[PPT] - Uncertain systems and robust control LMI methods Dimitri PEAUCELLE PowerPoint Presentation

SLIDE 1

Uncertain systems and robust control LMI methods

Dimitri PEAUCELLE Relaxation Approaches for Control of Uncertain Complex Systems: Methodologies and Tools Workshop at 52nd IEEE Conference on Decision and Control Monday December 9, 2013, Florence

SLIDE 2

Introduction ■ Objectives of this presentation:

Recall some existing results in robust control
Demonstrate how to test these with RoMulOC toolbox

http://projects.laas.fr/OLOCEP/romuloc/ ■ Underlying point of view:

Few techniques
Many results
Depend on modeling choices
D. Peaucelle

1 Florence, December 2013

SLIDE 3

Outline

Two classes of uncertain systems : polytopic & LFT

▲ Airplane example ▲ DEMETER satellite example ▲ Prospectives: descriptor uncertain modeling

Robust analysis

▲ Stability and performances - the well-posedness point of view ▲ System augmentation approach for sequences of SOS-like relaxations ▲ "Slack variable" results and "quadratic stability" as a special case

State-feedback design: multi-performance

▲ Based on dual system ▲ Almost LMI results in "slack variable" approach

D. Peaucelle

2 Florence, December 2013

SLIDE 4

Modeling of systems with uncertainties ■ Aircraft example

Complicated non-linear model - linearized around operation point (Vo, ... )

▲ 9 uncertain parameters:

(Not precisely known parameters such as inertia etc. & uncertainties on operating point)

˙ x =        1 Lp Lβ Lr g/(Vo + v) Yβ −1 N ˙

βg/(Vo + v)

Np Nβ + N ˙

βYβ

Nr − N ˙

β

       x + Bu ▲ Uncertainties given in intervals: Lp ≤ Lp ≤ Lp, Lβ ≤ Lβ ≤ Lβ ...

D. Peaucelle

3 Florence, December 2013

SLIDE 5

Modeling of systems with uncertainties ˙ x =        1 Lp Lβ Lr g/(Vo + v) Yβ −1 N ˙

βg/(Vo + v)

Np Nβ + N ˙

βYβ

Nr − N ˙

β

       x + Bu

Affine-dependent representation with bounds given by interval analysis

˙ x =        1 α22 α23 α23 α31 α33 −1 α41 α42 α43 α44        x + Bu Lp ≤ α22 ≤ Lp . . . N ˙

βg/(Vo + v) ≤ α41 ≤ N ˙ βg/(Vo + v) . . .

▲ Includes the original uncertain model but coupling between coefficients is lost ▲ Conservative: if a property is proved for polytopic model, it also holds for original one

D. Peaucelle

4 Florence, December 2013

SLIDE 6

Modeling of systems with uncertainties ˙ x =        1 α22 α23 α23 α31 α33 −1 α41 α42 α43 α44        x + Bu αij ≤ αij ≤ αij

This is an interval model: all coefficients are independent and in intervals
Sub-class of parallelotopic models (centered at A0 with deviations along axes A1, A2 etc.)

A(β) = A0 + β1A1 + β2A2 + . . . , βi ∈ [−1 1]

Sub class of polytopic models described as convex hull of vertices (in ex. v = 29 = 512 !)

A(ξ) =

v

v=1

ξvA[v] :

v

v=1

ξv = 1 , ξv ≥ 0 ▲ Demo in RoMulOC

D. Peaucelle

5 Florence, December 2013

SLIDE 7

Modeling of systems with uncertainties ˙ x =        1 Lp Lβ Lr g/(Vo + v) Yβ −1 N ˙

βg/(Vo + v)

Np Nβ + N ˙

βYβ

Nr − N ˙

β

       x + Bu

Linear-Fractional Transformation (LFT): make it linear in the uncertainties via feedback

z w

D. Peaucelle

6 Florence, December 2013

SLIDE 8

Modeling of systems with uncertainties ˙ x =        1 Lp Lβ Lr g/(Vo + v) Yβ −1 N ˙

βg/(Vo + v)

Np Nβ + N ˙

βYβ

Nr − N ˙

β

       x + Bu ▲ Handling the 1/(Vo + v) terms. z1 = 1/(Vo + v)x1 ⇔ Voz1 + vz1 = x1 ⇔    w1 = vz1 Voz1 + w1 = x1 ⇔    w1 = vz1 z1 = 1/Vox1 − 1/Vow1

D. Peaucelle

7 Florence, December 2013

SLIDE 9

Modeling of systems with uncertainties ˙ x =        1 Lp Lβ Lr g/(Vo + v) Yβ −1 N ˙

βg/(Vo + v)

Np Nβ + N ˙

βYβ

Nr − N ˙

β

       x + Bu ▲ Handling the 1/(Vo + v) terms, continued w1 = vz1   ˙ x z1   =           1 Lp Lβ Lr g/Vo Yβ −1 −g/Vo N ˙

βg/Vo

Np Nβ + N ˙

βYβ

Nr − N ˙

β

−N ˙

βg/Vo

1/Vo −1/Vo             x w1   +   B  

D. Peaucelle

8 Florence, December 2013

SLIDE 10

Modeling of systems with uncertainties   ˙ x z1   =           1 Lp Lβ Lr g/Vo Yβ −1 −g/Vo N ˙

βg/Vo

Np Nβ + N ˙

βYβ

Nr − N ˙

β

−N ˙

βg/Vo

1/Vo −1/Vo             x w1   +   B   ▲ Handling the N ˙

β term:

w1 = vz1 , w2 = N ˙

βz2

    ˙ x z1 z2     =              1 Lp Lβ Lr g/Vo Yβ −1 −g/Vo Np Nβ Nr 1 1/Vo −1/Vo g/Vo Yβ −1 −g/Vo                  x w1 w2     +   B   u

D. Peaucelle

9 Florence, December 2013

SLIDE 11

Modeling of systems with uncertainties ▲ In the end w = ∆z with ∆ = diag(v, N ˙

β, Yβ, Lp, Lβ, Lr, Np, Nβ, Nr) and

˙

x z

=

                              1 1 1 1 g/Vo −1 −g/Vo 1 1 1 1 1 1/Vo −1/Vo g/Vo −1 −g/Vo 1 1 1 1 1 1 1 1                              

x

w

+
B
u
LFT model is a feedback-loop of a purely uncertain matrix with purely certain system

▲ Can always be obtained if uncertainty enters rationally in the model ▲ Issue: having an LFT of minimal size (size of ∆)

D. Peaucelle

10 Florence, December 2013

SLIDE 12

Modeling of systems with uncertainties

Manipulation LFT made easy using the star-product

∆ ⋆   Md Mc Mb Ma   = Ma + Mb∆(I − Md∆)−1Mc ▲ Corresponds to the following loop: w = ∆z ⋆    z = Mdw + Mcu y = Mbw + Mau

u y w z

Always assumed to be well-posed: (I − Md∆) non singular for all uncertainties
D. Peaucelle

11 Florence, December 2013

SLIDE 13

Modeling of systems with uncertainties ▲ Elementary operations on LFTs ∆1 ⋆

Md

Mc Mb Ma

+ ∆2 ⋆
Nd

Nc Nb Na

=
∆1

∆2

⋆

 

Md Mc Nd Nc Mb Nb Ma + Na

  ∆1 ⋆

Md

Mc Mb Ma

· ∆2 ⋆
Nd

Nc Nb Na

=
∆1

∆2

⋆

 

Md McNb McNa Nd Nc Mb MaNb MaNa

 

∆ ⋆
Md

Mc Mb Ma

−1 = ∆ ⋆

Md − McM −1

a

Mb −McM −1

a

M −1

a

Mb M −1

a

Coded in Matlab’s Robust Control toolbox & in LFRToolbox

▲ Demo in Robust Control toolbox & RoMulOC

Allows also to manipulated complex control schemes

1

F Σ Δ Δ Σ1

1

Δ

2 2

K

3

D. Peaucelle

12 Florence, December 2013

SLIDE 14

Modeling of systems with uncertainties ■ DEMETER: a satellite of the MYRIAD family developed by CNES

All MYRIAD microsatellites share common plat-

form (including the control components), the load is different (and the gains of the control law are tuned accordingly).

On DEMETER the scientific load includes four

long appendices that study the ionospheric distur- bances (smsc.cnes.fr/DEMETER). Fine pointing towards earth is required.

▲ CNES: French national space center - governmental ▲ Accepted to provide data about DEMETER to the scientific community, [PA06] ▲ It is purely a benchmark: no possible implementation (no more on orbit)

D. Peaucelle

13 Florence, December 2013

SLIDE 15

Modeling of systems with uncertainties ■ LAAS studies for the attitude control of DEMETER

Uncertain modeling at small depointing errors
Mixed H2/H∞ reduced order control design (small depointing) [ADGH11]
Robustness analysis of the uncertain LTI model in closed-loop (small depointing) [PBG+10]
Periodic control law design using reaction wheels and magneto torquers (medium to large

depointing) [TAP+11]

Design of an adaptive control law replacing a commuting control (small to medium depointing)

[PDPM11]

D. Peaucelle

14 Florence, December 2013

SLIDE 16

Modeling of systems with uncertainties   J(∆) ˆ J(∆) ˆ JT (∆) I8     ¨ θ ¨ η   +   C(∆) K(∆)     ˙ η η   =   I   u ■ θ ∈ R3 is the attitude of the satellite close to an orientation defined as θ = 0 ■ η =

η1T

· · · η4T T ∈ R8 are the state of the flexible modes

4: number of appendices on DEMETER
each flexible mode is described by 2 states: bending & torsion

▲ Flexible modes in higher frequencies are neglected (including solar panel) ▲ All parameters are uncertain (cannot be measured on earth nor on orbit) ▲ 14 uncertainties enter the model in polynomial (2nd order) manner

D. Peaucelle

15 Florence, December 2013

SLIDE 17

Modeling of systems with uncertainties ■ Natural frequency and damping uncertainties ¨ η +

C(∆)

2ΩZ ˙

η +

K(∆)

Ω2 η = − ˆ

JT (∆)¨ θ

C(∆)

K(∆)

=
2ΩZ

Ω2

= Ω
2Z

Ω

Ω =

     ω1I2

...

ω4I2      , Z =      ζ1I2

...

ζ4I2     

Flexible modes of the four identical masts

▲ ωi ∈ [0.2 · 2π , 0.6 · 2π] ∀i ▲ ζi ∈ [5 · 10−4 , 5 · 10−3] ∀i

2 states per mast: bending and torsion
D. Peaucelle

16 Florence, December 2013

SLIDE 18

Modeling of systems with uncertainties ■ Natural frequency and damping uncertainties

C(∆)

K(∆)

= Ω
2Z

Ω

LFT modeling with Ω = Ωa + Ωb∆Ω and Z = Za + Zb∆Z

such that ∆Ω and ∆Z diagonal composed of |∆ωi| ≤ 1 and |∆ζi| ≤ 1.

Ω = ∆Ω ⋆

I

Ωb Ωa

, Z = ∆Z ⋆
I

Zb Za

2Z

Ω

=

∆Z

∆Ω

⋆
I
2Zb

Ωb

2Za

Ωa

Ω
2Z

Ω

=  

∆Ω ∆Z ∆Ω

  ⋆  

2Zb

Ωb

2Za

Ωa

I

Ωb

2ΩaZb

ΩaΩb

2ΩaZa

Ω2

a





D. Peaucelle

17 Florence, December 2013

SLIDE 19

Modeling of systems with uncertainties ■ Inertia of the satellite and contributions of the flexible modes   J(∆) ˆ J(∆) ˆ JT (∆) I   =   J1 + J1T + JT

2 ˜

J2J2 JT

2 ˜

JJ3 JT

3 ˜

JJ2 I   ■ cross inertia J1 =  

J12 J13 J23

 

Coupling between axes is unknown but limited:

▲ J12 ∈ [−x.xx , y.yy], J13 ∈ [−x.xx , y.yy], J23 ∈ [−x.xx , y.yy]

D. Peaucelle

18 Florence, December 2013

SLIDE 20

Modeling of systems with uncertainties

LFT modeling J1 =

 

J12 J13 J23

  = ∆J1 ⋆   J1c J1b J1a   with

J1a = J1b = ∆J1 = J1c =     J12c J13c J23c         J12b J13b J23b         ∆J12 ∆J13 ∆J23         1 1 1    

▲ and the normalized uncertainties are such that |∆Jij| ≤ 1. J1 + J1

T =

∆J1

∆J1

⋆

 

J1c JT

1b

J1b JT

1c

J1a + JT

1a

 

J1 + J1

T

=
∆J1

∆J1

⋆

  

J1c JT

1b

J1b JT

1c

J1a + JT

1a

  

D. Peaucelle

19 Florence, December 2013

SLIDE 21

Modeling of systems with uncertainties ■ Inertia of the satellite and contributions of the flexible modes   J(∆) ˆ J(∆) ˆ JT (∆) I   =   J1 + J1T + JT

2 ˜

J2J2 JT

2 ˜

JJ3 JT

3 ˜

JJ2 I   ■ square root of uncertainties on direct inertia ˜ J =  

J11 J22 J33

 

Inertia of each axis is positive unknown and bounded.
Coupling with flexible modes depend of these.

▲ J11 ∈ [xx.x , yy.y], J22 ∈ [xx.x , yy.y], J33 ∈ [xx.x , yy.y]

D. Peaucelle

20 Florence, December 2013

SLIDE 22

Modeling of systems with uncertainties   JT

2 ˜

J2J2 JT

2 ˜

JJ3 JT

3 ˜

JJ2 JT

3 J3

  =   JT

2 ˜

J JT

3

 

˜

JJ2 J3

LFT modeling with ˜

J = I + ∆ ˜

J ˜

Jc where |∆ ˜

Jii| ≤ 1:

JT

2 ˜

J JT

3

= ∆ ˜

J ⋆

 

I JT

2 ˜

JT

c

JT

2

JT

3

  ,

˜

JJ2 J3

= ∆ ˜

J ⋆

˜

JcJ2 I J2 J3

JT

2 ˜

J2J2 JT

2 ˜

JJ3 JT

3 ˜

JJ2 JT

3 J3

=
∆ ˜

J

∆ ˜

J

⋆

  

I J2 J3 ˜ JcJ2 JT

2 ˜

JT

c

JT

2

JT

2 J2

JT

2 J3

JT

3

JT

3 J2

JT

3 J3

  

D. Peaucelle

21 Florence, December 2013

SLIDE 23

Modeling of systems with uncertainties

LFT model of the inertia matrix
J(∆)

ˆ J(∆) ˆ JT (∆) I

=
∆J1

∆J1

⋆

      J1c JT

1b

J1b JT

1c

J1a + JT

1a

      +

∆ ˜

J

∆ ˜

J

⋆

      I J2 J3 ˜ JcJ2 JT

2 ˜

JT

c

JT

2

JT

2 J2

JT

2 J3

JT

3

JT

3 J2

JT

3 J3

      +

I − JT

3 J3

=

  

∆J1 ∆J1 ∆ ˜

J

∆ ˜

J

   ⋆       

J1c JT

1b

I J2 J3 ˜ JcJ2 J1b JT

1c

JT

2 ˜

JT

c

JT

2

J1a + JT

1a + JT 2 J2

JT

2 J3

JT

3

JT

3 J2

I

      

D. Peaucelle

22 Florence, December 2013

SLIDE 24

Modeling of systems with uncertainties

J(∆)

ˆ J(∆) ˆ JT (∆) I8 ¨ θ ¨ η

+
C(∆)

K(∆) ˙ η η

=
I
u
Descriptor LFT model with XT =
˙

θ

T

˙ ηT θT ηT

:

diag

  

∆J1 ∆J1 ∆ ˜

J

∆ ˜

J

   ⋆

Ed

Ec Eb Ea

˙

X = diag  

∆Ω ∆Z ∆Ω

  ⋆

Ad

Ac Ab Aa

X + Bu
Non-descriptor LFT model (taking the inverse of the left-hand side matrix):

˙ X = ∆ ⋆  

Ed − EcE−1

a

Eb −EcE−1

a

Ab −EcE−1

a

Aa −EcE−1

a

B Ad Ac E−1

a

Eb E−1

a

Ab E−1

a

Aa E−1

a

B

 

X

u

with ∆ = diag
∆J1

∆J1 ∆ ˜

J

∆ ˜

J

∆Ω ∆Z ∆Ω

.
D. Peaucelle

23 Florence, December 2013

SLIDE 25

Modeling of systems with uncertainties

Benchmark coded with possibility to choose

▲ 1, 2 or 3 axis model ▲ 0 to 4 appendices (each contributing to 2 flexibles dynamics) ▲ Identical or distinct models of appendices ▲ With or without models of reaction wheel actuators

Models with more or less numerical complexity

▲ Demo in RoMulOC

D. Peaucelle

24 Florence, December 2013

SLIDE 26

Modeling of systems with uncertainties ■ Prospective for RoMulOC future versions: descriptor modeling

LFT descriptor model of DEMETER - simple, closer to original equations

∆ ⋆  

    Ed Ec Eb Ea     ˙ X =     Ad Ac Ab Aa     X + Bu

  ▲ ∆ ∈ R36×36: exogenous LFT signals are z∆ ∈ R36, w∆ ∈ R36

Descriptor model also allows to model rationally dependent models in polytopic affine form

  

J1 + JT

1

˜ JJ2 J3

  

¨

θ ¨ η

+

  

JT

2 ˜

J JT

3

Ω −I I

  

πJ

πη

+

  

2Z Ω

  

˙

θ η

=

  

I

   u ▲ Includes fictive signals πJ ∈ R3 and πη ∈ R8 - only 11 exogenous signals!

[CTF02], [TD13] Descriptor models also appropriate for rationally non-linear systems,

▲ e.g. satellite J ˙ ω + ω×Jω = u , ˙ q = 1 2   −ω× ω − ωT   q

D. Peaucelle

25 Florence, December 2013

SLIDE 27

Outline

Two classes of uncertain systems : polytopic & LFT

▲ Airplane example ▲ DEMETER satellite example ▲ Prospectives: descriptor uncertain modeling

Robust analysis

▲ Stability and performances - the well-posedness point of view ▲ System augmentation approach for sequences of SOS-like relaxations ▲ "Slack variable" results and "quadratic stability" as a special case

State-feedback design: multi-performance

▲ Based on dual system ▲ Almost LMI results in "slack variable" approach

D. Peaucelle

26 Florence, December 2013

SLIDE 28

Robust analysis ■ Dynamical system viewed as an LFT ˙ x = Ax ⇔ s−1I ⋆ A ⇔    w = s−1z z = Aw

s−1: integrator. It is characterized by s−1 + s−∗ ≥ 0

▲ Property to be understood in the sense of scalar products of signals, with zero initial conditions: < ˙ x|s−1 ˙ x >τ=< ˙ x|x >τ= τ ˙ xT (t)x(t)dt = x(τ)T x(τ) ≥ 0

Well-posedness of the loop: (sI − A) non-singular for all s−1 + s−∗ ≥ 0

▲ i.e. all poles of A are in the open left-half of the complex plane

Stability is equivalent to the well-posedness problem

▲ Robust stability of LFT model equivalent to well-posedness of   s−1I ∆   ⋆   A B∆ C∆ D∆∆   , s−1 ∈ C+ ∆ ∈ ∆ ∆

D. Peaucelle

27 Florence, December 2013

SLIDE 29

Robust analysis ■ Dynamical system viewed as an LFT - discrete-time case xk+1 = Axk ⇔ z−1I ⋆ A ⇔    w = z−1z z = Aw

z−1: delay operator. It is characterized by z−∗z−1 ≤ 1.
Well-posedness of the loop: (zI − A) non-singular for all z∗z ≥ 1

▲ i.e. all poles of A are in the open unit disc ■ Same procedure applies to pole location & frequency-dependent specifications [IH05, IHF05]

Re

Im r Im Re

Re

Im

Im

Re

Re

Im

ω ω −j / −j / −ω −ω

D. Peaucelle

28 Florence, December 2013

SLIDE 30

Robust analysis ■ Input/Output performances can also be written as well-posedness problems

For H∞ (or induced L2) norm: small gain theorem

D + C(sI − A)−1B∞ ≤ γ ⇔   s−1I ∆   ⋆   A B C D   , s−1 ∈ C+ ∆ ≤ 1

γ

Similar well-posedness interpretations for H2 & impulse-to-peak performances [PBG09]

■ All robust analysis problems coded in RoMulOC boil down to well-posedness of ∇ ⋆ M

r

∇ ⋆ M(ξ)

with ∇ ∈ ∇

∇ a structured (block-diagonal) uncertain operator

and M(ξ) = ξvM [v] affine polytopic with respect to parametric uncertainties
D. Peaucelle

29 Florence, December 2013

SLIDE 31

Robust analysis ■ Deriving LMIs for guaranteeing well-posedness of ∇ ⋆ M , ∇ ∈ ∇ ∇

Done in the framework of topological separation [Saf80]
Quadratic separation in the context of linear systems with parametric uncertainties

[IH98, IS01, PAHG07, Pea07]

Integral quadratic separation in the context of linear systems with TV/NL operators

[PBG09, Pea09, PTGSB12]

Similar to results from Integral quadratic constraints literature [MR97, JM99, SK08]

■ Well-posedness iff exists a matrix Θ solution to one LMI + one IQC

I

M T

Θ

  I M   ≻ 0 ,   ∇ I  

Θ

  ∇ I  

≤ 0 , ∀∇ ∈ ∇

∇ ▲ IQC can be converted to finite set of LMI constraints F∇

∇(Θ) ≺ 0

▲ Conservative in general, except some lossless cases [MSF97]) ▲ Contains generalized KYP-lemma [IH05], S-procedure, DG-scalings etc.

D. Peaucelle

30 Florence, December 2013

SLIDE 32

Robust analysis ■ Well-posedness of ∇ ⋆ M , ∇ ∈ ∇ ∇

if exists a matrix Θ solution to LMIs

I

M T

Θ

  I M   ≻ 0 , F∇

∇(Θ) ≺ 0

■ No need to understand all that to use RoMulOC ▲ Demo in RoMulOC

Result extend to well-posedness of descriptor LFTs:

   w = ∇z M1z = M2w

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31 Florence, December 2013

SLIDE 33

Robust analysis ■ Reducing conservatism: the system augmentation approach [EPAH05, PAHG07]

Original robust stability problem: well-posedness of

   x = s−1 ˙ x w = ∆z ⋆    ˙ x = Ax + Bw z = Cx + Dw

First augmentation (assuming ˙

∆ = 0)              x = s−1 ˙ x w = s−1 ˙ w w = ∆z ˙ w = ∆˙ z ⋆              ˙ x = Ax + Bw ˙ w = ˙ w z = Cx + Dw ˙ z − C ˙ x = D ˙ w ▲ IQS applied to this well-posedness problem gives larger LMIs, proved to be less conservative ▲ Underlying Lyapunov function is parameter-dependent V (x, ∆) =   x w  

T

P1   x w   = xT   I ∆(I − ∆D)−1C  

T

P1   I ∆(I − ∆D)−1C   x = xT P(∆)x

D. Peaucelle

32 Florence, December 2013

SLIDE 34

Robust analysis

First augmentation (assuming ˙

∆ = 0) ▲ Demo in RoMulOC

Further augmentations (based on ˙

∆ = 0, ¨ ∆ = 0 ...) ▲ Provide sequence of less and less conservative LMIs (with increasing dimensions) ▲ Method directly related to Lasserre’s SOS relaxations [PS09] ▲ Method applies also for time-delay systems [GP06], saturations [PTGSB12]... ▲ Not yet coded in RoMulOC ▲ Tested on DEMETER model [PBG+10]

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SLIDE 35

Robust analysis ■ Well-posedness in the case of polytopic models ∇ ⋆ M(ξ)

with ∇ ∈ ∇

∇ a structured (block-diagonal) uncertain operator

and M(ξ) = ξvM [v] affine polytopic with respect to parametric uncertainties

■ Applying same method leads to solving for all ξ ∈ v

v=1 ξv = 1 ,

ξv ≥ 0

I

M T (ξ)

Θ(ξ)

  I M(ξ)   ≻ 0 , F∇

∇(Θ(ξ)) ≺ 0

Classical quadratic stability framework: solve for all vertices v = 1 . . . v
I

M [v]T

Θ

  I M [v]   ≻ 0 , F∇

∇(Θ) ≺ 0 , Θ22 0

▲ Conservatism due to the choice of Θ unique for all ξ ▲ Applies for convex sets ∇ ∇ (not applicable for pole location outside a disc)

D. Peaucelle

34 Florence, December 2013

SLIDE 36

Robust analysis

I

M T (ξ)

Θ(ξ)

  I M(ξ)   ≻ 0 , F∇

∇(Θ(ξ)) ≺ 0

Classical quadratic stability framework: solve for all vertices v = 1 . . . v
I

M [v]T

Θ

  I M [v]   ≻ 0 , F∇

∇(Θ) ≺ 0 , Θ22 0

The slack variable approach [OBG99, OGH99, PABB00, Pea09, PDSV09]

(A book is to come on the topic in 2014 - Y. Ebihara, D. Peaucelle)

Θ[v] ≻ G

M [v]

−I

+

  M [v]T −I   GT , F∇

∇(Θ[v]) ≺ 0

▲ Less conservative and with Θ(ξ) = v

v=1 ξvΘ[v]

▲ Smartly coded in RoMulOC to avoid unnecessary slack variables ▲ e.g. if Θ[v] = Θ unique and Θ22 ≺ 0 there is no need for G ▲ Demo in RoMulOC

D. Peaucelle

35 Florence, December 2013

SLIDE 37

Outline

Two classes of uncertain systems : polytopic & LFT

▲ Airplane example ▲ DEMETER satellite example ▲ Prospectives: descriptor uncertain modeling

Robust analysis

▲ Stability and performances - the well-posedness point of view ▲ System augmentation approach for sequences of SOS-like relaxations ▲ "Slack variable" results and "quadratic stability" as a special case

State-feedback design: multi-performance

▲ Based on dual system ▲ Almost LMI results in "slack variable" approach

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SLIDE 38

State-feedback design ■ Most well-posedness have a dual counterpart ∇ ⋆ M w.p. ⇔ ∇∗ ⋆ M T w.p. ▲ e.g. poles of ˙ x = Ax same as poles of ˙ xd = AT xd ▲ e.g. the H∞ and H2 norms D + C(sI − A)−1B∞ = DT + BT (s∗I − AT )−1CT ∞

Interesting feature, even for robust analysis problems

▲ LMIs for ∇ ⋆ M prove stability with Lyapunov matrix P(∆) quadratic in ∆ ▲ LMIs for ∇∗ ⋆ M T prove stability with Lyapunov matrix P −1(∆) quadratic in ∆ ▲ Demo in RoMulOC

D. Peaucelle

37 Florence, December 2013

SLIDE 39

State-feedback design ■ From analysis to state-feedback problems,

The stability case

˙ x = Ax → ˙ x = (A + BK)x

The general well-posedness case

∇ ⋆ M → ∇ ⋆ (MA + MB

K
)

■ Classical methodology to get LMIs:

Apply analysis conditions to dual model ∇∗ ⋆ (M T

A +

  KT   M T

B) to get a PMI

I

MA + MB

K

Θ  

I M T

A +

KT
M T

B

  ≻ 0

If Θ22 0 apply a Schur complement argument to get a BMI
Apply a linearizing change of variables XK = Y

⇒ FQS(Θ, Y ) ≺ 0 ▲ Demo in RoMulOC

D. Peaucelle

38 Florence, December 2013

SLIDE 40

State-feedback design ■ The slack variables case

BMI problem emerging from conditions on the dual model

Θ[v] ≻ G   M [v]T

A

+   KT   M [v]T

B

−I   + (...)T

Exist linearizing change of variables if structuring a priori as G =

  MG − I   H ⇒ FSV (MG, Θ[v], H, Y ) ≺ 0 ▲ ∇∗ ⋆ MG should be well-posed ▲ In some case there exists MG such that (SV) guaranteed to include (QS)

Some extensions of these results

▲ For periodic systems, MG has interesting non causal properties [EPA11] ▲ Applies to static output feedback design with minor modifications MG is then a stabilizing state-feedback, [PA01, AGPP10]

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SLIDE 41

State-feedback design ■ The multi-performance state-feedback problem, some examples

Find a unique K that stabilizes three different systems M1, M2, M3
Find K that locates the closed loop-poles of M1 in a disc and minimizes H∞ norm

■ In terms of LMI conditions : shaping paradigm [SGC97, ACP06]

Concatenate each LMI relative to each specification
Enforce common linearizing variable (X in QS case H in SV case)

▲ Slack variable allows different Lyapunov matrices for each performance (less conservative) ▲ Complex to find a priori MG for each performance ▲ Demo in RoMulOC

D. Peaucelle

40 Florence, December 2013

SLIDE 42

Outline

Two classes of uncertain systems : polytopic & LFT

▲ Airplane example ▲ DEMETER satellite example ▲ Prospectives: descriptor uncertain modeling

Robust analysis

▲ Stability and performances - the well-posedness point of view ▲ System augmentation approach for sequences of SOS-like relaxations ▲ "Slack variable" results and "quadratic stability" as a special case

State-feedback design: multi-performance

▲ Based on dual system ▲ Almost LMI results in "slack variable" approach

D. Peaucelle

41 Florence, December 2013

SLIDE 43

Conclusions ■ What RoMulOC does:

Provides LMI tests for many robust control problems
Analysis and state-feedback
Performances : stability, pole location, H∞, H2, impulse-to-peak
LFT and polytopic models
Two levels of results with different amount of conservatism

■ RoMulOC Prospectives

Further relaxations for reduced conservatism
Descriptor systems
Non-linearities and TV uncertainties
Output feedback design

■ What is R-RoMulOC ? ▲ The same but including randomization features from RACT, [TCD+, TCD13]

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SLIDE 44

REFERENCES

References

REFERENCES

References

[ACP06]

D. Arzelier, B. Clement, and D. Peaucelle, Multi-objective H2/H∞/impulse-to-peak control of a space launch vehicle, Euro-

pean J. of Control 12 (2006), no. 1. [ADGH11]

D. Arzelier, G. Deaconu, S. Gumussoy, and D. Henrion, H2 for HIFOO, International Conference on Control and Optimization

with Industrial Applications (Ankara), August 2011. [AGPP10]

D. Arzelier, E.N. Gryazina, D. Peaucelle, and B.T. Polyak, Mixed LMI/randomized methods for static output feedback control

design, American Control Conference (Baltimore), June 2010. [CTF02]

D. Coutinho, A. Trofino, and M. Fu, Guaranteed cost control of uncertain nonlinear systems via polynomial Lyapunov functions,

IEEE Trans. on Automat. Control 47 (2002), no. 9, 1575–1580. [EPA11]

Y. Ebihara, D. Peaucelle, and D. Arzelier, Periodically time-varying memory state-feedback controller synthesis for discrete-

time linear systems, Automatica 47 (2011), no. 1, 14–25. [EPAH05]

Y. Ebihara, D. Peaucelle, D. Arzelier, and T. Hagiwara, Robust performance analysis of linear time-invariant uncertain systems

by taking higher-order time-derivatives of the states, joint IEEE Conference on Decision and Control and European Control Conference (Seville, Spain), December 2005, In Invited Session "LMIs in Control". [GP06] F . Gouaisbaut and D. Peaucelle, Stability of time-delay systems with non-small delay, IEEE Conference on Decision and Control (San Diego), December 2006. [IH98]

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computations, IEEE Trans. on Automat. Control 43 (1998), no. 5, 619–630. [IH05]

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[IHF05]

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D. Peaucelle

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SLIDE 45

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[OBG99] M.C. de Oliveira, J. Bernussou, and J.C. Geromel, A new discrete-time stability condition, Systems & Control Letters 37 (1999), no. 4, 261–265. [OGH99] M.C. de Oliveira, J.C. Geromel, and L. Hsu, LMI characterization of structural and robust stability: The discrete-time case, Linear Algebra and its Applications 296 (1999), no. 1-3, 27–38. [PA01]

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Conference (Porto, Portugal), September 2001, pp. 3800–3805. [PA06]

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. Gouaisbaut, Quadratic separation for feedback connection of an uncertain matrix and an implicit linear transformation, Automatica 43 (2007), 795–804. [PBG09]

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D. Peaucelle

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SLIDE 46

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D. Peaucelle, A. Drouot, C. Pittet, and J. Mignot, Simple adaptive control without passivity assumptions and experiments on

satellite attitude control DEMETER benchmark, IFAC World Congress, August 2011, Paper in an invited session. [PDSV09] Goele Pipeleers, Bram Demeulenaere, Jan Swevers, and Lieven Vandenberghe, Extended LMI characterizations for stability and performance of linear systems, Systems & Control Letters 58 (2009), no. 7, 510 – 518. [Pea07]

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SLIDE 47

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