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

Authors: M. De la Sen
Journal: Discrete Dynamics in Nature and Society
 (2017)

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## Abstract

This paper formulates an “ad hoc” robust version under parametrical disturbances of the discrete version of the Kalman-Yakubovich-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 any member 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.


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## Body

## 1. Introduction

Continuous-time and discrete-time positive systems have been studied in detail in recent years [1–10]. 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 [13–15], which generalizes the most basic concept of stability of dynamic systems. See, for instance, [13, 14, 16–29] 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 input-output 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 closed-loop 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. Section2 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 continuous-time 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.


### 1.1. Notation and Terminology

(a)R+ is the set of nonnegative real numbers; R+p (p being a positive integer) is the Cartesian product p times of R+. The vector function vt∈R+p for some t≥0 if all its components are nonnegative at t. The matrix Q∈R+m×n if it is of order m×n, 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-p and Q∈R-m×n) 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 matrixQ∈R+m×n is said to be positive (denoted by Q>0) if it has at least a positive entry. A nonnegative matrix Q≥0 satisfies either Q>0 or Q=0. A matrix Q∈R-m×n, which has at least a negative entry, is said to be negative and denoted by Q<0 and, if all its entries are negative, then it is denoted by Q≪0.(c) A matrixQ∈R+m×n is said to be strictly positive (denoted by Q≫0) if all its entries are positive. Similarly, a vector v∈R+p 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 A≫B,  v≫w for matrices and vectors mean, respectively, A-B≫0 and v-w≫0. Interpretations of expressions like A>B,  v>w,  v≥w follow directly from the above ones.(d) We denoteQ≻0 (Q⪰0) if Q∈Rn×n is positive definite (positive semidefinite) and Q≺0 (Q⪯0) if Q∈Rn×n is negative definite (negative semidefinite).(e)In is the nth identity matrix.(f) A matrixQ∈R+n×n 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 Q (with respect to any norm) is μQ=limε→0+⁡I+εQ-I/ε. The spectrum of Q is the set of its eigenvalues (or spectrum) denoted by SpQ and its characteristic polynomial denoted by pQs = DetsI-Q, where s is a complex indeterminate and Det(·) stands for the determinant of the matrix (·). A subscript in the matrix measure μ(·)Q denotes the measure with respect to a particular (·)-norm. A matrix Q∈R+n×n is said to be convergent (or Schur), if all its eigenvalues lie in the strict unity circle. An H∞ complex function is Schur if its H∞-norm is bounded by unity while it is said to be strictly bounded real (SBR), if in addition its coefficients are real and its H∞-norm is strictly bounded by unity.(g)KpQ=PpPp-1∈1,∞ is the condition number of the matrix Q∈Rn×n with respect to the p-norm. It is infinity if and only if the matrix PTP is singular. In particular, K2Q=Q2Q-12 is the condition number of Q with respect to its l2 (or spectral) norm which is the quotient of its maximum and minimum eigenvalues in the case when it is square.(h) A matrixQ=Qij∈Rn×n is said to be a n-Metzler matrix, denoted by Q∈MEn×n, if and only if all its off-diagonal entries satisfy Qij≥0 for all i,j≠i∈n-≔1,2,…,n. A matrix Q=Qij∈Rn×n is said to be a M-matrix of order n, denoted by Q∈MMn×n, if and only if it is a Z-matrix; that is, all its entries satisfy Qii≥0 and Qij≤0 for all i,j≠i∈n-=1,2,…,n and, furthermore, all its eigenvalues are strictly unstable.(i)Qi and QiT denote, respectively, the ith column or row of the real Q-matrix, the superscripts “T” and “∗” denoting transpose and conjugate transpose, respectively. Qi, i being an integer number, denotes the ith power of the Q-matrix and provided that Q=Qij, QG=QijG is an associate matrix to Q defined as QijG=1 if Qij≠0 and QijG=0, otherwise. Note that Q≥0⇔QG≥0. vi denotes the ith component of the real vector v and v≥0⇔vG≥0. Thus, any positive system S has always an associate positive system SG which defines the pairwise relations input components/state-output components and state components/output components from its associate influence graph G [2, 3, 5], by defining all its parameterizing matrices according to the above criterion.(j)U(t)=1(t) is the unity step (Heaviside) function.(k)ej(n) is the unity vector of Rn whose unique nonzero component is the jth one which is unity.(l) The notationxk stands for a discrete/digital variable or vector x which is only defined as sampling instants tk=kT,  k∈Z+, with T being the sampling period. If x is a digital variable then it is only defined at sampling instants. If x is a discrete variable (i.e., that arising from the discretization of a continuous variable), then xk=xkT and any of both equivalent notations are used indistinctly in such a case.(m) The superscriptT stands for the transpose of a vector or matrix while KerO stands for the null-space of the operator O.


## 1.1. Notation and Terminology

(a)R+ is the set of nonnegative real numbers; R+p (p being a positive integer) is the Cartesian product p times of R+. The vector function vt∈R+p for some t≥0 if all its components are nonnegative at t. The matrix Q∈R+m×n if it is of order m×n, 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-p and Q∈R-m×n) 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 matrixQ∈R+m×n is said to be positive (denoted by Q>0) if it has at least a positive entry. A nonnegative matrix Q≥0 satisfies either Q>0 or Q=0. A matrix Q∈R-m×n, which has at least a negative entry, is said to be negative and denoted by Q<0 and, if all its entries are negative, then it is denoted by Q≪0.(c) A matrixQ∈R+m×n is said to be strictly positive (denoted by Q≫0) if all its entries are positive. Similarly, a vector v∈R+p 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 A≫B,  v≫w for matrices and vectors mean, respectively, A-B≫0 and v-w≫0. Interpretations of expressions like A>B,  v>w,  v≥w follow directly from the above ones.(d) We denoteQ≻0 (Q⪰0) if Q∈Rn×n is positive definite (positive semidefinite) and Q≺0 (Q⪯0) if Q∈Rn×n is negative definite (negative semidefinite).(e)In is the nth identity matrix.(f) A matrixQ∈R+n×n 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 Q (with respect to any norm) is μQ=limε→0+⁡I+εQ-I/ε. The spectrum of Q is the set of its eigenvalues (or spectrum) denoted by SpQ and its characteristic polynomial denoted by pQs = DetsI-Q, where s is a complex indeterminate and Det(·) stands for the determinant of the matrix (·). A subscript in the matrix measure μ(·)Q denotes the measure with respect to a particular (·)-norm. A matrix Q∈R+n×n is said to be convergent (or Schur), if all its eigenvalues lie in the strict unity circle. An H∞ complex function is Schur if its H∞-norm is bounded by unity while it is said to be strictly bounded real (SBR), if in addition its coefficients are real and its H∞-norm is strictly bounded by unity.(g)KpQ=PpPp-1∈1,∞ is the condition number of the matrix Q∈Rn×n with respect to the p-norm. It is infinity if and only if the matrix PTP is singular. In particular, K2Q=Q2Q-12 is the condition number of Q with respect to its l2 (or spectral) norm which is the quotient of its maximum and minimum eigenvalues in the case when it is square.(h) A matrixQ=Qij∈Rn×n is said to be a n-Metzler matrix, denoted by Q∈MEn×n, if and only if all its off-diagonal entries satisfy Qij≥0 for all i,j≠i∈n-≔1,2,…,n. A matrix Q=Qij∈Rn×n is said to be a M-matrix of order n, denoted by Q∈MMn×n, if and only if it is a Z-matrix; that is, all its entries satisfy Qii≥0 and Qij≤0 for all i,j≠i∈n-=1,2,…,n and, furthermore, all its eigenvalues are strictly unstable.(i)Qi and QiT denote, respectively, the ith column or row of the real Q-matrix, the superscripts “T” and “∗” denoting transpose and conjugate transpose, respectively. Qi, i being an integer number, denotes the ith power of the Q-matrix and provided that Q=Qij, QG=QijG is an associate matrix to Q defined as QijG=1 if Qij≠0 and QijG=0, otherwise. Note that Q≥0⇔QG≥0. vi denotes the ith component of the real vector v and v≥0⇔vG≥0. Thus, any positive system S has always an associate positive system SG which defines the pairwise relations input components/state-output components and state components/output components from its associate influence graph G [2, 3, 5], by defining all its parameterizing matrices according to the above criterion.(j)U(t)=1(t) is the unity step (Heaviside) function.(k)ej(n) is the unity vector of Rn whose unique nonzero component is the jth one which is unity.(l) The notationxk stands for a discrete/digital variable or vector x which is only defined as sampling instants tk=kT,  k∈Z+, with T being the sampling period. If x is a digital variable then it is only defined at sampling instants. If x is a discrete variable (i.e., that arising from the discretization of a continuous variable), then xk=xkT and any of both equivalent notations are used indistinctly in such a case.(m) The superscriptT stands for the transpose of a vector or matrix while KerO stands for the null-space of the operator O.


## 2. Hybrid System and Positivity and Controllability Properties

Consider the subsequent hybrid linear systemH: (1a)x˙ct=Acxct+Acsxck+Acdxdk+Bcut+Bcsuk,(1b)xdk+1=Adxdk+Adsxck+Bduk,(1c)yt=Ccxct+Ccsxck+Cdxdk+Dcut+Ddukfor all t∈kT,k+1T for any integer k≥0 with T>0 being the sampling period, where xc and xd are, respectively, the continuous and digital substates of respective dimensions nc and nd, and u∈Rm and y∈Rp are the input and output vectors. The continuous-time argument is denoted by (t) while the discrete-time argument is denoted by [k] and the associated continuous and digital variables are denoted correspondingly. That is, a continuous variable at sampling instants is denoted in the same way as a purely digital variable such that continuous and discrete (or digital) time arguments are, respectively, denoted with parenthesis (·), such as xct, yt, and ut, and brackets [·], such as xck=xckT,  xdk,  uk=ukT. On the other hand, the parameterization of (1a), (1b), and (1c) is as follows:(i)Ac and Ad are the matrix of continuous-time and of digital dynamics, respectively, and Acd and Ads are, respectively, the matrices of dynamics of couplings between the digital and continuous-time substates and continuous-time discretized and digital substates. The matrix Acs is the matrix of dynamics of coupling between the sampled continuous-time substate to its time evolution over the next sampling interval.(ii)Bc and Bd are continuous-time and digital control matrices and Bcs is a coupling control matrix from the sampled continuous-time control to the next intersample period continuous-time substate.(iii) The matricesCc,Ccs,Cd and Dc and Dd 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 xct,  xck, and xdk and inputs and outputs. Note that the hybrid system is driven by the control ut and by its samples ukT of period T acting as two independent control actions. At sampling instants, it follows by direct calculus from (1a), (1b), and (1c) that the hybrid system H is described by the following n=nc+ndth order extended discrete-time system of sampling period T driven by a fictitious extended input sequence vk⊂Rm+n whose element vk depends on u:kT,k+1T→Rm and since only finite input jumps happen at sampling instants, since impulsive jumps are not considered, vk depends on u:kT,k+1T→Rm since the updated value uk+1 at t=k+1T does not contribute to vk  DH: (2a)xk+1=Axk+B0uk+ξk,(2b)=Axk+Bvvk,(2c)yk=Cxk+Dukof state xk=xcTk,xdTkT for any integer k≥0, where (3)A=Ac′Ad′AdsAd;B0=eAcT∫0Te-AcτdτBcsBd,(4)Ac′=eAcTInc+∫0Te-AcτdτAcs;Ad′=eAcT∫0Te-AcτdτAcd,(5)C=Cc+Ccs⋮Cd;D=Dc+Dd,(6)Bv=B0⋮Inc0nd×nd(7)ξk=ξcTk,0TT;ξck=eAcT∫0Te-AcτBcukT+τdτ,(8)vk=uTk,ξcTkT,where vk∈Rm+n. The derivation of the extended discrete DH, (2a), (2b), and (2c), subject to (3)–(8), from the hybrid system H, (1a), (1b), and (1c), is direct from a time-integration of (1a), (1b), and (1c) on a sampling time interval kT,k+1T with initial conditions at t=kT. 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 systemH is positive if and only if Ac∈MEnc×nc,  Acs∈R+nc×nc,  Acd∈R+nc×nd,  Ad∈R+nd×nd,  Ads∈R+nd×nc,  Bc∈R+nc×m,  Bcs∈R+nc×m,  Bd∈R+nd×m,  Cc∈R+p×nc,  Ccs∈R+p×nc,  Cd∈R+p×nd,  Dc∈R+p×m,  Dd∈R+p×m.
Under the above given conditions,eAcT∈R+nc×nc,  Ac′∈R+nc×nc,  Ad′∈R+nc×nd,  B0∈R+n×2m,  C∈R+p×n,  D∈R+p×m,  ξk∈R+n, and vk∈R+m+n for k≥0, if u:0,∞→Rm, and then the extended discrete system DH is also positive.Theorem 2.
The following properties hold:
(i) Assume that
(a)Ac∈MEnc×nc, Acs∈R+nc×nc, Acd∈R+nc×nd, Ad∈R+nd×nd, Ads∈R+nd×nc, Bc∈R+nc×m, Bcs∈R+nc×m, Bd∈R+nd×m,
(b)Ac′, Ad, Ac′+Ad′Ind-Ad-1Ads and Ad+AdsInc-Ac′-1Ad′ are convergent.
Then,A is convergent and the unforced DH is globally asymptotically stable.
(ii) Under the hypotheses of Property (i), assume also thatA is convergent, A>0, and that 1/In-A-12>A-Aad2, where Aad=BlockDiagAc′,Ad. Then, Ac′ and Ad are convergent.
(iii)A is convergent, and then the unforced DH and the unforced H are both globally asymptotically stable, if the following conditions hold:
(
1)  Ac is Hurwitz and Ad is convergent such that ρd-1≤Ad2≤ρd1 and e-ρct≤eAct2≤e-ρc1t for some real constants ρc, ρc1, ρd1, and ρd, with ρc≥ρc1>0 and ρd-1≤ρd1<1.
(
2)  Acs2≤ε, Ads2≤ε, and Acd2≤ε for some ε∈0,ε∗, where ε∗=ρc/231-eρcT2+ρd2ρc21/2.
(iv)A is convergent if Ac′ and Ad are convergent and Acd2<1/Ac′-1∫0TeAcT-τdτ21+Ad-1Ads2.Proof.
First note thatA>0 and is convergent if and only if I-A is nonsingular and I-A-1>0 [9]. Direct calculations with (3) and the inverse of a 2 × 2 block partitioned matrix [36] yield in this case(9)I-A=In-Ac′-Ad′-AdsIn-Ad,I-A-1=Γ11Γ12Γ21Γ22,where(10)Γ11=Inc-Ac′-Ad′Ind-Ad-1Ads-1,Γ12=Inc-Ac′-1Ad′Ind-Ad-AdsInc-Ac′-1Ad′-1=Inc-Ac′-Ad′Ind-Ad-1Ads-1Ad′Ind-Ad-1,Γ21=Ind-Ad-1AdsInc-Ac′-Ad′Ind-Ad-1Ads-1=Ind-Ad-AdsInc-Ac′-1Ad′-1AdsInc-Ac′-1,Γ22=Ind-Ad-AdsInd-Ac′-1Ad′-1.Note that the above matrices exist with Γij>0,  i,j=1,2, since all the needed intermediate matrix inverses exist since
(a)Ad and Ac′ are convergent, Ad>0, and Ac′>0 (since Ac∈MEnc×nc, and then eAct>0 for any real t≥0) and Acs∈R+nc×nc, so that Ind-Ad and Inc-Ac′ are nonsingular with Ind-Ad-1>0 and Inc-Ac′>0,
(b)Ac′+Ad′Ind-Ad-1Ads and Ad+AdsInc-Ac′-1Ad′ are convergent with nonnegative entries so that Γ11 exists, Γ11>0, and Ind-Ad-AdsInc-Ac′-1Ad′-1 with Ind-Ad-AdsInc-Ac′-1Ad′-1>0.
As a result, there existsI-A-1 with A>0 and I-A-1≥0 which holds if and only if A is convergent so that the unforced DH is globally asymptotically stable. Property (i) has been proved. To prove Property (ii), note that A>0 and is convergent, so that In-A is a nonsingular M-matrix and In-A-1>0 exists. Since In-A is a nonsingular M-matrix, all its leading minors are positive. Thus, In-Aad≻0, then Inc-Ac′≻0 and Ind-Ad≻0 and, equivalently, Inc-Ac′-1≻0 and Ind-Ad-1≻0. On the other hand, one has(11)In-Aad-1=In-In-A-1A-Aad-1In-A-1.Since A≥Aad from the hypotheses, In-A-1A-Aad≥0 and if In-A-1A-Aad is convergent, that is, if 1>In-A-1A-Aad2 guaranteed if 1/In-A-12>A-Aad2, then In-In-A-1A-Aad-1>0. As a result, In-Aad-1>0,  Inc-Ac′>0, and Ind-Ad>0 and they have real positive eigenvalues since Inc-Ac′≻0 and Ind-Ad≻0. Thus, for any given nonzero vectors zc and zd of respective dimensions nc and nd, one concludes that zcTzc>zcTAc′zc and zdTzd>zcTAdzd. This implies that the spectral radii of Ac′ and Ad are less than unity so that Ac′ and Ad are positive and convergent.
To prove Property (iii), note that sinceAc is Hurwitz and Ad is convergent, A=A0+A~=A0I2n+A0-1A~, where(12)A0=Block DiageAcT⋮Ad(13)A~=∫0TeAcT-τdτAcs∫0TeAcT-τdτAcdAds0nd×nd=∫0TeAcT-τdτ0nc×nd-nc∫0TeAcT-τdτ0nc×nd-nc0nc×nd0nc×ndInd0nd×ndInd0nd×ndA-cs0nd×nc0nd×ndA-cdAds-A-cs0nd×nc0nd×nd0nd×ncwith(14)A-cs=Acs0nd-nc×nc∈Rnd×nc;A-cd=Acd0nd-nc×nc∈Rnd×nc,(15)A0-1A~=e-AcT0nc×nd0nd×ncAd-1L,L,0A-cs0nd×nc0nd×ndA-cdAds-A-cs0nd×nc0nd×nd0nd×nc=e-AcT0nc×nd0nd×ncAd-1∫0TeAcT-τdτ0nc×nd-nc∫0TeAcT-τdτ0nc×nd-nc0nc×nd0nc×ndInd0nd×ndInd0nd×ndA-cs0nd×nc0nd×ndA-cdAds-A-cs0nd×nc0nd×nd0nd×nc.Since Ac is Hurwitz and Ad and A0 are convergent, there exists a real constant K≥1, which is norm-dependent such that Ke-ρctIn≤eAct≤Ke-ρc1tIn for all t≥0. Note that 1≤Ad-12≤ρd and eρc1t≤e-Act2≤eρct for all t≥0 with -ρc1<0 being not less than the stability abscissa of Ac and ρd1 being not less than the convergence abscissa of Ad. Since max⁡Acs2,Ads2,Acd2≤ε, one gets from (15) that(16)A0-1A~2=A0-1A~TA0-1A~21/2≤2LTL21/23ε2≤231-eρcT2ρc2+ρd21/2ε.If ε∈0,ε∗, then A=A0I2n+A0-1A~ is nonsingular from Banach Perturbation Lemma [37] and then convergent since A0 is convergent from the continuity of the eigenvalues of matrix with respect to its entries. Then the unforced DH is globally asymptotically stable and the unforced H is also globally asymptotically stable since Ac is Hurwitz and xk→0 as k→∞ for any given initial condition. Property (iii) has been proved. Property (iv) follows by redefining A=A01+A~1=A01I2n+A01-1A~1(17)A01=Ac′0nc×ndAdsAd;A01-1=Ac′-10nc×nd-Ad-1AdsAc′-1Ad-1;A~1=0nd×nc∫0TeAcT-τdτAcd0nd×nc0nd×ndand then(18)A01-1A~1=Ac′-10nc×nd-Ad-1AdsAc′-1Ad-10nd×nc∫0TeAcT-τdτAcd0nd×nc0nd×nd=0nc×ncAc′-1∫0TeAcT-τdτAcd0nd×nc-Ad-1AdsAc′-1∫0TeAcT-τdτAcdso that A is nonsingular if Acd2<1/Ac′-1∫0TeAcT-τdτ21+Ad-1Ads2.Related to Theorem2, note that Ac′ is convergent by construction if Ac is Hurwitz and a guaranteed upper-bound of Acs is sufficiently small which increases as the sampling period T and the stability abscissa of Ac increase. The next result generalizes Theorem 1 if Ad is not necessarily convergent and Ac is not necessarily Hurwitz.Example 3.
Consider a positive hybrid system with scalar continuous-time and digital subsystems andAc=-ac such that Ac′=ac′=e-acT+1-e-acTacs/ac and Ad′=ad′=e-acT+1-e-acTacs/ac,  Ads=ads≥0,  Acs=acs≥0, and Ad=ad∈0,1, that is, nonnegative and convergent. From Theorem 2(i) if Ac′ is convergent then Ac is convergent implying that ac>0. Furthermore, for some ε∈0,1,  Ac′ is nonnegative and convergent if 0≤e-acT+1-e-acTacs/ac≤1-ε which implies that acs≤1-ε-e-acTac/1-e-acT. Note that acs<ac.
On the other hand, hypothesisac′+ad′Ind-ad-1ads being nonnegative and convergent holds if (19)0≤ac′+ad′Ind-ad-1ads≤1-ε1≤1-ε+ad′Ind-ad-1adsfor some ε1∈max⁡ε-ad′Ind-ad-1ads,0,1-ac′; that is, the left-hand side is nonnegative convergent, since (20)0≤ads≤ac1-ad1-ac′-ε1acd1-e-acTwhile ad+adsInc-ac′-1ad′ being convergent holds if (21)0≤ad≤1-ε2-acadsacd1-ac′ac=1-ε2-adsacd1-e-acT1-acs/ac-ε2for some ε2∈0,1-e-acT1-acs/ac. Particular numerical values which satisfy all the given joint constraints are, for instance, ac=0.1, so that the stable continuous dynamics has a small relative stability, T=0.1, ad=0, so that the digital dynamics has a maximum stability degree, and the forced system behavior is independent of the digital self-dynamics, ad′=0.995, ads=0.009.
It can be verified thatA is convergent with eigenvalues 0.9919 and -0.0089. The unforced DH is globally asymptotically stable [Theorem 2(i)]. Since, in this case with nc=1 and acs≥0, ac′ being convergent implies that ac>0 (i.e., Ac=-ac is Hurwitz), the unforced H is also globally asymptotically stable [Theorem 2(iii)].Corollary 4.
The following properties hold:
(i)A is convergent, and then the unforced DH is globally asymptotically stable, if
(
1)
A
0, (12), is nonsingular and there exists λ∈R∈-1,-1/2∪0,∞ such that 1+λA0 is convergent,
(
2) there exists R∋λ1∈λ,1+λ/K2A0 such that K2A0<1+λ/λ1 and A~2<λ1-λA02.
If, in addition,Ac is Hurwitz, then the unforced DH is globally asymptotically stable.
(ii) A sufficient condition for Property (i) to hold is(22)AcsAcdAds0nd×nd2<1+ρc1-eρcTλ1-λA02,where ρc∈R is the stability abscissa of Ac.Proof.
Note that, sinceA0 is nonsingular,(23)A=1+λA0+A~-λA0=1+λA0I2n+11+λA0-1A~-λA0and A is nonsingular from Banach’s Perturbation Lemma, under the condition A~2<λ1-λA02, if λ<λ1 and(24)1>λ11+λK2A0=λ11+λA02A0-12>11+λA0-12A~2+λA02>11+λA0-12A~-λA02.Since λ<λ1 and 1≤K2A0<1+λ/λ1, then λ<λ1<1+λ and, if λ<0, then λ<1/2 and λ<λ1<1-λ. Property (i) follows directly from (24). Property (ii) follows from the fact that (22) is a sufficient condition for A~2<λ1-λA02 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 B that B0uk+ξk=Bvvk=Bgk leading to the state system DH description driven by a real vector sequence:(25a)xk+1=Axk+Bvvk=Agxk+Bgvk+δvk=Agxk+Bggk+δkyk=Cxk+Dgk=Cgxk+Dggk+δykfor any integer k≥0, where Bv is reparameterized to some appropriate matrix B so as to drive the auxiliary control gkk∈Z0+⊂Rm+n of the form gk=uTk,gcTkT, where gckk∈Z0+⊂Rn, as follows:(25b)ukT+τ=VBcTeAcTT-τgck,∀τ∈0,T;ukT=ukfor some given prefixed m×m-matrix V=VT≥0, and(25c)δvk=A-Agxk+B-Bgvk;δk=A-Agxk+B-Bggk,δyk=C-Cgxk+D-Dggk,(25d)Bg=∫0TeAgcT-τdτBgcs∫0TeAgcT-τBgcVBgcTeAgcTT-τdτBgd0,B=∫0TeAcT-τdτBcs∫0TeAcT-τBcVBcTeAcTT-τdτBd0,Dg=Dgc+Dgd;D=Dc+Dd.Theorem 5.
DefineA~g=A-Ag and B~g=B-Bg such that Ag,Bg is a nominal controllable pair and Bg=X∈R+n×nc+m:B~g2≤Y,∀Y∈Rn×nc+m, where the control matrices of the nominal DHg, parameterized by Ag and Bg, and the current DH systems are those of the parameterization (25c) of (25a). Then, the following properties hold:
(i) The systemDH is controllable if and only if CA,B is full rank. If CA,B is not full rank then there exists a control sequence such that the system H is approximately controllable with state targeting error 0maxA-Ag2,B-Bg2.
(ii) The systemH is controllable if and only if rankCA,B=2n and rankCAc,Bc=n.
(iii) The systemDH is controllable if rankCAg,Bg=2n and(26)A-Ag2+B-Ag2<supz∈SpAgzI2n-Ag⋮Bg22+1supz∈SpAg⁡zI2n-Ag⋮Bg2-supz∈SpAgzI2n-Ag⋮Bg2.The system H is controllable if, furthermore, rankCAc,Bc=n.
(iv) The systemDH is reachable if it is controllable and, furthermore, A is nonsingular (in particular, if Property (iii) holds and, furthermore, Ag is nonsingular and A-Ag2<1/Ag-12=λmin1/2AgAgT). The system H is reachable if DH is reachable and Ac is nonsingular.Proof.
One gets by direct recursive calculation from (25a)(27)xk+i=Agixk+∑j=0i-1Agi-j-1Bggk+j+δk+j,(28)x~k+2n=xk+2n-Ag2nxk-∑j=02n-1Ag2n-j-1A-Agxk+j-∑j=02n-1Ag2n-j-1B-Bggk+j=Bg⋮AgBg⋮⋯⋮Ag2n-1Bggk+2n-1gk+2n-2⋮gk=CAg,BgCTAg,Bgg0kprovided that the input is generated from (29a)gTk+2n-1,gTk+2n-2,…,gTkT=gcTk+2n-1,uTk+2n-1,…,gcTk,uTkT=CTAg,Bgg0k=CTAg,BgCAg,BgCTAg,Bg-1x~k+2n,(29b)uk+iT+τ=UBcTeAcTT-τgck+iforτ∈0,T;i=0,1,…,2n-1for g0k∈R, where CA,B=B⋮AB⋮⋯⋮A2n-1B is the controllability matrix of the pair A,B. Note that since Ag,Bg is controllable, then CAg,BgCTAg,Bg is nonsingular. If A,B is controllable, then CA,B is full rank and fix Bg=B so that CAg,B-Bg=CAg,A-Ag=0 and the sufficiency part of the result follows by choosing a control law (29a) and (29b) subject to(30)g0k=CAg,BbCTAg,Bg-1xk+2n-Ag2nxk-∑j=02n-1Ag2n-j-1A-Agxk+jand the system H is then controllable. This proves the sufficiency part. The necessity part follows from (28) written in the equivalent form:(31)xk+2n-Ag2nxk=CA,Bgk+2n-1gk+2n-2⋮gk.If rankCA,B<2n, then, given xk, there exists x-∗ such that rankCA,B,x-∗>rankCA,B with xk+2n=x∗ and xk+2n-Ag2nxk=x-∗for some x∗∈R2n. Thus, the following linear algebraic system of equations (32)x-∗=CA,Bgk+2n-1gk+2n-2⋮gkresulting 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 DH is controllable if and only if CA,B is full rank. On the other hand, if the pair A,B is not controllable while the pair Ag,Bg is controllable, the system DH is approximately controllable with state targeting error -CAg,B-BgCTAg,Bgg0k=0A-Ag2,B-Bg2. Property (i) has been proved. On the other hand, note from (1a) that(33a)xckT+τ-∫0τeAcτ-σdσϑk-eAcTxck=∫0τeAcτ-σBcukT+σdσ,(33b)ϑk=Acsxck+Acdxdk+Bcsuk,∀τ∈0,T, ∀k∈Z+. If DH is controllable, then rankCAg,B=2n and xck and xdk are arbitrarily prefixed to xc∗k and xd∗k under some control sequence uk so that ϑk is prefixed as well to ϑ∗k=ϑ∗x∗k. Thus, from (33a) and (33b), xckT+τ is prefixed to any xc∗kT+τ, ∀τ∈0,T and any given k∈Z+ by a control law:(34)ukT+σ=∫0τeAcτ-σBcukT+σguk,guk=∫0τeAcτ-σBc∫0τeAcτ-σBcukT+σ-1xc∗kT+τ-∫0τeAcτ-σdσϑ∗k-eAcTxc∗kprovided that rankCAc,Bc=n and the system H is controllable. The sufficiency part of Property (ii) has been proved. The necessity follows by contradiction. Assume that rankCA,B<2n. Then, from Property (i), DH is not controllable so that H is not controllable at sampling points. Therefore, the system H is controllable only if rankCA,B=2n. From (33a) and (33b), H is controllable if DH is controllable, so that xk and ϑk can be prefixed in (33a) and (33b) from the controllability of H, only if rankCAc,Bc=n.
To prove Property (iii), note thatDH is reachable if and only if rankzI2n-A⋮B=2n, ∀z∈SpA from the Sze¨go-Kalman-Popov (or discrete Yakubovich-Kalman-Popov) controllability test. Equivalently, DH is reachable if and only if(35)zI2n-A⋮Bz∗I2n-ATBT=zI2n-Agz∗I2n-AgT+BgBgTI2n+zI2n-Agz∗I2n-AgT+BgBgT-1G~zis nonsingular, ∀z∈SpA, where zI2n-Agz∗I2n-AgT+BgBgT is nonsingular, ∀z∈SpAg, since Ag,Bg is controllable, and (36)G~z=Ag-AAgT-AT+B-BgBT-BgT+zI2n-AgAgT-AT+Ag-Az∗I2n-AgT+BgBT-BgT+B-BgBgTso that(37)G~z2≤Ag-A⋮B-Bg2Ag-A⋮B-Bg2+2zI2n-Ag⋮Bg2;∀z∈Cand supz∈SpAg⁡G~z2≤a2+2ab, where(38)a=Ag-A⋮B-Bg2;b=supz∈SpAgzI2n-Ag⋮Bg2so that DH is controllable from (36), since zI2n-Agz∗I2n-AgT++BgBgT is nonsingular ∀z∈SpAg, if (39)1>supz∈SpAgzI2n-Agz∗I2n-AgT++BgBgT-1G~zwhich is guaranteed if (40)a2b+a<1b=1zI2n-Agz∗I2n-AgT+BgBgT2;∀z∈SpAgsince(41)zI2n-Agz∗I2n-AgT+BgBgT-12≥1zI2n-Agz∗I2n-AgT++BgBgT2;∀z∈C.Condition (40) holds if a∈0,b2+b-1-b. This guarantees that rankCA,B=2n and DH is controllable. Since rankCAc,Bc=n, the system H is controllable from Property (ii). Property (iii) has been proved.
To prove Property (iv) note that reachability of the discreteDH is guaranteed from controllability to the origin and the nonsingularity of its matrix of dynamics A. Those conditions are guaranteed from the conditions of Property (iii) if Ag is nonsingular and A-Ag2<1/Ag-12 which guarantees that A=Ag+A-Ag=AgI2n+Ag-1A-Ag is nonsingular. On the other hand, if DH is reachable, then H is reachable if Ac is nonsingular, since the pair Ac,Bc is controllable.Note that if (26) is tested for z∈SpAg∩z∈C:z≥1 (i.e., for the unstable and critically stable modes of Ag), then it becomes a stabilizability test of the current DH provided that the nominal DHg 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.


## 3. 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 DH system in the event that the output matrix is identity and the input-output interconnection matrix is zero.Lemma 6.
The following properties hold:
(i) Assume thateiωI2n-A-1BI2m∗MeiωI2n-A-1BI2m⪯0 for some M=MT∈R3n+m×3n+m and all ω∈0,∞, where A=Ag+A~g and B=Bg+B~g.
Assume also that the discrete transfer matrixG~gz=zI2n-Ag-1A~g is strictly bounded real (SBR), that rankB~g=rankBg⋮B~g, and that I2m+KB is nonsingular, where KB is a real m+n×m+n matrix such that B~g=KBBg. Then, there exists Mg=MgT∈R3n+m×3n+m such that (42)eiωI2n-Ag-1BgI2m∗MgeiωI2n-Ag-1BgI2m⪯0.
(ii) Assume thateiωI2n-Ag-1BgI2m∗M′eiωI2n-Ag-1BgI2m⪯0 for some M′=M′T∈R3n+m×3n+m and all ω∈0,∞, where Ag=A-A~g and Bg=B-B~g with A≠Ag or B≠Bg.
Assume also that the discrete transfer matrixG~z=zI2n-A-1A~g is SBR, that rankB~g=rankB⋮B~g, and that I2m+KBg is nonsingular, where KBg is a real m+n×m+n matrix such that B~g=I2n+KBg-1Bg=-KBgB. Then, there exists MG=MGT∈R3n+m×3n+m such that (43)eiωI2n-A-1BI2n+m∗MGeiωI2n-A-1BI2n+m⪯0.Proof.
Note that there exists a realm+n×m+n matrix KB such that B~g=KBBg, from Rouché-Froebenious theorem, since rankB~g=rankBg⋮B~g and that Bg=Im+n+KB-1B since Im+n+KB is nonsingular. Also, since G~gz∞=supω∈R+⁡G~geiω<1,(44)eiωI2n-A-1B=eiωI2n-AgI2n-eiωI2n-Ag-1A~g-1Im+n+KBBg=L^-1iωI2n-eiωI2n-Ag-1A~g-1eiωI2n-Ag-1Bg,where (45)L^iω=eiωI2n-Ag-1Im+n+KBeiωI2n-Ag;∀ω∈0,∞so that (46)eiωI2n-Ag-1BgIm+n∗-MgeiωI2n-Ag-1BgIm+n⪰eiωI2n-Ag-1BgIm+n∗-M^giωeiωI2n-Ag-1BgIm+n=eiωI2n-A-1BIm+n∗-MeiωI2n-A-1BIm+n⪰0,where Mg=MgT∈R3n+m×3n+m has bounded entries and it is defined by (47)Mg=Block DiagL^-1iω-I2n-eiω-I2n-Ag-1A~g-1⋮I3n+m∗×MBlock DiagL^-1iω-I2n-eiω-I2n-Ag-1A~g-1⋮I3n+mfor ω-∈W, with W⊂R+ being nonempty, and such that the first matrix inequality of (46) holds. Property (i) has been proved. Property (ii) follows directly from duality considerations.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 Ag has to be a convergent matrix. Conversely, Lemma 6(ii) does not require for Ag to be a convergent matrix while A is a convergent matrix.
(
2
) If G~gz=zI2n-Ag-1A~g and G~z=zI2n-A-1A~g are SBR, then A and Ag are convergent matrices and the identities (48)zI2n-Ag=zI2n-AI2n+zI2n-A-1A~g=zI2n-AgI2n-zI2n-Ag-1A~gI2n+zI2n-A-1A~glead to(49)I2n-zI2n-Ag-1A~g=I2n+zI2n-A-1A~g-1.
(
3
) The identities(50)G~z=zI2n-Ag-A~g-1A~g=I2n-G~gz-1G~gz,G~gz=zI2n-A+A~g-1A~g=I2n+G~z-1G~zimply that G~z∞≤ε and G~gz∞≤εg are both SBR if εg∈0,1/2 and ε∈εg/1-εg,1, or if ε∈0,1/2 and εg∈ε/1-ε,1, and either Ag is convergent, resulting in A to be convergent, or A is convergent resulting in Ag to be convergent.
(4) Since Ag is convergent, then the condition G~gz=zI2n-Ag-1A~g being SBR of Lemma 6(i) is equivalent to I2n-Ag-1A~g2<1 in the case when Ag and A~g are positive and guaranteed if A~g2<1/I2n-Ag-12. See Theorem 3 of [9] for general linear discrete positive systems and the discrete KYP-Lemma. A similar conclusion arises for G~z=zI2n-A-1A~g being SBR of Lemma 6(ii) if A and A~g are positive. Then, such a condition is equivalent to I2n-A-1A~g2<1 and guaranteed if A~g2<1/I2n-A-12. Theorem 3 of [9] would also conclude that zI2n-Ag-1A~g and zI2n-A-1A~g are SBR if and only if there is a diagonal X∈R+n×n≻0 such that, respectively,(51)X00In-AgA~gIn0∗X00InAgA~gIn0≻0,X00In-AA~gIn0∗X00InAA~gIn0≻0.Theorem 8.
Assume the following:
(
1)
B
g
∈
R
+
n
×
2
m, Ag∈R+n×n is convergent, and A~g>-Ag with the pair Ag,Bg being controllable.
(
2)
G
~
z
=
z
I
2
n
-
A
g
-
A
~
g
-
1
A
~
g is SBR, and I2m+KBg is nonsingular, where KBg is a real m+n×m+n matrix such that B~g=I2n+KBg-1Bg>-Bg subject to rankB~g=rankBg+B~g⋮B~g.
(
3) Constraint (26) holds and there exists a matrix Mg=MgT∈R3n+m×3n+m, which is nonnegative in all entries except for the last 2m diagonal elements, such that (42) holds for all ω∈0,∞.
Then, the following properties hold:
(i) The pairA,B is controllable, where A∈R+2n×2n=Ag+A~g is convergent. Also, there exists MG=MGT∈R3n+m×3n+m, which is nonnegative in all entries except for the last m+n diagonal elements, such that (43) holds.
(ii) The following matrix inequalities hold which are, respectively, equivalent to (42) and (43):(52)I2n-Ag-1BgIm+nTMgI2n-Ag-1BgIm+n⪯0,(53)I2n-A-1BIm+nTMGI2n-A-1BIm+n⪯0.
(iii) There exist diagonal matricesPg⪰0 and PG⪰0 such that (54)Mg+Mg0⪯0;Mg0=AgTPgAg-PgAgTPgBgBgTPgAgBgTPgBg,(55)MG+MG0⪯0;MG0=ATPGA-PGATPGBBTPGABTPGB.
(iv) There existxg,pg≥0,  ug>0 and x,p≥0,  u>0 which satisfy(56)Ag-I2nxg+Bgug≤0,xgT,ugT⋮pgTMgAg-I2n⋮Bg≤0,(57)A-I2nx+Bu≤0,xT,uT⋮pTMGA-I2n⋮B≤0.
(v) Constraints (42) (for all ω∈0,∞), (52), (54), and (56) are mutually equivalent and, also, (43) (for all ω∈0,∞), (53), (55), and (57) are mutually equivalent. Furthermore, (42) implies (53), (55), and (57).
(vi) Assume thatG~z∞≤ε and G~gz∞≤εg are both SBR if εg∈0,1/2 and ε∈εg/1-εg,1, or if ε∈0,1/2 and εg∈ε/1-ε,1.Proof.
The first part of Property (i) follows from the controllability of the pairAg,Bg and the constraint (26) [Theorem 5(iii)], which guarantees the controllability of the pair A,B with A∈R+2n×2n being convergent and B∈R+2n×m+n from the two first assumptions since G~z=zI2n-A-1A~g is SBR, and B∈R+2n×m+n=Bg+B~g since Bg∈R+n×m+n. The second part follows from Lemma 6(ii) which concludes that (42) implies (43).
To prove Property (ii), note that constraint (52) holds since the equivalent constraint (42) (see [8]) holds from the given assumptions, Lemma 6(ii), and Property (i), and (53) holds since the equivalent constraint (43) holds. In the same way, Properties [(iii)–(v)] follow as well since constraints (42), (52), (54), and (56) (respectively, constraints (43), (53), (55), and (57)) are mutually equivalent [8], under the given assumptions, Lemma 6(ii), and Property (i).
Property (vi) follows since constraints (42), (52), (54), and (56) are equivalent to each other and constraints (43), (53), (55), and (57) are also equivalent to each other while (42) and (43) imply to each other since G~z∞≤ε and G~gz∞≤εg are both SBR (see Remarks 7(3)) since Ag and A are convergent.Remark 9.
If the semidefiniteness matrix inequalities in Theorem8 are replaced by definiteness counterparts then the equivalences hold without the controllability assumption of the pair Ag,Bg.
The following result is concerned with the positive realness of a discrete nominal transfer matrix of the extended discrete nominalDHg which guarantees that of the transfer matrix of a parametrical disturbed DHg 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 thatp=m+n,  Bg, Cg, and Dg are positive, Ag is positive, and the transfer matrix G1gz=Ggz-γ/2Ip is positive real for some real constant γ∈R+, where Ggz=CgzI2n-Ag-1Bg+Dg is strictly positive real, and that the triple Ag,Bg,Cg is controllable and observable. Assume that the parameterizing matrices of the DH (25a), (25b), (25c), and (25d) are subject to parametrical disturbances so that(58)A=Ag+A~;B=Bg+B~,C=Cg+C~;D=Dg+D~with the disturbance matrices being subject to A~≥-Ag,  B~≥-Bg,  C~≥-Cg, and D~≥-Dg. Assume, furthermore, that CAg,BgCTAg,Bg and CA,BCTA,B are monomial. Then, both DHg and DH are positive while the following properties hold:
(i) The transfer matrixGz=CzI2n-A-1B+D is strictly positive real (then Ais convergent), A,B,C,D>0 and G1z=Gz-γρ/2 is positive real if there exist matrices K~∈Rq×p,  L~∈R2n×q, where q is some arbitrary positive integer, P~≻-Pg∈R2n×2n (being diagonal), Q~≻-Qg∈R2n×2n satisfying the following set of matrix relations: (59)Q~=P~+LgL~T+L~LgT+L~L~T-AgTPgA~+A~TPgAg⪰-Q-g=-Qg-LgLgT-LgL~T-L~LgT-L~L~T,C~=BgTP~+B~TPg+P~Ag+BgT+B~TPg+P~A~+KgT+K~TL~T+K~TLgT,D~≥BgTPg+P~B~+KgTK~+12BgTP~Bg+B~TP~B~+B~TPgB~+K~TK~-γ1-ρIpfor some ρ∈0,1 and the given γ∈R+, for some existing matrices Qg≻0∈R2n×2n, Pg≻0diagonal∈R2n×2n,  Kg∈Rq×p,  Lg∈R2n×q which satisfy the following set of matrix identities:(60)AgTPgAg-Pg=-LgLgT-Qg,(61)BgTPgAg+KgTLgT=Cg,(62)KgTKg=Dg+DgT-γIp-BgTPgBg.
(ii) IfQg⪰0,  Q~⪰-Q-g, then Ggz and Gz are positive real even if Ag is critically stable.
(iii) IfQg⪰0,  Q~≻-Q-g, then Ggz is positive real, Gz is strictly positive real, and G1z is positive real even if Ag is critically stable.
(iv) IfG1gz, and then Ggz, (respectively, G1z and then Gz) are positive real then they do not satisfy Lemma 6, and conversely, for Mg=0-CT-CγIp-Dg+DgT.Proof.
Note thatCAg,BgCTAg,Bg-1 and CA,BCTA,B-1 are positive since CAg,BgCTAg,Bg and CA,BCTA,B are monomial so that the sequences ggck and gck 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 Ag is at least critically stable although nonnecessarily convergent. Note also that if γ>0 then Ggz is strictly positive real and if γ=0 then it is positive real if A is at least critically stable (rather than convergent) with eventual simple poles of positive semidefinite on z=1. Note that ReGgeiω+ReGgTe-iω-γIp⪰0 for any ω∈0,∞ since Ggz-γ/2Ip is positive real and p=m+n. From the equivalence between the Discrete Factorization Lemma and the Discrete Positive Real Lemma [38], there exist a positive definite real matrix Pg, which is diagonal since Ag is positive and convergent [3], and real matrices Kg, Lg, and Qg≻0 such that the matrix relations (61) hold implying from the Discrete Positive Factorization Lemma that (63)M^g=-Mg+Mg0=LgLgT+QgLgKgKgTLgTKgTKg⪰0,where (64)Mg=0-CT-CγIp-Dg+DgT;Mg0=AgTPgAg-PgAgTPgBgBgTPgAgBgTPgBgso that the following factorization holds:(65)ReG1geiω+ReG1gTe-iω=ReGgeiω+ReGgTe-iω-γIp=W1gTe-iωW1geiω⪰0;∀ω∈0,∞,where W1geiω=K+LTeiωI2n-Ag-1Bg. Thus, by invoking similar arguments of the equivalence between both lemmas, Gz=CzI2n-A-1B+D-γρ/2Ip is positive real for some given ρ∈0,1, if and only if, there exist a diagonal positive definite real matrix P and real matrices K, L, and Q≻0, subject to P~≻-Pg,  Q~⪰-Qg, satisfying(66)P=Pg+P~;K=Kg+K~L=Lg+L~;Q=Qg+Q~such that the following matrix relations hold:(67)ATPA-P=-LLT-Q,(68)BTPA+KTLT=C,(69)KTK=D+DT-γρIp-BTPB.Now, direct calculations show that (61) guarantee (68) if (59) hold under the given constraints while similar conditions to (63)–(65) hold by replacing Mg→MG, M^g→M^G=-MG+MG0⪯0 with the matrices being defined from their counterparts with the replacements γ→γρ,  Ag→A=Ag+A~,  Bg→B=Bg+B~,  Cg→C=Cg+C~,  Dg→D=Dg+D~,  Qg→Q=Qg+Q~,  Pg→P=Pg+P~,  Lg→L=Lg+L~,  Kg→K=Kg+K~. Property (i) has been proved. The proofs of Properties (ii)-(iii) are similar and they follow straightforwardly. The proof of Property (iv) follows by comparing Mg in (64) with (54) and by noting that it has negative entries except the last p=m+n diagonal entries because the entries of Cg≠0 are nonnegative since the system is positive. So, the state-space realizations of G1gz and Ggz do not fulfill Lemma 6 for Mg which has negative off-diagonal entries. A similar conclusion follows for G1z and Gz.Remark 11.
(
1
) Note that in Theorem 10(ii) Ag and A can be critically stable, since Ggz,  G1gz,  Gz, and G1z are positive real, so that they can eventually possess simple eigenvalues, such that the four resulting matrices Heiω+HTe-iω, with H 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 DHg is positive and Theorem 10 holds, then the extended discrete system DH 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 (70)∑k=k0k1yTkgk=ξxk1+1-ξxk0+∑k=k0k1λxk,gkwith gk=gcTk,uTkT and λxk,uk≥0 on any discrete-time interval k0,k1 with k0≥0. In particular, (71)∑k=k0k1yTkgk=ξxk1+1-ξxk0+∑k=k0k1λxk,gk=12xk1+1TPg+P~xk1+1-xk0TPg+P~xk0+∑k=k0k1xkgkTLg+L~LgT+L~T+Qg+Q~Lg+L~Kg+K~KgT+K~TLgT+L~TKgT+K~TKg+K~xkgk≥-12xk0TPg+P~xk0≥-12xk0TPaxk0for any integers k0≥0 and k1≥k0 with Pa=Pg-P~ and a close relation for the nominal DHg 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 state-space descriptions when dealing with the Continuous or Discrete Positive Real Lemmas or their equivalent Youla’s Factorization Lemmas.
Theorem10 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 DH being positive maintains the positivity and the positive realness property of the nominal DHg. 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 Theorem10 holds for the parametrical disturbance matrices. Then, both DHg and DH are positive and the following properties hold:
(i) The transfer matrixGz=CzI2n-A-1B+D is strictly positive real (then A is convergent), A,B,C,D>0, and G1z=Gz-γρ/2 is positive real if there exist matrices S~∈R2n×p,  P~≻Pg∈R2n×2n (being diagonal), R~∈Rp×p,  Q~∈R2n×2n satisfying the following set of matrix relations: (72)Q~=P~-AgTPgA~+A~TPgAg-AgTP~A~+A~TP~Ag-AgTP~Ag-A~TPgA~-A~TP~A~⪰-Q-g,C~=BgTP~+B~TPg+P~Ag+BgT+B~TPg+P~A~+S~T,D~≥BgTPg+P~B~+R~+12BgTP~Bg+B~TP~B~+B~TPgB~-γ1-ρIpfor some ρ∈0,1 and the given γ∈R+, for some existing matrices Qg≻0∈R2n×2n, Pg≻0diagonal∈R2n×2n,  Sg∈R2n×p and Rg∈Rp×p, which satisfy the following set of matrix identities:(73)AgTPgAg-Pg=-LgLgT-Qg,(74)BgTPgAg+SgT=Cg,(75)Rg=Dg+DgT-γIp-BgTPgBgprovided that(76)M^1g=LgLgT+QgSgSgTRg⪰0;M^~1g=Q~S~S~TR~⪰-M^1g.Furthermore, DH satisfies (71):(77)∑k=k0k1yTkgk=12xk1+1TPg+P~xk1+1-xk0TPg+P~xk0+∑k=k0k1xkgkTLg+L~LgT+L~T+Qg+Q~Lg+L~Kg+K~KgT+K~TLgT+L~TKgT+K~TKg+K~xkgk≥-12xk0TPg+P~xk0≥-12xk0TPaxk0for any integers k0≥0 and k1≥k0 with Pa=Pg-P~ and a close relation is satisfied by the nominal DHg with L~=0,  K~=0,  P~=Q~=0.
(ii) IfQg⪰0,  Q~⪰-Q-g, then Ggz and Gz are positive real even if Ag is critically stable.
(iii) IfQg⪰0,  Q~≻-Q-g, then Ggz is positive real, Gz is strictly positive real, and G1z is positive real even if Ag is critically stable.Proof.
Note that (74)-(76) lead to M^1G=M^1g+M^~1g⪰0 for the nominal discrete hybrid system if the parametrical disturbances satisfy (72). Then, both DHg and DH are positive, have positive real transfer matrices, and satisfy the respective passivity conditions (77) (see (65) and Remark 11(1)).Example 13.
A particular system of the studied hybrid class withn=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 nc=nd=2,  m=p=1,  T=0.01,(78)Ac=-110-2;Bc=bc1bc2=01;Cc=cc1cc2;Dc=dc,Acs=0.500.10.5;Acd=100.10.5;Ads=0.0001I4,Ad=0.510.40;Bd=0.010.1;Bcs=0.20.1,Cd=cdT=1,0;Dd=dd=1which leads to the following matrices:(79)C=Cc+Ccs⋮1,0;D=d=dc+1.One gets directly that (80)Φc0.01=eAc×0.01=0.9880190.00985100.990050;Γc0.01=∫00.01eAc0.01-τdτ=0.9900660.00495100.995017,Ac′=0.9890140.0098560.0009950170.991045;Ad′=0.65209980.09998940.04323320.216166,B0=0.1464030.04323320.250.4.If V=1, note that Ac is Hurwitz and Metzler and then Φc0.01 is positive and convergent. If dc≥0,  Bc>0, and Cc>0, then both the continuous-time subsystem and its discretized version to any sampling period are positive dynamic systems. Also, Ad is convergent.The transfer function of the uncoupled continuous-time subsystem is(81)Gcs=CcsI2-Ac-1Bc+dc=cc1bc1+cc2bc2s+γcs2+3s+2+dc;γc=2cc1bc1+cc1+cc2bc2,where s is the Laplace transform argument. It can be easily checked that the continuous transfer function is positive real if bc1=cc2=0; 0≤γc=cc1bc2≤3 and dc>0. If bc2>0 and cc1>0 with bc1=cc2=0, then the triple Ac,Bc,Cc is controllable and observable. If, in addition, γc>0, then Gcs is strictly positive real. If, furthermore, dc>0 or γc<3, then Gcs is strongly strictly positive real in the sense that ReGciω≥dc>0, ∀ω∈-∞,∞ [19]. We can choose bc1=cc2=0,  cc1=bc2>0,  dc≥0.
On the other hand, note that the uncoupled continuous-time subsystem is a mathematical model for some well-known linear dynamic systems as a damped mechanical system, or an RLC electric circuit, described by the differential equation:(82)v′′t+3v′t+2vt=ft=cc1bc2+2dcut+dcu′′t+3u′tforced by a term f(t) calculated from a primary control ut which is everywhere piecewise continuous if dc=0 and everywhere twice continuous-time differentiable with piecewise continuous second time-derivative if dc>0. Note that if Acs2<0.581976, then Ac being Hurwitz guarantees that Ac′ is convergent from the fulfillment of the stability constraint 1>Φc-10.01Γc0.012Acs2.The transfer function of the digital subsystem is Gdz-1=1-0.25z-1/1-0.5z-1-0.4z-2, where z is the discrete transfer function argument representing in the time delay a one-step advance operator q, and equivalently, q-1 is a one-step delay operator formally equivalent to z-1. It can be directly checked that the digital transfer function is strictly positive real with ReGdeiω≥dd=1 for ω∈0,2π. The extended DH has four stable eigenvalues, namely, 0.367879,0.135335,0.930074,-0.430074 in the free coupling case, that is, if Ads=0, Acd=0. By using a similar reasoning to that guaranteeing that Ac being Hurwitz implies that Ac′ is convergent, one concludes that the system matrix A of the DH is convergent if the sufficiency constraint below holds: (83)A-Block DiagAc′,Ad2≤max⁡Γc0.12Acd2,Ads2≤max⁡0.632121Acd2,Ads2<1Block DiagAc′-1,Ad-12=0.7406,Ag=0.9890140.0098560.65209980.09998940.0009950170.9910450.04323320.2161660.00010.00010.850.50.00010.000100.9,Bg=0.146403000.043233200.082419990.25000.400.Now, the Discrete Positive Real Lemma holds by defining an extended output y-k=yk,gcTkT=yk,u2k,u3kT equalizing the dimension of the input, that is, p-=m=3 while incorporating two extra components to the output to obtain the extended output with the corresponding replacements Cg→C-g,  Dg→D-g in (74) leading to(84)Pg=I4;Lg=0;C-g=Cg00000000=1+ccs1ccs21000000000;D-g=dd00010001,Pg-AgTPgAg⪰0.994093I4≻0,SgT=C-g-BgTAg=ccs1+0.855097ccs2-0.0443540.690162-0.1489840000-0.000082-0.0811681-0.003563-0.017816,Rg=2dd-γ-0.0334030-0.00356302-γ0-0.00356302-γ-0.006793=1.210-0.00356301.267930-0.00356301.2.Now, for γ=0.73207,  dd=0.9877365, (85)Qg=0.0217762-0.019591-1.1996452-0.517061-0.0019740.017794-0.08024-0.430494-1.19964524-0.08024-0.430494-0.875032-0.517061-0.430494-0.8750320.189968,(86)M^~1g=Q~S~S~TR~⪰-M^1g.(i) Note that if cs1=cs2=0,  dd=1,  γ=1, then (87)Rg=0.9565970-0.003563010-0.00356300.9932069,γ=0.956597,dd=0.9732985.The condition M^1g≻0 holds since 1>sg=Sg2/min⁡λminPg-AgTPgAg,λminRg with Sg2=1.109815. The positive realness condition of G-gz=Block DiagGgz-γ/2,1-γ/2I2 is guaranteed, where Ggz is the transfer function of the nominal DHg, and is ensured from the Discrete Positive Real Lemma under nominal parametrical conditions of the state-space realization. See Theorem 12(i), Eqs. (74). If the above basic constraint holds with a margin ε~>0 under the form 1>sg+ε~, then one gets from (72) admissible parametrical disturbances which keep the positive realness and the Discrete Positive Real Lemma. Alternative results can be got from Theorem 10 based on the hybrid version of Youla’s Factorization Lemma.
(ii) However, note that the positive realness ofDH is not fulfilled if, for instance, γ≤2dd-1 and Rg=1.966560-0.003563010-0.00356300.9932069 has two nonstable eigenvalues 1, 1.99933, then λminPg-AgTPgAg=1.93469, λminRg=0.9932069.
A direct extension of the two last theorems for the case whenp<m+n is the following.Corollary 14.
Assume that the hypotheses of Theorems10 and 12 hold except that p<m+n. Then, both theorems still hold for any fictitious extended output of 2m components defined by y-k=yTk⋮u-TkT, where u-k is any set of 2m-p input components, with the matrices Dp and D having an appropriate block submatrix with zero and unit entries.The following alternative result to Theorems10 and 12 follows with a direct proof, then omitted, for the case when the control input in-between sampling instants is generated directly from its value at sampling instants instead of from an independent discrete sequence.Corollary 15.
Assume that theggck=Δguk and gck=Δuk for the nominal DHg and current DH for some given matrices Δg,Δ∈R+m×m and that hypothesis of Theorem 10 holds with the dimensional change p=m for inputs and outputs resulting in the modified nominal and current control matrices of order 2n×p to those defined in (25c): (88)Bg=∫0TeAgcT-τBgcs+BgcVBgcTeAgcTT-τΔgdτBgd;B=∫0TeAcT-τBcs+BcVBcTeAcTT-τΔdτBd.Then, Theorems 10 and 12 hold.


## 4. Hyperstability Results

Hyperstability is a property associated with the nonnegativity for all time of the input-output energy for all nonlinear and time-varying output-feedback controller belonging to a certain class which satisfies Popov’s type inequality for any bounded initial conditions [11–13, 15] and which is linked to the positive realness of the transfer matrix of the linear dynamics.Thus, such a property is not associated, and it should not be confused, with the positivity of the trajectory solutions for any nonnegative initial conditions and controls. Both properties can be simultaneously inherent to some systems under certain conditions. The following hyperstability theorem holds as linked to Theorems 10 and 12 under controls got from any feedback nonlinear time-varying controller within the class satisfying Popov’s type inequality.Theorem 16.
Consider the discrete hybrid systemDH, (25a), (25b), (25c), and (25d) driven by a sequence gk, with p=m+n, and Ag,  Bg,  Cg, and Dg being positive matrices, and assume the following:
(
1) The sequence gk is generated by feedback from any nonlinear and eventually time-varying device of the form gk=-φyk,k which satisfies the following Popov’s inequality: (89)∑k=k0k1yTkφyk,k≥-γ02for some finite real number γ0≠0 and any given integers k0≥0 and k1>k0.
(
2) The transfer matrix Gg1z=CgzI2n-Ag-1Bg+Dg-γ/2Ip is positive real for some real constant γ∈R+ so that its state-space realization satisfies (74)-(76) (equivalently (61)) for some existing matrices Qg⪰0∈R2n×2n,  Pg≻0diagonal∈R2n×2n,  Sg∈R2n×p, and Rg∈Rp×p (equivalently with existing matrices Kg∈Rq×p, Lg∈R2n×q).
(
3) The parametrical disturbances satisfy the matrix constraints (72) or, equivalently, (59) with P~⪰-Pg. Then, the following properties hold:
(i) The given class of controls keeps the hyperstability of the closed-loopDH from that of DHg in the sense that xk2 is uniformly bounded for any given initial condition x0 and any nonlinear eventually time-varying controller satisfying Popov’s inequality (89). If, furthermore, Ag,Bg is controllable and (26) holds, then A,B is controllable and gk2 is uniformly bounded.
(ii) IfM^1G=M^1g+M^~1g≻0,  Q~≥-Qg,  R~⪰-Rg, then xk→0,  uk→0 (even if A,B is not controllable) under the given class of controls so that the closed-loop system is asymptotically hyperstable. Equivalently, G1gz=CgzI2n-Ag-1Bg+Dg-γ/2Ip is strictly positive real.
(iii) Assume thatG1gz is strictly positive real for some γ∈R+.Then, the closed-loop DH keeps the asymptotic hyperstability property from that of the nominal DHg in the sense that Property (i) holds and, furthermore, xk2,  gck2, and uk2 converge asymptotically to zero for any given initial condition x0.
(iv) Assume thatM^1G⪰0 with Qg≻0 and Q~≥-Qg (so that Gg1z is positive real). Assume also that either KgTKg≻0 or Ag,Bg is stabilizable while, furthermore and correspondingly, either KTK≻0 (equivalently R≻0) or A,B is stabilizable. Then, Property (iii) still holds.
(v) Properties (i)–(iii) hold for Corollary14  p<m+n and Corollary 15  p=m with the given modifications of the parameterizations.Proof.
One gets from (86) and (89)(90a)+∞>γ02+12xk0TPg+P~xk0≥∑k=k0k1yTkgk=12xk1+1TPg+P~xk1+1+∑k=k0k1xkgkTLg+L~LgT+L~T+Qg+Q~Lg+L~Kg+K~KgT+K~TLgT+L~TKgT+K~TKg+K~xkgk(90b)≥12xk1+1TPg+P~xk1+1for any integers k0≥0 and k1>k0 with the four-block partitioned matrices of (90a) being at least positive semidefinite (see Theorems 10 and 12). Since Pg+P~≻0, the sequence xk2 is uniformly bounded for any given initial condition x0 and any nonlinear eventually time-varying controller satisfying Popov’s inequality (89). Thus, the current closed-loop system DH is hyperstable as it is the nominal one. If Ag,Bg is controllable and (26) holds, then A,B is controllable for the parameterizations defined in (25c) and then the uniform boundedness of xk2 implies that of gk2. Property (i) has been proved. Under the conditions of Property (ii), since M^1g≻0, so that Qg≻0 and Rg≻0 (equivalently, KgTKg≻0), and M^~1g⪰-M^1g, so that Q~≥-Q-g,  R~⪰-Rg, then M^1G=M^1g+M^~1g≻0 and Pg≻0 and P~⪰-Pg, then it follows from the boundedness of the second inequality of (89) for all k1>k0 that the sampled state and input of the nominal and current DH converge asymptotically to zero without the need for any controllability assumption. This proves Property (ii). Under the additional conditions of Property (iii), A is convergent and xk→0 from the first identity of (68) which is a discrete Lyapunov matrix equation. Note that (74)-(76) hold with Qg≻0 and Q~⪰-Q-g (from the strict positive realness condition) and Rg≻0 as well as R≻0 (since R~⪰-Rg) and furthermore, KgTKg≻0 and KTK≻0. Otherwise, limω→∞⁡ReG1geiω+G1gTe-iω=0 and the nominal G1gz would not be strictly positive real as it would happen with the disturbed transfer matrix. This implies that xk→0 since otherwise limk1→∞⁡∑k0k1xTkQxk would be infinity from the system positivity. A similar argument concludes that gk→0, since KTK≻0, without requiring a controllability condition, since any eventual zero/pole cancellation in the transfer matrix is necessarily strictly stable (since the transfer matrix G1gz is strictly positive real) so that any eventual uncontrollable mode is asymptotically stable. This proves Property (iii). To prove Property (iv), note that if M^1G⪰0 with Qg≻0 and Q~≥-Q-g, then G1gz is positive real. From the system positivity and the finite upper-boundedness of (89), limk1→∞⁡∑k0k1xTkQg+Q~xk<∞ for any Q~⪰-Qg implying that xk→0 for any given finite x0. In the same way, limk1→∞⁡∑k0k1gTkKTKgk<∞. If KTK≻0, equivalently Q≻0, gk→0. Else, if the discrete pair A,B associated with DH is stabilizable, then there exists a matrix transformation T such that T-1AT=AccAcc-0Ac-c-,  T-1B=Bcc0 such that Acc,Bcc is controllable and Ac- (associated with uncontrollable but stable modes from the stabilizability assumption) is convergent. Therefore, the uncontrollable (but globally asymptotically stable) discrete substate of the transformed state sequence xc-c-k→0 while its forced controllable substate of dimension ncc≤n=nc+nd (the inequality being strict if there exists at least one uncontrollable mode) satisfies (91)xccfk=BccAccBcc⋯Accncc-1Bccgk-nccgk-ncc-1⋮gk-1so that xccfk→0 (and the whole controllable state xcck→0 since its homogeneous response vanishes as discrete time tends to infinity) if and only if gk→0 since its associated controllability matrix is full rank. Thus, xcck→0, xc-ck→0 since the state in the original coordinates satisfies xk→0 and then gk→0. Property (iv) has been proved. Property (v) follows directly from Properties (i)–(iii) and Corollaries 14-15.Note that the assumption(3) of Theorem 16 implies that Gz is positive real in Property (i) and strictly positive real in Property (ii).The solutions for any time satisfy the global stability properties proved in the following result.Corollary 17.
Assume thatV^ is such that the factorization V=V^V^T holds, the pair Ac,BcV^T is controllable, and all the hypotheses of Theorem 16(i) hold. Then, the current hybrid system H is globally asymptotically stable for any bounded initial conditions under any continuous-time control in-between samples (25b). As a result, the resulting hybrid SH is hyperstable. If the additional conditions of Theorem 16(ii) also hold, then H is, furthermore, asymptotically hyperstable.Proof.
From Theorem16(i), the sequences xk, uk, and gck are uniformly bounded for bounded initial conditions. If the continuous-time control is generated from (25b) and Ac,BcV^T is controllable, then ut is bounded for any t∈kT,k+1T,  ∀k∈Z0+. Also, xt is bounded with the proved bounded initial condition xk for any k∈Z0+ since the intersample control is bounded and the control at sampling instants is also bounded. Property (i) has been proved. Property (ii) is direct since, in addition, xk→0, uk→0 as k→∞, and gck→0 as k→∞ implies ut→0 for t∈kT,k+1T as k→∞ from (25b).Theorem16 implies under the controllability of Ac,Bc that the hybrid system H is globally stable as proved in the subsequent result provided that the control input is piecewise continuous on its definition domain. It does not require either the stability or the critical stability of Ac.Corollary 18.
Consider the hybrid systemH, (1a), (1b), and (1c), with p=m+n, and Ag, Bg, Cg, and Dg being positive matrices so that the pair Ac,Bc is controllable and that the controllability Gramian ∫0TeAcT-τBcBcTeAcTT-τdτ is monomial. Assume also that Ag,Bg is controllable, that (26) holds, that the control input u is everywhere piecewise continuous on its definition domain, and that an input driving sequence gk is generated by feedback from any nonlinear and eventually time-varying device of the form (92)gk=uTk,gcTkT=-φyk,k,(93)gck=∫0TeAcT-τBcBcTeAcTT-τdτ-1∫0TeAcT-τBcukT+τdτand satisfies Popov’s inequality (89) for some finite real number γ0≠0 and any given integers k0≥0 and k1>k0. Assume also that the assumptions(2) and(3) of Theorem 16 hold.
Then,xt2 and ut2 are uniformly bounded for any bounded initial conditions so that the hybrid system H is globally stable.Proof.
Since∫0TeAcT-τBcBcTeAcTT-τdτ is monomial, the sequence gck is nonnegative from (93) for any nonnegative input. Also, since G1gz=CgzI2n-Ag-1Bg+Dg-γ/2Ip is strictly positive real from assumption (2) of Theorem 16 the closed-loop nominal and current discrete DHg and DH are hyperstable from Theorem 16(i). Also, since the pair Ac,Bc is controllable, (93) is well posed since zI2n-Ag-1 exists since it is the inverse of the controllability Gramian on the time interval 0,T. Equation (1a) has the following unique solution for given initial conditions and control input: (94)xckT+σ=eAcσxck+∫0σeAcT-τBcukT+τ+Bcsukdτ+∫0σeAcT-τdτAcsxck+Acdxdk;∀τ∈0,T;∀k∈Z+.From Theorem 16(i) the sequences xk2,  gck2, and uk2 are uniformly bounded and ut is bounded on kT,k+1T, ∀k∈Z0+. Then, from (94), xct is bounded as well for t∈kT,k+1T,  ∀k∈Z0+.Example 19.
Consider Example13 with a choice of the auxiliary input satisfying gck→0 and under the given conditions guaranteeing that G-gz=BlockDiagGgz-γ/2,1-γ/2I2 is positive real. From Corollary 18, the closed-loop nominal system Hg and the discrete DHg are positive and asymptotically hyperstable, that is, globally asymptotically stable for the class of nonlinear and eventually time-varying nonlinear controllers within the class satisfying inequality (89). Thus, uk→0,  yk→0,  xk→0,  xt→0,  yt→0, and ut→0 as t→∞ if ut is generated from (25b) in the intersample time interval for any given initial conditions.Remark 20.
Note from (94) the important observation that Corollary 18 does not conclude that xt→0 as t→∞ for any piecewise control input ut, even if Ac is a stability matrix and xk→0 and uk→0, which has been got under the stronger conditions of Theorem 16(ii). If xk→0 and uk→0, then(95)limk→∞xckT+σ-∫0σeAcT-τBcukT+τdτ=0;∀σ∈0,T.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 input-output instantaneous power) converge asymptotically to zero except eventually for a set of time instants zero measure.Theorem 21.
Assume the following:
(
1) The pair Ac,Bc is controllable.
(
2) The nonlinear and eventually time-varying controller φyt,t is everywhere piecewise continuous with respect to y and continuous with respect to t in R+p×R+.
(
3) There is a partition of each interval kT,k+1T in a finite number sk of disjoint intervals of length 0<Tak≤T such that the sequences Pkjk, Qkjk, and Rkjk consist of positive definite elements for existing partitions of each interval kT,k+1T in a finite number sk of disjoint intervals of length 0<Tak≤T such that φyt,t is continuous on each interval kT+l-1Tak,k+1T+lTak for l=1,2,…,sk, ∀k∈Z+.
Then, the closed-loop systemH is asymptotically hyperstable in the mean so that ytkσ→0 and utkσ→0 as for tk+σ∈⋃k∈Z+⋃1≤lk≤skkT+lk-1Tak,k+1T+lkTak as k→∞. If φ0,t=0,  ∀t∈R+, then H is asymptotically hyperstable.Proof.
Consider an auxiliary eventually time-varying sampling periodTak=T/sk for the time interval kT,k+1T=∑j=0k-1sjTaj,∑j=0k-1sjTaj+skTak and sk∈Z+≥1,  ∀k∈Z+, is chosen so that ut is continuous on kT+lk-1Tak,kT+lkTak for l=1,2,…,sk,  ∀k∈Z+. Note that, in general, the nonunique choice of the eventually time-varying auxiliary sampling period subject to a maximum threshold period Tak≤T-a≤T,  ∀k∈Z+, can be always made since as φyt,t is assumed to be everywhere piecewise continuous with respect to y and continuous with respect to t, with eventual bounded isolated discontinuities, then ut=-φyt,t has the same property. Thus, one gets directly from (1a) that the solution of the continuous-time substate from kT,kT+Tak to kT,k+1T=kT,kT+skTa with initial conditions at kT,  ∀k∈Z+, is as follows:(96)xckT+lkTak=xc∑j=0k-1sjTaj+lkTak=eAcTakxc∑j=0k-1sjTaj+lk-1Tak+∫0TakeAcTak-τBcu∑j=0k-1sjTaj+lk-1Tak+τ+Bcsukdτ+∫0TakeAcTak-τdτAcsxck+Acdxdk=eAcTakxckT+lk-1Tak+∫0TakeAcTak-τdτAcsxck+Acdxdk+TakeAcTak-τckBc⋮TakeAcTak-τdkBcsu-kfor lk=1,2,…,sk, ∀k∈Z+, where (97)vclk-1,Tak=∫0TakeAcTak-τBcBcTeAcTTak-τdτ-1∫0TakeAcTak-τBcu∑j=0k-1sjTaj+lk-1Tak+τdτ,(98)u-k=u∑j=0k-1sjTaj+lk-1Tak+τckukτck=τcklk-1,Tak and τdk=τdkTak are vector functions from 0,Tak to R+n: ∀k∈Z+ after picking-up the values in the open intersample fictitious period searching its value leading to the integral mean value in the above identities, one per row of eAcTak-τckBc and one per row of eAcTak-τdkBcs and the corresponding input components, ∀k∈Z+. The matrix inverse in (97), which is the inverse of the controllability Gramian on 0,Tak, exists since the pair Ac,Bc of nonnegative matrices is controllable on any time interval 0,Tak of nonzero measure. Thus, one gets from (96) that(99)limk→∞xc∑j=0k-1sjTaj+lkTak-eAcTakxc∑j=0k-1sjTaj+lk-1Tak-TakeAcTak-τckBcukT+lk-1Tak+τck=0,limk→∞y∑j=0k-1sjTaj+lkTak-CceAcTakxc∑j=0k-1sjTaj+lk-1Tak-TakCceAcTak-τckBcukT+lk-1Tak+τck-Dcu∑j=0k-1sjTaj+lkTak=0since uk→0 and yk→0. Thus, for any given ε=ε1,ε2,…,εncT>0, there exists tk0=tk0ε∈R+ such that, for any k1>k0∈Z+ and associate tk1>tk0∈R+ by defining the auxiliary sampling instants t∑j=0k-1sj+lk, one gets from (90a) and (90b)(100)+∞>γ-02=γ02+12xtk0TPgck0+P~ck0xtk0≥∑k=k0k1∑ljk=0skyTtk+ljkTak+τcjkutk+ljkTak+τcjk=12xtk1+1TPgk1+P~k1xtk1+1+εtk+∑k=k0k1∑ljk=0skxtk+ljkTak+τcjkgtk+ljkTak+τcjkTLkjkLkjkT+QjkLkjkKkjkTLkjkTKkjkTKkjkxtk+ljkTak+τcjkgtk+ljkTak+τcjkwith εtk≤ε for k>k0, where yct=Ccxct+Dcut for t∈kT,k+1T=∑j=0k-1sjTaj,∑j=0k-1sjTaj+skTak and yctk=y∑j=0k-1sjTaj-yd∑j=0k-1sjTaj,  ∀k∈Z+. Note from (100) that since εtk→0 as k→∞, since ε can be taken to be arbitrarily small, and simultaneously since the closed-loop DH is asymptotically hyperstable since the matrices Qgk, Pgk, Rgk and Qk, Pk, Rk, are positive definite matrices for all k∈Z+, then xk→0,  gk→0,  uk→0,  yk→0,  xtk+ljkTak+τcjkgtk+ljkTak+τcjk→0, and utk+ljkTak+τcjk→0 as it follows from the direct extension Theorem 16(iv) to time-varying parameterizations under sufficiency type conditions of asymptotic hyperstability, provided that q=p and Kk is nonsingular for all k∈Z+ [13]. Since the state, input, and output have nonnegative components, one also gets that xct→0 and yt→0 as t→∞ except, eventually, at isolated time instants where the nonlinearity φyt,t 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:(101)ut=Fcxct+Gcrt,uk=Fdcxck+Fddxdk+Gdrk;∀t∈kT,k+1T;  ∀k∈Z+, where the various matrices are of the appropriate orders and nonnegative entries with rt=-φyt,t being an outer reference control generated via output-feedback by nonlinear and, eventually, a time-varying nonlinearity in the form φyt,t subject to Popov’s type inequality. If the control law is replaced in (1a)–(1c), then the closed-loop hybrid system H becomes Hcl described according to the following parametrical replacements:(102)Ac⟶A-c=Ac+BcFcc;Acs⟶A-cs=Acs+BcsFdc,Acd⟶A-cd=Acd+BcsFdd+BcFcd,Bc⟶B-c=BcGc;Bcs⟶B-cs=BcsGd;Bd⟶B-d=BdGd,Ads⟶A-ds=Ads+BdFdc;Ad⟶A-d=Ad+BdFdd,Cc⟶C-c=Cc+DcFcc;Ccs⟶C-cs=Ccs+DdFdc,(103)Cd⟶C-d=Cd+DdFdd;Dc⟶D-c=DcGd;Dd⟶D-d=DdGd.Consequently, (3) to (6) are modified, as driven by rt, according to(104)A⟶A-=eA-cTInc+∫0Te-A-cτdτA-cseA-cT∫0Te-A-cτdτA-cdA-dsA-d,B0⟶B-0=eA-cT∫0Te-A-cτdτB-csB-d,Bv⟶B-v=B-0⋮Inc0nd×nd,C⟶C-=C-c+C-cs⋮C-d;D-=D-c+D-d.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 Hcl.Theorem 22.
Assume thatAc,Bc, Ad,Bd, Ads,Bd, and Acd,Bc and Acd,Bcs are controllable pairs. Then, an appropriate feasible parameterization of the control gains in the fictitious discrete control law (92)-(93), with the replacement u·→r· 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 DHcl and then its hyperstability if rt=-φyt,t for any nonlinear and eventually time-varying nonlinearity φyt,t which satisfies Popov’s type inequality.
If, furthermore,DHcl is asymptotically hyperstable and the assumptions(2) and(3) of Theorem 21 hold, then the closed-loop hybrid Hcl is asymptotically hyperstable in the mean.Proof.
We refer with superscript bars to any matrices for either the parameterization or the Positivity Real Lemma (P-, Q-, L-, S-, and R-) after performing the control law (101). Note that, under controllability of any pair A,B, it is possible to choose a state-feedback control gain X for the achievement of any given arbitrarily prescribed stable closed-loop placement. Since Ac,Bc, Ad,Bd, Ads,Bd, and Acd,Bc and Acd,Bcs, so Acd+BcsFdd,Bc, are controllable pairs, it is feasible to choose the control gains Fcc and Fdd in such a way that A-c, and then eA-cT, and A-d have stable eigenvalues being as largely dominant, related to the spectral norms of A-cd and A-ds, as possible via the choices of Fcd and Fdc so that the dynamics A- of the closed-loop extended discrete hybrid system be a convergent matrix. On the other hand, one can choose the p-matrix Gd of sufficiently small nonnegative entries so that B- and C-, and then S- in the second constraint of the Discrete Positive Real Lemma has a sufficiently small spectral norm related to that of A while D-+D-T is dominant norm of order oGd2 over that of B-TP-B-, of order oGd22, so that R-⪰0 and Q-⪰0. In this way, the discrete modified closed-loop transfer matrix of DHcl, related to the new input rt, might be designed to be at least positive real. On the other hand, the asymptotic hyperstability in the mean of Hcl follows from Theorem 21 from the asymptotic hyperstability of DHcl and the assumptions (2) and (3) of Theorem 21 since the first assumption of such a theorem holds since the controllability of the pair Ac,Bc implies that of the pair A-c,B-c.


## 5. 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’s-type 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.



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*Source: 3928970-2017-12-13.xml*