State Estimation with Guaranteed Convergence Speed in the Presence of Sporadic Measurements
This paper deals with the problem of estimating the state of a linear time-invariant system in the presence of sporadically available measurements and external perturbations. An observer with a continuous intersample injection term is proposed. Such an intersample injection is provided by a linear dynamical system, whose state is reset to the measured output estimation error at each sampling time. The resulting system is augmented with a timer triggering the arrival of a new measurement and analyzed in a hybrid system framework. The design of the observer is performed to achieve global exponential stability with a given decay rate to a set wherein the estimation error is equal to zero. Robustness with respect to external perturbations and -external stability from the plant perturbation to a given performance output are considered. Moreover, computationally efficient algorithms based on the solution to linear matrix inequalities are proposed to design the observer. Finally, the effectiveness of the proposed methodology is shown in three examples.
In most real-world control engineering applications, measurements of the output of a continuous-time plant are only available to the algorithms at isolated times. Due to the use of digital systems in the implementation of the controllers, such a constraint is almost unavoidable and has lead researchers to propose algorithms that can cope with information not being available continuously. In what pertains to state estimation, such a practical need has brought to life a new research area aimed at developing observer schemes accounting for the discrete nature of the available measurements. When the information is available at periodic time instances, there are numerous design approaches in the literature that consist of designing a discrete-time observer for the discretized version of the process; see, e.g., [arcak2004framework, nevsic1999formulas], just to cite a few. Unfortunately, such an approach is limiting for several reasons. One reason stems from the fact that to precisely characterize the intersample behavior, one needs the exact discretized model of the plant, which may actually be impossible to obtain analytically in the case of nonlinear systems; see [nevsic1999formulas]. Furthermore, with such an approach no mismatch between the actual sampling time and the one used to discretize the plant is allowed in the analysis or in the discrete-time model used to solve the estimation problem. Very importantly, many modern applications, such as network control systems [hespanha2007survey], the output of the plant is often accessible only sporadically, making the fundamental assumption of measuring it periodically unrealistic.
To overcome the issues mentioned above, several state estimation strategies that accommodate information being available sporadically, at isolated times, have been proposed in the literature. Such strategies essentially belong to two main families. The first family pertains to observers whose state is entirely reset, according to a suitable law, whenever a new measurement is available, and that run open-loop in between such events – these are typically called continuous-discrete observers. The design of such observers is pursued, e.g., in [Ferrante2016state, mazenc2015design]. In particular, in [Ferrante2016state] the authors propose a hybrid systems approach to model and design, via Linear Matrix Inequalities (LMIs), a continuous-discrete observer ensuring exponential convergence of the estimation error and input-to-state stability with respect to measurement noise. In [mazenc2015design], a new design for continuous-discrete observers based on cooperative systems is proposed for the class of Lipschitz nonlinear systems.
The second family of strategies pertains to continuous-time observers whose output injection error between consecutive measurement events is estimated via a continuous-time update of the latest output measurement. This approach is pursued in [farza2014continuous, karafyllis2009continuous, postoyan2012framework, postoyan2014tracking, raff2008observer]. Specifically, the results in [karafyllis2009continuous, farza2014continuous] show that if a system admits a continuous-time observer and the observer has suitable robustness properties, then, one can build an observer guaranteeing asymptotic state reconstruction in the presence of intermittent measurements, provided that the time in between measurements is small enough. Later, the general approach in [karafyllis2009continuous] has been also extended by [postoyan2012framework] to the more general context on networked systems, in which communication protocols are considered. A different approach is pursued in [raff2008observer]. In particular, in this work, the authors, building on the literature of sampled-data systems, propose sufficient conditions in the form of LMIs to design a sampled-and-hold observer to estimate the state of a Lipschitz nonlinear system in the presence of sporadic measurements.
In this paper, we consider the problem of exponentially estimating the state of continuous-time Lipschitz nonlinear systems subject to external disturbances and in the presence of sporadic measurements, i.e., we assume the plant output to be sampled with a bounded nonuniform sampling period, possibly very large. To address this problem, we propose an observer with a continuous intersample injection and state resets. Such an intersample injection is provided by a linear time-invariant system, whose state is reset to the measured output estimation error at each sampling time.
Our contributions in the solution to this problem are as follows. Building on a hybrid system model of the proposed observer and of its interconnection with the plant, we propose results for the simultaneous design (co-design) of the observer and the intersample injection dynamics for the considered class of nonlinear systems. The approach we pursue relies on Lyapunov theory for hybrid systems in the framework in [goebel2012hybrid]; similar Lyapunov-based analyses for observers are also available in [postoyan2014tracking, Section VIII], [wang2017observer, ahmed2012high]. The use of the hybrid systems framework [goebel2012hybrid] can be seen as an alternative approach to the impulsive approach pursued, e.g., in [farza2014continuous]. The design we propose ensures exponential convergence of the estimation error with guaranteed convergence speed and robustness with respect to measurement noise and plant perturbations. More precisely, the decay rate of the estimation error can be specified as a design requirement cf. [fichera2012convex]. In addition, for a given performance output, we propose conditions to guarantee a particular -gain between the disturbances entering the plant and the desired performance output. The conditions in these results are turned into matrix inequalities, which are used to derive efficient design procedures of the proposed observer.
The methodology we propose gives rise to novel observer designs and allows one to recover as special cases the schemes presented in [karafyllis2009continuous, raff2008observer].
The remainder of the paper is organized as follows. Section II presents the system under consideration, the state estimation problem we solve, the outline of the proposed observer, and the hybrid modeling of the proposed observer. Section III is dedicated to the design of the proposed observer and to some optimization aspects. Finally, in an example, Section V shows the effectiveness of the results presented. A preliminary version of the results here appeared in the conference paper [ferrante2015hybrid].
Notation: The set is the set of positive integers including zero, the set is the set of strictly positive integers, represents the set of nonnegative real scalars, represents the set of the real matrices, and is the set of symmetric positive definite matrices. The identity matrix is denoted by , whereas the null matrix is denoted by . For a matrix , denotes the transpose of , , and . For a symmetric matrix , and ( and ) mean that () is, respectively, positive definite and positive semidefinite. In partitioned symmetric matrices, the symbol stands for symmetric blocks. Given matrices and , the matrix is the block-diagonal matrix having and as diagonal blocks. For a vector , denotes the Euclidean norm. Given two vectors , we denote . Given a vector and a closed set , the distance of to is defined as . For any function , we denote when it exists.
I-C Preliminaries on Hybrid Systems
We consider hybrid systems with state , input , and output of the form
In particular we denote, as the flow map, as the flow set, as the jump map, and as the jump set.
A set is a hybrid time domain if it is the union of a finite or infinite sequence of intervals , with the last interval (if existent) of the form with finite or . Given a hybrid time domain , we denote . A hybrid signal is a function defined over a hybrid time domain. Given a hybrid signal , then . A hybrid signal is called a hybrid input if is measurable and locally essentially bounded for each . In particular, we denote the class of hybrid inputs with values in . A hybrid signal is a hybrid arc if is locally absolutely continuous for each . In particular, we denote the class of hybrid arcs with values in . Given a hybrid signal , . A hybrid arc and a hybrid input define a solution pair to if and satisfies the dynamics of . A solution pair to is maximal if it cannot be extended and is complete if is unbounded; see [cai2009characterizations] for more details. With a slight abuse of terminology, given , in the sequel we say that leads to a solution to if , with for each , is a solution pair to .
Ii Problem Statement and Outline of Proposed Observer
Ii-a System Description
We consider continuous-time nonlinear time-invariant systems with disturbances of the form
where , , and are, respectively, the state, the measured output of the system, a nonmeasurable exogenous input, and the measurement noise affecting the output , while is a Lipschitz function with Lipschitz constant , i.e., for all
The matrices , and are constant and of appropriate dimensions. The output is available only at some time instances , , not known a priori. We assume that the sequence is strictly increasing and unbounded, and that (uniformly over such sequences) there exist two positive real scalars such that
The lower bound in condition (3) prevents the existence of accumulation points in the sequence , and, hence, avoids the existence of Zeno behaviors, which are typically undesired in practice. In fact, defines a strictly positive minimum time in between consecutive measurements. Furthermore, defines the Maximum Allowable Transfer Time (MATI) [postoyan2012framework].
Given a performance output , where is the estimate of to be generated, the problem to solve is as follows:
Design an observer providing an estimate of , such that the following three properties are fulfilled:
The set of points where the plant state and its estimate coincide (and any other state variables111The observer may have extra state variables that are used for estimation. In our setting, the sporadic nature of the available measurements of will be captured by a timer with resets. are bounded) is globally exponentially stable with a prescribed convergence rate for the plant (1) interconnected with the observer whenever the input and are identically zero;
The estimation error is bounded when the disturbances and are bounded;
-external stability from the input to the performance output is ensured with a prescribed -gain when .
Ii-B Outline of the Proposed Solution
Since measurements of the output are available in an impulsive fashion, assuming that the arrival of a new measurement can be instantaneously detected, inspired by [karafyllis2009continuous, postoyan2012framework, raff2008observer] to solve Problem 1, we propose the following observer with jumps
where and are real matrices of appropriate dimensions to be designed and represents the estimate of provided by the observer. The operating principle of the observer in (4) is as follows. The arrival of a new measurement triggers an instantaneous jump in the observer state. Specifically, at each jump, the measured output estimation error, i.e., , is instantaneously stored in . Then, in between consecutive measurements, is continuously updated according to continuous-time dynamics, and its value is continuously used as an intersample correction to feed a continuous-time observer. At this stage, we introduce the following change of variables which defines, respectively, the estimation error and the difference between the output estimation error and . Moreover, by defining as a performance output , where , we consider the following dynamical system with jumps:
where for each , and
Our approach consists of recasting (5) and the events at instants satisfying (3) as a hybrid system with nonunique solutions and then apply hybrid systems theory to guarantee that (5) solves Problem 1.
As a difference to [farza2014continuous, karafyllis2009continuous, postoyan2012framework], the results presented in the next two sections are based on the Lyapunov results for hybrid systems presented in [goebel2012hybrid] and, rather than emulation, consist of direct design methods of the proposed hybrid observer. Our design methods not only allow for completely designable intersample injection terms in the observer, but also allow for designs that cover the special cases of the schemes presented in [karafyllis2009continuous, raff2008observer]. Furthermore, as a difference to [postoyan2012framework], where an emulation-based approach is considered, our results provide constructive conditions for the design of the observer gains so as to enforce the desired convergence properties for a desired value of .
Iii Construction of the Observer and First Results
Iii-a Hybrid Modeling
The fact that the observer experiences jumps when a new measurement is available and evolves according to a differential equation in between updates suggests that the updating process of the error dynamics can be described via a hybrid system. Due to this, we represent the whole system composed by the plant (1), the observer (4), and the logic triggering jumps as a hybrid system. The proposed hybrid systems approach also models the hidden time-driven mechanism triggering the jumps of the observer.
To this end, in this work, and as in [FerranteIFAC2014], we augment the state of the system with an auxiliary timer variable that keeps track of the duration of flows and triggers a jump whenever a certain condition is verified. This additional state allows to describe the time-driven triggering mechanism as a state-driven triggering mechanism, which leads to a model that can be efficiently represented by relying on the framework for hybrid systems proposed in [goebel2012hybrid]. More precisely, we make decrease as ordinary time increases and, whenever , reset it to any point in , so as to enforce (3). After each jump, we allow the system to flow again. The whole system composed by the states , and , and the timer variable can be represented by the following hybrid system, which we denote , with state
with , input , , and output :
|where the flow set and the jump set are defined as follows|
The set-valued jump map allows to capture all possible sampling events occurring within or units of time from each other. Specifically, the hybrid model in (7a) is able to characterize not only the behavior of the analyzed system for a given sequence , but for any sequence satisfying (3).
Concerning the existence of solutions to system (7a) with zero input, by relying on the notion of solution proposed in [goebel2012hybrid], it is straightforward to check that for every initial condition every maximal solution to (7a) is complete. Thus, completeness of the maximal solutions to (7a) is guaranteed for any choice of the gains and , guaranteeing that provides an accurate model of the error dynamics in (5). In addition, one can characterize the domain of these solutions. Indeed for every initial condition , the domain of every maximal solution to (7a) can be written as follows:
where is the domain of the solution , which is a hybrid time domain; see [goebel2012hybrid] for further details on hybrid time domains.
Concerning solution pairs to (7a) with nonzero inputs, observe that given any solution pair , the definition of the sets and ensure that has the same structure illustrated in (8). Moreover, if is maximal then it is also complete222Completeness of maximal solution pairs can be shown by following similar arguments as in [goebel2012hybrid, Proposition 6.10.]. In particular, it is enough to observe that: , no finite escape time is possible (due to measurable and locally essentially bounded and Lipschitz uniformly in ), and solutions to from any initial condition in are nontrivial..
To solve Problem 1 our approach is to design the matrices and in the proposed observer in (7a) such that without disturbances, i.e., , the following set333By the definition of the system and of the set , for every , .
is exponentially stable and, when the disturbances are nonzero, the system is input-to-state stable with respect to . These properties are captured by the notions defined below:
( norm) Let be a hybrid signal and . The -truncated norm of is given by
where denotes the set of all such that ; see [cai2009characterizations] for further details. The norm of , denoted by is given by , where . When, in addition, is finite, we say that
Definition 2 (Pre-exponential input-to-state stability).
Let be closed. The system is pre-exponentially input-to-state-stable with respect to if there exist and such that each solution pair to with satisfies
for each . Whenever every maximal solution is complete, we say that is exponentially input-to-state-stable (eISS) with respect to .
Iii-B Sufficient conditions
In this section we provide a first sufficient condition to solve Problem 1. To this end, let us consider the following assumption, which is somehow driven by [goebel2009hybrid, Example 27] and whose role will be clarified later via Theorem 1.
Let and be given positive real numbers. There exist two continuously differentiable functions , , positive real numbers such that
the function satisfies for each
The following properties on the elements in the hybrid domain of solutions to will be used to establish our sufficient conditions.
Let , , , and . Then, each solution pair to satisfies
for every .
From (12), by rearranging the terms, one gets
then, for every strictly positive scalar , from the latter expression, and for every , one gets
Thus, being strictly positive, by selecting
yields (13), which concludes the proof. ∎
Consider the following Lyapunov function candidate for the hybrid system (7a) defined for every :
We prove first. To this end, notice that by setting and , in view of the definition of the set in (9), one gets
Moreover, from Assumption 1 item (A3) one has
and for each one has
Pick , let be a maximal solution pair to (7a), and pick . Furthermore, let be such that . Direct integration of thanks to (18) and (19), for each , yields444Given a sequence , we adopt the convention if .
which in turns gives
which, thanks to (17), implies that
Hence, for each one has555The first inequality is established by using the fact that for each and nonnegative real numbers, , while the second inequality follows from the fact that for any real numbers , and , implies .
Hence, since every maximal solution to is complete, is established.
To establish , we follow a similar approach as in [nevsic2013finite]. Pick and let be a maximal solution pair to . Pick any , then thanks to Assumption 1 item (A3), since, as shown in (18), is nonincreasing at jumps, direct integration of yields
where , which implies
Therefore, by taking the limit for approaching , thanks to (17), one gets with .
To show that the proposed observer solves Problem 1 as claimed in item , we show that (P1), (P2), and (P3) are fulfilled. Item already implies (P1) and (P2), since defines a lower bound on the decay rate with respect to the ordinary time ; see (23). To show that implies (P3), notice that since holds for any solution pair with and any hybrid signal, it holds in particular when the hybrid signal is obtained from a continuous-time signal of the original plant (1). Passing from hybrid signals and to right continuous signals , respectively, (see [Mayhew2013]), item leads to
hence concluding the proof. ∎
Iii-C Construction of the functions and in Assumption 1
With the aim of deriving constructive design strategies for the synthesis of the observer, we perform a particular choice for the functions in Assumption 1. Let , and be a positive real number. Inspired by [FGZN_Auto2014], we consider the following choice
The structure selected above for the functions and essentially allows to exploit the (quasi)-quadratic nature of the resulting Lyapunov function candidate to cast the solution to Problem 1 into the solution to certain matrix inequalities.
Let and be defined as in (27) and select , , and . Then, it turns out that items (A1) and (A2) of Assumption 1 are satisfied. Let , then, by straightforward calculations and by the definition of the flow map in (7b), it follows that for each , one has
Moreover, observe that thanks to (2), for any positive real number one has that
Therefore, by defining , for each one has , where the symmetric matrix is defined in (30).
Notice that, for it to be feasible, condition (28) requires the existence of such that , where and stands for the norm of its argument666To show this claim it suffices to observe that the satisfaction of (28) implies which turns out to be equivalent to ; see [Boyd].. Nevertheless, this condition is, in general, only necessary.
Although, for a given instance of Problem 1, the search of feasible solutions to (28) needs to be performed via numerical methods, it is worthwhile to provide minimum requirements to ensure, at least for suitable values of (small) and (large), the feasibility of (28). To this end, being the satisfaction of (28) equivalent to the satisfaction of item (A3) in Assumption 1 (for the particular choice of the functions and in (27)), one only needs to analyze under which conditions there exists a suitable selection of the real numbers that allows to fulfill (A3). This is illustrated in the result given next.
If there exist , , and such that
Then, there exist four positive real numbers , and such that the function satisfies
for each .
From (31), one has that there exist positive real numbers and a matrix such that for each
which, by squares completion, gives