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NRM 19 Planning the management: the policy

NRM 19 Planning the management: the policy. Andrea Castelletti. The control law. u t = m t (  t ). u t  M t (  t ). DM. Information available for t > 0. t.  t = ( I 0 ,….., I t , x 0 ,….., x t , u 0 ,….., u t - 1 ). Point-valued control law (PV).

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NRM 19 Planning the management: the policy

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  1. NRM 19Planning the management:the policy Andrea Castelletti

  2. The control law ut = mt( t ) ut Mt(t ) DM Information available for t > 0 t t = (I0,….., It, x0,….., xt ,u0 ,…..,ut - 1) Point-valued control law (PV) Set-valued control law (SP): Provides a set of control law producing the same performance.

  3. Set-valued control law A set-valued control law provides a set of controls producing the same performance ut Mt (t ) DM Mt (t ) = {mt( t )} It can be seen as a set of PV control laws

  4. Graphical representation Mt(t) ut set of controls proposed PV control law mt(•) Mt(•) SV control law mt(t) control t

  5. PV control policy Sequence of PV control laws over the project horizon p ={mt(•); t = 0, 1,…, h} If mt (•) = mt + k T(•) the policy is periodic of period T. SV control policy Sequence of SV control laws over the project horizon P ={ Mt(•); t = 0, 1, …, h} If Mt (•) = Mt + k T(•) the policy is periodic of period T.

  6. Information and state The argument of the policy is the information t = ( I0,….., It, x0,….., xt , u0 ,….., ut-1 ) However, the same performance could be obtained with less information. Sufficient statistics When: - the state is measurable - no deterministic disturbances Sufficient statistic =xt By properly redefining the state is always possible to fall back to this case

  7. The policy of the Piave system B5 B1 B4 B3 S2 C3 I2 S1 C2 B2 T2 B6 A2 C4 I1 C1 A1 S3 T1 I3 B7 B8 C5 C6 B9 T3 I4 C7 U1 I5 T4 U2 T5 U3 I6 B10 I7 C8 T6 T7 U4 I8 T8 U5 A8 U6 P

  8. x t2 The policy as a look-up table CONTROL LAW with 2 state variables CONTROL LAW with 1 state variable xt xt POLICY with 1 state variable t t mt(xt) CONTROL POLICY with 2 state variables

  9. The policy as a look-up table x3t+1 t+1 xt2 POLICY for a system with 3 state variables CONTROL LAW for a system with 3 state variables xt1 Hypercube xt3 t

  10. The PV design problem with Laplace

  11. The PV design problem with Laplaceproperties of the solution Without recursive decisions Pure planning problem Without planning decisions Control problem The two problems can be separated …

  12. The PV design problem with Laplaceproperties of the solution A planning problem ... Fix a value for up Fix a value for up Algorithms in Lec. 18 ... And evolve in the space of up ... which includes a Control Problem Compute J*(up) each time J*(up) is computed Compute J*(up) Optimal control algorithms

  13. Readings IPWRM.Theory Ch. 10 IPWRM.Practice Ch. 7

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