Uncertain Linear Dynamical Systems Research Articles

Tracking Control of a Class of Fuzzy Dynamical Systems under the Concept of Granular Differentiability

In this work, the tracking control of a class of uncertain linear dynamical systems is investigated. The uncertainty is considered to be represented as fuzzy numbers, and hence, these uncertain dynamical systems are referred to as fuzzy linear dynamical systems, which are presented in the form of fuzzy differential equations (FDEs). The solution of an FDE is found using an approach called relative-distance-measure fuzzy interval arithmetic and under the granular differentiability concept. The control objective is to provide a control law such that the output of the system tracks a desired reference input in the presence of uncertainties. To this end, a theorem is proposed, which suggests that the control law should take the form of a feedback of fuzzy states with fuzzy gains and a fuzzy pre-compensator. However, since the fuzzy states of the system may not always be measurable, a fuzzy observer is designed for the estimation of such fuzzy states. It is also clearly shown that the generalized Hukuhara differentiability concept is unable for solving the problem examined in this study. Finally, the efficiency of the approach is examined for a plane landing control problem.

Non-parametric identification of upper bound covariance matrices for min-sup Robust Kalman Filter: Application to the AR case

The min-sup type Robust Kalman Filter (RKF) introduced in Azhmyakov [2002] guarantees a robust estimate in uncertain linear dynamic systems under relatively weak assumptions related to the state and observation noises. In particular, it is supposed that the system and observation noises have some unknown probability distribution functions from some classes of centered distributions with bounded covariances with the known upper bound matrices. In our paper, we address the identification problem for upper-bound matrices in RKF, in the case of scalar observations. We use a novel Penalized Uncoverage (PU) function and an advanced optimization technique for this purpose. The novel PU-RKF methodology we develop in this paper is applied to robust state estimation in the stationary autoregressive model. We finally compare computationally our new PU-RKF algorithm with a classical approach involving a combination of the maximum-likelihood estimation and Kalman Filter (ML-KF) for Gaussian and some non-Gaussian noises.

Fuzzy tracking control of fuzzy linear dynamical systems

This paper deals with the fuzzy tracking control problem of a class of uncertain linear dynamical systems. In the uncertain linear dynamical system the uncertainty is considered as fuzzy numbers. This kind of uncertain linear dynamic systems is called fuzzy linear dynamic systems which are expressed in the form of a fuzzy differential equations system. The Mazandarani’s approach and the powerful concept of granular differentiability are utilized to deal with the fuzzy differential equations system. Fuzzy tracking control of fuzzy linear dynamic systems looks for a law for the fuzzy control signal by which the output of the system tracks the reference input. In this regard, two fuzzy control laws are proposed under two theorems. First, it is proved that the fuzzy control law is in the form of a fuzzy state feedback with fuzzy gains and an additional term which is a pre-compensator as a fuzzy number. Since in the presence of disturbances this fuzzy control law fails to tackle the problem then other fuzzy control law is proposed under the second theory. Thus, in the second theory, it is proved that the proposed fuzzy controller desirably leads to the output of the fuzzy linear dynamic system tracks the input and the fuzzy disturbance is rejected. In addition, two lemmas regarding the theories are also proved. Moreover, fuzzy tracking control of output of a two tanks in series system and landing jet aircraft are given showing the effectiveness of the proposed approaches.

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.

Intrusive Galerkin Projection for Model Order Reduction of Uncertain Linear Dynamic Systems

This paper deals with model order reduction of random parameter-dependent (RPD) linear time invariant (LTI) systems by using the RPD-balanced realization (BR), recently developed by the same authors. A novel way to deal with this crucial issue is presented. It is based on the intrusive Galerkin projection of the RPD-BR. Indeed, it is shown, through numerical simulations, that the intrusive Galerkin projection of the RPD-BR into the generalized polynomial chaos (GPC) space enables to generate a deterministic realization preserving stability properties for the original RPD-model. Moreover, the new deterministic state variables, which are the stochastic modes of the RPD-balanced state variables, are ordered with respect to their sub-contributions to the RPD-input/output behaviour, measured by deterministic Hankel values. Hence, a deterministic reduced order model is derived by deleting the state variables with small Hankel values and the complete RPD-reduced order model is then obtained. The feasibility of the proposed method is analyzed while its accuracy is compared to the RPD-truncated balanced realization (RPD-TBR).

A Short Note on Constrained Linear Control Systems With Multiplicative Ellipsoidal Uncertainty

This technical note revisits the classical robust control question of how to design linear control laws for uncertain linear dynamic systems. We formulate this robust control design problem from a modern optimal control perspective which allows us to take into account control and state constraints that have to be satisfied by the closed-loop system for all possible uncertainty scenarios. The contribution of this technical note is the derivation of a conservative but computationally tractable robust control design problem formulation for the case that multiplicative uncertainties are present in the linear dynamic systems, which render the robust control problem non-convex in general. Here, our approximation technique relies on propagating ellipsoidal bounds on the reachable states of the closed-loop system. The methods developed in this technical note can also be used as a building block for tube based model predictive control schemes, where robustly designed linear control laws can help to reduce the conservatism of open-loop predictions. We illustrate the proposed techniques with a numerical case study.

An efficient framework for the reliability-based design optimization of large-scale uncertain and stochastic linear systems

This paper is focused on the development of an efficient reliability-based design optimization algorithm for solving problems posed on uncertain linear dynamic systems characterized by large design variable vectors and driven by non-stationary stochastic excitation. The interest in such problems lies in the desire to define a new generation of tools that can efficiently solve practical problems, such as the design of high-rise buildings in seismic zones, characterized by numerous free parameters in a rigorously probabilistic setting. To this end a novel decoupling approach is developed based on defining and solving a limited sequence of deterministic optimization sub-problems. In particular, each sub-problem is formulated from information pertaining to a single simulation carried out exclusively in the current design point. This characteristic drastically limits the number of simulations necessary to find a solution to the original problem while making the proposed approach practically insensitive to the size of the design variable vector. To demonstrate the efficiency and strong convergence properties of the proposed approach, the structural system of a high-rise building defined by over three hundred free parameters is optimized under non-stationary stochastic earthquake excitation.

Stochastic adaptive event‐triggered control and network scheduling protocol co‐design for distributed networked systems

In a distributed ‘networked control system’ (NCS), multiple physical systems or agents are connected to their corresponding controllers through a shared packet-switched communication network. For such distributed NCS, periodic sampled controller design is unsuitable to handle packet-switched closed-loop control systems and a novel stochastic optimal adaptive event-sampled controller scheme is proposed in the application layer for each physical system or agent expressed as an uncertain linear dynamic system. Lyapunov stability analysis will be utilised to derive the event trigger condition. In addition, a network scheduling protocol is also required for such NCS. Traditional network scheduling protocols are unsuitable for such NCS since the behaviour of the physical systems is disregarded during the protocol design. Therefore in this study, a novel distributed network scheduling protocol via cross-layer approach is developed to improve the performance of distributed NCS by minimising an overall system cost function which consists of the information collected from both the event-triggered controller for each physical system in the application layer and the distributed scheduling protocol from the network layer. It will be demonstrated that the proposed co-design approach will not only allocate the network resources efficiently but also it will improve the performance of the overall distributed NCS. Simulation results are included to demonstrate the effectiveness of the proposed cross-layer co-design.

Set‐membership parity space approach for fault detection in linear uncertain dynamic systems

SummaryIn this paper, a set‐membership parity space approach for linear uncertain dynamic systems is proposed. First, a set of parity relations derived from the parity space approach is obtained by means of a transformation derived from the system characteristic polynomial. As a result of this transformation, parity relations can be expressed in regressor form. On the one hand, this facilitates the parameter estimation of those relations using a zonotopic set‐membership algorithm. On the other hand, fault detection is then based on checking, at every sample time, the non‐existence of a parameter value in the parameter uncertainty set such that the model is consistent with all the system measurements. The proposed approach is applied to two examples: a first illustrative case study based on a two‐tank system and a more realistic case study based on the wind turbine fault detection and isolation benchmark in order to evaluate its effectiveness. Copyright © 2014 John Wiley & Sons, Ltd.

An efficient reliability method for probabilistic H-infinity robust control of uncertain linear dynamic systems

Achieving balance between robustness and performance is always a fundamental challenge we are faced with in control design of systems in the presence of uncertainties. H∞ robust control theory has been extensively used to deal with the problems of robust stabilization and disturbance attenuation of uncertain systems. However, in traditional H∞ robust control, the most frequently used method for dealing with uncertainties is adopting the assumption of norm-bounded perturbations of arbitrary structure about a nominal plant and the design techniques are based on the deterministic worst-case scenarios that may never occur in a particular control system and thus often led to over-conservative results. In this paper we focus on developing a reliability method for probabilistic H∞ robust control of linear uncertain systems by describing the uncertain parameters as random variables. A new efficient reliability method for robust control of dynamic systems with probabilistic parametric uncertainties is presented systematically. Robust control design is carried out by solving a reliability-based optimization problem where the disturbance attenuation and control cost are minimized under the condition that reliability requirement is satisfied. One of the advantages of the presented reliability method is that it can be used directly for robust control design of parametric uncertain systems. Another advantage of the presented method is that it makes it possible to consider the factors such as system performance, control cost, and reliability simultaneously in an integrated framework and provides an essential basis for the coordination and tradeoffs between these important factors in control design of uncertain systems. Compared to traditional H∞ robust control, the presented method is less-conservative and more reasonable for dealing with probabilistic uncertainties. The presented formulations are within the framework of linear matrix inequality and thus can be carried out conveniently. Two numerical examples are investigated to demonstrate the effectiveness and feasibility of the presented method.

Concurrent learning adaptive control of linear systems with exponentially convergent bounds

SUMMARYConcurrent learning adaptive controllers, which use recorded and current data concurrently for adaptation, are developed for model reference adaptive control of uncertain linear dynamical systems. We show that a verifiable condition on the linear independence of the recorded data is sufficient to guarantee global exponential stability. We use this fact to develop exponentially decaying bounds on the tracking error and weight error, and estimate upper bounds on the control signal. These results allow the development of adaptive controllers that ensure good tracking without relying on high adaptation gains, and can be designed to avoid actuator saturation. Simulations and hardware experiments show improved performance. Copyright © 2012 John Wiley & Sons, Ltd.

Set-membership Parity Space Approach for Fault Detection in Linear Uncertain Dynamic Systems

In this paper, a set-membership parity space approach for linear uncertain dynamic systems is proposed. First, a set of parity relations derived from the parity space approach are obtained. Then, the parameter identification of those relations is performed using a set-membership algorithm. In particular, in this work the parametric uncertainty is bounded by a zonotope. Fault detection is then based on checking, at every sampling time, the non existence of a parameter value in the parameter uncertainty set such that the model is consistent with all the system measurements. The proposed approach will be applied to a case study based in a two-tank system in order to evaluate its effectiveness.

H∞ Subsystem Pairwise Distributed Control of Uncertain Linear Systems

The purpose of the paper is to present an algorithm of subsystem pairwise distributed control for one class of uncertain linear dynamical system. Regarding the interconnection in each subsystem pair as an unknown input, the control parameter design conditions are derived using an enhanced bounded real lemma, tying together stability of controlled subsystem pair and whole system stability. The validity of the proposed method is demonstrated by numerical example.

Minimizing Euclidian State Estimation Error for Linear Uncertain Dynamic Systems Based on Multisensor and Multi-Algorithm Fusion

In this paper, a multisensor linear dynamic system with model uncertainty and bounded noises is considered. Based on previously developed set-valued estimation methods in terms of convex optimization, we propose several efficient algorithms of centralized sensor fusion, distributed sensor fusion, and multi-algorithm fusion to minimize the Euclidian estimation error of the state vector. Obviously, an ellipsoid/box with a larger “size” cannot be in general guaranteed to contain another ellipsoid/box with a smaller “size” since centers and shapes of the two ellipsoids/boxes may be different from each other. This fact and the complementary advantages of multiple sensors and multiple algorithms motivate us to construct multiple estimation ellipsoids/boxes squashed along each entry of the state vector as much as possible respectively by using the technique of multiple differently weighted objectives. Then intersection fusion of these estimation ellipsoids/boxes yields a final Euclidian-error-minimized state estimate. Numerical examples show that the new method with multi-algorithm at both the sensors and the fusion center can significantly reduce the Euclidian estimation error of the state.

On Computing Envelopes for Discrete-time Linear Systems with Affine Parametric Uncertainties and Bounded Inputs

AbstractThe computation of envelopes enclosing the possible states and/or outputs of a class of uncertain linear dynamical systems is the subject of this paper. The resulting algorithm can be useful in several areas of control systems such as verification of safety properties and fault diagnosis (in order to choose suitable thresholds on some residuals, for instance). A particular class of polytopes, zonotopes, can be used to implicitly represent the computed sets. The related reachability algorithms have shown to be well suited to control the wrapping effect in the case of linear dynamical systems. However, parametric uncertainties, when taken into account, are often modeled by interval matrices which lead to a loss of parametric dependencies and result in the computation of rather pessimistic sets. The main contribution of this paper consists in extending an existing algorithm based on zonotopes so that it can efficiently propagate structured parametric uncertainties. A 6th order oscillating mass-spring system illustrates the resulting control of the wrapping effect by comparison with Monte-Carlo simulations.

Parity relations for linear uncertain dynamic systems

A new approach for the design of parity relations for linear dynamic systems with additive and multiplicative uncertainties is presented. Instead of cancelling uncertainties following the example of the so-called robust approaches, uncertain parity relations take uncertainties into account as bounded variables. The method is based on the analysis of zonotopes representing the uncertainties. It leads both to Boolean detection results and to an indicator representing the distance to the opposite decision.

Analytical approximation for stationary reliability of certain and uncertain linear dynamic systems with higher-dimensional output

An analytical approximation for the calculation of the stationary reliability of linear dynamic systems with higher-dimensional output under Gaussian excitation is presented. For systems with certain parameters theoretical and computational issues are discussed for two topics: (1) the correlation of failure events at different parts of the failure boundary and (2) the approximation of the conditional out-crossing rate across the failure boundary by the unconditional one. The correlation in the first topic is approximated by a multivariate integral, which is evaluated numerically by an efficient algorithm. For the second topic some existing semi-empirical approximations are discussed and a new one is introduced. The extension to systems with uncertain parameters requires the calculation of a multi-dimensional reliability integral over the space of the uncertain parameters. An existing asymptotic approximation is used for this task and an efficient scheme for numerical calculation of the first- and second-order derivatives of the integrand is presented. Stochastic simulation using an importance sampling approach is also considered as an alternative method, especially for cases where the dimension of the uncertain parameters is moderately large. Comparisons between the proposed approximations and Monte Carlo simulation for some examples related to earthquake excitation are made. It is suggested that the proposed analytical approximations are appropriate for problems that require a large number of consistent error estimates of the probability of failure, as occurs in reliability-based design optimization. Numerical problems regarding computational efficiency may arise when the dimension of both the output and the uncertain parameters is large. Copyright © 2006 John Wiley & Sons, Ltd.

Application of ellipsoidal estimation to satellite control design

The equations of motion of a small satellite moving along a prescribed trajectory under disturbances are analysed. Problems of this kind have been extensively investigated. The corresponding equations for relative motion errors, caused by the uncertainties in initial conditions and control implementation imperfections, are linearized. The linear equations are reformulated and the evolution equations for optimal ellipsoidal estimates of these errors are derived. It is shown that ellipsoidal bounding of reachable sets is an efficient approach to model uncertain linear dynamical systems. The procedure constructed in this paper allows one to take into account discrete observations and to design control aimed at compensating the disturbances between measurements. These measurements are assumed to be performed with small errors. A numerical example is given which illustrates that the presented control design algorithm is quite efficient and allows one to keep the error between the real and desired motion close to zero.

Design of fault diagnosis filters: A multi-objective approach

In this paper, fault detection and isolation (FDI) in linear uncertain dynamic systems is addressed. The main contributions consist of the formulation of the FDI problem as a filter-based multi-objective optimization problem and the study of the tools used for the solution. The design objectives are formulated in terms of H ∞ , H - and generalized H 2 performance specifications as well as regional constraints on filter poles. The problem is solved using linear matrix inequality (LMI) and generalized structured singular value ( μ g ) techniques. Special design features are illustrated through a simulation example and experimental results from a controlled hydraulic process are provided to demonstrate the potential of the proposed procedure.

New results of robust quadratically stabilizing control for uncertain linear time-delay systems

The problem of robust quadratic stabilization for a class of uncertain linear dynamic systems with time-delays in both state and control is discussed in this paper. The uncertain time-delay systems under consideration are described by state differential equations with unknown-but-bounded uncertain parameters that are confined into convex polyhedrons. A new necessary and sufficient condition for the existence of quadratically stabilizing control law and the corresponding synthesis method are derived, the results are given in terms of a set of linear matrix inequalities (LMIs). Finally, numerical examples are presented to demonstrate the effectiveness of the obtained results.