ADVANCED CONTROL THEORY

Dr. V. R. Jisha,
Associate Professor,
Dept. of Electrical Engg.,
CET
 Text Books
• Katsuhiko Ogata: “Modern Control Engineering”, fourth edition,
Pearson Education.

• Nagarath I. J and Gopal M, “Control System Engineering”, Wiley

Eastern.

• Gopal M, “Modern Control System Theory”, Wiley Eastern Ltd.,

New Delhi.

• Kuo B.C, “Analysis and Synthesis of Sampled Data Systems”,

Prentice Hall Publications.

• Norman S. Nise, “Control Systems Engineering”, John Wiley &

Sons Inc.

• Dorf & Bishop, “ Modern Control Systems”, Pearson Education.2

Syllabus

3
4
 Introduction: Control Systems
• Mars Pathfinder
• Aircraft autopilot
• Industrial automation

• A control system consisting of interconnected

components is designed to achieve a desired purpose.

• Control systems are used to achieve (1) improved

performance of a device or system (2) increased
productivity.

• Control Engineers play a critical role. 5

 Introduction (contd…)
• Already studied classical control

• Physical systems modeled in the form of a

transfer function

6
 Introduction (contd…)
• Time Domain Specifications

• Frequency domain Specifications

• Stability Analysis

• Compensators/ Controllers

7
 Introduction (contd…)
• Linear Systems: Systems that obey the “Principle of
superposition (homogeneity & additivity) ”
– Multiplying the inputs by a const ant α must multiply the
outputs by α
– The response to several inputs applied simultaneously must be
the sum of the individual responses to each input applied
separately

8
 Drawbacks: Classical Control
• Some drawbacks

– TF defined only under zero initial conditions

– TF model only applicable to Linear Time Invariant

(LTI) Systems and restricted to single input single
output systems (MIMO?)

9
 Drawbacks(contd…)
– It reveals only the system output for a given i/p and
provides no information regarding the internal
state of the system.
• Sometimes o/p of the system may be stable and yet
some of the system elements may have a tendency to
exceed their specified ratings.

– Sometimes it may be necessary and advantageous

to provide a feedback proportional to some of the
internal variables of a system, rather than o/p alone
for the purpose of stabilizing and improving the
performance of a system.
10
 Drawbacks(contd…)
– Classical design methods (root locus and
frequency domain based on TF model), are
essentially trial and error procedures.

– Such procedures are difficult to visualize and

organize in moderately complex systems and may
not lead to a control system which yields an
optimum performance.
• Modern Control

11
 State Variable Approach
• Need a more general mathematical representation of
a system which along with the o/ps, yields
information about the state of the system variables at
some predetermined points along the flow of signals.

• Such considerations have led to the development of

state variable approach.

• It is direct time domain approach which provides a

basis for modern control theory and system
optimization.
12
 State Variable Approach (contd…)
• It is a powerful approach for the analysis and design of
linear & nonlinear, time invariant & time variant , SISO &
MIMO systems.

• Incorrect to conclude that state variable approach can

completely replace TF approach.

• Classical approaches provide the control engineer, with a

deep physical insight into the system and greatly aid the
preliminary system design where a complex system is
approximated by a more manageable model.
13
 State Variable Approach (contd…)
• In transform domain analysis, Laplace transform is
needed for continuous time systems and z transform
needed for discrete time systems

• State variable approach offers us a way to look at

both continuous time and discrete time systems.

• Restrict our studies to LTI in this module

14
 State Variable Approach (contd…)
• With the ready availability of digital computers, it is
convenient to consider the time domain formulation
of equations representing control systems.

• The time domain is the mathematical domain that

incorporates the response and description of a
system in terms of time, t.

15
 Classical Control vs Modern Control
Classical Control Modern Control
Developed in 1920-1950 Developed in 1950-1980
Frequency domain analysis and design Time domain analysis and design
(based on transfer function )
Based on SISO models Based on MIMO models
Deals with input and output variables Deals with input, output and state
variables
Well developed robustness concepts Not well developed robustness concepts
No controllabilty/ Observability Inference Controllability/Observabilty can be
inferred
No optimality concerns Optimality issues incorporated

16
 Concept of state, state variable and
State space model
• A mathematical abstraction to represent or model
the dynamics of the system utilizes 3 types of
variables.
– i/p variable
– o/p variable
– State variable

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 Concept of state, state variable and
State space model (contd…)
• Let us consider the following circuit
• RL series circuit
• Variables?
– Inductor voltage
– Resistor voltage
– Current through the circuit
• Select a particular subset of all possible system
variables and call the variables in this subset as
“state variables”
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 Concept of state, state variable and
State space model (contd…)
• For an nth order system, write n simultaneous, first
order differential equations in terms of state
variables. Referred to as “state equations”.

• If we know the initial condition of all of the state

variables at t0, as well as the system input for t≥t0,
we can solve the simultaneous differential equations
for the state variables for t≥t0.

19
 Concept of state, state variable and
State space model (contd…)
• Algebraically combine the state variables with the
system’s input and find all of the other system
variables for t≥t0, this is referred to as output
equation.
• Consider an RL series circuit, with an initial current of
i(0).
• Can write the loop equation as

di
L  iR  v(t ) (a)
dt
20
 Concept of state, state variable and
State space model (contd…)
LsI ( s )  i (0)   RI ( s )  V ( s )

• Assuming the input v(t) to be unit step, u(t), whose

Laplace transform is V(s)=1/s and solving
 
1 1 1  i ( 0)
I ( s)    
Rs s R  s R
 
 L L
R R
1     t    t
L 
i (t )  1 e  i (0)e  L 
R  
 21
 Concept of state, state variable and
State space model (contd…)
• Can we represent all other variables of the system in
terms of i(t)?
vR (t )  Ri (t )
vL (t )  v(t )  Ri (t )
• Thus knowing the state variable i(t) and the input v(t)
we can find the value of any network variable at any
time t≥t0

22
Concept of state, state variable and
State space model (contd…)
di
L  iR  v(t )
dt
Let x1  i; u  v(t ); y  vR (t )
dx1 R 1

 x1   x1  u  State eqn
dt L L
y  Rx1  Output eqn

x1(t), x2(t),…xn(t) state vraibles

u1, u2,….,uminputs
y1.y2,…ypoutputs
23
 Concept of state, state variable and
State space model (contd…)
• Eqn (a) is not the unique way of expressing the
dynamics of the system.
• Can be written in terms of any other network
variable
vR
i
R
L dvR
 vR  v(t )
R dt
• This can be solved by knowing the initial condition
vR (0)  Ri(0) 24
 Concept of state, state variable and
State space model (contd…)
• RLC Series circuit
• Since the n/w is of 2nd order, 2 simultaneous first
order differential equations are required to
represent the complete dynamics

di 1
L  iR   idt  v(t )
dt C
i (t )  dq
dt
d 2q dq 1
L 2 R  q  v(t )
dt dt C 25
 Concept of state, state variable and
State space model (contd…)
• But an nth order differential equation can be
converted to n simultaneous differential equations
dxi
 ai1 x1  ai 2 x2    ain xn  bi f (t )
dt

• where each xi is a state variable

• aij ‘s and bi are constants for LTI systems
• RHS is a linear combination of the state variables
and input, f(t).
26
 Concept of state, state variable and
State space model (contd…)
dq
i (1)
dt
di 1 R 1
 q  i  v(t ) (2)
dt LC L L

• These equations are referred to as state equations and

can be solved simultaneously for the state variables q(t)
and i(t), using Laplace transform methods, if we know
the initial conditions for q(t) and i(t).

27
 Concept of state, state variable and
State space model (contd…)
• From the two state variables we can solve other
di 1
network variables L  iR   idt  v(t )
dt C
di 1 R 1 i (t )  dq
dt
L  q  i  v(t ) d 2q dq 1
dt LC L L L 2 R  q  v(t )
dt dt C
1 R 1
vL (t )   q  i  v(t ) (3)
LC L L
• The combined state equations (1-2) and the output
equation (3) form a viable representation of the n/w,
referred to as “State Space Representation”

28
Concept of state, state variable and
State space model (contd…)

If x1  q and x2  i
x1  x2
1 R 1
x2   x1  x2  v(t )
LC L L
 x1   0 1  x   0 
1
 x    1 R   x    1 u where u  v(t )
 2   LC 
L   2   L 
 1   x1 
y    R     1 u
 C   x2 
29
 Concept of state, state variable and
State space model (contd…)
• Another choice of two state variables can be made
for eg. vR(t) and vC(t).
• The resulting set of simultaneous first order
differential equations are
vR
i substituting in
R
di 1
L  iR   idt  v(t )
dt C
1
and differentiating vc (t )   idt
C
30
 Concept of state, state variable and
State space model (contd…)
dvR R R R
  vR  vC  v(t )
dt L L L
dvC 1
 vR
dt RC
• Again these differential equations can be solved for
the state variables if we know the initial conditions
along with v(t).

• All other n/w variable can be found as a linear

combination of these state variables.

31
 Concept of state, state variable and
State space model (contd…)
• In general for a linear system
X  AX  BU
Y  CX  DU
• X=state vector;
• X =derivative of the state vector wrt time
• Y=output vector
• U=input or control vector
• A= system matrix
• B=input matrix
• C=Output matrix and D=feedforward matrix 32
Block diagram of state space model

SISO system

MIMO system

33
 Concept of state, state variable and
State space model (contd…)
• State space representation consists of
– Simultaneous first order differential equations from which
the state variables can be solved
– Algebraic output equation from which all other system
variables can be found
• State Space representation is not unique, since a
different choice of state variables leads to difft
repr. of a system

34
 Definitions
• Linear combination
– A linear combination of n variables xi, for i= 1 to n is given
by the following sum S:
S  K n xn  K n 1 xn 1    K1 x1
• Linear independence
– A set of variables is said to be linearly independent if none
of the variables can be written as a linear combination of
the others
– For eg. given x1, x2 and x3 .If x2  5 x1  6 x2 then variables
are not independent

35
 Definitions(contd…)
• System variable: Any variable that responds to an
input or initial conditions in a system

• State variables: The smallest set of linearly

independent system variables such that the
knowledge of these variables at time t0 along with
known forcing functions (inputs) completely
determine the value of all system variables for all t≥0

36
 Definitions(contd…)
• State vector: A vector whose elements are the state
variables .
• State space: It is the n dimensional space whose axes
are the state variables .
• Eg. Illustrated in fig where vR and vC are the state
variables

37
 Definitions(contd…)
• These variables form the axes of the state space.
• For two dimensional cases state space reduces to the
state plane or phase plane.
• The state vector determines a point called state
point, in state space.
• The curve traced by the state point from t=t0 to t=t1,
in the direction of increasing time, is known as state
trajectory.

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State Space Representation: Nonlinear
Systems
• If we consider a mobile robot, the equations of
motion can be written as
x  v cos 
V
y  v sin 
  
If x1  x, x2  y, x3   , u1  v, u 2  
x1  u1 cos x3
x2  u1 sin x3
x3  u 2
X  f ( X , U )
In general a nonlinear system is
Y  g ( X ,U )
represented as 39
 A simple Mechanical System
If x(t0) and v(t0) known
t
1
v(t )   F (t )dt
M 
t0 t
1 1
  F (t ) dt   F (t )dt
M M t0
t
1
 v(t0 )   F (t )dt
d 1 M
v(t )  F (t ) t0

dt M t
x(t )   v(t )dt
d 
x(t )  v(t ) t
dt  x(t0 )   v(t )dt
t0
If x1 (t )  x(t ); x2 (t )  v(t )  x (t ); u (t )  F (t )

40
A simple Mechanical System(contd…)

 x1  0 1  x1   0 
 x   0 0  x    1 u
 2    2   M 
If y (t )  x(t )
 x1 
y  1 0 
 x2 

41
 Mechanical System2

d 2 x1  dx1 dx2  If states are x1 , x2 , x3 , x4

M 1 2  kx1  D2    0
dt  dt dt  x1 , x2 are displacements
dx2 d 2 x2  dx2 dx1  x1  x3 ;
D1  M2 2
 D2     f (t )
dt dt  dt dt  x 2  x4 ;
y1  x1 ; y2  x2 42
 Mechanical System2(contd…)
 0 0 1 0   0 
 x1     x1  
 x   0 0 0 1   x2   0 
 2    K D2 D2     f (t )
0  0 
 x3   M1 M1 M1   x3  
   D2 ( D1  D2 )    1 
 x 4   0 0
M2
 x
M 2   4   M 2 


 x1 
 
 y1  1 0 0 0  x2 
 y   0 1 0 0   x 
 2   3
 
 x4 

43
Mechanical System3

44
Mechanical System3(contd…)

45
Electrical Circuit 1

46
Electrical Circuit(contd…)

47
Armature Controlled DC Motor
  k f if
Tm  ia
Tm  k f i f ia
Tm  kT ia (1)

d
eb  kb (2)
dt
dia
va  ia Ra  La  eb (3)
dt
d 2 d
J 2 f  Tm  kT ia ( 4) f is the viscous friction coeft
dt dt 48
 Armature Controlled DC Motor(contd…)
• Let θ be the output and va be the input
Va ( s )  Eb ( s )
Eb ( s ) k b s ( s ) I a (s) 
La s  Ra
La sI a ( s )  Ra I a ( s )  Va ( s )  Eb ( s ) Va ( s)  kb s ( s )

Js 2 ( s )  fs ( s )  kT I a ( s ) La s  Ra

kT I a ( s )
 (s) 
Js 2  fs
k I ( s)
s  (s)  T a
Js  f

• Order of the system?

49
 Armature Controlled DC Motor(contd…)
• In mechanical system a natural choice of state
variables is position and speed
x1   ; x2  ; x3  ia
x1  x2
Jx2  fx2  kT x3
Va  kb x2  Ra x3  La x3

 
 x1  0 1 0   x1   0   x1 
 x   0  f kT   x    0 V y    x1  [1 0 0] x2 
 2  J J  2    a
 x3   kb Ra   x3  1 La   x3 
 0 
La La 
50