Robust Filtering for Linear Equality Constrained Systems

Abstract

This paper deals with the robust filtering problem for linear discrete-time constrained systems. The purpose is the design of a linear filter such that the resulting error system is bounded. An orthogonal factorization is used to decompose the original robust filtering problem into stochastic and deterministic parts, which are then solved separately. Finally, a numerical example is presented to demonstrate the applicability of the proposed method.

1. Introduction

Kalman filtering is one of the well-known 𝐻₂ filtering methods that is widely used in the fields of signal processing and automatic control [1]. It is noted that the Kalman filtering method is based on the assumption that the system has known model and its disturbances are Gaussian white noises with known statistics. In some applications, however, the statistics of the noises are not exactly known, and the standard Kalman filtering algorithms will generally not guarantee satisfactory performance and only can obtain the estimate value with great error. Also, the strict assumptions limit the application scope of this filtering especially when there are uncertainties in either the state model or the measurement model. To handle the above problem, an alternative regularized estimate method based on least square design technique has been proposed recently. The objective is to find a filter such that the resulting estimate error is bounded and the main idea of this method is to reduce the vector optimization problem to an equivalent scalar minimization problem [2, 3]. Compared with earlier studies, such as 𝐻_∞ method and guaranteed-cost method, the new method simultaneously uses regularization and weighting to deal with a class of uncertainties [3–5].

On the other hand, constrained filtering and control problem has drawn considerable attention over the past decades due to extensive application backgrounds. Actually, constraint formulation arises naturally in many fields such as target tracking, manufacturing production, engine health estimation, and vehicle motion [6, 7]. One of the features of these systems is that some components of the state are affected by some equations without noises. In conventional linear stochastic models with additive white process noise, filtering method for constrained systems has been investigated by many scholars. For example, Wen and Durrant-Whyte have considered the constrained problem by treating the set of constraint equations as additional accurate observations without noises [8]. Simon and Chia have shown that the solution of constrained problem can be obtained by treating the constraint equation as a constrained condition and solving a Lagrangian equation [9]. Moreover, Hewett et al. have presented a reduced null space method based on the null space decomposition to solve such problems [10].

Among the previous works on constrained estimation, the most popular approach is the projection method. This method enforces linear equality constraints on state space estimation, and the constrained estimate is merely a correction that forces the unconstrained estimate onto the constraint space [9]. In actual estimate and in value of objective function, the null space method often produces similar results as that of the project method. For the problem of robust filtering for constrained systems, however, there are still no results available in the literature. This motivates the present study. The regularized robust filtering is originally developed by Sayed to deal with the regularized dynamic system [2, 3] and Ishihara et al. use this method to present the robust filtering for uncertain singular system [11]. In this paper, we will give the regularized robust design method for uncertain constrained system.

In this paper, we deal with the robust filtering problem for uncertain constrained systems. Attention is focused on solving the least square problem, and the robust Kalman type recursion is developed. The remainder of this paper is organized as follows. Section 2 formulates the constrained systems and the problems to be solved. We review the filtering method for accurate constrained model in Section 3. In Section 4, the QR factorization is used to gain a new reduced system and the robust filtering is presented. We show numerical example that illustrates the new method performance in Section 5 and offer conclusion in Section 6.

The notation used in this paper is standard. 𝐴^T and 𝐴^† are the transpose and the pseudoinverse of the matrix 𝐴, respectively. 𝑃>0 (𝑃≥0) denotes a positive-definite (semidefinite) matrix. For a column vector 𝑥 and a positive matrix 𝑊, ‖𝑥‖² is the Euclidean norm of 𝑥, and ‖𝑥‖_𝑊 is the weighted form. diag{𝑥,𝑦} denotes a block diagonal matrix with entries 𝑥 and 𝑦.

2. Problem Formulation and Analysis

2.1. Problem Formation

Consider a uncertain linear constrained system described by following model: 𝑥_𝑘=𝐴_𝑘+𝛿𝐴_𝑘𝑥_𝑘−1+𝑤_𝑘−1,𝑦_𝑘=𝐻_𝑘+𝛿𝐻_𝑘𝑥_𝑘+𝑣_𝑘,𝐷_𝑘𝑥_𝑘=𝑑_𝑘, (2.1)(2.2)(2.3) where 𝑥_𝑘∈𝑅^𝑛 is the state vector satisfying equality constraints and 𝑦_𝑘∈𝑅^𝑚 is the measurement output. 𝐴_𝑘 is a 𝑛×𝑛 state update matrix, 𝐻_𝑘 is an 𝑚×𝑛 observation matrix, 𝐷_𝑘 is a 𝑠×𝑛 constraint matrix, and 𝑑_𝑘 is a known vector. 𝛿𝐴_𝑘 and 𝛿𝐻_𝑘 are time-varying uncertainties to the nominal system matrices. The initial state 𝑥₀, process noises sequence 𝑤_𝑘, and measurement noises sequence 𝑣_𝑘 are uncorrelated zero mean white noises with variance 𝐸𝑥₀𝑤_𝑘𝑣_𝑘𝑥₀𝑤_𝑙𝑣_𝑙^T=Π₀000Ω_𝑘𝛿_𝑘𝑙000𝑉_𝑘𝛿_𝑘𝑙, (2.4) where 𝛿_𝑘𝑙 is the Kronecker function, Π₀>0, Ω_𝑘>0, and 𝑉_𝑘>0. The uncertainties are assumed with the following structure: 𝛿𝐴_𝑘𝛿𝐻_𝑘=𝑀_𝑎𝑘𝑀_ℎ𝑘Δ_𝑘𝑁_𝑘,Δ_𝑘≤1, (2.5) where 𝑀_𝑎𝑘, 𝑀_ℎ𝑘, and 𝑁_𝑘 are known matrices, and Δ_𝑘 is a bounded matrix but otherwise arbitrary. We allow 𝑀_𝑎𝑘, 𝑀_ℎ𝑘, and 𝑁_𝑘 to vary with time.

The purpose of this paper is to find a recursive robust state estimate algorithm for this constrained system with modeling uncertainties. With the constrained condition (2.3), the system (2.1)–(2.3) is not a standard form and the robust filtering presented in [2] is not applicable, so we cannot directly use them to present the analysis. The key to solving this problem is to transfer the constrained system into some new systems without constraint.

On the other hand, the final estimate result of the state 𝑥_𝑘 should satisfy the additional constraint (2.3), which means that the estimate belongs to the space, denoted as Θ_𝑘, composed by the solutions of (2.3). The constraint matrix 𝐷_𝑘 and vector 𝑑_𝑘 are assumed to satisfy Rank[𝐷_𝑘𝑑_𝑘]=Rank[𝐷_𝑘] to make Θ_𝑘≠Ø. We assume that the constraint matrix 𝐷_𝑘 has full column row rank and 𝑠<𝑛.

3. Standard Constrained Filter

The constrained filter algorithm has some advantages compared with the standard Kalman filter, which are given in [9, 10]. In this section, we will review the constrained filtering method for accurate state-space model.

The accurate constrained system is 𝑥_𝑘=𝐴_𝑘𝑥_𝑘−1+𝑤_𝑘−1,𝑦_𝑘=𝐻_𝑘𝑥_𝑘+𝑣_𝑘,𝐷_𝑘𝑥_𝑘=𝑑_𝑘. (3.1)

In [10], it uses orthogonal factorization to decompose the original state into stochastic and deterministic parts. The 𝑄𝑅 factorization of 𝐷^T_𝑘𝐷^T_𝑘=𝑄_1,𝑘𝑄_2,𝑘𝑅_11,𝑘0 (3.2) and the initial state can be rewritten as 𝑥_𝑘=𝑥_𝑑,𝑘+𝑄_2,𝑘𝑧_𝑠,𝑘, (3.3) where 𝑥_𝑑,𝑘=𝑄_1,𝑘𝑅^−T_11,𝑘𝑑_𝑘. (3.4)

It also gives a new reduced constrained system 𝑧_𝑠,𝑘=𝑄^T_2,𝑘𝐴_𝑘𝑥_{𝑑,𝑘−1}+𝑄^T_2,𝑘𝐴_𝑘𝑄_2,𝑘−1𝑧_{𝑠,𝑘−1}+𝑄^T_2,𝑘𝑤_𝑘−1,𝑦_𝑘−𝐻_𝑘𝑥_𝑑,𝑘=𝐻_𝑘𝑄_2,𝑘𝑧_𝑠,𝑘+𝑣_𝑘. (3.5)

The recursive estimate algorithm for accurate constrained system can be summarized as

Step 0. Initialization
𝑄𝑅 decomposition: 𝑄_1,0𝑄_2,0𝑅_11,00=𝑄𝑅𝐷^T₀. (3.6)
The deterministic part is 𝑥_𝑑,0=𝑄_1,0𝑅^−T_11,0𝑑₀. (3.7)
Set 𝑧_{𝑠,0∣−1}=−𝑄^T_2,0𝑥_𝑑,0,𝑃_{𝑠,0∣−1}=𝑄^T_2,0Π₀𝑄_2,0. (3.8)

Step 1. Prediction
𝑄𝑅 decomposition: 𝑄_1,𝑘𝑄_2,𝑘𝑅_11,𝑘0=𝑄𝑅𝐷^T_𝑘. (3.9)
The deterministic part is 𝑥_𝑑,𝑘=𝑄_1,𝑘𝑅^−T_11,𝑘𝑑_𝑘. (3.10)
According to (3.5), it gives 𝑧_{𝑠,𝑘∣𝑘−1}=𝑄^T_2,𝑘𝐴_𝑘𝑄_2,𝑘−1𝑧_{𝑠,𝑘−1∣𝑘−1}+𝐴_𝑘𝑥_𝑑,𝑘,𝑃_{𝑠,𝑘∣𝑘−1}=𝑄^T_2,𝑘𝐴_𝑘𝑄_2,𝑘−1𝑃_{𝑠,𝑘−1∣𝑘−1}𝑄^T_2,𝑘−1𝐴^T_𝑘+Ω_𝑘−1𝑄_2,𝑘. (3.11)

Step 2. Measurement Update
One has𝐾_𝑧,𝑘=𝑃_{𝑠,𝑘∣𝑘−1}𝐻_𝑘𝑄_2,𝑘^T𝐻_𝑘𝑄_2,𝑘𝑃_{𝑠,𝑘∣𝑘−1}𝐻_𝑘𝑄_2,𝑘^T+𝑉_𝑘⁻¹,𝑧_{𝑠,𝑘∣𝑘}=𝑧_{𝑠,𝑘∣𝑘−1}+𝐾_𝑧,𝑘𝑦_𝑘−𝐻_𝑘𝑥_𝑑,𝑘−𝐻_𝑘𝑄_2,𝑘𝑧_{𝑠,𝑘∣𝑘−1},𝑃_{𝑠,𝑘∣𝑘}=𝑃_{𝑠,𝑘∣𝑘−1}−𝐾_𝑧,𝑘𝐻_𝑘𝑄_2,𝑘𝑃_{𝑠,𝑘∣𝑘−1}. (3.12)

Step 3. Reconstruction
Prediction reconstruction: 𝑥_{𝑘∣𝑘−1}=𝑥_𝑑,𝑘+𝑄_2,𝑘𝑧_{𝑠,𝑘∣𝑘−1},𝑃_{𝑘∣𝑘−1}=𝑄_2,𝑘𝑃_{𝑠,𝑘∣𝑘−1}𝑄^T_2,𝑘. (3.13)
Estimate reconstruction: 𝑥_𝑘∣𝑘=𝑥_𝑑,𝑘+𝑄_2,𝑘𝑧_{𝑠,𝑘∣𝑘},𝑃_𝑘∣𝑘=𝑄_2,𝑘𝑃_{𝑠,𝑘∣𝑘}𝑄^T_2,𝑘. (3.14)
The key for the above recursive estimation algorithm is finding the optimal estimation of 𝑧_{𝑠,𝑘∣𝑘}. With (3.5), the optimal estimate of 𝑧_{𝑠,𝑘∣𝑘} can be derived by solving the following regularized least-square problem: min_{𝑧𝑠,𝑘−1,𝑧𝑠,𝑘}𝑧_{𝑠,𝑘−1}−𝑧_{𝑠,𝑘−1∣𝑘−1}²_{𝑃−1𝑠,𝑘−1∣𝑘−1}+𝑧_𝑠,𝑘−𝑄^T_2,𝑘𝐴_𝑘𝑄_2,𝑘−1𝑧_{𝑠,𝑘−1}−𝑄^T_2,𝑘𝐴_𝑘𝑥_𝑑,𝑘²_{Ω−1𝑘−1}+𝑦_𝑘−𝐻_𝑘𝑥_𝑑,𝑘−𝐻_𝑘𝑄_2,𝑘𝑧_𝑠,𝑘²_𝑉𝑘−1. (3.15)

Next, we will present the robust filter for the uncertain constrained system also by solving a uncertain least-square problem.

4. Robust Filtering

Referring again to the state-space model (2.1)–(2.3), the optimum robust filtering method will be presented in this section. Firstly, we will decompose the original uncertain constrained system into two parts, and then solve them separately.

4.1. New Dimension Reduced Uncertain Model

With the state evolution equation (2.1) and measurement equation (2.3), we will give the optimal estimate 𝑥_𝑘∣𝑘 for state 𝑥_𝑘. Similar to the approach described in Section 3, we will use the null space method to deal with the uncertain model.

According to the uncertain model (2.1)–(2.3), we define the 𝑄𝑅 factorization of 𝐷^T_𝑘 and rewrite the initial state equation as 𝑥_𝑘=𝑥_𝑑,𝑘+𝑥_𝑠,𝑘=𝑄_1,𝑘𝜉_𝑘+𝑄_2,𝑘𝜂_𝑘, (4.1) where 𝑄^T_1,𝑘 is an 𝑠×𝑛 matrix whose columns form a basis for Span(𝐴), and 𝑄^T_2,𝑘 is an (𝑛−𝑠)×𝑛 matrix whose columns form a orthogonal basis for Span(𝐴)^⟂=Null(𝐴).

Substituting (4.1) into (2.3) gives 𝑅^T_11,𝑘0𝑄^T_1,𝑘𝑄^T_2,𝑘𝑄_1,𝑘𝜉_𝑘+𝑄_2,𝑘𝜂_𝑘=𝑑_𝑘, (4.2) then we have 𝜉_𝑘=𝑅^−T_11,𝑘𝑑_𝑘. (4.3)

Also substituting (4.1) into (2.1), we have 𝑄_1,𝑘𝜉_𝑘+𝑄_2,𝑘𝜂_𝑘=𝐴_𝑘+𝛿𝐴_𝑘𝑄_1,𝑘−1𝜉_𝑘−1+𝑄_2,𝑘−1𝜂_𝑘−1+𝑤_𝑘−1. (4.4)

Both sides of above equation multiplying 𝑄^T_2,𝑘 gives 𝑄^T_2,𝑘𝑄_1,𝑘𝜉_𝑘+𝑄^T_2,𝑘𝑄_2,𝑘𝜂_𝑘=𝑄^T_2,𝑘𝐴_𝑘+𝛿𝐴_𝑘𝑄_1,𝑘−1𝜉_𝑘−1+𝑄_2,𝑘−1𝜂_𝑘−1+𝑄^T_2,𝑘𝑤_𝑘−1. (4.5)

Since 𝑄^T_2,𝑘𝑄_1,𝑘𝑄=0,_T2,𝑘𝑄_2,𝑘=𝐼. (4.6)

We have 𝜂_𝑘=𝑄^T_2,𝑘𝐴_𝑘+𝛿𝐴_𝑘𝑄_1,𝑘−1𝜉_𝑘−1+𝑄^T_2,𝑘𝐴_𝑘+𝛿𝐴_𝑘𝑄_2,𝑘−1𝜂_𝑘−1+𝑄^T_2,𝑘𝑤_𝑘−1=𝑄^T_2,𝑘𝐴_𝑘+𝛿𝐴_𝑘𝑄_1,𝑘−1𝑅^−T_11,𝑘−1𝑑_𝑘−1+𝑄^T_2,𝑘𝐴_𝑘+𝛿𝐴_𝑘𝑄_2,𝑘−1𝜂_𝑘−1+𝑄^T_2,𝑘𝑤_𝑘−1. (4.7)

Similarly, substituting (4.1) into (2.2) gives 𝑦_𝑘=𝐻_𝑘+𝛿𝐻_𝑘𝑥_𝑘+𝑣_𝑘=𝐻_𝑘+𝛿𝐻_𝑘𝑄_1,𝑘𝜉_𝑘+𝑄_2,𝑘𝜂_𝑘+𝑣_𝑘=𝐻_𝑘+𝛿𝐻_𝑘𝑄_1,𝑘𝑅^−T_11,𝑘𝑑_𝑘+𝐻_𝑘+𝛿𝐻_𝑘𝑄_2,𝑘𝜂_𝑘+𝑣_𝑘, (4.8) that is, 𝑦_𝑘−𝐻_𝑘𝑄_1,𝑘𝑅^−T_11,𝑘𝑑_𝑘=𝛿𝐻_𝑘𝑄_1,𝑘𝑅^−T_11,𝑘𝑑_𝑘+𝐻_𝑘+𝛿𝐻_𝑘𝑄_2,𝑘𝜂_𝑘+𝑣_𝑘. (4.9)

The uncertain constrained state space model in 𝑥_𝑘 is converted into an unconstrained uncertain state space model in 𝜂_𝑘. Written together, (4.7) and (4.9) yield a new uncertain unconstrained state space model. 𝜂_𝑘=𝑄^T_2,𝑘𝐴_𝑘+𝛿𝐴_𝑘𝑄_1,𝑘−1𝑅^−T_11,𝑘−1𝑑_𝑘−1+𝑄^T_2,𝑘𝐴_𝑘+𝛿𝐴_𝑘𝑄_2,𝑘−1𝜂_𝑘−1+𝑄^T_2,𝑘𝑤_𝑘−1,𝑦_𝑘−𝐻_𝑘𝑄_1,𝑘𝑅^−T_11,𝑘𝑑_𝑘=𝛿𝐻_𝑘𝑄_1,𝑘𝑅^−T_11,𝑘𝑑_𝑘+𝐻_𝑘+𝛿𝐻_𝑘𝑄_2,𝑘𝜂_𝑘+𝑣_𝑘. (4.10)

4.2. Robust Filtering for the Uncertain Model

Reference [2] develops the framework for state estimation when the parameters of the state equations are subject to uncertainties. However, both the system matrix and measurement matrix in the system (4.7) and (4.9) have uncertainties, and the matrix defined in [2] cannot directly be used. In order to present the robust filtering for this system, some new matrices will be defined in next subsection.

Let us first introduce a lemma.

Lemma 4.1 (see [2]). Consider the following optimization problem: min_𝑥max_{𝛿𝐴,𝛿𝑏}‖𝑥‖²_𝑄+‖(𝐴+𝛿𝐴)𝑥−(𝑏−𝛿𝑏)‖²_𝑊, (4.11) where 𝐴 denotes the data matrix, 𝛿𝐴 denotes a perturbation matrix, 𝑏 denotes the measurement vector, and 𝛿𝑏 denotes a perturbation vector. 𝑥 is the unknown vector, 𝑄=𝑄^T>0 and 𝑊=𝑊^T>0 is a weighting matrix. 𝛿𝐴 and 𝛿𝑏 are assumed to satisfy a model 𝛿𝐴𝛿𝑏=𝐻Δ𝐸_𝑎𝐸_𝑏, (4.12) where Δ is an arbitrary contraction satisfying ‖Δ‖≤1. 𝐻, 𝐸_𝑎, and 𝐸_𝑏 are known quantities of appropriate dimensions.
The problem (4.11) has a unique solution, which is given by 𝑥=𝑄+𝐴^T𝑊𝐴⁻¹𝐴^T𝑊𝑏+𝜆𝐸^T_𝑎𝐸_𝑏, (4.13) where the modified weighting matrix {𝑄,𝑊} is defined by 𝑄∶=𝑄+𝜆𝐸^T_𝑎𝐸_𝑎,𝑊∶=𝑊+𝑊𝐻𝜆𝐼−𝐻^T𝑊𝐻^†𝐻^T𝑊, (4.14) and 𝜆 is a nonnegative scalar parameter obtained by following optimization problem: 𝜆=argmin_{𝜆≥‖𝐻T𝑊𝐻‖}𝐺(𝜆), (4.15) where 𝐺(𝜆)∶=‖𝑥(𝜆)‖²_𝑄+𝜆𝐸_𝑎𝑥(𝜆)−𝐸_𝑏²+‖𝐴𝑥(𝜆)−𝑏‖²_𝑊(𝜆). (4.16)
The auxiliary function are defined by 𝑊(𝜆)∶=𝑊+𝑊𝐻𝜆𝐼−𝐻^T𝑊𝐻^†𝐻^T𝑄(𝜆)𝑊,∶=𝑄+𝜆𝐸^T_𝑎𝐸_𝑎,𝑥(𝜆)∶=𝑄(𝜆)+𝐴^T𝑊(𝜆)𝐴⁻¹𝐴^T𝑊(𝜆)𝑏+𝜆𝐸^T_𝑎𝐸_𝑏. (4.17)

As mentioned in Section 3, the optimal estimate problem can be solved by minimizing the cost function (3.15). Similarly, the robust filtering problem for the dimension reduced model (4.7) and (4.9) can be turn to solve following least-square problem: min_{𝜂𝑘−1,𝜂𝑘}max_𝛿𝐴𝑘𝜂_𝑘−1−𝜂_{𝑘−1∣𝑘−1}²_{𝑃−1𝜂,𝑘−1∣𝑘−1}+𝜂_𝑘−𝑄^T_2,𝑘𝐴_𝑘+𝛿𝐴_𝑘𝑄_1,𝑘−1𝑅^−T_11,𝑘−1𝑑_𝑘−1−𝑄^T_2,𝑘𝐴_𝑘+𝛿𝐴_𝑘𝑄_2,𝑘−1𝜂_𝑘−1²_{Ω−1𝑘−1}+𝑦_𝑘−𝐻_𝑘𝑄_1,𝑘𝑅^−T_11,𝑘𝑑_𝑘−𝛿𝐻_𝑘𝑄_1,𝑘𝑅^−T_11,𝑘𝑑_𝑘−𝐻_𝑘+𝛿𝐻_𝑘𝑄_2,𝑘𝜂_𝑘²_𝑉𝑘−1, (4.18) where Ω_𝑘−1=𝑄^T_2,𝑘Ω_𝑘−1𝑄_2,𝑘. (4.19)

In (4.18), the parameters 𝐴_𝑘 and 𝐻_𝑘 contain uncertainties. With appropriate definition, the lest-square problem can be rewritten more compactly. Let us define 𝜂𝑥⟵^𝑘−1−𝜂_{𝑘−1∣𝑘−1}𝜂_𝑘,𝑄𝐴⟵^T2,𝑘𝐴_𝑘𝑄_2,𝑘−1−𝐼0𝐻_𝑘𝑄_2,𝑘,𝑄𝛿𝐴⟵^T2,𝑘𝛿𝐴_𝑘𝑄_2,𝑘−100𝛿𝐻_𝑘𝑄_2,𝑘,𝑄𝑏⟵^T2,𝑘𝐴_𝑘𝑄_1,𝑘−1𝑅^−T_11,𝑘−1𝑑_𝑘−1+𝑄^T_2,𝑘𝐴_𝑘𝑄_2,𝑘−1𝜂_{𝑘−1∣𝑘−1}𝐻_𝑘𝑄_1,𝑘𝑅^−T_11,𝑘𝑑_𝑘−𝑦_𝑘,𝑄𝛿𝑏⟵^T2,𝑘𝛿𝐴_𝑘𝑄_1,𝑘−1𝑅^−T_11,𝑘−1𝑑_𝑘−1+𝑄^T_2,𝑘𝛿𝐴_𝑘𝑄_2,𝑘−1𝜂_{𝑘−1∣𝑘−1}𝛿𝐻_𝑘𝑄_1,𝑘𝑅^−T_11,𝑘𝑑_𝑘,𝑃𝑄⟵^{−1𝜂,𝑘−1∣𝑘−1}0,𝑄00𝑊⟵^T2,𝑘Ω_𝑘−1𝑄_2,𝑘⁻¹00𝑉_𝑘⁻¹,𝑄𝐻⟵^T2,𝑘𝑀_ℎ𝑘00𝑄_2,𝑘,𝐸_𝑎⟵𝑁_𝑘𝑄_2,𝑘−100𝑄_2,𝑘,𝐸_𝑏⟵𝑁_𝑘𝑄_1,𝑘−1𝑅^−T_11,𝑘−1𝑑_𝑘−1+𝑄_2,𝑘−1𝜂_{𝑘−1∣𝑘−1}𝑁_𝑘𝑄_1,𝑘𝑅^−T_11,𝑘𝑑_𝑘. (4.20)

Let 𝜂_𝑘∣𝑘 and 𝑃_{𝜂,𝑘∣𝑘}𝑘=1,2,3,… be the estimate result and estimate error covariance of the stochastic vector 𝜂_𝑘, respectively. With Lemma 4.1 and above definition, the robust filter for 𝜂_𝑘 can be summarized as in the following theorem.

Theorem 4.2. Assume that the estimate 𝜂_{𝑘−1∣𝑘−1} and the estimate error covariance 𝑃_{𝜂,𝑘−1∣𝑘−1} of 𝜂_𝑘−1 have been known. At time index 𝑘, the robust filter of 𝑥 can be given by solving the following equation: 𝑄+𝐴^T𝑊𝐴𝑥=𝐴^T𝑊𝑏+𝜆_𝑘𝐸^T_𝑎𝐸_𝑏, (4.21) where 𝑃𝑄=^{−1𝜂,𝑘−1|𝑘−1}+𝜆_𝑘𝑁_𝑘𝑄_2,𝑘−1^T𝑁_𝑘𝑄_2,𝑘−100𝑄^T_2,𝑘𝑄_2,𝑘,Ω𝑊=^𝑘−100𝑉_𝑘⁻¹⁻¹,Ω_𝑘−1=𝑄^T_2,𝑘Ω_𝑘−1−𝜆⁻¹_𝑘−1𝑀_𝑎𝑘−1𝑀^T_𝑎𝑘−1𝑄_2,𝑘,𝑉_𝑘=𝑉_𝑘−𝜆⁻¹_𝑘−1𝐻_𝑘𝑄_2,𝑘𝑀_ℎ𝑘𝑀^T_ℎ𝑘𝐻_𝑘𝑄_2,𝑘^T, (4.22) and 𝜆_𝑘 is determined by minimizing the function 𝐺(𝜆) of (4.16) in the interval (𝜆_𝑙,𝑘,∞), where 𝜆_𝑙,𝑘∶=𝐻^T𝑊𝐻=diag𝑀^T_𝑎𝑘𝑄_2,𝑘𝑄^T_2,𝑘Ω_𝑘−1𝑄_2,𝑘⁻¹𝑄^T_2,𝑘𝑀_𝑎𝑘,𝑄^T_2,𝑘𝑉_𝑘⁻¹𝑄_2,𝑘. (4.23)

Proof. Analogous to [2, 3, 11], using Lemma 4.1 yields (4.21).

Theorem 4.2 gives the robust filter of 𝜂_𝑘, then, the robust constrained estimate of the full state 𝑥_𝑘 can be constructed by using the relationship 𝑥_𝑘∣𝑘=𝑄_1,𝑘𝜉_𝑘+𝑄_2,𝑘𝜂_𝑘∣𝑘. (4.24)

It is easily verified that 𝑥_𝑘∣𝑘 satisfies the constraint equation 𝐷_𝑘𝑥_𝑘∣𝑘=𝐷_𝑘𝑄_1,𝑘𝜉_𝑘+𝑄_2,𝑘𝜂_𝑘∣𝑘=𝐷_𝑘𝑄_1,𝑘𝜉_𝑘+𝐷_𝑘𝑄_2,𝑘𝜂_𝑘∣𝑘=𝑑_𝑘. (4.25)

Similarly, the constrained error covariance 𝑃_𝑘∣𝑘 can be computed by using 𝑃_𝑘∣𝑘=𝑄_2,𝑘𝑃_{𝜂,𝑘∣𝑘}𝑄^T_2,𝑘, (4.26) where 𝑃_{𝜂,𝑘∣𝑘} of (4.35) is the estimate error covariance of 𝜂_𝑘.

4.3. Recursive Form of Constrained Robust Filter

After some considerable algebra, similar to [2], the recursive robust estimate 𝑥_𝑘∣𝑘 can be summarized as follows.

Step 0. Initialization
𝑄𝑅 decomposition: 𝑅^T_11,00𝑄^T_1,0𝑄^T_2,0=𝑄𝑅𝐷₀. (4.27)
The deterministic part is 𝜉₀=𝑅^T_11,0𝑑₀. (4.28)
Set 𝜂_0∣0=𝑃_𝜂,0∣0𝐻₀𝑄_2,0^T𝑉₀⁻¹𝑦₀,𝑃_𝜂,0∣0=𝑄^T_2,0Π₀𝑄_2,0⁻¹+𝐻₀𝑄_2,0^T𝑉₀⁻¹𝐻₀𝑄_2,0⁻¹. (4.29)

Step 1. Determining 𝜆_𝑘
QR decomposition: 𝑅^T_11,𝑘0𝑄^T_1,𝑘𝑄^T_2,𝑘=𝑄𝑅𝐷_𝑘. (4.30)
It gives the deterministic part at time index 𝑘𝜉_𝑘=𝑅^T_11,𝑘𝑑_𝑘 (4.31) and the new dimension reduced robust system (4.7) and (4.9).
If 𝑀_𝑎𝑘≠0, then set 𝜆_𝑘=0. Otherwise, with the definitions of (4.20), determine the scalar parameter 𝜆_𝑘 by minimizing 𝐺(𝜆) over the interval (𝜆_𝑙,𝑘,∞).

Step 2. Replace Parameters
If 𝜆_𝑘≠0, the original parameters {Ω_𝑘−1,𝑉_𝑘,𝑃_{𝜂,𝑘−1∣𝑘−1},𝐴_𝑘} are replaced by Ω_𝑘−1=𝑄^T_2,𝑘Ω_𝑘−1−𝜆⁻¹_𝑘−1𝑀_𝑎𝑘−1𝑀^T_𝑎𝑘−1𝑄_2,𝑘,𝑉_𝑘=𝑉_𝑘−𝜆⁻¹_𝑘−1𝐻_𝑘𝑄_2,𝑘𝑀_ℎ𝑘𝑀^T_ℎ𝑘𝐻_𝑘𝑄_2,𝑘^T,𝑃_{𝜂,𝑘−1∣𝑘−1}=𝑃_{𝜂,𝑘−1∣𝑘−1}−𝑃_{𝜂,𝑘−1∣𝑘−1}𝑁_𝑘𝑄_2,𝑘−1^T×𝜆⁻¹_𝑘−1𝐼+𝑁_𝑘𝑄_2,𝑘−1𝑃_{𝜂,𝑘−1∣𝑘−1}𝑁_𝑘𝑄_2,𝑘−1^T−1×𝑁_𝑘𝑄_2,𝑘−1𝑃_{𝜂,𝑘−1∣𝑘−1},𝐴_𝑘=𝐴_𝑘𝐼−𝜆_𝑘−1𝑃_{𝑘−1∣𝑘−1}𝑁_𝑘𝑄_2,𝑘−1^T𝑁_𝑘𝑄_2,𝑘−1. (4.32)

Step 3. Prediction and Update
Prediction: 𝜂_{𝑘∣𝑘−1}=𝐴_𝑘𝜂_{𝑘−1∣𝑘−1}+𝑄^T_2,𝑘𝐴_𝑘𝑄_1,𝑘−1𝑅^−T_11,𝑘−1𝑑_𝑘−1,𝑃_{𝜂,𝑘∣𝑘−1}=𝐴_𝑘𝑃_{𝜂,𝑘∣𝑘−1}𝐴^T_𝑘. (4.33)
Update: 𝜂_𝑘∣𝑘=𝜂_{𝑘∣𝑘−1}+𝑃_{𝜂,𝑘∣𝑘−1}𝐻_𝑘𝑄_2,𝑘^T𝑉_𝑘⁻¹𝑒_𝑘,𝑃_{𝜂,𝑘∣𝑘}=𝑃_{𝜂,𝑘∣𝑘−1}−𝑃_{𝜂,𝑘∣𝑘−1}𝐻_𝑘𝑄_2,𝑘^T𝑉⁻¹_𝑒,𝑘𝐻_𝑘𝑄_2,𝑘𝑃_{𝜂,𝑘∣𝑘−1}, (4.34)(4.35) where 𝑒_𝑘=𝑦_𝑘−𝐻_𝑘𝑄_1,𝑘𝑅^−T_11,𝑘−1𝑑_𝑘−1−𝐻_𝑘𝑄_2,𝑘𝜂_{𝑘∣𝑘−1},𝑉_𝑒,𝑘=𝑉_𝑘+𝐻_𝑘𝑄_2,𝑘^T𝑃_{𝜂,𝑘∣𝑘−1}𝐻_𝑘𝑄_2,𝑘. (4.36)

Step 4. Reconstruction
Prediction reconstruction: 𝑥_{𝑘∣𝑘−1}=𝑄_1,𝑘𝜉_𝑘+𝑄_2,𝑘𝜂_{𝑘∣𝑘−1},𝑃_{𝑘∣𝑘−1}=𝑄_2,𝑘𝑃_{𝜂,𝑘∣𝑘−1}𝑄^T_2,𝑘. (4.37)
Estimate reconstruction: 𝑥_𝑘∣𝑘=𝑄_1,𝑘𝜉_𝑘+𝑄_2,𝑘𝜂_𝑘∣𝑘,𝑃_𝑘∣𝑘=𝑄_2,𝑘𝑃_{𝜂,𝑘∣𝑘}𝑄^T_2,𝑘. (4.38)

Steps 1–4 give the robust estimate of the full state 𝑥_𝑘.

Remark 4.3. From the definition of {Ω_𝑘−1,𝑉_𝑘,𝑃_{𝜂,𝑘−1∣𝑘−1},𝐴_𝑘} and the prediction and update process in Step 3, it is easy to verify that for the constrained system without uncertainties, the robust filtering algorithm reduces to the filtering result introduced in [10].

Remark 4.4. If 𝐷_𝑘∈𝑅^𝑛×𝑛 and Rank{𝐷_𝑘}=𝑛, the matrix 𝑄_2,𝑘 and the dimensional reduced model (4.7) will disappear, then we have 𝑥_𝑘∣𝑘=𝐷_𝑘⁻¹𝑑_𝑘.

5. Numerical Example

In this section, simulations are presented to verify the performance of the new algorithm. We consider an example described by (2.1)–(2.3), with 𝑥_𝑘=[𝑥¹_𝑘,𝑥²_𝑘,𝑥³_𝑘]^T. The parameters are given as follows: 𝐴_𝑘=110011001,𝐻_𝑘=0.810000.9,Ω_𝑘=,𝑉100030001_𝑘=1.2001,𝐷_𝑘=[](𝑘)𝑀1,0.3,0.2,𝑑=1,_𝑎𝑘=200,𝑀_ℎ𝑘=01,𝑁_𝑘=101. (5.1)

The initial state is 𝑥₀=⁰⁰₁, 𝑃₀=^0.10000.10_000.1, and we will take 𝐿=1000 sampling points. Figures 1 and 2 display the estimate error variance of 𝑥¹_𝑘 and 𝑥²_𝑘, respectively. The variance curves are computed via the ensemble-average 𝜀𝑥_𝑘−𝑥_𝑘≈1𝑇^𝑇_𝑖=1𝑥^𝑖_𝑘−𝑥^𝑖_𝑘. (5.2)

Figure 1 

Comparison of the estimate variance 𝑥1𝑘 with different methods.

Figure 2 

Comparison of the estimate variance 𝑥2𝑘 with different methods.

Each point at instant 𝑘 in each variance curve is the ensemble-average calculated over 𝑇=500 experiments. For each experiment 𝑖,Δ^𝑖_𝑘 with norm less or equal than one is selected randomly.

To demonstrate the performance of the new robust filter more clearly, we also present the variance curves of the Kalman filter for uncertain model and the system without uncertainties. The variances of these two filters are also shown in Figures 1 and 2.

From Figures 1 and 2, we see that the performance of new filter is better than that of Kalman filter when they are used to deal with the uncertain model, this is because Kalman filter does not consider the uncertain parameters. The variance of Kalman filter dealing with accurate model is smaller than that of new filter dealing with uncertain model.

Furthermore, the performance of an algorithm is often affected by the measurement noises, and larger noises variance always bring larger estimate error variance. Table 1 lists the error variance for 𝑥₁₀₀ with three different measurement noises variance to show the variation of performance.

Table 1 

Variation of error variance with different measurement noises variance.

From Table 1, we see that, for 𝑥¹_𝑘 and 𝑥²_𝑘, the larger is the noise variance, the larger is estimation error variance.

6. Conclusions

This paper has studied the robust constrained filtering problem for linear discrete uncertain systems. The original constrained system is transformed into a new uncertain unconstrained system. The state of the new system is derived by the least square method and then the optimal estimate is obtained similar to the update process of the robust Kalman filter. A numerical example is presented to show the effectiveness of the new filter. Next, we will consider the regularized filtering method for the case when network-induced phenomena are taken into account [12–15].

Acknowledgments

This work was supported by Natural Science Foundation of China under Grant (60801048, 61004088, 91016020), the Key Foundation for Basic Research from Science and Technology Commission of Shanghai (09JC1408000), the Key Foundation for Innovative Research from Education Commission of Shanghai (12ZZ197), and the Key Training Foundation from Shanghai Dianji University (12C102).