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					  Lars Imsland, ITK, NTNU
                       Veileder: Bjarne Foss


  • Robust (output) feedback of piecewise affine difference inclusions
       • Olav Slupphaug, Bjarne
  • Nonlinear MPC and output feedback: A “separation principle”
       • Rolf Findeisen, Frank Allgøwer, Bjarne
  • Control of a class of positive systems
       • Gisle Otto Eikrem, Bjarne
       • A general result on stabilization
       • Application to oil production: Stabilization of gas-lifted oil wells




Lars Imsland                       Prost årsmøte 2002          2002-06-11       1
                   Piecewise affine systems
•     Nonlinear, uncertain discrete
      time model




•     Known equilibrium input




•     Piecewise affine
      encapsulation




    Lars Imsland                  Prost årsmøte 2002   2002-06-11   2
               Problem statement
Find controller that stabilizes the difference inclusion




by output feedback




Lars Imsland                 Prost årsmøte 2002    2002-06-11   3
                Previous results
We have previously (Slupphaug, Imsland & Foss 2000) stated BMIs
  which upon feasibility gives

• Piecewise affine state feedback



• Piecewise affine dynamic output feedback




• The dynamic output feedback BMIs proved to be very hard
  to solve
Lars Imsland                Prost årsmøte 2002   2002-06-11       4
         Output feedback control
                structure
                  Process
    PA                                         PA
   State                                   Observer
 feedback                                Output Injection
               Observer model




                                           •   Nominal model
                                                    or
                                           •   Piecewise affine
                                               approximation
Lars Imsland        Prost årsmøte 2002    2002-06-11        5
       The synthesis inequalities
• LMIs guaranteeing a decreasing Lyapunov function everywhere




• LMIs guaranteeing region of attraction and conformance to
  constraints

• Low dimensional BMI




 Lars Imsland              Prost årsmøte 2002   2002-06-11    6
                       Example
• Nonlinear unstable system




• Partial state information (output)



• Uncertain system



• Constrained



Lars Imsland                Prost årsmøte 2002   2002-06-11   7
                 Nonlinearities

                                             “Real” nonlinearity
                                             p-a encapsulation




Observer nonlinearity
p-a approximation
Lars Imsland            Prost årsmøte 2002   2002-06-11    8
                    Controller and observer
2




1




0




-1




-2
     Lars Imsland          Prost årsmøte 2002   2002-06-11   9
                        Simulation
• State “constraints”

• Lyapunov level set

•

• Phase trajectory




    Lars Imsland           Prost årsmøte 2002   2002-06-11   10
                        MPC - prinsipp
             Past       Future



                                            Predicted outputs y(t+k|t)



                                   Manipulated
                                   inputs u(t+k)
                    t    t+1                     t+M               t+P
                           Input horizon

                                    Output horizon


Regn ut en optimal pådragsekvens som minimaliserer reguleringsfeil
samtidig som den tar hensyn til beskrankninger på pådrag og utganger.
   Lars Imsland                    Prost årsmøte 2002     2002-06-11     11
                     Receding horizon
 Past       Future

                                              •   Optimiser på tidspunkt t
                                                  (nye målinger)
                                              •   Bruk det første optimale
                                                  pådraget u(t)
                                              •   Gjenta optimalisering på
        t    t+1      t+M          t+P            tidspunkt t+1




                                                       Fordel med
                                                  “online optimization”:
                                                    TILBAKEKOBLING


             t+1       t+M+1          t+P+1

Lars Imsland                Prost årsmøte 2002        2002-06-11    12
       NMPC Open Loop Optimal
          Control Problem
Solve

subject to




with




Lars Imsland    Prost årsmøte 2002   2002-06-11   13
      The output feedback problem
• Problem: State information needed for prediction
• Often only output measurements available       u                             y
        – need to estimate system states                         System
                                                                      x


•      Many different observers for nonlinear systems
        – EKF, geometric, passivity based, extended Luenberger,
          optimization based, MHE…


• Questions:
        – How to guarantee stability of closed-loop with observer?
        – Which observer does facilitate solution?




    Lars Imsland                  Prost årsmøte 2002    2002-06-11        14
                 We have shown:
• For fast enough observer, short enough sampling time
    –   Closed loop is “practically” stable
    –   (Convergence to 0 under stronger conditions)
    –   Recover state feedback region of attraction
    –   Output feedback trajectories approach state feedback
        trajectories


• Results hold for general nonlinear system with required
  observability conditions (“uniform observability”)




Lars Imsland                   Prost årsmøte 2002   2002-06-11   15
               Gas-lifted oil wells
                 • Can have unstable production
                 • Instability caused by mechanisms related to
                   mass
                     – compressibility of gas
                     – gravity dominated flow
                 • Simple model based on mass balances
                   reproduce dynamic behavior
                 • Stabilization by simple controller based on
                   physical properties




Lars Imsland                 Prost årsmøte 2002   2002-06-11     16
     A class of positive systems
• Each state is measure of “mass” in a compartment - positive
• Dynamics (typically: mass balances) are
    – flow between compartments
    – external inflow to compartments
    – outflow from compartments
• Compartments can be divided into phases
• Each phase has one input
    – input either inflow or outflow to that phase
    – input has saturation
• Controllability assumptions
                                                              ...



Lars Imsland                   Prost årsmøte 2002    2002-06-11     17
           State feedback controller
•     Control objective: Stabilize total mass of each phase
•     Often: Equivalent to stabilization of an equilibrium
•     Controller: linearize “total mass dynamics” of each phase
•     Robustness properties

x2



                   x1+ x2=M*




                        x1     x1+ x2 +x3=M*


    Lars Imsland                   Prost årsmøte 2002   2002-06-11   18
                        Gas-lift
               • Control production choke and gas injection
                 choke to stabilize total mass of oil and gas
               • Stable total mass implies stable well production
               • Tuning knobs: setpoint for mass of oil and gas,
                 speed of controller
               • Steady state mass of oil decides well
                 performance (oil production)
                   – to a certain extent


               • Alternative: use only production choke
                   – Also obtains stability
                   – Less flexibility



Lars Imsland                Prost årsmøte 2002   2002-06-11   19
               Simulations on Olga




Lars Imsland          Prost årsmøte 2002   2002-06-11   20

				
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