Some Results on Simultaneous Input and State Estimation for Linear Systems⁎

July 9-11, 2018. Stockholm, Sweden on System Identification Proceedings,18th IFAC Symposium Proceedings,18th IFAC Symposium on System Identification July 9-11, 2018. Stockholm, Sweden July 9-11, 2018. Stockholm, Sweden Available online at www.sciencedirect.com

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Some Some Some Input Input Input

Results on Simultaneous Results Results on on Simultaneous Simultaneous and State Estimation for and for and State State Estimation Estimation for  Linear Systems  Linear Linear Systems Systems

Xinmin Song ∗,∗∗ ∗,∗∗ Xinmin Xinmin Song Song ∗,∗∗ ∗ ∗ ∗

Wei Xing Zheng ∗∗ ∗∗ Wei Wei Xing Xing Zheng Zheng ∗∗

School of Information Science and Engineering, Shandong Normal School of Science Engineering, Shandong Normal University, Jinan, Shandong, 250014, P. R. China (e-mail: School of Information Information Science and and Engineering, Shandong Normal University, Jinan, Shandong, 250014, P. R. China (e-mail: University, Jinan,[email protected]) Shandong, 250014, P. R. China (e-mail: [email protected]) ∗∗ School of Computing, Engineering and Mathematics, Western [email protected]) ∗∗ of Computing, Engineering Western ∗∗ School SydneyofUniversity, NSW and 2751,Mathematics, Australia (e-mail: School Computing,Sydney, Engineering and Mathematics, Western Sydney University, Sydney, NSW 2751, Australia (e-mail: [email protected]) Sydney University, Sydney, NSW 2751, Australia (e-mail: [email protected]) [email protected])

Abstract: This paper is concerned with the problem of simultaneous input and state estimation Abstract: This concerned with the of input and estimation for linear discrete-time with missing measurements. First, in order tostate simultaneously Abstract: This paper paper is issystems concerned with the problem problem of simultaneous simultaneous input and state estimation for linear discrete-time systems with missing measurements. First, in order to simultaneously estimate the input and state in the sense of unbiased minimum variance, a recursive estimator for linear discrete-time systems with missing measurements. First, in order to simultaneously estimate the input and state in the sense of unbiased minimum variance, a recursive estimator is designed terms one in Lyapunov and minimum one Riccati equation. Then some mild estimate theininput andofstate the senseequation of unbiased variance, a recursive estimator is designed in terms of and equation. Then mild conditions existence the infiniteequation horizon estimator are presented. Finally, simulation is designedfor in the terms of one oneofLyapunov Lyapunov equation and one one Riccati Riccati equation. Thena some some mild conditions the of infinite horizon are Finally, example is for provided to illustrate effectiveness of the proposed approach. conditions for the existence existence of the thethe infinite horizon estimator estimator are presented. presented. Finally, aa simulation simulation example is provided to illustrate the effectiveness of the proposed approach. example is provided to illustrate the effectiveness of the proposed approach. © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: State estimation, unknown input estimation, unbiased minimum variance, Keywords: State estimation, unknown input input estimation, estimation, unbiased unbiased minimum minimum variance, variance, convergence, missing measurements. Keywords: State estimation, unknown convergence, missing missing measurements. measurements. convergence, 1. INTRODUCTION al. (2009); Hsieh (2010); Li (2013); Su et al. (2015); Shi et 1. INTRODUCTION INTRODUCTION al. (2010); Li Su et (2015); Shi 1. (2016)).Hsieh However, compared with state estimation al. (2009); (2009); Hsieh (2010); Li (2013); (2013); Suthe et al. al. (2015); Shi et et al. (2016)). However, compared with the state estimation problem, theHowever, problemcompared of estimating unknown input (2016)). with the state estimation The estimation problem for linear systems with unknown al. problem, the problem of estimating input itself is also very important. Indeed, the someunknown important reproblem, the problem of estimating the unknown input The estimation problem for linear systems with unknown inputestimation has received considerable attention the past The problem for linear systemsduring with unknown itself is also very important. Indeed, some important results on this problem have been reported during the past input has hasand received considerable attention during the past past decades, various estimators attention have beenduring developed un- itself is also very important. Indeed, some important reinput received considerable the sults on this problem have been reported during the past decade, example, Gillijns & De Moor (2007); on see, thisfor problem have been reported during the Fang past decades, and assumptions various estimators estimators have been with developed un- sults der different on thehave systems unknown decades, and various been developed undecade, see, for example, Gillijns & De Moor (2007); Fang & De Callafon (2012); Yong et al. (2016), just to mention decade, see, for example, Gillijns & De Moor (2007); Fang der different different assumptions on the the systems systems with unknown input (see, for example, Kitanidis (1987);with Zheng et al. & De Callafon (2012); Yong et al. (2016), just to mention der assumptions on unknown a few. On the (2012); other hand, theal.study of just the to state estiDe Callafon Yong et (2016), mention input (see, for example, Kitanidis (1987); Zheng et al. al. & (1996);(see, Darouach & Zasadzinski (1997); Kerwin & Prince input for example, Kitanidis (1987); Zheng et aamation few. On the other hand, the study of the state estiproblem for systems with missing measurements few. On the other hand, the study of the state esti(1996); Darouach Darouach et & al. Zasadzinski (1997); Kerwin & Prince Prince (2000); (2003); Cheng et Kerwin al. (2009); Hsieh mation problem for systems with missing measurements (1996); & Zasadzinski (1997); & has exerted a tremendous fascination on a host of scholars problem for systems with missing measurements (2000); Darouach Darouach et Su al. (2003); (2003); Cheng et et al. et(2009); (2009); Hsieh mation (2010); Li (2013);et et al. (2015); Shi al. (2016); (2000); al. Cheng al. Hsieh has exerted aa tremendous aa host of (Nahi (1969); Sinopoli et fascination al. (2004); on Wang et al. (2009); has exerted tremendous fascination on host of scholars scholars (2010); Li (2013); Su et al. (2015); Shi et al. (2016); Gillijns Moor (2007); Fang et al. (2011); & De (2010); & LiDe (2013); Su et al. (2015); Shi et Fang al. (2016); (Nahi (1969); Sinopoli et al. (2004); Wang et al. (2009); Zhang et al. (2012)). It has come to our attention that Gillijns & &(2012); De Moor Moor (2007); Fang et al. al.Zhang (2011); Fang & De De (Nahi (1969); Sinopoli et al. (2004); Wang et al. (2009); Callafon Yong et al.Fang (2016); et Fang al. (2016); Gillijns De (2007); et (2011); & Zhang et al. (2012)). It has come to our attention the problem simultaneous estimationthat for et al.of(2012)). It hasinput comeand to state our attention that Callafon& (2012); (2012); Yong etand al. references (2016); Zhang Zhang et al. al.Most (2016); Zhang Zheng Yong (2018)et therein). of Zhang Callafon al. (2016); et (2016); the problem of simultaneous input and state estimation for linear systems with missing measurements has not been the problem of simultaneous input and state estimation for Zhang & Zheng Zhengresults (2018) and systems references therein). Most of the estimation about with unknown input Zhang & (2018) and references therein). Most of linear systems with missing measurements has not been fully investigated except for Zhang et al. (2016). systems with missing measurements has not been the estimation estimation results aboutwithout systems estimating with unknown unknown input linear focus on state results estimation unknown the about systems with input fully investigated except for Zhang et al. (2016). fully investigated except for Zhang et al. (2016). focus (Kitanidis on state state estimation estimation without&estimating estimating unknown input (1987); Darouach Zasadzinski (1997); focus on without unknown Based on the estimator design approach in Gillijns & input (Kitanidis (1987); Darouach & Zasadzinski (1997); Kerwin(Kitanidis & Prince (1987); (2000); Darouach et (2003); Cheng et Based on the estimator design approach in Gillijns & input & al. Zasadzinski (1997); De Moor Fang & design De Callafon (2012), the finite on (2007); the estimator approach in Gillijns & Kerwin & & Prince (2000); (2000); Darouach Darouach et et al. al. (2003); (2003); Cheng Cheng et et Based Kerwin  This workPrince De Moor (2007); & De (2012), the finite was supported in part by the Natural Science Founhorizon input state estimation De Moordistributed (2007); Fang Fang & and De Callafon Callafon (2012), is thestudied finite  This of work was supported part by the Natural Science Founhorizon input and state estimation studied dation Shandong Provincein Grant ZR2016FM17 and the  This work was supported inunder part by the Natural Science Founin Zhangdistributed et al. (2016) linear withis horizon distributed inputfor and statesystems estimation is missing studied dation of Shandong Provinceunder under Grant ZR2016FM17 and the Australian Research Council Grant DP120104986. in Zhang et al. (2016) for linear systems with missing dation of Shandong Province under Grant ZR2016FM17 and the in Zhang et al. (2016) for linear systems with missing Australian Research Council under Grant DP120104986.

Australian Research Council under Grant DP120104986. Copyright © 2018 IFAC 49 2405-8963 © © 2018 2018, IFAC IFAC (International Federation of Automatic Control) Copyright 49 Hosting by Elsevier Ltd. All rights reserved. Copyright © 2018 IFAC 49 Control. Peer review under responsibility of International Federation of Automatic 10.1016/j.ifacol.2018.09.089

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assumed to be of full column rank, which is necessary to ensure unbiased estimation.

measurements. In order to derive the estimator gains, the covariance matrix of the state, E{x(k)xT (k)}, is involved. Hence, when applying the design approach in Gillijns & De Moor (2007); Fang & De Callafon (2012); Zhang et al. (2016) to the infinite horizon input and state estimation for linear systems with missing measurements, there is a need to consider the convergence of the Lyapunov equation about E{x(k)xT (k)}. To meet the requirements of the convergence analysis, two conditions should be imposed: 1) the system matrix is stable; 2) the input covariance E{u(k)uT (k)} is equal to a constant matrix when k → ∞. In other words, if the system matrix is unstable, then the infinite horizon estimator cannot be designed based on the approach given in Gillijns & De Moor (2007); Fang & De Callafon (2012); Zhang et al. (2016).

For system (1), now we present the input and state estimator design by applying the same estimator design approach in Gillijns & De Moor (2007); Fang & De Callafon (2012); Zhang et al. (2016). Lemma 1. For the given system (1), the unbiased input and state estimator is given by x ˆ(k|k − 1) = A(k − 1)ˆ x(k − 1|k − 1) + B(k − 1)ˆ u(k − 1),

(2)

∗

u ˆ(k) = L (k)[y(k) − αC(k)ˆ x(k|k − 1)],

(3)

x ˆ(k|k) = xˆ(k|k − 1) x(k|k − 1)], + K ∗ (k)[y(k) − αC(k)ˆ

Motivated by the above-mentioned issues, we aim to present a new estimator design approach to the infinite horizon input and state estimation problem with missing measurements in this paper. Just like the stability of simultaneous input and state estimator (Fang & De Callafon (2012)), we develop a parallel to obtain the mean square stability properties of the proposed estimator; For systems with missing measurements, compared with the classical unbiased minimum variance estimator approaches in Gillijns & De Moor (2007); Fang & De Callafon (2012); Zhang et al. (2016), we provide a milder convergence condition on the system matrix (i.e., the system matrix is no longer required to be stable).

x ˆ(0| − 1) = x¯(0),

(4) (5)

where L∗ (k) = [DT (k)Q

−1

(k)D(k)]−1 DT (k)Q

K ∗ (k) = αP x (k|k − 1)C T (k)Q

−1

−1

(k),

(k)[I − D(k)L∗ (k)],

Q(k) = α(1 − α)C(k)S(k)C T (k) + R + α2 C(k)P x (k|k − 1)C T (k), and further S(k) and P x (k|k − 1) can be calculated via the following Lyapunov equation and Riccati equation: S(k + 1) = A(k)S(k)AT (k) + Q + B(k)E{u(k)uT (k)}B T (k),

(6)

T

P x (k + 1|k) = A(k)P x (k|k − 1)A (k) + Q T

+ B(k)RB (k) 2. PRELIMINARY AND PROBLEM STATEMENT

T

+ α(1 − α)B(k)C(k)S(k)C T (k)B (k), (7)

Consider the linear system described by

A(k) = A(k) − αB(k)C(k)

x(k + 1) = A(k)x(k) + B(k)u(k) + w(k),

(1a)

y(k) = η(k)C(k)x(k) + D(k)u(k) + v(k),

B(k) = A(k)K ∗ (k) + B(k)L∗ (k).

(1b)

where x(k) ∈ R , y(k) ∈ R and u(k) ∈ R are the state, measurement, and unknown input to be estimated, respectively. w(k) and v(k) are white noises with zero mean and covariances

Proof. The proof can be done by combining the results in Gillijns & De Moor (2007); Fang & De Callafon (2012); Zhang et al. (2016) together and thus is omitted here.

E{w(k)wT (j)} = Qδkj and E{v(k)v T (j)} = Rδkj ,

Remark 1. The simultaneous input and state estimator (2)–(5) can be found from Zhang et al. (2016). The estimation gains L∗ (k) and K ∗ (k) may be calculated in terms of the solutions to the Lyapunov equation (6) and Riccati equation (7). Therefore, when designing an infinite horizon estimator, the system matrix A is required to be stable and E{u(k)uT (k)} is required to be a constant matrix as k → ∞ due to the convergence requirements of the Lyapunov equation (6). That is to say, one cannot obtain the infinite horizon estimator from the estimator (2)–(5) when A is unstable. Motivated by this important

nx

ny

nu

respectively. The independent and identically distributed (i.i.d.) Bernoulli random variable η(k) is employed to describe the packet loss phenomenon with P r{η(k) = 1} = α, P r{η(k) = 0} = 1 − α. The initial state x(0) is a random vector with mean x ¯(0) and covariance matrix S(0). The random processes w(k), v(k), and η(k) for all k and the initial state x(0) are mutually independent. Throughout this paper, D(k) is 50

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51

Then the covariance matrix P u (k) is minimized and the minimum covariance matrix is given by

observation, we aim to design a new estimator without the constraint that A is stable.

P u (k) = [DT (k)Q Problem Statement: The aim of this paper is to find the parameter gains Lη (k) and Kη (k) which satisfy + B(k − 1)ˆ u(k − 1),

(k)D(k)]−1 .

(16)

Under the unbiasedness conditions (12)–(13), we have the following state estimation error covariance matrix

xˆ(k|k − 1) = A(k − 1)ˆ x(k − 1|k − 1) (8)

u ˆ(k) = Lη (k)[y(k) − η(k)C(k)ˆ x(k|k − 1)],

−1

P x (k) = P x (k|k − 1) + Kη (k)Q(k)KηT (k)

(9)

− αP x (k|k − 1)C T (k)KηT (k)

xˆ(k|k) = x ˆ(k|k − 1)

− αKη (k)C(k)P x (k|k − 1).

(17)

x(k|k − 1)], (10) + Kη (k)[y(k) − η(k)C(k)ˆ x ˆ(0| − 1) = x ¯(0).

(11)

Similar to Lemma 3, now we seek the optimal gain matrix Kη (k) to minimize the state estimation covariance matrix P x (k + 1). Consequently, we have the following lemma.

such that E{˜ u(k)} = 0, E{˜ x(k|k)} = 0, and

Lemma 4. Let Kη (k) be given by

E{˜ u(k)˜ uT (k)} and E{˜ x(k|k)˜ xT (k|k)}

−1

Kη∗ (k) = αP x(k|k−1)C T(k)Q (k)[I −D(k)L∗ (k)]. (18)

are minimized, where the estimation errors

Then the covariance matrix P x (k) of the state estimator is minimized and the minimum covariance matrix is given by (17) with the replacement of Kη (k) by Kη∗ (k).

u ˜(k) = u(k) − u ˆ(k), x ˜(k|k) = x(k) − x ˆ(k|k), and the expectation is taken over w, v and η. Furthermore, we will seek the existence conditions of the infinite horizon input and state estimator.

Assume that Y (k) ∈ S = {S ∈ Rnx ×nx | S = S T , S ≥ 0}, L(k) ∈ Rnu ×ny , K(k) ∈ Rnx ×ny . Then we consider the following operator

3. ESTIMATOR DESIGN

φ(L(k), K(k), Y (k))  T (k) + Q + B(k)R   T (k) = A(k)Y (k)A B

For illustration convenience, the covariance matrices of the estimation errors are defined as P u (k)  E{˜ u(k)˜ uT (k)},

where

x(k|k)˜ xT (k|k)}, P x (k)  E{˜

  T (k), + α(1 − α)B(k)C(k)Y (k)C T (k)B   A(k) = A(k) − αB(k)C(k),  B(k) = A(k)K(k) + B(k)L(k).

x(k|k − 1)˜ xT (k|k − 1)}, P x (k|k − 1)  E{˜ where x ˜(k|k − 1)  x(k) − x ˆ(k|k − 1).

The linear operator φ plays an important role in the convergence analysis of the input and state estimators, as will be seen subsequently.

3.1 Finite Horizon Estimation

Theorem 1. Under the conditions (12)–(13), we have the following input and state estimator:

Lemma 2. Under the initial condition (11), if the matrix D(k) in (1b) satisfies Lη (k)D(k) = I,

(12)

Kη (k)D(k) = 0,

(13)

x ˆ(k|k − 1) = A(k − 1)ˆ x(k − 1|k − 1) + B(k − 1)ˆ u(k − 1), u ˆ(k)=L∗η (k)[y(k)

then the input and state estimator given in (8)–(10) is unbiased. −1

(k)D(k)]−1 DT (k)Q

(20)

x ˆ(k|k) = xˆ(k|k − 1) x(k|k − 1)] (21) + Kη∗ (k)[y(k) − η(k)C(k)ˆ

Lemma 3. Let Lη (k) be given by L∗η (k) = [DT (k)Q

− η(k)C(k)ˆ x(k|k − 1)],

(19)

−1

(k),

x ˆ(0| − 1) = x¯(0),

(14)

L∗η (k)

Q(k) = αC(k)P x (k|k − 1)C T (k) + R.

(22)

and are given in (14) and (18), and the where estimator is calculated in terms of the following Riccati equation:

where (15) 51

Kη∗ (k)

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then the sequence {P x (k + 1|k)} in (23) is bounded for any k under any initial condition

P x (k + 1|k) = φ(L∗η (k), Kη∗ (k), P x (k|k − 1)) η (k)P x (k|k − 1)A Tη (k) + Q + B η (k)RB ηT (k) =A

0 ≤ P x (0| − 1) < ∞.

η (k)C(k)P x (k|k − 1)C T (k)B ηT (k), (23) + α(1 − α)B

For illustration convenience, let P Q0 ,x (k|k − 1), KηQ0 ,∗ (k) and LηQ0 ,∗ (k) stand for the covariance matrix satisfying with (23) and gain matrices abiding by (18), (14) with an arbitrary initial condition Q0 , respectively.

where

η (k) = A(k) − αB η (k)C(k), A η (k) = A(k)K ∗ (k) + B(k)L∗ (k). B η

η

Lemma 6. If there exist matrices L and K such that  is mean square stable, LD = I, KD =  [α(1−α)] 12 BC) (A, 1 0, and (AT , Q 2 ) is observable, then the sequence {P x (k + 1|k)} in (23) with the initial value

Proof. It is easy to see that (19)–(22) are based on (8)– (11), Lemmas 3 and 4. Now we deduce the Riccati equation (23). It follows from (19)–(21) that x ˜(k + 1|k)

P x (0| − 1) = 0

= [A(k) − αA(k)Kη∗ (k)C(k)

converges to a unique positive definite matrix P x . In other words, there holds

x(k|k − 1) − αB(k)L∗η (k)C(k)]˜ − [η(k) − α][A(k)Kη∗ (k) + −

x(k|k − 1) B(k)L∗η (k)]C(k)˜ ∗ [A(k)Kη (k) + B(k)L∗η (k)]v(k)

lim P x (k + 1|k) = P x ,

k→∞

+ w(k).

where P x is the solution of

(24)

P x = φ(L∗η , Kη∗ , P x ),

Thus the associated covariance matrix is given by

L∗η

P x (k + 1|k) = φ(L∗η (k), Kη∗ (k), P x (k|k − 1)),

x

D]

−1

−1

(29) T

D Q

−1

,

[I − DL∗η ],

T

Q = αCP C + R.

3.2 Infinite Horizon Estimation

(30) (31) (32)

Lemma 7. If there exist matrices L and K such that  is mean square stable, LD = I, KD =  [α(1−α)] 12 BC) (A, 1 0, and (AT , Q 2 ) is observable, then for any arbitrary but fixed initial nonnegative symmetric P x (0| − 1),

Consider the following linear system subject to multiplicative noise: y(k) = Cx(k).

= [D Q

−1

Kη∗ = αP x C T Q

which completes the proof.

x(k + 1) = Ax(k) + A0 x(k)ω(k),

T

lim P x (k + 1|k) = P x ,

(25)

k→∞

where P x is the unique positive definite solution of (30).

(26)

Moreover, (A, A0 ) is called mean square stable if for any initial value x(0),

On the basis of Lemmas 5–7 and Theorem 1, now we present the main result of the infinite horizon estimator as follows.

lim E{x(k)xT (k)} = 0;

k→∞

(A, A0 , C) is called exactly observable if Theorem 2. If there exist matrices L and K such that  is mean square stable, LD = I, KD =  [α(1−α)] 12 BC) (A, 1 T 0, and (A , Q 2 ) is observable, then the infinite horizon estimator is given by

y(k) ≡ 0, a.s. ∀ k ∈ {0, 1, . . .} =⇒ x(0) = 0. Here ω(k) in (25) is a wide sense stationary, second-order process with E{ω(k)} = 0 and E{ω(k)ω(j)} = δkj .

x ˆ(k|k − 1) = Aˆ x(k − 1|k − 1) + B u ˆ(k − 1), u ˆ(k) =

Lemma 5. If there exist matrices L and K such that  [α(1 − α)] 12 BC)  (A, is mean square stable and LD = I,

(27)

KD = 0,

(28)

L∗η [y(k)

− η(k)C x ˆ(k|k − 1)],

(33) (34)

x ˆ(k|k) = x ˆ(k|k − 1) ˆ(k|k − 1)], + Kη∗ [y(k) − η(k)C x x ˆ(0| − 1) = x ¯(0), where

L∗η

and

Kη∗

(35) (36)

can be calculated from (30)–(32).

where  = A − αBC,  A  = AK + BL, B

Proof. The proof can be done readily by combining the results of Lemmas 5–7 and Theorem 1. 52

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Remark 2. In Fang & De Callafon (2012), the asymptotic stability is analyzed for simultaneous input and state estimation in the unbiased minimum variance sense. As a parallel, we have obtained the mean square stability for simultaneous input and state estimation for systems with missing measurements in Theorem 2. Moreover, we have successfully removed the condition that the system matrix A is stable (Song et al. (2016)).

53





 

    

4. NUMERICAL EXAMPLE



Consider the linear discrete-time system     1.06 −0.3 0.8 x(k + 1) = x(k) + u(k) 0.2 −0.38 −0.4

 

+ ω(k), 

   0.4 1 1.0 y0 (k) = η(k) x(k) + u(k) −0.3 1.5 0.4





  





Fig. 1. The first state component x1 (k) and its estimator.

+ v(k), with





  −2 0 x(0) = , x ˆ(0| − 1) = , 2 0     0.49 0 0.36 0 Q= , R= , 0 0.49 0 0.36





 

 

S(0) = P (0| − 1) = I2 .



ω(k) and v(k) are whites noises with zero mean and covariance matrices Q and R, respectively. The unknown input, which needs to be estimated, is chosen as a Gaussian white noise with zero and variance 1. The random variable sequence {η(k)} is generated with α = 0.9. Noting that one of the eigenvalues of the matrix A is 1.0171, the approaches proposed in Gillijns & De Moor (2007); Fang & De Callafon (2012); Zhang et al. (2016) are thus inapplicable to this example. By employing the Matlab toolbox, our proposed estimator yields the following convergent solution for the Riccati equation (30):   1.5565 0.0725 x P = , 0.0725 0.7012

   





  





Fig. 2. The second state component x2 (k) and its estimator. 5. CONCLUSION

and the gain matrices L∗η and Kη∗ are obtained below:   L∗η = 1.0825 −0.2062 ,   0.1623 −0.4057 ∗ Kη = . −0.1882 0.4705

In this paper, we have investigated the problem of simultaneous input and state estimation for linear discrete-time systems with missing measurements. We have designed a recursive estimator in terms of one Lyapunov equation and one Riccati equation in the sense of unbiased minimum variance. The time-stamping technique has been utilized to develop a new estimator under milder conditions. Then the sufficient conditions for the convergence of the new estimator have been established. Finally, the usefulness of

It can be verified that the unbiased conditions (12) and (13) are satisfied. The simulation results are depicted in Figs. 1–3. It can be seen from Figs. 1–3 that the estimators can track the original state x(k) and unknown input u(k) well, which shows that the method presented in this paper is effective. 53

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Fig. 3. The unknown input u(k) and its estimator u ˆ(k). the proposed simultaneous input and state estimator has been demonstrated by a simulation example.

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