Lars ImslandProst

1

• 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

2
Piecewise affine systems

• Nonlinear, uncertain discrete time model
• Known equilibrium input
• Piecewise affine encapsulation

3
Problem statement

• Find controller that stabilizes the difference
  inclusion
• by output feedback

4
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

5
Output feedback control structure
Process
PA State feedback
PA Observer Output Injection
Observer model

• Nominal model
• or
• Piecewise affine approximation

6
The synthesis inequalities

• LMIs guaranteeing a decreasing Lyapunov function
  everywhere
• LMIs guaranteeing region of attraction and
  conformance to constraints
• Low dimensional BMI

7
Example

• Nonlinear unstable system
• Partial state information (output)
• Uncertain system
• Constrained

8
Nonlinearities
Real nonlinearity p-a encapsulation
Observer nonlinearity p-a approximation
9
Controller and observer
2
1
0
-1
-2
10
Simulation

• State constraints
• Lyapunov level set
• Phase trajectory

11
MPC - prinsipp
Past
Future
Predicted outputs y(tkt)
Manipulated inputs u(tk)
t
t1
tM
tP
Input horizon
Output horizon
Regn ut en optimal pådragsekvens som
minimaliserer reguleringsfeil samtidig som den
tar hensyn til beskrankninger på pådrag og
utganger.
12
Receding horizon
Past
Future

• Optimiser på tidspunkt t (nye målinger)
• Bruk det første optimale pådraget u(t)
• Gjenta optimalisering på tidspunkt t1

t
t1
tM
tP
Fordel med online optimization TILBAKEKOBLING
t1
tM1
tP1
13
NMPC Open Loop Optimal Control Problem

• Solve
• subject to
• with

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?

System
u
y
x
15
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)

16
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

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

...
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 x2M
x1
x1 x2 x3M
19
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

20
Simulations on Olga