Adaptive Kalman Filtering Developed from Recursive Least Squares Forgetting Algorithms

Recursive least squares (RLS) is derived as the recursive minimizer of the least-squares cost function. Moreover, it is well known that RLS is a special case of the Kalman filter. This work presents the Kalman filter least squares (KFLS) cost function, whose recursive minimizer gives the Kalman filter. KFLS is an extension of generalized forgetting recursive least squares (GF-RLS), a general framework which contains various extensions of RLS from the literature as special cases. This then implies that extensions of RLS are also special cases of the Kalman filter. Motivated by this connection, we propose an algorithm that combines extensions of RLS with the Kalman filter, resulting in a new class of adaptive Kalman filters. A numerical example shows that one such adaptive Kalman filter provides improved state estimation for a mass-spring-damper with intermittent, unmodeled collisions. This example suggests that such adaptive Kalman filtering may provide potential benefits for systems with non-classical disturbances.

SIFt-RLS: Subspace of Information Forgetting Recursive Least Squares

This paper presents subspace of information forgetting recursive least squares (SIFt-RLS), a directional forgetting algorithm which, at each step, forgets only in row space of the regressor matrix, or the \textit{information subspace}. As a result, SIFt-RLS tracks parameters that are in excited directions while not changing parameter estimation in unexcited directions. It is shown that SIFt-RLS guarantees an upper and lower bound of the covariance matrix, without assumptions of persistent excitation, and explicit bounds are given. Furthermore, sufficient conditions are given for the uniform Lyapunov stability and global uniform exponential stability of parameter estimation error in SIFt-RLS when estimating fixed parameters without noise. SIFt-RLS is compared to other RLS algorithms from the literature in a numerical example without persistently exciting data.

Convergence of Recursive Least Squares Based Input/Output System Identification with Model Order Mismatch

Discrete-time input/output models, also called infinite impulse response (IIR) models or autoregressive moving average (ARMA) models, are useful for online identification as they can be efficiently updated using recursive least squares (RLS) as new data is collected. Several works have studied the convergence of the input/output model coefficients identified using RLS under the assumption that the order of the identified model is the same as that of the true system. However, the case of model order mismatch is not as well addressed. This work begins by introducing the notion of \textit{equivalence} of input/output models of different orders. Next, this work analyzes online identification of input/output models in the case where the order of the identified model is higher than that of the true system. It is shown that, given persistently exciting data, the higher-order identified model converges to the model equivalent to the true system that minimizes the regularization term of RLS.

Efficient Batch and Recursive Least Squares for Matrix Parameter Estimation with Application to Adaptive MPC

Traditionally, batch least squares (BLS) and recursive least squares (RLS) are used for identification of a vector of parameters that form a linear model. In some situations, however, it is of interest to identify parameters in a matrix structure. In this case, a common approach is to transform the problem into standard vector form using the vectorization (vec) operator and the Kronecker product, known as vec-permutation. However, the use of the Kronecker product introduces extraneous zero terms in the regressor, resulting in unnecessary additional computational and space requirements. This work derives matrix BLS and RLS formulations which, under mild assumptions, minimize the same cost as the vec-permutation approach. This new approach requires less computational complexity and space complexity than vec-permutation in both BLS and RLS identification. It is also shown that persistent excitation guarantees convergence to the true matrix parameters. This method can used to improve computation time in the online identification of multiple-input, multiple-output systems for indirect adaptive model predictive control.

A Data-Driven Autopilot for Fixed-Wing Aircraft Based on Model Predictive Control

Autopilots for fixed-wing aircraft are typically designed based on linearized aerodynamic models consisting of stability and control derivatives obtained from wind-tunnel testing. The resulting local controllers are then pieced together using gain scheduling. For applications in which the aerodynamics are unmodeled, the present paper proposes an autopilot based on predictive cost adaptive control (PCAC). As an indirect adaptive control extension of model predictive control, PCAC uses recursive least squares (RLS) with variable-rate forgetting for online, closed-loop system identification. At each time step, RLS-based system identification updates the coefficients of an input-output model whose order is a hyperparameter specified by the user. For MPC, the receding-horizon optimization can be performed by either the backward-propagating Riccati equation or quadratic programming. The present paper investigates the performance of PCAC for fixed-wing aircraft without the use of any aerodynamic modeling or offline/prior data collection.

Adaptive Real-Time Numerical Differentiation with Variable-Rate Forgetting and Exponential Resetting

Digital PID control requires a differencing operation to implement the D gain. In order to suppress the effects of noisy data, the traditional approach is to filter the data, where the frequency response of the filter is adjusted manually based on the characteristics of the sensor noise. The present paper considers the case where the characteristics of the sensor noise change over time in an unknown way. This problem is addressed by applying adaptive real-time numerical differentiation based on adaptive input and state estimation (AISE). The contribution of this paper is to extend AISE to include variable-rate forgetting with exponential resetting, which allows AISE to more rapidly respond to changing noise characteristics while enforcing the boundedness of the covariance matrix used in recursive least squares.

Kinematics-Based Sensor Fault Detection for Autonomous Vehicles Using Real-Time Numerical Differentiation

Sensor fault detection is of extreme importance for ensuring the safe operation of vehicles. This paper introduces a novel approach to detecting and identifying faulty sensors. For ground vehicles confined to the horizontal plane, this technique is based on six kinematics-based error metrics that are computed in real time by using onboard sensor data encompassing compass, radar, rate gyro, and accelerometer measurements as well as their derivatives. Real-time numerical differentiation is performed by applying the adaptive input and state estimation (AIE/ASE) algorithm. Numerical examples are provided to assess the efficacy of the proposed methodology.

Real-Time Numerical Differentiation of Sampled Data Using Adaptive Input and State Estimation

Real-time numerical differentiation plays a crucial role in many digital control algorithms, such as PID control, which requires numerical differentiation to implement derivative action. This paper addresses the problem of numerical differentiation for real-time implementation with minimal prior information about the signal and noise using adaptive input and state estimation. Adaptive input estimation with adaptive state estimation (AIE/ASE) is based on retrospective cost input estimation, while adaptive state estimation is based on an adaptive Kalman filter in which the input-estimation error covariance and the measurement-noise covariance are updated online. The accuracy of AIE/ASE is compared numerically to several conventional numerical differentiation methods. Finally, AIE/ASE is applied to simulated vehicle position data generated from CarSim.

Generalized Forgetting Recursive Least Squares: Stability and Robustness Guarantees

This work present generalized forgetting recursive least squares (GF-RLS), a generalization of recursive least squares (RLS) that encompasses many extensions of RLS as special cases. First, sufficient conditions are presented for the 1) Lyapunov stability, 2) uniform Lyapunov stability, 3) global asymptotic stability, and 4) global uniform exponential stability of parameter estimation error in GF-RLS when estimating fixed parameters without noise. Second, robustness guarantees are derived for the estimation of time-varying parameters in the presence of measurement noise and regressor noise. These robustness guarantees are presented in terms of global uniform ultimate boundedness of the parameter estimation error. A specialization of this result gives a bound to the asymptotic bias of least squares estimators in the errors-in-variables problem. Lastly, a survey is presented to show how GF-RLS can be used to analyze various extensions of RLS from the literature.

Experimental Flight Testing of an Adaptive Autopilot with Parameter Drift Mitigation

This paper modifies an adaptive multicopter autopilot to mitigate instabilities caused by adaptive parameter drift and presents simulation and experimental results to validate the modified autopilot. The modified adaptive controller is obtained by including a static nonlinearity in the adaptive loop, updated by the retrospective cost adaptive control algorithm. It is shown in simulation and physical test experiments that the adaptive autopilot with proposed modifications can continually improve the fixed-gain autopilot as well as prevent the drift of the adaptive parameters, thus improving the robustness of the adaptive autopilot.