Two-Dimensional Transient Cycle Decomposition and Reduction (CDR) for Data Driven Nonlinear Dynamic System Modeling

Transient test cycles are an essential part of the development of both on and off-highway vehicles and machines. These cycles are used to test and validate the dynamic operation and performance of systems and they also form part of regulations such as Real-world Driving Emissions (RDE). A common approach to generating a transient cycle is to replicate recorded real-world transient operation. However, the dynamics of transient operation can vary significantly based on factors such as the operating environment, the use case, the user, and the type of vehicle or machine. The feasible duration of a transient test cycle is also often limited by development time and cost. Therefore, there is a need for a method to synthesize transient test cycles from real-world data in order to replicate a wide range of feasible dynamic operation in the shortest possible time. This paper proposes an innovative transient Cycle Decomposition and Reduction (CDR) method that generates a shortened representative driving cycle in speed-torque two-dimensional space. The steps include identifying micro-trajectories using critical points (e.g. minimum torque), identifying features of micro-trajectories, grouping similar micro-trajectories using k-means clustering and Gaussian process models, and using Markov chain models to efficiently splice together the micro-transients to form a new reduced cycle. The method is flexible and can be used to generate reduced transient cycles for both on-road and off-road applications. The proposed CDR method is demonstrated by using it to perturb a system and collect data for the identification of nonlinear dynamic system models. A 30-minute transient cycle is reduced to a 200-second representative cycle, which is used to train a Neuro-Fuzzy NOx emission model. The results show that this model can accurately predict NOx emissions behaviour over the original transient cycle. The proposed CDR method is easy to expand to higher dimensions and can be applied in a wide range of applications. Additionally, the information extracted during the process has useful physical meaning and can be further utilised, for example in driving pattern recognition, vehicle diagnosis, and autonomous vehicles.


I. INTRODUCTION
Standard test cycles have played an important role in both the automotive and off-road industries for controlling and optimising system performance and have formed part of legislation for several decades.For example, the New European Driving Cycle (NEDC) is a vehicle type-approval test which dates back to the 1980s and was last updated in 1997.In 2019 the NEDC was replaced in the European The associate editor coordinating the review of this manuscript and approving it for publication was Wentao Fan .
Union by the Worldwide harmonised Light vehicle Test Procedure (WLTP) which is longer and more dynamic.The WLTP uses the new World harmonised Light vehicle Test Cycles (WLTC).The World Harmonised Transient Cycle (WHTC) and Non-road Transient Cycle (NRTC) are two transient cycles for heavy-duty engines and for mobile offroad engines respectively used for certification and type approval [1], [2].
The European standard for type-approval test now adds the Real Driving Emissions (RDE) test to the WLTP.The purpose of this is to narrow the gap between laboratory testing and real-world fuel consumption and emissions [2], [3], [4].A representative driving cycle for the RDE test is also important for the design and analysis of Hybrid Vehicles (HVs), Plug-in Hybrid Vehicles (PHVs) and Electric Vehicles (EVs) [5], [6], [7], [8], [9], [10].
Over the last 15 years significant research has been undertaken into extracting representative drive cycles from real driving data.The main aim being to narrow the gap between laboratory results and the real world, whilst minimising development time and cost [11], [12], [13], [14].There are two popular approaches, one is based on Microtrips [11], [15], [16], [17].The other is based on Markov Chain Models (MCMs) together with Monte Carlo (MC) sampling methods [11], [18], [19].
The traditional definition of a Micro-trip is the duration between the start of a period of idling to the next occurrence of idling in the vehicle driving data [19].In the Micro-trip approach, the measured vehicle speed data is separated into Micro-trips.Any duplicated Micro-trips are removed.The remaining Micro-trips are connected to form a new much shorter representative drive cycle [19], [20], [21].
In the Markov chain model based approach, MCMs are developed from the recorded drive cycle.MC random sampling is then used to generate many candidate synthetic driving cycles that all satisfy some requirement(s), e.g.travelled distance.The most representative driving cycle (relative to the original driving cycle) is then selected from the group of candidate shortened driving cycles.This is typically done using one or more statistical metrics which quantify features of the drive cycle e.g.mean velocity, standard deviation of velocity, mean positive acceleration etc. [18], [22], [23].A global optimisation algorithm e.g. a Genetic Algorithm (GA) or equivalent can be used to select a representative driving cycle from many candidates [24].A comparison of these two approaches can be found in [17].This study concludes that the Micro-trip method is preferred since it generates more realistic cycles.
The majority of the literature related to drive cycle synthesis is based on real-world driving of on-road vehicles where the vehicle speed data is the main input.When a vehicle is in motion, in most cases a transmission system (e.g. a gearbox) changes the ratio between the wheel speed and the Power Unit (PU) (e.g.engine, hybrid system, batteryelectric-motor etc.) speed.This is normally achieved via a finite number of fixed ratios, or a continuously variable transmission ratio.Knowledge of the gear ratio and the drive wheel(s) circumference, enables the PU speed and torque to be determined for a given vehicle speed.For offroad vehicles or machines, the vehicle speed alone will often not be adequate to characterise the operation of the PU of the machine.For example, an excavator may be stationary and the load applied to the PU may be dominated by the hydraulic system pump(s).The efficiency of a PU normally changes with the PU speed and torque.Therefore, PU speed and torque are both necessary to describe the PU operation and performance.PU speed and torque together form a Two-Dimensional (2D) profile with respect to time.In PU testing on the dynamometer during development, PU output shaft speed and torque are typically controlled according to the 2D profile and are measured to determine PU power output.In such testing, a transmission system can be physically installed between the PU and dynamometer, or it can be emulated using a model which provides the speed and torque demands to the dynamometer-PU control system.
Although the MCM approach can synthesise transient cycles in more than one dimension [25], [26], [27], there are relatively few examples of synthesis of 2D speed-torque.The most relevant work is found in [24] where the authors propose an approach to develop a 2D dynamic cycle for an off-road Wheel-Loader (WL) application using hybrid Markov chain and a GA.Global Positioning System (GPS) data collected at a digging site is used as the input.A methodology is proposed for cycle decomposition and reduction on the speed-torque 2D profile.It focuses on 2D trajectory pattern categorisation and recognition.The aim of the work is to collect data for empirical nonlinear system dynamic modelling.
The authors have shown previously an Excitation Signal Design (ESD) methodology [28] for generating a multidimensional ESD to collect training data to identify dynamic Neuro-Fuzzy (NF) emissions models for vehicles equipped with a PU in the form of an Internal Combustion Engine (ICE).This earlier methodology was designed specially for training dynamic models to predict the ICE emissions behaviour in response to changes in the parameters used to define how the ICE is controlled within a short 2-minute transient manoeuvre.This was achieved by generating a five dimensional ESD which includes several ICE control system calibration variables in addition to a 2D speed-torque profile.This methodology was optimised for training a data driven model to perform well over a single short transient manoeuvre.
In this paper the authors generate excitation signals to most efficiently collect data for training nonlinear dynamic models to cover a wide range of the operating space.The authors propose a multi-step Cycle Decomposition and Reduction (CDR) methodology.This processes real-world transient cycle data to first identify 2D (speed-torque) micro-transients using critical points (e.g.minimum torque).Features of these micro-transients are then used in a multi-stage grouping process employing k-mean clustering and Gaussian process models to identify unique behaviour and to filter out repeat behaviour.Finally, MCMs are used to efficiently splice together identified micro-transients to form a new reduced cycle.
This paper presents several innovative approaches to the separation of a 2D (speed-torque) trajectory into microtrajectories.The innovations are: 1) the method for splitting the 2D trajectory into micro-trajectories, 2) the method for extracting features of 2D micro-trajectories, 3) the method for multi-level grouping and the measure of similarity used for the two trajectories to filter out similar micro-trajectories.This paper is organised as follows.In Section II, the procedure of 2D speed-torque cycle decomposition and reduction is introduced.There are seven main steps included in the procedure and each step is explained in detail through examples.In Section III a practical CDR case study demonstration is introduced and results presented and discussed.Conclusions and future work are presented in Section IV and Section V respectively.

II. METHODOLOGY A. CYCLE DECOMPOSITION AND REDUCTION PROCESS
The process of cycle decomposition and reduction to generate a shorter representative cycle starts with the extraction of useful information from real-world data.The key steps include identifying segments (i.e.micro-transients), cutting the segments, identifying and removing redundant segments and connecting the remaining segments to form a shortened cycle.The process is analogous to the compression of music or video file sizes using specialised lossy codecs (i.e. the original cycle cannot be fully reconstructed from the decomposed cycle as superfluous information is lost).
The proposed methodology for 2D (speed-torque) CDR includes seven main steps, these are summarised in Figure 1.Each of the seven steps are described in detail in the following sub-sections.The sub-figures in Figure 1 illustrate the decomposition of a 1-hour cycle (2D speed-torque) to a shorter 10-minute cycle, a six-fold reduction in cycle duration.The 2D speed-torque sub-plots show how the process yields a new cycle which includes similar trajectories around the boundary of the operating envelope but with a reduced density of points everywhere else since this is where there is greater repetition in the original cycle.The integration of this procedure with the development of a nonlinear dynamic model is illustrated in Figure 2.This shows how the seven step process is adapted to integrate the validation of the nonlinear dynamic model through feedback into the initial phase of the CDR process.Based on the validation performance of the identified nonlinear dynamic model, the CDR process can be flexibly adjusted to collect additional data to improve the model performance.The six-fold reduction in cycle time shown in Figure 1 (e.g.3600 seconds duration to 600 seconds duration) allows for several iterations whilst still taking less time for data collection.The authors found that sufficient model accuracy was achieved with a single pass through the CDR process as is shown later in Section III.The reduction in the cycle duration results in substantial improvements in data collection efficiency.This in turn can have other advantages, such as a reduced burden on resources and impact on the environment.This more efficient data collection can enable the nonlinear dynamic model to be identified over a wider region of the operating space than might otherwise be feasible, thus leading to more robust and better performing models.

B. CDR STEP 1: DATA COLLECTION
Operational transient cycle data is usually collected from on-road vehicles or off-road machines using data-logging systems.PU speed and torque data are required for the proposed CDR methodology and therefore both must be collected.As explained previously, the PU speed and torque data represent the operational state of the vehicle or machine more accurately compared to vehicle speed (or PU speed) or PU torque alone.For the CDR methodology described in this paper, a sample frequency between 1Hz and 10Hz is sufficient for the 2D speed-torque input data.

C. CDR STEP 2: MICRO-TRAJECTORY IDENTIFICATION
Once 2D speed-torque data is available, the next step of the CDR process is to break down the continuously sampled speed-torque trajectory into micro-trajectories.One possible method involves identifying critical points such as local minimum torque, local minimum speed, local maximum torque, local maximum speed, local minimum or maximum power or points with combinations of the critical values of speed, torque or power etc.The original trajectory can then be segmented into micro-trajectories based on the locations of VOLUME 12, 2024 these points with respect to time.The segments of the cycle located between these critical points then form the microtrajectories.
In this work, the authors selected minimum torque to identify critical points in a transient cycle.Figure 3

D. CDR STEP 3: MICRO-TRAJECTORY FEATURE EXTRACTION
There are many features for each micro-transient which can be extracted in 2D speed-torque space, examples include the length, the inner area, the centres etc.For each microtransient, the authors extract four features: area, centres, delta-speed and delta-torque.Figure 4 shows examples of two groups of micro-trajectories with similar values of these four features.
To compute the features of the micro-trajectories, all of the original raw data is first normalised in the speedtorque space.Figure 5 shows plots of three of the four features.Figure 5(a) shows normalised delta-speed vs normalised area.Figure 5(b) shows normalised delta-torque vs normalised delta-speed.The points highlighted in red are micro-trajectories that have Quasi-Steady-State (QSS) features (i.e.low values of normalised delta-speed and deltatorque).This illustrates how transient micro-trajectories can be separated from QSS micro-trajectories using features in 2D speed-torque space.If pure transient micro-trajectories are needed, then the QSS micro-trajectories can be easily filtered out.

E. CDR STEP 4: FIRST LEVEL OF GROUPING MICRO-TRAJECTORIES
The next two steps involve grouping the micro-trajectories.This is performed using k-means clustering and a Gaussian model based grouping method.The k-means clustering algorithm is an iterative algorithm that divides the unlabelled data-set into k different clusters in such a way that it minimises the sum of distances between the data point and it corresponding cluster centre.The value of k must be predetermined.
A common approach to finding the optimal number of clusters is the elbow method.In this method, a metric function is calculated for a selected number of clusters.After running the clustering algorithm, the metric function for each k clusters is plotted and forms a curve which typically reduces as the number of clusters increases.The optimal number of  clusters is where there is an obvious discontinuity in the curve.Figure 6 shows example results for three different metric functions.In Figure 6(a), the metric functions are the minimum distance between a pair of cluster centres (blue) and the maximum distance from points in the cluster to the cluster centre (red).In Figure 6(b), the metric function is the sum of each cluster's sum of distance to the cluster centre (SSUMD), the red circle denotes the optimum number of clusters.It can be seen that the SSUMD curve Figure 6(b) is more smooth compared to the two curves in Figure 6(a).The authors used the SSUMD metric function to identify the optimal number of clusters.

F. CDR STEP 5: SECOND LEVEL GROUPING OF MICRO-TRAJECTORIES
The next stage of the CDR process involves a second level grouping of the micro-trajectories.A Gaussian model based grouping method is used to group the micro-trajectories within the previously identified clusters or validate an existing group.For this step, a pair of micro-trajectories in the 2D space are first selected.For the micro-trajectory with a larger number of sample points, the distances of these sample points to the nearest sample point in the other micro-trajectory is computed.A Probability Distribution Function (PDF) of these distance values are then determined.A peak value of the PDF which is close to zero is a good indicator that the two micro-trajectories are close to each other.This second level grouping is used to identify and filter out overlapping micro-trajectories which have similar characteristics.Figure 7 shows an example of this second level grouping for two different pairs of similar micro-trajectories.Figure 7(a) shows one pair of similar micro-trajectories in 2D speed-torque space, and Figure 7(c) shows a second pair of micro-trajectories which are quite different to the first pair.The red and black lines in Figure 7(a) and Figure 7(c) are the two similar micro-trajectories, in each case the green lines highlight the distance between the nearest points of the micro-trajectories.The PDF for the two micro-trajectories in Figure 7(a) is shown in Figure 7(b).Figure 7(d) shows the PDF for the two micro-trajectories in Figure 7(c).This second-level grouping of micro-trajectories is more important for applications that have more than two dimensions.

G. CDR STEP 6: SELECTION OF MICRO-TRAJECTORIES
A representative micro-trajectory can be selected from similar micro-trajectories within a group.The selection can be based upon features such as the maximum rate of change, the length of micro-trajectory or operating region covered by the micro-trajectory, etc.Alternatively, similar microtrajectories within a group can be used to generate a single representative micro-trajectory using a merging or averaging approach.The selected or generated micro-trajectory can then be refined by smoothing and modifying it to fall within any imposed maximum rate of change in either speed or torque.From each micro-trajectory group, at least one microtrajectory should be identified by either approach.This ensures that all the different types of micro-trajectories are present in the shortened representative cycle.The authors used the maximum rate of change to select a micro-trajectory from a group.This was to ensure the fastest dynamics of the system where captured in the resulting nonlinear dynamic models.

H. CDR STEP 7: CONNECTING SELECTED MICRO-TRAJECTORIES USING A MARKOV CHAIN MODEL
The individual micro-transients identified in the CDR process up to this point cannot simply be connected together in time as there will be discontinuities between the speed and torque at the end of one micro-trajectory and the start of another.A simple connection approach is to order the microtrajectories in the order they occur in the original cycle and to form simple smoothing ramps between the adjacent end and start points.However, a more sophisticated approach involves using a MCM.
A MCM describes the possible transitions between a finite set of discrete states using a probability matrix.MCMs have been widely used in representative or surrogate vehicle cycle generation [29], [30], [31].In this work they are used to extract transition probabilities from the original raw drive cycle data and in reconstructing cycles, this information is then used to connect the selected micro-trajectories.
In order to exemplify the application of MCM in this study, a simple 1D time series illustration is used.Figure 8 shows a simple 1D time series of eight micro-trajectories consisting of three different patterns.The number above each microtrajectory in the figure is their pattern number.From Figure 8, a transition matrix can be formed as: Each element in matrix P, P(i, j) is the number of transition from pattern i to pattern j.The transition matrix in (1) can be converted to Markov Chain (MC) transition probability matrix P MC in (3) using the following calculation: In the reduced cycle, only one micro-trajectory is selected from each pattern of micro-trajectories.Therefore, the diagonal entries of P MC are forced to 0 to eliminate transition information within the same pattern.
Hence, the modified MC transition probability matrix becomes: An optimisation problem is then formed to find the optimal connection order of these three patterns: where, n p is a 1 × N row vector that includes integers from 1 to N inclusive to represent the connection order of N microtrajectories in time order.N 0 is used to pre-select the first micro-trajectory to force the representative cycle to start from a defined energy state.When there is no constraint (7), the solution to the optimisation problem in (6) for the CDR problem of simple time series in Figure 8 is shown in Figure 9.The connection order is 3-2-1.The sum of transition probability of this solution is: J = 1 + 0.5 = 1.5.When there is a constraint such as N 0 = 1, the solution to the optimisation problem in (6) for the CDR problem of the simple time series in Figure 8 is shown in Figure 10.The connection order is 1-2-3.The sum of transition probability of this solution is: J = 0.5 + 0.5 = 1.0.Once the connection order of all selected micro-trajectories has been determined, the next step is to decide whether or not extra points need to be added on the path from one micro-trajectory to the next micro-trajectory, and if so, how many.For example, when the end point of the first microtrajectory and the start point of the second micro-trajectory to be connected are not close to each other.In this scenario, these two points likely cannot be directly connected using one step in sample time.This is because either the rate change of speed or torque might exceed the system's physical capability.Hence, it is necessary to add extra points between.
The simplest way to add extra points is to use a straight line between these two points representing a rate of transition that is achievable.
An illustrative example is shown in Figure 11.Here, n p (i) and n p (i + 1) are the two micro-trajectories to be connected.The red line between these two micro-trajectories illustrates the straight line connecting one to the other.Defining the end point of the first micro-trajectory p(end), and the start point of the second-trajectory p(1), and r x,max and r y,max as the rate limits for x and y respectively.Then the number of points added in 2D space is the maximum of the following two integers, i.e. max(N x , N y ): where, R d is the round to ceiling function.

III. CDR CASE STUDY A. PROBLEM DESCRIPTION
To demonstrate the CDR methodology, a 30-minute transient cycle for an off-road Compact Wheel-Loader (CWL) machine was reduced to 200-seconds using steps 1-7 as described above.Figure 12 shows the speed and torque vs time for the original CWL cycle, sub-figures (a) and (c).
The corresponding reduced cycle generated using the CDR methodology is shown in sub-figures (b) and (d).
Figure 13 shows a 2D speed-torque scatter plot of both the original and reduced versions of the CWL cycle.The reduced speed-torque 2D cycle was tested on a 2.7 litre diesel ICE in a test facility in which the engine is coupled to a dynamometer.The test system controlled the enginedynamometer speed-torque to follow the reduced cycle.The collected 200-second reduced cycle data was then used to train a dynamic NF model of the engine exhaust measured NOx emissions.This NOx model was then validated using the same test system and engine running the original CWL 30-minute transient cycle.

B. EXPERIMENTAL SETUP
The test system comprises a 2.8 litre 55kW diesel engine connected to a 480kW dynamometer.The instrumentation system was developed for fast measurement of air, fuel and emissions to collect data for training nonlinear dynamic models.This includes the following specialised equipment: ultrasonic air mass flow meter, ultrasonic fuel meter, fast emissions analysers for CO2/CO and HC and PM measurement.In addition, specialised fast thermocouples developed at Loughborough University are used to enable accurate intake and exhaust system dynamic characterisation.The NOx emissions modelling results in this paper are based on measurements taken using a NOx analyser which has a response time of < 1 second.The test system and the control room are shown in Figure 14. Figure 15 is a simple schematic of the test system which highlights the speed-torque test parameters in relation to the system configuration.

C. CDR NONLINEAR DYNAMIC MODEL RESULTS
There are total of 640 variables logged during the experimental test with a sample time of 0.1 seconds, including Engine Control Module (ECM) sampled and calculated variables.Using a partial information based input selection algorithm, four inputs were down-selected as the inputs to the dynamic NOx emission NF model.They are fuel consumption (estimated by the ECM), engine rotational speed and EGR valve position (also provided by the ECM), and exhaust temperature which is measured using a fast thermocouple.The input orders for fuel, engine speed, EGR valve position and exhaust temperature are 3, 2, 7 and 4 respectively.The measured engine exhaust NOx emissions for the 200-second reduced cycle were used to train a nonlinear dynamic NF model.The number of the local models is 100.A more indepth description of the Neuro-Fuzzy modelling approach used in this study can be found in [32].The LOcal LInear MOdel Tree (LOLIMOT) algorithm was used for the training of the NF NOx model [33].
The resulting NF model training result is shown in Figure 16.There are several very large amplitude and short duration increases in NOx which are common in ICE transient operation with harsh micro-trajectories.The model is able to predict these events successfully, the smaller amplitude but more frequent events are also captured well.The R-square value for this training result is 0.9950.The original 30-minute cycle was also run on the same system and the nonlinear dynamic NF model NOx model was validated against the measured NOx emissions.The validation result is shown in Figure 17.The R-square value for this validation result is 0.9923.
The validation performance of the nonlinear dynamic NF NOx model shows that the CDR methodology described in this paper is very effective for reducing a transient cycle to a representative shorter cycle, and that this shorter cycle can be used to train nonlinear dynamic models which can predict the original cycle behaviour.The transient cycle duration was reduced by a factor of eight in this case study.
In summary, the CDR methodology presented in this paper can be used to significantly reduce the time and cost of testing and model training from transient data collected in the real world.The results show that the CDR reduced cycle can generate sufficiently rich data for efficient identification of nonlinear dynamic models of complex systems.Each step in the methodology can be adjusted in order to produce multiple different shortened cycles.Furthermore, useful metrics from the micro-trajectory identification and grouping steps can be used to categorise different real-world recorded cycles.

IV. CONCLUSION
This paper presents a seven-step transient cycle decomposition and reduction procedure (CDR) which produces a shorter representative transient cycle from a much longer original cycle.The first stage involves identifying 2D (speed-torque) micro-trajectories using critical points (e.g.minimum torque).Features of these micro-trajectories are then characterised and used in a multi-stage grouping process incorporating k-mean clustering and Gaussian process models to identify the unique micro-trajectory behaviour in the input data and to filter out repeat behaviour.Finally, Markov Chain Models are used to efficiently splice together identified micro-trajectories to form a new reduced cycle.A case study shows the generation of a reduced representative 200-second transient cycle which is 8 times shorter than the 30-minute original.A data driven nonlinear dynamic Neuro-Fuzzy model was trained on the data from the reduced cycle NOx emissions (R-square value = 0.9950).This model was then validated on the original cycle and good model performance was demonstrated (R-square value = 0.9923).This shows the CDR methodology can be used to generate efficient excitation signal designs for the identification of nonlinear dynamic models for complex systems.
Adjustments in each step of the methodology enables different reduced cycles to be generated, this is useful for generating different excitation designs for model training and validation purposes.The methodology can be further developed to integrate with online adaptive processes for system calibration, optimisation, control and also driving pattern recognition for autonomous vehicles.The methodology can be increased to higher dimensions.

V. FUTURE WORK
The CDR method discussed here aims for nonlinear-dynamic system modelling.It can be further developed into a CDR method integrated with the non-linearity information of the output variable which will result in a much shorter reduced cycle without compromising model performance.The statistical information of each grouped micro-trajectory can be utilised in generating a reduced cycle for other applications such as electric vehicle system level energy management.
The proposed work in this paper can also be further enhanced by integrating with other existing technologies such as statistical matrices, Monte Carlo (MC) sampling methods, and output feature selection.

FIGURE 1 .
FIGURE 1. Flow chart of the seven steps of the CDR methodology.

FIGURE 2 .
FIGURE 2. Flow chart of the CDR methodology integrated with nonlinear dynamic model identification.
shows an example of critical point based micro-trajectory identification for a short section of the transient cycle.It consists of two steps: Step 1 -identify the minimum torque points, Step 2 -identify transient micro-trajectory segments by finding the start and end points of each segment.The red points in Figure 3(a) are the local minimum torque points.

Figure 3 (
b) shows the start and end points of the microtrajectory segments.Figure 3(c) shows the four identified micro-trajectories in 2D speed-torque space, each microtrajectory is identified by a different colour, the arrows indicate the direction.

FIGURE 3 .
FIGURE 3. Example of micro-trajectory identification from a short PU speed and torque time series: (a) Step 1 -identify the minimum torque points.(b) Step 2 -identify transient micro-trajectory segments by finding the start and end point of each segment.(c) Four identified transient micro-trajectories in speed-torque 2D space.

FIGURE 4 .
FIGURE 4. Examples of two groups of micro-trajectories with similar features of area, centre, delta-speed and delta-torque in speed-torque 2D space.(a) Group 1.(b) Group 2.

FIGURE 5 .
FIGURE 5. Examples of scatter plots of three micro-trajectory features in normalised speed-torque 2D space.(a) Delta of normalised speed vs area computed using normalised speed and torque values.(b) Delta of normalise torque vs delta of normalised speed.

FIGURE 6 .
FIGURE 6. Examples of the elbow method to find the optimal number of clusters.(a) metric functions: the minimum distance between a pair of cluster centres (blue) and the maximum distance from points in the cluster to the cluster centre (red).(b) metric function: the sum of each cluster's sum of distance to the cluster centre (SSUMD).

FIGURE 7 .
FIGURE 7. Two examples of the identification of similar micro-trajectories within a cluster by determining the PDF of the distance between adjacent sample points for a pair of micro-trajectories.

FIGURE 8 .
FIGURE 8. Example: a simple 1D time series consisting of eight micro-trajectories with three identified patterns.

FIGURE 9 .
FIGURE 9. Example: Connection solution without constraint on the start pattern.(a) Original 1D cycle identified to have three patterns; (b) Connected reduced cycle without constraints on the start pattern.

FIGURE 10 .
FIGURE 10.Example: Connection solution with constraint on the start pattern: (a) Original 1D cycle identified to have three patterns; (b) Connected reduced cycle with constraints on the start pattern (N 0 = 1).
Figure 13 illustrates how the CDR methodology captures the unique transient behaviour around the periphery of the operating envelope whilst filtering out repeated transient behaviour in the middle of the operating envelope.The filtering out of nonunique micro-trajectories leads to the representative reduced cycle which is significantly shorter in duration.

FIGURE 14 .
FIGURE 14. Instrumented test engine and control system used for CDR method validation.

FIGURE 15 .
FIGURE 15.Test system schematic showing 2D speed-torque control and measurement.

FIGURE 16 .
FIGURE 16.Nonlinear dynamic NF model training result for NOx emissions for the CDR reduced cycle.

FIGURE 17 .
FIGURE 17. Nonlinear dynamic NF model validation result for NOx emissions.