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QIS COLLEGE OF ENGINEERING & TECHNOLOGY :: ONGOLE

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18G02P02S

M. Tech - I Semester (R18 Supplementary Examinations) June, 2019

MODERN CONTROL THEORY
(Common to PS & PE&PS)
th
Date: 28 June2019 Time: 3:00 Hours Max.Marks:60 Marks
Answer any FIVE Questions

1 a. What are the advantages and disadvantages of state space analysis?

State space analysis is an excellent method for the design and analysis of control systems.
The conventional and old method for the design and analysis of control systems is the transfer
function method. The transfer function method for design and analysis had many drawbacks.

 Transfer function is defined under zero initial conditions.

 Transfer function approach can be applied only to linear time invariant systems.
 It does not give any idea about the internal state of the system.
 It cannot be applied to multiple input multiple output systems.
 It is comparatively difficult to perform transfer function analysis on computers.

Any way state variable analysis can be performed on any type systems and it is very easy to
perform state variable analysis on computers. The most interesting feature of state space analysis
is that the state variable we choose for describing the system need not be physical quantities related
to the system. Variables that are not related to the physical quantities associated with the system
can be also selected as the state variables. Even variables that are immeasurable or unobservable
can be selected as state variables.

Advantages of state variable analysis.

 It can be applied to nonlinear system.

 It can be applied to tile invariant systems.
 It can be applied to multiple input multiple output systems.
 Its gives idea about the internal state of the system.
Develop the state model of linear system. Draw the block diagram of State model.
b.

State Equation Based Modeling Procedure The complete system model for a linear time-invariant
system consists of (i) a set of n state equations, defined in terms of the matrices A and B, and (ii)
a set of output equations that relate any output variables of interest to the state variables and inputs,
and expressed in terms of the C and D matrices. The task of modeling the system is to derive the
elements of the matrices, and to write the system model in the form: x˙ = Ax + Bu (11) y = Cx +
Du. (12) The matrices A and B are properties of the system and are determined by the system
structure and elements. The output equation matrices C and D are determined by the particular
choice of output variables. The overall modeling procedure developed in this chapter is based on
the following steps: 1. Determination of the system order n and selection of a set of state variables
from the linear graph system representation. 2. Generation of a set of state equations and the system
A and B matrices using a well-defined methodology. This step is also based on the linear graph
system description. 3. Determination of a suitable set of output equations and derivation of the
appropriate C and D matrices.

Block Diagram Representation of Linear Systems Described by State Equations The matrix-
based state equations express the derivatives of the state-variables explicitly in terms of the states
themselves and the inputs. In this form, the state vector is expressed as the direct result of a vector
integration. The block diagram representation is shown in Fig. 2. This general block diagram
shows the matrix operations from input to output in terms of the A, B, C, D matrices, but does not
show the path of individual variables. In state-determined systems, the state variables may always
be taken as the outputs of integrator blocks. A system of order n has n integrators in its block
diagram. The derivatives of the state variables are the inputs to the integrator blocks, and each state
equation expresses a derivative as a sum of weighted state variables and inputs. A detailed block
diagram representing a system of order n may be constructed directly from the state and output
equations as follows: Step 1: Draw n integrator (S−1) blocks, and assign a state variable to the
output of each block. 5 Figure 2: Vector block diagram for a linear system described by state-space
system dynamics. Step 2: At the input to each block (which represents the derivative of its state
variable) draw a summing element. Step 3: Use the state equations to connect the state variables
and inputs to the summing elements through scaling operator blocks. Step 4: Expand the output
equations and sum the state variables and inputs through a set of scaling operators to form the
components of the output.

Fig: Vector block diagram for a linear system described by state-space system dynamics

2 a. Define Controllability and observability. Explain with the help of Kalman’s test.

Controllability and observability are two important properties of state models which are to be studied
prior to designing a controller.
Controllability deals with the possibility of forcing the system to a particular state by application of a
control input. If a state is uncontrollable then no input will be able to control that state. On the other
hand whether or not the initial states can be observed from the output is determined using
observability property. Thus if a state is not observable then the controller will not be able to determine
its behavior from the system output and hence not be able to use that state to stabilize the system.

1. Controllability

Before going to any details, we would first formally define controllability. Consider a dynamical
system

(1)

where , , , .

Definition 1 The state equation (1) (or the pair (A,B) ) is said to be completey state controllable
or simply controllable if for any initial state x(0) and any final state x(N), there exists an input

sequence , , which transfers x(0) to x(N) for some finite N.

Otherwise the state equation (1) is uncontrollable.

Definition 2 Complete Output Controllability: The system given in equation (1) is said to be

completely output controllable or simply output controllable if any final output can be

reached from any initial state by applying an unconstrained input sequence

, , for some finite . Otherwise (1) is not output controllable.

1.1 Theorems on controllability

1. The state equation (1) or the pair (A,B) is controllable if and only if the controllability
matrix

has rank n, i.e., full row rank.

 2. The state equation (1) is controllable if the controllability grammian matrix

is nonsingular for any nonzero finite N.

3. If the system has a single input and the state model is in controllable canonical form then
the system is controllable.

4. When A has distinct eigenvalues and in Jordan/Diagonal canonical form the state model is
controllable if and only if all the rows of B are nonzero.

5. When A has multiple order eigenvalues and in Jordan canonical form, then the state model
is controllable if and only if

i. each Jordan block corresponds to one distinct eigenvalue and

ii. the elements of B that correspond to last row of each Jordan block are not all zero.

Output Controllability: The system in equation (1) is completely output controllable if and only

if the output controllability matrix

has rank , i.e., full row rank.

2 Observability

Definition 2 The state model (1) (or the pair (A,C) ) is said to be observable if any initial
state x(0) can be uniquely determined from the knowldge of output y(k) and input sequence u(k),

for , where N is some finite time. Otherwise the state model (1) is
unobservable.

2.1 Theorems on observability

1. The state model (1) or the pair (A,C) is observable if and only if the observability
matrix

has rank n, i.e., full column rank.

2. The state model (1) is observable if the observability grammian matrix

is nonsingular for any nonzero finite N.

3. If the state model is in observable canonical form then the system is observable.

4. When A has distinct eigenvalues and in Jordan/Diagonal canonical form, the state model is
observable if and only if none of the columns of C contain zeros.

5. When A has multiple order eigenvalues and in Jordan canonical form, then the state model is
observable if and only if

i. each Jordan block corresponds to one distinct eigenvalue and

ii. the elements of C that correspond to first column of each Jordan block are not all zero.

b. Determine controllability and observability of the system described by

 . 
 x1   0 1 0  x1  0
 .    x   0  u
 x2    0 0 1   2  
 .   6  11  6   x3  1 
 x3   
 
 x1 
Y  4 5 1  x 2 
 x3 
 3 a. What is a Non-linear system? What are the different types of Non-linearities? Explain each of
them in detail.
Non-linear system refers to the type of system where the output from the system does not vary directly
with respect to input to the system. The non-linear systems do not accompany the static linearity and
they are provided with threshold. Also, the fundamental of homogeneity is not accepted in non-linear
systems.

Common Non Linearities

In most types of control systems, we can not avoid the presence of certain types of non-
linearities. These can be classified as static or dynamic. A system for which there is a nonlinear
relationship between input and output, that does not involve a differential equation is called a
static nonlinearity. On the other hand, the input and output may be related through a nonlinear
differential equation. Such a system is called a dynamic nonlinearity.
Now we are going to discuss various types of non-linearities in a control system:
1. Saturation nonlinearity
2. Friction nonlinearity
3. Dead zone nonlinearity
4. Relay nonlinearity (ON OFF controller)
5. Backlash nonlinearity
Saturation Nonlinearity

Saturation nonlinearity is a common type of nonlinearity. For example see this nonlinearity in
the saturation in the magnetizing curve of DC motor. In order to understand this type of
nonlinearity let us discuss saturation curve or magnetizing curve which is given below:

From the above curve we can see that the output showing linear behavior in the beginning but
after that there is a saturation in the curve which one kind of non linearity in the system. We have
also shown approximated curve.
Same type of saturation non linearity also we can see in an amplifier for which the output is
proportional to the input only for a limited range of values of input. When the input exceeds this
range, the output tends to become non linearity.
Friction Nonlinearity

Anything which opposes the relative motion of the body is called friction. It is a kind of non
linearity present in the system. The common example in an electric motor in which we find
coulomb friction drag due to the rubbing contact between the brushes and the commutator.

Friction may be of three types and they are written below:

I. Static Friction : In simple words, the static friction acts on the body when the body is at
rest.
II. Dynamic Friction : Dynamic friction acts on the body when there is a relative motion
between the surface and the body.
III. Limiting Friction : It is defined as the maximum value of limiting friction that acts on the
body when it is at rest.
Dynamic friction can also be classified as (a) Sliding friction (b) Rolling friction. Sliding
friction acts when two bodies slides over each other while rolling acts when the bodies
rolls over another body.
In mechanical system we have two types of friction namely (a) Viscous friction (b) Static
friction.
Dead Zone Nonlinearity

Dead zone nonlinearity is shown in various electrical devices like motors, DC servo motors,
actuators etc. Dead zone non linearities refer to a condition in which output becomes zero when
the input crosses certain limiting value.

Relays Nonlinearity (ON/OFF Controller)

Electromechanical relays are frequently used in control systems where the control strategy
requires a control signal with only two or three states. This is also called as ON/OFF controller
or two state controller.

Relay Non-Linearity (a) ON/OFF (b) ON/OFF with Hysteresis (c) ON/OFF with Dead Zone. Fig
(a) shows the ideal characteristics of a bidirectional relay. In practice, relay will not respond
instantaneously. For input currents between the two switching instants, the relay may be in one
position or other depending upon the previous history of the input. This characteristic is called
ON/OFF with hysteresis that shows in Fig (b). A relay also has a definite amount of dead zone in
practice that show in Fig (c). The dead zone is caused by the fact that the relay field winding
requires a finite amount of current to move the armature.

Backlash Nonlinearity

Another important nonlinearity commonly occurring in the physical system is hysteresis in

mechanical transmissions such as gear trains and linkages. This nonlinearity is somewhat
different from magnetic hysteresis and is commonly referred to as backlash nonlinearities.
Backlash in fact is the play between the teeth of the drive gear and those of the driven gear.
Consider a gearbox as shown in below figure (a) having backlash as illustrated in fig (b).

Fig (b) shows the teeth A of the driven gear located midway between the teeth B1, B2 of the
driven gear. Fig (c) gives the relationship between input and output motions. As the teeth A is
driven clockwise from this position, no output motion takes place until the tooth A makes contact
with the tooth B1 of the driven gear after traveling a distance x/2. This output motion
corresponds to the segment mn of fig (c). After the contact is made the driven gear rotates
counterclockwise through the same angle as the drive gear if the gear ratio is assumed to be
unity. This is illustrated by the line segment no. As the input motion is reversed, the contact
between the teeth A and B1 is lost and the driven gear immediately becomes stationary based on
the assumption that the load is friction controlled with negligible inertia.
The output motion, therefore, causes till tooth A has traveled a distance x in the reverse direction
as shown in fig (c) by the segment op. After the tooth A establishes contact with the tooth B2, the
driven gear now mores in a clockwise direction as shown by segment pq. As the input motion is
reversed the direction gear is again at standstill for the segment qr and then follows the drive
gear along rn.
Describing Function Analysis of Nonlinear Systems
The describing function method in control system was invented by Nikolay Mitrofanovich
Kryloy and Nikolay Bogoliubov in year of 1930 and later it developed by Ralph Kochenburger.
The describing function method is used for finding out the stability of a non linear system of all
the analytical methods developed over the years for non linear control systems, this method is
generally agreed upon as being the most practically useful. This method is basically an
approximate extension of frequency response methods including Nyquist stability criterion to
non linear system.
The describing function method of a non linear system is defined to be the complex ratio of
amplitudes and phase angle between fundamental harmonic components of output to input
sinusoid. We can also called sinusoidal describing function. Mathematically,

Where,
N = describing function,
X = amplitude of input sinusoid,
Y = amplitude of the fundamental harmonic component of output,
φ1 = phase shift of the fundamental harmonic component of output.
Let us discuss the basic concept of describing the function of non linear control system.
Let us consider the below block diagram of a non linear system, where G1(s) and G2(s) represent
the linear element and N represent the non linear element.

Let us assume that input x to the non linear element is sinusoidal, i.e,

For this input, the output y of the non linear element will be a non sinusoidal periodic function
that may be expressed in terms of Fourier series as

Most of nonlinearities are odd symmetrical or odd half wave symmetrical; the mean value Y0 for
all such case is zero and therefore output will be,

As G1(s) G2(s) has low pass characteristics, it can be assumed to a good degree of approximation
that all higher harmonics of y are filtered out in the process, and the input x to the nonlinear
element N is mainly contributed by fundamental component of y i.e. first harmonics. So in the
describing function analysis, we assume that only the fundamental harmonic component of the
output. Since the higher harmonics in the output of a non linear system are often of smaller
amplitude than the amplitude of fundamental harmonic component. Most control systems are
low pass filters, with the result that the higher harmonics are very much attenuated compared
with the fundamental harmonic component.
Hence, y1 need only be considered.

We can write y1(t) in the form,

Where by using phasor,

The coefficient A1 and B1 of the Fourier series are given by-

From definition of describing function we have,

Let us find out describing function for these non-linearities.

Derive the describing function of Dead-zone and saturation Non linearity

Dead Zone Nonlinearity

Dead zone nonlinearity is shown in various electrical devices like motors, DC servo motors,
actuators etc. Dead zone non linearities refer to a condition in which output becomes zero when
the input crosses certain limiting value.

Relays Nonlinearity (ON/OFF Controller)

Electromechanical relays are frequently used in control systems where the control strategy
requires a control signal with only two or three states. This is also called as ON/OFF controller
or two state controller.

Relay Non-Linearity (a) ON/OFF (b) ON/OFF with Hysteresis (c) ON/OFF with Dead Zone. Fig
(a) shows the ideal characteristics of a bidirectional relay. In practice, relay will not respond
instantaneously. For input currents between the two switching instants, the relay may be in one
position or other depending upon the previous history of the input. This characteristic is called
ON/OFF with hysteresis that shows in Fig (b). A relay also has a definite amount of dead zone in
practice that show in Fig (c). The dead zone is caused by the fact that the relay field winding
requires a finite amount of current to move the armature.

Backlash Nonlinearity

Another important nonlinearity commonly occurring in the physical system is hysteresis in

mechanical transmissions such as gear trains and linkages. This nonlinearity is somewhat
different from magnetic hysteresis and is commonly referred to as backlash nonlinearities.
Backlash in fact is the play between the teeth of the drive gear and those of the driven gear.
Consider a gearbox as shown in below figure (a) having backlash as illustrated in fig (b).

Fig (b) shows the teeth A of the driven gear located midway between the teeth B1, B2 of the
driven gear. Fig (c) gives the relationship between input and output motions. As the teeth A is
driven clockwise from this position, no output motion takes place until the tooth A makes contact
with the tooth B1 of the driven gear after traveling a distance x/2. This output motion
corresponds to the segment mn of fig (c). After the contact is made the driven gear rotates
counterclockwise through the same angle as the drive gear if the gear ratio is assumed to be
unity. This is illustrated by the line segment no. As the input motion is reversed, the contact
between the teeth A and B1 is lost and the driven gear immediately becomes stationary based on
the assumption that the load is friction controlled with negligible inertia.
The output motion, therefore, causes till tooth A has traveled a distance x in the reverse direction
as shown in fig (c) by the segment op. After the tooth A establishes contact with the tooth B2, the
driven gear now mores in a clockwise direction as shown by segment pq. As the input motion is
reversed the direction gear is again at standstill for the segment qr and then follows the drive
gear along rn.
Describing Function Analysis of Nonlinear Systems
The describing function method in control system was invented by Nikolay Mitrofanovich
Kryloy and Nikolay Bogoliubov in year of 1930 and later it developed by Ralph Kochenburger.
The describing function method is used for finding out the stability of a non linear system of all
the analytical methods developed over the years for non linear control systems, this method is
generally agreed upon as being the most practically useful. This method is basically an
approximate extension of frequency response methods including Nyquist stability criterion to
non linear system.
The describing function method of a non linear system is defined to be the complex ratio of
amplitudes and phase angle between fundamental harmonic components of output to input
sinusoid. We can also called sinusoidal describing function. Mathematically,

Where,
N = describing function,
X = amplitude of input sinusoid,
Y = amplitude of the fundamental harmonic component of output,
φ1 = phase shift of the fundamental harmonic component of output.
Let us discuss the basic concept of describing the function of non linear control system.
Let us consider the below block diagram of a non linear system, where G1(s) and G2(s) represent
the linear element and N represent the non linear element.

Let us assume that input x to the non linear element is sinusoidal, i.e,

For this input, the output y of the non linear element will be a non sinusoidal periodic function
that may be expressed in terms of Fourier series as

Most of nonlinearities are odd symmetrical or odd half wave symmetrical; the mean value Y0 for
all such case is zero and therefore output will be,

As G1(s) G2(s) has low pass characteristics, it can be assumed to a good degree of approximation
that all higher harmonics of y are filtered out in the process, and the input x to the nonlinear
element N is mainly contributed by fundamental component of y i.e. first harmonics. So in the
describing function analysis, we assume that only the fundamental harmonic component of the
output. Since the higher harmonics in the output of a non linear system are often of smaller
amplitude than the amplitude of fundamental harmonic component. Most control systems are
low pass filters, with the result that the higher harmonics are very much attenuated compared
with the fundamental harmonic component.
Hence, y1 need only be considered.

We can write y1(t) in the form,

Where by using phasor,

The coefficient A1 and B1 of the Fourier series are given by-

From definition of describing function we have,

Let us find out describing function for these non linearities.

Describing Function for Saturation Non Linearity
We have the characteristic curve for saturation as shown in the given figure.

Let us take input function as

Now from the curve we can define the output as:

Let us first calculate Fourier series constant A1.

On substituting the value of the output in the above equation and integrating the function from 0
to 2π we have the value of the constant A1 as zero.
Similarly we can calculate the value of Fourier constant B1 for the given output and the value of
B1 can be calculated as,
The phase angle for the describing function can be calculated as

Thus the describing function for saturation is

Describing Function for Ideal Relay
We have the characteristic curve for ideal relay as shown in the given figure.

Let us take input function as

Now from the curve we can define the output as

The output periodic function has odd symmetry :

Let us first calculate Fourier series constant A1.

On substituting the value of the output in the above equation and integrating the function from 0
to 2π we have the value of the constant A1 as zero.
Similarly we can calculate the value of Fourier constant B1 for the given output and the value of
B1 can be calculated as

On substituting the value of the output in the above equation y(t) = Y we have the value of the
constant B1
And the phase angle for the describing function can be calculated as

Thus the describing function for an ideal relay is

4 a. What are singular points and how are they classified. Sketch them and explain.

The subject of the article is real planar vector fields x˙ = P (x, y), y˙ = Q(x, y), where P,Q are
polynomials. We consider two questions related to the second part of the Hilbert 16th problem
[Hi]. (1) What can be the number and arrangement of limit cycles of a vector field of degree d in
R2? (2) Given a classification of singular points of planar vector fields, how many singular points
of each type can a vector field of degree d in R2 have? Our approach to these problems comes
from the Viro method [V1] to [V4] (see also [IV] and [R]) invented in the framework of the first
part of the 16th problem, topology of real algebraic varieties.

This method, actually, consists in reducing a problem on polynomials with an arbitrary

Newtonpolyhedronto that on polynomials with smaller Newton polyhedra. Various applications
and developments related to the topology of real algebraic varieties and their singularities can be
found in [GKZ], [I1], [I2], [IV], [S1] to [S3], and [St]. Note also that Newton polyhedra and
diagrams have been used since the last century for the local study of singular points of differential
systems. (For the modernaccount, see, e.g., [Br].) In this paper we prove Viro-type “gluing”
theorems for planar polynomial vector fields (see Theorems 1.3.1, 1.4.1, and Corollary 1.4.2).
Using gluing theorems we construct vector fields with many limit cycles and vector fields with
given numbers of singular points of prescribed types.

It is known (see [E] and [Il2]) that a polynomial vector field has only finitely many limit cycles,
but no general upper bound (depending only on the degree) is found. On the other hand, one can
look for examples of fields with a large number of limit cycles. Among the known examples of
vector fields with many limit cycles, one can mention quadratic fields with four limit cycles (see
[An], [CW], and [Sh]), cubic fields with 11 limit cycles (see [Z]), vector fields of degree d close
to Hamiltonian ones and having (d2 +5d −14)/2 limit cycles (see [O]) or d(d +1)/2−1 limit cycles
(see [Il1] and [P]), and the vector fields of degrees d = 2k −1, k ≥ 2, with (1/2)d2 log2 d +O(d2)
limit cycles (see [CL]). The construction of C. J. Christopher and N. G. Lloyd [CL] provides an
asymptotic lower bound (1/8)d2 log2 d for the maximal number of limit cycles of a planar vector
field of degree d. We improve this asymptotic lower bound in the following way (see Theorem
2.1): For any integer d ≥ 3 there exists a planar vector field of degree d with at least (1/2)d2 log2
d −Cd2 log2 d limit cycles, where C is a positive number that does not depend on d. Let P0,P1,...,Pn
be polynomials in n variables. Under certain “general position” assumptions on Pi’s, A.
Khovanski˘i [Kh1] obtained sharp estimates (in terms of degrees of Pi’s) for the total index of the
vector field (P1,...,Pn) in Rn and in the regions P0 > 0 an d P0 < 0. In particular, for n = 2, he
proved that the total index I of singular points of a vector field of degree d, having no singularities
at infinity, satisfies |I | ≤ d, and, under the assumption that there are d2 real singular points (so all
the points have index ±1), all the values of I having the same parity as d in this range can be
realized by a suitable vector field (see also [Kh2]). We refine Khovanski˘i’s result for planar vector
fields in two steps. First, we distinguish between repellers and attractors (having the same index
+1). We call the corresponding classification topological and prove the following statement (see
Theorem 3.3): For any nonnegative integers d, s0, σ1, σ2, and σ3 satisfying 2s0 +σ1 +σ2 +σ3 =
d2, |σ1 +σ2 −σ3| ≤ d, there exists a planar vector field of degree d with 2s0 imaginary singular
points, σ1 attractors, σ2 repellers, and σ3 saddles. Second, we distinguish between a source and a
source-focus, and between a sink and a sink-focus. The corresponding classification is called thin
topological. The following theorem gives an asymptotically complete thin topological
classification of collections of singular points of vector fields satisfying some generality conditions
(see Subsection1.4, Section3, and Theorem 3.4): The last statement is proved by means of the
gluing Theorem 1.4.1. To prove the statement on topological classification, we use another gluing
theorem (see Theorem 4.1.1, Section4), which is based one torus action on (R∗)2 preserving only
topological types of singular points. The reason is that the proof of the result on topological
classification becomes much simpler and shorter than possible arguments with general gluing
theorems. We point out in this connection that gluing theorems may have various forms in
accordance with the problem studied. Remark. An answer to the question on singular points was
announced in [IS] with a sketch of the proof. However, that article contained minor flaws in its
assertions, and the proofs were incomplete. In the present article we give the corrected statements
and the complete proofs. Moreover, we add important details in the approach used. The paper
consists of several sections. In Section 1, we formulate and prove the gluing theorems. In the
subsequent sections, the gluing theorems are transformed into algorithmic procedures whose
outputs are vector fields with many limit cycles (see Section2) or with a givencollectionof types
of singular points (see Sections 4–7). Such a procedure contains two main steps: (1) analysis of
properties of vector fields with small Newton polygons; (2) subdivision of given(large) Newton
polygons into appropriate small sub polygons; search for suitable vector fields with these small
Newton polygons in order to provide the glued vector field with the required properties. For the
problem onlimit cycles, both the steps are carried out inSection2. For the problem on topological
classification of collections of singular points, both the steps are performed inSection4 by means
of the methods of [IS]. InSection5, we carry out the first step for the problem onthintopological
classificationof collections of singular points. In Section 6, we describe the basic construction and
give the thin topological classification for the case when all the singular points are real and the
total index is equal to −d. InSection7, we explainhow to complete the proof, but we skip the details,
because of the large number (more than ten) of cases, each of them requiring a slight
modificationof the mainalgorithm.