State-space Matrices Research Articles

H∞ control and filtering of discrete-time stochastic systems with multiplicative noise

Linear discrete-time systems with stochastic uncertainties in their state-space matrices are considered. The problems of finite-horizon filtering and output-feedback control are solved, taking into account possible cross-correlations between the uncertain parameters. In both problems, a cost function is defined which is the expected value of the relevant standard H ∞ performance index with respect to the uncertain parameters. A solution to the filtering problem is obtained first by applying the adjoint system and deriving a bounded real lemma for this system. This solution guarantees a prescribed estimation level of accuracy while minimizing an upper bound on the covariance of the estimation error. The solution of the filtering problem is also extended to the infinite-horizon case. The results of the filtering problem are used to solve the corresponding output-feedback problem. A filtering example is given where a comparison is made with the results obtained using bounded uncertainty design techniques.

Parameter-dependent state observer design for affine LPV systems

In this paper, the design of a parameter-dependent state observer that allows to estimate the state of an affine linear parameter-varying (LPV) system is investigated. The observer has the property of being parameter-dependent since the corresponding state space matrices are scheduled using an interpolation method. Moreover, the stability of the estimation error is based on the existence of an affine parameter-dependent Lyapunov function. The main contribution of this paper is that the problem of the observer design and the existence of such a Lyapunov function is interpreted as a flexible LMI feasibility condition. Some illustrative examples are presented at the end of this paper.

Robust ℋ/sub ∞/ filtering for a class of linear parameter-varying systems

In this paper, we investigate the problem of /spl Hscr//sub /spl infin// filtering for a class of linear parameter-varying (LPV) systems in which the state-space matrices depend affinely on time-varying parameters. We employ the notion of affine quadratic stability using parameter-dependent Lyapunov functionals. We develop a linear parameter-dependent filter such that the estimation error is affinely quadratically stable with a prescribed performance measure. It is established that the solvability conditions can be expressed by linear matrix inequalities which are then evaluated at the extreme points of the admissible parameter set. Simulation results of a typical example are presented.

Robust H Deconvolution and Its Application to Fault Detection

The problem of H∞ deconvolution of linear discrete-time stationary processes is considered where the parameters of the process are partially unknown. By the use of the state-space model of the system, the state-space matrices are assumed to reside in a given polytope. A stationary deconvolver is obtained that achieves a preassigned input estimation level for all of the matrices in the uncertainty polytope. Two types of deconvolvers are considered: The first one directly aims at estimating the selected inputs, whereas the second type tries to estimate a dynamically weighted version of the input. The theory presented is illustrated via an example of fault detection in a servosystem of an air vehicle.

A Flexible Identification Strategy for DABNet Models

Abstract The focus of this paper is to introduce a flexible identification strategy for a general class of models. We use DABNet models, which are composed of a linear state space system whose states, de- coupled by input, are mapped by a neural network. The linear state space matrices can be represented as loosely coupled first and second order sections. A set of conditions will be analyzed for this kind of model that allows an identification process to be as flexible as possible. By flexible we mean a set of features that would be desirable to find in an identification approach: constraint handling and optimal-input design.

Asymptotic Variance Analysis of Subspace Identification Methods

The class of subspace algorithms for system identification is an interesting complement to the maximum likelihood or prediction error methods, especially for multivariable systems. The statistical analysis of the subspace methods is difficult since the estimates depend on the data in a rather complicated manner. Previous results include proofs of generic consistency and asymptotic normality of the estimates. However, no explicit transparent expression for the covariance matrix of the limiting distribution has so far been reported because of the aforementioned difficulties. The main objective of this paper is to provide a methodology that simplifies the asymptotic analysis of subspace based estimation algorithms. The basic idea is illustrated by deriving the asymptotic covariance matrix corresponding to the estimates of the state-space matrices (or the transfer function estimate) of a quite general subspace algorithm.

Blind Localization by Subspace Method for a Scattering Model

This paper deals with the parameters estimation of a scattering environment modeled by a state-space equation, and the localization of a source emitting in it. Measurements are performed thanks to a multisensor network located in the monitored area. The problem is then to localize the source and to identify the propagation parameters using only the sensors signals. The blind aspect of this procedure requires the use of specific methods such as subspace one. This method is combined with the a priori information brought by the specific form of the state-space matrices to reduce the number of estimated parameters.

Linear parameter-varying discrete time-delay systems: Stability and l 2-gain controllers

In this paper, we investigate a class of linear parameter-varying discrete time-delay (LPVDTD) systems where the state-space matrices depend on time-varying parameters and the delay is unknown but bounded. We treat both notions of quadratic stability based on a single quadratic Lyapunov function and affine quadratic stability using parameter-dependent Lyapunov functions. In both cases, we develop LMI-based results of stability testing for time-delay as well as delayless discrete-time systems. Then, we design state-feedback controllers which guarantee quadratic stability and an induced l 2-norm bound. For the case of dynamic output feedback control, we use a parameter-independent quadratic Lyapunov-Krasovskii function to develop LMI-based solvability conditions which are evaluated at the extreme points of the admissible parameter set. Throughout the paper, complementary results for linear parameter-varying discrete (LPVD) systems without delay are presented.

Optimal model reduction via delay approximation

A model reduction method for SISO and MIMO time delay systems is studied in this paper. The proposed method is based on L2 minimality on the reduced order models via the approximation of the delay element by a finite dimensional model in the original system. The technique is developed in terms of state-space matrices and is formulated as an unconstrained optimization problem. To faciliate numerical implementation, analytical gradient formulas involved in the method are derived. A numerical example is presented to demonstrate the effectiveness of the proposed method.

Generic solvability of the failure detection and identification problem

Necessary and sufficient conditions for the generic solvability of ‘the extended fundamental problem in residual generation’ (EFPRG) (Massoumnia et al., 1989, IEEE Trans. Automat. Control 34, 316–321) in the case of simultaneous actuator and sensor failures are presented. In the process of deriving these conditions, the generic forms of supremal controllability and infimal unobservability subspaces of LTI systems having diagonal state-space matrices are obtained.

Decoupling of structured systems by parameter-independent precompensation and state feedback

In this paper, structured systems described by state-space models are considered. For these systems, the entries of the state-space model matrices are assumed to be either fixed zeros or free independent parameters. We study the dynamic decoupling of structured linear systems. It turns out that a large class of systems which is not state feedback decouplable can be made decouplable by the addition of a simple structured precompensator. Such a precompensator then has the property to be independent of the system parameters. We give a necessary and sufficient condition for a structured system to be made feedback decouplable using a parameter independent precompensator. Furthermore, we prove that when such a solution exists, the precompensator can be chosen to be diagonal. In this case the precompensator is constructed via a simple combinatorial algorithm. Our approach is based on a graph representation of the structured system. All the results of this paper are related to some particular shortest sets of input-output paths in the graph associated with the structured system.

An optimization approach to control reconfiguration

An optimization algorithm for the problem of control reconfiguration in response to operating condition changes or abrupt system component failures is presented here. The algorithm utilizes a version of the Broyden-Fletcher-Goldfarb-Shanno (BFGS) optimization method of conjugate directions. The algorithm provides an output feedback controller that not only stabilizes the new /impaired system, when possible, but also preserves much of the dynamics of the original /unfailed system. The design is such that the closed-loop system is robust with respect to inevitable uncertainties /modelling errors on the state-space matrices of the impaired system. The algorithm is applied to an aircraft longitudinal control system, for which two severe cases of failure are considered, first the loss of an actuator and then the loss of a sensor in addition to the actuator loss.

Optimal weighted L2 model reduction of delay systems

A model reduction method for stable delay systems under L2 optimality is introduced in this paper. The reduced models may take the form of either a stable finite dimensional system or a delay system with a reduced-order finite dimensional part. The two cases are studied under a unified framework. Contrary to many existing algorithms, the formulation is entirely given in terms of state-space matrices. The Routh canonical realization is used to parametrize the stable reduced models. The parameters which form the reduced order system are obtained by minimizing the weighted L2 norm of the approximation error through an unconstrained gradient-based minimization procedure. Numerical examples are used to illustrate the effectiveness of the proposed method.

Three-dimensional discrete-time lossless bounded real functions: minimal state-space realization

A simple technique for state-space realization of 3-dimensional discrete-time lossless bounded real functions is presented. The state-space matrices are of minimal dimension and are derived from the corresponding circuit implementation. The efficiency of the technique is due to the fact that in order to determine the 3D state-space model the circuit implementation is no longer needed because the solution is given in closed form.

Measurement-scheduled control for the RTAC problem: an LMI approach

The RTAC nonlinear control design benchmark problem is addressed using a multi-objective control methodology based on linear matrix inequalities and robust control. The approach hinges on the search for a common quadratic Lyapunov function ensuring various specifications (stability, L 2 -gain, command input and output peak bounds) for the closed-loop system. The resulting output-feedback controller is measurement-scheduled; precisely, its state-space matrices depend on the measurement vector, in a nonlinear fashion. We evaluate the performance of our design with various simulations and predicted trade-off curves. ( 1998 John Wiley & Sons, Ltd.

Sliding mode control with time-varying hyperplane for AMB systems

Deals with sliding mode hyperplane design for a class of linear parameter-varying (LPV) plants, the state-space matrices of which are an affine function of time varying physical parameters. The proposed hyperplane, involving a linear matrix inequality (LMI) approach, has continuous dynamics due to scheduling parameters and provides stability and robustness against parametric uncertainties. We have designed a time-varying hyperplane for a rotor-magnetic bearing system with a gyroscopic effect, which can be considered an LPV plant due to parameter dependence on rotational speed. The obtained hyperplane is continuously scheduled with respect to rotational speed. We successfully carried out experiments using a commercially available turbomolecular pump system and results were reasonable and good.

H∞ sampled-data synthesis and related numerical issues

In this paper we present a new, compact derivation of state-space formulae for the so-called discretisation-based solution of the H ∞ sampled-data control problem. Our approach is based on the established technique of continuous time-lifting, which is used to isometrically map the continuous-time, linear, periodically time-varying, sampled-data problem to a discretetime, linear, time-invariant problem. State-space formulae are derived for the equivalent, discrete-time problem by solving a set of two-point, boundary-value problems. The formulae accommodate a direct feed-through term from the disturbance inputs to the controlled outputs of the original plant and are simple, requiring the computation of only a single matrix exponential. It is also shown that the resultant formulae can be easily re-structured to give a numerically robust algorithm for computing the state-space matrices.

Companion based matrix functions: Description and minimal factorization

Companion based matrix functions are rational matrix functions admitting a minimal realization involving state space matrices that are first companions. Necessary and sufficient conditions are given for a rational matrix function to be companion based. Minimal factorization of such functions is discussed in detail. It is shown that the property of being companion based is hereditary with respect to minimal factorization. Also, the issue of minimal factorization is reduced to a division problem for pairs of monic polynomials of the same degree. In this context, a connection with the Euclidean algorithm is made. The results apply to canonical Wiener-Hopf factorization as well as to complete factorization. The analysis of the latter leads to a combinatorial problem involving the eigenvalues of the state space matrices. The algorithmic aspects of this problem are intimately related to the two machine flow shop problem and Johnson's rule from job scheduling theory.

Controllability and observability of heat exchanger networks in the time-varying parameter case

In this paper the structural controllability and observability of heat exchanger networks are determined from qualitative information about the heat exchangers and their network topology. The heat exchanger network is modelled as a time-varying linear system based on first engineering principles where the effect of the flowrate variations is described by unknown time-varying parameters, inlet temperature variations are regarded as disturbances and external heaters/coolers are considered as input variables. Necessary and sufficient conditions for structural controllability and observability of heat exchanger networks are derived, based on an extended Kalman-type rank criterion for linear systems with time-varying parameters. Determining the structural controllability and observability of heat exchanger networks needs only checks for the input and output connectability of the network in both constant and time-varying parameter cases. The results are extended to the more-practical case where bypass ratios are also used as control variables and where more than one time-varying parameter enters into the state-space matrices.

Self-scheduled [formula omitted]∞ control of linear parameter-varying systems: a design example

This paper is concerned with the design of gain-scheduled controllers with guaranteed H ∞ performance for a class of linear parameter-varying (LPV) plants. Here the plant state-space matrices are assumed to depend affinely on a vector θ of time-varying real parameters. Assuming real-time measurement of these parameters, they can be fed to the controller to optimize the performance and robustness of the closed-loop system. The resulting controller is time-varying and automatically ‘gain-scheduled’ along parameter trajectories. Based on the notion of quadratic H ∞ performance, solvability conditions are obtained for continuous- and discrete-time systems. In both cases the synthesis problem reduces to solving a system of linear matrix inequalities (LMIs). The main benefit of this approach is to bypass most difficulties associated with more classical schemes such as gain-interpolation or gain-scheduling techniques. The methodology presented in this paper is applied to the gain scheduling of a missile autopilot. The missile has a large operating range and high angles of attack. The difficulty of the problem is reinforced by tight performance requirements as well as the presence of flexible modes that limit the control bandwidth.