Fault detection for linear discrete-time periodic systems

An algebraic theory of linear periodic discrete-time (LPDT) systems is developed in terms of a periodic polynomial matrix description (PMD) and in a dynamics (module) structure defined over a non-commutative, non-integral and non-principal ideal ring /spl Rscr/ (the ring of periodic polynomials). Various properties of matrices and modules over /spl Rscr/ are explored, and links between algebraic properties of LPDT systems such as reachability, controllability are established directly in the periodic PMD and /spl Rscr/-module descriptions; i.e., without using the technique of lifting a LPDT system to its associated linear time-invariant (LTI) ones. The advantage of such characterizations is to show that LPDT systems are not index-invariant only for a class-that we characterize-of non-reversible time systems and that, except for this class of systems, the parametrization of all stabilizing controllers can be constructed similarly to LTI systems. Furthermore, one shows that for the class of non-reversible time systems above-mentioned, it is not possible to extract a canonical part like for LTI systems.

The fault detection problem is investigated for discrete-time linear uncertain systems. Instead of designing a fault detection system from the viewpoint of observer design for robust residual generation, an adaptive threshold approach is proposed to attain robustness against disturbance and norm-bounded model uncertainty. The main goal of the research is to develop a threshold design method that could establish an appropriate trade-off between false alarms and missed fault detection in the presence of model uncertainty. For this purpose, the H∞ optimization technique is adopted in the linear matrix inequality framework to compute the unknown parameters of an adaptive threshold. It is shown that the proposed fault detection system based on an adaptive threshold depends only on the system parameters and the control input of the monitored system. It is independent of robust residual generator designs in traditional observer-based fault detection systems. The effectiveness of the proposed approach is verified on two well-known benchmark systems: a direct-current motor and three tank systems. Several types of faults are successfully detected in both applications.

The problem of intermittent fault (IF) detection is studied for a class of linear discrete-time stochastic systems with unknown disturbance under the influence of a class of networked phenomena by using the moving horizon estimation (MHE). Aiming to normalise the transmission of data in network systems, the Round-Robin (RR) protocol is introduced for linear discrete-time stochastic systems, so the system is converted to a linear periodic system. By using the lifting technique and superposition principle, the linear periodic discrete-time system is transformed into a linear discrete time-invariant system. An MHE strategy is developed to cope with unknown disturbance by introducing provisional estimates of external unknown disturbances. By utilizing the moving horizon estimator, a scalar residual decoupled from the unknown disturbance is designed. Finally, the effectiveness of the proposed method is verified by a simulation example.

Zero-error regulation of discrete-time linear periodic systems

This paper studies the simultaneous fault detection and control problem for linear uncertain discrete-time systems. A design procedure dealing with polytopic uncertainties is proposed for residual generation, where both the fault detection objectives and some control objectives are considered simultaneously through certain performance indexes. To cope with the fault sensitivity performance index in a given frequency range directly, the generalised KYP lemma is introduced in this framework. A two-step procedure is adopted to obtain the solutions through satisfying a set of linear matrix inequalities. A numerical example is given to illustrate the effectiveness of the proposed methods.

Using the geometric approach, a solution is given to the problem of detecting faults in linear periodic discrete-time systems. The solution consists on developing an unknown input observer. The existing conditions are provided in terms of new geometric notion of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">outer</i> <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">observable</i> <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">subspace</i> that is the dual notion of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">inner</i> <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">reachable</i> <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">subspace</i> . Making use of the new notion, a residual generator is developed, such that the residual is made independent of unknown input disturbance, and its output-free response goes asymptotically to zero.

In this paper, the classical notion of a conditioned invariant subspace is extended to the case of linear periodic discrete-time systems, properly characterized and suitably specialized to the notion of an outer reconstructible subspace. These ideas are used in order to find necessary and sufficient conditions for the existence of a linear function dead-beat observer for such systems under unknown and unmeasurable disturbances and a synthesis procedure for such an observer.

In this paper the robust asymptotic tracking and disturbance rejection problem is studied for linear periodic discrete-time systems, and for general classes of disturbance functions and reference signals, consisting in free responses of periodic processes. The necessary and sufficient conditions for the existence of a solution, which involve the periodic notion of invariant zero, are found. A constructive design procedure of the periodic compensator is given; it is based on the internal model principle.

In this paper two problems are considered: that of finding a (possibly dynamic) linear periodic controller such that both the regulated output and the control input of a linear periodic discrete-time system become and remain zero, and that of designing a dead-beat observer of a linear function of the state of the system. For both these problems necessary and sufficient conditions for the existence of a solution are given, and synthesis procedures of both the controller and the observer are provided by the sufficiency proofs. Such conditions are very simple and concise when expressed in terms of controllable and/or unreconstructible states.

In this paper, we deal with residual vector generation for fault detection problems in linear systems via unknown input observer (UIO) when the so-called observer matching condition is not satisfied. Based on the relative degree between unknown input and output, a vector of the auxiliary output is introduced so that the observer matching condition is satisfied with respect to the vector. Auxiliary outputs are related to the derivatives of measured signals. However, differentiation leads to excessive amplification of measurement noise. A dynamically equivalent configuration of linear systems is developed using successive integrations to avoid differentiation. As such, auxiliary outputs are constructed without differentiation. Then, the equivalent dynamic system and its corresponding auxiliary outputs are used to generate the residual vector via an exponentially converging UIO. Fault detection in the generated residual vector is also investigated. Finally, the effectiveness of the proposed method is shown via numerical simulation.

Stability of Linear Discrete-Time Periodic Systems with Time-Delay

The classical row-by-row decoupling problem with stability by the use of a dynamic compensator is studied for linear periodic discrete-time systems. The necessary and sufficient conditions for the existence of a solution are found, and a design procedure of the dynamic compensator (including a feedback subcompensator and a pre-subcompensator) is provided by the sufficiency proof, also for the case of a dead-beat requirement for the state free response.

This article deals with a new representation of linear discrete-time periodic systems. This representation, based on the time-varying z-transform, turns out to be highly efficient in the field of automatic control, when an appropriate choice of the sampling period is made. This representation permits to extend many well-known theorems, in particular, the initial- and final-value theorems, to the case of these systems and to predict the presence of limit cycles. Finally, this representation can be used to establish an extension of the Nyquist criterion to the feedback discrete-time periodic systems.

This paper considers dynamical compensators design for purpose of pole assignment for discrete-time linear periodic systems. Similar to linear time-invariant systems, it is pointed out that the design of a periodic dynamical compensator can be converted into the design of a periodic output feedback controller for an augmented system. Utilizing the recent result on output feedback pole assignment, parametric solutions for this problem are obtained. The design approach can be used as a basis for the robust dynamical compensator design for this type of systems. Combined with a robustness index presented in this paper, robust dynamical compensator design problem is converted into a constrainted optimization problem. A numerical example is employed to illustrate the validity and feasibility of the methods.

A computational approach is proposed to solve the minimum norm and/or robust pole assignment problem for linear periodic discrete-time systems. The proposed approach uses a periodic Sylvester equation based parametrization of the periodic pole assignment problem and exploits the non-uniqueness of the problem by imposing conditions on the norm of the resulting periodic state feedback and/or on the condition numbers of the periodic eigenvector matrices of the closed-loop system. The solution method relies on using gradient search methods on suitably defined cost functions. Explicit expression of the gradients of cost functions are derived and the efficient evaluation of the cost functions and gradients is discussed. Numerical examples illustrate the effectiveness of the proposed approach.