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Modeli v Teorii Robastnogo Upravleni Dmitri i anoviq Konovalov - - PowerPoint PPT Presentation
Modeli v Teorii Robastnogo Upravleni Dmitri i anoviq Konovalov - - PowerPoint PPT Presentation
Modeli v Teorii Robastnogo Upravleni Dmitri i anoviq Konovalov Dmitry Peaucelle Dimitri i Posel Dimitri Peaucelle Tradicionna Xkola Upravlenie, Informaci i Optimizaci Pereslavl-Zalesski i In 2010
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Outline ➊ What is Robustness? ➋ Why concentrating on linear systems? ˙ x = f(x) − → ˙ x = A(∆)x + Bw(∆)w ➌ Affine polytopic-type models
K
[v] [2]
Σ Σ
[1]
Σ Σ(∆)
➍ LFT-type models
z
∆
w
∆
K
w z u y
Σ
∆ ➎ RoMulOC Toolbox
» sys = ussmodel( sys, delta )
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Pereslavlь-Zalesski i Inь 2010
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➊ What is Robustness?
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) and T(s)K(s) (for k = 10) phase margin > 90◦, gain margin = ∞ Robust !?
K
+−
Τ
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➊ What is Robustness?
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) and ˜ 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]. ▲ Stability of extremal values = Stability of interval
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➊ What is Robustness?
K
+−
Τ
k(s3+2.1s2+1.2s+0.1)(s2+0.1s+2.252) (s3+9s2+27s+27)s2(s2+0.2s+1.01)+k(s3+2.1s2+1.2s+0.1)(s2+0.1s+2.252)
=
k(s3+2.1s2+1.2s+0.1)(s2+0.1s+2.252) s7+a6s6+a5(k)s5+a4(k)s4+a3(k)s3+a2(k)s2+a1(k)s+a0(k)
▲ Uncertain characteristic polynomial included
in interval polynomial with independent coefficients: ai(k) = ai ≤ ai ≤ ai = ai(k)
s7 + a6s6 + a5(k)s5 + a4(k)s4 + a3(k)s3 + a2(k)s2 + a1(k)s + a0(k) = s7 + a6s6 + a5s5 + a4s4 + a3s3 + a2s2 + a1s + a0 ■ Kharitonov’s Theorem: stability of interval polynomial iff stability of four polynomials s7 + a6s6 + a5s5 + a4s4 + a3s3 + a2s2 + a1s + a0 s7 + a6s6 + a5s5 + a4s4 + a3s3 + a2s2 + a1s + a0 s7 + a6s6 + a5s5 + a4s4 + a3s3 + a2s2 + a1s + a0 s7 + a6s6 + a5s5 + a4s4 + a3s3 + a2s2 + a1s + a0 ▲ Conservative test when coefficients not independent
- Extendend for feedback loop of interval polynomials: 16/32 polynomials test
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➊ What is Robustness?
- The controller for k = 1 or k = 200 stabilizes both models T(s) and ˜
T(s). ▲ Does it hold in between ? T(s, ∆) =
s2+0.1s+∆ s2(s2+0.2s+1.01) = y u
K(s) = k s3+2.1s2+1.2s+0.1
s3+9s2+27s+27
=
u yr−y
K
+−
Τ
Define w∆ = ∆z∆ and z∆ = u, the feedback loop for yr = 0 writes as
z∆ w
∆
∆
Σ
Σk=1(s) =
s3+2.1s2+1.2s+0.1 s7+9.2s6+30.81s5+43.69s4+34.08s3+27.49s2+0.01s
Σk=200(s) =
200s3+420s2+240s+20 s7+9.2s6+229.8s5+481.5s4+314.7s3+71.27s2+2s
- Circle criterion for 1.002 ≤ ∆(t) ≤ 2.252
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➊ What is Robustness?
Step and frequency responses of closed-loop systems K ⋆ T and K ⋆ ˜
T for k = 200:
- K ⋆ ˜
T less "robust" to disturbances at frequencies ∈ [1 2]rad/s. ▲ Faulty behavior can be related to narrow peaks in the frequency domain
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➊ What is Robustness?
■ Two ‘definitions’ of robustness for non-linear systems
- Guarantee some characteristics of an output for a given class of input perturbations
z
Σ
y u w
K
z ∈ Z , ∀w ∈ W ▲ Many results using L2 norm: Z =
- z2 =
∞ z∗(t)z(t)dt ≤ γ2 ▲ Defines induced L2 norm of systems: Σ
u,y
⋆ K = max ||z||
||w|| =
min
||z||/||w||≤γ γ
▲ Invariant set problem (and others ?) can also be defined in this way
- Guarantee stability of
K (∆)
Σ
for a given set of uncertainties ∆ ∈ ∆
∆ ▲ Parametric uncertainty: ∆ is a vector of scalar, bounded, constant parameters ▲ Time-Varying: ∆(t) grasps TV and Non-Linear characteristics ▲ E.g. ∆ is a sector bounded operator defined on exogenous signals
e.g.
w∆(t) = [∆z∆](t) : ρz∆ ≤ w∆ ≤ ρz∆
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Outline ➊ What is Robustness? ➋ Why concentrating on linear systems? ˙ x = f(x) − → ˙ x = A(∆)x + Bw(∆)w ➌ Affine polytopic-type models
K
[v] [2]
Σ Σ
[1]
Σ Σ(∆)
➍ LFT-type models
z
∆
w
∆
K
w z u y
Σ
∆ ➎ RoMulOC Toolbox
» sys = ussmodel( sys, delta )
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➋ Why concentrating on linear systems?
■ 1st Lyapunov Method viewpoint ˙ x = f(x)
Linearization around equilibrium xe
= ⇒ ˙ X = AX
Locally asymptotically stable
⇐ =
Asymptotically stable
- Properties on approximated model kept true for original model: robustness
▲ Holds because asymptotic stability kept true for small variations ˙ X = (A + ∆)X , ||∆|| ≪ 1.
- Need to define precisely ∆ and ||∆|| ≪ 1
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➋ Why concentrating on linear systems?
■ Including model viewpoint ˙ x = f(x) : x ∈ X = ⇒ ˙ x = A(∆(t))x + w(t) : (∆, ˙ ∆) ∈ ∆ ∆ × ∆ ∆d , w ∈ W
- Example of a 1 axis rotating mechanical system with input u and perturbation θ
¨ ǫ(t) = −a1 ˙ ǫ(t) − a0 sin(ǫ(t) − ǫ0) + b cos θ(t) · u(t)
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➋ Why concentrating on linear systems?
■ Including model viewpoint (appropriate for Lipschitz function, or at least locally) ˙ x = f(x) : x ∈ X = ⇒ ˙ x = A(∆(t))x + w(t) : (∆, ˙ ∆) ∈ ∆ ∆ × ∆ ∆d , w ∈ W
- Example of a 1 axis rotating mechanical system with input u and perturbation θ
¨ ǫ(t) = −a1 ˙ ǫ(t) − a0 sin(ǫ(t) − ǫ0) + b cos θ(t) · u(t) ▲ Assumption: Model is valid only for ǫ ∈ [−10◦ 20◦], θ ∈ [−45◦ 45◦]. ▲ Errors in identification (parametric uncertainty): a1 ∈ [0.02 0.15], a0 ∈ [2 5], ǫ0 ∈ [−5◦ 5◦], b ∈ [0.1 0.3].
- Uncertain time-varying linear model
¨ ǫ(t) = −a1 ˙ ǫ(t) − ˜ a0(t)ǫ(t) + w(t) + ˜ b(t)u(t) ˜ a0(t) = a0 cos ǫ0
sin ǫ(t) ǫ(t)
∈ [1.95 5] , w(t) = a0 cos ǫ(t) sin ǫ0 ∈ [−0.0131 0.0131] ˜ b(t) = b cos θ(t) ∈ [0.0707 0.3] ▲ May be possible to add information on the derivatives of the uncertainties
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➋ Why concentrating on linear systems?
■ Input-to-output viewpoint ˙ x = f(x, 0) ˙ x = f(x, w) , z = g(x)
Locally asymptotically stable
= ⇒ z ≤ γ if w ≤ α ≪ 1
Bounded output for small perturbations
▲ How to characterize the performances?
- Many criteria for linear systems
- H∞, H2 with interpretations in time-domain, frequency domain, stochastic
- Other criteria such as impulse-to-peak (invariant set)
- Pole location: information on dynamics of modes
▲ Much attention has been paid to robust performance problems of linear MIMO systems ˙ x = f(x, w) , z = g(x) ˙ X = A(∆)X + Bw(∆)w , z = C(∆)X = ⇒ : ||z||/||w|| ≤ γ
guaranteed performance: max
∆ min γ
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Outline ➊ What is Robustness? ➋ Why concentrating on linear systems? ˙ x = f(x) − → ˙ x = A(∆)x + Bw(∆)w ➌ Affine polytopic-type models
K
[v] [2]
Σ Σ
[1]
Σ Σ(∆)
➍ LFT-type models
z
∆
w
∆
K
w z u y
Σ
∆ ➎ RoMulOC Toolbox
» sys = ussmodel( sys, delta )
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➌ Affine polytopic-type models
■ Interval polynomial SISO models (in Kharitonov’s like results) b0 + b1s + b2s2 · · · + bmsm a0 + a1s + a2s2 · · · + ansn
,
bi ≤ bi ≤ bi ai ≤ ai ≤ ai ▲ Restricted to SISO systems ▲ Coefficients assumed independent one from the other ▲ Only for parametric uncertainty: Constant uncertainties ▲ Structure not preserved by a change of basis a0 + a1s + a2s2 + a3s3 = a3(s + 1)3 + (a2 − 3a3)(s + 1)2 + (a1 − 2a2 + 3a3)(s + 1) + (a0 − a1 + a2 − a3)
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➌ Affine polytopic-type models
■ Interval polynomial SISO models (in Kharitonov’s like results) b0 + b1s + b2s2 · · · + bmsm a0 + a1s + a2s2 · · · + sn
,
bi ≤ bi ≤ bi ai ≤ ai ≤ ai
- Controller canonical form
˙ x = Ax + Buu y = Cyx 1 · · ·
. . . ... ... . . .
1 −a0 · · · −an−2 −an−1 A 1 · · ·
. . . ... ... . . .
1 −a0 · · · −an−2 −an−1
- b0
· · · bn−2 bn−1
- Cy
- b0
· · · bn−2 bn−1
- ▲ Element-wize matrix inequalities
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➌ Affine polytopic-type models
■ State space MIMO interval model ˙ x z y = A Bw Bu Cz Dzw Dzu Cy Dyw Dyu x w u = M x w u , M M M ▲ Coefficients assumed independent one from the other
- Time-varying uncertainties can be considered: M d ˙
M M d 0 ≤ ˙ mij ≤ 0 for constant (parametric) uncertainties −∞ ≤ ˙ mij ≤ +∞ if no information on the derivative ▲ Structure not preserved by a change of basis:
Coefficients of TAT −1 are not independent
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➌ Affine polytopic-type models
- M M M with ¯
p uncertain parameters, also writes as M = 1 2(M + M)
- M |0|
+
¯ p
- p=1
δp 1 2(mij − mij)Eij
- M |p|
, |δp| ≤ 1 ■ State space MIMO parallelotopic model ˙ x z y = M(δ) x w u , M(δ) = M |0| +
¯ p
- p=1
δpM |p| , |δp| ≤ 1
- Extends interval models to models with ‘axes’ M |p| of rank = 1.
- ‘Nominal’ model M |0| at the center of the set.
- Structure preserved by a change of basis.
- Time-varying uncertainties can be considered.
- δp may be directly related to physical values.
▲ Limited geometry of M(δ).
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➌ Affine polytopic-type models
- M(δ) = M |0| + ¯
p p=1 δpM |p| with ¯
p uncertain parameters, also writes as M(ζ) =
¯ v
- v=1
ζvM [v] : ζv=1...¯
v ≥ 0 , ¯ v
- v=1
ζv = 1
where M [v] = M |0| + ¯
p p=1 ±M |p| are the ¯
v = 2¯
p extremal points of the parallelotope.
■ State space MIMO polytopic model ˙ x z y = M(ζ) x w u , M(ζ) ∈ CO
- M [1], M [2], . . . M [¯
v]
- Convex hull of ¯
v vertices (¯ v may be < 2¯
p: see partitioning techniques).
- Natural modeling of systems identified on a finite number of operating points.
- Many control methods: tests on vertices asses robust stabilities of the whole polytope
(see next lecture)
▲ Physical meaning of parameters is lost. ▲ Time-varying uncertainties difficult to define.
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➌ Affine polytopic-type models
- Example: Uncertain modeling of longitudinal motion of aircraft.
Centre de gravit´ e Distance (L) Profondeur Force (F) Mouvement r´ esultant Stabilisateur horizontal
Vc (kts) Mach Mach MAX Vc MAX Altitude min A l t i t u d e M A X Vc min
Non-linear model around flight point i, approximated by uncertain linear model defined as convex hull of the neighbors in the parameter space:
Mθi(ζ) = CO
- Mθj : ||θj − θi|| ≤ α
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➌ Affine polytopic-type models
▲ Main limitation: modeling limited to affine parametric dependence.
- Proposed solution for rational models [Trofino]: descriptor-type modeling
- Example
˙ x =
δ1
2
1+δ2 x ⇔
1 ˙ x + −δ1 1 −δ1 1 + δ2 π1 π2 = 1 x
- Descriptor-like polytopic descriptor models
N(ζ) ˙ x π z y = M(ζ) x w u ,
- N(ζ)
M(ζ)
- ∈ CO
N [v] M [v]
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Outline ➊ What is Robustness? ➋ Why concentrating on linear systems? ˙ x = f(x) − → ˙ x = A(∆)x + Bw(∆)w ➌ Affine polytopic-type models
K
[v] [2]
Σ Σ
[1]
Σ Σ(∆)
➍ LFT-type models
z
∆
w
∆
K
w z u y
Σ
∆ ➎ RoMulOC Toolbox
» sys = ussmodel( sys, delta )
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➍ LFT-type models
- Example of Linear-Fractional Transformation (LFT)
˙ x =
δ1
2
1+δ2 x ⇒
˙ x = w∆1 z∆1 = w∆2 z∆2 = x − w∆3 z∆3 = x − w∆3 w∆1 = δ1z∆1 w∆2 = δ1z∆2 w∆3 = δ2z∆3
- Feedback-loop representation
˙ x = Ax + B∆w∆ z∆ = C∆x + D∆∆w∆ , w∆ = ∆z∆
- ∆ can always be found diagonal
- δi elements may be repeated several times
- δi elements may be time-varying
▲ LFT model is not unique. Open issue: finding ‘minimal’ representations. ˙ x = w∆ , z∆ = x −w∆ , w∆ =
- δ2
1
δ2
- z∆
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➍ LFT-type models
- Example of Linear-Fractional Transformation (LFT) - continued
˙ x = w∆ , z∆ = x −w∆ , w∆ =
- δ2
1
δ2
- z∆
▲ With the change of variables ˜ δ1 = δ2
1 − 1 it gives also
˙ x = x + w∆ , z∆ = x −x − w∆ , w∆ =
- ˜
δ1 δ2
- z∆
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➍ LFT-type models
- Example of Linear-Fractional Transformation (LFT) - continued
˙ x = x + w∆ , z∆ = x −x − w∆ , w∆ =
- ˜
δ1 δ2
- z∆
▲ Ex1 of uncertainty - Components produced on an assembly line with unprecise characteristics: 1 0 ˜ δ1 0 1 δ2 −1 0.5 2 0.5 −1 −2 2 −2 20 1 0 0 1 ˜ δ1 δ2 ≤ 0
- 1
∆T
- Ψ
1 ∆ ≤ 0 ∆ is Ψ-dissipative
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➍ LFT-type models
- Example of Linear-Fractional Transformation (LFT) - continued
˙ x = x + w∆ , z∆ = x −x − w∆ , w∆ =
- ˜
δ1 δ2
- z∆
▲ Ex2 of uncertainty - Physical knowledge gives interval uncertainty set:
- ˜
δ1 δ2
- ˜
δ1 δ2
- ˜
δ1 δ2
- ▲ Ex3 of uncertainty - Defined by a finite set of extremal values:
- ˜
δ1 δ2
- ∈ CO
˜ δ[v]
1
δ[v]
2
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➍ LFT-type models
■ Rationally dependent parametric uncertainties can be modeled as
z
∆
w
∆
K
w z u y
Σ
∆
˙ x z∆ z y = A B∆ Bw Bu C∆ D∆∆ D∆w D∆u Cz Dz∆ Dzw Dzu Cy Dy∆ Dyw Dyu x w∆ w u , w∆ = ∆z∆ ∆ = 1r1 ⊗ ∆1 · · · 1r2 ⊗ ∆2
. . . ...
1r¯
q ⊗ ∆¯
q
: 1rq ⊗ ∆q = ∆q
...
∆q
- rq times
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➍ LFT-type models
■ Parametric uncertainties in LFT models ∆ =
1r1 ⊗ ∆1 · · · 1r2 ⊗ ∆2
. . . . . .
1r¯ q ⊗ ∆¯ q
- ∆q May be defined in polytopes, parallelotopes or intervals
∆q ∈ co
- ∆[v=1...¯
vq] q
∆q = ∆|0|
q +
- δqp∆|p|
q
, |δqp| ≤ 1 ∆q ∆q ∆q
- or by algebraic equality/inequality constraints
▲ Quadratic constraints
- 1
∆qT
- Ψq
1 ∆q ≤ 0 ← Ψq-dissipative Φq-structured →
- 1
∆qT
- Φq
1 ∆q = 0 ▲ Polynomial constraints
- 1
∆qT ∆qT ⊙ ∆qT · · ·
- ˆ
Ψq
1 ∆q ∆q ⊙ ∆q
. . .
≤ 0
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➍ LFT-type models
- Example of polynomial constraint
∆ = δ1 δ2 δ4
1 + δ4 2 − δ2 1δ2 2 − δ2 1δ2 + δ1 ≤ 2
- 1
δ1 δ2 δ2
1
δ2
2
-
−2
1 2 1 2
− 1
2
− 1
2
1 − 1
2
− 1
2
1 1 δ1 δ2 δ2
1
δ2
2
≤ 0
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➍ LFT-type models
■ Some properties of LFTs ˙ x z∆ z y = A B∆ Bw Bu C∆ D∆∆ D∆w D∆u Cz Dz∆ Dzw Dzu Cy Dy∆ Dyw Dyu x w∆ w u , w∆ = ∆z∆
- Rational matrix expression:
˙ x z y = M(∆) x w u M(∆) = A Bw Bu Cz Dzw Dzu Cy Dyw Dyu + B∆ Dz∆ Dy∆ ∆(1 − D∆∆∆)−1
- =(1−∆D∆∆)−1∆
- C∆
D∆w D∆u
- ▲ Well-posedness of algebraic loop is assumed: (1 − ∆D∆∆) non singular for all ∆ ∈ ∆
∆.
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➍ LFT-type models
■ Some properties of LFTs
˙ x z∆ z y
=
A B∆ Bw Bu C∆ D∆∆ D∆w D∆u Cz Dz∆ Dzw Dzu Cy Dy∆ Dyw Dyu
x w∆ w u
, w∆ = ∆z∆
- Change of coordinates of the uncertainty: ∆ = ∆0 + T ˜
∆S ˙ x ˜ z∆ z y = A B∆T Bw Bu SC∆ D∆∆T SD∆w SD∆u Cz Dz∆T Dzw Dzu Cy Dy∆T Dyw Dyu + B∆ Dz∆ Dy∆ ∆0(1 − D∆∆∆0)−1 C∆ D∆w D∆u
-
x ˜ w∆ w u
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➍ LFT-type models
■ Some properties of LFTs
- Interconnected LFTs are LFT
1
F Σ Δ Δ Σ1
1
Δ
2 2
K
3
≡
z
∆
w
∆
K
w z u y
Σ
∆
∆ = ∆1 ∆2 ∆3
- Best tool for manipulation LFTs: LFR toolbox by JF Magni
www.cert.fr/dcsd/idco/perso/Magni/toolboxes.html
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➍ LFT-type models
■ Elements of ∆ may as well be uncertain operators:
- Sector bounded non-linearity (circle criterion case extended to MIMO operators):
w∆(t) = [∆z∆](t) : ρz∆ ≤ w∆ ≤ ρz∆
- Well-known sub-case, norm-bounded operators:
w∆(t) = [∆z∆](t) : w∆ ≤ ρz∆ → ∆∗∆ ≤ ρ21
- Passive operators:
w∆(t) = [∆z∆](t) : ∞ wT
∆(t)z∆(t)dt ≤ 0
→ ∆∗ + ∆ ≤ 0 ▲ Example: integrator (s−1) with zero initial conditions x = I ˙ x : ∞ xT (t) ˙ x(t)dt = lim
θ→∞
1 2xT (θ)x(θ) ≥ 0
- Time-delay w∆(t) = z∆(t − τ)
- Saturations, dead-zones, hysteresis, other...
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➍ LFT-type models
■ Elements of ∆ may be used to define performances:
- 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
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➍ LFT-type models
■ 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
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➍ LFT-type models
■ Important case :
If ∆
∆ is bounded, the uncertainty block can (sometimes conservative) be converted to: ∆µ = 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
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Pereslavlь-Zalesski i Inь 2010
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➍ LFT-type models
■ µ theory [Doyle,Safonov]: well-posedness analysis of
M(j ) ∆ ω
- Structured singular value : µ(ω) = 1/km(ω)
km(ω) = min{ k : ∃∆µ det(1 − k∆µM(jω)) = 0 }
- Well-posedness for all frequencies: µ = maxω>0 µ(ω) < 1
- If M(s) is stable and µ < 1 ⇒
Robust stability
- km = µ−1: extended definition of modulus margin
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Pereslavlь-Zalesski i Inь 2010
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Outline ➊ What is Robustness? ➋ Why concentrating on linear systems? ˙ x = f(x) − → ˙ x = A(∆)x + Bw(∆)w ➌ Affine polytopic-type models
K
[v] [2]
Σ Σ
[1]
Σ Σ(∆)
➍ LFT-type models
z
∆
w
∆
K
w z u y
Σ
∆ ➎ RoMulOC Toolbox
» sys = ussmodel( sys, delta )
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Pereslavlь-Zalesski i Inь 2010
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➎ RoMulOC Toolbox
■ RoMulOC: Robust Multi-Objective Control toolbox
- Freely distributed software www.laas.fr/OLOCEP/romuloc
- Works in Matlab c
- may be extended to Scilab (Thanks to S. Solovyev)
- Gathers robust control results developed at LAAS
- may be extended for other issues (delays, non-linearities...)
- Quickly and simply solve problems with well established results
- experimented on ’industrial’ examples
- Includes uncertain modeling features
▲ Interval, Parallelotopic, Polytopic uncertain systems ▲ LFT uncertain systems with uncertain blocks among:
norm-bounded, positive-real, dissipative, interval, parallelotopic, polytopic
▲ Similar to Control Toolbox of Matlab c
- but with performance signals z/w.
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Pereslavlь-Zalesski i Inь 2010
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➎ RoMulOC Toolbox
■ Example of LFT modeling in RoMulOC
- Definition of four different uncertainties
>> Dm = udiss( Psi, ’Inertia’); >> Dd = unb( 2, 2, 0.25, ’Damping’); >> De = uinter(-0.25, 0.25, ’Input’); >> Dc = upoly( Dcv, ’Output’); >> Delta = diag(Dm,Dd,De,Dc);
- Definition of a state-space model
>> sys = ssmodel(’Demo Example 1’); >> sys.A = [ zeros(n) , eye(n) ; -iM0*D0 , -iM0*K ]; >> sys.Bw = [zeros(n,m) ; -iM0*E0 ];
and so on for matrices Bd, Cd, Ddd, Cz, Dzd etc.
- Definition of the uncertain LFT system
>> usys = ussmodel( sys, Delta )
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➎ RoMulOC Toolbox
>> usys = ussmodel( sys, Delta ) Uncertain model : LFT
- ------- WITH --------
name: Demo Example 1 n=6 md=6 mw=3 n=6 dx = A*x + Bd*wd + Bw*w pd=7 zd = Cd*x + Ddd*wd + Ddw*w pz=3 z = Cz*x + Dzd*wd + Dzw*w continuous time ( dx : derivative operator )
- ------- AND
- diagonal structured uncertainty
size: 6x7 | nb blocks: 4 | independent blocks: 4 wd = diag( #5 #6 #7 #8 ) * zd index size constraint name #5 1x2 dissipative real Inertia #6 2x2 norm-bounded by 0.25 real Damping #7 1x1 interval 1 param real Input #8 2x2 polytope 3 vertices real Output
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➎ RoMulOC Toolbox
■ ”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 )
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➎ RoMulOC Toolbox
- 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 )
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➎ RoMulOC Toolbox
- 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 )
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➎ RoMulOC Toolbox
- 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 );
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➎ RoMulOC Toolbox
- The declared robust multi-objective control problem
>> quiz control problem: STATE-FEEDBACK design Lyapunov function: UNIQUE (quadratic stability) Specified performances / systems: # Hinfty < 4 / Helicopter # D-stability / Helicopter # minimize I2P / Helicopter
- Solve the problem
>> K = solvesdp( quiz ) 0.436574 guaranteed I2P norm K = 0.0442 0.0091 0.0305
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Pereslavlь-Zalesski i Inь 2010
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➎ RoMulOC Toolbox
■ 2000-2003 PIROLA Project
- Robust performance analysis (LFT models)
z
∆
w
∆
w z
∆
Σ
- Multi-objective control design
✛ ✛ 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
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Pereslavlь-Zalesski i Inь 2010
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➎ RoMulOC Toolbox
■ 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
+ +
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➎ RoMulOC Toolbox
■ 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
“Robust Analysis of the Longitudinal Control of a Civil Aircraft using RoMulOC”,
- G. Chevarria, D. Peaucelle, D. Arzelier, G. Puyou,
IEEE-MSC, 8-10 September 2010
Centre de gravit´ e Distance (L) Profondeur Force (F) Mouvement r´ esultant Stabilisateur horizontal
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➎ RoMulOC Toolbox
■ 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 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.
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Conclusions ■ Robust control: relies on uncertain modeling
- Linear uncertain representations including
▲ Parametric uncertainties:
badly identified parameters, deteriorated components, mass-production imprecise components...
▲ Time-varying and non-linear components: more than 1st order approximation
- Modeling should be done taking into account control objectives
▲ It defines the bounds on some of the uncertainties ▲ Simultaneously to the modeling, desired performances are defined ■ Tools for uncertain modeling
- Control Toolbox of Matlab c
- LFR toolbox www.cert.fr/dcsd/idco/perso/Magni/toolboxes.html
- RoMulOC toolbox www.laas.fr/OLOCEP/romuloc
■ Prospective work
- Polynomial constraints on uncertainties
- Descriptor-type models
- D. Peaucelle