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.

Gaussian Process Regression is a probabilistic modeling approach. In a Gaussian process (GP), an uncountable number of random variables indexed by the vector x ∈ Rd are assigned a Gaussian prior and any subset of them follows a multivariate Gaussian distribution. We assume the underlying model (f) of the system to be modeled follows a GP
where (m(x), k(x, x′)) are the mean and covariance (or kernel) functions of the GP, respectively. The mean and kernel function are chosen based on the system to be modeled. Since the measurements (y) taken from a system can be subject to noise, it is modeled as additive white Gaussian noise with zero mean and variance σ2, y=f+ε. Usually, the mean of a GP model is assumed to be zero. The parameters in the kernel function (θ) along with σ are referred to as hyperparameters. So, the Gaussian process used in this work can be denoted as
where x = [CN, SOI, GPP] ∈ R3 are the inputs to the model. The regression in GPR comes in while finding the hyperparameters. This is done by solving an optimization problem that maximizes the likelihood of observed data (Y, X) with respect to the hyperparameters.
where Y ∈ RN and X ∈ RN×3. N(0,KXX(θ)) denotes multivariate Gaussian probability distribution function with zero mean vector and covariance matrix KXX(θ) ∈ RN×N, obtained by evaluating the covariance function over the observed data. For making predictions using the model, the posterior mean and covariance conditioned on the observed data are used as the estimate and variances at the desired input (x*). These are given by
where CA50predm(x*) and CA50predσ(x*) represent the predicted mean and variance at (x*), respectively. K*X=KX*T is the vector obtained by evaluating the covariance function between the observed data and x*. K** is the covariance function evaluated at x*. More information on GPR and the underlying calculations can be found in Ref. [11]. These posterior estimates of mean and variance, of predicted CA50 obtained from the GPR model at a given input, are then used in the feedforward control design.

2.2 Feedforward Control Design.

The developed GPR model from the earlier section is now inverted by solving an optimization problem that minimizes an objective or cost function subject to constraints.
(1)
where J(x) is the objective function and (A, b) describes the constraints of optimization. The objective function and constraints can be designed based on the performance requirements. For solving this optimization problem, a real parameter genetic algorithm is used. Look-Up Tables (LUTs), built by solving the optimization problem, are used for feedforward control. The feedforward controller takes the desired CA50 and fuel CN number as inputs and provides the actuator controls (GPP and SOI) as outputs. The genetic algorithm searches for actuator controls (GPP and SOI) that minimize the cost function subject to constraints for a given desired (CA50, CN) input pair. The cost function J(x) is designed such that the absolute error between, CA50 predicted using the GPR model and the desired CA50, is reduced as well as the points picked are from high-confidence or low-variance regions
(2)
where CA50predm(CN,x) and CA50predσ(CN,x) are the mean and variance estimates of CA50 for input x and CN, obtained from the GPR model. CA50desired is the desired CA50. w1 is a weight that is used for tuning the LUTs toward lesser error or more confidence. This search is done over a range of specific input pairs and stored in LUTs, to be used in real time. For values that lie between the input pairs in the LUTs, linear interpolation is used for getting the control inputs. Therefore, the LUTs found can be seen as a 2D matrix for each control input with CA50 and fuel CN as its dimensions. More information on the design process of the LUTs can be found in Ref. [5]. The fuel CN number is an operating condition that cannot be directly measured in real-time and needs to be estimated. A fuel CN estimator that uses the GPR model to numerically search for a fuel CN number, given the inputs and output of the engine, provides an estimate of fuel CN in real time. When the difference between the actual CA50 and predicted CA50 is greater than a threshold, the fuel CN estimator is triggered to get an updated fuel CN estimate. A diagram of the real-time control system used is shown in Fig. 1.
Fig. 1
The real-time control framework used for regulating the combustion phasing of the engine
Fig. 1
The real-time control framework used for regulating the combustion phasing of the engine
Close modal

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.

A limit has been put on the rate of increase/decrease of the glow plug power to prevent damaging it. This rate-limit constraint can be represented as
where RGPP is the rate limit on the glow plug power actuation. The LUT provides the actuator commands at each sampling instant and this command will be held for the entire sampling period (sample and hold). The physical rate-limiter present between the LUT and glow plug (as shown in Fig. 1) linearly changes the GPP to the commanded value with the said rate limit (RGPP). Therefore, on discretizing the above condition, we get
where Ts is the sampling time period and GPP(k) is the GPP commanded at time instant k. As mentioned earlier, LUT corresponding to the GPP would be a 2D matrix and operating conditions between the points in the 2D matrix are linearly interpolated. Any linear constraint satisfied at the indices in the table will automatically guarantee that points in between will also satisfy it. So, it would be sufficient for the above constraint to be valid at the operating conditions in the LUT, for it to be valid across the whole range covered by the LUT. Using this argument, the above constraint can be written as
where GPP[p, q] indicates the value of GPP at the index position (p, q) in the 2D LUT. It is assumed that it takes one sampling time period (Ts) to move one step along an input dimension. (p, q) Indicates the indices along fuel CN and CA50 input dimensions, respectively. On generalizing this across, both the input dimensions gives
where T[i,j]→[k,l] is the time taken for moving from indices (i, j) to (k, l).

Assumption: The inputs to the LUT only change along one dimension per sampling time period and will vary linearly with known rate limits.

Using this assumption, the time taken for the inputs to change can be decoupled and a direct relationship between the outputs and inputs can be derived
where RCN and RCA50 are the rate of change of inputs. Ideally, these constraints are required to be applied among all the operating points to keep the glow plug power commanded by the LUT within the rate limit. However, since the inputs would always move between the operating points, if the constraint is satisfied between neighboring operating points, then it would be satisfied for moving between any of the operating points. Therefore, the cost function along with constraints turn out to be
(3)
Subject to:
(4)
where (N, M) are the lengths of fuel CN and CA50 dimensions, respectively. The constraints in Eq. (4) can now be included as additional constraints to the genetic algorithm search performed for finding the LUTs, using (A, b) in Eq. (1). In Eq. (4), the actuator constraints have been moved to constraints on controller inputs, which can be set by the user. Therefore, while designing the LUTs the controller input rates can be tuned to get a desired LUT that satisfies the rate-limit constraints of the actuator.

3.2 Energy-Efficient Ignition Assistant Actuation.

Higher fuel CN can combust easily without an ignition assistant. However, powering ignition assistant and using a retarded start of injection can achieve the same CA50. These are valid solutions but are energy-inefficient. We would want the genetic algorithm to avoid these solutions and prefer the ones that use the ignition assistant, only when required. However, on looking at the other side, i.e., lower fuel CN, for steady combustion an ignition assistant is required, and usage of a glow plug is essential in these cases. One way to ensure this would be to add hard constraints ensuring that the GPP is not below or above a certain level for a given fuel CN number, to the genetic search algorithm. However, these constraints could lead to infeasible solutions and require a lot of tuning as this needs to be done for various fuel CN numbers. Also, in some cases, these constraints could make the rate-limit constraints on GPP inconsistent. So, we propose a more consistent way by adding a penalty term for using the glow plug power, to the cost used in the genetic search algorithm (Eq. (2)). The cost being added should be such that the penalty for using GPP should be less in low fuel CN range and high for high fuel CN range. The following cost term is used
(5)
where CNlow and CNhigh are the lower and upper bounds of fuel CN number. GPP is the glow plug power for the given operating condition, and w2 is the parameter that needs to be tuned during the design process for obtaining a desirable LUT.

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.

Fig. 2
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.
Fig. 2
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.
Close modal
Table 1

Engine specifications

ItemValue
Engine typeDiesel, four-cycle
ConfigurationInline
Displacement2.0 L
Bore83 mm
Stroke90.4 mm
Compression ratio15.37:1
Combustion systemCommon rail direct injection
ItemValue
Engine typeDiesel, four-cycle
ConfigurationInline
Displacement2.0 L
Bore83 mm
Stroke90.4 mm
Compression ratio15.37:1
Combustion systemCommon 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.

Fig. 3
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.
Fig. 3
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.
Close modal

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.

Table 2

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

MetricBLRLEE
Average glow plug power (W)56.7125.21
Mean absolute error (degCA)0.700.69
MetricBLRLEE
Average glow plug power (W)56.7125.21
Mean absolute error (degCA)0.700.69

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.

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