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

Authors: Jin Zhu; Junhong Park; Kwan-Soo Lee; Maksym Spiryagin
Journal: Mathematical Problems in Engineering
 (2008)

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## 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.


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## Body

## 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 usingLagrange 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,{ℱt}t≥0,P) a complete probability space with filtration {ℱt}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|=λmax1/2(XXT),
where λmax(⋅) 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=[πij] being the chain generator, an N×N matrix. The entries πij,i,j∈S, are interpreted as transition rates such
that (2.1)P(r(t+dt)=j∣r(t)=i)={πijdt+o(dt)ifi≠j,1+πijdt+o(dt)ifi=j, where dt>0 and lim⁡dt→0(o(dt)/dt)=0. Here, πij>0(i≠j) is the transition rate from i to j. Notice that the total probability axiom
imposes πii<0 and (2.2)∑j=1Nπij=0,∀i∈S. Consider the following continuous-time Markovian jump
nonlinear system with uncertain noise: (2.3)x˙=f(x,r(t))+ω0,y=h(x,r(t))+υ0, where x∈Rn is state vector, and y∈Rm is measurement output. f(⋅,⋅)∈Rn and h(⋅,⋅)∈Rm 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 times,t≥0,
there are
(1)
E[ωt0]=0,E[υt0]=0,(2)
Cov[ωt0,ωs0]=W0δt,s=(W+ΔW)δt,s,W≥0,ΔW≥0,(3)
Cov[υt0,υs0]=V0δt,s=(V+ΔV)δt,s,V>0,ΔV≥0,(4)
E[(wt0υt0)wt0υt0⋅(ws0Tυs0T)]=[W0δt,s00V0δt,s]. In Assumption 2.1, W0∈Rn×n and V0∈Rm×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 moder(t)=i∈S and vector σ∈Rn,
the nonlinear vector functions f(⋅,⋅),h(⋅,⋅) are assumed to satisfy f(0,i)=h(0,i)=0 and (2.4)|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|, where A(i),C(i) are Jacobian matrices of f(⋅,⋅),h(⋅,⋅),
and ΔA(i),ΔC(i) satisfy (2.5)ΔA(i)=H1(i)F(i)E(i),ΔC(i)=H2(i)F(i)E(i), where H1(i),H2(i),
and E(i),i∈S, are known constant matrices, and F(i),i∈S, is unknown matrix satisfying FT(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: (2.6)x˙=[A(r(t))+ΔA(r(t))]x+ω0,y=[C(r(t))+ΔC(r(t))]x+υ0. 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 Ai,H1i,H2i,Ei,ΔAi,Ci,
and ΔCi.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]): (2.7)x^˙=A^ix^+Ki[y−C^ix^], where filtering gain Ki is given by the following coupled Riccati
equations: (2.8)A^i=Ai+(1ϵiH1iH1iT+W)Pi−1,C^i=1ϵiH2iH1iPi−1+Ci,AiPi+PiAiT+ϵiPiEiTEiPi+∑j=1NπijPj+1ϵiH1iH1iT+W=0,Ki=(QiC^iT+1ϵiH1iH2iT)(1ϵiH2iH2iT+V)−1,(A^i−KiC^i)Qi+Qi(A^i−KiC^i)T+KiVKiT+∑j=1NπijQj+1ϵiH1iH1iT+W=0. Here,
matrices Pi>0,Qi>0 and scalar ϵi are chosen so that tr(Qi) reaches the minimum.
With the above standard Kalman filter gainKi adopted, the state estimation error
is (2.9)E{(x−x^)T(x−x^)}≤max⁡j∈Str(Qj).Define the estimation error performance as(2.10)J(K1,K2,…,KN,W,V)=max⁡j∈Str(Qj). 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 thatΔW≠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 Ki,
the new state estimation error should be Qi0,
which satisfies (2.11)(A^i−KiC^i)Qi0+Qi0(A^i−KiC^i)T+KiV0KiT+∑j=1NπijQj0+1ϵiH1iH1iT+W0=0. Therefore, the new estimation error
performance is (2.12)J(K1,K2,…,KN,W0,V0)=max⁡j∈Str(Qj0).According to
(2.10) and (2.12), the deviation of estimation error performance yielded by noise
uncertainty (ΔW,ΔV) can be written as (2.13)ΔJ(K1,K2,…,KN,ΔW,ΔV)=J(K1,…,KN,W0,V0)−J(K1,…,KN,W,V)=max⁡j∈S⁢tr(Qj0)−max⁡j∈Str(Qj)≤r, where r>0 is a parameter which is given according to
detailed precision request of practical dynamic process.Our design purpose is shown in Figures1 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 Ki adopted as shown in Figure 1. But now noise is
with uncertainty (ΔW,ΔV);
the former designed Kalman filter gain Ki 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 Ki 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 Ki and take some actions in noise control. Based
on this idea, the new estimation performance is J+ΔJ with Ki 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 (3.1)(A^i−KiC^i)ΔQi+ΔQi(A^i−KiC^i)T+∑j=1NπijΔQj+ΔW+KiΔVKiT=0, where ΔQi=Qi0−Qi.
From the above equation, it is easily seen that tr(ΔQi) 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,…,KN,ΔW,ΔV) is a mapping from Ξ to R1,
and it has the following facts.Fact 1.
For any given noise uncertainty(ΔWj,ΔVj)∈Ξ,j=1,2,
if ΔW1≤ΔW2 and ΔV1≤ΔV2,
one has (3.2)ΔJ(K1,K2,…,KN,ΔW1,ΔV1)≤ΔJ(K1,K2,…,KN,ΔW2,ΔV2).Fact 2.
Define the maximum
admissible deviation of estimation performancer as (3.3)r=max⁡(ΔW,ΔV)∈ΞΔJ(K1,K2,…,KN,ΔW,ΔV). Thus, r could be achieved only by maximum noise
uncertainty (ΔW*,ΔV*),
which means that (3.4)r=ΔJ(K1,K2,…,KN,ΔW*,ΔV*). 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 (3.5)tr(Qi0)≤r+max⁡j∈Str(Qj). Therefore, for each mode i∈S,
there is (3.6)tr(ΔQi)=tr(Qi0)−tr(Qi)≤r+max⁡j∈Str(Qj)−tr(Qi). Define the maximum upper bound
of noise uncertainty as |ΔW|≤a,|ΔV|≤b;
thus (3.7)0≤ΔW≤aIn,0≤ΔV≤bIm. Substituting the above
inequalities into (3.1) and (3.6), one has (3.8)atr(Di)+btr(Gi)≤r+max⁡j∈Str(Qj)−tr(Qi),∀i∈S, where matrices Di,Gi>0,i∈S, satisfy the following coupled Riccati
equations: (3.9)(A^i−KiC^i)Di+Di(A^i−KiC^i)T+∑j=1NπijDj+In=0,(A^i−KiC^i)Gi+Gi(A^i−KiC^i)T+∑j=1NπijGj+KiKiT=0. 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: (3.10)maxa⋅bs.t.a⋅tr(Di)+b⋅tr(Gi)≤r+max⁡j∈Str(Qj)−tr(Qi),a≥0,b≥0,i∈S. 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

SinceΞ={(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: (3.11)maxa1⋅b1s.t.a1⋅tr(D1)+b1⋅tr(G1)≤r+max⁡j∈Str(Qj)−tr(Q1)maxa2⋅b2s.t.a2⋅tr(D2)+b2⋅tr(G2)≤r+max⁡j∈Str(Qj)−tr(Q2)⋮maxaN⋅bNs.t.aN⋅tr(DN)+bN⋅tr(GN)≤r+max⁡j∈Str(Qj)−tr(QN). By using Lagrange method, we have the optimal analytical solution
for each subproblem as (3.12)ai*=r+max⁡j∈Str(Qj)−tr(Qi)2tr(Di),bi*=r+max⁡j∈Str(Qj)−tr(Qi)2tr(Gi). Thus, the analytical solution
for the original nonlinear programming problem (see (3.10)) is given
as (3.13)a*=min⁡i∈Sai*=min⁡i∈S{r+max⁡j∈Str(Qj)−tr(Qi)2tr(Di)},b*=min⁡i∈Sbi*=min⁡i∈S{r+max⁡j∈Str(Qj)−tr(Qi)2tr(Gi)}.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 Ki,
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 Ki can still be effective though it is not
optimal. Moreover, the deviation of estimation performance is ensured to be
within a desirable precision r.


## 3.1. Mathematical Expression of Upper
Bound

According to
(2.8) and (2.11), we have (3.1)(A^i−KiC^i)ΔQi+ΔQi(A^i−KiC^i)T+∑j=1NπijΔQj+ΔW+KiΔVKiT=0, where ΔQi=Qi0−Qi.
From the above equation, it is easily seen that tr(ΔQi) 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,…,KN,ΔW,ΔV) is a mapping from Ξ to R1,
and it has the following facts.Fact 1.
For any given noise uncertainty(ΔWj,ΔVj)∈Ξ,j=1,2,
if ΔW1≤ΔW2 and ΔV1≤ΔV2,
one has (3.2)ΔJ(K1,K2,…,KN,ΔW1,ΔV1)≤ΔJ(K1,K2,…,KN,ΔW2,ΔV2).Fact 2.
Define the maximum
admissible deviation of estimation performancer as (3.3)r=max⁡(ΔW,ΔV)∈ΞΔJ(K1,K2,…,KN,ΔW,ΔV). Thus, r could be achieved only by maximum noise
uncertainty (ΔW*,ΔV*),
which means that (3.4)r=ΔJ(K1,K2,…,KN,ΔW*,ΔV*). 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 (3.5)tr(Qi0)≤r+max⁡j∈Str(Qj). Therefore, for each mode i∈S,
there is (3.6)tr(ΔQi)=tr(Qi0)−tr(Qi)≤r+max⁡j∈Str(Qj)−tr(Qi). Define the maximum upper bound
of noise uncertainty as |ΔW|≤a,|ΔV|≤b;
thus (3.7)0≤ΔW≤aIn,0≤ΔV≤bIm. Substituting the above
inequalities into (3.1) and (3.6), one has (3.8)atr(Di)+btr(Gi)≤r+max⁡j∈Str(Qj)−tr(Qi),∀i∈S, where matrices Di,Gi>0,i∈S, satisfy the following coupled Riccati
equations: (3.9)(A^i−KiC^i)Di+Di(A^i−KiC^i)T+∑j=1NπijDj+In=0,(A^i−KiC^i)Gi+Gi(A^i−KiC^i)T+∑j=1NπijGj+KiKiT=0. 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: (3.10)maxa⋅bs.t.a⋅tr(Di)+b⋅tr(Gi)≤r+max⁡j∈Str(Qj)−tr(Qi),a≥0,b≥0,i∈S. 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

SinceΞ={(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: (3.11)maxa1⋅b1s.t.a1⋅tr(D1)+b1⋅tr(G1)≤r+max⁡j∈Str(Qj)−tr(Q1)maxa2⋅b2s.t.a2⋅tr(D2)+b2⋅tr(G2)≤r+max⁡j∈Str(Qj)−tr(Q2)⋮maxaN⋅bNs.t.aN⋅tr(DN)+bN⋅tr(GN)≤r+max⁡j∈Str(Qj)−tr(QN). By using Lagrange method, we have the optimal analytical solution
for each subproblem as (3.12)ai*=r+max⁡j∈Str(Qj)−tr(Qi)2tr(Di),bi*=r+max⁡j∈Str(Qj)−tr(Qi)2tr(Gi). Thus, the analytical solution
for the original nonlinear programming problem (see (3.10)) is given
as (3.13)a*=min⁡i∈Sai*=min⁡i∈S{r+max⁡j∈Str(Qj)−tr(Qi)2tr(Di)},b*=min⁡i∈Sbi*=min⁡i∈S{r+max⁡j∈Str(Qj)−tr(Qi)2tr(Gi)}.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 Ki,
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 Ki 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

LetK1*,K2*,…,KN* 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 K1*,K2*,…,KN* satisfy (4.1)AiPi*+Pi*AiT+ϵiPi*EiTEiPi*+∑j=1NπijPj*+1ϵiH1iH1iT+W*=0,Ki*=(Qi*C^iT+1ϵiH1iH2iT)(1ϵiH2iH2iT+V*)−1,(A^i−Ki*C^i)Qi*+Qi*(A^i−Ki*C^i)T+Ki*V*Ki*T+∑j=1NπijQj*+1ϵiH1iH1iT+W*=0. According to the least-square
quality of standard Kalman filtering, we have (4.2)ΔJ(K1*,K2*,…,KN*,ΔW*,ΔV*)≤ΔJ(K1,K2,…,KN,ΔW*,ΔV*). On the other hand, with the
establishment of Fact 1, there
is (4.3)ΔJ(K1*,K2*,…,KN*,ΔW,ΔV)≤ΔJ(K1*,K2*,…,KN*,ΔW*,ΔV*). Thus, we have the following
saddle-point inequality: (4.4)ΔJ(K1*,K2*,…,KN*,ΔW,ΔV)≤ΔJ(K1*,K2*,…,KN*,ΔW*,ΔV*)≤ΔJ(K1,K2,…,KN,ΔW*,ΔV*). By game theory, we have (4.5)min⁡Kimax⁡(ΔW,ΔV)∈ΞΔJ(K1,K2,…,KN,ΔW,ΔV)=max⁡(ΔW,ΔV)∈Ξmin⁡KiΔJ(K1,K2,…,KN,ΔW,ΔV). 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*In,b*Im),
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*In,b*Im).
In practical dynamic process, when the noise uncertainty reaches the maximum,
the ideal deviation of performance is ΔJ(K1*,K2*,…,KN*,ΔW*,ΔV*),
and this deviation is less than the worst case ΔJ(K1,K2,…,KN,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 moder(t)=1 be given by (5.1)x˙1=−0.6x1+0.5x2+0.01sin(x1+x2)+ω10,x˙2=0.7x1+0.02cos(x1−x2)+ω20,y=x1+0.5x2+υ0. Let the system mode r(t)=2 be given by (5.2)x˙1=−x1+0.6x2+0.02sinx2+ω10,x˙2=0.8x1−1.1x2+0.02cosx1+ω20,y=x1+υ0, where uncertain state and
measurement noise are ω0=[ω10ω20]Tω10ω20]T and υ0;
its stationary covariance matrix is known as W=[1001],V=1; system mode transition matrix is Π=[−0.60.60.9−0.9]; the admissible bound of performance deviation
is r=0.3.The detailed algorithm is as follows.(1)
By applying Assumption2.1, we have A1=[−0.60.50.70],C1=[10.5]10.5,A2=[−10.60.8−1.1],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 thatΔCi≡0;
thus H21=H22=0.(2)
Solve (2.8) and get Q1,Q2 and K1,K2:Q1=[2.60581.42071.42073.1934],Q2=[2.65731.41811.41812.8473],K1=[0.87481.0921],K2=[0.65670.4866].(3)
Substitute the result to (3.10), using Lagrange method; the upper bound of noise uncertainty
is given as (5.3)a*=0.1014,b*=0.1701.(4)
Let the new noise covariance
matrix correspond to the maximum uncertainty:(5.4)W*=W+ΔW=W+a*⋅I2,V*=V+ΔV=V+b*⋅I1.(5)
Repeat step (2), and we have the correspondentQ1*,Q2*,K1*,K2* for new noise covariance matrix (W*,V*): (5.5)Q1*=[2.76411.44171.44173.2688],Q2*=[2.67881.44581.44582.9011],K1*=[0.89771.3021],K2*=[0.70120.5041].(6)
Applying the robust Kalman filtering, there is
saddle-point inequality:(5.6)ΔJ(K1*,K2*,ΔW,ΔV)≤ΔJ(K1*,K2*,ΔW*,ΔV*)=max{tr(Q1*),tr(Q2*)}−max{tr(Q1),tr(Q2)}=0.2337<0.3=ΔJ(K1,K2,ΔW*,ΔV*). 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 usingLagrange method. The simulation results show the
validity of this design.



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*Source: 583947-2008-09-15.xml*