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# Uncertainty Problem in Control & Decision Theory

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AACIMP 2010 Summer School lecture by Viktor Ivanenko. "Applied Mathematics" stream. "On the Models of Uncertainty in Decision and Control Problems" course. Part 1.

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### Uncertainty Problem in Control & Decision Theory

1. 1. Uncertainty Problem in Control & Decision Theory I. Introduction Control Theory: 1. Dynamics; 2. Uncertainty Decision Theory: 1. Uncertainty; 2. Dynamics Output Controller Plant Control System Consiquences Decision Decision Maker Situation Decision System
2. 2. Two examples DP Example 1. Action (Decision) L H, N S B, N L H, B, G, N R G, N S H, B, G, N R H, B, G, N Student 1: B>H, B>G, H>G Student 2: G>B, G>H, H>B Uncertainty Problem in Control & Decision Theory
3. 3. Two examples DP Example 2. Experiment (Observation) Experiment Observation H E S H C11 C12 C13 E C21 C22 C23 S C31 C32 C33 C11> C12, C11> C13 C22> C21, C22> C23 C33> C31, C33> C32 Example 1 є Class of non-parametric DSituations. Example 2 є Class of parametric DSituations. Uncertainty Problem in Control & Decision Theory
4. 4. II. Mathematic Model of DSituation. U - the set of decisions u U ; ( A) C - the set of consequences c C ( ) - the multivalued mapping :U C, (u) C, u U - the set of parameters g ( , ) - some mapping g :( U) C U , C, ( ) , Л U , C , Cu C , u U ; Л - Lottery Scheme (non-parametric DS) M ( , U , C, g ( , )), g ( , u) C, ,u U М - Matrix Scheme (parametric DS) Uncertainty Problem in Control & Decision Theory
5. 5. Л М !????? 1) 2) М Л, ,U , C , g ( , ) ; Л М ; (U , C , ( )); ( М ) Л (u) g ( , u) : , u U; C : (u) (u), u U : ZМ ( , , , ) ZЛ ( , , , ) g ( , u) (u), , u U М Л : Л М ( Л ) Л T1. Class of DS whose schemes representable in matrix form coinsides with the class of DS whose schemes representable in a lottery form. (V.I. Ivanenko, B. Munier, 2000) (V.I. Ivanenko, V. Mikhalevich, 2007) Uncertainty Problem in Control & Decision Theory
6. 6. Uncertainty, necessary condition ( ) but not sufficient! P in Z М MМ ( Z М , P) Data on the uncertainty Q in Z Л MЛ ( Z Л , Q) Q Qu , u U Strict certainty Strict uncertainty Stochastic uncertainty Uncertainty Problem in Control & Decision Theory
7. 7. For Stochastic Uncertainty: MЛ U , (C , ), u ,u U MЛ MМ MМ ( , , ),U , (C, ), g ( , ) MМ MЛ (M Л ) МЛ T.2. Class of DS whose mathematic models representable in matrix form coinsides with the class of DS whose mathematic models representable in a lottery form. (V. Ivanenko, V. Mikhalevich, 2007) M DS { M , P} M , L( , ) - utility function, L( , u ) R1 C M ( ,U , L, P). P - some regularity of uncertainty Uncertainty Problem in Control & Decision Theory
8. 8. III. Mathematic Model of Decision Maker ~ 1. c C C c - the Binary Relation on C o 2. First Optimization Problem c C ~ 3. u U U 4. Second Optimization Problem uo U №3 is the essence of Decision Making under Uncertainty. Uncertainty Problem in Control & Decision Theory
9. 9. Strict certainty: g :U C; co C 1 g :C U ; co uo U Strict uncertainty: The choice of u is not unique! M - the Set of DS ~ C U - Projector or Criterion Choice Rule (CCR) - the set of all possible projectors ~ ~ C, U, Uncertainty Problem in Control & Decision Theory
10. 10. IV. General Decision Problem Definition. CCR is any mapping ( ) Z define on , , * and associate to any Z some real function LZ ( ) define on U . Class of all CCR denote by o( ) ( ) all CCR’s that satisfied to the next three conditions: C1. If Zi ( ,U i , Li ) ( ) (i 1,2), U1 U 2 , L1 ( , u ) L2 ( , u ) at all u U1 , , then L* (u ) L* (u ), at all u U1. Z1 Z2 C2. If Z ( ,U , L) ( ) u1, u2 U , then from the inequality L( , u1 ) L( , u2 ) * at all , follows LZ (u1 ) L* (u2 ), and from a, b R, a 0, Z * L( , u1 ) a L( , u2 ) b at all , follows Lz (u1 ) a L* (u2 ) b. z C3. If Z ( ), u1 , u 2 , u3 U and L( , u1 ) L( , u2 ) 2 L( , u3 ), then L* (u1 ) L* (u2 ) z z 2 L* (u3 ). Z Uncertainty Problem in Control & Decision Theory
11. 11. E.Borel A.Kolmogorov Random 60 in Broad Sense Random Stochastic Events Events 30 XX Statistically frequency unstable events. 0011000000 111111 000000000000 }f (k) the set of points f(1)=f(0)=1/2 f(1)=1/4 f(0)=3/4 V. Ivanenko, B. Munier, I.Zorich (2000) Uncertainty Problem in Control & Decision Theory
12. 12. PF ( ) { p (2 [0,1]) : p( ) 1, p( A B) p( A) p( B A) A, B } p( ) - the set of all closed (in some topology) subsets p PF ( ) - the regularities of uncertainty. : p( ) ( ) Z if p P( ), Z ( p), Z ( ) ( Z ) L* ( ), then p Z P L* ( ) sup L( , u ) p(d ) Z u U p P T.3. p( ) o( ) ( Z , P) S General DP (V. Ivanenko, V. Labkovsky, 1986,2005) L* ( ) Z (V. Ivanenko, B. Munier, 2000) Uncertainty Problem in Control & Decision Theory