Multi-objective optimization for control design James Whidborne - - PowerPoint PPT Presentation

Multi-objective optimization for control design

James Whidborne

Centre for Aeronautics Cranfield University Sensors & their Application XVIII Queen Mary University of London 12–13 September 2016

Outline

• Conflicts and Trade-offs in Control Systems
• Multi-objective Optimization Problem
• Method of Inequalities (MOI)
• Multi-Objective Genetic Algorithms (MOGA)
• Mixed Optimization
• Convex Multi-objective Optimization
• Design Example - Maglev Suspension Controller Design
• References

Slides will be available from

Multi-objective optimization for control design — 1/63

Abstract Control system design solutions require the right balance between conflicting requirements such as cost, complexity, robustness and

• performance. Hence to quantify the design process, it is often

appropriate to formulate control system design problems as multi-objective optimization problems. There are a number of different approaches to solving such problems. The presentation will introduce the concept of multiobjective optimization, and outline several approaches familiar to the presenter, including the Method of Inequalities, the Multi-Objective Genetic Algorithm and multiobjective convex optimization. Various tradeoffs in control design will also be explored, in particular those between the quality

• f the sensors and the control system performance. The idea of

mixed-optimization is also introduced. The methods are illustrated with examples from flight control and maglev suspension control.

Multi-objective optimization for control design — 2/63

Conflicts and Trade-offs in Control Systems

• In general, engineering design consists of obtaining the right balance

between conflicting cost, design and performance requirements

• There are trade-offs to be made between conflicting requirements
• Hence to quantify design, it is appropriate to formulate problem as a

multi-objective problem

• There exist trade-offs in control systems design

Multi-objective optimization for control design — Conflicts and Trade-offs in Control Systems 3/63

Example - simple servo system

Consider a simple servo system with a proportional controller k with negative feedback

1 s(s+a)

k

✲ ✲ ✲ ✲ ✻ controller plant/motor reference

• utput

+ − Design parameter is controller gain k

Multi-objective optimization for control design — Conflicts and Trade-offs in Control Systems 4/63

Example - simple servo system

Plant: G(s) =

1 s(s+a)

Proportional controller k gives closed loop system: T(s) =

k s2+as+k

Fractional overshoot to step response: Mp = e−aπ/

√ 4k−a2

Time-to-peak tp =

2π √ 4k−a2 for k > (a2/4)

Time (sec.) Amplitude Step Response

5 10 15 20 25 0.5 1 1.5 From: U(1) To: Y(1)

Typical step response of second order system showing peak overshoot and time-to-peak

Multi-objective optimization for control design — Conflicts and Trade-offs in Control Systems 5/63

Example - simple servo system

There is a trade-off between Mp and tp

10

−1

10 10

1

10

2

0.2 0.4 0.6 0.8 1

Mp k (log-scale)

10

−1

10 10

1

10

2

2 4 6 8 10

tp (sec)

Overshoot Mp and time-to-peak tp against k

Multi-objective optimization for control design — Conflicts and Trade-offs in Control Systems 6/63

Example - simple servo system

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 3 4 5 6 7 8 9 10

Mp tp (sec) Trade-off curve showing Mp against tp

Multi-objective optimization for control design — Conflicts and Trade-offs in Control Systems 7/63

Sensor Noise versus Disturbance Trade-off

• Most feedback control problems can be cast into architecture below
• Considers effects of reference input, plant disturbance & sensor noise

R(s)

reference

✲

+ −

✲ K(s)

controller

✲ U(s) G(s)

plant

✲

+ +

❄ D(s)

disturbance

✲ Y(s)

• utput

❄ ✛

+ + M(s) sensor noise

✻

measured

• utput

Multi-objective optimization for control design — Conflicts and Trade-offs in Control Systems 8/63

The fundamental conflict

R(s)

✲

+ −

✲ K(s) ✲

U(s) G(s)

✲

+ +

❄

D(s)

✲ Y(s) ❄ ✛

+ + M(s)

✻

• Error is difference between reference and output

E(T) = R(T) − Y(T)

• Define sensitivity function S(s) & closed-loop transfer function T(s)

S(s) = 1 1 + G(s)K(s) T(s) = G(s)K(s) 1 + G(s)K(s) giving E(s) = S(s) [R(s) − D(s)] + T(s)M(s)

Multi-objective optimization for control design — Conflicts and Trade-offs in Control Systems 9/63

The fundamental conflict

• Taking magnitudes

|E(s)| = |S(s)| (|R(s)| + |D(s)|) + |T(s)||M(s)|

• To reduce error cause by R(s) and D(s) make |S(s)| small (in some sense)
• To reduce error cause by M(s) make |T(s)| small (in some sense)
• BUT

S(s) =

1 1+G(s)K(s)

and T(s) =

G(s)K(s) 1+G(s)K(s)

⇒ T(s) + S(s) = 1

• Hence

|T(jω)| + |S(jω)| ≥ 1

Re Im T(s) S(s) 1

Multi-objective optimization for control design — Conflicts and Trade-offs in Control Systems 10/63

Fundamental conflict — frequency response

• For design in frequency domain

|T(jω)| + |S(jω)| ≥ 1

• If |S(jω)| is made nearly zero, |T(jω)| becomes nearly unity (often greater

than unity)

• Conversely, if |T(jω)| is nearly zero, |S(jω)| must be at least nearly unity or

greater

• Unavoidable trade-off between attenuating plant disturbances, D(s), and

filtering out measurement error, M(s)

• Can also be shown that making |T(jw)| small means stability robustness &

small control effort plant disturbance rejection & reference tracking versus measurement noise filtering & stability robustness & control effort

Multi-objective optimization for control design — Conflicts and Trade-offs in Control Systems 11/63

Fundamental conflict example - simple servo system

10-2 10-1 100 101 102

frequency, ω (rad/s)

0.5 1 1.5 2 2.5

magnitude, |·| |S(jω)| |T(jω)| |S(jω)|+|T(jω)| Sensitivities for Servo System (with 10% overshoot)

Multi-objective optimization for control design — Conflicts and Trade-offs in Control Systems 12/63

Another Control Design Trade-off — Waterbed effects

Bode’s Sensitivity Integral states that the average of the logarithm

• f the sensitivity is conserved.

If the sensitivity function is decreased at low frequencies, it must be traded-off for a larger sensitivity function at higher frequencies ✻ ✲ω

log |S(jω)|

❄ ◆ ❘ ✻ ✻ ✍

push down pop up

• log|S(jω)|dω ≥ 0

Multi-objective optimization for control design — Conflicts and Trade-offs in Control Systems 13/63

Multi-objective Optimization

• Express design aims quantitatively as a set of n design objective functions

{φi(p) : i = 1 . . . n}, where p denotes the design parameter vector chosen by the designer

• Formulate design problem as a multi-objective optimization problem

Problem

min

p∈P {φi(p), for i = 1 . . . n}

where P denotes the set of possible design parameters p

Multi-objective optimization for control design — Multi-objective Optimization 14/63

Pareto-optimal solutions

• In most cases, objective functions, φi, are in conflict, so the reduction of one
• bjective function leads to increase in another
• Result of multi-objective optimization known as a Pareto-optimal solution
• Pareto-optimal solution has

property that it is not possible to reduce any one φi without increasing at least one other φi

• Point lying in the interior of the

attainable set is sub-optimal, since both φ1 and φ2 can be both reduced

• A solution to a multi-objective
• ptimization problem is hence

generally not unique ✻ ✲ ✻ Attainable set φ1(p) φ2(p) Pareto

• ptimal set

Multi-objective optimization for control design — Multi-objective Optimization 15/63

Multi-Objective Design Approaches

• Many approaches suggested in several disciplines
• Non-linear programming — many approaches proposed, in particular, work of

Polak, Mayne and coauthors

• Interactive multi-objective programming
• Convex optimization
• Some early software
• DELIGHT — many of Polak and Mayne approaches
• ANDECS — whole CACSD environment
• QDES —- Boyd & Barratt approach to convex optimization
• MOPS — Multi-Objective Parameter Synthesis — from DLR
• MODCONS — mixed optimization — MATLAB Toolbox
• MATLAB Optimization Toolbox (Goal Attainment method)
• CONVEX CONTROL DESIGN TOOLBOX — MATLAB Toolbox from ONERA
• Q-Synthese — convex optimization MATLAB Toolbox from Universität Kassel
• Many modern packages have been developed for general problems

Multi-objective optimization for control design — Multi-objective Optimization 16/63

Weighted Sum Gap

To solve problem, it is common to convert to a single objective optimization problem by use of a weighted sum objective function & use non-linear programming Φ(p) =

n

• i=1

wiφi(p)

• If some φi are not convex, then

parts of the Pareto optimal set may not be found for any wi

• Convex problems pose few

difficulties in converting to single-objective problems ✻ ✲ φ1 φ2

Pareto

• ptimal

curve duality gap weighted sum solutions

❃ ✾ ✙ ✌ ❄

Multi-objective optimization for control design — Multi-objective Optimization 17/63

Minimax Problem

Also common to use a max objective function Φ(p) = max

i

wiφi(p)

• Objective function is not everywhere differentiable — can cause problems
• Many other formulations have been proposed — some maintain the genuine

multi-objective nature of problem

Multi-objective optimization for control design — Multi-objective Optimization 18/63

Interactive Multi-objective Programming (IMOPS)

• Generally, the multi-objective design process is interactive, with computer

providing information to designer about conflicting design requirements, and designer adjusting problem specification to explore the various possible solutions to problem

• The design process is thus a two way process, with
• 1. computer providing information to designer about conflicting design requirements
• 2. and designer making decisions about ‘trade-offs’ between design requirements

based on this information as well as on the designer’s knowledge, experience and intuition about the particular problem

• Designer supported by various graphical displays which provide information

about progress of the search algorithm and about conflicting design requirements

Multi-objective optimization for control design — Multi-objective Optimization 19/63

Convex optimization

• Many multi-objective problems can be posed as convex optimization

problems by using a Youla parameterization (Boyd & Barratt)

• Efficient methods have been developed to solve such problems
• Some multi-objective problems can be posed as linear matrix inequalities

(LMI’s) — some others as linear programming problems (Elia & Dahleh)

Multi-objective optimization for control design — Multi-objective Optimization 20/63

Method of Inequalities (MOI)

Problem is expressed as a set of algebraic inequalities which need to be satisfied for a successful design

Problem

Find p such that inequalities φi(p) ≤ εi for i = 1 . . . n are satisfied, where εi are design goals chosen by the designer and represent the largest tolerable values of the objective functions φi

• The aim of the design is to find a p that simultaneously satisfies the set of

inequalities

• Generally used interactively in an IMOPS environment
• Designer iteratively tightens and relaxes εi with the aid of graphical tools
• In the limit, a Pareto-optimal solution is obtained — but in practise,

near-optimal solutions work better (see John Doyle’s work on Highly Optimized Systems and Robust Yet Fragile systems, for example)

Multi-objective optimization for control design — Method of Inequalities (MOI) 21/63

MOI — Admissible Set

Any solution to the problem lies in the admissible set (the set of all solutions): {p : φi(p) ≤ εi, i = 1 . . . n} ✻ ✲ φ1 φ2

Pareto

• ptimal

curve

ε1 ε2

admissible set

✾ ②

Multi-objective optimization for control design — Method of Inequalities (MOI) 22/63

MOI — IMOPS Interface

Multi-objective optimization for control design — Method of Inequalities (MOI) 23/63

Multi-Objective Genetic Algorithms (MOGA)

• Design philosophy of MOGA differs

from MOI, in that a set of simultaneous solutions is sought, and designer then selects best solution from the set

• Idea is to develop a population of

Pareto-optimal or near Pareto-optimal solutions ✻ ✲ φ1 φ2

Pareto

• ptimal

curve

✾

non-dominated solution dominated solution

× × × × × × ×

Multi-objective optimization for control design — Multi-Objective Genetic Algorithms (MOGA) 24/63

Multi-Objective Genetic Algorithms (MOGA)

• Design philosophy of MOGA differs

from MOI, in that a set of simultaneous solutions is sought, and designer then selects best solution from the set

• Idea is to develop a population of

Pareto-optimal or near Pareto-optimal solutions

• However, to restrict the size of the

near Pareto-optimal set design goals are also incorporated in a similar way to the MOI

• This formulation maintains the

genuine multi-objective nature of the problem ✻ ✲ φ1 φ2

Pareto

• ptimal

curve

✾

non-dominated solution dominated solution

× × × × × × × ε1 ε2

solution

Multi-objective optimization for control design — Multi-Objective Genetic Algorithms (MOGA) 25/63

MOGA Ranking

• The MOGA is set into a

multi-objective context by means of the fitness function

• Individuals are ranked on the basis
• f the number of other individuals

they are dominated by for the unsatisfied inequalities

• GAs are naturally parallel and

hence lend themselves well to multi-objective settings

• Also work well on non-smooth
• bjective functions – it is very easy

to extend GA’s to solve mixed continuous/integer problems

φ ε φ

i h j g f e d c a b

2

ε

2 1 1 Multi-objective optimization for control design — Multi-Objective Genetic Algorithms (MOGA) 26/63

Trade-off Diagram

1 2 3 4 5 6 7 Objective Cost

Multi-objective optimization for control design — Multi-Objective Genetic Algorithms (MOGA) 27/63

Example Solution

10

−5

10

−4

10

−3

10

−2

10

−1

10 20 40 60 80 100 120 140 ||R−Rq||∞ (log scale) Total number of bits for implementation 1 2 Multi-objective optimization for control design — Multi-Objective Genetic Algorithms (MOGA) 28/63

Mixed Optimization

Analytical optimization techniques (e.g. H∞, LQG) generally

• 1. have non-explicit closed-loop

performance,

• 2. are single-objective,
• 3. are robustly stable,
• 4. provide high-order controllers,
• 5. are not very flexible,
• 6. provide a global optimum,
• 7. can deal with relatively large

multivariable problems; Parameter optimization based methods (based on hill-climbing, GA’s etc)

• 1. have explicit closed-loop performance,
• 2. are often multi-objective,
• 3. are not implicitly robustly stable,
• 4. provide simple controllers,
• 5. are flexible,
• 6. are often non-convex resulting in local

minima,

• 7. can deal with small problems only,
• 8. may have difficulty stabilizing the

system.

Multi-objective optimization for control design — Mixed Optimization 29/63

Mixed Optimization

• A combination of analytical optimization and parameter search methods may

be able to overcome some of the limitations of using just one approach.

• The basic idea is simple, and that is to use search methods to automate the

iterative procedure of designing the weighting functions and other design parameters to achieve the performance specifications for the system.

Multi-objective optimization for control design — Mixed Optimization 30/63

Mixed Optimization – Overview

• Design parameters in analytical optimization are often weighting functions —

e.g. LQR cost function: ∞ (x′(t)Qx(t) + u′(t)Ru(t)) dt design parameters are Q and R

• Parameterize weighting functions by p and use MOI or MOGA to design p in

combination with optimal controller synthesis in mixed optimization approach

• e.g. for nominal plant G(s) augmented by a set of nw weighting functions

W(s) = (W1(s), W2(s) . . . Wnw ) — controller Kmin(s, G, W) which is optimal in some sense can be synthesized — set of closed-loop performance functions φ of optimal control system can be computed

• Parameterize weighting functions by the design vector p formulate as for MOI
• r MOGA and use search algorithm to design p such that set of performance

criteria are satisfied

Multi-objective optimization for control design — Mixed Optimization 31/63

Mixed Optimization - Early Examples

• J.S. Baras, M.K.H. Fan, W.T. Nye, and A.L. Tits. DELIGHT.LQG: A CAD system for control

systems design using LQG controller structure. In 18th Annual Conf. on Inf. Sci and Syst., Princeton, NJ, 1984.

• J.F. Whidborne, I. Postlethwaite, and D.-W. Gu. Robust controller design using H∞ loop-shaping

and the method of inequalities. Technical Report 92-33, Leicester University Engg Dept, Leicester, U.K., 1992.

• W. Tych. Multi-objective optimisation technique for mapping the technical design objectives into

the values of the weighting matrices in the linear quadratic regulator design problem. Technical Report TR-117, Univ. Lancaster Centre for Research on Environmental Systems and Statistics, Lancaster, UK, 1994.

• J.E. Paddison, R.M. Goodall, J. Bals, and G. Grübel. Multi-objective design study for a maglev

suspension controller using the databased ANDECS-MATLAB environment. In Proc. IEEE/IFAC

• Symp. on Comp. Aided Contr. Syst. Design (CACSD’94), pages 239–246, Tuscon, USA, March

1994.

• D. Haessig. Selection of LQR/LTR weighting matrices through constrained optimisation. In Proc.

1995 Amer. Contr. Conf., pages 458–460, Seattle, USA, 1995.

• N.V. Dakev, J.F. Whidborne, and A.J. Chipperfield. H∞ design of an EMS control system for a

maglev vehicle using evolutionary algorithms. In Proc. GALESIA 95, pages 226–231, Sheffield U.K., September 1995.

• J. King and B.A. White. Robust controller design using H∞ loop shaping with a genetic algorithm

multiobjective optimization. In Proc. 13th IFAC World Congress, page , San Fransisco, CA, June 1996.

Multi-objective optimization for control design — Mixed Optimization 32/63

Mixed Optimization — Conclusions

• Resulting controllers are moderately high order — controller order reduction

stage could be introduced

• Problem is non-linear — local minima only found
• Hill-climbing methods require accurate calculation of indices
• GA’s inefficient but indices can be approximate (problem can be non-smooth)

e.g. mixed sensitivity H∞-optimal

Multi-objective optimization for control design — Mixed Optimization 33/63

Convex Optimization

A function φ(p) is convex if for all p1, p2 ∈ P φ(λ1p1 + λ2p2) ≤ λ1φ(p1) + λ2φ(p2), λ1 + λ2 = 1,

• Convex function
• ptimization problems can

be solved relatively efficiently

• Norms of convex functions

are also convex

• Hence multiobjective

problems that are convex can be solved efficiently (by weighted sums, minimax, weighted squares etc) ✻ ✲ φ p p1 p2

φ(λ1p1 +λ2p2) λ1φ(p1) +λ2φ(p2)

Multi-objective optimization for control design — Convex Multi-objective Optimization 34/63

Convex Problems

Many performance/robustness indices are convex e.g.

• H∞-norms,

H2-norms, L1-norms, . . .

• Time domain

specifications envelope

• Frequency domain

specifications envelope

5 10 15

t

0.2 0.4 0.6 0.8 1 1.2 1.4

y(t) y1(t) y2(t) (y1(t)+y2(t))/2 bounds

Multi-objective optimization for control design — Convex Multi-objective Optimization 35/63

Q-parametrization

• Standard closed loop control

arrangement z = P11w + P12u e = P21w + P22u

• Closed loop transfer function:

Hzw = P11 + P12K(I − P22K)−1P21

• Not convex unless P22 = 0
• Convexify by Q-parametrization

w ✲ z u K y ✛ ✲ ✲ P11 P12 P21 P22

Multi-objective optimization for control design — Convex Multi-objective Optimization 36/63

Q-parametrization

• stabilizing nominal controller Knom is

added

• Controller F(s) is given as

F(s) = LFTl(Knom, ˜ Q)

• But to make the problem convex,

transfer function Hev from v to ˜ e must be zero

• Plant and nominal controller are

combined into a new transfer matrix ˜ T = LFTl(P, Knom) =:

• T1

T2 T3 T4

• P(s)

✲ Knom(s) ✛ ✲ ✲ ✲ ˜ Q(s)✛ w z u y v ˜ e ˜ T(s)

Multi-objective optimization for control design — Convex Multi-objective Optimization 37/63

Convex Q-parametrization

• Replace lower dotted block with

Q = (I − ˜ QT4)−1 ˜ Q = ˜ Q(I − T4 ˜ Q)−1

• Final structure is an LFT,

Hzw = LFTl(T, Q), and because T4 is zero, the closed loop transfer matrix is Hzw = T1 + T2QT3 and is convex in Q ˜ T(s) ✲ −T4(s) Σ ✲ Σ ✲ T4(s) ˜ Q(s) ✲ v ˜ e ✲ ❄ w z e ✛ ✲ ✛ T(s) Q(s)

Multi-objective optimization for control design — Convex Multi-objective Optimization 38/63

Observer structure Q-parametrization

˜ T(s) ✲ −T4(s) Σ ✲ Σ ✲ T4(s) ˜ Q(s) ✲ v ˜ e ✲ ❄ w z e ✛ ✲ ✛ T(s) Q(s)

Multi-objective optimization for control design — Convex Multi-objective Optimization 39/63

Convex Control Problem Problem

min

Q Φ

where Φ = [φ1, φ2, . . . , φn] All that remains is to parameterize Q(s) so that the convexity is maintained. For example Q(s, p) =

∞

• 1

piψi(s) can characterize all stable Q(s) (e.g. Ritz-Galerkin approximation) where {ψi(s)} provides an orthogonal basis. A finite truncation of the sequence can be used, i.e. Q(s, p) =

m

• 1

piψi(s)

Multi-objective optimization for control design — Convex Multi-objective Optimization 40/63

Convex Optimization — Conclusions

• Resulting controllers are very high order (in theory infinite dimensional) and
• ften not practical
• Performance indices φi must be calculated to very high accuracy — time

consuming

• Method provides “Limitations of Performance”

Multi-objective optimization for control design — Convex Multi-objective Optimization 41/63

Some References

• Boyd, S. & Barratt, C. Linear Controller Design: Limits of Performance,

Prentice-Hall, 1991 (convex MO optimization for control systems – can download from Boyd’s website)

• Stephen Boyd and Lieven Vandenberghe Convex Optimization Cambridge

University Press 2004 (can download from Boyd’s website)

• A. Linnemann. Convergent Ritz approximations of the set of stabilizing
• controllers. Syst. Control Lett., 36:151–156, 1999.
• G. Ferreres and G. Puyou. Flight control law design for a flexible aircraft:

Limits of performance. J. Guid. Control Dyn., 29(4):870–870, 2006.

• C. Scherer, P. Gahinet, and M. Chilali. Multiobjective output-feedback control

via LMI optimization. IEEE Trans. Autom. Control, 42(7):896–911, 1997.

Multi-objective optimization for control design — Convex Multi-objective Optimization 42/63

Example — Maglev Suspension Controller Design

• DC electro-magnetic

suspension uses attractive forces of electro-magnets acting upwards to levitate vehicle towards steel guideway

• High speed frictionless

transport — up to 500 km/hour

• Several operating Maglevs

including Shanghai-Pudong, Aichi Linimo and Incheon Airport

Multi-objective optimization for control design — Design Example — Maglev Suspension Controller 43/63

Control aims

• EMS inherently unstable —

needs active control

• must maintain airgap

between vehicle and guideway

• ensure quality of ride
• avoid actuator saturation

passenger cabin chassis levitation magnet guidance magnet guideway secondary suspension

Multi-objective optimization for control design — Design Example — Maglev Suspension Controller 44/63

Maglev Model m2 m1

❄ ✻ ❄ ❄ ❄ ❄ x2 h x1 F z

c k

mg

guideway passenger cabin chassis

• dc electromagnet force:

F(i, z, t) = K 2 i(t) z(t) 2 i is current

• Control voltage:

v(t) = Ri(t) + K d dt i(t) z(t)

• .

R is total resistance

• Secondary suspension consisting of airsprings and hydraulic shock

absorbers — assumed linear

• h(t) is disturbance resulting from variations in guideway profile

Multi-objective optimization for control design — Design Example — Maglev Suspension Controller 45/63

Performance for Maglev Suspension

Major disturbance from variations in guideway height: D = sup

• ˙

h(t)

• : t ≥ 0
• = 30 mm/s

For all possible h(t) such that sup{

• ˙

h(t)

• : t ≥ 0} ≤ D

sup {|yi(t, h)| : t ≥ 0} = D ∞ |yi(τ, 1)| dτ yi(τ, 1) is unit step response of ith output of linearised closed-loop system Nominal performance functions are defined on the airgap, passenger acceleration and control voltage

Multi-objective optimization for control design — Design Example — Maglev Suspension Controller 46/63

Performance for Maglev Suspension

For non-linear system performance, responses of airgap, passenger acceleration and control voltage to test input, htest(t), are calculated, and maximum absolute values determined

2 4 6 8 10 12 −50 50 100 150 200 Time (sec) Guideway disturbance (mm)

Test input htest

Multi-objective optimization for control design — Design Example — Maglev Suspension Controller 47/63

Performance for Maglev Suspension

A maximum power spectral density Φmax(ω) of passenger cabin acceleration has been recommended by US Department of Transportation as minimum ride quality standard — performance functional is defined based on Φmax(ω)

• The p.s.d. of track variations:

Φhh(ω) = Av/ω2 A depends on track quality, v is speed

• p.s.d. of passenger cabin:

Φ¨

x2¨ x2(ω) = |T¨ x2h(ω)|2 Av ω2 .

10 10

1

10

2

10 10

1

10

2

10

3

10

4

10

5

Frequency (rad/sec) Acceleration psd

Max permitted power spectral density

Multi-objective optimization for control design — Design Example — Maglev Suspension Controller 48/63

The Design

• Air gap measurement z, and passenger cabin acceleration, ¨

x2, used for feedback

• Weighting function configurations :

W1 = p1

(s2+p2s+p3) (s2+p4s+p5)

W2 =

• p6

(s+p7) (s+p8), p9

• Secondary suspension stiffness and damping factors c and d are included as

design parameters p10 and p11

Multi-objective optimization for control design — Design Example — Maglev Suspension Controller 49/63

Objective functions

Seven performance objective functions defined: φ1 = D ∞ |z(τ, 1)|dτ φ2 = D ∞ |¨ x2(τ, 1)|dτ φ3 = D ∞ |v(τ, 1)|dτ φ4 = max

t

{|z′(t, htest)|} φ5 = max

t

{|¨ x′

2(t, htest)|}

φ6 = max

t

{|v′(t, htest)|} φ7 = max

ω {Φ¨ x2¨ x2(ω) − Φmax(ω)}

Performance goals set: εγ = 5 ε1 = ε4 = 5 mm ε2 = ε5 = 500 mm/s2 ε3 = ε6 = 600 V ε7 = 0

Multi-objective optimization for control design — Design Example — Maglev Suspension Controller 50/63

Problem

Problem for mixed optimization design of EMS control system is to find p satisfying the inequalities: γ0(p) ≤ ǫγ and φi(p) ≤ εi, i = 1, 2, . . . , 7 where γ0 is optimal H∞-norm used by McFarlane & Glover’s LSDP

W1 G W2 Gs

✲ ✲ ✲ ✲

u y W1 G W2 Gs

✲ ✲ ✲

u y Ks ✛ W1 G W2 K

✲ ✛ ✛ ✛

u y Ks

Multi-objective optimization for control design — Design Example — Maglev Suspension Controller 51/63

Solution

• Multi-objective simulated annealing was used to solve problem
• Design vector found which met all design goals except φ1 and φ7, which were
• nly marginally exceeded
• Final performance objective functions:

γ0 = 2.64 φ1 = 5.06 mm φ2 = 391.4 mm/s2 φ3 = 201.9 V φ4 = 4.38 mm φ5 = 291.1 mm/s2 φ6 = 33.6 V φ7 = 0.075

• Designed weighting functions:

W1 = 903.2 (s2+169.6s+485.4)

(s2+69.5s+488.3)

W2 =

• 426.9 (s+256.2)

(s+379.2), 3.23

• Secondary suspension stiffness c = 90.3

Secondary suspension damping factors d = 20.0

Multi-objective optimization for control design — Design Example — Maglev Suspension Controller 52/63

Nonlinear system responses to test input htest

2 4 6 8 10 12 −50 50 100 150 200 Time (sec) Guideway disturbance (mm) Test input 2 4 6 8 10 12 −6 −4 −2 2 4 Airgap response to test input Time (sec) Airgap (mm)

Multi-objective optimization for control design — Design Example — Maglev Suspension Controller 53/63

Nonlinear system responses to test input htest

2 4 6 8 10 12 −300 −200 −100 100 200 Passenger acceleration response to test input Time (sec) Passenger acceleration (mm/sec/sec) 2 4 6 8 10 12 −20 20 40 60 Control voltage response to test input Time (sec) Control voltage (volts)

Multi-objective optimization for control design — Design Example — Maglev Suspension Controller 54/63

Power Spectral Densities

10 10

1

10

2

10 10

1

10

2

10

3

10

4

10

5

Frequency (rad/sec) Acceleration psd

Passenger cabin psd (—–) and ride quality standard psd (· · · )

Multi-objective optimization for control design — Design Example — Maglev Suspension Controller 55/63

Conclusions

Analytical optimization combined with MOI (mixed optimization):

• Explicit closed loop performance
• Multi-objective
• Robustly stable
• Flexible / interactive / trade-offs easily made
• High order controllers

Multi-objective optimization for control design — Design Example — Maglev Suspension Controller 56/63

Example References

• Dakev, N.V., J.F. Whidborne, A.J. Chipperfield & P.J. Fleming. 1997. “H∞

design of an EMS control system for a maglev vehicle using evolutionary algorithms.” Proc. IMechE, Part I: J. Syst. & Contr. 311(4):345–355.

• J.F. Whidborne and S. Arunsawatwong. Design of a critical control system

using simulated annealing and mixed optimization. In 2nd Asian Control Conf., pages I288–I289, Seoul, S. Korea, July 1997.

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Summary

• Considered the problem of trade-off in control design
• Leads to design being formulated as a multi-objective optimization problems
• Problem can be tackled using several methods including MOI, MOGA,

Convex Optimization & Mixed Optimization

• Example of Mixed Optimization given

Slides will be available from

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