In this paper Riccati and filter difference equations are obtained as an approximative solution to a reverse-time optimum control job specifying the set-valued province calculator. In order to obtain a solution to the set-valued province appraisal job, the discrete-time system kineticss are modeled backwards in clip. Besides a new distinct clip robust extended Kalman filter for unsure systems with uncertainnesss are described in footings of amount quadratic restraints and built-in quadratic restraints. The robust filter is an approximative set-valued province calculator which is robust in the sense that it can manage any uncertainnesss. A new attack through the re-organization of measurings is proposed to better the efficiency of calculation. A sufficient status for the being of a robust Kalman filter is derived.

Key Words – Time-varying system, Extended Kalman filters, hardiness, Riccati Difference Equation.

We Will Write a Custom Essay Specifically
For You For Only $13.90/page!

order now

AMS Subject Classification: 39A10, 39A11, 39A20.

1. Introduction

The Kalman filter is most widely used methods for tracking and appraisal due to its simpleness, optimality, tractableness and hardiness. However the application of the Kalman filter to non additive systems can be hard. The most common attack is to utilize the drawn-out Kalman filter which merely linearise all non additive theoretical accounts [ 14 ] so that the traditional additive Kalman filter can be applied [ 6 ] . The job of province appraisal which includes filtering, anticipation and smoothing has been one of the cardinal research subjects of control community. The Kalman filter, which addresses the minimisation of the filtrating mistake covariance, emerged as a major tool of province appraisal in 1960 & A ; acirc ; ˆ™s and mentions in this. A Kalman filter is an optimum calculator i-e it infers parametric quantities of involvement from indirect, inaccurate and unsure observations. It is recursive so that new measurings can be processed as they arrive. The standard Kalman filtrating assumes that the system theoretical account and the statistics of the noises are known precisely [ 2 ] . However, it may non be realistic in pattern. When the uncertainness exists within the system parametric quantities and/or the statistics of the noises, the public presentation of the standard Kalman filtering could be greatly degraded [ 7 ] . This has motivated many surveies of robust filtering for systems with uncertainnesss [ 8 ] . The job of robust appraisal of systems with norm-bounded parametric quantity uncertainnesss under the public presentation was foremost studied under the impression of guaranteed cost filtering. The jobs of parametric quantity and province appraisal can be found in many Fieldss of technology and scientific discipline. In mold of existent systems, it is non ever possible to gauge all necessary system parametric quantities merely by cognition of their physical base. In this instance the parametric quantities and provinces must be estimated by appropriate designation methods [ 3 ] . Many general analytical and experimental methods are suited for designation of additive system, but in the instance of nonlinear system we need to take more sophisticated methods. The Kalman filter and its alterations are really suited for methods of signal processing, optimisation, system designation and parametric quantity appraisal [ 10 ] . The guaranteed cost filtering is concerned with the design of a filter to guarantee an upper edge on the appraisal mistake discrepancies for all admissible parametric quantity uncertainnesss. More research mentions on this subject can be found for unsure continuous-time systems and unsure discrete-time systems [ 1 ] . Necessary and sufficient status for the being of a robust quadratic filter is given in footings of two robust difference equations. The optimisation of the filter involves seeking for appropriate grading parametric quantities. The traditional system augmentation [ 6 ] can be applied to turn to the hold job, which may ensue in a system of much higher dimension, particularly when the hold is big. To better the computational efficiency, we propose a new attack without fall backing to system augmentation. A figure of province appraisal techniques are available for both additive every bit good as nonlinear systems, of which the Kalman filter and its discrepancies such as the drawn-out Kalman filter ( EKF ) have found broad countries of application. In malice of their easiness of execution, Kalman filters are known to diverge in some instances under the influence of nonlinearities and uncertainnesss [ 2 ] . In order to get the better of this job, robust versions of the Kalman filter have been derived for assorted clauses of uncertainness description. Among the many attacks developed for province appraisal [ 13 ] , the set rank province appraisal attack of [ 9 ] provides a deterministic reading of the Kalman filter in footings of a set-valued province calculator. The set-valued province appraisal job involves happening the set of all provinces consistent with given end product measurings for a time-varying system with norm bounded noise input. Besides we extended the set rank province appraisal attack in order to suit additive unsure uninterrupted systems with an built-in quadratic restraint. A bulk of the province appraisal methods used in practical technology jobs are based on the usage of discrete-time procedures and measurings owing to the usage of digital detectors and digital signal processing. This has motivated continued attempts in the literature directed towards developing suited linear every bit good as nonlinear filters for distinct clip systems.

In this paper we describe the job preparation of the reverse-time discrete-time nonlinear unsure system and present the construct of set-valued province appraisal which is based on sum quadratic restraint and built-in quadratic restraints in Section 2. This set-valued province appraisal job is so expressed in footings of matching optimum control jobs which are discussed in Section 3. Section 4 and 5 provides an approximative solution to the optimum control job which leads to the Riccati and filter difference equations that define the discrete-time robust extended Kalman filter. The feasibleness and convergence are reached through algebraic Riccati equation. The sufficient status for the unsure systems is besides derived through algebraic and filter difference equations. We provide the simulation consequences in Section 6 and Section 7 concludes the paper.

2 Problem Formulation

We consider a reverse-time discrete-time unsure nonlinear system which is derived from a forward-time unsure nonlinear system. The uncertainnesss in the discrete-time non additive system are described by a sum quadratic restraint, which is derived from the corresponding continuous-time uncertainness description in the signifier of an built-in quadratic restraint. Furthermore, the construct of set-valued province appraisal is introduced in the signifier of difference equation which is related to a corresponding optimum control job.

2.1 Reverse-Time Discrete-Time Uncertain Nonlinear System:

We begin with a forward-time continuous-time unsure nonlinear system of the signifier

( 1 )

where is the province appraisal, is the known control input, and are the procedure and measurement uncertainness inputs severally. Besides is the uncertainness end product and Y is the mensural end product. Now consider the nonlinear maps and is a matrix map. The uncertainness associated with system ( 1 ) can be described in footings of an built-in quadratic restraint every bit defined as,

Where denotes the Euclidian norm. is the initial province value and is the nominal initial province value. Now we have to happen a difference between initial province and the nominal initial province. This finite difference is allowed by a non-zero value of the end product uncertainness or the changeless. If and, so. Besides, ( . ) and ( . ) represent admissible uncertainnesss described by,

( 3 )

where is a nonlinear time-varying dynamic uncertainness map and represents the map for all blink of an eyes of clip. Besides we can noted that and depend on in ( 3 ) indicates that they are allowed to depend dynamically on the system province. Besides, N = NT & A ; gt ; 0 is a matrix, is a province vector, is a given invariable and are given positive-definite, symmetric matrix maps of clip. In order to deduce a discrete-time robust set-valued province calculator, it is necessary to obtain this continuous-time unsure system. However, as mentioned in the debut, the discrete-time set-valued province calculator is most straightforward to deduce if this system is discretized in reverse-time instead than in forward-time. This will take to a nonlinear reverse- clip discrete-time unsure system described by the province equations as,

( 4 )

Where and stand for discrete-time nonlinear maps and is a given time-varying matrix. The uncertainness associated with the reverse-time discrete-time system ( 4 ) can be obtained by,


and the admissible uncertainnesss and are described as

( 6 )

and is a nonlinear time-varying dynamic uncertainness map. The unsure system ( 4 ) , with the matching sum quadratic restraint uncertainness description ( 5 ) is used to deduce the robust filter and Riccati equations, which define the discrete-time robust Extended Kalman Filter.

3. Optimal Control Problem

Very late, the control job in [ 6 ] for a category of systems with both stochastic mold uncertainness and deterministic mold uncertainness is investigated, where the stochastic uncertainness has been expressed as a multiplicative noise. It should be pointed out that, compared to the control instance, the corresponding robust filtering job for systems with stochastic and deterministic uncertainnesss has gained much less attending. This state of affairs motivates our present probe that to transport in a dynamic mold. As per the dynamic mold consider to be a fixed measured end product and a known control input for the unsure system ( 4 ) , ( 5 ) , for. The set-valued province appraisal job involves happening the set of all possible provinces at clip measure T for the system in ( 4 ) with initial conditions and uncertainness restraints defined in ( 5 ) which is consistent with the measured end product sequence and input sequence, so the end product sequence, follows from the definition of,

( 7 )

if and merely if there exists an unsure input sequence such that, besides the cost functional is derived from the Sum Quadratic Constant ( 5 ) as [ 9 ]



Here the vector is the solution to the reverse-time discrete- clip system ( 4 ) , with input uncertainness and terminal status. Hence

The optimisation job

( 10 )

for the system ( 4 ) , defines a nonlinear optimum control job with a mark indefinite quadratic cost map. The discrete-time robust Extended Kalman Filter is derived by happening an approximative solution to this optimum control job.


The corresponding discrete-time equation for this optimum control job is given by,

( 11 )

with the initial status

This is the point at which the linearization is performed corresponds to the current estimation as in the instance of the criterion Extended Kalman Filter. Now replacing the linearized footings to the above equation, an approximative solution to this nonlinear difference equation is obtained as follows,

( ,

( 12 )

Comparing the left manus side footings with the right manus side of the above equations, the undermentioned recursive equations are obtained.

Riccati Difference Equation

( 13 )


This solution of the Riccati equation in a clip invariant system converges to steady province ( finite ) covariance which is wholly discernible. Besides we can acquire the filter province difference equation.

Filter Difference Equation

( 14 )


The Kalman filter is applied to a linearized version of these equations without loss of optimality. The estimation is refined by re-evaluating the filter around the new estimated province runing point. This refinement process is iterated until small excess betterment is obtained which is called an iterated EKF.

5. Feasibility and convergence analysis of robust Filter

The feasibleness and convergence belongingss of the solutions of the RDE ( Riccati Difference Equation ) is associated with ARE ( Algebraic Riccati Equation ) . The difference from the uninterrupted clip instance and the distinct clip instance, the non-existence of the robust Fillter over fnite skyline is no longer needfully associated with the solution of the RDE and it & amp ; acirc ; ˆ™s going an boundless solution. Hence the being of the filter requires the fulfilment at each measure of a suited matrix inequality ( feasibility status ) . In such a instance, we are presenting the amount quadratic restraint and built-in quadratic restraint for happening the conditions associating to the initial province of uncertainness and the parametric quantity, under which we can guarantee feasibleness of the solutions of RDEs over an randomly long clip interval, and convergence towards the steady province robust Kalman filter in the signifier of Algebraic Riccati Equation [ 15 ] .

Feasible solution of a Kalman Filter

A existent positive definite solution of of RDE is termed a executable solution of RDE ( Riccati Difference Equation ) that if it satisfies the status at each measure. It can be shown utilizing monotonicity consequences on the algebraic Riccati equation that if a system is quadratically stable, so there exists an such that for any, there exists a stabilising solution to Algebraic Riccati Equation which leads to a bulging optimisation solution.

The feasibleness and convergence analysis to be studied can be stated as follows: Given an randomly big N, suited conditions on the initial province such that the solutions and executable solutions at every measure and and converge, severally to the stabilizing

solutions and as.

We shall now present two Lyapunov equations which is to take a sufficient status for bing feasibleness and convergence of the solution of algebraic difference equation. The first 1 is,

( 15 )




The definition of is, where and the nonzero Eigen values of ( the non zero characteristic root of a square matrixs of ) are the positive and negative characteristic root of a square matrixs of M.

The 2nd Lyapunov equation is,

( 16 )




( 17 )

where is the solution of Lyapunov equations ( 15 ) and ( 16 ) . Mi and are known existent matrices as defined in ( 15 ) and ( 16 ) in such instance, are positive semi definite. This is a sufficient status for guaranting feasibleness and convergence of the solutions of RDE over. The solutions of RDE and of RDE ( Riccati Difference Equation ) are executable over, and converge to the stabilising solutions and severally. As for a sufficiently little scalar, the positive initial province satisfies

( 18 )

where is defined as in ( 17 ) .

Convergence and executable solutions of a disputing optimum control job

Now see a Riccati filter difference equation for a given, the convergence and feasibleness of and depend on the initial province covariance edge and parametric quantity. Because is normally given a priori, it is necessary to analyze the undermentioned job: Given & A ; gt ; 0 and an obtained optimal which minimizes, is it possible to drive and from & A ; gt ; 0 to and by agencies of a suited time-varying map?

In order to work out this job, we shall use a scheme which varies harmonizing to a piecewise-constant form. We shall suggest a methodological analysis to happen a sequence

and a set of exchanging times such that the solution and with

( 19 )

are executable and converge to and severally, and the minimal hint is so obtained. In position of the above sufficient status, we present the undermentioned algorithm for finding -switching scheme.

Measure 1: Given a priori initial province and Lashkar-e-Taiba. If, so halt.

Measure 2: Let initial province be and, so happen an which satisfies ( 18 ) and minimizes | & A ; acirc ; ? ‘ | . Then the stable solutions ( and ( of ARE ( Algebraic Riccati Equation ) are obtained.

Measure 3: Let, and, besides iteratively compute and of RDE ( Riccati difference equation ) until they about approach to stable solutions and, severally. We denote Pk ( ) and Sk ( ) at this blink of an eye as K ( ) and k ( ) , severally.

Measure 4: If, so halt. Otherwise, travel to Step 5.

Measure 5: Let k ( ) and k ( ) be new initial provinces and, severally and allow, so travel to Step 2.

The constructs behind the exchanging scheme can be are as follows. If the initial covariance is sufficiently little and satisfies ( 18 ) , the optimum value with minimal hint ( ) can be computed straight so the convergence and feasibleness conditions are besides satisfied over On the other manus, if the initial covariance is really big, we can non near instantly. So we must choose a suited which satisfies ( 18 ) such that feasibleness and convergence are guaranteed with this. Then could near to through a finite figure of stairss happening at suited blink of an eyes. The choice of such blink of an eyes and the alterations in must fulfill ( 18 ) and the -switching algorithm at any clip. This leads to the convergence and the feasibleness status of the unsure systems.


When all necessary input Fieldss are known and designation job is executable, so we can continue through Matlab designation tool [ 4,7 ] . If the book reads all Fieldss from input GUI so the non additive designation tool will automatically bring forth the maps for job resolution. Then the user defines inputs, existent provinces, known changeless parametric quantities and unknown parametric quantities to be estimated [ 12 ] . Initial values have to be set for all provinces and unknown parametric quantities. The extra inputs are the weight matrices Q and R represents, which provinces have been measured and are included in the input informations for designation ( Fig 1 ) .

Fig 1. Non additive Identification theoretical account tool

User defines text characters stand foring inputs

, provinces and values of known parametric quantities are identified ( Fig. 2 ) [ 5,11 ] .

Fig 2 Input and State values

Unknown parametric quantities of Kalman addition which will be estimated into field estimation parametric quantities ( Fig. 3 ) i-e converted to known parametric quantities.

Fig 3 Initialization of unknown parametric quantity

Input informations can be read organize the leaden matrix Q and R ( Fig 4 ) . State inputs are converted the unknown uncertainnesss in to cognize certainties. This leads the efficient calculation.

Fig 4 Measured informations from works and weight matrices

7 Decision

In this paper an algebraic and Riccati difference equations are derived as an approximative set-valued province calculator solution, obtained from a corresponding optimum control job. We have analyzed the feasibleness and convergence belongingss of such robust filters through sum quadratic restraints and built-in quadratic restraints. A robust Extended Kalman Filter has been designed in this for the unsure systems with a province appraisal. We have reached the sufficient status for robust Kalman filtering job for unsure systems with algebraic and Riccati difference equations.