On the existence and design of functional observers for linear systems

Darouach [1] recently derived necessary and sufficient conditions for the existence and stability of functional observers with an order, p, equals to the dimension m of the vectors to be estimated. In general, these conditions are difficult to satisfy and when they are not, the only available option is to increase the order of the functional observers. This note presents new conditions for the existence of a general pth-order functional observer. Systematic procedures for the synthesis of reduced-order functional observers are given. A numerical example is given to illustrate the design procedures.

A common disturbance decoupled observer for linear systems with unknown inputs

This paper presents a solution to the problem of designing a common linear functional observer that can observe a partial set of the state vector of two linear systems with unknown inputs. A new structure of a decoupled linear functional observer is proposed for systems subject unknown disturbances, using only the available output information. Existence conditions as well as a design procedure are given for constructing the proposed observer. A numerical example is given to illustrate the effectiveness of the new observer structure.

Optimal multisensor data fusion for linear systems with missing measurements

Multisensor data fusion has attracted a lot of research in recent years. It has been widely used in many applications especially military applications for target tracking and identification. In this paper, we will handle the multisensor data fusion problem for systems suffering from the possibility of missing measurements. We present the optimal recursive fusion filter for measurements obtained from two sensors subject to random intermittent measurements. The noise covariance in the observation process is allowed to be singular which requires the use of generalized inverse. Illustration example shows the effectiveness of the proposed filter in the measurements loss case compared to the available optimal linear fusion methods.

Adaptive fast finite-time multiple-surface sliding control for a class of uncertain non-linear systems

This paper concerns the adaptive fast finite-time multiple-surface sliding control (AFFTMSSC) problem for a class of high-order uncertain non-linear systems of which the upper bounds of the system uncertainties are unknown. By using the fast control Lyapunov function and the method of so-called adding a power integrator merging with adaptive technique, a recursive design procedure is provided, which guarantees the fast finite-time stability of the closed-loop system. Further, it is proved that the control input is bounded.

Partial state estimation for linear systems with output and input time delays.

This paper deals with the problem of partial state observer design for linear systems that are subject to time delays in the measured output as well as the control input. By choosing a set of appropriate augmented Lyapunov-Krasovskii functionals with a triple-integral term and using the information of both the delayed output and input, a novel approach to design a minimal-order observer is proposed to guarantee that the observer error is ε-convergent with an exponential rate. Existence conditions of such an observer are derived in terms of matrix inequalities for the cases with time delays in both the output and input and with output delay only. Constructive design algorithms are introduced. Numerical examples are provided to illustrate the design procedure, practicality and effectiveness of the proposed observer.

A new algorithm to design minimal multi-functional observers for linear systems

Designing minimum possible order (minimal) observers for Multi-Input Multi-Output (MIMO) linear systems have always been an interesting subject. In this paper, a new methodology to design minimal multi-functional observers for Linear Time-Invariant (LTI) systems is proposed. The approach is applicable, and it also helps in regulating the convergence rate of the observed functions. It is assumed that the system is functional observable or functional detectable, which is less conservative than assuming the observability or detectability of the system. To satisfy the minimality of the observer, a recursive algorithm is provided that increases the order of the observer by appending the minimum required auxiliary functions to the desired functions that are going to be estimated. The algorithm increases the number of functions such that the necessary and sufficient conditions for the existence of a functional observer are satisfied. Moreover, a new methodology to solve the observer design interconnected equations is elaborated. Our new algorithm has advantages with regard to the other available methods in designing minimal order functional observers. Specifically, it is compared with the most common schemes, which are transformation based. Using numerical examples it is shown that under special circumstances, the conventional methods have some drawbacks. The problem partly lies in the lack of sufficient numerical degrees of freedom proposed by the conventional methods. It is shown that our proposed algorithm can resolve this issue. A recursive algorithm is also proposed to summarize the observer design procedure. Several numerical examples and simulation results illustrate the efficacy, superiority and different aspects of the theoretical findings.

Reduced order functional observers with application to partial state estimation of linear systems with input delay

Reduced order multi-functional observer design for multi-input multi-utput (MIMO) linear time-invariant (LTI) systems with constant delayed inputs is studied. This research is useful in the input estimation of LTI systems with actuator delay, as well as system monitoring and fault detection of these systems. Two approaches for designing an asymptotically stable functional observer for the system are proposed: delay-dependent and delay-free. The delay-dependent observer is infinite-dimensional, while the delay-free structure is finite-dimensional. Moreover, since the delay-free observer does not require any information on the time delay, it is more practical in real applications. However, the delay-dependent observer contains less restrictive assumptions and covers more variety of systems. The proposed observer design schemes are novel, simple to implement, and have improved numerical features compared to some of the other available approaches to design (unknown-input) functional observers. In addition, the proposed observers usually possess lower order than ordinary Luenberger observers, and the design schemes do not need the observability or detectability requirements of the system. The necessary and sufficient conditions of the existence of an asymptoticobserver in each scenario are explored. The extensions of the proposed observers to systems with multiple delayed-inputs are also discussed. Several numerical examples and simulation results are employed to support our theories.

New results in common functional state estimation for two linear systems with unknown inputs

This paper presents new developments in common functional observers for two systems. We improve an existing common functional observer scheme by reducing its order, and then investigate its existence conditions in terms of the original system matrices. These conditions have never been explored and they enable the users to know at the outset the class of systems for which the scheme is applicable. They also show that both observers can be designed independently of each other which significantly simplifies the design process. A numerical simulation verifies the findings.

On unknown-input functional observability of linear systems

Finding the least possible order of a stable Unknown-Input Functional Observer (UIFO) has always been a challenge in observer design theory. A practical recursive algorithm is proposed in this technical note to design a minimal multi-functional observer for multi-input multi-output (MIMO) linear time-invariant (LTI) systems with unknown-inputs. The concept of unknown-input functional observability is introduced,and it is used as a certificate of the convergence of our algorithm. The proposed procedure looks for a number of additional auxiliary functions to be augmented to the original functions desired for reconstruction. The resulting UIFO is proper, and minimal (of minimum possible order). Moreover, the algorithm does not need the system to be unknown-input observable. A numerical example shows the procedure as well as the effectiveness of the proposed algorithm.

Minimal multi-functional observers for linear systems using a direct approach

The direct approach in designing functional observers was first presented in [1] for estimating a single function of the states of a Linear Time-Invariant (LTI) system. One of the benefits of the direct scheme is that it does not require solving the interconnected Sylvester equations that appear in the other observer design approaches. In the present paper, the direct approach is extended to reconstruct multiple functions of the states in such a way that the minimum possible order of the observer is achieved. The observer is designed so that an asymptotic functional observer can be obtained with arbitrary convergence rate. In the proposed methodology, it is not necessary that a reduced order observer exists for the desired functions to be estimated. To release this limitation, an algorithm is employed to find some auxiliary functions in the minimum required number to be appended to the desired functions. This method assumes that the system is functional observable. This assumption however is less restrictive than the observability and detectability conditions of the system. A numerical example and simulation results explain the efficacy and the benefits of the proposed algorithm.

H-infinity control design for time-delay linear systems: a rational transfer function based approach

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Stochastic stability for Markovian jump linear systems associated with a finite number of jump times

This paper deals with a stochastic stability concept for discrete-time Markovian jump linear systems. The random jump parameter is associated to changes between the system operation modes due to failures or repairs, which can be well described by an underlying finite-state Markov chain. In the model studied, a fixed number of failures or repairs is allowed, after which, the system is brought to a halt for maintenance or for replacement. The usual concepts of stochastic stability are related to pure infinite horizon problems, and are not appropriate in this scenario. A new stability concept is introduced, named stochastic tau-stability that is tailored to the present setting. Necessary and sufficient conditions to ensure the stochastic tau-stability are provided, and the almost sure stability concept associated with this class of processes is also addressed. The paper also develops equivalences among second order concepts that parallels the results for infinite horizon problems. (C) 2003 Elsevier B.V. All rights reserved.

Robust state-derivative pole placement LMI-based designs for linear systems

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)