Robust analysis in a quadratic separation framework and application to Demeter satellite attitude control system Denis Arzelier, Alberto Bortott, Fr´ ed´ eric Gouaisbaut, Dimitri Peaucelle Christelle Pittet, Catherine Charbonnel CCT SCA & MOSAR - Toulouse - October 14th, 2009
Introduction ■ Results are part of a joint project involving CNES, LAAS-CNRS and Thales Alenia Space ● Flexible satellite attitude control benchmark Demeter developed at CNES ● Robust analysis methodology developed at LAAS-CNRS and coded in a Matlab toolbox : RoMulOC www.laas.fr/OLOCEP/romuloc ● Dedicated codes for Demeter and tests realized at LAAS-CNRS ● Further software developments done at Thales Alenia Space ● Large scale tests done at CNES and Thales Alenia Space 1 Toulouse - October 14th, 2009
Outline ➊ Demeter satellite ➋ Integral Quadratic Separation (IQS) ➌ Heuristic algorithm for optimization of stable domains ➍ Application to Demeter - 1 axis, 1 flexible mode, 3 uncertainties ➎ Conclusions 2 Toulouse - October 14th, 2009
➊ Demeter satelite ■ M-C-K uncertain model δ ¨ δ ˙ θ θ 0 0 = M ( δ J ) C − ¨ S ( δ ω , δ ξ ) ˙ η η 0 δθ 0 0 1 + U + K − S ( δ ω ) η 0 0 ▲ θ , ˙ θ : satellite orientation (3D) ▲ η , ˙ η : states of the flexible modes (up to 4 for each axis) ▲ δ J : 6 scalar uncertainties on the inertia ▲ δ ω , δ ξ : scalar uncertainties on the frequency and damping of flexible modes ● State space model: ˙ X = A ( δ ) X + B ( δ ) U A ( δ ) , B ( δ ) are rational w.r.t. uncertainties. 3 Toulouse - October 14th, 2009
➊ Demeter satelite ■ LFT model ˙ X = AX + B ∆ w ∆ + B u u , w ∆ = ∆ z ∆ z ∆ = C ∆ X + D ∆∆ w ∆ + D ∆ u u y = C y X + D y ∆ w ∆ + D yu u ▲ ∆ : diagonal matrix with δ J , δ ω and δ ξ elements ▲ Some elements δ are repeated ▲ The problem is normalized: δ ∈ [ − 1 1 ] ● Modeling is made possible in the following toolboxes � ) Control (Matlab c LFR (J.F RoMulOC (LAAS) . Magni) 4 Toulouse - October 14th, 2009
➊ Demeter satelite >> Ax = [1]; Fm = 1; model_type = 2; >> usys = demeter2romuloc(Ax,Fm,model_type) Uncertain model : LFT -------- WITH -------- n=4 md=5 mu=1 n=4 dx = A*x + Bd*wd + Bu*u pd=5 zd = Cd*x + Ddd*wd + Ddu*u py=1 y = Cy*x continuous time ( dx : derivative operator ) -------- AND -------- diagonal structured uncertainty size: 5x5 | nb blocks: 5 | independent blocks: 3 wd = diag( #1 #1 #2 #2 #3 ) * zd index size constraint name #1 1x1 interval 1 param real dJ11 #2 1x1 interval 1 param real dW1 #3 1x1 interval 1 param real dX1 5 Toulouse - October 14th, 2009
➊ Demeter satelite ● RoMulOC allows also polytopic models size: 5x5 | nb blocks: 1 | independent blocks: 1 wd = diag( #4 ) * zd index size constraint name #4 5x5 polytope 8 vertices real dJ11,dW1,dX1 ( − 1,1) (1,1) (0,0) ( − 1, − 1) (1, − 1) ▲ Aim: Guaranteed closed-loop robust stability for the ”biggest” polytope 6 Toulouse - October 14th, 2009
➊ Demeter satelite ▲ Secondary problem: Robust guaranteed H 2 norm (consumption) 7 Toulouse - October 14th, 2009
➊ Demeter satelite ▲ Secondary problem: Robust guaranteed H ∞ norm (robustness to unmodeled dynamics) 8 Toulouse - October 14th, 2009
➊ Demeter satelite ▲ Secondary problem: Robust guaranteed impulse-to-peak performance (control input saturation w.r.t. to initial depointing) 9 Toulouse - October 14th, 2009
➊ Demeter satelite Uncertain model : closed-loop satellite with performances -------- WITH -------- n=4 md=5 mw=1 n=4 dx = A*x + Bd*wd + Bw*w pd=5 zd = Cd*x + Ddd*wd + Ddw*w pz=1 z = Cz*x continuous time ( dx : derivative operator ) -------- AND -------- wd = diag( #4 ) * zd index size constraint name #4 5x5 polytope 8 vertices real dJ11,dW1,dX1 10 Toulouse - October 14th, 2009
Outline ➊ Demeter satellite ➋ Integral Quadratic Separation (IQS) ➌ Heuristic algorithm for optimization of stable domains ➍ Application to Demeter - 1 axis, 1 flexible mode, 3 uncertainties ➎ Conclusions 11 Toulouse - October 14th, 2009
➋ Integral Quadratic Separation (IQS) ■ Well-posedness w G (z, w)=0 w Bounded ( ¯ w, ¯ z ) z w ⇒ unique bounded ( w, z ) F (w, z)=0 z z ■ Considered case ● Linear implicit application: E and A are matrices (possibly not square) ● ∇ ∈ ∇ ∇ is bloc-diagonal. Contains scalar, full-bloc, LTI and LTV uncertainties w z w z ● For E = 1 and A = H ( jω ) one recovers IQC framework 12 Toulouse - October 14th, 2009
➋ Integral Quadratic Separation (IQS) ■ For dynamic systems ˙ x = Ax : well posedness ≡ internal stability G � �� � F � t � �� � z ( t ) = Aw ( t ) + ¯ z ( t ) , w ( t ) = z ( τ ) dτ + ¯ w ( t ) ���� ���� ���� 0 x ( t ) x ( t ) ˙ x (0) ▲ ¯ w contains information on initial conditions ● Well-posedness � � � � � � � � ¯ w w � � � � ⇔ ∀ ( ¯ w, ¯ z ) ∈ L 2 , ∃ !( w, z ) ∈ L 2 : ≤ γ � � � � � � � � ¯ z z � � � � ⇒ for zero initial conditions and zero perturbations: w = z = 0 is the unique solution (equilibrium point). ⇒ whatever bounded perturbations the state remains close to equilibrium (global stability) 13 Toulouse - October 14th, 2009
➋ Integral Quadratic Separation (IQS) ■ Integral Quadratic Separation [Automatica’08, CDC’07, ROCOND’09, ECC’09] ● For the case of linear application with uncertain operator E z ( t ) = A w ( t ) , w ( t ) = [ ∇ z ]( t ) ∇ ∈ ∇ ∇ where E = E 1 E 2 with E 1 full column rank, ● Integral Quadratic Separator (IQS) : ∃ Θ , matrix, solution of LMI � � ⊥∗ � � ⊥ Θ > 0 E 1 −A E 1 −A and Integral Quadratic Constraint (IQC) ∀∇ ∈ ∇ ∇ ∗ � ∞ E 2 z ( t ) E 2 z ( t ) dt ≤ 0 Θ [ ∇ z ]( t ) [ ∇ z ]( t ) 0 14 Toulouse - October 14th, 2009
➋ Integral Quadratic Separation (IQS) ∇ , ∃ LMI conditions for Θ solution to IQC ● For some given ∇ ∗ � ∞ E 2 z ( t ) E 2 z ( t ) dt ≤ 0 Θ [ ∇ z ]( t ) [ ∇ z ]( t ) 0 ▲ Θ is build out of IQS for elementary blocs of ∇ ▲ Improved DG -scalings, full-bloc S-procedure, vertex separators... ▲ Building Θ and related LMIs is tedious but can be automatized ( RoMulOC ) ▲ It is conservative except in few special cases [Meinsma et al., 1997]. 15 Toulouse - October 14th, 2009
➋ Integral Quadratic Separation (IQS) ■ Robust analysis in IQS framework: ● 1- Write the robust analysis problem as a well-posedness problem E z = A w , w = ∇ z ● 2- Build Integral Quadratic Separators for each elementary bloc of ∇ ● 3- Apply the IQS results to get (conservative) LMIs 16 Toulouse - October 14th, 2009
➋ Integral Quadratic Separation (IQS) ■ Demeter analysis problems: ● Well-posedness of � ▲ 1 n integrator ▲ ∆ matrix of uncertainties ▲ ∇ perf operator related to performances 17 Toulouse - October 14th, 2009
➋ Integral Quadratic Separation (IQS) ■ Other problem modeling produce other LMI conditions ● Dual system: z d = A T w d , w d = ∇ ∗ z d ● System augmentation (produces systems in descriptor form) x = Ax + ˙ B ∆ w ∆ z ∆ = C ∆ x + D ∆∆ w ∆ x = Ax + ˙ B ∆ w ∆ z ∆ = C ∆ ˙ ˙ x + D ∆∆ ˙ w ∆ z ∆ = C ∆ x + D ∆∆ w ∆ ⇒ w ∆ = ∆ z ∆ w ∆ = ∆ z ∆ =0 ���� ˙ w ∆ = ∆ ˙ ˙ z ∆ + ∆ z ∆ ▲ Equivalent to increasing the dependency of the (implicit) Lyapunov function � � � � ∗ ˆ V 0 ( x ) = x ∗ Px ⇒ V 1 ( x, ∆) = x ∗ z ∗ x ∗ z ∗ P ∆ ∆ 18 Toulouse - October 14th, 2009
Outline ➊ Demeter satellite ➋ Integral Quadratic Separation (IQS) ➌ Heuristic algorithm for optimization of stable domains ➍ Application to Demeter - 1 axis, 1 flexible mode, 3 uncertainties ➎ Conclusions 19 Toulouse - October 14th, 2009
➌ Heuristic algorithm for optimization of stable domains ● Values of parameters making the system stable and unstable (unkown) (−1,1) (1,1) (−1,−1) (1,−1) 20 Toulouse - October 14th, 2009
➌ Heuristic algorithm for optimization of stable domains ● Biggest squares and rectangles possibly obtained (−1,1) (1,1) (−1,−1) (1,−1) 21 Toulouse - October 14th, 2009
➌ Heuristic algorithm for optimization of stable domains ● Biggest polytope possibly obtained (if LMIs are not conservative) (−1,1) (1,1) (−1,−1) (1,−1) 22 Toulouse - October 14th, 2009
➌ Heuristic algorithm for optimization of stable domains ● Due to conservatism some polytopes may give feasible LMIs (full), others not (dotted) (−1,1) (1,1) (−1,−1) (1,−1) 23 Toulouse - October 14th, 2009
➌ Heuristic algorithm for optimization of stable domains ● One solution: pave the feasible set (−1,1) (1,1) (−1,−1) (1,−1) 24 Toulouse - October 14th, 2009
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