1. 05/05/2009
Fall 2008 Advanced Topics (EENG 4010) Control Systems Design (EENG 5310)

2. What is a Control System?
System- a combination of components that act together and perform a certain objective
Control System- a system in which the objective is to control a process or a device or environment
Process- a progressively continuing operations/development marked by a series of gradual changes that succeed one another in a relatively fixed way and lead towards
a particular result or end.

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Control Theory
Branch of systems theory (study of interactions and behavior of a complex assemblage)
Open Loop Control System
Control System
Manipulated Variable(s)
Control Variable(s)
Manipulated Variable(s)
Closed Loop Control System
Control System
Control Variable(s)
Feedback function

4. Classification of Systems
Classes of Systems
Lumped Parameter
Distributed Parameter (Partial Differential Equations, Transmission line example)
Deterministic
Stochastic
Continuous Time
Discrete Time
Linear
Nonlinear
Constant Coefficient
Time Varying
Homogeneous (No External Input; system behavior depends on initial conditions)
Non-homogeneous

5. Example Control Systems
Mechanical and Electo-mechanical (e.g. Turntable) Control Systems
Thermal (e.g. Temperature) Control System
Pneumatic Control System
Fluid (Hydraulic) Control Systems
Complex Control Systems
Industrial Controllers
On-off Controllers
Proportional Controllers
Integral Controllers
Proportional-plus-Integral Controllers
Proportional-plus-Derivative Controllers
Proportional-plus-Integral-plus-Derivative Controllers

6. Mathematical Background
Why needed? (A system with differentials, integrals etc.)
Complex variables (Cauchy-Reimann Conditions, Euler Theorem)
Laplace Transformation
Definition
Standard Transforms
Inverse Laplace Transforms
Z-Transforms

7. Laplace Transform Definition Condition for Existence
Laplace Transforms of exponential, step, ramp, sinusoidal, pulse, and impulse functions
Translation of and multiplication by
Effect of Change of time scale
Real and complex differentiations, initial and final value theorems, real integration, product theorem
Inverse Laplace Transform

8. Inverse Laplace Transform
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Inverse Laplace Transform
Definition
Formula is seldom or never used; instead, Heaviside partial fraction expansion is used.
Illustration with a problem:
Initial conditions: y(0) = 1, y’(0) = 0, and
r(t) = 1, t >= 0. Find the steady state response
Multiple pole case with
Use the ideas to find and

9. Applications Spring-mass-damper- Coulomb and viscous damper cases
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Applications
Spring-mass-damper- Coulomb and viscous damper cases
RLC circuit, and concept of analogous variables
Solution of spring-mass-damper (viscous case)
DC motor- Field current and armature current controlled cases
Block diagrams of the above DC-motor problems
Feedback System Transfer
functions and Signal flow graphs

10. Block Diagram Reduction
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Block Diagram Reduction
Combining blocks in a cascade
Moving a summing point ahead of a block
Moving summing point behind a block
Moving splitting point ahead of a block
Moving splitting point behind a block
Elimination of a feedback loop H2
Y(s) - +
R(s) G1 G2 G3 G4 + - + H1 H3

11. Signal Flow Graphs Mason’s Gain Formula
Solve these two equations and generalize to
get Mason’s Gain Formula r1 x1
a21
a12 x2 r2
a22 H2 H3 G1 G2 G3 G4
R(s)
Y(s) G6 G7 G8 G5
Find Y(s)/R(s) using the formula H8 H7

12. Control System Stability: Routh-Hurwitz Criterion
Why poles need to be in Right Hand Plane
Necessary condition involving Characteristic Equation (Polynomial) Coefficients
Proof that the above condition is not sufficient Ex: s3+s2+2s+8.
Routh-Hurwitz Criterion- Necessary & Sufficient

13. Routh-Hurwitz Criterion: Some Typical Problems
2nd and 3rd order systems
q(s)=s5+2s4+2s3+4s2+11s+10 (first element of a row 0; other elements are not)
q(s)=s4+s3+s2+s+K (Similar to above case)
q(s)=s3+2s2+4s+K (for k = 8, first element of a row 0; so are other elements of the row)
q(s)= s5+s4+2s3+2s2+s+1: Repeated roots on imaginary axis; Marginally stable case

14. Root-Locus Method: What and Why?
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Root-Locus Method: What and Why?
Plotting the trajectories of the poles of a closed loop control system with free parameter variations
Useful in the design for stability with out sacrificing much on performance
Closed Loop Transfer Function
Let Open Loop Gain
Roots of the closed loop characteristic
equation depend on K.
R(s)
Y(s) + -
G(s)
H(s)

15. Relationship between closed loop poles and open loop gain
When K=0, closed loop poles match open loop poles
When closed loop poles match open loop zeros.
Hence we can say, the closed loop poles start at open loop poles and approach closed loop zeros as K increases and thus form trajectories.

16. Mathematical Preliminaries of Root Locus Method
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Mathematical Preliminaries of Root Locus Method
Complex numbers can be
expressed as (absolute value,
angle) pairs.
Now,
The loci of closed loop poles can
be determined using the above
constraints (particularly, the
angle constraint) on G(s)H(s).
|s|
s=s+jw w q s
s=|s|.ejq
=|s| q
|s+s1| w
s=s+jw s f
-s1=-s1-jw1
s+s1=|s+s1|ejf

17. Root Locus Method- Step1 thru 3 of a 7-Step Procedure
Step-1: Locate poles and zeros of G(s)H(s). Step 2: Determine Root Locus on the real-axis using angle constraint. Value of K at any particular test point s can be calculated using the magnitude constraint. Step 3: Find asymptotes by using angle constraint in . Find asymptote centroid . This formula may be obtained by setting Illustrative Problem: -1 -2

18. Root Locus Method- Step 4
Step 4: Determine breakaway points (points where two or more loci coincide giving multiple roots and then deviate). Now, from the characteristic equation
where , we get, at a multiple pole s1, , because at s1,
. Thus we get at s1,
Since at s = s1. Thus, we get break points by setting dK/ds=0. In the
example, we get s = or (invalid).

19. Root Locus Method- Step 5
Step 5: Determine the points (if any) where the root loci cross the imaginary axis using Roth-Hurwitz Stability Criterion. Illustration with the Example Problem Characteristic equation for the problem:s3+3s2+2s+K From the array, we know that the system is marginally stable at K=6. Now, we can get the value of w (imaginary axis crossing) either by solving the second row 3s2+6 =0 or the original equation with s=jw. S3 1 2 S2 3 K S1
(6-K)/3 S0

20. Root Locus Method- Step 6 and 7
Step 6: Determine angles of departure at complex poles and arrival at complex zeros using angle criterion. Step 7: Choose a test point in the broad neighborhood of imaginary axis and origin and check whether sum of the angles is an odd multiple of +180 or If it does not satisfy, select another one. Continue the process till sufficient number of test points satisfying angle condition are located. Draw the root loci using information from steps 1-5.

21. Root Locus approach to Control System Design
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Root Locus approach to Control System Design
Effect of Addition of Poles to Open Loop Function: Pulls the root locus right; lowers system’s stability and slows down the settling of response.
Effect of Addition of Zeros to Open Loop Function: Pulls the Root Locus to Left; improves system stability and speeds up the settling of response s jw x s jw x s jw x x x x jw jw jw o x x x x o x x x x o x s s s

22. Performance Criteria Used In Design
We consider 2nd order systems here, because higher order systems with 2 dominant poles can be approximated to 2nd order systems e.g.
when
For 2nd order system
For unit step input
Where
Two types of performance criteria (Transient and Steady State)
Stability is a validity criterion (Non-negotiable).

23. Transient Performance Criteria
overshoot
ess
y(t)
1.0
Rise Time TR= Time to reach Value 1.0
Rise Time Tr1= Time from 0.1 to 0.9
Empirical Formula is
for
Settling time (Time to settle to within 98% of 1.0)=4/xwn
Peak Time
Percentage Overshoot = TP t TR TS
wnTr1
2.0 x
0.6

24. Series Compensators for Improved Design
RC OP-Amp Circuit for phase lead (or lag) compensator
Lead Compensator for Improved Transient Response; Example: Required to
reduce rise time to half keeping x = 0.5.
Lag Compensator for Improved steady-state performance. Example:

25. Frequency Response Analysis
Response to x(t) = X sin(wt)
G(s) = K/(Ts+1) and G(s)=(s+1/T1)/(s+1/T2) cases
Frequency response graphs- Bode, and Nyquist plots of
Resonant frequency and peak value
Nichols Chart
Nyquist Stability Criterion

26. Control System Design Using Frequency Response Analysis
Lead Compensation
Lag Compensation
Lag-Lead Compensation

27. State Space Analysis
State-Space Representation of a Generic Transfer Function in Canonical Forms:
Controllable Canonical Form
Observable Canonical Form
Diagonal Canonical Form
Jordan Canonical Form
Eigenvalue Analysis

28. Solution of State Equations
Solution of Homogeneous Equations
Interpretation of
Show that the state transition matrix is given by
Properties of
Solution of Nonhomogeneous Equations
Cayley-Hamilton Theorem

29. Controllability and Observability
Definitions of Controllable and Observable Systems
Controllabililty and Obervability Conditions
Principle of Duality

30. Control System Design in State Space
Necessary and Sufficient Condition for Arbitrary Pole Placement
Determination of Feedback Gain Matrix by Ackerman’s formula
Design of Servo Systems

31. Introduction to Sampled Data Control Systems
Z-transform and Inverse Z-transform
Properties of Z-Transform and Comparison with the Corresponding Laplace Transform Properties
Transfer Functions of Discrete Data Systems

32. Analysis of Sampled Data Systems
Input and Output Response of Sampled Data Systems
Differences in the Transient Characteristics of Continuous Data Systems and Corresponding Discrete (Sampled) Data Systems
Root Locus Analysis of Sampled Data Systems