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ANALYSIS OF MULTIRATE DIGITAL CONTROL BASED ON
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Wanna-9.1

ANALYSIS OF MULTIRATE DIGITAL
CONTROL BASED ON HIGH GAIN
OBSERVER AND SINGULAR
PERTURBATION THEORY

By

Muhammad Yasin

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Electrical Engineering

1996

ABSTRACT

ANALYSIS OF MULTIRATE DIGITAL CONTROL BASED ON I-HGH
GAIN OBSERVER AND SINGULAR PERTURBATION THEORY

By

Muhammad Yasin

A multirate control structure allows the designer to accommodate multiple
information rates and implement required control computations within the finite
computational capabilities of an onboard computer. The purpose of this study was to
analyze the multirate scheme of a stable, linear, and time invariant system. The output
feedback control is so designed that it uses a very high gain observer in its feedback path.
The dynamics of the control are slow as compared to the dynamics of the observer. The
behavior of the system is also analyzed using singular perturbation theory to see how the
fast and slow modes behave in first cycle. Also, singular perturbation theory is used to
show that the performance of the system under state feedback control can be recovered by
using the output feedback control which uses a very high gain observer in its feedback
path. The performance of the system is also studied by changing different parameters
like, initial conditions, value of e, and sampling times. In computer simulations, it is seen
that saturating the control helps in reducing the peaking phenomenon, that occurs due to

different initial conditions.

I dedicate this work to my biggest supporters in this world; my family; my

parents, brothers, and sisters.

ACKNOWLEDGMENTS

First of all, I am thankful to Allah Almighty for providing me with the
opportunity to undertake and complete this task. I am thankful to the Pakistan Army and
the National University of Science and Technology (NUST), Pakistan, for sponsoring this
study at Michigan State University.

I would like to express my deep thanks and sincere appreciation to all persons
who have contributed by inspiration, counseling, and assistance in the completion of this
study. The advice, professional guidance, and supervision of Dr. Hassan Khalil, have
been invaluable in the completion of this study. He has been very kind in sparing his
valueable time and making me understand through even minutest details of the subject.
Also, I would like to extend my deepest thanks to Dr Shanblatt, Who helped me in the
completion of my studies somehow or the other. I also want to convey my special thanks
to Lt Col Rahat Khan, Muhammad Jamal Khattak, Abdul Mujeer Qureshi, graduate
students at MSU, for helping me and providing their sincere advice during the course of
my studies. I will take this opportunity to thank the Pakistan Student Association and

Pakistani community for their care and concern.

My deepest love and thanks to my parents, who brought me up in a best manner,
for their unconditional support from the day I was born, and for my brothers and sisters
for their prayers that do not stop.

Above all, I deeply believe that my accomplishment and success are bestowed

upon me by the Almighty to whom I am very thankful.

Muhammad Yasin

TABLE OF CONTENTS

List of figures .................................................................................................................... viii
List of Symbols ................................................................................................................... ix

CHAPTER 1 INTRODUCTION AND HISTORICAL BACKGROUND

Introduction ............................................................................................................. 1
Historical Background ............................................................................................. 2
1.2.1 Multirate Digital Control ............................................................................. 2
1.2.2 Singular Perturbation Theory ....................................................................... 3
1.3 Objectives of The Thesis ......................................................................................... 4
1.4 Scheme of The Thesis .............................................................................................. 5

CHAPTER 2 HIGH GAIN OBSERVER

2.1 Observers in General ............................................................................................... 6
2.1 . 1 Introduction ................................................................................................. 6
2.1 .2 Formal Definition ........................................................................................ 7
2.1.3 Mathematical Description of Observability ................................................ 7
2.2 Observers in Linear Systems ................................................................................. 10
2.2.1 Task of Observer ....................................................................................... 10
2.2.2 Observer Design ........................................................................................ 10
2.2.3 Determination of System Matrices of The Observer ................................. 13
2.3 High Gain Observer ............................................................................................... 14
2.3.1 Output Feedback Control ........................................................................... 14
2.3.2 Design of Observer Gain Matrix, L ........................................................... 18
2.4 The Peaking Phenomenon ...................................................................................... 20
2.4.1 Reasons of Peaking .................................................................................... 20
2.4.2 Methods to Reduce Peaking ....................................................................... 22

CHAPTER 3 ANALYSIS BASED ON SINGULAR PERTURBATION

3. 1 Introduction ............................................................................................................ 24
3.2 Derivation of Singularly Perturbed System ........................................................... 25
3.2.1 Continuous Time Singularly Perturbed System ......................................... 25

vi

3.3

3.2.2 Discretization of Singularly Perturbed System .......................................... 29
Multirate Implementation of Singularly Perturbed Difference Equation .............. 32

CHAPTER 4 SIMULATION RESULTS

4.1 Construction of The System .................................................................................. 43
4.1.1 The Controlled System: A dc Servomotor ................................................ 43
4.1.2 Poles of The System .................................................................................. 45
4.1.3 Design of State Feedback Control ............................................................. 47
4.1.4 Construction of an Observer ...................................................................... 49
4.1.5 Combined System - Single Rate Implementation ...................................... 51
4.2 Simulations ............................................................................................................ 53
4.2.1 State Feedback Control .............................................................................. 5 3
4.2.2 Combined System - Single Rate Implementation ...................................... 54
4.2.3 Multirate Implementation .......................................................................... 56
4.2.4 Performance of Multirate Scheme Under State Feedback Control ............ 58
4.2.5 Behavior of Multirate scheme in First Cycle ............................................. 58
4.3 Effects of Changing Different Parameters ............................................................. 59
4.3.1 Effects of Changing The Initial Conditions ............................................... 59
4.3.1a Performance of Multirate Under State Feedback Control ......................... 59
4.3.1b Performance of Multirate Under Output Feedback Control ...................... 60
4.3.2 Effects of Saturating The Control .............................................................. 61
4.3.3 Effects of Changing e ................................................................................ 62
4.3.4 Effects of Changing Sampling Times ........................................................ 65
4.4 Observations and Results ....................................................................................... 67
4.5 Recommendations .................................................................................................. 68
CONCLUSION ............................................................................................................. 69
APPENDICES ............................................................................................................... 70
BIBLIOGRAPHY

vii

2.1:

2.2

2.3 :

LIST OF FIGURES

System with an observable subsystem SU] and an

unobservable subsystem Suz ....................................................................................... 8

: Two connected dynamic systems: Sl the original system

and 82 the observer .................................................................................................... 11

Block diagram showing implementation of equation (1 7) ........................................ 15

viii

A, B, C, D

Am 809 C09 D0
AA, BB, CC, DD
Ac

2, 17

a, b, c, d

a.» b.» co, do

31, b1, Ci, d1

(1)

L

K

D(e)

D-l

LIST OF SYMBOLS

Continuous time state space matrices
Continuous time observer matrices
System matrices of combined system

Canonical form of A

Continuous time matrices of singularly perturbed system
Discrete time state space matrices
Discrete time observer matrices
Discrete time system matrices

State transition matrix

Observer gain matrix

Gain matrix

Identity matrix

Matrix dependent on e

Inverse of D

Pole vector

Output vector of the state space system

Input vector of state space system

H)

8)

fobs

Plant state vector

Observer state vector
Observability grammian

Fast variable of singularly perturbed system, z-transform
Time

Settling time

Initial time

Final time

Sampling time of the control
Sampling time of observer
Angular position of motor
Angular velocity

Estimate of x

estimate of (1)

Parameter of high gain observer
estimation error

error term

Perturbation terms

Ratio of Tc and To, fast time scale
Slow time scale

Sampling frequency

Observer frequency

Eigenvalue
Moment of inertia
Resistance

Back emf constant
Torque constant

Mechanical time constant

Damping ratio

xi

CHAPTER 1

INTRODUCTION AND HISTORICAL BACKGROUND

1.11NIRQDIEIIQN

Multiple sample-rate digital control systems are of prominent interest in current
control research, development and applications. A multirate control structure allows the
designer to accommodate multiple information rates and implement required control
computations within the finite computational capabilities of an onboard computer.
Multirate digital control is a significant area of research and applications that is motivated
by practical implementation techniques. The motivation for multirate control has
traditionally been in aerospace applications where guidance and control laws must be
designed to accommodate multiple rates of sensor measurements and finite throughput
capabilities of onboard computers. Modern aerospace vehicles and systems are described
by high order dynamic models which typically include phenomena covering a wide range
of characteristic frequencies and instrumentation measurements available at multiple
rates. Multirate design techniques should soon find further utility in control application
for highly distributed systems, such as, communication networks, and power plant /
power distribution networks where the characteristic frequencies and time constants of a
local station’s dynamics may differ significantly from those of the network as a whole.
The historical development of multirate digital control and singular perturbation theory is

outlined in next section.

1.2HISIQRICALBACKGRQUND

1.2.1 MultirateDigitalEnntrQl

The use of digital control, or more precisely, sample data control, originated in
radar applications in world war 11. Because the rotating antenna of a radar system
illuminates a target intermittently, early radar aided tracking and fire control systems had
to be designed to utilize data in sampled form. Methods of effective design of control
systems using sampled data were under initial development during the later 1940’s, and
multirate system theory followed these efforts in the early 1950’s. Initially, researchers
developed multirate techniques as a method of evaluating more conventional types of
controllers such as continuous systems and single rate sampled data systems. For
example, one could study the inter sample behavior of a signal or output of a single rate
control system by introducing a phantom sampler, i.e., a fictitious sampler that operates at
a rate some integer ratio higher than that of the controller. A significant early contribution
to this general method of analysis, known as frequency decomposition, was made by
Shlansy and Ragazzini [1], who described the use of this technique in error sampled
control system development. Shortly following the origin of the frequency decomposition
technique, a similar frequency domain technique, known as switch decomposition, was
developed.

The switch decomposition technique, attributed to Kranc [2], provided a means of
representing a multirate control structure as an equivalent single rate controller, this

representation accomplished , the controller could be designed and analyzed using single

3
rate technique. In the late 1960’s Jury [3] showed an equivalence of the switch

decomposition technique and the frequency decomposition technique.

Time domain methods of multirate stability analysis and design were initiated by
Kalman and Bertram [4] with the publication of their state space stability analysis
technique in 1959. Barry [5] published a paper in 1975 in which he described the design
of multirate regulator and showed that its performance was superior to a single rate
regulator having the slow sampling rate. During 1979-81, research at “ The Analytic
Sciences Corporation ” ( TASC ) developed a new multirate control design technique
based on an optimal estimation and control formulation. In parallel with the work at
TASC, Amit and Powel [6], independently investigated a similar optimal control
formulation, among other things, some practical considerations for implementing
multirate control laws and a highly efficient method for solving periodic Riccatti equation
related to the design technique. Different design techniques for multirate digital control

are discussed in [7].

1.2.2 Singflarflerturbationlhm

Singularly perturbed systems, and more generally, multi time scale systems, often
occur naturally due to the presence of small parasitic parameters, typically small time
constants, or masses etc., multiplying time derivatives, or in more disguised form due to
the presence of large feedback gains and week coupling. While singular perturbation
theory, a traditional tool of fluid dynamics and nonlinear mechanics, embraces a wide
variety of dynamic phenomenon possessing slow and fast modes, its assimilation in

control is recent and rapidly developing. The methods of singular perturbation for initial

4

and boundary value problem approximations and stability were already largely
established in 1960’s, when they first became a means for simplified computations of
optimal trajectories. It was soon recognized that singular perturbations are present in most
classical and modern control schemes based on reduced order models which disregard
high frequency parasitics.

This led to the development of time scale methods for a variety of applications
including state feedback, output feedback, filter and observer design. Singular
perturbation methods also proved useful for the analysis of high gain feedback systems
and interpretation of other model order reduction techniques. More recently they have
been applied to modeling and control of dynamic networks and certain classes of large
scale systems. This versatility of singular perturbation methods is due to their use of time
scale properties which are common to both linear and nonlinear dynamic systems.

The first survey [8] of control theory applications of singular perturbations in
1976, included 130 references. The period of 1976-83 have witnessed an even faster
growth of this research area both in theoretical depth and breadth of applications, as
evidenced by surveys and books given in section A of the references of [9]. A brief
overview of the 80 years of traditional singular perturbations by O’Malley (1982A), [10],
lists 64 major references during that period. An analysis of singularly perturbed adaptive

system appears in Ioannou and Kokotovic (1983A), [11].
1.3 W

I was assigned to accomplish the following objectives.

a. Design a state feedback control using slow sampling period.

5

b. Design of observer based output feedback control using the multirate scheme.

0. Carry out analysis of multirate scheme using singular perturbation theory.

d. Study the behavior of the system under output feedback control.

e. Study the effect of changing different parameters in multirate implementation.

f. Study the effect of saturation of the output feedback control.

g. Study the behavior of slow and fast variables using singular perturbation
theory.

h. Study the behavior of slow and fast variables in one cycle while keeping the

control constant.
1.4 W

Chapter 1 deals with the general introduction and background of multirate digital
control and singular perturbation theory. In chapter 2, design of high observer is
discussed. Also the peaking phenomenon and its remedies are discussed in the same
chapter. In chapter 3, analysis of multirate scheme, using singular perturbation theory, is
done. First the system is brought into standard singularly perturbed form, and then the
approach of [12] is followed to study the behavior of slow variables for one cycle. It is
also shown that the output feedback control recovers the performance under state
feedback control using singular perturbation theory. Chapter 5 deals with the simulations

and their results, and finally the recommendations and conclusion is given.

CHAPTER 2

HIGH GAIN OBSERVER

2.1.QBSERYERSINQENERAL

2.1.1 Introduction

One of the objectives in control systems design is the achievement of suitable
eigenvalue locations in order to ensure satisfactory dynamic performance. In modern
control theory, linear state feedback provides an appropriate compensation technique to
meet this objective under the assumption that all state variables can be used in forming
feedback signals. Unfortunately, this assumption is not always valid in practice. A well
known approach to overcome this difficulty is to generate the feedback control law via an
estimate of the state vector. The estimation is performed using an asymptotic state
estimator, called an observer, which employs only the available directly measurable input
and output signals. Hence, the problem of designing controllers for systems with
incomplete state measurements is equivalent to constructing observer based controllers.

The idea of Observability and controllability was introduced by RE Kalman in
the 1950’s. Proceeding from a figurative description, it can be said that a system is
observable if it is possible to observe the states by using the output of the system, that is,
to look inside the system from the output. That means, all the states must have a
connection to the output. A system is said to be completely observable if all of its states

are observable. It is some times advantageous to have an observable system, especially

7

when the system to be controlled has unstable states. Of course, an observable system is
innocuous if the system is asymptotically stable.

The idea of Observability is explained with the help of figure 2.]. The figure
describes a system with input 11, output y and two subsystems with the states x1 and x2 .
The subsystem Sul is observable because the state x1 has a connection to the output of the
system, whereas the subsystem Suz is not observable because the state x2 is not connected

to the output of the system.
2.1.2 Ecrmalllefiniticn

The formal definition of Observability is as given below:-
“ A linear system is said to be observable at time to if x(tO) can be determined from the
output function y(to , 1) within a finite time interval to < r < tf. If this is true for all t and
x(t), then the system is said to be completely observable.” Referring to the formal
definition of Observability, it is clear that the Observability depends on the output y and
the state x.

An equivalent definition of Observability states that:-
“A system is observable on interval [to , tf] if any x(to) is uniquely determined by the

corresponding response y(t) for t < [to , tf] , where tr > to .”

2.1.3MathematicaLDescripticnnfflbserxabilitx

The Observability phenomenon can be described mathematically by using the
theorem given below. The theorem states that, given the system:-
x(t) = A x(t) + Bu(t) (l)

y(t) = C W) (2)

 

Figure 2.1:

 

’y

 

 

 

 

 

 

 

 

 

 

System with an observable subsystem 8,1 and an unobservable
subsystem Sag,

9
The system (1) - (2) is observable on interval [to , tf ], if and only if, the Observability

grammian

M(t0,tf) = :[CDT(t,t0)CT(t)C(t)CD(t,to)dt

is invertible ( i.e. nonsingular ). Also M( to , tf) is symmetric and positive semi definite.
Now consider the output given by :-
y(t) = C¢(t,to)X(to)

and assume that the matrix M( to , tf) is noninvertible ( i.e. singular ). If this is true, there
will exist an initial state x(to) different from zero which can produce a zero output for
interval of any length. Therefore, the matrix C (D( t , to) is not of full rank n, the order of
the system. Therefore, all the states are not transferred to the output and the system is not
observable.

In most cases, the calculations of the Observability grammian is very complicated.
Therefore, as an alternate method, we can use a simpler algebraic theorem to test the

Observability of the system. The theorem states that a linear time invariant system

:0) = A x(t)
with the output given by:-
y(t) = C W)

is observable, if and only if, the rank r(N) of the Observability test matrix, given by:-
N=[AT ATCT ____(An—I)TCT]

is equal to n, the order of the system.

10

2.2. W

2.2.1 Iaskcfflbscmer

The basic task of an observer is to reconstruct one or more missing states of a
given system by using the system output, provided that the system is completely
observable. The observer has to estimate the state such that the error between the actual
state x(t) and the estimated state 56(t) , that is

8(1) = 16(1) - f0) (3)
tends to zero. To build up an observer, a second dynamic system 82 is connected to the
output of the original system S. . Such a system is shown in figure 2.2. The system 82 has
almost the same structure as the original system S 1 . Both SI and 82 are assumed to be
linear and time invariant. The output of the system S1 is y, which is the input of the
second system 82 , u is the input to the system Sl . The output of the system 82 is called

the estimated state 1?.

2.2.2 Qbsemedlesign

Consider following linear, time invariant system S, , derived from (l)-(2)
5r(t) = A x(t) (4)
when system 82 is added to S, , it becomes
in) = 2 520) + chu) (5)
with y(t) = Cx(t) and L is an n x 1 gain matrix. Now suppose the transformation T, which

satisfies

LC=TA—21T (6)

 

 

 

ll

y=x

 

 

 

Figure 2.2: Two connected dynamic systems: SI the original system

and S; the observer.

 

 

5:

 

 

 

12
Also if 12(0) = T x(O) then £(t) = Tx(t). The general solution of (5) is given by:-

£(t) = Tx(t) + e?“ (2(0) - Tx(0)) (7)
To prove this solution subtract (4), after premultiplying by T, from (5), we get

at) — Tm) = 21 2(1) + LCx(t) — TA x(t) (8)
now using (6) in (8), we get

in) — Mt) = 21 (£0) — Mr» (9)
The solution of differential equation (9) is given with (7).

Now for a forced system, apply the transformation T on the input u. So we get
in) = .74 £(t) + LCx(t) + T1} u(t) (10)
The system 82 given by equation (5) or (10) is called as an observer and the matrix L is

called the observer gain. The poles of the observer should be chosen on the following

basis:-
a. They should be chosen arbitrarily.
b. They are independent of the poles of the original system.
c. They should be chosen so as to be in the left half of the s-plane, or
equivalently, inside the unit circle of z-plane, to meet the stability conditions.
(1. They should be placed at such locations so that the observer system is faster

than the control.
To get an observer with the same order as that of the original system, we let the

transformation T be the identity matrix 1. Therefore, (10) becomes:-

55(1) = Ii £(t) + LCx(t) + Bu(t) (11)

13

2.2.3 DetennmatroncflsttemMamcemflIheflbscryer

To determine the system matrices A , B and the gain matrix L, we consider the

error e(t) defined by (3) . Taking derivative of (3), we get

e(t) = x(t) — in) (12)

Substituting (1) and (11) in (12) , we get
é(t) = Ax(t) + Bu(t) — [Aim + LCx(t) + Bu(t)]
m
e'(t) = Ax(t) + Bu(t) — 21in) — LCx(t) — Bu(t)
Now substituting in) from (3) in above, we get
é(t) = Ax(t) + Bu(t) — 21(x(t) — e(t)) — LCx(t) — 311(1)
m
a0=fidO+GLJLth0MIB~flM0 (B)

When the error e(t) tends to zero, the coefficient matrices of (13) must also be zero,

therefore,

A—i—LC=0
which gives

fizA—LC 0%
mm B—é=o

which gives

B=B as

14

It is seen that B is the matrix B of the original system. The observer gain L can be
chosen arbitrarily to get the desired eigenvalues of the system. The matrix L can be
determined by control design methods, e.g., pole placement.

Now substituting (14) and (15) in (11), we get the observer equation depending on the

system matrices of the original system.

in) = (A — LC)SE(t) + LCx(t) + Bu(t) (16)
or

§(t) = A£(t) + Bu(t) + L(y(t) — C£(t)) (17)
The block diagram showing implementation of (17) is shown in figure 2.3.

2.3. W

2.3.1QutpuLEeedhackLQntml

Consider a fully linearizable system, refer to [l 3], represented by:-
x(t) = Ax(t) + 13[fI (x) + GI (x)u(t)] (1 8)
YO) = CW) (19)
In equation (18), the input term is a fimction of the state x and the input u(t) with a
coefficient matrix G,(x). The basic structural property of a linearizable system is the fact
that it can be transformed into coordinates where the nonlinearity of the system ( or a part
of it), satisfies the matching conditions. This permits the cancellation of the nonlinearity
by the state feedback. However, exact cancellation is almost impossible due to parameter
uncertainty, model simplification etc. The fact that the nonlinearity satisfies the matching

condition implies that error in modeling the nonlinearity will also satisfy the matching

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 2.3: Block diagram showing implementation of equation (17).

l6

condition. Therefore, robust control techniques like variable structural control, min-max
control, and high gain feedback can be used to robustify the linearization.

To implement controllers using output feedback, we need to estimate the
unmeasured state variables. The state estimators will have to be designed to preserve the
robustness achieved with state feedback. The output feedback controller is designed in
two steps. In first step, a state feedback control is designed to achieve the control task,
and in the second step, an observer is designed to recover the performance achieved with
state feedback. To get a description of the system which can be treated well, it is assumed
that the system can be linearized perfectly at a chosen state depending upon system

requirements. The linearizing state feedback control:-

u = G," (x)[—fl (x) + v] (20)
cancels the nonlinearity of the system. This cancellation, however requires perfect
knowledge of the nonlinearities f1(.) and G,(.). Since perfect knowledge is almost
impossible, it is more realistic to assume that we only know the models f,” (.) and Gm (.)
of the actual nonlinearities f,(.) and Gl(.). To consider the nonlinearity, a perturbation
term 8 (x,u), depending on the state x and the input u is added. Therefore, the system
(18) can be represented as:-

i(t) = Ax(t) + B[—f,,, (x) + G (x)u(t)] + 35 (x,u) (21)

ln
The key feature of this perturbation is the fact that it satisfies the matching conditions,

that is, it enters the state equation exactly at the same point where the control enters.

Suppose now that a state feedback control has been designed as:-

17
14(1) = Gl-nl (X)[-fu.(X) + 490,30] (22)

where (D(t,x) is linear or nonlinear control that ensures robust stabilization or tracking in
the presence of the uncertainty 5 . Now substituting (22) in (21), we get

J'c(t) = Ax(t) + BCD(t,x) + BS (x,u) (23)
Next task is to estimate the state x. Consider the full order observer given by:-

i(t) = A£(t) + B[f,,,(i) + G,,,(i)u(t)] + L(y(t) — 02(1)) (24)
which reflects the nominal nonlinear model of the state equation together with an error

driving term. Now using the estimate J? , (22) can be written as:-

um = G]: (f)[-f,,.(£) + arm] (25)
Substituting (25) in (24) , we get

in) = A£(t) + Btb(t,5c‘) + L(y(t) — 02(1)) (26)
Defining the estimation error as:-

e(t) = X(t) - £0) (27)
gives

e'(t) = 5c(t) — in) (28)
Substituting (23) and (26) in (28), we get

é(t) = Ax(t) — Aim + B<D(t,x) — Bcb(:,i) + 88 (x,u) — L(y — arm) (29)

We see that the term B<D(t,£) and B<D(t,x) do not cancel each other perfectly like they
do in linear control. Therefore, we define another term 81 dependent on time , the state

of the original system, and the state of the observer as:-

5 ,(t,x,£) = (D(t,x) — <b(t,i) (30)

18
Thus, using (30) and (27), (29) can be written as:-

é(t) = (A —- LC)e(t) + B[5 (x,u) +5 , (t,x,£)] (31)
where 8 (x,u) is the perturbation term which represents uncertainty in state equation,

and 5 1 (t,x,£) is an error induced by the presence of the nonlinearities.

2.3.2 DesignnflflhservcrfiainMatrixL

We need to design the observer gain matrix L to stabilize the matrix (A-LC) while
rejecting the effect of the perturbation term. This is ideally achieved if we could design L
such that the transfer function of the observer given by:-

H(s) = (31 — A + Lc)" B (32)
is identically zero. Since this ideal situation can be achieved under very restrictive
conditions, we try to design L as a function of some parameter a , (8 > O ), such that
H(s) approaches zero asymptotically as 8 tends to zero, point wise in- s.

The challenging task in nonlinear problems is performing closed loop stability
analysis with such observer design to ensure that the closed loop system with the
observer recovers robustness properties achieved with state feedback. In [14] and [15]
Esfandiari and Khalil use singular perturbation to design the observer gain L. The
singular perturbation analysis will be discussed in chapter 3. By representing the closed
loop system as a singularly perturbed one, they show the recovery of robustness
properties in the presence of nonvanishing perturbations. In particular, they show that the
output feedback controller recovers ultimate boundedness achieved under state feedback

control. Saberi and Sannuti [16] use singular perturbations and show recovery of robust

19
global stabilization of the origin. Petersen and Holot [17] use an H,o approach to design

L. Gibbens and Fu [18] also use the idea of Petersen and Holot to show the recovery of
robust tracking in the presence of nonvanishing perturbations. Teel and Praly [19] use
singular perturbation approach to achieve semi global stabilization.

A common feature of all these observer designs, when the relative degree of the
system is higher than one, is that they are asymptotic approximators of the output
derivatives. Now consider a second order system having system matrices, A as 2 x 2 ,
and C as l x 2 , and let the observer gain for this system be given by:-

L = [a b]r
For (A-LC) to be Hurwitz, we must choose a and b to be positive. The transfer function
H(s), given by (32), approaches zero asymptotically, if and only if, its Hm norm does so.
By calculating the H00 norm of H(s), it can be seen that a necessary condition for making
the norm arbitrarily small is that b >> a >> 1. The various observer designs reviewed
earlier arrive at this condition using different procedures. The singular perturbation

approach of [14] and [16] starts from the outset by choosing the observer gain L as :-

one
L=L2[ ]
s 1

for some or >0 .

All the procedures used achieved the condition b >> a >> 1 by choosing a = 0(1/ 8) and

b = 0(1/ 82) for a small positive parameter a. To see that these observers are

approximate differentiators, refer back to the observer equation (26), and notice that for

20
sufficiently large a and b/a , the transfer function from (I) to SE is almost zero while the

transfer function from y to 56 is given by:-

 

1 1 3 +1 I
(SI—A+LC)_L= bs z|:]
2 a
S +I;S+1 S

Q-nlt—t

where the approximation is valid on any compact frequency interval for sufficiently large
a and b/a. Keeping in view the above discussion, for our thesis we design the observer

gain for second order system as:-

1 1 T
L: — —
I8 82I

We use this form of observer gain matrix, which can produce very large numbers. That is

why, it is called as high gain observer.

2.4..THE_EEAKING_BHENQMENQN

2.4.IReasonscfpeaking

There are some intrinsic difficulties with controllers which use high gain
observers. First, such controllers amplify measurement noise. Second, they produce large
overshoots in the transient response of the closed loop system, which we refer to as
peaking phenomenon. These large overshoots, in the transient response, can even
destabilize the system completely.

The observer gain L is designed to bring the transfer function H(s), given by (32),
as close to zero as possible. This requires the observer gain to be very large. The high
gain L assigns the eigenvalues of (A-LC) far to the left in the left half of the s-plane.

Typically, L depends on a small parameter a , such that as 8 tends to zero, the eigenvalues

21
approach infinity. The corresponding modes in the state transition matrix exp[(A-LC)t]

take the form exp(-at/e) for some a > 0. Hence they decay to zero with in an 0(8) time
interval. For a relative degree one system, the coefficients of these exponential modes
will be bounded as L approaches infinity because the initial states are totally bounded.

For the systems of relative degree higher than one, the coefficients of the
exponential modes are necessarily driven to infinity as L increases towards infinity. Let
us use the second order system, considered previously, to illustrate this point. We already
know that reducing the transfer function H(s) to zero requires designing the observer gain
L = [a b]T such that b >> a >> 1. By calculating the state transition matrix, it can be seen
that the 2-1 element is given by:

,, 2
b e ‘lzsin[,Ib—g—]t
2 4

b-5‘.
4

 

when 4b > a2 , and

 

 

when 4b < a2.

The magnitude of the coefficient of the exponential mode is greater than JD in
the first case and b/a in the second case. Thus, as we increase a and b/a , we drive this
coefficient towards infinity. This causes an impulsive behavior in the transient response

of the estimation error. It is to be mentioned here that, the peaking phenomenon is

22.

associated with the process of designing L to reduce H(s) to zero, irrespective of the

particular design procedure used to design L.

2.4.2 MetthchducePeaking

Following methods help in reducing the peaking phenomenon.

a. In the presence of peaking, we have a bigger incentive to use singular perturbation
theory. This process involves scaling some of the variables by a dependent factor.
Once the system is brought into the standard singularly perturbed form, then it is
known from singular perturbation theory that, there is no peaking when the initial
states are of the order of 0(1). Peaking is induced only by initial conditions
which are of the order of 0( 8’8 ) with B > 0. Thus the mechanism, through which
the peaking is induced, has been represented as the value of the initial conditions
of some of the state variables. Knowing the scaling factor used to arrive at the
singularly perturbed form, we can clearly see how the initial conditions are scaled.
Through this process we can see how the region of attraction of the origin will
shrink as 8 decreases. We also see that any destabilization effect, as 8 tends to
zero, will be induced by initial conditions of the fast variables which take the form
0( 8’5 ). Although the results of singular perturbation theory are limited to cases
when the initial states are of order 0(1), realizing the role of the initial conditions
of the fast variable points in the direction which lead to the breakthrough result on

globally bounded control.

b. We can drive the controller and/or the observer into saturation so that it does not

exceed a certain value. But the limits of saturation will have to be defined. very

23
carefully. The bounded output feedback control recovers the performance of the

bounded state feedback controller as 8 tends to zero.
As a second possibility, we can use multirate sampling where we use, in digital

control, different sampling times for the control and the observer.

CHAPTER3

ANALYSIS BASED ON SINGULAR PERTURBATION

3.1.INIRQDUCIIQN

The singular perturbation approach to the analysis design of continuous time
systems possessing multiple time scales has matured over the past 25 years. Recently,
there has been interest in extending singular perturbation ideas to multiple time scale
discrete time systems. Singularly perturbed systems, and more generally, multi time scale
systems often occur naturally due to the presence of small parasitic parameters, typically
small time constants, or masses etc., multiplying time derivatives, or in more disguised
form, due to the presence of large feedback gains and weak coupling.

Various mathematical forms have been used to represent a two time scale discrete
time systems. Discretization and sampled data control of systems described by singularly
perturbed differential equations result in singularly perturbed difference equations.
Singularly perturbed difference equations might also arise naturally in inherently discrete
time systems. In this chapter, the two time scale nature of the system is exploited to
represent the system in the singularly perturbed form. At the end of the chapter, it
confirms the intuitive idea that the slow variables can be measured at a rate slower than

that of fast variables without degrading the performance of the system.

24

25
32.: 5'.\. I; 1;: '

r.
b
r.
.O
-
-

3.2.1 Conunumrslrmefimgularlxflertmheismcm

To motivate the form of the singularly perturbed difference equation, we start by
considering its continuous time analog first. Consider the continuous, linear, time
invariant system defined as:-

J'c(t) = Ax(t) + Bu(t) (l)

y(t) = CW) (2)
and the state feedback control is given by:-

u(t) = —Kx(t) + Nr(t) (3)
where A, B, C are system matrices in canonical form. A is n x n, B is n x m, and C is

p x n . K is the gain matrix. Now consider the observer equation

51(1) = A220) + L(y(t) — 02(1)) (4)
and the output feedback control, given by:-

u(t) = —K£(t) + Nr(t) (5)
where £(t) is the estimate of x(t). Now consider the error difference between the actual
and the estimated states, given by:-

e(’) = W) - fit) (6)
which gives

é(t) = 1(1)— 56(1) (7)
Using (1) and (4) in (7), we get

e'(t) = Ax(t) + Bu(t) — A£(t) — L(y(r) — Cx"(1))

26

Now using (2) in above equation, we get
e'(t) = Ax(t) + 811(1) — A£(1)— LCx(t) + LC£(1)
or
e(t) = (A — LC)(x(t) — 2(1)) + 311(1)
Now using (6) in above equation, we get
e'(t) = (A — LC) e(t) + Bu(t) (8)
The system matrix A can be written as:-
A = Ac + BA (9)
where AC is in canonical form of n x n , and A is some arbitrary matrix.
Therefore
A—LC=(AC +BA)—LC
using above equation in (8), we get
e(t) = (Ac + BA — LC)e(t) + Bu(t) (10)

Now choose D(s) such that D"(e) exists and e > O is a small parameter. In general

— q

8"_' 0 ... O

0 ° 3

D“): 82
i 8 O

 

 

27

 

rI—l 0 CI
8
O
D“(e)= , _1_
. 82
l 0
e
L0 O 1_

 

 

where n is the order of the system. We define another variable as
z(t) = D"(e)e(t)
Taking derivative of (1 1), we get
z'(t) = D"(s)e’(t)
Substituting (10) in (12), we get
2(1) = D" (e)[(Ac + BA — LC)e(1) + 311(1)]
Using (11) in above equation, we get
2(1) = D" (e)[(Ac + BA — LC)D(1—;)z(t)+ 311(1)]

01'

2(1) = D“(e)(A, — LC)D(e)z(1) + D“(e)BAD(e)z(1)+ D"(e)Bu(1)

(11)

(12)

(13)

Also choose the observer gain matrix L, such that (A-LC) is Hurwitz. ( The design

procedure for L has already been discussed in chapter 2 ). In general,

 

T
L- 9.1. 9; or”
_ 2 n
8 8 8

The general canonical form of the system matrices is:-

Therefore

 

 

Similarly we can prove that:-

D'1 (8)3 =

I-

8

 

 

 

o 1 0
0 0 1
[0
=[1 0
'31 1
8
—a2
0
8.2
Ac—LC= =
*1." 0
_8

D" (e)(Ac — LC) D(e) = l

l

n—l

O

 

28

 

8

 

 

 

 

 

 

 

 

 

 

(14)

(15)

29
Using the results (14) and (15) in (13), we get

2(1) = éAozU) + BAD(e)z(t) + Bu(t)

01'
22(1) = (A0 +e BAD(c))z(t) +e 1311(1) (16)

Now writing (1) and (16) in matrix form, we get

 

1“”) I" ° 11"“) [BI
. = + u(t) (17)
sz(t) 0 A0+eBAD(e) z(t) SB
or
2(1) _ A A (BAD [21(1) BI
I2(1)I’I0 0+: (8) 120) +[BJu(t) (18)

Equations (17) and (18) are the standard form of continuous time singularly perturbed
system. x(t) is the slow variable and z(t) is the fast variable. Now we define output

feedback control that consists of both slow and fast variables. Using (6) in (5),we get
u(t) = —K(x(t) — e(t)) + N r(t) (l9)
substituting (11) in (19),we get
u(t) = —K(x(t) — D(e)z(t))+ N r(t) (20)
3.2.2 DiscretizationnflSingularlxRQmMsttem
We can write equation (18) in the form:-
5'c‘(t) = Isl—fit) + Bu(t) (21)

__ A 0 2 B
Where A = A0 +8 BAD(8) B :[ ]

8

 

3O

. _ 1(1) _ _ x(t)
x(t) — [z(t)] x(t) — [z(t)]

Now we discretize (21) at a fast sampling period 8T, to get

and

f(n +1) = A, f(n) + Bd u(n) (22)
where n is the fast time scale and Ad and B, are discrete time matrices . In general
A, = e”
I;
B, = [#111 B
0

Therefore, using above equations, we get

_ ._ _ exp(Ae 7) O I
A“ ’ CXPMS T) ‘I 0 exp((A0 + e BAD(e))T)I

Now taking these matrix elements one by one:-

 

exp<Aen=1+eIAT+e1427?"+ ...... I

or we can write this as:-

exp(A3 T) = I + 8 A,(s) (23)
From (23), we have

A. (0)=AT
Similarly, we get

exp((A0 +e BAD(s))T)=e"°T + 0(8) (24)

Therefore,

31

[1+8 A,(e) O I
A, = 2
0 e"0 +0(8)

Now consider Bd , given by:-

— cT 1
[eA'dtB
Bd = er 0

[exp(( A0 + 8 EAIXS )) t] dt B

_ J

 

 

 

Consider the first matrix element of B,

 

8 T 'l‘
[eA’dthe [e‘mdsBzeBl (25)
0 0
where
T
B, = [emdsB (25A)
0
Similarly
. T A + e BAD(s) "
exp“ 0 It) dt B = e [exp((A0 + c BAD(8)) s)ds B = 8 B2 (26)
O 8 0
where

B2 = [exp((A0 + e BAD(s )) s) ds B (26A)

0

Therefore B, is written as:-

[8 l]
d
8B2

Substituting Ad and Ed in (22), we get

32
[x(n + 1)] = [I + e A, (e) 1 0 IIXWI + [a B, I”) (27)
z(n + l) 0 8‘4" + 0(8) z(n) 8 B2

or we can write it as

x(n +1) = (1 + e A, (8))x(n) + e B, u(n) (28)

z(n +1) = (e "° T + 0(8))z(n) + e B2 u(n) (29)
where

u(n) = —K(x(n) - D(8)z(n)) + Nr(n) (30)

Equation (27) is the discrete time version of (18) sampled at a fast sampling period 8T,
with the output feedback control given by (30). x(n) is called slow variable and z(n) is
fast variable. Equation (27) is expressed in fast time scale 11 and is called as singularly
perturbed difference equation. Equation (27) shows that the eigenvalues of the system

take the form:-
1~1(€)=1+8p.(8) i=1,2, ------ m1
Aj+m,(8)=qj(8) j:1, 2, """ m2

where p, (8) and q j. (8) are continuous at 8 = O and q j. (0) are the eigenvalues of e“ .

Since Rel(Ao)< 0, then Ik(e"""‘)|< 1, the fast modes in (27) are asymptotically stable
and reach steady state.
3.3.U. ,- 110‘11‘11010 13... "1.0%.. "1‘ .1-..”

It is possible to obtain a description of the system in a slow time scale if the fast

modes are asymptotically stable, see [12]. A slow time scale k is defined by the

33

relationship n= k/8, where k = O,1,2,---. Now consider the interval 5 s n < 5:1 such
8 8

that the control (30) is kept constant over this interval. Then for this interval, the control

law is written as:-

11(5) = -KIx(5) — D<e>z<£>I + N115) (31)
8 8 8 8
and
u(n) = 11(5) 1‘- s n < E (32)
8 8 8

Now substituting (31) in (28) and (29), respectively, results in a closed loop system,

described for the given interval, given by

x(n +1) =(1 + e A, (8))x(n) + e B, I—K(x(1‘—) — D(8)z(£)} + Nr(5)I (33)
S 8 8

Now starting from 5 , we calculate the state at w , therefore (33) gives
8 8

[1+1 1
11+]

x(k—flI =(1+eA,(e))“x(£)+ Z (1+e A,(8)) e‘HeB,
8 8 It

./=
8

I’KIX(£) — Dte>z(1‘—)} + Nr(£)I
8 3 8

Ir+1_l
t: k+l

= (1+eA,(e))“ — Z (I+8A,(8))
k

j:
8

"’e B,K 11(5) +
8

EL,
8 [1+1

2 (He/11(2)) . "”eB.KD<s> 2(51+
8

, It
[=-

34

field!

Z(1+eA,(e))k§ "eB,N r(-:— )

i=5
8

xII—flI = 1 + 11 + 111 (34)
8

Now solving for I, II, III separately. Rewriting I, we get

[(+1

I k+l

1= (1+8A, (e))' +e Z I1+eA, (e) . ”’(—B,K)I 11(5) (35)

I-
8

The expressions given below, which have been used in [12],

(I + e A(e))” e“) + 0(a) (36)
8 a (1 +8 24(8)): j=eI 4(OI’a't + 0(8) (37)

hold for any matrix A that is analytic at 8 = 0. Now using (36) and (37), equation (35)

gives:-

1 = [(12419) + 0(a)) + I I 1101111 + o<s>IBon<§>

where BO = -BK

Now using the fact that A,(O) = AT, we have eA' (O) = e‘” and therefore, we have

1 l
IeAl(0)’dt = Jeri/Id!

0 O

and after making change of variables, we get

Ie4"°"d1=—l-Ie’"d1
T0

lherefor

()1

N08 1:

01

there

Now (

Of

“her:

New

35

Therefore, I can now be written as:-

I=[(e‘"+0(8))+{1T:[e’“dt+0(8)}B,,:Ix( )

01'

1 T ,, k
1:.[AH .7 Oje 1113,, +0(8)Ix(;) (38)

Now take the second part of (34), i.e., II, and using the same procedure as before, we get

”=11

1 T ,, k
or II = |:? [e dth + 0(8)I2(;) (39)

0

[eA'dt + 0(8)}B,KD(8)I2(£)
0 8

Nil-—

where Bf= B ,KD(8)

Now consider III of (34) and use (37) as before; we get

1 T A1
111: IE1: Oje dt+0(8)}B, NIr(: )

'~]I-'

or 111 =[ [ta/“111B, + 0(8)Ir(£) (40)
0 8

where B, = B,N

Now substituting (38) , (39) , and (40) in (34), we get

425.1)- I. ,. i 0;. .11...,..(.)I.g).

I

l
T

~1|~

[eA'dth + 0(8)I2(‘:)+[ [eA'dtB, + 0(8)Ir(—:—) (41)

36
Taking into account the fast variable z(n), substituting (31) in (29) results in closed loop

system:-

z(n +1) = (e 4.1 + 0(8))z(n) + 3 B2 [—K{x(£) — D(e)z(§)} + Nr(§)] (42)

. k
As before, starting from — , we calculate the state at —k—+—1 .

 

 

8 8
k+l_l
k+1 _ AOT 1" E e AOT 2"“!
{—8 )_(e +0(s)) z(s)+ I; (e +0(e)) 83,
k
[— K{x(—) — D<e>z(5)l + N45]
8 8 J 8
= (e’l’r +0(8))le + 8: (em +0(e)) ‘ "jeBzKD(e) z(£)+
.=k 8
[:14 .. k+I-l-j . k
(e"‘" +0(a)) . s(-B,K)Jx(—)+
.=k 8
:(e’w +0(e)) '5 +18(B,N) r(—:—)
" J
z(k+1)=1+11+111 (43)
8

Now consider I, II, III separately.

k+l
k+l

1 = (eA’o +0(e))" + Z (e‘o’ +0(e)) . ""’aB,KD(e) 2(5)
k

37
Since Re MAO) < O , exp(A0T) is bounded, and the fact that:-

1

(e’w +0(s))”

 

 

 

 

Sl‘y'l‘fifikl O<e<8l O<y <1
8
is established, therefore,

(e-‘v’ + 0(a))1 . = 0(2)

Now for the second part ofl, we let e“ + 0(3) = A2 i = j — E , so we get
a

_Ir+l__l

82: (e‘u + 0(8)

j=~~
8

k

1

+1 . 8_‘ l_ i

. "=Z(A,)e' =0(1) e—>0
i=0

Therefore,

k+|

2 (6’4” + 0(a)) . "”eB,KD(e) = 0(2)

, k
j:
8

So for I, we get
k k
1 = (0(3) + 0(8))z(;) = 0(8)Z(;) (44)

Now consider part II of equation (43), i.e.,

It+l_l 1

11 = 2 (e a)" + 0(a)) S "”e(—B. K) 9x(£)
.gk 3

and using the same argument as above, we get
k
11 = 0(s)x(—) (45)
a

and proceeding in similar way for 111, we get

38

111 = 0(a)r(§) (46)
Now substituting (44), (45), and (46) in (43), we get
k + 1
Z( 8 -—-)= 0(8 )Z(: )+ 0(8)X(: )+ 0(8)r(: ) (47)

Now writing (41) and (47) in matrix form, we get

     

[(+1 I I k
x(—) _ e“‘+ ije ’“dtB +0(e) lje ’“dtB +0(s) x(g)
k+1 To To +
Z 0(2) 0(2) z(g)
1’ .,
?6le d:B,+0(e)r(§)
0(8)
01'
k
=|:A11+0(3) A12 +0(3):| x(g) +[Bn +0(8)]r(£) (48)
0(a) 0(2) 2(5) 0(2) e
8
where
All = e" +1T JeA'dtBO (49)
IT .,
A12 =;!e d113, . (50)
I.
= ~17; IeA'dtBi (51)

For sufficiently small a , equation (48) can be written as slow sub system, given by:-

f(-k-:—l) [An An] fig) [Bu] (k) (52)
= + I’ -
2(531) 0 0 2(5) 0 a

with same initial conditions.

Sampling (18) at fast sampling period 8T resulted in (48) where the control is

kept constant over the interval 5 S n < 511 . However dropping the 0(8) term from (48)
a 8

results in (52) which is almost same system, with some error, that can be obtained by
sampling the original system (1)-(3) at slower sampling period T. That is, the singularly
perturbed system (18) sampled at fast sampling period ST is O(s)+v close to the slow

system (1)-(3) sampled at slow sampling period T, i.e.,
k _ k
x(—)—x(—) = 0(e)+v k > O (53)
8 s

where v is the error term given by A12 .

For zero initial conditions, there is no error and the difference between the two
models, the model under state feedback and the model under output feedback, is exactly
0(8). However, for different initial conditions, the error v is induced which is given by
A12 . To see as to why this error is induced, consider the first cycle, i.e., the interval from
0 to 1/8. Due to different initial conditions, the two models start from different states
having more difference, and by the time they reach 1/8 from 0, the difference between the
two states will have decreased, thus decreasing the error. Now, the starting conditions for
the two models will almost be same but not exactly the same. In fact, after one cycle the

difference between the two models will be 0(Tc), i.e.,

4O

1 1
x( ;) stale feedback — x( ;) output feedback = 0( 7:: ) (5 4)

This is an error which will be carried along after one cycle. However, this error can be
reduced significantly by having small values of Tc and e . Therefore, for smaller values of
Tc and a, the error induced will be very small and we shall have closeness between the
two models. To see the closeness between the two models we proceed as following:-

From (48) , we have

 

4" +1) = (A,, + O(s)x(§) + (.4,2 + 0(8))z(§) + (3,, + 0(8))r(§) (55)

8

 

and z( +1) is given by (47). Now tracing back the different matrices we get

7‘ 1 T T
A,,=eA +—T— je A’dz(— [e “‘"dsjBK
0 0

1 T ,
A,2 = .7: jeA dtB,KD(e)

O

and

Bu = 1T iemdt[ ire'nstJBN

0 0

Substituting the expressions of A” , A12 and B“ in (55) and rearranging, we get

8 0 0

x(k—L1)=elrx()£ +-1—lje ’"dt —ef[’mdsBKx(—:- —)+7je A“dsBNr(: -—+)
8 TO

B. fwd: B, Kn(e)]z(5) + 0(.)[x(1‘.) + 2(5) + 45)] (56)
0 g e s 8

Now for small values of e , i.e., for a tending to zero, we drop out the 0(8) term, and

4]

I e "“ds = T D(e) = 0(0)

Therefore, (56) now becomes:-

xUC—f—l) = e”x(£) + lewd! B[—Kx(£) + Nr(£)] + [J— IeA'dt B,KD(O):|2(£) (57)
e s O a a T 0 3

Now using (3) for t = k / a in (57), we get

x(fl) = e”x(£) + ie’“dtBu(£)+[—1—38’"dtB,KD(O)]z(£) (58)
a a 0 a T 0 8

Looking at (58), the first two terms represent the same model that is obtained by
sampling (1) at slower sampling period T. The third term in (58) represent the error v
present due to fast variable having different initial conditions. Now since I X (en) | < 1,

and | A (exp(A0T))| < l, the system (48) is asymptotically stable for sufficiently small a .

Hence, the difference in deviation from steady state will become zero, i.e.,

3?[£)—>0 E(£)—)O as k—)oo
e a

That means:-

. . . k k +
Since we have consrdered an interval —
e

S n < ——1 , during which x(n) and z(n) are
a

bounded functions of x(k) and 2(5) , therefore
a 8

3?(n)—>O 2'(n)—-)0 as n—>oo

Hence, x(n) and z(n) are also bounded for all n 2 0 .

42

As a result of all this discussion, we conclude that we can recover the performance
of the system under state feedback using output feedback control. We see that the
system is also asymptotically stable and bounded and reach steady state value after the

first cycle. However, the closeness of the two systems is followed after first cycle when

the initial conditions are different.

CHAPTER 4

SIMULATION RESULTS

In this chapter, the formulation of the state feedback control, output feedback
control using high gain observer, and results of different simulations are discussed. The
concept of multirate control is used to design output feedback control. Then the effects of
changing different parameters like initial conditions, a, and sampling time are discussed.
Also the peaking phenomenon and saturation of control is discussed. Finally the results,

and future recommendations are given at the end.

4.1W

4J.lIhLCQflL[QH§.d.S¥SI§HLA.d§..S§IX.QIHQLQI

For my thesis, I took the dc servomotor as the system to be controlled [20]. The
motor can be described as a linear, time invariant system. We have the following second

order model for the dc servomotor:
®.(r)=w(t) (1)

t kam

"I

(6(1) = (2)

where
(9",(1) = position of the motor
ua (t) = input voltage

(u(t) = angular velocity

43

44

 

J = moment of inertia of motor and load : 2.6 x 10‘5 Kgm2
R = Resistance : 1.6 ohms
Kb = Back emf constant : 0.0782 Vs
KT = Torque constant of motor : 0.076 Nm/A
. . JR
1 m = Mechanical time constant of motor :
K, K .,

Now writing (1) and (2) in matrix form, we get

- 0 1 0
{Am}. ‘l@"‘”l+ .1.
03(t) "I: (u(t) KIT... "

_l

 

9 "I,
y(t) = [1 0][ 553)] (4)

It is to be noted that there is only one output of the motor,®m . Now replacing the
variables by their specific values of the motor, (3) is written as:

IMH“ . M ° 1
(u(t) 0 —142.86 (u(t) 1826.85

or, we may write (4) and (5) in the standard form of state space model, given by:
12(1) = Ax(t) + Bu(t) (6)
y(t) = CW) (7)

where

x(t) {9.0)}: [x(t)]
(”(0 x2 (I)

t _ ®m(t) _ xl(t)
x”' mm ‘ x.<t)

45

and the system matrices are given by:

0 l O
A = B =
0 -142.86 1826.85

C = [l 0] D = [0]
Equations (6) and (7) represent continuous time state space model. To get a discrete time
model, we have to discretize the given system. Now selecting ‘ zero order hold ‘ and the
sampling time as Tc = 5 ms, and using the Matlab command ‘ c2dm ’, we get the
discretized system as given below:

x(k +1) = ax(k) + bu(k) (8)

y(k) = cx(k) (9)

where a, b, and c are discrete time matrices, and are given by:
_ 1 0.0036 B _ 0.0182
0 - 0 0.4895 ' 65277
c = [1 O]

4.1.2RQlesofIhesttem

In control system design, we can use the pole placement method to place our
poles, if all the states are available. However , even if some of the states are not available,
we can use an observer in the feedback path. Using pole placement allows us to choose
the poles anywhere in the s-plane or z-plane, depending on the desired behavior of the
system. To find the pole location, we assume a second order system with the following

transfer function:

46

(02

GS = " 10
() sz+2§wns+wi ( )

 

where g is the damping ratio and a)" is the undamped natural frequency. Now if we

specify the relative overshoot to be maximum 5%, then using the relation:

_gn

{1; 7A
rel. overshoot = e‘ F’

the damping ratio is found to be g: i . Also, for the system, the settling time is

J2

assumed to be t5 = Is. For a second order system, we have

4.6
t, =—
‘ £60,.

and after substituting the values, we get to n = 68". Now the poles can be calculated by

setting the characteristic equation of our assumed transfer function (10) to zero. With the

above results, we get

1
sA +2-——-6s+ 6 A :0
.2 ()

which gives

I—H—J

The poles 51,2 lie on 45 degree axis, in the left half of s-plane, so they meet the stability

criterion. Now the poles can be discretized using the z-transform relation:

z = e“T

By specifying the values, Matlab directly calculates the poles in z-transform. The discrete

time poles are given by:

47
2,, = 0.9788 i 10.0208

For using in Matlab program, we define the pole vector as:

P = [0.9788+j0.0208 0.9788-j0.0208]

4.1.3I2esign_Q£SIate_Ee_edhack_C_o_ntml

The assumed control law is given by:

u(t) = -Kx(t) + Nr(t) (11)
This is for continuous time system. Since we are dealing with discrete time system, so the
control law for such system is defined as:

u(k) = —Kx(k) + Nr(k) (12)
where K is a 1x 2 gain matrix, x(k) is 2 x 1 state matrix. The state matrix contains two
states x1 and x2 . State x, is the output which is the difference between the reference
position and the actual position of the motor, and state x2 is the angular velocity 00 . The
Nr(k) is the weighted step forward function, which is needed to force the tracking error to
zero when the motor reaches the steady state value.

Now the gain K has to be designed such that the matrix (a-bK) meets the Hurwitz
stability criterion. Matlab command ‘place’ is used to calculate the gain K. We have
found out the gain K to be:

K = [0.0270 -0.0718]

The eigenvalues of (a-bK) are by:
eig(a — bK) = 0.9788 i j0.0208

which lie inside the unit circle of the z-plane.

48

To find the value of N, consider a discrete system with a transfer function H(z)
and step input r(k) with its z-transform given by:

Z

—.r
z—l

r(z) =

To force the output Y(z) to approach the value of the step input for discrete time k —> oo ,

that is, lim Y (k) , we get the following equation:

limY(k) = [lim(z-1)Y(z) = linll(Z-1)'H(Z)°ilr
am am :__’ z‘

: H(1)r (13)
To reach now the value of the step input, we have to multiply the step output r by the
inverse of the limit of the transfer function, which is H(l)". Now substituting (12) in (8),
we get

x(k +1) = (a — bK)x(k) + bNr(k) (14)
Now applying z-transform, we get

zlx(z) = (a — bK)x(z) + bN r(z)
Solving this equation for x(z) and then substituting in (9), we get

Y(z) = c(zI — a + bK)‘l bN r(z)
Now taking limits, we get

lzi—rrllflz) =c(I—a+bK)"bNr(1) (15)
Now comparing (15) with (13), we get

N = H(l)“ = (e(I—a+bK)-'b)" (16)

And after substituting the values, we get N = 0.0270.

49

4.1.4C0nstmctimfanflhsemr

To check, whether our system is observable or not, we use the method described

in chapter 2. We find the rank of Observability test matrix given by:
N = [A T A TCT]

Now using the given matrices of the plant, we get

“13 ‘31

It is obvious that the rank of N is 2, equal to the order of the system, therefore, the given
system is completely observable.
The observer structure has the same order as the original system. Therefore, the

observer equations are given by:

. . a .
(9,, =0) +—é'—(o,,, —o,,,) (17)
c3 = Sg—(om 43),) (18)

The parameters a, and a2 are chosen to be one. The observer gain matrix L is chosen to

be:

The equations (17) and (18) can be written in matrix form as:

A _l 1 A .1
(Z) = i (Z? + fi 9.. (19)
0) -7 0 0) __

8 82

50

where 6)," is the output of the motor. Now if we choose the value of 8 = 0.01667, which

corresponds to l = 60 , the gain matrix L is given by L = [60 3600]T which corresponds
8

to very high gain. With this value of 8, (19) can be written as:

[é]=[ ‘60 1][®]+[ 60 ]o,,, (20)
(,3 —3600 0 (,3 3600

The reason for choosing this value of 8 are:

a. The poles of the high gain observer are located relatively farther to the lefi in the
left half of the s-plane than the poles of the control. This means that the convergence of
the estimation error is faster than the actual control. The characteristic equation of the

unforced observer system is given by:

det[sI — A] = 52 + 605 + 3600 = 0
Therefore, the poles of the observer are located at:

s,, = —30 i 13073
b. Increasing the gain, that is making 8 -—> 0, means also increasing the frequency
band of the observer. Practical experience shows that the frequency of the observer
should be at least 10 times smaller than the sampling frequency. We assume that fobs = 60
Hz. The sampling frequency is assumed to be fs = 2000 Hz, which corresponds to a
sampling time of 0.5 ms. Therefore, fob, << f5 . We denote the sampling time of the

observer by To , then To = 0.5 ms. Comparing the sampling time Tc of the control, and To

51

. . . . . T
of the high gain observer, we see that the ratio IS g1ven by n = F‘ = 10. Thus, each tenth

0

observer update is pushed to the control.

Now consider (20). This is a continuous time observer equation. We can write it

0'0) = 4.00) + 8.9.0) (21)
y. 0) = C.0(t) (22)

where ‘0’ stands for observer state, and

0(1) = (3(1) = [9(1)]
co(t) 02 (I)
A 0 t
and 0(1): (:90) =[ ‘0]
(0(1) 02 (I)
Now we discretize the observer system (21) - (22) using ‘ zero order hold ‘ and sampling
time of T0 = 0.5 ms, using the Matlab command ‘c2dm’. We get
00 +1) = 4.000 + b.,9,,.(k> (23)

Y(k) = 6.0(k) (24)

where

0

0.7042 0.0043 F 02955
a - 0 = c0 = [I 0]
—15.3134 0.9595 15.3134

4.1.SComhinedestemfiingleRateJmplemsntation

The term ‘ combined ‘ is used to indicate that the observer is brought into the
feedback loop, but both the control and the observer use the same sampling rate.

Equations (8), (9) and (12) are rewritten as:

52

x(k +1) = ax(k) + bu(k) (8)
y(k) = 636(k) (9)
u(k) = -—Kx(k) + Nr(k) (12)

Now the output of the observer 0(k) is used in the control law (12) instead of the state
x(k). Therefore, (12) takes the form of output feedback control given by:

u(k) = —K0(k) + Nr(k) (25)
In (23), ®m(k) is the input to the observer. Since the input of the observer is, in fact,
output of the actual system, therefore, substituting (9) in (23), we get the observer
equation based on the output of the system as:

0(k +1) = b0 cx(k) + (100(k) (26)
Now substituting (25) in (8), we get the closed loop system as:

x(k +1) = ax(k) — bK0(k) + bNr(k) (27)

Now writing (26) and (27) in matrix form, we get

[x(kn) _ a —bK x(k) + bN k 28
L0(k+1) " boc a0 0(k) 0 r” ( )
and
k
y(k)=[c 0(2213] (29)

or it can be written as a new closed loop system as:
H(k +1) = AA H(k) + BBr(k) (30)
y(k) = CC H(k) (31)

where

53

0(k +1)

x(k +1)
H(k +1) =[ ] 0(k)

H(k) {1410]

and after substituting the values, we get

1 0.0036 -0.0005 0.0013q
0 0.4895 —-0.1762 0.4685

 

 

AA =
02958 0 0.7042 0.0043
_15.3134 0 -15.3134 0.9595_
”0.0005‘

0.1762

882 0 CC=[1 0 0 0]

.. O _

 

 

It is to be seen that zero padding is done in BB and CC to make the dimensions

compatible with the system, since the order of the combined system is 4.
4.2 SIMLLAIIQNS

After constructing the system, now we go to the computer simulations. I used

UNIX system to implement all the simulations.
4.2.1 StateEeedhackLQntrol

The Matlab program, that implements the state feedback control discussed in
section 4.1.1 - 4.1.3, is given in Appendix A. In this Matlab program, the continuous time
system matrices are defined first, then the system is discretized at sampling time Tc = 5
ms, and using the Matlab command ‘02drn’. Then the poles are calculated using 2-
transform formula 2 = exp(sTc). The gain matrix K is then calculated based on the poles
calculated earlier, by using the Matlab command ‘place’. The weighting factor N is

calculated. Then the matrices for the closed loop system are defined as A1 and B1. Then

54

defining r as a step input for k = 1000, (or it can be varied), the closed loop system is
implemented using ‘dstep’ command from Matlab. The output y and the states x, and x2
are stored in a vector [y,x]. The state feedback control is then calculated using the state
vector x. To fix the x-axis to time scale, a time vector is defined, and finally the ‘plot’
command is used to get the plots for desired states / output.

The resulting plots for the control U, and the states x1 and x2 and the output y are
given in Appendices A1, A2, A3 and A4, respectively. Looking at the plot for U, we see
that it has a spurious peak in the transient period, it reaches steady state value after about
Is, which is very close to our assumption of t5 = Is as given in section 4.1.2. The control
finally settles down to zero. Note that the maximum amplitude reached by the control in
its transient period is about 0.22. Now looking at the plot for the state x,, which is the
same as the output y, we see that this also reaches its steady state value after about Is,
which means that the output is exactly tracking the input which is a step input in this
case. The state x2 is following the control U, except for the difference in the transient

period, where the amplitude of x2 is much more than the amplitude of control U.

4.2.2 CombineinstemJngleRatelmplementanon

The Matlab program that implements the combined system, discussed in section
4.1.5, is given in Appendix B. In addition to the explanation given in section 4.2.1, the
continuous time observer matrices dependent on 8, given by A0 , B0 ,C0 and D0 are also
defined. For single rate implementation, both the control and the observer are discretized
at the same sampling time Tc , by using the Matlab command ‘c2dm’. Then the closed

loop system, with the observer placed in the feedback path, is simulated using ‘dstep’

55

command from Matlab. The output y and the states x1 , x2 , O, , 02 are stored in a vector
[y,x]. The states x, and x2 are the states of the controller, and 0,, 02 are the states of the
observer which represent the estimates of the states x, and x2 , respectively. Time vector
is also defined for x-axis, and finally the ‘plot’ command is used to get the desired graphs
of the states.

The resulting plots for the output feedback control U, states x1, x2, 01, 02 and the
output y are given in Appendices B], B2, B3, B4, BS and B6, respectively. Looking at
the graph for U, we see that it attains its steady state value after about two seconds, and
finally settles down to zero. This delay in settling time is caused by the observer in the
feedback path. However, it is to be noted that this control graph is almost similar to the
one described in section 4.2.1. Now looking at the graph for state x], which is also the
output of the system, it is seen that it also reaches its steady state value of one afier about
two seconds. Comparing this one, with the graph for x, of section 4.2.1, we note that both
are similar except for the small difference in the overshoots in transient period and
difference in settling time. The state x2 is following the control U, with large overshoot in
the transient period. The plots for O, and 02 are same as that for x, and x2, since 01 is the
estimate of x, and 02 is the estimate of x2. This is because the initial conditions for both
are the same and also the sampling time is the same. The point to note is that, the
performance under output feedback control using single rate implementation is similar to
the performance under state feedback control. Since zero initial conditions are assumed
for both the systems, no peaking is observed in this implementation. The peaking will be

observed when the initial conditions are different, as already discussed in chapter 2.

56

4.2.3 Multiratelmnlementation

Now we develop a Matlab program that will simulate the system (30) - (31) using
two different sampling rates. The dynamics of the control are slow and the dynamics of
the observer are fast. The Matlab program that simulates this scheme is given in
Appendix C. Note that the two sampling times are now different, the dynamics of the
observer are ten times faster than that of the control. The value of 8 is assumed to be
0.01667 as before. Now since the dynamics of the control and the observer are different,
so they cause an unstable behavior of the system. Therefore, before they enter the plant,
both should have the same (fast) dynamics. To cater for this point, we sample the actual
system given by continuous time matrices A, B, C, and D, at fast sampling period To .
Therefore, we get discrete time matrices al, b1, c1, and d], and

x(k +1) = a,x(k) + b,u(k) (32)
and (26) will now become:

0(k +1) = b0 c,x(k) + a00(k) (33)
with u(k) defined by (25), the output feedback control. Now we want to keep the control
(25) constant for ten points and after the tenth point, we want to update the control, while
the states of the observer are calculated at every point. So we define ‘n’ as fast time scale,

and select ‘n’ as a ratio of the sampling time of the control to the sampling time of the

. T . .
observer, i.e., n = T‘ . Since Tc = 5 ms and To = 0.5 ms, therefore the who comes out to

0

be n = 10. Also we define k as the slow time scale. The choice for the number of points k

depends on how long it takes for the output feedback control to reach its steady state

57

value. In our case, we have chosen the number of points of k for multirate to be equal to
the number of points used for the state feedback control. Then we need to make a loop in
the Matlab program, which, will carry out n x k calculations using (25), (32) and (33).
The Matlab command ‘dlsim’ is used to implement this scheme. The initial conditions are
also required to run the loop, and these initial conditions are defined as x0 and 00 for the
control and the observer, respectively. The results are stored in a vector [h,g], where h is
the output and g represents the states.

Now one cycle consists of ten such calculations before the control is updated for
the next cycle. Therefore, there is a need to store the results for each cycle. The program
is so designed such that after the very first cycle, the last value of each cycle becomes the
initial conditions for the next cycle. Therefore, to avoid double value storage in the
vectors gg and uu, we leave the last value of the results of each cycle and store them up to
n-l values. By this way, we shall be stacking the results of all the cycles in one vector
without having any of the double values. However, there is a problem observed in this
procedure, that is, instead of storing ten values for each cycle, we are, in fact, storing nine

values. That way, we loose information which corresponds to one complete cycle of fast

sampling period, i.e., To . To take care of this problem, we choose 71 = %+1 , so that,

0

now after storing n-l points, we shall have stored, in fact, ten points for each cycle,
instead of storing nine points for each cycle as previously. This procedure is shown in

Appendix C6.

58

This Matlab program for multirate scheme is run for same initial conditions for
both the control and the observer. The initial conditions x0 and 0,, are assumed to be zero
in this case. We can see, from Appendices C1, C2, C3, C4, and C5, that the behavior of
the output feedback control uu, and the states x, , x2 , O, ,and 02 is exactly the same as it
was for single rate implementation of the combined system discussed in section 4.2.2,
which means , that, for same (zero) initial conditions, the performance under output
feedback control using multirate scheme, is same as the performance under state feedback
control using single rate scheme.

4.2.4" 011141 '0 U. -.‘. 1‘11“ 11‘ .-.‘ “no. \ 01.0

This Matlab program can be modified to obtain the results if we use state
feedback control instead of the output feedback control inside the loop of the program. To
implement this scheme, we replace the equation for u,, inside the loop, by the equation:

uo = —Kxo + Nr (34)
This scheme is shown in Appendix D. This program is run for the same (zero) initial
conditions for both the systems. Now looking at the Appendices D1, D2, D3, D4, and D5,
we observe that these graphs are very much similar, in fact, the same, to the one obtained
by the state feedback control designed in section 4.2.1. This was the case when the initial
conditions are the same and assumed to be zero for both systems. The effects of different

initial conditions will be discussed later in this chapter.

4.2.5 BehammofMulhrateSchemunhrstflycle

Now to see the behavior of multirate scheme in first cycle, we modify the Matlab

program given in Appendix C. We keep n = 10 and change k from k = 100 to k = 1, so

59
that the output feedback control is constant over this one cycle. We also assume that the

initial conditions are the same (zero) for both systems. This scheme is shown in Appendix
B. We see from the Appendix E1 that the control is constant for one cycle. The
Appendices E2, E3 show that the states x, and x2 are rising in the first cycle. From
Appendix E4, E5 it can be seen that the state 0, is trying to follow the state x], but state
02 has more steeper trajectory; however it is also rising to attain its maximum value. The
important point to be noted here is that all the states attain very low amplitude value in
the first cycle.

4.3 70 nit I; I { kl ’aiaul I:

All the cases discussed in section 4.2 correspond to the same (zero) initial
conditions, the value of 8 = 0.01667 and Tc = 5 ms, T0 = 0.5 ms. Now let us see as to

what happens if we change these parameters.

4.3.1 Effects of Changing the Initial Conditions

4.3.1a' 0111-..1‘0 14...: 10.1- 2‘, “0.1.. r 01,0

Now the initial conditions are made different for the plant and the observer, i.e.,
Xo=[0;01 and 06=[1;1I
This scheme is shown in Appendix F. The resulting graphs for uu, x1 , x2 , shown in
Appendices F1, F2, and F3, respectively, are similar to the one for the case when the
initial conditions were the same (zero). However, we can notice the change in the
behavior of O, and 02 given in Appendices F4, F5, respectively. We can see the change

in the transient period of both the observer states. This is, because the initial conditions

60

for the observer are different from the initial conditions of the plant, 0,, = [1 ; 1] in this
case. The two systems starting at different initial states will naturally exhibit different
behavior. The overshoots in the transient response of 01 and 02 do not cause any change
in the control uu, and the states x, and x2 , because the control law (34) is not using any
of the states 0, and 02 , i.e., the observer is bypassed. However, it is seen that all the

states reach their steady state values after about Is. The overshoots in the transient period

of 02 is of the order of 0(1).
8

4.3.1bBerf0nnanceilndeLQmputfieedbackE0ntrol

The performance of the system under output feedback control, with the same
(zero) initial conditions, was discussed in section 4.2.3. In this section, we consider the
performance of the same system, but with different initial conditions for the two systems.
The initial conditions for the observer are assumed to be 0,, = [1 ; 1], whereas the initial
conditions for the plant are x0 = [0 ; 0]. This scheme is represented in Appendix G, and
the resulting graphs for control uu, and the states x, , x2 , 0, , and 02 are given in
Appendices G1, G2, G3, G4, and G5, respectively. Now the results are quite different
from all the previous cases. From the graphs, we can observe the large overshoots in the
transient response, which we call as peaking phenomenon, which was discussed in
chapter 2. Now since the initial conditions for the observer are 0,, = [1 ; 1], there is an
impulsive behavior, i.e., large overshoot, in the transient period. These overshoots are fed
to the control given by (25). Therefore, the control (25) also exhibits the large overshoot

in its transient period, which is obvious from the graph for uu, given in Appendix G1.

61

These overshoots are then fed to the plant via the control (25), which causes the states x,
and x2 to exhibit peaking in their transient period. Looking at the graphs for different
states, it can be seen that the transient response of 0, is transmitted to state x, and the
transient response of 02 is transmitted to state x2 . Since the output of plant is the input to
the observer, these states disrupted with peaking are fed to the observer, which like
before, enter the plant via the control (25). This process continues and we observe very

large overshoots in the transient period. The overshoots in the states x2 and 02 are of the
1

order of 0(—).
8

However, all the states reach the steady state value after the first cycle. This is
because, the fast modes, i.e., O, and 02 decay rapidly in the first cycle as discussed in
chapter 3. After the first cycle, the response of the system is due to the slow modes, i.e.,
x, and x2 . The behavior of the system in the first cycle, for different initial conditions, is
given in Appendix H. We can see from Appendices H4 and H5, that the fast modes are
decaying rapidly from their starting position. This proves the idea discussed in chapter 3,
where we used singularly perturbed approach to prove that, the response of the system,

after first cycle, is due to the slow modes as the fast modes decay to zero in first cycle.

4.3.2 EffectsofSaturatinglheLQntml

As discussed earlier in chapter 2, we can reduce peaking by saturating the control.
In this scheme, we set the limits of output feedback control so that it can swing between
specified values. This scheme of saturating the control is shown in Appendix I. A subloop

is inserted in the outer loop of the Matlab program, and the maximum swing limits are set

62

to :05. This value of swing limits is chosen keeping in view the maximum swing of the
state feedback control. After setting the saturation limits of the output feedback control,
the program is run.

The resulting graphs for saturated control, and the states are given in Appendices
II, 12, I3, 14 and 15, respectively. Looking at the graph for control, we see that, the
maximum swing has been limited to i0.5. It is important to note the behavior of the
output x, . Due to the output feedback control saturated, the negative part of overshoot,
which had an amplitude of about -5 in case of nonsaturated control, has been significantly
reduced to an amplitude of -0.4. Also the positive part of overshoot has been reduced to
about 1.3 from about 2.5. As a result of saturation, the graph for x, has come closer to the
one obtained under state feedback control. Similarly, due to saturation, the overshoots in
the transient period of x2 has been reduced significantly. Also, the behavior of x2 under
saturated output feedback control is close to the one under state feedback control. The
effect of saturation is also obvious on the observer state 0, , which exhibits similar
behavior as state x, . However, the observer state 02 , refer to Appendix 15, does not
saturate much as the negative part of overshoot remains unchanged, however, the positive
part of overshoot reduces from about 7 to about 3. This is because, the saturation does not
eliminate peaking form 02 . Therefore, the idea of reducing peaking phenomenon by

saturating the control, discussed in chapter 2, is confirmed here.

4.3.3 Effectmehangingj

Now to see the effects of changing the value of 8, we keep the other parameters

like Tc , T,, , initial conditions, same as before and change only 8. Now we assume the

63

value of 8 ten times smaller than the previous value, i.e., now we have 8 = 0.001667,

which corresponds to l = 600. The observer gain matrix L, now has large numbers, i.e.,
8

L = [600 360000]T

The poles of the observer are now located, even farther to the left in the left half of the s-
plane than the poles of the control. The poles of the observer with this value of 8 are

located at:
s,, = —300 i 13006

This means that the convergence of the estimation error will be even more faster than the
control as compared to the previous case. The Matlab program that simulates this scheme
for multirate case is given in Appendix J. The resulting graphs for the control and the
states are given in Appendices J1, J2, J3, J4, and 15, respectively. In this simulation, the
initial conditions are assumed to be the same (zero). We can see that, the control and all
the states attain their steady state value after about Is. Whereas for the same program
with 8 = 0.01667, as given in Appendix C, it took about 23 for the control and the states
to reach their steady state values. The graphs for 8 = 0.001667 are very much close to the
one obtained under state feedback control given in Appendix A.

Now let us see the effect of having different initial conditions. In the Matlab
program given in Appendix K, the initial conditions for the observer are set to 0,, = [1 ; 1]
and for the plant x,, = [0 ; 0]. The resulting graphs are given in Appendices K1, K2, K3,
K4, and K5, respectively. We can observe the peaking phenomenon, but the point to note

is that, the peaking is observed for a very short period in the transient period. It seems to

54
be a sort of pulse with very short time period and large amplitude in the first cycle, after

which the system reaches its steady state value. The effect of decreasing the value of 8,
has caused relatively larger overshoots in the control and the states x2 and 02 , as can be
seen from the graphs given in Appendices H1, H3, and H5, respectively. Since, due to
different initial conditions, state of the observer 02 exhibits more peaking in the transient
period, which is being fed to the control, therefore the same effect appears in the transient
response of the control. Now at the output, we have only one state x], which is fed back
to the observer, therefore the state 02 remains unaffected thereafier. However, the control
and ultimately the state x2 , will be effected by the state 02 . That is why, we see the large
overshoots in the transient periods of control and x2 . We also observe that, as 8 is made
smaller, the overshoots in the transient periods of x, and 0, have been reduced as
compared to the case of previous value of 8. Also notice that, the amplitude of the
overshoot of 02 is about ten times larger as compared to the case when 8 was ten times
larger.

Now let us saturate the control to reduce the peaking. The saturation level for the
control is fixed to i0.5 as before. This scheme is given in Appendix L. The resulting
graphs for control and the states are given in Appendices L1, L2, L3, L4, and L5,
respectively. Looking at the graph for the output x, , we note that the response has been
improved a lot as compared to the case when 8 = 0.01667. The negative peak in the
transient period has been reduced significantly, and this graph is closer to the one given in

Appendix A2. Therefore, by reducing the value of 8, we get the response under output

65

feedback control closer to the one under state feedback control. The effect of saturation is
also obvious from the graphs given in Appendices L3 and L4.

Now let us see the behavior in the first cycle for 8 = 0.001667. For the same (zero)
initial conditions, the response in first cycle is almost same as discussed earlier for the
case when 8 = 0.01667. However, for initial conditions different from the previous case,
i.e., for 0,, = [1 ; 1] and x,, = [0 ; 0], the response is different. The scheme when the initial
conditions are the same (zero) is given in Appendix M and the corresponding graphs for
control and the states are given in Appendices M1, M2, M3, M4, and M5, respectively.
And for different initial conditions, the scheme is given in Appendix N, with
corresponding graphs given in Appendices N1, N2, N3, N4, and N5, respectively. In the
first cycle, the states x, and x2 are still rising, but as is seen from Appendix N4, the state
0, is reaching its steady state value, after decaying rapidly. Also, observing the state 02
from Appendix N5, we see that, after decaying very rapidly and attaining the maximum
overshoot, the state is trying to reach the steady state value. This shows that the decay of
fast modes in the first cycle is even more rapid as compared to the case when 8 =

0.01667. This also confirms the idea discussed in chapter 3.

4.3.4 Effectmfflhanginmnplinglimes

Now let us assume a sampling time of 10 ms for the control, i.e., T,, = 10 ms. To
keep up the ratio of 100 between the sampling times of the control and the observer, we
assume sampling time for observer to be 0.1 ms, i.e., T,, = 0.1 ms. The Matlab program
for the simulation of state feedback control, using new sampling time, is shown in

Appendix P. Since the poles of the system are dependent on the sampling time due to the

66
relation given by z = exp(sTc), therefore they will change depending on the value of the

sampling time. The program for new sampling time is run for 500 sampling points, and
the corresponding graphs for the control U, and the states x1, x2 are given in Appendices
P1, P2, and P3, respectively. Looking at these graphs, we see that they are exactly similar
to the one given in Appendix A. It takes about Is to reach the steady state value as for the
case when the sampling time was T, = 5 ms.

For multirate implementation, we keep the control constant for n = 100. Now if

we sample the control at T, = 10 ms, then to get 11 = 100, we sample the observer at

T,,=0.l ms as n =% . That means, the dynamics of the observer are about 100 times

0

faster than the dynamics of the control. Using these sampling times means that every
100th update of observer is being fed to the control. In this case when n = 100, we shall
repeat the computations for k = 500 to get the interval of 55 on x-axis. This scheme of
multirate implementation is given in Appendix Q. The resulting graphs for the control
and the states are given in Appendices Q1, Q2, Q3, Q4, and Q5, respectively. These
simulations are for the case when the initial conditions for the two systems are different,
i.e., 0,, = [l ; l], and x,, = [0 ; 0]. Looking at these graphs, we see that, the behavior of the
control and the states is almost similar to the one given in Appendix G. The only
difference is in the amplitude of the overshoots in the transient period, i.e., the peaking of
little less amplitude has been introduced. This peaking can be reduced by saturating the
control as before. It is also seen that, the steady state value is reached after about 1.53

which is exactly the same as was found in case of previous values of the sampling times.

67

Therefore, changing the sampling times have not produced any significant change in the

behavior of the system.

4.4 W

After going through all the simulations, as given in Appendices A-Q, we conclude

the following results.

a.

The state feedback control designed was stable and bounded, and reaches its
steady state value.

The system worked good under state feedback control. All the states were stable
and bounded and reached their steady state value.

The output feedback control designed also achieved stability and boundedness
and reached the steady state value. As a result, all the states were stable and
bounded and reached their steady state values.

The performance of the system under output feedback control was as good as
under state feedback control. In other words, we could recover the performance
of the system under state feedback control by using the output feedback control.
In the first cycle, the fast modes decay rapidly and afier the first cycle, the
response of the system is due to the slow modes with small error induced by the
fast mode.

As 8 —) 0, the convergence of the estimation error is much faster, and the
system’s performance under output feedback control is very much closer to the

performance under state feedback control.

j.

68

Changing the initial conditions, i.e., having different initial conditions for the
plant and the observer induces peaking.

Saturating the control helps in reducing the peaking phenomenon significantly.
The effect of changing the sampling times is not very significant, and ultimately
it does not effect much the performance of the system, as long as the sampling
time is chosen small enough.

The ideas discussed in chapter 2 and 3 were confirmed.

4.5 RECQMMENDAIIQNS

P

I have the following recommendations for future work.

In my simulations, l have used only one input, that is, the step input. These
simulations should be run with different inputs, such as, impulse input, ramp
input, etc. to see the behavior of the system.

In the analysis given in chapter 3, the observer is assumed to be continuous time,
whereas [12] suggests that the observer should be in discrete time. Therefore, the
same analysis should be done with the observer converted in discrete time.

In my simulations, the saturation of the control could not eliminate peaking
significantly from the observer state 02 . Therefore, efforts may be done to
reduce it.

The simulations are performed for a stable system. However, the same program

should be tested for an unstable system and compare the results.

69

CQNCLIISIQN.

The performance of a linear, time invariant system was tested using the output
feedback control that uses a very high gain observer in its feedback path. The behavior of
the system was checked, theoratically, in the first cycle using singular perturbation
theory. The simulation results later, proved that the fast modes decay in the first cycle
rapidly and the response of the system after the first cycle, is due to the slow modes.
Also, simulation results proved that, the performance of the system under output feedback
control was as good as under state feedback control. In other words, we could recover the
performance of the system under state feedback control by using the output feedback
control. It was also proved that, the saturation of the control helps in reducing the peaking
that is caused by having different initial conditions for the two systems. It is also seen that
by making the value of 8 very small, the system’s performance under output feedback

control can be brought very close to the performance under state feedback control.

APPENDICES

70

APPENDIX A

Matlab program for State Feedback Control

A = [0 1;0 -l42.86];

B = [0;1826.85];

C = [1 01;

D = [0];

To = 0.005;

[a,b,c,d] = c2dm(A,B,C,D,Tc,'zoh')

S, = (-6/sqrt(2))+i*(6/sqrt(2));

32 = (-6/Sqrt(2))-i*(6/Sqrt(2));

Z1 = e"19(SI'WI‘C);

Z2 = CXP(32*TC);

Pc = [Z1 Z2]

K = place(a,b,Pc)

1= [1 0;0 1];

N = inv(c*inv(I-a+b*K)*b)

A1 = [a-b*K];

Bl = [b*N];

1U = [1];

k = 1000;

step = [k];

NN = N*ones(k,1);

[y,x] = dstep(A1, B1, c, d, IU, step);

U = (x*(-l)*K')+NN;

x1 = X(Z,1);

X2 = X(I,2);

for k = l:length(U)
t(k,1)=k*Tc;

end

whos;

plot(t,U); title( ' plot for U , k = 1000, Tc = 0.005, state feedback control ' )

xlabel ( ' time (sec) ' );

ylabel ( ' amplitude ' );

end

71
APPENDIX Al

plot for U 11:11:00. Tcaooos. Sate feedback control

 

0.25 l r l I I r I I I

0.2-

0.15-1

 

0.06 r

 

 

I—
p
_

 

 

005 L l 1 l l
O . .

72
APPENDIX A2

pbt b1 :1 ,k .1m0.Tc-n 0.005

 

 

 

 

1.2 1 1 1 1 1 1 1 1
‘ -

0.0 -

0.0 '-

0.‘ '-

0.2 r
0 1 1 1 1 1 1 1 1
0 0.5 I 15 2 2.5 3 3.5 4.5

the Inc]

 

mpkude

2.5

73

APPENDIX A3

pbt 101:2 .k a IMOJ'ca 0.005

 

 

 

I

I

j

 

 

4.5

 

74
APPENDIX A4

plot for} .11 - 1000. T1: - 0.006

 

 

 

 

1.2 1 r 1 1 T f 1 1
1 1.
0.0 .
0.6 '
0.4 -
02 ’-
0 1 1 1 1 L 1 1 1
0 0.5 1 1.5 2 2.5 3 3.5 4.5

inc {soc}

 

75

APPENDIX B

Combined System - Single Rate Implementation

A = [0 1;0 -142.86];
B = [0;1826.85];
C = [1 0];
D = [0];
Tc = 0.005;
[a,b,c,d] = c2dm(A,B,C,D,Tc,'zoh');
S, = (-6/sqrt(2))+i*(6/sqrt(2));
$2 = (-6/Sqrt(2))-i*(6/Sqrt(2));
Z, = exp(s1 *Tc);
Z2 = exp(s2*Tc);
PC = [Zr Zz];
K = place(a,b,Pc);
I = [l 0;0 1];
N = inv(c*inv(I-a+b*K)*b);
8 = 0.01667;
A0 = {-1/8 1;-1/(8"2) 0];
Bo = [1/8;1/(8"2)];
Co = [l 0];
D0 = [0];
[ao,bo,co,do] = c2dm(Ao,Bo,Co,Do,Tc,'zoh');
AA = [a -b*K;bo*c ao];
BB = [b*N;0;0];
CC = [c 0 0];
DD = [0];
k = 1000;
step = [k];
ID = [11;
NN = N*ones(k,1);
[y,x] = dstep(AA, BB, CC, DD, IU, step);
K = [0 0 K];
U = ( x*(-l)*K' )+NN;
x1 = X(I,1);
X2 = X(I,2);
0, = x(:,3);
02 = x(:,4);
for k = l:length(U)
t(k,1) = k*Tc;
end

76

plot(t,U); title ( ' graph for U, 8 = 0.01667, Tc = 0.005, k = 1000, single rate combined
system ' )

xlabel ( ' Time(sec) ' );

ylabel ( ' Amplitude ' );

end

Ampwde

0.10

APPENDIX B1

77

Plottor Ll .epeo.01667.Tc-0.005.k-10w.e1n¢e are combined system

 

 

I

I

I

I

 

T

I

I

 

 

2.5
Tineteec]

 

78
APPENDIX B2

Plot bl x1.epe - 0.01007.Tc - 0.005. 11 . 1&0

 

I I I I I I I

 

 

 

 

 

 

l I L L l l l
1 .5 2 2.5 3 3.5 4 4.5
Tineteec]

‘FLJ .

Aliphatic

2.5

79
APPENDIX BS

131131le2. we. 0.01667.Tc- 0.M5.k .1000

 

 

 

T I I I I

 

-
.—

l I I

-

 

1 .5 2 2.5 a 3.5
Theteec]

4.5

 

 

APPENDIX B4

80

Plotter 01.epe- 0.01567.Tc a 0.005. It .

1000

 

 

I

I

 

l

I

 

 

 

2.5
Theteec]

4.5

 

Amanda

2.5

81
APPENDIX BS

P101101 02.0p8: 0.01667.Tc- 0.005.1t-1000

 

 

I

 

I I I I I

r

 

 

l I l l l

 

1 .5 2 2.5 a 3 .5
Theteec]

4.5

 

 

82
APPENDIX B6

P1011011. epe :- 0.01667.Tc= 0.005.11 21000

I I T I I I I

 

 

 

 

 

l L I

1.5 2 2.5 3 3.5 4 4.5 5
T'eneteec]

)—
-
-
_

 

 

83

APPENDIX C

Multirate Implementation - Zero Initial Conditions

A = [0 1;0 -142.86];
B = [0;1826.85];
C = [1 0];
D=Wk
1= [1 0;0 1];
e = [0 0;0 0];
Tc = 0.005;
[a,b,c,d] = c2dm(A,B,C,D,Tc,'zoh');
S, = (-6/sqrt(2))+i*(6/sqrt(2));
52 = (-6/Sqrt(2))-i*(6/Sqrt(2));
Z, = exp(S,*Tc);
Z2 = exp(S2*Tc);
PC = [Zr Z21;
K = place(a,b,Pc);
N = inv(c*inv(I-a+b*K)*b);
8 = 0.01667;
A0 = [-l/8 l;-1/(8"2)0];
Bo = [1/8;1/(8"2)];
Co = [1 0];
D0 = [0];
To = 0.0005;
[ao,bo,co,do] = chm( Ao, Bo, Co, Do, To, 'zoh' );
[a,,b,,c,,d,] = c2dm( A, B, C, D, To, 'zoh' );
a = [a, e;bo*c, ao];
bb = [b1;0;01;
cc = [c, 0 0];
dd = [0];
X6 = [0:0];
06 = [0;0];
r = l;
86 = [25.806];
88 = [1;
W = I];
n = 11;
kmax = 1000;
for k = lzkmax
if k = 1
81 = go;
elseif k > 1

 

84

gl = (g(n,:))';
end
00: gl(3:4);
u,, = -K*0,,+N*r;
u = u,,*ones(n,1);
[h,g] = dlsim(aa, bb, cc, dd, u, g,);
88 = [88;8(1:n-1,:)];
uu = [uu;u(1 :n-1)];
X = 880.112);
0 = 880,3:4);
x1 = gg(:al);
x2 = gg(92);
0| = gg(:93);
02 = gg(:’4);
end
for k = 1:1ength(0,)
t(k,1) = k*To;
end
whos;
plot(t,01); title( ' Graph for 0,, 8 = 0.01667, 11 = 11, k = 1000, 0,, = [0;0], To = 0.0005 ' )
xlabel ('Time (sec)’);
ylabel ('Amplitude’);
end

Ampltude

0.2

0.15

0.1

85
APPENDIX C 1

Graph b1 uu.epe= 001667.11 - 11.11: 1000.00 3 p.01.To :- 0.0005

 

I I r I I I I I I

 

I

 

L

 

2.5
The {sec}

0.5 1

 

0.4

02

86

APPENDIX C2

crew 10! x1. epe - 001007.11 a 11.11-10”. 00 - 10.0].1'0 :- 0.0005

 

 

I

I

I

f

I

I

r

 

l

_

 

2.5
The {eecl

4.5

 

Al'lpIIJdd

2.5

APPENDIX C3

87

(31301110! :2. one :- 0.01067.n = 11.11: 1000. 00 - I0.0].To a 0.0005

 

 

r

I

I

I

r

f

r

I

I

 

 

2.5
Time (sec)

 

 

88
APPENDIX C4

anh 5:101 . Q0:- 0.01667. 11 1:11.11 3 1000.00 3 D.0].To - 0.0005

1.4

1.2

0.4

0.2

 

 

I

I

r

I

I

I

I

I

 

l

 

2.5
Time {sec}

4.5

 

Maude

89
APPENDIX C5

Gllph hr 02. epe - 001067.11 - 11.14 s 1&0. 00 : p.0].T0 a 0.0005

 

2.6 T I I I I I I fl I

 

 

 

1 1 1 1 1 1 1 r 1
-0.5

 

 

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
Time {sec}

00
C

'_‘II COCO 1|

u:

0.0287
0.0287
0.0287
0.0287
0.0287
0.0287
0.0287
0.0287
0.0287
0.0287
0.0287

1 .0e—03

0.0064
0.0250
0.0550
0.0956
0.1461
0.2058
0.2740
0.3501
0.4337
0.5242

0
0.0000
0.0000
0.0001
0.0001
0.0001
0.0002
0.0003
0.0004
0.0004
0.0005

0
0.0253
0.0489
0.0709
0.0913
0.1 103
0.1281
0.1446
0.1599
0.1742
0.1875

0

0
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000

0

0
0.0000
0.0001
0.0002
0.0003
0.0006
0.0009
0.0014
0.0019
0.0027

90
APPENDIX C6

gg=

0
0.0000
0.0000
0.0001
0.0001
0.0001
0.0002
0.0003
0.0004
0.0004

1.11]:

0.0287
0.0287
0.0287
0.0287
0.0287
0.0287
0.0287
0.0287
0.0287
0.0287

0.0289
0.0289
0.0289
0.0289
0.0289
0.0289
0.0289
0.0289
0.0289
0.0289
0.0289

0.0005
0.0006
0.0007
0.0008
0.0009
0.001 1
0.0012
0.0013
0.0014
0.0016
0.0017

0
0.0253
0.0489
0.0709
0.0913
0.1 103
0.1281
0.1446
0.1599
0.1742

0

0
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000

0

0
0.0000
0.0001
0.0002
0.0003
0.0006
0.0009
0.0014
0.0019

91

0.0005
0.0006
0.0007
0.0008
0.0009
0.001 1
0.0012
0.0013
0.0014
0.0016
0.0017

g:

0.0000
0.0000
0.0001
0.0001
0.0001
0.0002
0.0003
0.0004
0.0004
0.0005
0.0006
0.0007
0.0008
0.0009
0.001 1
0.0012
0.0013
0.0014
0.0016

“1.1:

0.0287
0.0287
0.0287
0.0287
0.0287
0.0287
0.0287
0.0287
0.0287
0.0287
0.0289
0.0289
0.0289
0.0289
0.0289
0.0289
0.0289
0.0289
0.0289
0.0289

0.1875
0.2001
0.21 18
0.2227
0.2328
0.2423
0.251 1
0.2593
0.2669
0.2740
0.2806

0.0253
0.0489
0.0709
0.0913
0.1103
0.1281
0.1446
0.1599
0.1742
0.1875
0.2001
0.21 18
0.2227
0.2328
0.2423
0.251 1
0.2593
0.2669
0.2740

0.0000
0.0001
0.0001
0.0001
0.0001
0.0002
0.0002
0.0002
0.0003
0.0003
0.0003

0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0001
0.0001
0.0001
0.0001
0.0002
0.0002
0.0002
0.0003
0.0003

0.0027
0.0035
0.0045
0.0056
0.0069
0.0084
0.0100
0.01 17
0.0137
0.0158
0.0180

0.0000
0.0001
0.0002
0.0003
0.0006
0.0009
0.0014
0.0019
0.0027
0.0035
0.0045
0.0056
0.0069
0.0084
0.0100
0.01 17
0.0137
0.0158

92

93
APPENDIX D

Multirate Implementation - Under State Feedback

A = [0 1;0 -142.86];
B = [0;1826.85];
C = [1 0];
D = [0];
1= [1 0;0 1];
e = [0 0;0 0];
Tc = 0.005;
[a,b,c,d] = c2dm(A,B,C,D,Tc,'zoh');
31 = (-6/Sqrt(2))+i*(6/Sqrt(2));
$2 = (-6/Sqrt(2))-i*(6/Sqrt(2));
Z1 = exp(S,*Tc);
Z2 = eXP(Sz*TC);
PC = [Zr 22];
K = place(a,b,Pc);
N = inv(c*inv(I-a+b*K)*b);
8 = 0.01667;
A0 = [-1/8 1;-1/(8"2) 0];
Bo = [1/8;1/(8"2)];
Co = [l 0];
D0 = [0];
To = 0.0005;
[ao,bo,co,do] = c2dm( Ao, Bo, Co, Do, To, 'zoh' );
[a,,b,,c,,d,] = c2dm( A, B, C, D, To, 'zoh' );
a = [a, e;bo*c, ao];
bb = 1b1;0;01;
cc = [c, 0 0];
dd = [0];
X6 = [0:01;
0o = [0;O];
r = 1;
8. = [xo;Oo];
88 = [1;
W = [1;
n = 11;
kmax = 1000;
for k = 1:kmax
if k = 1
81 = go;
elseif k > 1

94
gl = (g(n9:))';

end
xo= 81(1 :2);
u,, = -K*x,,+N*r;

u = u,,*ones(n,1);
[h,g] = dlsim(aa, bb, cc, dd, u, g,);
88 = [88;8(1:n-1.:)1;
uu = [uu;u(1:n-1)];
X = 880.112);
0 = 88(:.3:4);
x1 = 88(311);
x2 = 880.2);
01 = gg(:93);
02 = gg('94)9
end
for k = l:length(0,)
t(k,1) = k*To;
end
whos;
plot(t,0,); title( ' Graph for 0,, 8 = 0.01667, 11 = 11, k = 1000, 0,, = [0;0], To = 0.0005 ' )
xlabel ('Time (sec)');
ylabel ('Amplitude');
end

Ampltme

95

APPENDIX D1

Grub {or uu.epes 001557.11 - 11.11 =1000.0o :- p.0].To - c.0005

 

0.25 r r 1 1 1

0.2-

0.1 '

I

I

I

I

 

 

 

-005 L l I l L
0 0.5 1 1 .5 2 2.5

Time Ieec]

4.5

 

111111111116.

96

APPENDIX D2

81.1111 161 111.1111. . 001007.11 =.11.11.101:o.011 . 10.01.11: . c.0005

 

 

 

 

1 1 f 1 1 1 1 1 1
1
0.0
0.6
0.4
0.2
0 1 1 1 1 1 1 1 1 1
0 0.5 1 1 .5 2 2.5 3 0.5 4 4.5

The teec]

 

Ampiflde

97
APPENDIX D3

Gum for :2. ops - 001667.11 = 11 . 11:10:10,011 = I0.0].To - 0.0005

3 I I I I T I I I I

 

2.5 - -

 

05* r
.

 

 

 

 

0.5 1 1.5 2 25 3 3.5 4 4.5 5

Amplaida

0.4

98
APPENDIX D4

Graph b101.apa= 001557.11: 11.11-1w0.

0° :1 D.0].T0 - 0.0005

 

 

I I I I I I

I I I

 

L l l l l J

 

0 .5 1 1 .5 2 2.5 3
Time (sec)

3.5 4 4.5

 

99
APPENDIX D5

Graph 10! 02. ops . 0.01667. 11 = 11.11 = 1000.01: a p.01.1'o . c.0005

 

3 I I I I I I T I I

 

0.5-1

 

 

1 1 1 1 1 1 I 4 1
-O.5

 

 

0 0.6 1 1.5 2 2.6 3 3.5 4 4.5
Time {sec}

100
APPENDIX E

Multirate Implementation - Behavior in First Cycle

A = [O 1;O -142.86];
B = [0;1826.85];
C = [1 0];
D = [0];
1= [1 O;O 1];
e = [O 0;0 0];
To = 0.005;
[a,b,c,d] = c2dm(A,B,C,D,Tc,'zoh');
S, = (-6/sqrt(2))+i*(6/sqrt(2));
32 = (~6/Sqrt(2))-i*(6/Sqrt(2));
2] 7‘ e"13(Sl"'1“3);
22 = exp(Sz*Tc);
PC = [Z1 Z2];
K = place(a,b,Pc);
N = inv(c*inv(I-a+b*K)*b);
a = 0.01667;
A0 = [-1/8 1;-1/(£"2) 0];
B0 = [1/8;1/(8"2)];
Co = [1 0];
D0 = [0];
To = 0.0005;
[ao,bo,co,do] = c2dm( A0, Bo, C0, Do, To, 'zoh' );
[a,,b,,cl,d,] = c2dm( A, B, C, D, To, 'zoh' );
a = [a1 e;bo*c, ao];
bb = [b1;0;0];
cc = [c1 0 O];
dd = [0];
x0 = [Wk
00 = [090];
r = 1;
go = [X1500];
gg = l];
W = l];
n = 11;
kmax = l;
for k = 1:kmax
if k = 1
g1 = go;
elseif k > 1

101

S1 = (8(n,=))';
end
00: g1(3:4);
no = -K*OO+N*r;
u = uo*ones(n,l);
[h,g] = dlsim(aa, bb, cc, dd, u, g);
8% = [gg;g(12n-1,I)];
uu = [uu;u(1:n-1)];
x = gg(:,1 :2);
O = gg(:,3:4);
X1 = ggcal);
X2 = gg(92);
01= gg(:,3);
02 = gg(:94);
end
for k = l:length(Ol)
t(k,1) = k*To;
end
whos;
plot(t,01); title( ' Graph for 01, a = 0.01667, 11 = 1 1, k = 1, 00 = [0;0], To = 0.0005 ' )
xlabel ('Time (sec)');
ylabel ('Amplitude');
end

102
APPENDIX El

G-ph for muons-0.01 66711-11 k=1.00-|0.0].To..0005

 

would.

 

 

 

 

1.5 I I I I I I l
1 - _
05 - _
0 - _
-0.5 ~ -

_‘ l I 1 l L l 1 l
0.5 1.5 2 2.5 a 3.5 4 4.5 5
Time (see) -I

Aliphatic

103

 

 

 

 

 

APPENDIX E2
‘ 5 x104 Graph 1o: x1.cps=0.01667.n-11 11:1 .Oo=|0.0].To=-.0005
‘ -
3.5 -
3 -
2.5 -
2 -
1.5 r _
i '- _,
05 - _
$5 1 1.15 2L 2:5 ; sis 41 is 5
Tlmo (sec) 4

104
APPENDIX E3

anh 100 x2.cps-0.01667.ns1 1 had .Oo-|0.0].To-.0005
0.1. I I v r I r r r

 

0.16'- ..
0.14 - _

0.12 - -

0.04 - -

0.02 - j

 

 

 

1 J A l 1 l A l
‘06 1 1.5 2 2.5 0 3.5 4 4.5 6
Time {sec} -l

105

 

 

 

 

 

APPENDIX E4
x 10" anh for O1 .opo-0.01667.n-1 1 11.1 .Oo.p,o],ro.,ooos
3.5 ' ' V I I 17 W I
3 ..
2.5 '
2 -
1 .5 "
‘ .—
05 >- _
o 1 M1 1 1 1 1 1
0.6 1 1.5 2 2.5 3 3.5 4 4.5 5
Time {“0}! -l

Amplude

106
APPENDIX E5

x 10" anh 101 02.eps=0.01 “7.11:1 1 k=1.0o-D.0].To=.0005

2 I I I I I I fl I

 

1.0?

15'-

1.41-

00'-
.

06'
.

0.2 -

 

 

 

 

l
(0.6 1 1.6 2 2.5 3 3.6 4 4.6 6
Time (sec) at 10"

107
APPENDIX F

Multirate Implementation Under State Feedback Control
For Different Initial Conditions

A = [O 1;O -142.86];
B = [0;1826.85];
C = [1 0];
D = [0];
1= [1 O;O 1];
e = [O O;O 0];
Tc = 0.005;
[a,b,c,d] = c2dm(A,B,C,D,Tc,'zoh');
SI = (-6/sqrt(2))+i*(6/sqrt(2));
32 = (-6/Sqrt(2))-i*(6/Sq11(2));
Zn = eX13(Sn"'TC);
22 = eXP(Sz*TC);
PC = [Z1 Z2];
K = place(a,b,Pc);
N = inv(c*inv(I-a+b*K)*b);
a = 0.01667;
A0 = {-1/8 1;-1/(e"2) 0];
Bo = [l/e;1/(e"2)];
Co = [1 0];
D0 = [0];
To = 0.0005;
[ao,bo,co,do] = c2dm( A0, B0, Co, Do, To, 'zoh' );
[a,,bbchdl] = c2dm( A, B, C, D, To, 'zoh' );
a = [a, e;bo*c. ao];
bb = [b1;0;0];
cc = [c, O O];
dd = [0];
X0 = [0:0];
00 = [1;1];
r = 1;
go 2 [xo;00];
gg=fl;
W = fl;
n=1h
kmax = 1000;
for k = 1:kmax
if k = l
g1 = go;

108

elseif k > 1
g1 = (g(n,=))';
end
xo= g1(1:2);
u0 = -K*x0+N*r;
u = uo*ones(n,1);
[h,g] = dlsim(aa, bb, cc, dd, u, g.);
gg = [gg;g(1 :n-1,:)];
uu = [uu;u(1:n-1)];
X = gg(=,1 :2);
0 = gg(t,3:4);
X1 = gg(:91);
X2 = gg(92)9
01 = gg(93)9
02 = gg(:,4);
end
for k = l:length(Ol)
t(k,1) = k*To;
end
whos;
plot(t,01); title( ' Graph for 0,, a = 0.01667, 11 = 11, k = 1000, 00 = [1;1], To = 0.0005 ' )
xlabel ('Time (sec)');
ylabel ('Amplitude');
end

Amplude

109
APPENDIX F1

Graph 1o: uu.epe. 001667.11 -11, 14 -1000.00 - l1 ;1]. To . 0.0006

 

0.25 I r f T I I I I r

0.15-

0.1 ’

 

 

:-
b
1—
p—

 

 

-005 I I I I I
0 0.5 1 1 .5 2 2.5 3 3.5 ‘ ‘5

Time nee)

Ampltida

1 10
APPENDIX F2

crew 10! K1,“).- 001067.111- 11. k-1000.0o-|1;1].To n0.0005

 

 

 

 

 

1.2 r 1 1 1 x 1 1 I u
1 -

0.0 b

0.0 -

0.4 '-

02 h
0 1 1 1 1 1 1 1 4 1
0 0.5 1 15 2 2.5 3 3.5 4 4.5

Time {sec}

1 11
APPENDIX F3

Gum 10nd. eps- 0.01667.n =11. It: 1000. 00: I1 ;1]. To: 0.0005

 

a I I T I I I I I I

2.5 '-

 

 

 

_-
h
-
1—

 

 

“0.6 I I I l l
0 0.5 1 1 .5 2 2.5 3 3.5 4 4.5

Time (sec)

1 12
APPENDIX F4

anh blOi. 00- 0.01067. 11 -11. 11.1000. 00 - |1;1].To - c.0005

 

1.2 I I I I I I I I I

 

 

I 1 I
1.5 2 2.6 a 3.5 4 4.5
Time {sec}

1-
u-
h
h—

 

 

1 13
APPENDIX F5

Graph bl02.ops- 001667.11 =11. k=1000.0o : I1;1].To 30.0005

 

10

-25.

-30-

 

 

 

I

I

I

r

I

I

I

 

 

2.5
Time {sec}

4.5

 

1 14
APPENDIX G

Multirate Implementation Under Output Feedback Control
For Different Initial Conditions

A = [O 1;O -142.86];
B = [0;1826.85];
C = [1 0];
D = [01;
1= [1 O;O 1];
e = [O 0;0 0];
Tc = 0.005;
[a,b,c,d] = c2dm(A,B,C,D,Tc,'zoh');
S1 = (-6/sqrt(2»+i*(6/sqrt(2»;
82 = (-6/sqrt(2»—i*(6/sqrt(2»;
Z1 = exp(S,*Tc);
Z2 = exp(Sz*Tc);
PC = [21 Z21;
K = place(a,b,Pc);
N = inv(c*inv(I-a+b*K)*b);
e = 0.01667;
A0 = [-1/8 1;-1/(e"2) 0];
B0 = [l/s;1/(e"2)];
Co = [1 0];
D0 = [0];
To = 0.0005;
[ao,bo,co,do] = c2dm( A0, B0, C0, Do, To, 'zoh' );
[a1,bl,c1,d1] = c2dm( A, B, C, D, To, 'zoh' );
a = [a, e;bo*c, ao];
bb = 1b1;0;0];
cc = [c, O O];
dd = [0];
X0 = [Wk
00 = 11;”;
r = 1;
go = [Xo;Ool;
gg = [1;
W = l];
n = 11;
kmax = 1000;
for k = 1:kmax
if k = 1
g] = go;

115

elseif k > 1
81 = (g(n,=))';
end
00: gl(3:4);
110 = -K*OO+N*r;
u = uo*ones(n,l);
[h,g] = dlsim(aa, bb, cc, dd, u, g);
as = [gg;g(1:n-1,:)];
uu = [uu;u(l :n-1)];
x = gg(:,l :2);
O = gg(:,3:4);
x1 = gg(:,l);
x2 = gg(:,Z);
01 = gg(93)9
02 = gg(94)9
end
for k = l:length(Ol)
t(k,1) = k*To;
end
whos;
plot(t,01); title( ' Graph for 0,, e = 0.01667, 11 = 11, k = 1000, 00 = [1;1], To = 0.0005 ')
xlabel ('Time (sec)');
ylabel ('Amplitude');
end

1 16
APPENDIX G1

Graph bruu.epe- 001667.11 :11. 11 .- 1000.011 -|1;1]. To s 0.0005

 

1.5 I I I T I I I I I

 

 

 

_3 I l l 1 l 1 I I J
0 0.6 1 1.6 2 2.6 3 3.6 4 4.6
The {sec}

 

1 17
APPENDIX G2

Graph 10! than. a 001687.11: 11. k: 1000.00- |1;1].To-0.0005

a T r I I I I I I I

 

 

 

 

 

-6 1 1 1 I l 1 1 l I
0 0.6 1 1 .6 2 2.6 3 3.6 4 4.6 6
Time {sec}

 

1 18
APPENDIX G3

Grew for :2. Opt-0.01667J1- 11. 14- 1000. Oo- I1 ;1]. To- 000“

I5 r I I I I I I I I

 

 

 

 

 

 

 

_” I l l I I 1 l l I
0 0.6 1 1 6 2 2.6 3 3 .6 4 4 .6 6
Time (sec)

 

l 19
APPENDIX G4

Gllph bl01. m3: 0.01667. 11 = 11. It. 1000.00 a |1;1].To a 0.0005

 

 

I I I I I I I

T I

 

I I I I I I A

 

0.6 1 1 .5 2 2.6 3 3.6
Time (sec)

4 4.5

 

120
APPENDIX G5

Glaph bl02. eps- 0.01667. [1 :11. 1131000, 00 - I1;1].To a: 0.0005
20 I I I I l I I I I

 

 

 

 

 

I

0 0.6 1 16 2 2.6 3 3.5 4 4.6 6
Time (see)

b—
h
—
p

121
APPENDIX H

Multirate Implementation - Behavior in First Cycle
For Different Initial Conditions

A = [O 1;O -l42.86];
B = [0;1826.85];
C = [1 0];
D = [0];
I = [1 0;0 1];
e = [O O;O 0];
Tc = 0.005;
[a,b,c,d] = c2dm(A,B,C,D,Tc,'zoh');
81 = (-6/sqrt(2»+i*(6/sqrt(2»;
32 = (-6/Sqr1(2))-i*(6/Sqrt(2));
Z. = exp<sl*'rc);
22 = exp(Sz*Tc);
Pc = [Z] 22];
K = place(a,b,Pc);
N = inv(c*inv(I-a+b*K)*b);
e = 0.01667;
A0 = [-1/8 l;-1/(8"2) 0];
Bo = [1/8;1/(e"2)];
Co = [l 0];
D0 = [0];
To = 0.0005;
[ao,bo,co,do] = c2dm( A0, B0, C0, Do, To, 'zoh' );
[a,,b.,c,,d1] = c2dm( A, B, C, D, To, 'zoh' );
a = [a, e;bo*cl ao];
Db = [b1;0;0];
cc = [c, O O];
dd = [0];
X0 = [0:0];
00 = [1:1];
r = 1;
go = [MOO];
gg=fl;
1111 = [1;
n=lh
kmax = 1;
for k = 1:kmax
if k = 1
g1 : go;

122

elseif k > l
g] = (g(n9:))';
end
00: g1(3:4);
u0 = -K*OO+N*r;
u = uo*ones(n,1);
[h,g] = dlsim(aa, bb, cc, dd, 11, g,);
gg = [gg;g(1=n-1,=)];
uu = [uu;u(1:n-1)];
X = gg(:,l =2);
0 = gg(:,3:4);
x1 = gg(:,l);
x2 = gg(:,Z);
01= gg(:,3);
02 = gg(:,4);
end
for k = l:length(Ol)
t(k,1) = k*To;
end
whos;
plot(t,O,); title( ' Graph for 01, a = 0.01667, n = 11, k = 1, 00 = [1;1],To = 0.0005 ')
xlabel ('Time (sec)');
ylabel ('Amplitude');
end

123
APPENDIX H1

Grephiowu.epe-0.01667.n=11. k=1.0o=|1;1].To= 0.0005

Ampliude

 

 

 

 

 

1.5 I I I I I I I I
1 - _
06 - d
o 1- -1
-06 1- -.

-1 I I I I L I I I
0.6 1 1.6 2 2.6 3 3.6 4 4.6 6
Time {sec} -1

Amplwde

124
APPENDIX H2

1: ”,4 Graph for x1.epaa0.01667.n-11.k=1.0o=|1 ;1].To-.0006

I.2 I I I I I— I I I

 

00"

0.4 -

0.2-

 

 

 

 

0
0.6 1 1.6 2 2.6 3 3.6 4 4.6 6
Time (sec) -1

125
APPENDIX H3

Graph for x2.eps-0.01667.n=11.k=1.00=|1;1].To-.0w6

 

0.45

0.4 '-

Amplaide

0.16”

0.11-

0.05 P

 

T

I I

_1_ I

I

I

 

 

2.6 3
Time {sec}

126
APPENDIX H4

Graph ior O1.epa=0.01667.n=1 1.11:1 .Oo=|1 ;1 ].To=.0006

Amplaide

 

 

 

1 1 r u 1 fl 1
0.36 ~ _
0.0 - ..
0.36 - -1
0.3 - -
0.75 - -

0.7 ‘ 1 L ‘ ‘ ' 1
0.6 1.6 2 2.6 3 3.6 4 4.6 6
Time {sec} .1

x10

 

Amplmde

127
APPENDIX H5

Graph for 02.epa=0.01367.n=11.k=1.0o-|1;1].To-u.w06

2 I I 17 I I I I I

 

-4

~10

 

 

 

 

_1‘ I l I I I I I I
0.6 1 1.6 2 2.6 3 3.6 4 4.6
Time (see) a: '04

128
APPENDIX I

Multirate Implementation - Control Saturated

A = [0 1;O -142.86];
B = [0;1826.85];
C = [1 0];
D = [0];
I = [1 O;O 1];
e = [O 0;O 0];
Tc = 0.005;
[a,b,c,d] = c2dm(A,B,C,D,Tc,'zoh');
S, = (-6/sqrt(2))+i*(6/sqrt(2));
$2 = (-6/Sqrt(2))-i*(6/Sqrt(2));
Z1 = eXP(S1*TC);
22 = exp(Sz*Tc);
PC = [21 Z21;
K = place(a,b,Pc);
N = inv(c*inv(I-a+b*K)*b);
e = 0.01667;
A0 = [~l/e 1;-1/(s"2) 0];
Bo = [l/e;l/(s"2)];
Co = [1 0];
D0 = [0];
To = 0.0005;
[ao,bo,co,do] = c2dm( Ao, Bo, Co, Do, To, 'zoh' );
[a,,b,,c,,dl] = c2dm( A, B, C, D, To, 'zoh' );
a = [a1 e;bo*c1 ao];
DD = [b1;0;0];
cc = [c, O O];
dd = [0];
X0 = [W]:
0o = [111];
r = 1;
go = [Xo;Ool;
gg = [1;
W = [1;
m = 0.5;
n=lh
kmax = 1000;
for k = 1:kmax
if k = 1
31 = go;

129

elseif k > 1
g1 = (g(n9:))';
end
00: gl(334);
u0 = -K*OO+N*r;
if abs(uo) <= m
111 = no;
else if abs(uo) > m
u] = m*sign(uo);
end
11 = u,*ones(n,l);
[h,g] = dlsim(aa, bb, cc, dd, u, g,);
gg = [gg;g(1:n-1,I)];
uu = [uu;u(l :n-1)];
x = gg(:,l :2);
0 = gg(:,3:4);
x1 = gg(:,l);
x2 = gg(:,Z);
01 = gg(:93)9
02 = gg(94)9
end
for k = l:length(Ol)
t(k,1) = k*To;
end
whos;
plot(t,01); title( ' Graph for 0,, e = 0.01667, n = 11, k = 1000, 00 = [1;1], To = 0.0005 ')
xlabel ('Time (sec)');
ylabel ('Amplitude’);
end

0.3

0.2

0.1

M00311“
1':

-0.2

130

APPENDIX 11

gram for uu, epea 001667.11 = 11.11: 1000.00 =|1;1].COIII“I

 

I

 

 

fi—

I

I

I

I

I

I

I

I

 

 

2.6
Tineisec]

4.5

 

Amplaide

1.4

APPENDIX 12

131

mph brx1.epe-0.01337.n. 11.11. 10m.OOs |1;1].conteat

 

 

I

‘T

f

I

I

I

I

T

 

 

I

 

2.6
Theieec]

4.5

 

132

APPENDIX I3

mph iorx2.epe- 001337.11 a 11 . 11- 1000.00 . |1;1].contsat

 

Ampliude
1'3

-11

 

 

I

I

I

I

I

 

h

 

2.6
Theieec]

4.5

 

133
APPENDIX 14

graph 10101. epe - 0.01667. 11 a 11.11 - 1000.011 - l1;1].Co111 eat

 

I.‘ I I I I I I I I I

 

 

 

 

 

I I I I I I
-05 1 1 i

0 .6 1 1 .5 2 2.6 3 3 .6 4 .5
Time(sec)

&

Amplwde
I
a

-20

-26

134

APPENDIX IS

graph 10102. epa . 0.01337. 11 =11.11 - 11110. 00 a |1;1].Com sat

 

T

 

 

I

I

I

I

 

 

2.6
Tineiaec]

4.5

 

135
APPENDIX J

Multirate Implementation Under Output Feedback Control
With Different Value of 8

A = [O 1;O -142.86];
B = [0;1826.85];
C = [l O];
D = [0];
I: [1 0;0 l];
e = [O 0;O 0];
Tc = 0.005;
[a,b,c,d] = c2dm(A,B,C,D,Tc,'zoh');
S, = (-6/sqrt(2))+i*(6/sqrt(2));
8: = (-6/sqrt(2»-i*(6/sqrt(2»;
Z1 = eXp(31"'T<=);
22 = eXp(Sz*TC);
PC = [Zn 22];
K = place(a,b,Pc);
N = inv(c*inv(I-a+b*K)*b);
e = 0.001667;
A0 = [-1/8 l;-1/(8"2) 0];
Bo = [1/8;1/(e"2)];
Co = [l 0];
Do = [0];
To = 0.0005;
[ao,bo,co,do] = c2dm( Ao, Bo, Co, Do, To, 'zoh' );
[a,,b1,c,,d,] = c2dm( A, B, C, D, To, 'zoh' );
a = [a, e;bo*c1 ao];
bb = [b1;0;0];
cc = [cl 0 O];
dd = [0];
X0 = [Wk
00 = [010];
r = 1;
go = [M00];
as = U;
ml = fl;
n=1h
kmax = 1000;
for k = 1:kmax
if k = 1
g1 = go;

136

elseif k > 1
g1 = (gm):
end
00: gl(3:4);
u0 = -K*OO+N*r;
u = uo*ones(n,1);
[h,g] = dlsim(aa, bb, cc, dd, u, g,);
gg = [gg;g(12n-1,z)];
uu = [uu;u(1:n-l)];
X = gg(:,l :2);
0 = gg(:,3:4);
x1 = gg(:,l);
x2 = gg(92)9
01= gg(:,3);
02 = gg(:,4);
end
for k = l:length(Ol)
t(k,1) = k*To;
end
whos;
plot(t,01); title( ' Graph for 0,, a = 0.001667, 11 = l 1, k = 1000, 00 = [0;0], To = 0.0005 ' )
xlabel ('Time (sec)');
ylabel ('Amplitude');
end

Amplaide

137
APPENDIX J1

Gmph ior uu.epa-.001337.11a11.k=1000.0o-|0;0].To=.0006

 

 

 

 

 

0.2 w - I ' T ' r ' '
0.15 - I
0.1 1 ‘
0.05 - —
0 r
-006 0 015 ,1 1 is 21 2.} 3 316 4 416 6

Time {sec}

133
APPENDIX J2

Gram for x1.epe-.001337.1h1 1 .k-1w0.0o-p,0].To-.0WG

 

Amplaide

0.4

02

 

I

I

I

I

I

 

I

.-

 

2.6
Time {sec}

0.5

4.5

 

139
APPENDIX J3

Graph for x2.epe=.001337.11=1 1 .11.1ooo.0o.p;01.1'o..ooos

Amplaide

25

1.5

-0.6
0

 

 

 

I

I

I

I

 

I

 

2.6
Time (sec)

4.5

 

140
APPENDIX J4

Graph b1 01 ape-001337 .11=1 1.11.1000.Oo-p ,0].To- .ooos

Amplaide

1.2

0.4

02

 

 

I

I

I

I

 

I

1—

 

2.6
The {sec}

4.5

 

141
APPENDIX J5

Graph 101 02.eps-.001337.11-1 1.k=1000.00=|0;0].T0:-.0005

 

a I I I I I I I I I

III
I

 

Amplflda

‘
I

05"

 

 

11—
—
11—
i—n

 

 

_oos I I I I I
O 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Time 1'ch

142
APPENDIX K

Multirate Implementation Under Output Feedback Control
With Different Value of 8 & Different Initial Conditions

A = [O 1;O -142.86];
B = [0;1826.85];
C = [1 0];
D = [0];
I = [1 0;0 1];
e = [0 O;O 0];
Tc = 0.005;
[a,b,c,d] = c2dm(A,B,C,D,Tc,'zoh');
S, = (-6/sqrt(2))+i*(6/sqrt(2));
$2 = (-6/sqrt(2))-i*(6/sqrt(2));
Z, = exp(S,*Tc);
22 = e1111(Szu'TC);
PC = [Z] Z2];
K = place(a,b,Pc);
N = inv(c*inv(I-a+b*K)*b);
e = 0.001667;
A0 = [-l/8 l;-l/(e"2) 0];
B0 = [l/s;1/(s"2)];
Co = [l 0];
D0 = [0];
To = 0.0005;
[ao,bo,co,do] = c2dm( Ao, Bo, C0, Do, To, 'zoh’ );
[a,,b,,c,,d,] = c2dm( A, B, C, D, To, 'zoh' );
aa = [a, e;bo*c, ao];
bb = [b1;0;0];
cc = [c, 0 O];
dd = [0];
X0 = [0;0];
0o = 11;”;
= 1;

go = [X1500];
gg = I];
IN = [1;
n = 11;
kmax = 1000;
for k = 1:kmax

if k = 1

g, = go;

I43

elseif k > 1
81 = (80113”;
end
00: gl(3:4);
u0 = -K*OO+N*r;
u = uo*ones(n,1);
[h,g] = dlsim(aa, bb, cc, dd, u, g,);
as = [gg;g(1 :n-1,:)];
uu = [uu;u(1:n-1)];
X = gg(:,ltz);
0 = gg(:,3:4);
X, = gg(o9l)9
X2 = gg(92)9
01 = gg(:93)9
02 = gg(94)9
end
for k = l:length(O,)
t(k,1) = k*To;
end
whos;
plot(t,O,); title( ' Graph for 0,, e = 0.001667, 11 = 11, k = 1000, 00 = [1;1], To = 0.0005 ')
xlabel (’Time (sec)');
ylabel (’Amplitude');
end

-4

144
APPENDIX K1

Graph bruu.epa-.001337.11-11.k-1000.0o-|1;1].To-.0006

 

 

I I I I I I I I

 

I I I I I

1—
—
-

 

0.6 1 16 2 2.6 3 3.6 4
Time (sec)

4.6

 

Amplaide

145
APPENDIX K2

Gram for x1.epa-.001337.11-1 1.11:1wo.Oe-|1;1].To..0006

 

 

 

 

Time {sec}

1.5 . . . . . . . . .
,_
0.5 -
°[
_115 1
-1-
-15 -

_2 I I I I I I I I I

o 0.5 1 1.5 2 2.5 a 3.5 4 4.5

 

146
APPENDIX K3

Gram for Kama-001337.11“ 1 .11-1000.0o.|1;11.1o..ooos

10 I I I I I I I I I

 

 

 

 

 

 

 

 

-25 ~ .1
-3o - -
45 - -
-40 1 1 1 1 L 1 1 L 1

0 0.5 1 15 2 2.5 3 3.5 4 4.5 5

Time (sec)

147
APPENDIX K4

Graph 101 O1 .epe- .001 337.11-1 1.k-1000.0o-|1 ;1 ].To- .0006

 

I.6 I l I I I j I I

 

-15..

 

 

-2 I I I I g i I I
0 0.6 1 1.6 2 2.6 3 3.6 4
Time (sec)

4.6

 

148
APPENDIX K5

Graph b1 O2.epea.001337.11-11.k=1000.Oo=|1;1].To=.0006

Ampiaide

-150

 

I

I

I

I

I

I

 

 

I
F
1-

1-
.
.-
1.
1-
I

:—

 

2.6
Time isec]

0.5

4.5

 

149
APPENDIX L

Multirate Implementation - Control Saturated
With Different Value of a

A = [0 1;O -142.86];
B = [0;1826.85];
C = [1 0];
D = [0];
I = [1 O;0 1];
e = [O O;0 0];
Tc = 0.005;
[a,b,c,d] = c2dm(A,B,C,D,Tc,'zoh');
S, = (-6/sqrt(2))+i*(6/sqrt(2));
82 = (-6/sqrt(2))-i*(6/sqrt(2));
Z, = exp(S,*Tc);
22 = eXp(Sz*TC);
PC = [Z1 22];
K = place(a,b,Pc);
N = inv(c*inv(I-a+b*K)*b);
e = 0.001667;
A0 = [-1/8 1;-1/(e"2) 0];
Bo = [1/8;1/(8"2)];
Co = [1 0];
D0 = [0];
To = 0.0005;
[ao,bo,co,do] = c2dm( Ao, Bo, C0, Do, To, 'zoh' );
[a,,b,,c,,d,] = c2dm( A, B, C, D, To, 'zoh' );
a = [a, e;bo*c, ao];
DD = [b1;0;01;
cc = [c, O O];
dd = [0];
X0 = [0;0];
00: [1 :1];
r = 1;
go = [X1000];
gg=fl;
uu=fl;
m = 0.5;
n = 11;
kmax = 1000;
for k = 1:kmax
if k = 1

150

g, = go;
elseif k > 1
g1 = (e(n,=))';
end
00: gl(3:4);
u0 = -K*OO+N*r;
if abs(uo) <= m
u, = 110;
else if abs(uo) > m
u, = m*sign(uo);
end
11 = u,*ones(n,1);
[h,g] = dlsim(aa, bb, cc, dd, 11, g,);
8g = [gg;g(1=n-1,=)];
uu = [uu;u(1:n-1)];
x = gg(:,1:2);
O = gg(:,3:4);
x1 = gg(:9l)9
x2 = gg(:,Z);
0| = gg(93)9
02 = gg(:,4);
end
for k = l:length(O,)
t(k,1) = k*To;
end
whos;
plot(t,01); title( ' Graph for O,, a = 0.001667, 11 = 11, k = 1000, 00 = [1;1], To = 0.0005 ')
xlabel ('Time (sec)');
ylabel ('Amplitude');
end

0.4

0.2

0.1

APPENDIX L1

151

graph 101 uu.epes 0.001337.11 = 11.11 =1000.0o . |1;1].Comea1

 

I

 

 

I

I

I

I

I

I

I

 

I

_

 

0 0.5

2.6
Tineiaec]

4.5

 

gram 101 111

152
APPPENDIX L2

.epa-0.001337.11- 11.14111 1000.00sl1 ;1]. 03111331

 

 

I I

 

I I I I I I I

 

I I I I I I I

 

1 6 2 2.6 3 3.6 4 4.6
Theieec]

 

Amplmde

153
APPENDIX L3

graph 101 112. epa - 0.0013611! 211.11: 1000.00 2 I1 ;1]. 00mm

 

 

 

I I I I I I I I I

 

 

 

-4

0.6 1 16 2 2.6 3 3.6 4 4.6
Theieec]

 

154
APPENDIX L4

mph 10101.epe- 0.m1337.11. 11.11. 1000.00-l1;1].001110at

 

I.2 I I I I I I I I I

 

 

 

1 1 1 1 1 1 1
-0.4 1 1

 

 

0.6 1 16 2 2.6 3 3.6 4 4.6
Theiuc]

100

155

APPENDIX L5

maph 10102. epes 0.w1637.n = 11.11. 1000. 00 . |1;1].001110a1

 

I

I

I

I

I

I

I

 

 

_—

 

2.6
Time(sec)

3.5

4.5

 

156
APPENDIX M

Multirate Implementation - Behavior in First Cycle
With Different Value of a

A = [0 1;0 -142.86];
B = [0;1826.85];
C = [1 O];
D = [0];
1= [1 O;O 1];
e = [O O;0 0];
Tc = 0.005;
[a,b,c,d] = c2dm(A,B,C,D,Tc,'zoh');
S, = (-6/sqrt(2))+i*(6/sqrt(2));
32 = (-6/Sqrt(2))-i*(6/Sqrt(2));
21 = exp(S,*Tc);
Z2 = exp(Sz*Tc);
PC = [Z] 22];
K = place(a,b,Pc);
N = inv(c*inv(I-a+b*K)*b);
e = 0.001667;
A0 = [-1/8 1;-1/(e"2) 0];
B0 = [1/8;1/(e"2)];
Co = [1 0];
D0 = [0];
To = 0.0005;
[ao,bo,co,do] = chm( A0, B0, C0, Do, To, 'zoh' );
[a,,b,,c,,d,] = c2dm( A, B, C, D, To, 'zoh' );
aa = [a, e;bo*c, ao];
bb = [b1;0;0];
cc = [c, O O];
dd = [0];
X6 = [0;0];
0o = [090];
r = 1;
so = [X600];
gg=fl;
W = fl;
n=1h
kmax = l;
for k = 1:kmax
if k = 1
g1 : go;

157

elseif k > 1
g1 = (g(n9:)).;
end
00: gl(34)9
u0 = -K*OO+N*r;
u = uo*ones(n,l);
[h,g] = dlsim(aa, bb, cc, dd, 11, g,);
gg = [gg;g(1:n-1,:)];
uu = [uu;u(1:n-1)];
X = gg(:,l 22);
0 = gg(:,3:4);
X1 = gg(91)9
X2 = gg(:,Z);
01 = gg(:,3);
02 = gg(:94)9
end
for k = l:length(O,)
t(k,1) = k*To;
end
whos;
plot(t,01); title( ' Graph for O,, s = 0.001667, 11 = 11, k = 1, 00 = [0;0], To = 0.0005 ' )
xlabel ('Time (sec)');
ylabel ('Amplitude');
end

Amphmu

158

APPENDIX M1

Graph 101w.epe- 0.N1067.n- 11.11: 1.001: p.01.To-0.0006

 

 

 

 

 

1.5 I I I I I I I I
1 v- -1
0.5 - J
o .— —1
-06- -

_' I I I I I I I I
06 1 16 2 26 3 36 4 46 6
Time ieec] -1

x10

Amplaide
10
I

159
APPENDIX M2

11 10" Graph 101 x1.epa-.001337.11-11.11.1.oo.|0.0].To..oms

‘5 j I I I I I I I

 

 

 

 

 

(11.6 1 1.6 2 2.6 3 3.6 4 4.6 6
Time {sec} 4

Ampliude

160
APPENDIX M3

Graph for lama-001337.031 1.11:1 .0040 311.163.0005

 

 

 

 

0.10 1 1 r 1 1 1 1 r
0.16 - -1
0.14 - -
0.12 - ..
0.1 '- -
0.“ '- .1
0.06 '- '1
0.04 - -
0.02 *- 7

1 1 1 1 1 1 1 1
%.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Tha {333] -a

x10

2.5

16]

 

 

 

APPENDIX M4
1: 10" Graph 10101 .epen.001337.11-1 1.11:1 .Oo.pp],ro.,mos
J I I I I I I
6 1 1 6 2 2.6 3 3.6 4 4.6 6
Time 1333] -3

 

Amplaide

162
APPENDIX M5

Graph 101 02.epa-.001337 .11=11.i1111.00-p;0].T0-.0006
0.00 I I I I I T I I

 

000-
.

000*
.

003-
.

0.02 -

0.01 -

 

 

 

 

0.6 1 1.6 2 2.6 3 3.6 4 4.6 6
Time (sec) 11 10"

163
APPENDIX N

Multirate Implementation - Behavior in First Cycle
With Different Value of 8 & Different Initial Conditions

A = [0 l;O -142.86];
B = [0;1826.85];
C = [1 0];
D = [0];
I = [l 0;O l];
e = [0 O;O 0];
Tc = 0.005;
[a,b,c,d] = c2dm(A,B,C,D,Tc,'zoh');
S1 = (-6/Sqrt(2))+i*(6/Sqrt(2));
32 = (-6/ S(1111(2))-i"'(6/ Sqrt(2));
2] = eXP(Sl"TC);
Zz = eXp(Sz*TC);
PC = [Z1 Z2];
K = place(a,b,Pc);
N = inv(c*inv(I-a+b*K)*b);
a = 0.001667;
A0 = [-l/e 1;-1/(e"2) 0];
B0 = [1/8;1/(e"2)];
Co = [l 0];
D0 = [0];
To = 0.0005;
[ao,bo,co,do] = c2dm( Ao, B0, C0, Do, To, 'zoh' );
[a,,b,,c,,d,] = c2dm( A, B, C, D, To, 'zoh' );
aa = [a, e;bo*c, ao];
bb = [135010];
cc = [c, O 0];
dd = [0];
x0 = [090];
00 = [1 ;1];
r = 1;
go = [X1600];
gg=fl;
W = U;
n=lh
kmax = l;
for k = 1:kmax
if k = l
g, = go;

164

elseif k > l
g] = (g(n9:))';
end
00: gl(34)9
u0 = -K*OO+N*r;
u = uo*ones(n,1);
[h,g] = dlsim(aa, bb, cc, dd, u, g,);
gs = [gg;g(1:n-1,=)];
uu = [uu;u(1:n-1)];
x = gg(:,122);
O = gg(:,3:4);
X, : gg(91)9
X2 = gg(:,Z);
0| = gg(93)9
02 = gg(:,4);
end
for k = l:length(O,)
t(k,1) = k*To;
end
whos;
plot(t,0,); title( ' Graph for 0,, a = 0.001667, 11 = 11, k = 1, 00 = [1;1], To = 0.0005 ')
xlabel ('Time (sec)');
ylabel ('Amplitude');
end

165
APPENDIX N1

Gmph 101 uu.ep0-.m1337.n-11.11-1.00-|1;1].T0-.0006

Amplaide

 

1.6

05"

I r I I I

I

I

 

 

I L I I I

 

 

1 1.5 2 2.5 0

Time (sec)

3.5

Amplaide

166
APPENDIX N2

1: ”,4 Graph for x1.epa-.001337.11=11.k-1.00.|1;1],To.,ooos

I.2 I I I I I I I I

 

00"

0.4‘

02'-

 

 

 

 

0.6 1 1.6 2 2.6 3 3.6 4 4.6 6
Time {sec} 11 10"

Amplmde

167
APPENDIX N3

Gram for x2.epe- 001337.11-1 1 .1121 .Oo-I1 ;1].To-.0m5

 

0.45

0.4 -

0.15“

0.1 1-

 

I I I I I I I

 

I I I I I I l

 

1.6 2 2.6 3 3.6 4 4.6 6
Time (sec) -1

 

Amplwde

168
APPENDIX N4

Graph 101 01.epea.001337.11=1 1.11-1.00=|1;1].T0=.0006

 

I I I I I I I I I

0.3 -

0.4 -

02r

—0.2 -

 

b

_o" I I I I I I I

 

 

0.5 1 1.5 2 2.5 3 3.5 4 4.5

Time ieec] x ,0

169
APPENDIX N 5

Graph 101 02.epea.001337.11=1 1.11:1 .00-l1 ;1 1.16.0005

 

50

-250

 

"“11.

I

 

I I I r I I

 

 

2 2.6 3 3.6 4 4.6 6
Time (sec) 11 10"

170
APPENDIX P

Matlab program for State Feedback Control

With New Value of Tc

A = [O 1;O -l42.86];

B = [0;1826.85];

C = [1 0];

D = [0];

Tc = 0.01;

[a,b,c,d] = c2dm(A,B,C,D,Tc,'zoh')

S, = (-6/sqrt(2))+i*(6/sqrt(2));

S2 = (-6/ Sqrt(2))-i*(6/ SC01(2));

Z1 = 6XP(S1*TC);

22 = eXP(Sz*TC);

Pc = [Z, Z2]

K = place(a,b,Pc)

I = [1 O;O 1];

N = inv(c*inv(I-a+b*K)*b)

A1 = [a-b*K];

Bl = [b*N];

ID = [1];

k = 500;

step = [k];

NN = N*ones(k,l);

[y,x] = dstep(Al , B1, c, d, 1U, step);

U = (x*(-1)*K')+NN;

X1 : X(I,I);

x2 = x(:,2);

for k = l:length(U)
t(k,1)=k*Tc;

end

whos;

plot(t,U); title( ’ plot for U , k = 500, Tc = 0.01, state feedback control ' )

xlabel ( ' time (sec) ' );

ylabel ( ' amplitude ' );

end

 

171
APPENDIX P1

“II“ 1.1 .k- 50°.ICI 0.01
0125 I I I I I I I

 

02"
O

0.15 7

 

0.05 *-

 

 

1 I 1 1
43.05 J—

1

 

‘r

4.5

 

172
APPENDIX P2

[301101 111.11- 600.Tc. 0.01

 

1.2 I I I I I I I I 7

 

 

 

 

 

 

2 2.6 3 3.6 4 4.6
he (see)

amplitude

1 73
APPENDIX P3

plotfor :2 .k- 500.1'c. 0.01

 

a I I I I I I

 

0.5 '

 

 

 

 

"T

174
APPENDIX Q

Multirate Implementation Under Output Feedback Control
With New Values of Tc and TO

A = [0 1;0 -142.86];
B = [0;1826.85];
C = [1 O];
D = [0];
1= [1 O;O 1];
e = [0 0,0 0];
To = 0.01;
[a,b,c,d] = c2dm(A,B,C,D,Tc,'zoh');
SI = (-6/sqrt(2))+i*(6/sqrt(2));
32 = (-6/sqrt(2»-i*(6/sqrt(2»;
Zn = eXP(Sl*TC);
22 = eXP(Sz*TC);
PC = [Zn 22];
K = place(a,b,Pc);
N = inv(c*inv(I-a+b*K)*b);
e = 0.01667;
A0 = [-1/8 1;-1/(e"2) 0];
Bo = [1/8;1/(8"2)];
Co = [l 0];
D0 = [0];
To = 0.0001;
[ao,bo,co,do] = c2dm( A0, B0, C0, Do, To, 'zoh' );
[a,,b1,cl,dl] = c2dm( A, B, C, D, To, 'zoh' );
aa = [a, e;bo*c, ao];
bb = [b1;0;0];
cc = [c, O O];
dd = [0];
x0 = [0:0];
00 = [1:1];
r = 1;
go = [MOO];
gg = [1;
W = l];
n = 101;
kmax = 500;
for k = 1:kmax
if k = 1
g] = go;

175

elseif k > 1
gl = (g(n’:))';
end
Oo= gl(3:4);
u0 = -K*O°+N*r;
u = uo*ones(n,1);
[h,g] = dlsim(aa, bb, cc, dd, u, g,);
gg = [gg;g(1=n-1,I)];
uu = [uu;u(1:n-1)];
x = gg(:,l 22);
0 = gg(:,3:4);
X1 = gg(:al);
x2 = gg(:,Z);
0: = gg(:,3);
02 = gg(:a4);
end
for k = l:length(Ol)
t(k,1) = k*TO;
end
whos;
plot(t,01); title( ' Graph for 0,, e = 0.01667, n = 101, k = 500, 00 = [1;1], To = 0.0001 ')
xlabel ('Time (sec)');
ylabel ('Amplitude');
end

l 76
APPENDIX Q1

Gum for w.opo-.01867.n-10t.k-GO0.0o-|1 ;1 1.13-0001

 

I I I I I T I I I I

 

 

 

1 l l 1 l
2.5 a 3.5 t 4.5
The (sec)

 

 

177
APPENDIX Q2

Graph br :1 fill-.01 6673-1 01 Jt-SOODo-u ;1],To-.m01

 

 

 

 

 

2 I I I I I I I I I
'—
OI-
.
u
a
-_1~
a.
5
‘
-2-
-3-
_‘ l l J 1 l l l l I
0 05 1 1.5 2 25 3 35 ‘ ‘5

178
APPENDIX Q3

Graph to! x2.ops-.01667,n-101 .k-SOO.00.|1 ;t ].To-.w01

 

Is I I I r I I I r I

 

 

 

26 9 3.5 4 4.5

 

I79
APPENDIX Q4

G-IJI'I bl’ 01 .ops- .01 007 n: 1 01 ,k-SMOo-II ;1].Tos .0001

 

2 I II I I I

 

 

 

I I T I
1-
I
o .-
O
u
2-. .
3
-2-
-3-
_‘ l I l l l l l l l
0 0.6 1 15 2 2.5 0 0.5 4 4.5

Time {soc}

 

l 80
APPENDIX Q5

Glph b! 023902.01“? 11.-101 .k-SMQo-Ii ;1].To- .0001

 

Is I I T I I I I I I

 

 

l l l l l

 

 

 

2.5 a 3.5 4 L5
The (ac)

BIBLIOGRAPHY

 

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