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Principles of Information Science

Chapter 8

Principles of Information Execution: Control Theory

1, Classical Theory of Control

Simplest Model of Control System

N. Wiener: Control theory in engineering, whether it is concerned with man, animal or machine, can only be regarded as a part of information science.

I(G) Controller I(O,N) Object Noise I(O, N, G, K)

Description of Controlled Object

Object Description: The states, The Ways of states vary

s1 … t1 . . . tM x1 xM

sN

s1 s1

s2

s4 y1 yN

s3

T

P(11) … P(14) … … P(41) … P(44)

Description of Goal and Effect

The initial condition, s, of the controlled object and the final condition, g, are s two states of the object in the state space of control problem. The path connecting all the states from s to g is the one the possible solutions for the control problem. In an N dimensional space, the states and the control effect can be described as s={s1, …, sN}, g={g1, …, gN} __ 2 ε = [ ∑ (gn – g’n)2 ]1/2

n

g

The Mechanism of Control

Mechanism of Control: from Information to Action

Strategy Information Execution Action Object

The Categories of Control

Open-Loop

Noise

Controlling

Execution

Object

Closed-Loop Goal Noise Controlling Execution Object

Effect

Deterministic Strategy to Control Strategy

Mathematical Programming: Strategy Producing

X – N dimensional column vector f(X) – dependence relationship between goal and the system states g(X) – environment constraints of the system

The optimum control strategy can be produced through the maximizing (minimizing) the goal function under the given constraints: Max(Min) f(X)

{g(X)}

An Example: Linear Programming

Goal function: f = 5 x1 + x2 Constraints: (1) x1 ≥ 0; (2) x2 ≥ 0; (3) x1 + x2 ≤ 6; (4) 3 x1 + x2 ≤ 12; (5) x1 – x2 ≤ 2. Solution: x2 A (2) (3) B (4) (5) D C x1

f

Fmax = 136/7

Point C: x1 = 26/7 x2 = 6/7

O (1)

Other Approaches

Non-Linear Programming Integer Programming One dimensional Search Higher dimensional search Dynamic Programming, etc.

2, Intelligent Control

Intelligent Approach to Control Strategy

Goal

Comm. Cog. + DM Comm.

Sensing

Object Noise

Controller

Adaptation & Learning: Concepts

Terminologies & Definitions E—environment; P—performance; S—system; T– time span. E changes while P stays in a prescribed range, S is then stable. S changes while P stays in a prescribed range, S is reliable. E & structure of S changed while P is still satisfactory, S is said adaptive. E changed and P meets requirements after T, S is said a learning system. Structure changed yet P meets requirements after T, S is said a self-repaired system.

Adaptation: Model

It is request that X(t) properly responds to an unknown V(t). V(t) Adaptive Algorithm W(t) V(t) Variable Controller u(t) Object X(t) P(t) Performance Judgment

X(t) is compared to V(t) and producing P(t), then controller is adjusted based on W(t), which is the output of adaptive algorithm driven by P(t).

Information Threshold

The goal of control: H(X|Y) = 0 From I(X;Y) = H(X) – H(X|Y) we have X H(X|Y) = H(X) – [H(Y) – H(Y|X)] = H(Y|X) + H(X) – H(Y) Object Controller Y

N Since H(Y|X) ≥ 0, in order to make H(X|Y) = 0, there must be H(Y) ≥ H(X) = H(N)

Information Criterion for Control Strategy

Assume that among M possible states in search space, the present state is Xmo, and goal state is Xmg. If I(ζm(k+1) ) > I(ζmk) > … > I(ζmo ), search continues; ζ ζ ζ Otherwise, search stops. The optimum solution can be obtained by the calculation: I(ζmk0 ) = Max I(ζmk ) ζ ζ

k

Partial Summary

Processing Ch.6

Conversions-1 Ch. 7-1

Conversions-2 Ch. 7-2

Transferring Ch.5

Ch.9, 10

Transferring Ch.5

Acquisition Ch.4

The World Ch. 1, 2, 3

Execution Ch. 8

Principles of Information Execution

1. The essence of information execution is to converse strategic information into corresponding action in practice. 2. The strategy generation is feasible in most cases if the Goal is rational and the knowledge for problem-solving are necessary and sufficient. 3. As the product of information, strategy is theoretically executable as the strategy clearly tells what states the system should stay and how to move the current state to the one desired. 4. The strategy is practically executable via the mapping between strategy to action – the information-to-energy conversion. 5. Whether the control system can successfully perform its function under disturbance depends on the system’s stability, adaptability, knowledge availability and the criterion of Information Threshold maintainability.

Exercises

1. Please give your own explanation on the mechanism for information-action conversion. 2. Try to prove the “Information Threshold Criterion” in the designing of control systems. 3. Do you agree the saying that control theory is a part of information science? Why and why not?

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