Design and MATLAB Simulation: Dead Beat Controller for First Order System

Summary

The video discusses the design and functionality of a deadbeat digital controller. It explains the assumptions made while designing the controller for a system with step input, emphasizing the sampling periods and time delays inherent to the process. The script covers the mathematical derivation of the transfer functions, conversion between s-domain and z-domain, and provides insights into the controller's output in a discrete system. The focus is on achieving a step response with a one-sampling period delay, illustrating the accuracy of the deadbeat controller in maintaining stability.

Takeaways

• 🎯 The script discusses the design and implementation of a deadbeat controller, a type of digital controller used in control systems.
• 🔍 The deadbeat controller is designed with the assumption that the set point is changed to S = 1, and the controller should respond to this change.
• 📊 The script explains the use of the Z-transform to analyze the system's response, particularly focusing on the step input and its transformation.
• 🛠️ The process of the system is given as G(s) = 1/(3s+1), and the zero-order hold is introduced with a transfer function H(s) = (1 - e^(-ts))/s where t is the sampling period.
• #️⃣ The script details the calculation of the digital controller's transfer function, D(z), which is derived from the process and hold circuit transfer functions.
• 🔢 The script provides a step-by-step guide to finding the controller's transfer function in the Z-domain, emphasizing the cancellation of terms and the final form of the controller.
• 📉 The response of the system to a step input is expected to have a delay of one sampling period, which is a key assumption in the design of the deadbeat controller.
• 🔄 The script also covers the conversion of the controller's transfer function from Z-domain to time domain, which is necessary for implementation in a digital system.
• 🔧 The script concludes with a simulation example to verify the effectiveness of the designed deadbeat controller, demonstrating its response to a step change in input.
• 📝 The importance of converting the controller's coefficients to positive values for simulation purposes is highlighted, ensuring the controller's practical application.

Q & A

What is a deadbeat controller in digital control systems?

A deadbeat controller is a type of digital controller designed to achieve the desired output in the minimum possible time, typically after one sampling period, by minimizing the error between the set point and the process output.

What assumption is made when designing a deadbeat controller?

The assumption made when designing a deadbeat controller is that the system can respond to a step input immediately after one sampling period without any delay, which is why the controller is designed to achieve the set point after one sampling period.

What is the significance of the set point being equal to 1 in the context of the deadbeat controller?

In the context of the deadbeat controller, setting the set point to 1 represents the desired steady-state output that the controller aims to achieve. It's a standard way to design the controller to handle a unit step input.

How does a zero-order hold affect the system when implementing a digital controller?

A zero-order hold introduces a delay in the system's response, which is considered when designing the deadbeat controller. It ensures that the controller's output is held constant between sampling periods, affecting how the controller is designed to respond to inputs.

What is the process transfer function mentioned in the script?

The process transfer function mentioned in the script is '1/(3s + 1)', which represents the dynamic behavior of the process being controlled in the Laplace domain.

Why is it necessary to consider the sampling period when designing a digital controller?

The sampling period is crucial in designing a digital controller because it determines the frequency at which the controller updates its output. This directly impacts the controller's ability to respond to changes in the system and achieve the desired set point.

What is the role of the hold circuit transfer function in the digital control system?

The hold circuit transfer function, often represented as '1 - e^(-sT)' where T is the sampling period, is used to model the behavior of a zero-order hold. It is essential for converting the continuous output of the digital controller into a form that can be used by the discrete-time process.

How does the deadbeat controller respond to a step input in the time domain?

The deadbeat controller is designed to respond to a step input by achieving the set point after one sampling period. This means that the output of the process should reach the desired value immediately after one sampling period, with minimal overshoot or error.

What is the significance of the term 'E^(-Ts)' in the context of the controller design?

The term 'E^(-Ts)' represents the effect of the delay introduced by the zero-order hold in the Laplace domain. It is used to account for the time shift that occurs due to the sampling and holding of the controller's output between sampling periods.

How is the digital controller's transfer function derived from the process and hold circuit transfer functions?

The digital controller's transfer function is derived by taking into account the process transfer function and the hold circuit transfer function. The controller is designed such that the product of the process and hold circuit transfer functions, when multiplied by the controller transfer function, results in a unity gain system with a step response that reaches the set point after one sampling period.

Outlines

00:00

🔍 Introduction to Deadbeat Controller

The paragraph introduces the concept of a deadbeat controller, a type of digital controller used in control systems. The controller is designed to respond to a set point change, specifically aiming for the set point to be 1. The discussion revolves around the mathematical representation of the controller's behavior, including the use of the Z-transform to model the system's response to a step input. The controller's expectation is that after one sampling period, the system output should reach the desired set point, which is a key assumption in the design of the deadbeat controller. The paragraph also touches on the system's process and the role of a zero-order hold in shaping the input signal.

05:02

📐 Derivation of the Deadbeat Controller

This section delves into the mathematical derivation of the deadbeat controller. It starts with the assumption of a system's transfer function and the hold circuit's transfer function. The paragraph explains the process of finding the controller's transfer function by manipulating the expected system response and the given input. The focus is on canceling out terms to simplify the expression and arriving at the digital controller's formula. The importance of understanding the system's dynamics and the role of the sampling period in the design of the controller is emphasized. The paragraph concludes with the assumption of specific values for the system's transfer function and the hold circuit's transfer function to illustrate the calculation.

10:02

🛠️ Implementing the Deadbeat Controller

The paragraph discusses the practical implementation of the deadbeat controller. It explains the conversion of the controller's transfer function from the Z-domain to the time domain, which is necessary for applying it in a real-world scenario. The focus is on the algebraic manipulation required to express the controller's output in terms of past and present error signals. The paragraph provides a step-by-step guide on how to convert the controller's formula into a form that can be used in simulations. It also touches on the need to ensure the coefficients are in positive form, which is crucial for the controller's implementation in a digital system.

15:07

🔬 Simulation of the Deadbeat Controller

The final paragraph describes the simulation process to validate the deadbeat controller's effectiveness. It outlines the setup for the simulation, including the system's transfer function and the digital controller's parameters. The paragraph emphasizes the expected system response after one sampling period, which should be in line with the design goals of the deadbeat controller. The simulation results are discussed, confirming that the controller performs as expected, achieving the desired output after one sampling period. The paragraph concludes by suggesting further exploration of the deadbeat controller in subsequent discussions.

Keywords

💡Deadbeat Controller

A deadbeat controller is a type of digital controller designed to bring a system to its desired state in the shortest possible time, usually within one sampling period. In the video, the deadbeat controller is designed with the assumption that the system’s output reaches the desired value after one sampling period. It minimizes the delay in system response, which is critical in digital control systems.

💡Set Point

The set point refers to the desired target value that a system aims to achieve. In the video, the set point is represented mathematically as '1/S' and is used as the basis for designing the deadbeat controller. The controller is tasked with ensuring that the system follows the set point as closely as possible after each sampling period.

💡Step Input

A step input is a type of signal input that changes from one value to another instantaneously. It is often used to test system responses. In the video, a step input is provided to the digital controller to observe how the system responds, especially after one sampling period, where the expected outcome is a delayed response.

💡Sampling Period

The sampling period is the time interval at which the system is evaluated or measured. In the context of the video, the sampling period is key to understanding how the deadbeat controller works, as the response of the system is expected to occur after exactly one sampling period due to the inherent delay in processing through the system blocks.

💡Z-Transform

The Z-Transform is a mathematical tool used in digital signal processing to convert a discrete-time signal into the frequency domain. It plays a critical role in the video’s explanation of the deadbeat controller, as it allows the analysis of the system’s behavior in response to inputs like the step signal in a digital context.

💡Zero Order Hold (ZOH)

Zero Order Hold (ZOH) is a method used to convert a digital signal into an analog signal by holding the last sample value constant until the next sample is received. In the video, it is mentioned as a critical component in the control process, as it helps bridge the gap between discrete digital control signals and continuous processes.

💡Transfer Function

A transfer function is a mathematical representation of the relationship between the input and output of a system. In the video, the transfer function of the system (given as 1/(3S + 1)) is used to determine how the system behaves in response to inputs, and it is essential in designing the controller.

💡Process Delay

Process delay refers to the time lag between when an input is applied to a system and when the output response occurs. In the video, the deadbeat controller accounts for a process delay of one sampling period, meaning that the system’s output responds to a step input only after this delay.

💡Error Signal

The error signal is the difference between the desired set point and the actual output of the system. In the video, the error signal is used as the input to the controller, and the controller adjusts its output based on the magnitude of this error to bring the system back to the set point.

💡Digital Controller

A digital controller is a control system that operates on digital signals, using algorithms and digital logic to adjust system outputs. In the video, the digital controller (D(z)) is designed based on the system’s transfer function, aiming to regulate the system’s behavior with minimal delay in response to a step input.

Highlights

Introduction to the concept of a deadbeat controller in digital control systems.

Explanation of the assumption that the set point of the system should change to 1.

Designing a controller with a set point change to achieve the desired system response.

The importance of understanding the system's step input and its digital representation.

Description of the zero-order hold in the system and its impact on the process.

The process transfer function and its relation to the system's dynamics.

Expectation of the system's response after a step input and the significance of the first sampling period.

Assumption made in designing the deadbeat controller regarding the system's output after a step change.

The concept of delay in the system's response due to the sampling period.

Derivation of the deadbeat controller's transfer function based on the system's expected response.

The role of the hold circuit transfer function in the overall system's behavior.

Conversion of the continuous-time process to discrete time for digital controller design.

The formula for converting a continuous-time transfer function to its discrete-time equivalent.

The derivation of the digital controller's transfer function for a given process.

The impact of the sampling period on the system's response and controller design.

Simulation of the deadbeat controller to verify its effectiveness in achieving the desired system response.

The final form of the digital controller and its implementation in the simulation.

Conclusion and summary of the deadbeat controller's design and its practical application.

Transcripts

deadbeat controller

it is a digital controller

so now we will come to that deadbeat  controller deadbeat controller

the deadbeat controller is the Assumption we made  is we are designing a controller first set point  

change that is why set point of visit S equal  to 1 by S why is the point of s equal to 1 by  

S for that we are going to design a controller we  are going to give a set point change further this  

controller should work for that we are going to  design therefore why is a point of s equal to 1  

by S therefore y set point of visit equal to 1  by 1 minus Z power minus one if you take is a  

transform for 1 by S you will get one by one minus  Z power minus 1. this and all you have to memorize

we know the set point

suppose I am giving step input  to the system this is the system  

this is the system I am giving y set  point of s equal to 1 by S I am giving

a digital controller is the D opposite  is there a zero order hold is there  

then a process is there  process is 1 by 3 S Plus One

for that I have to find a Dr wizard  here also one sampler is there

here what is y f is that I am I am expecting  that if I give you a step input like this  

in digital how it appears this is one by S what  is 1 by 1 minus is it for minus 1 if you plot  

it will be like this this is the step input  1 by 1 minus Z power minus one digital step

that is given if this is given I am expecting a  response of like this I'm expecting a response  

like this what is the meaning this is first  sampling period this is first sampling period

this is second sampling period zero of sampling  period third sampling period if I give at zeroth  

instant if I give you a step change the process  output will be like this this is the Assumption  

we made while designing the deadbeat controller  the input is like this here the input is like this  

at zeroth instead the output is like this  the output is like this after one sampling  

period I am getting the response that is the  Assumption we made why one sample repair in  

delay we cannot expect at the same time that's  why we are giving one sampling period delay  

we can expect after one sampling period  because it has to pass through all these blocks  

that's why it has to pass through the process  process as the time time constant that's why  

we can expect at least after one sampling period  so after one sampling period the response is like  

step input therefore what is the Y of here yes is  one by S it is also step two but not one by S the  

radius delay how much one sampling period That's  why E power minus t s by S this is in s domain  

what is in Easy domain for a step y opposite equal  to 1 by 1 minus is power minus one this is that is  

that domain step signal one by one minus Z power  minus one the delay is that delay by one sampling  

period that's why we have to multiply by Z power  minus one that's all so here we said is equal to  

One Step signal but delay by one sampling period  suppose two sampling period 11 is z for minus two

here we are assuming that the response is  available after one sampling period the response  

is available after one sampling period therefore  E Z power minus one upon 1 minus Z power minus one  

so here uh what is what is uh we have visited by  way off he said therefore the transfer function  

output by input equal to what output is the Z  power minus 1 upon 1 minus Z power minus one  

input is one by one minus Z power minus one these  two getting canceled that is equal to E Z power  

minus 1 reset power minus one therefore y f  is it by set point of visit equal to what is  

the response we are expecting a step output is  expecting but a delay by one sampling period  

what is the input we are giving step input  that is one pi 1 by 1 minus Z power minus 1.  

so the denominator getting canceled therefore E  Z power minus one so now we can find the digital  

controller what is digital controller  D opposite equal to 1 by h g p of Z  

multiplied by y opposite by

therefore the controller is 1 by hgp of z e z  power minus 1 divided by 1 minus Z power minus  

1 that's up if you know HTTP of is at that you can  find the control so if you find HTTP of his ad you  

can find the controller let us assume that the  G of s equal to G of s equal to 1 by 3 S Plus 1

this is G of s let us assume what is hold circuit  transfer function of volt circuit is H of s equal  

to 1 minus E power minus t s upon s t equal  to 1 sampling period equal to one therefore  

one minus E power minus s upon s therefore  h g p of s this is GP process transaction  

HC GP of s equal to 1 minus E power minus s t  s sound like when it is 1 upon s multiplied by  

transfer function is one by three S Plus One  any delay if you find the transfer function  

is a in either domain if you convert this integer  domain you will get reset power minus 1 that's a  

E power minus c s equal to E Z power  minus one if sampling period is one  

therefore what is HTTP of z h g p f is it  because to find digital controller you need  

HTTP result so we have to convert this yes domain  user domain therefore 1 minus we said power minus  

1 for the numerator and is a transform of 1 upon  s multiplied by three S Plus 1 that you've got  

so we have to find that is a transform of is a  transform of 1 upon s multiplied by three s plus  

one that is HTTP officer that is HTTP officer so  that the standard formula is available that is  

is a transform of 1 by S multiplied  by Tab S Plus 1. equal to

1 by Tau 1 minus E power 1 by Tau

T by Tau actually T by T is sampling period  yes sampling period equal to one therefore E  

power 1 by T E Z power minus 1 upon 1  minus Z power minus 1 multiplied by 1  

minus E power minus 1 by T by Tau  t equal to 1. this is the formula

this is let us put this as Tau Tau this is  Tau this is Tau sampling period equal to  

sampling period equal to T therefore T by Tau

T by Tau here also T by t equal to one  that's why I wrote one by two already

what is h of s 1 minus E power minus t s divided  by s s only we brought here the remaining thing  

is 1 minus Z power minus 1 therefore HTTP of is  that equal to 1 minus Z power minus 1 multiplied  

by 1 minus E power minus 1 by 3 because what we  assumed G of s equal to 1 upon 3 S Plus 1 1 by 3  

huh multiplied by E Z power minus 1  whole divided by 1 minus Z power minus  

1 multiplied by 1 minus E power 1 by 3 is a  power minus 1. so these two getting canceled  

these two getting canceled therefore h g p are  visit equal to 1 minus E power minus 0.33 is

minus 1 whole divided by point seven one six

one minus

0.716 is that power minus 1.

this is HTTP officer HTTP of visit equal to  um equal to 1 minus 0.716 is 0.283 is a power  

minus one whole divided by 1 minus 0.716 Z power  minus one this is http therefore B of is it equal  

to water digital controller d a visit equal to  one by h g p of visit we are assuming that for  

starter system that's why we have is Z power minus  one whole divided by 1 minus Z power minus 1. HTTP  

of is it equal to what just now we found 1 minus  1 minus 0.716 Z power minus 1 whole divided by 0.2  

83 E Z power minus 1 the numerator is a power  minus 1 the denominator 1 minus Z power minus 1.  

that is equal to E Z power minus 1 minus 0.716

the Z power minus 2 whole divided by 0.283 E  Z power minus 1 minus 0.283 E Z power minus 2.  

this is the digital controller if you write  this way this will be convenient to convert  

into time domain for that it will be  useful you can enter like that also  

how to convert this into time domain  cross multiply E Z power minus 1 means  

if you convert this into time  domain is it for minus 1 means

DF K minus 1.

1 means one means D of K this is B of K minus 2.71  D of K minus 1 but B of K minus 1 equal to e of  

uh M of K controller output divided by E of K  means if we cross multiply you can write like this  

M of K upon e of K suppose you are giving the  error as input to the controller controller output  

is M of K that is E Z power minus 1 minus 0.716  is that power minus 2 upon 0.283 is that power  

minus 1 minus 0.283 is that power minus two if you  cross multiply you will get the M of K minus 1.  

we can take this uh 0.283 outside M of  K minus 1 minus M of K minus 2 equal to  

1 by 0.283 multiplied by if you bring the D  of K this side you will get e of a minus 1.  

minus 0.716 E of K minus two if you shift by a H  one sampling period you will get M of k equal to  

M of K minus M of K minus 1 equal to 1 by 0.283  e of K present error minus 0.716 E of K minus one  

so this is why I am telling this is what is the  controller output M of K is the percent controller  

out equal to you bring this to the right hand  side you bring this to the right hand side  

if you if you bring this to the right hand side in  plus sign what is the controller output pressure  

previous controller output plus 1 by 283 times  percent error minus 0.716 times divided by 0.283  

previous error previous sample instant error  like that we can find the controller output  

this is to understand what is DF is  it but in simulation you can enter  

this value directly but you have to convert  this into positive coefficients so multiply  

on both sides scissor power plus 2 you  will get one Z power 1 minus 0.716 whole  

divided by here also you multiply by E Z  power 2 you will get 0.283 is that minus

this is the controller in positive powers is  at minus 0.716 divided by 0.283 Z minus 0.283  

so the controller is D opposite  we got B of is it equal to  

all right Z minus point seven  on six point seven on six upon

0.283

Z minus 0.283 this is the digital controller for  R did we controller for 1 upon 3 S Plus 1 times  

let us simulate and check whether  it is working or not let us check

so the transfer function is 1 upon 3s plus  one the digital controller we found is  

the digital controller we found is

one minus point seven minus 0.716 coefficient of  numerator is entered we have to select a discrete  

transfer function in the Toyota denominator  is 0.283 minus 0.28 coefficients we entered  

so that's all about digital controller  we put this is the zero order hold

domain so between these two zero order  voltage input must be there we simulate the

this is for step change so we gave a step change  in the input but the response is after one  

sampling liquidity is settled what way we designed  we are expecting the responses it should be one  

after one sampling period the response must be one  after one sampling period that is correct only now  

it is at a settled at one so what we designed is  after one sampling query the response must be one  

it is there you understand this so we will see  more about deadbeat controller in the next case