Linear Time-varying Stochastic Systems Research Articles

Regret optimal control for uncertain stochastic systems

We consider control of uncertain linear time-varying stochastic systems from the perspective of regret minimization. Specifically, we focus on the problem of designing a feedback controller that minimizes the loss relative to a clairvoyant optimal policy that has foreknowledge of both the system dynamics and the exogenous disturbances. In this competitive framework, establishing robustness guarantees proves challenging as, differently from the case where the model is known, the clairvoyant optimal policy is not only inapplicable, but also impossible to compute without knowledge of the system parameters. To address this challenge, we embrace a scenario optimization approach, and we propose minimizing regret robustly over a finite set of randomly sampled system parameters. We prove that this policy optimization problem can be solved through semidefinite programming, and that the corresponding solution retains strong probabilistic out-of-sample regret guarantees in face of the uncertain dynamics. Our method naturally extends to include satisfaction of safety constraints with high probability. We validate our theoretical results and showcase the potential of our approach by means of numerical simulations.

Linear Quadratic Optimal Control Problem for Linear Stochastic Generalized System in Hilbert Spaces

A finite-horizon linear stochastic quadratic optimal control problem is investigated by the GE-evolution operator in the sense of the mild solution in Hilbert spaces. We assume that the coefficient operator of the differential term is a bounded linear operator and that the state and input operators are time-varying in the dynamic equation of the problem. Optimal state feedback along with the well-posedness of the generalized Riccati equation is obtained for the finite-horizon case. The results are also applicable to the linear quadratic optimal control problem of ordinary time-varying linear stochastic systems.

LTV stochastic systems stabilization with large and variable input delay

In this paper we propose a solution to the state-feedback and output-feedback stabilization problem for linear time-varying stochastic systems affected by arbitrarily large and variable input delay. It is proved that under the proposed controller the underlying stochastic process is exponentially centered and mean square bounded. The solution is given through a set of delay differential equations with cardinality proportional to the delay bound. The predictor is based on the semigroup generated by the closed-loop system in absence of delay, and its computation is described by a numerically reliable and robust method. In the deterministic case this method generates the same optimal trajectories as in the delay-less case.

A new look at the statistical identification of nonstationary systems

The paper presents a new, two-stage approach to identification of linear time-varying stochastic systems, based on the concepts of preestimation and postfiltering. The proposed preestimated parameter trajectories are unbiased but have large variability. Hence, to obtain reliable estimates of system parameters, the preestimated trajectories must be further filtered (postfiltered). It is shown how one can design and optimize such postfilters using the basis function framework. The proposed solution to adaptive tuning of postfilter settings is based on parallel estimation and cross-validatory analysis. When compared with the classical solutions to the problem of parameter tracking, the new approach offers, without compromising good tracking performance, significant computational savings, higher numerical robustness and greater flexibility.

A Two-Dimensional Approach to Iterative Learning Control with Randomly Varying Trial Lengths

In this paper, iterative learning control (ILC) is considered to solve the tracking problem of time-varying linear stochastic systems with randomly varying trial lengths. Using the two-dimensional Kalman filtering technique, the authors can establish a recursive framework for designing the learning gain matrix along both time and iteration axes by optimizing the trace of input error covariance matrix. It is strictly proved that the input error converges to zero asymptotically in mean square sense and thus the tracking error covariance converges. The extensions to that prior distribution of nonuniform trial lengths is unknown are also investigated with an asymptotical estimation method. Numerical simulations are provided to verify the effectiveness of the proposed framework.

JOINT ESTIMATION OF STATES AND PARAMETERS OF LINEAR SYSTEMS WITH PARAMETER FAULTS UNDER NON-GAUSSIAN NOISES

Joint estimation of states and time-varying parameters of linear state space models is of practical importance for the fault diagnosis and fault tolerant control. Previous works on this topic consider the joint estimation in the Gaussian noise environment, but not in the presence of outliers. The known fact is that the measurements have inconsistent observations with the largest part of the observation population (outliers). They can significantly make worse the properties of linearly recursive algorithms which are designed to work in the presence of Gaussian noises. This paper proposes the strategy of the joint parameter-state robust estimation of linear state space models in the presence of non-Gaussian noises. The case of parameter-dependent matrices is considered. Because of its good features in robust filtering, the extended Masreliez-Martin filter represents a cornerstone for realization of the robust algorithms for joint state-parameter estimation of linear time-varying stochastic systems in the presence of non-Gaussian noises. The good features of the proposed robust algorithm for joint estimation of linear time-varying stochastic systems are illustrated by intensive simulations.

<italic>H</italic>_ Index for Linear Time-Varying Markov Jump Stochastic Systems and Its Application to Fault Detection

In this paper, the H _ index for Markov jump linear time-varying stochastic systems and its application to robust H _ fault detection filter (FDF) are under consideration. First, a set of finite horizon backward generalized differential Riccati equations (GDREs) and a set of matrix inequalities are introduced. Based on the introduced backward GDREs and matrix inequalities, for nominal Markov jump linear time-varying stochastic systems, two necessary and sufficient conditions for the finite horizon H _ index larger than a given prescribed level $\beta > 0$ are given. Second, for norm uncertain Markov jump linear time-varying stochastic systems, sufficient conditions are presented in terms of matrix inequalities. As applications, two equivalent conditions for the existence of robust H _ FDF are obtained for nominal linear stochastic systems. In particular, under the case of norm uncertainties, a robust H _ FDF is designed based on the feasibility of linear matrix inequalities.

Kalman Filter Reinforced by Least Mean Square for Systems with Unknown Inputs

This paper addresses the state estimation problem of linear discrete-time time-varying stochastic systems with unknown inputs (UIs). It is shown that the globally optimal unbiased minimum-variance filters may not satisfy the minimum-variance property, and hence they cannot eliminate noises appropriately. If this is the case, the well-known Kalman filter may give a better solution, which however may also not be the best one due to that the imbedded unknown input model may not be practical. To remedy the filtering degradation problem, a robust filter named as the KFLMS, which has good noise rejection property for such systems, is developed in this paper, where the UI estimates are obtained by using least mean square algorithm and the state estimation is achieved via the previous proposed two-stage Kalman filtering approach. Numerical examples are provided to show the effectiveness of the proposed results. Specifically, simulation results illustrate the goodness of the new method in the sense of lower root mean square error and better noise rejection property.

Robust Kalman Filtering for Discrete-Time Time-Varying Systems With Stochastic and Norm-Bounded Uncertainties

In this paper, a new robust Kalman filter is proposed for discrete-time time-varying linear stochastic systems. The system under consideration is subject to stochastic and norm-bounded uncertainties in all matrices of the system model. In the proposed approach, the filter is first achieved by solving a stochastic min–max optimization problem. Next, we find an upper bound on the estimation error covariance, and then, by using a linear matrix inequality (LMI) optimization problem, unknown parameters of the filter are determined such that the obtained upper bound is minimized. Finally, two numerical examples are given to demonstrate the effectiveness and performance of the proposed filtering approach compared to the existing robust filters.

Partial-State Stabilization and Optimal Feedback Control for Stochastic Dynamical Systems

In this paper, we develop a unified framework to address the problem of optimal nonlinear analysis and feedback control for partial stability and partial-state stabilization of stochastic dynamical systems. Partial asymptotic stability in probability of the closed-loop nonlinear system is guaranteed by means of a Lyapunov function that is positive definite and decrescent with respect to part of the system state which can clearly be seen to be the solution to the steady-state form of the stochastic Hamilton–Jacobi–Bellman equation, and hence, guaranteeing both partial stability in probability and optimality. The overall framework provides the foundation for extending optimal linear-quadratic stochastic controller synthesis to nonlinear-nonquadratic optimal partial-state stochastic stabilization. Connections to optimal linear and nonlinear regulation for linear and nonlinear time-varying stochastic systems with quadratic and nonlinear-nonquadratic cost functionals are also provided. Finally, we also develop optimal feedback controllers for affine stochastic nonlinear systems using an inverse optimality framework tailored to the partial-state stochastic stabilization problem and use this result to address polynomial and multilinear forms in the performance criterion.

Detectability, observability and Lyapunov-type theorems of linear discrete time-varying stochastic systems with multiplicative noise

ABSTRACTThe objective of this paper is to study detectability, observability and related Lyapunov-type theorems of linear discrete-time time-varying stochastic systems with multiplicative noise. Some new concepts such as uniform detectability, -exact detectability (resp. -exact detectability, -exact detectability, -exact detectability) and -exact observability (resp. -exact detectability, -exact detectability, -exact detectability) are introduced, respectively, and nice properties associated with uniform detectability, exact detectability and exact observability are also obtained. Moreover, some Lyapunov-type theorems associated with generalised Lyapunov equations and exponential stability in mean square sense are presented under uniform detectability, -exact observability and -exact detectability, respectively.

Robust Stability and Robust Stabilization of a Class of Discrete-Time Time-Varying Linear Stochastic Systems

In this paper, a small gain result is presented for a class of discrete-time time-varying Markovian jump systems with state-dependent and input-dependent noise. Using the stabilizing solutions of some generalized Riccati equations, the result is obtained based on a Lyapunov type argument. As an application of the proposed small gain result, we discuss the problem of robust stability and robust state-feedback stabilization of the considered class of stochastic systems via the stability radius concept.

Robust consensus for multi-agent systems over unbalanced directed networks

In this paper, a distributed consensus protocol is proposed for discrete-time single-integer multi-agent systems with measurement noises under general fixed directed topologies. The time-varying control gains satisfying the stochastic approximation conditions are introduced to attenuate noises, thus the closed-loop multi-agent system is intrinsically a linear time-varying stochastic difference system. Then the mean square consensus convergence analysis is developed based on the Lyapunov technique, and the construction of the Lyapunov function especially does not require the typical balanced network topology condition assumed for the existence of quadratic Lyapunov function. Thus, the proposed consensus protocol can be applicable to more general networked multi-agent systems, particularly when the bidirectional and/or balanced information exchanges between agents are not required. Under the proposed protocol, it is proved that the state of each agent converges in mean square to a common random variable whose mathematical expectation is the weighted average of agents’ initial state values; meanwhile, the random variable’s variance is bounded.

Almost Periodic Parameters for the Second Moment Stability of Linear Stochastic Systems

The technical note presents conditions to assure the uniform second moment stability for a class of linear time-varying stochastic systems. The system parameters are assumed to be almost periodic, a concept that is weaker than the periodic one. Under the existence of the long-run average cost associated with the stochastic system, we apply the almost periodicity to prove the desired stability result. An application illustrates the usefulness of the approach by implementing an almost periodic state-feedback strategy to control the velocity of a DC motor device.

A generalised minimum variance controller for MIMO time-varying systems

The problem of generalised minimum variance control of linear multi-input multi-output time-varying systems is studied. The plants are described using a time-varying controlled autoregressive moving average model and the cost functional is the sum of plant output tracking error variances plus squared control variables. Both the output errors and the input variables are weighted using time-varying matrices. A generalised minimum variance controller is developed in a transfer operator framework without using a pseudocommutation technique. The minimum variance controller is extended and the stably invertible condition is removed for multi-input multi-output time-varying systems. The controller can be applied to a large class of linear time-varying stochastic systems.

Information fusion estimators for systems with multiple sensors of different packet dropout rates

In this paper, the optimal centralized and distributed fusion estimation problems in the linear minimum variance (LMV) sense are investigated for multi-sensor systems with multiple packet dropouts. For discrete time-varying linear stochastic systems with multiple sensors of different packet dropout rates, the LMV centralized fusion estimators (CFEs) including filter, predictor and smoother are presented in virtue of the method of innovation analysis. However, CFEs can bring expensive computational cost and poor reliability due to augmentation. To reduce the computational cost and improve the reliability, the distributed fusion estimators (DFEs) are given based on the well-known optimal fusion estimation algorithm weighted by scalars in the LMV sense, which have the parallel structures. Estimation error cross-covariance matrices between any two sensor subsystems are derived to obtain the distributed fusion estimators. A numerical example shows the effectiveness of the proposed algorithms.

Stochastic Observability Test for Discrete-Time Kalman Filters

Stochastic observability refers to the existence of a filter for which the errors of the estimated state mean vector have bounded variance. In this paper, we derive a test to assess the stochastic observability of a Kalman filter implemented for discrete linear time-varying stochastic systems. This test is derived with the assumptions that the system matrices consist of known deterministic parameters and that there is complete uncertainty in the statistics of the initial state vector. This test can also be used to assess the stochastic observability of extended Kalman filters implemented for nonlinear stochastic systems linearized about the true state vector trajectory. We illustrate the utility of the stochastic observability test using an aided inertial navigation system. We also provide a counterexample to illustrate that observability is a necessary, but not sufficient, condition for the stochastic observability of a Kalman filter implemented for a system.

적응 이동 구간 칼만 필터를 이용한 무인 잠수정의 항법 시스템에 관한 연구

In this paper, an underwater navigation system with adaptive receding horizon Kalman filter (ARHKF) is studied. It is well known that incorrect statistical information and temporal disturbance invoke errors of any navigation systems with Kalman filter, which makes the autonomous navigation difficult in real underwater environment. In this context, two kinds of problems are herein considered. The first one is the development of an algorithm, which estimates the noise covariance of a linear discrete time-varying stochastic system. The second one is the implementation of ARHKF to underwater navigation systems. The performance of the derived estimation algorithm of noise covariance and the ARHKF are verified by simulation and experiment in the towing tank of Seoul National University.

Subspace-based prediction of linear time-varying stochastic systems

In this paper, a new subspace method for predicting time-varying stochastic systems is proposed. Using the concept of angle between past and present subspaces spanned by the extended observability matrices, the future signal subspace is predicted by rotating the present subspace in the geometrical sense, and time-varying system matrices are derived from the resultant signal subspace. Proposed algorithm is improved for fast-varying systems. Furthermore, recursive implementation of both algorithms is developed.

Mean Square Exponential Stability for some Stochastic Linear Discrete Time Systems

The problem of the mean square exponential stability for a class of discrete-time linear stochastic systems subject to independent random perturbations and Markovian switching is investigated. Four different definitions of the concept of exponential stability in mean square are introduced and it is shown that they are not always equivalent. One definition of the concept of mean square exponential stability is done in terms of the exponential stability of the evolution defined by a sequence of linear positive operators on an ordered Hilbert space. The other three definitions are given in terms of different types of exponential behavior of the trajectories of the considered system. In our approach the Markov chain is not prefixed. The only available information about the Markov chain is the sequence of probability transition matrices and the set of its states. In this way one obtains that if the system is affected by Markovian jumping the property of exponential stability is independent of the initial distribution of the Markov chain. The definition expressed in terms of exponential stability of the evolution generated by a sequence of linear positive operators, allows us to characterize the mean square exponential stability based on the existence of some quadratic Lyapunov functions. Unlike the continuous time framework, for the discrete time linear stochastic systems with Markovian jumping two types of Lyapunov operators are introduced. Therefore in the case of discrete time linear stochastic systems subject to Markovian perturbations one obtains characterizations of the mean square exponential stability which do not have an analogous in the continuous time. One of the aim of this paper is to show that in the general case of discretetime time-varying linear stochastic systems subject to an homogeneous or an inhomogeneous Markov chain, exponential stability mean square defined in terms of state space trajectories of the systems cannot be always characterized via quadratic Lyapunov functions. The results developed in this paper may be used to derive some procedures for designing stabilizing controllers for the considered class of discrete-time linear stochastic systems.