Study of Robust 𝐻_∞ Filtering Application in Loosely Coupled INS/GPS System

Abstract

Since a celebrate linear minimum mean square (MMS) Kalman filter in integration GPS/INS system cannot guarantee the robustness performance, a 𝐻_∞ filtering with respect to polytopic uncertainty is designed. The purpose of this paper is to give an illustration of this application and a contrast with traditional Kalman filter. A game theory 𝐻_∞ filter is first reviewed; next we utilize linear matrix inequalities (LMI) approach to design the robust 𝐻_∞ filter. For the special INS/GPS model, unstable model case is considered. We give an explanation for Kalman filter divergence under uncertain dynamic system and simultaneously investigate the relationship between 𝐻_∞ filter and Kalman filter. A loosely coupled INS/GPS simulation system is given here to verify this application. Result shows that the robust 𝐻_∞ filter has a better performance when system suffers uncertainty; also it is more robust compared to the conventional Kalman filter.

1. Introduction

Inertial navigation system (INS) is a self-contained system that had been widely used in military, mining, and tunnel construction in the recent decades. However, the INS parameter update functions are nonlinear, the process biases are coupled, and twice integration procedure definitely enlarges the noise influence, the dynamic error of INS keeps increasing with the time going on. Due to hardware and sensor limitation, low cost INS device can only provide an accurate navigation solution for a short period. To conquer this hardship, INS/GPS integrate system is proposed. The GPS is a time bias-free and precise system under less interference circumstance. Measurement of GPS can contribute a calibration to INS, which maintains the accuracy constant.

Estimation theory is widely used in integrate navigation field that blends and fuses information to get an optimal solution. For a typical INS/GPS system, the filter dynamic equation is normally chosen by the INS error state for a linear propagation property. It calls up various investigations based on the modification of a Kalman filter to improve the accuracy or to reduce the computation load of the system [1–3].

However, Kalman filter is the optimal estimation of minimum mean square error (MSE) for a linear time invariant (LTI) system; it has a strict requirement for the process noise covariance 𝐐 and measurement noise covariance 𝐑. If the noise is Gaussian, Kalman filter is the linear minimum variance estimator and if the noise is not Gaussian, it is linear minimum variance estimator. In practical engineering problem, we cannot absolutely assure either the statistical character of the noise or of the dynamic model. And if the objective is to achieve a minimum worst-case estimator rather than minimum variance estimator, Kalman filter will not meet the requirement. In other words, Kalman filter cannot guarantee the robustness if the system contains structure or model uncertainty and lack of noise statistical character.

Motivated by this problem, precursors tried to improve the robustness performance although at the sacrifice of somewhat accuracy such as an EKF approach for parameter identification system [4]. 𝐻_∞ filter is undoubtedly the best choice since it calls for no prior knowledge of the noise distribution. Also we see application in INS/GPS area [5–7], including bounded filter theory [8], and we will review the game theory which is also called the minimax theory used in this area. It is well known that no models can all the way precisely describe a system; huge efforts have been put into the research of Robust 𝐻_∞ filer. This theory assumes that there exists some uncertainty in the system and tries to solve this problem by a quadratic stabilized constrain. Riccati equation and linear matrix inequality (LMI) are two main approaches to deal with robust problems. Compared to Riccati process, LMI does not make any assumptions about the parameters and can be solved efficiently with various MATLAB toolboxes [9, 10]. 𝐻_∞ theory was first applied in the control area [11]; state space realization is a bond connecting it with a estimation theory. In fact, most of research of robust 𝐻_∞ filter is based on the “conservatism,” an important index in robust control. To the authors’ knowledge, there is no investigation of the robust 𝐻_∞ filter considered unstable factor in INS/GPS model; here we merely discuss this application and how it can be implemented.

In this paper, we firstly analyse the relationship between 𝐻_∞ filter and Kalman filter and then modify the LMI design approach, making it possible to apply the robust 𝐻_∞ filter for INS/GPS model; a classical 15-element (incorporating errors of attitude, velocity, position, and gyro and accelerator biases) state vector is used to check filter performance. The purpose of this paper is to make a summary and an easy access to 𝐻_∞ filtering implementation in INS/GPS system.

The rest of the paper is organized as follows. Derivation of recursive 𝐻_∞ filter formulation based on “game theory” [12] is introduced in Section 2; in Section 3 gives designing procedure for robust 𝐻_∞ filter based on LMI; Section 4 discusses the Kalman divergence issue with uncertain system and method in dealing with unstable model when using LMI; in Section 5, we compare the robust 𝐻_∞ filter and Kalman filter performance in different situation for loosely coupled INS/GPS model.

The notation used here is fairly standard: ‖⋅‖₂ is a 𝐿₂ norm; 𝜎(𝐷) denotes the largest singular value of 𝐷; 𝑇_𝑎𝑏 denotes a transform operator from 𝑎 to 𝑏; and 𝐑^𝐧 is 𝑛th dimensional Euclidian state space. Tr(𝑀) stands for the trace of matrix 𝑀, symbol ∗ represents the symmetry term in a symmetric matrix, sup means the supremum, and 𝐴^𝑇 represents the transposed matrix 𝐴.

2. Game Theory Recursive 𝐻_∞ Filter

Since many previous authors had utilized 𝐻_∞ filter in INS/GPS model, we discuss the game theory 𝐻_∞ approach in this section as a fundamental illustration of the robust 𝐻_∞ filter. In order to give a definition of 𝐻_∞ norm. We begin with the 𝐿₂ norm. For a continuous infinite systems during the period (0,𝑇), 𝐿₂ norm is defined as ‖𝑥(𝑡)‖₂=^𝑇₀𝑥(𝑡)𝑥^𝑇(𝑡)𝑑𝑡, (1) where to a discrete system, suppose the sampling number is 𝑁 in the period 𝑇; then ‖𝑥(𝑘)‖₂=^𝑁_𝑘=1𝑥(𝑘)𝑥^𝑇(𝑘). (2)

𝐿₂ reflects the meaning of “energy” of the variable. Then the 𝐻_∞ norm of a linear, time-invariant operator 𝑔=𝑧(𝑤) is defined as 𝑇_𝑔𝑤∞‖𝑧‖=sup²‖𝑤‖₂=sup𝜎𝑇_𝑔𝑤𝑗𝜔, (3) where 𝜎(⋅) denotes the max singular value of the square matrix 𝑇∗𝑇^𝑇 and 𝑗𝜔 is the frequency parameter in the range of [−𝜋,𝜋]. In frequency domain, 𝐻_∞ norm means a peak value of all the frequency ranges, while in time domain, it means the largest energy “transfer” form variable 𝑤 to variable 𝑧; in other words, 𝑤 has the largest influence on 𝑧. So what is the 𝐻_∞ filter problem? Consider the following state-space system: 𝑥()(𝑘)(𝑘)(𝑘)(𝑘)(𝑘),𝑧(𝑘)(𝑘)(0)𝑘+1=𝐴𝑥+𝐵𝑤,𝑦=𝐶𝑥+𝑣=𝐿𝑥,𝑥=𝑥₀, (4) where 𝑥∈𝐑^𝐧 is the dynamic state, 𝑦∈𝐑^𝐯 is the measurement, 𝑧∈𝐑^𝐫 is the output to be estimated, 𝐿𝑥(𝑡) is linear combination of 𝑥, and 𝑤∈𝐑^𝐪 and 𝑣∈𝐑^𝐩 are the process and measurement noise, respectively. (𝐴,𝐵) is controllable and (𝐴,𝐶) is detectable. Note that it is detectable but not “observable”; in an INS/GPS model, the measurement is often “detectable” not “observable.” Now, consider such a cost function: ∑𝐽=^𝑁𝑘=1‖𝑧(𝑘)−𝑧(𝑘)‖²_2𝑆𝑘𝑥₀−𝑥₀²_{2𝑃𝑘−1}+∑^𝑁_𝑘=1‖𝑤‖²_{2𝑄𝑘−1}+‖𝑣‖²_{2𝑅𝑘−1}, (5) where 𝑃_𝑘, 𝑄_𝑘, 𝑅_𝑘, and 𝑆_𝑘 are symmetric, positive definite weighting matrices, and formulation ‖𝑥‖²_2𝑃𝑘=𝑥𝑃_𝑘𝑥^′. Note that these weighting matrix are not noise covariance matrix in a kalman filter, they can be set arbitrary by the filter designer, of cause, can set to be equal with Kalman filter’s 𝑄, 𝑅, 𝑃.

The 𝐻_∞ filter problem is to find an estimation strategy 𝑧(𝑡) which minimizes 𝐽. The famous game theory depicted the estimation of cost function as a dynamic, two-person game: one is the engineering and the other say the nature. The goal of engineering is to find a proper estimation that minimizes 𝐽, while his opponent, the nature tries to maximize 𝐽 from the disturbances and initial state adversely. As a contrast to Kalman filter (KF), what KF pursuits is to minimize ∑^𝑁_𝑘=1‖𝑧(𝑘)−𝑧(𝑘)‖²_2𝑆𝑘, but it cannot guarantee the 𝐽 under some fixed extent, meaning that there may be a huge noise influence on the estimation result when achieves the minimum MSE.

Substituting, ‖𝑣‖²_{2𝑅𝑘−1}=‖𝑦(𝑘)−𝐶𝑥(𝑘)‖²_{2𝑅𝑘−1}; this problem can turn to seeking a saddle point 𝐽(𝑥^∗₀,𝑤^∗,𝑦^∗,𝑥^∗), satisfying𝐽𝑥₀,𝑤,𝑦,𝑥<𝐽𝑥^∗₀,𝑤^∗,𝑦,𝑥<𝐽𝑥^∗₀,𝑤^∗,𝑦^∗,𝑥^∗. (6) In fact, this saddle point, that is, the smallest 𝐽, is not tractable in normal case. In general, people look for a closed form solution to the suboptimal estimation such that in Figure 1: the sole saddle point, which lies in the crossing of the upside and bottom surfaces, is difficult to achieve, while suboptimal question guarantees the solution in the range of the two planes that contains the saddle point.

Figure 1 

Game theory 𝐻∞ filter model.

A summary for the game theory 𝐻_∞ filter approach is: fix a prescribed parameter 𝛾 satisfying 𝐽=𝛾², 𝛾>0; try to achieve minimum ∑^𝑁_𝑘=1‖𝑧(𝑘)−𝑧(𝑘)‖²_2𝑆𝑘 subject to (5). This derivation process can be implemented by using the Lagrange multiplier [13, 14]. The steady-state 𝐻_∞ filter formulations are given by 𝑥_𝑘+1=𝐴_𝑘𝑥_𝑘+𝐴_𝑘𝐾_𝑘𝑦_𝑘−𝐶_𝑘𝑥_𝑘,𝐾_𝑘=𝑃_𝑘𝐼−𝛾²𝑃_𝑘+𝐶^𝑇_𝑘𝑅_𝑘⁻¹𝐶_𝑘𝑃_𝑘⁻¹𝐶^𝑇_𝑘𝑅_𝑘⁻¹,𝑃_𝑘+1=𝐴_𝑘𝑃_𝑘𝐼−𝛾²𝑃_𝑘+𝐶^𝑇_𝑘𝑅_𝑘⁻¹𝐶_𝑘𝑃_𝑘⁻¹𝐴^𝑇_𝑘+𝑄_𝑘. (7) For an intuitive contrast with recursive Kalman filter, some transformations are carried out here. A typical Kalman gain 𝐾_𝑘 can be written as 𝐾_𝑘=𝐼+𝑃⁻_𝑘𝐶^𝑇_𝑘𝑅_𝑘⁻¹𝐶_𝑘⁻¹𝑃⁻_𝑘𝐶^𝑇_𝑘𝑅_𝑘⁻¹. (8) Premultiplying outside by 𝑃⁻_𝑘 and postmultiplying each term inside by the inverse of 𝑃⁻_𝑘, we obtain 𝐾_𝑘=𝑃⁻_𝑘𝑃⁻_𝑘+𝑃⁻_𝑘𝐶^𝑇_𝑘𝑅_𝑘⁻¹𝐶_𝑘𝑃⁻_𝑘⁻¹𝑃⁻_𝑘𝐶^𝑇_𝑘𝑅_𝑘⁻¹. (9) Postmultiplying outside the parentheses by the inverse of 𝑃⁻_𝑘 and premultiplying each term inside parentheses by the inverse of 𝑃⁻_𝑘, we have 𝐾_𝑘=𝑃⁻_𝑘𝐼+𝐶^𝑇_𝑘𝑅_𝑘⁻¹𝐶_𝑘𝑃_𝑘⁻¹⁻¹𝐶^𝑇_𝑘𝑅_𝑘⁻¹. (10) So the reformed Kalman filter is given as 𝑥⁻_𝑘+1=𝐴_𝑘𝑥⁻_𝑘+𝐴_𝑘𝐾_𝑘𝑦_𝑘−𝐶_𝑘𝑥⁻_𝑘,𝐾_𝑘=𝑃⁻_𝑘𝐼+𝐶^𝑇_𝑘𝑅_𝑘⁻¹𝐶_𝑘𝑃⁻_𝑘⁻¹𝐶^𝑇_𝑘𝑅_𝑘⁻¹,𝑃⁻_𝑘+1=𝐴_𝑘𝑃⁻_𝑘𝐼+𝐶^𝑇_𝑘𝑅_𝑘⁻¹𝐶_𝑘𝑃⁻_𝑘⁻¹𝐴^𝑇_𝑘+𝑄_𝑘. (11)

Note that there is 𝑃⁻_𝑘 in (10) while 𝑃_𝑘 in (11). For a LTI system, it is easy to prove that the 𝑃⁻_𝑘 and 𝑃⁺_𝑘 converse to the same stable constant when 𝑇→∞, proof procedure is omitted here for brief. Now suppose that 𝛾²=0; the 𝐻_∞ filter reduces to Kalman filter fantastically. Again, analogous to Figure 1, if the up half-surface represents the performance of the Kalman filter, this filter will pursuit the minimum MMS point, that is, ellipsoid endpoint without concerning the noise transform level. We treat the 𝐻_∞ as a filter criterion, and minimax game theory provides a guarantee about this, and the noise attenuation parameter 𝛾 is also the key index in robust 𝐻_∞ filter.

3. Robust 𝐻_∞ Filter

Consider a system 𝑥̇(𝑡)(𝑡)(𝑡)(𝑡)(𝑡)(𝑡),𝑧(𝑡)(𝑡)(0)=𝐴𝑥+𝐵𝑤,𝑦=𝐶𝑥+𝐷𝑤=𝐿𝑥,𝑥=𝑥₀. (12)

Note that the difference with system (1) lies in the description of noise. Setting 𝐵=[𝐵0], 𝐷=[0𝐷], and 𝑤=[𝑤𝑣]^′ we can get the above formula. In this term, the disturbance process and measurement noise are seen as a united element.

For a system, it is impossible to obtain an exact system model due to the presence of inherent parameter uncertainty instinct. The matrices 𝐴, 𝐵, 𝐶, 𝐷 are not exactly known or depend on some time-varying parameter that vary in a fixed bound set. This is called “polytopic model” with the character ∈𝐴(𝛼)𝐵(𝛼)𝐶(𝛼)𝐷(𝛼)=𝑆=𝐴𝐵𝐶𝐷_𝑁𝑖=1𝛼(𝑖)𝐴(𝑖)𝐵(𝑖)𝐶(𝑖)𝐷(𝑖)=Λ. (13)

The vector time-varying parameter 𝛼(𝑡) belongs to the unit simplex as Ω=𝛼(𝑖)∈𝑅^𝑛,^𝑁_𝑖=1𝛼(𝑖)=1;𝛼(𝑖)≥0. (14)

Any uncertain matrices set 𝑆(𝛼) belongs to a polytopic model with convex bound uncertain domain Λ; hence any 𝑆(𝛼)∈Λ, 𝑆(𝛼) can be written as a combination of vertices 𝑆(𝑖) of the convex polytope. And the model is supposed to be entirely known if the vertices number 𝑁=1.

Another important descriptor is the norm-bounded uncertainty. The mode 𝑆 subject to norm-bounded parameter uncertainty is written in the form 𝑆=𝑆₀+HFE (15) for the uncertain matrix parameter 𝐹 such that 𝐹𝐹^′=𝐼, 𝐻, and 𝐸 are known matrices.

The purpose of the filter problem is to design an estimation 𝑧 which is given by 𝑧=ϝ⋅𝑦, where ϝ is an operator filter set with the form 𝑥̇(𝑡)=𝐴_𝑓𝑥(𝑡)+𝐵_𝑓𝑦(𝑡),𝑧(𝑡)=𝐶_𝑓𝑥(𝑡),𝑥(0)=0 (16) for all the admissible uncertainty Λ such that {𝐴_𝑓,𝐵_𝑓,𝐶_𝑓}∈ϝ. The estimation error 𝑒 can easily be yielded as 𝑒=𝑧−𝑧. Construct state variable 𝜉=[𝑥𝑥]^𝑇, the closed loop state-space form is 𝜉̂(𝑡)=𝐴_𝑎𝜉(𝑡)+𝐵_𝑎𝑤(𝑡);𝑒(𝑡)=𝐶_𝑎𝑥𝑖(𝑡), (17) where 𝐴_𝑎=𝐾𝐴0_𝑓𝐶𝐴_𝑓,𝐵_𝑎=𝐵𝐾_𝑓𝐷,𝐶_𝑎=𝐿−𝐶_𝑓. (18)

Defining the transfer function from the noise input 𝑤 to estimation error 𝑒 is 𝑇_𝑤𝑒(𝑠); hence 𝑇_𝑤𝑒(𝑠)=𝐶_𝑎(𝑠𝐼−𝐴_𝑎)⁻¹𝐵_𝑎. Robust 𝐻_∞ filtering problem: determining 𝐴_𝑓, 𝐵_𝑓, 𝐶_𝑓, for all the admissible uncertainty Λ, satisfy the following:(i)model (17) is asymptotic stabilization;(ii)transfer function infinity norm holds that ‖𝑇_𝑤𝑒(𝑠)‖_∞≤𝛾.According to the bounded real lemma, the closed loop system (17) with the upper bounded 𝛾 is asymptotically stable if and only if there exists a matrix 𝑃=𝑃^𝑇 to such the algebraic Riccati inequality: 𝐴^𝑇_𝑎𝑃_𝑎+𝑃_𝑎𝐴_𝑎+𝛾⁻²𝑃_𝑎𝐵_𝑎𝐵^𝑇_𝑎𝑃_𝑎+𝐶^𝑇_𝑎𝐶_𝑎<0. (19) Implement Schur complement lemma, this inequality is equivalent with the LMI: 𝐴_𝑎𝑃+𝑃𝐴^𝑇𝑃_𝑎𝐶^𝑇_𝑎𝐵_𝑎𝐶_𝑎𝑃_𝑎𝐵−𝛾𝐼0_𝑇𝑎0−𝛾𝐼<0. (20)

3.1. LMI Approach to Robust 𝐻_∞ Filtering

The LMI can be solved efficiently by the Matlab LMI Toolbox; however, two problems have to be considered.(i)Solutions to LMI (20) are 𝐴_𝑎, 𝐵_𝑎, 𝐶_𝑎, so the relationship between inequality and variables𝐴_𝑓, 𝐵_𝑓, 𝐶_𝑓 must be founded.(ii)The inequality should be transformed into linear convex optimisation problem, so it can be addressed by LMI approach.Considering the formulation (20), partition 𝑃 together with its inverse as 𝑃𝑃=¹¹𝑃₁₂𝑃^𝑇₁₂𝑃₂₂,𝑃⁻¹=𝑃₁₁𝑃₁₂𝑃^𝑇₁₂𝑃₂₂, (21) substituting the partitioned matrix 𝑃 in (17) with 𝑃₁₁, 𝑃₂₂, 𝑃₁₁, 𝑃₂₂; the inequality changes into𝐴𝑃₁₁+𝑃₁₁𝐴^𝑇𝐵∗∗∗_𝑓𝐶𝑃₁₁+𝐴_𝑓𝑃^𝑇₁₂+𝑃^𝑇₁₂𝐴^𝑇𝐵_𝑓𝐶𝑃₁₂+𝐴_𝑓𝑃₂₂+𝑃^𝑇₁₂𝐶^𝑇𝐵^𝑇_𝑓+𝑃₂₂𝐴^𝑇_𝑓∗∗𝐿𝑃₁₁−𝐶_𝑓𝑃^𝑇₁₂𝐿𝑃₁₂−𝐶_𝑓𝑃₂₂𝐵−𝛾𝐼∗_𝑇𝐷^𝑇𝐵^𝑇_𝑓,0−𝛾𝐼 (22)where ∗ denotes the symmetric block. The next step is converting this joint nonlinear matrix into LMI form [15]. Making an equivalent transform by permuting third and fourth columns and rows, we obtain𝐴𝑃₁₁+𝑃₁₁𝐴^𝑇𝐵∗∗∗_𝑓𝐶𝑃₁₁+𝐴_𝑓𝑃^𝑇₁₂+𝑃^𝑇₁₂𝐴^𝑇𝐵_𝑓𝐶𝑃₁₂+𝐴_𝑓𝑃₂₂+𝑃^𝑇₁₂𝐶^𝑇𝐵^𝑇_𝑓+𝑃₂₂𝐴^𝑇_𝑓𝐵∗∗_𝑇𝐷^𝑇𝐵^𝑇_𝑓−𝛾𝐼∗𝐿𝑃₁₁−𝐶_𝑓𝑃^𝑇₁₂𝐿𝑃₁₂−𝐶_𝑓𝑃₂₂.0−𝛾𝐼 (23)

Define the nonsingular matrix 𝑃Ψ=⁻¹¹¹𝑃000₁₁𝑃₁₂.0000𝐼0000𝐼 (24) Applying the congruence transformation, pre- and postmultiplying Ψ^𝑇 and Ψ, the inequality can be obtained as 𝐴𝑋+𝑋𝐴^𝑇𝐴∗∗∗_𝑇𝑋+𝑃₁₁𝐴+𝑌𝐶+𝐾𝑃₁₁𝐴+𝐴^𝑇𝑃₁₁+𝑌𝐶+𝐶^𝑇𝑌^𝑇𝐵∗∗_𝑇𝑋𝐵^𝑇𝑃₁₁+𝐷^𝑇𝑌^𝑇−𝛾𝐼∗𝐿−𝑍𝐿0−𝛾𝐼>0, (25) where 𝑋, 𝑌, 𝑍, 𝐾 are matrix variables defined as 𝑋=𝑃⁻¹₁₁,𝑌=𝑃₁₂,𝑍=𝐶_𝑓𝑃^𝑇₁₂𝑃𝑋,𝐾=₁₂𝐴_𝑓𝑃^𝑇₁₂𝑋𝑃𝑋,𝑋𝑋₁₁>0. (26)

The objective is to determine the feasible solutions to the LMI; hence the filter matrices can be obtained: 𝐴𝑓=𝑃⁻¹₁₂𝐾𝑃^𝑇₁₂𝑋⁻¹,𝐵_𝑓=𝑃⁻¹₁₂𝑌,𝐶𝑓=𝑍𝑃^𝑇₁₂𝑋⁻¹ (27) and matrices 𝑃₁₂, 𝑃₁₂ can be set arbitrarily to be nonsingular as 𝑃₁₂𝑃^𝑇₁₂=𝐼−𝑃₁₁𝑃₁₁. (28) Here the filtering problem can be solved by a convex LMI optimisation approach. Give a summary to this filter problem. LMI 𝐻_∞ filtering: minimise 𝛾 subject to inequality (26) hold. Based on the inherent properties of polytopic system, this approach can be extended to a robust filter problem. Substitute 𝐴, 𝐵, 𝐶, 𝐷 by the vertex matrices 𝐴_𝑖, 𝐵_𝑖, 𝐶_𝑖, 𝐷_𝑖 of Λ, and guaranteeing that all the vertex LMIs hold, the robust 𝐻_∞ problem can be solved.

It should be highlighted that this formulation can reduced to standard Kalman filter when restricting some special case. Suppose that 𝐵𝐷^𝑇=0 and 𝐷𝐷^𝑇=𝐼; a standard Kalman state description (4) can recovered from (12) when replace this strict LMI (21) by a non strict one through choosing that 𝑋→0, 𝐾=−𝑌𝐶−𝑃₁₁𝐴, and 𝐿=𝑍; then the remaining variable should hold the LMI as 𝑃₁₁𝐴+𝐴^𝑇𝑃₁₁+𝑌𝐶+𝐶^𝑇𝑌^𝑇𝐵∗∗_𝑇𝑃₁₁+𝐷^𝑇𝑌^𝑇−𝛾𝐼∗.𝐿0−𝛾𝐼 (29) Define the new variable 𝑃=𝑃⁻¹₁₁ and 𝑌=−𝐶^𝑇. Applying the Schur complement to (29), we obtain 𝐴𝑃+𝑃𝐴^𝑇−𝑃𝐶^′𝐶−𝛾⁻²𝐿^𝑇𝐿𝑃+𝐵𝐵^𝑇<0. (30) If the parameter 𝛾→∞ (30) is reduced to 𝐴𝑃+𝑃𝐴^𝑇−𝑃𝐶^′𝐶𝑃+𝐵𝐵^𝑇<0, (31) the optimal solution 𝑃 is arbitrarily close to the solution of this algebraic Riccati equation when replacing the inequality by a same equality. And this Riccati equation solution is our familiar estimation error covariance 𝑃(𝑘), when 𝑘→∞, also called stable state of the Kalman filter for a LTI system. Defining 𝑃₁₁=𝑃^𝑇₁₂ and 𝑃⁻¹₁₂=−𝑃⁻¹₁₁ and holding (28) we have 𝐶_𝑓=𝑍𝑃^𝑇₁₂𝑋⁻¹=𝑍𝑃^𝑇₁₂𝑃⁻¹₁₁⁻¹𝐵=𝑍=𝐿,_𝑓=𝑃⁻¹₁₂𝑌=−𝑃⁻¹₁₁𝐴𝑌=𝑃𝐶,_𝑓=𝑃⁻¹₁₂𝐾𝑃^𝑇₁₂𝑋⁻¹=𝑃⁻¹₁₂𝐾=𝑃⁻¹₁₂−𝑌𝐶−𝑃₁₁𝐴=−𝑃⁻¹₁₂𝑃₁₁𝐴−𝑃⁻¹₁₂𝑌𝐶=𝐴−𝐵_𝑓𝐶. (32)

4. Robust 𝐻_∞ to INS/GPS Loosely Coupled Model

4.1. INS Error Model

The estimation objective in GPS/INS system is the INS error. Filter for this INS dynamic error model consists of 15 elements such as𝑥=𝛿𝜙_𝑥𝛿𝜙_𝑦𝛿𝜙_𝑧𝛿𝑉_𝑥𝛿𝑉_𝑦𝛿𝑉_𝑧𝛿𝐿𝛿𝜆𝛿ℎ𝜀_𝑥𝑏𝜀_𝑦𝑏𝜀_𝑧𝑏∇_𝑥𝑏∇_𝑦𝑏∇_𝑧𝑏, (33)where 𝛿𝜙_𝑥, 𝛿𝜙_𝑦, 𝛿𝜙_𝑧 are the pitch, roll, and yaw errors, 𝑉_𝑥, 𝛿𝑉_𝑦, 𝛿𝑉_𝑧 are the east, north, and up velocity errors, 𝛿𝐿, 𝛿𝜆, 𝛿ℎ are the latitude, longitude, and height errors, 𝜀_𝑥𝑏, 𝜀_𝑦𝑏, 𝜀_𝑧𝑏, and ∇_𝑥𝑏, ∇_𝑦𝑏, ∇_𝑧𝑏 are the gyroscope and accelerator bias errors in subscript orientation. Assume that the gyroscope and accelerator have a Gaussian white noise. The transition matrix can be written as 𝐹(𝑡)=𝐹_𝑁(𝑡)_9×9𝐹_𝑆(𝑡)_9×60_6×9𝐼_6×6. (34)𝐹_𝑠(𝑡) is the 9 fundamental parameter transfer matrix of which element detailed definition can be found in [16], 𝐼 is identity matrix, and 𝐹_𝑆(𝑡) is the transfer matrix between fundamental parameter and INS sensors bias, written as 𝐹_𝑆(𝑡)=0_3×3𝐶^𝑛_𝑏𝐶^𝑛_𝑏0_3×30_3×30_3×3. (35)𝐶^𝑛_𝑏 is the rotation matrix to transform the states from body frame to navigation frame. It is obviously that 𝐹(𝑡) is a time-varying matrix determined by the real attitude 𝜙, velocity 𝑉, and position 𝑃, although can be seen as a piecewise approximate constant matrix when implementing filtering for each epoch. 𝐻_∞ estimation calls for no prestatistical character of noise, but accurate system matrices like 𝐴, 𝐵, 𝐶, 𝐷. If the INS output is corrupt, then 𝐹_𝑆 cannot be guaranteed, neither is the estimation state accuracy 𝑋. And that is why a robust 𝐻_∞ filter is designed for such a case.

4.2. Alternative LMI Approach to Unstable Model

It can be easy found that in a LMI approach to solve the 𝐻_∞ filter context, a precondition that transfer matrix 𝐴(𝐹_𝑆) be stable must be satisfied. The reason for this is that since the state-space model cannot be completely cancelled in the estimation error, they must be stable to guarantee the boundness of estimation error. And to apply LMI (21), it must guarantee diagonal element strictly less than 0. If 𝐴 is unstable, 𝑃>0, 𝐴^𝑇𝑃+𝑃𝐴<0 the first element of (21) is infeasible. To deal with this tough situation, an alternative LMI approach is proposed for coping with unstable model [17, 18]. Consider system (12), its state vector can be partitioned as 𝑥(𝑡)=𝑥_𝑠(𝑡)𝑥_𝑢(𝑡)^𝑇 (36) and the system matrices partitions are 𝐴(𝛼)=𝐴_𝑠(𝛼)0𝐴_𝑢𝑠(𝛼)𝐴_𝑢𝐵,𝐵=^𝑠𝐵_𝑢,𝐶(𝛼)=𝐶_𝑠(𝛼)𝐶_𝑢,𝐿=𝐿_𝑠(𝛼)𝐿_𝑢, (37) where the subscript “𝑠” denotes the stable component and “𝑢” denotes the unstable component.

Assumption. Consider the following.(i)The subsystem 𝑥̇_𝑠(𝑡)=𝐴_𝑠𝑥_𝑠(𝑡) is quadratically stable.(ii)The matrix 𝐴_𝑢 is unstable.(iii)(𝐶_𝑠,𝐴_𝑠) is detectable and (𝐴,𝐵) is controllable.Next, partition the filter matrices, and we obtain 𝑥(𝑡)=𝑥_𝑠(𝑡)𝑥_𝑢(𝑡),𝐴_𝑓=𝐴_𝑓𝑠𝐴_𝑓1𝐴_𝑓2𝐴_𝑓𝑢,𝐵_𝑓=𝐵_𝑓𝑠𝐵_𝑓𝑢,𝐶_𝑓=𝐶_𝑓𝑠𝐶_𝑓𝑢. (38) Change the variable 𝜉 with a augmented state, 𝜉 will become 𝜉=𝑥_𝑠𝑥_𝑥𝑥_𝑢−𝑥_𝑢. (39) Equation (17) is reformed as 𝜉̇(𝑡)=𝐴_𝑎(𝛼)𝜉(𝑡)+𝐴_𝑎𝑢𝑥_𝑢(𝑡)+𝐵_𝑎𝑤(𝑡),𝑒(𝑡)=𝐶_𝑎(𝛼)𝜉(𝑡)+𝐶_𝑎𝑢𝑥_𝑢(𝑡), (40) where 𝐴_𝑎(𝛼)=𝐴_𝑠(𝛼)𝐵00_𝑓𝑠𝐶_𝑠(𝛼)𝐴_𝑓𝑠−𝐴_𝑓1𝐴_𝑢𝑠−𝐵_𝑓𝑢𝐶_𝑠(𝛼)−𝐴_𝑓2𝐴_𝑓𝑢,𝐴_𝑎𝑢=0𝐴_𝑓1+𝐵_𝑓𝑠𝐶_𝑢𝐴_𝑢−𝐵_𝑓𝑢𝐶_𝑢−𝐴_𝑓𝑢,𝐵_𝑎=𝐵_𝑠𝐵_𝑓𝑠𝐷𝐵_𝑢−𝐵_𝑓𝑢𝐷,𝐶_𝑎(𝛼)=𝐿_𝑠−𝐶_𝑓𝑠𝐶_𝑓𝑢,𝐶_𝑎𝑢=𝐿_𝑢−𝐶_𝑓𝑢. (41) So the partial state 𝑥_𝑠 is bounded while 𝑥_𝑢 is unbounded. If 𝜉(𝑡) and 𝑒(𝑡) are bounded, there must be assured that 𝐴_𝑎𝑢=0 and 𝐶_𝑎𝑢=0, also meaning that 𝐴_𝑓1=−𝐵_𝑓𝑠𝐶_𝑢,𝐴_𝑓𝑢=𝐴_𝑢−𝐵_𝑓𝑢𝐶_𝑢,𝐶_𝑓𝑢=𝐿_𝑢. (42) Now, we have solved the unstable problems and the remaining processor can use a typical LMI approach to the robust 𝐻_∞ filtering. Since we cannot remove the unstable part from an entire system, the only choice is to try a decomposition of the stable and unstable parts; the key factor lies in eliminating the unbounded estimation influence rendering that the final result is not corrupted.

4.3. Uncertainty to Kalman Filter Divergency Issue

Another drawback when applying Kalman filter to an uncertain and unstable system is divergency issue. The steady state of (4) is 𝑥(𝑘+1)=(𝐴−𝐾𝐶)𝑥(𝑡)+𝐾𝑦(𝑡). (43) The estimation error dynamic function is 𝑥(𝑘+1)=(𝐴−𝐾𝐶)𝑥(𝑘+1)+𝑤(𝑡)+𝐾𝑣(𝑡). (44)

If (𝐴−𝐾𝐶) is stable, then error dynamic function is stable, meaning KF is stable. Now, suppose that 𝐴 is uncertain, and the real matrix is 𝐴^′, and 𝐴 is the nominal matrix in filtering. We obtain 𝑥(𝑘+1)=𝑥(𝑘+1)−𝑥(𝑘+1)=𝐴^′𝑥(𝑘)𝑥(𝑘)𝑥(𝑘)(𝑘)(𝑘)=𝐴−𝐴+𝐾𝐶−𝐾𝑦+𝑤_′𝑥(𝑘)(𝑥(𝑘)−𝑥(𝑘(𝑥(𝑘)(𝑘(𝑘)(𝑘)=𝐴−𝐴+𝐴))+𝐾𝐶−𝑥))−𝐾𝑣+𝑤_′𝑥(𝑘)+()𝑥(𝑘)(𝑘)(𝑘).−𝐴𝐴−𝐾𝐶+𝑤−𝐾𝑣 (45) The above formulation can be written as 𝑥()𝑥()=𝐴𝑘+1𝑘+1𝐴0_′𝑥(𝑘)𝑥(𝑘)+𝑤(𝑘)𝑤(𝑘)(𝑘).−𝐴𝐴−𝐾𝐶−𝐾𝑣 (46) If 𝐴^′ and 𝐴 are both unstable, when 𝑘→∞, 𝑥(𝑡) can be arbitrarily large, so even 𝐴^′−𝐴 is small; hence 𝑥(𝑘) can be arbitrarily large, and the filer is divergent.

5. Simulation Result

The state model is as (34), observations are the difference between INS and GPS position measurement. This simulation chooses a static condition, and its period lasts for 500 s. Gyros constant drift and random white noise 1 sigma are 0.02^∘/h and 0.01^∘/h; accelerator constant drift and random white noise 1 sigma are 100 μg and 10 μg. GPS position measurement white noise 1 sigma is 1 m/s. Corresponding correlation noise matrices are set as 𝑄=diag0_6×1;[0.01m,0.01m,0.01m];0.01^∘/h,0.01^∘/h,0.01^∘/h;10𝜇g,10𝜇g,10𝜇g²,𝑅=diag([1m,1m,1m])². (47) Initial estimation error correlation matrix can be given as 𝑃=diag1^∘,1^∘,1^∘,[1m/s,1m/s,1m/s],[10m,10m,10m],0.1^∘/h,0.1^∘/h,0.1^∘/h,100𝜇g,100𝜇g,100𝜇g. (48) This simulation software details can be found in [19], since compared to INS, GPS measurement is more easily interfered; two scenarios are considered. (i)Normal situation means that there is an accurate prior information for the noise character.(ii)Interfered situation means that the GPS measurement noise is 5 times compared to nominal noise character during 100–200 s.Results are shown in Figures 2 and 3; pink curve represents the real value. Subpictures show the error attitude, error velocity, error position, gyro bias, and accelerator bias in turn. Because the up direction is divergent, only north and east results are shown here.

Figure 2 

Normal situation Kalman filter and robust 𝐻∞ filter results.

Figure 3 

Interfered situation Kalman filter and robust 𝐻∞ filter results.

Triple INS compensate parameters 𝛿𝜙, 𝛿𝑉, 𝛿𝑃 statistical index STD is shown in Tables 1 and 2 for these two situations.

Table 1 

Normal situation STD index.

Table 2 

Interfered situation STD index.

In normal situation, KF shows a more “stable” and “accurate” performance than 𝐻_∞ filter; the 𝐻_∞ filter exhibits a jitter character in the tracking process of real value, whereas in interfered situation, 𝐻_∞ filter is more robust, takes account of the north 𝛿𝑉, 𝐻_∞ filter STD varies from 0.0366 to 0.0372 while KF from 0.0296 to 0.0473. Note that 𝐻_∞ filter contributes a better performance to 𝛿𝜙 in both situations because 𝐻_∞ filter has a “weak” effect on estimation error matrix 𝑃 which means that the initial value 𝑃 has a weak influence. In the follow scenario, we extend the simulation time to 1500 s without calibration to INS using the filter output, result is shown in Figure 4. The Kalman filter suffers a divergence. Although there is a large trend of the estimation error, 𝐻_∞ is stable during this period.

Figure 4 

Normal situation Kalman filter and robust 𝐻∞ filter results for 1500 s.

6. Conclusion

In this paper, we design a robust 𝐻_∞ filter for a loosely coupled INS/GPS model. Firstly, a game theory gives a 𝐻_∞ approach in a Kalman form; it can be easily transformed by adjusting the noise attenuation parameter 𝛾. Next, a robust 𝐻_∞ filter based LMI is discussed; the unstable case is considered. We also illustrate that the robust 𝐻_∞ filter based on LMI approach can convert into standard KF in special conditions. For a direct performance contrast, simulation result shows the INS error estimate state that is not compensated for INS. This proposed approach may be applied in a system under uncertainty such as low-cost IMU and interfered GPS environment.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research was funded by the State Scholarship Fund of the China Scholarships Council (CSC). The authors would like to thank all the editors and anonymous reviewers for improving this paper.