
This document describes a sensor fault diagnosis scheme for a DC/DC converter used in hybrid electric vehicles. The scheme uses a bank of extended Kalman filters to generate residuals by comparing estimated and actual sensor measurements. A generalized likelihood ratio test evaluates the residuals to detect faults. The diagnosis scheme was tested on a hardware prototype of a bidirectional DC/DC converter. Modeling of the power converter system and details of the proposed residual-based fault diagnosis algorithm are provided.
Applying Smoothing Techniques to Passive Target Tracking.pptxismailshaik2023
Β
The main objective of this project is to track a under water target using Sound Navigation and Ranging (SONAR) measurements in passive mode, in twoβdimensional space making use of bearing angle measurements. An Extended Kalman filter algorithm is considered for processing noise altered measurements along with smoothers algorithms to reduce the errors in the estimates of target parameters (range, course, and speed of the target). Details of mathematical modelling for simulating and implementation of the target and observer paths and outcomes are presented in this work.
This document provides an overview of PID controllers, including:
- The basic feedback loop and proportional, integral, and derivative algorithms
- Implementation issues like set-point weighing, windup, and digital implementation
- Practical operational aspects like bumpless transfer between manual and automatic modes
The document summarizes different types of digital counters, including asynchronous counters, synchronous counters, ring counters, and Johnson counters. Asynchronous counters have each flip-flop triggered by the previous one, limiting speed, while synchronous counters trigger all flip-flops simultaneously using a common clock, increasing speed. Ring counters circulate a single '1' bit around the register. Johnson counters are like ring counters but with the inverted output of the last flip-flop connected to the first. Examples and applications of each type are provided.
Control Signal Flow Graphs lecture notesabbas miry
Β
Signal flow graphs consist of:
π Nodes βrepresent signals
π Branches βrepresent system blocks
Branches labeled with system transfer functions
Signal flow graphs are an alternative to block diagrams for graphically describing systems. They consist of nodes to represent signals and branches to represent system blocks labeled with transfer functions. To convert a block diagram to a signal flow graph, identify and label all signals, place a node for each, connect nodes with branches in place of blocks while maintaining direction, and label branches with transfer functions. Mason's rule provides a formula to calculate the overall transfer function of a system represented by a signal flow graph based on terms like forward path gains, loop gains, and non-touching loop gains. Controller design uses feedback to modify a system's response to meet performance specifications by placing the closed-loop poles through selection of controller parameters.
This document discusses digital IIR filter design. It describes different types of analog filters like Butterworth, Chebyshev, Elliptic and Bessel filters. It then explains how to design digital IIR filters using analog filter design principles with impulse invariant and bilinear transformations from the s-domain to the z-domain. Step-by-step procedures are provided for designing low-pass Butterworth and Chebyshev filters using these two transformation methods. The concepts of aliasing and pre-warping frequencies are also covered in the bilinear transformation section.
The document discusses digital filters and their design. It begins with an introduction to filters and their uses in signal processing applications. It then covers linear time-invariant filters and their transfer functions. It discusses the differences between non-recursive (FIR) and recursive (IIR) filters. The document presents various filter structures for implementation, including direct form I and direct form II structures. It also discusses designing FIR and IIR filters as well as issues in their implementation.
This document summarizes techniques for designing finite impulse response (FIR) filters and digital controllers. It discusses truncating the impulse response of an ideal filter to design FIR filters. It also presents a method for digital controller design that matches the step response of an analog controller to approximate it digitally. Examples are provided of designing an FIR lowpass filter and approximating an analog feedback controller with a digital controller using step response matching.
This document summarizes techniques for designing finite impulse response (FIR) filters and digital controllers. It discusses truncating the impulse response of an ideal filter to design FIR filters. It also presents a method for digital controller design that matches the step response of an analog controller to approximate it digitally. Examples are provided of designing an FIR lowpass filter and approximating an analog feedback controller with a digital controller using step response matching.
Harnessing WebAssembly for Real-time Stateless Streaming PipelinesChristina Lin
Β
Traditionally, dealing with real-time data pipelines has involved significant overhead, even for straightforward tasks like data transformation or masking. However, in this talk, weβll venture into the dynamic realm of WebAssembly (WASM) and discover how it can revolutionize the creation of stateless streaming pipelines within a Kafka (Redpanda) broker. These pipelines are adept at managing low-latency, high-data-volume scenarios.
Understanding Inductive Bias in Machine LearningSUTEJAS
Β
This presentation explores the concept of inductive bias in machine learning. It explains how algorithms come with built-in assumptions and preferences that guide the learning process. You'll learn about the different types of inductive bias and how they can impact the performance and generalizability of machine learning models.
The presentation also covers the positive and negative aspects of inductive bias, along with strategies for mitigating potential drawbacks. We'll explore examples of how bias manifests in algorithms like neural networks and decision trees.
By understanding inductive bias, you can gain valuable insights into how machine learning models work and make informed decisions when building and deploying them.
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This document discusses PID control. It begins by explaining that PID controllers are the most common type used in process industries. It then provides the general feedback control loop model and shows how to derive the closed-loop transfer functions. Next, it defines the characteristic equation and provides an example using a P-only controller. The document continues by presenting the position and velocity forms of the PID algorithm as well as digital equivalents. It concludes by discussing different PID configurations available in distributed control systems.
The document discusses real time implementation of active noise control. It begins with an introduction to noise and different noise reduction techniques. It then discusses active noise control in more detail, including its structure, block diagram, and adaptive algorithm techniques like the steepest descent and least mean square algorithms. The document presents simulation results of noise cancellation using MATLAB. It describes implementing active noise control on a Texas Instruments C6713 DSK board using Code Composer Studio and MATLAB Simulink. Issues faced during the hardware implementation are also discussed.
The document discusses digital filters and their design process. It explains that the design process involves four main steps: approximation, realization, studying arithmetic errors, and implementation.
For approximation, direct and indirect methods are used to generate a transfer function that satisfies the filter specifications. Realization generates a filter network from the transfer function. Studying arithmetic errors examines how quantization affects filter performance. Implementation realizes the filter in either software or hardware.
The document also outlines the basic building blocks of digital filters, including adders, multipliers, and delay elements. It introduces linear time-invariant digital filters and explains their input-output relationship using difference equations and the z-transform.
This chapter introduces analog computing techniques. It discusses the components of an analog computer including operational amplifiers, resistors, capacitors and inductors. It describes how operational amplifiers can be used to simulate linear systems using inverting amplifiers, non-inverting amplifiers, summer amplifiers, integrators and differentiators. The chapter also covers how to apply magnitude and time scaling to model systems within the voltage range of an analog computer. An example shows how to derive the scaled dynamic model of a system and realize it using operational amplifiers.
This document describes an automatic agriculture assistance system that uses autonomous vehicles guided by GPS navigation to help farmers cultivate crops on a large scale. The system uses commercial tractors outfitted with encoders, ultrasonic sensors, an onboard computer and PLC controller. The vehicle's motion is controlled through parameters like driving speed and steering angle. A Kalman filter is used to estimate the vehicle's state based on encoder readings and reduce errors between the actual and planned paths. The system is intended to help save farmer's time and money during crop cultivation.
Advanced Nonlinear PID-Based Antagonistic Control for Pneumatic Muscle Actuatorsmanikuty123
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a seminar on "Advanced Nonlinear PID-Based Antagonistic Control for Pneumatic Muscle Actuators" for control enginerring,It involves the control of pneumatic mucle actuators using Advanced nonlinear PID control.It is a model less control.These actuators are mainly used in humanoid robborts.this leads to more easier and robust control.
Development of Digital Controller for DC-DC Buck ConverterIJPEDS-IAES
Β
This paper presents a design & implementation of 3P3Z (3-pole 3-zero)
digital controller based on DSC (Digital Signal Controller) for low voltage
synchronous Buck Converter. The proposed control involves one voltage
control loop. Analog Type-3 controller is designed for Buck Converter using
standard frequency response techniques.Type-3 analog controller transforms
to 3P3Z controller in discrete domain.Matlab/Simulink model of the Buck
Converter with digital controller is developed. Simualtion results for steady
Keyword: state response and load transient response is tested using the model.
- The group aims to filter ECG signals acquired from patients in an MRI environment to improve signal quality which is currently lacking.
- Objectives are to filter 80% of MRI machine interference on ECG signals, convert analog ECG/MRI signals to digital for processing, and convert back to analog for legacy devices.
- Goals are to further adaptive filtering in medicine and improve ECG signal quality to potentially save lives.
This file contains slides that explains the IIR filter design techniques. Especially the time invariance and bilinear transformations. The material found in this presentation was taken from Oppenheim second edition reference book, I hope that anyone who read this presentation to leave a feedback that mention its suitability
This document describes a sensor fault diagnosis scheme for a DC/DC converter used in hybrid electric vehicles. The scheme uses a bank of extended Kalman filters to generate residuals by comparing estimated and actual sensor measurements. A generalized likelihood ratio test evaluates the residuals to detect faults. The diagnosis scheme was tested on a hardware prototype of a bidirectional DC/DC converter. Modeling of the power converter system and details of the proposed residual-based fault diagnosis algorithm are provided.
Applying Smoothing Techniques to Passive Target Tracking.pptxismailshaik2023
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The main objective of this project is to track a under water target using Sound Navigation and Ranging (SONAR) measurements in passive mode, in twoβdimensional space making use of bearing angle measurements. An Extended Kalman filter algorithm is considered for processing noise altered measurements along with smoothers algorithms to reduce the errors in the estimates of target parameters (range, course, and speed of the target). Details of mathematical modelling for simulating and implementation of the target and observer paths and outcomes are presented in this work.
This document provides an overview of PID controllers, including:
- The basic feedback loop and proportional, integral, and derivative algorithms
- Implementation issues like set-point weighing, windup, and digital implementation
- Practical operational aspects like bumpless transfer between manual and automatic modes
The document summarizes different types of digital counters, including asynchronous counters, synchronous counters, ring counters, and Johnson counters. Asynchronous counters have each flip-flop triggered by the previous one, limiting speed, while synchronous counters trigger all flip-flops simultaneously using a common clock, increasing speed. Ring counters circulate a single '1' bit around the register. Johnson counters are like ring counters but with the inverted output of the last flip-flop connected to the first. Examples and applications of each type are provided.
Control Signal Flow Graphs lecture notesabbas miry
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Signal flow graphs consist of:
π Nodes βrepresent signals
π Branches βrepresent system blocks
Branches labeled with system transfer functions
Signal flow graphs are an alternative to block diagrams for graphically describing systems. They consist of nodes to represent signals and branches to represent system blocks labeled with transfer functions. To convert a block diagram to a signal flow graph, identify and label all signals, place a node for each, connect nodes with branches in place of blocks while maintaining direction, and label branches with transfer functions. Mason's rule provides a formula to calculate the overall transfer function of a system represented by a signal flow graph based on terms like forward path gains, loop gains, and non-touching loop gains. Controller design uses feedback to modify a system's response to meet performance specifications by placing the closed-loop poles through selection of controller parameters.
This document discusses digital IIR filter design. It describes different types of analog filters like Butterworth, Chebyshev, Elliptic and Bessel filters. It then explains how to design digital IIR filters using analog filter design principles with impulse invariant and bilinear transformations from the s-domain to the z-domain. Step-by-step procedures are provided for designing low-pass Butterworth and Chebyshev filters using these two transformation methods. The concepts of aliasing and pre-warping frequencies are also covered in the bilinear transformation section.
The document discusses digital filters and their design. It begins with an introduction to filters and their uses in signal processing applications. It then covers linear time-invariant filters and their transfer functions. It discusses the differences between non-recursive (FIR) and recursive (IIR) filters. The document presents various filter structures for implementation, including direct form I and direct form II structures. It also discusses designing FIR and IIR filters as well as issues in their implementation.
This document summarizes techniques for designing finite impulse response (FIR) filters and digital controllers. It discusses truncating the impulse response of an ideal filter to design FIR filters. It also presents a method for digital controller design that matches the step response of an analog controller to approximate it digitally. Examples are provided of designing an FIR lowpass filter and approximating an analog feedback controller with a digital controller using step response matching.
This document summarizes techniques for designing finite impulse response (FIR) filters and digital controllers. It discusses truncating the impulse response of an ideal filter to design FIR filters. It also presents a method for digital controller design that matches the step response of an analog controller to approximate it digitally. Examples are provided of designing an FIR lowpass filter and approximating an analog feedback controller with a digital controller using step response matching.
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This presentation explores the concept of inductive bias in machine learning. It explains how algorithms come with built-in assumptions and preferences that guide the learning process. You'll learn about the different types of inductive bias and how they can impact the performance and generalizability of machine learning models.
The presentation also covers the positive and negative aspects of inductive bias, along with strategies for mitigating potential drawbacks. We'll explore examples of how bias manifests in algorithms like neural networks and decision trees.
By understanding inductive bias, you can gain valuable insights into how machine learning models work and make informed decisions when building and deploying them.
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The aquaponic system of planting is a method that does not require soil usage. It is a method that only needs water, fish, lava rocks (a substitute for soil), and plants. Aquaponic systems are sustainable and environmentally friendly. Its use not only helps to plant in small spaces but also helps reduce artificial chemical use and minimizes excess water use, as aquaponics consumes 90% less water than soil-based gardening. The study applied a descriptive and experimental design to assess and compare conventional and reconstructed aquaponic methods for reproducing tomatoes. The researchers created an observation checklist to determine the significant factors of the study. The study aims to determine the significant difference between traditional aquaponics and reconstructed aquaponics systems propagating tomatoes in terms of height, weight, girth, and number of fruits. The reconstructed aquaponics systemβs higher growth yield results in a much more nourished crop than the traditional aquaponics system. It is superior in its number of fruits, height, weight, and girth measurement. Moreover, the reconstructed aquaponics system is proven to eliminate all the hindrances present in the traditional aquaponics system, which are overcrowding of fish, algae growth, pest problems, contaminated water, and dead fish.
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Embedded machine learning-based road conditions and driving behavior monitoringIJECEIAES
Β
Car accident rates have increased in recent years, resulting in losses in human lives, properties, and other financial costs. An embedded machine learning-based system is developed to address this critical issue. The system can monitor road conditions, detect driving patterns, and identify aggressive driving behaviors. The system is based on neural networks trained on a comprehensive dataset of driving events, driving styles, and road conditions. The system effectively detects potential risks and helps mitigate the frequency and impact of accidents. The primary goal is to ensure the safety of drivers and vehicles. Collecting data involved gathering information on three key road events: normal street and normal drive, speed bumps, circular yellow speed bumps, and three aggressive driving actions: sudden start, sudden stop, and sudden entry. The gathered data is processed and analyzed using a machine learning system designed for limited power and memory devices. The developed system resulted in 91.9% accuracy, 93.6% precision, and 92% recall. The achieved inference time on an Arduino Nano 33 BLE Sense with a 32-bit CPU running at 64 MHz is 34 ms and requires 2.6 kB peak RAM and 139.9 kB program flash memory, making it suitable for resource-constrained embedded systems.
Embedded machine learning-based road conditions and driving behavior monitoring
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Lecture_8-Digital implementation of analog controller design.pdf
1. Digital implementation of Analog Controllers
Direct control design - Analytical Method
Dr. Amin Danial
2. References
β Katsuhiko Ogata Discrete-Time Control
Systems 2nd edition,1995
β M. Sami Fadali, Antonio Visioli - Digital
Control Engineering, Analysis and Design-
Second Edition_Academic Press (2019)
β Gene F. Franklin, J. David Powell, Michael L.
Workman - Digital Control of Dynamic
Systems-Prentice Hall (1998)
β A. V. OPPENHEIM, A. S. WILLSKY and S. H.
NAWAB , Signals & Systems, PRENTICE HALL,
1996.
3. Digital implementation of analog controller design
β’This lecture introduces an indirect approach to digital
controller design.
β’The approach is based on designing an analog controller
for the analog subsystem and then obtaining an
equivalent digital controller and using it to digitally
implement the desired control.
β’The digital controller can be obtained using a number of
methods that are well known in the field of signal
processing, where they are used in the design of digital
filters.
β’In fact, a controller can be viewed as a filter that
attenuates some dynamics and enhance others so as to
obtain the desired time response.
4. Procedure:
1. Design a controller πΆ(π ) for the analog subsystem to
meet the desired design specifications.
2. Map the analog controller to a digital controller πΆ(π§)
using a suitable transformation.
3. Tune the gain of the transfer function meet the design
specifications.
4. Check the sampled time response of the digital control
system and repeat steps 1 to 3, if necessary, until the
design specifications are met.
Digital implementation of analog controller design
5. β’ The transformation from an analog to a digital
filterβmust satisfy the following requirements:
1. A stable analog filter (poles in the left half plane
(LHP)) must transform to a stable digital filter.
2. The frequency response of the digital filter
must closely resemble the frequency response
of the analog filter in the frequency range 0 β
ππ
2
where ππ is the sampling frequency.
β’ Most filter transformations satisfy these two
requirements to varying degrees.
β’ However, this is not true of all analog-to-digital
transformations
Digital implementation of analog controller design
6. β’ The forward differencing approximation of the derivative is:
αΆ
π¦(ππ) β
π¦ π + 1 π β π¦ ππ
π
β’ In the same way
α·
π¦ ππ β
αΆ
π¦ π + 1 π β αΆ
π¦(ππ)
π
β
1
π
π¦ π + 2 π β π¦ π + 1 π
π
β
π¦ π + 1 π β π¦ ππ
π
β
1
π2
π¦ π + 2 π β 2π¦ π + 1 π + π¦ ππ
β’ This yields the mapping of π to π§ as follows:
π π π β
π§ β 1
π
π(π§)
β’ Therefore, the direct transformation of an s-transfer function to a z-transfer
function is possible using the substitution
π β
π§ β 1
π
Digital implementation of analog controller design
Differencing methods - Forward differencing
7. β’ Ex1: Apply the forward difference approximation of the
derivative to the second-order analog filter:
and examine the stability of the resulting digital
filter for a stable analog filter.
Digital implementation of analog controller design
Differencing methods - Forward differencing
8. β’ EX1(Cont.): Solution: (Note π¦(ππ) β‘ π¦(π))
β’ The differential equation from the given transfer function is:
β’ Then
β’ By multiply both sides by π2and rearrange the equation
β’ Equivalently, we obtain the transfer function of the filter using the
simpler transformation
Digital implementation of analog controller design
Differencing methods - Forward differencing
9. β’ EX1(Cont.): Solution:
β’ For a stable analog filter, we have π > 0 and ππ > 0
(positive denominator coefficients are sufficient for a
second-order polynomial)
β’ From Jury test, the instability condition ππ > π0 :
β’ If the sampling period of 0.2 s and an undamped natural
frequency of 10 rad/s yield unstable filters for any
underdamped analog filter.
Digital implementation of analog controller design
Differencing methods - Forward differencing
10. β’ The backward differencing approximation of the derivative is:
(Note π¦(ππ) β‘ π¦(π))
β’ Similarly
β’ This yields the substitution
Digital implementation of analog controller design
Differencing methods - Backward differencing
11. β’ Ex2: Apply the backward difference approximation of the
derivative to the second-order analog filter.
and examine the stability of the resulting digital filter for a stable
analog filter.
Solution:
Digital implementation of analog controller design
Differencing methods - Backward differencing
12. β’ Ex2: (cont.)
β’ The stability conditions for the digital filter (Jury test) are
β’ The conditions are all satisfied for π > 0 and ππ > 0βthat is, for
all stable analog filters.
Digital implementation of analog controller design
Differencing methods - Backward differencing
ππ < π0 β
π 1 > 0 β
π β1 > 0 β
13. β’ In pole-zero matching, a discrete approximation is obtained from an
analog filter by mapping both poles and zeros using ππ = πππ π
or
ππ§ = πππ§π
, where ππ or ππ§ is a pole or a zero in the z-domain and the
s-domain, respectively.
β’ If the analog filter has π poles and π zeros, then we say that the
filter has π β π zeros at infinity.
β’ For π β π zeros at infinity, we add π β π β 1 digital filter zeros at
β 1. Leading to the computation of the output requires values of the
input at past sampling points
β’ If the zeros are not added, it can be shown that the resulting system
will include a time delay.
β’ Finally, we adjust the gain of the digital filter so that it is equal to that
of the analog filter at a critical frequency dependent on the filter. For
a low-pass filter, πΌ is selected so that the gains are equal at DC; for a
bandpass filter, they are set equal at the center of the pass band.
Digital implementation of analog controller design
Pole-zero matching
14. β’ For analog filter:
πΊπ π = πΎ
Οπ=1
π
(π β ππ)
Οπ=1
π
(π β ππ)
β’ We get the corresponding digital filter as follows:
πΊ π§ = πΌπΎ
π§ + 1 πβπβ1 Οπ=1
π
(π§ β ππππ)
Οπ=1
π
(π§ β ππππ
)
β’ Where Ξ± is a constant selected for equal filter gains at a critical
frequency. For example, for a low-pass filter, Ξ± is selected to match
the DC gains using πΊ 1 = πΊπ(0).
β’ For a high-pass filter, it is selected to match the high-frequency
gains using πΊ β1 = πΊπ β (Setting π§ = π πππ = β1 (i.e., ππ =
π) is equivalent to selecting the folding frequency ππ /2, which is
the highest frequency allowable without aliasing)
Digital implementation of analog controller design
Pole-zero matching
15. β’ Ex3: Find a pole-zero matched digital filter
approximation for the analog filter.
If the damping ratio is equal to 0.5 and the undamped
natural frequency is 5 rad/s, determine the transfer
function of the digital filter for a sampling period of 0.1 s.
Digital implementation of analog controller design
Pole-zero matching
16. Ex3: (cont.) Solution
β’ The analog filter has two zeros at infinity and complex conjugate
poles at π = βπππ Β± πππ.
β’ using the pole-zero matching transformation we get
πΊ π§ = πΌ
π§ + 1
π§ β πβπππππππππ ( π§ β πβπππππβππππ
πΊ π§ = πΌ
π§ + 1
π§2 β 2πβππππ cos πππ + πβ2ππππ
For π = 0.5, ππ = 5, πππ π = 0.1. then ππ = ππ 1 β π2, the gain πΌ
is determined from πΊ 1 = πΊπ 0
Therefore,
πΊ π§ =
0.0963(π§ + 1)
π§2 β 1.414π§ + 0.6065
Digital implementation of analog controller design
Pole-zero matching
17. β’ Using the relation :
π§ = ππ π
=
π
π π
2
πβ
π π
2
β
1 +
π π
2
1 β
π π
2
Then
π =
2
π
π§ β 1
π§ + 1
β’ The bilinear transformation maps points in the
LHP to points inside the unit circle and thus
guarantees the stability of a digital filter for a
stable analog filter.
Digital implementation of analog controller design
Bilinear transformation
18. β’ Ex4: Design a digital filter by applying the bilinear transformation
to the analog filter
πΆπ π =
1
0.1π + 1
with π = 0.1 s.
Solution:
Digital implementation of analog controller design
Bilinear transformation
19. β’ Ex5: design a PI controller for the following
system
πΊπ π =
1
π + 2
to meet the following specs:
1. Damping ratio is 0.5
2. Natural undamped frequency is 5rad/sec
Then implement it digitally using Backward
difference ππ = 0.01π .
And then draw the whole system block diagram.
Digital implementation of analog controller design
20. β’ Ex5 (cont.)
β’ It is required that π = 0.5 and ππ = 5
β’ The controller TF is πΆπ π =
πΎππ +πΎπ
π
β’ The characteristic equation of the closed loop system
is 1 + πΆπ π πΊπ π = 0
1 +
πΎππ + πΎπ
π
.
1
π + 2
= 0
π 2
+ πΎπ + 2 π + πΎπ = 0
The required characteristic equation is:
π 2
+ 2πππs + ππ
2
= 0
Therefore
π 2
+ πΎπ + 2 π + πΎπ = π 2
+ 2πππs + ππ
2
Digital implementation of analog controller design
21. β’ Ex5 (cont.)
β’ By comparing the coefficients, we get:
πΎπ + 2 = 2 β 0.5 β 5
πΎπ = 3
πΎπ = ππ
2
= 25
Therefore, the PI controller is
πΆπ π =
3π + 25
π
By using the backward difference method
π =
π§ β 1
π§ππ
Digital implementation of analog controller design
24. β’ Ex5 (cont.)
Digital implementation of analog controller design
Analog Controller
Digital Controller
25. β’ For the following system:
Where
πΊ π§ = ππ‘ππππ
1 β πβπ π
π
πΊπ π
Direct control design - Analytical Method
26. β’ In this approach, it is required to find the desired
closed loop pulse transfer function πΉ π§ (This
approach to design is known as synthesis)
β’ Then
πΆ π§
π π§
=
πΊπ· π§ πΊ π§
1 + πΊπ· π§ πΊ π§
= πΉ(π§)
Consequently, the controller is found as follows:
πΊπ· π§ =
πΉ π§
πΊ(π§) 1 β πΉ π§
Direct control design - Analytical Method
27. β’ The designed system must be physically
realizable. The conditions for physical realizability
are as follows:
1. The order of numerator of πΊπ· π§ must be equal or
lower than the order of the denominator. Otherwise,
the controller will be noncausal, i.e. it needs future
input to produce the current output.
2. If the plant has a delay πβππ then the designed closed
loop system must have at least the same delay.
Otherwise, the closed loop system would have to
respond before an input was give, which is physically
impossible.
Direct control design - Analytical Method
28. 3. We must avoid of canceling any unstable pole of the plant be a
zero of the controller because any error in the cancellation will
cause instability.
If πΊ(π§) has a pole at πΌ then:
πΊ π§ =
πΊ1 π§
π§ β πΌ
Where πΊ1 π§ has no term that cancels π§ β πΌ.
Consequently,
πΉ π§ =
πΊπ· π§
πΊ1 π§
π§ β πΌ
1 + πΊπ·(π§)
πΊ1 π§
π§ β πΌ
1 β πΉ π§ =
1
1 + πΊπ· π§
πΊ1 π§
π§ β πΌ
=
π§ β πΌ
π§ β πΌ + πΊπ· π§ πΊ1 π§
Direct control design - Analytical Method
29. Therefore, πΌ is zero of 1 β πΉ π§ , hence:
πΉ πΌ = 1
4. The poles of the controller πΊπ·(π§) do not cancel zeros of
πΊ π§ which lies outside the unit circle.
Therefore, if there is a zero (π§ β π½)of πΊ(π§) which is outside
the unit circle. Then, we can write πΊ π§ as follows:
πΊ π§ = π§ β π½ πΊ2(π§)
Consequently, in case of (π§ β π½) is not cancelled:
πΉ π§ =
πΊπ· π§ π§ β π½ πΊ2(π§)
1 + πΊπ·(π§) π§ β π½ πΊ2(π§)
Hence,
πΉ π½ = 0
Direct control design - Analytical Method
30. 5. An additional condition can be imposed to
address steady-state accuracy
requirements.
In particular, if zero steady-state error due
to a step input is required
lim
πββ
π ππ = lim
π§β1
π§ β 1 πΉ π§ .
π§
π§ β 1
= 1
Therefore,
πΉ 1 = 1
Direct control design - Analytical Method
31. β’ Steps of design
1.Select the desired settling time Ts and the desired
maximum overshoot.
2.Select a suitable continuous-time closed-loop first-
order or second-order closed-loop system with unit
gain.
3.Obtain by converting the s-plane pole location to the
z-plane pole location using pole-zero matching.
4.Verify that πΉ(π§) meets the conditions for causality,
stability, and steady-state error. If not, modify πΉ(π§)
until the conditions are met.
Direct control design - Analytical Method
32. β’ EX6: Design a digital controller for a DC motor speed control
system where the (type 0) analog plant has the transfer function
πΊπ =
1
(π + 1)(π + 10)
To obtain zero steady-state error due to a unit step, ππ =
1.15 πππ/π , and π = 0.88. The sampling period is chosen as π =
0.02 s
β’ Solution
Since in digital control system a ZOH is always inserted between the
digital controller and the system to be controlled, hence
πΊ π§ = ππ‘ππππ
1 β πβπ π
π
πΊπ π = (1 β π§β1)ππ‘ππππ
πΊπ π
π
Direct control design - Analytical Method
33. β’ EX6 (cont.)
πΊ π§ = 1.86 Γ 10β4
π§ + 0.9293
(π§ β 0.8187)(π§ β 0.9802)
Then, based on the requirements πΉ π is:
πΉ π =
ππ
2
π 2 + 2ππππ + ππ
2 =
1.322
π 2 + 2024π + 1.322
Using zero-pole matching
πΉ π§ = 0.25921 Γ 10β3
π§ + 1
π§2 β 1.95981π§ + 0.96033
Direct control design - Analytical Method
34. EX6(cont.):
Then from
πΊπ· π§ =
πΉ π§
πΊ(π§) 1 β πΉ π§
We get
πΊπ· π§ =
1.3932(π§ β 0.8187)(π§ β 0.9802)(π§ + 1)
(π§ β 1)(π§ + 0.9293)(π§ β 0.9601)
Direct control design - Analytical Method