Guaranteed Performance Robust Kalman Filter for Continuous-Time Markovian Jump Nonlinear System with Uncertain Noise

Abstract

Robust Kalman filtering design for continuous-time Markovian jump nonlinear systems with uncertain noise was investigated. Because of complexity of Markovian jump systems, the statistical characteristics of system noise and observation noise are time-varying or unmeasurable instead of being stationary. In view of robust estimation, maximum admissible upper bound of the uncertainty to noise covariance matrix was given based on system state estimation performance. As long as the noise uncertainty is limited within this bound via noise control, the Kalman filter has robustness against noise uncertainty, and stability of dynamic systems can be ensured. It is proved by game theory that this design is a robust mini-max filter. A numerical example shows the validity of this design.

1. Introduction

Optimal filtering problem has been a hot topic in past decades, within which Kalman filtering is one of the most popular estimation approaches and considerable effort has been devoted to its theory and applications. The applications of Kalman filtering theory may be found in a large spectrum of different fields ranging from various engineering problems to biology, geoscience, economics, and management [1]. For standard Kalman filtering, one of the key assumptions is that system noise and observation noise are Gaussian, whose covariances are known and stationary. However, in many actual problems, the statistical characteristics (covariances) of noise may be time-varying instead of being stationary, and in some cases it is impossible to get the exact measurement values, which means that the noise covariances are uncertain instead of being exactly known; for this reason, the stochastic noise is called “uncertain." Consequently, the standard Kalman filter may not be robust against modeling uncertainty and disturbances. Thus, the study of a robust state estimation approach is of practical importance and has been attracting more interest over the past few years. A useful approach is to use a game-theoretic formulation with which one minimizes the worst performance stimulated by uncertain factors, and some corresponding results of robust filtering for linear systems with uncertain noise have been addressed in [2–4].

On the other hand, Markovian jump systems, which are convenient tools for representing many real-world systems [5], have aroused much attention in recent years. In the case of fault detection, fault-tolerant control, and multimodal control, discrete jumps in the continuous dynamics are used to model component failures and sudden switch of system dynamics. With further study of Markovian jump systems, many achievements have been made in the last decade on stability analysis [6, 7], filtering [8, 9], and controller design [10, 11]. Among the efforts towards filtering, Shi et al. [12] and Mahmoud et al. [13] gave Kalman filtering equations for continuous-time and discrete-time Markovian jump linear systems with structure uncertainty, respectively. However, in the above-referred contributions, all the research work is facing the same problem as that of nonjump systems; both the state equation and output measurement are subjected to stationary Gaussian noise so that an optimal filtering gain is obtained based on the stationary noise covariance matrix. But this is only an ideal assumption for Markovian jump systems. As we know, Markovian jump systems are used to represent a class of systems which are usually accompanied by sudden changes of working environment or system dynamics. For this reason, noise uncertainty occurs more frequently or with more probability than in nonjump systems. Moreover, with the uncertainty to noise covariance matrix increasing, the estimation of system state tends to be inaccurate or false, which may cause errors in control signals and in worst case may lead to breakdown of the whole dynamic systems.

To avoid this tragical situation, a direct way dealing with this problem is to redesign Kalman filter for jump systems by using new noise covariance matrix. But as we have pointed out above, it is almost impossible to get the real-time information of noise covariance matrix since it is time-varying or unmeasurable; therefore we could not update Kalman filter gain online. Another feasible way is to give an admissible bound for estimation performance of system state so that the predesigned Kalman filter will remain effective and the system operates in the course of nature as long as the real-time estimation error is within this precision. To achieve this purpose, we perform the following design method. By using the view of robust estimation, a maximum bound of noise covariance matrix uncertainty is obtained through calculation according to admissible bias for estimation performance of system state. If we could ensure the noise uncertainty to be within this bound via technical method such as noise control, the estimation of system state can be within a desired precision, and thus stability of the whole dynamic process can be achieved. It should be noted that in this research work, we do not mean to eliminate the effect of noise entirely because it is almost impossible or highly costly to do so in practical environment. Our work focuses on the upper bound of noise change level; thus it means only that the change to noise covariance matrix is required to be limited within this bound no matter what the stationary covariance matrix is.

In this paper, robust Kalman filtering for continuous-time Markovian jump nonlinear systems with uncertain noise is considered. Firstly, we give some assumptions so that the nonlinear jump systems can be modeled as a linear one by local linearization. Secondly, we seek the maximum upper bound of nonstructural uncertainty to noise covariance matrix such that the deviation of performance can be within a prescribed precision. Then, we discuss the analytical solution of maximum bound using 𝐿𝑎𝑔𝑟𝑎𝑛𝑔𝑒 method. Finally, we prove the establishment of saddle inequality, and show that this filter design is a mini-max robust filter using game theory. At the end of the paper, an illustrative example is used to show the validity of our method.

2. Problem Description

Throughout the paper, unless otherwise specified, we denote by (Ω,ℱ,{ℱ_𝑡}_𝑡≥0,𝐏) a complete probability space with filtration {ℱ_𝑡}_𝑡≥0 satisfying the usual conditions (i.e., it is right-continuous, and ℱ₀ contains all 𝑝-null sets). Let |𝑥| stand for the usual Euclidean norm for a vector 𝑥, and let |𝑋| denote the Frobenius norm of a matrix 𝑋 defined by |𝑋|=𝜆^1/2_max(𝑋𝑋^𝑇), where 𝜆_max(⋅) is the maximum eigenvalue of matrix and the superscript 𝑇 represents transpose. Operator Tr(⋅) denotes the matrix trace, and we denote by 𝑋>0(≥0) that matrix 𝑋 is positive definite (semipositive definite). Let {𝑟(𝑡),𝑡≥0} be a right-continuous Markov process on the probability space taking values in finite state space 𝑆={1,2,…,𝑁} with Π=[𝜋_𝑖𝑗] being the chain generator, an 𝑁×𝑁 matrix. The entries 𝜋_𝑖𝑗,𝑖,𝑗∈𝑆, are interpreted as transition rates such that 𝜋𝐏(𝑟(𝑡+𝑑𝑡)=𝑗∣𝑟(𝑡)=𝑖)=^𝑖𝑗𝑑𝑡+𝑜(𝑑𝑡)if𝑖≠𝑗,1+𝜋_𝑖𝑗𝑑𝑡+𝑜(𝑑𝑡)if𝑖=𝑗, (2.1) where 𝑑𝑡>0 and lim_𝑑𝑡→0(𝑜(𝑑𝑡)/𝑑𝑡)=0. Here, 𝜋_𝑖𝑗>0(𝑖≠𝑗) is the transition rate from 𝑖 to 𝑗. Notice that the total probability axiom imposes 𝜋_𝑖𝑖<0 and ^𝑁_𝑗=1𝜋_𝑖𝑗=0,∀𝑖∈𝑆. (2.2) Consider the following continuous-time Markovian jump nonlinear system with uncertain noise: 𝑥̇=𝑓(𝑥,𝑟(𝑡))+𝜔⁰,𝑦=ℎ(𝑥,𝑟(𝑡))+𝜐⁰, (2.3) where 𝑥∈𝐑^𝑛 is state vector, and 𝑦∈𝐑^𝑚 is measurement output. 𝑓(⋅,⋅)∈𝐑^𝑛 and ℎ(⋅,⋅)∈𝐑^𝑚 are nonlinear vector functions. 𝜔⁰ and 𝜐⁰ are 𝑛-dimensional and 𝑚-dimensional white noises and satisfy the following assumption.

Assumption 2.1. For any given time 𝑠,𝑡≥0, there are
(1)𝐸[𝜔⁰_𝑡]=0,𝐸[𝜐⁰_𝑡]=0,(2)Cov[𝜔⁰_𝑡,𝜔⁰_𝑠]=𝑊⁰𝛿_𝑡,𝑠=(𝑊+Δ𝑊)𝛿_𝑡,𝑠,𝑊≥0,Δ𝑊≥0,(3)Cov[𝜐⁰_𝑡,𝜐⁰_𝑠]=𝑉⁰𝛿_𝑡,𝑠=(𝑉+Δ𝑉)𝛿_𝑡,𝑠,𝑉>0,Δ𝑉≥0,(4)𝐸[(^𝑤0𝑡_𝜐0𝑡)^𝑤0𝑡_𝜐0𝑡⋅(𝑤_𝑠^0𝑇𝜐_𝑠^0𝑇)]=[^{𝑊0𝛿𝑡,𝑠0}_{0𝑉0𝛿𝑡,𝑠}]. In Assumption 2.1, 𝑊⁰∈𝐑^𝑛×𝑛 and 𝑉⁰∈𝐑^𝑚×𝑚 consist of two parts, where 𝑊 and 𝑉 denote the stationary noise covariance matrix, whose values are exactly known. Δ𝑊 and Δ𝑉 denote the noise uncertainty caused by time-varying or sudden switch of system dynamics; they are unknown but norm-bounded. 𝛿(⋅,⋅) is a 𝐷𝑖𝑟𝑎𝑐 function taking values in {0,1}. For the deduction of robust Kalman filter, we introduce the following assumption.

Assumption 2.2. For any fixed system mode 𝑟(𝑡)=𝑖∈𝑆 and vector 𝜎∈𝐑^𝑛, the nonlinear vector functions 𝑓(⋅,⋅),ℎ(⋅,⋅) are assumed to satisfy 𝑓(0,𝑖)=ℎ(0,𝑖)=0 and |𝑓(𝑥+𝜎,𝑖)−𝑓(𝑥,𝑖)−𝐴(𝑖)𝜎|≤|Δ𝐴(𝑖)||𝜎|,|ℎ(𝑥+𝜎,𝑖)−ℎ(𝑥,𝑖)−𝐶(𝑖)𝜎|≤|Δ𝐶(𝑖)||𝜎|, (2.4) where 𝐴(𝑖),𝐶(𝑖) are Jacobian matrices of 𝑓(⋅,⋅),ℎ(⋅,⋅), and Δ𝐴(𝑖),Δ𝐶(𝑖) satisfy Δ𝐴(𝑖)=𝐻₁(𝑖)𝐹(𝑖)𝐸(𝑖),Δ𝐶(𝑖)=𝐻₂(𝑖)𝐹(𝑖)𝐸(𝑖), (2.5) where 𝐻₁(𝑖),𝐻₂(𝑖), and 𝐸(𝑖),𝑖∈𝑆, are known constant matrices, and 𝐹(𝑖),𝑖∈𝑆, is unknown matrix satisfying 𝐹^𝑇(𝑖)𝐹(𝑖)≤𝐼. Establishing Assumption 2.2, the Markovian jump nonlinear system could be transformed to a nominal linear model via local linearization technique: 𝑥̇=[𝐴(𝑟(𝑡))+Δ𝐴(𝑟(𝑡))]𝑥+𝜔⁰,𝑦=[𝐶(𝑟(𝑡))+Δ𝐶(𝑟(𝑡))]𝑥+𝜐⁰. (2.6) For simplification, we denote 𝐴(𝑟(𝑡)=𝑖),𝐻₁(𝑟(𝑡)=𝑖),𝐻₂(𝑟(𝑡)=𝑖),𝐸(𝑟(𝑡)=𝑖),Δ𝐴(𝑟(𝑡)=𝑖),𝐶(𝑟(𝑡)=𝑖), and Δ𝐶(𝑟(𝑡)=𝑖) by 𝐴_𝑖,𝐻_1𝑖,𝐻_2𝑖,𝐸_𝑖,Δ𝐴_𝑖,𝐶_𝑖, and Δ𝐶_𝑖.

Theorem 2.3. Consider stochastically stable Markovian jump system (2.6), and assume that the noise is stationary, which means that Δ𝑊=Δ𝑉=0. Then, one has the following standard Kalman filter (see [12]): 𝑥̇=𝐴_𝑖𝑥+𝐾_𝑖[𝑦−𝐶_𝑖𝑥], (2.7) where filtering gain 𝐾_𝑖 is given by the following coupled Riccati equations: 𝐴_𝑖=𝐴_𝑖1+(𝜖_𝑖𝐻_1𝑖𝐻^𝑇_1𝑖+𝑊)𝑃_𝑖⁻¹,𝐶_𝑖=1𝜖_𝑖𝐻_2𝑖𝐻_1𝑖𝑃_𝑖⁻¹+𝐶_𝑖,𝐴_𝑖𝑃_𝑖+𝑃_𝑖𝐴^𝑇_𝑖+𝜖_𝑖𝑃_𝑖𝐸^𝑇_𝑖𝐸_𝑖𝑃_𝑖+∑^𝑁_𝑗=1𝜋_𝑖𝑗𝑃_𝑗+1𝜖_𝑖𝐻_1𝑖𝐻^𝑇_1𝑖𝐾+𝑊=0,_𝑖=(𝑄_𝑖𝐶^𝑇_𝑖+1𝜖_𝑖𝐻_1𝑖𝐻^𝑇_2𝑖1)(𝜖_𝑖𝐻_2𝑖𝐻^𝑇_2𝑖+𝑉)⁻¹,(𝐴_𝑖−𝐾_𝑖𝐶_𝑖)𝑄_𝑖+𝑄_𝑖(𝐴_𝑖−𝐾_𝑖𝐶_𝑖)^𝑇+𝐾_𝑖𝑉𝐾^𝑇_𝑖+∑^𝑁_𝑗=1𝜋_𝑖𝑗𝑄_𝑗+1𝜖_𝑖𝐻_1𝑖𝐻^𝑇_1𝑖+𝑊=0. (2.8) Here, matrices 𝑃_𝑖>0,𝑄_𝑖>0 and scalar 𝜖_𝑖 are chosen so that tr(𝑄_𝑖) reaches the minimum.
With the above standard Kalman filter gain 𝐾_𝑖 adopted, the state estimation error is 𝐸{(𝑥−𝑥)^𝑇(𝑥−𝑥)}≤max_𝑗∈𝑆tr(𝑄_𝑗). (2.9)

Define the estimation error performance as 𝐽(𝐾₁,𝐾₂,…,𝐾_𝑁,𝑊,𝑉)=max_𝑗∈𝑆tr(𝑄_𝑗). (2.10) According to Theorem 2.3 and the quality of Kalman filtering, if the noise is stationary (Δ𝑊=Δ𝑉=0), the estimation error performance can achieve the minimum value by adopting standard Kalman filtering (2.7).

Now, we consider that the noise is not stationary, which means that Δ𝑊≠0 and Δ𝑉≠0; thus the new covariance matrix of noise is 𝑊⁰,𝑉⁰. If we still adopt the former predesigned standard Kalman filter gain 𝐾_𝑖, the new state estimation error should be 𝑄⁰_𝑖, which satisfies (𝐴_𝑖−𝐾_𝑖𝐶_𝑖)𝑄⁰_𝑖+𝑄⁰_𝑖(𝐴_𝑖−𝐾_𝑖𝐶_𝑖)^𝑇+𝐾_𝑖𝑉⁰𝐾^𝑇_𝑖+^𝑁_𝑗=1𝜋_𝑖𝑗𝑄⁰_𝑗+1𝜖_𝑖𝐻_1𝑖𝐻^𝑇_1𝑖+𝑊⁰=0. (2.11) Therefore, the new estimation error performance is 𝐽(𝐾₁,𝐾₂,…,𝐾_𝑁,𝑊⁰,𝑉⁰)=max_𝑗∈𝑆tr(𝑄⁰_𝑗). (2.12)

According to (2.10) and (2.12), the deviation of estimation error performance yielded by noise uncertainty (Δ𝑊,Δ𝑉) can be written as Δ𝐽(𝐾₁,𝐾₂,…,𝐾_𝑁,Δ𝑊,Δ𝑉)=𝐽(𝐾₁,…,𝐾_𝑁,𝑊⁰,𝑉⁰)−𝐽(𝐾₁,…,𝐾_𝑁,𝑊,𝑉)=max_𝑗∈𝑆tr(𝑄⁰_𝑗)−max_𝑗∈𝑆tr(𝑄_𝑗)≤𝑟, (2.13) where 𝑟>0 is a parameter which is given according to detailed precision request of practical dynamic process.

Our design purpose is shown in Figures 1 and 2. Suppose that the noise is stationary with covariance matrix 𝑊,𝑉, and that the system filtering performance is 𝐽 with standard filtering gain 𝐾_𝑖 adopted as shown in Figure 1. But now noise is with uncertainty (Δ𝑊,Δ𝑉); the former designed Kalman filter gain 𝐾_𝑖 will no longer be an optimal one. If we still want to get the precise estimation of system state so that the dynamic system could remain stable, there are two choices. One is to update the Kalman filter gain 𝐾_𝑖 according to new noise covariance matrix (𝑊+Δ𝑊,𝑉+Δ𝑉), but this way is impossible or highly costly. Another way is to still adopt the former designed filtering gain 𝐾_𝑖 and take some actions in noise control. Based on this idea, the new estimation performance is 𝐽+Δ𝐽 with 𝐾_𝑖 adopted, and a deviation Δ𝐽 occurs resulting from noise uncertainty (see Figure 2). For the robustness of system, which means that Δ𝐽 is less than an admissible precision 𝑟, there must be some limitation to noise uncertainty. Using the view of robust estimation, we are trying to find a maximum upper bound 𝑎,𝑏 for the uncertainty to noise covariance matrix. As long as the noise uncertainty is controlled to satisfy |Δ𝑊|≤𝑎,|Δ𝑉|≤𝑏 via noise control, we will achieve deviation of estimation performance to be within the admissible precision 𝑟, which means that Δ𝐽≤𝑟; thus the general system has robustness to noise uncertainty, and stability of the whole dynamic process can be maintained whatever the original stationary noise covariance matrix (𝑊,𝑉) is. In the following part of this paper, we seek the solution of maximum upper bound 𝑎,𝑏.

Figure 1 

Standard Kalman filter with stationary noise.

Figure 2 

Robust Kalman filter with uncertain noise.

3. Upper Bound of Nonstructural Noise Uncertainty

3.1. Mathematical Expression of Upper Bound

According to (2.8) and (2.11), we have (𝐴_𝑖−𝐾_𝑖𝐶_𝑖)Δ𝑄_𝑖+Δ𝑄_𝑖(𝐴_𝑖−𝐾_𝑖𝐶_𝑖)^𝑇+^𝑁_𝑗=1𝜋_𝑖𝑗Δ𝑄_𝑗+Δ𝑊+𝐾_𝑖Δ𝑉𝐾^𝑇_𝑖=0, (3.1) where Δ𝑄_𝑖=𝑄⁰_𝑖−𝑄_𝑖. From the above equation, it is easily seen that tr(Δ𝑄_𝑖) is a linear mapping of (Δ𝑊,Δ𝑉). Define a compact convex set as Ξ={(Δ𝑊,Δ𝑉)∶0≤Δ𝑊≤Δ𝑊^∗,0≤Δ𝑉≤Δ𝑉^∗}; thus the deviation of performance Δ𝐽(𝐾₁,𝐾₂,…,𝐾_𝑁,Δ𝑊,Δ𝑉) is a mapping from Ξ to 𝐑¹, and it has the following facts.

Fact 1. For any given noise uncertainty (Δ𝑊_𝑗,Δ𝑉_𝑗)∈Ξ,𝑗=1,2, if Δ𝑊₁≤Δ𝑊₂ and Δ𝑉₁≤Δ𝑉₂, one has Δ𝐽(𝐾₁,𝐾₂,…,𝐾_𝑁,Δ𝑊₁,Δ𝑉₁)≤Δ𝐽(𝐾₁,𝐾₂,…,𝐾_𝑁,Δ𝑊₂,Δ𝑉₂). (3.2)

Fact 2. Define the maximum admissible deviation of estimation performance 𝑟 as 𝑟=max_{(Δ𝑊,Δ𝑉)∈Ξ}Δ𝐽(𝐾₁,𝐾₂,…,𝐾_𝑁,Δ𝑊,Δ𝑉). (3.3) Thus, 𝑟 could be achieved only by maximum noise uncertainty (Δ𝑊^∗,Δ𝑉^∗), which means that 𝑟=Δ𝐽(𝐾₁,𝐾₂,…,𝐾_𝑁,Δ𝑊^∗,Δ𝑉^∗). (3.4) The purpose of the following work is to construct a maximal compact convex set Ξ^∗, as long as the noise uncertainty satisfies (Δ𝑊,Δ𝑉)∈Ξ^∗, (2.13) is sure to establish. According to the finity of mode 𝑆, (2.13) is equivalent to tr(𝑄⁰_𝑖)≤𝑟+max_𝑗∈𝑆tr(𝑄_𝑗). (3.5) Therefore, for each mode 𝑖∈𝑆, there is tr(Δ𝑄_𝑖)=tr(𝑄⁰_𝑖)−tr(𝑄_𝑖)≤𝑟+max_𝑗∈𝑆tr(𝑄_𝑗)−tr(𝑄_𝑖). (3.6) Define the maximum upper bound of noise uncertainty as |Δ𝑊|≤𝑎,|Δ𝑉|≤𝑏; thus 0≤Δ𝑊≤𝑎𝐼_𝑛,0≤Δ𝑉≤𝑏𝐼_𝑚. (3.7) Substituting the above inequalities into (3.1) and (3.6), one has 𝑎tr(𝐷_𝑖)+𝑏tr(𝐺_𝑖)≤𝑟+max_𝑗∈𝑆tr(𝑄_𝑗)−tr(𝑄_𝑖),∀𝑖∈𝑆, (3.8) where matrices 𝐷_𝑖,𝐺_𝑖>0,𝑖∈𝑆, satisfy the following coupled Riccati equations: (𝐴_𝑖−𝐾_𝑖𝐶_𝑖)𝐷_𝑖+𝐷_𝑖(𝐴_𝑖−𝐾_𝑖𝐶_𝑖)^𝑇+∑^𝑁_𝑗=1𝜋_𝑖𝑗𝐷_𝑗+𝐼_𝑛(𝐴=0,_𝑖−𝐾_𝑖𝐶_𝑖)𝐺_𝑖+𝐺_𝑖(𝐴_𝑖−𝐾_𝑖𝐶_𝑖)^𝑇+∑^𝑁_𝑗=1𝜋_𝑖𝑗𝐺_𝑗+𝐾_𝑖𝐾^𝑇_𝑖=0. (3.9) According to the above analysis, seeking a maximum upper bound of noise uncertainty (Δ𝑊,Δ𝑉) is equivalent to obtaining the optimal solution of 𝑎,𝑏: max𝑎⋅𝑏s.t.𝑎⋅tr(𝐷_𝑖)+𝑏⋅tr(𝐺_𝑖)≤𝑟+max_𝑗∈𝑆tr(𝑄_𝑗)−tr(𝑄_𝑖),𝑎≥0,𝑏≥0,𝑖∈𝑆. (3.10) Thus, seeking the optimal solution 𝑎,𝑏 is transformed to a nonlinear programming problem with linear inequalities' constraints. Now, we discuss how to find the analytical solution of such problem.

3.2. Analytical Solution

Since Ξ={(Δ𝑊,Δ𝑉)} is a compact convex set and the inequalities' constraints in (3.10) compose a compact set on which 𝑎⋅𝑏 is defined as a continuous function, thus the nonlinear programming problem must have optimal solution 𝑎^∗,𝑏^∗ and the existence of solution is proved. Next, we will seek the analytical solution 𝑎^∗,𝑏^∗.

Decompose the original nonlinear programming problem (see (3.10)) into 𝑁 subproblems: max𝑎₁⋅𝑏₁s.t.𝑎₁⋅tr(𝐷₁)+𝑏₁⋅tr(𝐺₁)≤𝑟+max_𝑗∈𝑆tr(𝑄_𝑗)−tr(𝑄₁)max𝑎₂⋅𝑏₂s.t.𝑎₂⋅tr(𝐷₂)+𝑏₂⋅tr(𝐺₂)≤𝑟+max_𝑗∈𝑆tr(𝑄_𝑗)−tr(𝑄₂)⋮max𝑎_𝑁⋅𝑏_𝑁s.t.𝑎_𝑁⋅tr(𝐷_𝑁)+𝑏_𝑁⋅tr(𝐺_𝑁)≤𝑟+max_𝑗∈𝑆tr(𝑄_𝑗)−tr(𝑄_𝑁). (3.11) By using 𝐿𝑎𝑔𝑟𝑎𝑛𝑔𝑒 method, we have the optimal analytical solution for each subproblem as 𝑎^∗_𝑖=𝑟+max_𝑗∈𝑆tr(𝑄_𝑗)−tr(𝑄_𝑖)2tr(𝐷_𝑖),𝑏^∗_𝑖=𝑟+max_𝑗∈𝑆tr(𝑄_𝑗)−tr(𝑄_𝑖)2tr(𝐺_𝑖). (3.12) Thus, the analytical solution for the original nonlinear programming problem (see (3.10)) is given as 𝑎^∗=min_𝑖∈𝑆𝑎^∗_𝑖=min_𝑖∈𝑆{𝑟+max_𝑗∈𝑆tr(𝑄_𝑗)−tr(𝑄_𝑖)2tr(𝐷_𝑖)𝑏},_∗=min_𝑖∈𝑆𝑏^∗_𝑖=min_𝑖∈𝑆{𝑟+max_𝑗∈𝑆tr(𝑄_𝑗)−tr(𝑄_𝑖)2tr(𝐺_𝑖)}. (3.13)

Remark 3.1. The analytical solution of the nonlinear programming problem is given by the above analysis; however, it is only an optimal solution for each subproblem. This analytical solution in (3.13) is local optimal, but global suboptimal. For the global optimal solution, we could only get the numerical solution using “fmincon" function in MATLAB software. The optimal analytical solution of such nonlinear programming problem is still an open problem in mathematics for further exploration.

Theorem 3.2. Consider Markovian jump system (2.6). If one adopts state estimator (2.7) and Kalman filter gain (2.8), there exists a maximum admissible compact set Ξ. As long as the uncertainty to noise covariance matrix satisfies (Δ𝑊,Δ𝑉)∈Ξ, the deviation of system state estimation error performance Δ𝐽 is within a given precision 𝑟.

Remark 3.3. Take into account the existence of noise uncertainty, and the new noise covariance matrix is given as (𝑊+Δ𝑊,𝑉+Δ𝑉). Thus, the former predesigned optimal Kalman filter gain 𝐾_𝑖, which is deduced from stationary noise (𝑊,𝑉), will no longer be optimal and may cause distortion of control signals. But, this does not mean that we need to redesign Kalman filter. According to the above analysis, if we can successfully limit noise uncertainty (Δ𝑊,Δ𝑉) to be within an admissible compact set Ξ, the predesigned Kalman filter gain 𝐾_𝑖 can still be effective though it is not optimal. Moreover, the deviation of estimation performance is ensured to be within a desirable precision 𝑟.

4. Mini-Max Robust Filter

Let 𝐾^∗₁,𝐾^∗₂,…,𝐾^∗_𝑁 denote the standard Kalman filtering gain according to new noise covariance matrix pair (𝑊^∗,𝑉^∗)=(𝑊+Δ𝑊^∗,𝑉+Δ𝑉^∗), which corresponds to the maximum admissible noise uncertainty (Δ𝑊^∗,Δ𝑉^∗); thus 𝐾^∗₁,𝐾^∗₂,…,𝐾^∗_𝑁 satisfy 𝐴_𝑖𝑃^∗_𝑖+𝑃^∗_𝑖𝐴^𝑇_𝑖+𝜖_𝑖𝑃^∗_𝑖𝐸^𝑇_𝑖𝐸_𝑖𝑃^∗_𝑖+∑^𝑁_𝑗=1𝜋_𝑖𝑗𝑃^∗_𝑗+1𝜖_𝑖𝐻_1𝑖𝐻^𝑇_1𝑖+𝑊^∗𝐾=0,_∗𝑖=(𝑄^∗_𝑖𝐶^𝑇_𝑖+1𝜖_𝑖𝐻_1𝑖𝐻^𝑇_2𝑖1)(𝜖_𝑖𝐻_2𝑖𝐻^𝑇_2𝑖+𝑉^∗)⁻¹,(𝐴_𝑖−𝐾^∗_𝑖𝐶_𝑖)𝑄^∗_𝑖+𝑄^∗_𝑖(𝐴_𝑖−𝐾^∗_𝑖𝐶_𝑖)^𝑇+𝐾^∗_𝑖𝑉^∗𝐾_𝑖^∗𝑇+∑^𝑁_𝑗=1𝜋_𝑖𝑗𝑄^∗_𝑗+1𝜖_𝑖𝐻_1𝑖𝐻^𝑇_1𝑖+𝑊^∗=0. (4.1) According to the least-square quality of standard Kalman filtering, we have Δ𝐽(𝐾^∗₁,𝐾^∗₂,…,𝐾^∗_𝑁,Δ𝑊^∗,Δ𝑉^∗)≤Δ𝐽(𝐾₁,𝐾₂,…,𝐾_𝑁,Δ𝑊^∗,Δ𝑉^∗). (4.2) On the other hand, with the establishment of Fact 1, there is Δ𝐽(𝐾^∗₁,𝐾^∗₂,…,𝐾^∗_𝑁,Δ𝑊,Δ𝑉)≤Δ𝐽(𝐾^∗₁,𝐾^∗₂,…,𝐾^∗_𝑁,Δ𝑊^∗,Δ𝑉^∗). (4.3) Thus, we have the following saddle-point inequality: Δ𝐽(𝐾^∗₁,𝐾^∗₂,…,𝐾^∗_𝑁,Δ𝑊,Δ𝑉)≤Δ𝐽(𝐾^∗₁,𝐾^∗₂,…,𝐾^∗_𝑁,Δ𝑊^∗,Δ𝑉^∗)≤Δ𝐽(𝐾₁,𝐾₂,…,𝐾_𝑁,Δ𝑊^∗,Δ𝑉^∗). (4.4) By game theory, we have min_𝐾𝑖max_{(Δ𝑊,Δ𝑉)∈Ξ}Δ𝐽(𝐾₁,𝐾₂,…,𝐾_𝑁,Δ𝑊,Δ𝑉)=max_{(Δ𝑊,Δ𝑉)∈Ξ}min_𝐾𝑖Δ𝐽(𝐾₁,𝐾₂,…,𝐾_𝑁,Δ𝑊,Δ𝑉). (4.5) This means that the optimal estimation under the worst situation is a mini-max filter. It cannot only minimize the estimation performance under the largest noise uncertainty (𝑎^∗𝐼_𝑛,𝑏^∗𝐼_𝑚), but can also ensure the deviation to be within a given precision 𝑟. For this reason, this Kalman filter design is a robust mini-max filter.

Remark 4.1. Traditional Kalman filtering design is performed on the basis that noise covariance matrix is stationary and exactly known, and it will fail when the noise covariance matrix is unknown or has uncertainty. In our method, the filter design could be divided into two steps. Firstly, we design standard Kalman filter according to the stationary noise covariance matrix (𝑊,𝑉), then via some technical methods such as noise control we impose the noise uncertainty to be within the given bound (Δ𝑊^∗,Δ𝑉^∗), which could be presented in the form of nonstructural (𝑎^∗𝐼_𝑛,𝑏^∗𝐼_𝑚). In practical dynamic process, when the noise uncertainty reaches the maximum, the ideal deviation of performance is Δ𝐽(𝐾^∗₁,𝐾^∗₂,…,𝐾^∗_𝑁,Δ𝑊^∗,Δ𝑉^∗), and this deviation is less than the worst case Δ𝐽(𝐾₁,𝐾₂,…,𝐾_𝑁,Δ𝑊^∗,Δ𝑉^∗)≤𝑟, which ensures the estimation of system states and control signals to be precise to some extent, and the synthetical system to be robust and stable. For this reason, the Kalman filter design has robustness to noise uncertainty, and according to (4.5), this filter is also a mini-max filter .

5. Simulation

Consider the following two-mode Markovian jump system.

Let the system mode 𝑟(𝑡)=1 be given by 𝑥̇₁=−0.6𝑥₁+0.5𝑥₂+0.01sin(𝑥₁+𝑥₂)+𝜔⁰₁,𝑥̇₂=0.7𝑥₁+0.02cos(𝑥₁−𝑥₂)+𝜔⁰₂,𝑦=𝑥₁+0.5𝑥₂+𝜐⁰. (5.1) Let the system mode 𝑟(𝑡)=2 be given by 𝑥̇₁=−𝑥₁+0.6𝑥₂+0.02sin𝑥₂+𝜔⁰₁,𝑥̇₂=0.8𝑥₁−1.1𝑥₂+0.02cos𝑥₁+𝜔⁰₂,𝑦=𝑥₁+𝜐⁰, (5.2) where uncertain state and measurement noise are 𝜔⁰=[𝜔⁰₁𝜔⁰₂]𝑇𝜔⁰₁𝜔⁰₂]^𝑇 and 𝜐⁰; its stationary covariance matrix is known as 𝑊=[¹⁰₀₁],𝑉=1; system mode transition matrix is Π=[^−0.60.6_0.9−0.9]; the admissible bound of performance deviation is 𝑟=0.3.

The detailed algorithm is as follows.

(1)By applying Assumption 2.1, we have 𝐴₁=[^−0.60.5_0.70],𝐶₁=[10.5]10.5,𝐴₂=[^−10.6_0.8−1.1],𝐶₂=[10]10,𝐻₁₁=[0.10.2]𝑇0.10.2]^𝑇,𝐸₁=[0.10.1]0.10.1,𝐻₁₂=[0.10.1]𝑇0.10.1]^𝑇, 𝐸₂=[0.20.2]0.20.2. Notice that Δ𝐶_𝑖≡0; thus 𝐻₂₁=𝐻₂₂=0.(2)Solve (2.8) and get 𝑄₁,𝑄₂ and 𝐾₁,𝐾₂∶𝑄₁=[^2.60581.4207_1.42073.1934],𝑄₂=[^2.65731.4181_1.41812.8473],𝐾₁=[^0.8748_1.0921],𝐾₂=[^0.6567_0.4866].(3)Substitute the result to (3.10), using 𝐿𝑎𝑔𝑟𝑎𝑛𝑔𝑒 method; the upper bound of noise uncertainty is given as 𝑎^∗=0.1014,𝑏^∗=0.1701. (5.3)(4)Let the new noise covariance matrix correspond to the maximum uncertainty: 𝑊^∗=𝑊+Δ𝑊=𝑊+𝑎^∗⋅𝐼₂,𝑉^∗=𝑉+Δ𝑉=𝑉+𝑏^∗⋅𝐼₁. (5.4)(5)Repeat step (2), and we have the correspondent 𝑄^∗₁,𝑄^∗₂,𝐾^∗₁,𝐾^∗₂ for new noise covariance matrix (𝑊^∗,𝑉^∗): 𝑄^∗₁=2.76411.44171.44173.2688,𝑄^∗₂=,𝐾2.67881.44581.44582.9011_∗1=0.89771.3021,𝐾^∗₂=.0.70120.5041 (5.5)(6)Applying the robust Kalman filtering, there is saddle-point inequality: Δ𝐽(𝐾^∗₁,𝐾^∗₂,Δ𝑊,Δ𝑉)≤Δ𝐽(𝐾^∗₁,𝐾^∗₂,Δ𝑊^∗,Δ𝑉^∗)=max{tr(𝑄^∗₁),tr(𝑄^∗₂)}−max{tr(𝑄₁),tr(𝑄₂)}=0.2337<0.3=Δ𝐽(𝐾₁,𝐾₂,Δ𝑊^∗,Δ𝑉^∗). (5.6) From the above simulation, it is seen that with the noise uncertainty being limited within the upper bound 𝑎^∗,𝑏^∗ via noise control, the deviation of system estimation performance is less than the admissible precision 𝑟. Because the analytical solution 𝑎^∗,𝑏^∗ is global suboptimal, the deviation of system estimation performance (0.2337) is obviously less than admissible precision (0.3), which means that this solution is a conservative one and the global optimal solution of 𝑎,𝑏 could be a little greater than 𝑎^∗,𝑏^∗. Thus, this method allows flexibility to the designer to some extent.

6. Conclusion

In this paper, robust Kalman filter for continuous-time Markovian jump nonlinear systems with uncertain noise is considered. For the stability of dynamic system when statistical information of noise is unavailable, a new design method is given by obtaining the maximum admissible bound of uncertainty to noise covariance matrix. Based on this, the deviation of system estimation performance is thus guaranteed to be within a given precision. Furthermore, the worst performance yielded by noise uncertainty can be minimized by this method since it is a mini-max robust filter. The analytical solution of the bound to noise uncertainty is also discussed in this paper, which is a global, suboptimal, and conservative solution using 𝐿𝑎𝑔𝑟𝑎𝑛𝑔𝑒 method. The simulation results show the validity of this design.

Acknowledgment

This work has been funded by BK21 research project: Switching Control of Systems with Structure Uncertainty and Noise.