Variations on anti-windup synthesis Theoretical aspects & application to PIO alleviation
Jean-Marc Biannic & Sophie Tarbouriech ONERA-DCSD & LAAS-CNRS – Toulouse, France
[email protected],
[email protected]
Workshop Airbus, November 2011
1. Introduction
2. Anti-Windup I
3. Anti-Windup II
4. Application
Outline 1
Introduction Motivations Anti-windup
2
Anti-Windup I : formulation & state space approach
3
Anti-Windup II : Frequency domain approach
4
PIO alleviation
5
Conclusions
Variations on anti-windup synthesis (J-M. Biannic & S. Tarbouriech)
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1. Introduction
2. Anti-Windup I
3. Anti-Windup II
4. Application
Introduction : Motivations Saturations (position, rate, acceleration) are : present in almost every systems (in particular in the actuators), especially active as the level of performance required is high (small time answer, high level reference input) considering in plus that the actuator is calibrated to the nearest, often responsible of : performance degradation, stability degradation (size of the attraction region) non invertible The control law synthesis requires to take into account such nonlinearities either a priori or a posteriori... which then makes first necessary to have efficient and performant analysis tools.
Variations on anti-windup synthesis (J-M. Biannic & S. Tarbouriech)
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1. Introduction
2. Anti-Windup I
3. Anti-Windup II
4. Application
Introduction : Anti-windup One can distinguish two classes of approaches which allow to deal with control design in presence of saturations : saturation avoidance control : the control input amplitudes and rates are small enough such that saturations are never active. One remains in the linear framework. saturation allowance control : the control inputs are now likely to saturation and one has to deal with the nonlinear behavior induced by the saturations. The approach proposed in this presentation is based on the optimization of a well-known structure of compensators : the anti-windup scheme. This approach behaves to the saturation allowance control class which allows to exploit at the most the actuators capabilities. It has also to be noted the the saturations are taken into account a posteriori.
Variations on anti-windup synthesis (J-M. Biannic & S. Tarbouriech)
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1. Introduction
2. Anti-Windup I
3. Anti-Windup II
4. Application
Outline 1
Introduction
2
Anti-Windup I : formulation & state space approach Structure Anti-Windup Design objectives Solution
3
Anti-Windup II : Frequency domain approach
4
PIO alleviation
5
Conclusions
Variations on anti-windup synthesis (J-M. Biannic & S. Tarbouriech)
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1. Introduction
2. Anti-Windup I
3. Anti-Windup II
4. Application
Anti-Windup structure
v
J(s)
w −
r
K(s)
+
A(s)
G(s)
y
Remark : In the absence of saturations (ǫ = 0), the anti-windup action disappears (immediately or progressively according to the dynamics of J(s)), and the closed loop retrieves it nominal properties (linear). Variations on anti-windup synthesis (J-M. Biannic & S. Tarbouriech)
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1. Introduction
2. Anti-Windup I
3. Anti-Windup II
4. Application
Design objectives Considering a compensator K(s), designed without saturations, the objective is to compute a (possibly dynamic) anti-windup gain J(s), such that : The nonlinear closed loop remains stable for high amplitude reference inputs (optimization of the stability region), The nonlinear behavior remains as close as possible to the linear behavior described by L(s) which acts as a reference model (minimization of the energy of the error signal zp ) v
J(s)
w −
r
˜ K(s) :
+
~
K(s)
A(s)
y
+
r + v1 x˙ K = AK xK + BK y r + v2 u = C K x K + DK y zp
G(s) −
L(s)
Variations on anti-windup synthesis (J-M. Biannic & S. Tarbouriech)
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1. Introduction
2. Anti-Windup I
3. Anti-Windup II
4. Application
Preliminary transformations In the previous scheme : saturations are replaced by dead-zone, the reference r of exogenous inputs is removed and substituted with a new state associated to the dynamics of a reference input generator Example : Step ≈ slowly decreasing exponential échelon
r(t)
t
R(s) : r(t) ˙ = −ǫr(t) , r(0) = r0 Variations on anti-windup synthesis (J-M. Biannic & S. Tarbouriech)
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1. Introduction
2. Anti-Windup I
3. Anti-Windup II
4. Application
Preliminary transformations v
J(s)
w −
+
r ~
K(s)
y
A(s)
zp
+
G(s) −
R(s)
− +
L(s)
Φ
w
J(s)
v
z
M(s)
zp
Σ Variations on anti-windup synthesis (J-M. Biannic & S. Tarbouriech)
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1. Introduction
2. Anti-Windup I
3. Anti-Windup II
4. Application
Preliminary transformations Remarks : The augmented state model M (s) is, by construction and without restriction, stable since it includes the compensator K(s) which, for Φ ≡ 0, guarantees nominal stability and performance As soon as the actuator A(s) is strictly proper, this is generally also the case for the augmented system M (s). However, in presence notably of multiple saturation on a same axe (position/rate), non-null element are susceptible to appear on the direct transmission of M (s). These elements express the presence of nested saturations... A specific approach has in that case to be undertaken. One can also eliminate them by filtering. Variations on anti-windup synthesis (J-M. Biannic & S. Tarbouriech)
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1. Introduction
2. Anti-Windup I
3. Anti-Windup II
4. Application
Nonlinear closed loop Under mild assumptions, the augmented plant M (s) has no feedthrough terms and then reads : CΦ M (s) = (sI − A)−1 [BΦ Bv ] (1) Cp while the gain J(s) is searched under the general form : J(s) = CJ (sI − AJ )−1 BJ + DJ
(2)
which results in the nonlinear CL state-space representation Σ : x˙ = A Bv CJ x + Bφ + Bv DJ Φ(z) 0 AJ BJ Σ: z = Cφ 0 x , zp = Cp 0 x w ∈ IR m
Φ
Σ (s)
(3)
z ∈ IR m zp ∈ IR p
Variations on anti-windup synthesis (J-M. Biannic & S. Tarbouriech)
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1. Introduction
2. Anti-Windup I
3. Anti-Windup II
4. Application
Stability and nonlinear performance A classical way to evaluate the stability (inside a region) and the performance of the nonlinear system Σ consists in establishing the existence of a Lyapunov function : V (x) = x′ P x such that, along the system trajectories : 1 ∀x ∈ E = {x/V (x) ≤ 1} , V˙ (x) + zp′ zp < 0 γ
(4)
By integration, one then concludes that in the region E, the energy of the signal zp remains bounded by γ. By describing the nonlinearity Φ using generalized sector conditions (Tarbouriech, Gomes da Silva), not much conservative, the test (4), via the S-procedure, is transformed into a much simpler test based on matrix inequalities... Variations on anti-windup synthesis (J-M. Biannic & S. Tarbouriech)
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1. Introduction
2. Anti-Windup I
3. Anti-Windup II
4. Application
Stability and nonlinear performance If there exist matrices Q ∈ IR n×n , S = diag(s1 , . . . , sm ), and Z ∈ IR m×n and a positive scalar γ such that : A Ba C J Bφ S 0 Q BJ 0 AJ 0, i = 1...m (6) Zi + C φ i 0 Q 1 then Σ is stable in E = x/x′ Q−1 x ≤ 1 . Moreover, zp verifies :
Z
∞
zp (t)′ zp (t) dt ≤ γ
(7)
0
Variations on anti-windup synthesis (J-M. Biannic & S. Tarbouriech)
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1. Introduction
2. Anti-Windup I
3. Anti-Windup II
4. Application
Synthesis conditions The matrix inequalities (5)-(6) are linear in the analysis variables (Q, S, Z, γ) and in the synthesis (AJ , BJ , CJ , DJ ) variables considered separately. A priori, the computation of the anti-windup compensator, which requires to simultaneously optimize the analysis and synthesis variables, is not convex. The inequality (5) becomes a BMI. However, the convexity may be recovered in two cases : “full-order” synthesis : nJ = nM fixed dynamics synthesis : matrices AJ and CJ of the compensator J(s) are fixed a priori. Remark : Without any negative effect on the convexity, additional contraints on Q−1 may be easily considered such as to enlarge the region of stability in certain favored directions the state space... Variations on anti-windup synthesis (J-M. Biannic & S. Tarbouriech)
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1. Introduction
2. Anti-Windup I
3. Anti-Windup II
4. Application
“Full-order” synthesis (1/2) There exists a compensator J(s) satisfying (5)-(6) iff ∃X = X T , Y = Y T ∈ IR nM ×nM , S = diag(s1 , . . . , sm ), U, V ∈ IR m×nM such that the following LMI conditions hold :
AT X + XA ⋆ Cp −γIp
