Optimal Finite-precision Implementations of Linear Parameter Varying Controllers James F Whidborne Department of Aerospace Sciences, Cranfield University, UK Philippe Chevrel IRCCyN, Nantes, France IFAC World Congress 2008 – p. 1/20
Introduction — the FWL Effects • set of real numbers that can be stored in a digital computer is a subset of all real numbers because of Finite-Word-Length (FWL) • constants and variables in a digital computer are subject to rounding and have a finite range (the FWL effects) There are three main problems arising from FWL effects:. • (i) coefficient sensitivity problem — errors resulting from finite precision in the controller coefficients • (ii) round-off noise problem — errors resulting from rounding of variables after each arithmetic computation • (iii) overflow/underflow problem — limitations imposed by the finite range of variables and constants (also the scaling problem) We consider coefficient sensitivity problem for LPV digital controllers Often important to reduce wordlength to reduce controller complexity (Roger Brockett plenary) IFAC World Congress 2008 – p. 2/20
Introduction — LPV Controller Implementation There appears to be little previous work on LPV controller implementation: • Apkarian (1997) considers discretization problem • Kelly and Evers (1997) recommends balanced realizations for gain-scheduling problems • ‘resilience’ problem for periodically varying linear state feedback controllers has been studied by Farges et al. (2007) FWL problems for LPV controllers seem not to have been studied IFAC World Congress 2008 – p. 3/20
Coefficient Sensitivity Problem FWL effects are strongly dependent on the controller realization LPV state-space controller has form x ( k + 1) = A ( θ ( k )) x ( k ) + B ( θ ( k )) y ( k ) u ( k ) = C ( θ ( k )) x ( k ) + D ( θ ( k )) y ( k ) . All equivalent state-space realizations are given by x ( k + 1) = T − 1 A ( θ ( k )) T ˜ x ( k ) + T − 1 B ( θ ( k )) y ( k ) ˜ u ( k ) = C ( θ ( k )) Tx ( k ) + D ( θ ( k )) y ( k ) where T is non-singular Problem: determine T such that closed-loop LPV system is insensitive to rounding in controller coefficients IFAC World Congress 2008 – p. 4/20
LPV Systems Discrete-time LPV plant x p ( k + 1) = A p ( θ ( k )) x p ( k ) + B p u ( k ) y ( k ) = C p x p ( k ) where A p depends affinely on the time-varying parameter vector, θ ( k ) , and θ ( k ) is known at the sample instant, k (i.e. the measurement is available in real time) Assume LPV controller x ( k + 1) = A ( θ ( k )) x ( k ) + B ( θ ( k )) y ( k ) u ( k ) = C ( θ ( k )) x ( k ) + D ( θ ( k )) y ( k ) . has been designed, and where R ( θ ( k )) depends affinely on θ where � � A ( θ ( k )) B ( θ ( k )) R := C ( θ ( k )) D ( θ ( k )) IFAC World Congress 2008 – p. 5/20
LPV Systems Closed loop system matrix is given by � � A ( θ ( k )) B ( θ ( k )) C p A c = B p C ( θ ( k )) A p ( θ ) + B p D ( θ ( k )) C p Defining � � � � � � 0 0 I 0 I 0 A 0 := B I := C I := 0 A p 0 B p 0 C p we get A c = A 0 ( θ ( k )) + B I R ( θ ( k )) C I , which is also affinely dependent on θ IFAC World Congress 2008 – p. 6/20
LTI System Coefficient Sensitivity Due to rounding of coefficients, controller matrix R is perturbed to R + ∆ and closed loop system matrix is perturbed to A c + B I ∆ C I Let maximum perturbation be given by the max norm � ∆ � max := max i,j | ∆ i,j | and define the FWL stability margin as � � η 0 := inf � ∆ � max : A c + B I ∆ C I is unstable η 0 is hard to compute IFAC World Congress 2008 – p. 7/20
LTI System Coefficient Sensitivity Fialho and Georgiou (1994) propose using spectral norm �� λ i : λ i are the eigenvalues of ∆ T ∆ � � ∆ � 2 := max with FWL stability margin given by complex stability radius � � η c := inf � ∆ � 2 : A C + B I ∆ C I is unstable which can be easily computed by 1 η c = � C I ( zI − A c ) − 1 B I � ∞ and � · � ∞ denotes the H ∞ -norm Since � ∆ � max ≤ � ∆ � 2 , then η c provides an upper bound on η 0 IFAC World Congress 2008 – p. 8/20
Minimal Sensitivity for LTI Controller Problem : find T that maximizes η c Equivalent to the H ∞ minimization problem � C I ( zI − A T ( T )) − 1 B I � � min � ∞ T non singular where � � � � T − 1 0 T 0 A T ( T ) := A c 0 I 0 I Can be solved by solving a sequence of LMI problems as proposed by Fialho and Georgiou (2001) IFAC World Congress 2008 – p. 9/20
LPV quadratic H ∞ Following Apkarian et al. (1995b), we will consider polytopic LPV systems Define a matrix polytope as the convex hull of r matrices, N 1 , N 2 , . . . , N r , � r r � � � Co { N i , i = 1 , . . . , r } := α i N i : α i ≥ 0 , α i = 1 . i =1 i =1 We assume that the discrete time varying parameter, θ ( k ) , is confined to the the polytope, Θ , with vertices ˆ θ 1 , ˆ θ 2 , . . . , ˆ θ r , that is θ ( k ) ∈ Θ , where Θ := Co { ˆ θ 1 , ˆ θ 2 , . . . , ˆ θ r } and that the dependence of the state space matrices on θ is affine IFAC World Congress 2008 – p. 10/20
LPV quadratic H ∞ A polytopic system has quadratic H ∞ performance (Apkarian et al., 1995b) of γ if and only if there exists a Lyapunov function V ( x ) = x T Px with X > 0 that establishes global stability and ensures that the L 2 gain of the system is bounded by γ . That is � y � 2 < γ � u � 2 along all possible parameter trajectories θ ( k ) ∈ Θ . IFAC World Congress 2008 – p. 11/20
Coefficient Sensitivity Minimization for LPV Systems The closed loop LPV system is affine in θ and is hence a polytopic LPV system So if we replace the stability radius maximization problem of the LTI case by a quadratic H ∞ performance (these are equivqlent for LTI system) we can solve a minimal coefficient sensitivity problem for the LPV system So we just need to solve a system of LMIs that minimizes the H ∞ performance measure at each vertex— thus the following is proposed Proposition The optimal quadratic H ∞ performance, γ opt is the minimum γ for which there exists a P = P T > 0 of the form P 1 0 0 P = 0 P 2 0 0 0 I such that M T i ( γ ) PM i ( γ ) < P, for i = 1 , 2 , . . . , r where � � A c ( ˆ θ i ) B I /γ M i ( γ ) := C I 0 The optimal nonsingular transformation matrix is obtained from P 2 = T T opt T opt IFAC World Congress 2008 – p. 12/20
Example Continuous-time state space plant x = A g ( α 1 , α 2 ) x + B g u, ˙ y = C g x α 1 + α 2 = 1 , α 1 ≥ 0 , α 2 ≥ 0 , with − 1 / 100 0 0 1 / 100 � � A g = 0 0 1 B g = 0 C g = 0 1 0 (0 . 2 α 1 + 2 α 2 ) − (0 . 2 α 1 + 2 α 2 ) − (0 . 1 α 1 + α 2 ) 0 Weighting functions ( s + 1 / 5) ( s/ 50 + 1) W 1 ( s ) = and W 2 ( s ) = 1 . 8( s + 1 / 5000) ( s/ 10000 + 10) The MATLAB LMI Toolbox function, hinfgs , with SKS H ∞ -criterion is used to obtain an LPV controller IFAC World Congress 2008 – p. 13/20
Example Controller at each vertex is discretized using the Tustin transformation with a sampling rate of 500Hz The LMI M T i ( γ ) PM i ( γ ) < P, for i = 1 , 2 is repeatedly solved with a bisection search to obtain γ opt = 2 . 736 × 10 3 For comparison, modal and balanced gramian realizations also calculated IFAC World Congress 2008 – p. 14/20
Frozen θ performance 2800 2780 2760 2740 � M opt � ∞ , γ opt 2720 2700 2680 2660 2640 2620 2600 0 0.2 0.4 0.6 0.8 1 α 1 Frozen- α 1 H ∞ -norm against α 1 — optimum quadratic performance γ opt is shown as the dashed line IFAC World Congress 2008 – p. 15/20
Comparison −4 x 10 4 3 η c 2 1 original modal balanced optimal 0 0 0.2 0.4 0.6 0.8 1 α 1 Complex stability radius, η c , against α 1 for frozen α 1 IFAC World Congress 2008 – p. 16/20
Eigenvalue sensitivity A measure of the closed -loop poles sensitivity LTI systems is n + m 1 � Ψ = 1 − | λ k | Ψ k k =1 where { λ i : i = 1 , . . . , m + n } represents the set of unique closed-loop poles/ eigenvalues and � ∂λ k n x � 2 � Ψ k = ∂x i i =1 and { x i : i = 1 , . . . , n x } are the controller parameters IFAC World Congress 2008 – p. 17/20
Eigenvalue sensitivity 6 10 original modal balanced optimal 5 Ψ 10 4 10 0 0.2 0.4 0.6 0.8 1 α 1 Complex stability radius, η c , against α 1 for frozen α 1 — note singularity at α 1 ≃ 0 . 634 IFAC World Congress 2008 – p. 18/20
Conclusions • Singularity at α 1 ≃ 0 . 634 results from closed loop eigenvalues branching from real to complex pair Problem does not arise in practise for LTI control design — pole multiplicities avoided • Hence eigenvalue sensitivity is not generally suitable for LPV systems coefficient sensitivity • LMI problems are easily solved — hence quadratic H ∞ measure suitable for LPV systems IFAC World Congress 2008 – p. 19/20
Further work • harder test case • observer -controller structures • effect of rounding on the scheduling parameter, θ • closed loop transfer function sensitivity • the round-off noise problem and the scaling problem Acknowledgements to Ecole Centrale Nantes for supporting this work IFAC World Congress 2008 – p. 20/20
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