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State Estimation for Stochastic Time Varying Systems with Disturbance Rejection⁎
Proceedings,18th Proceedings,18th IFAC IFAC Symposium Symposium on on System System Identification Identification July 9-11, 2018. Stockholm, Sweden on Proceedings,18th IFAC System Proceedings,18th IFAC Symposium Symposium System Identification Identification Available online at www.sciencedirect.com July 9-11, 2018. Stockholm, Sweden on July 9-11, Sweden Proceedings,18th IFAC Symposium July 9-11, 2018. 2018. Stockholm, Stockholm, Sweden on System Identification July 9-11, 2018. Stockholm, Sweden
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IFAC PapersOnLine 51-15 (2018) 55–59
State State Estimation Estimation for for Stochastic Stochastic Time Time Varying Varying State Estimation for Stochastic Time Varying Systems with Disturbance Rejection State Estimation for Stochastic Time Varying Systems with Disturbance Rejection Systems with Disturbance Rejection ∗ ∗∗ Systems with Disturbance Rejection Qinghua Qinghua Zhang Zhang ∗ Liangquan Liangquan Zhang. Zhang. ∗∗
∗∗ Qinghua Zhang Zhang ∗∗ Liangquan Liangquan Zhang. Zhang. ∗∗ Qinghua ∗ ∗∗ Qinghua Zhang Liangquan Zhang. Inria, Inria, IFSTTAR, IFSTTAR, Universit´ Universit´ee de de Rennes, Rennes, Campus Campus de de Beaulieu, Beaulieu, Inria, IFSTTAR, Universit´ ee de Rennes, Campus de 35042 Rennes Cedex, France. Email: [email protected] Inria, IFSTTAR, Universit´ de Rennes, Campus de Beaulieu, Beaulieu, 35042 Rennes Cedex, France. Email: [email protected] ∗ ∗∗ Inria, IFSTTAR, Universit´ e de Rennes, Campus de Beaulieu, 35042 Rennes Cedex, France. Email: [email protected] of Science, Beijing University of Posts and ∗∗ School 35042 Rennes Cedex, France. Email: [email protected] School of Science, Beijing University of Posts and ∗∗ ∗∗ School 35042 Rennes France. Email: [email protected] ofCedex, Science, Beijing University ofChina. Posts and and Telecommunications, Beijing 100876, School of Science, Beijing University of Posts Telecommunications, Beijing 100876, China. ∗∗ School of Science, Beijing University Posts and Telecommunications, Beijing 100876,ofChina. China. Email: [email protected] Telecommunications, Beijing 100876, Email: [email protected] Telecommunications, Beijing 100876, China. Email: Email: [email protected] [email protected] Email: [email protected] Abstract: State State estimation estimation in in the the presence presence of of unknown unknown disturbances disturbances is is useful useful for for the the design design of of Abstract: Abstract: State in unknown disturbances useful for the design robust systems systems inestimation different engineering engineering fields.of Most results available on onis this topic are restricted Abstract: State estimation in the the presence presence ofMost unknown disturbances isthis useful for are the restricted design of of robust in different fields. results available topic Abstract: State in systems, the presence ofMost unknown disturbances isthis useful for are thehave design of robust systems inestimation different engineering fields. Most results available on this topic are restricted to linear linear time invariant (LTI) systems, whereas linear time varying on (LTV) systems have been robust systems in different engineering fields. results available topic restricted to time invariant (LTI) whereas linear time varying (LTV) systems been robust systems in different engineering fields. Most results available on this topic are restricted to linear time invariant (LTI) systems, whereas linear time varying (LTV) systems have been studied to a lesser extent. Existing results on LTV systems are mainly based on the minimization to linear time invariant (LTI) systems, whereas linear time varying (LTV) systems have been studied to a lesser extent. Existing results on LTV systems are mainly based on the minimization to linear invariant (LTI) systems, whereas linear time (LTV) systems been studied to a lesser extent. Existing results on systems are mainly based on the of the the state estimation error covariance, ignoring the important issue of of the stability stability ofhave the state state studied totime a estimation lesser extent. Existing resultsignoring on LTV LTVthe systems arevarying mainly based on the minimization minimization of state error covariance, important issue the of the studied to a lesser extent. Existing results on LTV systems are mainly based on the minimization of the state estimation error covariance, ignoring the important issue of the stability of the state estimation error dynamics, which has been a main focus of the studies in the LTI case. The of the state error estimation error which covariance, ignoring the important issue of theinstability the state estimation dynamics, has been a main focus of the studies the LTIofcase. The of the state estimation error covariance, ignoring the important issue of the stability of the state estimation error dynamics, which has been a main focus of the studies in the LTI case. The purpose of this paper is to propose a numerically efficient algorithm for state estimation with estimation error dynamics, which has been a main focus of the studies in the LTI case. The purpose of this paper is to propose a numerically efficient algorithm for state estimation with estimation dynamics, which has been a main focus of the studies in theincluding LTI case.linear The purpose of this paper is propose a efficient algorithm for estimation with disturbance rejection, into the general framework of LTV stochastic stochastic systems, including linear purpose of error this paper isin tothe propose a numerically numerically efficient algorithm for state state estimation with disturbance rejection, general framework of LTV systems, purpose of this paper is to propose a numerically efficient algorithm for state estimation with disturbance rejection, in the general framework of LTV stochastic systems, including linear parameter varying (LPV) systems, with easily checkable conditions guaranteeing the stability disturbance rejection, in the general framework of LTV conditions stochastic guaranteeing systems, including linear parameter varying (LPV) systems, with easily checkable the stability disturbance rejection, in the general framework of LTV stochastic systems, including linear parameter varying (LPV) systems, with easily conditions guaranteeing the of the the algorithm. algorithm. The design method is conceptually simple: disturbance is first first rejected from parameter varying (LPV) systems, with easily checkable checkable conditions guaranteeing the stability stability of The design method is conceptually simple: disturbance is rejected from parameter varying (LPV) systems, with easily checkable conditions guaranteeing the stability of the algorithm. The design method is conceptually simple: disturbance is first rejected from the state equation by appropriate output injection, then the Kalman filter is applied to the of the algorithm. The design method is conceptually simple: disturbance is first rejected from the state equation by appropriate output injection, then the Kalman filter is applied to the of the algorithm. The design method is conceptually simple: disturbance is first rejected from the state equation by appropriate output injection, then the Kalman filter is applied to the resulting state-space model after the output injection. the state equation by appropriate output injection, then the Kalman filter is applied to the resulting state-space model after the output injection. the state state-space equation bymodel appropriate resulting after output injection. resulting state-space model after the theoutput outputinjection, injection.then the Kalman filter is applied to the © 2018, IFAC (International Federation ofoutput Automatic Control) Hosting by Elsevier Ltd. All rights reserved. resulting state-space model after the injection. Keywords: disturbance disturbance rejection, rejection, state state estimation, estimation, LTV/LPV system, system, Kalman Kalman filter. filter. Keywords: LTV/LPV Keywords: disturbance disturbance rejection, rejection, state state estimation, estimation, LTV/LPV LTV/LPV system, system, Kalman Kalman filter. filter. Keywords: Keywords: disturbance rejection, state estimation, LTV/LPV system, Kalman filter. 1. INTRODUCTION Compared 1. INTRODUCTION Compared to to LTI LTI systems, systems, linear linear time time varying varying (LTV) (LTV) 1. Compared to LTI linear time varying (LTV) systems have studied to aa lesser 1. INTRODUCTION INTRODUCTION Compared to been LTI systems, systems, linear time extent. varyingExisting (LTV) systems have been studied to lesser extent. Existing 1. aINTRODUCTION Compared to LTI systems, linear time varying (LTV) systems have been studied to a lesser extent. Existing State estimation is common task in different engineerresults on LTV systems have been mainly focused on studied a lesser State estimation is a common task in different engineer- systems results onhave LTVbeen systems haveto been mainlyextent. focusedExisting on the the State estimation is a common task in different engineersystems have been studied to a lesser extent. Existing results on LTV systems have been mainly focused on the ing fields where dynamic systems are described in statedesign of algorithms minimizing the state estimation State estimation is a common task in different engineerresults on LTV systems have been mainly focused on the ing fields where dynamic systems are described in state- design of algorithms minimizing the state estimation error error State estimation is a common taskare incontrol different ing fields where dynamic described in onalgorithms LTV systems have Darouach been focused onerror the design of minimizing the state estimation space form, for instance, instance, insystems automatic control for engineerfeedback covariance (Kitanidis, 1987; Zasadzinski, ing fields where dynamic systems are described in statestate- results design of algorithms minimizing themainly stateand estimation error space form, for in automatic for feedback covariance (Kitanidis, 1987; Darouach and Zasadzinski, ing fields where dynamic systems are described in statespace form, for instance, in automatic control for feedback design of algorithms minimizing the state estimation error covariance (Kitanidis, 1987; Darouach and Zasadzinski, control, in fault diagnosis for detecting abnormal state 1997; Hsieh, 2000; Gillijns and De Moor, 2007), ignoring space form, for instance, in automatic control for feedback 1987; Darouach Zasadzinski, control, in fault diagnosis for detecting abnormal state covariance 1997; Hsieh,(Kitanidis, 2000; Gillijns and De Moor,and 2007), ignoring space form, for in automatic control for feedback control, in diagnosis for abnormal state 1987; Darouach and Zasadzinski, 1997; Hsieh, 2000; Gillijns and De Moor, 2007), ignoring trajectories, andinstance, in chemical or detecting bioprocess engineering as covariance the important issue of the stability the state estimation control, in fault fault diagnosis for detecting abnormal state 1997; Hsieh,(Kitanidis, 2000; Gillijns and De of Moor, 2007), ignoring trajectories, and in chemical or bioprocess engineering as the important issue of the stability of the state estimation control, in fault diagnosis for detecting abnormal state trajectories, and in chemical or bioprocess engineering as 1997; Hsieh, 2000; Gillijns and De Moor, 2007), ignoring the important issue of the stability of the state estimation soft sensors for production monitoring. Among classical error equations. In (Darouach and Zasadzinski, 1997), the trajectories, and in chemical or bioprocess engineering as the important issue of the stability of the state estimation soft sensors for production monitoring. Among classical error equations. In (Darouach and Zasadzinski, 1997), the trajectories, and in chemical or bioprocess engineering as soft sensors for production monitoring. Among classical the important issue of the stability of the state estimation error equations. In (Darouach and Zasadzinski, 1997), the state estimation methods, the Kalman filter (Kalman, stability is for only, the soft sensors for production monitoring. classical error equations. In (Darouach and Zasadzinski, the state estimation methods, the Kalman Among filter (Kalman, stability is studied studied for LTI LTI systems systems only, though though1997), the algoalgosoft sensors for production monitoring. Among classical state estimation methods, the Kalman filter (Kalman, error equations. In (Darouach and Zasadzinski, 1997), the stability is studied for LTI systems only, though the algo1963) and the Luenberger observer (Luenberger, 1971) are rithm proposed in the paper is formulated for general LTV state and estimation methods,observer the Kalman filter (Kalman, is studied for paper LTI systems only, though the algo1963) the Luenberger (Luenberger, 1971) are stability rithm proposed in the is formulated for general LTV state estimation methods, the Kalman filter (Kalman, 1963) and the Luenberger observer (Luenberger, 1971) are stability is studied for LTI systems only, though the algorithm proposed in the paper is formulated for general LTV well known algorithms. systems. This lack of stability result is due to the difficulty 1963) and the Luenberger observer (Luenberger, 1971) are rithm proposed in the paper is formulated for general LTV well known algorithms. This lack of stability result is due to the difficulty 1963) and the Luenberger (Luenberger, 1971) are systems. well algorithms. rithm proposed in the paper is formulated for general LTV systems. This lack of stability result is to the difficulty for analyzing As aa matter transfer well known algorithms. systems. This LTV lack ofsystems. stability result is due due of to fact, the difficulty Moreknown recently, some studies studiesobserver are focused focused on state state estimation estimation for analyzing LTV systems. As matter of fact, transfer More recently, some are on well known algorithms. systems. This lack of stability result is due to the difficulty for analyzing LTV systems. As a matter of fact, transfer functions, as a widely used tool for LTI system analysis, analyzing systems. As afor matter of fact, analysis, transfer More studies focused in the therecently, presencesome of (unknown) (unknown) disturbances, alsoestimation known as as for functions, as aLTV widely used tool LTI system More recently, some studies are aredisturbances, focused on on state state estimation in presence of also known for analyzing systems. As afor matter of Nevertheless, fact, analysis, transfer functions, as aaLTV widely used tool LTI system are not available for general LTV systems. More recently, some studies are focused on state estimation functions, as widely used tool for LTI system analysis, in the presence of (unknown) disturbances, also known as unknown inputs, in order to develop algorithms robust to are not available for general LTV systems. Nevertheless, in the presence of (unknown) disturbances, also known as unknown inputs, in order to develop algorithms robust to functions, as a widely used tool for LTI system analysis, are not available for general LTV systems. Nevertheless, it is important to study LTV systems, including in the presence of (unknown) disturbances, also known as are not available for general LTV systems. Nevertheless, unknown inputs, in order to develop algorithms robust to disturbances or unknown unknown inputs. The earliest earliest works on this this is important to study LTV systems, including linear linear unknown inputs, in orderinputs. to develop algorithms robust to it disturbances or The works on are available general LTV systems. Nevertheless, it is important to study LTV systems, including linear parameter varying systems, since aa large of unknown inputs, in order to develop algorithms robust to it isnot important tofor(LPV) study LTV systems, linear disturbances or unknown inputs. The earliest works on this topic were mainly about linear time invariant (LTI) devarying (LPV) systems, since including large class class of disturbances or unknown works on this topic were mainly aboutinputs. linear The timeearliest invariant (LTI) de- parameter it is important to study LTV systems, including linear parameter varying (LPV) systems, since a large class of nonlinear systems can be addressed through LTV/LPV disturbances or unknown inputs. The earliest works on this parameter varying (LPV) systems, since a large class topic were mainly about linear time invariant (LTI) deterministic systems, for example, (Yang and Wilde, 1988; nonlinear systems can be addressed through LTV/LPV topic were mainly linear time invariant (LTI) de- parameter varying (LPV) systems, since a large class of terministic systems,about for example, (Yang and Wilde, 1988; of nonlinear systems can be addressed through LTV/LPV reformulation and approximation (T´ o th, 2010). topic were et mainly about linearand time invariant (LTI) de- nonlinear systems can be addressed through terministic systems, for (Yang and Wilde, 1988; Darouach et al., 1994; 1994; Chen and Patton, 1999). These reformulation and approximation (T´oth, 2010).LTV/LPV terministic systems, for example, example, (Yang and 1999). Wilde, 1988; Darouach al., Chen Patton, These nonlinear systems can be addressed through LTV/LPV reformulation and approximation (T´ o th, 2010). terministic systems, for example, (Yang and Wilde, 1988; reformulation and approximation (T´ o th, 2010). Darouach et al., 1994; Chen and Patton, 1999). These works aimed aimed at designing designing observers under 1999). some matrix matrix model corresponds to an LTI Darouach et al., 1994; Chen and Patton, These Because works at observers under some Because an an LTV LTV corresponds an LTI model model at at andmodel approximation (T´oto th, 2010). Darouach etensuring al., 1994; Chen and estimation Patton, These works aimed at observers under some Because an LTV an LTI at constraints, ensuring stable state estimation errormatrix equa- reformulation every aa naive solution LTV would works aimed at designing designing observers under 1999). some matrix Because aninstant, LTV model model corresponds to an systems LTI model model at constraints, stable state error equaevery time time instant, naivecorresponds solution for forto LTV systems would works aimed at designing observers under some matrix constraints, ensuring stable state estimation error equaBecause an LTV model corresponds to an LTI model at every time instant, a naive solution for LTV systems would tions, which are not affected by disturbances. To explore be to apply any method developed for LTI systems, by constraints, stable state estimation error equa- every time instant, a naive developed solution forfor LTV systems would tions, which ensuring are not affected by disturbances. To explore be to apply any method LTI systems, by constraints, ensuring stable state estimation error equations, which are not affected by disturbances. To explore every time instant, a naive solution for LTV systems would be to apply any method developed for LTI systems, by the degrees of freedom left by such matrix constraints, renewing the designed algorithm at every time instant. tions, which are not affected by disturbances. To explore be to apply any method developed for LTI systems, by the degrees of freedom left by such matrix constraints, renewing the designed algorithm at every time instant. tions, whichof are notmatrix affected by such disturbances. To explore the degrees freedom left by matrix to apply any method developed for LTI systems, by renewing the designed algorithm at every time instant. techniques ofof linear matrix inequalities (LMI) constraints, are used in in be However, the fact at time instant an LTV the degrees of freedom leftinequalities by such matrix constraints, renewing every time instant. techniques linear (LMI) are used However, the the designed fact that thatalgorithm at every every at time instant an LTV the degrees of freedom left by such matrix constraints, techniques of linear matrix inequalities (LMI) are used in renewing the designed algorithm at every time instant. However, the fact that at every time instant an LTV some works to minimize criterions based on H and L system is stable in the LTI sense does not ensure the ∞ and techniques matrix inequalities (LMI) used Lin22 However, factin that an LTV some worksoftolinear minimize criterions based on Hare system is the stable the at LTIevery sensetime doesinstant not ensure the ∞ techniques ofto linear matrix inequalities (LMI) used some works minimize criterions based on H and L the fact at every time instant anknown LTV system is stable in the LTI sense does not ensure the norms (Gao et al., Bezzaoucha et Some stability the whole LTV system. is well ∞ some based on 2017). Hare Lin22 However, system isof stable in that the LTI senseThis doesfact not ensure the ∞ and normsworks (Gao to et minimize al., 2016; 2016; criterions Bezzaoucha et al., al., 2017). Some stability of the whole LTV system. This fact is well known some minimize based on 2017). H L normsworks (Gao to ethave al., considered 2016; criterions Bezzaoucha et al., al., 2017). Some system isof stable in the LTI sense doesfact not ensure the stability of the whole LTV system. This fact is well known other studies LTI stochastic systems, by ∞ and 2 in the classical system theory. It is thus a non trivial task norms (Gao et al., 2016; Bezzaoucha et Some stability the whole LTV system. This is well known other studies have considered LTI stochastic systems, by in the classical system theory. It is thus a non trivial task norms (Gao ethave al., considered 2016; Bezzaoucha al., 2017). Some other LTI systems, by of theresults whole LTV system. This fact is well known in the classical system theory. It is thus aasystems non trivial task designing state estimators based on the variance to generalize developed for LTI to LTV other studies studies have considered LTI stochastic systems, by stability in the classical system theory. It is thus non trivial task designing state estimators based on stochastic theetminimum minimum variance generalize results developed for LTI systems to LTV other studies considered LTI stochastic systems, by to designing state estimators based on the variance in the classicalresults systemdeveloped theory. It for is thus non trivial task to generalize LTI to criterion et 1995; Keller et systems. designing statehave estimators the minimum minimum variance to generalize results developed for LTI asystems systems to LTV LTV systems. criterion (Darouach (Darouach et al., al., based 1995; on Keller et al., al., 1998). 1998). designing state estimators based on the minimum variance criterion (Darouach et al., 1995; Keller et al., 1998). to generalize results developed for LTI systems to LTV systems. criterion (Darouach et al., 1995; Keller et al., 1998). systems. The purpose The purpose of of this this paper paper is is to to propose propose aa numerically numerically effiefficriterion (Darouach et al., 1995; Keller et al., 1998). systems. The purpose purpose of for thisstate paper is to to propose propose adisturbance numericallyrejecefficient algorithm estimation with L. Zhang acknowledges the financial support partly by the NaThe of this paper is a numerically efficient algorithm for state estimation with disturbance rejec L. Zhang acknowledges the financial support partly by the NaThe purpose of for thisstate paper is to of propose adisturbance numerically efficient algorithm estimation with rejection, in the general framework LTV stochastic systems, L. Zhang cient algorithm for state estimation with disturbance rejectional Nature Science Foundation of China (11701040) and Innovaacknowledges the financial support partly by the Na tion, in the general framework of LTV stochastic systems, L. Zhang the financial support partly and by the National Natureacknowledges Science Foundation of China (11701040) Innovacient algorithm for state estimation with disturbance rejection, in the general framework of LTV stochastic systems, with easily checkable conditions guaranteeing the stability tion, in the checkable general framework LTV stochastic tion of BUPT for tional Natureacknowledges Science Foundation of (500417024). China (11701040) and InnovaL. Foundation Zhang theYouth financial support partly and by the Nawith easily conditionsofguaranteeing the systems, stability tional Nature Science Foundation of China (11701040) Innovation Foundation of BUPT for Youth (500417024). tion, in the checkable general framework LTV stochastic with easily easily checkable conditionsofguaranteeing guaranteeing the systems, stability tion of for tional Nature Science Foundation of (500417024). China (11701040) and Innovawith conditions the stability tion Foundation Foundation of BUPT BUPT for Youth Youth (500417024). easily checkable conditions guaranteeing the stability 2405-8963 © 2018, Federation of Automatic Control) with Hosting by Elsevier Ltd. All rights reserved. tion Foundation of IFAC BUPT(International for Youth (500417024). ∗ ∗ ∗ ∗
Copyright © 2018 IFAC 55 Copyright © under 2018 IFAC 55 Control. Peer review responsibility of International Federation of Automatic Copyright © 55 Copyright © 2018 2018 IFAC IFAC 55 10.1016/j.ifacol.2018.09.090 Copyright © 2018 IFAC 55
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of the algorithm. Notably, this algorithm is applicable to LPV systems, which are viewed as a particular class of LPV systems. The method for designing this algorithm is conceptually simple: disturbances are rejected from the state equation by output injection, then the classical Kalman filter is applied to the new state-space model obtained after the output injection. To the best of our knowledge, this is the first result for LTV systems with disturbance rejection ensuring the stability of the state estimation error equation. This paper is organized as follows. In Section 2 the considered problem is formulated. In Section 3 the output injection for disturbance rejection is presented. In Setion 4 the Kalman filter is applied to the system after output injection. Numerical examples are presented in Section 5. Finally, conclusions are drawn in Section 6.
Disturbance subspace assumptions. (iv) q ≤ m (no more disturbances than output sensors). (v) There exists a positive constant γ such that, for all k = 0, 1, 2 . . . , the smallest singular value of the matrix product Ck+1 Ek is not smaller than γ. These assumptions are necessary for numerically reliable rejection of disturbances. Notice that Assumptions (iv) and (v) imply that Ck+1 Ek has a full column rank. In the classical Kalman filter theory, the stability of the Kalman filter mainly relies on observability and controllability assumptions. Similar assumptions will be formulated in the next section, after the disturbance rejection by means of output injection.
2. PROBLEM FORMULATION
The purpose this section is to reject the disturbance term Ek dk from the state equation by means of output injection, so that the Kalman filter can be applied without being affected by the disturbance. In (Kitanidis, 1987), linear filters in the form of x ˆk+1|k+1 = Ak x ˆk|k + Lk+1 (yk+1 − Ck+1 Ak x ˆk|k ) (2) were considered, neglecting the known input term Bk uk , which is not essential in linear filter design problems. The filter gain Lk+1 was determined by minimizing the state estimation error covariance, under the constraint (3) Lk+1 Ck+1 Ek − Ek = 0. See the equation (8) in (Kitanidis, 1987), formulated with different notations. This constraint ensures that the state estimation error is not affected by the disturbance term Ek dk . The solution to this constrained optimization problem leads to a filter gain that is considerably more sophisticated than the gain of the Kalman filter, and there is no obvious way to analyze the stability of the resulting filter. The main idea of the present paper is to reject the disturbance term before the filter design. In the first step, the disturbance rejection is achieved with an output injection satisfying a condition similar to constraint (3). Then the filter design is made as if there was no disturbance term in the considered problem, simplifying the stability analysis of the resulting filter. Let Gk ∈ Rn×m be a bounded matrix sequence to be specified later. The following equation holds due to (1b): (4) 0 = Gk+1 (yk+1 − Ck+1 xk+1 − vk+1 ). Add each side of this equation to the corresponding side of (1a), then xk+1 = Ak xk + Bk uk + Ek dk + wk + Gk+1 (yk+1 − Ck+1 xk+1 − vk+1 ). (5) For the purpose of disturbance rejection in the following steps, it is important that this output injection is made with the output equation at instant k + 1, instead of k. At the right hand side of equation (5), substitute xk+1 with (1a), then, after some rearrangement (with In denoting the n × n identity matrix), xk+1 = (In − Gk+1 Ck+1 )Ak xk + (In − Gk+1 Ck+1 )Bk uk + (In − Gk+1 Ck+1 )Ek dk + Gk+1 yk+1 + (In − Gk+1 Ck+1 )wk − Gk+1 vk+1 . (6)
3. OUTPUT INJECTION FOR DISTURBANCE REJECTION
Consider LTV stochastic systems in the form of xk+1 = Ak xk + Bk uk + Ek dk + wk (1a) (1b) y k = Ck x k + v k where xk ∈ Rn is the state, yk ∈ Rm the output, uk ∈ Rp the (known) input, dk ∈ Rq the disturbance (or unknown input), wk ∈ Rn the state noise, vk ∈ Rm the output noise, and Ak , Bk , Ck , Ek are known matrices of appropriate sizes. The purpose of this paper is to design an algorithm for the estimation of the state xk from the known input uk , the output yk and the known matrices Ak , Bk , Ck , Ek . While the noises wk , vk are random variables, the disturbance dk is completely arbitrary, notably, no probability distribution, nor upper bound of dk , is assumed. The fact that Ek is known, but dk is completely unknown, means that, at each instant k, a completely arbitrary disturbance can only affect a known subspace of the statespace, and this subspace may vary with k. The arbitrary character of dk means that there is a void subspace in the state equation. Typically, Ek is filled with 0’s and 1’s. For example, at the instant k, if Ek = [1, 0, . . . , 0]T , then the first component of the state equation (1a) is void. It may appear impossible to estimate the state xk when part of the state equation is missing. Of course, compared to the classical case where the term Ek dk did not exist, here some extra assumptions are required so that state estimation remains feasible, like the last two assumptions listed below. Basic assumptions. (i) Ak , Bk , Ck , Ek are bounded matrix sequences. (ii) The initial state x0 ∈ Rn is a random vector following the Gaussian distribution N (¯ x0 , P0 ), with a mean vector x ¯0 and a positive definite covariance matrix P0 . (iii) wk and vk are zero mean white Gaussian noises independent of each other and of x0 , with bounded covariance matrices E(wk wkT ) = Qk and E(vk vkT ) = Rk . These assumptions are usually made in the classical LTV system Kalman filter theory, apart from the involved matrix Ek that did not exist in the classical case. 56
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By treating Gk+1 yk+1 as a known input term, the classical Kalman filter can be applied to the state-space model (16), yielding a state estimator for the system originally described by (1). It is indeed an application of the classical Kalman filter as originally published in (Kalman, 1963), as no disturbance term appears in the state-space model (16). This is an indirect way for addressing the initial state estimation problem formulated with (1). For the sake of simplicity, let us neglect the correlation between w ¯k and vk . Then the classical Kalman filter applied to (16) is as follows. After the initialization ¯0 (17a) x ˆ0|0 = x (17b) P0|0 = P0 , the state estimate x ˆk|k and its covariance matrix Pk|k are then recursively computed, for k = 0, 1, 2, . . . , ¯k (18a) Pk+1|k = A¯k Pk|k A¯Tk + Q
If the matrix sequence Gk is chosen such that, at every time instant k, (7) (In − Gk+1 Ck+1 )Ek = 0, then equation (6) becomes ¯k uk + Gk+1 yk+1 + w ¯k , (8) xk+1 = A¯k xk + B with (9) A¯k (In − Gk+1 Ck+1 )Ak ¯ (10) Bk (In − Gk+1 Ck+1 )Bk (11) w ¯k (In − Gk+1 Ck+1 )wk − Gk+1 vk+1 . Now the disturbance term Ek dk has disappeared from equation (8).
Remark 1. Condition (7) that should be satisfied by Gk is the same as (3), which was imposed to the Kitanidis filter gain Lk . However, here more degrees of freedom are left to the choice of Gk , which is not determined by the minimum covariance criterion, unlike the Kitanidis filter gain Lk . The covariance matrix of the new state noise w ¯k is ¯ k (In − Gk+1 Ck+1 )Qk (In − Gk+1 Ck+1 )T Q + Gk+1 Rk+1 GTk+1 .
57
T + Rk+1 Σk+1 = Ck+1 Pk+1|k Ck+1
(18b)
T Pk+1|k Ck+1 Σ−1 k+1
(18c) (18d) (18e) (18f) (18g)
Kk+1 = Pk+1|k+1 = (In − Kk+1 Ck+1 )Pk+1|k ¯k uk + Gk+1 yk+1 x ˆk+1|k = A¯k x ˆk|k + B ˆk+1|k y˜k+1 = yk+1 − Ck+1 x ˆk+1|k + Kk+1 y˜k+1 . x ˆk+1|k+1 = x
(12)
Rewrite equation (7) as (13) Gk+1 Ck+1 Ek = Ek , which should be solved for the unknown matrix Gk+1 . Due to Assumption (iv), the matrix product Ck+1 Ek has no more columns than rows, thus in most situations, solutions of Gk+1 exist. A sufficient and necessary condition for the existence of solutions is that each row of Ek is a linear combination of the rows of the matrix product Ck+1 Ek . This condition is ensured by assumption (v). Given an m × n matrix M , another n × m matrix M g is a generalized inverse of M if M M g M = M . With this notation, the solution (14) Gk+1 = Ek (Ck+1 Ek )g with any generalized inverse (Ck+1 Ek )g satisfies equation (13). The particular solution Gk+1 = Ek [(Ck+1 Ek )T (Ck+1 Ek )]−1 (Ck+1 Ek )T (15) will be adopted in the remaining part of this paper. The boundedness of this solution Gk+1 is then ensured by Assumptions (i) and (v). Consequently, with this particular ¯ k defined respectively sequence Gk , the matrices A¯k and Q in (9) and (12) are also bounded. Now it is time to formulate the last assumptions ensuring the stability of the state estimator proposed in this paper.
In order to ensure the stability of this Kalman filter, two more assumptions will be formulated below, in addition to the assumptions already made in Section 2. These assumption are slightly different from those assumed in the classical Kalman filter theory, which is not suitable for the particular case considered here, as explained later in the two remarks following the statements of Properties 1 and 2. Observability and controllability assumptions. (vi) The matrix sequence pair [A¯k , Ck ] is uniformly completely observable. 1 ¯ 2 ] is (vii) The matrix sequence pair [A¯k (In−Kk Ck ), Q k 1 ¯ 2 is the uniformly completely controllable, where Q k ¯ matrix square root of Qk . The uniform complete observability and controllability are defined with the aid of Gramian matrices. See (Kalman, 1963; Jazwinski, 1970). Notice that these observability and controllability assumptions involve the matrix A¯k depending on Gk , which in turn depends on Ek . Like in the classical Kalman filter theory, if the involved matrices are known in advance (typically constant or periodical), then these assumptions can be checked in advance, otherwise they have to be checked in real time (e.g. for LPV systems). Property 1. (Boundedness). Under Assumptions (i)-(vii), the recursively computed matrices Pk|k and Pk+1|k are bounded, so are the innovation covariance Σk and the Kalman gain Kk .
4. KALMAN FILTER, STABILITY AND OPTIMALITY Consider the state-space model composed of the state equation (8) and the output equation (1b), which are grouped below for ease of later references: ¯k uk + Gk+1 yk+1 + w ¯k , (16a) xk+1 = A¯k xk + B (16b) y k = Ck x k + v k . The state xk , the input uk and the output yk of this new model are the same as in the original system model (1). In other words, these two models represent the same system. In this sense, the two models are equivalent.
This property ensures the boundedness of all the variables involved in the recursive computations of the Kalman filter except the states estimates. As Gaussian noises are not bounded, the state estimation errors are (in principle) not 57
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5. NUMERICAL EXAMPLES
bounded, but the error covariance matrices are bounded. Obviously, boundedness is crucial for any recursive algorithm in real time applications. Property 2. (Stability). Under Assumptions (i)-(vii), the error dynamics of the state estimate x ˆk|k of the Kalman filter (18) is exponentially stable. Remark 2. Recall that the boundedness of the sequence Gk is ensured by Assumptions (i) and (v). When applying the Kalman filter to (16), Gk+1 yk+1 is treated as a known bounded input term. Then it may appear that Properties 1 and 2 can be ensured by the classical Kalman filter theory (Kalman, 1963; Jazwinski, 1970). In fact, one of the important conditions required by the classical Kalman filter theory is not satisfied in the present case: when applied to system (16), the classical theory would assume that the matrix A¯k is invertible for all time instant k, whereas this matrix defined in (9) is always singular, because of equality (7). Fortunately, a new proof of the Kalman filter stability, ensuring Properties 1 and 2, has been worked out, without requiring invertible A¯k . As this complete remake of the Kalman filter stability proof is much longer than the present paper, it will be reported elsewhere. Remark 3. Despite the fact that the correlation between w ¯k and vk has been neglected by the Kalman filter applied to system (16), Properties 1 and 2 hold for the following reasons. The computations of Pk+1|k , Pk+1|k+1 , Σk+1 and Kk+1 ¯ k and Rk+1 through are fully determined by A¯k , Ck+1 , Q (18a)-(18d), hence Property 1 is a consequence of the ¯ k , Rk , no matter w assumptions made on A¯k , Ck , Q ¯k and vk are correlated or not. Let the state estimation error be denoted by x ˜k|k xk − x ˆk|k ,
In order to illustrate the state estimator proposed in this paper, the results of its application to two examples are presented below. 5.1 LTI example Let us borrow the example presented in Section 3.2.2, page 76, of (Chen and Patton, 1999). The example was originally in continuous time, with −1 −1 1 0 1 0 0 , Ec = 0 . Ac = −1 0 0 , Cc = 0 0 1 0 0 −1 −1 To apply the discrete time algorithm, this continuous time system is discretized with the sample time T = 1, then 0.1262 0.5335 0 Ac T 0 ≈ −0.5335 0.6597 (22) A=e 0.2417 −0.5335 0.3679 1 0 0 (23) C = Cc = 0 0 1 T
E = A−1 c (A − In )Ec ≈ [−0.5335 0.3403 −0.0986] . The noise covariance matrices are Q = 0.1I3 , R = 0.05I2 . The disturbance is simulated with a deterministic part and a random part as dk = 5 sin(0.1k) + ωk (24) where ωk is an independent and identically distributed random sequence following the Gaussian distribution N (0, 1). For the system model after output injection, 0.9670 0.1787 T −1 T G = E[(CE) (CE)] (CE) ≈ −0.6168 −0.1140 0.1787 0.0330 −0.0390 0.1130 −0.0658 A¯ = (In − GC)A ≈ −0.4281 0.9279 0.0419 0.2111 −0.6112 0.3557
(19)
then the error dynamics equation is ˜k|k x ˜k+1|k+1 = (In − Kk+1 Ck+1 )A¯k x ¯k − Kk+1 vk+1 . + (In − Kk+1 Ck+1 )w
¯ = (In − GC)Q(In − GC)T + GRGT Q 0.0517 −0.0308 −0.0089 ≈ −0.0308 0.1590 −0.0057 . −0.0089 −0.0057 0.0983
(20) (21)
The initial state x ¯0 = 0 and the initial state estimate covariance P0 = I3 (the 3 × 3 identity matrix). The simulated state and the estimated state are displayed in Figure 1. The state vector is correctly estimated, despite the presence of the disturbance.
Like in (Kalman, 1963; Jazwinski, 1970), the stability of the error dynamics concerns only the deterministic part of this error equation, fully characterized by the matrix (In − Kk+1 Ck+1 )A¯k . As the computation of Kk+1 is fully ¯ k and Rk+1 through (18a)determined by A¯k , Ck+1 , Q (18d), the stability of the error dynamics is fully ensured ¯ k , Rk . Therefore, by the assumptions made on A¯k , Ck , Q Property 2 is not affected by the correlation between w ¯k and vk .
5.2 LTV example The previous example is now modified so that the system becomes time varying. A sinusoid term is added to the central entry of the matrix A, and a triangular wave is added to the second component of the vector E. More precisely, 0.1262 0.5335 0 0 (25) Ak = −0.5335 0.6597 + sin(k) 0.2417 −0.5335 0.3679 T −0.5335 Ek = 0.3403 + ∆(k) (26) −0.0986
Another well known property of the Kalman filter is its optimality. As the correlation between the noises w ¯k and vk was neglected when formulating the Kalman filter (18), the optimality does not exactly hold, but the result should not be far from being optimal. The correlation between the state noise and the output noise is often ignored in Kalman filter applications. In the present case, it is not trivial to take into account the correlation caused by output injection. Further investigations are necessary in order to improve the optimality of the proposed algorithm. 58
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been established in the original work by Kalman. For LTV systems with disturbance rejection, to our knowledge, no result ensuring the stability of state estimators has been reported, hence we believe that the result presented in this paper represents an important progress in studies on this topic.
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Bezzaoucha, S., Voos, H., and Darouach, M. (2017). A new polytopic approach for the unknown input functional observer design. International Journal of Control, online version, 1–20. doi:10.1080/00207179.2017.1288299. Chen, J. and Patton, R. (1999). Robust Model-Based Fault Diagnosis for Dynamic Systems. Kluwer Academic Publishers, London. Darouach, M. and Zasadzinski, M. (1997). Unbiased minimum variance estimation for systems with unknown exogenous inputs. Automatica, 33(4), 717–719. Darouach, M., Zasadzinski, M., Bassong Onana, A., and Nowakowski, S. (1995). Kalman filtering with unknown inputs via optimal state estimation of singular systems. International Journal of Systems Science, 26(10), 2015– 2028. Darouach, M., Zasadzinski, M., and Xu., S.J. (1994). Fullorder observers for linear systems with unknown inputs. IEEE Transactions on Automatic Control, 39(3), 606– 609. Gao, N., Darouach, M., Voos, H., and Alma, M. (2016). New unified h-infinity dynamic observer design for linear systems with unknown inputs. Automatica, 65, 43–52. Gillijns, S. and De Moor, B. (2007). Unbiased minimumvariance input and state estimation for linear discretetime systems with direct feedthrough. Automatica, 43, 934–937. Hsieh, C.S. (2000). Robust two-stage kalman filters for systems with unknown inputs. IEEE Transactions on Automatic Control, 45(12), 2374–2378. Jazwinski, A.H. (1970). Stochastic Processes and Filtering Theory, volume 64 of Mathematics in Science and Engineering. Academic Press, New York. Kalman, R.E. (1963). New methods in Wiener filtering theory. In J.L. Bogdanoff and F. Kozin (eds.), Proceedings of the First Symposium on Engineering Applications of Random Function Theory and Probability. John Wiley & Sons, New York. Keller, J.Y., Darouach, M., and Caramelle, L. (1998). Kalman filter with unknown inputs and robust twostage filter. International Journal of Systems Science, 29(1), 41–47. Kitanidis, P.K. (1987). Unbiased minimum variance linear state estimation. Automatica, 23(6), 775–778. Luenberger, D. (1971). An introduction to observers. IEEE Transactions on Automatic Control, 16(6), 596– 602. T´oth, R. (2010). Modeling and Identification of Linear Parameter-Varying Systems. Springer. Yang, F. and Wilde, R. (1988). Observers for linear systems with unknown inputs. IEEE Transactions on Automatic Control, 33(7), 677–681.
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Fig. 2. LTV (time varying) system example. The 3 components of the simulated state vector and their estimates are plotted. with the triangular wave x 1 x + − 1, ∆(k) = 4 − 10 10 2 where · denotes the floor function, i.e.. for any x ∈ R, x is the largest integer less than or equal to x. The two noise covariance matrices are multiplied by 10, because the signal magnitudes become larger due to the added time varying terms, resulting in Q = I3 , R = 0.5I2 . The other data are the same as in the previous example. The simulated state and the estimated state are displayed in Figure 2. Again the state vector is correctly estimated, despite the presence of the disturbance. 6. CONCLUSION The stability of state estimators is of primary importance, as demonstrated by classical studies on LTI system state observers. For LTV systems, the stability of state estimators is less often discussed, including in the case of the Kalman filter, despite the fact that its stability has 59
ScienceDirect
IFAC PapersOnLine 51-15 (2018) 55–59
State State Estimation Estimation for for Stochastic Stochastic Time Time Varying Varying State Estimation for Stochastic Time Varying Systems with Disturbance Rejection State Estimation for Stochastic Time Varying Systems with Disturbance Rejection Systems with Disturbance Rejection ∗ ∗∗ Systems with Disturbance Rejection Qinghua Qinghua Zhang Zhang ∗ Liangquan Liangquan Zhang. Zhang. ∗∗
∗∗ Qinghua Zhang Zhang ∗∗ Liangquan Liangquan Zhang. Zhang. ∗∗ Qinghua ∗ ∗∗ Qinghua Zhang Liangquan Zhang. Inria, Inria, IFSTTAR, IFSTTAR, Universit´ Universit´ee de de Rennes, Rennes, Campus Campus de de Beaulieu, Beaulieu, Inria, IFSTTAR, Universit´ ee de Rennes, Campus de 35042 Rennes Cedex, France. Email: [email protected] Inria, IFSTTAR, Universit´ de Rennes, Campus de Beaulieu, Beaulieu, 35042 Rennes Cedex, France. Email: [email protected] ∗ ∗∗ Inria, IFSTTAR, Universit´ e de Rennes, Campus de Beaulieu, 35042 Rennes Cedex, France. Email: [email protected] of Science, Beijing University of Posts and ∗∗ School 35042 Rennes Cedex, France. Email: [email protected] School of Science, Beijing University of Posts and ∗∗ ∗∗ School 35042 Rennes France. Email: [email protected] ofCedex, Science, Beijing University ofChina. Posts and and Telecommunications, Beijing 100876, School of Science, Beijing University of Posts Telecommunications, Beijing 100876, China. ∗∗ School of Science, Beijing University Posts and Telecommunications, Beijing 100876,ofChina. China. Email: [email protected] Telecommunications, Beijing 100876, Email: [email protected] Telecommunications, Beijing 100876, China. Email: Email: [email protected] [email protected] Email: [email protected] Abstract: State State estimation estimation in in the the presence presence of of unknown unknown disturbances disturbances is is useful useful for for the the design design of of Abstract: Abstract: State in unknown disturbances useful for the design robust systems systems inestimation different engineering engineering fields.of Most results available on onis this topic are restricted Abstract: State estimation in the the presence presence ofMost unknown disturbances isthis useful for are the restricted design of of robust in different fields. results available topic Abstract: State in systems, the presence ofMost unknown disturbances isthis useful for are thehave design of robust systems inestimation different engineering fields. Most results available on this topic are restricted to linear linear time invariant (LTI) systems, whereas linear time varying on (LTV) systems have been robust systems in different engineering fields. results available topic restricted to time invariant (LTI) whereas linear time varying (LTV) systems been robust systems in different engineering fields. Most results available on this topic are restricted to linear time invariant (LTI) systems, whereas linear time varying (LTV) systems have been studied to a lesser extent. Existing results on LTV systems are mainly based on the minimization to linear time invariant (LTI) systems, whereas linear time varying (LTV) systems have been studied to a lesser extent. Existing results on LTV systems are mainly based on the minimization to linear invariant (LTI) systems, whereas linear time (LTV) systems been studied to a lesser extent. Existing results on systems are mainly based on the of the the state estimation error covariance, ignoring the important issue of of the stability stability ofhave the state state studied totime a estimation lesser extent. Existing resultsignoring on LTV LTVthe systems arevarying mainly based on the minimization minimization of state error covariance, important issue the of the studied to a lesser extent. Existing results on LTV systems are mainly based on the minimization of the state estimation error covariance, ignoring the important issue of the stability of the state estimation error dynamics, which has been a main focus of the studies in the LTI case. The of the state error estimation error which covariance, ignoring the important issue of theinstability the state estimation dynamics, has been a main focus of the studies the LTIofcase. The of the state estimation error covariance, ignoring the important issue of the stability of the state estimation error dynamics, which has been a main focus of the studies in the LTI case. The purpose of this paper is to propose a numerically efficient algorithm for state estimation with estimation error dynamics, which has been a main focus of the studies in the LTI case. The purpose of this paper is to propose a numerically efficient algorithm for state estimation with estimation dynamics, which has been a main focus of the studies in theincluding LTI case.linear The purpose of this paper is propose a efficient algorithm for estimation with disturbance rejection, into the general framework of LTV stochastic stochastic systems, including linear purpose of error this paper isin tothe propose a numerically numerically efficient algorithm for state state estimation with disturbance rejection, general framework of LTV systems, purpose of this paper is to propose a numerically efficient algorithm for state estimation with disturbance rejection, in the general framework of LTV stochastic systems, including linear parameter varying (LPV) systems, with easily checkable conditions guaranteeing the stability disturbance rejection, in the general framework of LTV conditions stochastic guaranteeing systems, including linear parameter varying (LPV) systems, with easily checkable the stability disturbance rejection, in the general framework of LTV stochastic systems, including linear parameter varying (LPV) systems, with easily conditions guaranteeing the of the the algorithm. algorithm. The design method is conceptually simple: disturbance is first first rejected from parameter varying (LPV) systems, with easily checkable checkable conditions guaranteeing the stability stability of The design method is conceptually simple: disturbance is rejected from parameter varying (LPV) systems, with easily checkable conditions guaranteeing the stability of the algorithm. The design method is conceptually simple: disturbance is first rejected from the state equation by appropriate output injection, then the Kalman filter is applied to the of the algorithm. The design method is conceptually simple: disturbance is first rejected from the state equation by appropriate output injection, then the Kalman filter is applied to the of the algorithm. The design method is conceptually simple: disturbance is first rejected from the state equation by appropriate output injection, then the Kalman filter is applied to the resulting state-space model after the output injection. the state equation by appropriate output injection, then the Kalman filter is applied to the resulting state-space model after the output injection. the state state-space equation bymodel appropriate resulting after output injection. resulting state-space model after the theoutput outputinjection, injection.then the Kalman filter is applied to the © 2018, IFAC (International Federation ofoutput Automatic Control) Hosting by Elsevier Ltd. All rights reserved. resulting state-space model after the injection. Keywords: disturbance disturbance rejection, rejection, state state estimation, estimation, LTV/LPV system, system, Kalman Kalman filter. filter. Keywords: LTV/LPV Keywords: disturbance disturbance rejection, rejection, state state estimation, estimation, LTV/LPV LTV/LPV system, system, Kalman Kalman filter. filter. Keywords: Keywords: disturbance rejection, state estimation, LTV/LPV system, Kalman filter. 1. INTRODUCTION Compared 1. INTRODUCTION Compared to to LTI LTI systems, systems, linear linear time time varying varying (LTV) (LTV) 1. Compared to LTI linear time varying (LTV) systems have studied to aa lesser 1. INTRODUCTION INTRODUCTION Compared to been LTI systems, systems, linear time extent. varyingExisting (LTV) systems have been studied to lesser extent. Existing 1. aINTRODUCTION Compared to LTI systems, linear time varying (LTV) systems have been studied to a lesser extent. Existing State estimation is common task in different engineerresults on LTV systems have been mainly focused on studied a lesser State estimation is a common task in different engineer- systems results onhave LTVbeen systems haveto been mainlyextent. focusedExisting on the the State estimation is a common task in different engineersystems have been studied to a lesser extent. Existing results on LTV systems have been mainly focused on the ing fields where dynamic systems are described in statedesign of algorithms minimizing the state estimation State estimation is a common task in different engineerresults on LTV systems have been mainly focused on the ing fields where dynamic systems are described in state- design of algorithms minimizing the state estimation error error State estimation is a common taskare incontrol different ing fields where dynamic described in onalgorithms LTV systems have Darouach been focused onerror the design of minimizing the state estimation space form, for instance, instance, insystems automatic control for engineerfeedback covariance (Kitanidis, 1987; Zasadzinski, ing fields where dynamic systems are described in statestate- results design of algorithms minimizing themainly stateand estimation error space form, for in automatic for feedback covariance (Kitanidis, 1987; Darouach and Zasadzinski, ing fields where dynamic systems are described in statespace form, for instance, in automatic control for feedback design of algorithms minimizing the state estimation error covariance (Kitanidis, 1987; Darouach and Zasadzinski, control, in fault diagnosis for detecting abnormal state 1997; Hsieh, 2000; Gillijns and De Moor, 2007), ignoring space form, for instance, in automatic control for feedback 1987; Darouach Zasadzinski, control, in fault diagnosis for detecting abnormal state covariance 1997; Hsieh,(Kitanidis, 2000; Gillijns and De Moor,and 2007), ignoring space form, for in automatic control for feedback control, in diagnosis for abnormal state 1987; Darouach and Zasadzinski, 1997; Hsieh, 2000; Gillijns and De Moor, 2007), ignoring trajectories, andinstance, in chemical or detecting bioprocess engineering as covariance the important issue of the stability the state estimation control, in fault fault diagnosis for detecting abnormal state 1997; Hsieh,(Kitanidis, 2000; Gillijns and De of Moor, 2007), ignoring trajectories, and in chemical or bioprocess engineering as the important issue of the stability of the state estimation control, in fault diagnosis for detecting abnormal state trajectories, and in chemical or bioprocess engineering as 1997; Hsieh, 2000; Gillijns and De Moor, 2007), ignoring the important issue of the stability of the state estimation soft sensors for production monitoring. Among classical error equations. In (Darouach and Zasadzinski, 1997), the trajectories, and in chemical or bioprocess engineering as the important issue of the stability of the state estimation soft sensors for production monitoring. Among classical error equations. In (Darouach and Zasadzinski, 1997), the trajectories, and in chemical or bioprocess engineering as soft sensors for production monitoring. Among classical the important issue of the stability of the state estimation error equations. In (Darouach and Zasadzinski, 1997), the state estimation methods, the Kalman filter (Kalman, stability is for only, the soft sensors for production monitoring. classical error equations. In (Darouach and Zasadzinski, the state estimation methods, the Kalman Among filter (Kalman, stability is studied studied for LTI LTI systems systems only, though though1997), the algoalgosoft sensors for production monitoring. Among classical state estimation methods, the Kalman filter (Kalman, error equations. In (Darouach and Zasadzinski, 1997), the stability is studied for LTI systems only, though the algo1963) and the Luenberger observer (Luenberger, 1971) are rithm proposed in the paper is formulated for general LTV state and estimation methods,observer the Kalman filter (Kalman, is studied for paper LTI systems only, though the algo1963) the Luenberger (Luenberger, 1971) are stability rithm proposed in the is formulated for general LTV state estimation methods, the Kalman filter (Kalman, 1963) and the Luenberger observer (Luenberger, 1971) are stability is studied for LTI systems only, though the algorithm proposed in the paper is formulated for general LTV well known algorithms. systems. This lack of stability result is due to the difficulty 1963) and the Luenberger observer (Luenberger, 1971) are rithm proposed in the paper is formulated for general LTV well known algorithms. This lack of stability result is due to the difficulty 1963) and the Luenberger (Luenberger, 1971) are systems. well algorithms. rithm proposed in the paper is formulated for general LTV systems. This lack of stability result is to the difficulty for analyzing As aa matter transfer well known algorithms. systems. This LTV lack ofsystems. stability result is due due of to fact, the difficulty Moreknown recently, some studies studiesobserver are focused focused on state state estimation estimation for analyzing LTV systems. As matter of fact, transfer More recently, some are on well known algorithms. systems. This lack of stability result is due to the difficulty for analyzing LTV systems. As a matter of fact, transfer functions, as a widely used tool for LTI system analysis, analyzing systems. As afor matter of fact, analysis, transfer More studies focused in the therecently, presencesome of (unknown) (unknown) disturbances, alsoestimation known as as for functions, as aLTV widely used tool LTI system More recently, some studies are aredisturbances, focused on on state state estimation in presence of also known for analyzing systems. As afor matter of Nevertheless, fact, analysis, transfer functions, as aaLTV widely used tool LTI system are not available for general LTV systems. More recently, some studies are focused on state estimation functions, as widely used tool for LTI system analysis, in the presence of (unknown) disturbances, also known as unknown inputs, in order to develop algorithms robust to are not available for general LTV systems. Nevertheless, in the presence of (unknown) disturbances, also known as unknown inputs, in order to develop algorithms robust to functions, as a widely used tool for LTI system analysis, are not available for general LTV systems. Nevertheless, it is important to study LTV systems, including in the presence of (unknown) disturbances, also known as are not available for general LTV systems. Nevertheless, unknown inputs, in order to develop algorithms robust to disturbances or unknown unknown inputs. The earliest earliest works on this this is important to study LTV systems, including linear linear unknown inputs, in orderinputs. to develop algorithms robust to it disturbances or The works on are available general LTV systems. Nevertheless, it is important to study LTV systems, including linear parameter varying systems, since aa large of unknown inputs, in order to develop algorithms robust to it isnot important tofor(LPV) study LTV systems, linear disturbances or unknown inputs. The earliest works on this topic were mainly about linear time invariant (LTI) devarying (LPV) systems, since including large class class of disturbances or unknown works on this topic were mainly aboutinputs. linear The timeearliest invariant (LTI) de- parameter it is important to study LTV systems, including linear parameter varying (LPV) systems, since a large class of nonlinear systems can be addressed through LTV/LPV disturbances or unknown inputs. The earliest works on this parameter varying (LPV) systems, since a large class topic were mainly about linear time invariant (LTI) deterministic systems, for example, (Yang and Wilde, 1988; nonlinear systems can be addressed through LTV/LPV topic were mainly linear time invariant (LTI) de- parameter varying (LPV) systems, since a large class of terministic systems,about for example, (Yang and Wilde, 1988; of nonlinear systems can be addressed through LTV/LPV reformulation and approximation (T´ o th, 2010). topic were et mainly about linearand time invariant (LTI) de- nonlinear systems can be addressed through terministic systems, for (Yang and Wilde, 1988; Darouach et al., 1994; 1994; Chen and Patton, 1999). These reformulation and approximation (T´oth, 2010).LTV/LPV terministic systems, for example, example, (Yang and 1999). Wilde, 1988; Darouach al., Chen Patton, These nonlinear systems can be addressed through LTV/LPV reformulation and approximation (T´ o th, 2010). terministic systems, for example, (Yang and Wilde, 1988; reformulation and approximation (T´ o th, 2010). Darouach et al., 1994; Chen and Patton, 1999). These works aimed aimed at designing designing observers under 1999). some matrix matrix model corresponds to an LTI Darouach et al., 1994; Chen and Patton, These Because works at observers under some Because an an LTV LTV corresponds an LTI model model at at andmodel approximation (T´oto th, 2010). Darouach etensuring al., 1994; Chen and estimation Patton, These works aimed at observers under some Because an LTV an LTI at constraints, ensuring stable state estimation errormatrix equa- reformulation every aa naive solution LTV would works aimed at designing designing observers under 1999). some matrix Because aninstant, LTV model model corresponds to an systems LTI model model at constraints, stable state error equaevery time time instant, naivecorresponds solution for forto LTV systems would works aimed at designing observers under some matrix constraints, ensuring stable state estimation error equaBecause an LTV model corresponds to an LTI model at every time instant, a naive solution for LTV systems would tions, which are not affected by disturbances. To explore be to apply any method developed for LTI systems, by constraints, stable state estimation error equa- every time instant, a naive developed solution forfor LTV systems would tions, which ensuring are not affected by disturbances. To explore be to apply any method LTI systems, by constraints, ensuring stable state estimation error equations, which are not affected by disturbances. To explore every time instant, a naive solution for LTV systems would be to apply any method developed for LTI systems, by the degrees of freedom left by such matrix constraints, renewing the designed algorithm at every time instant. tions, which are not affected by disturbances. To explore be to apply any method developed for LTI systems, by the degrees of freedom left by such matrix constraints, renewing the designed algorithm at every time instant. tions, whichof are notmatrix affected by such disturbances. To explore the degrees freedom left by matrix to apply any method developed for LTI systems, by renewing the designed algorithm at every time instant. techniques ofof linear matrix inequalities (LMI) constraints, are used in in be However, the fact at time instant an LTV the degrees of freedom leftinequalities by such matrix constraints, renewing every time instant. techniques linear (LMI) are used However, the the designed fact that thatalgorithm at every every at time instant an LTV the degrees of freedom left by such matrix constraints, techniques of linear matrix inequalities (LMI) are used in renewing the designed algorithm at every time instant. However, the fact that at every time instant an LTV some works to minimize criterions based on H and L system is stable in the LTI sense does not ensure the ∞ and techniques matrix inequalities (LMI) used Lin22 However, factin that an LTV some worksoftolinear minimize criterions based on Hare system is the stable the at LTIevery sensetime doesinstant not ensure the ∞ techniques ofto linear matrix inequalities (LMI) used some works minimize criterions based on H and L the fact at every time instant anknown LTV system is stable in the LTI sense does not ensure the norms (Gao et al., Bezzaoucha et Some stability the whole LTV system. is well ∞ some based on 2017). Hare Lin22 However, system isof stable in that the LTI senseThis doesfact not ensure the ∞ and normsworks (Gao to et minimize al., 2016; 2016; criterions Bezzaoucha et al., al., 2017). Some stability of the whole LTV system. This fact is well known some minimize based on 2017). H L normsworks (Gao to ethave al., considered 2016; criterions Bezzaoucha et al., al., 2017). Some system isof stable in the LTI sense doesfact not ensure the stability of the whole LTV system. This fact is well known other studies LTI stochastic systems, by ∞ and 2 in the classical system theory. It is thus a non trivial task norms (Gao et al., 2016; Bezzaoucha et Some stability the whole LTV system. This is well known other studies have considered LTI stochastic systems, by in the classical system theory. It is thus a non trivial task norms (Gao ethave al., considered 2016; Bezzaoucha al., 2017). Some other LTI systems, by of theresults whole LTV system. This fact is well known in the classical system theory. It is thus aasystems non trivial task designing state estimators based on the variance to generalize developed for LTI to LTV other studies studies have considered LTI stochastic systems, by stability in the classical system theory. It is thus non trivial task designing state estimators based on stochastic theetminimum minimum variance generalize results developed for LTI systems to LTV other studies considered LTI stochastic systems, by to designing state estimators based on the variance in the classicalresults systemdeveloped theory. It for is thus non trivial task to generalize LTI to criterion et 1995; Keller et systems. designing statehave estimators the minimum minimum variance to generalize results developed for LTI asystems systems to LTV LTV systems. criterion (Darouach (Darouach et al., al., based 1995; on Keller et al., al., 1998). 1998). designing state estimators based on the minimum variance criterion (Darouach et al., 1995; Keller et al., 1998). to generalize results developed for LTI systems to LTV systems. criterion (Darouach et al., 1995; Keller et al., 1998). systems. The purpose The purpose of of this this paper paper is is to to propose propose aa numerically numerically effiefficriterion (Darouach et al., 1995; Keller et al., 1998). systems. The purpose purpose of for thisstate paper is to to propose propose adisturbance numericallyrejecefficient algorithm estimation with L. Zhang acknowledges the financial support partly by the NaThe of this paper is a numerically efficient algorithm for state estimation with disturbance rejec L. Zhang acknowledges the financial support partly by the NaThe purpose of for thisstate paper is to of propose adisturbance numerically efficient algorithm estimation with rejection, in the general framework LTV stochastic systems, L. Zhang cient algorithm for state estimation with disturbance rejectional Nature Science Foundation of China (11701040) and Innovaacknowledges the financial support partly by the Na tion, in the general framework of LTV stochastic systems, L. Zhang the financial support partly and by the National Natureacknowledges Science Foundation of China (11701040) Innovacient algorithm for state estimation with disturbance rejection, in the general framework of LTV stochastic systems, with easily checkable conditions guaranteeing the stability tion, in the checkable general framework LTV stochastic tion of BUPT for tional Natureacknowledges Science Foundation of (500417024). China (11701040) and InnovaL. Foundation Zhang theYouth financial support partly and by the Nawith easily conditionsofguaranteeing the systems, stability tional Nature Science Foundation of China (11701040) Innovation Foundation of BUPT for Youth (500417024). tion, in the checkable general framework LTV stochastic with easily easily checkable conditionsofguaranteeing guaranteeing the systems, stability tion of for tional Nature Science Foundation of (500417024). China (11701040) and Innovawith conditions the stability tion Foundation Foundation of BUPT BUPT for Youth Youth (500417024). easily checkable conditions guaranteeing the stability 2405-8963 © 2018, Federation of Automatic Control) with Hosting by Elsevier Ltd. All rights reserved. tion Foundation of IFAC BUPT(International for Youth (500417024). ∗ ∗ ∗ ∗
Copyright © 2018 IFAC 55 Copyright © under 2018 IFAC 55 Control. Peer review responsibility of International Federation of Automatic Copyright © 55 Copyright © 2018 2018 IFAC IFAC 55 10.1016/j.ifacol.2018.09.090 Copyright © 2018 IFAC 55
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of the algorithm. Notably, this algorithm is applicable to LPV systems, which are viewed as a particular class of LPV systems. The method for designing this algorithm is conceptually simple: disturbances are rejected from the state equation by output injection, then the classical Kalman filter is applied to the new state-space model obtained after the output injection. To the best of our knowledge, this is the first result for LTV systems with disturbance rejection ensuring the stability of the state estimation error equation. This paper is organized as follows. In Section 2 the considered problem is formulated. In Section 3 the output injection for disturbance rejection is presented. In Setion 4 the Kalman filter is applied to the system after output injection. Numerical examples are presented in Section 5. Finally, conclusions are drawn in Section 6.
Disturbance subspace assumptions. (iv) q ≤ m (no more disturbances than output sensors). (v) There exists a positive constant γ such that, for all k = 0, 1, 2 . . . , the smallest singular value of the matrix product Ck+1 Ek is not smaller than γ. These assumptions are necessary for numerically reliable rejection of disturbances. Notice that Assumptions (iv) and (v) imply that Ck+1 Ek has a full column rank. In the classical Kalman filter theory, the stability of the Kalman filter mainly relies on observability and controllability assumptions. Similar assumptions will be formulated in the next section, after the disturbance rejection by means of output injection.
2. PROBLEM FORMULATION
The purpose this section is to reject the disturbance term Ek dk from the state equation by means of output injection, so that the Kalman filter can be applied without being affected by the disturbance. In (Kitanidis, 1987), linear filters in the form of x ˆk+1|k+1 = Ak x ˆk|k + Lk+1 (yk+1 − Ck+1 Ak x ˆk|k ) (2) were considered, neglecting the known input term Bk uk , which is not essential in linear filter design problems. The filter gain Lk+1 was determined by minimizing the state estimation error covariance, under the constraint (3) Lk+1 Ck+1 Ek − Ek = 0. See the equation (8) in (Kitanidis, 1987), formulated with different notations. This constraint ensures that the state estimation error is not affected by the disturbance term Ek dk . The solution to this constrained optimization problem leads to a filter gain that is considerably more sophisticated than the gain of the Kalman filter, and there is no obvious way to analyze the stability of the resulting filter. The main idea of the present paper is to reject the disturbance term before the filter design. In the first step, the disturbance rejection is achieved with an output injection satisfying a condition similar to constraint (3). Then the filter design is made as if there was no disturbance term in the considered problem, simplifying the stability analysis of the resulting filter. Let Gk ∈ Rn×m be a bounded matrix sequence to be specified later. The following equation holds due to (1b): (4) 0 = Gk+1 (yk+1 − Ck+1 xk+1 − vk+1 ). Add each side of this equation to the corresponding side of (1a), then xk+1 = Ak xk + Bk uk + Ek dk + wk + Gk+1 (yk+1 − Ck+1 xk+1 − vk+1 ). (5) For the purpose of disturbance rejection in the following steps, it is important that this output injection is made with the output equation at instant k + 1, instead of k. At the right hand side of equation (5), substitute xk+1 with (1a), then, after some rearrangement (with In denoting the n × n identity matrix), xk+1 = (In − Gk+1 Ck+1 )Ak xk + (In − Gk+1 Ck+1 )Bk uk + (In − Gk+1 Ck+1 )Ek dk + Gk+1 yk+1 + (In − Gk+1 Ck+1 )wk − Gk+1 vk+1 . (6)
3. OUTPUT INJECTION FOR DISTURBANCE REJECTION
Consider LTV stochastic systems in the form of xk+1 = Ak xk + Bk uk + Ek dk + wk (1a) (1b) y k = Ck x k + v k where xk ∈ Rn is the state, yk ∈ Rm the output, uk ∈ Rp the (known) input, dk ∈ Rq the disturbance (or unknown input), wk ∈ Rn the state noise, vk ∈ Rm the output noise, and Ak , Bk , Ck , Ek are known matrices of appropriate sizes. The purpose of this paper is to design an algorithm for the estimation of the state xk from the known input uk , the output yk and the known matrices Ak , Bk , Ck , Ek . While the noises wk , vk are random variables, the disturbance dk is completely arbitrary, notably, no probability distribution, nor upper bound of dk , is assumed. The fact that Ek is known, but dk is completely unknown, means that, at each instant k, a completely arbitrary disturbance can only affect a known subspace of the statespace, and this subspace may vary with k. The arbitrary character of dk means that there is a void subspace in the state equation. Typically, Ek is filled with 0’s and 1’s. For example, at the instant k, if Ek = [1, 0, . . . , 0]T , then the first component of the state equation (1a) is void. It may appear impossible to estimate the state xk when part of the state equation is missing. Of course, compared to the classical case where the term Ek dk did not exist, here some extra assumptions are required so that state estimation remains feasible, like the last two assumptions listed below. Basic assumptions. (i) Ak , Bk , Ck , Ek are bounded matrix sequences. (ii) The initial state x0 ∈ Rn is a random vector following the Gaussian distribution N (¯ x0 , P0 ), with a mean vector x ¯0 and a positive definite covariance matrix P0 . (iii) wk and vk are zero mean white Gaussian noises independent of each other and of x0 , with bounded covariance matrices E(wk wkT ) = Qk and E(vk vkT ) = Rk . These assumptions are usually made in the classical LTV system Kalman filter theory, apart from the involved matrix Ek that did not exist in the classical case. 56
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By treating Gk+1 yk+1 as a known input term, the classical Kalman filter can be applied to the state-space model (16), yielding a state estimator for the system originally described by (1). It is indeed an application of the classical Kalman filter as originally published in (Kalman, 1963), as no disturbance term appears in the state-space model (16). This is an indirect way for addressing the initial state estimation problem formulated with (1). For the sake of simplicity, let us neglect the correlation between w ¯k and vk . Then the classical Kalman filter applied to (16) is as follows. After the initialization ¯0 (17a) x ˆ0|0 = x (17b) P0|0 = P0 , the state estimate x ˆk|k and its covariance matrix Pk|k are then recursively computed, for k = 0, 1, 2, . . . , ¯k (18a) Pk+1|k = A¯k Pk|k A¯Tk + Q
If the matrix sequence Gk is chosen such that, at every time instant k, (7) (In − Gk+1 Ck+1 )Ek = 0, then equation (6) becomes ¯k uk + Gk+1 yk+1 + w ¯k , (8) xk+1 = A¯k xk + B with (9) A¯k (In − Gk+1 Ck+1 )Ak ¯ (10) Bk (In − Gk+1 Ck+1 )Bk (11) w ¯k (In − Gk+1 Ck+1 )wk − Gk+1 vk+1 . Now the disturbance term Ek dk has disappeared from equation (8).
Remark 1. Condition (7) that should be satisfied by Gk is the same as (3), which was imposed to the Kitanidis filter gain Lk . However, here more degrees of freedom are left to the choice of Gk , which is not determined by the minimum covariance criterion, unlike the Kitanidis filter gain Lk . The covariance matrix of the new state noise w ¯k is ¯ k (In − Gk+1 Ck+1 )Qk (In − Gk+1 Ck+1 )T Q + Gk+1 Rk+1 GTk+1 .
57
T + Rk+1 Σk+1 = Ck+1 Pk+1|k Ck+1
(18b)
T Pk+1|k Ck+1 Σ−1 k+1
(18c) (18d) (18e) (18f) (18g)
Kk+1 = Pk+1|k+1 = (In − Kk+1 Ck+1 )Pk+1|k ¯k uk + Gk+1 yk+1 x ˆk+1|k = A¯k x ˆk|k + B ˆk+1|k y˜k+1 = yk+1 − Ck+1 x ˆk+1|k + Kk+1 y˜k+1 . x ˆk+1|k+1 = x
(12)
Rewrite equation (7) as (13) Gk+1 Ck+1 Ek = Ek , which should be solved for the unknown matrix Gk+1 . Due to Assumption (iv), the matrix product Ck+1 Ek has no more columns than rows, thus in most situations, solutions of Gk+1 exist. A sufficient and necessary condition for the existence of solutions is that each row of Ek is a linear combination of the rows of the matrix product Ck+1 Ek . This condition is ensured by assumption (v). Given an m × n matrix M , another n × m matrix M g is a generalized inverse of M if M M g M = M . With this notation, the solution (14) Gk+1 = Ek (Ck+1 Ek )g with any generalized inverse (Ck+1 Ek )g satisfies equation (13). The particular solution Gk+1 = Ek [(Ck+1 Ek )T (Ck+1 Ek )]−1 (Ck+1 Ek )T (15) will be adopted in the remaining part of this paper. The boundedness of this solution Gk+1 is then ensured by Assumptions (i) and (v). Consequently, with this particular ¯ k defined respectively sequence Gk , the matrices A¯k and Q in (9) and (12) are also bounded. Now it is time to formulate the last assumptions ensuring the stability of the state estimator proposed in this paper.
In order to ensure the stability of this Kalman filter, two more assumptions will be formulated below, in addition to the assumptions already made in Section 2. These assumption are slightly different from those assumed in the classical Kalman filter theory, which is not suitable for the particular case considered here, as explained later in the two remarks following the statements of Properties 1 and 2. Observability and controllability assumptions. (vi) The matrix sequence pair [A¯k , Ck ] is uniformly completely observable. 1 ¯ 2 ] is (vii) The matrix sequence pair [A¯k (In−Kk Ck ), Q k 1 ¯ 2 is the uniformly completely controllable, where Q k ¯ matrix square root of Qk . The uniform complete observability and controllability are defined with the aid of Gramian matrices. See (Kalman, 1963; Jazwinski, 1970). Notice that these observability and controllability assumptions involve the matrix A¯k depending on Gk , which in turn depends on Ek . Like in the classical Kalman filter theory, if the involved matrices are known in advance (typically constant or periodical), then these assumptions can be checked in advance, otherwise they have to be checked in real time (e.g. for LPV systems). Property 1. (Boundedness). Under Assumptions (i)-(vii), the recursively computed matrices Pk|k and Pk+1|k are bounded, so are the innovation covariance Σk and the Kalman gain Kk .
4. KALMAN FILTER, STABILITY AND OPTIMALITY Consider the state-space model composed of the state equation (8) and the output equation (1b), which are grouped below for ease of later references: ¯k uk + Gk+1 yk+1 + w ¯k , (16a) xk+1 = A¯k xk + B (16b) y k = Ck x k + v k . The state xk , the input uk and the output yk of this new model are the same as in the original system model (1). In other words, these two models represent the same system. In this sense, the two models are equivalent.
This property ensures the boundedness of all the variables involved in the recursive computations of the Kalman filter except the states estimates. As Gaussian noises are not bounded, the state estimation errors are (in principle) not 57
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5. NUMERICAL EXAMPLES
bounded, but the error covariance matrices are bounded. Obviously, boundedness is crucial for any recursive algorithm in real time applications. Property 2. (Stability). Under Assumptions (i)-(vii), the error dynamics of the state estimate x ˆk|k of the Kalman filter (18) is exponentially stable. Remark 2. Recall that the boundedness of the sequence Gk is ensured by Assumptions (i) and (v). When applying the Kalman filter to (16), Gk+1 yk+1 is treated as a known bounded input term. Then it may appear that Properties 1 and 2 can be ensured by the classical Kalman filter theory (Kalman, 1963; Jazwinski, 1970). In fact, one of the important conditions required by the classical Kalman filter theory is not satisfied in the present case: when applied to system (16), the classical theory would assume that the matrix A¯k is invertible for all time instant k, whereas this matrix defined in (9) is always singular, because of equality (7). Fortunately, a new proof of the Kalman filter stability, ensuring Properties 1 and 2, has been worked out, without requiring invertible A¯k . As this complete remake of the Kalman filter stability proof is much longer than the present paper, it will be reported elsewhere. Remark 3. Despite the fact that the correlation between w ¯k and vk has been neglected by the Kalman filter applied to system (16), Properties 1 and 2 hold for the following reasons. The computations of Pk+1|k , Pk+1|k+1 , Σk+1 and Kk+1 ¯ k and Rk+1 through are fully determined by A¯k , Ck+1 , Q (18a)-(18d), hence Property 1 is a consequence of the ¯ k , Rk , no matter w assumptions made on A¯k , Ck , Q ¯k and vk are correlated or not. Let the state estimation error be denoted by x ˜k|k xk − x ˆk|k ,
In order to illustrate the state estimator proposed in this paper, the results of its application to two examples are presented below. 5.1 LTI example Let us borrow the example presented in Section 3.2.2, page 76, of (Chen and Patton, 1999). The example was originally in continuous time, with −1 −1 1 0 1 0 0 , Ec = 0 . Ac = −1 0 0 , Cc = 0 0 1 0 0 −1 −1 To apply the discrete time algorithm, this continuous time system is discretized with the sample time T = 1, then 0.1262 0.5335 0 Ac T 0 ≈ −0.5335 0.6597 (22) A=e 0.2417 −0.5335 0.3679 1 0 0 (23) C = Cc = 0 0 1 T
E = A−1 c (A − In )Ec ≈ [−0.5335 0.3403 −0.0986] . The noise covariance matrices are Q = 0.1I3 , R = 0.05I2 . The disturbance is simulated with a deterministic part and a random part as dk = 5 sin(0.1k) + ωk (24) where ωk is an independent and identically distributed random sequence following the Gaussian distribution N (0, 1). For the system model after output injection, 0.9670 0.1787 T −1 T G = E[(CE) (CE)] (CE) ≈ −0.6168 −0.1140 0.1787 0.0330 −0.0390 0.1130 −0.0658 A¯ = (In − GC)A ≈ −0.4281 0.9279 0.0419 0.2111 −0.6112 0.3557
(19)
then the error dynamics equation is ˜k|k x ˜k+1|k+1 = (In − Kk+1 Ck+1 )A¯k x ¯k − Kk+1 vk+1 . + (In − Kk+1 Ck+1 )w
¯ = (In − GC)Q(In − GC)T + GRGT Q 0.0517 −0.0308 −0.0089 ≈ −0.0308 0.1590 −0.0057 . −0.0089 −0.0057 0.0983
(20) (21)
The initial state x ¯0 = 0 and the initial state estimate covariance P0 = I3 (the 3 × 3 identity matrix). The simulated state and the estimated state are displayed in Figure 1. The state vector is correctly estimated, despite the presence of the disturbance.
Like in (Kalman, 1963; Jazwinski, 1970), the stability of the error dynamics concerns only the deterministic part of this error equation, fully characterized by the matrix (In − Kk+1 Ck+1 )A¯k . As the computation of Kk+1 is fully ¯ k and Rk+1 through (18a)determined by A¯k , Ck+1 , Q (18d), the stability of the error dynamics is fully ensured ¯ k , Rk . Therefore, by the assumptions made on A¯k , Ck , Q Property 2 is not affected by the correlation between w ¯k and vk .
5.2 LTV example The previous example is now modified so that the system becomes time varying. A sinusoid term is added to the central entry of the matrix A, and a triangular wave is added to the second component of the vector E. More precisely, 0.1262 0.5335 0 0 (25) Ak = −0.5335 0.6597 + sin(k) 0.2417 −0.5335 0.3679 T −0.5335 Ek = 0.3403 + ∆(k) (26) −0.0986
Another well known property of the Kalman filter is its optimality. As the correlation between the noises w ¯k and vk was neglected when formulating the Kalman filter (18), the optimality does not exactly hold, but the result should not be far from being optimal. The correlation between the state noise and the output noise is often ignored in Kalman filter applications. In the present case, it is not trivial to take into account the correlation caused by output injection. Further investigations are necessary in order to improve the optimality of the proposed algorithm. 58
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been established in the original work by Kalman. For LTV systems with disturbance rejection, to our knowledge, no result ensuring the stability of state estimators has been reported, hence we believe that the result presented in this paper represents an important progress in studies on this topic.
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Bezzaoucha, S., Voos, H., and Darouach, M. (2017). A new polytopic approach for the unknown input functional observer design. International Journal of Control, online version, 1–20. doi:10.1080/00207179.2017.1288299. Chen, J. and Patton, R. (1999). Robust Model-Based Fault Diagnosis for Dynamic Systems. Kluwer Academic Publishers, London. Darouach, M. and Zasadzinski, M. (1997). Unbiased minimum variance estimation for systems with unknown exogenous inputs. Automatica, 33(4), 717–719. Darouach, M., Zasadzinski, M., Bassong Onana, A., and Nowakowski, S. (1995). Kalman filtering with unknown inputs via optimal state estimation of singular systems. International Journal of Systems Science, 26(10), 2015– 2028. Darouach, M., Zasadzinski, M., and Xu., S.J. (1994). Fullorder observers for linear systems with unknown inputs. IEEE Transactions on Automatic Control, 39(3), 606– 609. Gao, N., Darouach, M., Voos, H., and Alma, M. (2016). New unified h-infinity dynamic observer design for linear systems with unknown inputs. Automatica, 65, 43–52. Gillijns, S. and De Moor, B. (2007). Unbiased minimumvariance input and state estimation for linear discretetime systems with direct feedthrough. Automatica, 43, 934–937. Hsieh, C.S. (2000). Robust two-stage kalman filters for systems with unknown inputs. IEEE Transactions on Automatic Control, 45(12), 2374–2378. Jazwinski, A.H. (1970). Stochastic Processes and Filtering Theory, volume 64 of Mathematics in Science and Engineering. Academic Press, New York. Kalman, R.E. (1963). New methods in Wiener filtering theory. In J.L. Bogdanoff and F. Kozin (eds.), Proceedings of the First Symposium on Engineering Applications of Random Function Theory and Probability. John Wiley & Sons, New York. Keller, J.Y., Darouach, M., and Caramelle, L. (1998). Kalman filter with unknown inputs and robust twostage filter. International Journal of Systems Science, 29(1), 41–47. Kitanidis, P.K. (1987). Unbiased minimum variance linear state estimation. Automatica, 23(6), 775–778. Luenberger, D. (1971). An introduction to observers. IEEE Transactions on Automatic Control, 16(6), 596– 602. T´oth, R. (2010). Modeling and Identification of Linear Parameter-Varying Systems. Springer. Yang, F. and Wilde, R. (1988). Observers for linear systems with unknown inputs. IEEE Transactions on Automatic Control, 33(7), 677–681.
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Fig. 2. LTV (time varying) system example. The 3 components of the simulated state vector and their estimates are plotted. with the triangular wave x 1 x + − 1, ∆(k) = 4 − 10 10 2 where · denotes the floor function, i.e.. for any x ∈ R, x is the largest integer less than or equal to x. The two noise covariance matrices are multiplied by 10, because the signal magnitudes become larger due to the added time varying terms, resulting in Q = I3 , R = 0.5I2 . The other data are the same as in the previous example. The simulated state and the estimated state are displayed in Figure 2. Again the state vector is correctly estimated, despite the presence of the disturbance. 6. CONCLUSION The stability of state estimators is of primary importance, as demonstrated by classical studies on LTI system state observers. For LTV systems, the stability of state estimators is less often discussed, including in the case of the Kalman filter, despite the fact that its stability has 59