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

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.


Introduction
e traditional Kalman filter [1] and its extension can recursively estimate the state of the linear system with process noise and measurement noise. e 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. e 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][6][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. en, 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). is method transforms the output equation by singular value decomposition of the distribution matrix. en 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. is paper put forward the novel recursive threestep 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. e 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.
is paper is organized as follows: in Section 2, 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. e structure of NRTSF is also obtained. Finally, Section 5 demonstrates the effectiveness of the proposed filters through simulation.

Problem Formulation
Consider the linear discrete-time-varying system: where x k ∈ R n is the state vector, d k ∈ R m is an unknown input vector, and y k ∈ R p is the measurement. e process noise w k ∈ R n and the measurement noise v k ∈ R p are assumed to be mutually uncorrelated, zero-mean, white random signals with known covariance matrices, respectively. e time-varying matrices A k , G k , C k , and H k are known with an appropriate dimension. roughout the paper, the conditions that (A k , G k ) is observable and that x 0 is independent of w k and v k are satisfied. And the unbiased estimate is known. e optimal filtering problem of the above system is to obtain the unbiased optimal filtering sequence of unknown input d 0|0 , . . . , d k|k and state x 0|0 , . . . , x k|k recursively based on the initial estimate x 0 , the covariance matrix P x 0 , and the sequence of measurement y 0 , y 1 , . . . , y k . If H k � 0, the system is transformed into a linear discrete-time-varying system without direct feedthrough of unknown input to output. en, 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.

NRTSF-DF
e RTSF proposed by Steven Gilljins in [16] can solve the state and unknown input estimation problem of linear system (1)-(2) while rank(H k ) � m, k � 0, 1, . . .. When the unknown input distribution matrix in the output equation is not of full column rank, that is, rank(H k ) � r k < m, we consider a NRTSF-DF design method. e following is the derivation process.
If rank(H k ) � r k ≤ m, perform full rank decomposition: where H k ∈ R p×r k , T k ∈ R r k ×m , and rank(H k ) � rank (T k ) � r k . e full rank decomposition steps are given in Appendix.
Defining the virtual unknown input by d k � T k d k , then H k d k � H k d k . If the estimation value of d k is expressed as d k|k , the minimal norm estimation value of unknown input d k is where T + k is the Moore-Penrose inverse of T k . en, original output equation (2) is rewritten as where d k ∈ R r k is the virtual unknown input vector.

Mathematical Problems in Engineering
Based on the system state equation (1) and output equation (5), we consider NRTSF-DF of the form where the matrices M k ∈ R r k ×p and L k ∈ R n×p still have to be determined.

Time Update.
Let x k−1|k−1 and d k−1|k−1 denote the optimal unbiased estimates of x k−1 and d k−1 given measurement sequence y 0 , y 1 , · · · , y k−1 ; then, the time update is e error in the estimate x k|k−1 is given by where Consequently, the covariance matrix of x k|k−1 is given by with

Virtual Unknown Input Estimation.
In this section, the estimation of the virtual unknown input d k is considered.

Unbiased Virtual Unknown Input Estimation.
Defining the innovation y k ≜ y k − C k x k|k−1 , it follows from (5) that where e k is given by Owing to is process is similar to the proof of eorem 1 in [16], so it is omitted.
From eorem 1, rank(H k ) � r k is a necessary and sufficient condition for an unbiased virtual unknown input estimator of form (7).
corresponding to the least-squares (LS) solution of (12) satisfies eorem 1. But from the Gauss-Markov theorem, the LS solution is not necessarily minimum-variance as a result of where c is a positive real number.
(4) is the MVU estimator of d k . e variance of the optimal virtual unknown input estimate is Proof. is process is similar to the proof of eorem 2 in [16], so it is omitted.
We use d * k|k to express the optimal virtual unknown input estimate corresponding to M * k and let d * In the last step, we use measurement y k to update x k|k−1 . Using (8) and (12), we find that Consequently, (8) is unbiased for all d k if and only if L k contents Suppose L k satisfy (19), from (18): So, we can calculate L k by minimizing the trace of (20) under the unbiasedness constraint of (19).

Mathematical Problems in Engineering 3
Theorem 3. L k is given by where k minimizes the trace of (20) under the constraint of (19).

Proof.
is process is similar to the proof of eorem 3 in [16], so it is omitted.
We use x * k|k to express the state estimate corresponding to L * k . From (7) and (21), en, we consider the expressions of From (20) and (21), we obtain Using (23) and (17), it follows that Furthermore, by d k|k � T + k d k|k , we can get Also, note that if H k is of full column rank, letting H k � H k and T k � I m , RTSF is obtained.

NRTSF
If H k � 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 e classical filter proposed by Gilljins and De Moor in [15] can solve the state estimation problem when H k � 0, but the application conditions to use this filter are that the unknown input distribution matrix G k−1 in the system equation must meet rank(G k−1 ) � m, k � 1, 2, . . ..
When the unknown input distribution matrix is not of full column rank, that is, rank(G k−1 ) � r k < m, then the classical filter cannot be used. Similar to Section 3, we can also consider an NRTSF design method. e following is the NRTSF derivation process.
If rank(G k−1 ) � r k ≤ m, perform full rank decomposition: where G k−1 ∈ R n×r k , T k−1 ∈ R r k ×m , and rank(G k−1 ) � rank(T k−1 ) � r k . e full rank decomposition steps are given in Appendix.

Defining the virtual unknown input by
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 d k−1 is where T + k−1 is the Moore-Penrose inverse of T k−1 . en, original system equation (30) is rewritten as where d k−1 ∈ R r k is the virtual unknown input vector. Based on the system state equation (34) and output equation (31), we consider NRTSF of the form where the matrices M k ∈ R r k ×p and L k ∈ R n×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 .

Time Update.
Let x k−1|k−1 express the optimal unbiased estimates of x k−1 given measurement sequence y 0 , y 1 , . . . , y k−1 ; then, the time update is Similarly, the covariance matrix of x k|k−1 is given by

Virtual Unknown Input
Estimation. e derivation idea in this section is the same as Section 3.2 except that the time index of unknown input is different.

Unbiased Virtual Unknown Input Estimation.
Defining the innovation y k ≜ y k − C k x k|k−1 , it follows from (31), (34), and (35) that where e k is given by Due to the fact that x k|k−1 is unbiased, we can obtain an unbiased estimate of the virtual unknown input d k−1 from y k .
is process is similar to the proof of eorem 1, so it is omitted.
e matrix corresponding to the LS solution of (40) satisfies eorem 4. But from the Gauss-Markov theorem, it is not necessarily minimum-variance as a result of where c is a positive real number.
where F k ≜ C k G k−1 , (33) is the MVU estimator of d k−1 . e variance of the corresponding input estimate is Proof. is process is similar to the proof of eorem 2, so it is omitted.
Consequently, (37) is unbiased for all possible d k−1 if and only if L k satisfies Let L k satisfy (46); then, P x k|k is given by So, we can calculate L k by minimizing the trace of (47) under the unbiasedness constraint of (46).

Mathematical Problems in Engineering 5
Theorem 6. L k is given by where K k � P x k|k−1 C T k R −1 k minimizes the trace of (47) under the constraint of (46).

Proof.
is process is similar to the proof of eorem 3, so it is omitted.

Estimation of Virtual Unknown Input.
Calculate the rank of G k−1 , make full rank decomposition of G k−1 , and calculate the virtual unknown input estimates d k−1|k and corresponding variance matrix at time instant k:

Measurement Update.
Calculate the state estimate x k|k and corresponding variance matrix at time instant k:

Example
In this section, we consider the state and unknown input estimation problem when the system is interfered by d k as well as zero-mean Gaussian white noise. Specifically, the estimation problem we consider were given in Du [23]. e parameters for the linear system are given by T used in this example is given in Figure 1.
e results of using NRTSF-DF to estimate the unknown input are presented in Figure 2. From Figure 2, NERTSF-DF can estimate the unknown inputs d 2k , d 3k but has no effect on d 1k . e reason is that the direct feedthrough matrix H k is not of full column rank and, therefore, there is no information about the unknown input d 1k in the measurement.
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 vector x k in Figure 3. And the estimation errors of x 1k , x 2k , and x 3k are shown in Figure 4. It can be seen from the figure that the state estimation value can track the true value.
; then, the system is transformed into a linear discrete system without direct feedthrough. e results of using NRTSF to estimate the unknown input are shown in Figure 5. e NRTSF has no effect on d 3k because G k−1 is not of full column rank. And there is no information about the unknown input d 3k in the system state. Furthermore, there is no information about the d 3k in measurement. Figures 6 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.

Conclusion
is 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.
is 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. e simulation results show that both of the proposed filters can effectively estimate the unknown input and system state.