input-to-state stability: a unifying framework for robust mpc eduardo camacho... · microsoft...

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Input-to-state stability:a unifying framework

for robust MPC

Daniel LimonT. Alamo, D.M. Raimondo, D. Muñoz de la

PeñaJ.M. Bravo and E.F. Camacho

Dpto. Ingeniería de Sistemas y AutomáticaEscuela Superior de Ingenieros

Universidad de Sevilla2D. Limon. Keynote: ISS: an unifying framework for robust MPC. NMPC’08 Pavia (Italy)

Outline

Problem statement

Input-to-state stability

Nominal Model Predictive Control

Robust Model Predictive Control

Min-max Model Predictive Control

Conclusions

3D. Limon. Keynote: ISS: an unifying framework for robust MPC. NMPC’08 Pavia (Italy)

Problem statement


Problem statement


Problem statement


Problem statement

2


Outline

Problem statement

Input-to-state stabilityWhy ISS? A gentle motivation

Input-to-state practical stability (ISpS)




Conclusions


Input-to-state stability (ISS)


A gentle motivation for ISS


Some definitions

Function of class K (K-function)

We say that γ : ℜ+→ℜ+ is a function of class K if it constinuously strictly increasing and γ(0)=0.

If it is unbounded then it is of class K∞

Function of class KL (KL-function)

We say that β : ℜ+x ℜ+→ℜ+ is a function of

class if KL is of class for each fixed t and for each fixed s decreases in t with limt→∞β(s,t)=0



Closely related with the Ultimately bounded notion



3







Example (derived from Kellet’02)

1 2 3 4

1

2

3

0 2 4 6 8 10

0

1

2

3

4

5







Example

1 2 3 4

1

2

3

0 2 4 6 8 100

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

4


Input-to-state practical stability


Input-to-state practical stability


Outline

Problem statement



Stabilizing MPC

ISS of MPC



Conclusions




Stabilizing nominal MPC


ISS of nominal MPC

5


ISS of nominal MPC


Outline

Problem statement




A Practical formulation of the problem

Robust constraint satisfaction based on reachable sets

ISS of robust MPC


Conclusions


OptimalConservativePerformance

HighTractableComplexity

Control lawsControl actions Decision variables

Closed-LoopOpen-Loop


Robust predictive controllers:Uncertainty is taken into account in the design







6








Illustrative example

-1.2 -1.1 -1 -0.9 -0.80.8

0.9

1

1.1

1.2

-0.8 -0.75 -0.7 -0.65 -0.6 -0.55

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

Interval arithmeticsZonotope inclusion DC-programming


-1.5 -1 -0.5 0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.5


Sequence of reachable sets

Zonotope inclusion DC-programming


ISS of robust MPC

Robust constraintsatisfaction

Nominalprediction

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ISS of robust MPC


ISS of robust MPC


Outline

Problem statement





Practical stability of min-max MPC

Min-max formulation for input-to-state stability guarantee

Conclusions


Min-max model predictive control


Min-max model predictive control

Robust constraintsatisfaction

Worst-case scenario



8




Min-max MPC with ISS guarantee



(Raimondo’08)





UncertaintiesPosition

SpeedControl input


Conclusions

Input-to-state stability as a suitable framework for robuststability

Well-established theoretical frameworkLyapunov-like conditionsInteresting properties: stability margins, small-gain resultsFits MPC stability results

Uniform continuity plays a relevant role in ISS

Sufficient conditions for MPC with ISS guaranteedNominal MPCRobust MPC with nominal predictionsMin-max MPC

Robust constraint satisfaction by means of guaranteed

range estimators

input-to-state stability: a unifying framework for robust mpc eduardo camacho... · microsoft...

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