Recursive Engine In-Cylinder Pressure Reconstruction Using Sensor-Fused Engine Speed

6.1. Calculated Speed-Based Cylinder Pressure Reconstruction

Step 1:

Use system identification approaches to identify the model between cylinder pressure and vibration signal, and denote the identified model as $\widehat{\mathit{G}}$. The model $\widehat{\mathit{G}}$ has four inputs (i.e., four cylinder pressure signals $P_m\left(k\right)\in \mathbb{R}$, $m=1,2,3,4$) and one output (i.e., the vibration signal $y\left(k\right)$). The state-space representation of the model $\widehat{\mathit{G}}$ is as follows:

$$\left\{\begin{array}{cc}\hfill \mathit{x}(k+1)& =\mathit{A}\mathit{x}\left(k\right)+\mathit{B}\mathit{u}\left(k\right),\hfill \\ \hfill y\left(k\right)& =\mathit{C}\mathit{x}\left(k\right)+\mathit{D}\mathit{u}\left(k\right)+v_{\mathrm{g}}\left(k\right),\hfill \end{array}\right.$$

(27)

where $\mathit{x}\left(k\right)\in {\mathbb{R}}^n$ denotes the state vector, the matrices $\mathit{A}$, $\mathit{B}$, $\mathit{C}$, and $\mathit{D}$ denote the state matrix, the input matrix, the output matrix, and the feedthrough matrix, respectively, the term $v_{\mathrm{g}}\left(k\right)$ denotes the output error, and the input $\mathit{u}\left(k\right)\in {\mathbb{R}}^4$ is represented as follows:

$$\mathit{u}\left(k\right)=\left(\begin{array}{c}{P}_1\left(k\right)\\ P_2\left(k\right)\\ P_3\left(k\right)\\ P_4\left(k\right)\end{array}\right).$$

(28)

Step 2:

Under stationary engine operating conditions, similar to the modeling of the vibration signal in Equation (5), the cylinder pressure signal $P_1\left(k\right)$ can be expressed as the output of the following state-space model:

$$\left\{\begin{array}{cc}\hfill {\mathit{x}}_{\mathrm{p}}(k+1)& ={\mathit{A}}_{\mathrm{p}}\left(f\left(k\right)\right){\mathit{x}}_{\mathrm{p}}\left(k\right),\hfill \\ \hfill P_1\left(k\right)& ={\mathit{C}}_{\mathrm{p}}{\mathit{x}}_{\mathrm{p}}\left(k\right)+v_{\mathrm{p}}\left(k\right),\hfill \end{array}\right.$$

(29)

where the vector ${\mathit{x}}_{\mathrm{p}}\left(k\right)\in {\mathbb{R}}^{n_{\mathrm{p}}}$ denotes the state vector, $v_{\mathrm{p}}\left(k\right)$ represents the output error, the value of $v_{\mathrm{p}}\left(k\right)$ should be guaranteed to be a small value, and the matrix ${\mathit{A}}_{\mathrm{p}}\left(f\left(k\right)\right)$ and the matrix ${\mathit{C}}_{\mathrm{p}}$ denote the state matrix and the output matrix, respectively.

Step 3:

As displayed in Figure 5, D1, D2, and D3 denote the symbols of three single-input, single-output delay blocks. The model of the each delay block can be expressed as follows:

$$\left\{\begin{array}{cc}\hfill {\mathit{x}}_{\mathrm{d}}(k+1)& ={\mathit{G}}_{\mathrm{d}}\left(f\left(k\right)\right){\mathit{x}}_{\mathrm{d}}\left(k\right)+{\mathit{H}}_{\mathrm{d}}\left(f\left(k\right)\right)u_{\mathrm{d}}\left(k\right),\hfill \\ \hfill y_{\mathrm{d}}\left(k\right)& ={\mathit{C}}_{\mathrm{d}}\left(f\left(k\right)\right){\mathit{x}}_{\mathrm{d}}\left(k\right)+v_{\mathrm{d}}\left(k\right),\hfill \end{array}\right.$$

(30)

where the vectors ${\mathit{x}}_{\mathrm{d}}\left(k\right)\in {\mathbb{R}}^{n_{\mathrm{d}}}$, $u_{\mathrm{d}}\left(k\right)\in \mathbb{R}$, and $y_{\mathrm{d}}\left(k\right)\in \mathbb{R}$ denote the state, input, and output signal of the delay system, respectively. The matrices ${\mathit{G}}_{\mathrm{d}}\left(f\left(k\right)\right)$, ${\mathit{H}}_{\mathrm{d}}\left(f\left(k\right)\right)$, and ${\mathit{C}}_{\mathrm{d}}\left(f\left(k\right)\right)$ denote the state matrix, the input matrix, and the output matrix, respectively.

We denote the conceptual time-varying transfer operator of each delay block as $G_{\mathrm{d}}(q^{-1},f\left(k\right))$ with q the forward shift operator.

Step 4:

As shown in Figure 5, based on the models of three delay blocks, the other cylinder pressure signals can be obtained by knowing the cylinder No. 1 pressure signal; thus, we can obtain a single-input single-output model between the cylinder No. 1 pressure signal and the vibration signal, and the model can be formulated as follows:

$$\left\{\begin{array}{cc}\hfill {\mathit{x}}_{\mathrm{s}}(k+1)& ={\mathit{A}}_{\mathrm{s}}\left(f\left(k\right)\right){\mathit{x}}_{\mathrm{s}}\left(k\right)+{\mathit{B}}_{\mathrm{s}}\left(f\left(k\right)\right)P_1\left(k\right),\hfill \\ \hfill y\left(k\right)& ={\mathit{C}}_{\mathrm{s}}\left(f\left(k\right)\right){\mathit{x}}_{\mathrm{s}}\left(k\right)+{\mathit{D}}_{\mathrm{s}}{P}_1\left(k\right)+v_{\mathrm{s}}\left(k\right),\hfill \end{array}\right.$$

(31)

where ${\mathit{x}}_{\mathrm{s}}\left(k\right)\in {\mathbb{R}}^{n_{\mathrm{s}}}$ denotes the state vector, the term $v_{\mathrm{s}}\left(k\right)$ denotes the output error, and the matrices ${\mathit{A}}_{\mathrm{s}}\left(f\left(k\right)\right)$, ${\mathit{B}}_{\mathrm{s}}\left(f\left(k\right)\right)$, ${\mathit{C}}_{\mathrm{s}}\left(f\left(k\right)\right)$, ${\mathit{D}}_{\mathrm{s}}$, and the state vector ${\mathit{x}}_{\mathrm{s}}\left(k\right)$ are given as follows:

$${\mathit{A}}_{\mathrm{s}}\left(f\left(k\right)\right)=\left(\begin{array}{cc}\mathit{A}& \mathit{B}\left(\begin{array}{ccc}\mathbf{0}& \mathbf{0}& \mathbf{0}\\ {\mathit{C}}_{\mathrm{d}}\left(f\left(k\right)\right)& \mathbf{0}& \mathbf{0}\\ \mathbf{0}& {\mathit{C}}_{\mathrm{d}}\left(f\left(k\right)\right)& \mathbf{0}\\ \mathbf{0}& \mathbf{0}& {\mathit{C}}_{\mathrm{d}}\left(f\left(k\right)\right)\end{array}\right)\\ \mathbf{0}& \left(\begin{array}{ccc}{\mathit{G}}_{\mathrm{d}}\left(f\left(k\right)\right)& \mathbf{0}& \mathbf{0}\\ {\mathit{H}}_{\mathrm{d}}\left(f\left(k\right)\right){\mathit{C}}_{\mathrm{d}}\left(f\left(k\right)\right)& {\mathit{G}}_{\mathrm{d}}\left(f\left(k\right)\right)& \mathbf{0}\\ \mathbf{0}& {\mathit{H}}_{\mathrm{d}}\left(f\left(k\right)\right){\mathit{C}}_{\mathrm{d}}\left(f\left(k\right)\right)& {\mathit{G}}_{\mathrm{d}}\left(f\left(k\right)\right)\end{array}\right)\end{array}\right),$$

(32)

$${\mathit{B}}_{\mathrm{s}}\left(f\left(k\right)\right)={\left(\begin{array}{ccc}\left(\begin{array}{cccc}1& 0& 0& 0\end{array}\right){\mathit{B}}^{\mathrm{T}}& {\left({\mathit{H}}_{\mathrm{d}}\left(f\left(k\right)\right)\right)}^{\mathrm{T}}& \mathbf{0}\end{array}\right)}^{\mathrm{T}},$$

(33)

$${\mathit{C}}_{\mathrm{s}}\left(f\left(k\right)\right)=\left(\begin{array}{cc}\mathit{C}& \mathit{D}\left(\begin{array}{ccc}\mathbf{0}& \mathbf{0}& \mathbf{0}\\ {\mathit{C}}_{\mathrm{d}}\left(f\left(k\right)\right)& \mathbf{0}& \mathbf{0}\\ \mathbf{0}& {\mathit{C}}_{\mathrm{d}}\left(f\left(k\right)\right)& \mathbf{0}\\ \mathbf{0}& \mathbf{0}& {\mathit{C}}_{\mathrm{d}}\left(f\left(k\right)\right)\end{array}\right)\end{array}\right),$$

(34)

$${\mathit{D}}_{\mathrm{s}}=\mathit{D}\left(\begin{array}{c}1\\ 0\\ 0\\ 0\end{array}\right),$$

(35)

and

$${\mathit{x}}_{\mathrm{s}}\left(k\right)=\left(\begin{array}{c}\mathit{x}\left(k\right)\\ {\mathit{x}}_{\mathrm{d}}\left(k\right)\\ {\mathit{x}}_{\mathrm{d}}\left(k\right)\\ {\mathit{x}}_{\mathrm{d}}\left(k\right)\end{array}\right),$$

(36)

respectively.

Step 5:

By augmenting the state of the model (29) with the state of the model (31), the augmented model ${\mathit{G}}_{\mathrm{a}}$ can be obtained as follows:

$$\left\{\begin{array}{cc}\hfill {\mathit{x}}_{\mathrm{a}}(k+1)& ={\mathit{A}}_{\mathrm{a}}\left(f\left(k\right)\right){\mathit{x}}_{\mathrm{a}}\left(k\right),\hfill \\ \hfill y\left(k\right)& ={\mathit{C}}_{\mathrm{a}}\left(f\left(k\right)\right){\mathit{x}}_{\mathrm{a}}\left(k\right)+v_{\mathrm{a}}\left(k\right),\hfill \end{array}\right.$$

(37)

where ${\mathit{x}}_{\mathrm{a}}\left(k\right)\in {\mathbb{R}}^{n_{\mathrm{a}}}$ represents the state vector. The state matrix ${\mathit{A}}_{\mathrm{a}}\left(f\left(k\right)\right)$, the output matrix ${\mathit{C}}_{\mathrm{a}}\left(f\left(k\right)\right)$, and the state vector can be denoted as follows:

$${\mathit{A}}_{\mathrm{a}}\left(f\left(k\right)\right)=\left(\begin{array}{cc}{\mathit{A}}_{\mathrm{s}}\left(f\left(k\right)\right)& {\mathit{B}}_{\mathrm{s}}\left(f\left(k\right)\right){\mathit{C}}_{\mathrm{p}}\\ \mathbf{0}& {\mathit{A}}_{\mathrm{p}}\left(f\left(k\right)\right)\end{array}\right),$$

(38)

$${\mathit{C}}_{\mathrm{a}}\left(f\left(k\right)\right)=\left(\begin{array}{cc}{\mathit{C}}_{\mathrm{s}}\left(f\left(k\right)\right)& {\mathit{D}}_{\mathrm{s}}{\mathit{C}}_{\mathrm{p}}\end{array}\right),$$

(39)

and

$${\mathit{x}}_{\mathrm{a}}\left(k\right)=\left(\begin{array}{c}{\mathit{x}}_{\mathrm{s}}\left(k\right)\\ {\mathit{x}}_{\mathrm{p}}\left(k\right)\end{array}\right),$$

(40)

respectively.

$\left\{v_{\mathrm{a}}\left(k\right)\right\}$ is assumed to be a scalar white noise process, of which the covariance function is ${\sigma }_{\mathrm{a}}{\delta }_{kj}$, and the value of ${\sigma }_{\mathrm{a}}$ is tunable.

Step 6:

Based on the augmented model (37), the specific algorithm for reconstructing the cylinder No. 1 pressure signal is briefly displayed as a group of the following recursive equations:

$$\begin{array}{cc}{\widehat{\mathit{x}}}_{\mathrm{a}}^{-}\left(k\right)\hfill & ={\mathit{A}}_{\mathrm{a}}\left(f\left(k\right)\right){\widehat{\mathit{x}}}_{\mathrm{a}}(k-1),\hfill \end{array}$$

(41)

$$\begin{array}{cc}{\widehat{y}}^{-}\left(k\right)\hfill & ={\mathit{C}}_{\mathrm{a}}\left(f\left(k\right)\right){\widehat{\mathit{x}}}_{\mathrm{a}}^{-}\left(k\right),\hfill \end{array}$$

(42)

$$\begin{array}{cc}e\left(k\right)\hfill & =y\left(k\right)-{\widehat{y}}^{-}\left(k\right),\hfill \end{array}$$

(43)

$$\begin{array}{cc}{\widehat{\mathit{x}}}_{\mathrm{a}}\left(k\right)\hfill & ={\widehat{\mathit{x}}}_{\mathrm{a}}^{-}\left(k\right)+\mathit{K}\left(k\right)e\left(k\right),\hfill \end{array}$$

(44)

$$\begin{array}{cc}{\widehat{\mathit{x}}}_{\mathrm{p}}\left(k\right)\hfill & \leftarrow {\widehat{\mathit{x}}}_{\mathrm{a}}\left(k\right),\hfill \end{array}$$

(45)

$$\begin{array}{cc}{\widehat{P}}_1\left(k\right)\hfill & ={\mathit{C}}_{\mathrm{p}}{\widehat{\mathit{x}}}_{\mathrm{p}}\left(k\right),\hfill \end{array}$$

(46)

where $\mathit{K}\left(k\right)$ denotes the Kalman filter gain.

It should be noted that under non-stationary operating conditions, a forgetting factor should be involved in the above Kalman filter. In addition to the cylinder No. 1 pressure estimate ${\widehat{P}}_1\left(k\right)$, three other cylinder pressure signals can be simultaneously reconstructed using the model (30) of the delay block.

6.2. Sensor Fusion-Based Cylinder Pressure Reconstruction

Algorithm 3: Sensor-fused engine speed-based cylinder pressure reconstruction.

7.1. Sensor Fusion Results

7.2. Cylinder Pressure Reconstruction Results