ℋ_∞ Estimates for Discrete-Time Markovian Jump Linear Systems

Abstract

This paper deals with the problem of ℋ_∞ filtering for discrete-time Markovian jump linear systems. Predicted and filtered estimates are obtained based on the game theory. Both filters are solved through recursive algorithms. The Markovian system considered assumes that the jump parameters are not accessible. Necessary and sufficient conditions are provided to the existence of the filters. A numerical example is provided in order to show the effectiveness of the approach proposed.

1. Introduction

Filtering of Markovian jump linear systems (MJLSs) has been subject of intensive study in the last years. Different and creative approaches to deal with this class of filter have been considered in the literature. One of the main alternatives to solve this kind of problem is based on Kalman filter algorithms, see for instance [1, 2]. Filters proposed in these references are based on linear minimum mean square error estimates of discrete-time MJLS. They present an interesting feature related with recursiveness; however, they are not robust in nature.

When the robustness is relevant to the filtering process and demands an extra performance of the filtering approach, ℋ_∞ techniques are always considered as one of the best solutions to be adopted. In this way, a closed-loop transfer function from the unknown disturbances to the estimation error is designed in order to satisfy a prescribed ℋ_∞-norm constraint. In general, algorithms developed to deduce ℋ_∞ Markovian filters are based on linear matrix inequalities (LMIs), see for instance [3–12]. In particular, [4, 6] assume that the jump parameter of the Markov chain is not available.

In this paper, we propose ℋ_∞ filters for discrete-time MJLS which are calculated in terms of recursive algorithms based on Riccati equations. Following the approach considered in [13], we develop predicted and filtered estimates based on the two-players game theory. The idea of game theory was also adopted in sthochastic ℋ₂/ℋ_∞ control, see for instance [14].

We define the status of the players in the filtering problems considered in this paper in order to reach an equilibrium between two contradictory objectives. The first player can be interpreted as the maximizer of the estimation cost, whereas the second player tries to find an estimate that brings the quadratic cost to a minimum. A solution exists for a specified 𝛾-level if the resulting cost is positive. We assume in these Markovian filtering problems that the jump parameter is not accessible. The recursiveness of the approach we are proposing is the main advantage of these Markovian filters if compared with the filters aforementioned. As a by-product of this approach, we provide necessary conditions for the existence of them based on only known parameters of the Markovian system.

This paper is organized as follows: in Section 2, we present the problem statement; in Section 3, ℋ_∞ filters for discrete-time MJLS are presented; and in Section 4, a comparative study, based on numerical example, between the approach we are proposing and the filter developed in [4] is performed.

2. Problem Definition

The ℋ_∞ recursive filters developed in this paper are based on the following discrete-time MJLS: 𝑥_𝑖+1=𝐹_𝑖,Θ𝑖𝑥_𝑖+𝐺_𝑖,Θ𝑖𝑢_𝑖𝑦,𝑖=0,1…,_𝑖=𝐻_𝑖,Θ𝑖𝑥_𝑖+𝐷_𝑖,Θ𝑖𝑤_𝑖,𝑠_𝑖=𝐿_𝑖,Θ𝑖𝑥_𝑖+𝑅_𝑖,Θ𝑖𝑣_𝑖, (1) where 𝑥_𝑖∈ℜ^𝑛 is the valued state, 𝑦_𝑖∈ℜ^𝑚 is the valued output sequence, 𝑠_𝑖∈ℜ^𝑝 is the valued signal to be estimated, 𝑢_𝑖∈ℜ^𝑝, 𝑤_𝑖∈ℜ^𝑞, and 𝑣_𝑖∈ℜ^𝑞 are random disturbances; Θ_𝑖 is a discrete-time Markov chain with finite state space {1,…,𝑁} and transition probability matrix 𝑃=[𝑝_𝑗𝑘]. We set 𝜋_𝑖,𝑗∶=𝑃(Θ_𝑖=𝑗), 𝐹_𝑖,𝑘∈ℜ^𝑛×𝑛,  𝐺_𝑖,𝑘∈ℜ^𝑛×𝑝,  𝐻_𝑖,𝑘∈ℜ^𝑚×𝑛,  𝐷_𝑖,𝑘∈ℜ^𝑚×𝑞,  𝐿_𝑖,𝑘∈ℜ^𝑝×𝑛, and 𝑅_𝑖,𝑘∈ℜ^𝑝×𝑞, 𝑘∈{1,…,𝑁} for 𝑖≥0. The random disturbances {𝑢_𝑖}, {𝑤_𝑖}, and {𝑣_𝑖} are assumed to be null mean with finite second order moments, independent wide sense stationary sequences mutually independent with covariance matrices equal to the 𝑈_𝑖, 𝑊_𝑖, and 𝑉_𝑖, respectively. 𝑥₀1_{Θ0=𝑘} are random vectors with 𝔼{𝑥₀1_{Θ0=𝑘}}=𝜇_𝑘 (where 1_{⋅} denotes Dirac measure) and 𝔼{𝑥₀𝑥^𝑇₀1_{Θ0=𝑘}}=𝑉_𝑘; 𝑥₀ and {Θ_𝑖} are independent of {𝑢_𝑖}, {𝑤_𝑖} and {𝑣_𝑖}. A scalar 𝛾>0 and a sequence of sets of observations 𝑦₀,𝑦₀,𝑦₁,…,𝑦₀,…,𝑦_𝑙,… (2) are defined to find at each instant 𝑙 a filtered estimate 𝑠_𝑙|𝑙 of 𝑠_𝑙 such that sup_𝑥0𝑠_0∣0−𝐿_0,Θ0𝑥₀²_𝑉0−1𝑥₀−𝐹₋₁𝑥₀²_𝑃0−1+𝑦₀−𝐻_0,Θ0𝑥₀²_𝑊0−1<𝛾², (3) is satisfied for 𝑙=0, and sup_{𝑥𝑖𝑙𝑖=0}∑^𝑙_𝑖=0𝑠_𝑖|𝑙−𝐿_𝑖,Θ𝑖𝑥_𝑖²_𝑉𝑖−1𝑥₀−𝐹₋₁𝑥₀²_𝑃0−1+∑^𝑙_𝑖=0𝑦_𝑖−𝐻_𝑖,Θ𝑖𝑥_𝑖²_𝑊𝑖−1+∑^𝑙−1_𝑖=0𝑥_𝑖+1−𝐹_𝑖,Θ𝑖𝑥_𝑖²_𝑈𝑖−1<𝛾² (4) is satisfied for 𝑙>0. For the predicted estimate, given 𝛾>0 and a sequence of observations (2), the problem is to find at each instant 𝑙 a prediction 𝑠_𝑙+1|𝑙 of 𝑠_𝑙+1 such that sup_𝑥0𝑠_0∣−1−𝐿_0,Θ0𝑥₀²_𝑉0−1𝑥₀−𝐹₋₁𝑥₀²_𝑃0−1<𝛾² (5) is satisfied for 𝑙=−1 and sup_{𝑥𝑖𝑙+1𝑖=0}∑^𝑙+1_𝑖=0𝑠_𝑖∣𝑙−𝐿_𝑖,Θ𝑖𝑥_𝑖²_𝑉𝑖−1𝑥₀−𝐹₋₁𝑥₀²_𝑃0−1+∑^𝑙_𝑖=0𝑦_𝑖−𝐻_𝑖,Θ𝑖𝑥_𝑖²_𝑊𝑖−1+∑^𝑙_𝑖=0𝑥_𝑖+1−𝐹_𝑖,Θ𝑖𝑥_𝑖²_𝑈𝑖−1<𝛾² (6) is satisfied for 𝑙≥0. The sequence of solutions 𝑠_𝑙|𝑙 and 𝑠_𝑙+1|𝑙 is outputs of the respective ℋ_∞ filters. Due to the hybrid nature of this class of system, at each instant of time a new model of a countable set of models is used to calculate the functionals (4) and (6). In general, to synthesize recursive filtering algorithms for this class of problems is not an easy task. Thanks to the augmented model of (1) proposed in [1] we can redefine these functionals in order to develop ℋ_∞ recursive filters similar to those we find in the literature for standard state-space systems, without jumps. The augmented model of (1) can be written as follows: 𝑧_𝑖+1=ℱ_𝑖𝑧_𝑖+𝜓_𝑖𝑦,𝑖=0,1…_𝑖=ℋ_𝑖𝑧_𝑖+𝜑_𝑖,𝑠_𝑖=ℒ_𝑖𝑧_𝑖+𝜎_𝑖, (7) where the parameter matrices are given by ℱ_𝑖𝑝∶=¹¹𝐹_𝑖,1⋯𝑝_𝑁1𝐹_𝑖,𝑁𝑝⋮⋱⋮_1𝑁𝐹_𝑖,1⋯𝑝_𝑁𝑁𝐹_𝑖,𝑁,ℋ_𝑖∶=𝐻_𝑖,1⋯𝐻_𝑖,𝑁,ℒ_𝑖∶=𝐿_𝑖,1⋯𝐿_𝑖,𝑁, (8) whose dimensions are defined by ℱ_𝑖∈ℜ^{𝑁𝑛×𝑁𝑛}, ℋ_𝑖∈ℜ^{𝑚×𝑁𝑛}, and ℒ_𝑖∈ℜ^{𝑝×𝑁𝑛}; and the state variable is defined as 𝑧_𝑖∶=𝑧^𝑇_𝑖,1⋯𝑧^𝑇_𝑖,𝑁^𝑇∈ℜ^𝑁𝑛,𝑧_𝑖,𝑘∶=𝑥_𝑖1_{{Θ𝑖=𝑘}}∈ℜ^𝑛. (9) We define for 𝑖≥0 and 𝑘∈{1,…,𝑁}, 𝑍_𝑖,𝑘∶=𝔼𝑧_𝑖,𝑘𝑧^𝑇_𝑖,𝑘∈ℜ^𝑛×𝑛,𝑍_𝑖∶=𝔼𝑧_𝑖𝑧^𝑇_𝑖=diag𝑍_𝑖,𝑘∈ℜ^{𝑁𝑛×𝑁𝑛}, (10) where 𝑍_𝑖,𝑘 is given by the following recursive equation: 𝑍_𝑖+1,𝑘=^𝑁_𝑗=1𝑝_𝑗𝑘𝐹_𝑖,𝑗𝑍_𝑖,𝑗𝐹^𝑇_𝑖,𝑗+^𝑁_𝑗=1𝑝_𝑗𝑘𝜋_𝑖,𝑗𝐺_𝑖,𝑗𝑈_𝑖𝐺^𝑇_𝑖,𝑗,𝑍_0,𝑘=𝑉_𝑘. (11)

We define also in the next lemma, weighting matrices Λ_𝑖, Π_𝑖, and Γ_𝑖 as variances of random disturbances given by 𝜓_𝑖, 𝜑_𝑖, and 𝜎_𝑖, respectively.

Lemma 1. Let 𝜓_𝑖, 𝜑_𝑖, and 𝜎_𝑖 defined by 𝜓_𝑖∶=ℳ_𝑖+1𝑧_𝑖+𝜗_𝑖,𝜑_𝑖∶=𝐷_𝑖,Θ𝑖𝑤_𝑖,𝜎_𝑖∶=𝑅_𝑖,Θ𝑖𝑣_𝑖, (12) for 𝑖≥0 and 𝑗,𝑘∈{1,…,𝑁}, where ℳ^𝑇_𝑖+1∶=ℳ^𝑇_𝑖+1,1⋯ℳ^𝑇_𝑖+1,𝑁,ℳ_𝑖+1,𝑘∶=1_{{Θ𝑖+1=𝑘}}−𝑝_1𝑘𝐹_𝑖,11_{Θ𝑖=1}⋯1_{{Θ𝑖+1=𝑘}}−𝑝_𝑁𝑘𝐹_𝑖,𝑁1_{{Θ𝑖=𝑁}},𝜗_𝑖1∶=^{{𝜃(𝑖+1)=1}}𝐺_𝑖,Θ𝑖𝑢_𝑖⋮1_{{𝜃(𝑖+1)=𝑁}}𝐺_𝑖,Θ𝑖𝑢_𝑖. (13) The variances of 𝜓_𝑖, 𝜑_𝑖, and 𝜎_𝑖 can be obtained by the following equations: Γ_𝑖∶=diag^𝑁_𝑗=1𝑝_𝑗𝑘𝐹_𝑖,𝑗𝑍_𝑖,𝑗𝐹^𝑇_𝑖,𝑗−ℱ_𝑖𝑍_𝑖ℱ^𝑇_𝑖+diag^𝑁_𝑗=1𝑝_𝑗𝑘𝜋_𝑖,𝑗𝐺_𝑖,𝑗𝑈_𝑖𝐺^𝑇_𝑖,𝑗,Π_𝑖∶=𝒟_𝑖𝒟^𝑇_𝑖,Λ_𝑖∶=ℛ_𝑖ℛ^𝑇_𝑖, (14) respectively, where 𝒟_𝑖∶=𝐷_𝑖,1𝜋^1/2_𝑖,1𝑊_𝑖^1/2⋯𝐷_𝑖,𝑁𝜋^1/2_𝑖,𝑁𝑊_𝑖^1/2,ℛ_𝑖∶=𝑅_𝑖,1𝜋^1/2_𝑖,1𝑉_𝑖^1/2⋯𝑅_𝑖,𝑁𝜋^1/2_𝑖,𝑁𝑉_𝑖^1/2. (15)(16)

Proof. The proof follows the arguments proposed in [15], Chapter 3, and in [1].

3. ℋ_∞ Estimates for DMJLS

In this section, we propose recursive ℋ_∞ estimates for discrete-time MJLS (DMJLS). It is known that for standard state-space systems, this kind of filtering approach is difficult to be implemented online due to the fact that we do not know the minimum 𝛾 for each step of the recursion. In general, it depends on the estimate error variance matrix which should be calculated at the same instant of time. In this sense, the existence condition of this filter is not known a priori. To ameliorate this limitation we provide, for this Markovian problem, necessary conditions to tune this parameter depending on the variances of the random disturbances 𝜓_𝑖, 𝜑_𝑖, and 𝜎_𝑖 and on the known parameter matrices of the augmented model of (1) given in (7).

3.1. ℋ_∞ Filter

Consider (3) and (4) for the augmented model (7): sup_𝑧0𝑠_0|0−ℒ₀𝑧₀²_Λ0−1𝑧₀−ℱ₋₁𝑧₀²𝑍₀⁻¹+𝑦₀−ℋ₀𝑧₀²_Π0−1<𝛾²_𝑎, (17) for 𝑙=0, and sup_{𝑧𝑖𝑙𝑖=0}∑^𝑙_𝑖=0𝑠_𝑖∣𝑙−ℒ_𝑖𝑧_𝑖²_Λ𝑖−1𝑧₀−ℱ₋₁𝑧₀²𝑍₀⁻¹+∑^𝑙_𝑖=0𝑦_𝑖−ℋ_𝑖𝑧_𝑖²_Π𝑖−1+∑^𝑙−1_𝑖=0𝑧_𝑖+1−ℱ_𝑖𝑧_𝑖²_Γ𝑖−1<𝛾²_𝑎, (18) for 𝑙>0. We can rewrite this ℋ_∞ filtering problem in terms of the following optimization problem: min_{𝑧𝑖∣𝑙𝑙𝑖=0}max_{𝑠𝑖∣𝑙𝑙𝑖=0}𝐽^𝑓_𝑙𝑦_𝑖^𝑙_𝑖=0,𝑠_𝑖∣𝑙^𝑙_𝑖=0,𝑧_𝑖∣𝑙^𝑙_𝑖=0>0, (19) where 𝐽^𝑓₀∶=𝑧_0∣0−ℱ₋₁𝑧₀²𝑍₀⁻¹+𝑦₀−ℋ₀𝑧_0∣0²_Π0−1−𝛾_𝑎⁻²𝑠_0∣0−ℒ₀𝑧_0|0²_Λ0−1,𝑙=0,𝐽^𝑓_𝑙∶=𝑧_0|𝑙−ℱ₋₁𝑧₀²𝑍₀⁻¹+^𝑙_𝑖=0𝑦_𝑖−ℋ_𝑖𝑧_𝑖|𝑙²_Π𝑖−1+^𝑙−1_𝑖=0𝑧_𝑖+1|𝑙−ℱ_𝑖𝑧_𝑖|𝑙²_Γ𝑖−1−𝛾_𝑎^𝑙−2_𝑖=0𝑠_𝑖∣𝑙−ℒ_𝑖𝑧_𝑖∣𝑙²_Λ𝑖−1,𝑙>0. (20)(21)

Notice that it is not necessary to maximize 𝐽^𝑓_𝑙 over {𝑠_𝑖|𝑙}^𝑙_𝑖=0 in the optimization problem (19). The solution of the ℋ_∞ filter is guaranteed if, and only if, there exists a {𝑠̂_𝑖|𝑙}^𝑙_𝑖=0 for which 𝐽^𝑓_𝑙({𝑦_𝑖}^𝑙_𝑖=0,{𝑠̂_𝑖|𝑙}^𝑙_𝑖=0,{𝑧_𝑖|𝑙}^𝑙_𝑖=0)>0 has a minimum {𝑧_𝑖|𝑙}^𝑙_𝑖=0. Therefore, in order to deal with only the minimization problem (19), for each 𝑙≥0 it is easy to show that (21) can be rewritten as 𝐽^𝑓_𝑙∶=𝔘_𝑙𝔛_𝑙|𝑙−𝔅_𝑙^𝑇ℜ_𝑙𝔘_𝑙𝔛_𝑙|𝑙−𝔅_𝑙, (22) where 𝔛_𝑙|𝑙𝑧∶=^𝑙|𝑙⋮𝑧_1|𝑙𝑧_0|𝑙,𝔅_𝑙𝒵∶=^𝑙⋮𝒵₁𝒵₀,ℜ_𝑙ℛ∶=^𝑙000⋱000ℛ₀,𝔘_𝑙ℰ∶=^𝑙𝒜_𝑙−10000ℰ_𝑙−1⋱0000⋱𝒜₁0000ℰ₁𝒜₀0000ℰ₀,ℰ_𝑖𝐼∶=ℋ_𝑖ℒ_𝑖,ℛ_𝑖Γ∶=^−1𝑖−1000Π_𝑖⁻¹000−𝛾_𝑎⁻²Λ_𝑖⁻¹,𝒜_𝑖−1∶=−ℱ_𝑖−100,𝒵_𝑖0∶=𝑦_𝑖𝑠_𝑖|𝑙ℤ,1≤𝑖≤𝑙,₀ℱ∶=⁻¹𝑧₀𝑦₀𝑠_0|𝑙,Γ₋₁∶=𝑍₀. (23) For all 𝑙≥0, we can conclude that the following recurrent relations are valid: ℜ_𝑙ℛ∶=^𝑙00ℜ_𝑙−1,𝔅_𝑙𝒵∶=^𝑙𝔅_𝑙−1,𝔛_𝑙|𝑙𝑧∶=^𝑙|𝑙𝔛_{𝑙−1|𝑙−1},𝔘_𝑙ℰ∶=^𝑙𝛼_𝑙−10𝔘_𝑙−1,𝛼_𝑙−1∶=𝒜_𝑙−10⋯0. (24)

According to [16], there exists a minimum for (22) if and only if the positiveness of 𝔘^𝑇_𝑙ℜ_𝑙𝔘_𝑙 is guaranteed.

Lemma 2. Consider matrices 𝔘_𝑙 and ℜ_𝑙 and column vectors 𝔅_𝑙 and 𝔛_𝑙|𝑙 of appropriate dimensions with ℜ_𝑙 symmetric. For any 𝔅_𝑙 we have inf_𝑥𝐽^𝑓_𝑙>−∞, (25) if and only if 𝔘^𝑇_𝑙ℜ_𝑙𝔘_𝑙≥0 and Ker(𝔘^𝑇_𝑙ℜ_𝑙𝔘_𝑙)⊂Ker(ℜ_𝑙𝔘_𝑙). If the minimum is attained, it is unique if and only if 𝔘^𝑇_𝑙ℜ_𝑙𝔘_𝑙>0 and the optimal solution is given by 𝔛_𝑙|𝑙=(𝔘^𝑇_𝑙ℜ_𝑙𝔘_𝑙)⁻¹𝔘^𝑇_𝑙ℜ_𝑙𝔅_𝑙.

With this fundamental lemma in mind, we can obtain an existence condition for the filter we are proposing in the following, based on the known parameter matrices of the Markovian model. For 𝑙>0, the term 𝔘^𝑇_𝑙ℜ_𝑙𝔘_𝑙 of 𝔛_𝑙|𝑙 can be written as ℰ^𝑇_𝑙ℛ_𝑙ℰ_𝑙ℰ^𝑇_𝑙ℛ_𝑙𝛼_𝑙−1𝛼^𝑇_𝑙−1ℛ_𝑙ℰ_𝑙𝔘^𝑇_𝑙−1ℜ_𝑙−1𝔘_𝑙−1+𝛼^𝑇_𝑙−1ℛ_𝑙𝛼_𝑙−1, (26) and a necessary condition for 𝔘^𝑇_𝑙ℜ_𝑙𝔘_𝑙>0 is the positiveness of ℰ^𝑇_𝑙ℛ_𝑙ℰ_𝑙, which in terms of the augmented Markovian model of (1) can be established as follows: Γ⁻¹_𝑙−1+ℋ^𝑇_𝑙Π_𝑙⁻¹ℋ_𝑙−𝛾_𝑎⁻²ℒ^𝑇_𝑙Λ_𝑙⁻¹ℒ_𝑙>0. (27)

Remark 3. A lower bound for 𝛾_𝑎 can be calculated through (27) based on the augmented model (7) and Lemma 1. Notice that (27) is a natural and interesting extension of the filtering of standard state-space systems [13], without jumps, to the DMJLS we are dealing with in this paper.
Now we are in a position to deduce the ℋ_∞ recursive filtered estimate for DMJLS. The next theorem provides the solution for this problem based on the augmented model (7) and on the sequence of measurements (2) of the original Markovian system (1).

Theorem 4. Consider the augmented Markovian model (7). There exists a recursive ℋ_∞ filter for this system, defined by the following recursive equations: 𝑍⁻¹_0|0∶=𝑍₀⁻¹+ℋ^𝑇₀Π₀⁻¹ℋ₀−𝛾_𝑎⁻²ℒ^𝑇₀Λ₀⁻¹ℒ₀,𝑍⁻¹_𝑙|𝑙∶=Γ_𝑙−1+ℱ_𝑙−1𝑍_{𝑙−1|𝑙−1}ℱ^𝑇_𝑙−1⁻¹+ℋ^𝑇_𝑙Π_𝑙⁻¹ℋ_𝑙−𝛾_𝑎⁻²ℒ^𝑇_𝑙Λ_𝑙⁻¹ℒ_𝑙,𝑧_0|0∶=𝑃⁻¹_0|0+𝛾_𝑎⁻²ℒ^𝑇₀Λ₀⁻¹ℒ₀⁻¹×𝑍₀⁻¹ℱ^𝑇₋₁𝑧₀+ℋ^𝑇₀Π₀⁻¹𝑦₀,𝑧_𝑙|𝑙∶=𝑍⁻¹_𝑙|𝑙+𝛾_𝑎⁻²ℒ^𝑇_𝑙Λ_𝑙⁻¹ℒ_𝑙⁻¹×𝑋⁻¹_𝑙−1ℱ_𝑙−1𝑧_{𝑙−1|𝑙−1}+ℋ^𝑇_𝑙Π_𝑙⁻¹𝑦_𝑙,𝑋_𝑙−1∶=Γ_𝑙−1+ℱ_𝑙−1𝑍_{𝑙−1|𝑙−1}ℱ^𝑇_𝑙−1,𝑠̂_𝑙|𝑙∶=ℒ_𝑙𝑧_𝑙|𝑙, (28)(29)(30) if and only if 𝑍_𝑙|𝑙>0, 𝑙=0,1,…, and the attenuation level 𝛾_𝑎 is positive.

Proof. This proof follows standard arguments used to deduce Kalman filters and basically consists in checking the positiveness of (26) and in finding the equation of the filter through the Lemma 2. In order to have 𝔘^𝑇_𝑙ℜ_𝑙𝔘_𝑙>0, we can rewrite this term as (26). The (2,2) sub block of (26) must be positive definite. Assuming that the positiveness of 𝔘^𝑇_𝑙−1ℜ_𝑙−1𝔘_𝑙−1 is guaranteed and 𝛼^𝑇_𝑙−1ℛ_𝑙𝛼_𝑙−1=ℱ^𝑇_𝑙−1Γ⁻¹_𝑙−1ℱ_𝑙−1000≥0, (31) the (2,2) term of (26) is positive definite and 𝔘^𝑇_𝑙ℜ_𝑙𝔘_𝑙>0 if and only if the Schur complement of the (2,2) block, 𝑀⁻¹_𝑙|𝑙∶=ℰ^𝑇_𝑙ℛ_𝑙⁻¹+𝛼_𝑙−1𝔘^𝑇_𝑙−1ℜ_𝑙−1𝔘_𝑙−1⁻¹𝛼^𝑇_𝑙−1⁻¹ℰ_𝑙, (32) is positive definite. For 𝑙−1, 𝑀_{𝑙−1|𝑙−1} is the (1,1) block of (𝔘^𝑇_𝑙−1ℜ_𝑙−1𝔘_𝑙−1)⁻¹. With 𝛼_𝑙−1 defined in (24), we obtain 𝑀⁻¹_𝑙|𝑙∶=ℰ^𝑇_𝑙ℛ_𝑙⁻¹+𝒜_𝑙−1𝑀_{𝑙−1|𝑙−1}𝒜^𝑇_𝑙−1⁻¹ℰ_𝑙. (33) Considering 𝑍_𝑙|𝑙∶=𝑀_𝑙|𝑙 in terms of the original data (23), we obtain (28). From Lemma 2 we have the solution for the minimization of (22). Based on the recurrent relations (23) and (24), considering 𝛼_𝑙−1𝔛_{𝑙−1|𝑗}=𝒜_𝑙−1𝑥_{𝑙−1|𝑗} for 𝑗≥𝑙−1, and introducing 𝑍_𝑙|𝑙𝑃_12,𝑙𝑃_21,𝑙𝑃_22,𝑙ℰ∶=^𝑇𝑙ℛ_𝑙ℰ_𝑙ℰ^𝑇_𝑙ℛ_𝑙𝛼_𝑙−1𝛼^𝑇_𝑙−1ℛ_𝑙ℰ_𝑙𝔘^𝑇_𝑙−1ℜ_𝑙−1𝔘_𝑙−1+𝛼^𝑇_𝑙−1ℛ_𝑙𝛼_𝑙−1⁻¹, (34) holds (29) taking into account that 𝑠_𝑙|𝑙=𝑠̂_𝑙|𝑙∶=ℒ_𝑙𝑧_𝑙|𝑙.

3.2. ℋ_∞ Predictor

In this subsection, we present the recursive ℋ_∞ predictor filter for DMJLS. The original ℋ_∞ problem (5)-(6) can be rewritten in terms of the following optimization problem: min_{𝑧𝑖|𝑙𝑙+1𝑖=0}max_{𝑠𝑖|𝑙𝑙+1𝑖=0}𝐽^𝑝_𝑙𝑦_𝑖^𝑙_𝑖=0,𝑠_𝑖|𝑙^𝑙+1_𝑖=0,𝑧_𝑖|𝑙^𝑙+1_𝑖=0>0, (35) where 𝐽^𝑝₋₁∶=𝑧_0|−1−ℱ₋₁𝑧₀²𝑍₀⁻¹−𝛾_𝑎⁻²𝑠_0|−1−ℒ₀𝑧_0|−1²_Λ0−1𝐽,𝑙=−1,_𝑝𝑙∶=𝑧_0|𝑙−ℱ₋₁𝑧₀²𝑍₀⁻¹+^𝑙_𝑖=0𝑦_𝑖−ℋ_𝑖𝑧_𝑖|𝑙²_Π𝑖−1+^𝑙_𝑖=0𝑧_𝑖+1|𝑙−ℱ_𝑖𝑧_𝑖|𝑙²_Γ𝑖−1−𝛾_𝑎^−2𝑙+1_𝑖=0𝑠_𝑖|𝑙−ℒ_𝑖𝑧_𝑖|𝑙²_Λ𝑖−1,𝑙≥0. (36)

The ℋ_∞ predictor filter exists at the instant 𝑙 if and only if there exists a sequence {𝑠̂_𝑖|𝑙}^𝑙+1_𝑖=0 such that 𝐽^𝑝_𝑙({𝑦_𝑖}^𝑙_𝑖=0,{𝑠̂_𝑖|𝑙}^𝑙+1_𝑖=0,{𝑧_𝑖|𝑙}^𝑙+1_𝑖=0) has a minimum {𝑧_𝑖|𝑙}^𝑙+1_𝑖=0 for which 𝐽^𝑝_𝑙𝑦_𝑖^𝑙_𝑖=0,𝑠̂_𝑖|𝑙^𝑙+1_𝑖=0,𝑧_𝑖|𝑙^𝑙+1_𝑖=0>0. (37)

The prediction optimization problem is equivalent in nature to the filter problem, aforementioned. In the next theorem we present the ℋ_∞ predictor filter whose proof follows the line of the filtered version.

Theorem 5. The Markovian ℋ_∞ prediction problem (35) is solvable if, and only if, 𝑍⁻¹_𝑙+1|𝑙−𝛾_𝑎⁻²ℒ^𝑇_𝑙+1Λ⁻¹_𝑙+1ℒ_𝑙+1>0, where the sequence {𝑍_𝑙+1|𝑙} and the predictor filter are calculated by the recursions 𝑍_0|−1∶=𝑍₀,𝑍_𝑙+1|𝑙∶=Γ_𝑙+ℱ_𝑙𝑍_{𝑙|𝑙−1}ℱ^𝑇_𝑙−ℱ_𝑙𝑍_{𝑙|𝑙−1}ℋ_𝑙ℒ_𝑙^𝑇×𝑊⁻¹_𝑒∞,𝑙ℋ_𝑙ℒ_𝑙𝑍_{𝑙|𝑙−1}ℱ_𝑙,𝑊_𝑒∞,𝑙Π∶=^𝑙00−𝛾²_𝑎Λ_𝑙+ℋ_𝑙ℒ_𝑙𝑍_{𝑙|𝑙−1}ℋ^𝑇_𝑙ℒ^𝑇_𝑙,𝑧_𝑙+1|𝑙∶=ℱ_𝑙𝑧_{𝑙|𝑙−1}+ℱ_𝑙𝑀_{𝑙|𝑙−1}ℋ^𝑇_𝑙Π_𝑙+ℋ_𝑙𝑀_{𝑙|𝑙−1}ℋ^𝑇_𝑙⁻¹×𝑦_𝑙−ℋ_𝑙𝑧_{𝑙|𝑙−1},𝑀⁻¹_{𝑙|𝑙−1}∶=𝑍⁻¹_{𝑙|𝑙−1}−𝛾_𝑎⁻²ℒ^𝑇_𝑙Λ_𝑙⁻¹ℒ_𝑙,𝑠̂_𝑙+1|𝑙∶=ℒ_𝑙+1𝑧_𝑙+1|𝑙. (38)

In order to recover the original predicted and filtered state estimates of System (1), we calculate the following summations: 𝑥_{𝑖|𝑖−1}=^𝑁_𝑗=1𝑧_{𝑖,𝑗|𝑖−1},𝑥_𝑖|𝑖=^𝑁_𝑗=1𝑧_{𝑖,𝑗|𝑖}, (39) where 𝑧_{𝑖|𝑖−1}∶=[𝑧^𝑇_{𝑖,1|𝑖−1}⋯𝑧^𝑇_{𝑖,𝑁|𝑖−1}]^𝑇∈ℜ^𝑁𝑛 and 𝑧_{𝑖,𝑘|𝑖−1}∶=𝑥_{𝑖|𝑖−1}1_{{Θ𝑖=𝑘}}∈ℜ^𝑛. 𝑥_𝑖|𝑖 is defined in the same way.

Remark 6. To establish the stability of the stationary ℋ_∞ prediction filter, it is assumed that all matrices of (1) and the transition probabilities 𝑝_𝑗𝑘 are time invariant; System (1) is mean square stable (MSS) and its Markov chain {Θ_𝑖} is ergodic. There exists a unique positive-definite solution 𝑍 (with 𝑖→∞) to the algebraic Riccati equation 𝑍∶=Γ+ℱ𝑍ℱ^𝑇𝑍ℋℒ−ℱ^𝑇𝑊⁻¹_𝑒∞ℋℒ𝑊𝑍ℱ,_𝑒∞∶=Π00−𝛾²_𝑎Λ+ℋℒ𝑍ℋ^𝑇ℒ^𝑇, (40) providing that 𝑍⁻¹−𝛾_𝑎⁻²ℒ^𝑇Λ⁻¹ℒ>0, for any 𝛾_𝑎 fixed, which guarantees that 𝑊_𝑒∞>0 and thus the above inverse is well defined. Following the guidelines defined in [1], the stability of the predictor filter is assured with 𝑟_𝜎ℱ−ℱ𝑀ℋ^𝑇Π+ℋ𝑀ℋ^𝑇−1ℋ<1, (41) where 𝑟_𝜎(⋅) denotes spectral radius of the dynamic matrix of the filter with stationary gain. The asymptotic stability of the filtered version can also be assured following this line of argumentation.

Remark 7. Similar to the filtered case, we can also find a necessary existence condition to the Markovian ℋ_∞ prediction filter given by 𝑍₀⁻¹−𝛾_𝑎⁻²ℒ^𝑇₀Λ₀⁻¹ℒ₀Γ>0,𝑙=−1,_𝑙−1−𝛾_𝑎⁻²ℒ^𝑇_𝑙+1Λ⁻¹_𝑙+1ℒ_𝑙+1>0,𝑙≥0. (42)

Remark 8. If we consider 𝛾_𝑎→∞, the ℋ_∞ filter proposed in this paper reduces to the Kalman filter for MJLS proposed in [1]. For the case with no jumps (𝑁=1), the predicted and the filtered ℋ_∞ estimates reduce to the ℋ_∞ filters for state-space systems given in [17].

4. Numerical Example

In this section, we compare the filtering approach proposed with the filter developed in [4]. With two Markovian states, the model (1) is defined based on the following parameters: 𝑃=0.90.10.90.1,𝐹₁=,𝐹0.700.10.1₂=,𝐺0.600.10.2₁=𝐺₂=,𝐻0.8731000.2089₁=𝐻₂=,𝐷0.10₁=𝐷₂=,𝐿0.0080₁=𝐿₂=,𝑅0.50₁=𝑅₂=0.10.3. (43) We compare both filters in the predicted form, written in the following way: 𝑧_𝑙+1|𝑙=𝒜_𝑙𝑧_{𝑙|𝑙−1}+ℬ_𝑙𝑦_𝑙,𝑠̂_𝑙+1|𝑙=ℒ_𝑙+1𝑧_𝑙+1|𝑙, (44) where 𝒜_𝑙=ℱ_𝑙−ℱ_𝑙𝑀_{𝑙|𝑙−1}ℋ^𝑇_𝑙Π_𝑙+ℋ_𝑙𝑀_{𝑙|𝑙−1}ℋ^𝑇_𝑙⁻¹ℋ_𝑙,ℬ_𝑙=ℱ_𝑙𝑀_{𝑙|𝑙−1}ℋ^𝑇_𝑙Π_𝑙+ℋ_𝑙𝑀_{𝑙|𝑙−1}ℋ^𝑇_𝑙⁻¹. (45) The filter of [4] is considered in the strictly proper form as 𝑥_𝑓(𝑘+1)=𝐴_𝑓𝑥_𝑓(𝑘)+𝐵_𝑓𝑦(𝑘),𝑧_𝑓(𝑘)=𝐶_𝑓𝑥_𝑓(𝑘), (46) whose solution is given in terms of linear matrix inequalities. We show in Figure 1 the root-mean-square errors (rms) of both filters. We performed 1000 Monte Carlo simulations from 𝑖=0,…,6 with the values of Θ_𝑖 generated randomly. The initial condition 𝑥₀ is considered Gaussian with mean [0.1960.295]^𝑇 and variance 0.03840.05780.05780.870; Θ_𝑖∈{1,2}, 𝑢_𝑖, 𝑤_𝑖, and 𝑣_𝑖 are independent sequences of noises with 𝑈_𝑖, 𝑊_𝑖, and 𝑉_𝑖 identity matrices with appropriate dimensions, 𝜋₁(0)=0.05, and 𝜋₂(0)=0.95. We obtained 𝛾_𝑎=1.9523 for the filter proposed in this paper and 𝛾=0.2886 for the filter proposed in [4]. The parameter matrices of our filter were computed as 𝒜_𝑙=,ℬ0.01160−0.078400.00030.090.00030.180.00130−0.0087000.0100.02_𝑙=6.18430.89670.68710.0996,ℒ_𝑙+1=0.500.50, (47) and for the filter of [4] were computed as 𝐴_𝑓=,𝐵−0.2154−0.0122−0.0342−0.3654_𝑓=9.41962.4656,𝐶_𝑓=0.47900.0962. (48)

Figure 1 

Root-mean-square errors for ℋ∞ filters based on Riccati recursive equation and linear matrix inequalities.

In spite of the smaller 𝛾 provided by the filter of [4], the rms of both filters are equivalent. This difference in 𝛾 is due to the fact that the functionals of both approaches are different in nature. However, if we consider the same numerical value of all parameters but with matrices 𝑅₁ and 𝑅₂ multiplied by 10⁶, we obtain for the proposed recursive filter 𝛾_𝑎=0.001. On the other hand, we obtain an infeasible solution with the LMI proposed in Section IV of [4] (computed through Matlab 7.4.0). The mode-independent filter provided in [4] deals with only sufficient conditions. Considering offline computations, our approach provides necessary and sufficient conditions for the existence of the filter.

It is important to recall that the proposed approach enables all parameter matrices of (1) to be time varying. It also provides a practical estimate for the minimum admissible 𝛾 through the necessary condition (27), which depends on only known parameters of the Markovian system.

5. Conclusion

This paper developed ℋ_∞ predicted and filtered estimates for DMJLS. They were deduced based on the assumption that jump parameter is not accessible. The numerical example showed the effectiveness of this approach. Through an augmented model of the standard Markovian system, these filters were deduced based on the game theory. It allows us to deal with recursive algorithms to solve this kind of filtering problem where the parameter matrices can be time varying for each operation mode of the Markovian system. As future works, we intend to solve nonlinear filtering and control problems for DMJLS based on the problems raised by [12, 18].