Uncertain Linear Time-varying 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.

New conditions on finite-time stability of linear discrete-time system

This paper is concerned with the problems of set-based finite-time stability (SFTS) and set-based finite-time boundedness (SFTB) for both certain and uncertain linear time-varying systems. The concepts of SFTS and SFTB are defined. Different from existing results, sufficient conditions for SFTS and SFTB are directly derived from the basic definitions of finite-time stability (FTS) and finite-time boundedness (FTB) by using the convex hull technique rather than utilizing the weighted quadratic functions. Thus, more practical constraints on the system states can be dealt with. Furthermore, intervals, zonotopes and polytopes are employed to describe the typical compact convex sets. For linear uncertain systems, the uncertain time-varying state sets are assumed to be represented by interval matrices and matrix zonotopes, respectively. Finally, numerical examples are provided to illustrate the effectiveness of the main results.

Finite-time Boundedness of Polytopic Uncertain Systems with Guaranteed Cost Output Feedback Control

A problem of designing robust guaranteed cost reduced-order output feedback controllers for finite-time boundedness of linear time-varying uncertain systems is considered in this paper. Assuming the system parameters to lie in a polytopic set, differential linear matrix inequality (DLMI) conditions for the design of reduced-order dynamic controllers to handle worst-case exogenous disturbances are derived. The designed controller guarantees the cost on considered disturbance attenuation performance on the system to be below an upper bound, while simultaneously satisfying the conditions for finite-time boundedness and finite L2 gain. Numerical simulations are carried out to demonstrate the applicability of the proposed method in bounding the system trajectories to a predefined set.

Adaptive control of LTV systems with uncertain periodic coefficients

The paper considers the control problem for a class of uncertain linear time-varying systems. The time-varying (TV) parameters are represented by multisinusoidal functions with a priori unknown amplitudes, phases, and frequencies of harmonics. The maximum numbers of the harmonics are known. To compensate the influence of the TV parameters on the closed-loop system, based on internal model principle, TV parameters observers are designed, and the plant model is represented in the parameterized form with constant unknown parameters. This form is used for design of adaptive backstepping controller with modular identifiers with improved parameters tuning.

Reduced-Order Min-Max Guaranteed Cost Control for Finite-Time Boundedness of Uncertain Linear Systems Using Output Feedback

This letter considers a problem of designing reduced-order dynamic output feedback controllers for finite-time boundedness of uncertain linear time-varying systems subjected to exogenous disturbances. Differential linear matrix inequality conditions are derived for finite-time boundedness with output feedback, such that the uncertain system simultaneously satisfies finite <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathscr {L}_{2}$ </tex-math></inline-formula> gain in the presence of worst-case exogenous disturbances. The designed min-max controller minimizes an upper bound on the considered performance index, providing a guaranteed cost for disturbance attenuation performance. Finite-time bounded state feedback controller design is shown to be a special case of the proposed method. The performance efficacy and applicability of the controller designed using the proposed iterative algorithm are compared with some of the recent methods from the literature.

An adaptive observer for uncertain linear time-varying systems with unknown additive perturbations

In this paper we are interested in the problem of adaptive state observation of linear time-varying (LTV) systems where the system and the input matrices depend on unknown time-varying parameters. It is assumed that these parameters satisfy some known LTV dynamics, but with unknown initial conditions. Moreover, the state equation is perturbed by an additive signal generated from an exosystem with uncertain constant parameters. Our main contribution is to propose a globally convergent state observer that requires only a weak excitation assumption on the system, which is satisfied in the present context of regulation.

Delay-dependent criteria for robust stability and stabilization of fractional-order time-varying delay systems

In this paper, the robust stability and stabilization problems for fractional-order time-varying delay systems are investigated. Firstly, a new fractional-order Razumikhin theorem is given. By using the proposed fractional-order Razumikhin theorem, novel delay-dependent stability conditions for both nominal and uncertain linear fractional-order time-varying delay systems are derived. The results are in form of linear matrix inequalities, which are convenient for application and calculation. Then, the obtained stability conditions are utilized to derive a state feedback stabilization controller. To tackle the computational difficulty of the controller design method, a local optimization algorithm is proposed. Finally, three examples are provided to illustrate that the proposed results are valid and less conservative.

Event-triggered control for linear time-varying systems using a positive systems approach

This paper presents a new design for the event-triggered output feedback control of uncertain linear time-varying systems. Departing from the traditional treatments of event-triggered linear time-invariant systems, it is shown that factoring of fundamental solutions as products of solutions of matrix differential equations, together with small-gain arguments and the notions of interval observers and positive systems, lead to a new class of robust event-triggered output-feedback controllers. A key innovation is our use of new event triggers, which use vectors of absolute values instead of the usual Euclidean 2-norms. An illustrative example of a curve tracking system from marine robotics validates the efficacy of the proposed design scheme.

Distributed fault detection and isolation for uncertain linear discrete time-varying heterogeneous multi-agent systems

The distributed fault detection and isolation of linear discrete time-varying multi-agent systems subject to heterogeneous dynamics and norm bounded model uncertainties is investigated. By combining the model uncertainties and external disturbances into a new generalized disturbance, a distributed closed-loop residual generator is constructed based on the estimate of the generalized disturbance. Then, the concerned fault detection is transformed into an indefinite quadratic minimum problem by using the finite-horizon robust H∞ filtering method, for which necessary and sufficient minimum conditions are provided by the Krein-space theory. Afterwards, a computationally efficient recursive algorithm is further developed to determine the residual for individual agent. Based on the resulting residuals, the fault occurrence can be alerted by devising a novel distributed fault detection scheme, which is then applied for the fault isolation by some appropriate transformation of the output measurements. Finally, both numerical and practical simulations are proposed to verify the effectiveness of the proposed algorithm.

Blending based multiple-model adaptive control of multivariable systems with application to lateral vehicle motion control

This paper develops multiple fixed model blending based adaptive parameter identification schemes for multi-input multi-output (MIMO) systems with polytopic parameter uncertainty. The developed identification schemes are proven to be asymptotically stable for uncertain linear time-invariant (LTI) MIMO systems, and is shown to provide fast adaptation for even uncertain linear time-varying (LTV) systems. Furthermore, utilizing the proposed parameter identification schemes, a linear-quadratic (LQ) optimal multiple-model adaptive control (MMAC) scheme is developed for linear MIMO systems with polytopic uncertainties. The proposed MMAC scheme is proven to be asymptotically stable for LTI MIMO systems and applied to tracking control of uncertain lateral vehicle dynamics. A set of simulation test results are presented to verify the stability, effectiveness, and comparison of the proposed MMAC scheme.

Fault diagnosis of coal-mine-gas charging sensor networks using iterative learning-control algorithm

To detect and estimate the faults of discrete linear time-varying uncertain systems, a discrete learning strategy is applied to fault diagnosis, and a new fault-detection and estimation algorithm is proposed. The algorithm adopts the threshold-limit technology. In the selected optimal time domain, a residual signal is used to perform iterative learning correction for the introduced virtual faults so that the virtual faults in an actual system approach the actual faults. The same method is repeated in the remaining optimal time domain to achieve the objective of fault diagnosis. The algorithm not only completes the fault detection and estimation of a discrete linear time-varying uncertain system but also improves the reliability of fault detection and reduces the false alarm rate. Finally, the simulation results verify the effectiveness of the proposed algorithm.

Robust decentralized switched controller design for linear continuous-time systems

Abstract The paper is devoted to obtain original equivalent subsystem method to design of decentralized controller for linear large scale systems. On the theoretical example a new robust decentralized PID switched controller design procedure is obtained for linear time-varying (gain scheduled plant model) uncertain complex system with decentralized output and input structure. Controller design procedure to decentralized controller design performs on the subsystem level. The designed decentralized switched controller ensures the robust stability of closed-loop complex polytopic system with performance H 2 quadratic cost function (QSR). The proposed practical examples with ideal or non-ideal switch of switching parameters show the effectiveness of equivalent subsystem approach.

Characteristic Model-Based Adaptive Control for a Class of MIMO Uncertain Nonaffine Nonlinear Systems Governed by Differential Equations

This paper addresses the difficulty of designing a controller for a class of multi-input multi-output uncertain nonaffine nonlinear systems governed by differential equations. We first derive the first-order characteristic model composed of a linear time-varying uncertain system for such nonaffine systems and then design an adaptive controller based on this first-order characteristic model for position tracking control. The designed controller exhibits a simple structure that can effectively avoid the controller singularity problem. The stability of the closed-loop system is analyzed using the Lyapunov method. The effectiveness of our proposed method is validated with a numerical example.

Finite horizon robustness analysis of LTV systems using integral quadratic constraints

The goal of this paper is to assess the robustness of an uncertain linear time-varying (LTV) system on a finite time horizon. The uncertain system is modeled as an interconnection of a known LTV system and a perturbation. The input/output behavior of the perturbation is described by time-domain, integral quadratic constraints (IQCs). Typical notions of robustness, e.g. nominal stability and gain/phase margins, can be insufficient for finite-horizon analysis. Instead, this paper focuses on robust induced gains and bounds on the reachable set of states. Sufficient conditions to compute robust performance bounds are formulated using dissipation inequalities and IQCs. The analysis conditions are provided in two equivalent forms as Riccati differential equations and differential linear matrix inequalities, and an algorithm is developed leveraging both forms.

On robust Kalman filter for two-dimensional uncertain linear discrete time-varying systems: A least squares method

The robust Kalman filter design problem for two-dimensional uncertain linear discrete time-varying systems with stochastic noises is investigated in this study. First, we prove that the solution to a certain deterministic regularized least squares problem constrained by the nominal two-dimensional system model is equivalent to the generalized two-dimensional Kalman filter. Then, based on this relationship, the robust state estimation problem for two-dimensional uncertain systems with stochastic noises is interpreted as a deterministic robust regularized least squares problem subject to two-dimensional dynamic constraint. Finally, by solving the robust regularized least squares problem and using a simple approximation, a recursive robust two-dimensional Kalman filter is determined. A heat transfer process serves as an example to show the properties and efficacy of the proposed filter.

Krein-space based robust [formula omitted] fault estimation for two-dimensional uncertain linear discrete time-varying systems

In this study, the robust H∞ fault estimation problem for two-dimensional linear time-varying systems with norm-bounded unknown input, measurement noise, and time-varying process uncertainty is investigated. By introducing an equivalent auxiliary system and a new certain indefinite quadratic form performance function, the system uncertainty can be appropriately considered into the new performance function and the fault estimator design is converted to the minimization problem of a quadratic form. Based on the partially equivalence property between the deterministic quadratic form problem and the Krein space estimation theory, the two-dimensional H∞ fault estimation problem can be solved via signal deconvolution in Krein space. Through employing projection operation and Riccati-like difference equation in two dimensions, both the recursive form fault estimator and the explicit condition for existence of the estimator are derived. One Darboux equation example is provided to illustrate the effectiveness of the proposed fault estimator.

Robust Fusion Filtering for Multisensor Time-Varying Uncertain Systems: The Finite Horizon Case

The robustH∞fusion filtering problem is considered for linear time-varying uncertain systems observed by multiple sensors. A performance index function for this problem is defined as an indefinite quadratic inequality which is solved by the projection method in Krein space. On this basis, a robust centralized finite horizonH∞fusion filtering algorithm is proposed. However, this centralized fusion method is with poor real time property, as the number of sensors increases. To resolve this difficulty, within the sequential fusion framework, the performance index function is described as a set of quadratic inequalities including an indefinite quadratic inequality. And a sequential robust finite horizonH∞fusion filtering algorithm is given by solving this quadratic inequality group. Finally, two simulation examples for time-varying/time-invariant multisensor systems are exploited to demonstrate the effectiveness of the proposed methods in the respect of the real time property and filtering accuracy.

BMI-Based Stabilization of Linear Uncertain Plants With Polynomially Time Varying Parameters

This technical note considers the robust stabilization of uncertain linear time-varying continuous-time systems with a mode-switch dynamics. Each mode is characterized by a theoretically unbounded dynamical matrix containing elements whose time behavior over bounded time intervals is sufficiently smooth to be well described by interval polynomials of arbitrary degree. Using a parameter dependent Lyapunov function polynomially depending on time, the stabilizing controller for each single mode is directly obtained by the solution of some BMI's, which become LMI's by fixing two positive scalars. The stability conditions of the switching closed loop system are derived defining a switched Lyapunov function and involving the permanence time interval of the switching plant over each single mode. A salient feature of the technical note is that, unlike all the other existing methods, each plant mode can be stabilized over arbitrarily large uncertain domains of parameters and their derivatives.

Robustification of time varying linear quadratic optimal control based on output integral sliding modes

A hierarchical observer for an uncertain linear time-varying system is designed generalising the output integral sliding-mode approach. The observer is based on a step by step reconstruction of the output and its derivatives, to reconstruct the state theoretically exactly right after the initial time, eliminating completely the effects of the uncertainties/perturbations. The observed state is used in linear quadratic control with an output integral sliding-mode technique, allowing the uncertain system to behave as a nominal (uncertainties/perturbations free) one. The proposed approach is tested by computer simulations.

Robust stability on averaging behaviour of linear time-varying uncertain systems

This research proposes a new type of robust stability – mean robust asymptotic stability for linear time-varying (LTV) uncertain systems and its extension in robust asymptotic stability analysis. The system will feature this kind of stability if its state mean varies towards an equilibrium point. Robust input–output finite-time stability of the LTV uncertain system over a bounded time interval is first analysed by proposing a criterion to characterize the stability condition. The possible cases of steady-state error and state oscillation are then considered and tackled. Finally, some case studies are presented to demonstrate superiority of the proposed work.