Principles of Information Science Chapter 8 Information execution Control Theory
Principles of Information Science Chapter 8 Information Execution Control Theory
1 Fundamentals of Control Theory N. Wiener: Control theory in engineering whether it is Concerned with Man, animal or machine, can only be regarded as a part of the theory of information. I(G) Controller (O,E) F(S) Obiect
§1 Fundamentals of Control Theory Controller Object N. Wiener: Control theory in engineering, whether it is Concerned with Man, animal or machine, can only be regarded as a part of the theory of information. I(O,E) F(S(I)) I(G)
82.1 Description of Controlled Object Object Description: The states, The Ways tI S3 P(11)…P(14) ●● MT P(41)…P(44)
s1 §2.1 Description of Controlled Object Object Description: The states, The Ways s1 s2 s4 s3 P(11) … P(14) … … P(41) … P(44) s1 … sN t1 . . tM . T x1 xM yN y1
82.2 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 8 problem. The path connecting all the states from s to g is one the possible solutions for the control problem In an ndimensional space the states and the control effect can be described as s={S1,…,sN},g={g1,…,g} 2(gn-g
§2.2 Description of Goal and Effect s g The initial condition, s, of the controlled object and the final condition, g, are two states of the object in the state space of control problem. The path connecting all the states from s to g is 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} = [ (gn – g’n ) ] 2 __ n 2 1/2 ' n g
83 The Mechanism of Control Mechanism of control: from Information to Action Strategy Information Execution Action Object
§3 The Mechanism of Control Mechanism of Control: from Information to Action Strategy Execution Object Information Action
83.1 The Categories of Control Open-Loop Closed-Loop Controlling Execution Object Goal Noise- Controlling Execution Obiect Effect
§3.1 The Categories of Control Open-Loop & Closed-Loop Noise Controlling Execution Object Noise Controlling Execution Object Goal Effect
83.2 Control Strategy Producing Mathematical Programming: Strategy Producing X-ndimensional 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)}
§3.2 Control Strategy Producing 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=5x1+x2 Constraints:(1)x1≥0;(2)x2≥0;(3)x1+x2≤6; (4)3x1+x2s12;(5)x1-x2≤2. Solution: 3) B → Point o: max=136/7 x1=26/7 x2=6/7 (2)
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: (1) (2) (3) (5) (4) O A B C D Point C: x1 = 26/7 x2 = 6/7 f = 136/7 max
Other Approaches Non-Linear programming Integer programming One dimensional search Higher dimensional search Dynamic Programming, etc
Other Approaches Non-Linear Programming Integer Programming One dimensional Search Higher dimensional search Dynamic Programming, etc
§33 The Stability Issue Typical solution- Lyapunoff Criteria If the state equation of a system is: dxn /dt= anmXm, nE(1, N then its eigen-equation is: AA-n=0 (1) If the real part of all roots of the eigen-equation are negative, then the undisturbed motion is always asymptotically stable. (2)If there exists at least one root with positive real part, then the undisturbed motion is always unstable (3)Direct Lyapunove Criterion(see references)
§3.3 The Stability Issue Typical solution – Lyapunoff Criteria If the state equation of a system is: dx /dt = a x , n m (1,N) nm m=1 N n then its eigen-equation is: |A-|=0 (1) If the real part of all roots of the eigen-equation are negative, then the undisturbed motion is always asymptotically stable. (2) If there exists at least one root with positive real part, then the undisturbed motion is always unstable. (3) Direct Lyapunove Criterion (see references)