INTERNATIONAL JOURNAL OF MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES Volume 12, 2018

LQ Problem in Stabilization of Linear Metzlerian Continuous-time Systems

D. Krokavec and A. Filasova

represented by state-space equations, the synthesis of Abstract—The paper present a consistent set of linear stabilizing state-feedback controllers, guaranteeing the closed- inequalities which guaranties asymptotic stability of the closed-loop loop system is asymptotically stable and internally positive, is system, warranties strictly Metzlerian system structure, and adjusts conditionally supported by linear programming to meet the the state and output variables coincident with prescribed quadratic closed-loop system positive structure [11], [12]. In order to limits. To realize with a positive control law gain, the diagonal stabilizability of strictly Metzlerian linear continuous-time systems is reduce the number of constraints entering the solution in linear approved, and the related closed-form expression of design condi- programming methods, an alternative synthesis procedure with tions is provided. The results are illustrated using a particular LQ is proposed in [13], where the system parameter boundaries problem, for which numerical examples are given. are defined by n linear matrix inequalities (LMI), if the system is strictly Metzlerian. Because a solution of such defined base Keywords—asymptotic stability, linear matrix inequalities, linear set of LMIs only assures that the closed-loop system matrix is quadratic control, Metzlerian continuous-time systems, state feedback strictly Metzler, the design conditions are complemented by stabilization. another LMI that imposes a stable asymptotic solution. Since the applied LMI variables are of structure, it I. INTRODUCTION can be refereed about diagonal stabilizability of the strictly Positive systems indicate the processes whose variables Metzlerian continuous-time linear systems. represent quantities that do not have meaning unless they are Constraining the class of controller matrix gains to be nonnegative [1]. Since, in the relevant continuous-time state- positive, it does not alleviate the complexity of the solutions space description, the system matrix of a positive system is for non strictly Metzlerian systems. Proceeding along the same Metzler, theory of Metzler matrices is naturally applied to this lines, and pursuing the formal system analogy, some appli- kind of dynamical systems [2]. Additionally limited in the way cable extensions of the above formulations for strictly positive that the system input and output matrices are at least discrete-time linear systems can be found in [14], [15]. nonnegative matrices [3], system stabilization means strictly Analyzing the challenging problem of state-feedback defined task to design a positive gain matrix of control law so stabilization of strictly Metzlerian linear continuous-time that the closed-loop system matrix is Metzler and Hurwitz [4]. systems, the main motivation of this paper are design Therefore, most of techniques applicable to ordinary linear conditions formulated for infinite-time horizon control with systems can not be straightly nominated to positive linear linear quadratic cost functions. Since, at defined constraints on systems [5], [6]. Mainly the books [7], [8] treat a considerable elements of a strictly Metzlerian system matrix structure, the number of the approaches to positive system analysis, and task cannot be formulated using a Riccati equation form, the include illustrative algorithms for many specific tasks, but matrices of cost function are used to extend that one LMI, there still remains a wide variety of related problems which reflects stability condition in overall completion of the (controllability, observability, speed of response, robustness) LMIs set in design conditions. The configuration chosen corres- which need to be solved and addressed to Metzlerian linear ponds a way exploiting the minimizing of the quadratic cost systems. A more detailed treatments of problems are given, criterion subject to a closed-loop stability constraint, the e.g. in [9], [10]. framework used is standard and convenient because other additive The trend in synthesis of feedback control of Metzlerian constraints may be included into design formulation. systems tends to simplify and disambiguate the strictly defined Used notations are conventional so that xT , X T denote design conditions. Supposing that the Metzlerian systems is transpose of the vector x , and matrix X , respectively, x+ , X+ indicate a nonnegative vector and a ,

≻ means that X is a symmetric positive , The work presented in this paper was supported by VEGA, the Grant X 0 Agency of Ministry of Education and Academy of Science of Slovak ρ()X reports the eigenvalue spectrum of the square matrix Republic, under Grant No. 1/0608/17. This support is very gratefully i X , the symbol In marks the n-th order unit matrix, diag[ ] acknowledged. ℝ n ℝ n× r The both authors are with the Department of Cybernetics and Artificial enters up a diagonal matrix, + , + signify the set of all n Intelligence, Faculty of Electrical Engineering and Informatics, Technical dimensional real non-negative vectors and n× r real non- University of Kosice, Letnna 9/B, 042 00 Kosice, Slovakia (phone: ++421 negative matrices, respectively. 55 602 4213; e-mail: [email protected], [email protected])

ISSN: 1998-0140 236 INTERNATIONAL JOURNAL OF MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES Volume 12, 2018

II. LINEAR CONTINUOUS-TIME POSITIVE SYSTEMS Lemma1: [15] Within the basic notations as above and To define the system positive structures, and to extend their applying the vector input variable formal stabilizability properties, it is preferred in the follo- kT  1  wing the state-space system description defined in the standard u()()()t= − Kq t = − ⋮  q t (10) way as T  kr  ɺq()()()t= Aq t + Bu t (1) on the strictly Metzler MIMO system (1), (2), while the y()()t= Cq t (2) × positive gain matrix K ∈ ℝ r n is prescribed to force the where the equations (1), (2) belong to the Metzlerian class of closed-loop system matrix n r m positive systems if q()t ∈ ℝ + , u()t ∈ ℝ + , y()t ∈ ℝ + (all r ≥ = − = − T variables are nonnegative) for all t 0 . Ac A BK A∑ bk k k (11) n× n In the general case, the matrix A∈ ℝ is restricted to k =1 being strictly Metzler (its diagonal elements are negative and then the matrix Ac is Metzler, if for given non-negative matrix its off-diagonal elements are positive) and the matrices ∈ ℝ n× r ∈ ℝ n× n n× r m× n B + and a strictly Metzler matrix A + there exist B ∈ ℝ , C ∈ ℝ are nonnegative (all its entries are × + + positive definite diagonal matrices PR, ∈ ℝ n n such that for nonnegative and at least one is positive). Satisfying these k + h=1,2, … n − 1, k=1,2, … r restrictions, the system (1), (2) is referred as a linear strictly T Metzlerian system. Note, a strictly Metzler matrix is stable if it PP= ≻ 0 (12) = T ≻ is Hurwitz. RRk k 0 (13) Terminating the class of admissible controllers to be linear r − ≺ and considering, for simplicity, a SISO linear strictly A(,)i i (1↔n ) ∑ BR dk k 0 (14) Metzlerian system (1), (2) controlled by the dimensionally k =1 compatible control, constrained to use a linear function of the r h+ hT− h hT ≻ state measurements, and a strictly positive real vector k such TA(,)j j h (1↔n )/ nTPTBTR∑ dk k 0 (15) that k =1 subject to the notations u( t )= −kT q ( t ), k ∈ ℝ n (3) 0 0⋯ 0 1  then the state-space enrollment of the closed-loop system is ⋯  =1 0 0 0 −1 = T given as T ⋱  , TT (16)   ɺ = −T = ⋯ q()()()()t A bk qt Ac q t (4) 0 0 1 0  y()()t= Cq t (5) + =  ⋯ ⋯  A(j , j h )(1↔n )/ n diag a1,1+h a n − h, n a n − h + 1,1 a n , h  where (17) A= A − bkT (6) ⋯ c b11 b 1r  ⋯  has to be a strictly Metzler matrix. Consequently, the closed- = [ ⋯ ] = b21 b 2r B b1 b 2 br ⋮  (18) loop system matrix structure (5) prescribes the algebraic   ⋯ inequalities corresponding to the strictly Metzler matrix Ac as bn1 b nr  follows = ⋯ B dk diagb1k b 2 k b nk  (19) a= a − b k <0 for alli = 1,2, … , n (7) cii ii i i Then, if there are satisfied the above conditions for = − > = … ≠ acij a ij b i k j 0 for alli , j 1,2, , n , i j (8) prescribed set of variables, the control gain is = −1 TTT= = ⋯ where the detailed formats of the Metzler system matrix Kdk R k P, k k l K dk , l [ 1 1 1 ] (20) parameters, as well as the state controller gain vector structure are Remark 1: Since the rows and columns of an n× n square matrix are indexed from 1 to n, the addition modulo n +1 on a a⋯ a  b  k  11 12 1n 1 1 the set of residues S is considered in the following as a a⋯ a  b  k  = 21 22 2n =2 = 2 (9) + = + A  ⋮  ,,b⋮  k  ⋮  (j h )mod n+1 r 1, where r is the element of S to which the       result of the usual sum of integers j and k is congruent modulo a a ⋯ a  b  k   n1 n 2 nn  n  n  n +1. The used shorthand symbolical notation is + = + Although the structure of the state feedback control law (3) (j h )(1↔n )/ n r 1. is simple, it should be noted that the positiveness constraint for Comment 1: As it is seen from Lemma 1, the resulting 2 ∈ ℝ n× n the solvability of the gain k is extended by the set of n conditions prescribe the Metzlerian structure of Ac + scalar inequalities (6), (7). but do not guarantee that Ac is [14]. To solve Generalizing for MIMO (multiple input, multiple output) the stabilization problem for a strictly Metzlerian system (1), systems, the following lema yields. (2) with a diagonally stabilizable pair (,)AB the following theorem is proposed.

ISSN: 1998-0140 237 INTERNATIONAL JOURNAL OF MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES Volume 12, 2018

Theorem 1: [16] The control law (3) stabilizes the linear III. ENHANCED CONTROL DESIGN strictly Metzlerian system (1), (2) if for given positive definite The following theorem gives a more general version of the × × diagonal matrices Q ∈ ℝ n n , U ∈ ℝ r r there exist positive design conditions, implying from the slack matrix decoupling ∈ ℝ n× n definite diagonal matrices PR, k such that for principle. h=1,2, … n − 1, k=1,2, … r , Theorem 2: The control law (3) stabilizes the linear strictly = T PP ≻ 0 (21) Metzlerian system (1), (2) if for given positive definite = T ≻ ∈ ℝ n× n ∈ ℝ r× r RRk k 0 (22) diagonal matrices Q , U and given positive  r r  δ ∈ ℝ there exist positive definite diagonal matrices APPA+T − BllRT − RllBT ∗ ∗ ∈ ℝ n× n = … − = …  ∑dk k ∑ k dk  PVR,, k such that for h1,2, n 1, k1,2, r ,  k=1 k=1  r PP= T ≻ 0 (29)  T −−1 ∗  ≺ ∑hk l R k U 0 T  =  VV= ≻ 0 (30) k 1 −  P 0− Q 1 = T ≻   RRk k 0 (31)    r r  +T − T − T ∗ ∗ ∗ (23) APPA∑ BllRdk k ∑ RllB k dk  k=1 k=1 r  r  − ≺  V− P +δ AP − δ ∑ B l lT R−2δ P ∗ ∗  A(,)i i (1↔n ) ∑ BR dk k 0 (24) dk k ≺ 0  k=1  k =1  r  T −−1 ∗ r  ∑hk l R k 0 U  TAh(,)j j+ h TPTBTR hT− h hT ≻ 0 (25)  k=1  (1↔n )/ n∑ dk k − −1 k =1  P 0 0 Q  + T where T , A(,)j j h (1↔n )/ n , Bdk , bk , Rk , l are introdu- (32) ced in (16)-(19) and r − ≺ A(,)i i (1↔n ) ∑ BR dk k 0 (33) T = ⋯ ⋯ hk 0 01k 0 0 (26) k =1 is the vector with the value 1 on the k -th position. r TAh(,)j j+ h TPTBTR hT− h hT ≻ 0 (34) When the above conditions hold, the positive control gain (1↔n )/ n∑ dk k × = K ∈ ℝ r n is given in (9), where k 1 + T − where T , A(,)j j h (1↔n )/ n , Bdk , bk , Rk , l are introdu- K= R P1, kTT= l K (27) T dk k k dk ced in (16)-(19) and hk in (26). This encompass the design conditions with connection to When the above conditions hold, the positive control gain ∈ ℝ r× n LQ problem defined in Theorem 1. In this viewpoint, the K is given in (9), where condition unifies the design with quadratic constraints and = −1 TT= Kdk R k P, k k l K dk (35) feedback full state control for a class of strictly Metzlerian linear dynamical systems with performance appraisals of Proof: Reflecting the fact that the system (1), (2) is linear, infinite time horizon and quadratic costs, focusing perfect the Lyapunov function is chosen in the form state-feedback measurements and addressing the benefits of v(q ( t ))= qT ( t ) Sq ( t ) (36) feedback in multi-input/multi-output Metzlerian linear dynami- × with S ∈ ℝ n n taking the diagonal positive definite structure. cal systems. × Considering positive definite diagonal matrices Q ∈ ℝ n n , Applying the above given positive definite diagonal × × ∈ ℝ r r matrices PR, ∈ ℝ n n , k=1,2, … r , corroborated in theorem U , the Lyapunov function derivative in the sense of k the Krasovskii theorem [17] is predefined as formulation, different matrix inequalities can be used instead ɺv(())q t=ɺ qT () t Sq () t + qT () t Sqɺ () t of the complex inequality structure (23) to ensure stability in (37) Lyapunov sense while, if the set of inequalities is affirmative, ≤ −(qT ()t Qq () t + uT () t Uu ())0 t < an asymptotically stable closed-loop system is obtained. and substituting (10) it yields The simplest applicable matrix inequality which can replace ɺv(())q t=ɺ qT ()() t Sq t + qT ()() t Sqɺ t + qT ()( t QKUKq +T )()0t < inequality (23), but with no constraint on the system state and (38) input variables, takes the form Writing (4) with (11) as r r APPA+T − BllRT − RllBT ≺ 0 (28) −ɺ = ∑dk k ∑ k dk Ac q(t ) q ( t ) 0 (39) k =1 k =1 then with an arbitrary positive definite diagonal matrix while the control input is generated by the closed-loop control n× n M ∈ ℝ+ and with a positive scalar δ ∈ ℝ + it yields policy (10). Evidently, (28) can be simply derived from (23) n× n r× r T + ɺT δ −ɺ = prescribing the zero matrices Q ∈ ℝ and U ∈ ℝ . (q(t)M q (t) M )( Ac q (t) q (t)) 0 (40) Therefore, adding (36) and its transposition to (34) it is

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obtained Remark 2: Theorem 2 solves the state-feedback control vɺ((t))q= qɺ T () t Sq () t + qT () t Sqɺ () t problem for linear strictly Metzlerian system (1), (2), given by the diagonally stabilizable pair (,)AB with T T +q(t )( Q + K UK ) q ( t ) quadratic constraints represented by the couple of positive (41) (qT (t)M++ qɺ T (t)δ M )( Aq (t)− qɺ (t)) definite diagonal weighting matrices (,)QU of c appropriate dimensions. Introducing the symmetric slack + T TT− ɺ +δ ɺ < (q (t)Ac q (t))(Mq (t) Mq (t) ) 0 matrix variable P , the system parameter matrices (,)AB and with are strictly decoupled in the LMIs (30)-(32) from the Lyapunov matrix V , while the matrix V verifying the TTT= ɺ  qv ()()()t q t q t  (42) closed-loop stability remains symmetric positive definite then (41) can be rewritten as and diagonal. By this procedure, the control problem is parameterized in such LMIs structure, which admits more ɺ = T < v(qv ( t )) q v ( t ) T vq v ( t ) 0 (43) freedom in the controller design for Metzlerian systems where since except free defining weighting matrices (,)QU there δ ∈ℝ  + T + +T ∗  is a free tuning parameter + . = MAc Ac M Q K UK ≺ Tv   0 (44) Corollary 1: Defining the T as  S− M+δ MA −2δ M  p  c  follows

Because of the strictly structure of Ac , the inequality (44) is 0 I n  a bilinear inequality, it is necessary to define the I 0  −1 T = n , TT= (52) transformation matrix T for transformation (44) into the p  I  p p c  r  linear form so that  I n  = = −1 and premultiplying the left side and postmultiplying the right TPPPMc diag[ ] , (45) side, then (47) implies Pre-multiplying from the left side and post-multiplying from −δ − +δ  2 P V P Ac P 0 0  the right side by Tc then (44) implies  − +δ T + TT  = VPAAAPcc PP c PK P ≺  + T + + T ∗  H  − −1  0 (53) Ac PP Ac PQP PK UKP 0 KP U 0   ≺ 0 (46)  −1 − +δ − δ 0 P 0 −Q  PSPc PPA 2 P   The inequality (53) can be factorized as and using the Schur complement property, it can write  −δ − +δ   + T ∗ ∗ ∗  2 P V P Ac P 0 0 AAcPP c  − +δ T   − +δ − δ ∗ ∗  = V PP Ac 0 0 0 VPAPcP 2 H    −1  ≺ 0 (47) 0 0 0 0 KP 0− U ∗    −1  0 0 0 0  P 0 0− Q  0 0 0 0  (54) TT × 0 APP+ A PK P  ∈ ℝ n n + c c − ≺ 0 where a positive definite diagonal matrix V is denoted 0 KP− U1 0   −  as follows 0 P 0 −Q 1 = VPSP (48) Setting δ = 0 ,VP= then, evidently, Hence, with (27), (26) and with the A structure given in (11), c 0 0 0 0  it can write  + TT  r = 0 APP Acc PK P ≤ H  − −1  0 (55) A P= AP − b kT P 0 KP U 0 c∑ k k  − −1 k =1 0 P 0 Q  r = − T and it is obvious, to be satisfied (55), the following inequality AP∑ bk r k (49) k =1 has to yield r = − T  + T ∗ ∗  AP∑ Bdk ll R k AAPP cc  −1  k =1 KP− U ∗ ≺ 0 (56)  − −1 r r r  P 0 Q  KP= h kT P = h rT = h lT R (50) ∑k k ∑k k ∑ k k Thus, it can be finally observed using (49)-(51) that (56) k =1 k =1 k =1 provides componentwise interlinking with (23). where Note, the enhanced formulation gives substantiation for TT= =TT = different solutions obtaining in dependency on linear matrix rk k k P,, b k B dk l r k l R k (51) inequalities (23) or (30), (32), respectively, when combining and the relations (49), (50) are used to modify (47) as (32). them with (12)-(15) to construct the control design conditions Then, combining (32) with (21), (22), (24), (25) concludes the with connection to LQ task for strictly Metzler linear systems. proof. ∎

ISSN: 1998-0140 239 INTERNATIONAL JOURNAL OF MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES Volume 12, 2018

IV. ILLUSTRATIVE EXAMPLE The strictly Metzlerian system (1), (2) is concretized for the system parameters [13] −3.3800 0.2080 6.7150 5.6760   0.5810− 4.2900 2.0500 0.6750 A =    1.0670 4.2730− 6.6540 5.8930    0.0480 2.2730 1.3430− 2.1040 0.0400 0.0189    0.0568 0.0203  4 0 1 0  B = , C =   0.0114 0.0315  0 0 0 1    0.0114 0.0170

Since the algebraic manipulation provides that the eigenva- Fig. 1: Closed-loop system state response lue spectrum of the strictly Metzler matrix A is ρ(A )={ 1.9761, − 9.4392, − 4.4824 ± 1.2499i} such defined Metzler system is unstable. From the system parameter expressions the auxiliary design frameworks follow that = − − − − A(i , i )(1↔ 4) diag[ 3.3800 4.2900 6.6540 2.1040] + = A(i , i 1)(1↔ 4)/4 diag[ 0.2080 2.0500 5.8930 0.0480] + = A(,i i 2)(1↔ 4)/4 diag[ 6.7150 0.67501.0670 2.2730] + = A(i , i 3)(1↔ 4)/4 diag[ 5.6760 0.5810 4.2730 1.3430] = Bd1 diag[ 0.0400 0.0568 0.0114 0.0114] = Bd 2 diag[ 0.0189 0.0203 0.0315 0.0170] Fig. 2: Closed-loop system output response while the T of the forth length and the vectors h1 , h2 are Evidently, the control law gain matrix K is a strictly positi- 0 0 0 1  ve matrix. 1 0 0 0  1   0  T =   , h=, h = It can simple show that the solution is the stable Metzler 0 1 0 0 1 0 2  1        matrix Ac of the form 0 0 1 0 -3.5197 0.0541 5.0103 5.1583   Theorem 1 gives the possibility to synthesize state feedback 0.3844 -4.4813 0.0549 0.0829 =   gains by solving this diagonal LMI problem. Within defined Ac  1.0204 4.1203 -8.8438 5.1602 design framework, prescribing the weight matrices QI= 10 ,   4  0.00522.1806 0.0976 -2.5128 UI= 0.01 and solving (21)-(25) rewritten for the SeDuMi 2 such that its eigenvalue spectrum is package [18], all LMI conditions of the above theorem are ρ = { ± } satisfied and asymptotical stability of the closed-loop system is (Ac ) -0.9580 -9.6020 -4.3988 1. 1567i Defining the system forced mod by the control policy guaranteed by a stable strictly Metzler matrix Ac , if the values of the LMI variables are u(t )= − Kq ( t ) + w ( t ), wT = [ 1.36 0.68] P = diag[ 0.2083 0.0142 0.0742 0.0113] and setting q(0) = 0 , the simulation results for control of the R = diag[0.7011 0.0267 0.8732 0.0273] system are shown in Fig. 1 and Fig. 2, where the state variables 1 vector q()t , as well as the output variables vector y()t are = R2 diag[0.0555 0.0595 4.8489 0.2536] positive, when the parameters of the control input are set up as This gives, prescribed by (10), (27), the control law matrix mentioned above. Because Ac is stable, then q()t , y()t tend gain K that to constant values as t → ∞ .   Note, closed loop is asymptotically stable and externally 3.3661 1.8754 11.7690 2.4101 × K =   positive if A is stable Metzler matrix and C ∈ ℝ m n is a 0.2667 4.1775 65.3565 22.4240 c + non-negative matrix.

ISSN: 1998-0140 240 INTERNATIONAL JOURNAL OF MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES Volume 12, 2018

Fig. 3: Closed-loop system state enhanced response Fig. 5: Closed-loop system state standard response

Fig. 4: Closed-loop system output enhanced response Fig. 6: Closed-loop system output standard response

Solving the set of LMIs (29)--(34) with the above prescri- Using the standard algorithms [13], where in the set of bed Q , U and the tuning parameter δ = 1.8 the result is (21)-(25) is inequality (23) replaced by inequality (28), then P = diag[0.0216 0.0006 0.0125 0.000 5] P = diag[0.6618 0.0590 0.1336 0.053 2] = V = diag[0.1152 0.0037 0.1757 0.001 8] R1 diag[1.2310 0.0903 2.8440 0.0716] = R = diag[0.5664 0.3015 4.2142 1.4121] R1 diag[0.0906 0.0001 0.1308 0.0001] 2 = and the closed-loop system parameters are R2 diag[0.0002 0.0066 0.8966 0.0160] 1.8601 1.5308 21.2869 1.3463 In this case the gain K and the matrix Ac are such that K =   0.8559 5.1131 31.5423 26.5640 4.1996 0.1699 10.4438 0.1346 K =   -3.4706 0.0502 5.2680 5.1206 0.0104 10.4867 71.6190 32.7143   0.4580 -4.4807 0.2008 0.0593 =   -3.5482 0.0032 4.9451 5.0530 Ac    1.0189 4.0948 -7.8881 5.0420  0.3423 -4.5125 0.0030 0.0033   A =  0.0123 2.1686 0.5646 -2.5711 c  1.0190 3.9412 -9.0258 4.8623   Since the eigenvalue spectrum of A is  0.0001 2.0927 0.0061 -2.6620 c ρ = ± (Ac ) {-0.3771 -9.0806 -4.4764 1. 2684i} while the stable eigenvalue spectrum of Ac is ρ = { ± } it is evident that the closed-loop system dynamics is substan- (Ac ) -1.2896 -9.7062 -4.3764 1. 1173i tially slower. It is evident from the eigenvalues spectrum that the closed- To compare all responses in terms of steady-state loop system dynamics is in this case faster. Running under the variable values, the setting point vector of the control policy in same simulation conditions as are given above, the closed-loop simulation is changed as wT = [0.4 0.2] . The closed-loop system responses are presented in Fig. 3 and Fig 4. system responses are presented in Fig. 5 and Fig 6.

ISSN: 1998-0140 241 INTERNATIONAL JOURNAL OF MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES Volume 12, 2018

Note, as can be seen [16], [19], the direct use of the [9] J. E. Feng, J. Lam, P. Li, and Z. Shu, “Decay rate constrained principle of static decoupling [19] can lead to that the closed- stabilization of positive systems using static output feedback," Int. J. Robust and Nonlinear Control, vol. 21, no. 1, 2011, pp. 44-54 loop system being not internally positive. Since this means that [10] J. Shen and J Lam, "On static output-feedback stabilization for multi- not for every non-negative initial system state vector the output input multi-output positive systems," Int. J. Robust and Nonlinear of the system will be positive, when setting the operating point Control, vol. 25, no. 16, 2015, pp. 3154–3162. [11] M. Ait Rami and F. Tadeo, "Linear programming approach to impose of the system in the simulations, the principle of static positiveness in closed-loop and estimated states," in Proc. 16th Int. decoupling was not used. Symp. Mathematical Theory of Networks and Systems, Kyoto, Japan, 2006, pp. 2470-2477. [12] M. Ait Rami and F. Tadeo, "Controller synthesis for positive linear V. CONCLUDING REMARKS systems with bounded controls," IEEE Trans. Circuits and Systems, vol. The paper is concentrated specifically on effective design of 54, no. 2, 2007, pp. 151-155. [13] D. Krokavec and A. Filasova, "LMI based principles in strictly the full state feedback control for strictly Metzlerian continu- Metzlerian systems control design," Mathematical Problems in ous-time systems, to accomplish that the closed-loop system Engineering, vol. 2018, ID 9590253, 14 p. matrix be strictly Metzler and Hurwitz and the control gain be [14] D. Krokavec and A. Filasova, "On control of discrete-time LTI positive a positive matrix. systems," Applied Mathematical Sciences, vol. 11, no. 50, 2017, pp. 2459-2476. The principle combines the algebraic constraints, implying [15] D. Krokavec and A. Filasova, "Stabilization of discrete-time LTI from the predefined closed-loop system strictly Metzler matrix positive systems," Archives of Control Sciences, vol. 27, no. 4, 2017, structure, and defined as a basic set of LMIs with an additive pp. 575-594. [16] D. Krokavec and A. Filasova, "Stabilization of linear Metzlerian stabilizing matrix inequality, containing moreover the systems with connection to LQ problem," in Proc. 2nd Int. Conf. prescribed state and input signal amplitude quadratic con- Control, Artificial Intelligence, Robotics and Optimization ICCAIRO strains, to guarantee closed-loop system asymptotic stability 2018, Prague, Czech Republic, 2018 (in press). [17] W. M. Haddad and V. Chellaboina, Nonlinear Dynamical Systems and and dynamics. Control. A Lyapunov-Based Approach, Princeton: Princeton University Whilst a solution can be obtained only via positive definite Press, 2008. and diagonal matrix variables entering this set of inequalities, [18] D. Peaucelle, D. Henrion, Y. Labit, and K. Taitz, User's Guide for progress is made in incorporating an LMI structure to realize SeDuMi Interface, Toulouse: LAAS-CNRS, 2002. [19] B. Shafai, R. Ghadami and A. Oghbaee, "Constrained stabilization with the diagonal stabilizability of strictly Metzlerian linear conti- maximum stability radius for linear continuous-time systems," in Proc. nuous-time systems. As it is illustrated by the numerical 52nd IEEE Conf. Decision and Control, Florence, Italy, 2013, pp. 3415- example, hopefully of interest to researchers, the proposed 3420. [20] Q. G. Wang, Decoupling Control, Berlin: Springer-Verlag, 2003. fulfilment provides numerically effective computational frameworks with potential adaptation to not strictly Metzlerian systems.

Many problems concerning the non unique solutions, such as design of the non-negative gain matrix for not strictly Metzlerian linear systems, or for discrete-time not strictly D. Krokavec received M.Sc. degree in automatic control in 1967 and Ph.D. positive linear systems, are still open. These problems are degree in technical cybernetics in 1982 from the Faculty of Electrical Engineering, Slovak University of Technology in Bratislava, Slovakia. In challenging and interesting research works in the future, while 1984 he was promoted Associated Professor from the Technical University in the proposed LMI structure variant seems promising in further Kosice, Slovakia, and in 1999 he was appointed Full Professor in automation aspects to be subjects to future work. and control. From 1968 to 1971 he was a Research Assistant at the Research Institute of Automation and Mechanization in Nove Mesto nad Vahom, Slovakia, REFERENCES since 1971 he is with the Department of Cybernetics and Artificial Intelligence, Faculty of Electrical Engineering and Informatics, Technical [1] Positive Systems, F. Cacace, L. Farina, R. Setola and A. Germani, Eds. University of Kosice. In the long term, he specializes in stochastic processes Lecture Notes in Control and Information Sciences, vol. 471, Cham: in dynamic systems, digital control systems and digital signal processing, and Springer, 2016. in dynamic system fault diagnosis. [2] J. M. Carnicer, J. M. Pena, and R .A. Zalik, “Strictly totally positive systems," J. Approximation Theory, vol. 92, 1998, pp. 411-441. Prof. Krokavec is a member of the IFAC Technical Committee TC 1.4. [3] P. De Leenheer and D. Aeyels, "Stabilization of positive linear systems," Stochastic Systems and the IFAC Technical Committee TC 6.4. Fault Systems & Control Letters, vol. 44, no. 4. 2001, pp. 259-271. Detection, Supervision & Safety of Technical Processes – SAFEPROCESS. [4] J. Back and A. Astolfi, "Design of positive linear observers for positive linear systems via coordinate transformations and positive realizations," SIAM J. Control and Optimization, vol. 47, no. 1, 2008, pp. 345-373. [5] H. Gao, J. Lam, C. Wang, and S. Xu,"Control for stability and Anna Filasova graduated in technical cybernetics and received M.Sc. degree positivity. Equivalent conditions and computation," IEEE Trans. in 1975, and Ph.D. degree in 1993, both from the Faculty of Electrical Circuits and Systems, vol. 52, no. 9, 2005, pp. 540-544. Engineering and Informatics, Technical University of Kosice, Slovakia. In [6] B. Shafai and A. Oghbaee, "Positive quadratic stabilization of uncertain 1999 she was appointed Associated Professor from the Technical University linear system," in Proc. 2014 IEEE Multi-Conference on Systems and in Kosice in technical cybernetics. Control, Antibes, France, 2014, pp. 1412-1417. She is with the Department of Cybernetics and Artificial Intelligence, [7] L. Farina and S. Rinaldi, Positive Linear Systems. Theory and Faculty of Electrical Engineering and Informatics, Technical University of Applications, New York: John Wiley & Sons, 2000. Kosice, and she has been working there as an Assistant Professor from 1975 [8] T. Kaczorek and L. Sajewski, The Realization Problem for Positive and to 1999. Her main research interests are in robust control, decentralized Fractional Systems, Cham: Springer, 2014. control, large-scale system optimization, and control reconfiguration.

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