1. 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

2. 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

3. 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

4. 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

5. 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

6. 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

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

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

9. Controller and observer2 1 -1 -2
Lars Imsland Prost årsmøte 2002

10. 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

11. 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

12. 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

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

14. 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

15. 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

16. 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

17. 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

18. 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

19. 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

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