Uncertain systems and robust control LMI methods Dimitri PEAUCELLE Relaxation Approaches for Control of Uncertain Complex Systems: Methodologies and Tools Workshop at 52nd IEEE Conference on Decision and Control Monday December 9, 2013, Florence
Introduction ■ Objectives of this presentation: ● Recall some existing results in robust control ● Demonstrate how to test these with RoMulOC toolbox http://projects.laas.fr/OLOCEP/romuloc/ ■ Underlying point of view: ● Few techniques ● Many results ● Depend on modeling choices D. Peaucelle 1 Florence, December 2013
Outline ● Two classes of uncertain systems : polytopic & LFT ▲ Airplane example ▲ DEMETER satellite example ▲ Prospectives: descriptor uncertain modeling ● Robust analysis ▲ Stability and performances - the well-posedness point of view ▲ System augmentation approach for sequences of SOS-like relaxations ▲ "Slack variable" results and "quadratic stability" as a special case ● State-feedback design: multi-performance ▲ Based on dual system ▲ Almost LMI results in "slack variable" approach D. Peaucelle 2 Florence, December 2013
Modeling of systems with uncertainties ■ Aircraft example ● Complicated non-linear model - linearized around operation point ( V o , ... ) ▲ 9 uncertain parameters: (Not precisely known parameters such as inertia etc. & uncertainties on operating point) 0 1 0 0 0 L p L β L r x = ˙ x + Bu g/ ( V o + v ) 0 Y β − 1 N ˙ β g/ ( V o + v ) N p N β + N ˙ β Y β N r − N ˙ β L p ≤ L p ≤ L p , L β ≤ L β ≤ L β ... ▲ Uncertainties given in intervals: D. Peaucelle 3 Florence, December 2013
Modeling of systems with uncertainties 0 1 0 0 0 L p L β L r x = ˙ x + Bu g/ ( V o + v ) 0 Y β − 1 N ˙ β g/ ( V o + v ) N p N β + N ˙ β Y β N r − N ˙ β ● Affine-dependent representation with bounds given by interval analysis 0 1 0 0 0 α 22 α 23 α 23 x = ˙ x + Bu α 31 0 α 33 − 1 α 41 α 42 α 43 α 44 L p ≤ α 22 ≤ L p . . . N ˙ β g/ ( V o + v ) ≤ α 41 ≤ N ˙ β g/ ( V o + v ) . . . ▲ Includes the original uncertain model but coupling between coefficients is lost ▲ Conservative: if a property is proved for polytopic model, it also holds for original one D. Peaucelle 4 Florence, December 2013
Modeling of systems with uncertainties 0 1 0 0 0 α 22 α 23 α 23 x = ˙ x + Bu α 31 0 α 33 − 1 α 41 α 42 α 43 α 44 α ij ≤ α ij ≤ α ij ● This is an interval model: all coefficients are independent and in intervals ● Sub-class of parallelotopic models (centered at A 0 with deviations along axes A 1 , A 2 etc.) A ( β ) = A 0 + β 1 A 1 + β 2 A 2 + . . . , β i ∈ [ − 1 1] ● Sub class of polytopic models described as convex hull of vertices (in ex. v = 2 9 = 512 !) v v � � ξ v A [ v ] A ( ξ ) = : ξ v = 1 , ξ v ≥ 0 v =1 v =1 ▲ Demo in RoMulOC D. Peaucelle 5 Florence, December 2013
Modeling of systems with uncertainties 0 1 0 0 0 L p L β L r x = ˙ x + Bu g/ ( V o + v ) 0 Y β − 1 N ˙ β g/ ( V o + v ) N p N β + N ˙ β Y β N r − N ˙ β ● Linear-Fractional Transformation (LFT): make it linear in the uncertainties via feedback � w z � � � D. Peaucelle 6 Florence, December 2013
Modeling of systems with uncertainties 0 1 0 0 0 L p L β L r x = ˙ x + Bu g/ ( V o + v ) 0 Y β − 1 N ˙ β g/ ( V o + v ) N p N β + N ˙ β Y β N r − N ˙ β ▲ Handling the 1 / ( V o + v ) terms. z 1 = 1 / ( V o + v ) x 1 ⇔ V o z 1 + vz 1 = x 1 w 1 = vz 1 ⇔ V o z 1 + w 1 = x 1 w 1 = vz 1 ⇔ z 1 = 1 /V o x 1 − 1 /V o w 1 D. Peaucelle 7 Florence, December 2013
Modeling of systems with uncertainties 0 1 0 0 0 L p L β L r x = ˙ x + Bu g/ ( V o + v ) 0 Y β − 1 N ˙ β g/ ( V o + v ) N p N β + N ˙ β Y β N r − N ˙ β ▲ Handling the 1 / ( V o + v ) terms, continued w 1 = vz 1 0 1 0 0 0 0 L p L β L r 0 ˙ x x B = + g/V o 0 Y β − 1 − g/V o z 1 w 1 0 N ˙ β g/V o N p N β + N ˙ β Y β N r − N ˙ − N ˙ β g/V o β 1 /V o 0 0 0 − 1 /V o D. Peaucelle 8 Florence, December 2013
Modeling of systems with uncertainties 0 1 0 0 0 0 L p L β L r 0 ˙ x x B = + g/V o 0 Y β − 1 − g/V o z 1 w 1 0 N ˙ β g/V o N p N β + N ˙ β Y β N r − N ˙ − N ˙ β g/V o β 1 /V o 0 0 0 − 1 /V o ▲ Handling the N ˙ β term: w 1 = vz 1 , w 2 = N ˙ β z 2 0 1 0 0 0 0 0 L p L β L r 0 0 x ˙ x g/V o 0 Y β − 1 − g/V o 0 B u = + z 1 w 1 0 N p N β N r 0 1 0 z 2 w 2 1 /V o 0 0 0 − 1 /V o 0 g/V o 0 Y β − 1 − g/V o 0 D. Peaucelle 9 Florence, December 2013
Modeling of systems with uncertainties ▲ In the end w = ∆ z with ∆ = diag ( v, N ˙ β , Y β , L p , L β , L r , N p , N β , N r ) and 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 g/V o 0 0 − 1 − g/V o 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 1 /V o 0 0 0 − 1 /V o 0 0 0 0 0 0 0 0 g/V o 0 0 − 1 − g/V o 0 1 0 0 0 0 0 0 � � � � � � x ˙ x B = + u 0 0 1 0 0 0 0 0 0 0 0 0 0 z w 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 ● LFT model is a feedback-loop of a purely uncertain matrix with purely certain system ▲ Can always be obtained if uncertainty enters rationally in the model ▲ Issue: having an LFT of minimal size (size of ∆ ) D. Peaucelle 10 Florence, December 2013
Modeling of systems with uncertainties ● Manipulation LFT made easy using the star-product M d M c = M a + M b ∆( I − M d ∆) − 1 M c ∆ ⋆ M b M a ▲ Corresponds to the following loop: z = M d w + M c u w = ∆ z ⋆ y = M b w + M a u � w z � u y ● Always assumed to be well-posed: ( I − M d ∆) non singular for all uncertainties D. Peaucelle 11 Florence, December 2013
Modeling of systems with uncertainties ▲ Elementary operations on LFTs � � � � � � M d M c 0 M d M c N d N c ∆ 1 0 ∆ 1 ⋆ + ∆ 2 ⋆ = ⋆ N d N c 0 M b M a N b N a ∆ 2 0 M b N b M a + N a � � � � � � M d M c N b M c N a M d M c N d N c ∆ 1 0 ∆ 1 ⋆ · ∆ 2 ⋆ = ⋆ N d N c 0 M b M a N b N a ∆ 2 0 M b M a N b M a N a � � �� − 1 � � M d − M c M − 1 − M c M − 1 M d M c M b ∆ ⋆ = ∆ ⋆ a a M − 1 M − 1 M b M a M b a a ● Coded in Matlab’s Robust Control toolbox & in LFRToolbox ▲ Demo in Robust Control toolbox & RoMulOC Δ Δ 1 2 F Σ 1 Σ 2 1 ● Allows also to manipulated complex control schemes Δ 3 K D. Peaucelle 12 Florence, December 2013
Modeling of systems with uncertainties ■ DEMETER: a satellite of the MYRIAD family developed by CNES ● All MYRIAD microsatellites share common plat- form (including the control components), the load is different (and the gains of the control law are tuned accordingly). ● On DEMETER the scientific load includes four long appendices that study the ionospheric distur- bances ( smsc.cnes.fr/DEMETER ). Fine pointing towards earth is required. ▲ CNES: French national space center - governmental ▲ Accepted to provide data about DEMETER to the scientific community, [PA06] ▲ It is purely a benchmark: no possible implementation (no more on orbit) D. Peaucelle 13 Florence, December 2013
Modeling of systems with uncertainties ■ LAAS studies for the attitude control of DEMETER ● Uncertain modeling at small depointing errors ● Mixed H 2 /H ∞ reduced order control design (small depointing) [ADGH11] ● Robustness analysis of the uncertain LTI model in closed-loop (small depointing) [PBG + 10] ● Periodic control law design using reaction wheels and magneto torquers (medium to large depointing) [TAP + 11] ● Design of an adaptive control law replacing a commuting control (small to medium depointing) [PDPM11] D. Peaucelle 14 Florence, December 2013
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