Besides the torque model given in equation (4.17), a nonlinearλmodel is used as the other
component of the 2×2 MIMO model:
y(t) = θ1+θ2u2(t−10)2+θ3u1(t−10)u3(t−10) +θ4u1(t)u3(t−7) (4.53) +θ5u2(t−10)u3(t) +θ6u1(t−10)u2(t−10) +θ7u1(t)u3(t−10) +θ8u1(t)2 +θ9u1(t−10) +θ10u3(t) +θ11u3(t)2+θ12u1(t−10)2+θ13u2(t−10)u3(t−7) +θ14u3(t−10) +θ15u3(t)u3(t)u3(t−7) +θ16u1(t)u1(t−10) +θ17u1(t)u2(t−10) +θ18u3(t−7)2+θ19u2(t−10) +θ20u3(t−4) +θ21u3(t)u3(t−4) z(k) = y(t) +ϵ(t) The parameters are:
θ = [θ1, θ2, ..., θ21]
= [5.69,10.75,1.86×10−7,1.75×10−8,0.0046,0.00031,−2.37×10−7,5.05×10−8,
−0.00066,0.0031,−1.28×10−6,2.27×10−8,0.003,−8.69×10−5,−1.37×10−6,
−1.50×10−8,−0.0002,3.22×10−7,−23.36,−0.00023,3.18×10−7]
CHAPTER 4. OPTIMAL INPUT DESIGN FOR SYSTEM IDENTIFICATION 102 Table 4.16: Preference vector of optimal input design for MIMO system
w1 w2
D optimum 477.91 802.84
AI optimum 3945.2 3621.8
Since each MISO model does not include any regressor of the output of the other model, it is feasible to treat this MIMO model as two independent MISO models which can be identified separately therefore two set of optimal input can be design for each of them. However, for the purpose of saving experimental time, we propose an approach to developing a composite objective function by weighting the objectives of two MISO models. One set of optimal test signals can be developed accordingly and used as the identification signal for the two models with the expectation of exciting the behaviours of both models.
To determine the weightings, firstly the magnitudes of the values of sub-objective func- tions need to be scaled. For a function for which the minimum and maximum value is given,
the resulting value can be normalized in [0,1]. However, the minimum values of the sub-
objective functions of optimal input design are not provided initially and a large amount of computation will be required in order to find the minimum value of each sub-objective func- tion. Thus in this work a trade-off approach of scaling is proposed. Applying a white noise
signal to all of the models, the corresponding absolute value of each sub-objective function,v
is computed and used as one component of the weighting factors of the other sub-objectives.
In an optimization which hask sub-objectives, the weighting factor of the ith sub-objective
wi is given by:
wi =v1v2. . . vi−1vi+1. . . vk (4.54)
where vi is the values of the ith sub-objective function. The catalytic converter converts
harmful emission of an gasoline IC engine into less harmful substances. However, it works
effectively provided that the λof the emissions is 1 with a small tolerance of approximately
1%. The accuracy of theλmodel is thus considered more important than the torque model
and it is weighted relatively by 2:1 for importance in the following experiment.
Optimal input designs with D-optimal criterion and the proposed AI-optimal criterion are carried out in order to minimize the estimated parameters and output prediction of the MIMO model. The determined weights are shown in Table 4.16.
Each type of input design is carried out 10 times with different initial conditions and then utilized for model identification. The model estimated by optimal and non-optimal
inputs are compared and the results of statistical validation measured byeandR2 are shown
in Table 4.17 and 4.18. It is indicated that the optimal input design with proper weightings are able to minimize the function value of each individual sub-objective and correspondingly
CHAPTER 4. OPTIMAL INPUT DESIGN FOR SYSTEM IDENTIFICATION 103 Table 4.17: Validation results of torque model
e R2P RBS R2AP RBS RU DRN2 MDop 2.08 76.54% 64.00% 48.14% MAIop 2.76 77.80% 65.85% 50.50% MP RBS 2.45 -449.75% -892.30% -1416% MAP RBS 2.24 72.09% 61.63% 47.54% MU DRN 4.50 57.36% 52.00% 42.08%
Table 4.18: Validation results of λmodel
e RP RBS2 R2AP RBS R2U DRN MDop 0.31 85.82% 75.10% 59.24% MAIop 0.34 86.64% 76.63% 61.76% MP RBS 2.40 -6210% -11732% -13961% MAP RBS 0.54 82.75% 73.95% 59.81% MU DRN 0.59 72.68% 67.05% 55.81%
improved accuracy is obtained in all identified models. Since the required experimental time for the optimization of the composite objective function is close to the time cost of optimizing a single sub-objective function, this approach is more efficient with a large number of sub-
objectives. Moreover the model obtained by D-optimum gives the smallestebut the second
bestR2. As argued in [55] [101] , the D-optimal criterion is consistent with the G-optimal
criterion in principle so that it should also be a sensible criterion for optimization of output prediction. However differently from most output space criteria, the D-optimal criterion does
not take the selection ofU0 which is discussed in Section 4.8.1 into account so should not be
considered as the best choice of output prediction based input design for black box models.
4.12
Conclusions
Technologies of optimization are implemented for optimal test signal design with the purpose of improving the quality of identified models. An iterative procedure for constrained optimal input design for black box systems is developed. Commonly used excitation signals for initial estimation of models are discussed and a white noise signal is applied in experiments on a 1.6L 4 cylinder SI PFI Zetec engine. An original MISO torque model is identified which is subsequently used as the basis of experiments on optimal input design in this chapter.
Experiments of input design are firstly implemented to a known system for the conve- nience of comparing the parameters and regulating the disturbance. An implementation on a black box modelling of the virtual engine is given subsequently in order to demonstrate the effectiveness in industrial applications. Various algorithms for optimization are tested for the optimal input design and the deterministic PS algorithm is recognized to be the most
CHAPTER 4. OPTIMAL INPUT DESIGN FOR SYSTEM IDENTIFICATION 104 appropriate particularly for reasons of the repeatability.
For the optimization of parameter estimation, A-optimal and D-optimal criteria are em- ployed. The experimental results indicate that A-optimal criterion is effective if the regressors of model have similar scales of magnitude but may lose efficiency if significant diversity exists in scales. However the D-optimum method is not affected and provides more accurate estima- tion of parameter in all cases. A weighted A-optimum is proposed as an alternative approach to the D-optimum. This criterion weights the parameter variance by corresponding squared output sensitivity terms and gives an estimation with similar accuracy to the D-optimum.
As the true parameters of a black box system is generally unknown, the optimization of output prediction is more suitable for practical applications. Objective functions can be designed according to classic G-optimal and I-optimal criteria. A new criterion based on a minimization of a simplified sum of output error is proposed and illustrated to be the most effective for an improved output prediction since it gives the best computing efficiency.
The statistical validation shows the advantages of optimal inputs in identifying an ac- curate model for a known system and a unknown virtual engine. In applications of MIMO model identification, methodologies of input design can be applied to generate a set of optimal inputs by minimising a comprehensive objective function which is composed of the weighted values of sub-objective functions. The optimal inputs are effective to improve the accuracy of all sub-models with less computational burden.
The proposed methodology of optimal input design is used in the later chapter of dynamic model-based calibration and control. The optimal inputs are designed to further improve the accuracy of polynomial engine models.
Chapter 5
Selection of Parameter Estimation
Methods
5.1
Introduction
The quality of system identification is known to be affected by two main factors: model structure and parameter values. Techniques of DoE such as input design have been developed with the purpose of reducing the error of parameter estimation before selecting the estimator. However, the estimation method does have a significant influence on the estimation results, which thus should be selected sensibly according to the prior knowledge of the system.
In this chapter, the model types which should be determined by eventual application of the model are introduced and estimation methods for different types of models are discussed and subsequently evaluated by examples. A simulation error method is developed from a traditional prediction error method. The proposed estimation method for simulation models is initially demonstrated with an identification of a known system and then applied to identify a black box model of the virtual engine.