## Abstract

This technical brief presents anti-windup adaptation algorithms for a look-up table (LUT), widely used in data-driven engine control systems to accurately model complex features while minimizing computational demand. Engine control systems are prone to uncertain variations due to aging, faults, and manufacturing tolerances, which can impact performance and emissions unless effectively managed. Therefore, there is a growing demand for adaptive features in these systems to maintain robust performance and emissions over their lifespan. This study develops computationally efficient adaptive look-up table (ALUT) algorithms using anti-windup recursive parameter estimation and covariance matrix resetting, ensuring robust and rapid adaptation under various operating conditions. The effectiveness of these algorithms is demonstrated through adapting an engine-out nitrogen oxides (NO_{x}) concentration map, which is crucial for tailpipe emission controls in compression-ignition (CI) engines.

## 1 Introduction

Over the past decades, engine control systems have witnessed significant improvements and increased complexity, driven by the response of the automotive industry to stringent regulations on tailpipe emissions and fuel efficiency. Among others, adaptive features in engine control systems are becoming increasingly essential to address uncertain system variations, ensuring the maintenance of performance and tailpipe emissions levels throughout the lifespan of vehicles [1,2].

Meanwhile, look-up tables (LUTs) have found extensive use in data-driven engine control systems due to their ability to model complicated systems with high accuracy at a low computational cost. LUTs are calibrated using large amounts of experimental data and stored in an engine control unit (ECU) for reference based on operating conditions through linear or bilinear interpolation. However, a drawback of conventional LUTs is their lack of adaptability to uncertain system variations once implemented. Consequently, engine performance and tailpipe emissions may degrade in the presence of nontrivial system variations.

In contrast, adaptive look-up tables (ALUTs), which can dynamically adjust to uncertain system variations such as aging, faults, and manufacturing tolerance, have the capability to maintain modeling and control accuracy over the lifespan. Additionally, ALUTs have the potential to substantially reduce the calibration effort and become self-calibrating. This is particularly advantageous as the complexity of engine control systems continues to grow, placing greater demands on calibration efforts.

Adaptive look-up tables have been the subject of extensive study in recent years. In the literature, a recursive parameter estimation problem has been formulated, and various methods have been developed, which can be technically classified into three categories: (1) recursive least squares (RLS); (2) Kalman filter (KF); and (3) ad hoc estimation methods. First, RLS-based algorithms have found application in adaptive volumetric efficiency maps critical for air–fuel ratio control in spark-ignition (SI) engines [3], map-based adaptive feedforward control of active engine mount systems to dampen engine vibrations [4], and adaptive map-based spark timing control in SI engines to accommodate fuel variation [5]. Second, KF-based algorithms have been employed for adaptive engine-out nitrogen oxides (NO_{x}) models crucial for NO_{x} control of compression-ignition (CI) engines [6,7], and adaptive mass air flow sensor drift correction maps to accurately estimate mass air flow rates [8]. Finally, ad hoc algorithms have been used for adaptive engine friction models to enhance torque tracking performance [9] and adaptive map-based knock control in SI engines [10].

However, RLS or KF-based algorithms have a notable limitation: their strong dependence on input excitation, namely, the operating condition. When the input excitation is not persistent, as in steady-state engine operation at idle, the covariance matrix in RLS or KF can increase unboundedly, a phenomenon referred to as *covariance windup*. If the input excitation becomes persistent again after a prolonged covariance windup, the estimation algorithm becomes sensitive to measurement noise, resulting in a long transient with poor estimation. Notably, this issue for an ALUT algorithm has not been well addressed. Different anti-windup RLS or KF methods have been developed, including: (1) directional forgetting recursive least squares (DFRLS) [11,12]; (2) variable forgetting recursive least squares (VFRLS) [13,14]; (3) regularized recursive least squares [15,16]; (4) adaptive Kalman filter (AKF) [17,18]; and (5) regularized Kalman filter [19].

Adaptive look-up table algorithms based on DFRLS and VFRLS have been developed and successfully demonstrated, motivated by their anti-windup capability and computational efficiency in the previous work of the authors [20]. In this study, AKF is further included, coupled with an enhancement in the convergence rate through covariance resetting with memory. This application is illustrated through the adaptation of an engine-out NO_{x} concentration map, which plays a crucial role in tailpipe emission controls for CI engines.

The remainder of this paper is outlined as follows: Sec. 2 introduces the background of look-up table interpolation and adaptation. Section 3 provides the details of the anti-windup adaptation algorithms, including anti-windup recursive parameter estimation and covariance matrix resetting. In Sec. 4, the developed algorithms are demonstrated and compared in terms of convergence rate and anti-windup capability. Finally, Sec. 5 presents the concluding remarks.

## 2 Background

### 2.1 Look-Up Table Interpolation.

Bilinear interpolation is introduced for a constant two-dimensional look-up table (2D LUT) that takes two model inputs indicating operating conditions and produces one model output. It is worth nothing that bilinear interpolation can be simplified to a one-dimensional look-up table (1D LUT) [20] or extended to a higher dimensional look-up table (nD LUT) where $n\u22653$ with minor modifications, although higher dimensions are rarely used in practice due to complexity.

Figure 1 illustrates the 2D LUT, consisting of two constant input breakpoint vectors: $X={Xi}$ and $Y={Yj}$ in ascending order (i.e., *X _{m}* >

*X*for

_{n}*m*>

*n*and

*Y*>

_{p}*Y*for

_{q}*p*>

*q*), and one constant output array: $Z={Zij}$ for $i\u2208{1,\u20092,\u20093,\u20094}$ and $j\u2208{1,\u20092,\u20093,\u20094,\u20095}$. At time

*k*, depending on two inputs,

*x*(

*k*) and

*y*(

*k*), two active adjacent input breakpoints are determined in each direction: $XL(k)$ and $XR(k)$ such that $XL(k)\u2264x(k)<XR(k)$, and $YL(k)$ and $YR(k)$ such that $YL(k)\u2264y(k)<YR(k)$. Correspondingly, four active output elements: $ZLL(k),\u2009ZLR(k),\u2009ZRR(k)$, and $ZRL(k)$ are determined. It is important to note that the active input breakpoints and output elements of the constant 2D LUT can change with time, but they are taken from the constant vectors

*X*and

*Y*, and constant array

*Z*.

*k*is given by

Note that both the regression vector and parameter vector are known and available for LUT interpolation.

### 2.2 Look-Up Table Adaptation.

Look-up table adaptation is an inverse problem of LUT interpolation as illustrated in Fig. 3. Interpolation computes *z*(*k*) using known $\phi (k)$ and $\theta (k)$ (see Fig. 3(a)). Conversely, adaptation estimates the unknown $\theta (k)$, denoted by $\theta \u0302(k)$, using known $\phi (k)$ and *z*(*k*) (see Fig. 3(b)).

The recursive parameter estimation problem is formulated for LUT adaptation as follows:

*Assumptions*

Constant input breakpoints vectors:

*X*and*Y*.Time-varying output array:

*Z*(*k*).Modeling output

*z*(*k*) (i.e., interpolation) and regression vector $\phi (k)$ (i.e., operating condition) are available.

*Parameter Estimation Problem*

- Estimate the parameter vector $\theta \u0302(k)$, a subset of
*Z*(*k*), that minimizes the following cost function:$J(k)=\u2211i=0k\lambda k\u2212i(z(i)\u2212\phi T(i)\theta \u0302(k))2$

The constant forgetting factor $0<\lambda <1$ is introduced to account for the time-varying output array *Z*(*k*) by discounting older information and giving stronger influence to newer information. It is worth noting that the ALUT is stored in the nonvolatile memory of an ECU, ensuring it retains the latest learning and is ready for use at the start of the next driving cycle.

## 3 Anti-Windup Adaptation Algorithm

A standard RLS with a fixed forgetting factor, namely, *exponential forgetting recursive least squares* (EFRLS), is briefly reviewed. Then, anti-windup methods will follow to address its drawbacks.

### 3.1 Exponential Forgetting Recursive Least Squares.

*k*is in the form of

*G*(

*k*) is given by

*P*(

*k*) is the inverse of the information matrix

*R*(

*k*), which is recursively updated by exponentially forgetting of the previous information matrix in all directions uniformly and incorporating the newly entering information

Note that the second term in Eq. (8), representing the current operating condition, is a rank-one positive semidefinite matrix. Consequently, if the operating condition remains unexcited for an extended period, such as during constant idle operation, the information matrix becomes ill-conditioned (i.e., approaching singularity) due to the forgetting factor less than 1. This ill-conditioning results in unbounded increases in the covariance matrix, known as covariance windup. This study aims to prevent covariance windup in RLS or KF-based LUT adaptation, even during periods of weak input excitation, to reduce noise sensitivity and avoid prolonged transients in estimation when the input becomes excited again.

### 3.2 Directional Forgetting Recursive Least Squares.

where $r(k)=\phi T(k)P(k\u22121)\phi (k)$. This ensures that the information matrix *R*(*k*) is positive definite, and in turn, the covariance matrix *P*(*k*) is upper-bounded regardless of input excitation, given that *P*(0) is finite and positive definite [11]. The parameter is then recursively estimated using Eqs. (5) and (6), but with 1 in place of *λ* in Eq. (6).

### 3.3 Variable Forgetting Recursive Least Squares.

where $0<\lambda min<1$ is the constant minimum forgetting factor, $e=z(k)\u2212\phi (k)T\theta \u0302(k\u22121)$ is the interpolation error, and Γ_{max} is the constant maximum trace. The forgetting factor varies based on the interpolation error within the boundary: $\lambda min\u2264\lambda (k)\u22641$. When the interpolation error is zero, it stops forgetting old information (i.e., $\lambda (k)=1$), ensuring the information matrix never becomes singular and remains positive definite. Consequently, the covariance matrix is upper-bounded even during periods of low excitation. Conversely, when the interpolation error becomes significant, the forgetting factor approaches the minimum, allowing for a faster convergence rate. Finally, parameters are estimated using Eqs. (5) and (6), but with Eq. (14) replacing *λ* in Eq. (6).

### 3.4 Adaptive Kalman Filter.

*w*(

*k*) and

*e*(

*k*) are a random walk uncertainty and interpolation error, respectively, and their constant covariance matrices are given by $Q=Cov(w(k))$ and $R=Cov(e(k))$. The formula of the standard Kalman filter (KF) is given by

where $\theta \u0302(k)$ is the parameter estimate, *G*(*k*) is the gain, and *P*(*k*) is the covariance matrix, respectively. If the state-space model of Eqs. (15) and (16) is observable, the above standard KF guarantees optimal estimation of $\theta (k)$. However, in fact, it is conditionally observable, i.e., observable only if $\phi (k)$ persistently changes [17]. Otherwise, the covariance matrix *P*(*k*) will diverge unboundedly, i.e., covariance windup, leading to poor transients and/or bias in estimation.

*Q*, the AKF employs

*Q*(

*k*), which is adaptively updated

where *P _{d}* is the desired symmetric positive definite matrix.

The convergence behavior of the covariance matrix *P*(*k*) is summarized. Let $Ep(k)=P(k)\u2212Pd$ denote the covariance matrix error. By applying the matrix inversion lemma to $Pd\u2212Q(k)$ and $P(k+1)\u2212Q(k)$, the covariance matrix error at time *k *+* *1 is derived as $Ep(k+1)=Ap(P(k))\u22121Ep(k)Ap(Pd)\u2212T$, where $Ap(P(k))=I+P(k)\phi (k)R\u22121\phi T(k)$. Thus, $Ep(k)$ converges to zero exponentially [17,18]. In other words, in AKF *P*(*k*) converges to *P _{d}*, meaning anti-windup capability. One can balance between noise immunity and convergence rate through tuning

*P*.

_{d}### 3.5 Covariance Matrix Resetting.

In Eq. (2), $\theta (k)$ changes over time, influenced by uncertain variations and varying operating conditions. It is noted that there is an overlap of either one or two active output elements during transition as illustrated in Fig. 4. This implies partial correlation between *P*(*k*) and $P(k+1)$. Therefore, appropriately resetting the covariance matrix during transition is crucial. Two resetting methods are proposed to avoid suboptimal and slow convergence rates of estimation during transition.

#### Resetting Without Memory.

The first resetting method neglects the correlation as in Ref. [20]. When a transition of active output elements occurs, the covariance matrix is reset with a constant diagonal matrix: $P0=diag(v0,v0,v0,v0)\u2208\u211c4\xd71$. Since it is reset without memory, this approach may lead to a long transient in estimation. Note that identical diagonal elements are used because no prior information is available.

#### Resetting With Memory.

*k*+

*1, when a transition occurs,*

*P*(

*k*) is reset with an adaptive diagonal matrix: $P(k)=diag(vLL(k),vRL(k),vLR(k),vRR(k))\u2208\u211c4\xd71$, where the diagonal elements are retrieved from the adaptive variance matrix

for $Z(k)\u2208\u211cm\xd7n$, depending on operating condition. Only the active elements out of *V*(*k*) are adaptively updated with the diagonal elements of *P*(*k*).

*P*(

*k*) is assumed to be given as

At this moment, $v22(k),\u2009v32(k),\u2009v23(k)$, and $v33(k)$ are updated with $p11(k),\u2009p22(k),\u2009p33(k)$, and $p44(k)$, respectively. At time *k *+* *1, when the transition occurs, *P*(*k*) is reset as $P(k)=diag(v33(k),v43(k),v34(k),v44(k))$ to compute $P(k+1)$ using Eqs. (9) or (11) or (19). And $v33(k+1),\u2009v43(k+1),\u2009v34(k+1)$, and $v44(k+1)$ are updated with $p11(k+1),\u2009p22(k+1),\u2009p33(k+1)$, and $p44(k+1)$ which are the diagonal elements of $P(k+1)$. Note that the inactive elements of *V*(*k*) remain the same as the previous ones. The second resetting method with memory prevents *P*(*k*) from a drastic change at transition, leading to a smooth transition and, consequently, better transient behavior in parameter estimation.

## 4 Algorithm Validation

The ALUT adaptation algorithms, including recursive parameter estimation methods (EFRLS, DFRLS, VFRLS, and AKF), and covariance resetting strategies (without memory and with memory), are applied and numerically demonstrated with adaptation of an engine-out NO_{x} concentration map for CI engines. The proposed algorithms are compared with the sample mean method, a baseline approach available in Ref. [21], to assess their effectiveness and performance.

Figure 5 depicts a modern turbocharged diesel engine and aftertreatment system widely used for heavy-duty applications due to its high performance and efficiency. Due to lean combustion at high temperature and pressure, a diesel engine inevitably emits a significant amount of NO_{x} posing a serious threat to the environment and human health. Therefore, active aftertreatment control systems, such as a urea-selective catalytic reduction (urea-SCR) system, are needed to convert toxic NO_{x} into nontoxic water (H_{2}O), carbon dioxide (CO_{2}), and nitrogen (N_{2}).

Accurate NO_{x} sensing is crucial for urea dosing control in the urea-SCR system [22,23]. However, since NO_{x} sensors are inactive at low temperatures, alternative virtual NO_{x} sensors, like neural network models [24] and polynomial models [25], have been developed. Yet, these lack adaptation due to complexity. A NO_{x} LUT, as shown in Fig. 6, is preferred in production ECUs for its accuracy and computational efficiency. With the 2027 emissions regulations demanding significant NO_{x} reduction during SCR light-off [27], NO_{x} modeling has become more crucial.

An ALUT offers a valuable solution for accommodating system variations over time. Specifically, a NO_{x} ALUT is adaptively updated using a NO_{x} sensor signal in hot conditions and aids urea dosing control during sensor inactivity in cold conditions. Engine speed and torque are assumed available from the ECU. In this study, intentionally a 25% bias from the true map is introduced. The heavy-duty federal test procedure transient cycle (Fig. 7) is simulated to validate the developed algorithms, with multiplicative random noise up to ±0.1% added to the model output *z*(*k*).

respectively. Note that inactive elements of the ALUT, corresponding to operating conditions not visited during the entire driving cycle, are excluded when computing $Em(k)$.

As evidenced by the absolute output interpolation errors in the first rows and mean absolute ALUT estimation errors in the second rows, the baseline fails to converge to the true map. In contrast, EFRLS, DFRLS, VFRLS, and AKF exhibit general convergence trends, indicated by the small absolute output prediction errors in the first rows and decaying mean absolute ALUT errors in the second rows. However, EFRLS becomes sensitive to even small noise when the input is excited again after long covariance windup due to the overwhelmingly large covariance matrix, as seen in the disturbed solid greens in the second rows. All anti-windup parameter estimation methods, including DFRLS, VFRLS, and AKF, demonstrate anti-windup capability, showcasing monotonously decreasing mean absolute ALUT errors in the second rows and bounded covariance matrix in the third rows, leading to significantly improved robustness against noise.

By employing covariance matrix resetting with memory, preventing unnecessary increase in the covariance matrix during transition is shown in the third rows in Figs. 8 and 9, thereby averting potential instability in estimation. The RMS of the prediction errors (upper) and ALUT errors (lower) is given in Fig. 10. The RMS of the ALUT errors is reduced by up to 4.5% by resetting with memory.

## 5 Conclusion

This brief presents an ALUT algorithm integrating recursive parameter estimation and covariance matrix resetting. Four estimation methods (EFRLS, DFRLS, VFRLS, and AKF) are compared against a baseline sample mean method. Two resetting methods are developed: one without memory, resetting to fixed values, and one with memory, recalling diagonal elements corresponding to active ALUT elements in an adaptive variance matrix. The algorithms are simple, computationally efficient, and broadly applicable to adaptive engine control systems. Evaluation involves different estimation and resetting methods, assessing their capacity to adapt NO_{x} emission concentration maps. Numerical validation highlights DFRLS, VFRLS, and AKF as effective anti-windup methods. DFRLS requires lower calibration effort, while VFRLS and AKF demonstrate rapid convergence. Resetting with memory offers faster adaptation but increases memory usage.

## Funding Data

URC Faculty Research Grant at Oakland University.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

## References

_{2}Regulations