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Robust Control Using Convex Optimization Kyoto University March 12, - - PowerPoint PPT Presentation

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Robust Control Using Convex Optimization Kyoto University March 12, 2010 Dimitri Peaucelle LAAS-CNRS, Toulouse, FRANCE peaucelle@laas.fr http://homepages.laas.fr/peaucell Introduction Robustness in control theory Robustness properties

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Robust Control Using Convex Optimization

Kyoto University March 12, 2010 Dimitri Peaucelle LAAS-CNRS, Toulouse, FRANCE peaucelle@laas.fr http://homepages.laas.fr/peaucell

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Introduction ➊ Robustness in control theory

  • Robustness properties of the feedback loop
  • Some classical measures of robustness
  • Disturbance rejection and robustness to parametric uncertainty - a modeling issue
  • Robustness: tradeoff between complexity of systems (non-linear, etc.) and simplicity of models

➋ Optimization based tools

  • Linear Matrix Inequalities (LMI) framework [1990’s]
  • Efficient fast solvers and nice parser for Matlab [2000’s]
  • Adapted tool for robustness issues - examples of Lyapunov-based results

➌ Robust control theory at LAAS

  • RoMulOC - A Matlab toolbox
  • Applications in aerospace
  • Integral Quadratic Separation
  • D. Peaucelle

1 Seminar at Kyoto Univ. - March 12, 2010

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➊ Robustness in control theory

Let the following double integrator with a flexible mode (attitude of satellite with solar pannel)

T(s) = 1 s2 s2 + 0.1s + 1.002 s2 + 0.2s + 1.01

for which the following controller is designed

K(s) = k s3 + 2.1s2 + 1.2s + 0.1 s3 + 9s2 + 27s + 27

The Nichols plots of T(s)K(s) (for k = 10) phase margin > 90◦, gain margin = ∞ Robust !?

  • D. Peaucelle

2 Seminar at Kyoto Univ. - March 12, 2010

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➊ Robustness in control theory

Assume uncertainty on the flexible mode (damaged solar panel)

T(s) = s2 + 0.1s + 1.002 s2(s2 + 0.2s + 1.01) → ˜ T(s) = s2 + 0.1s + 2.252 s2(s2 + 0.2s + 1.01)

for the same control (k = 10) the Nichols plots of ˜

T(s)K(s)

indicates that the closed-loop is unstable for k = 10 but it is stable for k = 1 and k = 200.

▲ The controller stabilizes ˜ T(s) for k = 1 and k = 200 but not for k ∈ [4 100]. ▲ The controller for k = 1 or k = 200 stabilizes both models T(s) and ˜ T(s).

  • D. Peaucelle

3 Seminar at Kyoto Univ. - March 12, 2010

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➊ Robustness in control theory

Step and frequency responses of closed-loop systems for k = 200:

▲ Small variations in the parametric space can have large impact. ▲ Faulty behavior can be related to narrow peaks in the frequency domain

  • D. Peaucelle

4 Seminar at Kyoto Univ. - March 12, 2010

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➊ Robustness in control theory

■ Two ‘definitions’ of robustness

  • Guarantee some characteristics for a given class of perturbations

e.g.

z

Σ

y u w

K

z ≤ γw , ∀w : w < ∞ ▲ Many results using L2 norm: w2 = ∞ w∗(t)w(t)dt ▲ Defines induced L2 norm of the system: Σ

u,y

⋆ K = min γ

(also known as H∞ norm for linear systems)

  • D. Peaucelle

5 Seminar at Kyoto Univ. - March 12, 2010

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➊ Robustness in control theory

■ Two ‘definitions’ of robustness

  • Guarantee some characteristics of an output for a given class of input perturbations

e.g.

z

Σ

y u w

K

z ≤ γw , ∀w : w < ∞ ▲ Many results using L2 norm: w2 = ∞ w∗(t)w(t)dt ▲ Defines induced L2 norm of the system: Σ

u,y

⋆ K = min γ

(also known as H∞ norm for linear systems)

  • Guarantee stability of

K (∆)

Σ

for a given set of uncertainties ∆ ∈ ∆

∆ ▲ Parametric uncertainty: ∆ is a vector of scalar, bounded, constant parameters ▲ Linear Time-Varying (LTV): Σ is linear , ∆(t) grasps TV and Non-Linear characteristics ▲ Uncertain dynamic uncertainties: ∆ is a constrained operator

e.g.

w∆(t) = [∆z∆](t) : w∆ ≤ ρz∆

  • D. Peaucelle

6 Seminar at Kyoto Univ. - March 12, 2010

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➊ Robustness in control theory

  • Example of uncertain modeling

▲ Non-linear system with unknown parameter ˙ x(t) = −αx(t) + sin(x(t)) , α ∈ [−2.5 − 1.5] ▲ Included in the following uncertain model ˙ x(t) = −2x(t) + δ1x(t) + δ2(t)x(t) , δ1 ∈ [−0.5 0.5] , δ2(t)(= sin x

x ) ∈ [−0.2 1]

▲ Included itself in ˙ x(t) = −2x(t) +

  • 1

1

  • ∆(t)x(t)

∆(t) =   δ1 δ2(t)  

  • 1

∆(t)

T   

−0.7576 −0.6061 1.8182 −0.6061 1.5152

  

  • 1

∆(t)

  • ≤ 0
  • D. Peaucelle

7 Seminar at Kyoto Univ. - March 12, 2010

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➊ Robustness in control theory

  • Example of uncertain modeling (followed)

▲ Non-linear system with unknown parameter ˙ x(t) = −αx(t) + sin(x(t)) , α ∈ [−2.5 − 1.5] ▲ Included in the following uncertain model ˙ x(t) = −1.6x(t)+0.5˜ δ1x(t)+0.6˜ δ2(t)x(t) , ˜ δ1 ∈ [−1 1] , ˜ δ2(t)(= sin x

x ) ∈ [−1 1]

▲ Included itself in ˙ x(t) = −1.6x(t) +

  • 0.7416

0.8124

  • ˜

∆(t)x(t) ˜ ∆(t) =   ˜ δ1 ˜ δ2(t)  

: ∆(t)T ∆(t) ≤ 1

  • D. Peaucelle

8 Seminar at Kyoto Univ. - March 12, 2010

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➊ Robustness in control theory

■ The case of norm-bounded uncertainties

  • Non-structured uncertain norm-bounded operator - enters affinelly in the model

˙ x = A(∆)x = (A + B∆∆C∆)x : ∆T ∆ ≤ ρ1

  • More general: the uncertain operator enters rationally in the model

˙ x = A(∆)x = (A + B∆∆(1 − D∆∆∆)−1C∆)x : ∆T ∆ ≤ ρ1

  • Linear-Fractional-Transform: equivalent modeling using exogenous signals

˙ x = Ax + B∆w∆ z∆ = C∆x + D∆∆w∆ , w∆ = ∆z∆ : ∆T ∆ ≤ ρ1

z∆ w

∆

∆

Σ

Σ

w∆,z∆

⋆ ∆

  • Generalizes to the case of bounded operators

w∆(t) = [∆z∆](t) : w∆ ≤ ρz∆

  • D. Peaucelle

9 Seminar at Kyoto Univ. - March 12, 2010

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➊ Robustness in control theory

■ Robustness w.r.t. uncertain parameters v.s. disturbance rejection

  • Stability of Σ

w∆,z∆

⋆ ∆ for all w∆(t) = [∆z∆](t) : w∆ ≤ ρz∆

equivalent to

  • L2 induced norm Σ ≤ γ = 1

ρ

z∆ w

∆

∆

Σ

stable for w∆ ≤ ρz∆

⇔

∆

Σ

w=w

∆

z=z

z ≤ γw ▲ Robust stability optimization problem:

find maximal ρ ≥ 1 such that the closed loop is stable for all admissible uncertainties

▲ L2-Performance optimization problem:

find minimal γ ≤ 1 that guarantees performance rejection

  • D. Peaucelle

10 Seminar at Kyoto Univ. - March 12, 2010

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➊ Robustness in control theory

■ Modulus margin example

K

+−

Τ

L2-gain given by maximal gain of the closed-loop

for all frequencies (see Bode plot)

K

− +

∆ Τ

Robustness to norm-bounded uncertainties

■ Extends to MIMO case: Bode plot replaced by sigma plot σ(K(jω)T(jω)) = maximal singular value of matrix K(jω)T(jω) for each frequency ω

  • D. Peaucelle

11 Seminar at Kyoto Univ. - March 12, 2010

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➊ Robustness in control theory

■ µ theory

M(j ) ∆ ω

structured singular value :

µ(ω) = 1/km(ω) < 1 , ∀ω km(ω) = min{ k : ∃∆ det(1 − k∆M(jω)) = 0 } ∆ =                        δ11r1

...

δpR1rpR ˆ δ11ˆ

r1

...

ˆ δpC1ˆ

rpC

∆1

...

∆pF                        δp ∈ R : |δp| ≤ 1

,

ˆ δp ∈ C : |ˆ δp| ≤ 1

,

∆p ∈ Cmp×lp : ∆p ≤ 1

  • D. Peaucelle

12 Seminar at Kyoto Univ. - March 12, 2010

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➊ Robustness in control theory

  • Example of structured singular value [Skogestad 96]

M(jω) =   a a b b   ▲ If ∆ = δ1 then µ(ω) = max |λ(M(jω))| = |a + b| ▲ If ∆ =   δ1 δ2   then µ(ω) = |a| + |b| ▲ If ∆ is full block then µ(ω) = σ(M(jω)) = √ 2a2 + 2b2

  • D. Peaucelle

13 Seminar at Kyoto Univ. - March 12, 2010

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➊ Robustness in control theory

■ Standard robust analysis problem:

z∆ w

∆

∆

Σ

prove stability of (Σ

w∆,z∆

⋆ ∆)

for all ∆ ∈ ∆

∆ (block diagonal operator with LTI and/or TV uncertainties) ■ Standard robust design problem:

y

∆

w

∆

K

u z

Σ

∆

Find K that guarantees stability of ((Σ

w∆,z∆

⋆ ∆)

u,y

⋆ K)

for all ∆ ∈ ∆

∆ (block diagonal operator with LTI and/or TV uncertainties)

  • ∆ contains unknown parameters, scheduling parameters,

approximations of non-linearities, delays...

  • Σ is a linear model: crude but simple representation of the system
  • D. Peaucelle

14 Seminar at Kyoto Univ. - March 12, 2010

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➊ Robustness in control theory

■ Generalizes to robust performance problems

  • z

∆

w

∆

w z

∆

Σ

Guarantee an input/output property for all uncertainties

  • z

∆

w

∆

K

w z u y

Σ

∆

Find a controller that guarantees input/output properties for all ∆ ∈ ∆

∆

  • Find a controller that guarantees simultaneously several robust specifications

(possibly defined on different models)

✛ ✛ K

1 3

K

1 1

Π Π K Π

2 3 2 2 3

∆

Σ

∆

Σ

∆

Σ

  • D. Peaucelle

15 Seminar at Kyoto Univ. - March 12, 2010

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➊ Robustness in control theory

✛ ✛ K

1 3

K

1 1

Π Π K Π

2 3 2 2 3

∆

Σ

∆

Σ

∆

Σ

  • Naturally defined as existence (feasibility) problem over several constraints
  • While a nominal performance Σ ⋆ K = γ can be defined by an equality
  • Robust performance (Σ ⋆ ∆) ⋆ K ≤ γ, ∀∆ ∈ ∆

∆ can only be an inequality

  • Finding the ’best’ robust controller: optimization problem over inequality constraints

▲ What type of optimization problem? Unique optimum? Convex? Convergence time? ...

  • D. Peaucelle

16 Seminar at Kyoto Univ. - March 12, 2010

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SLIDE 18

Outline ➊ Robustness in control theory

  • Robustness properties of the feedback loop
  • Some classical measures of robustness
  • Disturbance rejection and robustness to parametric uncertainty - a modeling issue
  • Robustness: tradeoff between complexity of systems (non-linear, etc.) and simplicity of models

➋ Optimization based tools

  • Linear Matrix Inequalities (LMI) framework [1990’s]
  • Efficient fast solvers and nice parser for Matlab [2000’s]
  • Adapted tool for robustness issues - examples of Lyapunov-based results

➌ Robust control theory at LAAS

  • RoMulOC - A Matlab toolbox
  • Applications in aerospace
  • Integral Quadratic Separation
  • D. Peaucelle

17 Seminar at Kyoto Univ. - March 12, 2010

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➊ Robustness in control theory

■ Special case of

z

∆

w

∆

K

w z u y

Σ

∆

The affine polytopic model

K

[v] [2]

Σ Σ

[1]

Σ Σ(∆)

Convex hull of ¯

v vertices A(∆) = ¯

v v=1 ξvA[v] , Bw(∆) = ¯ v v=1 ξvB[v] w

. . . : ξv ≥ 0 , ¯

v v=1 ξv = 1

▲ Example: Linear combination of linear models identified on different operating points. ▲ Usually assumed to be parametric uncertainties (constant) ▲ Example: ˙ x(t) = −αx(t) + sin(x(t)) , α ∈ [−2.5 − 1.5]

  • D. Peaucelle

18 Seminar at Kyoto Univ. - March 12, 2010

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SLIDE 20

Outline ➊ Robustness in control theory

  • Robustness properties of the feedback loop
  • Some classical measures of robustness
  • Disturbance rejection and robustness to parametric uncertainty - a modeling issue
  • Robustness: tradeoff between complexity of systems (non-linear, etc.) and simplicity of models

➋ Optimization based tools

  • Linear Matrix Inequalities (LMI) framework [1990’s]
  • Efficient fast solvers and nice parser for Matlab [2000’s]
  • Adapted tool for robustness issues - examples of Lyapunov-based results

➌ Robust control theory at LAAS

  • RoMulOC - A Matlab toolbox
  • Applications in aerospace
  • Integral Quadratic Separation
  • D. Peaucelle

19 Seminar at Kyoto Univ. - March 12, 2010

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SLIDE 21

➋ Optimization based tools

■ Semi-Definite Programming and LMIs

  • Extension of LP to semi-definite matrices

min cx : Ax = b , xi ≥ 0 (LP) |

mat(x) ≥ 0 (SDP)

  • Convexity, duality, polynomial-time algorithms (O(n6.5 log(1/ǫ))).

max bT y : AT y − cT = z ,

mat(z) ≥ 0

  • 1st developments and 1st results : LMI formalism & Control Theory

min

  • giyi

: F0 +

  • Fiyi ≥ 0

▲ The H∞ norm computation example for G(s) ∼ (A, B, C, D) : G(s)2

∞

= min γ : P > 0 ,   AT P + PA + CT

z Cz

BwP + CT

z Dzw

PBT

w + DT zwCz

−γ1 + DT

zwDzw

  ≤ 0

  • D. Peaucelle

20 Seminar at Kyoto Univ. - March 12, 2010

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SLIDE 22

➋ Optimization based tools

■ SDP solvers and parsers

  • LMI Control Toolbox ➾ Control Toolbox

1st solver, dedicated to LMIs issued from Control Theory, Matlab, owner.

  • SDP solvers: SP

, SeDuMi, SDPT3, CSDP , DSDP , SDPA... Active field, mathematical programing, C/C++, free.

  • Parsers: tklmitool, sdpsol, SeDuMiInterface, YALMIP

Convert LMIs to SDP solver format, Matlab (Scilab), free.

http://users.isy.liu.se/johanl/yalmip http://www.laas.fr/OLOCEP/SciYalmip

  • D. Peaucelle

21 Seminar at Kyoto Univ. - March 12, 2010

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SLIDE 23

➋ Optimization based tools

■ SDP-LMI issues and prospectives

  • Any SDP representable problem is ”solved” (numerical problems due to size and structure)

▲ Find ”SDP-ables” problems

(linear systems, performances, robustness, LPV, saturations, delays, singular systems...)

▲ Equivalent SDP formulations ➾ distinguish which are numerically efficient ▲ New SDP solvers: faster, precise, robust (need for benchmark examples)

  • D. Peaucelle

22 Seminar at Kyoto Univ. - March 12, 2010

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SLIDE 24

➋ Optimization based tools

■ SDP-LMI issues and prospectives

  • Any SDP representable problem is ”solved” (numerical problems due to size and structure)

▲ Find ”SDP-ables” problems

(linear systems, performances, robustness, LPV, saturations, delays, singular systems...)

▲ Equivalent SDP formulations ➾ distinguish which are numerically efficient ▲ New SDP solvers: faster, precise, robust (need for benchmark examples)

  • Any ”SDP-able” problem has a dual interpretation

▲ New theoretical results (worst case) ▲ New proofs (Lyapunov functions = Lagrange multipliers; related to SOS) ▲ SDP formulas numerically stable (KYP-lemma)

  • D. Peaucelle

23 Seminar at Kyoto Univ. - March 12, 2010

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SLIDE 25

➋ Optimization based tools

■ SDP-LMI issues and prospectives

  • Any SDP representable problem is ”solved” (numerical problems due to size and structure)

▲ Find ”SDP-ables” problems

(linear systems, performances, robustness, LPV, saturations, delays, singular systems...)

▲ Equivalent SDP formulations ➾ distinguish which are numerically efficient ▲ New SDP solvers: faster, precise, robust (need for benchmark examples)

  • Any ”SDP-able” problem has a dual interpretation

▲ New theoretical results (worst case) ▲ New proofs (Lyapunov functions = Lagrange multipliers; related to SOS) ▲ SDP formulas numerically stable (KYP-lemma)

  • Non ”SDP-able” : Robustesse & Multi-objective & Relaxation of NP-hard problems

▲ Optimistic / Pessimistic (conservative) results ▲ Reduce the gap (upper/lower bounds) while handling numerical complexity growth.

  • D. Peaucelle

24 Seminar at Kyoto Univ. - March 12, 2010

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SLIDE 26

➋ Optimization based tools

■ SDP-LMI issues and prospectives

  • Any SDP representable problem is ”solved” (numerical problems due to size and structure)

▲ Find ”SDP-ables” problems

(linear systems, performances, robustness, LPV, saturations, delays, singular systems...)

▲ Equivalent SDP formulations ➾ distinguish which are numerically efficient ▲ New SDP solvers: faster, precise, robust (need for benchmark examples)

  • Any ”SDP-able” problem has a dual interpretation

▲ New theoretical results (worst case) ▲ New proofs (Lyapunov functions = Lagrange multipliers; related to SOS) ▲ SDP formulas numerically stable (KYP-lemma)

  • Non ”SDP-able” : Robustesse & Multi-objective & Relaxation of NP-hard problems

▲ Optimistic / Pessimistic (conservative) results ▲ Reduce the gap (upper/lower bounds) while handling numerical complexity growth.

  • Develop software for ”industrial” application / adapted to the application field

➾ RoMulOC toolbox

  • D. Peaucelle

25 Seminar at Kyoto Univ. - March 12, 2010

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SLIDE 27

➋ Optimization based tools

■ Nominal performance analysis: V (x) = xT Px Lyapunov function (P > 0)

  • Stability

AT P + PA < 0

  • Regional pole placement
  • 1

s∗

 r11 r12 r∗

12

r22     1 s   < 0 ⇔

  • 1

A∗

 r11P r12P r∗

12P

r22P     1 A   < 0

  • H∞ norm

  AT P + PA + CT

z Cz

PBw + CT

z Dzw

BT

wP + DT zwCz

−γ21 + DT

zwDzw

  < 0

  • H2 norm

AT P + PA + CT

z Cz < 0

trace(BT

wPBw) < γ2

  • Impulse-to-peak

AT P + PA < 0 BT

wPBw < γ21

CT

z Cz < P

DT

zwDzw < γ21

  • D. Peaucelle

26 Seminar at Kyoto Univ. - March 12, 2010

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SLIDE 28

➋ Optimization based tools

■ Robust performance analysis: V (x, ∆) parameter-dependent Lyapunov function. ▲ Nominal analysis (LMI) → Robust analysis (NP-hard) ∃ P : LΣ(P) < 0 → ∀ ∆ ∈ ∆ ∆ , ∃ P(∆) : LΣ(∆)(P(∆)) < 0 ▲ Test over sample values {∆1...N} ∈ ∆ ∆ gives optimistic results

(some results exist if {∆1...N} is uniform distribution of ∆

∆ and large N)

  • D. Peaucelle

27 Seminar at Kyoto Univ. - March 12, 2010

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SLIDE 29

➋ Optimization based tools

■ Robust performance analysis: V (x, ∆) parameter-dependent Lyapunov function. ▲ Nominal analysis (LMI) → Robust analysis (NP-hard) ∃ P : LΣ(P) < 0 → ∀ ∆ ∈ ∆ ∆ , ∃ P(∆) : LΣ(∆)(P(∆)) < 0 ▲ Test over sample values {∆1...N} ∈ ∆ ∆ gives optimistic results

(some results exist if {∆1...N} is uniform distribution of ∆

∆ and large N)

  • Choice of P(∆) for having a finite number of decision variables :

➙ “Quadratic Stability”: P(∆) = P ➙ Polytopic PDLF: P(∆) = ξvP [v] ➙ P(∆) polynomial w.r.t. ξv ➙ Quadratic-LFT PDLF: P(∆) =

  • 1

∆T

C

  • ˆ

P

  • 1

∆C

  • ∆C = (1 − ∆D∆∆)−1∆C∆

➙ P(∆) polynomial w.r.t. ∆C

K

[v] [2]

Σ Σ

[1]

Σ Σ(∆)

z

∆

w

∆

K

w z u y

Σ

∆

  • D. Peaucelle

28 Seminar at Kyoto Univ. - March 12, 2010

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SLIDE 30

➋ Optimization based tools

■ Robust performance analysis: V (x, ∆) parameter-dependent Lyapunov function. ▲ Nominal analysis (LMI) → Robust analysis (NP-hard) ∃ P : LΣ(P) < 0 → ∀ ∆ ∈ ∆ ∆ , ∃ P(∆) : LΣ(∆)(P(∆)) < 0 ▲ Test over sample values {∆1...N} ∈ ∆ ∆ gives optimistic results

(some results exist if {∆1...N} is uniform distribution of ∆

∆ and large N)

  • Choice of P(∆) for having a finite number of decision variables :

➙ “Quadratic Stability”: P(∆) = P ➙ Polytopic PDLF: P(∆) = ξvP [v] ➙ Quadratic-LFT PDLF: P(∆) =

  • 1

∆T

C

  • ˆ

P

  • 1

∆C

  • ▲ LMIs over infinite number of variables

∀ ∆ ∈ ∆ ∆ , ∃ P(∆) : LΣ(∆)(P(∆)) < 0 ⇐ ∃ P [v]or ˆ P : ∀ ∆ ∈ ∆ ∆ , LΣ(∆)(P(∆)) < 0

  • D. Peaucelle

29 Seminar at Kyoto Univ. - March 12, 2010

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SLIDE 31

➋ Optimization based tools

■ Conservative LMIs for polytopic models (Example of stability analysis) ˙ x = A(∆)x with A(∆) = ¯

v v=1 ξvA[v] : ξ ∈ Ξ = {ξv ≥ 0, ¯ v v=1 ξv = 1}

  • D. Peaucelle

30 Seminar at Kyoto Univ. - March 12, 2010

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SLIDE 32

➋ Optimization based tools

■ Conservative LMIs for polytopic models (Example of stability analysis) ˙ x = A(∆)x with A(∆) = ¯

v v=1 ξvA[v] : ξ ∈ Ξ = {ξv ≥ 0, ¯ v v=1 ξv = 1}

  • “Quadratic Stability”: P(∆) = P

˙ V (x) < 0 ⇔ AT (∆)P + PA(∆) < 0 ⇔ A[v]T P + PA[v] < 0

  • D. Peaucelle

31 Seminar at Kyoto Univ. - March 12, 2010

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SLIDE 33

➋ Optimization based tools

■ Conservative LMIs for polytopic models (Example of stability analysis) ˙ x = A(∆)x with A(∆) = ¯

v v=1 ξvA[v] : ξ ∈ Ξ = {ξv ≥ 0, ¯ v v=1 ξv = 1}

  • “Quadratic Stability”: P(∆) = P

˙ V (x) < 0 ⇔ AT (∆)P + PA(∆) < 0 ⇔ A[v]T P + PA[v] < 0

  • Polytopic PDLF: P(∆) = ζiP [i]

  x ˙ x  

T 

 P(∆) P(∆)     x ˙ x   < 0 :

  • A(∆)

−1

 x ˙ x   = 0 ⇔

Finsler Lemma

  P(∆) P(∆)   + G(∆)

  • A(∆)

−1

  • +

  AT (∆) −1   GT (∆) < 0 ⇐ G(∆) = G & convexity   P [v] P [v]   + G

  • A[v]

−1

  • +

  A[v]T −1   GT < 0

  • D. Peaucelle

32 Seminar at Kyoto Univ. - March 12, 2010

slide-34
SLIDE 34

➋ Optimization based tools

■ Conservative LMIs for LFT models (Example of stability analysis) ˙ x = Ax + B∆w∆ with w∆ = ∆z∆ = ∆C∆x + ∆D∆∆w∆

  • D. Peaucelle

33 Seminar at Kyoto Univ. - March 12, 2010

slide-35
SLIDE 35

➋ Optimization based tools

■ Conservative LMIs for LFT models (Example of stability analysis) ˙ x = Ax + B∆w∆ with w∆ = ∆z∆ = ∆C∆x + ∆D∆∆w∆

  • “Quadratic Stability”: P(∆) = P

˙ V (x) =   x w∆  

∗ 

 1 A B∆  

∗ 

 0 P P     1 A B∆     x w∆   < 0 :

  • ∆

−1

 C∆ D∆∆ 1     x w∆   = 0 ⇔

Finsler Lemma

M ∗

A

  0 P P   MA < τM ∗

C

  ∆∗ −1  

  • ∆

−1

  • MC
  • D. Peaucelle

34 Seminar at Kyoto Univ. - March 12, 2010

slide-36
SLIDE 36

➋ Optimization based tools

■ Conservative LMIs for LFT models (Example of stability analysis) ˙ x = Ax + B∆w∆ with w∆ = ∆z∆ = ∆C∆x + ∆D∆∆w∆

  • “Quadratic Stability”: P(∆) = P

˙ V (x) =   x w∆  

∗ 

 1 A B∆  

∗ 

 0 P P     1 A B∆     x w∆   < 0 :

  • ∆

−1

 C∆ D∆∆ 1     x w∆   = 0 ⇔

Finsler Lemma

M ∗

A

  0 P P   MA < M ∗

CΘMC ≤ τM ∗ C

  ∆∗ −1  

  • ∆

−1

  • MC

with

  • 1

∆∗

  • Θ

  1 ∆   ≤ 0

  • D. Peaucelle

35 Seminar at Kyoto Univ. - March 12, 2010

slide-37
SLIDE 37

➋ Optimization based tools

■ Conservative LMIs for LFT models (Example of stability analysis) ˙ x = Ax + B∆w∆ with w∆ = ∆z∆ = ∆C∆x + ∆D∆∆w∆

  • “Quadratic Stability”: P(∆) = P

˙ V (x) =   x w∆  

∗ 

 1 A B∆  

∗ 

 0 P P     1 A B∆     x w∆   < 0 :

  • ∆

−1

 C∆ D∆∆ 1     x w∆   = 0 ⇔

Finsler Lemma

M ∗

A

  0 P P   MA < M ∗

CΘMC ≤ τM ∗ C

  ∆∗ −1  

  • ∆

−1

  • MC

with

  • 1

∆∗

  • Θ

  1 ∆   ≤ 0

  • Quadratic-LFT PDLF - same methodology (yet needs many matrix manipulations).
  • D. Peaucelle

36 Seminar at Kyoto Univ. - March 12, 2010

slide-38
SLIDE 38

Outline ➊ Robustness in control theory

  • Robustness properties of the feedback loop
  • Some classical measures of robustness
  • Disturbance rejection and robustness to parametric uncertainty - a modeling issue
  • Robustness: tradeoff between complexity of systems (non-linear, etc.) and simplicity of models

➋ Optimization based tools

  • Linear Matrix Inequalities (LMI) framework [1990’s]
  • Efficient fast solvers and nice parser for Matlab [2000’s]
  • Adapted tool for robustness issues - examples of Lyapunov-based results

➌ Robust control theory at LAAS

  • RoMulOC - A Matlab toolbox
  • Applications in aerospace
  • Integral Quadratic Separation
  • D. Peaucelle

37 Seminar at Kyoto Univ. - March 12, 2010

slide-39
SLIDE 39

➌ Robust control theory at LAAS

■ ”Helicopter“ example

  • System defined at maximal value of parameters

>> sysmax = ssmodel( ’Helicopter’ ); >> sysmax.A = [0 1 0 ;0 0 1;0 -2.8 -0.14]; >> sysmax.Bw = [0;0;-14]; >> sysmax.Bu = [0;0;8]; >> sysmax.Dzu = 1 name: Helicopter n=3 mw=1 mu=1 n=3 dx = A*x + Bw*w + Bu*u pz=1 z = Dzu*u continuous time ( dx : derivative operator )

  • D. Peaucelle

38 Seminar at Kyoto Univ. - March 12, 2010

slide-40
SLIDE 40

➌ Robust control theory at LAAS

  • System defined at maximal value of parameters

>> sysmax = ssmodel( ’Helicopter’ ); >> sysmax.A = [0 1 0 ;0 0 1;0 -2.8 -0.14]; >> sysmax.Bw = [0;0;-14]; >> sysmax.Bu = [0;0;8]; >> sysmax.Dzu = 1;

  • System defined at minimal value of parameters

>> sysmin = ssmodel( ’Helicopter’ ); >> sysmin.A = [0 1 0 ;0 0 1;0 -3 -0.2]; >> sysmin.Bw = [0;0;-14]; >> sysmin.Bu = [0;0;8]; >> sysmin.Dzu = 1 name: Helicopter n=3 mw=1 mu=1 n=3 dx = A*x + Bw*w + Bu*u pz=1 z = Dzu*u continuous time ( dx : derivative operator )

  • D. Peaucelle

39 Seminar at Kyoto Univ. - March 12, 2010

slide-41
SLIDE 41

➌ Robust control theory at LAAS

  • Uncertain system defined as interval of max and min

>> usys = uinter( sysmin, sysmax ) Uncertain model : interval 2 param

  • ------- WITH --------

name: Helicopter n=3 mw=1 mu=1 n=3 dx = A*x + Bw*w + Bu*u pz=1 z = Dzu*u continuous time ( dx : derivative operator )

  • Interval model converted to polytopic model

>> usys = u2poly( usys ) Uncertain model : polytope 4 vertices

  • ------- WITH --------

name: Helicopter n=3 mw=1 mu=1 n=3 dx = A*x + Bw*w + Bu*u pz=1 z = Dzu*u continuous time ( dx : derivative operator )

  • D. Peaucelle

40 Seminar at Kyoto Univ. - March 12, 2010

slide-42
SLIDE 42

➌ Robust control theory at LAAS

  • Declare a state-feedback design problem

>> quiz = ctrpb( ’state-feedback’, ’Lyap-unique’ ) control problem: STATE-FEEDBACK design Lyapunov function: UNIQUE (quadratic stability) No specified performance

  • Add an H∞ performance objective

>> quiz = quiz + hinfty( usys, 4 );

  • Add a pole location performance objective

>> r = region( ’plane’, -0.1 ) Half-plane such that: Re(z)<-0.1 >> quiz = quiz + dstability( usys, r )

  • Add an impulse-to-peak performance minimization objective

>> quiz = quiz + i2p( usys ) control problem: STATE-FEEDBACK design Lyapunov function: UNIQUE (quadratic stability) Specified performances / systems: # Hinfty < 4 / Helicopter # D-stability / Helicopter # minimize I2P / Helicopter

  • D. Peaucelle

41 Seminar at Kyoto Univ. - March 12, 2010

slide-43
SLIDE 43

➌ Robust control theory at LAAS

  • The quiz object

>> quiz control problem: STATE-FEEDBACK design Lyapunov function: UNIQUE (quadratic stability) Specified performances / systems: # Hinfty < 4 / Helicopter # D-stability / Helicopter # minimize I2P / Helicopter

  • Contains decision variables

>> quiz.vars [3x3 sdpvar] ’Lyapunov matrix’ [1x3 sdpvar] ’S=-K*P’ [1x1 sdpvar] ’S-procedure scaling’ [1x1 sdpvar] ’g > (I2P cost)ˆ2’

  • D. Peaucelle

42 Seminar at Kyoto Univ. - March 12, 2010

slide-44
SLIDE 44

➌ Robust control theory at LAAS

  • Constrained by LMIs

>> quiz.lmi ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ | ID| Constraint| Type| Tag| ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ | #1| Numeric value| Matrix inequality 3x3| Lyap >0| | #2| Numeric value| Matrix inequality 4x4| Var Lyap <0| | #3| Numeric value| Matrix inequality 4x4| Var Lyap <0| | #4| Numeric value| Matrix inequality 4x4| Var Lyap <0| | #5| Numeric value| Matrix inequality 4x4| Var Lyap <0| | #6| Numeric value| Matrix inequality 3x3| Var Lyap <0| | #7| Numeric value| Matrix inequality 3x3| Var Lyap <0| | #8| Numeric value| Matrix inequality 3x3| Var Lyap <0| | #9| Numeric value| Matrix inequality 3x3| Var Lyap <0| | #10| Numeric value| Matrix inequality 3x3| Constraint 1| | #11| Numeric value| Matrix inequality 4x4| Constraint 2| | #12| Numeric value| Matrix inequality 3x3| Constraint 3| | #13| Numeric value| Element-wise 1x1| Constraint 4| | #14| Numeric value| Matrix inequality 3x3| Constraint 1| | #15| Numeric value| Matrix inequality 4x4| Constraint 2| | #16| Numeric value| Matrix inequality 3x3| Constraint 3| | #17| Numeric value| Element-wise 1x1| Constraint 4|

  • D. Peaucelle

43 Seminar at Kyoto Univ. - March 12, 2010

slide-45
SLIDE 45

➌ Robust control theory at LAAS

| #18| Numeric value| Matrix inequality 3x3| Constraint 1| | #19| Numeric value| Matrix inequality 4x4| Constraint 2| | #20| Numeric value| Matrix inequality 3x3| Constraint 3| | #21| Numeric value| Element-wise 1x1| Constraint 4| | #22| Numeric value| Matrix inequality 3x3| Constraint 1| | #23| Numeric value| Matrix inequality 4x4| Constraint 2| | #24| Numeric value| Matrix inequality 3x3| Constraint 3| | #25| Numeric value| Element-wise 1x1| Constraint 4| ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

  • D. Peaucelle

44 Seminar at Kyoto Univ. - March 12, 2010

slide-46
SLIDE 46

➌ Robust control theory at LAAS

  • And can be solved (SeDuMi solver by default)

>> K = solvesdp( quiz ) SeDuMi 1.1R3 by AdvOL, 2006 and Jos F. Sturm, 1998-2003. Alg = 2: xz-corrector, theta = 0.250, beta = 0.500 eqs m = 11, order n = 76, dim = 250, blocks = 22 nnz(A) = 282 + 0, nnz(ADA) = 117, nnz(L) = 64 it : b*y gap delta rate t/tP* t/tD* feas cg cg prec 0 : 6.96E+01 0.000 1 :

  • 1.79E+02 1.85E+01 0.000 0.2657 0.9000 0.9000
  • 0.09

1 1 5.0E+02 2 :

  • 1.05E+02 5.96E+00 0.000 0.3223 0.9000 0.9000

1.55 1 1 1.1E+02 3 :

  • 2.56E+01 1.38E+00 0.000 0.2312 0.9000 0.9000

1.73 1 1 1.9E+01 4 :

  • 5.54E+00 2.62E-01 0.000 0.1902 0.9000 0.9000

1.21 1 1 3.2E+00 5 :

  • 1.84E+00 8.00E-02 0.000 0.3050 0.9000 0.9000

1.29 1 1 8.3E-01 6 :

  • 7.08E-01 2.90E-02 0.000 0.3621 0.9000 0.9000

1.35 1 1 2.6E-01 7 :

  • 2.95E-01 1.05E-02 0.000 0.3637 0.9000 0.9000

1.27 1 1 8.3E-02 8 :

  • 2.30E-01 3.57E-03 0.000 0.3393 0.9000 0.9000

1.12 1 1 2.7E-02 9 :

  • 1.97E-01 6.73E-04 0.000 0.1882 0.9000 0.9000

1.00 1 1 5.1E-03 10 :

  • 1.91E-01 2.02E-05 0.000 0.0300 0.9900 0.9900

0.98 1 1 1.6E-04 11 :

  • 1.91E-01 1.13E-06 0.000 0.0558 0.9900 0.9900

1.00 1 1 8.7E-06 12 :

  • 1.91E-01 3.08E-07 0.000 0.2737 0.9000 0.9000

1.00 1 1 2.4E-06 13 :

  • 1.91E-01 1.33E-08 0.000 0.0433 0.9900 0.9900

1.00 1 1 1.0E-07 14 :

  • 1.91E-01 3.01E-09 0.000 0.2261 0.9000 0.9000

1.00 2 2 2.3E-08

  • D. Peaucelle

45 Seminar at Kyoto Univ. - March 12, 2010

slide-47
SLIDE 47

➌ Robust control theory at LAAS

15 :

  • 1.91E-01 7.53E-10 0.000 0.2498 0.9000 0.9000

1.00 2 2 5.8E-09 16 :

  • 1.91E-01 4.50E-11 0.087 0.0598 0.9900 0.9900

1.00 2 2 3.5E-10 iter seconds digits c*x b*y 16 0.4 Inf -1.9059654950e-01 -1.9059654919e-01 |Ax-b| = 3.6e-10, [Ay-c]_+ = 2.6E-11, |x|= 5.0e-01, |y|= 3.7e+02 Detailed timing (sec) Pre IPM Post 1.800E-01 4.000E-01 7.000E-02 Max-norms: ||b||=1, ||c|| = 196, Cholesky |add|=0, |skip| = 0, ||L.L|| = 42153.4. Feasibility is not strictly determined Worst constraint residual is -2.59066e-11 < 0 0.436574 (=sqrt(double(CTRPB.vars{4}))) may be a guaranteed I2P norm K = 0.0442 0.0091 0.0305

  • D. Peaucelle

46 Seminar at Kyoto Univ. - March 12, 2010

slide-48
SLIDE 48

➌ Robust control theory at LAAS

■ Welcome to new RoMulOC users http://www.laas.fr/OLOCEP/romuloc

  • Version 1 - started in june 2005

Contains robust analysis results presented in this talk. LMIs are coded using YALMIP parser, all available SDP solvers can be used.

  • Version 2 - started in february 2007

Includes design facilities for robust multi-objective state-feedback. Coding LMI results for full order output-feedback design is planned.

  • Version 3 - maybe this year

Would include heuristic tools for solving static output-feedback design problems.

  • D. Peaucelle

47 Seminar at Kyoto Univ. - March 12, 2010

slide-49
SLIDE 49

Outline ➊ Robustness in control theory

  • Robustness properties of the feedback loop
  • Some classical measures of robustness
  • Disturbance rejection and robustness to parametric uncertainty - a modeling issue
  • Robustness: tradeoff between complexity of systems (non-linear, etc.) and simplicity of models

➋ Optimization based tools

  • Linear Matrix Inequalities (LMI) framework [1990’s]
  • Efficient fast solvers and nice parser for Matlab [2000’s]
  • Adapted tool for robustness issues - examples of Lyapunov-based results

➌ Robust control theory at LAAS

  • RoMulOC - A Matlab toolbox
  • Applications in aerospace
  • Integral Quadratic Separation
  • D. Peaucelle

48 Seminar at Kyoto Univ. - March 12, 2010

slide-50
SLIDE 50

➌ Robust control theory at LAAS

■ 2000-2003 PIROLA Project

  • Robust performance analysis (LFT models & PDLF)

z

∆

w

∆

w z

∆

Σ

  • Multi-objective control design (”dilated LMI” method)

✛ ✛ K

1 1

Π Π K Π

2 2 3 3

K

Σ Σ Σ

”Multi-objective H2/H∞/impulse-to-peak control of a space launch vehicle”, D. Arzelier, B. Clement, D. Peaucelle, European Journal of Control, Vol 12, nb 1, pages 57-70, 2006

  • D. Peaucelle

49 Seminar at Kyoto Univ. - March 12, 2010

slide-51
SLIDE 51

➌ Robust control theory at LAAS

■ 2006 SNECMA Turbofan engines

  • Robust performance analysis (LFT models)

z

∆

w

∆

w z

∆

Σ

  • Analysis of given operating points (uncertain linear models)

▲ Results using RoMulOC

− + +

z w

+− −

z w

+ +

  • D. Peaucelle

50 Seminar at Kyoto Univ. - March 12, 2010

slide-52
SLIDE 52

➌ Robust control theory at LAAS

■ 2009 Airbus civil aircraft - longitudinal motion

  • Analysis of given operating points (uncertain linear models)
  • Robust performance analysis (LFT & polytopic models)

[v]

Σ

[2]

Σ Σ

[1]

Σ(∆)

z

∆

w

∆

w z

∆

Σ

▲ Results using RoMulOC ▲ Two types of PDLF P(∆) = ξvP [v]

; P(∆) =

  • 1

∆T

  • ˆ

P   1 ∆  

Centre de gravit´ e Distance (L) Profondeur Force (F) Mouvement r´ esultant Stabilisateur horizontal

  • D. Peaucelle

51 Seminar at Kyoto Univ. - March 12, 2010

slide-53
SLIDE 53

➌ Robust control theory at LAAS

  • Results to be presented in Yokohama 2010

“Robust Analysis of the Longitudinal Control of a Civil Aircraft using RoMulOC”,

  • G. Chevarria, D. Peaucelle, D. Arzelier, G. Puyou,

IEEE International Symposium on Computer-Aided Control System Design, 8-10 September 2010

▲ Stability analysis

633 flight points 12 not robustly stable (unstable vertex), 548 proved robustly stable using the quad-poly test (P(∆) = P ) Remaining 73 are proved to be robustly stable using either PDLF tests. Table 1: LMI sizes and times for stability tests

  • No. of vars
  • No. of rows

Mean time quad-poly 45 64 0.34s PDLF-poly 432 118 0.94s PDLF-LFT 456 135 2.08s

  • D. Peaucelle

52 Seminar at Kyoto Univ. - March 12, 2010

slide-54
SLIDE 54

➌ Robust control theory at LAAS

▲ Robust pole location: sector (to measure damping)

Table 2: Results for damping criterion

ψ%

Mean time Less conservative quad-poly 37.889% 2.77s PDLF-poly 10.590% 6.84s 136 PDLF-LFT 10.639% 9.85s 143 Table 3: Damping criterion for two particular flight points

ψ∗(i) i ψm(i)

quad-poly PDLF-poly PDLF-LFT 565 0.5932 0.3001 0.5182 0.5417 252 0.7384 0.3858 0.6640 0.6048

  • D. Peaucelle

53 Seminar at Kyoto Univ. - March 12, 2010

slide-55
SLIDE 55

➌ Robust control theory at LAAS

▲ Robust H2 performance

Table 4: Results for robust H2 cost

γ2%

Mean time Less conservative quad-poly 130.10% 0.63s PDLF-poly 5.97% 2.77s 605 PDLF-LFT 15.01% 7.22s 16

▲ Robust H∞ performance

Table 5: Results for robust H∞ cost

γ∞%

Mean time Less conservative quad-poly 781.41% 0.80s PDLF-poly 3.56% 2.43s 315 PDLF-LFT 5.81% 34.65s 32

  • D. Peaucelle

54 Seminar at Kyoto Univ. - March 12, 2010

slide-56
SLIDE 56

➌ Robust control theory at LAAS

■ 2008-2009 CNES - attitude control DEMETER satellite

  • Uncertain model from 1 to 3 axes and from 1 to 5 flexible modes
  • Robust performance analysis (LFT models)

z

∆

w

∆

w z

∆

Σ

  • Find ’largest’ uncertain domains - improve controller validation

▲ Results using RoMulOC

  • Results to be presented in Nara 2010

“Robust analysis of Demeter benchmark via quadratic separation”,

  • D. Peaucelle, A. Bortott, F. Gouaisbaut, D. Arzelier, C. Pittet,

IFAC Symposium on Automatic Control in Aerospace, 6-10 September 2010.

  • D. Peaucelle

55 Seminar at Kyoto Univ. - March 12, 2010

slide-57
SLIDE 57

➌ Robust control theory at LAAS

▲ Tests for axis 1 with 1 flexible mode ▲ 3 uncertainties: inertia δJ, damping δζ, natural frequency δω ; scaled ∈ [−1 1] ▲ Heuristic algorithm to find ’biggest’ polytope for which robust stability is guaranteed

  • Test with P(∆) = P
  • D. Peaucelle

56 Seminar at Kyoto Univ. - March 12, 2010

slide-58
SLIDE 58

➌ Robust control theory at LAAS

  • Test with P(∆) = P
  • Test with P(∆)

=

  • 1

∆T

  • ˆ

P   1 ∆  

  • D. Peaucelle

57 Seminar at Kyoto Univ. - March 12, 2010

slide-59
SLIDE 59

➌ Robust control theory at LAAS

  • Test with P(∆)

=

  • 1

∆

T ˆ P

  • 1

∆

  • Test with P(∆)

=

  • 1

∆

T ˆ P

  • 1

∆

  • modified heuristic
  • D. Peaucelle

58 Seminar at Kyoto Univ. - March 12, 2010

slide-60
SLIDE 60

➌ Robust control theory at LAAS

  • Test with P(∆)=
  • 1

∆

T ˆ P

  • 1

∆

  • & destabilizing values obtained by grid

▲ Results are conservative: choice of P(∆) & due to method for solving finite dimensional pb ▲ Expected improvement for better choices of P(∆) (higher order polynomial)

  • D. Peaucelle

59 Seminar at Kyoto Univ. - March 12, 2010

slide-61
SLIDE 61

➌ Robust control theory at LAAS

  • Test with P(∆)=
  • 1

∆ ∆2

T ˆ P

  • 1

∆ ∆2

  • and P(∆)=

 

1 ∆ ∆2 ∆3

 

T

ˆ P  

1 ∆ ∆2 ∆3

  ▲ LMI computation time drastically increased ▲ Heuristic algorithm converges in fewer step (global time not significantly impacted)

  • D. Peaucelle

60 Seminar at Kyoto Univ. - March 12, 2010

slide-62
SLIDE 62

Outline ➊ Robustness in control theory

  • Robustness properties of the feedback loop
  • Some classical measures of robustness
  • Disturbance rejection and robustness to parametric uncertainty - a modeling issue
  • Robustness: tradeoff between complexity of systems (non-linear, etc.) and simplicity of models

➋ Optimization based tools

  • Linear Matrix Inequalities (LMI) framework [1990’s]
  • Efficient fast solvers and nice parser for Matlab [2000’s]
  • Adapted tool for robustness issues - examples of Lyapunov-based results

➌ Robust control theory at LAAS

  • RoMulOC - A Matlab toolbox
  • Applications in aerospace
  • Integral Quadratic Separation
  • D. Peaucelle

61 Seminar at Kyoto Univ. - March 12, 2010

slide-63
SLIDE 63

➌ Robust control theory at LAAS

■ Topological Separation

[Safonov 80] G (z, w)=0

z w

z z w w F (w, z)=0

■ LTI case: Quadratic Separation

[Iwasaki, Hara 95]

■ Today: Quadratic Separation for implicit linear applications

[Automatica 2007]

z w w z

  • Well-posedness

∃¯ γ > 0 : ∀(¯ z, ¯ w) ∀∇ ∈ ∇ ∇ ,

 Ez w  

  • ≤ ¯

γ

 ¯ z ¯ w  

  • D. Peaucelle

62 Seminar at Kyoto Univ. - March 12, 2010

slide-64
SLIDE 64

➌ Robust control theory at LAAS

  • Robust performance

z

∆

w

∆

w z

∆

Σ

⇔

Well-posedness

z w w z

▲ H∞ performance of ˙ x = Ax + Bww , z = Czx + Dzww is smaller than γ

equivalent to well-posedness of

1

  • E

  ˙ x z   =   A Bw Cz Dzw  

  • A

  x w   ,   x w   =   I ∆  

  • ∇

  ˙ x z   : ∆ ≤ 1 γ ▲ Other performances? - Yes for impulse-to-norm and impulse-to-peak

[ECC’09]

  • ∇ can contain delays

w(t) = [δτz](t) = z(t − τ)

[Gouaisbaut]

▲ Other operators? - dead-zone, saturation ... ■ Quadratic separation is a versatile framework

  • D. Peaucelle

63 Seminar at Kyoto Univ. - March 12, 2010

slide-65
SLIDE 65

➌ Robust control theory at LAAS

■ Integral Quadratic Separation

[Automatica’07, CDC’08, ROCOND’09] Well posedness of

Ez(t) = Aw(t) , w(t) = [∇z](t) ∇ ∈ ∇ ∇

where E = E1E2 with E1 full column rank, iff ∃Θ Integral Quadratic Separator, solution of LMI

  • E1

−A ⊥∗ Θ

  • E1

−A ⊥ > 0

and Integral Quadratic Constraint (IQC) ∀∇ ∈ ∇

∇ ∞   E2z(t) [∇z](t)  

∗

Θ   E2z(t) [∇z](t)   dt ≤ 0 ▲ Conservatism comes in the choice of Θ solution to the IQC ▲ Conservatism reduction? - Possible using redundant system modeling

  • D. Peaucelle

64 Seminar at Kyoto Univ. - March 12, 2010

slide-66
SLIDE 66

➌ Robust control theory at LAAS

■ Links between Lyapunov based results and Quadratic Separation

  • P defines the separator with respect to integrator I (with zero initial conditions)

∞   ˙ x(t) [I ˙ x](t)  

T 

 −P − P     ˙ x(t) [I ˙ x](t)   dt = limτ→∞ τ

0 − ˙

xT (t)Px(t) − xT (t)P ˙ x(t)dt = limτ→∞ xT (0)Px(0)

  • =0

−xT (τ)Px(τ) ≤ 0

  • In PDLF results with P(∆) =
  • 1

∆T

C

  • ˆ

P

  • 1

∆C

  • ,

ˆ P defines a separator with respect to I on an augmented state   x z∆   = I   ˙ x ˙ z∆  

  • Conservatism reduction methods in Lyapunov framework ⇔ System augmentation in QS

▲ System augmentation implies to work with descriptor systems

  • D. Peaucelle

65 Seminar at Kyoto Univ. - March 12, 2010

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SLIDE 67

Conclusions ■ Robustnes: important issue in control theory

  • Robustness results have impact for many other problems

■ LMI: central tool for robustness

  • Effective to transfer theory to industrial applications with software

■ Quadratic Separation framework not yet fully exploited

  • Extensions to non-linearities, time-varying operators...
  • Descriptor version of RoMulOC: Romuald
  • D. Peaucelle

66 Seminar at Kyoto Univ. - March 12, 2010