Abstract
Surrogate-model or data-driven model-based control frameworks are becoming increasingly popular in recent years due to their ease of model development and enhanced computational power, making them suitable for real-time use. However, when it comes to modeling aspects related to time, difficulties arise as many of the models deal with quasi-static systems. In this paper, we propose a method to model time-dependent actuator constraints in a surrogate-model-based control framework for controlling the combustion phasing in a multi-fuel UAS engine. Along with this, a conducive method for designing an energy-efficient ignition assistant control is discussed. The developed methods are then tested on a diesel engine, and the results show a more robust and energy-efficient combustion phasing control as the fuel property varies in real-time.
1 Introduction
In this paper, we focus on designing a controller for a Diesel engine that is used in multi-fuel Unmanned Aircraft Systems (UAS). Some of the requirements in these systems include being able to control the combustion phasing of the engine with varying operating conditions. Usually, the operating conditions include engine speed, load, intake air temperature and pressure, and fuel used. In the case of diesel fuel, the fuel combustion properties are characterized using its Cetane (CN) number. The various available controls include pilot and main injection timings, and their corresponding duration, power supplied to ignition assistant, and common rail pressure. In this paper, we use the time taken for 50% of the injected fuel to burn (CA50), to characterize the combustion phasing of the engine. Given the number of controls and operating conditions, it would be an industrious task to manually tune a controller by solely conducting experiments on the engine. So, a model-based approach is preferred. A physics-based model using computational fluid dynamics (CFD) [1] can be developed that describes the combustion process thoroughly but it can be quite complex leading to large computation overheads. On the other hand, data-driven models don't require detailed physical modeling and can be computationally efficient. Control frameworks based on data-driven models are getting more recognition in recent years [2]. Artificial Neural Networks (ANNs) were used to model the emissions from a diesel engine [3]. A support vector machine (SVM)-based model for NOx emissions was used in the study by Aliramezani et al. [4]. Kriging or Gaussian Process Regression (GPR) model-based control frameworks were used by Dong et al. and Xia et al. [5,6]. GPR models, along with an estimate of the prediction, also give an estimate of the variance (or confidence) for the prediction. This makes them quite attractive for designing control frameworks and hence has been used in this paper too. However, in spite of the numerous benefits offered by them, capturing the time dynamics inherently by the GPR model would not be possible as the inputs and outputs of the system are described in a quasi-static manner. When controllers are developed using this model, a lack of time-dependent constraints could lead to deterioration in their performance. Methods for updating the GPR model to adapt to changing operating conditions with time have also been proposed [7]. Methods have been proposed for modeling time dynamics in GPR models [8,9], which involve developing multiple models at various fixed time instances, with the underlying model at each discrete time instant being quasi-static. However, these might not be suitable for cases when the system inherently has continuous-time constraints.
One area where time-dependent constraints could be present is at the actuators. For example, limitations on the frequency or rate of change of actuation, delay in actuation, etc. If these are not modeled, then it could lead to a performance deficit or might even damage the actuator. In order to prevent damaging the actuator, one can add a physical constraint to it, but this could deteriorate the performance of the controller, as this constraint would prevent the actuator from following the commanded control signal. Therefore, a method is required for modeling these constraints in the control design process. One such method is described in this paper. The data-driven model is not modified but modifications are made in the control design to incorporate the time-dependent constraints. The proposed method is then applied to a multi-fuel UAS engine. It has been shown in Ref. [10] that using an ignition assistant, combustion in a diesel engine using lower CN fuels can be done more reliably. The power supplied to the ignition assistant needs to be actuated continuously by the controller to maintain stable combustion across varying fuel CN numbers. However, the rate of change of power supplied to the ignition assistant is bound to keep it safe. This constraint is time-dependent and cannot be directly modeled into the GPR model used in this framework. A method for modeling this into the control design process is discussed in the following sections. Along with this, a conducive design method for reducing the amount of power used by ignition assistants is also discussed in this paper.
The remainder of this paper is organized as follows: Sec. 2 gives a brief description of the control framework. Section 3 describes the design methodology for including actuator constraints and reducing the energy consumption of the actuator, into the controller design. Section 4 describes the experiments performed and discusses the results obtained. Finally, Sec. 5 provides the concluding remarks.
2 Control Framework
2.1 System Model.
In this work, the objective is to control the 50% combustion timing (CA50) in a Diesel engine with variable fuel. A Gaussian Process Regression (GPR) model [11] is used as a surrogate model for the CA50 of the engine; with fuel Cetane (CN) number, Start Of main Injection (SOI), and power supplied to the ignition assistant (or Glow Plug Power, GPP) as inputs. All the other parameters and controls are kept constant. This model is constructed based on the experimental data at various test points, apriori.
2.1.1 Gaussian Process Regression.
2.2 Feedforward Control Design.
2.3 Effect of Rate-Limiting Constraints.
For our system of interest, time-dependent constraints are added to the ignition assistant. To prevent damaging it, a rate-limit is enforced on the power supplied to it. This is done by using a physical rate-limiter as shown in Fig. 1. The effects of adding a physical rate-limit to the overall system are discussed in the following sections.
2.3.1 Transient Time.
It takes a certain amount of time for the actuator to reach a commanded value and only after this can we expect the system to give the desired response. In cases, where additional physical constraints (such as rate-limit) are put on the system, the time taken for the system to reach steady-state or transient time can be longer. For the current system of interest, if the rate-limiter constraints are not modeled in the design process, the actual glow plug power can take varying amounts of time to reach the commanded value. This time depends on the rate-limit, current, and commanded actuator values. By modeling or including these constraints in the control design, one can put a bound on the amount of time taken by the controls to reach the commanded value. In turn, this leads to a more reliable response for a given change in inputs to the LUT.
2.3.2 Robustness of the System.
One area where robustness comes into play for our system of interest is when there are modeling errors in the GPR model. The fuel CN estimator uses the GPR model to numerically estimate a fuel CN value given the inputs and outputs of the engine. It uses the actual GPP, i.e., after rate-limiting, supplied to the engine as shown in Fig. 1. Since the points in the LUT are optimized to have relatively lower variance or high confidence, fuel CN estimation near these regions will be more reliable. In cases when the commanded GPP violates the rate-limit, during the transient (i.e., when the GPP commanded and rate-limited GPP are not equal), the actual inputs to the engine will be away from the points in the LUTs. The confidence of the model in these regions is less, making it prone to modeling errors. If a fuel CN estimation is triggered during this time, the fuel CN estimates might not match the actual fuel CN value. Upon using this fuel CN value, the inputs provided by the LUT might not produce a CA50 that is close to the CA50 predicted by the model, and this triggers a fuel CN estimation again. This cycle would keep on going until the operating points are in a more reliable region of the model. During this period, the performance of the feedforward control would be unreliable. Also, these repeated estimations can cause undesirable oscillations in the control inputs. However, if rate-limit constraints are included in the design process, the operating points would always be close to the LUT aiding in triggering the fuel CN estimation in more reliable parts of the model. In some cases, certain points in the LUT itself could have modeling errors, and these could trigger a fuel CN estimation cycle. However, as operating points will be close to LUT when the constraints are included, stable or more reliable regions can be reached quickly. This was seen in the tests performed and is discussed in Sec. 4.
3 Design Methodology
3.1 Inclusion of Glow Plug Power Rate Limit.
Assumption: The inputs to the LUT only change along one dimension per sampling time period and will vary linearly with known rate limits.
3.2 Energy-Efficient Ignition Assistant Actuation.
4 Experimental Results
4.1 Experimental Setup.
Experiments were performed on a four-cylinder diesel engine, and its specifications are given in Table 1. All the experiments were performed at the Engine Research Center of the University of Wisconsin-Madison at a fixed operating condition of 1200 RPM, 4.5 bar IMEP load, and fixed injection duration of 66 ms. A total of 375 data points were collected sweeping through fuel CN, GPP, and SOI timings, with constant spacing after identifying boundaries for stable combustion. A Gaussian Process regression model using a squared exponential kernel function with a separate length scale for each predictor was constructed using the collected data. This model was then used for developing the LUTs as well as in the fuel CN estimator. The LUTs without adding the rate-limit constraints and energy costs developed in this paper are referred to as Baseline LUTs. A plot of the LUTs obtained for several fuel CN numbers is shown in Fig. 2(a). The LUTs designed after including the GPP rate-limiter constraint and energy cost as described in Sec. 3 are referred to as Rate-Limited Energy Efficient (RLEE) LUTs. For these LUTs, the following rate limits were used, RGPP = 10 W/s, RCN = 1 s−1, and RCA50 = 1 degCA/s. A plot of the LUTs obtained for several fuel CN numbers is shown in Fig. 2(b). On comparing Figs. 2(a) and 2(b), one can see that the points picked in the RLEE LUTs are distributed uniformly across the GPP (vertical) axis. This indicates that the change in GPP is not drastic between neighboring operating points and is always bounded by the applied rate-limit constraints.

Plots of the points picked up in the look-up table at fuel CN numbers (25, 30, 35, 42, and 48). Glow plug power is plotted along the vertical axis and the start of injection is along the horizontal axis. The labels on the plots indicate the corresponding desired CA50. (a) Baseline look-up table and (b) rate-limited energy-efficient look-up table.

Plots of the points picked up in the look-up table at fuel CN numbers (25, 30, 35, 42, and 48). Glow plug power is plotted along the vertical axis and the start of injection is along the horizontal axis. The labels on the plots indicate the corresponding desired CA50. (a) Baseline look-up table and (b) rate-limited energy-efficient look-up table.
Engine specifications
Item | Value |
---|---|
Engine type | Diesel, four-cycle |
Configuration | Inline |
Displacement | 2.0 L |
Bore | 83 mm |
Stroke | 90.4 mm |
Compression ratio | 15.37:1 |
Combustion system | Common rail direct injection |
Item | Value |
---|---|
Engine type | Diesel, four-cycle |
Configuration | Inline |
Displacement | 2.0 L |
Bore | 83 mm |
Stroke | 90.4 mm |
Compression ratio | 15.37:1 |
Combustion system | Common rail direct injection |
4.2 Testing.
Real-time control strategy was implemented using dSPACE-based control hardware developed at the University of Minnesota-Twin Cities in conjunction with the engine. A fuel switch test was conducted on the engine to verify the performance of the designed LUTs. In this test, the fuel in the engine is switched from fuel CN 25 → 48 in real-time. The objective of the controller is to regulate the control inputs (GPP and SOI) to maintain a constant desired CA50 of 9 degCA, with the fuel CN varying.
The results obtained using the Baseline LUTs are shown in Fig. 3(a). It can be seen that the commanded and actual GPP, indicated using dash-dotted and solid lines, respectively, in the plot, do not overlap in regions past 450 s. This is seen when the GPP commanded by the feedforward control violates the rate limit, causing the rate limiter to limit the actual GPP being supplied. During the same time instances, oscillations in CA50, fuel CN, GPP, and SOI can be observed. As discussed in Sec. 2.3.2, fuel CN estimation triggered in regions where the commanded and actual GPP are not equal can lead to estimation in unreliable regions of the model. The oscillations observed are due to the multiple unreliable CN estimations that occur before reaching a steady-state. Figure 3(b) shows the results using RLEE LUT. The commanded GPP and actual GPP are very close if not overlapping for majority of the time, when compared to the Baseline case, satisfying our requirement for the commanded GPP to be within the rate-limit constraints. For both cases (Figs. 3(a) and 3(b)), the CN estimates during the fuel switch tests were close to the expected trends based on the physical sizing of the common rail system, validating the performance of CN estimator.

Results obtained while tracking CA50 = 9 degCA and switching the fuels. In both the figures, the first plot (from the top) shows the measured CA50 (solid line) and desired CA50 (dash-dotted line); the second plot shows the fuel CN estimated by fuel CN estimator, third plot shows the GPP commanded (dash-dotted line) by the LUT as well as the actual GPP (solid line) after rate limiting it, and the fourth plot shows the SOI commanded. The black lines in the first plot (from the top) in both figures indicate the ±1 degCA bounds about the desired CA50: (a) fuel switch from 25 to 48 using Baseline LUTs and (b) fuel switch from 25 to 48 using RLEE LUTs.

Results obtained while tracking CA50 = 9 degCA and switching the fuels. In both the figures, the first plot (from the top) shows the measured CA50 (solid line) and desired CA50 (dash-dotted line); the second plot shows the fuel CN estimated by fuel CN estimator, third plot shows the GPP commanded (dash-dotted line) by the LUT as well as the actual GPP (solid line) after rate limiting it, and the fourth plot shows the SOI commanded. The black lines in the first plot (from the top) in both figures indicate the ±1 degCA bounds about the desired CA50: (a) fuel switch from 25 to 48 using Baseline LUTs and (b) fuel switch from 25 to 48 using RLEE LUTs.
Oscillations due to model inaccuracies can be seen in Fig. 3(b), in the low fuel CN range. However, these oscillations reach a steady-state quicker than the Baseline case. Ideally, if the assumptions are held, then the commanded and actual GPP would exactly overlap one another. However, it can be seen that there are cases where they don't exactly overlap. This is because of the numerical instabilities in the fuel CN estimator. The fuel CN estimates, change in steps, and this breaks the assumption on inputs to vary linearly within a rate-limit, leading to small differences between the actual and commanded GPP. The average glow plug power consumed and mean absolute error in tracking during these tests are shown in Table 2. A reduction of 55.5% in the average glow plug power consumed was observed without any reduction in the tracking performance, by using the RLEE LUTs when compared to baseline LUTs during the tests.
Average glow plug power consumed and mean absolute error in CA50 during the fuel switch tests shown in Fig. 3 using baseline (BL) and RLEE LUTs
Metric | BL | RLEE |
---|---|---|
Average glow plug power (W) | 56.71 | 25.21 |
Mean absolute error (degCA) | 0.70 | 0.69 |
Metric | BL | RLEE |
---|---|---|
Average glow plug power (W) | 56.71 | 25.21 |
Mean absolute error (degCA) | 0.70 | 0.69 |
5 Conclusion
In this paper, a method for incorporating time-dependent actuator constraints into GPR-based control framework was presented. Along with this, a conducive implementation for designing energy-efficient control strategy for a multi-fuel UAS engine was also presented. The developed methods were tested by performing fuel switches in real-time. Experimental results show the effectiveness of the proposed method by reducing the actuator oscillation and energy consumption while maintaining precise combustion phasing tracking.
Acknowledgment
The research was sponsored by the DEVCOM Army Research Laboratory and was accomplished under Cooperative Agreement Number W911NF-20-2-0161. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the DEVCOM Army Research Laboratory of the U.S. Government. The authors would also like to thank the team at the Engine Research Center of the University of Wisconsin-Madison for providing us with the engine data for this work.
Conflict of Interest
There are no conflicts of interest.
Data Availability Statement
The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.