Barmuta, P.

Couckuyt, Ivo

Dhaene, Tom

Knudde, Nicolas

Lukasik, K.

Schreurs, D.

Spina, Domenico

Finding the optimal working conditions for non-linear electrical components under large signal stimuli can be challenging, mainly due to the high number of input dimensions and multiple local minima of the goal function. In this paper a Bayesian optimization method is applied in order to limit the number of evaluations by a commercial harmonic balance simulator. The method is applied to amplifier optimization utilizing Wolsfspeed CGH40010F GaN HEMT, for which input power, bias voltages and load at fundamental harmonic frequencies are changed in order to maximize for combined efficiency, gain, and output power. The optimum is found already after 80 iterations

Ghent University Academic Bibliography

1Data-Efficient Bayesian Optimization withConstraints for Power Amplifier DesignNicolas Knudde∗, Ivo Couckuyt∗, Domenico Spina∗, Konstanty Łukasik†‡,Paweł Barmuta†‡, Dominique Schreurs† and Tom Dhaene∗∗Ghent University - imec, Department of Information Technology, 9052 Ghent, Belgium†KU Leuven, Department of Electrical Engineering, 3000 Leuven, Belgium‡Warsaw University of Technology, Faculty of Electronics and Information Technology, 00-665 Warsaw, PolandEmail: nicolas.knudde@ugent.beAbstract—Finding the optimal working conditions for non-linear electrical components under large signal stimuli can bechallenging, mainly due to the high number of input dimensionsand multiple local minima of the goal function. In this papera Bayesian optimization method is applied in order to limitthe number of evaluations by a commercial harmonic balancesimulator. The method is applied to amplifier optimization utiliz-ing Wolfspeed CGH40010F GaN HEMT, for which input power,bias voltages and load at fundamental harmonic frequencies arechanged in order to maximize for combined efficiency, gain, andoutput power. The optimum is found already after 80 iterations.Index Terms—Bayesian optimization, Gaussian process, har-monic balance, load-pull, transistorI. INTRODUCTIONModern microwave power amplifiers have to fulfill multiplevery stringent requirements, e.g., high output power, highPower Added Efficiency (PAE), etc. This makes the designprocess very cumbersome, and often leads to extensive useof optimization tools in the design process. Unfortunately, thecommercial CAD environments lack efficient optimizers, es-pecially when the objective function has a highly-dimensionalparameter space and multiple local minima, which is thetypical case for power amplifier design. This problem has beenaddressed in [1], [2], in which Bayesian Optimization (BO) isapplied to the design of high-efficiency power amplifiers [3].The optimal Large Signal Operating Point (LSOP), i.e., set ofinput variables’ values, was found already within only a fewtens of simulations.Due to its efficiency, BO is also used in this work. How-ever, the efficiency is further increased by employing a re-cently introduced sampling strategy, called Max-Value Entropysearch [4]. We also incorporate the constraints, e.g., powerdissipation or breakdown voltage, in the optimization processthrough so-called Probability of Feasibility (PoF) [5]. Thework is organized as follows. In Sec. II, Bayesian optimizationis explained with a particular focus on sampling strategies.Next, in Sec. III, the optimization results of the transistor’slarge signal operating point are shown and discussed, whichleads to the conclusions drawn in Sec. IV.Initialdesignparameters[xk]Kk=1f(xi), cj(xi)StoppingCriterionSurrogatemodels off(x), ci(x)Bayesian OptimizationReturn bestparametersNewparameter xExpensivesimulationAccuracy CheckOptimize α(x)ExpensiveSimulationNoGP ModelingYesFig. 1. Flowchart of the Bayesian optimization algorithm.II. METHODOLOGYThe idea behind BO is to perform optimization on surrogatemodels, which mimic the real optimization problem, butcontrary to the latter, are very cheap to evaluate [6]. Theflowchart of BO algorithm is shown in Fig. 1. First, theobjective function is evaluated over an initial set of valuesof the design parameters [xk]Kk=1 via simulation of the deviceunder test. This allows to extract the first surrogate modelof the objective function f(x). Since the surrogate model isvery cheap to evaluate, it is used by the optimizer to findthe global minimum. This optimum is then evaluated via anew simulation, and when none of the stop criteria is met,the surrogate model is updated. Therefore, each additionalsimulation refines the surrogate model, which increases thechance of finding the global optimum of the true problem byusing surrogate models.As one can see, BO is very general and several mathematicaltechniques can be adopted to build the desired surrogatemodel, such as Random Forest [7] or Bayesian neural network[8]. In this paper, we have chosen Gaussian Processes (GPs)[9], thanks to their accuracy and modeling power. A GPrepresents a distribution over functions f : X → R, that iscompletely defined by a suitable mean function m : X → R78-1-5386-5204-6/18/$31.00 c©2018 IEEE2and covariance function k : X × X → R, where the typeof mean and covariance functions depends on the particularproblem under study.A sampling strategy is defined based on the surrogatemodel that is computed so far, which is called acquisitionfunction. Typical acquisition functions include Expected Im-provement (EI) [3] and Probability of Improvement (POI)[10]. In this work, the recently introduced Max-Value EntropySearch (MES) is adopted [4], which automatically balancesexploration and exploitation. In particular, the MES aims tomaximize the information gain about the minimum value fmin,which is quantified by the differential entropy:αMES(x) = H(p(fGP |D,x))− E(H(p(fGP |D,x, fmin))),(1)where D represents the already observed data {xi, yi}Ni=1, Hdenotes the entropy and the expectation E is computed over therandom variable fmin. Note that the distribution of fmin cannotbe calculated analytically, but can be sampled via Gumbelsampling [11]. Furthermore, p(fGP |D,x, fmin) is a truncatedGaussian, since all y values have to be greater then fmin.However, in many engineering problems it is not sufficientto find a global maximum of the objective, but some physicalconstraints must be satisfied as well. Hence, the acquisitionfunction has to be modified accordingly. In practice, this isdone through the Probability of Feasibility (PoF) [5], whereeach constraint is often formulated as a suitable function c :X → R that has to be negative.In our problem setting, the constraints depend on the phys-ical characteristics of the transistor considered and, typically,include limitations on the power balance and the maximumDC currents of the device. Unfortunately, it is not possibleto know upfront (before simulating the device) if a particularcombination of design parameters will lead to a constraintsviolation. Hence, first a suitable GP-based surrogate modelis built for each function ci for i = 1, . . . , N , where N isthe total number of constraints, then the corresponding PoF iscalculated:αPoF,i(x) = p(cGPi (x) < 0). (2)Note that αPoF,i is equal to the probability that the i−thconstraint is satisfied and such probability is estimated withincreasing precision as the number of iterations of the opti-mization algorithm increases.Finally, in order to combine the ability of selecting feasibledesign points via PoF and the capability of the MES to findthe optimum, the overall acquisition function is given byα(x) = αMES(x)N∏iαPoF,i(x) (3)III. RESULTS AND DISCUSSIONThe device under test is the GaN HEMT transistorCGH40010F 10 W from Wolfspeed, which is simulated inADS1. The gate VG ∈ [−2, 0] V and drain voltage VD ∈[16, 42] V, the input power Pin ∈ [0.01, 1] W and the complexload impedance ZL constitute LSOP x. Furthermore, the load1Advanced Design System (ADS), Keysight Technologies, Santa Rosa, CA.reflection coefficient ΓL must have a magnitude less thanone and it is parameterized in polar coordinates to efficientlyenforce this constraint. A typical design objective is to have thegain G, PAE, and output power Pout as high as possible. Sincethese quantities are highly correlated, in order to reach thisgoal it is sufficient to minimize a suitable objective functionf , defined as:f(x) = −G(x)PAE(x)Pout(x), (4)where the vector x is LSOP. Additionally, the followingconstraints must hold:Pdc − Pout + Pin − 14 W ≤ 0, , (5)Idsi − 4.5 A ≤ 0, , (6)Igsi − 4 mA ≤ 0., (7)where Pdc and Pin are the DC and input power, respectively,and Idsi, Igsi are the maximum instantaneous drain and gatecurrents, respectively. Note that all these conditions can beexpressed as ci < 0.The constraints and the objective f are modeled separately.In particular, a periodic kernel is used to model the phaseof ΓL, indicated as φ, which is a periodic function. For theremaining design parameters x˜, a Mate´rn 3/2 kernel is usedto compute the GP model:k(x,x′) = kM (x˜, x˜′)kP (φ, φ′) (8)kM (x˜, x˜′) = σ2(1 +√3‖x˜− x˜′‖`)exp(−√3‖x˜− x˜′‖`)(9)kP (φ, φ′) = exp(− sin(φ− φ′)2λ)(10)where the mean function is considered equal to zero. Theinitial set of values of the design parameters (needed tocompute the GP models) are generated by a minimax Latinhypercube of 10 points. The kernel (9) is adopted to model theconstraints as well. Next, the acquisition function is optimizedby evaluating the GP models for 500 randomly sampledpoints and by considering the best result as starting pointof the Limited-memory Broyden–Fletcher–Goldfarb–Shannonoptimizer (L-BFGS). Finally, the Bayesian optimization wasperformed by GPflowOpt, a Bayesian optimization package inPython [12].Furthermore, the proposed method is compared with a regu-lar gradient optimizer (the Sequential Least Squares Program-ming (SLSQP)) and two Bayesian optimization approachesbased on acquisition functions computed via EI and POI, usedin [1]. Note that, in order to show statistically relevant results,each optimization algorithm was executed ten times and themean results are shown in Figs. 2 and 3.In particular, Fig. 2 shows that the proposed MES-basedtechnique has superior performance with respect to similarBayesian optimization approaches, thanks to its trade-off be-tween exploration and exploitation; while the SLSQP gradientoptimizer is unable to find a feasible minimum of the objectivefunction. For the Bayesian optimization methods considered,30 20 40 60 80 100Iterations500040003000200010000Objective valueMESPOIEISLSQPFig. 2. Feasible minimum of the objective function with respect to thenumber of iterations. The thick lines are the means of ten runs of the methodssurrounded by their 98% confidence interval, calculated by bootstrapping.0 20 40 60 80 100Iterations0.20.40.60.81.0Fraction of Infeasible EvaluationsMESPOIEIFig. 3. Fraction of infeasible samples with respect to the number of iterations.The thick lines are the means of ten runs of the methods surrounded by their98% confidence interval, calculated by bootstrapping.the number of evaluations on design points resulting in con-straint violations is shown in Fig. 3: as expected, the numberof infeasible points decreases with respect to the number ofiterations and drops to a fraction of approximately 0.4 fairlyquick after 40 samples. As can be seen EI and MES haveoverlapping confidence intervals, but on average, MES stillperforms better.TABLE IOPTIMAL LARGE-SIGNAL OPERATING POINT WITH CORRESPONDINGPERFORMANCE AND CONSTRAINT METRICS.Transistor CharacteristicsG 1810.55PAE 0.43Pout 0.053 WDesign parametersVG 0.16 VVD 0.22 VPin 0.16 WΓL 0.053Constraintsc1 -0.056 Wc2 -2.25 Ac3 -0 mAFinally, the overall minimum value of the objective f issummarized in Table I, where c1, c2 and c3 correspond toequations (5), (6) and (7), respectively.IV. CONCLUSIONSIn this work, a Bayesian optimization strategy was usedfor the optimization of a simulated device. The Max-ValueEntropy search shows good results in combination with theProbability of Feasibility to take into account constraints.Using this combination allowed to achieve lower values of theobjective function despite the number of algorithm iterationsin comparison to Probability of Improvement and ExpectedImprovement. It also has to be noted that the ratio of infeasiblesamples with respect to the number of iterations drops veryquickly to around 0.4 for already 40 samples, and is expectedto decrease further as more LSOPs are evaluated.ACKNOWLEDGEMENTSThis work was supported by FWO-Vlaanderen. P. Barmutais supported by the C2 project (C24/15/015). I. Couckuyt is apostdoctoral research fellow of FWO-Vlaanderen.REFERENCES[1] P. Chen, B. M. Merrick, and T. J. Brazil, “Bayesian optimization forbroadband high-efficiency power amplifier designs,” IEEE Transactionson Microwave Theory and Techniques, vol. 63, no. 12, pp. 4263–4272,2015.[2] P. Chen, J. Xia, B. M. Merrick, and T. J. Brazil, “Multiobjective bayesianoptimization for active load modulation in a broadband 20-w gan dohertypower amplifier design,” IEEE Transactions on Microwave Theory andTechniques, vol. 65, no. 3, pp. 860–871, 2017.[3] D. R. Jones, M. Schonlau, and W. J. Welch, “Efficient global optimiza-tion of expensive black-box functions,” Journal of Global optimization,vol. 13, no. 4, pp. 455–492, 1998.[4] Z. Wang and S. Jegelka, “Max-value entropy search for efficientbayesian optimization,” in International Conference on Machine Learn-ing (ICML), 2017.[5] M. Schonlau, “Computer experiments and global optimization,” Ph.D.dissertation, Waterloo, Ont., Canada, Canada, 1997, aAINQ22234.[6] I. Couckuyt, T. Dhaene, and P. Demeester, “oodace toolbox: A flexibleobject-oriented kriging implementation,” Journal of Machine LearningResearch, vol. 15, pp. 3183–3186, 2014.[7] F. Hutter, H. H. Hoos, and K. Leyton-Brown, “Sequential model-based optimization for general algorithm configuration,” in InternationalConference on Learning and Intelligent Optimization. Springer, 2011,pp. 507–523.[8] J. T. Springenberg, A. Klein, S. Falkner, and F. Hutter, “Bayesianoptimization with robust bayesian neural networks,” in Advances inNeural Information Processing Systems 29. Curran Associates, Inc.,2016, pp. 4134–4142.[9] C. E. Rasmussen, “Gaussian processes in machine learning,” in Ad-vanced lectures on machine learning. Springer, 2004, pp. 63–71.[10] H. J. Kushner, “A new method of locating the maximum point of anarbitrary multipeak curve in the presence of noise,” Journal of BasicEngineering, vol. 86, no. 1, pp. 97–106, 1964.[11] R. A. Fisher, The genetical theory of natural selection: a completevariorum edition. Oxford University Press, 1999.[12] N. Knudde, J. van der Herten, T. Dhaene, and I. Couckuyt,“GPflowOpt: A Bayesian Optimization Library using TensorFlow,”arXiv preprint – arXiv:1711.03845, 2017. [Online]. Available:https://arxiv.org/abs/1711.03845

Data-efficient bayesian optimization with constraints for power amplifier design

Crossref

Data-Efficient Bayesian Optimization with Constraints for Power Amplifier Design

https://biblio.ugent.be/publication/8573080/file/8573083.pdf

Data-efficient bayesian optimization with constraints for power amplifier design

Abstract

Similar works

Full text

Available Versions

Ghent University Academic Bibliography

Crossref