Lars Imsland Prost årsmøte 2002 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 My name is Lars Imsland, I come from the Norwegian University of Science and Technology, Department of Engineering Cybernetics. I will talk about robust output feedback for a class of nonlinear systems using piecewise affine observers. Lars Imsland Prost årsmøte 2002

Piecewise affine systems Nonlinear, uncertain discrete time model Known equilibrium input Piecewise affine encapsulation The systems we look at are nonlinear, uncertain discrete-time models of this type. The one major assumption we make, is that we now the equilibrium input. We find upper and lower bounds on the uncertain vector field in different subsets of the statespace. We call these subsets local model validity sets. Using these upper and lower bounds, we can formulate a piecewice affine parameter dependent model. Further, the system matrices are affine in the parameters. I(x) tells us which local model validity set is active. We have previously shown how to obtain such models in a fairly systematic manner Note that the new uncertainty parameter theta captures the original uncertainty, in addition to the nonlinearity. Mention constraints Note that the new uncertainty parameter theta takes care of both the original uncertainty and the nonlinearity within the local model validity set. Lars Imsland Prost årsmøte 2002

Lars Imsland Prost årsmøte 2002 Problem statement Find controller that stabilizes the difference inclusion by output feedback Our problem is then to find an output feedback controller that stabilizes this difference inclusion. And in this talk, the output feedback controller will be observer based Lars Imsland Prost årsmøte 2002

Lars Imsland Prost årsmøte 2002 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 That is, different state feedback matrices in the different local model validity sets. But while the formulation of piecewise affine dynamic output feedback was rather straight forward, the... Last point: Meaning finding a feasible solution. Even for very small problems, where we knew solutions in advance, the implemented algorithms did not succeed. This motivated us to consider output feedback by observers in this framework. Lars Imsland Prost årsmøte 2002

Output feedback control structure Process PA State feedback PA Observer Output Injection Observer model The process is as described earlier. The observer injection matrices are here parameterised in j, that is, indicating a partition of the output space - another approach is to parameterise it in the observer state space, as for the state feedback - this will often simplify the procedure. The observer model could be a nominal model of the nonlinear system. Then the observer model should be encapsulated in the same way as the system. However, note that we can choose these functions piecewise affine, with a similar partition of the observer state space, and then the encapsulation is exact. This reduces conservatism of the procedure. Nominal model or Piecewise affine approximation Lars Imsland Prost årsmøte 2002

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

Lars Imsland Prost årsmøte 2002 Example Nonlinear unstable system Partial state information (output) Uncertain system Constrained Lars Imsland Prost årsmøte 2002

Lars Imsland Prost årsmøte 2002 Nonlinearities “Real” nonlinearity p-a encapsulation Observer nonlinearity p-a approximation Lars Imsland Prost årsmøte 2002

Controller and observer 2 1 -1 -2 Lars Imsland Prost årsmøte 2002

Lars Imsland Prost årsmøte 2002 Simulation State “constraints” Lyapunov level set Phase trajectory Note that Ra can be given in the combined state and observer statespace, or the combined state and observer error space. It might seem that the trajectory is not crossing Lyapunov level sets at all times, but note that this is a projection from a 4D total state space. Lars Imsland Prost årsmøte 2002

MPC - prinsipp Past Future Predicted outputs y(t+k|t) Manipulated inputs u(t+k) t t+1 t+M t+P Input horizon Nevn tilstandsestimator 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

Receding horizon Fordel med “online optimization”: TILBAKEKOBLING Past Future Optimiser på tidspunkt t (nye målinger) Bruk det første optimale pådraget u(t) Gjenta optimalisering på tidspunkt t+1 t t+1 t+M t+P Fordel med “online optimization”: TILBAKEKOBLING t+1 t+M+1 t+P+1 Lars Imsland Prost årsmøte 2002

NMPC Open Loop Optimal Control Problem Solve subject to with Ikke vis! Lars Imsland Prost årsmøte 2002

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? u System y x Lars Imsland Prost årsmøte 2002

Lars Imsland Prost årsmøte 2002 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”) Forklar practically stable Bevis bruker high gain observer Lars Imsland Prost årsmøte 2002

Lars Imsland Prost årsmøte 2002 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

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

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

Lars Imsland Prost årsmøte 2002 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

Lars Imsland Prost årsmøte 2002 Simulations on Olga Lars Imsland Prost årsmøte 2002