System Identification Using Kalman Filters

Abstract

The present study focuses on Model Order Reduction (MOR) methods of non-intrusive nature that can be seen as belonging to the category of system identification techniques. Indeed, whereas the system to analyze is considered as a black box, the accurate modeling of the relationship between its input and output is the aim of the proposed techniques. In this framework, the paper deals with two different methodologies for the system identification of thermal problems. The first identifies a linear thermal system by means of an Extended Kalman Filter (EKF). The approach starts from an a priori guessed analytical model whose expression is assumed to describe appropriately the response of the system to identify. Then, the EKF is used for estimating the model transient states and parameters. However, this methodology is not extended to the processing of nonlinear systems due to the difficulty related to the analytical model construction step. Therefore, a second approach is presented, based on an Unscented Kalman Filter (UKF). Finally, a Finite Element (FE) model is used as a reference, and the good agreement between the FE results and the responses produced by the EKF and UKF methods in the linear case fully illustrate their interest.

Appendices

Model Construction Step Using EKF

The solution of Eq. (55.3) on the time interval t _i  t _f is given by [1]:

$$\displaystyle{ x_{r}(t_{f}) = x_{r}(t_{i})\,{e}^{\tilde{A}\,(t_{f}-t_{i})} +\int \limits _{ t_{i}}^{t_{f} }{e}^{\tilde{A}\,(t_{f}-\tau )}\tilde{B}\,u(\tau )\,d\tau }$$

(55.7)

where the exponential matrix is defined as ${e}^{\tilde{A}t} =\sum \limits _{ k=0}^{\infty }\frac{1} {k!}{(\tilde{A}t)}^{k}$ With t _i  = t _k and $t_{f} = t_{k+1}$, (55.7) becomes:

$$\displaystyle{ x_{r}(t_{k+1}) = x_{r}(t_{k})\,{e}^{\tilde{A}\,(t_{k+1}-t_{k})} +\int \limits _{ t_{k}}^{t_{k+1} }{e}^{\tilde{A}\,(t_{k+1}-\tau )}\tilde{B}\,u(\tau )\,d\tau }$$

(55.8)

Simplifying the notation by writing k instead of t _k and supposing u(t) constant over the sampling interval t _k  t _k + 1 , the discrete state space model is written as follows:

$$\displaystyle{ \left.\begin{array}{l} x_{r_{k}} =\tilde{ A}_{d}x_{r_{k-1}} +\tilde{ B}_{d}u_{k-1} \\ y_{r_{k}} =\tilde{ C}_{d}x_{r_{k}}\\ \end{array} \right \}\,\tilde{A}_{d} = {e}^{\tilde{A}\,T}\,\,;\,\,\tilde{B}_{ d} =\int \limits _{ t_{k}}^{t_{k+1} }{e}^{\tilde{A}\,(t_{k+1}-\tau )}\tilde{B}d\tau \, =\int \limits _{ 0}^{T}{e}^{\tilde{A}\tau \,}Bd\tau \,;\,\,\tilde{C}_{ d} =\tilde{ C} }$$

(55.9)

where $x_{r_{k}}$ is the state vector of internal variables at time k, y _k the observation vector at time k, u _k − 1 the input data at time k − 1, and $(\tilde{A}_{d}\,\tilde{B}_{d}\,\tilde{C}_{d})$ the constitutive matrices of the discrete reduced-order model: $\tilde{A}_{d} = \left [\begin{array}{*{20}c} {e}^{a_{1}T}&& \\ &\ddots & \\ &&{e}^{a_{n_{r}}T}\\ \end{array} \right ]$; $\tilde{B}_{d} = ({e}^{\tilde{A}_{d}T}-I)\left [\begin{array}{l} b_{1}\\ \vdots \\ b_{n_{r}}\\ \end{array} \right ] = \left [\begin{array}{l} \frac{b_{1}} {a_{1}} ({e}^{a_{1}T} - 1)\\ \vdots \\ \frac{b_{n_{r}}} {a_{n_{r}}}({e}^{a_{n_{r}}T} - 1) \\ \end{array} \right ]$; $\tilde{C}_{d} =\tilde{ C} = \left [\begin{array}{*{20}c} c_{11} & \cdots & c_{1n_{r}}\\ \vdots & \ddots & \vdots \\ c_{n_{r}1} & \cdots &c_{n_{r}n_{r}}\\ \end{array} \right ]$.

The objective of our procedure being the identification of parameters, they have to be included in the state vector. The functions $\tilde{A}_{d}$ and $\tilde{C}_{d}$ are thereby nonlinear and will be denoted $\tilde{f}_{d}$ and $\tilde{h}_{d}$, respectively. The discrete model is then given by:

$$\displaystyle{ \left \{\begin{array}{lcl} x_{k}& =&\left [\begin{array}{l} x_{r_{k}} \\ \theta _{k}\\ \end{array} \right ] = \left [\begin{array}{l} \tilde{f}_{d_{k}}\left (x_{r_{k-1}},u_{k-1},\theta _{k-1}\right )\\ \end{array} \right ] \\ y_{k} & =&\tilde{h}_{d}(x_{r_{k}},\theta _{k-1})\\ \end{array} \right. }$$

(55.10)

EKF and UKF Algorithms

55.2.1 Extended Kalman Filter (EKF)

Extended Kalman Filter algorithm

Description:

• 1: Initialization:

  State mean and covarianc at k = 0: $\hat{x}_{0} = E\left [x_{0}\right ]$ and $P_{0} = E\left [(x_{0} -\hat{ x}_{0}){(X_{0} -\hat{ x}_{0})}^{T}\right ]$

• 2: Prediction phase

  • (a) The process model Jacobian: $F_{k} = \dfrac{\partial f_{k}} {\partial x} _{x=\hat{x}_{k-1}}$

  • (b) Predicted state mean and covariance: $\hat{x}_{k}^{-} = f_{k}(\hat{x}_{k-1},u_{k-1})$ and $P_{k}^{-} = F_{k}P_{k}F_{k}^{T} + Q$

• 3: Correction phase

  • (a) Measurement model Jacobian: $H_{k} = \dfrac{\partial h_{k}} {\partial x} _{x=\hat{x}_{k}^{-}}$

  • (b) Measurement update:

    • Measurement prediction: $\hat{y}_{k}= h_{k}\left (\hat{x}_{k}^{-}\right )$

    • Innovation (Residual term): $\tilde{y}_{k} = y_{k} -\hat{ y}_{k}$

    • Innovation covariance matrix: $M_{k} = \mathop{cov}\left (\tilde{y}_{k}\right )\,=\,H_{k}P_{k}^{-}H_{k}^{T} + R$

  • (c) Updated state mean and Covariance:

    • Kalman Gain matrix: $K_{k} = P_{k}^{-}H_{k}^{T}M_{k}^{-1}$

    • State update: $\hat{x}_{k} =\hat{ x}_{k}^{-} + K_{k}\tilde{y}_{k}$

    • Covariance update: $P_{k} = \left (I - K_{k}H_{k}\right )P_{k}^{-}$

55.2.2 Unscented Transform (UT)

Unscented Transform

Let x ∈ ℝ ⁿ be a Gaussian random vector and y = g(x) a general nonlinear function, g : ℝ ⁿ  → ℝ ^m; $y = g(x);\,\,E\left [x\right ] =\bar{ x};\,E[(x -\bar{ x})$ ${(x -\bar{ x})}^{T}]\,=\,P_{xx}$

1: Decomposition of the distribution in 2n + 1 sigma-points $\{\chi _{i,\,\omega _{i}}\}_{i=0\,\ldots \,2n} = UT(\bar{x},P_{xx})$ where $\chi _{0} =\bar{ x}\ \ \ ;\ \ \ \omega _{0} = \frac{\kappa } {(n+\kappa )}$ $\left.\chi _{i} =\bar{ x} + [\sqrt{(n+\kappa )P_{xx}}]\ \ \ ;\ \ \ \omega _{i} = \frac{1} {2(n+\kappa )} \chi _{i+n} =\bar{ x} - [\sqrt{(n+\kappa )P_{xx}}]\ \ \ ;\ \ \ \omega _{i+n} = \frac{1} {2(n+\kappa )}\right \}\,i = 1\,\ldots \,n$ N.B. The term $\left [\sqrt{(n+\kappa )P_{xx}}\right ]_{i}$ represents the ith column vector of the matrix square root (n + κ)P _xx and is derived via the Cholesky factorisation. The parameter κ is a scaling parameter and ω _i an associated weight of each sigma-point.

55.2.3 Unscented Kalman Filter (UKF)

Unscented Kalman Filter algorithm

Description:

• 1: Initialization:

  State mean and covarianc at k = 0: $\hat{x}_{0} = E\left [x_{0}\right ]$ and $P_{0} = E\left [(x_{0} -\hat{ x}_{0}){(x_{0} -\hat{ x}_{0})}^{T}\right ]$

• 2: Prediction phase

  • (a) Generation of 2n + 1 sigma-points $\{\chi _{_{i},k-1},\omega _{i}\}_{i=0\,\ldots \,2n} = UT(\hat{x}_{k-1},P_{x_{k-1}})$

  • (b) Predicted state: $\chi _{_{i},k}^{-} = f_{k}(\chi _{i,k-1},u_{k-1})$ and $\hat{x}_{_{k}}^{-} =\sum \limits _{ i=0}^{2n}\omega _{i}\chi _{_{i},k}^{-}$

  • (c) Predicted covariance: $P_{x_{k}}^{-} =\sum \limits _{ i=0}^{2n}\omega _{i}(\chi _{_{i},k}^{-}-\hat{ x}_{_{k}}^{-}){(\chi _{_{i},k}^{-}-\hat{ x}_{_{k}}^{-})}^{T} + Q$

• 3: Correction phase

  • (a) Measurement update: $Y _{i,k} = h_{k}(\chi _{_{i,k}}^{-})$

  • (b) Measurement prediction: $\hat{y_{k}} =\sum \limits _{ i=0}^{2n}\omega _{i}Y _{_{i},k}$

  • (c) Innovation (Residual term): $\tilde{y}_{k} = Y _{i,k} -\hat{ y_{k}}$

  • (d) Innovation covariance: $P_{y_{k}} =\sum \limits _{ i=0}^{2n}\omega _{i}\tilde{y}_{k}\tilde{y}_{k}^{T} + R$

  • (e) Cross covariance: $P_{x_{k}y_{k}} =\sum \limits _{ i=0}^{2n}\omega _{i}(\chi _{_{i,k}}^{-}-\hat{ x}_{_{k}}^{-}){(Y _{i,k} -\hat{ y_{k}})}^{T} + R$

  • (f) Updated state mean and Covariance:

    • Kalman Gain matrix: $K_{k} = P_{x_{k}y_{k}}P_{_{y_{ k}}}^{-1}$

    • State update: $\hat{x}_{k} =\hat{ x}_{k}^{-} + K_{k}\tilde{y}_{k}$

    • Covariance update: $P_{x_{k}} = P_{x_{k}}^{-}- K_{k}P_{y_{k}}K_{k}^{T}$