H0K03a : Advanced Process Control Model-based Predictive Control 3 : - - PowerPoint PPT Presentation

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H0K03a : Advanced Process Control Model-based Predictive Control 3 : - - PowerPoint PPT Presentation

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H0K03a : Advanced Process Control Model-based Predictive Control 3 : Stability Bert Pluymers Prof. Bart De Moor Katholieke Universiteit Leuven, Belgium Faculty of Engineering Sciences Department of Electrical Engineering (ESAT) Research Group

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Bert Pluymers

  • Prof. Bart De Moor

Katholieke Universiteit Leuven, Belgium Faculty of Engineering Sciences Department of Electrical Engineering (ESAT) Research Group SCD-SISTA

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

H0K03a : Advanced Process Control

Model-based Predictive Control 3 : Stability

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1

Overview

  • Introduction
  • Example
  • Stability Theory
  • Set Invariance
  • Implementations

Lecture 3 : Stability

  • Introduction
  • Example
  • Stability Theory
  • Set Invariance
  • Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

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2

MPC Paradigm

  • Introduction
  • Example
  • Stability Theory
  • Set Invariance
  • Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

  • At every discrete time instant , given information about

the current system state , calculate an ‘optimal’ input sequence over a finite time horizon :

  • Apply the first input to the real system
  • Repeat at the next time instant , using new state

measurements / estimates.

N N

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3

Optimality of input sequence

  • Introduction
  • Example
  • Stability Theory
  • Set Invariance
  • Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

computed at to be applied at

Optimal input sequences Input sequence applied to the system

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4

Stability Analysis

  • Introduction
  • Example
  • Stability Theory
  • Set Invariance
  • Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

  • Classical Way :

Analyse poles/zeros of and associated transfer functions.

+ -

plant linear controller u x r

+ -

plant MPC controller u x r

  • Modelbased Predictive Control :

Lyapunov theory for stability.

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5

Inverted Pendulum

  • Introduction
  • Example
  • Stability Theory
  • Set Invariance
  • Implementations
  • 1 input :
  • 4 states :
  • open loop unstable system

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

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6

Inverted Pendulum

  • Introduction
  • Example
  • Stability Theory
  • Set Invariance
  • Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

  • 4 different horizon lengths
  • 3 different MPC variants (to be defined later)

Non-minimum phase behaviour

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7

Stability Theory

  • Introduction
  • Example
  • Stability Theory
  • Set Invariance
  • Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

  • Explicit vs. Optimization-based controller
  • Transfer functions → Lyapunov theory

Stability is obtained / proven in 2 steps :

  • 1. Recursive feasibility

i.e. controller well-defined for all k

  • 2. Lyapunov function construction

i.e. trajectories converge to equilibrium

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8

Stability Theory

  • Introduction
  • Example
  • Stability Theory
  • Set Invariance
  • Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

Limited validity of MPC stability framework :

  • only for ‘stabilization’ problems :
  • initial state
  • system steered towards
  • no disturbances allowed (but extension possible)
  • no general stability framework for ‘tracking’ problems
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9

Stability Theory

  • Introduction
  • Example
  • Stability Theory
  • Set Invariance
  • Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

Recursive Feasibility If the optimization problem is feasible for time , then it is also feasible for time . (and hence for all ) Feasible Region The region in state space, defined by all states for which the MPC optimization problem is feasible. → Recursive feasibility proven : all states within feasible region lead to trajectories for which the MPC-controller is feasible and hence well-defined.

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10

Stability Theory

  • Introduction
  • Example
  • Stability Theory
  • Set Invariance
  • Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

→ Recursive feasibility proven : all states within feasible region lead to trajectories for which the MPC-controller is feasible and hence well-defined. feasible region

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11

MPC Stability Measures

  • Introduction
  • Example
  • Stability Theory
  • Set Invariance
  • Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

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MPC Stability Measures

  • Introduction
  • Example
  • Stability Theory
  • Set Invariance
  • Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

recursive feasibility (terminal constraint) Lyapunov stability (terminal cost)

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Step 1 : Recursive Feasibility

  • Introduction
  • Example
  • Stability Theory
  • Set Invariance
  • Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

Problem : given the optimal (and hence feasible) solution to the optimization at time , construct a feasible solution for the optimization at time .

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14

Step 1 : Recursive Feasibility

  • Introduction
  • Example
  • Stability Theory
  • Set Invariance
  • Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

Given : To be found : Observe / Choose :

?

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15

Step 1 : Recursive Feasibility

  • Introduction
  • Example
  • Stability Theory
  • Set Invariance
  • Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

Plant state at time predicted at time : Real plant state at time : Assumption : No plant model mismatch, i.e. Hence, reusing the overlapping part of the input sequence will also result in an identical state sequence

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16

Step 1 : Recursive Feasibility

  • Introduction
  • Example
  • Stability Theory
  • Set Invariance
  • Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

OK OK OK OK OK ??? ??? ???

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17

Step 1 : Recursive Feasibility

  • Introduction
  • Example
  • Stability Theory
  • Set Invariance
  • Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

Condition 1 : Satisfied if Condition 2 : Condition 3 : How to choose and ?

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18

Step 1 : Recursive Feasibility

  • Introduction
  • Example
  • Stability Theory
  • Set Invariance
  • Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

Assume we know a locally stabilizing controller : i.e. such that is locally stable. How to choose and ? Then choose

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19

Step 1 : Recursive Feasibility

  • Introduction
  • Example
  • Stability Theory
  • Set Invariance
  • Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

OK OK OK OK OK ??? OK ???

Condition 2 : Condition 3 :

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Step 1 : Recursive Feasibility

  • Introduction
  • Example
  • Stability Theory
  • Set Invariance
  • Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

Condition 2 : Condition 3 : Since we know that … Condition 2 is satisfied if Condition 3 is satisfied if

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Step 1 : Recursive Feasibility

  • Introduction
  • Example
  • Stability Theory
  • Set Invariance
  • Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

Summary & Interpretation

Recursive feasibility is guaranteed if 1) 2) 3)

Terminal constraint is feasible w.r.t state constraints Terminal constraint is feasible w.r.t input constraints Terminal constraint is a positive invariant set w.r.t

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Step 2 : Lyapunov stability

  • Introduction
  • Example
  • Stability Theory
  • Set Invariance
  • Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

  • Lyapunov stability :

If such that for some region around 0 then all trajectories starting within asymptotically evolve towards 0.

  • In the MPC context :
  • is chosen as the feasible region,
  • is chosen as the optimal cost value of the MPC
  • ptimization problem for the given

Under which conditions is a Lyapunov function ???

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Step 2 : Lyapunov stability

  • Introduction
  • Example
  • Stability Theory
  • Set Invariance
  • Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

Under which conditions is a Lyapunov function ??? We have to prove that Or in other words that This is satisfied if

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Step 2 : Lyapunov stability

  • Introduction
  • Example
  • Stability Theory
  • Set Invariance
  • Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

Special relationship between the two cost expressions : should be < 0 Satisfied if Condition 4 :

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25 Lyapunov inequality i.e. should ‘overbound’ cost of terminal controller

  • Introduction
  • Example
  • Stability Theory
  • Set Invariance
  • Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

Recursive feasibility and asymptotic stability is guaranteed if 1) 2) 3) 4)

Terminal constraint is feasible w.r.t state constraints Terminal constraint is feasible w.r.t input constraints Terminal constraint is a positive invariant set w.r.t

Summary & Interpretation

Iff the optimization problem is feasible at time !!!!!

  • conditions are sufficient, but not necessary
  • is only used implicitly !
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26

Set Invariance

  • Introduction
  • Example
  • Stability Theory
  • Set Invariance
  • Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

“Given an autonomous dynamical system, then a set is (positive) invariant if it is guaranteed that if the current state lies within , all future states will also lie within .”

  • 1.5
  • 1
  • 0.5

0.5 1 1.5

  • 1.5
  • 1
  • 0.5

0.5 1 1.5

  • 1.5
  • 1
  • 0.5

0.5 1 1.5

  • 1.5
  • 1
  • 0.5

0.5 1 1.5

not invariant invariant

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  • Useful tool for analysis of controllers for constrained systems
  • Example :

– linear system – linear controller

Set Invariance

  • Introduction
  • Example
  • Stability Theory
  • Set Invariance
  • Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

– state constraints

  • 1.5
  • 1
  • 0.5

0.5 1 1.5

  • 1.5
  • 1
  • 0.5

0.5 1 1.5

‘feasible region’ of closed loop system

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Consider an autonomous time-invariant system as defined previously A set is …

Set Invariance

  • Introduction
  • Example
  • Stability Theory
  • Set Invariance
  • Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

… feasible iff Problem : Given an autonomous dynamical system subject to state constraints, find the feasible invariant set of maximal size. … invariant iff

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Given an LTI system subject to linear constraints then the largest size feasible invariant set can be found as with a finite integer.

  • Introduction
  • Example
  • Stability Theory
  • Set Invariance
  • Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

Given an LTI system subject to linear constraints then the largest size feasible invariant set can be found as with a finite integer.

  • is constructed by simple forward prediction
  • can be proven to be the largest feasible invariant set
  • is called the Maximal Admissible Set (MAS)
  • is constructed by simple forward prediction
  • can be proven to be the largest feasible invariant set
  • is called the Maximal Admissible Set (MAS)

Invariant sets for LTI systems

(Gilbert et al.,1991, IEEE TAC)

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30

  • Introduction
  • Example
  • Stability Theory
  • Set Invariance
  • Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

Implementations

How to choose such that conditions are satisfied ? Different possibilities, depending of

  • type of system (linear, non-linear)
  • stability of the system
  • presence of state constraints
  • horizon length
  • time constraints during design !
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31

  • Introduction
  • Example
  • Stability Theory
  • Set Invariance
  • Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

Implementations

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32

  • Introduction
  • Example
  • Stability Theory
  • Set Invariance
  • Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

Implementations

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33

  • Introduction
  • Example
  • Stability Theory
  • Set Invariance
  • Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

Implementations

2 1

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34

  • Introduction
  • Example
  • Stability Theory
  • Set Invariance
  • Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

Implementations

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35

  • Introduction
  • Example
  • Stability Theory
  • Set Invariance
  • Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

Implementations

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36

  • Introduction
  • Example
  • Stability Theory
  • Set Invariance
  • Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

Implementations

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  • Introduction
  • Example
  • Stability Theory
  • Set Invariance
  • Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

Implementations

Terminal constraint set determines feasible region

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38

  • Introduction
  • Example
  • Stability Theory
  • Set Invariance
  • Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

Implementations

Solid : standard MPC, dashed : terminal cost, constraint, dotted : terminal equality constr.

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39

  • Introduction
  • Example
  • Stability Theory
  • Set Invariance
  • Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

Conclusion

  • stability of standard MPC not guaranteed
  • pole/zero analysis impossible
  • recursive feasibility
  • Lyapunov stability
  • general stability framework for stabilization problems
  • different implementations
  • stability measures allow the use of shorter horizons
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40

  • Introduction
  • Example
  • Stability Theory
  • Set Invariance
  • Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

References