Introduction to mathematical control theory - Dr. Purnima Pandit

Introduction to
Mathematical Control Theory
Dr. Purnima Pandit
Department of Applied Mathematics
Faculty of Technology and Engineering
The M. S. University of Baroda

1.Introduction
• State-Space and frequency domain representation of control
systems.
• Discrete-Time and Continuous-Time Systems.
• Linear and nonlinear systems with examples.

Introduction
• Mathematical control theory is the area of application-oriented
mathematics that deals with the basic principles underlying the
analysis and design of control systems.
• To control an object means to influence its behavior so as to
achieve a desired goal.
• In order to implement this influence, engineers build devices that
incorporate various mathematical techniques.

Introduction
• These devices range from:
• Watt’s steam engine governor, designed during the English Industrial
Revolution
• The sophisticated microprocessor controllers found in consumer
items —such as CD players and automobiles
• Industrial robots and airplane autopilots.

Introduction
• The objective of this subject is the study of these devices and
their interaction with the object being controlled.

Introduction
• While on the one hand one wants to understand the fundamental
limitations that mathematics imposes on what is achievable,
irrespective of the precise technology being used, it is also true that
technology may well influence the type of question to be asked and
the choice of mathematical model.
• An example of this is the use of difference rather than
differential equations when one is interested in digital
control.

Introduction
• There have been two main lines of work in control theory:
• One of these is based on the idea that a good model of the object to
be controlled is available and that one wants to somehow
optimize its behavior.
• For instance, physical principles and engineering specifications can be
—and are— used in order to calculate that trajectory of a spacecraft
which minimizes total travel time or fuel consumption.
• The techniques here are closely related to the classical calculus of
variations and to other areas of optimization theory.

Introduction
• The other main line of work is that based on the constraints imposed
by uncertainty about the model or about the environment in which
the object operates. The central tool here is the use of feedback
in order to correct for deviations from the desired behavior.
• For instance, various feedback control systems are used during actual
space flight in order to compensate for errors from the pre-computed
trajectory.
• Mathematically, stability theory, dynamical systems, and especially
the theory of functions of a complex variable, have had a strong
influence on this approach.

Introduction
System – An interconnection of elements and devices for a desired purpose.
Control System – An interconnection of components forming a system
configuration that will provide a desired response.
Process – The device, plant, or
system under control. The input and
output relationship represents the
cause-and-effect relationship of the
process.

Introduction
Multivariable Control System
Open-Loop Control Systems utilize a
controller or control actuator to obtain the
desired response.
Closed-Loop Control Systems utilizes
feedback to compare the actual output to
the desired output response.

History
18th Century James Watt’s centrifugal governor for the speed control of a steam engine.
1920s Minorsky worked on automatic controllers for steering ships.
1930s Nyquist developed a method for analyzing the stability of controlled systems
1940s Frequency response methods made it possible to design linear closed-loop control
systems
1950s Root-locus method due to Evans was fully developed
1960s State space methods, optimal control, adaptive control and
1980s Learning controls are begun to investigated and developed.
Present and on-going research fields. Recent application of modern control theory
includes such non-engineering systems such as biological, biomedical, economic and
socio-economic systems
???????????????????????????????????

Watt’s Flyball Governor
(18th century)
Greece (BC) – Float regulator mechanism
Holland (16th Century)– Temperature regulator
Examples Control Systems

Water-level float regulator
Examples Control Systems

Examples of Modern Control Systems

(a) Automobile steering control
system.
(b) The driver uses the difference
between the actual and the
desired direction of travel to
generate a controlled
adjustment of the steering
wheel.
(c) Typical direction-of-travel
response.
Examples of Modern Control Systems

The Future of Control Systems and Robotics

The Control System Design Process

Mathematical Models of Control System
SPRING, MASS DAMPER SYSTEM:
Mathematical models of dynamic processes are often derived using physical laws such as Newton’s As
an example consider first a simple mechanical system, a spring/mass/damper. It consists of a weight m
on a spring with spring constant k, its motion damped by friction with coefficient f
If y(t) is the displacement from the resting position and u(t)
is the force applied, it can be shown using Newton’s law
that the motion is described by the following linear,
ordinary differential equation with constant coefficients:

StateVariable Descriptions
Let x1(t) = y(t), x2(t) = (t) be new variables, called state variables. Then the system is equivalently described by
the equations
with initial conditions x1(0) = y0 and x2(0) = y1. Since y(t) is of interest, the output equation y(t) = x1(t) is also
added. These can be written as
which are of the general form
Here, x(t) is a 2x1 vector (a column vector) with elements
the two state variables x1(t) and x2(t). It is called the state
vector. The variable u(t) is the input and y(t) is the output
of the system.
The first equation is a vector differential equation called the state equation. The second equation is an algebraic
equation called the output equation.
In the above example D = 0; D is called the direct link, as it directly connects the input to the output, as opposed
to connecting through x(t) and the dynamics of the system. The above description is the state variable or state
space description of the system

Mathematical Models of Control System
RLC CIRCUIT :
For a second example consider an electric RLC circuit with i(t) the input current of a current source, and
v(t) the output voltage across a load resistance R.
Using Kirchhoff’s laws one may derive:
which describes the dependence of the output voltage v(t) to
the input current i(t). Given i(t) for t ≥ 0, the initial values v(0)
and (0) must also be given to uniquely define v(t) for t ≥ 0.
Although the interpretation of the variables is completely
different, their relations described by the same linear, second-
order differential equation with constant coefficients.
This fact is well understood and leads to the study of
mechanical, thermal, fluid systems via convenient electric
circuits.

If we start with the second-order differential equation and set
we derive an equivalent state variable description, namely,

Note that the choice of state variables is not unique. A state variable description of a system can sometimes be
derived directly, and not through a higher-order differential equation.
To illustrate, consider the circuit example presented above: using Kirchhoff’s current law
and from the voltage law
If the state variables are selected to be x1 = vc, x2 = iL, then the equations may be written as
where v = RiL = Rx2 is the output of interest.

The linear models studied here are very useful not only because they describe linear dynamical processes, but
also because they can be approximations of nonlinear dynamical processes in the neighborhood of an
operating point. The idea in linear approximations of nonlinear dynamics is analogous to using Taylor series
approximations of functions to extract a linear approximation. A simple example is that of a simple pendulum
where for small excursions from the equilibrium at zero, sin x1 is approximately equal to x1 and the equations
become linear, namely,
Nonlinear Models - Linearization:

Note that in the next subsection the trajectory y(t) is determined,
in terms of:
• the system parameters
• the initial conditions and
• the applied input force u(t)
using a methodology based on Laplace transform.

TransferFunctions:
The transfer function of a linear, time-invariant system is
the ratio of the Laplace transform of the output Y(s) to the Laplace
transform of the corresponding input U(s) with all initial conditions
assumed to be zero.
Recalling Laplace transform is given by

From Differential Equations to Transfer Functions:
Let the equation
with some initial conditions
describe a process of interest, for example, a spring/mass/damper system. Taking Laplace transform of both
sides we obtain
where Y(s) = L{y(t)} and U(s) = L{u(t)}. Combining terms and solving with respect to Y(s) we obtain:
Assuming the initial conditions are zero,
where G(s) is the transfer function of the system defined above.

We are concerned with transfer functions G(s) that are
rational functions, that is, ratios of polynomials in s,
G(s) = n(s)/d(s).
We are interested in proper G(s) where,
lim
𝑠→∞
𝐺 𝑠 < ∞.
Proper G(s) have degree n(s) ≤ degree d(s).
In most cases degree n(s) < degree d(s), which case G(s) is
called strictly proper. Consider the transfer function

From State Space Descriptions to Transfer Functions:
Consider
with x(0) initial conditions; x(t) is in general an n-tuple, that is a (column) vector with n elements. Taking Laplace
transform of both sides of the state equation:
where In is the n  n identity matrix; it has 1 on all diagonal elements and 0 everywhere else, e.g.,
Then
Taking now Laplace transform on both sides of the output equation we obtain Y(s) = CX(s) + DU(s). Substituting
we obtain,
The response y(t) is the inverse Laplace of Y(s). Note that the second term on the right-hand side of the
expression depends on x(0) and it is zero when the initial conditions are zero, i.e., when x(0) = 0. The first term
describes the dependence of Y on U and it is not difficult to see that the transfer function G(s) of the systems is

Consider the spring/mass/damper example discussed previously with state variable description
If m = 1, f = 3, k = 2, then
Example:

Transfer Function to State Space Representation:
We have seen above how to go from a state space description to the
corresponding transfer function. The converse operation leads to the
following question:
Given a transfer function G(s), how can a state representation for this
system be obtained?

Using the state space description and properties of Laplace transform an explicit expression for y(t) in
terms of u(t) and x(0) may be derived. To illustrate, consider the scalar case
Using Laplace transform:
initial condition.
from which
Note that the second term is a convolution integral.

Similarly in the vector case, given
it can be shown that
and
Notice that
The matrix exponential eAt is defined by the (convergent) series
as
as

Let x1(t) be the water level in Tank 1 and x2(t) be the water level in Tank 2. Let be the
rate of outflow from Tank 1 and be rate of outflow from Tank 2. Let u be the supply of
water to the system. The system can be modelled into the following differential
equations:
Example: Tank Problem 1

Synthesis of Ethyl acetate
Subprocesses:
•Mixtank - the fresh and recycled Acetic Acid is mixed
•Reactor - the ethanol is mixed with Acetic Acid in the ratio 2:3
•Separator - Ethyl Acetate and the unused Acetic Acid are separated
Model of Nonlinear System:

Mixing Tank Sub Process:
h’
V
Q2,C2Q1,C1
Q,C
Inflows:
Fresh Acetic Acid and Recycled Acetic Acid with concentrations C1 and C2 with constant
rates Q1 and Q2.
Outflow:
Mixed Acetic Acid at a rate Q(t) with concentration C(t).

V(t) - the volume of the fluid in the tank at time t.
The mass balance and mole balance equations are :
)()()()())()((
)()()()(
2211
21
tQtCtQCtQCtVtC
dt
d
tQtQtQtV
dt
d
+=
+=
The outflow Q(t) is characterized by the turbulent
flow relation
A
tV
kthktQ
)(
)()( ==
h(t) – height at time t , A - the cross sectional area
k - the constant
1
2

Putting (2) in (1) we get:
   
)(
)(
)(
)(
))()((
)(
2
2
1
1
21
tV
(t)Q
t-CC
tV
(t)Q
tCC
dt
tVtCd
A
tV
(t) -kQ(t)Q
dt
dV(t)
+=
+=
The above equation can be put in the standard form:
))(),(()()( tutxftButx +=
.
00
11
,
)(
)(
)(,
)(
)(
)(
2
1






=





=





= B
tQ
tQ
tu
tC
tV
tx
where,
The above equation (4) depicts the continuous time invariant semilinear
dynamical system representing the state of the process.
3
4
     













+


=
)(
)()(
)(
)()(
)(
)(),(
2211
tV
tQtCC
tV
tQtCC
A
tV
k
tutxf

Controllability
The system as (4) is said to be controllable if for given any two states x0
and xf there exist a control sequence u(t) which steers the statex0, at time
t = 0, to xf in finite time tf .
In case of semilinear systems:
•global controllability – hard
•local controllability - suffices
The practical requirement is to steer the perturbed state to the
equilibrium state (x0,u0).
Equilibrium state: (x0,u0) is the desired flow with desired
concentration

Controllability
This semilinear system is controllable if the linearized system is
controllable in the neighborhood of equilibrium and the nonlinearity
is of Lipschitz type.
The nonlinearity involved in system (4) given by
is Lipschitz except at (0, 0).
     













+


=
)(
)()(
)(
)()(
)(
)(),(
2211
tV
tQtCC
tV
tQtCC
A
tV
k
tutxf

Introduction to mathematical control theory - Dr. Purnima Pandit

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