A novel methodology to update table in air data system of a high performance fighter aircraft

Abstract

Accurate air data parameters and measurements are critical for the successful flight missions of any modern fighters. Externally mounted vanes and probes measure the local flow angles, pressure, and temperature. These local measurements are to be corrected through a calibration process to obtain the free stream flow angles, pressure, and ambient temperature, etc. The preflight calibration of these sensors are carried out using wind tunnel tests/CFD computations. This paper presents a novel methodology to update air data table of these sensors, post-flight. The Air Data Computer (ADC) of the aircraft discussed in this paper hosts air data tables that are pre-calibrated using the Maximum Likelihood Estimation (MLE) method. However, challenges have been experienced in extending MLE methods for unsteady flights, when dynamic effects prevail. Unsteady conditions during wind up turns are inevitable to perform calibration at High Angles of Attack (AoA) regimes. Hence, an extended Kalman filter based methodology is proposed for calibration and table update. A complete process is tested for an entire flight envelope having 200 and odd flights. The results demonstrate the strength of the technique for air data calibration and table update in ADC.

Appendix 1

1.1 Extended Kalman Filter (EKF) Principle

Considering a generalized n-dimensional nonlinear model of the system, the state vector ${\mathbf{x}} \in {\mathbb{R}}^{n}$ is evolved according to the following discrete-time system model:

$$ {\mathbf{x}}_{k} = {\mathbf{f}}({\mathbf{x}}_{k - 1} ,{\mathbf{u}}_{k} ) + {\mathbf{w}}_{k} , \, $$

where ${\mathbf{w}} \in {\mathbb{R}}^{n}$ is a Gaussian process noise vector with covariance matrix ${\mathbf{Q}} \in {\mathbb{R}}^{n \times n}$.

These measurements are related to the state vector via the observation model as

$$ {\mathbf{z}}_{k} = {\mathbf{h}}({\mathbf{x}}_{k} ) + {\mathbf{v}}_{k} \, $$

where the measurement function ${\mathbf{h}}({\mathbf{x}})$ is a nonlinear function of the system state x and ${\mathbf{v}} \in {\mathbb{R}}^{r}$ represents associated Gaussian or non-Gaussian measurement noise vector with covariance matrix ${\mathbf{R}} \in {\mathbb{R}}^{r \times r}$.

The EKF is based on the analytical Taylor series expansion of the nonlinear systems and observation equations about the current estimated value and then, along with imperfect measurements EKF updates the estimates of the state vector and the covariance matrix. The update is accomplished through the Kalman gain matrix.

The governing equations of EKF are given by:

1.2 Time update step

Predicted state estimate

$$ {\mathbf{x}}_{{{k \mathord{\left/ {\vphantom {k {k - 1}}} \right. \kern-\nulldelimiterspace} {k - 1}}}} = {\mathbf{f}}({\mathbf{x}}_{{{{k - 1} \mathord{\left/ {\vphantom {{k - 1} {k - 1}}} \right. \kern-\nulldelimiterspace} {k - 1}}}} ,{\mathbf{u}}_{k} ) $$

Predicted covariance estimate:

$$ {\mathbf{P}}_{{{k \mathord{\left/ {\vphantom {k {k - 1}}} \right. \kern-\nulldelimiterspace} {k - 1}}}} {\mathbf{ = F}}_{k - 1} {\mathbf{P}}_{{{{k - 1} \mathord{\left/ {\vphantom {{k - 1} {k - 1}}} \right. \kern-\nulldelimiterspace} {k - 1}}}} {\mathbf{F}}_{k - 1}^{T} + {\mathbf{Q}}_{k - 1} $$

1.3 Measurement update step

Innovation covariance:

$$ {\mathbf{S}}_{k} {\mathbf{ = H}}_{k} {\mathbf{P}}_{{{k \mathord{\left/ {\vphantom {k {k - 1}}} \right. \kern-\nulldelimiterspace} {k - 1}}}} {\mathbf{H}}_{k}^{T} + {\mathbf{R}}_{k} $$

Near optimal Kalman gain:

$$ {\mathbf{K}}_{k} {\mathbf{ = P}}_{{{k \mathord{\left/ {\vphantom {k {k - 1}}} \right. \kern-\nulldelimiterspace} {k - 1}}}} {\mathbf{H}}_{k}^{T} {\mathbf{S}}_{k}^{ - 1} $$

Updated state estimate:

$$ {\mathbf{x}}_{{{k \mathord{\left/ {\vphantom {k k}} \right. \kern-\nulldelimiterspace} k}}} {\mathbf{ = x}}_{{{k \mathord{\left/ {\vphantom {k {k - 1}}} \right. \kern-\nulldelimiterspace} {k - 1}}}} + {\mathbf{K}}_{k} ({\mathbf{z}}_{k} - {\mathbf{h}}({\mathbf{x}}_{{{k \mathord{\left/ {\vphantom {k {k - 1}}} \right. \kern-\nulldelimiterspace} {k - 1}}}} )) $$

Updated covariance estimate:

$$ {\mathbf{P}}_{{{k \mathord{\left/ {\vphantom {k k}} \right. \kern-\nulldelimiterspace} k}}} {\mathbf{ = }}({\mathbf{I}} - {\mathbf{K}}_{k} {\mathbf{H}}_{k} ){\mathbf{P}}_{{{k \mathord{\left/ {\vphantom {k {k - 1}}} \right. \kern-\nulldelimiterspace} {k - 1}}}} $$

where the state transition and observation matrices are defined by the following Jacobians:

$$ \left. {{\mathbf{F}}_{k - 1} = \frac{{\partial {\mathbf{f}}}}{{\partial {\mathbf{x}}}}} \right|_{{{\mathbf{x}}_{{{{k - 1} \mathord{\left/ {\vphantom {{k - 1} {k - 1}}} \right. \kern-\nulldelimiterspace} {k - 1}}}} }} \;{\text{and}}\;\left. {{\mathbf{H}}_{k} = \frac{{\partial {\mathbf{h}}}}{{\partial {\mathbf{x}}}}} \right|_{{{\mathbf{x}}_{{{k \mathord{\left/ {\vphantom {k {k - 1}}} \right. \kern-\nulldelimiterspace} {k - 1}}}} }} $$

where ${\mathbf{f}}({\mathbf{x}}_{{{{k - 1} \mathord{\left/ {\vphantom {{k - 1} {k - 1}}} \right. \kern-\nulldelimiterspace} {k - 1}}}} ,{\mathbf{u}}_{k} ) \,$ is nonlinear state transition functions. It describes the transition of state vector x from one time instant to another in response to a control input ${\mathbf{u}} \in {\mathbb{R}}^{p} \,$

Time update step equations are used in the time update (prediction) process to propagate state and error covariance matrix further in time. Predicted values ${\mathbf{x}}_{{{k \mathord{\left/ {\vphantom {k {k - 1}}} \right. \kern-\nulldelimiterspace} {k - 1}}}}$ and ${\mathbf{P}}_{{{k \mathord{\left/ {\vphantom {k {k - 1}}} \right. \kern-\nulldelimiterspace} {k - 1}}}}$ are then passed to the measurement update to calculate Kalman gain ${\mathbf{K}}_{k}$, corrected system state ${\mathbf{x}}_{{{k \mathord{\left/ {\vphantom {k k}} \right. \kern-\nulldelimiterspace} k}}}$ and error covariance ${\mathbf{P}}_{{{k \mathord{\left/ {\vphantom {k k}} \right. \kern-\nulldelimiterspace} k}}}$ in measurement update step.