Preferences in Evolutionary Multi-Objective Optimisation with Noisy Fitness Functions: Hardware in the Loop Study

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1 Proceedings of the International Multiconference on ISSN Computer Science and Information Technology, pp PIPS Preferences in Evolutionary Multi-Objective Optimisation with Noisy Fitness Functions: Hardware in the Loop Study Piotr Woźniak Technical University of Lodz, Control Theory Group, 18/22 Stefanowskiego St., Lodz, Poland Abstract. Multi-objective optimisation (MOO) is an important class of problem in engineering. The conflict of objectives in MOO places the issue of compromise in a central position. Since no single solution optimises all objectives, decision-making based on human preference is a part in solving MOO problems. In this paper application of the evolutionary MOO to the dynamic system controller design by use of the hardware in the loop is presented. Thanks to this approach problems of un-modelled plant dynamics and uncertainty of parameters are alleviated because no mathematical model is needed. The a-priori search of one solution does not require knowledge of a whole Pareto front. 1 Introduction Real-world optimisation problems are characterised by the existence of multiple, often conflicting, objectives. In the absence of information about the relative importance of the various criteria, such problems typically admit multiple optimal solutions where any improvement in one objective can only be obtained at the expense of degradation in other objectives. Edgeworth and Pareto captured this notion mathematically in the criterion widely known as Pareto optimality. The general MOO problem may be stated as follows. Without loss of generality, the minimisation of objectives is considered in the discussion throughout this paper. Definition 1 Find the vector of decision variables X = {x 1, x 2,, x n} which satisfies the m inequality constraints: p equality constraints: g i X 0, i=1,2,., m h i X 0, i=1,2,., p and minimises the k objective functions: f X =[ f 1 X f 2 X f k X ]. In general the objectives are non-commensurable and in conflict with each other. 337

2 338 Piotr Woźniak Definition 2. A vector of decision variables X j dominates X i in the Pareto-optimality sense if and only if X j performs better in at least one objective and at least as good as X i in the rest X i X j f n j f n i for 1 n k and f m j f m i for m [1, k ], s.t. m n. If neither of the two candidate solutions dominates the other, they are nondominated. The Pareto-optimal set of a MOO problem consists of solutions which dominate other solutions in the feasible region. The image of this set in the objective space is the Pareto-front. The immediate effect, when applying the criterion of Pareto optimality to highdimensional problems, is that a large number of solutions within any generation are non-dominated and, therefore, qualitatively indistinguishable from each other. It is empirically evidenced in [7, pp ]. Therefore, when the number of objectives is large, Pareto dominance-based ranking procedures become ineffective in sorting out the quality of solutions and selection operators play a minor role to compensate for this effect, which translates into a lack of selective pressure. The dimensionality reduction techniques may be applied to alleviate this issues. This approach takes advantage of possibility that some objectives may behave in a non-conflicting manner near the Pareto-optimal region although a conflict exists elsewhere. In such cases, the Pareto-optimal front will be of dimension lower than the number of objectives. This technique, has been recently successfully applied to the EMOO electric synchronous machine design [1, 2]. Another possibility, typical for engineering approach to the MOO, is by search for set of solutions in a region of interest to the DM without approximating points on the complete high-dimensional Pareto-optimal front. Thus, the final solution of the Evolutionary MMO (EMOO) problem results, not only from the search process, but also from the decision process. As such, this paper provides a basis for online solutions selection with hardware in the loop (HiL) setup with Permanent Magnet Synchronous Machine (PMSM). The DM preferences in the EMOO are discussed in the next Section. In Section 3 the robustness issues of the EMOO of the HiL setup. Design problem is introduced in Section 4 along the hardware overview. Results of the research are presented in Section 5. Section 6 concludes the presentation. 2 Issues in the EMOO algorithms Evolutionary algorithms are global parallel search and optimisation methods based around Darwinian principles, working on a population of potential solutions to a problem. Every individual in the population represents a particular solution to the EMOO problem. The population is evolved over a series of generations (iterations of the algorithm) to produce better solutions. The quality of the solution is evaluated by the fitness assignment i.e. functions which characterises the individual s performance in the problem domain. Most of published results on EMOO algorithms base on explicit objective functions. In this paper we consider the approach when each fitness evaluation is run on the hardware (as introduced in Section 4).

3 Preferences in Evolutionary Multi-Objective Optimisation 339 The relative degree of the fitness value determines the level of propagation of the individual s genes to the next iteration. In the EMOO algorithm NSGA-II [7] in use here, the rank of a certain individual corresponds to the number of individuals in the current population by which it is dominated. All non-dominated individuals are assigned rank zero, while dominated ones are penalised according to the population density of the corresponding region of the trade-off surface. 2.1 Preferences in the EMOO Solutions on the Pareto-optimal set perform better in some objectives and worse in other. Since no single solution optimises all objectives, Decision-Making (DM) based on subjective human preference is a natural aspect in solving EMOO problems. In real problems only a set of solutions in a region of interest to the DM is required. One may consider introducing some component to overall fitness measure for each solution. This preference related fitness component zeroes on a subset of the Paretooptimal solutions considered of high relevance to the DM for the particular EMOO problem. The vagueness and context-dependence of the DM s preference structure result, unsurprisingly, in various mathematical models as well as techniques proposed for the incorporation of preferences into EMOO [3-5, 8, 10]. The integration of the DM preferences in the EMOO process must allow the selection. Two different classes of methods can be defined, taking into account the way how a search and decision processes are integrated with EMOO [3]: i) A-priori methods: The DM preferences are defined in terms of an aggregating function which combines objectives values into a single utility value. ii) A-posteriori methods: The approximation of the Pareto optimal set is supplied to the DM, which selects the most preferred among the alternatives using e.g. Weighted Metrics approach [6]. In this paper we consider the a-priori approach based on Reference Point Method [8]. We consider it as most suitable in the problem under investigation although for this methodology it is necessary to know beforehand the criteria space, and also the DM needs to have some knowledge about the location of regions of the best solutions. No need for the whole Pareto front approximation is the main advantage over all a-posteriori approaches. 2.2 Reference Point Method In the Reference Point Method the aim is to reach a Pareto front region located near a specific pre-defined reference point or points. This method can be applied using or not a weight vector. The main ideas of this method are the following: i) Solutions closer to the reference points, in the objective space, are to be more preferred, ii) Solutions within a ε-neighbourhood to a near-reference point solution are to be preferred, in order to maintain a diverse set of solutions near each reference point.

4 340 Piotr Woźniak To allow the identification of the solutions located near the reference point the normalised Euclidian distance (d ij) between each solution of the best nondominance level and the reference point is calculated 2 k f i (X) z i ij = (1) i max min i= 1 f i f i d w max min where : f i and f i are the maximum and the minimum value of the objective function for criterion i, and zi and w i are the i-th component of the reference point and weight vector, respectively. The distance (1) takes into account simultaneously the relative importance of the different criteria, quantified through the weight vector, and the distance between the solution and the reference point. The solutions with lower distance to the reference point and, simultaneously, with high weight vector, will be selected preferentially. This procedure was been implemented [9] through the modification of the NSGA-II algorithm [5] niching strategy: Step 1: For each reference point, the distance (1) of each solution of the front is calculated and the solutions are sorted in ascending order of distance. This way, the solution closest to the reference point is assigned a rank of one, Step 2: After computations in Step 1 are performed for all reference points, the solutions with a smaller preference distance are preferred in the tournament selection, Step 3: To control the extent of obtained solutions, an ε - neighbourhood idea is used in the niching operator. First, a random solution is picked from the nondominated set. Thereafter, all solutions having a sum of normalised difference in objective values of ε or less from the chosen solution are assigned an artificial large preference distance. This way, only one solution within an ε-neighbourhood is emphasised. Then, another solution from the nondominated set is picked and the above procedure is performed. The use of the ε-based selection strategy ensures a spread of solutions near the preferred Pareto-optimal regions. The main change in NSGA-II was made on the crowding operator, since in this case the aim is not to obtain a great diversity of solutions along the Pareto frontier but only a subset of these solutions that minimise the distance to the reference point. 3 Robustness issues in the EMOO based control system design The performance of a particular control design is fundamentally tied to the accuracy of the model upon which it is based. This is especially true for iterative control design and optimisation procedures. The substitution of Hhardware in the loop (HiL) for the software model in the EMOO opens up new possibilities for design based on real world performance indices. In real optimisation problems, a wide range of uncertainties have to be taken into account. Generally, uncertainties in EMOO can be categorised into four classes. Each

5 Preferences in Evolutionary Multi-Objective Optimisation 341 of them is presented with additional comments concerning the HiL setup to be presented in Section 4.2. i. Noise; in the HiL it comes from measurement errors from sensors (e.g. the Sensors Block from the system presented on Fig. 2), ii. Errors in fitness approximations based on mathematical modelling; introducing the HiL approach we eliminate this type of uncertainty, iii. Time-varying parameters of fitness; in the considered HiL based control system design changes in environment may be neglected because in this study evaluations are available in less than 1 [s] that is in time shorter than all time constants in the system [11], iv. Perturbations of parameters and environmental parameters after the optimal solution has been determined; this type is not considered in this paper, however it may be easily included in the HiL by the appropriate design of experiments sampling approaches. In general noisy or uncertain scenario the proximity of other fitness values, even if only close on one objective, can influence how the rank is assigned. Limits on objectives, constraints on the chromosomes, and sharing can all be implemented easily within this ranking framework, allowing interactive DM with any noisy systems possible. At the current stage of research no disturbance was introduced. 4 Hardware in the Loop in the EMOO of the Permanent Magnet Synchronous Motor speed control design The structure of a PI speed controller considered it this paper is given by 1 t u(t)= k pe( t ) e( )d (2) T i 0 where u(t) is the output of the controller at time t, e(t) is the error between the actual and the desired speed of the motor shaft, k p, T i are constants of the controller, and decision parameters of the EMOO problem. Depending on the choice of k p, T i the response of a controller to a given error signal will vary significantly. Various methods exist to tune the gains of a PI controller off-line to attain the prescribed transient response and steady-state error criteria [12] because there is no analytical way of finding the optimal set of constants (k p, T i). Empirical methods such as Ziegler-Nichols tuning can be used. Up-to-date search methods generally involve some form of iterative approach to achieve performance criteria such as risetime, overshoot and settling-time [12]. For that reason it was decided to use a EMOO approach to find several possible parameter settings that would result in good performance.

6 342 Piotr Woźniak 4.1 Objective functions When a controller is designed for a specific system it is often desired that multiple design requirements are met. The drive system is no exception. When evaluating the performance of the PI controller parameters for the speed control loop a step response is considered and different performance measures are investigated [11]. The fitness functions to be minimised are of the form: i) the speed tracking error denoted as ITAE [12] ITAE e (t) tdt (3) p ii) composite objective based on the time period when output current of the converter is on constraints. This objective (denoted hereafter as I constr) is related to the power consumption of the system [11]. Int(i) i(t) dt (4) In this way, it is required the search be performed basing on two noncommensurable objectives. They may be visualised in form of fitness landscapes over constrained parameters space as presented on Fig.1. The values for the sample visualisation are obtained from a sequence of step response tests for 0 t 1 [s]. Fig. 1. Fitness landscape for objective functions (3) and (4). 4.2 Hardware overview In the application under consideration here, the software implemented PI controller to be optimised (Fig. 2) performs tracking of a speed demand for the PMSM. It is achieved by calcultion of a current demand for Power Module. Its value depends on the shaft speed (from the Speed Sensor), and the Load (from the Manometer).

7 Preferences in Evolutionary Multi-Objective Optimisation 343 Software Hardware-in-the-Loop EMOO Algorithm dspace Control System Power Module PI MLIB NO STOP Conditions YES Position and Speed Sensors Results Decision Maker MATLAB Kollmorgen PMSM Momentometer DutymAx DS Load Fig. 2. On-line PI speed controller EMOO design with the hardware in the loop The actual view of the experimental installation is presented on Fig. 3. Fig. 3. Actual view of the experimental installation. The measurement of fitness functions were realised in each iteration of the EMOO as a set of step response tests. There is one test for each member of population under consideration. Thanks to the symmetry in the PMSM operation it is possible to alter the sign of successive tests. It enables cutting down the time of experiments. Sample sequence of six, 0,8 [s] long tests is presented on Fig. 4.

8 344 Piotr Woźniak Fig. 4. Registered sequence of six 0,8 [s] long, speed step-response tests with altering sign for v desired = 25 [rad/s] t 4.3 EMOO setup The parameters of the optimisation algorithm NSGA-II implemented during EMOO of the problem defined by (2)-(4) under constraints: i) in the parameters space defined by inequalities 0,25 < k p < 20 ; 25< 1/T i < 200, ii) in the H-i-L installation (not presented; for details see [11]), are presented in Table 1. Table 1. Basic parameters of the NSGA-II algorithm Algorithm parameter Value Population N 60 Generations G 50 Pool size N/2 30 Tour size 2 Crossover 0.9 probability Mutation probability Results The result of measurement s noise influence on the EMOO speed control design is presented after normalisation to [0,1] on Fig. 5 in form of a Pareto set. This approximation obtained using NSGA-II [4] without DM preferences modifications (as discussed in Section 2.2) presents significant influence of noise which prevents even spread of non-dominated solutions along the front. To alleviate this effect further research on relation between the bounded noise and quality of the whole Pareto set approximation is needed.

9 Preferences in Evolutionary Multi-Objective Optimisation 345 Fig. 4. Single solution selected from the Pareto front approximation (stars) with all elite solutions from 100 iterations. There are several techniques for selection of one exact, or with some acceptable tolerance, Pareto-optimal solution. For this study the Reference Point Method modified NSGA-II algorithm is used to select single Pareto-optimal solution k p=2.01, 1/T i=29.3 [1/s]. The step response for final values gives the speed step response presented in Fig.5. Fig. 5. Comparison of speed step response for the selected single Pareto-optimal solution (bold line) and four extreme decision pairs. Further research is needed to evaluate influence of the measurement noise on the radius of solution set under DM preferences.

10 346 Piotr Woźniak 6 Conclusions This paper investigates the design of a dynamic controller for the drive system directly onto hardware. A multi-objective evolutionary algorithm is applied to the task of controller development, while an objective function ranks the system response to find the Pareto-optimal set of controllers. The measurement noise present during each evaluation at run-time deteriorate the quality of Pareto set approximation. To prevent decline of results the Decision Maker preferences are included in the EMOO process in form of the a-priori type Reference Point Method. This approach demonstrates very good properties providing the designer the set of optimisation parameters in the area surrounding the reference point. References 1. Woźniak P., Witczak P.: Dimensionality Reduction in Evolutionary Multiobjective Design of Permanent Magnet Generator, In: Proc.of Fourth Int. Conference on Evolutionary Multi- Criterion Optimization EMO 2007 (2007) Woźniak P.: Dimensionality Reduction in Evolutionary Multiobjective Design: Case Study, In: Proc.of ACM-SIGEVO on Genetic and Evolutionary Computation GECCO 07 (2007) Coello C.A.: Handling Preferences in Evolutionary Multiobjective Optimization: A Survey, in Proc. Congr. Evolutionary Computation (2000) Cvetkovic D., Parmee I. C.: Preferences and Their Application in Evolutionary Multiobjective Optimization, IEEE Trans. on Evol. Comput., 6, 1 (2002) Branke J., Deb K.: Integrating User Preferences into Evolutionary Multi-Objective Optimization, in Yaochu Jin (editor), Knowledge Incorporation in Evolutionary Computation, Springer (2005) Miettinen K. M.: Nonlinear Multiobjective Optimization, Kluwer, Boston (1999). 7 Deb K.: Multi-Objective Optimization using Evolutionary Algorithms, Wiley, Chichester (2001). 8 Wierzbicki A. P.: The use of reference objectives in multiobjective optimization, In Fandel, G., Gal, T. Eds, Multiple Criteria Decision Making Theory and Applications. Springer, Berlin (1980) Deb K., Sundar J., Bhaskara U., Chaudhuri S.: Reference Point Based Multi-Objective Optimization Using Evolutionary Algorithms, ISSN International Journal of Computational Intelligence Research, 2, 3 (2006) Ferreira J. C., Fonseca C. M., Gaspar-Cunha A.: Methodology to Select Solutions from the Pareto-Optimal Set: A Comparative Study, In: Proc.of ACM-SIGEVO on Genetic and Evolutionary Computation GECCO 07 (2007) Woźniak P., Sobieraj T.: Multi-objective control design by experiment-based evolutionary methods : case study, approved for presentation at the Conference on Control for Power Electronics and Electrical Drives, SENE 07 (2007). 12 Ogata, K.: Modern Control Engineering, 3 rd Ed., Prentice Hall, 1997.