Development of Frequency Weighted Model Order Reduction Techniques for Discrete-Time One-Dimensional and Two-Dimensional Linear Systems with Error Bounds

Enns’ frequency weighted model reduction method yields an unstable reduced model. Many stability-preserving techniques for one-dimensional and two-dimensional reduced-order systems have been demonstrated; however, these methods produce significant truncation errors. This article presents a frequency weighted stability preserving framework, which addresses Enns’ main problem concerning reduced-order model instability. Unlike other stability-preserving techniques, the offered frameworks provide an easily computable a priori error-bound expression. The simulation results show that the proposed frameworks outperform existing stability-preserving approaches, demonstrating effectiveness.

In this article, following acronyms/abbreviations are used: MOR Model order reduction ROM Reduced order model ODEs Ordinary differential equations PDEs Partial differential equations m-D Multi-dimensional 1-D One-dimensional 2-D Two-dimensional BT Balanced Truncation HNA Hankel norm approximation CRSD Causal recursive separable denominator GJ Gawronski & Juang GA Gugercin & Antoulas The of the original large-scale model, such as stability/passivity and input-output response. The MOR has made a significant contribution to the control system, mainly in the simulation of complex systems such as large-scale complex integrated circuits, robotics systems, communication systems, and controller reduction, etc., A record of the user's interactions with the recommendation system [1]- [6].
A substantial amount of study in the MOR of large-scale systems has been done recently, and various methods to MOR have been proposed [7]- [12]. Mathematical modeling aims to analyze dynamical systems, which is an important part of control systems engineering. The need for more rigorous mathematical models is growing as models get more complicated. Simulation becomes computationally tedious largescale systems containing lumped parameter systems, such as ODEs, distributed parameter systems, such as PDE, etc. Dealing with these conditions is made easier with MOR. The study of these complex and large-scale systems is a difficult task; as a result, ROMs are needed to make the analysis easier.
m-D systems are those models in control system theory where several independent variables occur (like time). The Roesser 2-D model is a sub-class of m-D systems, and it has vast applications in control systems theory. Due to their applications in various key areas such as IP, SP, SDP, water steam heating, DSP filters, etc., 2-D systems have been a continually increasing research interest area in recent years. The Roesser 2-D model contributes in the following various fields such as: • Automated irrigation channels [13].
• Linear repetitive processes [21]. • Iterative learning control [22], [23]. The researcher focused on fundamental problems such as decomposition, factorization, stability, and model reduction, etc. As decomposition, factorization, stability, and model reduction are not straightforward extensions for 1-D models, the fundamental theorem of the algebra does not apply to m-D systems directly [24]- [27].

B. LITERATURE REVIEW
The most often used MOR approach is BT [28], while using BT [28] methods, it is necessary to balance the system, which is equivalent to determining the system's controllability and observability Gramians in a unique diagonal form. The Cholesky factors of these Gramians can be efficiently computed as dual Lyapunov equation solutions for systems with few inputs and outputs. BT [28] provides ROMs for the 1-D LTI continuous and the discrete-time systems that guarantee stability and yield error bounds. However, an entire frequency interim is used to execute MOR operations, while the particular frequency band is concerned only in practical applications, i.e., the controller reduction case. Similarly, Glover [29] used an optimal HNA to perform the MOR operation. The HNA is a model reduction method that offers the best Hankel semi-norm approximation. These promote the usage of frequency weights in MOR. Therefore, Enns [30], [31] provided the frequency weighted MOR approach for the 1-D LTI continuous and the discrete-time systems by inducing frequency weights (i.e., input, output and doublesided) in BT [28] approach. However, this approach [30] generates unstable ROMs in the case of double-sided weightings [32]. Similarly, the limited frequency interval [33] is of concern for some applications (i.e., controller and filter reduction). The frequency-limited intervals Gramians based MOR approach for the 1-D LTI continuous and the discretetime systems were implemented by the GJ [34] and WZ [35], respectively; however, it does often result in unstable ROMs at certain frequency-intervals [36], [37] and there exist no a priori error bound expressions for these techniques [30], [34], [35].
Recently, a significant amount of research has been conducted on the MOR of large-scale systems, and a number of different MOR approaches have been developed. [7]- [12]. In [7], second-order dynamical systems using structurepreserving balanced truncation approaches are provided, which deals with first-order constrained balanced truncation approaches and apply them to second-order systems utilizing various second-order balanced truncation formulas. The work presented in [8] is based on a balanced truncation model order reduction for discrete-time systems that preserves stability after reduction. However, due to the iterative nature of this method, it becomes more complicated when the order of the original system increases. Correspondingly, [9] presents MOR based on cross Gramians; this method [9] uses Sylvester equations rather than Lyapunov equations as described by BT [28], Enns [30], GJ [34], and WZ [35]. Furthermore, this method is only applicable to bilinear systems that employ a truncated cross Gramian projection approach. A similar work based on interpolation is presented in [10]. It proposes adaptive techniques for computing time delay systems' reduced-order model. The algorithms use greedy iterations to choose expansion locations and interpolate the transfer function. Similarly, another interpolation-based approach is presented in [11]. It focuses on dominated and temporal moment retention. It condenses the large-scale complete order model into a lower order system, allowing approximate computation denominator by employing generalized pole clustering. The factors division procedure yields the approximate numerator, which results in the ROM. In [12], the MOR for 2-D discrete-time system MOR is presented. This method ensures the stability of the filtering error system and H ∞ performance when the noise frequency ranges are known beforehand. Using the gKYP lemma, Finsler's lemma, and some independent matrices yield fewer conservative findings. The research briefly discussed above are based on cross Gramians, interpolation, and Kalman filtering. Furthermore, to overcome the shortcomings as appeared in Enns [30], GJ [34], and WZ [35] substantial amount of research have been conducted over the couple of decades [36]- [46], which are briefly discussed as follows with their drawbacks.
To overcome the main drawback as appeared in [30], the Lin & Chiu [38] introduced strictly proper two-sided weights to ensure the stability of ROMs; however, this method cannot be used in controller reduction applications due to no polezero cancellation assumption required in the method. Later on, VA [39] introduced an alternative approach to ensure the stability of the ROM for the continuous-time frequency weighted systems. Since the main weakness of Lin and Chiu's [38] technique is the requirement that no pole-zero cancellation occurs when forming the augmented systems (input augmented and output augmented). This prevents the applicability of this method when solving controller reduction problems involving weights; however, this technique [39] is only valid for strictly proper original systems.
The instability problem in [30] is related to the indefiniteness of the corresponding input and output matrices; CB [40] provided the stability-preserving frequency weighted MOR method by ensuring the input and output matrices are positive/semi-positive definite. As a result, some eigenvalues have significant variations while others have slight variations. Dissimilar effects on each eigenvalue of the input and output matrices result in a significant approximation error in the ROM. The GS method [41] combines unweighted balanced and partial-fraction-based frequency weighted balanced re-duction techniques, ensuring ROM stability but being parameterized. The GS [42] also proposed a MOR technique for 2-D discrete-time weighted systems. However, truncating negative eigenvalues causes a significant approximation error in 2-D ROM. The stability-preserving frequency-weighted MOR approach introduced by IG [43] involves varying the input and output matrices, but subtracting all eigenvalues from minor eigenvalues results in zeroing the last eigenvalue, resulting in an unequal effect to eigenvalues and a significant approximation error in the ROM.
Together with the use of positive/semi-positive definiteness of input and output matrices, GA [36] established stability preserving frequency limited Gramians based MOR approach. However, the asymmetrical impacts on all eigenvalues cause significant approximation error [36]. By using frequency-limited intervals, GS [41] developed ROM stability. GS's approach [41] produces a large approximation error due to the significant variation in the original system. In later work, IG [44] adjusted the eigenvalues matrix by subtracting the least dependent negative eigenvalue from all the eigenvalues; nonetheless, the modified eigenvalues cause significant changes to the original systems and large approximation error. Similarly, [45] offers three techniques to maintain ROM stability; however, [45] is iterative, which is inefficient when the original system's order rises.
Similarly, to overcome the main drawback as appeared in [35], GS [37] ensures the stability of the ROM by improvising the eigenvalues matrix; however, due to the truncation of negative eigenvalues and absolute of all the eigenvalues, it increases a distance from the eigenvalues matrix of the original systems, which leads to a large approximation in the ROM. Similarly, IG [46] also introduced frequency limited MOR approach for the discrete-time systems; however, this approach results in significant truncation errors in the desired discrete frequency intervals due to the significant variance from the original system and zeroing the effect of the last eigenvalue.
Recently, a significant amount of research has been conducted on the MOR of large-scale systems based on balanced approach [47]- [51]. In [47], weighted and limited interval discrete-time 1-D systems are provided. The frequency limited intervals for 1-D and 2-D systems are given in [48]. Similarly, frequency weighted and limited MOR approaches for power systems are given in [49]- [51].
The BT [28], Enns [30], GJ [34], and WZ [35] yield unstable ROM and do not provide a priori error-bound expressions. Further, their successive stability preserving approaches [36]- [46] ensure stability in some conditions and generate significant truncation error due to the substantial variation to the original systems (i.e., pole-zero cancellation, absolute of negative eigenvalues, truncation of all negative eigenvalues, zeroing the effect of the last eigenvalue, etc.).

C. MAIN CONTRIBUTION AND PAPER ORGANIZATION
A novel method for 1-D and 2-D discrete-time systems is proposed. For 1-D and 2-D discrete-time systems, the VOLUME 4, 2016 suggested method offers a new discrete frequency weighted strategy exhibiting small truncation error. The square root of all eigenvalues with similar effects prevents the zeroing of the last eigenvalues, provides an equal impact on all eigenvalues, and preserves the eigenvalues' structure of some input and output matrices. Compared to other stability-preserving model reduction frameworks based on frequency-weighted Gramians, the proposed method provides small variation to the original system.
The main contributions of this paper are as follows: • Decomposition of the discrete-time 2-D CRSD model based on frequency weightings into two decomposed 1-D sub-models is attained by using the minimal rankdecomposition conditions. • Modifications to associated input and output matrices are performed for 1-D models and corresponding decomposed 1-D sub-models to assure positive and semipositive definiteness of associated input and output matrices. • The controllability and observability Gramians for 1-D models and decomposed 1-D sub-models in the given frequency weights are computed, corresponding to modified input and output matrices. The MOR framework based on frequency weighted for linear time-invariant discrete-time 1-D and 2-D systems is presented in this paper. The 1-D and 2-D un-weighted and weighted models are discussed in Section II, and the 2-D model decomposition via minimal rank-decomposition conditions. The balance truncation approach, as well as frequency weighted MOR approaches, are discussed in Section III. The existing stability-preserving frequency weighted balancing related techniques for 1-D and 2-D discrete-time systems are also discussed in this part. Section IV lays out the proposed work for 1-D and 2-D discrete-time systems and the a priori error-bound expressions for 1-D and 2-D cases. In addition, the numerical simulation results are presented in section V, where a comparison is made between existing 1-D, and 2-D frequency weighted MOR techniques and proposed techniques, demonstrating the proposed techniques' efficacy.

II. PRELIMINARIES
This section presents the corresponding un-weighted and frequency weighted 1-D and 2-D state space systems.

A. 1-D STATE SPACE SYSTEM
Here we provide a brief overview of un-weighted, and frequency weighted 1-D state-space discrete-time systems.

2) Frequency Weighted 1-D State-Space System
Consider a transfer function form of a stable discrete-time input-weighting model be given as: where A iw ∈ ℜ (ni)×(ni) , B iw ∈ ℜ (ni)×(mi) , C iw ∈ ℜ (pi)×(ni) , D iw ∈ ℜ (pi)×(mi) and {A iw , B iw , C iw , D iw } is its n i th order minimal realization. Similarly, consider a transfer function form of a stable discrete-time outputweighting model where A ow ∈ ℜ (no)×(no) , B ow ∈ ℜ (no)×(mo) , C ow ∈ ℜ (po)×(no) , D ow ∈ ℜ (po)×(mo) and {A ow , B ow , C ow , D ow } is its n o th order minimal realization. The input-augmented and the output-augmented systems are given by: where

B. 2-D STATE SPACE SYSTEMS
Here we provide a brief overview of un-weighted and frequency weighted 2-D systems with its decomposition based on minimal rank-decomposition criteria and weighted 2-D state-space discrete-time systems.
Similarly, the minimal rank-decomposition of Roesser's state-space realization subject to A 2 = 0 can be written as: consequently, 2-D separable denominator state-space can be written as: Similarly, the decomposed 1-D systemF 1 Remark 1: The 2-D models, as in (9), generally don't exist in CRSD form; however, existing 1-D MOR schemes are only applicable to the 2-D systems when it exists in 2-D CRSD form. In addition, we need minimal rank-decomposition criteria to obtain decomposed 1-D sub-models as in (14)- (15) and (19)- (20).
obtained by using 1-D BT [28], then the 2-D discrete-time ROM F r [z 1 , z 2 ] is asymptotically stable. Moreover, the frequency response truncation error is bounded by: Alternatively, whereρ i andφ i are the Hankel Singular-values of the decomposed sub- obtained by using 1-D an optimal Hankel norm approximation [29], then the 2- Moreover, the frequency response truncation error is bounded by: Alternatively, whereρ i andφ i are the optimal Hankel Singular-values of the decomposed sub-systemsF 1 [z 1 ] andF 2 [z 2 ], respectively.

2) Frequency Weighted 2-D State-Space System
The 2-D weighted discrete-time systems arrangement is shown in Figure 1. Consider a transfer function stable 2-D linear time-invariant discrete-time input weighted system [42] be given as: where } is its (n 1i + n 2i ) th dimensional minimal realization with p i number of inputs and q i number of outputs. Similarly, consider a transfer function stable 2-D linear time-invariant discrete-time output weighted system [42] be given as: where } is its (n 1o +n 2o ) th dimensional minimal realization, p o and q o are the number of inputs and outputs respectively.   Figure 2) [42]. where

III. 1-D MODEL REDUCTION TECHNIQUES
Here we provide a brief overview of un-weighted [28] and frequency weighted [30] model reduction techniques for the discrete-time 1-D systems.

A. UN-WEIGHTED 1-D MODEL REDUCTION TECHNIQUE
Let the controllability Gramians P c * and the observability Gramians Q o * for the entire frequency interim be given as [28]: that are the solution of the following Lyapunov equations: Let a similarity transformation matrix T b be given as: The ROM is attained as [28], [29]:

B. FREQUENCY WEIGHTED 1-D MODEL REDUCTION TECHNIQUE
Let the controllability Gramians P ai and the observability Gramians Q ao for the corresponding input-augmented (5) and the output-augmented (6) realization respectively, that satisfy the following Lyapunov equations: that satisfy the following Lyapunov equations: Truncating 1 st and 4 th block of (33) and (34), respectively, we have the following Lyapunov equations: where (38) By using the eigenvalues decomposition of X E and Y E we have the following: where S E1 and R E1 have (l − 1) and (k − 1) numbers of positive eigenvalues respectively; similarly, S E2 and R E2 have (n−l) and (n−k) numbers of negative eigenvalues respectively. Let T E be the transformation matrix obtained as: The transformation matrix T E transforms the original stable large-scale system realization into a balanced realization. The ROM F r * [z] =D r * +Ĉ r * [zI −Â r * ] −1B r * is acquired by truncating the transformed balanced realization.

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Remark 2: This technique [30] provide unstable ROMs because input/output associated matrices X E and Y E respectively are indefinite (i.e., X E ≤ 0 and Y E ≤ 0) [32] when both-sided weights are used.

IV. EXISTING STABILITY PRESERVING FREQUENCY WEIGHTED MOR TECHNIQUES
Here we provide a brief overview of existing frequency weighted model reduction techniques for the 1-D [39], [40], [43] and 2-D [42] systems.

A. EXISTING 1-D STABILITY PRESERVING FREQUENCY WEIGHTED MOR TECHNIQUES
CB [40], VA [39], and IG [43] improvised Enns's [30] input and output associated matrices X E and Y E , respectively, to yield positive and positive-semi definiteness of these matrices, which consequently yield stability of the ROM. These techniques also offer an error bounds formula. The controllability and observability Gramians P ex and Q ex , respectively, satisfying the following Lyapunov equations: The improvisation by CB [40], VA [39], and IG [43] introduced fictitious input and output associated matrices B ex ∈ {B ex1 [40], B ex2 , [39], B ex3 [43]} and C ex ∈ {C ex1 [40], C ex2 [39], C ex3 [43]}, respectively, can be computed as: Let T ex ∈ {T ex1 , T ex2 , T ex3 } a transformation matrix be obtained as: whereξ j ≥ξ j+1 , j = 1, 2, 3, . . . , n − 1,ξ r >ξ r+1 where r is the order of the ROM. The ROM F r * [z] =D r * +C r * [zI − A r * ] −1B r * is acquired as: consequently, yield minimal and stable ROMs. These techniques offer formula for the error bounds. Remark 4: The following error-bound expression exists [40]: with the existence of the rank conditions [37] [39], B ex3 [43]} and for each output related matrix C ex ∈ {C ex1 [40], C ex2 [39], C ex3 [43]} grant positive and positive-semi definite of the original system's input and the original system's output associated matrices, respectively; which results into the positive and positivesemi definite of the controllability matrices P ex ∈ {P ex1 [40], P ex2 , [39], P ex3 [43]} and the observability matrices Q ex ∈ {Q ex1 [40], Q ex2 , [39], Q ex3 [43]} in a unique way. This leads to the existence of the different transformation matrices T ex ∈ {T ex1 [40], T ex2 , [39], T ex3 [43]}. As a consequence, three existing stability-preserving model order reduction techniques are established. [30] matrices X E and Y E and applied these matrices for 2-D MOR case (by using minimal rank-decomposition conditions) to grant positive and positive-semi definite of these input and output associated matrices, which consequently grant stable ROMs for the decomposed two 1-D systems and also yield error bounds. For decomposed systemsF 1

V. MAIN RESULTS
The stability preserving strategies for 1-D discrete-time systems proposed by CB [40], GS [37], and IG [46] modified X E and Y E to ensure the stability of the ROM by making positive and semi-positive definite of the associated input and the associated output matrices. However, these methods induce significant truncation errors in some distinct frequency weights due to significant variance form the original systems.
This paper presents a stability preserving frequencyweighted MOR technique for discrete-time 1-D and 2-D systems. For the 1-D and 2-D systems, the ROM's stability is ensured by inserting some fictitious input and output matrices. The fictitious matrices are created by square-rooting eigenvalues that have identical effects on each eigenvalue of 1-D and 2-D discrete-time input and output matrices to construct stable ROMs with low truncation errors at specified frequency weights. Decomposition is performed first for the discrete-time 2-D weighted system using the minimal rankdecomposition condition as illustrated in (11,16); then, the controllability and the observability Gramians are computed based on modified associated input and output matrices for decomposed 1-D sub-systems. The proposed scheme also provides an a priori error bound expressions by using the BT and an optimal Hankel norm approximation approaches, respectively, for the 1-D and 2-D discrete-time frequency weighted systems. A comparison among different existing frequency weighted MOR techniques (including 1-D and 2-D systems) with proposed techniques are presented, which show the efficacy of proposed methods.

A. 1-D FREQUENCY WEIGHTED MODEL REDUCTION TECHNIQUE FOR DISCRETE-TIME SYSTEMS
Let a new fictitious controllability Gramians matrixP m and the observability Gramians matrixQ m for 1-D discrete-time systems are computed as whereX m =B mB T m andȲ m =C T mCm . By eigenvalues decomposition ofX m andȲ m we have the following:

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The new fictitiousB m andC m are given as input and output associated matrices respectively, wherē Let the similarity transformation matrixT m is calculated as: CT m =C= C r * C2 , D =D r * .
The above MOR procedure can be viewed in the context of non-minimum phase systems.

Lemma 5 ( [54]):
If the n th order square discrete-time 1-D minimal realization be given as: ; then, A i = A * − B * D −1 * C * has k eigenvalues outside the unit circle. Let λ l [A i ]λ j [A i ] ̸ = 1 ∀ l, j; then, there exist a unique controllability and the observability matrices, P c * and Q o * , respectively, which are the solution to the Lyapunov equation as in (27) and , respectively. Further, Q o * contain k and n − k negative and positive eigenvalues, respectively. Remark 9: The realization F * [z] can be decomposed into two sub-systems as: The realization F k [z] has exactly k zeros outside of the unit disk; whereas, the rest of the zeros are inside the unit disc. Similarly, the above MOR procedure can be viewed in the context of unstable minimum phase systems. Lemma 6 ( [54]): If the n th order square discrete-time 1-D realization be given as: with no eigenvalues on the unit circle and let P c * = P T c * be the solution to the Lyapunov equation as in (53) with P c * = diag{Σ c1 , Σ c2 }, where Σ c1 is non-singular matrix and Σ c2 > 0 is a diagonal matrix. Then, A k and A * have k unstable poles (eigenvalues) outside the unit circle; also, A k has no eigenvalues inside the unit circle. Assume that F * [π] = D * is a nonsingular and λ l [A * ]λ j [A * ] ̸ = 1 ∀ l, j. Further, P c * contain k and n − k negative and positive eigenvalues, respectively. Remark 10: The realization F c * [z] can be decomposed into two sub-systems as: The realization F k [z] has exactly k poles (eigenvalues) outside of the unit disk; whereas, the rest of the poles are inside the unit disc. Furthermore, the proposed MOR procedure can be employed for the marginally stable systems by decomposing the original systems into sub-systems (i.e., asymptotically stable + marginally stable). Lemma 7 ( [55]): There exists a similarity transformation matrix T sm that satisfies: such that the full-order-model as in (1) is marginally stable and matrix A * has a full rank. Remark 11: The decomposition as in Lemma. 7 enables each sub-system to be reduced in a manner that preserves its particular notion of stability. Further, the MOR for each sub-systems are obtained in a similar way as in (59)- (60).
Therefore, the transformed-realization {A * ,B m ,C m } is minimal and the stability of the ROM is guaranteed. Lemma 8: The fictitious input associated matrices X E ≤ B mB T m ≥ 0 and the fictitious output associated matrices Y E ≤C T mCm ≥ 0, likewise, the controllability matrices P E <P m > 0 and the observability matrices Q E <Q m > 0. Therefore, the transformed-realization {A * ,B m ,C m } obtained is minimal and stable which also guaranteed the ROM's stability in the desired frequency-intervals. Proof of Lemma 8: we will demonstrate that the realization A * ,B m ,C m is minimal (i.e., controllable and observable). Since the controllability Gramians matrixP m and the observability Gramians  (53) and (54) respectively, sō Since for P E ≥ 0; consequently,P m ≥ 0 [37]. Similarly, for Q E ≥ 0; consequently,Q m ≥ 0. As a consequence, the original-system matrix A * is stable. Resultantly, the pair (A * ,B m ) is controllable and the pair (A * ,C m ) is observable (i.e., A * ,B m ,C m is minimal). Lemma 9: [56] Since the pair (A,B m ) satisfy the following Lyapunov equation (53), T m , forP m ≥ 0; then, the original large-scale system is asymptotically stable if f it is controllable. Suppose the original system is not asymptotically stable. In that case, eigenvalues of the original large-scale system (i.e., eig|A * |) are outside of the unit circle, not on the inside of the unit circle. Proof of Lemma 9: The first part is obvious. To proof the second part, let A * and ν * have eigenvalue λ and corresponding left eigenvector respectively; then, ν * A * = ν * λ and A T * ν =λν. Appropriately pre-multiplying and post-multiplying the Lyapunov equation (53) by ν * and ν respectively; consequently, gives with the existence of the rank conditions rank B m B * = rank B m and rank C m C * = rank C m , wherē Since rank B m B * = rank B m and rank C m C * = rank C m , the relationships B * =B mKm and C = L mCm holds: By partitioningB m = B m1 B m2 ,C m = C m1Cm2 and substitutingB r * =B m1Km ,C r * = L mCm1 , respectively, yields: If {Ā r * ,B m1 ,C m1 } is the ROM attained after reduction of the large-scale original transformed system {A * ,B m ,C m }. Then, Therefore, Theorem 2: The following error-bound expression exists: with the existence of the rank conditions rank B mh B * = rank B mh and rank C mh C * = rank C mh , wherē The proof of above-mentioned Theorem 2 is similar to the proof of Theorem 1; hence, omitted for the brevity. C m C * = rank C m (which follows from [57]) are satisfied. Remark 13: When X E ≥ 0 and Y E ≥ 0; then, P E = P ex = P m and Q E = Q ex =Q m ; consequently, ROMs obtained by using [30], [40], [39], [43], and suggested technique are the equivalent. Otherwise P E <P m and Q E <Q m .
Furthermore, the frequency-weighted Hankel singular-values satisfy : (λ j [P E Q E ]) 1/2 ≤ (λ j [P mQm ]) 1/2 . Remark 14: When X E ≥ 0 and Y E ≥ 0; then, ROMs obtained using Enns [30] and suggested framework are the equivalent.  (53)-(54) causes difficulty in computing the ROM based on Gramians of sampleddata models for smaller sampling periods. The numerical results are distorted by errors up to a particular limit for the sampling step. To get over this limitation, an "approximately" balanced realization of the sampled-data system is obtained straight from its continuous-time counterpart's balanced realization. When the sample time is reduced to zero, this realization comes "near" to be exactly balanced for "extremely small" (i.e., δ[T ] = T 2 − T 1 = ι) sample steps (i.e., considerably less than the systems' time constants), where T is sampling time, and ι is a very small number. Similarly, the error based on the Hankel singular values (i.e.,ρ j ) and frequency response error will be the same. It's also worth noting that the bilinear mapping (i.e., z −→ (1 + s)/(1 − s)) produces a balanced continuous-time equivalent system if the original discrete-time approach was similarly balanced [29]. Theorem 3: The following Lyapunov equation for the suggested framework holds: Proof of Theorem 3: Using (39), (41), (57) and (58) we have the following: where matricesB (ext) andC (ext) are obtained by (57 − 40) and (58 − 42), respectively.
substitute (67 and 53) in (63) and (68 and 54) in (64) we have the following: Remark 18: Note that by applying stability robustness theorem [59] to the frequency weighted model reduction problem, the combine weighted systems is stable if the following inequalities hold (see chapter 3 of [31] for more detail) The above inequalities also provide the criteria for the choice of weightings (i.e., input weightings and output weightings).

B. 2-D FREQUENCY WEIGHTED MODEL REDUCTION TECHNIQUE FOR DISCRETE-TIME SYSTEMS
Let the controllability Gramians P ia and the observability Gramians Q oa for the corresponding input-augmented (25) and the output-augmented (26) realization, respectively, be given as: that are the solution of the following Lyapunov equations: Truncating (3,3) and (1, 1) block of (65) and (66), respectively, we have the following Lyapunov equations: where The stability is ensured for 2-D discrete-time system by making the input X ϵ2 = B ϵ2 ϵ 2 (69) associated matrices positive and positive semi definite. The fictitious matrices B mϵ 2 andC mϵ 2 are obtained by improvising B ϵ2 = U ϵ2 S 1/2 ϵ2 for r n ≥ 0 (72) Remark 19: When the following rank conditions holds: then, the following relationship holds for the fictitious input and the fictitious output matrices. whereK Remark 20: It can be seen in [40] that (73) Remark 21: Assumptions rank[B 1 B 2 ] = rank [B 2 ] and rank C 1 C 2 = rank [C 1 ] will always be satisfied for B 2 and C 1 be full column rank and row rank, respectively. Using (75), (76), (79) and (80), we can derive new matrices B mϵ 1 andC mϵ 2 as follows: then, Theorem 4: The following rank conditions are always hold: Theorem 5: The realization {A,B mϵ ,C mϵ , D} is minimal, stable, and separable denominator.

Proof of Theorem 5:
The proof of above Theorem 5 follows from the minimality, stability, and separability of the 2-D discrete-time system realization {A, B, C, D}.
The minimal rank-decomposition of new realization {A,B mϵ ,C mϵ , D} subject to A 3 = 0 can be written as: Remark 22: The equation (86) can be solvable forD mϵ if f one of the following equivalent conditions holds [60]: 2) There exist some matrices Y ϵ and Z ϵ such that D = L mϵ 1 Y ϵ and D = Z ϵKmϵ 2 . Remark 23: The requirements for the existence of (86) for strictly proper original systems is immediately met. This requirement will be met when the full row rankL mϵ 1 and the full column rankK mϵ 2 is exist. We notice that even by settingD mϵ = 0 we can get rid of this assumption. Remark 24: The realizations {A 1 ,B mϵ 1 * ,C mϵ 1 ,D mϵ 1 * } and {A 4 ,B mϵ 2 ,C mϵ 2 * ,D mϵ 2 * } are minimal and stable.

VI. NUMERICAL SIMULATIONS
To highlight the comparison of existing frequency weighted models ( [30], [37], [46]), a numerical example of a multiinput multi-output doubly-fed induction generator (DFIG) based variable-speed wind turbine (double-cage induction generator) for the power system (current model) is presented in Example-1. Furthermore, the 2-D discrete-time system is demonstrated in Example-2. Figs. 3, 4, and 7 depicted the frequency response error for the entire frequency-weights of the approximated model obtained by using existing ( [30], [37], [46]) and suggested frameworks. In addition, Figs. 5 and 6 depict the original 2-D model, and ROMs acquired using the existing and suggested methods, in the specified frequency-weights, of the ROMs acquired through the use of different existing ( [30], [42]) and suggested techniques. 1. Induction Generator Parameters: Base voltage = 690V , Base power = 2M W , Angular velocity = 2πf m , f m = 50Hz, Stator resistance = 0.00488 p.u., Doublecage reactance = 0.0453 p.u." Stator leakage reactance = 0.09241 p.u., Rotor resistance = 0.00549 p.u., Rotor leakage reactance = 0.09955 p.u., Rotor to doublecage mutual reactance = 0.02 p.u., Magnetizing reactance = 3.95279 p.u., Load inertia constant = 3.5, Double-cage resistance = 0.2696 p.u.. 2. DFIG Control Parameters: Speed limit=1800 r/min, Cut-in speed = 1000 r/min, Shutdown Speed=2000 r/min. Example 1: Consider a stable LTI 6 th order DFIG model (current model) as given in [61], the discretized sampling time is T s = 0.001sec, with the following input weights and the output weights: The frequency-response error comparison is given in Fig. 3 and 4 of 2 nd and 3 rd order ROMs, respectively. The pole locations of existing ( [30], [40], [39], [43]) and proposed techniques are provided in Table. 2, it can also be observed that [30] produces unstable 2 nd and 3 rd order ROMs along  with the pole locations at z = −1.12469 ± 1.5327i and z = 1.12133, 1.001579 ± 1.002044i, respectively. However, in the given frequency-weights, proposed techniques produce low frequency-response truncation error with stable ROMs comparable to existing stability-preserving algorithms ( [40], [39], [43]). Example 2: Consider a 6 th order stable 2-D discrete-time system [37]: with the desired frequency-weights as given in [42]. Fig. 5 and 6 show the stable 2-D original and ROMs obtained using the existing [42] and proposed techniques, respectively. The frequency-response error comparison in the desired frequency-weights is given in Fig. 7. The pole locations of [30] and proposed techniques are provided in Table. 2, it can also be observed that [30] produce unstable 3 rd dimension ROMs along with the pole locations at z 1 = 1.00889, posed method to linear time-variant, descriptor, and bilinear systems. This method can also be used to analyze continuous systems in the time domain. Moreover, the proposed methodology may be expended for other variants of 2-D systems, such as positive 2-D continuous delayed systems based on L 1 -gain control design.
ACKNOWLEDGMENT "Authors would like to thank the National University of Sciences & Technology (NUST) for supporting this research work."