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 Stability analysis Dept of Inst Technology Andhra 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 Technology Andhra university college of Engineering

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(1860-1940) Norbert wiener - positive and negative feed back in biology (1885-1964) Dept of Inst Technology Andhra university college of Engineering

Dept of Inst Technology Andhra university college of Engineering 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

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

Effects of feedback System dynamics Effect of disturbance 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 Technology Andhra 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 Technology Andhra university college of Engineering

Steady state errors for various types of instruments Dept of Inst Technology Andhra university college of Engineering

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

Locus of the Roots of Characteristic Equation Dept of Inst Technology Andhra university college of Engineering

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

Performance Indices

Frequency response characteristics - Polar plots

Bode plots

Phase & gain margins

Nyquist plots Third order system First order system Second 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 basis Not applicable to all types of in puts Don'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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