LMI Based Principles in Strictly Metzlerian Systems Control Design

Hindawi Mathematical Problems in Engineering Volume 2018, Article ID 9590253, 14 pages https://doi.org/10.1155/2018/9590253

Research Article LMI Based Principles in Strictly Metzlerian Systems Control Design

Dušan Krokavec and Anna Filasová

Department of Cybernetics and Artifcial Intelligence, Faculty of Electrical Engineering and Informatics, Technical University of Koˇsice, Letna9,04200Ko´ ˇsice, Slovakia

Correspondence should be addressed to Duˇsan Krokavec; [email protected]

Received 5 August 2017; Revised 15 January 2018; Accepted 22 April 2018; Published 17 July 2018

Academic Editor: Marzio Pennisi

Copyright © 2018 Duˇsan Krokavec and Anna Filasova.´ Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Te paper is concerned with the design requirements that relax the existing conditions reported in the previous literature for continuous-time linear positive systems, reformulating the linear programming approach by the linear matrix inequalities principle. Incorporating an associated structure of linear matrix inequalities, combined with the Lyapunov inequality guaranteeing asymptotic stability of positive system structures, the conditions are presented, with which the state-feedback controllers and the system state observers can be designed. A numerical example illustrates the proposed conditions.

1. Introduction design. Te second group of methods represents [8], where LMI conditions to state observer design for positive linear Positive systems are ofen found in the modeling and control systems are presented, but the observer is not of Luenberger of engineering and industrial processes, whose state variables type since the standard observer gain matrix does not specify represent quantities that do not have meaning unless they the observer dynamics and an additive matrix is used to are nonnegative [1]. Te mathematical theory of Metzler defne the observer stability. matrices has a close relationship to the theory of positive Te most similar idea to the one presented in this linear continuous-time dynamical systems, since in the state- paper is the approach given in [9, 10], using formulation space description form the system dynamics matrix of a based on the set of LMIs. Tese approaches are considered positive systems is Metzler and the system input and output for both positive and nonpositive systems. However, since matrices are nonnegative matrices. In abbreviated terms, such nondiagonal matrix variables are exploited, they do not in continuous-time linear systems are denoted in the following general guarantee diagonally dominant Metzler structure of as Metzlerian systems. Stability and stabilization of such sys- the closed-loop system matrix. Te approach of the present temsarereported,e.g.,in[2–6]andvariousmethodsbased paper makes it possible to achieve this additional constraint, on linear programming (LP) and linear matrix inequalities if required. (LMI) are considered for positive stabilization and observer Te linear programming approach was efectively used designs, as it is further elaborated in subsequent paragraphs. forstaticoutputfeedbackdesignofpositivesystemsin[11, Te task of the state observer design for positive linear 12], and [13, 14] consider both positive observer and control system is solved in [7] using the coordinates transformation. design with bounded control constraint. If the coordinate transform matrix is a solution of the Te main motivation of this paper is to give design specifed Silvester equation and its inverse is nonnegative, conditions for stabilization of linear positive continuous-time the positive observer is constructed only for transformed systems by the state-feedback, as well as the conditions for state coordinates. Another proposed way is to transform Lunberger state observers design in the same systems struc- system into controllable and uncontrollable parts to obtain tures, using the formulation based on strict LMIs. Analyzing a reduced order realization, suitable for positive observer structures of the closed-loop system matrix, and the observer 2 Mathematical Problems in Engineering system matrix, the resulting algebraic constraints, which lead A Metzler matrix � is called strictly Metzler if its diagonal to strictly Metzler matrices, are formulated as a set of LMIs elements are negative and its of-diagonal elements are with diagonal matrix variables. Tis set is extended by an positive.AMetzlermatrixisstableifitisstrictlyMetzlerand LMI, refecting the Lyapunov stability condition [15]. Since Hurwitz. the stability is posed as an LMI, constraints implying from the �×� strictly Metzler matrix structure in the form of LMIs results Proposition 4 (see [21]). AMetzlermatrix� ∈ R is stable also in the LMI formulation [16]. Advantages may also be ifandonlyifallprincipalminorsofthematrix−� are positive. added to the fact that these conditions can be adapted to the Proposition 5 � ∈ R�×� synthesis of nonnegative gain matrices of the regulator or (see [22, 23]). Asquarematrix is estimator, as well as for the synthesis resulting in a Metzler said to be strictly diagonally dominant if matrix structure, if the conditions for its zero nondiagonal � � � � �� > ∑ �� ∀� ∈ ⟨1, �⟩ , elements are satisfed. Preferring LMI structure, the proofs � �� =� ̸ (3) areofstandardwayinthesenseofLyapunovprincipleand refect the facts when optimization in positive systems has where �� denotes the entry in the �th row and �th column. LMIs representation. If the matrix is strictly diagonally dominant and all its Te paper is organized as follows. Afer the Introduction, diagonal elements are negative, then the real parts of all its Section 2 presents some preliminaries, including the char- eigenvalues are negative. A strictly Metzler matrix is stable if acterization of positive systems. A newly introduced set of it is strictly diagonally dominant. LMIs, describing the design conditions for strictly Metzlerian SISO systems, is theoretically substantiated and proven in To highlight some features of positive systems, presented Section 3 and, subsequently, the design conditions for strictly exposition is placed within the framework introduced by the Metzlerian MIMO systems are generalized in Section 4. An monograph [17], p. 8 and 41. example is provided to demonstrate the proposed approach in Section 5, while Section 6 draws some conclusions. Proposition 6 �(�) �≥0 � � (see [17]). Asolution of (1) for is Used notations are conventional so that � , � denote asymptotically stable and positive if � is stable Metzler matrix, � � � ∈ �×� � transpose of the vector and matrix ,respectively, � ∈ R+ is nonnegative matrix and �(�) ∈ R+ for given �(�) ∈ � �×� � R+, � ∈ R+ denote nonnegative vector and nonnegative R+ and �(0) ∈ R+.Telinearsystem(1),(2)isasymptotically � �×� � = � ≻0 � stable and positive if � is stable Metzler matrix, � ∈ R+ , � ∈ matrix, means that is symmetric positive �×� � defnite matrix, �(∗) indicates the eigenvalue spectrum of R+ are nonnegative matrices and �(�) ∈ R+ for all �(�) ∈ R� �(0) ∈ R asquarematrix,thesymbol�� marks the �-th order unit + and +.Telinearsystem(1),(2)isasymptotically [⋅] R� R�×� stableandexternallypositiveif� is stable Metzler matrix and matrix, diag enters up a diagonal matrix, , refer to �×� �×� the set of all �-dimensional real vectors and �×�real matrices, � ∈ R+ , � ∈ R+ are nonnegative matrices. � �×� respectively, and R�, R+ refertothesetofall�-dimensional real nonnegative vectors and �×�real nonnegative matrices, In process control, (2) defnes the measurement sub- �(�) respectively. system on the plant and defnes the output variables, whose values must be complied. If the � matrix would be not nonnegative, some of the output variables could be negative. 2. System Description Tus, considering the nonnegative state variables and � ∈ R�×� In the following the Metzlerian class of positive linear dynam- + ,theoutputvariablesarealsononnegative. ical systems is considered where state-space description is Proposition 7 ([24], Lyapunov inequalities). Autonomous �̇ (�) = �� (�) + �� (�) , (1) system (1) is asymptotically stable if there exist symmetric �×� �×� positive defnite matrices �, � ∈ R or �, � ∈ R such � (�) = �� (�) , (2) that � � � where �(�) ∈ R , �(�) ∈ R , �(�) ∈ R stand for state, � �×� �×� � = � ≻0, control input, and measurable output, � ∈ R , � ∈ R , �×� � � ∈ R . � = � ≻0, (4) Defnition 1 ([17], positive linear system). Te linear system �� + �� + � ≺0, (1), (2) is said to be positive if and only if for every nonnegative initial state and for every nonnegative input its state and � = �� ≻0, output are nonnegative. � = �� ≻0, �×� � (5) Defnition 2. Amatrix� ∈ R+ and a vector � ∈ R+ are � nonnegative if all its entries are nonnegative and at least one �� + � � + � ≺0. is positive. Note that the equivalent statements (4) and (5) are given �×� Defnition 3 (see [18–20]). A square matrix � ∈ R is to circumvent formulations with inverse matrices in the Metzler matrix if its of-diagonal elements are nonnegative. following sections, since direct use (4) leads to bilinear matrix Mathematical Problems in Engineering 3 inequality in the design of the observer, and direct use (5) 3. SISO Strictly Metzlerian Systems leads to bilinear matrix inequality in the synthesis of the regulator. It is easily verifable that premultiplying the lef side Te systems under consideration in this section are linear −1 and postmultiplying the right side by the matrix � = � and single-input, single-output (SISO) continuous-time dynamic −1 −1 setting � = � �� ; then (5) implies (4). systems represented in state-space form as

�×� Lemma 8. If � ∈ R is a symmetric positive defnite matrix, �̇ (�) = �� (�) + �� (�) , (9) it yields the following relation: � � (�) = � � (�) , (10)

� � = �� ≻0⇐⇒ where �(�) ∈ R is the vector of the system state variables, �(�) ∈ R, �(�) ∈ R are the input and output variables, (6) �×� � � 1 1 � 1 � ∈ R is Metzler matrix, � ∈ R+, � ∈ R� are nonnegative � + � = � + (∗) ≻0, � 2 2 2 vectors, the pair (�, �) is controllable, and the pair (�, � ) is observable. �×� � Note that if � ∈ R is Metzler matrix, and � ∈ R+, � � ∈ R are nonnegative vectors, (9) and (10) are Metzlerian ∗ (1/2)�� (1/2)� + where denotes the symmetric item to . system. Proof. Since it yields 3.1. State Controller Design. Considering system (9), (10), the full state control law is defned as

� � = � ≻0, � (7) � (�) =−� � (�) , (11) 1 1 1 1 � � = � + � = � + � ≻0, (8) � 2 2 2 2 where � ∈ R+ is the gain vector. Tus, using (9), (11), it yields

� �̇ (�) =(� − �� ) � (�) = �� (�) , (12) (7) and (8) imply (6). Tis concludes the proof. � � (�) = � � (�) , (13) Defnition 9 ([25], congruent modulo �). Let � be a fxed positive integer. Two integers � and ℎ are congruent modulo � if they difer by an integral multiple of the integer � (they where leave the same remainder when divided by �). If � and ℎ are � congruent modulo �, the expression (� = ℎ)mod � is called a �� = � − �� (14) congruence, and the number � is called the modulus of the congruence. is the closed-loop system dynamics matrix. Note that the controller has to be designed not only to Note that the statement (� = ℎ)mod � is equivalent to the statement “(� − ℎ) is divisible by �” or to the statement “there stabilize the system, but also to render the stable closed-loop is an integer � for which �−ℎ=��” and the word “modulo” system matrix strictly Metzlerian. generally means “to the modulus”. Te following theorem defnes the LMI conditions to obtain a strictly Metzler matrix �� but does not guarantee that � Defnition 10 (see [26]). Let � = {0,1,2,...,� − 1} be � is stable strictly Metzler matrix. the complete set of residues for any positive integer �.Te Teorem 12. Te closed-loop system (12), (13) is strictly addition modulo � on � is (ℎ + �)mod � =�,where� is the � Metzlerian if system (9), (10) is strictly Metzlerian and there element of to which the result of the usual sum of integers � ∈ R�×� ℎ and � is congruent modulo �. exists a positive defnite diagonal matrix such that for ℎ = 1, 2, . . . � − 1, Corollary 11. Te problem of indexing in this paper is that the rows and columns of a square matrix of dimension �×�are 1 (� (�, �) − � �) + (∗) ≺0, (15) generally denoted from 1 to � and not from 0 to �−1.Fromthis 2 (1←→�) � reason let �={0,1,2,...,�}be the complete set of residues for 1 ℎ ℎ� ℎ ℎ� any positive integer �+1. Ten the addition modulo �+1on � (� � (�, � + ℎ) � − � � � �)+(∗) 2 (1←→�)/� � is in the following defned as (�+ℎ)mod �+1 =�+1,where� is the (16) element of � to which the result of the usual sum of integers ℎ ≻0, and � is congruent modulo �+1. Te used shorthand symbolical notation for (1 + ℎ)mod �+1 =�+1is so (� + ℎ)(1←→�)/� =�+1. � = diag [�1 �2 ⋅⋅⋅ ��]≻0, (17) 4 Mathematical Problems in Engineering

where respectively, where � = diag[{��}�=1,...,�] is the diagonal 00⋅⋅⋅01 matrix variable. [ ] Moreover, (26) implies [10⋅⋅⋅00] [ ] � = [ ] , [ d ] �11 (18) [ ] [ � ] [00⋅⋅⋅10] [ 22 ] [ ] [ d ] �−1 = ��, [ ��] � � =[�1 �2 ⋅⋅⋅ ��], (19) (28) �1 �1 [ ] [ ] �� = diag [�1 �2 ⋅⋅⋅ ��]=diag [{��} ], (20) [ � ] [ � ] �=1,...,� [ 2 ] [ 2 ] − [ ] [ ] ≺0, [ d ] [ d ] � (�, �)1←→� = diag [�11 �22 ⋅⋅⋅ ��] (21) [ ��] [ ��] = diag [{��}�=1,...,�],

� (�, � + ℎ)(1←→�)/� Tat is, (27) is a symmetric matrix. Tus, exploiting (6), then (22) (27) can be written as (15) using the notations (17) and (20), = diag [�1,1+ℎ �2,2+ℎ ⋅⋅⋅ ��−ℎ,� ��−ℎ+1,1 ⋅⋅⋅ ��,ℎ], (21). Rewriting (23) as �×� while �(�, �)1←→�, �(�, � + ℎ)(1←→�)/�, �, ��, � ∈ R . Hereafer, ∗ denotes the symmetric item in a symmetric �12 �13 ⋅⋅⋅ �1� �11 matrix. [ ] [� � ⋅⋅⋅ � � ] [ 22 23 2� 21] � [ ] Proof. Writing the closed-loop dynamics matrix � as fol- [ . ] lows: [ . ] � � ⋅⋅⋅ � � �1 [ �2 �3 �� 1] �11 �12 ⋅⋅⋅ �1� [ ] [ ] (29) [� � ⋅⋅⋅ � ] [�2] � [ 21 22 2�] [ ] 1 − [ ] [�1 �2 ⋅⋅⋅ ��]≺0, (23) [ ] [ d ] [ . ] [� ] [ ] [ . ] [ 2] − [ . ] [�2 �3 ⋅⋅⋅ �� 1], [��1 ��2 ⋅⋅⋅ ��] [ . ] [��] [ . ] � it is evident that �� is a strictly Metzler matrix if all of its main [ �] diagonal elements are negative, i.e., it can set for the diagonal elements of (29) �� −�� <0 ∀�=1,2,...,� (24) andallitsofdiagonalelementsarepositive,i.e., �12 [ ] [ � ] �� −�� >0 ∀�,�=1,2,...,�, �=�.̸ (25) [ 23 ] [ ] [ d ] To solve by an LMI solver, LMIs have to be symmetric, � and so (24) can be rewritten in the following diagonal matrix [ �1] structure: (30) �1 �2 � [ ] [ ] 11 [ � ] [ � ] [ ] [ 2 ] [ 3 ] [ � ] − [ ] [ ] ≻0, [ 22 ] [ d ] [ d ] [ ] [ d ] [ ��] [ �1] [ ��] (26) which leads to �1 �1 [ ] [ ] [ � ] [ � ] [ 2 ] [ 2 ] � (�, � + 1) − � � ≻0, − [ ] [ ] ≺0, (1←→�)/� � �1 (31) [ d ] [ d ] � � [ ��] [ ��] where �1 is the diagonal matrix with one circular shif of its diagonal elements. It is evident that (31) is also a symmetric � (�, �)(1←→�) − �� ≺0, (27) matrix. Mathematical Problems in Engineering 5

�×� Repeating this procedure ℎ-times, it can be obtained from existpositivedefnitediagonalmatrices�, �, � ∈ R such (23) that that for ℎ=1,2,...�−1, � � � �� + �� − �� − �� + � ≺0, (38) �1,1+ℎ ⋅⋅⋅ �1,� �1,1 ⋅⋅⋅ �1,ℎ [ ] 1 [� ⋅⋅⋅ � � ⋅⋅⋅ � ] (� (�, �) � − � �) + (∗) ≺0, [ 2,�+ℎ 2,� 2,1 2,ℎ] (1←→�) � (39) [ ] 2 [ . . . . ] [ . d . . d . ] 1 (�ℎ� (�, � + ℎ) �ℎ�� − �ℎ� �ℎ��)+(∗) 2 (1←→�)/� � [��,�+ℎ ⋅⋅⋅ ��,� ��,1 ⋅⋅⋅ ��,ℎ] (40) (32) ≻0, �1 [ ] [� ] � = diag [�1 �2 ⋅⋅⋅ ��]≻0, (41) [ 2] − [ ] [� � ⋅⋅⋅ � ], [ . ] 1+ℎ 2+ℎ ℎ � = [� � ⋅⋅⋅ � ]≻0, [ . ] diag 1 2 � (42)

[��] � = diag [�1 �2 ⋅⋅⋅ ��]≻0, (43)

� �� =[� � ⋅⋅⋅ � ]=��, � =[11⋅⋅⋅1] , (44) and, consequently, 1 2 � where the parameters are given in (18)–(22). � When the above conditions hold, the control law gain vector 1,1+ℎ � [ ] � is given as [ � ] [ 2,2+ℎ ] [ ] � = ��−1, �� = ��. [ d ] (45)

[ ��,ℎ] Proof. Inserting (14) into (4) gives (33) � (� − ��) � + � (� − ��) + � ≺0. �1 �1+ℎ (46) [ ] [ ] [ � ] [ � ] � [ 2 ] [ 2+ℎ ] Since with a positive diagonal matrix it also yields − [ ] [ ] ≻0, d d � � � [ ] [ ] �� + �� − �� − �� + � ≺0, (47) [ � ] [ � ] � ℎ using the notation �� = �� which gives the symmetric matrix inequality (48) (47) implies (38). � � (�, � + ℎ)(1←→�)/� − ��,ℎ ≻0, (34) Multiplying the right side of (27) by gives � (�, �)(1←→�) � − �� ≺0, (49) � � ℎ where �ℎ is the diagonal matrix with circular shifs of and with the notation its diagonal elements. Using the permutation matrix � of the structure (58) [22] � = �� (50) it can be easily verifed that for ℎ=1,2,...�−1it yields (49) implies (39). Multiplying the right side of (37) by � gives ℎ −ℎ ℎ ℎ� � = � ��,ℎ� = � ��,ℎ� . (35) ℎ ℎ� ℎ ℎ� � � (�, � + ℎ)(1←→�)/� � � − � �� ≻0 (51)

ℎ� and with the notation (50) then (51) implies (40). Tis Tus, premultiplying the right side by � and postmultiply- ℎ concludes the proof. ing the lef side by � then (34) leads to Note that conditions (38)-(43) are all LMIs; that is, they ℎ ℎ� ℎ ℎ� ℎ ℎ� areconvexinthedefnedmatrixvariables. � � (�, � + ℎ)(1←→�)/� � − � �� ,ℎ� ≻0, (36) Remark 14. It can be noted that the necessary diagonal matrix �ℎ� (�, � + ℎ) �ℎ� − �ℎ� �ℎ�� ≻0, (1←→�)/� � (37) variable structure of �,presentedinTeorem12,directly implies the diagonal matrix variable structures of �, � in respectively,andusing(6),then(37)implies(16).Tis Teorem13.Since,inthetermsoftheKrasovskiitheorem concludes the proof. [27], the matrices �, � in Proposition 7 can be zero matrices, this way can be also applied in (38), and in the inequalitieswe Teorem 13. Te closed-loop system (12), (13) is stable strictly exploited the proposition properties in the following parts of Metzlerian if system (9), (10) is strictly Metzlerian and there the paper. 6 Mathematical Problems in Engineering

3.2. Luenberger Observer Design. Considering the observable it is evident that �� is a strictly Metzler matrix if system (9), (10), the Luenberger observer is given as � −�� <0 ∀�=1,2,...,�, �̇� (�) = �� (�) + �� (�) + � (� (�) −�� (�)) , (52) �� (64)

� �� −�� >0 ∀�,�=1,2,...,�, �=�.̸ (65) �� (�) = � �� (�) (53) and using (9), (10) and (52), (53) it yields Writing (64) in the diagonal matrix structure � �̇ (�) =(� − �� ) � (�) = �� (�) , (54) �11 where [ ] [ �22 ] � (�) = � (�) − �� (�) , [ ] [ d ] � (55) [ ] �� = � − �� , [ ��] � ��(�) ∈ R is the vector of the system state estimate, ��(�) ∈ (66) � � � � R is the output variable estimate, and � ∈ R is the observer 1 1 [ ] [ ] gain vector. [ � ] [ � ] [ 2 ] [ 2 ] Note that the observer has to be designed not only to be − [ ] [ ] ≺0, [ d ] [ d ] stable, but also to render the observer system matrix stable strictly Metzlerian. [ ��] [ ��]

Teorem 15. Te Luenberger observer (52), (53) is strictly � (�, �)(1←→�) − �� ≺0, (67) Metzlerian if system (9), (10) is strictly Metzlerian and there �×� exists a positive defnite diagonal matrix � ∈ R such that � = [{� } ] ℎ=1,2,...�−1 respectively, where diag � �=1,...,� is the diagonal matrix for variable, it is evident that such structure is symmetric and 1 (� (�, �) − �� )+(∗) ≺0, (67) can be written as (56) using notations (60), (61). 2 (1←→�) � (56) Overwriting inequality (63) as

1 ℎ ℎ� ℎ ℎ� (� � (�+ℎ,�) � − �� )+(∗) � � ⋅⋅⋅ � � 2 (1←→�)/� � 21 22 2� 2 (57) [ ] [ ] [� � ⋅⋅⋅ � ] [� ] ≻0, [ 31 32 3�] [ 3] [ ] [ ] [ . ] [ . ] [ . ] − [ . ] [�1 �2 ⋅⋅⋅ ��], (68) � = diag [�1 �2 ⋅⋅⋅ ��] ≻0, (58) [ ] [ ] [ ] [ ] [� � ⋅⋅⋅ � ] [� ] where � is defned in (18), �1 �2 �� [�11 �12 ⋅⋅⋅ �1�] [�1] � =[�1 �2 ⋅⋅⋅ ��], (59) � � ⋅⋅⋅ � �� = diag [ 1 2 �]=diag [{��}�=1,...,�], (60) ithastobeforthediagonalelementsof(68) � (�, �) = [� � ⋅⋅⋅ � ] 1←→� diag 11 22 �� 21 [ ] (61) [ � ] = diag [{��} ], [ 32 ] �=1,...,� [ ] [ d ] � (�+ℎ,�) (1←→�)/� � (62) [ 1�] = diag [�1+ℎ,1 �2+ℎ,2 ⋅⋅⋅ ��,�−ℎ �1,�−ℎ+1 ⋅⋅⋅ �ℎ,�], (69) �2 �1 �(�, �) , �(� + ℎ, �) �, � , � ∈ R�×� [ ] [ ] while 1←→� (1←→�)/�, � . [ � ] [ � ] [ 3 ] [ 2 ] − [ ] [ ] ≻0, Proof. Since the duality principle has to be applied on the [ d ] [ d ] structure (23), to avoid some misinterpretation the proof is [ � ] [ � ] giveninpartlyfulldualsense. 1 � Noting down the observer matrix �� structure (55) as which leads to the symmetric inequality �1 �11 �12 ⋅⋅⋅ �1� [ ] [ ] [�2] � (�+1,�) − � � ≻0, [�21 �22 ⋅⋅⋅ �2�] [ ] (1←→�)/� �1 � (70) [ ] − [ ] [� � ⋅⋅⋅ � ]≺0, [ ] [ . ] 1 2 � (63) [ d ] [ . ] where ��1 is the diagonal matrix � with one circular shif of [��1 ��2 ⋅⋅⋅ ��] [��] its diagonal elements. Mathematical Problems in Engineering 7

�×� Repeating this procedure ℎ-times, it can obtain the there exist positive defnite diagonal matrices �, �, � ∈ R following: such that for ℎ=1,2,...�−1

� � � �� + � � − �� − �� + � ≺0, (77) � � ⋅⋅⋅ � � 1+ℎ,1 1+ℎ,2 1+ℎ,� 1+ℎ 1 [ ] [ ] (�� (�, �) − �� )+(∗) ≺0, [ . ] [ . ] (1←→�) � (78) [ . ] [ . ] 2 [ ] [ ] [ ] [ ] 1 [ � � ⋅⋅⋅ � ] [ � ] (��ℎ� (�+ℎ,�) �ℎ� − ��ℎ� �ℎ�)+(∗) [ �,1 �,2 �,� ] [ � ] (1←→�)/� � [ ] − [ ] [�1 �2 ⋅⋅⋅ ��], (71) 2 [ � � ⋅⋅⋅ � ] [ � ] (79) [ 11 12 1� ] [ 1 ] [ ] [ ] ≻0, [ . ] [ . ] [ . ] [ . ] � = diag [V1 V2 ⋅⋅⋅ V�] ≻0, (80) [ �ℎ,1 �ℎ,2 ⋅⋅⋅ �ℎ,� ] [ �ℎ ] � = diag [�1 �2 ⋅⋅⋅ ��]≻0, (81) �1+ℎ,1 [ ] � = [� � ⋅⋅⋅ � ]≻0, [ � ] diag 1 2 � (82) [ 2+ℎ,2 ] [ d ] � � � [ ] � =[�1 �2 ⋅⋅⋅ ��]=� �, � =[11⋅⋅⋅1] , (83)

[ �ℎ,�] (72) where the parameters are given in (18), (59)–(62). �1+ℎ �1 When the above conditions hold, the observer gain vector � [ ] [ ] [ � ] [ � ] is given as [ 2+ℎ ] [ 2 ] − [ ] [ ] ≻0, [ d ] [ d ] −1 � = � �, � = ��. (84) [ �ℎ] [ ��] Proof. Inserting (55) into (5) gives for a positive defnite diagonal matrix � that respectively. Tus, analogously, (72) results in � � (� − ��) + (� − ��) � + � ≺0, (85)

�� + �� − �� − �� + � ≺0, � (�+ℎ,�)(1←→�)/� − ��,ℎ�� ≻0. (73) (86)

respectively. Terefore, using the notation � Using the permutation matrix of the structure (58) it �� = ��, yields for ℎ=1,2,...�−1that (87)

(86) implies (77). � ℎ −ℎ ℎ ℎ� Multiplying the lef side of (67) by gives � = � ��,ℎ� = � ��,ℎ� (74) �� (�, �)(1←→�) − �� ≺0, (88)

ℎ and premultiplying the lef side by � and postmultiplying and with the notation ℎ� the right side by � then the symmetric structure (73) � = �� (89)

(88) implies (78). �ℎ� �+ℎ,� �ℎ� − �ℎ� �ℎ��ℎ� �ℎ� ≻0, ( )(1←→�)/� �,ℎ � (75) Multiplying the lef side of (76) by � gives ℎ ℎ� ℎ ℎ� � � (�+ℎ,�)(1←→�)/� � − �� ≻0, (76) ℎ ℎ� ℎ ℎ� �� (�, � + ℎ)(1←→�)/� � − �� ≻0 (90)

and with the notation (89) then (90) implies (79). Tis respectively, and using (6), then (76) implies (57). Tis concludes the proof. concludes the proof.

Teorem 16. Te Luenberger observer (52), (53) is stable Also conditions (77)-(82) are all LMIs that is they are strictly Metzlerian if system (9), (10) is strictly Metzlerian and convex in the defned matrix variables. 8 Mathematical Problems in Engineering

3.3. Supporting Matrix Structures. To simplify obtaining where �(�, � + ℎ) �(� + ℎ, �) (1←→�)/� or (1←→�)/�,itispossibleto � (�) = � (�) − � (�) , construct the following matrices: � (100) � = � − ��. �∘ =[��] � � ∈ R�×� � ∈ � � ⋅⋅⋅ � � � ⋅⋅⋅ � Naturally, if is Metzler matrix, and 11 12 1� 11 12 1� R�×� � ∈ R�×� [ ] + , + are nonnegative matrices, system (1), (2) is [� � ⋅⋅⋅ � � � ⋅⋅⋅ � ] [ 21 22 2� 21 22 2�] (91) Metzlerian system. Tus, it is necessary to render the closed- = [ ] , � � [ . . ] loop system matrix � and observer system matrix � as [ . d . d ] stable strictly Metzler ones. [� � ⋅⋅⋅ � � � ⋅⋅⋅ � ] 11 12 1� 11 12 1� Teorem 17. Te closed-loop system (94), (95) is strictly Metzlerian if system (1), (2) is strictly Metzlerian and there exist �11 �12 ⋅⋅⋅ �1� �×� [ ] positive defnite diagonal matrices �� ∈ R such that for [� � ⋅⋅⋅ � ] [ 21 22 2�] ℎ=1,2,...�−1, �=1,2...,�, [ ] [ . ] [ . d ] � [ ] 1 [ ] (� (�, �)1←→� − ∑��)+(∗) ≺0, (101) � [� � ⋅⋅⋅ � ] 2 ⬦ [ �1 �2 ��] �=1 � =[ ]=[ ] , (92) � [�11 �12 ⋅⋅⋅ �1�] � [ ] 1 ℎ ℎ� ℎ ℎ� [ ] (� � (�, � + ℎ)(1←→�)/� � − ∑� �� ) [�21 �22 ⋅⋅⋅ �2�] 2 [ ] �=1 [ ] (102) [ . ] [ . d ] + (∗) ≻0,

[��1 ��2 ⋅⋅⋅ ��] �� = diag [��1 ��2 ⋅⋅⋅ ��]≻0, (103) respectively. � �(�, �) �(�, � + ℎ) �−1 where is defned in (18) and 1←→�, (1←→�)/� Ten, using the set of upper subdiagonals of are introduced in (21), (22), respectively, dimension of �,thematrices�(�, � + ℎ)(1←→�)/� can be ℎ = 1,2,...�−1 sequentially constructed for from (91) and, �11 �12 ⋅⋅⋅ �1� exploiting the set of �−1lower subdiagonals of dimension [ ] [� � ⋅⋅⋅ � ] of �,thematrices�(� + ℎ, �)(1←→�)/� canbesequentially [ 21 22 2�] � = [ ] , constructed for ℎ=1,2,...�−1refecting (92). [ . ] (104) [ . ] � � ⋅⋅⋅ � 4. MIMO Strictly Metzlerian Systems [ �1 �2 ��] � � ⋅⋅⋅ � Linear multiple-input, multiple-output (MIMO) continuous- �� = diag [ 1� 2� ��]=diag [{��}�=1,...,�], (105) time closed-loop dynamical systems obtained from the con- �×� trollable system (1), (2) by using the state control law while ��, �� ∈ R .

�×� � (�) =−�� (�) , � ∈ R , (93) Proof. Inanalogywith(24)itisevidentthat�� is Metzler matrix if all of its main diagonal elements satisfy the condi- are described by the state-space equations tions � �̇ (�) = (� − ��) � (�) = �� (�) , (94) �� − ∑�� <0 ∀�=1,2,...,� (106) �=1 � (�) = �� (�) , (95) andallitsofdiagonalelementssatisfytheinequalities where � � − ∑� � >0 ∀�,�=1,2,...,�, �=�.̸ �� = � − ��. (96) �� (107) �=1 Te MIMO Luenberger observer, associated with the Tus, using the above notations and (103), (105), then (106) observable system (1), (2), is given as implies

�̇� (�) = �� (�) + �� (�) + � (� (�) − �� (�)) , (97) � � (�, �)(1←→�) − ∑�� ≻0 (108) �� (�) = �� (�) , (98) �=1

�̇ (�) = (� − ��) � (�) = �� (�) , (99) and, consequently, (108) implies (101). Mathematical Problems in Engineering 9

Inthesameway,(107)canbeinterpretedas Since multiplying the right side of (108), (109) by � leads to � ℎ ℎ� ℎ ℎ� � � (�, � + ℎ)(1←→�)/� � − ∑� �� ≻0 (109) �=1 � � (�, �)(1←→�) � − ∑�� ≺0, (121) and (109) implies (102). Tis concludes the proof. �=1 � Teorem 18. Te closed-loop system (12), (13) is stable strictly ℎ ℎ� ℎ ℎ� � � (�, � + ℎ)(1←→�)/� � � − ∑� �� ≻0, (122) Metzlerian if system (1), (2) is strictly Metzlerian and there exist �×� �=1 positive defnite diagonal matrices �, ��, � ∈ R such that for ℎ=1,2,...�−1, �=1,2,...�,

� then, with the notation � � � �� + �� − ∑ (�� + �� ) + � ≺0, (110) �=1 �� = ��, (123) 1 � (� (�, �) � − ∑� � ) + (∗) ≺0, 2 (1←→�) �� (111) �=1 (121) and (122) imply (111), (112), respectively. Tis concludes 1 � the proof. (�ℎ� (�, � + ℎ) �ℎ�� − ∑�ℎ� �ℎ�� ) 2 (1←→�)/� �� =1 (112) Remark 19. Te point was fnding conditions for the synthesis + (∗) ≻0, of the state control law to stabilize positive systems that would be sufciently general. Te stabilizability is bound to a particular system pair (�, �) and the strictly Metzler matrix � = diag [�1 �2 ⋅⋅⋅ ��]≻0, (113) � does not a priori generate any boundaries on elements of � � = diag [�1 �2 ⋅⋅⋅ ��]≻0, (114) the nonnegative matrix to obtain that the matrix of closed- loop dynamics �� = � − �� is (strictly) Metzler matrix if � �� = diag [��1 ��2 ⋅⋅⋅ ��] ≻0, (115) is positive matrix. If the Metzler matrix element �� =0, �=� ̸ it follows for � � � �� = [��1 ��2 ⋅⋅⋅ ��] = � ��, � = [1 1 ⋅⋅⋅ 1] , (116) given � from condition (107) that all elements of row � of the matrix � must be zero. With such a boundary on the structure � � � � where �� = � �� and the remaining parameters are given in of the matrix , the positive gain matrix , if exists, can be (18), (21), and (22). designed according to Teorem 18. When the above conditions hold, the control gain matrix � Te second way is to design � as nonnegative matrix. In � =0 �=� ̸ is given as this case, if �� , , it is necessary to defne the diagonal matrices �� in(115)asstructuredmatrixvariables,whose�� −1 � �� = �� , elements are zero for given , and then the nonnegative gain matrix �, if exists, can be designed according to Teorem 18. � � �� = � ��, Potentially,bothwayscanbecombined. Analogously the above-mentioned approaches can be �� (117) applied in Luenberger observer synthesis for Metzlerian [ 1 ] [ . ] systems. � = [ . ] . [ . ] Teorem 20. Te Luenberger observer (97), (98) is strictly �� [ � ] Metzlerian if system (1), (2) is strictly Metzlerian and there �×� exist positive defnite diagonal matrices �� ∈ R such that Proof. Inserting (96) into (4) gives for ℎ = 1, 2, . . . � − 1, � = 1, 2, . . . �,

� (� − ��) � + � (� − ��) + � ≺0. (118) 1 � � � � (� (�, �) − ∑� � )+(∗) ≺0, � � 2 1←→� � �� (124) (� − ∑�� ) � + � (� − ∑�� ) + � ≺0. (119) �=1 �=1 �=1 1 � (�ℎ� (�+ℎ,�) �ℎ� − ∑� �ℎ� �ℎ�) respectively, and using the notation 2 (1←→�)/� � � �=1 (125) � � �� = �� , (120) + (∗) ≻0,

(119) implies (110). �� = diag [�1� �2� ⋅⋅⋅ ��]≻0, (126) 10 Mathematical Problems in Engineering

where � is defned in (18), �(�, �)1←→�, �(� + ℎ, �)(1←→�)/� is there exists a positive defnite diagonal matrices �, �, �� ∈ �×� introduced in (61), (62), respectively, R such that for ℎ = 1, 2, . . . � − 1, �=1,2,...�,

� � � � �� + � � − ∑ (�� + �� ) + � ≺0, (133) �=1 �11 �12 ⋅⋅⋅ �1� [ ] 1 � [�21 �22 ⋅⋅⋅ �2� ] [ ] (�� (�, �)(1←→�) − ∑��)+(∗) ≺0, (134) � = [ ] , 2 [ . ] (127) �=1 [ . ] 1 ℎ ℎ� [��1 ��2 ⋅⋅⋅ ��] (�� (�+ℎ,�) � 2 (1←→�)/� �� = diag [��1 ��2 ⋅⋅⋅ ��] , �=1,2...,�, (128) (135) � ℎ ℎ� − ∑�� )+(∗) ≻0, �=1

�×� � = [V V ⋅⋅⋅ V ] ≻0, while �� ∈ R . diag 1 2 � (136)

� = [�1 �2 ⋅⋅⋅ ��]≻0, Proof. In analogy with (63) it is evident that �� is Metzler diag (137) matrix if �� = diag [��1 ��2 ⋅⋅⋅ ��]≻0, (138)

� � �� =[��1 ��2 ⋅⋅⋅ ��]=� ��, � (139) � =[11⋅⋅⋅1]� , �� − ∑�� <0 ∀�=1,2,...,�, (129) �=1 � � � where �� = � �� and the remaining parameters are given in �� − ∑�� >0 ∀�,�=1,2,...,�, �=�.̸ (130) (18), (61), and (62). �=1 When the above conditions hold, the observer gain matrix � is given as

−1 �� = � ��, Tus, using the above notations and (126), (127) then (129) � = � �, implies � � (140)

� =[�1 ⋅⋅⋅ ��].

� Proof. Inserting (100) into (5) gives � (�, �)(1←→�) − ∑�� ≻0 (131) � �=1 � (� − ��) + (� − ��) � + � ≺0, (141)

� � � � � � (� − ∑�� )+(� − ∑�� ) � + � ≺0, (142) and, consequently, (131) implies (124). �=1 �=1 Inthesameway,(130)canbeinterpretedas respectively. Terefore, using the notation

� � �� = �� , (143) � �ℎ� (�, � + ℎ) �ℎ� − ∑� �ℎ� �ℎ� ≻0 (1←→�)/� � �� (132) (142) implies (133). �=1 Multiplying the lef side of (131) and (132) by � gives

� �� (�, �)(1←→�) − ∑�� ≺0, (144) and (132) implies (125). Tis concludes the proof. �=1 � Teorem 21. Te Luenberger observer (97), (98) is stable ℎ ℎ� ℎ ℎ� �� (�, � + ℎ)(1←→�)/� � − ∑�� ≻0, (145) strictly Metzlerian if system (1), (2) is strictly Metzlerian and �=1 Mathematical Problems in Engineering 11 respectively, and with the notation � (�, �)(1←→4) −3.3800 [ ] [ −4.2900 ] � = �� [ ] � � (146) = [ ] [ −6.6540 ] [ −2.1040] (144), (145) imply (134), (135), respectively. Tis concludes the � (�, � + 1) proof. (1←→4)/4 0.2080 Remark 22. If the boundary conditions can be formulated [ ] [ 2.0500 ] as quadratic boundaries (e.g., LQ constraints on state and [ ] � � = [ ] , inputs, ∞-norm, -stability circle area, etc.), they can be [ 5.8930 ] included in the presented design conditions for positive systems because they only modify the structure of the LMIs [ 0.0480] (110), (133), forcing the system stability or add new LMSs. � (�, � + 2)(1←→4)/4 6.7150 5. Illustrative Example [ ] [ 0.6750 ] [ ] = [ ] , Te considered strictly Metzlerian system is represented by [ 1.0670 ] the model (1), (2) with the parameters [ 2.2730]

� (�, � + 3)(1←→4)/4 −3.3800 0.2080 6.7150 5.6760 [ ] 5.6760 [ 0.5810 −4.2900 2.0500 0.6750 ] [ ] [ ] � = [ ] , [ 0.5810 ] [ 1.0670 4.2730 −6.6540 5.8930 ] [ ] = [ ] . [ 4.2730 ] [ 0.0480 2.2730 1.3430 −2.1040] [ 1.3430] 0.0400 0.0189 [ ] (147) (149) [0.0568 0.0203] [ ] � = [ ] , Using the SeDuMi package [28] to solve the given set of [0.0114 0.0315] LMIs (110)–(116) the LMI variables are [0.0114 0.0170] � = diag [0.7015 0.0570 0.1416 0.0521], 4010 � =[ ]. � = diag [0.1937 0.0262 0.0795 0.0094], 0001 (150) �1 = diag [1.5314 0.0853 2.9122 0.0469],

�2 = diag [0.5444 0.3043 5.2296 1.3593]. It is possible to verify that the Metzler matrix �is unstable with the eigenvalue spectrum Te control law gain matrix is computed by using (117) as � �1 =[2.1831 1.4967 20.5689 0.9003], � (�) ={1.9761, −9.4392, −4.4824 ± 1.2499 }. � i (148) �2 =[0.7761 5.3383 36.9371 26.0694], (151) 2.1831 1.4967 20.5689 0.9003 � =[ ], To solve the stabilization task, the auxiliary parameters 0.7761 5.3383 36.9371 26.0694 are constructed as which imply the stable strictly Metzler matrix of closed-loop dynamics (96) 0001 −3.4820 0.0473 5.1949 5.1478 [ ] [ ] [1000] [ 0.4413 −4.4834 0.1321 0.0947 ] [ ] [ ] � = [ ] , �� = [ ] , (152) [0100] [ 1.0178 4.0881 −8.0497 5.0626 ] [0010] [ 0.0100 2.1652 0.4810 −2.5577] 12 Mathematical Problems in Engineering

0.14 To design the strictly Metzlerian observer, the auxiliary 0.12 parameters are constructed as

0.1 � (�+1,�)(1←→4)/4 0.08 0.5810 q(t) 0.06 [ ] [ 4.2730 ] [ ] 0.04 = [ ] , [ 1.3430 ] 0.02 [ 5.6760] 0 0 12345678910 � (�+2,�) t [s] (1←→4)/4 1.0670 Ｋ1(Ｎ) Ｋ3(Ｎ) [ ] Ｋ2(Ｎ) Ｋ4(Ｎ) [ 2.2730 ] (154) [ ] = [ ] , Figure 1: Te system state variable responses. [ 6.7150 ] [ 0.6750] 0.7 � (�+3,�)(1←→4)/4 0.6 0.0480 0.5 [ ] [ 0.2080 ] [ ] 0.4 = [ ] . [ 2.0500 ] y(t) 0.3 [ 5.8930] 0.2

0.1 Terefore, the set of LMIs (133)–(139) is satisfed with the LMI 0 variables 0 12345678910 t [s] � = diag [0.1869 0.5680 0.3436 0.3567], y1(Ｎ) � = [0.2032 0.5450 0.3678 0.5954], y (Ｎ) diag 2 (155) � = [0.0294 0.0583 0.0676 0.0016], Figure 2: Te system output variable responses. 1 diag

�2 = diag [0.8587 0.2377 1.6965 0.9121]. with the eigenvalue spectrum Te observer gain matrix is computed by using (140) as

0.1573 �(��)={−0.5008, −9.1379, −4.4670 ± 1.2281i}. (153) [ ] [0.1026] [ ] �1 = [ ] , Analyzing the numerical results, it is evident that in [0.1967] this case the stabilization of the unstable strictly Metzlerian [0.0046] system principally changes the value of the unstable mode, whilestablemodesaremodifedverylittle.Itisalsoobvious 4.5958 that the strictly Metzler matrix �� is stable but not diagonally [ ] [0.4186] dominant. [ ] �2 = [ ] , (156) Te obtained results are illustrated in Figures 1 and 2, [4.9373] where the state variables vector �(�) as well as the output 2.5572 variables vector �(�) are positive when the input of the closed- [ ] loop system is �(�) = −��(�) + �(�) with the positive forced � 0.1573 4.5958 vector � (�) = [0.5 0.3]. [ ] [0.1026 0.4186] One can verify that these results are perfectly comparable [ ] � = [ ] , with the ones obtained by the methodology constructed [0.1967 4.9373] forthediscrete-timepositiveLTIsystemsfortheassociated 0.1967 2.5572 continuous-time and discrete-time system models [29]. [ ] Mathematical Problems in Engineering 13

0.14 observer gain that establish the stable strictly Metzlerian 0.12 structures of Luenberger observers. Based on a stable strictly Metzler matrix, algebraic constraints implying from linear 0.1 programming approach are reformulated as a set of LMIs 0.08 and replenished by the Lyapunov matrix inequality in the (Ｎ)

？ sense of the second Lyapunov method. It is derived that all Ｋ 0.06 matrixvariablesassociatedwiththisLMIsensemblehave 0.04 to be positive defnite and diagonal. Te proposed approach provides a numerically reliable computational framework, 0.02 as illustrated using the numerical example, and might be 0 extended to other particular cases. 0 12345678910 Teexamplealsoillustratesthatatthelargepositive t [s] values of some elements outside the diagonal of the strictly Ｋ 1(Ｎ) Ｋ 3(Ｎ) Metzler matrix of the system �, neither the closed-loop ？？ � � Ｋ？2(Ｎ) Ｋ？4(Ｎ) system matrix � nor the observer system matrix � is diagonally dominant, although they are stable strictly Metzler Figure 3: Te system state variable estimates. matrices. Te results of our other simulations show that if the system matrix � is diagonally dominant strictly Metzler matrix, the matrices �� and �� are diagonally dominant which leads to the stable strictly Metzler matrix of the strictly Metzler matrix. Further research topics include this observer dynamics (100) problemaswellastheproblemsofforcedmodeinpositive systems, discrete-time linear positive structures, and nonlin- −4.0092 0.0507 6.5577 1.0802 ear positive systems. [ ] [ 0.1706 −4.3926 1.9474 0.2564 ] [ ] �� = [ ] , (157) [ 0.2802 4.0763 −6.8507 0.9557 ] Conflicts of Interest [ 0.0026 2.2684 1.3384 −4.6612] Te authors declare that there are no conficts of interest regarding the publication of this paper. with the eigenvalue spectrum �(� )={−1.3666, −4.6855, −4.9957 −8.8659}. � (158) Acknowledgments Analyzing the results, it is evident that the stable strictly Te work presented in this paper was supported by VEGA, Metzlerian observer is designed to the unstable strictly the Grant Agency of the Ministry of Education, and the Metzlerian system that is the observer system matrix �� is Academy of Science of Slovak Republic, under Grant no. stable strictly Metzler, but not diagonally dominant. 1/0608/17. Tis support is very gratefully acknowledged. InFigure3,theresponseofthestrictlyMetzlerian observer in estimating the closed-loop system state variable is presented, acting under the same simulation conditions References as the above presented for the closed-loop system. Since the [1] F. Cacace, L. Farina, R. Setola, and A. Germani, Eds., Positive state variable estimation errors can be practically neglected, systems,vol.471ofLecture Notes in Control and Information no other responses of this stable Metzlerian observer are Sciences,Springer,Cham,2016. presented. In addition, the example illustrates that at the large [2]J.M.Carnicer,J.M.Pena,andR.A.Zalik,“Strictlytotally positive systems,” Journal of Approximation Teory,vol.92,no. positive values of some elements outside the diagonal of the 3, pp. 411–441, 1998. strictly Metzler matrix of the system �, neither the closed- � loop system matrix �� nor the observer system matrix �� is [3] O. Pastravanu and M. H. Matcovschi, “M, -stability of positive diagonally dominant, although they are stable strictly Metzler linear systems,” Mathematical Problems in Engineering,vol. matrices. However, the results of our other simulations show 2016, Article ID 9605464, p. 11, 2016. that if the system matrix � is diagonally dominant strictly [4] X. Chen, J. Lam, P. Li, and Z. Shu, “l1-induced norm and Metzler matrix, even the matrices �� and �� are diagonally controller synthesis of positive systems,” Automatica,vol.49,no. dominant to the strictly Metzler matrix. 5,pp.1377–1385,2013. [5] N. K. Son and D. Hinrichsen, “Robust stability of positive 6. Concluding Remarks continuous time systems,” Numerical Functional Analysis and Optimization,vol.17,no.5-6,pp.649–659,1996. A novel approach is presented in the paper to address the [6] T. Kaczorek, “Stability and stabilization of positive frac- problem of efectively computing a state-feedback control tional linear systems by state-feedbacks,” Bulletin of the Polish law gain that makes the strictly Metzlerian system in closed- Academy of Sciences—Technical Sciences,vol.58,no.4,pp.537– loop to be positive and stable, as well as of recounting the 554, 2010. 14 Mathematical Problems in Engineering

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