optimal control theory
DESCRIPTION
Optimal Control Theory. Prof .P.L.H .Vara Prasad. Dept of Instrument Technology Andhra university college of Engineering. Overview of Presentation. What is control system Darwin theory Open and closed loops Stages of Developments of control systems Mathematical modeling - PowerPoint PPT PresentationTRANSCRIPT
Optimal Control Theory
Dept of Instrument TechnologyAndhra university college of Engineering
Prof .P.L.H .Vara Prasad
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 TechnologyAndhra university college of Engineering
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 TechnologyAndhra university college of Engineering
Darwin (1805)Feedback over long time periodsis responsible for the evolution of species.
Dept of Inst TechnologyAndhra university college of Engineering
vito volterra - Balance between two populations of fish(1860-1940)
Norbert wiener - positive and negative feed back in biology (1885-1964)
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 TechnologyAndhra university college of Engineering
Open loop Accuracy depends
on calibration. Simple. Less stable. Presence of non-
linearities cause malfunctions
Open loop Accuracy depends
on calibration. Simple. Less stable. Presence of non-
linearities cause malfunctions
Closed loop
Due to feed back
Complex
More stable
Effect of non-linearity can be minimized by selection of proper reference signal and feed back components
Closed loop
Due to feed back
Complex
More stable
Effect of non-linearity can be minimized by selection of proper reference signal and feed back components
Effects of feedback
System dynamics normal improved Time constant 1/a 1/(a+k) Effect of disturbance
Direct -1/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
Robot using pattern- recognition process
Temperature control system
Analogous systems
Mathematical model of gyro
Mathematical modeling of physical systems
Stages of Developments of control systems
Dept of Inst TechnologyAndhra university college of Engineering
Example of 2nd order system
optimization
Maximize the profit or to minimize the cost dynamic programming .
Non linear optimal control
Nature of response -poles
Unit step response of a control system
Dept of Inst TechnologyAndhra university college of Engineering
Steady state errors for various types of instruments
Dept of Inst TechnologyAndhra university college of Engineering
For Higher order systems Rouths –Hurwitz stability criterion & its application
Dept of Inst TechnologyAndhra university college of Engineering
Locus of the Roots of Characteristic Equation
Dept of Inst TechnologyAndhra university college of Engineering
Root Contour
Dept of Inst TechnologyAndhra university college of Engineering
Performance Indices
Frequency response characteristics- Polar plots
Bode plots
Phase & gain margins
Nyquist plots
First order system Second order system Third order system
Nyquist stability
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 basisNot applicable to all types of in putsDon't include initial conditions
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
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
State space representation
State Space Representation
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)
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
Optimal Control Systems Criteria
Selection of Performance Index Design for Optimal Control within
constraints
Performance Indices
Magnitudes of steady state errors Types of systems Dynamic error coefficients Error performance indexes
Optimization of Control System State Equation and Output Equation Control Vector Constraints of the Problem System Parameters Questions regarding the existence of
Optimal control
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.
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 .
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
Stability in the Sense of Liapunov
Stable Equilibrium state Asymptotically Stable Unstable state
Liapunov main stability theorem
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