Frequency Weighted Model Order Reduction Technique and Error Bounds for Discrete Time Systems

Model reduction is a process of approximating higher order original models by comparatively lower order models with reasonable accuracy in order to provide ease in design, modeling and simulation for large complex systems. Generally, model reduction techniques approximate the higher order systems for whole frequency range. However, certain applications (like controller reduction) require frequency weighted approximation, which introduce the concept of using frequency weights inmodel reduction techniques. Limitations of some existing frequency weighted model reduction techniques include lack of stability of reduced order models (for two sided weighting case) and frequency response error bounds. A new frequency weighted technique for balanced model reduction for discrete time systems is proposed. The proposed technique guarantees stable reduced order models even for the case when two sided weightings are present. Efficient technique for frequency weighted Gramians is also proposed. Results are compared with other existing frequency weighted model reduction techniques for discrete time systems. Moreover, the proposed technique yields frequency response error bounds.


Introduction
Model reduction has played a significant role in modern control system design and caught a lot of attention in the last few decades [1][2][3][4][5].It is desirable that reduced order model preserves the fundamental properties of original system like stability, passivity, and so forth.Moreover, the approximation error between original and reduced order system is required to be small.Balanced truncation [6] is prominent model reduction scheme, which not only ensures stability of reduced order systems but also provides frequency response a priori error bounds.Other schemes like Hankel norm approximation [7], Pade approximation [8], Krylov technique [9,10], linear matrix inequality (LMI) technique [11,12], and so forth, are also useful for model reduction.
Hankel norm approximation has complex implementation and does not preserve steady state [13].Pade and Krylov approximation sometimes provide unstable reduced order models and there exist no global error bounds [14,15].LMI technique is based on mathematical iterative methods (i.e., convex optimization and bisection algorithm); therefore, much computational power is consumed [16].However, LMI technique has been applied on various model reduction problems for different types of systems (including time delay [17], discrete state delay [18,19], switched hybrid [20], and nonlinear stochastic [21]).Applications of model reduction are not only restricted to control engineering but also find utility in medical [22] and text summarization [23] areas, and so forth.Most model reduction algorithms tend to minimize the reduction error over the whole frequency range; however there are certain situations (like controller reduction [24]), wherein the approximation error is more critical over a certain band of frequencies.
Enns [25] has enhanced the balanced truncation [6] to incorporate frequency weights for model reduction of higher order systems.This technique can work with input, output, and two sided weightings.However, in two sided weighting case, this technique may give unstable reduced order models [26].To circumvent instability issue in the presence of two sided weightings, many techniques appear in the literature (including Lin and Chiu [27], Wang et al. [28], Varga and Anderson [29], Ghafoor and Sreeram [30], etc.).
In Lin and Chiu [27] technique weightings are strictly proper.It was improved to include more general proper weightings in [26].Varga and Anderson [29] technique yields proper model for strictly proper systems also.Ghafoor and Sreeram [30] technique is a parameterized technique.Wang et al. 's technique is relatively useful, since other techniques [27,29] are not applicable for controller order reduction.Moreover, Wang et al. 's technique yields easily computable expression for a priori error bounds.
Various partial fraction based techniques appear in the literature [31][32][33][34][35] that works for continuous as well as discrete time systems.Unfortunately, most of these techniques yield large frequency response error [30] as compared to Enns technique.However, [35] incorporates free parameters to reduce the approximation error.
Most frequently, frequency weighted model reduction problem is treated in continuous time; however, there are few papers (like [36][37][38]) which deal with this problem explicitly in discrete time.Sahlan et al. 's [36] technique (discrete time version of [31]) is a modified version of Lin and Chiu [27] technique.Campbell et al. 's [37] technique is discrete time counterpart of Wang et al. 's [28] technique, which not only provides stability for two sided case but also gives easily computable error bounds.Campbell et al. 's [37] technique involves taking absolute values of eigenvalues.This may introduce a large change to the system and hence a larger error if it contains some negative eigenvalues.Varga and Anderson's [38] tend to minimize the distance between Enns and Campbell et al. 's techniques Gramians by eliminating negative eigenvalues.
In this work, a technique is developed for frequency weighted balanced model order reduction for discrete time systems.The large change in eigenvalues is avoided by yielding similar effect on all eigenvalues.The stability is guaranteed even for double sided weighting.The proposed technique provides comparable frequency response error and yields easily computable a priori error bounds.Numerical examples are given to show the usefulness and comparison of proposed technique with the existing frequency weighted balanced reduction techniques.

Preliminaries
In this section we review some of the existing frequency weighted model reduction techniques for discrete time systems which include Enns [25], Campbell et al. 's [37], and Varga and Anderson's [38].
Consider a stable full order original system with transfer function () = ( − ) −1  + , a stable input weighting system with input transfer function   () =   ( −   ) −1   +   , and a stable output weighting system with output transfer function   () =   ( −   ) −1   +   ; the augmented systems are given by where Let the Gramians satisfy the following Lyapunov equations: Expanding the (1,1) and (2,2) blocks of above equations, we get where 2.1.Enns Technique [25].Let  be contragredient matrix obtained as where   ≥  +1 ,  = 1, 2, 3, . . .,  − 1 and   >  +1 .By transforming and then partitioning the original system, we have where  11 ∈  × .The reduced order system is obtained as follows: Remark 1.It is not guaranteed to ensure that   ≥ 0 and   ≥ 0; the reduced order models obtained using Enns technique may not remain stable for both sided weightings.[37].Campbell et al. 's [37] (a discrete time version of [28]) guarantees the positive semidefiniteness of symmetric matrices   and   to ensure stability.Let the new controllability   and observability   Gramians, respectively, be calculated by solving the following Lyapunov equations:

Campbell et al. 's Technique
which are used to obtain contragredient matrix  as where The reduced order systems are calculated by transforming and then partitioning the original system.
Remark 2. The stability of reduced order models in the presence of both input and output weightings is guaranteed and the following error bound holds [37]       ( where [38].Note that the Gramians satisfy   ≤   and   ≤   .For minimizing the distances between the Gramians,   −   and   −   , Varga and Anderson proposed the following technique.

Varga and Anderson's Technique
Let new controllability and observability Gramians   and   , respectively, be calculated as the solutions to Lyapunov equations which are used obtain contragredient matrix  as where Σ = diag{ 1 , where The reduced order systems are calculated by transforming and then partitioning the original system.Remark 3. The stability of the reduced system is guaranteed and the following error bound holds [38] where and

Main Results
In Campbell et al. 's [37] technique, the symmetric matrices   and   are guaranteed positive/semipositive definite by taking the square root of absolute values of the eigenvalues obtained by eigen decomposition of symmetric matrices   and   .This may lead to a large change in some eigenvalues and may not affect other eigenvalues.Although in Varga and Anderson's [38] technique, this large change was slightly improved by eliminating negative eigenvalues, but the problem persists with the other eigenvalues.In the following, a new technique is proposed in which a similar effect on all eigenvalues of indefinite matrices   and   guarantees stability, error bound, and improved frequency response error.

Proposed Technique.
Let a new controllability   and observability   Gramians, respectively, be calculated by solving the following Lyapunov equations: The matrices   and   are the new fictitious input and output matrices, respectively, and are defined as The terms   ,   ,   , and   are calculated by the orthogonal eigen decomposition of symmetric matrices   =        and   =        , where Let a contragradient transformation matrix  be obtained as where Σ = diag{ 1 ,  2 ,  3 , . . .,   }, and   ≥  +1 ,  = 1, 2, . . .,  − 1,   >  +1 .The reduced order system is calculated by transforming and partitioning the original system.

Theorem 5. The following error bound for the proposed technique holds if the rank conditions rank [𝐵
where Proof.We show proof of (i) (whereas (ii) and (iii) are special cases of (i) Remark 7.For the case when input   () weights are co-inner and output   () weights are inner [40], then  =   =   =   =   and  =   =   =   =   , where  and  are unweighted Gramians defined as Remark 8.For the case when the symmetric matrices   ≥ 0 and   ≥ 0, the reduced order models obtained using Enns [25], Campbell et al. 's [37], Varga and Anderson's [38], and proposed technique are same.

Computational Aspects.
The frequency weighted balanced truncation model reduction techniques balance the original system and then truncate the balanced realization to get the desired reduced order system.The balancing procedure involves computation of transformation matrix from controllability and observability Gramians.Sometimes these matrices become numerically low rank especially in large scale systems (possibly) due to rapid decay of their eigenvalues [3].Due to this reason, balancing procedure becomes inefficient.Accuracy enhancing techniques for different frequency weighted model reduction techniques appear in [4,29].
For unweighted case, Hammarling's technique [41] is used to obtained Cholesky factors of Gramian matrices from original system realization without actually computing controllability and observability Gramian matrices, respectively.
In frequency weighted techniques, Cholesky factors of the Gramian matrices are obtained from the augmented system realizations.Let  and  be the Cholesky factors of the augmented system Gramians matrices   and   of ( 4) and ( 5 By making use of Cholesky factors  and  calculated above, the Cholesky factors corresponding to Gramians in frequency weighted model reduction techniques like Enns [25], Campbell et al. 's [37], Varga and Anderson's [38], and proposed technique can be obtained as follows: (1) Enns Technique.In the following we establish a relationship between Cholesky factors of Gramian matrices used in Enns and proposed techniques.Equations ( 18) and ( 19) can be expressed as Since   = ( −   ) 1/2 ( −   )

Illustrative Examples
with the following weightings The first order reduced model obtained by Enns [25] technique is unstable while reduced order model obtained by with the following output weighting: Table 3 gives the comparison of error and error bounds for reduced order systems obtained by Enns, Campbell et al. 's Varga and Anderson's and proposed techniques for output weighting.Note that, the proposed technique compares well and yields relatively lower error as compared to other techniques.

Conclusion
A new frequency weighted technique for model reduction of discrete time systems is explored.The reduced order models obtained in the presence of input, output, and two sided weightings are stable.A comparison with existing schemes shows that proposed technique provides comparable results (mostly produces lower error) for reduced order models.

Table 1 :
Frequency weighted errors and error bounds comparison for reduced order models. =   +   =   22  22 +   12  12 +      Remark 9. Note that, Cholesky factors for Enns and proposed technique are computed directly from augmented system realization using Hammarling technique without calculating augmented system realization Gramian matrices   and   .

Table 1
Campbell et al. 's, Varga and Anderson's,and proposed techniques is stable yielding frequency response errors 112.9338, 100.8739, and 94.116, respectively.Note that, proposed technique provides stability with relatively lower error when compared to other techniques.gives the comparison of error and error bounds for reduced order systems obtained by Enns, Campbell et al. 's, Varga and Anderson's, and proposed techniques for the input and two sided weighting cases.Note that, the proposed technique mostly yields lower error as compared to other techniques.

Table 2
gives the comparison of error and error bounds for reduced order systems obtained by Enns, Campbell et al. 's Varga and Anderson's and proposed techniques for the input weighting case.Note that, the proposed technique compares well and yields relatively lower error as compared to other techniques.