About a Class of Positive Hybrid Dynamic Linear Systems and an Associate Extended Kalman-Yakubovich-Popov Lemma

This paper formulates an “ad hoc” robust version under parametrical disturbances of the discrete version of the KalmanYakubovich-Popov Lemma for a class of positive hybrid dynamic linear systems which consist of a continuous-time system coupled with a discrete-time or a digital one. An extended discrete system, whose state vector contains both the digital one and the discretization of the continuous-time one at sampling instants, is a key analysis element in the formulation. The hyperstability and asymptotic hyperstability properties of the studied class of positive hybrid systems under feedback from anymember of a nonlinear (and, eventually, time-varying) class of controllers, which satisfies a Popov’s-type inequality, are also investigated as linked to the positive realness of the associated transfer matrices.


Introduction
Continuous-time and discrete-time positive systems have been studied in detail in recent years [1][2][3][4][5][6][7][8][9][10].In particular, if both the state and output possess such a property, the positivity is said to be internal or, simply, the system is positive.If the output possesses such a property, the system is said to be externally positive.Therefore, positive systems are intrinsically interesting to describe some problems like Markov chains, queuing problems, certain distillation columns, and biological and other physical compartmental problems where populations or concentrations cannot be negative [2,3].A related property is that time-invariant dynamic linear systems which are externally positive, while they have positive real or strictly positive real transfer matrices, are, in addition, hyperstable or asymptotically hyperstable, that is, globally Lyapunov stable for any nonlinear and/or time-varying feedback device satisfying a Popov's-type inequality for all time [11,12].Such a property of asymptotic hyperstability generalizes that of absolute stability [13][14][15], which generalizes the most basic concept of stability of dynamic systems.See, for instance, [13,14,[16][17][18][19][20][21][22][23][24][25][26][27][28][29] and references therein.The hyperstability property, which has a frequency-based physical interpretation in terms of positive realness of the transfer function of a feed-forward linear block, is also related to external positivity of the inputoutput relation rather than to (internal) positivity of the state-trajectory solution what is equivalent to positivity of the instantaneous input-output power and the input-output energy [2,3,13,15,30].It is well known that closed-loop hyperstability is, by nature, a powerful version of closedloop stability since it refers to the stability of an hyperstable linear feed-forward plant (in the sense of positive realness of the associate transfer matrix) under a wide class of feedback controllers applied.The above important properties make very attractive potential research issues for kind of more complex dynamic systems with applied projection including those lying in the class of continuous/digital hybrid systems.On the other hand, the class of hybrid systems consisting of continuous-time and discrete-time (or digital) systems are of an increasing interest since many existing industrial installations combine both kinds of systems.An elementary well-known case is when a discrete-time controller is used for a continuous-time plant.Another case is related to teleoperation systems where certain variables evolve in a discrete-time or digital fashion.A background literature and related relevant results are given in [1,7,11,16,17,26,31,32] and some of the references therein.The objective of this paper is to address appropriate versions of the Kalman-Yakubovich-Popov Lemma (KYP-Lemma) for a class of hybrid systems consisting of coupled linear continuous-time and digital dynamic subsystems, firstly proposed in [31], provided that they are, furthermore, positive [7], in the sense that, for any initial condition and any admissible controls both with nonnegative components, all the components of the state and output trajectory solutions are nonnegative for all time [33].General related results on positivity of wide usefulness are available in [34,35].
The paper is organized as follows.Firstly, a notation and terminology subsection is allocated below in this introductory section.Section 2 characterizes the class of hybrid systems dealt with and formulates with explicit results its positivity and some of its stability and asymptotic stability properties.A relevant auxiliary system for those studies is the so-called extended discrete hybrid system for which only the signals at sampling points are relevant and whose state is composed of both the digital substate and the discretized version of the continuous-time subsystem at sampling instants.Some of the obtained results display how the stability is kept under small coupling between the continuous-time and the discrete-time digital substates provided that the continuoustime and digital dynamics are stable.The section contains also controllability results provided that a nominal system version keeps that property.Section 3 is devoted to the continuous and discrete versions of the KYP-Lemma for a simplified version related to the relevant pairs of the system and control matrices and for the general version related to the whole state-space realization.The relationships between the positive realness of the transfer matrix to the state-space realization are characterized for both the positive extended discrete hybrid system and the whole hybrid system through the KYP-Lemma and Youla's factorization lemma.The obtained results are formulated in terms of robustness in the sense that the positive realness and the system's positivity of a nominal version of the hybrid system are kept under certain explicit conditions for the parametrical disturbances which deviate the hybrid system from its nominal parameterization.Section 4 relates the former results of positive realness and the hyperstability and asymptotic hyperstability properties of the auxiliary extended discrete hybrid system and to those of the whole hybrid system for the case when the plant input is got via feedback from a nonlinear and eventually time-varying device which satisfies a Popov's type inequality.Some further study is also provided in Section 5 related to the design of a stabilizing linear control scheme which either simply stabilizes the dynamics or improves its relative stability degree of the hybrid system in an internal control loop prior to the operation via any member of the given class of nonlinear and time-varying control controllers so as to ensure the hyperstability of the whole closed-loop system.Finally, conclusions end the paper.if it is of order  × , with all its entries being nonnegative.R − = R/R + is the set of nonpositive real numbers.Note that R = R + ∪ R − and 0 ∈ (R + ∩ R − ).Vectors and matrices are nonpositive (being, respectively, in R  − and  ∈ R × − ) if they have nonpositive entries.Z, Z + , and Z − are the set of integer numbers and its subsets of nonnegative and nonpositive real parts, respectively.
(b) A matrix  ∈ R × + is said to be positive (denoted by  > 0) if it has at least a positive entry.A nonnegative matrix  ≥ 0 satisfies either  > 0 or  = 0.A matrix  ∈ R × − , which has at least a negative entry, is said to be negative and denoted by  < 0 and, if all its entries are negative, then it is denoted by  ≪ 0.
(c) A matrix  ∈ R × + is said to be strictly positive (denoted by  ≫ 0) if all its entries are positive.Similarly, a vector V ∈ R  + is said to be positive (denoted by V > 0) if it has at least a positive component.It is said to be strictly positive (denoted by V ≫ 0) if all its components are positive.Also, the notations  ≫ , V ≫  for matrices and vectors mean, respectively,  −  ≫ 0 and V −  ≫ 0. Interpretations of expressions like  > , V > , V ≥  follow directly from the above ones.
(d) We denote  ≻ 0 (e)   is the th identity matrix.(f) A matrix  ∈ R × + is said to be stable, or a stability matrix, if its characteristic polynomial is Hurwitz or, equivalently, if all its eigenvalues have negative real parts.The matrix measure of the matrix  (with respect to any norm) is () = lim →0 + ((‖ + ‖ − ‖‖)/).The spectrum of  is the set of its eigenvalues (or spectrum) denoted by Sp and its characteristic polynomial denoted by   () = Det( − ), where  is a complex indeterminate and Det(⋅) stands for the determinant of the matrix (⋅).A subscript in the matrix measure  (⋅) () denotes the measure with respect to a particular (⋅)-norm.A matrix  ∈ R × + is said to be convergent (or Schur), if all its eigenvalues lie in the strict unity circle.An  ∞ complex function is Schur if its  ∞norm is bounded by unity while it is said to be strictly bounded real (SBR), if in addition its coefficients are real and its  ∞ -norm is strictly bounded by unity. (g is the condition number of the matrix  ∈ R × with respect to the -norm.It is infinity if and only if the matrix    is singular.In particular,  2 () = ‖ 2 ‖‖ −1 ‖ 2 is the condition number of  with respect to its ℓ 2 (or spectral) norm which is the quotient of its maximum and minimum eigenvalues in the case when it is square.
(i)   and    denote, respectively, the th column or row of the real -matrix, the superscripts "" and " * " denoting transpose and conjugate transpose, respectively.  ,  being an integer number, denotes the th power of the -matrix and provided that  = (  ),  () = ( ()   ) is an associate matrix to  defined as  ()   = 1 if   ̸ = 0 and  ()  = 0, otherwise.Note that  ≥ 0 ⇔  () ≥ 0. V  denotes the th component of the real vector V and V ≥ 0 ⇔ V () ≥ 0. Thus, any positive system  has always an associate positive system  () which defines the pairwise relations input components/state-output components and state components/output components from its associate influence graph  [2,3,5], by defining all its parameterizing matrices according to the above criterion.
(k)  ()  is the unity vector of R  whose unique nonzero component is the th one which is unity.
(l) The notation [] stands for a discrete/digital variable or vector  which is only defined as sampling instants   = ,  ∈ Z + , with  being the sampling period.If  is a digital variable then it is only defined at sampling instants.If  is a discrete variable (i.e., that arising from the discretization of a continuous variable), then [] = () and any of both equivalent notations are used indistinctly in such a case.
(m) The superscript  stands for the transpose of a vector or matrix while Ker() stands for the null-space of the operator .

Hybrid System and Positivity and Controllability Properties
Consider the subsequent hybrid linear system : (i)   and   are the matrix of continuous-time and of digital dynamics, respectively, and   and   are, respectively, the matrices of dynamics of couplings between the digital and continuous-time substates and continuoustime discretized and digital substates.The matrix   is the matrix of dynamics of coupling between the sampled continuous-time substate to its time evolution over the next sampling interval.
(ii)   and   are continuous-time and digital control matrices and   is a coupling control matrix from the sampled continuous-time control to the next intersample period continuous-time substate.
(iii) The matrices   ,   ,   and   and   in (1c) are the various output and input-output interconnection matrices generating the output of the hybrid system from its continuous-time substate, its discretized value at sampling instants, the digital substate, and the continuous-time input and its sampled value.
The orders of all the real constant system parameterizing matrices displayed in (1a), (1b), and (1c) agree with the corresponding dimensions of the continuous, discrete, and digital substates   (),   [], and   [] and inputs and outputs.Note that the hybrid system is driven by the control () and by its samples () of period  acting as two independent control actions.At sampling instants, it follows by direct calculus from (1a), (1b), and (1c) that the hybrid system  is described by the following  =   +   th order extended discrete-time system of sampling period  driven by a fictitious extended input sequence {V[]} ⊂ R + whose element V[] depends on  : [, ( + 1)] → R  and since only finite input jumps happen at sampling instants, since impulsive jumps are not considered, V[] depends on  : [, ( + 1)) → R  since the updated value [ + 1] at  = ( + 1) does not contribute to V[] : ]  for any integer  ≥ 0, where where V[] ∈ R + .The derivation of the extended discrete , (2a), (2b), and (2c), subject to (3)-( 8), from the hybrid system , (1a), (1b), and (1c), is direct from a time-integration of (1a), (1b), and (1c) on a sampling time interval [, ( + 1)) with initial conditions at  = .The following positivity result holds as a direct extension from the SISO (single-input single-output case) hybrid parameterization of [7,11,31].
Theorem 1.The system  is positive if and only if   ∈ Under the above given conditions,     ∈ R for  ≥ 0, if  : [0, ∞] → R  , and then the extended discrete system  is also positive.

Theorem 2. The following properties hold:
(i) Assume that (a)   ∈ Then,  is convergent and the unforced  is globally asymptotically stable.
(2) ‖  ‖ 2 ≤ , ‖  ‖ 2 ≤ , and ‖  ‖ 2 ≤  for some  ∈ [0,  * ), where and   are convergent and Proof.First note that  > 0 and is convergent if and only if (−) is nonsingular and (−) −1 > 0 [9].Direct calculations with (3) and the inverse of a 2 × 2 block partitioned matrix [36] yield in this case where As a result, there exists ( − ) −1 with  > 0 and ( − ) −1 ≥ 0 which holds if and only if  is convergent so that the unforced  is globally asymptotically stable.Property (i) has been proved.To prove Property (ii), note that  > 0 and is convergent, so that (  − ) is a nonsingular -matrix and (  − ) −1 > 0 exists.Since (  − ) is a nonsingular matrix, all its leading minors are positive.Thus, (  −  ) ≻ 0, then (   −    ) ≻ 0 and (   −   ) ≻ 0 and, equivalently, (   −    ) −1 ≻ 0 and (   −   ) −1 ≻ 0. On the other hand, one has Since  ≥   from the hypotheses, (  − ) −1 ( −   ) ≥ 0 and if To prove Property (iii), note that since   is Hurwitz and   is convergent,  =  0 + Ã =  0 ( 2 +  −1 0 Ã), where with Since   is Hurwitz and   and  0 are convergent, there exists a real constant  ≥ 1, which is norm-dependent such that for all  ≥ 0 with − 1 < 0 being not less than the stability abscissa of   and  1 being not less than the convergence abscissa of   .Since max(‖  ‖ 2 , ‖  ‖ 2 , ‖  ‖ 2 ) ≤ , one gets from (15) that [37] and then convergent since  0 is convergent from the continuity of the eigenvalues of matrix with respect to its entries.Then the unforced  is globally asymptotically stable and the unforced  is also globally asymptotically stable since   is Hurwitz and [] → 0 as  → ∞ for any given initial condition.Property (iii) has been proved.Property (iv) follows by redefining  =  01 + Ã1 =  01 ( 2 +  −1 01 Ã1 ) and then Related to Theorem 2, note that    is convergent by construction if   is Hurwitz and a guaranteed upper-bound of ‖  ‖ is sufficiently small which increases as the sampling period  and the stability abscissa of   increase.The next result generalizes Theorem 1 if   is not necessarily convergent and   is not necessarily Hurwitz.
Example 3. Consider a positive hybrid system with scalar continuous-time and digital subsystems and On the other hand, hypothesis    +    (   −   ) −1   being nonnegative and convergent holds if for some while for some  2 ∈ (0, (1 −  −   )(1 −   /  )).Particular numerical values which satisfy all the given joint constraints are, for instance,   = 0.1, so that the stable continuous dynamics has a small relative stability,  = 0.1,   = 0, so that the digital dynamics has a maximum stability degree, and the forced system behavior is independent of the digital self-dynamics,    = 0.995,   = 0.009.

Corollary 4. The following properties hold:
(i)  is convergent, and then the unforced  is globally asymptotically stable, if (1)  0 , (12), is nonsingular and there exists If, in addition,   is Hurwitz, then the unforced  is globally asymptotically stable.
(ii) A sufficient condition for Property (i) to hold is where   ∈ R is the stability abscissa of   .
Proof.Note that, since  0 is nonsingular, and  is nonsingular from Banach's Perturbation Lemma, under the condition Property (i) follows directly from (24).Property (ii) follows from the fact that ( 22) is a sufficient condition for ‖ Ã‖ 2 < ( 1 − ||)‖ 0 ‖ 2 in view of the first identity of (13).
The following theorem refers to "controllability" as the property of controllability to the origin and to "reachability" as that of controllability from the origin.Note from (2a), (3), and ( 6)-( 8) the structure of matrix  that  0 [] + [] =  V V[] = [] leading to the state system  description driven by a real vector sequence: for any integer  ≥ 0, where  V is reparameterized to some appropriate matrix  so as to drive the auxiliary control for some given prefixed  × -matrix  =   ≥ 0, and (26 The system  is controllable if, furthermore, rank (  ,   ) = .
(iv) The system  is reachable if it is controllable and, furthermore,  is nonsingular (in particular, if Property (iii) holds and, furthermore,   is nonsingular and The system  is reachable if  is reachable and   is nonsingular.
Proof.One gets by direct recursive calculation from (25a) x provided that the input is generated from for  ∈ (0, ) ;  = 0, 1, . . ., 2 − 1 and the system  is then controllable.This proves the sufficiency part.The necessity part follows from (28) written in the equivalent form: If rank (, ) < 2, then, given [], there exists . Thus, the following linear algebraic system of equations resulting from ( 31) is an incompatible one from Rouché-Froebenius theorem of Linear Algebra.This leads to the proof of the necessity part of the first part of Property (i).Then, system  is controllable if and only if (, ) is full rank.On the other hand, if the pair (, ) is not controllable while the pair (  ,   ) is controllable, the system  is approximately controllable with state targeting error Property (i) has been proved.On the other hand, note from (1a) that is nonsingular, ∀ ∈ Sp, where so that  is controllable from (36), since which is guaranteed if Condition (40) holds if  ∈ [0, √  2 +  −1 −).This guarantees that rank (, ) = 2 and  is controllable.Since rank (  ,   ) = , the system  is controllable from Property (ii).Property (iii) has been proved.
To prove Property (iv) note that reachability of the discrete  is guaranteed from controllability to the origin and the nonsingularity of its matrix of dynamics .Those conditions are guaranteed from the conditions of Property (iii) if   is nonsingular and Note that if (26) is tested for  ∈ (Sp  ) ∩ { ∈ C : || ≥ 1} (i.e., for the unstable and critically stable modes of   ), then it becomes a stabilizability test of the current  provided that the nominal   is stabilizable.In other words, stabilizability is the property implying that any uncontrollable mode is asymptotically stable while any unstable or critically stable mode is controllable.

The Kalman-Yakubovich-Popov Lemma
The following technical result will be then used for deriving a simplified but useful version of the KYP-Lemma (see [8,37] and references therein) for the given  system in the event that the output matrix is identity and the input-output interconnection matrix is zero.Lemma 6.The following properties hold: ] ⪯ 0 for some  =   ∈ R (3+)×(3+) and all  ∈ [0, ∞], where  =   + Ã and  =   + B .

Remarks 7.
(1) Note that Lemma 6(i) does not require for A to be a convergent matrix (i.e., a stability matrix on the discrete framework) while   has to be a convergent matrix.Conversely, Lemma 6(ii) does not require for   to be a convergent matrix while  is a convergent matrix. ( Ã and G() = ( 2 −) −1 Ã are SBR, then  and   are convergent matrices and the identities lead to (3) The identities (3) Constraint (26) holds and there exists a matrix   =    ∈ R (3+)×(3+) , which is nonnegative in all entries except for the last 2 diagonal elements, such that (42) holds for all  ∈ [0 , ∞].
The following result is concerned with the positive realness of a discrete nominal transfer matrix of the extended discrete nominal   which guarantees that of the transfer matrix of a parametrical disturbed   under a set of structured parametrical perturbations of the dynamics, output, control, and interconnection matrices.The result is based on the equivalence between the positive realness of a transfer matrix and the associated state-space realization, namely, the Positive Real Lemma, so-called alternatively Kalman-Szëgo-Popov Lemma or KSP Lemma, being a discrete version of the KYP-Lemma and of those ones with the Discrete Positive Factorization Lemma (so-called alternatively Youla's Factorization Lemma) Theorem 10.Assume that  =  + ,   ,   , and   are positive,   is positive, and the transfer matrix  1 () =   () − (/2)  is positive real for some real constant  ∈ R + , where   () =   ( 2 −   ) −1   +   is strictly positive real, and that the triple (  ,   ,   ) is controllable and observable.Assume that the parameterizing matrices of the  (25a), (25b), (25c), and (25d) are subject to parametrical disturbances so that with the disturbance matrices being subject to Ã ≥ −  , B ≥ −  , C ≥ −  , and D ≥ −  .Assume, furthermore, that (  ,   )  (  ,   ) and (, )  (, ) are monomial.Then, both   and  are positive while the following properties hold: (i) The transfer matrix () = ( 2 − ) −1  +  is strictly positive real (then  is convergent), , , ,  > 0 and  1 () = () − /2 is positive real if there exist matrices K ∈ R × , L ∈ R 2× , where  is some arbitrary positive integer, satisfying the following set of matrix relations: for some  ∈ (0 , 1] and the given  ∈ R + , for some existing matrices which satisfy the following set of matrix identities: are positive since (  ,   )  (  ,   ) and (, )  (, ) are monomial so that the sequences {  []} and {  []} are nonnegative for any nonnegative control.Note also that from the conditions on the parameterizing matrices both extended discrete systems describing the given hybrid system are positive.Note that   is at least critically stable although nonnecessarily convergent.Note also that if  > 0 then   () is strictly positive real and if  = 0 then it is positive real if A is at least critically stable (rather than convergent) with eventual simple poles of positive semidefinite on || = 1.Note that Re(  (  ) + Re    ( − )) −   ⪰ 0 for any  ∈ [0, ∞] since   () − (/2)  is positive real and  =  + .From the equivalence between the Discrete Factorization Lemma and the Discrete Positive Real Lemma [38], there exist a positive definite real matrix   , which is diagonal since   is positive and convergent [3], and real matrices   ,   , and   ≻ 0 such that the matrix relations (61) hold implying from the Discrete Positive Factorization Lemma that where so that the following factorization holds: where  1 (  ) = +  (   2 −   ) −1   .Thus, by invoking similar arguments of the equivalence between both lemmas, () = ( 2 − ) −1  +  − (/2)  is positive real for some given  ∈ (0, 1], if and only if, there exist a diagonal positive definite real matrix  and real matrices , , and  ≻ 0, subject to P ≻ −  , Q ⪰ −  , satisfying such that the following matrix relations hold: Now, direct calculations show that (61) guarantee ( 68 54) and by noting that it has negative entries except the last  =  +  diagonal entries because the entries of   ̸ = 0 are nonnegative since the system is positive.So, the state-space realizations of  1 () and   () do not fulfill Lemma 6 for   which has negative off-diagonal entries.A similar conclusion follows for  1 () and ().

Remark 11.
(1) Note that in Theorem 10(ii)   and  can be critically stable, since   (),  1 (), (), and  1 () are positive real, so that they can eventually possess simple eigenvalues, such that the four resulting matrices (  ) +   ( − ), with  being any of the four above ones, have positive semidefinite residuals at such simple critical poles.
(2) Note that if the nominal extended discrete system   is positive and Theorem 10 holds, then the extended discrete system  is positive and it is also positive in the input-output positivity (or "passivity") sense of [13,15] (see also [30]) since positive realness of transfer matrices is equivalent in the discrete-time domain to )  and (  ,   ) ≥ 0 on any discrete-time interval [ 0 ,  1 ] with  0 ≥ 0. In particular, for any integers  0 ≥ 0 and  1 ≥  0 with   =   − | P| and a close relation for the nominal   with L = 0, K = 0, P = Q = 0.
(3) Usually, the positive real and positive factorization lemmas are stated for minimal (i.e., simultaneously controllable and observable) state-space realizations in order to exclude from the analysis eventual unstable and critically stable (in the nonstrict positive realness case) zero-pole cancellations in the transfer matrices [13,15].The intuitive reason is that the state-space realization is got as a minimal one from the given transfer matrix so that it does not give information about eventual cancellations removed from the transfer matrices and its implication in the statespace descriptions when dealing with the Continuous or Discrete Positive Real Lemmas or their equivalent Youla's Factorization Lemmas.
Theorem 10 states a characterization of the admissible structured perturbations for the dynamics, output, control, and interconnection matrices of a state-space realization associated with the discrete nominal positive real transfer matrix which guarantee that the perturbed system  being positive maintains the positivity and the positive realness property of the nominal   .Based on the Discrete Positive Real Lemma without invoking the factorization result, we now establish a parallel result to be applicable for nonstructured parametrical disturbances at the expense of testing the positive definiteness of an associated matrix.
Theorem 12. Assume that the hypothesis of Theorem 10 holds for the parametrical disturbance matrices.Then, both   and  are positive and the following properties hold: (i) The transfer matrix () = ( 2 − ) −1  +  is strictly positive real (then  is convergent), , , ,  > 0, and  1 () = () − /2 is positive real if there exist matrices satisfying the following set of matrix relations: for some  ∈ (0 , 1] and the given  ∈ R + , for some existing matrices and   ∈ R × , which satisfy the following set of matrix identities: Furthermore,  satisfies (71): for any integers  0 ≥ 0 and  1 ≥  0 with   =   − | P| and a close relation is satisfied by the nominal   with L = 0, K = 0, P = Q = 0.
(iii) If   ⪰ 0, Q ≻ −  , then   () is positive real, () is strictly positive real, and  1 () is positive real even if   is critically stable.
Example 13.A particular system of the studied hybrid class with  = 4 is now discussed with some of the parameters being fixed "a prior" while others are primarily left undetermined in order to find the needed positive realness conditions.Consider the following the hybrid system (1a), (1b), and (1c) with   =   = 2,  =  = 1,  = 0.01, =    = (1, 0) ; which leads to the following matrices: If  = 1, note that   is Hurwitz and Metzler and then Φ  (0.01) is positive and convergent.If   ≥ 0,   > 0, and   > 0, then both the continuous-time subsystem and its discretized version to any sampling period are positive dynamic systems.Also,   is convergent.The transfer function of the uncoupled continuous-time subsystem is where  is the Laplace transform argument.It can be easily checked that the continuous transfer function is positive real if  1 =  2 = 0; 0 ≤   =  1  2 ≤ 3 and   > 0. If  2 > 0 and  1 > 0 with  1 =  2 = 0, then the triple (  ,   ,   ) is controllable and observable.If, in addition,   > 0, then   () is strictly positive real.If, furthermore,   > 0 or   < 3, then   () is strongly strictly positive real in the sense that Re On the other hand, note that the uncoupled continuoustime subsystem is a mathematical model for some wellknown linear dynamic systems as a damped mechanical system, or an RLC electric circuit, described by the differential equation: forced by a term () calculated from a primary control () which is everywhere piecewise continuous if   = 0 and everywhere twice continuous-time differentiable with piecewise continuous second time-derivative if   > 0. Note that if ‖  ‖ 2 < 0.581976, then   being Hurwitz guarantees that    is convergent from the fulfillment of the stability constraint , where  is the discrete transfer function argument representing in the time delay a one-step advance operator , and equivalently,  −1 is a one-step delay operator formally equivalent to  −1 .It can be directly checked that the digital transfer function is strictly positive real with Re   (  ) ≥   = 1 for  ∈ [0, 2].The extended  has four stable eigenvalues, namely, 0.367879, 0.135335, 0.930074, −0.430074 in the free coupling case, that is, if   = 0,   = 0.By using a similar reasoning to that guaranteeing that   being Hurwitz implies that    is convergent, one concludes that the system matrix  of the  is convergent if the sufficiency constraint below holds: (iii) Assume that  1 () is strictly positive real for some  ∈ R + .Then, the closed-loop  keeps the asymptotic hyperstability property from that of the nominal   in the sense that Property (i) holds and, furthermore, {‖[]‖ 2 }, {‖  []‖ 2 }, and {‖[]‖ 2 } converge asymptotically to zero for any given initial condition [0].
Proof.One gets from ( 86) and ( 89) for any integers  0 ≥ 0 and  1 >  0 with the fourblock partitioned matrices of (90a) being at least positive semidefinite (see Theorems 10 and 12).Since   + P ≻ 0, the sequence {‖[]‖ 2 } is uniformly bounded for any given initial condition [0] and any nonlinear eventually time-varying controller satisfying Popov's inequality (89).Thus, the current closed-loop system  is hyperstable as it is the nominal one.If (  ,   ) is controllable and (26) holds, then (, ) is controllable for the parameterizations defined in (25c) and then the uniform boundedness of {‖[]‖ 2 } implies that of {‖[]‖ 2 }.Property (i) has been proved.Under the conditions of Property (ii), since M1 ≻ 0, so that   ≻ 0 and   ≻ 0 (equivalently,      ≻ 0), and M1 ⪰ − M1 , so that Q ≥ −  , R ⪰ −  , then M1 = M1 + M1 ≻ 0 and   ≻ 0 and P ⪰ −  , then it follows from the boundedness of the second inequality of (89) for all  1 >  0 that the sampled state and input of the nominal and current  converge asymptotically to zero without the need for any controllability assumption.This proves Property (ii).Under the additional conditions of Property (iii),  is convergent and {[]} → 0 from the first identity of (68) which is a discrete Lyapunov matrix equation.Note that (74)-(76) hold with   ≻ 0 and Q ⪰ −  (from the strict positive realness condition) and   ≻ 0 as well as  ≻ 0 (since R ⪰ −  ) and furthermore,      ≻ 0 and    ≻ 0.
Otherwise, lim →∞ Re( 1 (  ) +   1 ( − )) = 0 and the nominal  1 () would not be strictly positive real as it would happen with the disturbed transfer matrix.This implies that {[]} → 0 since otherwise lim  1 →∞ ∑  1  0   [][] would be infinity from the system positivity.A similar argument concludes that {[]} → 0, since    ≻ 0, without requiring a controllability condition, since any eventual zero/pole cancellation in the transfer matrix is necessarily strictly stable (since the transfer matrix  1 () is strictly positive real) so that any eventual uncontrollable mode is asymptotically stable.This proves Property (iii).To prove Property (iv), note that if M1 ⪰ 0 with   ≻ 0 and Q ≥ −  , then  1 () is positive real.From the system positivity and the finite upperboundedness of (89), lim  We now introduce the concept of asymptotic hyperstability in the mean in the sense that the system is globally stable and, furthermore, the input and output power (and then its inputoutput instantaneous power) converge asymptotically to zero except eventually for a set of time instants zero measure.
(2) The nonlinear and eventually time-varying controller ((), ) is everywhere piecewise continuous with respect to  and continuous with respect to  in R  + × R + .
Note from (100) that since (  ) → 0 as  → ∞, since  can be taken to be arbitrarily small, and simultaneously since the closedloop  is asymptotically hyperstable since the matrices   ,   ,   and   ,   ,   , are positive definite matrices for all  ∈ Z + , then { [  +ℓ     +   ] } → 0, and {[  + ℓ     +    ]} → 0 as it follows from the direct extension Theorem 16(iv) to timevarying parameterizations under sufficiency type conditions of asymptotic hyperstability, provided that  =  and   is nonsingular for all  ∈ Z + [13].Since the state, input, and output have nonnegative components, one also gets that   () → 0 and () → 0 as  → ∞ except, eventually, at isolated time instants where the nonlinearity ((), ) is not continuous.
If the linear part of the system is not positive real, or even stable, or if it is suited to improve its relative stability, a linear feedback law can be injected prior to the operation by the nonlinear device towards the achievement of positive realness or strictly positive realness of the transfer matrix describing the linear feed-forward block.In particular, assume that following state-feedback linear control law is given: ( The following result is related to the achievement of the hyperstability of the closed-loop extended discrete hybrid system under the given control law as well as the asymptotic hyperstability in the mean of  ℓ . Theorem 22. Assume that (  ,   ), (  ,   ), (  ,   ), and (  ,   ) and (  ,   ) are controllable pairs.Then, an appropriate feasible parameterization of the control gains in the fictitious discrete control law ( 92)-( 93), with the replacement (⋅) → (⋅) and the system closed-loop reparameterization (103), associated with the feedback control law (101), may lead to a positive real transfer matrix of the positive closed-loop system  ℓ and then its hyperstability if () = −((), ) for any nonlinear and eventually time-varying nonlinearity ((), ) which satisfies Popov's type inequality.If, furthermore,  ℓ is asymptotically hyperstable and the assumptions ( 2) and ( 3) of Theorem 21 hold, then the closedloop hybrid  ℓ is asymptotically hyperstable in the mean.
Proof.We refer with superscript bars to any matrices for either the parameterization or the Positivity Real Lemma (, , , , and ) after performing the control law (101).Note that, under controllability of any pair (, ), it is possible to choose a state-feedback control gain  for the achievement of any given arbitrarily prescribed stable closed-loop placement.Since (  ,   ), (  ,   ), (  ,   ), and (  ,   ) and (  ,   ), so (  +     ,   ), are controllable pairs, it is feasible to choose the control gains   and   in such a way that   , and then     , and   have stable eigenvalues being as largely dominant, related to the spectral norms of   and   , as possible via the choices of   and   so that the dynamics  of the closed-loop extended discrete hybrid system be a convergent matrix.On the other hand, one can choose the -matrix   of sufficiently small nonnegative entries so that  and , and then  in the second constraint of the Discrete Positive Real Lemma has a sufficiently small spectral norm related to that of  while  +   is dominant norm of order (‖  ‖ 2 ) over that of   , of order (‖  ‖ 2 2 ), so that  ⪰ 0 and  ⪰ 0. In this way, the discrete modified closed-loop transfer matrix of   , related to the new input (), might be designed to be at least positive real.On the other hand, the asymptotic hyperstability in the mean of  ℓ follows from Theorem 21 from the asymptotic hyperstability of  ℓ and the assumptions (2) and (3) of Theorem 21 since the first assumption of such a theorem holds since the controllability of the pair (  ,   ) implies that of the pair (  ,   ).

Conclusions
This paper has investigated a class of hybrid systems dealt with and characterized with explicit results its positivity and some of its stability properties.The hybrid system consists of a dynamic system which has a continuous-time substate and a digital one with mutual coupled dynamics.An extended discrete hybrid system which describes any hybrid system in the given class at sampling instants is investigated to establish the stability and controllability properties of the discretized system.The state of the extended discrete hybrid system contains the discretized substate of the continuous-time subsystem at sampling instants and the digital substate.The paper studies the stability and controllability, in a robustness context for parametrical disturbance, of such an extended discrete system whose state is defined by both the digital substate and the discretized version of the continuous-time subsystem at sampling instants.Two discrete versions of the KYP-Lemma are given for (a) a simplified version of the hybrid system related to the relevant pairs of the system and control matrices and (b) for a more general version of such a lemma related to the whole state-space realization involving the output and input-output interconnection matrices as well.The relationships of the positive realness of the transfer matrix to the state-space realization are explicitly characterized related to the discrete KYP-Lemma and Youla's factorization Lemma.The obtained results on positive realness are related to the hyperstability and asymptotic hyperstability properties of the hybrid system for any member of a class of nonlinear and perhaps time-varying controller device satisfying Popov'stype inequality.Finally, some extensions are given for the case where there is a supplementary stabilizing linear control scheme which stabilizes the dynamics hybrid system prior to the nonlinear and time-varying control law operation to establish the hyperstability of the closed-loop system.

1. 1 .
Notation and Terminology.(a) R + is the set of nonnegative real numbers; R  + ( being a positive integer) is the Cartesian product  times of R + .The vector function V() ∈ R  + for some  ≥ 0 if all its components are nonnegative at .The matrix  ∈ R × +

)
Theorem 5. Define Ã =  −   and B =  −   such that (  ,   ) is a nominal controllable pair and   = { ∈ ≤ , ∀ ∈ R ×(  +) }, where the control matrices of the nominal   , parameterized by   and   , and the current  systems are those of the parameterization (25c) of (25a).Then, the following properties hold:(iii) The system  is controllable if rank (  ,   ) = 2 and (i)The system  is controllable if and only if (, ) is full rank.If (, ) is not full rank then there exists a control sequence such that the system  is approximately controllable with state targeting error 0[max(‖ −   ‖ 2 , ‖ −   ‖ 2 )].(ii)The system  is controllable if and only if rank (, ) = 2 and rank (  ,   ) = .