ℋ_∞ Model Reduction for Discrete-Time Markovian Jump Systems with Deficient Mode Information

Abstract

This paper investigates the problem of ℋ_∞ model reduction for a class of discrete-time Markovian jump linear systems (MJLSs) with deficient mode information, which simultaneously involves the exactly known, partially unknown, and uncertain transition probabilities. By fully utilizing the properties of the transition probability matrices, together with the convexification of uncertain domains, a new ℋ_∞ performance analysis criterion for the underlying MJLSs is first derived, and then two approaches, namely, the convex linearisation approach and iterative approach, for the ℋ_∞ model reduction synthesis are proposed. Finally, a simulation example is provided to illustrate the effectiveness of the proposed design methods.

1. Introduction

Markovian jump linear systems (MJLSs), initially proposed by Krasovskiĭ and Lidskiĭ [1], have been attracting increasing attention over the past decades. The appeal for the study of this class of hybrid systems is that MJLSs are powerful to model plants whose structures are subject to random abrupt changes, for example, the fault-prone manufacturing systems [2], power systems [3], economic systems [4], networked control systems (NCSs) [5–10], and so on. As a critical factor, the transition probabilities (TPs) in the Markov chain determine the jump rules of different modes and are assumed to be completely known in most of the existing literature [11–14].

However, it is noted that, in many practical engineering systems, not all the mode transition information can be obtained exactly. Recently, there have appeared some results on the analysis and synthesis of MJLSs with uncertain TPs [15–18] or partially unknown TPs [19–21]. To mention a few, the authors in [15] considered the robust stochastic stability analysis problem for a class of MJLSs with norm-bounded uncertainties in the TPs; the author in [16] dealt with the robust stability analysis and stabilization problems for MJLSs with polytopic-type uncertain TPs; the authors in [17] studied the robust stability analysis for a class of discrete-time Markovian jumping neural networks with defective statistics of mode transitions; the authors in [19] investigated the ℋ_∞ filtering design for MJLSs with partially unknown TPs. In particular, it is noted that the introduction of deficient mode information, which simultaneously contains the exactly known, partially unknown, and uncertain TPs, is very important and useful in both theoretical advances and practical applications of MJLSs.

On the other hand, in many engineering applications, high- or even infinite-order complex mathematical models are frequently used to describe physical systems and processes [22–24]. These high-order models bring serious difficulties to analysis and synthesis of the underlying systems. Therefore, model simplification/reduction with respect to certain criteria has received considerable attention in recent years. Correspondingly, many important results on model reduction have been obtained, involving various efficient approaches such as the Hankel norm approximation method [25], the ℋ₂ approach [26, 27], the ℋ_∞ approach [28–30], and the ℒ₂-ℒ_∞ approach [31]. More recently, the linear matrix inequality (LMI) technique has been exploited to solve the model reduction problem for different classes of systems, such as bilinear systems [27], time-delay systems [28], and T-S fuzzy systems [32–36]. Specifically, the authors in [4, 20] considered the ℋ_∞ model reduction problem for a class of MJLSs with completely known TPs and partially unknown TPs, respectively. However, to the authors’ best knowledge, few results have been reported on ℋ_∞ model reduction for MJLSs with deficient mode information, which simultaneously consists of the exactly known, partially unknown, and uncertain TPs. This motivates us for the present study.

According to the issues mentioned previously, in this paper, we will make an attempt to tackle the ℋ_∞ model reduction problem for a class of discrete-time MJLSs with deficient mode information. Such deficient mode information simultaneously contains the exactly known, partially unknown, and uncertain TPs, which is a more practical scenario. By fully utilizing the properties of the transition probability matrices, together with the convexification of uncertain domains, a new ℋ_∞ performance analysis criterion for MJLSs with deficient mode information will be firstly derived. To solve the model reduction problem, two distinctly different approaches will be then proposed. The first approach is based on a linearisation procedure, which casts the model reduction into a convex optimization problem. The second one, which is based on the cone complementarity linearisation (CCL) idea [37, 38], casts the model reduction into a sequential minimization problem subject to LMI constraints. The effectiveness and superiority of the proposed approaches will be illustrated by a simulation example.

Notations. The notations used throughout the paper are standard. 𝐑^𝑛 and 𝐑^𝑚×𝑛 denote, respectively, the 𝑛-dimensional Euclidean space and the set of all 𝑚×𝑛 real matrices; ℕ⁺ represents the set of positive integers; the notation 𝑃>0 means that 𝑃 is real symmetric and positive definite; 𝐈 and 𝟎 represent the identity matrix and a zero matrix, respectively; ‖⋅‖ refers to the Euclidean vector norm; 𝑙₂[0,∞) is the space of square summable infinite sequence and, for 𝑤={𝑤(𝑘)}∈𝑙₂[0,∞), its norm is given by ‖𝑤‖₂=∑^∞_𝑘=0‖𝑤(𝑘)‖²; (𝒮,ℱ,𝒫) denotes a complete probability space, in which 𝒮 is the sample space, ℱ is the 𝜎 algebra of subsets of the sample space, and 𝒫 is the probability measure on ℱ; ℰ[⋅] stands for the mathematical expectation and, for the sequence 𝑧={𝑧(𝑘)}∈𝑙₂((𝒮,ℱ,𝒫),[0,∞)), its norm is given by ‖𝑧‖_ℰ2=ℰ[∑^∞_𝑘=0‖𝑧(𝑘)‖²]. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations.

2. Problem Formulation and Preliminaries

A discrete-time Markovian jump linear system (MJLS) on a complete probability space (𝒮,ℱ,𝒫) can be described as (Σ):𝑥()(𝑟(𝑘𝑥(𝑘)(𝑟(𝑘𝑢(𝑘),𝑧(𝑘)(𝑟(𝑘𝑥(𝑘)(𝑟(𝑘𝑢(𝑘),𝑘+1=𝐴))+𝐵))=𝐶))+𝐷)) (1) where 𝑥(𝑘)∈𝐑^𝑛𝑥 is the state vector; 𝑢(𝑘)∈𝐑^𝑛𝑢 is the control input vector which belongs to 𝑙₂[0,∞); 𝑧(𝑘)∈𝐑^𝑛𝑧 is the output vector. The process {𝑟(𝑘),𝑘≥0} is described by a discrete-time homogeneous Markov chain, which takes values in a finite set ℐ∶={1,…,𝑁} with mode transition probabilities (TPs) Pr(𝑟(𝑘+1)=𝑗∣𝑟(𝑘)=𝑖)=𝜋_𝑖𝑗, where 𝜋_𝑖𝑗≥0, for all 𝑖,𝑗∈ℐ, and ∑^𝑁_𝑗=1𝜋_𝑖𝑗=1. For 𝑟(𝑘)=𝑖, 𝑖∈ℐ, the system matrices of the 𝑖th mode are denoted by (𝐴_𝑖,𝐵_𝑖,𝐶_𝑖,𝐷_𝑖), which are real and known. Throughout the paper, we assume that system (Σ) is stochastically stable, which is a prerequisite for model reduction design.

In addition, the TPs of the jumping process are considered to be polytopic-type uncertain and partially accessed; that is, the transition probability matrix (TPM) Π=[𝜋_𝑖𝑗]_𝑁×𝑁 is assumed to belong to a given polytope P_Π∶={Π∣Π=∑^𝑀_𝑠=1𝛼_𝑠Π_𝑠;𝛼_𝑠≥0,∑^𝑀_𝑠=1𝛼_𝑠=1}, where vertices Π_𝑠=[𝜋_𝑖𝑗]_𝑁×𝑁, 𝑖,𝑗∈ℐ, 𝑠=1,2,…,𝑀 are given TPMs containing unknown elements still. For instance, for system (Σ) with four operation modes, the TPM may be as 𝜋₁₁𝜋₁₂𝜋₁₃𝜋₁₄𝜋₂₁𝜋₂₂𝜋₂₃𝜋₂₄𝜋₃₁𝜋₃₂𝜋₃₃𝜋₃₄𝜋₄₁𝜋₄₂𝜋₄₃𝜋₄₄, (2)where the elements labeled with “” and “” represent the unknown information and polytopic uncertainties on TPs, respectively, and the others are known TPs. For notational clarity, for all 𝑖∈ℐ, we denote ℐ=ℐ_𝒦^(𝑖)∪ℐ^(𝑖)_𝒰𝒞∪ℐ^(𝑖)_𝒰𝒦 as follows: ℐ_𝒦^(𝑖)∶=𝑗∶𝜋_𝑖𝑗isknown,ℐ^(𝑖)_𝒰𝒞∶=𝑗∶𝜋_𝑖𝑗isuncertain,ℐ^(𝑖)_𝒰𝒦∶=𝑗∶𝜋_𝑖𝑗isunknown. (3)Moreover, if ℐ_𝒦^(𝑖)≠∅ and ℐ^(𝑖)_𝒰𝒞≠∅, it is further described as ℐ_𝒦^(𝑖)∶=𝒦₁^(𝑖),…,𝒦_𝑡^(𝑖)_𝑖,∀1≤𝑡_𝑖ℐ≤𝑁−2,_{(𝑖)𝒰𝒞}∶=𝒰₁^(𝑖),…,𝒰_𝑣^(𝑖)_𝑖,∀1≤𝑣_𝑖≤𝑁, (4)where 𝒦_𝑡^(𝑖)_𝑖∈ℕ⁺ represents the 𝑡_𝑖th known element with the index 𝒦_𝑡^(𝑖)_𝑖 in the 𝑖th row of TPM and 𝒰_𝑣^(𝑖)_𝑖∈ℕ⁺ represents the 𝑣_𝑖th uncertain element with the index 𝒰_𝑣^(𝑖)_𝑖 in the 𝑖th row of TPM, respectively. Obviously, 1≤𝑡_𝑖+𝑣_𝑖≤𝑁. Also, we denote 𝜋^(𝑖𝑠)_𝒰𝒦∶=_{𝑗∈ℐ(𝑖)𝒰𝒦}𝜋_𝑖𝑗=1−_{𝑗∈ℐ𝒦(𝑖)}𝜋_𝑖𝑗−_{𝑗∈ℐ(𝑖)𝒰𝒞}𝜋^(𝑠)_𝑖𝑗, (5) where 𝜋^(𝑠)_𝑖𝑗 represents an uncertain TP in the 𝑠th polytope, for all 𝑠=1,…,𝑀.

To approximate the original MJLS (1), in this paper, we are interested in designing the following mode-dependent reduced-order model: Σ:𝑥(𝑘+1)=𝐴_𝑟𝑖𝑥(𝑘)+𝐵_𝑟𝑖𝑢(𝑘),𝑧(𝑘)=𝐶_𝑟𝑖𝑥(𝑘)+𝐷_𝑟𝑖𝑢(𝑘), (6) where 𝑥(𝑘)∈𝐑^𝑛𝑟(𝑛_𝑟<𝑛_𝑥), 𝑧(𝑘)∈𝐑^𝑛𝑧, and 𝐴_𝑟𝑖∈𝐑^{𝑛𝑟×𝑛𝑟}, 𝐵_𝑟𝑖∈𝐑^{𝑛𝑟×𝑛𝑢}, 𝐶_𝑟𝑖∈𝐑^{𝑛𝑧×𝑛𝑟}, and 𝐷_𝑟𝑖∈𝐑^{𝑛𝑧×𝑛𝑢} are the gains of the reduced-order models to be determined.

Define 𝑥(𝑘)∶=𝑥^𝑇(𝑘)𝑥^𝑇(𝑘)^𝑇 and 𝑧(𝑘)∶=𝑧(𝑘)−𝑧(𝑘). Then, by augmenting (1) and (6), the model error dynamics can be described as Σ:𝑥(𝑘+1)=𝐴_𝑖𝑥(𝑘)+𝐵_𝑖𝑢(𝑘),𝑧(𝑘)=𝐶_𝑖𝑥(𝑘)+𝐷_𝑖𝑢(𝑘), (7) where 𝐴_𝑖𝐴∶=^𝑖𝟎𝟎𝐴_𝑟𝑖,𝐵_𝑖𝐵∶=^𝑖𝐵_𝑟𝑖,𝐶_𝑖∶=𝐶_𝑖−𝐶_𝑟𝑖,𝐷_𝑖∶=𝐷_𝑖−𝐷_𝑟𝑖. (8)

Therefore, the purpose of this paper is to design a mode-dependent reduced-order model in the form of (6), such that the model error system (Σ) in (7) with deficient mode information is stochastically stable and the induced 𝑙₂-norm of the operator from 𝑢(𝑘) to the model error 𝑧(𝑘) is less than 𝛾; that is, ‖𝑧(𝑘)‖_ℰ2<𝛾‖𝑢(𝑘)‖₂, under zero initial conditions for any nonzero 𝑢(𝑘)∈𝑙₂[0,∞).

Before ending the section, we give the following lemma on the ℋ_∞ performance analysis of system (7) with completely known TPs, which will be used in the proof of our main results.

Lemma 1 (see [4]). For the MJLS (7) with completely known TPs and a given scalar 𝛾>0, if the coupled inequalities 𝐴^𝑇_𝑖𝓟_𝑖𝐴_𝑖−𝑃_𝑖+𝐶^𝑇_𝑖𝐶_𝑖𝐴^𝑇_𝑖𝓟_𝑖𝐵_𝑖+𝐶^𝑇_𝑖𝐷_𝑖∗−𝛾²𝐈−𝐵^𝑇_𝑖𝓟_𝑖𝐵_𝑖−𝐷^𝑇_𝑖𝐷_𝑖<0,∀𝑖∈ℐ, (9) where 𝓟_𝑖∶=∑^𝑁_𝑗=1𝜋_𝑖𝑗𝑃_𝑗, have a feasible solution 𝑃={𝑃₁,𝑃₂,…,𝑃_𝑁} with 𝑃_𝑖>0, then the MJLS (7) with completely known TPs is stochastically stable with an ℋ_∞ performance 𝛾.

3. Main Results

In this section, we will first derive the ℋ_∞ performance analysis criterion for the model error system (Σ) in (7) with deficient mode information. Then, two distinctly different approaches will be proposed to solve the ℋ_∞ model reduction problem formulated in the previous section.

3.1. ℋ_∞ Performance Analysis

In the following, by exploiting the properties of the transition probability matrices (TPMs), together with the convexification of uncertain domains, an ℋ_∞ performance analysis criterion for the model error system (Σ) in (7) with deficient mode information is presented, which will play an instrumental role in solving the ℋ_∞ model reduction problem.

Proposition 2. The MJLS (7) with deficient mode information is stochastically stable with a guaranteed ℋ_∞ performance 𝛾 if there exist positive-definite symmetric matrices 𝑃_𝑖∈𝐑^{(𝑛𝑥+𝑛𝑟)×(𝑛𝑥+𝑛𝑟)}, 𝑖∈ℐ, such that the following matrix inequalities hold: Φ^(𝑠)_𝑖𝑗∶=𝐴^𝑇_𝑖𝓟_𝑗^(𝑖𝑠)𝐴_𝑖−𝑃_𝑖+𝐶^𝑇_𝑖𝐶_𝑖𝐴^𝑇_𝑖𝓟_𝑗^(𝑖𝑠)𝐵_𝑖+𝐶^𝑇_𝑖𝐷_𝑖∗−𝛾²𝐈−𝐵^𝑇_𝑖𝓟_𝑗^(𝑖𝑠)𝐵_𝑖−𝐷^𝑇_𝑖𝐷_𝑖<0,∀𝑖∈ℐ,𝑗∈ℐ^(𝑖)_𝒰𝒦,𝑠=1,…,𝑀, (10) where 𝓟_𝑗^(𝑖𝑠)∶=_{𝑗∈ℐ𝒦(𝑖)}𝜋_i𝑗𝑃_𝑗+_{𝑗∈ℐ(𝑖)𝒰𝒞}𝜋^(𝑠)_𝑖𝑗𝑃_𝑗+𝜋^(𝑖𝑠)_𝒰𝒦𝑃_{𝑗𝑗∈ℐ(𝑖)𝒰𝒦},𝜋^(𝑖𝑠)_𝒰𝒦∶=1−_{𝑗∈ℐ𝒦(𝑖)}𝜋_𝑖𝑗−_{𝑗∈ℐ(𝑖)𝒰𝒞}𝜋^(𝑠)_𝑖𝑗. (11)

Proof. Based on Lemma 1, it is known that system (7) with completely known transition probabilities (TPs) is stochastically stable with an ℋ_∞ performance 𝛾 if (9) holds. Now due to ∑^𝑁_𝑗=1𝜋_𝑖𝑗=1 and with deficient mode information, we rewrite the term ∑^𝑁_𝑗=1𝜋_𝑖𝑗𝑃_𝑗 as ^𝑁_𝑗=1𝜋_𝑖𝑗𝑃_𝑗=_{𝑗∈ℐ𝒦(𝑖)}𝜋_𝑖𝑗𝑃_𝑗+_{𝑗∈ℐ(𝑖)𝒰𝒞}^𝑀_𝑠=1𝛼_𝑠𝜋^(𝑠)_𝑖𝑗𝑃_𝑗+_{𝑗∈ℐ(𝑖)𝒰𝒦}𝜋_𝑖𝑗𝑃_𝑗. (12) Considering the fact that 0≤𝛼_𝑠≤1, ∑^𝑀_𝑠=1𝛼_𝑠=1, and 0≤𝜋_𝑖𝑗/𝜋^(𝑖𝑠)_𝒰𝒦≤1, ∑_𝑗∈ℐ^(𝑖)_𝒰𝒦(𝜋_𝑖𝑗/𝜋^(𝑖𝑠)_𝒰𝒦)=1, (12) can be rewritten as ^𝑁_𝑗=1𝜋_𝑖𝑗𝑃_𝑗=^𝑀_𝑠=1𝛼_{𝑠𝑗∈ℐ(𝑖)𝒰𝒦}𝜋_𝑖𝑗𝜋^(𝑖𝑠)_{𝒰𝒦𝑗∈ℐ𝒦(𝑖)}𝜋_𝑖𝑗𝑃_𝑗+_{𝑗∈ℐ(𝑖)𝒰𝒞}𝜋^(𝑠)_𝑖𝑗𝑃_𝑗+𝜋^(𝑖𝑠)_𝒰𝒦𝑃_𝑗. (13) Thus, with deficient mode information, the left-hand side (LHS) of inequality (9) can be rewritten as LHS(9)=^𝑀_𝑠=1𝛼_{𝑠𝑗∈ℐ(𝑖)𝒰𝒦}𝜋_𝑖𝑗𝜋^(𝑖𝑠)_𝒰𝒦Φ^(𝑠)_𝑖𝑗, (14) where Φ^(𝑠)_𝑖𝑗 is defined in (10). Then (9) holds if and only if Φ^(𝑠)_𝑖𝑗<0, which implies that, in the presence of unknown and uncertain TPs, inequality (9) is equivalent to (10). This completes the proof.

Remark 3. By fully exploiting the properties of the TPMs, together with the convexification of uncertain domains, a sufficient condition for the ℋ_∞ performance analysis has been derived for the model error system (Σ) with deficient mode information in Proposition 2. It is noted that there exist product terms between the Lyapunov matrices and system matrices in condition (10), which will bring some difficulties in the solutions of model reduction problem. By applying decoupling techniques, in the following, two distinctly different approaches to solve the ℋ_∞ model reduction problem will be proposed.

3.2. Model Reduction via Convex Linearisation Approach

In this subsection, based on a linearisation procedure, the model reduction problem described in the previous section will be cast into a convex optimization problem. The following theorem presents a sufficient condition for the existence of admissible reduced-order models.

Theorem 4. Consider MJLS in (1) with deficient mode information and reduced-order model in the form of (6). The model error system in (7) is stochastically stable with an ℋ_∞ performance 𝛾 if there exist positive-definite symmetric matrices 𝑃_𝑖=^{𝑃𝑖(1)𝑃𝑖(2)}_{∗𝑃𝑖(3)}∈𝐑^{(𝑛𝑥+𝑛𝑟)×(𝑛𝑥+𝑛𝑟)} and matrices 𝐺_𝑖(1)∈𝐑^{𝑛𝑥×𝑛𝑥}, 𝐺_𝑖(2)∈𝐑^{𝑛𝑟×𝑛𝑟}, 𝐺_𝑖(3)∈𝐑^{𝑛𝑥×𝑛𝑟}, 𝐴_𝑟𝑖∈𝐑^{𝑛𝑟×𝑛𝑟}, 𝐵_𝑟𝑖∈𝐑^{𝑛𝑟×𝑛𝑢}, 𝐶_𝑟𝑖∈𝐑^{𝑛𝑧×𝑛𝑟}, and 𝐷_𝑟𝑖∈𝐑^{𝑛𝑧×𝑛𝑢}, 𝑖∈ℐ, such that the following LMIs hold:𝓟^(𝑖𝑠)_𝑗(1)−𝐺_𝑖(1)−𝐺^𝑇_𝑖(1)𝓟^(𝑖𝑠)_𝑗(2)−𝐻𝐺_𝑖(2)−𝐺^𝑇_𝑖(3)𝟎𝐺_𝑖(1)𝐴_𝑖𝐻𝐴_𝑟𝑖𝐺_𝑖(1)𝐵_𝑖+𝐻𝐵_𝑟𝑖∗𝓟^(𝑖𝑠)_𝑗(3)−𝐺_𝑖(2)−𝐺^𝑇_𝑖(2)𝟎𝐺_𝑖(3)𝐴_𝑖𝐴_𝑟𝑖𝐺_𝑖(3)𝐵_𝑖+𝐵_𝑟𝑖∗∗−𝐈𝐶_𝑖−𝐶_𝑟𝑖𝐷_𝑖−𝐷_𝑟𝑖∗∗∗−𝑃_𝑖(1)−𝑃_𝑖(2)𝟎∗∗∗∗−𝑃_𝑖(3)𝟎∗∗∗∗∗−𝛾²𝐈<0,𝑖∈ℐ,𝑗∈ℐ^(𝑖)_𝒰𝒦,𝑠=1,…,𝑀, (15)
where 𝓟^(𝑖𝑠)_𝑗(𝑚)∶=_{𝑗∈ℐ𝒦(𝑖)}𝜋_𝑖𝑗𝑃_𝑗(𝑚)+_{𝑗∈ℐ(𝑖)𝒰𝒞}𝜋^(𝑠)_𝑖𝑗𝑃_𝑗(𝑚)+𝜋^(𝑖𝑠)_𝒰𝒦𝑃_{𝑗(𝑚)𝑗∈ℐ(𝑖)𝒰𝒦},𝜋𝑚=1,2,3,_{(𝑖𝑠)𝒰𝒦}∶=1−_{𝑗∈ℐ𝒦(𝑖)}𝜋_𝑖𝑗−_{𝑗∈ℐ(𝑖)𝒰𝒞}𝜋^(𝑠)_𝑖𝑗,𝐻∶=𝐈_{𝑛𝑟×𝑛𝑟}𝟎_{𝑛𝑟×(𝑛𝑥−𝑛𝑟)}^𝑇. (16)
Moreover, if the above conditions have a set of feasible solutions (𝑃_𝑖(1),𝑃_𝑖(2),𝑃_𝑖(3),𝐺_𝑖(1),𝐺_𝑖(2),𝐺_𝑖(3),𝐴_𝑟𝑖,𝐵_𝑟𝑖,𝐶_𝑟𝑖,𝐷_𝑟𝑖), then an admissible 𝑛_𝑟-order approximation model in the form of (6) can be constructed as 𝐴_𝑟𝑖=𝐺⁻¹_𝑖(2)𝐴_𝑟𝑖,𝐵_𝑟𝑖=𝐺⁻¹_𝑖(2)𝐵_𝑟𝑖,𝐶_𝑟𝑖=𝐶_𝑟𝑖,𝐷_𝑟𝑖=𝐷_𝑟𝑖. (17)

Proof. It follows from Proposition 2 that if we can show (10), then the claimed results follow. By Schur complement, inequality (10) is equivalent to 𝓟_𝑗^{(𝑖𝑠)−1}𝟎𝐴_𝑖𝐵_𝑖∗−𝐈𝐶_𝑖𝐷_𝑖∗∗−𝑃_𝑖𝟎∗∗∗−𝛾²𝐈<0,∀𝑖∈ℐ,𝑗∈ℐ^(𝑖)_𝒰𝒦,𝑠=1,…,𝑀, (18) where 𝓟_𝑗^(𝑖𝑠)∶=_{𝑗∈ℐ𝒦(𝑖)}𝜋_𝑖𝑗𝑃_𝑗+_{𝑗∈ℐ(𝑖)𝒰𝒞}𝜋^(𝑠)_𝑖𝑗𝑃_𝑗+𝜋^(𝑖𝑠)_𝒰𝒦𝑃_{𝑗𝑗∈ℐ(𝑖)𝒰𝒦},𝜋^(𝑖𝑠)_𝒰𝒦∶=1−_{𝑗∈ℐ𝒦(𝑖)}𝜋_𝑖𝑗−_{𝑗∈ℐ(𝑖)𝒰𝒞}𝜋^(𝑠)_𝑖𝑗. (19) Perform a congruent transformation to (18) by diag{𝐺_𝑖,𝐈,𝐈,𝐈}, and consider the following bounding inequality: −𝐺_𝑖𝓟_𝑗^{(𝑖𝑠)−1}𝐺^𝑇_𝑖≤𝓟_𝑗^(𝑖𝑠)−𝐺_𝑖−𝐺^𝑇_𝑖. (20) It is thus easy to see that the following inequality implies (18), 𝓟_𝑗^(𝑖𝑠)−𝐺_𝑖−𝐺^𝑇_𝑖𝟎𝐺_𝑖𝐴_𝑖𝐺_𝑖𝐵_𝑖∗−𝐈𝐶_𝑖𝐷_𝑖∗∗−𝑃_𝑖𝟎∗∗∗−𝛾²𝐈<0,∀𝑖∈ℐ,𝑗∈ℐ^(𝑖)_𝒰𝒦,𝑠=1,…,𝑀. (21)
Furthermore, for model reduction synthesis purpose, we choose the slack matrix 𝐺_𝑖 as 𝐺_𝑖𝐺∶=^𝑖(1)𝐻𝐺₍₂₎𝐺_𝑖(3)𝐺₍₄₎, (22) where 𝐻∶=𝐈_𝑛𝑟𝟎_{𝑛𝑟×(𝑛𝑥−𝑛𝑟)}^𝑇, 𝐺_𝑖(1)∈𝐑^{𝑛𝑥×𝑛𝑥}, 𝐺_𝑖(3)∈𝐑^{𝑛𝑟×𝑛𝑥}, 𝐺₍₂₎∈𝐑^{𝑛𝑟×𝑛𝑟}, and 𝐺₍₄₎∈𝐑^{𝑛𝑟×𝑛𝑟}. Then, similar to [22], performing a congruent transformation to 𝐺_𝑖(1)+𝐺^𝑇_𝑖(1)𝐻𝐺₍₂₎+𝐺^𝑇_𝑖(3)∗𝐺₍₄₎+𝐺^𝑇₍₄₎ (23) by 𝐈𝟎𝟎𝐺₍₂₎𝐺⁻¹₍₄₎ yields 𝐺_𝑖(1)+𝐺^𝑇_𝑖(1)𝐻𝐺₍₂₎𝐺^−𝑇₍₄₎𝐺^𝑇₍₂₎+𝐺^𝑇_𝑖(3)𝐺^−𝑇₍₄₎𝐺^𝑇₍₂₎∗𝐺₍₂₎𝐺^−𝑇₍₄₎𝐺^𝑇₍₂₎+𝐺₍₂₎𝐺⁻¹₍₄₎𝐺^𝑇₍₂₎𝐺∶=^𝑖(1)+𝐺^𝑇_𝑖(1)𝐻𝐺₍₂₎+𝐺^𝑇_𝑖(3)∗𝐺₍₂₎+𝐺^𝑇₍₂₎. (24) Thus, instead of (22), one can directly specify matrix 𝐺_𝑖 of the following form without loss of generality: 𝐺_𝑖=G_𝑖(1)𝐻𝐺_𝑖(2)𝐺_𝑖(3)𝐺_𝑖(2),𝑖∈ℐ. (25) It is noted that in this way the matrix variable 𝐺_𝑖(2) can be absorbed by the reduced-order model gain variables 𝐴_𝑟𝑖 and 𝐵_𝑟𝑖 by introducing 𝐴_𝑟𝑖∶=𝐺_𝑖(2)𝐴_𝑟𝑖,𝐵_𝑟𝑖∶=𝐺_𝑖(2)𝐵_𝑟𝑖. (26) This feature enables one to make no congruent transformation to the original matrix inequality, and all the slack variables can be set as Markovian switching.
Then, substituting matrices 𝐺_𝑖 given in (25) into (10), together with the consideration of the matrices defined in (26), leads to (15) exactly.
On the other hand, the conditions in (15) imply that −𝐺_𝑖(2)−𝐺^𝑇_𝑖(2)<0, which means that 𝐺_𝑖(2) is nonsingular. Thus, the reduced-order model gains can be constructed by (17). The proof is completed.

Remark 5. Theorem 4 provides a sufficient condition for the solvability of ℋ_∞ model reduction synthesis problem for the MJLS (1) with deficient mode information. A desired reduced-order model can be determined by solving the following convex optimization problem.

Problem MRLA (Model Reduction via Linearisation Approach). Minimize 𝛾 subject to (15) for 𝑃_𝑖(1), 𝑃_𝑖(2), 𝑃_𝑖(3), 𝐺_𝑖(1), 𝐺_𝑖(2), 𝐺_𝑖(3), 𝐴_𝑟𝑖, 𝐵_𝑟𝑖, 𝐶_𝑟𝑖, 𝐷_𝑟𝑖, 𝑖∈ℐ.

From the proof of Theorem 4, it is easy to see that the matrix inequality linearisation procedure is based on the bounding inequality in (20) with slack variables of a structural constraint in (25). This inevitably brings some degree of design conservatism. Another avenue to solve the nonlinear matrix inequality problem in (10) is by utilizing the cone complementarity linearisation (CCL) technique [37]. To this end, we will resort to an iterative approach to solve the model reduction problem.

3.3. Model Reduction via Iterative Approach

In this subsection, we will solve the model reduction problem by using the CCL technique. The following theorem gives another sufficient condition for the existence of admissible reduced-order models.

Theorem 6. Consider MJLS in (1) with deficient mode information and reduced-order model in the form of (6). The model error system in (7) is stochastically stable with an ℋ_∞ performance 𝛾 if there exist positive-definite symmetric matrices {P_𝑖,𝑋_𝑖}∈𝐑^{(𝑛𝑥+𝑛𝑟)×(𝑛𝑥+𝑛𝑟)} and matrices 𝐴_𝑟𝑖∈𝐑^{𝑛𝑟×𝑛𝑟}, 𝐵_𝑟𝑖∈𝐑^{𝑛𝑟×𝑛𝑢}, 𝐶_𝑟𝑖∈𝐑^{𝑛𝑧×𝑛𝑟}, and 𝐷_𝑟𝑖∈𝐑^{𝑛𝑧×𝑛𝑢}, 𝑖∈ℐ, such that 𝒳_𝑗^(𝑖)𝟎ϝ_𝑖^(𝑠)𝒜_𝑖+𝐸𝒢_𝑟𝑖𝑅ϝ_𝑖^(𝑠)ℬ_𝑖+𝐸𝒢_𝑟𝑖𝑆∗−𝐈𝓒_𝑖+𝐹𝒢_𝑟𝑖𝑅𝒟_𝑖+𝐹𝒢_𝑟𝑖S∗∗−𝑃_𝑖𝟎∗∗∗−𝛾²𝐈<0,𝑖∈+𝐹𝒢_𝑟𝑖𝑅ℐ,𝑗∈ℐ^(𝑖)_𝒰𝒦,𝑠=1,…,𝑀,𝑃_𝑖𝑋_𝑖=𝐈,𝑖∈ℐ, (27)(28) where 𝒳_𝑗^(𝑖)∶=diag𝑋_𝒦1^(𝑖),…,𝑋_𝒦𝑡𝑖^(𝑖),𝑋_𝒰1^(𝑖),…,𝑋_𝒰𝑣𝑖^(𝑖),𝑋_𝑗,ϝ_𝑖^(𝑠)∶=𝜋_𝑖𝒦1^(𝑖)𝐈,…,𝜋_{𝑖𝒦𝑡𝑖}^(𝑖)𝐈,𝜋^(𝑠)_{𝑖𝒰1(𝑖)}𝐈,…,𝜋^(𝑠)_{𝑖𝒰𝑣𝑖(𝑖)}𝐈,𝜋^(𝑖𝑠)_𝒰𝒦𝐈^𝑇,𝒢_𝑟𝑖𝐴∶=^𝑟𝑖𝐵_𝑟𝑖𝐶_𝑟𝑖𝐷_𝑟𝑖,𝒜_𝑖𝐴∶=^𝑖𝟎𝟎𝟎,ℬ_𝑖𝐵∶=^𝑖𝟎,𝓒_𝑖∶=𝐶_𝑖𝟎,𝒟_𝑖∶=𝐷_𝑖,𝟎𝐈,𝜋,𝐸∶=𝟎𝟎𝐈𝟎𝐹∶=𝟎−𝐈,𝑅∶=𝟎𝐈𝟎𝟎,𝑆∶=_{(𝑖𝑠)𝒰𝒦}∶=1−_{𝑗∈ℐ𝒦(𝑖)}𝜋_𝑖𝑗−_{𝑗∈ℐ(𝑖)𝒰𝒞}𝜋^(𝑠)_𝑖𝑗. (29)

Proof. From Proposition 2, we know that for 𝑖∈ℐ there exists a reduced-order model in the form of (6) such that the model error system (7) is stochastically stable with a guaranteed ℋ_∞ error performance 𝛾 if there exist matrices 𝑃_𝑖 satisfying (10). First, rewrite the matrices defined in (8) in the following form: 𝐴_𝑖∶=𝒜_𝑖+𝐸𝒢_𝑟𝑖𝑅,𝐵_𝑖∶=ℬ_𝑖+𝐸𝒢_𝑟𝑖𝑆,𝐶_𝑖∶=𝓒_𝑖+𝐹𝒢_𝑟𝑖𝑅,𝐷_𝑖∶=𝒟_𝑖+𝐹𝒢_𝑟𝑖𝑆, (30) where 𝒢_𝑟𝑖𝐴∶=^𝑟𝑖𝐵_𝑟𝑖𝐶_𝑟𝑖𝐷_𝑟𝑖,𝒜_𝑖𝐴∶=^𝑖𝟎𝟎𝟎,ℬ_𝑖𝐵∶=^𝑖𝟎,𝓒_𝑖∶=𝐶_𝑖𝟎,𝒟_𝑖∶=𝐷_𝑖,𝟎𝐈.,𝐸∶=𝟎𝟎𝐈𝟎𝐹∶=𝟎−𝐈,𝑅∶=𝟎𝐈𝟎𝟎,𝑆∶= (31)
Then, by Schur complement to (10), and with the notations ℐ_𝒦^(𝑖)∶={𝒦₁^(𝑖),…,𝒦_𝑡^(𝑖)_𝑖}, ℐ^(𝑖)_𝒰𝒞∶={𝒰₁^(𝑖),…,𝒰_𝑣^(𝑖)_𝑖}, we have that (10) is equivalent to 𝓟_𝑗^(𝑖)𝟎ϝ_𝑖^(𝑠)𝒜_𝑖+𝐸𝒢_𝑟𝑖𝑅ϝ_𝑖^(𝑠)ℬ_𝑖+𝐸𝒢_𝑟𝑖𝑆∗−𝐈𝓒_𝑖+𝐹𝒢_𝑟𝑖𝑅𝒟_𝑖+𝐹𝒢_𝑟𝑖𝑆∗∗−𝑃_𝑖𝟎∗∗∗−𝛾²𝐈<0,𝑖∈ℐ,𝑗∈ℐ^(𝑖)_𝒰𝒦,𝑠=1,…,𝑀, (32) where 𝓟_𝑗^(𝑖)∶=diag𝑃_𝒦⁻¹_1(𝑖),…,𝑃_𝒦⁻¹_{𝑡𝑖(𝑖)},𝑃_𝒰⁻¹_1(𝑖),…,𝑃_𝒰⁻¹_{𝑣𝑖(𝑖)},𝑃_𝑗⁻¹,ϝ_𝑖^(𝑠)∶=𝜋_𝑖𝒦1^(𝑖)𝐈,…,𝜋_{𝑖𝒦𝑡𝑖}^(𝑖)𝐈,𝜋^(𝑠)_{𝑖𝒰1(𝑖)}𝐈,…,𝜋^(𝑠)_{𝑖𝒰𝑣𝑖(𝑖)}𝐈,𝜋^(𝑖𝑠)_𝒰𝒦𝐈^𝑇. (33) Setting 𝑋_𝑖∶=𝑃_𝑖⁻¹ and considering (33), it is easy to see that (32) is equivalent to (27) and (28). This completes the proof.

Remark 7. Theorem 6 provides another sufficient condition for testing the solvability of 𝑛_𝑟-order ℋ_∞ model reduction synthesis for MJLS (1) with deficient mode information. It is noted that the conditions in Theorem 6 are not strict LMI-based conditions due to the matrix equality in (28). However, with the CCL algorithm in [37, 38], we can resolve this nonconvex feasibility problem by formulating it into a sequential optimization problem subject to LMI constraints.

Based on the previous discussions and using a CCL technique, the nonconvex feasibility problem given in (27) and (28) is converted into the following nonlinear minimization problem that involves LMI conditions.

Problem MRIA (Model Reduction via Iterative Approach). Minimize Trace (∑^𝑁_𝑖=1𝑃_𝑖𝑋_𝑖) subject to (27) and 𝑃_𝑖𝐈∗𝑋_𝑖≥0,∀𝑖∈ℐ. (34)

Then, the suboptimal performance of 𝛾 can be found by the following algorithm. The convergence of this algorithm is guaranteed in terms of similar results in [37, 38].

Algorithm MRIA: Suboptimal Performance of 𝛾

Step 1. Choose a sufficiently large initial 𝛾>0, such that there exists a feasible solution to (27) and (34). Set 𝛾₀=𝛾.

Step 2. Find a feasible set (𝑃_𝑖⁽⁰⁾,𝑋_𝑖⁽⁰⁾,𝐴⁽⁰⁾_𝑟𝑖,𝐵⁽⁰⁾_𝑟𝑖,𝐶⁽⁰⁾_𝑟𝑖,𝐷⁽⁰⁾_𝑟𝑖, for  all 𝑖∈ℐ) that satisfies the conditions in (27) and (34). Set 𝑞=0.

Step 3. Solve the following LMI problem over the variables 𝑃_𝑖, 𝑋_𝑖, 𝐴_𝑟𝑖, 𝐵_𝑟𝑖, 𝐶_𝑟𝑖, and 𝐷_𝑟𝑖: MinimizeTrace^𝑁_𝑖=1𝑃_𝑖^(𝑞)𝑋_𝑖+𝑃_𝑖𝑋_𝑖^(𝑞) (35) subject  to (27)  and  (34).
Set 𝑃_𝑖^(𝑞+1)=𝑃_𝑖 and 𝑋_𝑖^(𝑞+1)=𝑋_𝑖.

Step 4. Substituting the gains 𝐴_𝑟𝑖, 𝐵_𝑟𝑖, 𝐶_𝑟𝑖, and 𝐷_𝑟𝑖 obtained in Step 3 into (10) and if the LMIs in (10) are feasible with respect to the variables 𝑃_𝑖, then set 𝛾₀=𝛾 and return to Step 2 after decreasing 𝛾 to some extent. If (10) are infeasible within the maximum number of iterations allowed, then exit. Otherwise, set 𝑞=𝑞+1, and go to Step 3.

Remark 8. It is noted that the ℋ_∞ model reduction problem for discrete-time MJLSs has been considered in [20]. However, there are some remarkable differences between our results and those in [20]. Firstly, it is noted that in this paper the deficient mode information, which simultaneously contains the exactly known, partially unknown, and uncertain transition probabilities (TPs), is considered for MJLS (1), which is more general and practical for engineering applications, whereas the TPs considered in [20] are only partially unknown. Secondly, by fully utilizing the properties of the TPM, including the relation ∑_𝑗∈ℐ^(𝑖)_𝒰𝒦𝜋_𝑖𝑗=1−∑_{𝑗∈ℐ𝒦}^(𝑖)𝜋_𝑖𝑗−∑_𝑗∈ℐ^(𝑖)_𝒰𝒞𝜋^(𝑠)_𝑖𝑗, Theorem 4 gives a unified formulation of ℋ_∞ model reduction design for MJLS (1), while in [20] the above relation is neglected, which may cause possible conservatism. In addition, two different approaches, namely, the convex linearisation approach and iterative approach, to solve the model reduction problem are proposed, while in [20] only the linearisation approach was presented. It will be shown in the simulation part that the previous points are crucial to reduce the conservatism of model reduction design for MJLS in (1).

Remark 9. It is worth mentioning that the conditions given in Theorem 4 are convex and thus can be readily solved with commercially available software. The design conservatism of Theorem 4 mainly comes from the bounding inequality in (20) with the slack variable of a structural constraint in (25). In the iterative approach, the conditions given in (27) and (34) are equivalent to the corresponding performance analysis results. This is the main advantage of Theorem 6 over Theorem 4. However, the numerical computation cost involved in Algorithm MRIA is also much larger than that involved in Theorem 4 (MRLA), especially when the number of iterations increases.

4. An Illustrative Example

In this section, we will present an illustrative example to demonstrate the effectiveness and superiority of the proposed approaches.

Consider a Markovian jump linear system (MJLS) described by (Σ) with parameters as follows: 𝐴₁𝐵₁𝐶₁𝐷₁=0.05−0.270.440.390.50.550.330.380.550.20.10.170.270.440.30.050.220.160.110.110.10.2−0.3,𝐴0.5₂𝐵₂𝐶₂𝐷₂=0.11−0.170.270.44−0.80.550.060.220.55−0.20.050.170.280.44−0.10.170.050.06−0.11−10.5−0.80.30.5,𝐴0.5₃𝐵₃𝐶₃𝐷₃=0.160.06−0.020.180.20.04−0.370.53−0.04−0.2−0.08−0.32−0.05−0.11−0.1−0.170.400.040.290.11.40.70.2−0.8,𝐴0.5₄𝐵₄𝐶₄𝐷₄=0.230.01−0.55−0.380.9−0.330.36−0.48−0.10−1.1−0.20−0.450.1−0.19−0.70.230.160.5−0.30−1.2−0.71.21.2−0.6.0.5 (36) Four different cases for the transition probability matrix (TPM) are given in Table 1, where the transition probabilities (TPs) labeled “” and “” represent the unknown and uncertain elements, respectively. Specifically, Case  1, Case  2, Case  3, and Case  4 stand for the completely known TPs, deficient mode information (including known, partially unknown, and uncertain TPs), partially unknown TPs, and completely unknown TPs, respectively.

Table 1 

Four different TPMs.

For Case 2 shown in Table 1, the uncertain TPs comprise two vertices Π_𝑠, 𝑠=1,2, where the third rows Π_𝑠(3), 𝑠=1,2, are given by Π₁₍₃₎=𝜋₃₁0.1𝜋₃₃,Π0.3₂₍₃₎=𝜋₃₁0.3𝜋₃₃0.4, (37) and the other rows in the two vertices are defined with the same elements; that is, Π_𝑠(1)=,Π0.30.20.10.4_𝑠(2)=𝜋₂₁𝜋₂₂,Π0.30.2_𝑠(4)=0.2𝜋₄₂𝜋₄₃𝜋₄₄,𝑠=1,2. (38)

The objective is to design an 𝑛_𝑟-order model of the form (6) to approximate the above system such that the model error system (7) is stochastically stable with an ℋ_∞ performance index 𝛾. Here, we first assume that the MJLS is subject to the partially unknown TPs in Case 3, and then a detailed comparison between the minimum ℋ_∞ performance indices 𝛾_min obtained based on Theorem 4 in this paper and the results in [20] is summarized in Table 2. It is observed from Table 2 that the design approach proposed in Theorem 4 is less conservative than [20].

Table 2 

Comparison of minimum ℋ_∞ performance with deficient TPM2 in Case 3.

In the following, we will further demonstrate the advantage and less conservatism of Theorem 6 over Theorem 4 proposed in this paper. By solving the problems MRLA and MRIA, another detailed comparison of the minimum ℋ_∞ performance indices 𝛾_min is listed in Table 3, where the bold numbers in the brackets represent the number of iterations. By inspection of Tables 1 and 2, it is easy to see that the results in Theorem 6 are less conservative than those in Theorem 4. It is also shown that the more information on TPs is available, the better ℋ_∞ performance can be obtained, which is effective to reduce the conservatism of model reduction design. Therefore, the introduction of the polytopic-type uncertain TPs is significant.

Table 3 

Comparison of minimum ℋ_∞ performance for different TPMs.

Specifically, with 𝑛_𝑟=3, we obtain 𝛾_min=1.1687 by Theorem 4 with deficient TPM1 shown in Table 1, and the three-order model parameters are given by 𝐴_𝑟1𝐵_𝑟1𝐶_𝑟1𝐷_𝑟1=−0.05400.25290.2176−0.84900.32740.62850.48620.1883−0.15430.32500.2913−0.8224−0.94250.1661−0.0302,𝐴0.6282_𝑟2𝐵_𝑟2𝐶_𝑟2𝐷_𝑟2=0.17240.10950.38940.87420.36540.46740.27850.3566−0.04840.34570.24160.1962−0.64960.6678−0.7150,𝐴0.6001_𝑟3𝐵_𝑟3𝐶_𝑟3𝐷_𝑟3=0.26550.2048−0.1861−0.25370.1033−0.16180.24050.1205−0.1747−0.39360.10180.3127−1.2638−0.0046−0.0662,𝐴0.4385_𝑟4𝐵_𝑟4𝐶_𝑟4𝐷_𝑟4=0.1162−0.5423−0.5004−0.7796−0.37670.0416−0.35891.18590.0724−0.37650.05330.65050.5008−0.9345−1.1617.0.3712 (39)

With 𝑛_𝑟=3, we obtain 𝛾=0.6 by solving the Algorithm MRIA after 39 iterations with deficient TPM1 shown in Table 1, and the three-order model parameters are given by 𝐴_𝑟1𝐵_𝑟1𝐶_𝑟1𝐷_𝑟1=0.28000.07720.6328−0.43520.52610.23070.7350−0.15210.04620.24680.3092−0.0833−1.52390.20640.4439,𝐴0.4937_𝑟2𝐵_𝑟2𝐶_𝑟2𝐷_𝑟2=0.3132−0.00480.72630.55100.49000.02640.7604−0.24550.08260.1022−0.00380.5862−0.72150.8753−0.3862,𝐴0.4679_𝑟3𝐵_𝑟3𝐶_𝑟3𝐷_𝑟3=0.1946−0.09210.0870−0.08260.1133−0.2445−0.50270.2327−0.14520.19800.3471−0.0378−1.9205−0.36850.8553,𝐴0.4941_𝑟4𝐵_𝑟4𝐶_𝑟4𝐷_𝑟4=−0.0592−0.2981−0.6252−0.3605−0.5600−0.00610.01200.51090.1490−0.0012−0.31140.83360.5634−2.4550−0.7422.0.4789 (40)

The feasible solutions for the other cases are omitted for brevity.

In order to further illustrate the effectiveness of the designed approximation models, simulations have been carried out. Specifically, choose the zero initial condition and input 𝑢(𝑘) as 𝑢(𝑘)=𝑒^{(−0.2𝑘+1)}()()sin0.2𝑘,10≤𝑘≤30,0.3cos0.25𝑘,50≤𝑘≤90,0,otherwise. (41) With the above obtained approximation models under Case 2 given in Table 1, and one of the possible realizations of the Markovian jumping mode shown in Figure 1, the output trajectories of the original system and approximation models obtained based on Theorem 4 (MRLA) and Theorem 6 (MRIA) are shown in Figures 2 and 3, respectively. It can be clearly observed from the simulation curves that, despite the deficient TPs, the designed reduced-order models can approximate the original system very well.

Figure 1 

One possible system mode evolution.

Figure 2 

Output trajectories of the original system and reduced-order models with deficient TPM1 based on Theorem 4 (MRLA).

Figure 3 

Output trajectories of the original system and reduced-order models with deficient TPM1 based on Theorem 6 (MRIA).

5. Conclusions

This paper has studied the problem of robust ℋ_∞ model reduction for a class of discrete-time Markovian jump linear systems (MJLSs) with deficient mode information, which simultaneously consists of the exactly known, partially unknown, and polytopic-type uncertain transition probabilities and is thus more general and practical. By fully utilizing the properties of the transition probability matrices, together with the convexification of uncertain domains, a new ℋ_∞ performance analysis criterion has been developed. Two distinctly different approaches, namely, the convex linearisation approach and iterative approach, have been proposed to solve the model reduction problem. A simulation example has been given to illustrate the less conservatism and effectiveness of the proposed approaches.

Acknowledgments

The authors are grateful to the guest editors and anonymous reviewers for their constructive comments based on which the presentation of this paper has been greatly improved. This work was supported in part by the National Natural Science Foundation of China (61004038), in part by the National 973 Project of China (2009CB320600), in part by the Program for New Century Excellent Talents in University, in part by the Fundamental Research Funds for the Central Universities (HIT.BRETIII.201214), and in part by the Alexander von Humboldt Foundation of Germany.