Uncertain Linear Systems Research Articles

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.

Robust Stability Analysis of Uncertain Linear Fractional-Order Systems With Time-Varying Uncertainty for 0 < α < 2

This paper focuses on the stability analysis of linear fractional-order systems with fractional-order 0<α<2, in the presence of time-varying uncertainty. To obtain a robust stability condition, we first derive a new upper bound for the norm of Mittag-Leffler function associated with the nominal fractional-order system matrix. Then, by finding an upper bound for the norm of the uncertain fractional-order system solution, a sufficient non-Lyapunov robust stability condition is proposed. Unlike the previous methods for robust stability analysis of uncertain fractional-order systems, the proposed stability condition is applicable to systems with time-varying uncertainty. Moreover, the proposed condition depends on the fractional-order of the system and the upper bound of the uncertainty matrix norm. Finally, the offered stability criteria are examined on numerical uncertain linear fractional-order systems with 0<α<1 and 1<α<2 to verify the applicability of the proposed condition. Furthermore, the stability of an uncertain fractional-order Sallen–Key filter is checked via the offered condition.

Improved Results on Robust Energy-to-Peak Filtering for Continuous-Time Uncertain Linear Systems

This paper addresses the problem of energy-to-peak filtering for continuous-time systems with polytope uncertainties. The aim is to design a new robust filtering that guarantees the filtering error system is asymptotically stable with a prescribed energy-to-peak performance. Parameter-dependent Lyapunov approach and Finsler’s lemma are used to achieve this aim. Sufficient condition for the existence of energy-to-peak filtering is presented in terms of linear matrix inequalities. The proposed method is less conservative than the earlier results by adding more slack matrices. A simulation is given to show the effectiveness of the proposed method.

Static output feedback fault tolerant control using control allocation scheme

SummaryThis paper describes two novel schemes for fault tolerant control using robust suboptimal static output feedback design methods. These schemes can also be employed as actuator redundancy management for overactuated uncertain linear systems. In contrast to many existing methods in the literature that assume the control input matrix (i) is not of full‐rank such that it can be factorized into two matrices and (ii) it does not involve uncertainty, these schemes can be applied to systems whose control input matrix cannot be factorized and/or involve uncertainty. The so‐called virtual control, in these schemes, is calculated using suboptimal ‐based static output feedback design schemes constructed to be robust against uncertainties emanating from inherent input matrix uncertainty and visibility of the control allocator to the controller. Then, using two proposed control allocation schemes (fixed and on‐line), the obtained virtual control signal is redistributed among remaining (redundant or nonfaulty) set of actuators. As the proposed schemes are modular‐based, they can be employed as real‐time fault tolerant control schemes with no need to reconfigure the controller in the case of actuator faults or failures. The effectiveness of the proposed schemes is discussed and compared with numerical examples.

Networked Control System Design for Turbofan Aeroengines with Aging and Deterioration

This paper is concerned with designing a networked controller for a mixed flow two-spool turbofan aeroengine with aging and deterioration. Firstly, the state-space representation of the aeroengine considering aging and deterioration is identified, by which the engine system with aging and deterioration is modeled as an uncertain linear system. Then based on this uncertain linear system, theoretical results from the networked control systems and the regional pole assignment are introduced to formulate the networked engine control design in the form of linear matrix inequalities (LMIs). By solving these LMIs simultaneously, a networked engine controller is obtained which guarantees both the robustness against delay/dropout and the satisfactory dynamic performance. Finally, the proposed method is applied to an aerothermodynamic component-level engine simulator to demonstrate its validity and applicability. The corresponding delay/dropout margin is also calculated, which provides reference for the future development of the distributed engine control system.

A novel robust non-fragile control approach for a class of uncertain linear systems with input constraints

This paper addresses the non-fragile control problem for a class of uncertain linear systems subject to model uncertainty, controller perturbations, fault signals and input constraints. The controller to be designed is supposed to have additive gain perturbations. A novel state feedback controller is proposed based on the exact available expectation of a Bernoulli random variable, which is introduced to model the feature of the controller gain perturbation that randomly occurs. By using Lyapunov stability theory, new sufficient conditions are derived to design non-fragile controller for a class of uncertain linear systems considering input constraints. Compared with the existing non-fragile state feedback controller methods, the non-fragile property is fully considered to improve the tolerance of uncertainties in the controller, where the conservativeness can be reduced via the Bernoulli random variable. The effectiveness of the proposed control strategy is illustrated by two numerical examples.

Event-triggered sliding mode control for uncertain linear systems with time-varying delay and stochastic disturbance

ABSTRACTThis paper investigates the control issue of uncertain systems in presenting of time-varying delay and stochastic disturbance. The state information is real-timely measured by the sensor and directly sent to an event-trigger, then only necessary data is transmitted by the event-trigger to the controller while the rest redundant information is simply discarded. Since the controller receives a discontinuous-time data sequence, an observer is designed to restructure a substitutional continuous-time state and an observer-state-dependent sliding mode control law is proposed. By the proposed method, not only the mean square exponential stability of closed-loop system can be guaranteed almost surely but also the data transmission amount can be reduced. In addition, the chattering behaviour can be alleviated, meanwhile, the match conditions of uncertainty and disturbance are not required. More significantly, the Zeno phenomena of the event-trigger can be conveniently excluded and the minimal event-triggering interval is also given. A numerical example is provided to demonstrate the effectiveness of the proposed method.

Robust consensus of uncertain linear multi-agent systems via dynamic output feedback

This paper systematically deals with robust consensus of uncertain linear multi-agent systems via dynamic output-feedback protocols. Agents are assumed to have identical nominal linear time-invariant dynamics but are subject to heterogeneous additive stable perturbations. Dynamic output-feedback protocols with or without controller state information exchange between neighboring controllers are studied in a unified framework. Two methods are proposed for protocol design, which need to solve an algebraic Riccati equation and some scalar/matrix inequalities. The first method characterizes some key parameters by scalar inequalities related to the nonzero eigenvalues of the Laplacian, which requires the diagonalizability of the Laplacian, while the second method characterizes the parameters by linear matrix inequalities, which circumvents the requirement of the diagonalizability of the Laplacian. Compared with existing results, the proposed approach can simultaneously cope with heterogeneous uncertainties and directed communication graphs. Numerical results verify the advantages of the proposed design methods.

Combinatorial methods for invariance and safety of hybrid systems

Inspired by Switching Systems and Automata theory, we investigate how combinatorial analysis techniques can be performed on a hybrid automaton in order to enhance its safety or invariance analysis. We focus on the particular case of Constrained Switching Systems, that is, hybrid automata with linear dynamics and no guards. We follow two opposite approaches, each with unique benefits: First, we construct invariant sets via the ‘Reduced’ system, induced by a smaller graph which consists of the essential nodes, called the unavoidable nodes. The computational amelioration of working with a smaller, and in certain cases the minimum necessary number of nodes, is significant. Second, we exploit graph liftings, in particular the Iterated Dynamics Lift (T-Lift) and the Path-Dependent Lift (P-Lift). For the former case, we show that invariant sets can be computed in a fraction of the iterations compared to the non-lifted case, while we show how the latter can be utilized to compute non-convex approximations of invariant sets of a controlled complexity.We also revisit well studied problems, highlighting the potential benefits of the approach. In particular, we apply our framework to (i) invariant sets computations for systems with dwell-time restrictions, (ii) fast computations of the maximal invariant set for uncertain linear systems and (iii) non-convex approximations of the minimal invariant set for arbitrary switching linear systems.

Consensus-Based Distributed Robust Filtering for Multisensor Systems With Stochastic Uncertainties

This paper proposes a distributed robust Kalman filter for time-varying uncertain linear multisensor systems subjected to stochastic uncertainties. A consensus algorithm is utilized to compromise on a single data among information of all nodes in a multisensor system. Distributed filtering based on consensus remarkably stands out from other algorithms because each node not only estimates its local states correctly, but also reaches an agreement with other nodes over a sensor network as well. The distributed filtering problem is formulated here using consensus on estimation (CE) algorithm. The optimal parameters of the filter are computed by minimizing the covariance of the estimation error. Moreover, the stability of the touched upon algorithm is proved by the Lyapunov stability theorem. Simulation results for a multisensor system with 100 nodes are presented to show the effectiveness and performance of the proposed CE-based distributed robust filtering approach.

Linear output regulation with dynamic optimization for uncertain linear over-actuated systems

This paper considers the linear output regulation problem for uncertain over-actuated plants. The general form of input redundancy considered in this work implies the existence of multiple control inputs and state trajectories compatible with a prescribed reference for the output. On-line selection, according to certain performance criteria, of the most suitable of these inputs-state trajectories leads to a linear output regulation problem with dynamic redundancy allocation. We present a solution that augments the well known internal model control scheme with two additional dynamical systems. The first one, named annihilator, parametrizes the inputs and the corresponding state trajectories that are invisible from the output. The second one, named redundancy allocator, dynamically selects the best solution according to a predefined performance criterion. We derive explicit solutions for the performance criterion equal to relaxed 1, 2, and ∞- norms of the plant input. This set-up is a particular case of the dynamic redundancy allocation problem named dynamic input allocation. The proposed solutions can be implemented in an error feedback form and are especially suitable for optimizing sparsity, power and amplitude of the control input. Finally, structural stability, robustness and existence of a unique steady-state are proven.

Adaptive output tracking of linear uncertain discrete-time networked systems over switching topologies

ABSTRACTIn this paper, we study the asymptotic output tracking problem for a class of minimum phase uncertain linear discrete-time multi-agent systems over jointly connected digraphs. As the systems contain arbitrarily large uncertain parameters, the robust feedback control technique does not work. We develop a distributed adaptive control law composed of a distributed observer and a purely decentralised adaptive control law. We first establish a stability result for a class of linear switched systems, which extends the existing results. This stability result leads to a novel distributed observer for the leader system which estimates the leader's signal using the output of the leader system locally and passes the estimated leader's signal to each follower. We then further show that the distributed adaptive control law solves our problem under some mild conditions. An example is used to illustrate the effectiveness and generality of our approach.

Fuzzy Fractional Quadratic Regulator Problem Under Granular Fuzzy Fractional Derivatives

In this paper, a class of uncertain linear dynamical systems called fuzzy fractional linear dynamical systems is investigated. The aim is to find control inputs to keep the states of the fuzzy fractional dynamical systems near the zero in an optimal manner. The optimality criterion is in a form of a granular fuzzy integral whose integrand is a quadratic function of the state variables and control inputs. The fuzzy fractional dynamical system is described using fuzzy fractional differential equations (FFDEs). In order to achieve the aim, an effective approach for solving FFDEs should be at disposal. Due to some restrictions imposed by the previous approaches dealing with FFDEs, a new approach is proposed. The proposed approach is based on the granular derivative and the so-called relative-distance-measure fuzzy interval arithmetic. New definitions of fuzzy fractional derivatives and integral called left and right granular Riemann–Liouville fuzzy fractional derivatives, left and right granular Caputo fuzzy fractional derivatives, and the left and right granular fuzzy fractional integral are also presented. In addition, the concepts of granular fuzzy partial derivative and granular fuzzy chain rule are introduced. By the approximations of the granular fuzzy fractional integral and the granular Caputo fuzzy fractional derivative, the approximation solution to the FFDEs is obtained. Consequently, based on the new concepts and theorems, the solution to the fuzzy fractional quadratic regulator problem is given by a theorem. This paper closes with an example of regulating the motion of Boeing 747 in longitudinal direction with the presence of uncertainty in the initial conditions and the coefficients.

Robust Control of a Chemical Multivariable System in the Presence of Strong Uncertainties in the Model Parameters

This paper deals with control method for chemical multivariable systems with time delays and interactions between loops, which is a significant problem of uncertain linear control systems. The distillation column model is required to satisfy certain robustness conditions. The design of robust H∞ controller for this system provides new and important results. The distillation column was tested with a nominal model and various uncertainties. Within uncertainties in the process will be modeled variations of gains alone, and gains and delays simultaneously for different set point changes, disturbance rejection. The simulation results of the distillation model ensure a robust stability and performance. An excellent decoupled response with zero tracking error is done in all simulations using Matlab/Simulink software.

A novel semi-active control strategy based on the quantitative feedback theory for a vehicle suspension system with magneto-rheological damper saturation

This paper presents a robust controller for a semi-active suspension system with actuator saturation. It addresses the vehicle vibration attenuation problem under two cases: (i) without actuator fault (magneto-rheological (MR) damper) and (ii) with base oil leakage in MR damper. The solution is obtained using the quantitative feedback theory (QFT) design. The actuator (MR damper) dynamics is well approximated by a first order model with an uncertain time constant. The superior performance of the proposed QFT controller, for fault free case, is compared with the sky-hook, H∞ control and passive suspension in time as well as in frequency domains. To address the system with faulty MR damper, a multiple model approach based QFT design is then proposed. The multiplicative type fault is considered as an MR damper fault and the faulty damper output delivering capability is then reduced by 80% to 75% due to the oil leakage in the damper. The proposed idea is centered on dividing the large uncertain system into a set of small uncertain linear system and then design the QFT controller corresponding to each of the generated linear models. The global controller is formulated by aggregating the local controller with the gap metric based weights. This new approach produces the similar performance as that of the passive fault tolerant control (FTC)-QFT design with less control effort. Extensive comparative simulations are provided to show the efficacy of the proposed method over passive FTC-QFT, passive FTC-H∞ design and sky-hook control.

Design of robust nonfragile fault detection filter for uncertain dynamic systems with quantization

This paper investigates the fault detection problem for uncertain linear systems with respect to signal quantization. The measurement output transmitted via the digital communication link is considered to be quantized by a dynamic quantizer. Moreover, different from most of existing results on fault detection where the residual generator is assumed to be realized perfectly as the designed one, this study takes the inaccuracy and uncertainty on the implementation of residual generator into account. This paper pays much attention to designing a fault detection filter with quantization as the residual generator and formulates the design problem into the H∞ framework. The objective is to guarantee the asymptotical stability and prescribed performance of the residual system. The S-procedure and a two-step approach are adopted to handle the effects of quantization and uncertainties on residual system. Corresponding design conditions of a robust fault detection filter and a robust nonfragile ones are derived in the form of linear matrix inequalities. Finally, the efficiency of the theoretical results is illustrated by the numerical example.

Robust cooperative output regulation under exosystem detectability constraint

ABSTRACTThis paper revisits the robust cooperative output regulation problem for a class of linear uncertain multi-agent systems. In particular, it is assumed here that the agents in the system can only access incomplete measurements of the exosystem, and the local regulated error signals are not available to the agents to be used in control. Under these assumptions, the agents in the system cannot independently reconstruct the exosystem dynamics, or rely on their own local measurements to achieve the objectives of the output regulation problem. The solution to the regulation problem proposed in this paper is a distributed dynamic control law that reconstructs the exosystem states, given a mild collective detectability assumption. Furthermore, the proposed distributed control law incorporates an internal model of the exosystem to allow for uncertain dynamics of the multi-agent system. A numerical example is offered to illustrate the effectiveness of the proposed control solution.

Synthesis of adaptive gain robust model-following/tracking controllers for a class of uncertain linear systems

This paper proposes a new adaptive gain robust model-following/tracking controller for a class of uncertain linear systems. The proposed adaptive gain robust controller is composed of state feedback laws with fixed gains, feedfoward inputs for the reference model and nonlinear compensation inputs with adjustable time-varying parameters. Moreover, the proposed control strategy can achieve good transient performance and avoid the excessive control input by means of the design parameter. In this paper, linear matrix inequality-based sufficient conditions for the existence of the proposed adaptive gain controller are given, and two design strategies are presented. Finally, simple illustrative examples are included to show the effectiveness of the proposed adaptive gain robust controller.

Robust Tracking and Disturbance Rejection for Linear Uncertain System With Unknown State Delay and Disturbance

A robust tracking control method is presented in this paper for an uncertain plant with an unknown state delay and an exogenous disturbance. The effects of the uncertainties, the delay, and the exogenous disturbance are treated as a total disturbance; thus, the construction of the observer does not need the delay information. The system design is divided into the design of the gains of the state-feedback controller as well as the design of the gains of the observer and the improved equivalent-input-disturbance (EID) estimator. The pole-assignment method is used to design the gains of the state-feedback controller of a simplified system. A robust-stability condition in the form of a linear matrix inequality is derived to determine the gains of the observer and the improved EID estimator. Since the devised Lyapunov functional is of a more general form than those in existing EID-based methods and the restrictive commutative condition is avoided in this design, the developed design method is less conservative. Finally, comparisons of the developed method with a sliding-mode control method for a matched disturbance and a conventional EID-based method for an unmatched disturbance illustrate the validity and superiority of the developed method.

Cooperative robust output regulation of linear uncertain multiple multivariable systems with performance constraint

This paper is to pursue a general investigation of cooperative robust output regulation for linear continuous-time multiple multivariable systems with unknown system parameters and unmodeled external disturbances. We show that, under standard minimum-phase and relative degree like assumptions, an internal model principle based output-feedback protocol can be constructed by incorporating suitable dynamic compensators, even when the parametric uncertainties are arbitrarily large in some sense. Moreover, we are able to establish a redesigned protocol by means of adapting the H∞ control method. It assures a desired robustness property for the closed-loop system of attenuating external unmodeled disturbances. Hence, our study offers a performance-constrained robust control solution in a distributed control fashion.