Interval observers design for systems with ostensible Metzler system matrices

Abstract

This paper attempts to resolve the problem concerning the interval observers design for linear systems with ostensible Metzler system matrices. Because system dynamics matrices are partially different from strictly Metzler structures, a solution is achieved by constructing a composed system matrix representation, which combines pre-compensated interval matrix structures fixed with a prescribed region of D-stability and the reconstructed strictly Metzler matrix structure, related to the original interval system matrix parameter definition. A novel design procedure is presented, which results in a strictly positive observer gain matrix and guarantees that the lower estimates of the positive state variables are non-negative when considering the given system structure and the non-negative system state initial values. The design is computationally simple since it is reduced to the feasibility of the set of linear matrix inequalities.

1 Introduction

Interval observers have appeared as an alternative technique for robust state estimation (Moisan et al., 2009). Whilst, when using the technique based on classical observers, only the initial condition is assumed to be unknown (Luenberger, 1971), interval observers structures are constructed assuming that the upper and lower bounds of the initial conditions are known (Raïssi and Efimov, 2018; Khan et al., 2020). The main limitation to the interval observers theory is that the trajectories of the system that start from an internally bounded initial condition will enclose the stable system trajectory only if the system is positive that its system matrix is Metzler and Hurwitz and that other matrix parameters are non-negative (Farina and Rinaldi, 2000). Thus, the positivity of interval estimation error dynamics is one of the most restrictive assumptions for interval observers design. When restricted to the Metzler structure of system matrices, as well as to non-negative input and output matrices, such systems are referred to as Metzler systems (Nikaido, 1968; Smith, 1995; Liu et al., 2011), with a stringent approach that reflects the diagonal stabilization principle. Although a certain class of systems can be transformed through a change of coordinates into positive cooperative systems (Mazenc and Bernard, 2011; Mazenc and Bernard, 2014), no general technique exists for such construction.

When maintaining platforms for positive systems with nonnegative states (Nikaido, 1968; Smith, 1995; Moisan et al., 2009), the theory of Metzler matrices (Berman et al., 1989) implies some additional parametric constraints to reflect the system positiveness (Shorten et al., 2009) and to construct the system representation (Son and Hinrichsen, 1996; Gao et al., 2005; Liu et al., 2017; Ito and Dinh, 2020). Since the linear time-invariant system theory cannot be directly used for linear positive systems, various combinations of linear programming and linear matrix inequalities (LMI) are generally used to represent Metzler systems (Ait Rami and Tadeo, 2006; Shu et al., 2008; Anderson and Murray, 2018; Guo et al., 2020). The benefits of a potential unification are presented (Krokavec and Filasová, 2018) when reflecting diagonal stabilization and associated Metzler system matrix parametric representations by a specific set of LMIs.

The system matrix parametric constraints give rise to substantially complex design methods when applied to positive systems with interval-defined model parameters (Ganesan, 2007). To demarcate the object of study in this field, Metzler matrix transforms are reflected for interval observers analysis (Mason, 2012; Chambon et al., 2015). Interval observers design for linear time-varying (LTV) systems, as well as for a class of non-linear time-varying systems with output specifications, exploits static coordinate transformation (Efimov et al., 2013; Raïssi and Efimov, 2018) when translating a stable LPV system to another stable and cooperative LPV system. The LMI-based conditions applicable in interval observers design for positive Metzler systems have also been studied (Krokavec and Filasová, 2020b). The utilization of interval observers for interconnected schemes is often applied in relation to distributed interval estimation and distributed feedback control (Wang et al., 2020; Wang X L et al., 2022; Zhang et al., 2022); this also reflects that their application for continuous linear large-scale systems is limited due to the system’s complexity (Wang T et al., 2022). These problems are still open in distributed applications since it is difficult to ensure that the system state will be enclosed by the cooperative estimated upper and lower bounds of the observed system state (Huong, 2022; Li et al., 2022), as well as in interval estimation strategy for anti-disturbance control of drones (Yong et al., 2020).

This paper contributes to the properties of the interval state estimation for linear systems with ostensible Metzler system matrices. It outlines a new LMI-based approach to determine interval observers with positive observer gains using a combined representation of the ostensible Metzler system matrix (Krokavec and Filasová, 2022). Design conditions are formulated using LMIs, respecting the diagonal stabilization principle, Metzler system matrix parametric constraints, and given interval matrix bounds. Because linear systems with ostensible Metzler matrices are not positive, even if their matrix parameters are non-negative, the main limitation to the solution is that the interval estimation only works for system state variables whose trajectories for a non-negative initial state are non-negative.

The outline of this paper is as follows. Following an introduction in Section 1, the basic preliminaries are discussed in Section 2. Section 3 presents the LMI structures necessary for observer stability and the positive gain, and the design method of interval observers for a given class of positive systems is presented in Section 4. To illustrate the design task, its efficiency is demonstrated by numerical solutions in Section 5; in Section 6, conclusions are briefly presented.

Throughout the paper, X ≺ 0 conveys briefly that a real square matrix X is a symmetric and negative definite, notations x^T and X^T identify the transpose of a vector or a matrix, I_n indicates the nth order unit matrix, function ρ(⋅) reflects the eigenvalue spectrum of a real square matrix, diag[ ⋅ ] enters a block diagonal matrix, the symbol * is used as ellipsis in a symmetric matrix, is the set of real function z(t) with the property ‖z(t)‖ < ∞, the relations x₁ ≤ x₂ and X₁ ≤ X₂ operate on corresponding elements element-wise, qualifies the set of (non-negative) real numbers, refers to the set of n × r (non-negative) real matrices, and denotes the set of strictly Metzler square matrices.

2 Basic preliminaries

To explain the technique used, the main question can be illustrated using the linear Metzler systems given as follows:where , , and are non-negative vectors of the system, input and output, disturbance , is norm-bounded and non-negative, and , , and are non-negative matrices.

Definition 1. Berman et al. (1989). A square matrixis a strictly Metzler matrix if all its diagonal elements are negative and all its off-diagonal elements are positive.

This study also uses the term “a purely Metzler matrix ” if all diagonal elements of A are negative and all its off-diagonal elements are non-negative, and “an ostensible Metzler matrix ” if all diagonal elements of A are negative and at least one off-diagonal element is negative while the number of non-negative off-diagonal elements is prevalent.

2.1 Positive continuous-time linear systems

The following assumptions make it possible to cover constraints on the parameters of system (1) and (2) when studying the system positivity and the conditions of its diagonal stabilizability.

Lemma 1. Tanaka and Langbort (2011). Disturbance-free system(1), (2)is internally positive if and only ifAis (strictly, purely) Metzler andB,Care entry-wise non-negative.

In consequence, any solution of an autonomous and disturbance-free linear system with a Metzler matrix is element-wise non-negative for all t ≥ 0, provided that q (0) ≥0. The output solution y(t) for such a defined solution is non-negative if . Such dynamical systems are called non-negative only if initial conditions in are considered.

Remark 1. A strictly Metzler matrixmakes ofn²constraintsThese parametric constraints imply the strict application of the diagonal stabilization principle (Shorten et al., 2009; Mason, 2012) in analysis. If a strictly Metzleris represented (in relation to the observer design) in the following rhombic form, where the diagonal items reflected by the column index define multiple circular shifts of elements of the columns ofAas (Krokavec and Filasová, 2020b)the diagonal stabilization principle can be appropriately respected using the derived diagonal matrix structures related toA_Θasforh = 0, 1, … , n−1.

Remark 2. Defining the matrixin the circulant permutation form (Horn and Johnson, 1995)and considering a diagonal matrix, where, then

The aforementioned results can be combined and reflected by the following lemma.

Lemma 2. Krokavec and Filasová (2020a)If a positive matrixforces the strictly Metzler matrix, whereis strictly Metzler andis non-negative, thenA_eis parameterized as

whereh = 1, … , n − 1 and the diagonal matricesare composed in the following ways:

The proof of the aforementioned lemma is based on the fact that only diagonal matric representations are applicable for the diagonal stabilization of positive systems.

2.2 Ostensible Metzler matrices

Given a system with the dynamical model (1) and (2) and considering that is an ostensible Metzler matrix, then inclusion of the negative off-diagonal elements of A into the design task is built on the following basic facts from the theory of matrices.

Definition 2. Shores (2007)Matrixis similar to the matrixif there exists an invertible similarity transform matrixsuch that

If X and Λ are similar, then they have the same eigenvalues, their algebraic multiplicities are the same, their characteristic polynomials are the same, and their determinants and traces are the same.

Remark 3. Letbe the set of eigenvectors for a matrixandis the associated set of eigenvalues ofXsuch that eigenvalues are all distinct, then(14)impliesandare linearly independent.

2.3 Intervally defined ostensible Metzler matrices

In this case, is assumed that q(0) and the ostensible Metzler system parameter A are unknown but bounded by constant bounding vectors and constant bounding matrices of appropriate dimensions in such a way that (these inequalities being understood element-wise (Jaulin et al., 2001))

Since the main goal is the design of an interval observer of the state, it is considered that , , and are known non-negative matrices, the system input u(t) is norm-bounded, and the following assumption is adopted.

Assumption 1. The function boundsandare given such that

This assumption states that the disturbance is known up to some interval error.

Corollary 1. Strictly Metzlerand an entry-wise negative and Hurwitzto the composed forms of the ostensible Metzler matricesexist if there exist positive scalarssuch that, withit yieldswhereΣis from (20) and,,, and are constructed by the rules defined in(21)and(22).

3 General interval observers structure

Under these introduced assumptions, the interval observers equations for systems with intervally given ostensible Metzler matrices can be defined as follows:where and are, respectively, the lower and upper interval estimates for the state q(t) and

Using the observation errorsit follows from (1), (33), and (34) that

To construct a Hurwitz stable , guaranteeing also strictly Metzler and Hurwitz matrices when implementing for ostensible Metzler , , and , then (38) can be rewritten aswhere

To apply the parametrization principle in designing this class of observer, the following corollary is objective.

Corollary 2. State observation error dynamics(40)entail the parameterizations of the strictly Metzler matrices and as follows:while the parameterizations(12)and(13)stay unchanged.

Provided that (25) is satisfied, then for all the estimates and are bounded with the limit properties, illustrated by the following remark.

Remark 6. Performing an inner adjustment for(38)asrespectively, and substituting(1)in(47)yieldsand, ifis Hurwitz,is nonnegative,is positive, and, then the lower system state estimate produced by the interval observer constructed on the system model with the ostensible Metzler matrix converges to a non-negative trajectory if. Consequently, provided that, then for allthe estimatesandgiven by(33)and(34)produce the interval bounds only to those system state variablesq_i(t), i = 1, … , nwhich are non-negative.

4 Interval observers design

The design goals are Hurwitz stable matrices and and strictly Metzler and Hurwitz matrices and when implementing for ostensible Metzler and . A solution method, resulting in positive matrix gain , is given in Theorem 2.

5 Illustrative examples

In this section, two examples are presented to demonstrate the effectiveness of the interval observers design.

Example 1. To illustrate the proposed design principles, the stable interval ostensible strictly Metzler systems (1) and (2) are constructed on the matrices

To apply Theorem 2 conditions, the derived design parameters are selected asand the related matrix structures are constructed from the system matrix bounds as follows:

Thus, using Σ and η = 0.005 yields for A°_m that

Setting means and , and so , take the values

Furthermore, and are computed aswhich are strictly Metzler, and their rhombic representations imply the diagonal matrices for the observer synthesiswhilst straightforward calculations give

Using LMIs defined by Theorem 2, the feasible matrix variables result in the non-negative gain matrix when applying the SeDuMi package (Peaucelle et al., 2002)

This infuses the strictly Metzler and Hurwitz matrices and aswhere . In addition, it can be seen that, due to the structure of matrix C, the elements on the third columns of matrices and have not changed compared to and .

Applying the same gain matrix to the ostensible Metzler matrices yields

It should be noted that the positions of the negative off-diagonal elements in and , as well as in and , have been preserved. In addition, in the considered case, .

By simulating the response of the autonomous system with considered interval ostensible Metzler parameters to better illustrate the ostensible Metzler phenomena, the dynamics of the system werethe initial system state was set as q(0) = [ 0.5 7.5 0 ]^T, , and . The simulation is executed in the MATLAB framework using Simulink.

Figure 1 depicts the time responses of the first system state variable and its upper and lower estimations; Figure 2 shows the time responses for the third system state variable. Although the given system is not positive, it can be seen from Figures 1 and 2 that the behaviors of these state variables are correctly intervally estimated by the proposed interval observer if the components and , indicated in (48), are compensated by suitably choosing the observer initial states, satisfying conditions . Since the state variable q₂(t) is undefined in sign, its interval estimation is also undefined in sign. This case is trivial and is not presented.

FIGURE 1

FIGURE 2

Moreover, considering the effect of the fixed uncompensated part with prescribed D-stability region related to , , the proposed approach leads to a structure that has the properties of a stable system. By using the tuning parameter δ, the D-stability region can be analytically continued.

Example 2. To demonstrate the application validity of the suggested interval observer, the second example is presented on the linearized dynamic model of a U.S. Navy F-404 engine which powers the F/A-18 aircraft (Kwon et al., 1999). The corresponding dynamic model is a stable interval ostensible purely Metzler system (1), (2), written asSince the interval matrices of the system are purely Metzler, due to the structure of their second column, it is advantageous if the measurement system corresponds the following matrix C

Applying analogously as the aforementioned conditions of Theorem 2, the resulting matrix representations arewhere , , , , and .

Analogously constructing the diagonal matrices for the interval observer synthesis from the rhombic representations of the interval system matrices and for the used matrix C, the feasible matrix variables resulting from the conditions defined by Theorem 2 are

This infuses the strictly Metzler and Hurwitz matrices and aswhere . In addition, it can be seen that, due to the structure of matrix C, the elements on the second columns of matrices and have not changed compared to and .

Using these ostensible purely Metzler matrices results in stable, purely Metzler structures

By simulating the response of the observer in the forced mode, it is set as

Figure 3 depicts the time responses of the first system state variable and its upper and lower estimations; Figure 4 shows the time responses for the third system state variable. Although the given system is not positive, it can be seen from Figure 3 and Figure 4 that the behaviors of these state variables are correctly intervally estimated by the proposed interval observer.

FIGURE 3

FIGURE 4

Note for both examples, since , are scalar variables defined directly by a feasible solution of LMIs, they can be indicated as the values at the disturbance attenuation levels. The scalar variable p provides an additional degree of freedom in solving the problem of the dynamics of an interval observer, which should generally be faster than the dynamics of the system. Because the synthesis method is a two-step procedure, it is possible to sequentially define the locations of the stable regions first for the uncontrolled stable dynamics of and by defining the D-region of stability using the parameter p > 0 (ν > 0 is just some small positive value by means of which Σ is regularized) and then, indirectly via LMIs, finding a solution that guarantees the required rate of estimation error convergence. Both tasks are parametrically dependent, while mutual interaction in the resulting dynamics of the observer is defined by the parameter p > 0, and its interactive setting is, as a rule, sufficient.

6 Concluding remarks

This paper presents new results concerning the interval state estimation of intervally defined ostensible Metzler systems. It proposes how this problem can be formulated using a positive parametric representation and how a constructive procedure based on LMIs can be used respecting the diagonally stabilization principle. It is therefore proven that the gain matrix of the interval observer can be constructed for strict positivity when the stability of the interval observer is defined for a strictly Metzler approximation of the ostensible Metzler system matrix in combination with its stable complement, having a prescribed region of D-stability. The intention was to define the synthesis conditions based only on the quadratic Lyapunov function and to suppress the influence of disturbance in the state estimation by setting the upper bounds of the H_∞ norm of its transfer function matrix. The proposed synthesis conditions are not singular, ensuring fast enough convergence of estimation errors, and do not require prior knowledge of the disturbance boundary. With a constant output matrix and the fact that only the upper and lower bounds of the system dynamics matrix are required, such interval observers have relatively high robustness to changes in system parameters. No comparable results in the field of interval estimators for systems with Metzler dynamics seem to have been published so far.

The use of the class of application models was strictly limited by the occurrence of the description of dynamics in the form of Metzler matrices, a class which also includes models of turbo engines applied in the field of networked aircraft fault tolerant control and diagnosis (Jin and Chen, 2014; Li et al., 2020). The goal of the idea was to derive a method for application in the context of interval observer-based methodology for aircraft engine diagnosis and fault-tolerant control (Lamouchi et al., 2022). It is still left as an open question.

This approach requires further theoretical investigation, especially if the considered continuous-time systems have ostensible Metzler system matrices that have a dominant number of negative and zero elements outside the main diagonal. Further research is thus envisaged on both theoretical and applied aspects in anti-disturbance tracking control for unmanned aerial vehicles and drones considering ostensible Metzler and Hurwitz model parameter setting (Yong, 2022; Song et al., 2023).

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