# Simultaneous Unknown Input and State Estimation for the Linear System with a Rank-Deficient Distribution Matrix

Authors: Yu Hua; Na Wang; Keyou Zhao
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

The classical recursive three-step filter can be used to estimate the state and unknown input when the system is affected by unknown input, but the recursive three-step filter cannot be applied when the unknown input distribution matrix is not of full column rank. In order to solve the above problem, this paper proposes two novel filters according to the linear minimum-variance unbiased estimation criterion. Firstly, while the unknown input distribution matrix in the output equation is not of full column rank, a novel recursive three-step filter with direct feedthrough was proposed. Then, a novel recursive three-step filter was developed when the unknown input distribution matrix in the system equation is not of full column rank. Finally, the specific recursive steps of the corresponding filters are summarized. And the simulation results show that the proposed filters can effectively estimate the system state and unknown input.


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## Body

## 1. Introduction

The traditional Kalman filter [1] and its extension can recursively estimate the state of the linear system with process noise and measurement noise. The time-domain recursive filter brings greater convenience for continuously processing input data, so it can play a more important role in control theory and engineering. The Kalman filter requires the noise to be stationary white noise, but this supposition is sometimes not feasible because unknown input may not be white noise and cannot be measured.In the fields of environmental monitoring [2] and disturbance suppression [3, 4], the system equation or output equation contains unknown input owing to environmental impacts and improper selection of model parameters. In recent decades, the problem of state estimation with unknown input has received extensive attention.For continuous-time systems, the necessary and sufficient conditions for the existence of optimal state filters have been established [5–7]. Furthermore, the steps to reconstruct unknown input are also quite complete [8, 9]. For the state estimation problem of discrete-time systems, an early solution was to add an unknown input vector to the system state vector. Then, the Kalman filter was used to estimate the augmented state. However, the scenarios of using this solution are limited to that the dynamical evolution of unknown input is known [10, 11]. In order to reduce computation costs of the augmented state filter, Friedland [11] proposed the two-stage Kalman filter in which the state estimation and unknown input estimation are decoupled. Although this filter has been successfully applied in some instances, it is still limited to the requirement that the dynamic evolution of unknown input is available. When the unknown input only affects the system equation, Kitanidis [5] developed an optimal recursive state filter which can estimate the system state without prior knowledge of the unknown input. And the stability and convergence conditions of the above filter were raised by Darouach and Zasadzinski [12]. Further, Darouachet al. [13] extended this filter. So, the filter is valid when unknown input is directly feedthrough to the output equation; that is, the unknown input affects both the system equation and output equation.Although the above methods can get the estimation value of the system state, they all ignore obtaining the estimation value of unknown input, which is necessary in some practical applications.Hsieh [14] established a robust two-stage Kalman filter (RTSKF). For systems without direct feedthrough of unknown input to output, it can give the joint state and unknown input estimation. But the optimality of the unknown input estimation has not been proven. Furthermore, Gillijns and De Moor [15] proposed a recursive three-step filter (RTSF), which gave a proof that the unknown input estimation is optimal. And the form of the unknown input estimation obtained by RTSF is consistent with that of RTSKF. On the other hand, Gillijns and De Moor [16] extended RTSF so that it is still valid for linear discrete-time systems with direct feedthrough.Despite the fact that the above filters can solve the problem of simultaneously estimating system state and unknown input, they are based on a precondition: the distribution matrix of unknown input must be of full column rank.For systems with direct feedthrough, if the distribution matrix of unknown input in output equation is not of full column rank, Cheng et al. [17] presented an unbiased minimum-variance state estimation (UMVSE). This method transforms the output equation by singular value decomposition of the distribution matrix. Then UMVSE is applied to address the problem of state estimation for the new system. However, this method omitted the estimation of unknown input. Hsieh [18] used an extension of RTSF (ERTSF) to estimate the unknown input and state under the assumption that the distribution matrix is not of full column rank, but some parameters in ERTSF were obtained by experience. This paper put forward the novel recursive three-step filter with direct feedthrough (NRTSF-DF) which can give estimation value of the state and unknown input under the same assumption. And compared with ERTSF, the parameter of NRTSF-DF can be exactly obtained. For systems without direct feedthrough, the problem of filter design with unknown input still exists though there are few related literature studies about it. Similar to NRTSF-DF, the novel recursive three-step filter (NRTSF) is proposed in this paper. The novel filters can achieve a simultaneous estimation of the system state and unknown input under the condition that the unknown input distribution matrix is not of full column rank.In recent years, the research of estimating system state with unknown input is concentrated on nonlinear systems. Based on the EKF structure, the filters estimating the state of the nonlinear system were designed in [19, 20]. Furthermore, by making some improvements on UKF [21], study [22] obtained the filter with RTSF form. And the filter can estimate the state and unknown input simultaneously.This paper is organized as follows: in Section2, the problem is formulated. Section 3 deals with the design of the optimal filter for the system with direct feedthrough, and the specific structure of NRTSF-DF is summarized. Next, the optimal filter for the system without feedthrough is established in Section 4. The structure of NRTSF is also obtained. Finally, Section 5 demonstrates the effectiveness of the proposed filters through simulation.


## 2. Problem Formulation

Consider the linear discrete-time-varying system:(1)xk=Ak−1xk−1+Gk−1dk−1+wk−1,(2)yk=Ckxk+Hkdk+vk,where xk∈Rn is the state vector, dk∈Rm is an unknown input vector, and yk∈Rp is the measurement. The process noise wk∈Rn and the measurement noise vk∈Rp are assumed to be mutually uncorrelated, zero-mean, white random signals with known covariance matrices, Qk=EwkwkT≥0, and Rk=EvkvkT>0, respectively. The time-varying matrices Ak,Gk,Ck, and Hk are known with an appropriate dimension. Throughout the paper, the conditions that Ak,Gk is observable and that x0 is independent of wk and vk are satisfied. And the unbiased estimate x^0=Ex0 with P0x=Ex0−x^0x0−x^0T is known.The optimal filtering problem of the above system is to obtain the unbiased optimal filtering sequence of unknown inputd^0|0,…,d^k|k and state x^0|0,…,x^k|k recursively based on the initial estimate x^0, the covariance matrix P0x, and the sequence of measurement y0,y1,…,yk. If Hk=0, the system is transformed into a linear discrete-time-varying system without direct feedthrough of unknown input to output. Then, the corresponding optimal filtering problem is transformed to obtain the unbiased optimal filtering sequence of unknown input d^0|1,…,d^k−1|k and state x^0|0,…,x^k|k under the corresponding conditions.


## 3. NRTSF-DF

The RTSF proposed by Steven Gilljins in [16] can solve the state and unknown input estimation problem of linear system (1)-(2) while rankHk=m,k=0,1,…. When the unknown input distribution matrix in the output equation is not of full column rank, that is, rankHk=rk<m, we consider a NRTSF-DF design method. The following is the derivation process.IfrankHk=rk≤m, perform full rank decomposition:(3)Hk=H¯kTk,where H¯k∈Rp×rk,Tk∈Rrk×m, and rankH¯k=rankTk=rk. The full rank decomposition steps are given in Appendix.Defining the virtual unknown input byd¯k=Tkdk, then Hkdk=H¯kd¯k. If the estimation value of d¯k is expressed as d¯^k|k, the minimal norm estimation value of unknown input dk is(4)d^k|k=Tk+d¯^k|k,where Tk+ is the Moore–Penrose inverse of Tk. Then, original output equation (2) is rewritten as(5)yk=Ckxk+H¯kd¯k+vk,where d¯k∈Rrk is the virtual unknown input vector.Based on the system state equation (1) and output equation (5), we consider NRTSF-DF of the form(6)x^k|k−1=Ak−1x^k−1|k−1+Gk−1d^k−1|k−1,(7)d¯^k|k=M¯kyk−Ckx^k|k−1,(8)x^k|k=x^k|k−1+L¯kyk−Ckx^k|k−1,where the matrices M¯k∈Rrk×p and L¯k∈Rn×p still have to be determined.


### 3.1. Time Update

Letx^k−1|k−1 and d^k−1|k−1 denote the optimal unbiased estimates of xk−1 and dk−1 given measurement sequence y0,y1,⋯,yk−1; then, the time update is(9)x^k|k−1=Ak−1x^k−1|k−1+Gk−1d^k−1|k−1.The error in the estimatex^k|k−1 is given by(10)x˜k|k−1≜xk−x^k|k−1=Ak−1x˜k−1|k−1+Gk−1d˜k−1+wk−1,where x˜k|k≜xk−x^k|k and d˜k≜dk−d^k|k. Consequently, the covariance matrix of x^k|k−1 is given by(11)Pk|k−1x≜Ex˜k|k−1x˜k|k−1T=Ak−1Gk−1Pk−1|k−1xPk−1|k−1xdPk−1|k−1dxPk−1|k−1dAk−1TGk−1T+Qk−1,with Pk|kx≜Ex˜k|kx˜k|kT,Pk|kd≜Ed˜kd˜kT,Pk|kxdT=Pk|kdx≜Ed˜kx˜k|kT.


### 3.2. Virtual Unknown Input Estimation

In this section, the estimation of the virtual unknown inputd¯k is considered.


#### 3.2.1. Unbiased Virtual Unknown Input Estimation

Defining the innovationy˜k≜yk−Ckx^k|k−1, it follows from (5) that(12)y˜k=H¯kd¯k+ek,where ek is given by(13)ek=Ckx˜k|k−1+vk.Owing tox˜k|k−1 is unbiased, Eek=0 and Ey˜k=H¯kEd¯k. So, we can obtain an unbiased estimate of the virtual unknown input d¯k from y˜k.Theorem 1.
Supposex^k|k−1 is unbiased; then, (6) and (7) calculate the unbiased value of d¯k if and only if M¯k satisfies M¯kH¯k=Irk.Proof.
This process is similar to the proof of Theorem1 in [16], so it is omitted.
From Theorem1, rankH¯k=rk is a necessary and sufficient condition for an unbiased virtual unknown input estimator of form (7). The matrix M¯k=H¯kTH¯k−1H¯kT corresponding to the least-squares (LS) solution of (12) satisfies Theorem 1. But from the Gauss–Markov theorem, the LS solution is not necessarily minimum-variance as a result of(14)R˜k≜EekekT=CkPk|k−1xCkT+Rk≠cI,where c is a positive real number.


#### 3.2.2. MVU Virtual Unknown Input Estimation

An MVU estimate ofd¯k is calculated by weighted LS (WLS) estimation.Theorem 2.
Letx^k|k−1 be unbiased, and let R˜k and H¯kTR˜k−1H¯k be nonsingular; then, for(15)M¯k∗=H¯kTR˜k−1H¯k−1H¯kTR˜k−1,(4) is the MVU estimator of d¯k. The variance of the optimal virtual unknown input estimate is(16)Pk|k∗d¯=H¯kTR˜k−1H¯k−1.Proof.
This process is similar to the proof of Theorem2 in [16], so it is omitted.
We used¯^k|k∗ to express the optimal virtual unknown input estimate corresponding to M¯k∗ and let d¯˜k∗≜d¯k−d¯^k|k∗. Then, d¯˜k∗ is given by(17)d¯˜k∗=I−M¯k∗H¯kd¯k−M¯k∗ek=−M¯k∗ek.


### 3.3. Measurement Update

In the last step, we use measurementyk to update x^k|k−1. Using (8) and (12), we find that(18)x˜k|k=I−L¯kCkx˜k|k−1−L¯kH¯kd¯k−L¯kvk.Consequently, (8) is unbiased for all d¯k if and only if L¯k contents(19)L¯kH¯k=0.SupposeL¯k satisfy (19), from (18):(20)Pk|kx=I−L¯kCkPk|k−1xI−L¯kCkT+L¯kRkL¯kT.So, we can calculateL¯k by minimizing the trace of (20) under the unbiasedness constraint of (19).Theorem 3.
L¯k is given by(21)L¯k∗=Kk∗I−H¯kM¯k∗,where Kk∗=Pk|k−1xCkTR˜k−1 minimizes the trace of (20) under the constraint of (19).Proof.
This process is similar to the proof of Theorem3 in [16], so it is omitted.
We usex^k|k∗ to express the state estimate corresponding to L¯k∗. From (7) and (21),(22)x^k|k∗=x^k|k−1+Kk∗I−H¯kM¯k∗yk−Ckx^k|k−1=x^k|k−1+Kk∗yk−Ckx^k|k−1−H¯kd¯^k|k∗.
Then, we consider the expressions ofPk|k∗x≜Ex˜k|k∗x˜k|k∗T and Pk|k∗xd¯≜Ex˜k|k∗d¯˜k∗T, where(23)x˜k|k∗≜xk−x^k|k∗=I−L¯k∗Ckx˜k|k−1−L¯k∗vk.
From (20) and (21), we obtain(24)Pk|k∗x=Pk|k−1x−Kk∗R˜k−H¯kPk|k∗d¯H¯kTKk∗T.
Using (23) and (17), it follows that(25)Pk|k∗xd¯=−Pk|k−1xCkTM¯k∗T=−Kk∗H¯kPk|k∗d¯.
Furthermore, byd^k|k=Tk+d¯^k|k, we can get(26)Pk|kd=Tk+Pk|k∗d¯Tk+TPk|kxd=Pk|kdxT=−K¯kH¯kPk|k∗d¯Tk+T.


### 3.4. Summary of NRTSF-DF Equations

In order to reflect NRTSF-DF clearly, summarize it as follows:


#### 3.4.1. Time Update

Based on the unbiased estimatesd^k−1|k−1 and x^k−1|k−1, the state estimates and corresponding variance matrix from the time instant k−1 to k are obtained:(27)x^k|k−1=Ak−1x^k−1|k−1+Gk−1d^k−1|k−1,Pk|k−1x=Ak−1Gk−1Pk−1|k−1xPk−1|k−1xdPk−1|k−1dxPk−1|k−1dAk−1TGk−1T+Qk−1.


#### 3.4.2. Estimation of Virtual Unknown Input

Calculate the rank ofHk, make full rank decomposition of Hk, and calculate the virtual unknown input estimates d¯^k|k and corresponding variance matrix at time instant k:(28)R˜k=CkPk|k−1xCkT+Rk,M¯k=H¯kTR˜k−1H¯k−1H¯kTR˜k−1,d¯^k|k=M¯kyk−Ckx^k|k−1,Pk|kd¯=H¯kTR˜k−1H¯k−1.


#### 3.4.3. Measurement Update

Calculate the state estimatex^k|k and corresponding variance matrix at time instant k:(29)Kk=Pk|k−1xCkTR˜k−1,L¯k=KkI−H¯kM¯k,x^k|k=x^k|k−1+L¯kyk−Ckx^k|k−1,Pk|kx=Pk|k−1x−KkR˜k−H¯kPk|kd¯H¯kTKkT,Pk|kxd¯=Pk|kd¯xT=−KkH¯kPk|kd¯,d^k|k=Tk+d¯^k|k,Pk|kd=Tk+Pk|kd¯Tk+T,Pk|kxd=Pk|kdxT=−KkH¯kPk|kd¯Tk+T.Also, note that ifHk is of full column rank, letting H¯k=Hk and Tk=Im, RTSF is obtained.


## 3.1. Time Update

Letx^k−1|k−1 and d^k−1|k−1 denote the optimal unbiased estimates of xk−1 and dk−1 given measurement sequence y0,y1,⋯,yk−1; then, the time update is(9)x^k|k−1=Ak−1x^k−1|k−1+Gk−1d^k−1|k−1.The error in the estimatex^k|k−1 is given by(10)x˜k|k−1≜xk−x^k|k−1=Ak−1x˜k−1|k−1+Gk−1d˜k−1+wk−1,where x˜k|k≜xk−x^k|k and d˜k≜dk−d^k|k. Consequently, the covariance matrix of x^k|k−1 is given by(11)Pk|k−1x≜Ex˜k|k−1x˜k|k−1T=Ak−1Gk−1Pk−1|k−1xPk−1|k−1xdPk−1|k−1dxPk−1|k−1dAk−1TGk−1T+Qk−1,with Pk|kx≜Ex˜k|kx˜k|kT,Pk|kd≜Ed˜kd˜kT,Pk|kxdT=Pk|kdx≜Ed˜kx˜k|kT.


## 3.2. Virtual Unknown Input Estimation

In this section, the estimation of the virtual unknown inputd¯k is considered.


### 3.2.1. Unbiased Virtual Unknown Input Estimation

Defining the innovationy˜k≜yk−Ckx^k|k−1, it follows from (5) that(12)y˜k=H¯kd¯k+ek,where ek is given by(13)ek=Ckx˜k|k−1+vk.Owing tox˜k|k−1 is unbiased, Eek=0 and Ey˜k=H¯kEd¯k. So, we can obtain an unbiased estimate of the virtual unknown input d¯k from y˜k.Theorem 1.
Supposex^k|k−1 is unbiased; then, (6) and (7) calculate the unbiased value of d¯k if and only if M¯k satisfies M¯kH¯k=Irk.Proof.
This process is similar to the proof of Theorem1 in [16], so it is omitted.
From Theorem1, rankH¯k=rk is a necessary and sufficient condition for an unbiased virtual unknown input estimator of form (7). The matrix M¯k=H¯kTH¯k−1H¯kT corresponding to the least-squares (LS) solution of (12) satisfies Theorem 1. But from the Gauss–Markov theorem, the LS solution is not necessarily minimum-variance as a result of(14)R˜k≜EekekT=CkPk|k−1xCkT+Rk≠cI,where c is a positive real number.


### 3.2.2. MVU Virtual Unknown Input Estimation

An MVU estimate ofd¯k is calculated by weighted LS (WLS) estimation.Theorem 2.
Letx^k|k−1 be unbiased, and let R˜k and H¯kTR˜k−1H¯k be nonsingular; then, for(15)M¯k∗=H¯kTR˜k−1H¯k−1H¯kTR˜k−1,(4) is the MVU estimator of d¯k. The variance of the optimal virtual unknown input estimate is(16)Pk|k∗d¯=H¯kTR˜k−1H¯k−1.Proof.
This process is similar to the proof of Theorem2 in [16], so it is omitted.
We used¯^k|k∗ to express the optimal virtual unknown input estimate corresponding to M¯k∗ and let d¯˜k∗≜d¯k−d¯^k|k∗. Then, d¯˜k∗ is given by(17)d¯˜k∗=I−M¯k∗H¯kd¯k−M¯k∗ek=−M¯k∗ek.


## 3.2.1. Unbiased Virtual Unknown Input Estimation

Defining the innovationy˜k≜yk−Ckx^k|k−1, it follows from (5) that(12)y˜k=H¯kd¯k+ek,where ek is given by(13)ek=Ckx˜k|k−1+vk.Owing tox˜k|k−1 is unbiased, Eek=0 and Ey˜k=H¯kEd¯k. So, we can obtain an unbiased estimate of the virtual unknown input d¯k from y˜k.Theorem 1.
Supposex^k|k−1 is unbiased; then, (6) and (7) calculate the unbiased value of d¯k if and only if M¯k satisfies M¯kH¯k=Irk.Proof.
This process is similar to the proof of Theorem1 in [16], so it is omitted.
From Theorem1, rankH¯k=rk is a necessary and sufficient condition for an unbiased virtual unknown input estimator of form (7). The matrix M¯k=H¯kTH¯k−1H¯kT corresponding to the least-squares (LS) solution of (12) satisfies Theorem 1. But from the Gauss–Markov theorem, the LS solution is not necessarily minimum-variance as a result of(14)R˜k≜EekekT=CkPk|k−1xCkT+Rk≠cI,where c is a positive real number.


## 3.2.2. MVU Virtual Unknown Input Estimation

An MVU estimate ofd¯k is calculated by weighted LS (WLS) estimation.Theorem 2.
Letx^k|k−1 be unbiased, and let R˜k and H¯kTR˜k−1H¯k be nonsingular; then, for(15)M¯k∗=H¯kTR˜k−1H¯k−1H¯kTR˜k−1,(4) is the MVU estimator of d¯k. The variance of the optimal virtual unknown input estimate is(16)Pk|k∗d¯=H¯kTR˜k−1H¯k−1.Proof.
This process is similar to the proof of Theorem2 in [16], so it is omitted.
We used¯^k|k∗ to express the optimal virtual unknown input estimate corresponding to M¯k∗ and let d¯˜k∗≜d¯k−d¯^k|k∗. Then, d¯˜k∗ is given by(17)d¯˜k∗=I−M¯k∗H¯kd¯k−M¯k∗ek=−M¯k∗ek.


## 3.3. Measurement Update

In the last step, we use measurementyk to update x^k|k−1. Using (8) and (12), we find that(18)x˜k|k=I−L¯kCkx˜k|k−1−L¯kH¯kd¯k−L¯kvk.Consequently, (8) is unbiased for all d¯k if and only if L¯k contents(19)L¯kH¯k=0.SupposeL¯k satisfy (19), from (18):(20)Pk|kx=I−L¯kCkPk|k−1xI−L¯kCkT+L¯kRkL¯kT.So, we can calculateL¯k by minimizing the trace of (20) under the unbiasedness constraint of (19).Theorem 3.
L¯k is given by(21)L¯k∗=Kk∗I−H¯kM¯k∗,where Kk∗=Pk|k−1xCkTR˜k−1 minimizes the trace of (20) under the constraint of (19).Proof.
This process is similar to the proof of Theorem3 in [16], so it is omitted.
We usex^k|k∗ to express the state estimate corresponding to L¯k∗. From (7) and (21),(22)x^k|k∗=x^k|k−1+Kk∗I−H¯kM¯k∗yk−Ckx^k|k−1=x^k|k−1+Kk∗yk−Ckx^k|k−1−H¯kd¯^k|k∗.
Then, we consider the expressions ofPk|k∗x≜Ex˜k|k∗x˜k|k∗T and Pk|k∗xd¯≜Ex˜k|k∗d¯˜k∗T, where(23)x˜k|k∗≜xk−x^k|k∗=I−L¯k∗Ckx˜k|k−1−L¯k∗vk.
From (20) and (21), we obtain(24)Pk|k∗x=Pk|k−1x−Kk∗R˜k−H¯kPk|k∗d¯H¯kTKk∗T.
Using (23) and (17), it follows that(25)Pk|k∗xd¯=−Pk|k−1xCkTM¯k∗T=−Kk∗H¯kPk|k∗d¯.
Furthermore, byd^k|k=Tk+d¯^k|k, we can get(26)Pk|kd=Tk+Pk|k∗d¯Tk+TPk|kxd=Pk|kdxT=−K¯kH¯kPk|k∗d¯Tk+T.


## 3.4. Summary of NRTSF-DF Equations

In order to reflect NRTSF-DF clearly, summarize it as follows:


### 3.4.1. Time Update

Based on the unbiased estimatesd^k−1|k−1 and x^k−1|k−1, the state estimates and corresponding variance matrix from the time instant k−1 to k are obtained:(27)x^k|k−1=Ak−1x^k−1|k−1+Gk−1d^k−1|k−1,Pk|k−1x=Ak−1Gk−1Pk−1|k−1xPk−1|k−1xdPk−1|k−1dxPk−1|k−1dAk−1TGk−1T+Qk−1.


### 3.4.2. Estimation of Virtual Unknown Input

Calculate the rank ofHk, make full rank decomposition of Hk, and calculate the virtual unknown input estimates d¯^k|k and corresponding variance matrix at time instant k:(28)R˜k=CkPk|k−1xCkT+Rk,M¯k=H¯kTR˜k−1H¯k−1H¯kTR˜k−1,d¯^k|k=M¯kyk−Ckx^k|k−1,Pk|kd¯=H¯kTR˜k−1H¯k−1.


### 3.4.3. Measurement Update

Calculate the state estimatex^k|k and corresponding variance matrix at time instant k:(29)Kk=Pk|k−1xCkTR˜k−1,L¯k=KkI−H¯kM¯k,x^k|k=x^k|k−1+L¯kyk−Ckx^k|k−1,Pk|kx=Pk|k−1x−KkR˜k−H¯kPk|kd¯H¯kTKkT,Pk|kxd¯=Pk|kd¯xT=−KkH¯kPk|kd¯,d^k|k=Tk+d¯^k|k,Pk|kd=Tk+Pk|kd¯Tk+T,Pk|kxd=Pk|kdxT=−KkH¯kPk|kd¯Tk+T.Also, note that ifHk is of full column rank, letting H¯k=Hk and Tk=Im, RTSF is obtained.


## 3.4.1. Time Update

Based on the unbiased estimatesd^k−1|k−1 and x^k−1|k−1, the state estimates and corresponding variance matrix from the time instant k−1 to k are obtained:(27)x^k|k−1=Ak−1x^k−1|k−1+Gk−1d^k−1|k−1,Pk|k−1x=Ak−1Gk−1Pk−1|k−1xPk−1|k−1xdPk−1|k−1dxPk−1|k−1dAk−1TGk−1T+Qk−1.


## 3.4.2. Estimation of Virtual Unknown Input

Calculate the rank ofHk, make full rank decomposition of Hk, and calculate the virtual unknown input estimates d¯^k|k and corresponding variance matrix at time instant k:(28)R˜k=CkPk|k−1xCkT+Rk,M¯k=H¯kTR˜k−1H¯k−1H¯kTR˜k−1,d¯^k|k=M¯kyk−Ckx^k|k−1,Pk|kd¯=H¯kTR˜k−1H¯k−1.


## 3.4.3. Measurement Update

Calculate the state estimatex^k|k and corresponding variance matrix at time instant k:(29)Kk=Pk|k−1xCkTR˜k−1,L¯k=KkI−H¯kM¯k,x^k|k=x^k|k−1+L¯kyk−Ckx^k|k−1,Pk|kx=Pk|k−1x−KkR˜k−H¯kPk|kd¯H¯kTKkT,Pk|kxd¯=Pk|kd¯xT=−KkH¯kPk|kd¯,d^k|k=Tk+d¯^k|k,Pk|kd=Tk+Pk|kd¯Tk+T,Pk|kxd=Pk|kdxT=−KkH¯kPk|kd¯Tk+T.Also, note that ifHk is of full column rank, letting H¯k=Hk and Tk=Im, RTSF is obtained.


## 4. NRTSF

IfHk=0, systems (1) and (2) are transformed into a linear discrete-time-varying system without direct feedthrough of unknown input to output. It can be expressed as(30)xk=Ak−1xk−1+Gk−1dk−1+wk−1,(31)yk=Ckxk+vk.The classical filter proposed by Gilljins and De Moor in [15] can solve the state estimation problem when Hk=0, but the application conditions to use this filter are that the unknown input distribution matrix Gk−1 in the system equation must meet rankGk−1=m,k=1,2,….When the unknown input distribution matrix is not of full column rank, that is,rankGk−1=rk<m, then the classical filter cannot be used. Similar to Section 3, we can also consider an NRTSF design method. The following is the NRTSF derivation process.IfrankGk−1=rk≤m, perform full rank decomposition:(32)Gk−1=G¯k−1Tk−1,where G¯k−1∈Rn×rk,Tk−1∈Rrk×m, and rankG¯k−1=rankTk−1=rk. The full rank decomposition steps are given in Appendix.Defining the virtual unknown input byd¯k−1=Tk−1dk−1, then Gk−1dk−1=G¯k−1d¯k−1. Because the unknown input is estimated with one step delay, the estimation value of d¯k−1 is expressed as d¯^k−1|k and the minimal norm estimation value of the unknown input dk−1 is(33)d^k−1|k=Tk−1+d¯^k−1|k,where Tk−1+ is the Moore–Penrose inverse of Tk−1. Then, original system equation (30) is rewritten as(34)xk=Ak−1xk−1+G¯k−1d¯k−1+wk−1,where d¯k−1∈Rrk is the virtual unknown input vector.Based on the system state equation (34) and output equation (31), we consider NRTSF of the form(35)x^k|k−1=Ak−1x^k−1|k−1,(36)d¯^k−1|k=M¯kyk−Ckx^k|k−1,(37)x^k|k=x^k|k−1+L¯kyk−Ckx^k|k−1,where the matrices M¯k∈Rrk×p and L¯k∈Rn×p still have to be determined. Compared with the previous section, the obvious difference is that the second step in NRTSF calculates the value of virtual unknown input d¯k−1|k, while the previous filter yields an estimate of virtual unknown input d¯k|k.


### 4.1. Time Update

Letx^k−1|k−1 express the optimal unbiased estimates of xk−1 given measurement sequence y0,y1,…,yk−1; then, the time update is(38)x^k|k−1=Ak−1x^k−1|k−1.Similarly, the covariance matrix ofx^k|k−1 is given by(39)Pk|k−1x≜Ak−1Pk−1|k−1xAk−1T+Qk−1,with Pk|kx≜Ex˜k|kx˜k|kT,x˜k|k≜xk−x^k|k.


### 4.2. Virtual Unknown Input Estimation

The derivation idea in this section is the same asSection 3.2 except that the time index of unknown input is different.


#### 4.2.1. Unbiased Virtual Unknown Input Estimation

Defining the innovationy˜k≜yk−Ckx^k|k−1, it follows from (31), (34), and (35) that(40)y˜k=CkG¯k−1d¯k−1+ek,where ek is given by(41)ek=CkAk−1x˜k−1|k−1+wk−1+vk.Due to the fact thatx^k|k−1 is unbiased, we can obtain an unbiased estimate of the virtual unknown input d¯k−1 from y˜k.Theorem 4.
Supposex^k−1|k−1 is unbiased; then, (35) and (36) calculate the unbiased value of d¯k−1 if and only if M¯k satisfies M¯kCkG¯k−1=Irk.Proof.
This process is similar to the proof of Theorem1, so it is omitted.
The matrix corresponding to the LS solution of (40) satisfies Theorem 4. But from the Gauss–Markov theorem, it is not necessarily minimum-variance as a result of(42)R˜k≜EekekT=CkAk−1Pk−1|k−1xAk−1T+Qk−1CkT+Rk=CkPk|k−1xCkT+Rk≠cI,where c is a positive real number.


#### 4.2.2. MVU Virtual Unknown Input Estimation

Through weighted LS estimation, we obtain an MVU estimate ofd¯k−1.Theorem 5.
Letx^k−1|k−1 be unbiased and let R˜k and FkTR˜k−1Fk be positive definite; then, for(43)M¯k=FkTR˜k−1Fk−1FkTR˜k−1,where Fk≜CkG¯k−1, (33) is the MVU estimator of d¯k−1. The variance of the corresponding input estimate is(44)Pk−1|kd¯=FkTR˜k−1Fk−1.Proof.
This process is similar to the proof of Theorem2, so it is omitted.


### 4.3. Measurement Update

First, the estimator must satisfyEx^k|k−xk=0, which can be expressed as(45)Ex^k|k−1+L¯kCkAk−1xk−1+CkG¯k−1d¯k−1+Ckwk−1+vk−Ckx^k|k−1−Ak−1xk−1−G¯k−1d¯k−1−wk−1=0.Consequently, (37) is unbiased for all possible d¯k−1 if and only if L¯k satisfies(46)L¯kCkG¯k−1−G¯k−1=0.LetL¯k satisfy (46); then, Pk|kx is given by(47)Pk|kx=I−L¯kCkAk−1Pk−1|k−1xAk−1T+Qk−1I−L¯kCkT+L¯kRkL¯kT=L¯kR˜kL¯kT−Pk|k−1xCkTL¯kT−L¯kCkPk|k−1x+Pk|k−1x.So, we can calculateL¯k by minimizing the trace of (47) under the unbiasedness constraint of (46).Theorem 6.
L¯k is given by(48)L¯k=Kk+I−KkCkG¯k−1M¯k,where Kk=Pk|k−1xCkTR˜k−1 minimizes the trace of (47) under the constraint of (46).Proof.
This process is similar to the proof of Theorem3, so it is omitted.
Substituting (48) in (37) yields the equivalent state update:(49)x^k|k=x^k|k−1+Kk+I−KkCkG¯k−1M¯kyk−Ckx^k|k−1.
Form (47) and (48),(50)Pk|kx=Pk|k−1x−KkR˜kKkT+I−KkCkG¯k−1Pk−1|kd¯G¯k−1TI−KkCkT.


### 4.4. Summary of NRTSF Equations

#### 4.4.1. Time Update

Based on the unbiased estimatesx^k−1|k−1, the state estimates and corresponding variance matrix from time instant k−1 to k are obtained:(51)x^k|k−1=Ak−1x^k−1|k−1,Pk|k−1x=Ak−1Pk−1|k−1xAk−1T+Qk−1.


#### 4.4.2. Estimation of Virtual Unknown Input

Calculate the rank ofGk−1, make full rank decomposition of Gk−1, and calculate the virtual unknown input estimates d¯^k−1|k and corresponding variance matrix at time instant k:(52)R˜k=CkPk|k−1xCkT+Rk,Fk=CkG¯k−1,M¯k=FkTR˜k−1Fk−1FkTR˜k−1,d¯^k−1|k=M¯kyk−Ckx^k|k−1,Pk−1|kd¯=FkTR˜k−1Fk−1.


#### 4.4.3. Measurement Update

Calculate the state estimatex^k|k and corresponding variance matrix at time instant k:(53)Kk=Pk|k−1xCkTR˜k−1,L¯k=Kk+I−KkCkG¯k−1M¯k,x^k|k=x^k|k−1+L¯kyk−Ckx^k|k−1,Pk|kx=Pk|k−1x−KkR˜kKkT+I−KkCkG¯k−1Pk−1|kd¯G¯k−1TI−KkCkT,d^k−1|k=Tk−1+d¯^k−1|k.


## 4.1. Time Update

Letx^k−1|k−1 express the optimal unbiased estimates of xk−1 given measurement sequence y0,y1,…,yk−1; then, the time update is(38)x^k|k−1=Ak−1x^k−1|k−1.Similarly, the covariance matrix ofx^k|k−1 is given by(39)Pk|k−1x≜Ak−1Pk−1|k−1xAk−1T+Qk−1,with Pk|kx≜Ex˜k|kx˜k|kT,x˜k|k≜xk−x^k|k.


## 4.2. Virtual Unknown Input Estimation

The derivation idea in this section is the same asSection 3.2 except that the time index of unknown input is different.


### 4.2.1. Unbiased Virtual Unknown Input Estimation

Defining the innovationy˜k≜yk−Ckx^k|k−1, it follows from (31), (34), and (35) that(40)y˜k=CkG¯k−1d¯k−1+ek,where ek is given by(41)ek=CkAk−1x˜k−1|k−1+wk−1+vk.Due to the fact thatx^k|k−1 is unbiased, we can obtain an unbiased estimate of the virtual unknown input d¯k−1 from y˜k.Theorem 4.
Supposex^k−1|k−1 is unbiased; then, (35) and (36) calculate the unbiased value of d¯k−1 if and only if M¯k satisfies M¯kCkG¯k−1=Irk.Proof.
This process is similar to the proof of Theorem1, so it is omitted.
The matrix corresponding to the LS solution of (40) satisfies Theorem 4. But from the Gauss–Markov theorem, it is not necessarily minimum-variance as a result of(42)R˜k≜EekekT=CkAk−1Pk−1|k−1xAk−1T+Qk−1CkT+Rk=CkPk|k−1xCkT+Rk≠cI,where c is a positive real number.


### 4.2.2. MVU Virtual Unknown Input Estimation

Through weighted LS estimation, we obtain an MVU estimate ofd¯k−1.Theorem 5.
Letx^k−1|k−1 be unbiased and let R˜k and FkTR˜k−1Fk be positive definite; then, for(43)M¯k=FkTR˜k−1Fk−1FkTR˜k−1,where Fk≜CkG¯k−1, (33) is the MVU estimator of d¯k−1. The variance of the corresponding input estimate is(44)Pk−1|kd¯=FkTR˜k−1Fk−1.Proof.
This process is similar to the proof of Theorem2, so it is omitted.


## 4.2.1. Unbiased Virtual Unknown Input Estimation

Defining the innovationy˜k≜yk−Ckx^k|k−1, it follows from (31), (34), and (35) that(40)y˜k=CkG¯k−1d¯k−1+ek,where ek is given by(41)ek=CkAk−1x˜k−1|k−1+wk−1+vk.Due to the fact thatx^k|k−1 is unbiased, we can obtain an unbiased estimate of the virtual unknown input d¯k−1 from y˜k.Theorem 4.
Supposex^k−1|k−1 is unbiased; then, (35) and (36) calculate the unbiased value of d¯k−1 if and only if M¯k satisfies M¯kCkG¯k−1=Irk.Proof.
This process is similar to the proof of Theorem1, so it is omitted.
The matrix corresponding to the LS solution of (40) satisfies Theorem 4. But from the Gauss–Markov theorem, it is not necessarily minimum-variance as a result of(42)R˜k≜EekekT=CkAk−1Pk−1|k−1xAk−1T+Qk−1CkT+Rk=CkPk|k−1xCkT+Rk≠cI,where c is a positive real number.


## 4.2.2. MVU Virtual Unknown Input Estimation

Through weighted LS estimation, we obtain an MVU estimate ofd¯k−1.Theorem 5.
Letx^k−1|k−1 be unbiased and let R˜k and FkTR˜k−1Fk be positive definite; then, for(43)M¯k=FkTR˜k−1Fk−1FkTR˜k−1,where Fk≜CkG¯k−1, (33) is the MVU estimator of d¯k−1. The variance of the corresponding input estimate is(44)Pk−1|kd¯=FkTR˜k−1Fk−1.Proof.
This process is similar to the proof of Theorem2, so it is omitted.


## 4.3. Measurement Update

First, the estimator must satisfyEx^k|k−xk=0, which can be expressed as(45)Ex^k|k−1+L¯kCkAk−1xk−1+CkG¯k−1d¯k−1+Ckwk−1+vk−Ckx^k|k−1−Ak−1xk−1−G¯k−1d¯k−1−wk−1=0.Consequently, (37) is unbiased for all possible d¯k−1 if and only if L¯k satisfies(46)L¯kCkG¯k−1−G¯k−1=0.LetL¯k satisfy (46); then, Pk|kx is given by(47)Pk|kx=I−L¯kCkAk−1Pk−1|k−1xAk−1T+Qk−1I−L¯kCkT+L¯kRkL¯kT=L¯kR˜kL¯kT−Pk|k−1xCkTL¯kT−L¯kCkPk|k−1x+Pk|k−1x.So, we can calculateL¯k by minimizing the trace of (47) under the unbiasedness constraint of (46).Theorem 6.
L¯k is given by(48)L¯k=Kk+I−KkCkG¯k−1M¯k,where Kk=Pk|k−1xCkTR˜k−1 minimizes the trace of (47) under the constraint of (46).Proof.
This process is similar to the proof of Theorem3, so it is omitted.
Substituting (48) in (37) yields the equivalent state update:(49)x^k|k=x^k|k−1+Kk+I−KkCkG¯k−1M¯kyk−Ckx^k|k−1.
Form (47) and (48),(50)Pk|kx=Pk|k−1x−KkR˜kKkT+I−KkCkG¯k−1Pk−1|kd¯G¯k−1TI−KkCkT.


## 4.4. Summary of NRTSF Equations

### 4.4.1. Time Update

Based on the unbiased estimatesx^k−1|k−1, the state estimates and corresponding variance matrix from time instant k−1 to k are obtained:(51)x^k|k−1=Ak−1x^k−1|k−1,Pk|k−1x=Ak−1Pk−1|k−1xAk−1T+Qk−1.


### 4.4.2. Estimation of Virtual Unknown Input

Calculate the rank ofGk−1, make full rank decomposition of Gk−1, and calculate the virtual unknown input estimates d¯^k−1|k and corresponding variance matrix at time instant k:(52)R˜k=CkPk|k−1xCkT+Rk,Fk=CkG¯k−1,M¯k=FkTR˜k−1Fk−1FkTR˜k−1,d¯^k−1|k=M¯kyk−Ckx^k|k−1,Pk−1|kd¯=FkTR˜k−1Fk−1.


### 4.4.3. Measurement Update

Calculate the state estimatex^k|k and corresponding variance matrix at time instant k:(53)Kk=Pk|k−1xCkTR˜k−1,L¯k=Kk+I−KkCkG¯k−1M¯k,x^k|k=x^k|k−1+L¯kyk−Ckx^k|k−1,Pk|kx=Pk|k−1x−KkR˜kKkT+I−KkCkG¯k−1Pk−1|kd¯G¯k−1TI−KkCkT,d^k−1|k=Tk−1+d¯^k−1|k.


## 4.4.1. Time Update

Based on the unbiased estimatesx^k−1|k−1, the state estimates and corresponding variance matrix from time instant k−1 to k are obtained:(51)x^k|k−1=Ak−1x^k−1|k−1,Pk|k−1x=Ak−1Pk−1|k−1xAk−1T+Qk−1.


## 4.4.2. Estimation of Virtual Unknown Input

Calculate the rank ofGk−1, make full rank decomposition of Gk−1, and calculate the virtual unknown input estimates d¯^k−1|k and corresponding variance matrix at time instant k:(52)R˜k=CkPk|k−1xCkT+Rk,Fk=CkG¯k−1,M¯k=FkTR˜k−1Fk−1FkTR˜k−1,d¯^k−1|k=M¯kyk−Ckx^k|k−1,Pk−1|kd¯=FkTR˜k−1Fk−1.


## 4.4.3. Measurement Update

Calculate the state estimatex^k|k and corresponding variance matrix at time instant k:(53)Kk=Pk|k−1xCkTR˜k−1,L¯k=Kk+I−KkCkG¯k−1M¯k,x^k|k=x^k|k−1+L¯kyk−Ckx^k|k−1,Pk|kx=Pk|k−1x−KkR˜kKkT+I−KkCkG¯k−1Pk−1|kd¯G¯k−1TI−KkCkT,d^k−1|k=Tk−1+d¯^k−1|k.


## 5. Example

In this section, we consider the state and unknown input estimation problem when the system is interfered bydk as well as zero-mean Gaussian white noise. Specifically, the estimation problem we consider were given in Du [23]. The parameters for the linear system are given by(54)A=0.5200000.2101000.3010000.7100000.1,G=100000010000000.1,C=I5,H=001000010000000,R=10−2×1000.5001000.3001000.5001000.3001,Q=10−4×10000010.50000.51000001000001.The unknown inputdk=d1kd2kd3kT used in this example is given in Figure 1.Figure 1
Actual values of the unknown inputdk.The results of using NRTSF-DF to estimate the unknown input are presented in Figure2. From Figure 2, NERTSF-DF can estimate the unknown inputs d2k,d3k but has no effect on d1k. The reason is that the direct feedthrough matrix Hk is not of full column rank and, therefore, there is no information about the unknown input d1k in the measurement.Figure 2
Estimation values of the unknown inputdk.Since the unknown input only affects the first three elements of the system state, we only plot the true value and estimated value of the first, second, and third element of state vectorxk in Figure 3. And the estimation errors of x1k,x2k, and x3k are shown in Figure 4. It can be seen from the figure that the state estimation value can track the true value.Figure 3
States and their estimations. (a) Estimation of statex1k. (b) Estimation of state x2k. (c) Estimation of state x3k.
(a)(b)(c)Figure 4
State estimation error.For the linear system without direct feedthrough of unknown input to output, letH=000000000000000 and G=100000010000000; then, the system is transformed into a linear discrete system without direct feedthrough.The results of using NRTSF to estimate the unknown input are shown in Figure5. The NRTSF has no effect on d3k because Gk−1 is not of full column rank. And there is no information about the unknown input d3k in the system state. Furthermore, there is no information about the d3k in measurement.Figure 5
Estimation values of the unknown inputdk.Figures6 and 7 show the true value, estimation value, and the estimation error of the first three elements of the state vector, respectively. From the figure, NRTSF is effective.Figure 6
States and their estimations. (a) Estimation of statex1k. (b) Estimation of state x2k. (c) Estimation of state x3k.
(a)(b)(c)Figure 7
State estimation error.


## 6. Conclusion

This paper discusses the problem of joint state and unknown input estimation for linear systems with an unknown input and proposes two novel filters, respectively, in accordance with the linear minimum-variance unbiased estimation criterion. For systems with direct feedthrough, a novel recursive three-step filter with direct feedthrough is proposed. This filter can solve the problem that the classical recursive three-step filter cannot be used when the unknown input distribution matrix is not of full column rank. For the situation that unknown input only affects system equation and the distribution matrix is not of full column rank, a novel recursive three-step filter is proposed. The simulation results show that both of the proposed filters can effectively estimate the unknown input and system state.



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