Multi-objective Controller Design:

Multi-objective Controller Design: Evolutionary algorithms and Bilinear Matrix Inequalities for a passive suspension A. Molina-Cristobal*, C. Papageorgiou**, G. T. Parks*, M. C. Smith**, P. J. Clarkson* *Engineering Design Centre **Control Group Department of Engineering, University of Cambridge IEEE Colloquium on Optimisation for Control, Sheffield, 24 April 2006 p. 1/2

Contents Introduction: Linear Matrix Inequalities Convex Optimisation Multi-objective Control Design Multi-objective Optimisation Design of passive vehicle suspensions Quarter-car model - Inerter Multiobjective optimisation using MOGA and BMIs Conclusions IEEE Colloquium on Optimisation for Control, Sheffield, 24 April 2006 p. 2/2

Linear Matrix Inequalities A linear matrix inequality (LMI) is an expression, (x) = 0 + x 1 1 + + x n n > 0 i = T i are real symmetric matrices, The inequality means that (x) is positive definite matrix LMI example problem: ¾ 2 0 1 ¾ 1 1 0 ¾ 0 1 0 (x 1, x 2 ) = 0 5 0 + x 1 1 0 0 + x 2 1 1 1 > 0 0 0 0 0 0 1 0 1 0 this is equivalent to (x 1, x 2 ) = ¾ 2 x 1 (x 1 + x 2 ) 0 (x 1 + x 2 ) 5 x 2 x 2 0 x 2 x 1 > 0 IEEE Colloquium on Optimisation for Control, Sheffield, 24 April 2006 p. 3/2

Geometry of LMIs (Convexity) (x 1, x 2 ) = ¾ 2 x 1 (x 1 + x 2 ) 0 (x 1 + x 2 ) 5 x 2 x 2 0 x 2 x 1 > 0 easible iff all principal minors are nonnegative polynomial inequalities f i (x) > 0 f 1 (x) = 2 x 1 > 0, f 2 (x) = 5 x 2 > 0, f 3 (x) = x 1 > 0, f 4 (x) = (2 x 1 )(5 x 2 ) (x 1 + x 2 ) 2 > 0, f 5 (x) = x 1 (2 x 1 ) > 0, f 6 (x) = x 1 (5 x 2 ) x 2 2 > 0, f 7 (x) = x 1 ((2 x 1 )(5 x 2 ) (x 1 + x 2 ) 2 ) x 2 2 (2 x 1) > 0 IEEE Colloquium on Optimisation for Control, Sheffield, 24 April 2006 p. 4/2

Intersection of LMIs If (x) and G(x) are LMIs, the intersection has the form H(x) = > 0 0 G(x) ¾ (x) 0 Example of intersections (x) = ¾ 2 x 1 (x 1 + x 2 ) 0 (x 1 + x 2 ) 5 x 2 x 2 0 x 2 x 1 > 0 G(x) = > 0 0 3x 1 + x 2 20 ¾ 3x 1 + x 2 + 3 0 IEEE Colloquium on Optimisation for Control, Sheffield, 24 April 2006 p. 5/2

Convex optimisation over Linear Matrix Inequalities (LMIs) / Semidefinite Programming (SDP) An SDP is an optimisation problem with linear objective and semidefinite constraints or LMI constraints. min x Ψ subject c T to x ( x) m = O + i= 1 x i i > 0 In 1994, the LMI solvers became widely available. i.e. Projective Algorithm of Nesterov and Nemirovski (interior-point methods), available in the LMI Control toolbox. In control theory, the idea is to reduce a multiobjective control problem into a convex optimisation or SDP problem.

Lyapunov s Inequality (1892) x & = Ax The system is asymptotically stable if there exist a matrix P>0 (positive definite) that satisfied the Lyapunov inequality A T P + PA < 0, 0 1 1 2 T p p 11 21 p p 12 22 + p p 11 21 p p 12 22 0 1 1 < 0 2 Henrion, 2003

Bilinear Matrix Inequalities A bilinear matrix inequality (BMI) is an expression, BMI example (Henrion,2005): (x) = 0 + i x i i + i j x i x j ij < 0 Nonconvex! (x) = ¾ 10 0.5 2 0.5 4.5 0 2 0 0 + x ¾ 1 9 0.5 0 0.5 0 3 + 0 3 1 ¾ 1.8 0.1 0.4 x 2 0.1 1.2 1 0.4 1 0 + x ¾ 1x 2 0 0 2 0 5.5 3 < 0 2 3 0 IEEE Colloquium on Optimisation for Control, Sheffield, 24 April 2006 p. 6/2

Mixed H 2 /H Control State-feedback close-loop system with two performance channels ẋ = (A + B w K)x + B w w z = (C + D u K)x + D w w z 2 = (C 2 + D 2u K)x and mixed performance specification T w z < γ and T w z2 2 < γ 2, BMIs for the H norm constraint AX + B u KX + ( ) + B w Bw T C X + D u KX + D w Bw T D w D w T γ2 < 0, X I > 0 ¾ BMIs for the H 2 norm constraint ¾ AX 2 + B u KX 2 + ( ) B w < 0, I W C 2X 2 + D 2u KX 2 > 0, tr(w) < γ2 X 2 ¾ Two terms KX and KX 2 cannot be linearized simultaneously, Remedy X = X = X 2, then Y = KX as change of variable to recover convexity, this leads to conservative results. IEEE Colloquium on Optimisation for Control, Sheffield, 24 April 2006 p. 7/2

Mechanical network: vehicle suspension quarter-car model ront suspension s m s z s k s K(s) m u z u k t z r IEEE Colloquium on Optimisation for Control, Sheffield, 24 April 2006 p. 8/2

Mechanical networks A mechanical network is a rigid interconnection of mechanical elements (springs, masses, dampers and levers). A mechanical one-port network with force-velocity pair (, v) is defined to be passive if, T 0 (t)v(t)dt 0. one port mechanical network v 1 v 2 Impedance is defined as a positive-real function Z(s) = across variable through variable = ˆvˆ Admittance is defined as a positive-real function Y (s) = 1 Z(s) = ˆ ˆv. IEEE Colloquium on Optimisation for Control, Sheffield, 24 April 2006 p. 9/2

orce-current analogy Mechanical Electrical v 2 v 1 d dt = k(v 2 v 1 ) Y (s) = k s spring i v 2 v 1 di dt = 1 L (v 2 v 1 ) i Y (s) = 1 Ls inductor v 2 = b d(v 2 v 1 ) dt Y (s) = bs i i v 2 v 1 v 1 mass? i = C d(v 2 v 1 ) dt Y (s) = Cs capacitor v 2 v 1 = c(v 2 v 1 ) Y (s) = c damper i v 2 v 1 i i = 1 R (v 2 v 1 ) Y (s) = 1 R resistor IEEE Colloquium on Optimisation for Control, Sheffield, 24 April 2006 p. 10/2

The Inerter Concept: A mechanical two-terminal device such that the relative acceleration between the terminals is proportional to the force applied at the terminals. M. C. Smith, Synthesis of Mechanical Networks: The Inerter, IEEE Transactions on Automatic Control, 47(10), 1648 1662, 2002 IEEE Colloquium on Optimisation for Control, Sheffield, 24 April 2006 p. 11/2

The Inerter Concept: A mechanical two-terminal device such that the relative acceleration between the terminals is proportional to the force applied at the terminals. Theory: v 2 v 1 = b( v 2 v 1 ) b : inertance Practice: Can one build an inerter? Rack-and-pinion inerter, mass 2 kg, inertance = 70 700 kg, travel = 80 mm IEEE Colloquium on Optimisation for Control, Sheffield, 24 April 2006 p. 11/2

Suspension for the quarter-car model s sprung and unsprung mass k s m s K(s) z s z s = s m s m s k s m s (z s z u ), z u = m u + k s m u (z s z u ) + k t m u (z r z u ), m u z u suspension k t ˆ = K(s)(sẑ s sẑ u ), z r Ride comfort: J 1 := 2π(V κ) (1/2) stẑr ẑ s 2 RMS body vertical acceleration in response to road disturbances IEEE Colloquium on Optimisation for Control, Sheffield, 24 April 2006 p. 12/2

Suspension for the quarter-car model s sprung and unsprung mass k s m s K(s) z s z s = s m s m s k s m s (z s z u ), z u = m u + k s m u (z s z u ) + k t m u (z r z u ), m u z u suspension k t ˆ = K(s)(sẑ s sẑ u ), z r Tyre grip: J 3 := 2π(V κ) 1/2 1 s T ẑ r k t (ẑ u ẑ r ) 2 RMS dynamic tyre load in response to road disturbances IEEE Colloquium on Optimisation for Control, Sheffield, 24 April 2006 p. 12/2

Suspension for the quarter-car model s sprung and unsprung mass k s m s K(s) z s z s = s m s m s k s m s (z s z u ), z u = m u + k s m u (z s z u ) + k t m u (z r z u ), m u z u suspension k t ˆ = K(s)(sẑ s sẑ u ), z r Rejection of external loads: J 5 := T ˆ s ẑ s IEEE Colloquium on Optimisation for Control, Sheffield, 24 April 2006 p. 12/2

Control problem formulation Objective: Synthesize a positive real admittance Y (s) to improve performance criterions. ormulation: Optimize a vector of T w z over positive real controllers K(s). z G(s) w v 2 v 1 velocity K(s) force Solution: Characterize and solve the problem using Linear Matrix Inequalities IEEE Colloquium on Optimisation for Control, Sheffield, 24 April 2006 p. 13/2

LMI ormulation Characterize H and H 2 performances (Scherer et al., 1997), T w z 2 2 = Tr(CXCT ), X solves a Lyapunov equation... Bounded real lemma of T w z = sup w L2 z 2 w 2... Positive Real Lemma (Boyd et al., 1994): Given that K(s) positive real, X > 0, AT X + XA XB C T 0 B T X C D T D IEEE Colloquium on Optimisation for Control, Sheffield, 24 April 2006 p. 14/2

multi-objective controller design Simultaneous J 1 and J 3 minimization, min K(s) positive real T ẑ r ˆż s 2 and Tẑr Ê (ẑ u ẑ r ) 2 Positive real constraint bilinear matrix inequality with respect to Lyapunov matrices X cl, X k and controller matrix K(s) The approach taken here is to minimise (1 λ) J2 1 Ĵ 1 2 + λ J2 3 Ĵ 3 2, for 0 < λ < 1 Solved locally with iterative convex optimization methods IEEE Colloquium on Optimisation for Control, Sheffield, 24 April 2006 p. 15/2

Multi-objetive optimisation with BMIs 600 BMI Pareto front 580 560 540 J 3 520 500 480 460 1.44 1.46 1.48 1.5 1.52 1.54 1.56 J 1 Lolcal Search: Relies on the intuitive choice of a feasible starting point IEEE Colloquium on Optimisation for Control, Sheffield, 24 April 2006 p. 16/2

MOGA-based method Parameter encoding:the decision variables controller Optimise simultaneously: K(s) = bs + c s2 + a 1 s + a 2 s 2 + b 1 s + b 2 Ride comfort J 1 := 2π(V κ) (1/2) stẑr ẑ s 2 Tyre grip J 3 := 2π(V κ) 1/2 1 s T ẑ r k t (ẑ u ẑ r ) 2 subject to K(s) been positive real (constraint) IEEE Colloquium on Optimisation for Control, Sheffield, 24 April 2006 p. 17/2

Multi-objetive optimisation with GAs 580 BMI Pareto front MOGA1 Pareto front 560 540 J 3 520 500 480 460 1.45 1.5 1.55 1.6 1.65 1.7 J 1 Converge to a Local Pareto optimum Deceptive problem: often the entire search favors the non-global optimum IEEE Colloquium on Optimisation for Control, Sheffield, 24 April 2006 p. 18/2

Multi-objetive optimisation with GAs 580 BMI Pareto front MOGA1 Pareto front MOGA2 Pareto front 560 540 J 3 520 500 480 460 1.45 1.5 1.55 1.6 1.65 1.7 J 1 Remedy: Increase the population size (from 200 to 600 individuals) IEEE Colloquium on Optimisation for Control, Sheffield, 24 April 2006 p. 19/2

MOGA-based method:three objectives Parameter encoding:the decision variables controller Optimise simultaneously: K(s) = bs + c s2 + a 1 s + a 2 s 2 + b 1 s + b 2 Ride comfort: J 1 := 2π(V κ) (1/2) stẑr ẑ s 2 Tyre grip: J 3 := 2π(V κ) 1/2 1 s T ẑ r k t (ẑ u ẑ r ) 2 Rejection to external roads: J 5 := T ˆs ẑ s subject to K(s) been positive real (constraint) IEEE Colloquium on Optimisation for Control, Sheffield, 24 April 2006 p. 20/2

Three-objective optimisation 560 12 540 10 520 8 J 3 500 J 5 6 480 4 460 1.4 1.5 1.6 1.7 1.8 1.9 J 1 2 550 500 J 3 450 1.4 1.6 1.8 J 1 12 12 10 10 8 8 J 5 J 5 6 6 4 4 2 450 500 550 600 2 1.4 1.5 1.6 1.7 1.8 1.9 J 3 J 1 IEEE Colloquium on Optimisation for Control, Sheffield, 24 April 2006 p. 21/2

3D Pareto front 12 11 10 9 8 J 5 7 6 5 4 3 2 560 540 520 500 J 3 480 460 1.4 1.5 1.6 J 1 1.7 1.8 1.9 IEEE Colloquium on Optimisation for Control, Sheffield, 24 April 2006 p. 22/2

3D Pareto front 12 11 10 9 8 J 5 7 6 5 4 3 1.8 2 1.6 560 550 540 530 520 510 500 490 480 470 460 450 1.4 J 1 J 3 IEEE Colloquium on Optimisation for Control, Sheffield, 24 April 2006 p. 22/2

Realisation of the "best" suspension 1 0.8 0.6 Cost 0.4 0.2 0 J_1 J_3 J_5 Objective 361 kg 14740 N/m 3496 Ns/m 2871 Ns/m 12 kg IEEE Colloquium on Optimisation for Control, Sheffield, 24 April 2006 p. 23/2

Conclusions BMI optimisation over positive real controllers has been shown to be effective for two-objective optimisation problem. However, the design of a single controller requires several attempts. MOGA-base method- The Pareto-front can be investigated in a single optimisation run. More than two objectives can be included straightforwardly Thusm the designer engineer has a choice from among the Pareto-optimal set. A Possible disadvantage of using MOGA is that some experience may be needed to choose appropriate parameter values, such as the population size. IEEE Colloquium on Optimisation for Control, Sheffield, 24 April 2006 p. 24/2

The end Thank you, e-mail:am664@cam.ac.uk Reference: A Molina-Cristobal, C Papageorgiou, G T Parks, M C Smith, P J Clarkson. Multi-objective Controller Design: Evolutionary Algorithms and Bilinear Matrix Inequalities for a Passive Suspension Proceedings of the IAC Workshop on Control Applications of Optimization, Cachan, rance, April 2006 IEEE Colloquium on Optimisation for Control, Sheffield, 24 April 2006 p. 25/2