CONTROL SYSTEM ENGINEERING-II (3-1-0)

121 Pages·2015·3.07 MB·English

Lecture Notes Control System Engineering-II

VEER SURENDRA SAI UNIVERSITY OF TECHNOLOGY BURLA, ODISHA, INDIA

DEPARTMENT OF ELECTRICAL ENGINEERING

CONTROL SYSTEM ENGINEERING-II (3-1-0)
Lecture Notes
Subject Code: CSE-II

For 6th sem. Electrical Engineering & 7th Sem. EEE Student

1
DISCLAIMER

COPYRIGHT IS NOT RESERVED BY AUTHORS. AUTHORS ARE NOT RESPONSIBLE
FOR ANY LEGAL ISSUES ARISING OUT OF ANY COPYRIGHT DEMANDS AND/OR
REPRINT ISSUES CONTAINED IN THIS MATERIALS. THIS IS NOT MEANT FOR ANY
COMMERCIAL PURPOSE & ONLY MEANT FOR PERSONAL USE OF STUDENTS
FOLLOWING SYLLABUS. READERS ARE REQUESTED TO SEND ANY TYPING
ERRORS CONTAINED, HEREIN.

2
Department of Electrical Engineering,

CONTROL SYSTEM ENGINEERING-II (3-1-0)

MODULE-I (10 HOURS)

State Variable Analysis and Design: Introduction, Concepts of State, Sate Variables and State Model,
State Models for Linear Continuous-Time Systems, State Variables and Linear Discrete-Time
Systems, Diagonalization, Solution of State Equations, Concepts of Controllability and Observability,
Pole Placement by State Feedback, Observer based state feedback control.

MODULE-II (10 HOURS)

Introduction of Design: The Design Problem, Preliminary Considerations of Classical Design,
Realization of Basic Compensators, Cascade Compensation in Time Domain(Reshaping the Root
Locus), Cascade Compensation in Frequency Domain(Reshaping the Bode Plot),
Introduction to Feedback Compensation and Robust Control System Design.
Digital Control Systems: Advantages and disadvantages of Digital Control, Representation of
Sampled process, The z-transform, The z-transfer Function. Transfer function Models and dynamic
response of Sampled-data closed loop Control Systems, The Z and S domain Relationship, Stability
Analysis.

MODULE-III (10 HOURS)

Nonlinear Systems: Introduction, Common Physical Non-linearities, The Phase-plane Method: Basic
Concepts, Singular Points, Stability of Nonlinear System, Construction of Phase-trajectories, The
Describing Function Method: Basic Concepts, Derivation of Describing Functions, Stability analysis
by Describing Function Method, Jump Resonance, Signal Stabilization.
Liapunov‟s Stability Analysis: Introduction, Liapunov‟s Stability Criterion, The Direct Method of
Liapunov and the Linear System, Methods of Constructing Liapunov Functions for Nonlinear
Systems, Popov‟s Criterion.

MODULE-IV (10 HOURS)

Optimal Control Systems: Introduction, Parameter Optimization: Servomechanisms, Optimal Control
Problems: State Variable Approach, The State Regulator Problem, The Infinite-time Regulator
Problem, The Output regulator and the Tracking Problems, Parameter Optimization: Regulators,
Introduction to Adaptive Control.

BOOKS

[1]. K. Ogata, “Modem Control Engineering”, PHI.
[2]. I.J. Nagrath, M. Gopal, “Control Systems Engineering”, New Age International Publishers.
[3]. J.J.Distefano, III, A.R.Stubberud, I.J.Williams, “Feedback and Control Systems”, TMH.
[4]. K.Ogata, “Discrete Time Control System”, Pearson Education Asia.

3
MODULE-I
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.
Advantages of state variable analysis.
 It can be applied to non linear 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.

State Variable Analysis and Design
State: The state of a dynamic system is the smallest set of variables called state variables such that
the knowledge of these variables at time t=t (Initial condition), together with the knowledge of input
o
for ≥ 𝑡 , completely determines the behaviour of the system for any time 𝑡 ≥ 𝑡 .
0 0
State vector: If n state variables are needed to completely describe the behaviour 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.
State space: The n-dimensional space whose co-ordinate axes consists of the x axis, x axis,.... x
1 2 n
axis, where x , x ,..... x are state variables: is called a state space.
1 2 n
State Model
Lets consider a multi input & multi output system is having
r inputs 𝑢 𝑡 ,𝑢 𝑡 ,…….𝑢 (𝑡)
1 2 𝑟
m no of outputs 𝑦 𝑡 ,𝑦 𝑡 ,…….𝑦 (𝑡)
1 2 𝑚
n no of state variables 𝑥 𝑡 ,𝑥 𝑡 ,…….𝑥 (𝑡)
1 2 𝑛
Then the state model is given by state & output equation
X t = AX t +BU t …………state equation
Y t = CX t +DU t ………output equation
A is state matrix of size (n×n)
B is the input matrix of size (n×r)
C is the output matrix of size (m×n)
4
D is the direct transmission matrix of size (m×r)
X(t) is the state vector of size (n×1)
Y(t) is the output vector of size (m×1)
U(t) is the input vector of size (r×1)

(Block diagram of the linear, continuous time control system represented in state space)
𝐗 𝐭 =𝐀𝐗 𝐭 +𝐁𝐮 𝐭
𝐘 𝐭 =𝐂𝐗 𝐭 +𝐃𝐮 𝐭

STATE SPACE REPRESENTATION OF NTH ORDER SYSTEMS OF LINEAR
DIFFERENTIAL EQUATION IN WHICH FORCING FUNCTION DOES NOT INVOLVE
DERIVATIVE TERM
Consider following nth order LTI system relating the output y(t) to the input u(t).
𝑦𝑛 +𝑎 𝑦𝑛−1 +𝑎 𝑦𝑛−2 +⋯+𝑎 𝑦1 +𝑎 𝑦 = 𝑢
1 2 𝑛−1 𝑛
Phase variables: The phase variables are defined as those particular state variables which are
obtained from one of the system variables & its (n-1) derivatives. Often the variables used is
the system output & the remaining state variables are then derivatives of the output.
Let us define the state variables as
𝑥 = 𝑦
1
𝑑𝑦 𝑑𝑥
𝑥 = =
2 𝑑𝑡 𝑑𝑡
𝑑𝑦 𝑑𝑥
2
𝑥 = =
3 𝑑𝑡 𝑑𝑡
⋮ ⋮ ⋮
𝑑𝑥
𝑥 = 𝑦𝑛−1 = 𝑛−1
𝑛 𝑑𝑡
5
From the above equations we can write
𝑥 = 𝑥
1 2
𝑥 = 𝑥
2 3
⋮ ⋮

𝑥 = 𝑥
𝑛−1 𝑛
𝑥 = −𝑎 𝑥 −𝑎 𝑥 −⋯………−𝑎 𝑥 +𝑢
𝑛 𝑛 1 𝑛−1 2 1 𝑛
Writing the above state equation in vector matrix form
X t = AX t +Bu t
0 1 0…… 0
𝑥
1
0 0 1…… 0
𝑥2
Where 𝑋 = , 𝐴 = ⋮ ⋮ ⋮ ⋮
⋮
⋮ 0 0 0…… 1
𝑥
𝑛 𝑛×1 −𝑎 −𝑎 −𝑎 ……. −𝑎
𝑛 𝑛−1 𝑛−2 1 𝑛×𝑛
Output equation can be written as
Y t = CX t
𝐶 = 1 0……. 0
1×𝑛

Example: Direct Derivation of State Space Model (Mechanical Translating)
Derive a state space model for the system shown. The input is f and the output is y.
a

We can write free body equations for the system at x and at y.
6
Freebody Diagram Equation

There are three energy storage elements, so we expect three state equations. The energy
storage elements are the spring, k , the mass, m, and the spring, k . Therefore we choose
2 1
as our state variables x (the energy in spring k is ½k x²), the velocity at x (the energy in
2 2
the mass m is ½mv², where v is the first derivative of x), and y (the energy in spring k is
1
½k (y-x)² , so we could pick y-x as a state variable, but we'll just use y (since x is already a
1
state variable; recall that the choice of state variables is not unique). Our state variables
become:

Now we want equations for their derivatives. The equations of motion from the free body
diagrams yield

or

7
with the input u=f .
a

Example: Direct Derivation of State Space Model (Electrical)
Derive a state space model for the system shown. The input is i and the output is e .
a 2

There are three energy storage elements, so we expect three state equations. Try
choosing i , i and e as state variables. Now we want equations for their derivatives. The
1 2 1
voltage across the inductor L is e (which is one of our state variables)
2 1

so our first state variable equation is

If we sum currents into the node labeled n1 we get

This equation has our input (ia) and two state variable (iL2 and iL1) and the current
through the capacitor. So from this we can get our second state equation

Our third, and final, state equation we get by writing an equation for the voltage across
L (which is e ) in terms of our other state variables
1 2

We also need an output equation:

8
So our state space representation becomes

State Space to Transfer Function
Consider the state space system:

Now, take the Laplace Transform (with zero initial conditions since we are finding a
transfer function):

We want to solve for the ratio of Y(s) to U(s), so we need so remove Q(s) from the
output equation. We start by solving the state equation for Q(s)

The matrix Φ(s) is called the state transition matrix. Now we put this into the output
equation

Now we can solve for the transfer function:

Note that although there are many state space representations of a given system, all
of those representations will result in the same transfer function (i.e., the transfer
function of a system is unique; the state space representation is not).
9
Example: State Space to Transfer Function
Find the transfer function of the system with state space representation

First find (sI-A) and the Φ=(sI-A)-1 (note: this calculation is not obvious. Details
are here). Rules for inverting a 3x3 matrix are here.

Now we can find the transfer function

To make this task easier, MatLab has a command (ss2tf) for converting from state space
to transfer function.
>> % First define state space system
>> A=[0 1 0; 0 0 1; -3 -4 -2];
>> B=[0; 0; 1];
>> C=[5 1 0];
>> [n,d]=ss2tf(A,B,C,D)

n =
10