Some results related to modeling systems with uncertainty Neskolko - - PowerPoint PPT Presentation

Some results related to modeling systems with uncertainty

Neskolьko rezulьtatov ob modelirovanie sistem s neopredel¨ ennostmi Dimitri PEAUCELLE - Dmitri i Жanoviq Poselь LAAS-CNRS - Université de Toulouse - FRANCE Sankt-Peterburg Dekabrь 2012

Introduction ■ Outline ❶ Motivations for uncertain descriptor modeling ❷ Affine polytopic models ❸ LFT models - frequency dependent models ❹ Augmented descriptor models and conservatism reduction

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❶ Motivations for uncertain descriptor modeling

■ Models issued from physics are naturally in descriptor form E ˙ x = Ax + Bu

• Example: mechanical system

M ¨ θ + C ˙ θ + Kθ = u M: inertia ; C: friction ; K: stiffness ; u external forces   M O O I     ¨ θ ˙ θ   =   −C −K I O     ˙ θ θ   +   I O   u

• Example: robotic systems with Lagrange formulations [MG89]

E may not be invertible

• Example: networks of systems with algebraic constraints describing links
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❶ Motivations for uncertain descriptor modeling

■ Descriptor models can be converted to usual models

• Example: mechanical system

¨ θ + M −1C ˙ θ + M −1Kθ = M −1u

• Assumes that M −1 is well-conditioned and known
• If some parameters are unknown: M(∆), C(∆), K(∆)

¨ θ + M −1(∆)C(∆) ˙ θ + M −1(∆)K(∆)θ = M −1(∆)u

• Increased complexity of the model

■ Descriptor models are preferable for describing systems with uncertainty

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❶ Motivations for uncertain descriptor modeling

■ Example: DC motor I ˙ ω = bu regulated in speed u = −ω

• Parameters are assumed uncertain I = 1 + δ1, b = 1 + δ2

(1 + δ1) ˙ ω = −(1 + δ2)ω ⇒ ˙ ω = −1 + δ2 1 + δ1 ω

• Model is rational w.r.t. uncertainties: exists an LFT representation

z∆ w

∆

∆

Σ

−1 + δ2 1 + δ1 = A + B∆∆(I − D∆∆)−1C∆ =   A B∆ C∆ D∆   ⋆ ∆ ▲ Can be build with Robust Control toolbox of Matlab or LFRT [Mag05]

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❶ Motivations for uncertain descriptor modeling

• Building the LFT for ˙

ω = − 1+δ2

1+δ1ω

▲ 1st step: descriptor form with no denominators ˙ ω + δ1 ˙ ω = −ω − δ2ω ▲ 2nd step: all multiplications correspond to a feedback ˙ ω + w1 = −ω − w2 : w1 = δ1z1 z1 = ˙ ω , w1 = δ2z2 z2 = ω ▲ 3rd step: descriptor LFT: ∆ =

• δ1

δ2

• , z∆ =
• z1

z2

• , w∆ =
• w1

w2

• ,

 

1 −1 1 1

 

• ˙

ω z∆

• =

 

−1 −1 −1 1

 

• ω

w∆

• , w∆ = ∆z∆

▲ last step: invert the left-hand side matrix

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❶ Motivations for uncertain descriptor modeling

• LFT for ˙

ω = − 1+δ2

1+δ1ω − 1+δ2

1+δ1 =

    1 −1 1 1    

−1 

   −1 −1 −1 1     ⋆   δ1 δ2   =     −1 −1 −1 −1 −1 −1 1     ⋆   δ1 δ2   = −1 +

• −1

−1

• 

 δ1 δ2    I −   −1 −1     δ1 δ2    

−1 

 −1 1   = −1 −

• δ1

δ2

• 

 1 + δ1 δ2 1  

−1 

 −1 1   = −1 −

• δ1

δ2

• 

 − 1+δ2

1+δ1

1   = −1 + δ1−δ2

1+δ1 = − 1+δ2 1+δ1

■ The example shows the interest of descriptor models,

even if only for technical manipulations

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❶ Motivations for uncertain descriptor modeling

■ All fractional models have affine descriptor representations [MAS03] ▲ Proof: All fractional models can be converted to an LFT   ˙ x z∆   =   A B∆ C∆ D∆     x w∆   , w∆ = ∆z∆ ▲ the LFT gives the affine descriptor form:   I −B∆∆ O I − D∆∆     ˙ x z∆   =   A C∆   x

• Can give representations of smaller dimensions

▲ Example  

1 δ1 δ2 1 + δ1 δ2 1

   

˙ ω z1 z2

  =  

−1 −1 1

  ω ⇔ (1 + δ1) ˙ ω = −(1 + δ2)ω

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❶ Motivations for uncertain descriptor modeling

■ General descriptor models Exx ˙ x + Exππ = Ax + Bu y + Exy ˙ x + Eyππ = Cx + Du

• x: state ; u: inputs
• π: linearly constrained signals
• Exx and A may not be square

▲ Can be converted to ˆ E ˙ ˆ x = ˆ Aˆ x+ ˆ Bu with ˆ E and ˆ A square and ˙ ˆ x =  

˙ x π λ

  ▲ Not recommend: increased size of the model

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Outline ❶ Motivations for uncertain descriptor modeling ❷ Affine polytopic models ❸ LFT models - frequency dependent models ❹ Augmented descriptor models and conservatism reduction

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❷ Affine polytopic models

■ All fractional models have affine descriptor representations Exx(∆) ˙ x + Exπ(∆)π = A(∆)x + B(∆)u y + Exy(∆) ˙ x + Eyπ(∆)π = C(∆)x + D(∆)u

• Models also used for polynomial non-linear models [CTF02]

■ General affine descriptor data   −Exx(∆) −Exπ(∆) A(∆) B(∆) −Eyx(∆) −Eyπ(∆) C(∆) D(∆)   = M(∆)

• Different classes of affine models

▲ intervals: M M(∆) M

(element-wise mij ≤ mij(∆) ≤ mij)

▲ parallelotopes: M(∆) = M |0| + ¯

p p=1 δpM |p|

: |δp| ≤ 1 ▲ polytopes: M(∆) ∈ co

• M [v=1...¯

v]

• intervals ⊂ parallelotopes ⊂ polytopes
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❷ Affine polytopic models

■ Example: two mass spring system

M1 M2 x1 u k f x2eq w1 w2

M1¨ x1 + f( ˙ x1 − ˙ x2) + k(x1 − x2) = u + w1 M2¨ x2 + f( ˙ x2 − ˙ x1) + k(x2 − x1) = w2

• General affine data model

  M1 f −f k −k M2 −f f −k k  

• =M(∆)

      

¨ x1 ¨ x2 ˙ x1 ˙ x2 x1 x2

       =

• u + w1

w2

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❷ Affine polytopic models

■ Example: two mass spring system - continued

• 4 uncertain parameters Mi ≤ Mi ≤ M i,

f ≤ f ≤ f, k ≤ k ≤ k M(∆) =

• M1

f −f k −k M2 −f f −k k

• Interval model generates conservatism (elements assumed independent)

  M1 f −f k −k M2 −f f −k k  

• 

 M1 f −f k −k M2 −f f −k k  

• 

 M1 f −f k −k M2 −f f −k k  

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❷ Affine polytopic models

■ Example: two mass spring system - continued M(∆) =

• M1

f −f k −k M2 −f f −k k

• Parallelotopic model

▲ Nominal model: M |0| (center of the intervals)

M(∆) = M|0| + δM1

1 2(M1 − M1)

  1  

. . .

+δk 1 2(k − k)   0 1 −1 −1 1  

• M|k|

▲ M |k| "axis" of variations along δk.

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❷ Affine polytopic models

■ Example: two mass spring system - continued M(∆) =

• M1

f −f k −k M2 −f f −k k

• Polytopic model: 24 = 16 vertices (δi = ±1 in parallelotopic model)

M [1] =

• M1

f −f k −k M2 −f f −k k

• M [2] =
• M1

f −f k −k M2 −f f −k k

• .

. .

M [16] =

• M1

f −f k −k M2 −f f −k k

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❷ Affine polytopic models

■ Some techniques to deal with affine uncertain models

• General type of analysis criteria

ηTΘ(∆)η < 0 , ∀η = 0 : M(∆)η = 0 ▲ Example: negative definite derivative of Lyapunov function V = xTP(∆)x ˙ V = ηT

• O

P(∆) P(∆) O

• η < 0, ∀η =
• ˙

x x

• = 0 :
• −I

A(∆)

• η = 0
• Usual LMI type result

(M ⊥(∆) matrix generating the null-space of M(∆))

M ⊥T(∆)Θ(∆)M ⊥(∆) < O ▲ Problem: M ⊥(∆) is a rational function of ∆ ▲ Solutions: SOS [HG05, Sch06], Polyá [CGTV09, OdOP08]

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❷ Affine polytopic models

■ Some techniques to deal with affine uncertain models

• General type of analysis criteria

ηTΘ(∆)η < 0 , ∀η = 0 : M(∆)η = 0

• Usual LMI type result

(M ⊥(∆) matrix generating the null-space of M(∆))

M ⊥T(∆)Θ(∆)M ⊥(∆) < O

• Slack variables results

(variables issued from the "creation" Finsler lemma)

∃F : Θ(∆) < FM(∆) + M T(∆)F T ▲ Conservative, but sufficient to test at the vertices ▲ Θ(∆) affine (parameter-dependent Lyapunov function) ▲ [OBG99, PABB00, EH04, PDSV09]

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Outline ❶ Motivations for uncertain descriptor modeling ❷ Affine polytopic models ❸ LFT models - frequency dependent models ❹ Augmented descriptor models and conservatism reduction

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❸ LFT models

■ Affine polytopic models well adapted for scalar parametric uncertainty ■ LFT suitable for other types of uncertainties:

• Frequency dependent uncertainties
• Linear / non-linear operators
• Time-varying parameters
• ...

z∆ w

∆

∆

Σ

■ Linear combination of repeated uncertainties [Sch06] ∆ =

• Jj(Irj ⊗ ∆j)Kj ,

[J1 · · · J¯

] orthonormal

▲ Example: bloc-diagonal uncertainties ∆ =  

∆1 δ2 δ2

 

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❸ LFT models

■ Dissipative structured parametric uncertainties

• An uncertain matrix ∆ is said to be {Ψ1, Ψ2, Ψ3}-dissipative if
• I

∆∗

• 

 Ψ1 Ψ2 Ψ∗

2

Ψ3     I ∆   ≤ O .

• An uncertain matrix ∆ is said to be {Φ1, Φ2, Φ3}-structured if
• I

∆∗

• 

 Φ1 Φ2 Φ∗

2

Φ3     I ∆   = O .

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❸ LFT models

• Examples of {Ψ1, Ψ2, Ψ3}-dissipative uncertainties

▲ Norm-bounded uncertainties are {−γ2I, O, I}-dissipative

• I

∆∗ −γ2I O O I I ∆

• = −γ2I + ∆∗∆ ≤ O .

w∆ = ∆z∆ ⇒ w∆ ≤ γz∆ ▲ Positive real uncertainties are {O, −I, O}-dissipative

• I

∆∗ O −I −I O I ∆

• = −(∆∗ + ∆) ≤ O .

w∆ = ∆z∆ ⇒ < w∆|z∆ > ≥ 0

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❸ LFT models

• Examples of {Ψ1, Ψ2, Ψ3}-dissipative uncertainties - continued

▲ Uncertainties in an "ellipsoid" are {∆∗

0Z∆0 − R, −∆∗ 0Z, Z}-dissipative

• I

∆∗ ∆∗

0Z∆0 − R

−∆∗

0Z

−Z∆0 Z I ∆

• = (∆ − ∆0)∗Z(∆ − ∆0) − R ≤ O .

(discs of the complex plane for scalar ∆ = δ)

▲ Uncertainties in a "half space" {−∆∗

0N + N∆0, N ∗, O}-dissipative

• I

∆∗ −∆∗

0N + N∆0

N∗ N O I ∆

• = (∆ − ∆0)∗N + N ∗(∆ − ∆0) ≤ O .

(half of the complex plane limited by a line for scalar ∆ = δ)

• {Φ1, Φ2, Φ3}-structured uncertainties: border of "ellipsoids" or of "half spaces"
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❸ LFT models

• {Ψ1, Ψ2, Ψ3}-dissipative {Φ1, Φ2, Φ3}-structured, examples

[IH05]

▲ δ = (jω)−1 with bounded frequencies ω ∈ [ ω ω ]: {2, −j(ω + ω), 2ωω}-dissipative {0, 1, 0}-structured ▲ δ = (jω)−1 with high frequencies ω ≥ ω ≥ 0: {0, −jω, 2ω2}-dissipative {0, 1, 0}-structured ▲ δ = e−jω with bounded frequencies ω ∈ [ ω ω ]: {2 cos ω−ω

2 , −ej ω+ω

2 , 0}-dissipative

{−1, 0, 1}-structured

• −ω
• ω

ω −j / −j / −ω

▲ δ ∈ R bounded δ ∈ [ δ δ ]: {2δδ, −(δ + δ), 2}-dissipative {0, j, 0}-structured

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❸ LFT models

■ System analysis in frequency intervals: s−1 = (jω)−1 , ω ∈ [ ω ω ]

• Allows loop-shaping type specifications
• Have interpretations in time domain [IHF05]
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❸ LFT models

■ Performances?

• Stability: no poles in C+ ⇔ well-posedness w.r.t. s−1 ∈ C+
• L2 perf: z2 ≤ γw2 ⇔ well-posedness w.r.t. w = ∇z : ∇2 ≤ 1

γ

▲ Can be tested for given frequencies or for frequency intervals

• Dissipativity:

∞

• z(t)

w(t)

∗

Ψ1 Ψ2 Ψ∗

2

Ψ3 z(t) w(t)

• dt ≥ 0

⇔ well-posedness w.r.t. w = ∇z : ∇ {Ψ1, Ψ2, Ψ3}-dissipative ■ Performances ⇔ well-posedness w.r.t. uncertain operators

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❸ LFT models

■ Well-posedness [Saf80]

z w w z

• For all bounded external signals ( ¯

w, ¯ z)

the internal signals (w, Ez) are unique and bounded

▲ Example: z = ˙ x, w = ∇ ˙ x = x (i.e. ∇ integrator).

Well posedness ⇔

∀ initial conditions ¯ w = x(0) and bounded disturbances ¯ z x is unique, no impulsive modes, no divergence (stability)

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❸ LFT models

■ Well-posedness [PAHG07]

z w w z

• Integral quadratic separation (IQS) result

▲ Assume E = E1E2 with E1 full column rank and let M =

• −E1

A

• ▲ The loop is well-posed iff there exists Θ such that

ηTΘη < 0 , ∀η = 0 : Mη = 0

(1) and for all “uncertainties"

∞   E2z(t) [∇z](t)  

∗

Θ   E2z(t) [∇z](t)   dt ≥ 0

(2)

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❸ LFT models

• IQS result

▲ (1): LMI-type constraint (M ⊥TΘM ⊥ < O) ▲ (2): Integral quadratic constraint (IQC)

• IQS applies easily to uncertainties defined, or included in IQCs

▲ Example: ∇ = Ir ⊗ ∆, ∆ is {Ψ1, Ψ2, Ψ3}-dissipative {Φ1, Φ2, Φ3}-structured Θ =   P ⊗ Ψ1 P ⊗ Ψ2 P ⊗ Ψ∗

2

P ⊗ Ψ3   +   R ⊗ Ψ1 R ⊗ Ψ2 R ⊗ Ψ∗

2

R ⊗ Ψ3   P > O R = R∗

(generalized version of D/G-scaling)

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❸ LFT models

■ IQC also defined in the frequency domain (Parseval) ∞   E2z(jω) [∇z](jω)  

∗

Θ   E2z(jω) [∇z](jω)   dω ≥ 0

(3)

• IQC formulas [MR97, JM99] apply for building IQS results
• ∇ can contain

▲ Saturations [FLR06, PTGSB12] ▲ Delays [JS01, GP07, PAHG07] ▲ ... to be continued ■ Parametrizing Θ solutions to the IQC is in general conservative [MSF97, IH05] ▲ Lossless formulas iff ∆ =

¯ 

• j=1

Jj(Irj⊗∆j)Kj    ¯  = 2 , r1 > 1 , r2 = 1 ¯  = 3 , r1 = r2 = r3 = 1

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Outline ❶ Motivations for uncertain descriptor modeling ❷ Affine polytopic models ❸ LFT models - frequency dependent models ❹ Augmented descriptor models and conservatism reduction

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❹ Augmented descriptor models and conservatism reduction

■ Except in special cases finding Θ solution to the IQCs is conservative

• Existing parameterizations of Θ apply both for constant and time-varying ∆
• But ∆

∆∞ ⊂ ∆ ∆d ⊂ ∆ ∆0 where ▲ ∆ ∆0 set of constant uncertainties such that Σ(∆) is stable ▲ ∆ ∆d set of uncertainties with ˙ ∆ ≤ d such that Σ(∆) is stable ▲ ∆ ∆∞ set of TV uncertainties with switches such that Σ(∆) is stable

• Reducing conservatism: can be achieved by introducing information about ˙

∆

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❹ Augmented descriptor models and conservatism reduction

■ Descriptor augmentation method - example

• Assume system in LFT form
• ˙

x = Ax + Bw∆ z∆ = Cx + Dw∆

⋆

• x = s−1 ˙

x w∆ = ∆z∆

• Add information about derivative of uncertainty:

˙ w∆ = ∆ ˙ z∆

• =w∆1

+ ˙ ∆z∆

• =w∆2
• Define derivative of exogenous signals:

˙ z∆ = C ˙ x + D ˙ w∆ = C ˙ x + Dw∆1 + Dw∆2 , w∆ = s−1 ˙ w∆

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❹ Augmented descriptor models and conservatism reduction

■ Descriptor augmentation method - example - continued

• ˙

x = Ax + Bw∆ z∆ = Cx + Dw∆

⋆

• x = s−1 ˙

x w∆ = ∆z∆

• All equations gathered separating linear constrains and "uncertainties"

                

˙ x = Ax + Bw∆ z∆ = Cx + Dw∆ ˙ z∆ − C ˙ x = Dw∆1 + Dw∆2 ˙ w∆ = w∆1 + w∆2 0 = w∆ − w∆

⋆                 

x = s−1 ˙ x w∆ = s−1 ˙ w∆ w∆ = ∆z∆ w∆1 = ∆ ˙ z∆ w∆2 = ˙ ∆z∆

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❹ Augmented descriptor models and conservatism reduction

■ Descriptor augmentation method - example - continued

• Descriptor LFT form of system augmented with information on derivatives

z =

• ˙

x ˙ w∆ z∆ ˙ z∆ T , w =

• x

w∆ w∆ w∆1 w∆2 T        

I O O O O O I O −C O O I O I O O O O O O

        z =        

A B O O O C D O O O O O O D D O O O I I O I −I O O

        w ⋆ w =     

s−1In+m O O O ∆ O O O ∆ O ˙ ∆ O

     z

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❹ Augmented descriptor models and conservatism reduction

■ IQS results applicable to the original & the augmented descriptor model

• LMI formulas for augmented model are less conservative
• IQS formulas implicitly contain Lyapunov certificates

▲ Lyapunov certificate in case of LMI for original model V = xTPx ▲ Parameter-dependent Lyapunov certificate in case of augmented system V =   x w∆  

T

P   x w∆   , w∆ = ∆(I − D∆)−1Cx

• The system augmentation technique is also proved efficient for systems with

delays [PAHG07], with saturations [PTGSB12]...

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Conclusions ■ Some advantages of descriptor models

• Natural models from physics
• More flexibility for system description
• Affine descriptor models are alternatives to LFT models
• Well-posedness of descriptor LFT loops

= general framework for robust analysis problems (includes frequency dependent specifications/uncertainties)

• System augmentation with derivatives = less conservative results

(made possible thanks to descriptor modeling)

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REFERENCES

References

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