Design and MATLAB Simulation: Dead Beat Controller for First Order System
Summary
TLDRThe 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
🔍 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.
📐 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.
🛠️ 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.
🔬 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.
Mindmap
Keywords
💡Deadbeat Controller
💡Set Point
💡Step Input
💡Sampling Period
💡Z-Transform
💡Zero Order Hold (ZOH)
💡Transfer Function
💡Process Delay
💡Error Signal
💡Digital Controller
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
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