Characterization of Input–Output Negative Imaginary Systems in a Dissipative Framework

In this article, we define the notion of stable input–output negative imaginary (IONI) systems. This new class captures and unifies all the existing stable subclasses of negative imaginary (NI) systems and is capable of distinguishing between the strict subclasses (e.g., strongly strictly negative imaginary, output strictly negative imaginary (OSNI), input strictly negative imaginary, etc.) in the literature. In addition to a frequency-domain definition, the proposed IONI class has been characterized in a time-domain dissipative framework in terms of a new quadratic supply rate <inline-formula><tex-math notation="LaTeX">$w(u,\bar{u},\dot{\bar{y}})$</tex-math></inline-formula>. This supply rate consists of the system’s input (<inline-formula><tex-math notation="LaTeX">$u$</tex-math></inline-formula>), an auxiliary input (<inline-formula><tex-math notation="LaTeX">$\bar{u}$</tex-math></inline-formula>) that is a filtered version of the system’s input, and the time-derivative of an auxiliary output of the system (<inline-formula><tex-math notation="LaTeX">$\dot{\bar{y}}$</tex-math></inline-formula>). This supply rate corrects earlier supply rate attempts in the literature, which were only expressed in terms of the input (<inline-formula><tex-math notation="LaTeX">$u$</tex-math></inline-formula>) and the time-derivative of the system’s output (<inline-formula><tex-math notation="LaTeX">$\dot{y}$</tex-math></inline-formula>). In this article, IONI systems are proved to be a class of dissipative systems with respect to the proposed supply rate <inline-formula><tex-math notation="LaTeX">$w(u,\bar{u},\dot{\bar{y}})$</tex-math></inline-formula>. Subsequently, an equivalent frequency-dependent <inline-formula><tex-math notation="LaTeX">$(Q(\omega), S(\omega), R(\omega))$</tex-math></inline-formula> dissipative supply rate is also proposed for IONI systems. These findings reveal the connections between the NI property and classical dissipativity in both the time domain and frequency domain. We also provide linear matrix inequality (LMI) tests on the state-space matrices to check whether a system belongs to the IONI class or any of its important subclasses. Finally, the derived results are specialized for OSNI systems since such systems exhibit interesting closed-loop stability properties when connected, in a positive feedback loop, to NI systems without poles at the origin. Several illustrative numerical examples are provided to make the results intuitive and useful.


I. INTRODUCTION
N EGATIVE imaginary (NI) systems theory was introduced in [1] and was primarily inspired by the "positive position feedback control" of highly resonant mechanical systems with colocated position sensors and force actuators [2]. NI theory has formalized and unified some well-known vibration control techniques (e.g., graphical techniques and integral resonant control schemes) developed for lightly damped flexible structures using positive position feedback [1], [3]. NI theory offers a stand-alone robust control analysis and synthesis framework, similar to passivity and small-gain methodologies [4]. The NI system property is closely related to counterclockwise input-output dynamics in a nonlinear setting [5] and input-output Hamiltonian systems in both a linear and a nonlinear setting [6], [7]. NI control theory can be considered to be an energy-based control methodology [8], and consequently, it has a strong connection with dissipative theory [9]. These connections will be investigated in detail in this article. NI systems theory has gained popularity owing to its simple robust stability condition that depends only on the dc loop gain. Hence, the theory can be easily applied to practical systems without having an exact mathematical model [10]- [13]. NI theory finds potential applications in vibration control of lightly damped flexible structures [1], cantilever beams [14], large space structures [15], and robotic manipulators [15], in control of nanopositioning systems [16], in control of large vehicle platoons [17], etc.
The connections between NI systems theory and classical dissipativity have not yet been thoroughly explored. In the case of passive systems, a complete characterization exists in the literature, which was built on Willems's dissipative framework [9] and Hill-Moylan's (Q, S, R)-dissipative framework [24]- [26].  All strict and nonstrict passive systems can be shown to be dissipative in respect of a supply rate w(u, y) that depends on the input (u) and the output (y) of the system. Different variants of the passivity theorem are available in the literature, which are proven using the (Q, S, R)-dissipative framework [25], [26]. In [27], Griggs et al. introduced a class of systems with "mixed" input-output passive and finite-gain properties. Griggs et al. [27], also proved finite-gain input-output stability of the closed-loop system having "mixed" properties using a frequency-domain dissipative approach. Inspired by the work presented in [27], Patra and Lanzon introduced in [19] the notion of "mixed" IONI and finite-gain properties along with a stand-alone frequency-domain definition for IONI systems on a finite frequency interval. Patra and Lanzon [19] also provided a frequency-domain (Q(ω), S(ω), R(ω))-dissipative supply rate to characterize such systems. Later, Das et al. [20], [21] pursued a similar approach alike [19] to establish internal stability conditions for interconnected systems with "mixed" NI, passive and finite-gain properties.
Unlike [19]- [21], in this article, it is shown that the IONI systems are dissipative with respect to a new time-domain supply rate w(u,ū,ẏ) = 2ẏ u − δẏ ẏ − εū ū by proving the existence of a positive semidefinite storage function V (x). An auxiliary outputȳ = y − Du is utilized to capture the full class of OSNI systems (i.e., including biproper cases), while the auxiliary inputū, which is a filtered version (as discussed later in Section V) of the actual input u, is used to capture an ISNI property. For a strictly proper OSNI system, this supply rate reduces to 2ẏ u − δẏ ẏ, which finds an interesting physical interpretation. For example, in the case of a spring-mass-damper system being OSNI, the termẏ u gives the mechanical power input [velocity (ẏ) × force (u)], while the termẏ ẏ represents the power dissipated in the damper (dẏ 2 ), and hence, the expression T 0 (2ẏ u − δẏ ẏ)dt gives the stored energy of the system, which is always nonnegative. However, for more general systems, the supply rate provides an abstraction of the net power inflow into the system, and often, it is not possible to find an exact physical interpretation.
Apart from the time-domain analysis, a frequency-domain (Q(ω), S(ω), R(ω))-dissipative framework is also proposed in this article to characterize IONI systems. Thereafter, an equivalence is established between the time-domain and frequencydomain dissipative frameworks via applying Parseval's theorem. Furthermore, LMI-based state-space characterizations are derived for the IONI systems and each of its subclasses. We also specialize the above results to OSNI systems since such systems exhibit interesting closed-loop stability properties when connected (in a positive feedback loop) with NI systems that may contain complex conjugate poles on the imaginary axis excluding the origin.

II. NOTATION AND MATHEMATICAL PRELIMINARIES
The notation is standard throughout. The set of all natural numbers (excluding 0) is denoted by N = {1, 2, 3, . . . }. R ≥0 denotes the set of all nonnegative real numbers. A − * and A − represent shorthand for (A −1 ) * and (A −1 ) respectively. λ max (A) denotes the maximum eigenvalue of a matrix A that has only real eigenvalues. Let R m×n be the set of all real, rational, and proper transfer function matrices of dimension m × n, and let RH m×n ∞ denote the set of all asymptotically stable transfer function matrices in R m×n . For M (s) ∈ R m×m , the real-Hermitian and imaginary-Hermitian frequency response parts are given by  [19], [28] under the inner product f, g = 1 2π A dynamical system is said to be initially relaxed if it has zero initial condition, i.e., x(0) = 0. The term "stable system" refers to an asymptotically stable system, i.e., the associated transfer function matrix belongs to RH ∞ . The space of all real-valued, absolutely square integrable, time-domain functions is defined by L m An energy supply rate function w(u, y) is an abstraction of the rate of energy inflow into a physical system that is expressed by the mapping w : U × Y → R, where the input space U ∈ L m 2e and the output space Y ∈ L p 2e , and satisfies the property 0 w(u, y) dt < ∞ for all admissible (u, y) ∈ U × Y and ∀T ∈ [0, ∞). In particular, Note that an energy supply rate can also be defined in the frequency domain for a stable system, and it remains equivalent to the corresponding time-domain supply rate via Parseval's theorem [4]. The symbol denotes the product operator. For a transfer function M (s) ∈ R m×m , L −1 [M (s)] represents the impulse response (also called the Kernel function), where L −1 denotes the inverse Laplace operator. The symbol denotes the time-domain convolution operator and the expression y(t) = L −1 [M (s)] u(t) indicates that the output signal y(t) is being generated by the time-domain convolution of the impulse response of the system and an input u(t). The spectral factorization [29], [30] of a transfer function matrix F (s) is given by where F s (s) denotes the stable, minimum phase spectral factor of F (s) and F ∼ s (s) = F s (−s) indicates the antistable, antiminimum phase spectral factor. Let S 1 and S 2 be two subsets of R, then

III. TECHNICAL PRELIMINARIES
In this section, essential technical preliminaries, definitions, and lemmas are presented, which underpin the proofs of the main results of this article.
The finite-dimensional, causal, LTI systems studied in this article are described by 2 The admissible inputs u(t) are considered to be in the space L m 2 such that the unique solution of the state trajectory x(t) exists forward in time t ≥ 0 and x ∈ L n 2e . Therefore, the output y(t) also exists forward in time t ≥ 0 and y ∈ L p 2e . Let us introduce the state transition function Φ, associated with M , being a mapping from R ≥0 × R ≥0 × R n × L m 2 to R n . Here, Φ(t 1 , t 0 , x(t 0 ), u(t)) denotes the state x(t 1 ) at time t 1 when the system M starts from an initial state x(t 0 ) ∈ R n at time t = t 0 , and an admissible input u(t) is applied on M for the time interval t ∈ [t 0 , t 1 ].

A. Dissipative Systems Notations and Definitions
Let us recall the notion of dissipativity of finite-dimensional, causal, LTI systems introduced in [9]. It is important to mention here that in the following definitions related to time-domain dissipativity, we have chosen to restrict the input space to L m 2 since, in this article, we aim to establish the equivalence between the time-domain dissipativity and frequency-domain dissipativity of stable NI systems where the frequency-domain dissipativity is characterized by only finite energy input signals U ∈ L 2 m (jR). Definition 1 (Dissipative systems) [9]: A dynamical system M , given in (1), is said to be dissipative with respect to an energy supply rate w(u, y) if there exists a function V : R n → R ≥0 , called the storage function, such that for any T ∈ [0, ∞), any initial condition x(0) ∈ R n and any admissible input u ∈ L m 2 where x(T ) = Φ(T, 0, x(0), u(t)) and w(u, y) has been evaluated along any trajectory of (1). Inequality (2) is known as the "dissipation inequality" in the sense of Willems. Note that for asymptotically stable LTI systems and for all input u ∈ L m 2 , lim t→∞ x(t) is finite and also x ∈ L m 2 ; hence, y ∈ L p 2 implying ∞ 0 w(u, y) dt < ∞. In such cases, Willems's dissipation inequality implies Furthermore, if V : R n → R ≥0 is a differentiable storage function, then the dissipation inequality (2) can be expressed in the differential form as where the "dot" represents the time-derivative. Note that for finite-dimensional LTI systems with minimal state-space realizations, the storage function V (x) can be characterized with a quadratic form x P x, without loss of generality, where P = P > 0 [9], [31]. Moreover, in an LTI setting, the storage function V (x) can always be assumed to be a differentiable function of x [24], [32]. For a dissipative system with a completely controllable state space, the "required supply" is defined as [33] where x * ∈ R n represents the point of minimum storage. In general, the origin of a state space is the point of minimum storage, where V (x * ) = V (0) = 0. The "required supply" is the least amount of energy required to excite a system to a desired state from the state of minimum energy level [34]. V r (x) is a possible storage function for any dissipative system with a reachable (from the origin) state space. Definition 2 ((Q, S, R)-dissipativity in Hill-Moylan's framework) [24]: A dynamical system M , given by (1) with x 0 = 0, is said to be (Q, S, R)-dissipative if there exist Q = Q T ∈ R p×p , S ∈ R p×m and R = R T ∈ R m×m such that T 0 y Qy + 2y Su + u Ru dt ≥ 0 for any T ∈ [0, ∞) and all u ∈ L m 2 . If the supply rate function in Willems's framework is considered to be w(u, y) = y Qy + 2y Su + u Ru where Q = Q T ∈ R p×p , S ∈ R p×m and R = R T ∈ R m×m , then (4) takes the form y Qy + 2y Su + u Ru ≥V (x).
So far we have discussed only time-domain dissipativity. However, dissipative characterization can also be expressed in the frequency domain. The following definition articulates the notion of frequency-domain (Q(ω), S(ω), R(ω))-dissipativity, which may be regarded as a frequency-domain counterpart of the Hill-Moylan's (Q, S, R)-dissipativity.
Definition 3 ((Q(ω), S(ω), R(ω))-dissipativity) [19], [27]: Let M (s) ∈ RH p×m ∞ be the transfer function matrix of a causal system M with the input-output relationship Y (s) = M (s)U (s), where U ∈ L 2 m (jR). Then, M is said to be (Q(ω), S(ω), R(ω))-dissipative with respect to the frequency- for all U ∈ L 2 m (jR). The following lemma on the characterization of output strictly passive (OSP) systems is recalled here, so that it can be used later in this article to define the OSNI systems property.
Lemma 1 [8], [18]: A system F (s) ∈ RH m×m ∞ with F (s) + F ∼ (s) having full normal rank is OSP if and only if there exists δ p > 0 such that

B. Definitions of Negative Imaginary Systems
In this section, we recall the definitions of NI and SNI systems. Definition 4 (NI System) [15], [35]: Let M (s) be the real, rational and proper transfer function matrix of a square and causal system without any poles in the open right-half plane. M (s) is said to be NI if the following three conditions hold:  [23], [36], [37] and NI theory has also been recently extended to discrete-time LTI systems [38]. However, in this article, we restrict our attention to only continuous-time, real, rational, and proper NI systems as per Definition 4.
Definition 5 (SNI System) [1]: Let M (s) be the real, rational, and proper transfer function matrix of a square and causal system. M (s) is said to be SNI if M (s) has no poles in {s ∈ C : [s] ≥ 0} and j[M (jω) − M (jω) * ] > 0 for all ω ∈ (0, ∞). We now present a necessary and sufficient condition for internal stability of an NI-SNI positive feedback interconnection, as shown in Fig. 1.
Theorem 1 [35]: Let M (s) be an NI system without poles at the origin and N (s) be an SNI system. Then, the positive feedback interconnection of M (s) and N (s), as shown in Fig. 1, is internally stable if and only if

C. Relationship Between the Transmission Zeros of a Transfer Function Matrix and the Rank Deficiency of Its Imaginary-Hermitian Part
In the following, we establish a relationship between the transmission zeros of an NI transfer function matrix on the imaginary axis and the rank deficiency of its imaginary-Hermitian part at that frequency. This result will be used later in Section VI to prove the closed-loop stability of a positive feedback interconnection of an NI system without poles at the origin and an OSNI system.
Lemma 2: Let G(s) ∈ R m×m be an NI system with full normal rank. Suppose s = jω z with ω z ∈ (0, ∞) is a transmission zero of G(s) but not a pole. Then, det[G(jω z ) − G(jω z ) * ] = 0.
Proof: As s = jω z with ω z ∈ (0, ∞) is a transmission zero of G(s), there exists a nonzero vector x ∈ C m such that G(jω z )x = 0. This then implies [39, p. 406]). Therefore, we have Lemma 3: Let G(s) ∈ R m×m be an NI system. Then, Proof: Suppose via contradiction that det[G(jω z )] = 0. Then, there exists a nonzero vector x ∈ C m such that G(jω z )x = 0, which ultimately implies x * [j{G(jω z ) − G(jω z ) * }]x = 0, as shown in the proof of Lemma 2. But, the result violates the supposition that

IV. IONI SYSTEMS
In this section, we define a unifying class of stable negative imaginary systems, termed as IONI systems, 3 that encompasses the existing strict forms of NI systems, namely, 1) SSNI systems introduced in [22] (denoted by SSNI (α=1,β=1) in this article), 2) a different class of SSNI systems defined in [23] (denoted by SSNI (α=2,β=1) in this article), and 3) OSNI systems defined in [18] and modified later in [8]. Moreover, the proposed IONI class also gives birth to two new subclasses of SNI systems, termed as the ISNI systems and very strictly negative imaginary (VSNI) systems in this article. The set-theoretic relationship among the subclasses of IONI systems is illustrated in the Venn diagram of Fig. 2.
Then, M (s) is said to be IONI with a level of output strictness δ ≥ 0, a level of input strictness ε ≥ 0, and having an arrival rate specified by α ∈ N and a departure rate specified by β ∈ N ( Remark 1: α ∈ N (resp. β ∈ N) is referred to as the arrival (resp. departure) rate as it determines the behavior of j[M (jω) − M (jω) * ] as ω → ∞ (resp. ω → 0).
We will now classify IONI systems on the basis of the values of the parameters δ, ε, α, and β.
has full normal rank}. The following lemma states the connections between the above classifications.
Lemma 5: The following five statements hold.
, then it is also ISNI. Proof: All five cases are trivial consequences of Definition 7.
The following lemma gives a simpler, yet equivalent, characterization for each of the classes in Definition 7.
has full normal rank and , sufficiency trivially follows on choosing ε 0 = 0 and any α 0 ∈ N and β 0 ∈ N whereas necessity follows from Lemma 4 on choosing ε = 0 in Lemma 4. Note that the pointwise frequency-domain condition (11) can equivalently be expressed on the open positive frequency interval, that is, for all ω ∈ (0, ∞), as shown in Lemma 7.
and taking the transpose throughout. On multiplying this last inequality byω and (12) by ω, which is clearly finite and positive semidefinite in the limit ω → ∞.
The following lemma shows that the SNI set contains the same elements as the ISNI set.
. This then implies that there exist a sufficiently small ε > 0 and sufficiently large α ∈ N and β ∈ The following lemma shows that the set of SSNI (α=1,β=1) systems is contained within the set of SSNI (α=2,β=1) systems and also within the set of VSNI systems.
Lemma 9: Let M (s) be SSNI (α=1,β=1) . Then, M (s) is also SSNI (α=2,β=1) and VSNI. Proof The following lemma states that SSNI (α=2,β=1) systems that have a strictly proper sM (s) are also VSNI. The interpretation of a strictly proper sM (s) is easy in a scalar setting as it would correspond to systemsM (s) with a relative degree of two.
The following lemma shows that in the scalar case, the set of SSNI (α=2,β=1) systems is contained within the set of VSNI systems.
In the sequel, we will present six numerical examples corresponding to each region of the Venn diagram in Fig. 2 to illustrate different examples of IONI systems. Note that it is possible to check the strict conditions separately, one at a time, as explained in the next lemma.
area 1 of the Venn diagram (see Fig. 2) because 30 289 ]; and .  Fig. 2 12.5 rad/s; and at  and M 6c (s) = 1 s 2 belong to area 6 of the Venn diagram (hence do not belong to the IONI class) since they are not asymptotically stable.

V. CONNECTIONS BETWEEN IONI SYSTEMS AND DISSIPATIVITY
Section V-A derives a stable spectral factor of a transfer function associated with the filter term in (11) for usage in the subsequent sections. Section V-B extends the classical notion of dissipativity to include supply rates that involve the time derivative of the system's output taking inspiration from [5], [25], [33] and introduces a new time-domain dissipative framework for characterizing the class of stable IONI systems, including its strict subclasses. In Section V-C, IONI systems are characterized in an equivalent frequency-domain framework with respect to a (Q(ω), S(ω), R(ω))-dissipative supply rate.

A. Analysis of the Filter Term Used in Definition 6
In order to establish the connections between the IONI system property (11) and dissipative theory, a new supply rate w(u,ū,ẏ) will be proposed in the sequel to characterize IONI systems in a time-domain dissipative framework. This supply rate involves the input to a physical system (u), an auxiliary input (ū) which is a filtered version of u, and the time-derivative of an auxiliary output (ẏ) where the auxiliary outputȳ = y − M (∞)u. In order to obtainū, a bandpass filter has to be constructed as the stable spectral factor of where α ∈ N and β ∈ N. Note that when s = jω, f (jω) = ω 2β 1 + ω 2(α+β−1) which is the frequency response function within the last term of (11) associated with ε.  given by i=0 s 2 + 2 sin (2i + 1)π 2(α + β − 1) s + 1 when α + β is even and α + β > 2.
Proof: Applying the rules of stable spectral factorization [29], [30]  I m ]U (jω) dω quantifies the input energy dissipation, and hence, it signifies the level of input strictness of an IONI system. Fig. 3 reveals that ISNI systems must have an imaginary-Hermitian frequency response, which is strictly less than zero for all ω ∈ (0, ∞) and can only become zero at ω = 0 and ω = ∞. Second, the arrival rate at ω = ∞ and the departure rate at ω = 0 are governed by the parameters α and β, respectively. The arrival (resp. departure) rate at ω = ∞ (resp. ω = 0) is the decay (resp. growth) rate of the imaginary-Hermitian frequency response toward (resp. away from) the real axis near ω = ∞ (resp. ω = 0).

B. IONI Systems in a Time-Domain Dissipative Framework
In this section, we will establish that for an initially relaxed IONI system with a controllable state-space, there always exists a positive semidefinite storage function V (x) such that the system satisfies the dissipation inequality (2) with a particular time-domain supply rate w(u,ū,ẏ) = 2ẏ u − δẏ ẏ − εū ū for some δ ≥ 0 and ε ≥ 0, whereȳ = y − M (∞)u is defined as an auxiliary output of M andū is a filtered auxiliary input chosen as the inverse Laplace ofŪ (s) = [f s (s)I m ]U (s) where U (s) = L [u(t)] and f s (s) ∈ RH ∞ is defined in (14). Note that in this section, the admissible inputs u are considered to be in the space L m 2 along with sufficient smoothness properties such that a unique solution of the state trajectory x(t) exists forward in time t ≥ 0 and also (since A will be assumed Hurwitz) x ∈ L n 2 . Hence,ẏ(t) = Cẋ(t) = CAx(t) + CBu(t) also exists for all t ≥ 0 andẏ ∈ L m 2 . Furthermore,ū ∈ L m 2 since f s (s) ∈ RH ∞ and since u ∈ L m 2 by assumption. Theorem 2: Let M be a finite-dimensional, causal and square system given by the minimal state-space equationsẋ = Ax + Bu and y = Cx + Du with zero initial condition. Let the associated transfer function matrix be M (s) ∈ RH m×m (14). Let δ ≥ 0, ε ≥ 0, α ∈ N and β ∈ N. Then, D = D and M is dissipative with respect to the supply rate w(u,ū,ẏ) = 2ẏ u − δẏ ẏ − εū ū if and only if M (s) is IONI (δ,ε,α,β) .
Proof: The proof has been divided into the sufficiency and necessary parts as follows.
(Sufficiency) First note that M (s) is IONI (δ,ε,α,β) implies that M (s) is stable NI, which in turn implies D = D [1]. To show that an IONI (δ,ε,α,β) system M is dissipative with respect to the supply rate w(u,ū,ẏ) = 2ẏ u − δẏ ẏ − εū ū, we have to establish that there exists a storage function V : R n → R ≥0 such that M satisfies the dissipation inequality (2). Since the state space is assumed to be completely controllable, there exists an admissible input u(t) defined as when t > 0, which steers the system from x(t −1 ) = 0 to any x(0) ∈ R n . Let y(t) be the corresponding output and Y (jω), Y (jω), U (jω) andŪ (jω) denote, respectively, the Fourier transform of the real-valued time-domain signals y(t), [since A is Hurwitz and applying Parseval's theorem [4]] Hence, for arbitrary t −1 ≤ 0 and x(t −1 ) = 0, we have 0 t −1 w(u,ū,ẏ) dt ≥ 0. We now construct the required supply function as where the origin is the point of minimum storage (i.e., x * = 0). Thus, V r (x) can be considered as a storage function candidate associated with the IONI (δ,ε,α,β) system M [33].
It remains to be shown that V r (x) satisfies the dissipation inequality (2). Note that in taking the system from x = 0 at t = 0 to x 1 ∈ R n at t = t 1 , we could first take it to x 0 ∈ R n at time t 0 while minimizing the energy and then take it to x 1 at time t 1 along the path for which the dissipation inequality is to be evaluated. This is possible since M is a causal and time-invariant system. As V r (x 1 ) represents the infimum amount of energy required to reach x 1 at t = t 1 from x = 0 at t = 0, the energy required to reach the same destination x 1 from the same starting point x = 0 via any other path will be greater than or equal to V r (x 1 ). Therefore, V r (x 0 ) + t 1 t 0 w(u,ū,ẏ) dt ≥ V r (x 1 ) follows. It can, hence, be concluded that the IONI (δ,ε,α,β) system M is dissipative with respect to the supply rate w(u,ū,ẏ) = 2ẏ u − δẏ ẏ − εū ū for the same δ, ε, α and β.
(Necessity) This part proceeds through a sequence of implications, where frequency-domain integrals with limits from −∞ to ∞ are considered taking inspiration from similar arguments used in [19] and [27]. For the same choice of δ ≥ 0, ε ≥ 0, and α, β ∈ N, and since M (s) ∈ RH m×m ∞ , M is dissipative with respect to The equivalence between (15) and (16)  The following corollary is an immediate consequence of Theorem 2 and establishes the time-domain dissipativity of all stable NI systems and also the strict subclasses (e.g., ISNI, SSNI (α=1,β=1) , SSNI (α=2,β=1) , VSNI, OSNI) under the NI systems class.
Corollary 1: Let M be a finite-dimensional, causal and square system given by the minimal state-space equationsẋ = Ax + Bu and y = Cx + Du with x(0) = 0. Let the associated transfer function matrix be M (s) ∈ RH m×m (14). Then The following lemma gives a necessary and sufficient condition for checking time-domain dissipativity of an IONI (δ,ε,α,β) system without involving a storage function.
Proof: The proof readily follows from the necessity part of the proof of Theorem 2 and Definition 6.
The following corollary provides the connection between the time-domain and frequency-domain dissipativity. Proof: Trivial from Theorems 2 and 3.

VI. STATE-SPACE CHARACTERIZATION OF IONI SYSTEMS
In this section, we provide a state-space characterization of the full class of IONI systems. The state-space realizations in the results of this section are not required to be minimal.
Proof: Trivial application of Theorem 4 with ε = 0 and by removing the states associated with f s (s). The LMI is just a Schur complement rearrangement.

VII. OSNI SYSTEMS IN A DISSIPATIVE FRAMEWORK
As with other strict subclasses (e.g., ISNI, SSNI (α=1,β=1) , SSNI (α=2,β=1) , VSNI) under the IONI class, the pointwise frequency-domain condition (11) defines the OSNI subclass when δ > 0. The OSNI class was originally proposed in [18] and generalized later in [8]. OSNI systems exhibit several interesting properties and also obey a simple internal stability condition when interconnected with a (not necessarily stable) NI system in a positive feedback loop. This section is dedicated solely to the OSNI class of systems to 1) develop a minimal state-space characterization for OSNI systems, 2) describe OSNI systems both in the time-domain and the frequency-domain dissipative frameworks, and 3) establish the equivalence between the OSNI lemma conditions and the time-domain dissipative characterization of OSNI systems.

A. Specialized OSNI Lemma for Minimal Systems
We first show the connection between OSNI and OSP systems. The following lemma provides a necessary and sufficient condition for a system given by a minimal state-space realization to be OSNI and is a generalization of [18,Lemma 6]. (23) Proof: Since the realization is minimal, A is Hurwitz and hence nonsingular. Then, LMI (22) in Corollary 8 is equivalent to by taking a Schur complement with respect to 1 δ I m . This condition then implies that X < 0 via XA + A X ≥ δA C CA. Note that the matrix inequality in (23) is not in LMI form but can be readily converted into an LMI by applying the Schur complement lemma [4, Appendix A.61].

B. Equivalence Between Time-Domain Dissipativity and State-Space Characterization of OSNI Systems
We have already established that OSNI systems are dissipative with respect to the time-domain supply rate w(u,ẏ) = 2ẏ u − δẏ ẏ with δ > 0 whereȳ = y − Du is selected as an auxiliary output of the system. In this subsection, we will show that for a stable LTI system with a minimal state-space realization, the conditions in OSNI Lemma 16 are equivalent to time-domain dissipativity with respect to the proposed supply rate w(u,ẏ) and a specific storage function given by

D. Internal Stability Condition of an NI-OSNI Interconnection
This section deals with internal stability of a positive feedback interconnection of NI and OSNI systems, as shown in Fig. 1. In order to prove the internal stability theorem of the NI-OSNI interconnection, we need the following technical lemma first.
We are now ready to state the feedback stability result for the positive feedback interconnection (see Fig. 1) of an NI system without poles at the origin and an OSNI system. Since M (s) is NI without poles at the origin and N (s) is OSNI, there exist real symmetric matrices Y 1 > 0 and Y 2 > 0 such that M (s) satisfies A 1 Y 1 + Y 1 A 1 ≤ 0 and B 1 + A 1 Y 1 C 1 = 0 [1], [41], while N (s) satisfies A 2 Y 2 + Y 2 A 2 + δ 2 (C 2 A 2 Y 2 ) (C 2 A 2 Y 2 ) ≤ 0 with some δ 2 > 0 and B 2 + A 2 Y 2 C 2 = 0 via Lemma 16. The second inequality implies A 2 Y 2 + Y 2 A 2 ≤ 0 since δ 2 > 0. Define U = and only if the corresponding system matrix A cl is Hurwitz. As in [35], it can be shown that A cl = ΦT . Then, following the proof of [35,Th. 9], except that Lemma 18 must be used instead of [35,Lemma 6] to take into account the fact that N (s) is an OSNI system here instead of an SNI system, internal stability is equivalent to det Cases I and II together complete the proof.
The following corollary specializes Theorem 5 when the systems satisfy additional constraints at infinite frequency.
Corollary 9: Let M (s) be a (not necessarily stable) NI system without poles at the origin and N (s) be an OSNI system. Let Proof: The proof readily follows from Theorem 5 by imposing the additional constraints that either M (s) is strictly proper, or else, M (∞)N (∞) = 0 and N (∞) ≥ 0.
The following numerical example illustrates the applicability of Lemma 18, Theorem 5 and Corollary 9.

VIII. CONCLUSION
In this article, we define the class of stable IONI systems. This new IONI class captures all stable NI systems and includes within it the existing strict subclasses (e.g., SSNI [22], SSNI [23], and OSNI [8], [18]). It also creates two new strict subclasses: ISNI and VSNI. This article also establishes the missing link between NI theory and classical dissipativity in the sense of Willems [9]. A new time-domain dissipative supply rate w(u,ū,ẏ) is introduced to characterize the full class of IONI systems which involves the system's input (u), an auxiliary filtered version of the input (ū) and the time-derivative of an auxiliary output of the system (ẏ). This article also proves that IONI systems belong to a class of dissipative systems defined with respect to the particular supply rate w (u,ū,ẏ). In addition to the time-domain dissipative framework, a (Q(ω), S(ω), R(ω))dissipative supply rate is also developed to characterize IONI systems. Most importantly, all these characterizations are shown to be equivalent and they are also consistent with the original pointwise frequency-domain definition of stable NI systems. Furthermore, necessary and sufficient state-space conditions are derived in LMI form to check whether a given system is IONI (δ,ε,α,β) or ISNI or OSNI or VSNI or SSNI (α=1,β=1) or SSNI (α=2,β=1) . Finally, a necessary and sufficient internal stability condition is also presented for a positive feedback interconnection of an NI system with an OSNI system when the NI system may contain poles on the jω-axis except at the origin.