Báo cáo nghiên cứu khoa học: Research on control methods of sampled data systems and application

Block diagram of transfer function of CRC system with zero-order hold Input signal Output signal... Furthermore, building upon the conceptualframework of time-delay systems, an advanced

VIET NAM NATIONAL UNIVERSITY, HA NOI

INTERNATIONAL SCHOOL

STUDENT RESEARCH REPORT

RESEARCH ON CONTROL METHODS OF SAMPLED DATA

SYSTEMS AND APPLICATION

<CN.NC.SV 23_01>

Hanoi, 15/4/2024

TEAM LEADER INFORMATION

- Program: Automation and Informatics li š

- Address: Tien Lang, Hai Phong

- Phone no /Email: 0979487546 — 21070370@vnu.edu.vn

II Academic Results (from the first year to now)

Academic year Overall score Academic rating

2021 3.2 Good

2022 3.36 Good

2023 3.37 Good

III Other achievements:

- GPA semester II, 2022-2023 academic year 4.0/4.0

- Academic encouragement scholarship for Excellent Class I at VNU-IS

Hanoi,15 thang 12 nam 2024

Advisor Team Leader

(Sign and write fullname) (Sign and write fullname)

` P4 Hưng

YAS Dao Quoc HungPham Ngoc Thanh

TABLE OF CONTENTS

1.

2.

Literature TCVICW: HH TH HH HH HH HH kệ 13Chapter I: Stability of digital control system: - c5 1x seesee 16

"` m— 162.2 Problem Statement 8n ẻ 16

2.2.1 General framework of digital computer control sySf€1S: - «+++ 16

2.2.2 Sampled data implementation: - «+ cee eseeseeseeeeeeeeeceeessesseeseeseeaeeaees 172.3 Zero-Order NOI! oo eee ee eee ẻẻ ố 182.4 General digital control system represented by sampled datfa: - 192.5 The Z-tramsfOrm: 0 eee 192.6 Closed-loop feedback sampled-data SWS€TNS: s6 5 S1 sssirsrssre 212.7 Stability of digital control SYSf€THS: - ĩ- 5 SH ng ng nệt 212.7.1 Stability CONCILION: - Q.1 11v TH HH HH HH ky 21

"Nhan ậš87ỮDU -.: 222.8 Engineering application: 1 23

Chapter II: Stability of time-delay System — Pade approximation: 30kbtddididididididiiiida: - 303.2 Problem Staf€IT€HIẨL - - 25 131191 11910191 ng HH nh nh cư 303.2.1 General framework Of control SySteMS: - 5 + E23 1E kE SE ri, 303.2.2 General control systems represented by transfer functIOnS: - 323.2 Structure controller ImpÏermenf4fIOI: - <5 + + SE E*ekEsskEseekeskeeeee 333.3 _ Time-delÌay issues! esc eesccesceesceessecesceesceceseceseesscessnecsaeesseesseeceaecsseeseeeaeenas 333.4 Model of feedback with time-delay inputs - 55555 £+s£+sv+seesske 333.5 | Model of feedback with time-delay happening at the measurements: 343.7 Stability of dynamical system with time-delay: ec eeseecceneeeeceeeeneeeeeeseeeneeaees 35

3

COMCIUSIONL 0001ẼẺ10577 62ADDIeVIALION! 0000Ẻ8ẺẺ86 62Reference? cee cesscscceeseceeesecneeessesseesesseseseeeseeneeas Error! Bookmark not defined.

[LIST OF TABLES]

s” An An-2 An—4 s1 An-1 an-3 An—s5

sh? Dn-1 Dn—3 bạ—s

sv Cn-1 Cn-3 Cn—s

s09 hn-1

Table 1: Routh Table

Step | Obtain K, A, B, A, = BKC from eq (3 — 11)

Step 2 Run LMI in eq (3-13) with Robust Control ToolBox if tin < 0, the stability is

guaranteed, otherwise the stability is not guaranteed and the system may be unstable

Step 3 Run simulation to check the resources

Table 2: Process to test the system’s stability

[LIST OF FIGURES]

joi (Digital) woe

Digital hà Digital-to- Actuator

Digital Digital-to- ' y(t)

an

computer

analog-converter Process

to-digital converter

Analog-Figure 2.4 Bock diagram of digital control of the studied systems

Figure 2.6 The block diagram of digital control system

Desired Zero-order Hold Controller Motor Reel Dynamics

output E(s) — Actual cable

response a Us) T(s) velocity

Destred Zero-order Hold Controller Motor Reel Dynamics

output — FÉ) ` Actual cable

response /^, T(s) 1 velocity

' Tachometer The feedbacl-loop

Figure 2.8 Block diagram of transfer function of CRC system with zero-order hold

Input signal Output signal

Desired Actuator Process

Figure 3.4 Block diagram of transfer functions of the studied systems

Desired Controller Actuator Process

output A

rospanse —— Actual process

V(s)

Measured velocity

Motor Reel Dynamics

Desired Controller —~ ——_ Actual cable

velocity E(s) U(s) T(s) 1 velocity

: hE

R(s) + [_ sai s+12

V(s)

Measured velocity

ị M(s)

— :

Tachometer The feedback-loop

Motor Reel Dynamics

—~ ——— Actualcable

T(s) 1 velocity

Measured velocity

Figure 3.7 Diagram of transfer function of the studied system

Controller Motor Reel Dynamics

Desired —— —— f ` Actual cable

velocity

V(s)

Measured velocity

The feedback-loop

Figure 3.8 Block diagram of transfer function of CRC system with time-delay in feedback

10

Controller Motor Reel Dynamics

Actual cable velocity T(s)

V(s)

Measured velocity

RESEARCH ON CONTROL METHODS OF SAMPLED DATA SYSTEMS AND APPLICATION

1. Project Code: CN.NC.SV 23 01

2 Member List:

Full Name Class ID

Dao Quoc Hung AAI2021B 21070370Nguyen Viet Tung AAI2021B 21070348

Advisor(s):

Pham Ngoc Thanh — Lecturer PhD — Faculty of applied sciences

Abstract (300 words or less):

This research project delves into the realm of control algorithms tailored forsampled data systems, with applications in engineering contexts Initially, attention

is directed towards elucidating the control structure inherent in sampled datasystems, alongside methodologies for their representation via z-transformations.This phase also encompasses a meticulous analysis of stability and transientresponse characteristics employing transfer functions Subsequently, the operation

of sampled data systems undergoes scrutiny, revealing parallels with time delaysystems, where the delay is confined within the bounds of the sampling time.Addressing the challenges posed by such time-delayed systems, the Padeapproximation emerges as a pivotal tool, facilitating the transformation of thesesystems into conventional dynamical structures Consequently, this transformationenables the application of established techniques such as the Routh-Hurwitz test toassess system performance effectively Furthermore, building upon the conceptualframework of time-delay systems, an advanced approach rooted in Lyapunovstability is explored to ensure the robust stability of the studied systems, therebypaving the way for the delineation of control regions essential for system operation

Keywords (3 — 5 words)Sampled data system, time-delay system, modern control, Pade approximation

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SUMMARY REPORT IN STUDENT RESEARCH,

2023-2024 ACADEMIC YEAR

1 Literature review:

In many contemporary engineering systems, the control of system variables overtime is imperative Controllers play a crucial role in ensuring desirable transient andsteady-state behaviors for these systems To achieve satisfactory performance despitedisturbances and model uncertainties, most controllers employ negative feedbackmechanisms Typically, a sensor is employed to measure the system's output, which isthen compared to a reference signal Control action is subsequently determined based

on the error signal, representing the disparity between the reference and actual values.Traditionally, controllers have been analog systems, comprising electrical, fluidic,pneumatic, or mechanical components These systems operate with analog inputs andoutputs, meaning their signals vary continuously over time and encompass a continuousrange of amplitudes However, in recent decades, analog controllers have beenprogressively supplanted by digital counterparts Digital controllers, operating withinputs and outputs defined at discrete time points, manifest as digital circuits, computers,

or microprocessors Firstly, one might assume that controllers continuously monitoringsystem outputs would outperform those relying on sampled output values It would seemlogical that control variables undergoing continuous changes would provide superiorcontrol compared to those changing periodically Indeed, this assumption holds true Ifall other factors were equal between digital and analog control, analog control wouldindeed be superior to digital control Digital control involves systems where controlactions are updated at specific discrete time intervals Discrete-time models establishmathematical relationships between system variables at these discrete time points Inthis chapter, we explore the mathematical properties of discrete-time models, whichserve as the foundation for subsequent discussions While this content may serve as areview for many readers familiar with basic control and system theory, it is self-contained, requiring no prior knowledge of discrete-time systems

To devise a digital control system, our objective is to derive either a z-domain transferfunction or a difference equation model for the controller that aligns with the specifieddesign criteria This model can either be derived from an analog controller that fulfillsthe same design specifications or directly designed in the z-domain using methodologiesclosely resembling those employed in analog controller design within the s-domain.Both of these approaches are elaborated upon within this chapter The Z-transformsignificantly simplifies the analysis and design processes of practical devices like filtersand controllers As we progress through the book, the benefits of utilizing the Z-transform in streamlining these tasks will become increasingly apparent Z-transformsplay a crucial role in simplifying operations involving difference equations Forinstance, the convolution of two discrete-time signals is streamlined into the product oftwo algebraic expressions through the application of Z-transforms To illustrate this, let's

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examine the convolution sum While this approach may not be particularlyadvantageous for short sequences due to comparable computational efforts involved inevaluating convolution and polynomial products, it proves beneficial for sequencescomprising an infinite number of elements Addressing two key concerns, firstly, wemust ascertain the convergence of a sum consisting of an infinite number of elements.Secondly, we need to determine whether we can uniquely recover the time domainelements, 1.e., the values taken by the sequence at different time intervals, from theproduct polynomial These issues will now be explored further.

Stability is a cornerstone attribute of any feedback control system, dictating its ability

to maintain equilibrium and desired performance over time Once stability is assured,attention naturally shifts to exploring the system's relative stability, which pertains to itsresilience to disturbances and variations in operating conditions To discern the stabilitycharacteristics of a system, engineers employ various methods capable of assessing bothits absolute and relative stability.By scrutinizing the arrangement of polynomialcoefficients, this method provides insights into the system's stability properties, allowingengineers to make informed decisions regarding system design and performanceoptimization However, while these methods offer valuable insights into the absolute andrelative stability of a system, they primarily operate in the frequency-independentdomain This limitation prompts the exploration of stability analysis techniques in thefrequency domain, where the system's response to sinusoidal inputs is scrutinized.Frequency response analysis provides a comprehensive understanding of a system'sbehavior across a range of frequencies, offering critical insights into its stability marginsand robustness.By experimentally exciting the system with sinusoidal input signals,engineers can obtain its frequency response characteristics, even in scenarios whereprecise parameter values are unknown This empirical approach to frequency responseanalysis empowers engineers to assess the relative stability of a system and identifypotential areas for improvement or optimization.Moreover, the development of afrequency-domain stability criterion by H Nyquist in 1932 revolutionized stabilityanalysis in linear control systems.In essence, stability analysis in both the time andfrequency domains plays a pivotal role in the design, analysis, and optimization ofcontrol systems By leveraging a diverse array of methods and techniques, engineers canensure that feedback control systems exhibit robust stability characteristics across abroad spectrum of operating conditions, thus fulfilling their intended functions reliablyand effectively

Arcoding to Ms Fridman's paper introduces an enhanced method for robust data control aimed at improving system performance This method conceptualizes thesystem as a continuous-time entity with a control input featuring continuous partial

sampled-14

delay By employing a descriptive approach to address time-delayed systems, succinctlinear matrix inequality (LMI) conditions are established to attain sampled-data statefeedback stability This novel approach is applied to tackle two distinct challenges:achieving sampled-data stability in systems characterized by multi-subject uncertaintyand ensuring regional stability through sampled-data saturation state response Central

to this method is the modeling of continuous-time systems with digital control ascontinuous-time systems featuring delayed control inputs This strategic modelingexpands the applicability and enhances the efficiency of the proposed method

In this project, we will use continuous approaches to solve problems with discretesystems, thereby comparing and evaluating the effectiveness of the system Thestructure of our project is as follows

e In the first chapter of our study, we delve deeply into the intricate control structure

inherent in sampled data systems This pivotal exploration is rooted in thefundamental understanding that sampled data systems can be effectivelyconceptualized and analyzed within the framework of discrete-time systems Hereinlies the crux of our approach: by considering sampled data systems through the lens

of discrete-time systems, we pave the way for the implementation of sophisticatedalgorithms Specifically, we harness the power of Z-transforms, a cornerstonemathematical tool in digital signal processing, to meticulously dissect the dynamics

of these systems Furthermore, our investigation extends to the realm of stabilityanalysis, where we navigate through the intricate stability conditions existing in theZ-domain By rigorously applying these stability criteria, we aim to ensure therobust performance and reliability of the sampled data systems under scrutiny Thus,through our comprehensive examination of control structures and stabilityconsiderations, we lay the groundwork for a profound understanding of theperformance dynamics inherent in sampled data systems

e In Chapter 2, an investigation is conducted into a novel approach aimed at

controlling sampled data systems This exploration is grounded in the recognitionthat such systems can be characterized as possessing inherent time-delay properties,wherein the delay is confined within the temporal bounds dictated by the samplinginterval Employing the Pade approximation methodology, these time-delaysystems can be effectively transformed into linear systems devoid of delaycomponents Subsequently, this transformation facilitates the application of adiverse array of control techniques, strategically employed to optimize theperformance and efficacy of the systems under study

e In Chapter 3, Moreover, extending upon the conceptual framework delineated by

time-delay systems, our investigation delves into an advanced methodologicalapproach grounded in Lyapunov stability theory This rigorous exploration aims

to ascertain the robust stability of the systems under scrutiny By leveraging

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Lyapunov stability theory, we endeavor to provide a comprehensiveunderstanding of the stability properties inherent in the studied systems, therebylaying the groundwork for the delineation of critical control regions essential forthe effective operation of these systems.

2 Chapter I: Stability of digital control system:

2.1 Abstract:

This chapter considers the stability of digital control systems At first, we haves presentedthe notice of a sampled-data system by discussing the z- transforms by s Due to the use of

z — transform of a transfer functions, we have analysed the stability and transient response

of a systems The basics of closed-sloop stability with a digital controller in the loop arecovered with a short presentation on the role of root locus in the design process

2.2 Problem Statement:

2.2.1 General framework of digital computer control systems:

The benefits of digital control encompass enhanced measurement sensitivity, facilitated bylow-energy signals utilized by digital sensors and devices Digital control also leveragesdigitally coded signals, digital sensors, transducers, and microprocessors, enablingwidespread application and communication

sot (Digital oe Analog

Digital gital) Digital-to- Actuator Process (Analog)

v A digital computer is a device that consists of a central processing unit (CPU),

input-output units and memory unit

Y DAC (digital-to-Analog converter) is a device that converts digital signals into

Analog signals

¥ ADC (Analog to Digital converter) is a device that converts Analog signals into

Digital signals

v Process is a componen which can be represented graphically.

Y Zero-order hold is a device which convert Analog signals into Digital signals have

called ADC

2.2.2 Sampled data implementation:

Sampled data are data obtained for the system variables only at discrete intervals and are

denoted as x(kT).A sampler is basically a switch that closes every T seconds for one

instant of time, called the sampling period (Richard , C Dorf, 2022)

Figure 2.2 An ideal sampler with input continuous signal r(t)

The figure 1.2 have describes the simple process for sampled-signal with input

continuous signal is r(t) and the output is the sampled signal r’(t), define as

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Figure 2.3 The response of a Zero-order hold to an sampled signal input r(kT)

The transfer function of the zero-order hold is

1-—e ST (2-3)

1 1

G(s) == 2s = 5

18

2.4 General digital control system represented by sampled data:

In this chapter, we use the sampled data to demonstrate the process of systems inputs andoutputs of as follows (see Fig 2.4)

Analog-Figure 2.4 Bock diagram of digital control of the studied systems

The sampled data is defined as a part from system For digital control systems represent in

Figure 2.4, the reference input is a sequences of samples values r(kT).

The variables of signal m(kT) and u(kT) are discrete signals in contrast to m(t) andy(t), which are continuous functions of time

with g,(t) are invert-Laplace transform of G;(s)

The next, we are sampling impulse response

2.6 Closed-loop feedback sampled-data systems:

We have a figure which is the closed-loop sampled-data system

Figure 2.5 The block diagram reduction of sampled-data control systems

Then, we consider a feedback control system with a digital controller

Digital Controller

EŒ) ~¬

| ?9Ƒ——

A linear continuous feedback control system is stable if all of the closed-loop

transfer function T(s) lie in the left half of the s-plane.

2.7 Stability of digital control systems:

V(z) _ T(z) = E(z)D(z)G(z) _ E(z)D(z)G(z) (2-16)

=Tữ) = E()+E(z)D(2)6()_ E(z) + E(z)G(z)D(z)

_ G(z)E(z)D(z) _ G(z)D(z)

_ E(z)(1+G(z)D(z))_ 1+6(z)DŒ)

A linear continuous feedback control system is stable if all of the closed-loop transfer

function T(s) lie in the left half of the s-plane.

To consider the stability of digital control system, we apply the well-known stabilityconditions for digital systems

(2-19)F(1) > 0,(1)

Let we also consider an example in chapter 01

Fist we consider the case that M(s) = 1

Desired Zero-order Hold Controller Motor Reel Dynamics

E(s)

output Actual cable

Figure 2.7 Block diagram of transfer function of CRC system with ideal measuremt

The zero-order hold is

(2-21)

23

Unity feedback, the closed-loop characteristic equation is

—0.06 < Kp < 0.067

26

Desired Tero-order Hold Controller Motor Reel Dynamics

output E(s) —— —_— — —A~ Actual cable

Figure 2.8 Block diagram of transfer function of CRC system with zero-order hold

The zero-order hold is

G = (1 —e7S7) x K x(Sxi+ 725 town tea)

The roots of the polynomial

F(z) = aszŸ + a;zZ” +a,zZ+ a

This chapter considers the stability of time-delay systems by using the Pade approximation.

At first, we establish a general architecture of a conventional feedback control systems.Due to the use of open communication infrastructure, the time delays can appear withinthe augmented closed-loop control systems, leading to degradation in the performance ofsystems To overcome this problem, we use Pade approximation to convert the systemswith time delays to the conventional structure of dynamical systems Based on the modelsobtained, Famous Routh-Hurwitz test can be used to consider the performance of studysystems Two practical examples are proposed to validate the effectiveness of our strategy

3.2 Problem statement:

3.2.1 General framework of control systems:

Input signal Output signal

> SYSTEM >

30

Figure 3.1 General architecture of systems

Figure 2 shows the general architecture of an engineering system In which, signalsrepresent some independent variable that contain some information about the be haviour

of some natural phenomenon Systems are defined as an interconnections of elements and

devices for a desired purpose of control engineering (Chaparro & Akan, 2019) In control

engineering, a input and output system (SISO) is a simple variable control system with one input and one output A control system is an

single-interconnection of components forming a system configuration that will provide a desired

system response.

A closed-loop control system utilizes an additional measure of the actual output to

compare the actual output with the desired output response We have the following

simple diagram of closed-loop feedback control system.

Figure 3.2 Block diagram of a closed-loop feedback control for engineering system.

Figure 3.2 describes a block diagram of a closed-loop feedback control for engineeringsystem The following elements are considered

An actuator is a device employed by the control system to alter or adjust the environment

vY A sensor is a device that provides a measurement of a desired external signal.

Y Controller is a piece of equipment or an instrument used to produce a desired

performance of the plant

Process is a component which can be represented graphically

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3.2.2 General control systems represented by transfer functions:

In this chapter, we use the transfer functions to demonstrate the relationship betweensystems inputs and outputs as follows (see Fig 3.3)

Desired C Il Actuator Process

be zero (Chaparro & Akan, 2019) For control systems presented in Figure , in general the

transfer function between output measurement, V (s), and input reference, R(s) as follows

G(s) => (3-1)

The transfer function of a linear system is defined as the ratio of the Laplace transform ofthe output variable to the Laplace transform variable, with all initial conditions assumed to

be zero (Chaparro & Akan, 2019) For control systems presented in Figure , in general the

transfer function between output measurement, V(s), and input reference, R(s) as follows

G(s) = >= (3-2)

3.2 Structure controller implementation:

One form of controller widely used in industrial process control is three-term, PID(Proportional-Integral-Derivative) controller

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