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 publi . ranfiel d. a . uk /eh 30 81 / 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 of 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 plant/motor controller ✲ ✲ ✲ output ✲ reference 1 k s ( s + a ) + ✻ − 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: Step Response From: U(1) 1.5 1 G ( s ) = s ( s + a ) Proportional controller k gives closed loop system: 1 Amplitude k To: Y(1) T ( s ) = s 2 + as + k Fractional overshoot to step 0.5 response: √ M p = e − a π/ 4 k − a 2 0 0 5 10 15 20 25 Time-to-peak Time (sec.) Typical step response of second order system showing 4 k − a 2 for k > ( a 2 / 4 ) 2 π t p = √ 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 M p and t p 1 10 0.8 8 0.6 6 t p (sec) M p 0.4 4 0.2 2 0 0 −1 −1 0 0 1 1 2 2 10 10 10 10 10 10 10 10 k (log-scale) Overshoot M p and time-to-peak t p against k Multi-objective optimization for control design — Conflicts and Trade-offs in Control Systems 6/63
Example - simple servo system 10 9 8 7 6 t p (sec) 5 4 3 2 1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 M p Trade-off curve showing M p against t p 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 D ( s ) disturbance + ❄ U ( s ) + ✲ ✲ K ( s ) ✲ ✲ + ✲ Y ( s ) R ( s ) G ( s ) reference output ✻ − + controller plant ❄ ✛ M ( s ) + sensor noise measured output Multi-objective optimization for control design — Conflicts and Trade-offs in Control Systems 8/63
The fundamental conflict D ( s ) + + ❄ ✲ ✲ K ( s ) U ( s ) G ( s ) ✲ ✲ ✲ Y ( s ) + R ( 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 ) G ( s ) K ( s ) 1 S ( s ) = T ( s ) = 1 + 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 G ( s ) K ( s ) 1 S ( s ) = and T ( s ) = ⇒ T ( s ) + S ( s ) = 1 1 + G ( s ) K ( s ) 1 + G ( s ) K ( s ) • Hence | T ( j ω ) | + | S ( j ω ) | ≥ 1 Im T ( s ) S ( s ) Re 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 ( j ω ) | + | S ( j ω ) | ≥ 1 • If | S ( j ω ) | is made nearly zero, | T ( j ω ) | becomes nearly unity (often greater than unity) • Conversely, if | T ( j ω ) | is nearly zero, | S ( j ω ) | 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 2.5 |S(j ω )| |T(j ω )| 2 |S(j ω )|+|T(j ω )| magnitude, | · | 1.5 1 0.5 0 10 -2 10 -1 10 0 10 1 10 2 frequency, ω (rad/s) 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 log | S ( j ω ) | ✻ Bode’s Sensitivity Integral states pop up that the average of the logarithm ✻ ✻ of the sensitivity is conserved. ✍ ✲ ω 0 If the sensitivity function is decreased at low frequencies, it ❘ must be traded-off for a larger � log | S ( j ω ) | d ω ≥ 0 sensitivity function at higher ◆ frequencies push down ❄ 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 { φ i ( 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 p ∈P { φ i ( p ) , for i = 1 . . . n } min 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, φ i , are in conflict, so the reduction of one objective function leads to increase in another • Result of multi-objective optimization known as a Pareto-optimal solution • Pareto-optimal solution has ✻ φ 2 ( p ) property that it is not possible to reduce any one φ i without Attainable set increasing at least one other φ i • Point lying in the interior of the attainable set is sub-optimal, since both φ 1 and φ 2 can be ✻ Pareto both reduced optimal set • A solution to a multi-objective ✲ φ 1 ( p ) optimization problem is hence generally not unique 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
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