Nonlinearity modeling for online estimation of industrial cooling fan speed subject to model uncertainties and state-dependent measurement noise

3.1 EKF

The EKF is a commonly used technique for estimating the state of a nonlinear dynamic system [20]. It extends the traditional KF to handle nonlinearity by linearizing the system dynamics with the Taylor series expansion.

Consider a nonlinear stochastic process model defined by a state transition model and an observation model

where x k ∈ ℝ n represents the state vector at time step k and u k ∈ ℝ m is the system input. Here, f ( ⋅ ) ∈ ℝ n and h ( ⋅ ) ∈ ℝ l represent the nonlinear transition and measurement functions, respectively. The process and measurement noise w k ∈ ℝ n and v k ∈ ℝ l are also assumed to be independent of each other, with zero-mean Gaussian distributions:

where Q k and R k are covariance matrices signifying system model uncertainty and observation model measurement noise, respectively. The zero-mean Gaussian distribution assumptions are valid in this research topic based on the prior work [14].

In the EKF, these nonlinear functions are approximated through linearization around the current state estimate, enabling the application of KF principles. The EKF algorithm can be summarized as follows:

where F k and H k are the Jacobian matrices defined as

In the prediction step, the algorithm estimates the next state x ˆ k and updates the associated uncertainty P k using the nonlinear transition function f ( ⋅ ) . The correction step refines the prediction by assimilating measurement information z k , adjusting the state estimate based on the Kalman gain K k , and refining the uncertainty estimate. This EKF algorithm provides a systematic approach to handling nonlinearities, extending the KF’s utility in estimating states for dynamic systems.

The EKF utilizes Taylor series expansion to approximate the KF, replacing system matrices with Jacobians. Notably, the EKF is suitable for differentiable models, as it relies on the existence of Jacobians. However, this approach is limited, as it may not work well with non-differentiable or highly nonlinear models, where Jacobians are hard to compute or involve high computational costs. In such cases, linearization may result in larger errors in mean and covariance, potentially leading to divergence of estimates. To better address these issues, UT and UKF will be introduced in the following section.

3.2 UKF

For state estimation of nonlinear models, EKF merely uses one single point to approximate the Gaussian distribution. If the function is highly nonlinear in this local region, linearization may lead to poor estimates. The UKF handles this problem with UT, which uses a set of sample points to describe the Gaussian model and propagates them through the nonlinear system [26].

To utilize the UKF, the weights of the sigma points need to be determined in advance. For every new-coming measurement z k , based on the given control input u k and the previous state x ˆ k − 1 and covariance P k − 1 , the sigma points are then generated. Next, similar to the KF and the EKF algorithms, the UKF has prediction and correction steps. In the prediction step, the sigma points are propagated through the transition model.

and the predicted mean (or the prior estimate of state) x ˆ k − and covariance P k − can be calculated by the propagated sigma points with their weights

In the correction step, the sigma points are transformed through the observation model,

then the predicted measurement mean and covariance can be calculated as

Note that the predicted covariance Eq. (15) and predicted measurement covariance Eq. (17) are made up of the uncertainty of the UT and the process noise Q k and measurement noise R k , respectively.

The cross-covariance matrix between the state space and measurement space is given as

Based on the definition of the error covariance matrix, P z , k can be expressed as

Similarly, P x z , k can be written as

Recalling the Kalman gain in EKF,

one can find out that Eqs. (19) and (20) compose the elements in Eq. (21), therefore yielding the Kalman gain as

The state update equation is given as

Substituting Eqs. (20) and (22) into the covariance matrix update in EKF gives the covariance matrix update of UKF

The algorithm of the UKF is summarized as the pseudocode in Algorithm 2.

In summary, the EKF is suitable for systems with mild non-linearity, while the UKF is more robust and suitable for systems with strong non-linearity. The UKF introduces sigma points to not only directly reflect the system evolution but also comprehensively capture system characteristics. Therefore, when dealing with nonlinear systems, the UKF holds a distinct advantage over the EKF, allowing for more accurate estimation by better capturing the characteristics of system nonlinearities. Therefore, this work integrates the UKF to deal with the modeling nonlinearity in the industrial cooling fan and also proposes a state-dependent measurement covariance update mechanism to enhance the fan speed estimation online.

4.1 Measurement noise modeling

In the previous sections, the observation disturbance is assumed to be homoscedastic, that is, the variance remains constant throughout the rotation speed, as shown in Figure 4(a). However, based on the practical experiment observations, it is revealed that the amount of noise becomes larger with a higher fan speed. In this case, the noise of the cooling fan should be assumed to be heteroscedastic, which means that the variance depends on the value of the speed. Figure 4(b) demonstrates that as the value becomes larger, the data points are more widely distributed due to its larger variance, showing the heteroscedasticity of the noise.

Figure 4

Noise behavior statistical analysis. (a) Homoscedasticity. (b) Heteroscedasticity.

Assume that the standard deviation is a function of speed

where a = a 1 a 2 T are the parameters to be determined, and σ k is the standard deviation at time step k .

The fitting problem is fitted into

Monitoring different values of the fan speed, collecting the measured data, and calculating the standard deviations yield

The parameter vector a can be solved using the well-known least-squares solution. Finally, the modulation of R can be established by

with a time-varying noise covariance R k ; the degree of belief in measurements can be adjusted and give a better approximation.

4.2 Cooling fan modeling error discussion

In a recent study on the estimation of cooling fan parameters [13], the real-time parameter estimation results show that for every stair change in the given input, the estimated values exhibit sharp fluctuations, which mainly results from the error of modeling. Recalling the cooling fan model Eq. (9), the transient response of the electrical dynamics is neglected since its time constant L / R is much smaller in magnitude than the mechanical time constant J m / B m . If the electrical dynamics is reconsidered, the overall real system can be described as a second-order system:

The system in Eqs. (40) and (41) become relatively complex and, unfortunately, the information of current i is not available for general industrial cooling fan systems. Thus, parameter identification for Eq. (40) cannot be conducted.

As a result, the second-order system is reduced by a first-order nonlinear dynamic equation, where the equivalent equation is expressed as Eq. (7). Note that the equivalent parameters Eq. (8) are based on the fact that the driver dynamics is completely ignored. The model reduction process is described in Section 2.

Since only the fan speed output is available, this work attempts to use Figure 5(b) to replace Figure 5(a) as a reduced-order model for equivalent parameter identification. Note that the equivalent first-order model reduction assumes that the electrical dynamics is much faster than the mechanical fan dynamics. This assumption is reasonable and is going to be illustrated by the following simulation and experimental comparison.

Figure 5

Cooling fan system representation and model reduction. (a) Electrical-circuit-derived second-order fan dynamics. (b) First-order equivalent cooling fan model reduction equivalent representation.

To demonstrate the model mismatch behavior caused by the electrical dynamics, L = 0.01 , R = 1 , K = 1 , J m = 0.15 , B m = 0.012 , and C d = 0.0025 . Based on the output speed, Eq. (7) was further considered to approximate the responses Eqs. (40) and (41), the associated estimated equivalent parameters are J = 0.1635 , α = 1.0106 , and C D = 0.0025 , respectively. The corresponding simulation is shown in Figure 6(a).

Figure 6

Cooling fan system transient speed response comparison between the electrical-circuit-derived second-order fan dynamics and the reduced order cooling fan representation. (a) Simulation observation. (b) Experimental validation.

Based on the given stair-liked excitation input signal, the simulation results considering Eqs. (40) and (41) show that for an abrupt change in the applied input, the speed response possesses a relatively smooth rising speed within a short period of time. However, the simulation result using Eq. (7) illustrates a sharp rising speed. This slight difference comes from the model reduction or the so-called modeling errors. The qualitative behavior can also be observed from the experiment, as illustrated in Figure 6(b), which further validates the assumption. Consequently, the mismatches between the real system and the equivalent system leads to fluctuation of estimates whenever the applied input changes abruptly.

In conclusion, the main reason this work considered a model reduction is that the current output, generated from Eq. (40), for the general industrial fans in the market is not available. To avoid the use of the current information for cooling fan modeling and fan speed prediction, this work applied a modeling reduction. Figure 6(a) and (b) verify the reasonability of this consideration.

To address this issue and further amend the estimated biases identified in the UKF, the parameter update method with UKF is introduced to refine the speed estimates and to pursue unbiased estimates, and the equivalent system Eq. (7) and the proposed parameter update mechanism are used to make up for the model discrepancy and the parameter identification error.

4.3 UKF with parameter update mechanism

Recalling the cooling fan model in Eqs. (25) and (26), through diverse scenarios encompassing different environments, time frames, and operational conditions, it has been observed from experience that the model parameters of the fan system may undergo slight perturbations, suggesting the potential for time-varying characteristics.

Consequently, we attribute the effects of model uncertainty or disturbances in the fan to the variation of system parameters. Based on this assumption, an online parameter estimation mechanism to enhance unbiased estimation of fan speed is conducted.

Consider the formulation of these time-varying parameters as follows.

where a ˆ , a ˆ N , and b ˆ are the estimated values obtained from the identification results by the low-pass filtering method [13] and a ˜ ( t ) , a ˜ N ( t ) , and b ˜ ( t ) are the time-varying parameter uncertainties, describing the effects of the modeling errors and external perturbations. The main strategy in this section is to consider the parameter uncertainties as variables, whose change rates a ˜ ̇ , a ˜ ̇ N , and b ˜ ̇ are assumed to be Gaussian distributions with zero means and standard deviations σ a , σ a N , and σ b , respectively. That is, the dynamics of the parameter uncertainties satisfy

Thus, the overall system can be described as follows:

1. Process model:

   (44) ω ̇ ( t ) = − [ a ˆ + a ˜ ( t ) ] ω ( t ) − [ a ˆ N + a ˜ N ( t ) ] ω 2 ( t ) + [ b ˆ + b ˜ ( t ) ] [ τ ( t ) + τ d ( t ) ] , a ˜ ̇ ( t ) = w a ( t ) , a ˜ ̇ N ( t ) = w a N ( t ) , b ˜ ̇ ( t ) = w b ( t ) .

2. Observation equation:

   (45) z ( t ) = ω ( t ) + v ( t ) ,

   of which ω , a ˜ , a ˜ N , and b ˜ are the states of the system, and τ d , w a , w a N , and w b are the perturbations.

   Similarly, to utilize the UKF algorithm, the discrete-time model representation of Eq. (44) is derived through the forward difference, given by

   (46) ω k + 1 = [ 1 − ( a ˆ + a ˜ k ) T ] ω k − ( a ˆ N + a ˜ N , k ) T ω k 2 + ( b ˆ + b ˜ k ) T ( τ k + τ d , k ) a ˜ k + 1 = a ˜ k + T w a , k a ˜ N , k + 1 = a ˜ N , k + T w a N , k b ˜ k + 1 = b ˜ k + T w b , k ,

   and

   (47) z k = ω k + v k .

   Concerning the discrete-time model Eq. (46), the actual change rates of uncertainties are unknown in the real world. Fortunately, based on practical experimental statistics, parameter uncertainties can be assumed to be slowly time-varying. Therefore, the practical framework of the UKF for online realization can be modeled as the following transition equation.

3. State transition equation:

   (48) ω k + 1 = [ 1 − ( a ˆ + a ˜ k ) T ] ω k − ( a ˆ N + a ˜ N , k ) T ω k 2 + ( b ˆ + b ˜ k ) T τ k , a ˜ k + 1 = a ˜ k , a ˜ N , k + 1 = a ˜ N , k , b ˜ k + 1 = b ˜ k ,

   where the initial condition of the parameters can be obtained from the estimation algorithms [13,14].

4. Observation equation:

   (49) z k = ω k .

5. The state vector, the process, and measurement noise are expressed as follows:

and

Note that τ d , k , w a , k , w a N , k , and w b , k are assumed to be statistically independent.

The simulation condition is mainly given the same as in the previous instance, where the simulation signal is obtained by the nominal parameters given by J = 0.15 , α = 0.012 , and C D = 0.0025 (i.e. a = 0.08 , a N = 0.0167 , and b = 6.6667 ). Based on the studies of Chen and Peng, Li et al., and Peng et al. [32,33,34], the parameters utilized in UKF are assumed to have 10% uncertainty. That is, a ˆ = 0.9 a , a ˆ N = 0.9 a N , and b ˆ = 0.9 b , and their variances σ a 2 = 0.1 a , σ a N 2 = 0.1 a N , and σ b 2 = 0.1 b are applied to construct Q k in Eq. (51). For the comparison of estimation performance, the covariance values for the standard UKF are set to Q = 2 and R = 2 .

Figure 7 shows the speed estimating results for the general UKF and the proposed method, which evidently shows that with the parameter update mechanism, the biased estimate issue from the parameter uncertainty is eliminated, and the effect of noise is attenuated as well. Figure 8 provides the numerical errors between the estimation results and the ground truth. One can observe that the error of the traditional UKF is not zero-mean and is noisier, while the proposed mechanism alleviates the bias phenomenon and efficiently filters out the noise. The RMS errors are quantified as 1.3393 for the traditional UKF and significantly reduced to 0.7485 for the UKF with the parameter update mechanism.

Figure 7

Speed estimate by UKF with the parameter update mechanism.

Figure 8

Speed estimation error by UKF with the parameter update mechanism.

Based on simulation results, the conclusion is drawn that under the influence of time-varying uncertainties in perturbation, the traditional UKF, while adjustable through tuning Q to modify estimation performance, still exhibits inherent biases in state estimation regardless of the adjustments made. In comparison, the UKF with a parameter update mechanism offers a more versatile solution, preventing biases in state estimation and effectively reducing the interference of measurement noise.

5.1 Experiments on R modulation

In preparation for applying UKF algorithms to the cooling fans, the modulation of the covariance R must be carried out first. As hypothesized, the variance R is a function of fan speed as described by Eq. (39). To determine the relationship between R and the speed, a series of different speed values should be collected, then the standard deviations at different speed levels can be calculated. However, the data collection is memory-requested. Aiming at this issue, the recursive mean and recursive standard deviation are applied [35].

then σ k = V a r k .

Applying a fixed step input, the speed will reach a steady state after 3 s, and the recursions start at this point. The embedded system only needs to store the final value of the recursion and repeat this process for several different levels of fixed input voltages. The relationship between R and the speed can therefore be obtained by Eq. (38) using a considerably smaller memory size. Figure 10 demonstrates the recursion process at PWM = 80%, and the overall statistical data are recorded in Table 1.

Figure 10

Recursive mean and recursive standard deviation.

Based on the assumption of heteroscedasticity and the prescribed formulation, the parameters describing the relationship between the speed and standard deviation can be derived by the least-squares solution, as shown in Figure 11.

Figure 11

Fitting line for speed and standard deviation of fan 1 with a 1 = 0.0039 and a 2 = 0.0045 .

5.2 Experiments on UKF with parameter update mechanism

The experiments on speed estimation were conducted using two input scenarios: (i) a stair input PWM (Figure 12) and (ii) manual modulation using knob 1 (Figures 13 and 14). In the case of the traditional UKF, a tradeoff arises where a smoother speed requires higher confidence in the mathematical model. However, inherent model and parameter uncertainties in the system lead to a biased speed estimate, as depicted in Figure 12(b) and (c). Conversely, prioritizing an unbiased estimate places greater emphasis on trusting the measurements, which can lead to a decrease in filter effectiveness. As seen in Figure 13(b), the traditional UKF estimate (red line) closely aligns with the measured value (gray line).

Figure 12

Speed estimates by UKF and UKF with parameter update for stair input. (a) Speed estimation comparison. (b) Responses – partial enlargement. (c) Speed estimation errors.

Figure 13

Speed estimates by UKF and UKF with parameter update for manual modulation input. (a) Speed estimation comparison. (b) Responses – partial enlargement. (c) Speed estimation errors.

Figure 14

Manual modulation of fan input commands.

In contrast, the proposed UKF with a parameter update mechanism demonstrates the ability to achieve an unbiased estimate while simultaneously filtering out undesirable noise for both types of input PWM. Figure 13(c) illustrates the corresponding speed estimation errors. Furthermore, Figures 12(c) and 13(c) highlight the proposed method’s capability to mitigate modeling errors that arise from neglecting the electrical driver model. A comparative analysis between Figures 12(b) and 13(b) leads to the conclusion that, for the traditional UKF, the adjustment of Q and R can be a cumbersome process. An undersized Q and R ratio results in biased estimation states, while an oversized Q and R ratio causes the estimated states to resemble noisy signals. In contrast, the proposed UKF with a parameter update mechanism adeptly avoids such issues.