Chapter one(1).pdf

Modern Control System (ECEg4321)
By:Yeshambel Fentahun
yeshfe21@gmail.com
Jigjiga University
Jigjiga Institute of Technology
School of Electrical and Computer Engineering
power stream
April 8, 2022

Modern Vs. Classical control Systems Open-Loop Vs. Closed-Loop Control Systems. Mathematical M
Chapter 1
PRESENTATION OUTLINES
1 Modern Vs. Classical control Systems
2 Open-Loop Vs. Closed-Loop Control Systems.
3 Mathematical Modeling of Control Systems.
(JJU) Modern Control System April 8, 2022 1/45

Modern Vs. Classical control
Systems Modern Vs. Classical control Systems Open-Loop Vs. Closed-Loop Control Systems. Mathematical M
Modern Vs. Classical control Systems
Modern control theory:
It is applicable to: multiple-input, multiple-output sys-
tems, which may be linear or nonlinear, time invariant or
time varying.
Also, modern control theory is essentially time-domain
approach and frequency domain approach (in certain cases
such as H-infinity control).
Classical control theory:
It is applicable only to linear, time-invariant, single-input,
single-output systems.It is also a complex frequency-domain
approach.

Open-Loop Vs. Closed-Loop
Control Systems. Modern Vs. Classical control Systems Open-Loop Vs. Closed-Loop Control Systems. Mathematical M
Open-Loop Vs. Closed-Loop Control Systems.
Open-Loop Control Systems: Asystem in which con-
trol action does not depend on output is known Open-
Loop Control Systems.
Figure:1 Open loop control system
The major advantages of open-loop control systems are
as follows:
1. Simple construction and ease of maintenance.
2. Less expensive than a corresponding closed-loop sys-
tem.

Cont...
Closed -Loop Control Systems: Asystem in which con-
trol action does depend on output is known clesd-Loop
Control Systems.
It is feed back control system.
Figure:2 Closed loop control system

Cont...
An advantage of the closed loop control system is:
The fact that the use of feedback makes the system response rela-
tively insensitive to external disturbances and internal variations in
system parameters.

Mathematical Modeling of Control
Systems. Modern Vs. Classical control Systems Open-Loop Vs. Closed-Loop Control Systems. Mathematical M
Mathematical Modeling of Control Systems:
State?
The state of a dynamic system is the smallest set of variables (called
state variables) such that knowledge of these variables at t=t0 , to-
gether with knowledge of the input for t > t0, completely determines
the behavior of the system for any time t > t0 .
State variable: The state variables of a dynamic system are the
variables making up the smallest set of variables that determine the
state of the dynamic system.
State vector: If n state variables are needed to completely describe
the behavior of a given system, then these n state variables can be
considered the n components of a vector x. Such a vector is called a
state vector.

Mathematical Modeling of Control Systems
(state space based)

(state space based)

(state space based)
State space:The n-dimensional space whose coordinate axes consist
of the x1 axis, x2 axis,...,xn axis, where x1 ,x2,..., xn are state
variables, is called a state space. Any state can be represented by a
point in the state space.
State space equations:State-space analysis is concerned with three
types of variables that are involved in the modeling of dynamic sys-
tems: input variables, output variables, and state variables.

(state space based)
If vector functions f and/or g involve time t explicitly, then the sys-
tem is called a time-varying system.
If eqn (1) & (2)are linearized about the operating state, then we have
the following linearized state equation and output equation:
Where is A(t) called state matrix, B(t) the input matrix,C(t) the out-
put matrix and D(t) direct transmission matrix.
Block diagram representation:

(state space based)
If vector functions f and g do not involve time t explicitly then the
system is called a time-invariant system. And eqn(1) & (2) is sim-
plified as:

In short;
• Nonlinear time variant system
• Nonlinear time invariant system

Cont...
• Linear time variant system
• Linear time invariant system

Example 1: For the mechanical system shown below
(assume that the system is linear), The external force u(t)
is the input to the system, and the displacement y(t) of
the mass is the output. The displacement y(t) is mea-
sured from the equilibrium position in the absence of the
external force.

Cont...
Figure 2

Cont...
The system equation from the diagram:
This system is of second order. which means that the system in-
volves two integrators. Let us define state variables x1(t) and x2(t)
as:

Cont...
The output equation is:
y=x1

The state equation in vector-matrix form:
and, the output equation can be written as:

Cont...
The above equation is in standard form:

.

Example 2: RLC circuit formulate the state space representation
,state space equation by vecter form and output equation. The state
of this system can be described in terms of a set of variables [x1
x2], where x1 is the capacitor voltage vc(t) and x2 is equal to the
inductor current iL(t).

Utilizing KCL at the junction, we obtain a first order differential
equation by describing the rate of change of capacitor voltage
. KVL for the right-hand loop provides the equation describing the
rate of change of inducator current as

The output of the system is represented by the linear algebraic equa-
tion
. Rearranging to write in the form of vector-matrix
.

The output signal is then
.

Exercise:1.The 2 mass system shown figure be-
low,find the state and output equation when the
state variabeles are the position and velocity of
each mass.

The state space representation from the abeve is called contrrollabel
canonical form and the out put equation is

Example 1:find the stae and out put equation for

Considering a differential equation:
One way to obtain a state equation and output equation for this case
is defining n state variables as:

Exercise:2.Obtain a state-space equation and output equa-
tion for the system defined by
Hint:

Transfer function of SISO system from state
space equations
If we have a state space equation as:
It can be calculated in the form of TF as:
Taking the Laplace transform of equation (1)

space equations
Substituting equation (6) into (4)

space equations
Exampel:Find the transfer function of the system from
the following stat space and output equation:

space equations
Exercise:3.Find the transfer function of the system from
the following stat space and output equation:
Answer

Eigenvalues and Eigenvectors
Definition:A nonzero vector x is an eigenvector. A square
matrix A if there exists a scalar λ such that Ax = λ. Then
λ is an eigenvalue.
Note:The zero vector can not be an eigenvector even
though A0 = λ0. But λ = 0 can be an eigenvalue. Let
x be an eigenvector of the matrix A. Then there must ex-
ist an eigenvalue λ such that Ax = λx or, equivalently,
Ax - λx = 0 or (A – λI)x = 0
Example 1: Find the eigenvalues and eigenvectors of

Computation of the State Transition Matrix
linear time-invariant state equation

linear time-invariant state equation

Example: calculate:
a.The state transition matrix φ(s) and φ(t)
b.The transient response of the state variabel fromm the
set of intial conditions for following system.

Chapter one(1).pdf

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Chapter one(1).pdf