On the Exponential Stability of Stochastic Perturbed Singular Systems in Mean Square

The approach of Lyapunov functions is one of the most efficient ones for the investigation of the stability of stochastic systems, in particular, of singular stochastic systems. The main objective of the paper is the analysis of the stability of stochastic perturbed singular systems by using Lyapunov techniques under the assumption that the initial conditions are consistent. The uniform exponential stability in mean square and the practical uniform exponential stability in mean square of solutions of stochastic perturbed singular systems based on Lyapunov techniques are investigated. Moreover, we study the problem of stability and stabilization of some classes of stochastic singular systems. Finally, an illustrative example is given to illustrate the effectiveness of the proposed results

The Bohl spectrum for nonautonomous differential equations

We develop the Bohl spectrum for nonautonomous linear differential equation on a half line, which is a spectral concept that lies between the Lyapunov and the Sacker--Sell spectrum. We prove that the Bohl spectrum is given by the union of finitely many intervals, and we show by means of an explicit example that the Bohl spectrum does not coincide with the Sacker--Sell spectrum in general. We demonstrate for this example that any higher-order nonlinear perturbation is exponentially stable, although this not evident from the Sacker--Sell spectrum. We also analyze in detail situations in which the Bohl spectrum is identical to the Sacker-Sell spectrum

Growth conditions for the stability of a class of time-varying perturbed singular systems

summary:In this paper, we investigate the problem of stability of linear time-varying singular systems, which are transferable into a standard canonical form. Sufficient conditions on exponential stability and practical exponential stability of solutions of linear perturbed singular systems are obtained based on generalized Gronwall inequalities and Lyapunov techniques. Moreover, we study the problem of stability and stabilization for some classes of singular systems. Finally, we present a numerical example to validate the effectiveness of the abstract results of this paper

Zero dynamics and stabilization for linear DAEs

We study linear differential-algebraic multi-input multi-output systems which are not necessarily regular and investigate the asymptotic stability of the zero dynamics and stabilizability. To this end, the concepts of autonomous zero dynamics, transmission zeros, right-invertibility, stabilizability in the behavioral sense and detectability in the behavioral sense are introduced and algebraic characterizations are derived. It is then proved, for the class of right-invertible systems with autonomous zero dynamics, that asymptotic stability of the zero dynamics is equivalent to three conditions: stabilizability in the behavioral sense, detectability in the behavioral sense, and the condition that all transmission zeros of the system are in the open left complex half-plane. Furthermore, for the same class, it is shown that we can achieve, by a compatible control in the behavioral sense, that the Lyapunov exponent of the interconnected system equals the Lyapunov exponent of the zero dynamics

Resilient Corner-Based Vehicle Velocity Estimation

© 2017 IEEE. Pirani, M., Hashemi, E., Khajepour, A., Fidan, B., Kasaiezadeh, A., Chen, S.-K., & Litkouhi, B. (2017). Resilient Corner-Based Vehicle Velocity Estimation. IEEE Transactions on Control Systems Technology, 1–11. https://doi.org/10.1109/TCST.2017.2669157This paper presents longitudinal and lateral velocity estimators by considering the effect of the suspension compliance (SC) at each corner (tire) for ground vehicles. The estimators are developed to be resilient to sensor measurement inaccuracies, model and tire parameter uncertainties, switchings in observer gains, and measurement failures. More particularly, the stability of the observer is investigated, and its robustness to road condition uncertainties and sensor noises is analyzed. The sensitivity of the observers' stability and performance to the model parameter changes is discussed. Moreover, the stability of the velocity observers for two cases of arbitrary and stochastic switching gains is investigated. The stochastic stability of the observer in the presence of faulty measurements is also studied, and it is shown that if the probability of a faulty measurement occurring is less than a certain threshold, the observer error dynamics will remain stochastically stable. The performance of the observer and the effect of the SC are validated via several road experiments.Automotive Partnership Canada || Ontario Research Fund || General Motors Co. [grant numbers APCPJ 395996-09 and ORF-RE-04-039

Robust stability of differential-algebraic equations

This paper presents a survey of recent results on the robust stability analysis and the distance to instability for linear time-invariant and time-varying differential-algebraic equations (DAEs). Different stability concepts such as exponential and asymptotic stability are studied and their robustness is analyzed under general as well as restricted sets of real or complex perturbations. Formulas for the distances are presented whenever these are available and the continuity of the distances in terms of the data is discussed. Some open problems and challenges are indicated

Robustness of stability of time-varying index-1 DAEs

We study exponential stability and its robustness for time-varying linear index-1 differential-algebraic equations. The effect of perturbations in the leading coefficient matrix is investigated. An appropriate class of allowable perturbations is introduced. Robustness of exponential stability with respect to a certain class of perturbations is proved in terms of the Bohl exponent and perturbation operator. Finally, a stability radius involving these perturbations is introduced and investigated. In particular, a lower bound for the stability radius is derived. The results are presented by means of illustrative examples

Reliable Vehicle State and Parameter Estimation

Diverse vehicle active safety systems including vehicle electronic stability control (ESC) system, anti-lock braking system (ABS), and traction control system (TCS) are significantly relying on information about the vehicle's states and parameters, as well as the vehicle's surroundings. However, many important states or parameters, such as sideslip angle, tire-road friction coefficient, road gradient and vehicle mass are hard to directly measure, and hence advanced estimation algorithms are needed. Furthermore, enhancements of sensor technologies and the emergence of new concepts such as {\it Internet of Things} and their automotive version, {\it Internet of Vehicles}, facilitate reliable and resilient estimation of vehicle states and road conditions. Consequently, developing a resilient estimation structure to operate with the available sensor data in commercial vehicles and be flexible enough to incorporate new information in future cars is the main objective of this thesis. This thesis presents a reliable corner-based vehicle velocity estimation and a road condition classification algorithm. For vehicle velocity estimation, a combination of vehicle kinematics and the LuGre tire model is introduced in the design of a corner-based velocity observer. Moreover, the observability condition for both cases of time-invariant and parameter varying is studied. The effect of suspension compliance on enhancing the accuracy of the vehicle corner velocity estimation is also investigated and the results are verified via several experimental tests. The performance and the robustness of the proposed corner-based vehicle velocity estimation to model and road condition uncertainties is analyzed. The stability of the observer is discussed, and analytical expressions for the boundedness of the estimation error in the presence of system uncertainties for the case of fixed observer gains are derived. Furthermore, the stability of the observer under arbitrary and stochastic observer gain switching is studied and the performances of the observer for these two switching scenarios are compared. At the end, the sensitivity of the proposed observer to tire parameter variations is analyzed. These analyses are referred to as offline reliability methods. In addition to the off-line reliability analysis, an online reliability measure of the proposed velocity estimation is introduced, using vehicle kinematic relations. Moreover, methods to distinguish measurement faults from estimation faults are presented. Several experimental results are provided to verify the approach. An algorithm for identifying (classifying) road friction is proposed in this thesis. The analytical foundation of this algorithm, which is based on vehicle response to lateral excitation, is introduced and its performance is discussed and compared to previous approaches. The sensitivity of this algorithm to vehicle/tire parameter variations is also studied. At the end, various experimental results consisting of several maneuvers on different road conditions are presented to verify the performance of the algorithm

Full Vehicle State Estimation Using a Holistic Corner-based Approach

Vehicles' active safety systems use different sensors, vehicle states, and actuators, along with an advanced control algorithm, to assist drivers and to maintain the dynamics of a vehicle within a desired safe range in case of instability in vehicle motion. Therefore, recent developments in such vehicle stability control and autonomous driving systems have led to substantial interest in reliable road angle and vehicle states (tire forces and vehicle velocities) estimation. Advances in applications of sensor technologies, sensor fusion, and cooperative estimation in intelligent transportation systems facilitate reliable and robust estimation of vehicle states and road angles. In this direction, developing a flexible and reliable estimation structure at a reasonable cost to operate the available sensor data for the proper functioning of active safety systems in current vehicles is a preeminent objective of the car manufacturers in dealing with the technological changes in the automotive industry. This thesis presents a novel generic integrated tire force and velocity estimation system at each corner to monitor tire capacities and slip condition individually and to address road uncertainty issues in the current model-based vehicle state estimators. Tire force estimators are developed using computationally efficient nonlinear and Kalman-based observers and common measurements in production vehicles. The stability and performance of the time-varying estimators are explored and it is shown that the developed integrated structure is robust to model uncertainties including tire properties, inflation pressure, and effective rolling radius, does not need tire parameters and road friction information, and can transfer from one car to another. The main challenges for velocity estimation are the lack of knowledge of road friction in the model-based methods and accumulated error in kinematic-based approaches. To tackle these issues, the lumped LuGre tire model is integrated with the vehicle kinematics in this research. It is shown that the proposed generic corner-based estimator reduces the number of required tire parameters significantly and does not require knowledge of the road friction. The stability and performance of the time-varying velocity estimators are studied and the sensitivity of the observers' stability to the model parameter changes is discussed. The proposed velocity estimators are validated in simulations and road experiments with two vehicles in several maneuvers with various driveline configurations on roads with different friction conditions. The simulation and experimental results substantiate the accuracy and robustness of the state estimators for even harsh maneuvers on surfaces with varying friction. A corner-based lateral state estimation is also developed for conventional cars application independent of the wheel torques. This approach utilizes variable weighted axles' estimates and high slip detection modules to deal with uncertainties associated with longitudinal forces in large steering. Therefore, the output of the lateral estimator is not altered by the longitudinal force effect and its performance is not compromised. A method for road classification is also investigated utilizing the vehicle lateral response in diverse maneuvers. Moreover, the designed estimation structure is shown to work with various driveline configurations such as front, rear, or all-wheel drive and can be easily reconfigured to operate with different vehicles and control systems' actuator configurations such as differential braking, torque vectoring, or their combinations on the front or rear axles. This research has resulted in two US pending patents on vehicle speed estimation and sensor fault diagnosis and successful transfer of these patents to industry