Lars ImslandProst årsmøte 20022002-06-111 Lars Imsland, ITK, NTNU Veileder: Bjarne Foss Robust (output) feedback of piecewise affine difference inclusions.

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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

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Piecewise affine systems

• Nonlinear, uncertain discrete time model

• Known equilibrium input

• Piecewise affine encapsulation

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Problem statementFind controller that stabilizes the difference inclusion

by output feedback

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Previous resultsWe 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

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Output feedback control structure

Process

Observer model

PAObserver

Output Injection

PAState

feedback

• Nominal model

or

• Piecewise affine approximation

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The synthesis inequalities• LMIs guaranteeing a decreasing Lyapunov function everywhere

• LMIs guaranteeing region of attraction and conformance to constraints

• Low dimensional BMI

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Example• Nonlinear unstable system

• Partial state information (output)

• Uncertain system

• Constrained

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Nonlinearities

“Real” nonlinearityp-a encapsulation

Observer nonlinearityp-a approximation

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Controller and observer

0

-1

-2

1

2

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Simulation

• State “constraints”

• Lyapunov level set

•

• Phase trajectory

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MPC - prinsippPast Future

t t+1 t+M t+PInput horizon

Output horizon

Predicted outputs y(t+k|t)

Manipulated inputs u(t+k)

Regn ut en optimal pådragsekvens som minimaliserer reguleringsfeil samtidig som den tar hensyn til beskrankninger på pådrag og utganger.

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• Optimiser på tidspunkt t (nye målinger)

• Bruk det første optimale pådraget u(t)

• Gjenta optimalisering på tidspunkt t+1t t+1 t+M t+P

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

Past Future

Receding horizon

Fordel med “online optimization”:

TILBAKEKOBLING

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NMPC Open Loop Optimal Control Problem

Solve

subject to

with

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The output feedback problem• Problem: State information needed for prediction• Often only output measurements available

– need to estimate system states

• 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?

Systemu y

x

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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”)

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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

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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

...

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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

x1

x2

x1+ x2=M*

x1+ x2 +x3=M*

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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

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Simulations on Olga