Stability Analysis of Transfer Alignment Filter Based on the µ Theory

Lihua Zhu,Yu Wang,Lei Wang and Zhiqiang Wu,

Abstract:The performance of the transfer alignment has great impact on inertial navigation systems.As the transfer alignment is generally implemented using a filter to compensate the errors,its accuracy,rapidity and anti-disturbance capability are key properties to evaluate the filtering process.In terms of the superiority in dealing with the noise,H∞ filtering has been used to improve the anti-disturbance capability of the transfer alignment.However,there is still a need to incorporate system uncertainty due to various dynamic conditions.Based on the structural value theory,a robustness stability analysis method has been proposed for the transfer alignment to evaluate the impact of uncertainty on the navigation system.The mathematical derivation has been elaborated in this paper,and the simulation has been carried out to verify the effectiveness of the algorithm.

Keywords:Robustness stability,transfer alignment,inertial navigation system,µ theory.

1 Introduction

In the recent literature,the issue of stability in various systems has received considerable attention.The transfer alignment of navigation system is one of the most popular methods for a moving base,which uses the observation difference between MINS (master inertial navigation system)and the SINS (slave inertial navigation system)to estimate the system errors by filters [Zhou,Lian,Yang et al.(2018)].Similar to the most systems in use today,there are usually uncertainties incorporated in the navigation systems,which may model a number of factors,including:dynamics that are neglected to make the model tractable,as with large scale structures;nonlinearities that are either too hard or too complicated to model;and parameters that are not known exactly,either because they are hard to measure or there are varying manufacturing conditions [Candès,Romberg and Tao (2006)].

Since the H∞ performance is robust with respect to the input and observation noises,it has attracted much attention since the 1980s.The H∞ filtering is a state estimation of minimizing the maximum energy in the estimation error over all the disturbance trajectories [Yang and Che (2008)].The state estimation based on this criterion is valid when a significant uncertainty exists in the disturbance statistics.Plus,the design for an H∞ filter does not require knowledge of the statistics of the system or the observation noise and it possesses the robustness against the systems and noise uncertainties.To reduce the effect of the uncertainties on the system,H∞ filter has been applied to deal with the parameter uncertainties by increasing the robustness of the system [Zhao (2018);Wang and Yang (2018);Liu,Wang,He et al.(2017);Li,Chen,Zhou et al.(2009)].

The most general and accurate means of analyzing and characterizing the effect of system uncertainty on robust performance and stability,is the structured singular value μframework developed by Doyle and other researchers [Doyle (1985);Packard and Doyle (1993);Zhou,Doyle and Glover (1996)].Allowing for the precise measurement of changing effects in operating conditions and uncertainty in model parameters on stability and performance robustness,this approach has been applied in various fields.For example,μ analysis was used to model the uncertainties and evaluate the stability of a power system [Liu (2018)].Bottura et al.[Bottura and Neto (2000)] investigated μanalysis to test robust stability and performance variations in speed control of an induction motor system.Zhao et al.[Zhao,Qiu and Feng (2016)] examined the robust stability of an integrated navigation system;He et al.[He,Wu and She (2004)] employed the μ analysis in exploring the robustness of the uncertain neutral systems with mixed delays.Pavel [Pavel (2004)] from University of Toronto had done research on stability of the optical communication networks with μ-analysis.Kim et al.[Kim and Cho (2016)] evaluated the robustness of a biochemical network through μ-analysis.

This paper presents an approach of robustness stability evaluation of transfer alignment for a moving base navigation system by using the structured singular value analysis(µ-analysis).It takes the input,the output,the transfer function,and the parameter variation into linear association and reconstruction,and then modifies the system to standard feedback system for the eventual analysis.This paper is organized as follows.Section 2 is devoted to introducing some basic notations and definitions of the structured singular value theory.Section 3 establishes the system model of the H∞ transfer alignment filter.After that,the derivation of alignment system model and uncertainty module with Linear fractional transformation (LFT)of feedback transfer alignment system is given in Section 4.Section 5 shows the simulation experiment with the proposed algorithm to test the validation of the algorithm.Concluding remarks are provided in Section 6.

2 Preliminary

Theoretically,the structured singular value is an extended concept of constant matrix singular value,which is also known as the m analysis.It is a powerful tool,to analyze the robust stability,nominal performance and the robust performance of dynamic systems.

The general feedback framework of m analysis with systemM(s)and the uncertaintyD(s)is shown in Fig.1.Any linear interconnection of inputs,outputs and commands along with perturbations and controller can be viewed in this context and rearranged to match this diagram.

Figure 1:General feedback framework

Where,ω1,ω2are the exogenous disturbing input vectors,e1,e2are the error vectors.Given an uncertainty with known structure,bound value,the setBΔ,a set of possibly real and/or complex uncertainties.

Where,(.)denotes the maximum singular value of a matrix and two non-negative integers S and F represent the number of repeated scalar blocks and full blocks,respectively.Consider the closed loop system with the constant matrixM(s)and the uncertainty Δ(s),the structured singular valueµΔ(M)is defined as:

In a word,the structured singular value is defined as the inverse of the smallest possible uncertainty.

The first step in the µ analysis is to derive a linear power converter model.The properties of µ,and the consequently the µ analysis results,refer to the LFT standard representation of the control problem.Linear fractional transformation is a matrix function,which is a useful way to standardize block diagrams for robust control analysis and design.Many control problems can be expressed within the framework of LFT,which is shown in Fig.2 and Fig.3.This framework can be used in describing and analyzing the uncertain system,whereMis assumed to be the invariable part of the control system,whileΔis the block diagonal matrix.Then,the matrix M is partitioned as in Definition 1.

Definition 1For a complex matrix M

And the complex matrices Δl=Cq2×p2and Δu=Cq1×p1of appropriate size define a lower LFT with respect to Δlas:

Provided the inverse matrices (I-M22Δl)-1exists.

And an upperLFTwith respect to Δuis defined as

Provided the inverse matrices (I-M11Δu)-1exists.

Fu(M,Δu)andFl(M,Δl)denote the upper LFT and the lowerLFTseparately,with the visual definition in Fig.2 and Fig.3,and the corresponding expressions are shown in Eqs.(7)and (8).

Figure 2:Lower LFT structure frame

Figure 3:Upper LFT structure frame

Eq.(7)and Eq.(8)correspond to Fig.2 and Fig.3,respectively.

The explanation ofLFT(takeFl(M,Δ)for example).A nominal mappingM11is perturbed by uncertainty Δ,whileM12,M21andM22reflect how the uncertainty influences the nominal mapping.

Robust Stability Theorem:Assumingβ＞0,the closed-loop system in Fig.1 is stable if and only if the condition in Eq.(9)is satisfied.

For all Δ,such that Δ(⋅)∈µ(Δ)while||Δ||∞＜1/β

3 The system model of the filter

Unlike standard Kalman filter,there is no assumption on the statistical properties of the interference signal in H∞ filtering.Targeting at the systems with uncertainties and external interference,it is to build a filter that makes the H∞ norm of the filtering error output from the interference input minimized.The definition of the H∞ filter is:

Given γ＞0，resolve the causality filterF(s)∈RH∞(if it exits)，make

Here,=F(s)y。

The transfer alignment system with the H∞ filter can be modeled by the following equations:

Fig.4 shows the framework of the H∞ filter

Figure 4:Framework of the H∞ filter

Where,zΔ=z-

Using LFT to describe the H∞ filter,we could obtain the LFT of the H∞ filter.

Figure 5:LFT description of the H∞ filter

Where,the systemP(s)shows as

Where,filterF(s)∈RH∞,and the Eq.(13)has to be satisfied for the H∞ filter.

According Fig.5,Eq.(14)can be obtained as follows.

Where,

In our caseD11=0 andD21=0，and the filter can be simplified as:

Where,Y∞is the positive definite solution of the equation in (17)

The system matrixP(s),the filter matrixF(s)can be divided into four blocks,as follows:

4 Robust stability analysis

According to the lower LFT,the transfer alignment system and the filter can be integrated into one representationM,as shown in Eq.(18)below:

Applying the system model to Eq.(18),the following results can be obtained:

In order to extract the structural perturbation and introduce it into the feedback system,it is necessary to apply a series of transformations toM.Firstly,it is converted into the form of a feedback form shown in Fig.6.

Assuming thatGcontains structural perturbationΔG,G=G0+ΔG,whereG0is the nominal system without structural perturbation.The extracted perturbationΔGis then introduced to the feedback structure to do the robust analysis,shown in Fig.7

Figure 7:The feedback system with structural perturbation

ΔGis generated by the structure disturbanceΔMof systemM,M0is the non-disturbance part,andG0=M0(I-M0)-1

Then we have the structural perturbation:

Therefore,the stability of the transfer alignment with H∞ filtering in Fig.4 is equivalent to robust stability of the closed loop system in Fig.7.Where,M=(I+G)-1G.

So,

According to therobust stability theorem,the infinite norm ||Δ||∞and the structured singular valueµΔ(G0)of the nominal systemG0are then calculated to judge whether the items,namely,||Δ||∞＜1/βandµΔ(G0)＜βare satisfied at the same time,so as to evaluate the robust stability of the feedback system.

5 Equations and Mathematical Expressions

In this section,the mathematical simulations have been carried out to test the robust stability of the transfer alignment filter in the uniform motion,at the speed of 10m/s.The discrete filtering model of the transfer alignment can be expressed in Eq.(27),and the parameters setting are summarized in Tab.1.

Where,Xkis the system state vector at timek,φk,k-1isthestate transition matrix,Γk,k-1is the system noise matrix,Wkis the system noise,Zkis the measurement,Hkis the measurement matrix,Vkis the system measurement noise,Ykis the output.

Table 1:parameters setting

The simulation is carried out in the North-East-Up (ENU)coordination system,and we simulated the velocity matching measurement for an illustration.The state vector isXk=[δVeδVnφeφnφu∇e∇nεeεnεu] and the measurement vector is consist of the northern velocity and eastern velocityZk=[δVeδVn].The system model is:

Where,cij(i,j=1,2,3)is the element of the attitude matrix,and

Considering the system structure perturbation caused by the dynamic influence,we have assigned different uncertainties with several parameters in the system matrix:F11(±10%),F12(±2%)andF22(±4%).As it is seen that,the uncertain parameters mainly cover velocity-relative factors,they are two element of the principal diagonal,one element with velocity and one element with position.Then,the bode diagrams in Fig.8-Fig.10 are of help to analyze the robustness stability of the transfer alignment system.

Figure 8:Frequency response of the attitudes

Figure 9:Frequency response of then northern and western velocities

Figure 10:Frequency response of the northern and western positions

In these figures,due to the uncertainty of the system,the frequency response of the plant is different from the nominal model.The red “+” indicates the nominal model and the blue dashed line indicates the results of 20 random samples for the model with uncertainty.Obviously,the uncertainty of the model changes the frequency response curve of the system.Because of the correlation and coupling of system parameters,the velocity uncertainties result in disturbances in the output of the alignment.It can be clearly seen that the perturbation on the attitudes are more serious,which easily lead to the operating frequency band.The positions are less perturbed by the uncertainties.

6 Concluding remarks

An algorithm of robustness stability analysis has been proposed for the transfer alignment to evaluate the impact of uncertainty on the navigation system.Based on the theory of the structured singular value analysis,a feedback structure of the transfer alignment filtering system has been developed,where the H∞filter is adopted to perform the process.Furthermore,simulations have been carried out to exemplify the useof the proposed algorithm.The significance of this research is to find a way to evaluate the performance of the transfer alignment process when the system incorporates structural uncertainties under complex dynamic conditions,so as to further guide and improve the system design.This paper preliminarily validates the proposed algorithm,while different conditions like the vehicle maneuver motion,time delay,vibration and swaying environment that could introduce system uncertainties need elaborated robustness stability analysis.Such issue will be further addressed in the future study.

Acknowledgement:This work is supported by National Natural Science Foundation of China,No.61803203,and the Fundamental Research Funds for the Central Universities,No.30918011305.