1. Optimal Control Theory
Prof .P.L.H .Vara Prasad
Dept of Instrument Technology
Andhra university college of Engineering

2. Overview of Presentation
What is control system
Darwin theory
Open and closed loops
Stages of Developments of control systems
Mathematical modeling
Stability analysis
Dept of Inst Technology
Andhra university college of Engineering

3. What is a control system ?
A control system is a device or set of devices to manage, command, direct or regulate the behavior of other devices or  systems.
Dept of Inst Technology
Andhra university college of Engineering

4. Dept of Inst Technology Andhra university college of Engineering
Darwin (1805)
Feedback over long time periods
is responsible for the evolution of species.
vito volterra - Balance between two populations of fish( )
Norbert wiener - positive and negative feed back in biology ( )
Dept of Inst Technology
Andhra university college of Engineering

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Open loop & closed loop
“… if every instrument could accomplish its own work, obeying or anticipating the will of others … if the shuttle weaved and the pick touched the lyre without a hand to guide them, chief workmen would not need servants, nor masters slaves.”
Hall (1907) : Law of supply and demand must distrait fluctuations
Any control system Letting is to fluctuate and try to find the dynamics.
Dept of Inst Technology
Andhra university college of Engineering

6. Closed loop Open loop Accuracy depends on calibration. Simple.
Due to feed back
Complex
More stable
Effect of non-linearity can be minimized by selection of proper reference signal and feed back components
Open loop
Accuracy depends on calibration.
Simple.
Less stable.
Presence of non-linearities cause malfunctions

7. Effects of feedback System dynamics Effect of disturbance
normal improved
Time constant
1/a /(a+k)
Effect of disturbance
Direct /g(s)h(s) reduced
Gain is high low gain G/(1+GH)
If GH= -1 , gain = infinity
Selection of GH is more important in finding stable
low Band width high band width

8. Robot using pattern- recognition process

9. Temperature control system

10. Analogous systems

11. Mathematical model of gyro

12. Mathematical modeling of physical systems

13. Stages of Developments of control systems
Dept of Inst Technology
Andhra university college of Engineering

14. Example of 2nd order system

15. optimization
Maximize the profit or to minimize the cost dynamic programming .
Non linear optimal control

16. Nature of response -poles

17. Unit step response of a control system
Dept of Inst Technology
Andhra university college of Engineering

18. Steady state errors for various types of instruments
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Andhra university college of Engineering

19. Dept of Inst Technology Andhra university college of Engineering
For Higher order systems Rouths –Hurwitz stability criterion & its application
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Andhra university college of Engineering

20. Locus of the Roots of Characteristic Equation
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Andhra university college of Engineering

21. Dept of Inst Technology Andhra university college of Engineering
Root Contour
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Andhra university college of Engineering

22. Performance Indices

23. Frequency response characteristics - Polar plots

24. Bode plots

25. Phase & gain margins

26. Nyquist plots Third order system First order system
Second order system

27. Nyquist stability

28. Limitations of Conventional Control Theory
Applicable only to linear time invariant systems.
Single input and single output systems
Don’t apply to the design of optimal control systems
Complex Frequency domain approach
Trial error basis
Not applicable to all types of in puts
Don't include initial conditions

29. State Space Analysis of Control Systems
Definitions of State Systems
Representation of systems
Eigen values of a Matrix
Solutions of Time Invariant System
State Transition Matrix

30. Definitions
State – smallest set of variables that determines the behavior of system
State variables – smallest set of variables that determine the state of the dynamic system
State vector – N state variables forming the components of vector
Sate space – N dimensional space whose axis are state variables

31. State space representation

32. State Space Representation

33. Solutions of Time Invariant System
Solution of Vector Matrix Differential Equation X|= Ax (for Homogenous System) is given by
X(t) = eAt X(0) (1)
Ø(t) = eAt = L -1 [ (sI-A)-1 ] (2)

34. Solutions of Time Invariant System…(Cont’d)
Solution of Vector Matrix Differential Equation X|= Ax+Bu
(for Non- Homogenous System) is given by
X(t) = eAt X(0) + ∫t0 e ^{A(t - T)} * Bu(T) dT

35. Optimal Control Systems Criteria
Selection of Performance Index
Design for Optimal Control within constraints

36. Performance Indices Magnitudes of steady state errors Types of systems
Dynamic error coefficients
Error performance indexes

37. Optimization of Control System
State Equation and Output Equation
Control Vector
Constraints of the Problem
System Parameters
Questions regarding the existence of Optimal control

38. Controllability
A system is Controllable at time t(0) if it is possible by means of an unconstrained control vector to transfer the System from any initial state Xt(0) to any other state in a finite interval of time.
Consider X| = Ax+Bu then system is completely state controllable if the rank of the Matrix
[ B | AB | …….An-1B ] be n.

39. Observability
A system is said to be observable at time t(0) if, with the system in state Xt(0) it is possible to determine the state from the observation of output over a finite interval of time.
Consider X| = Ax+Bu, Y=Cox then system is completely state observable if rank of N * M matrix [C* | A*C* | …… (A*)n-1 C*] is of rank n .

40. Liapunov Stability Analysis
Phase plane analysis and describing function methods – applicable for Non-linear systems
Applicable to first and second order systems
Liapunov Stability Analysis is suitable for Non-linear and|or Time varying State Equations

41. Stability in the Sense of Liapunov
Stable Equilibrium state
Asymptotically Stable
Unstable state

42. Liapunov main stability theorem

43. Thank you