Applications of Block Pulse Response Circulant Matrix and its Singular Value Decomposition to MIMO Control and Identification

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1 58 International Journal Kwang of Control, Soon Automation, Lee and Wangyun and Systems, Won vol. 5, no. 5, pp , October 27 Applications of Bloc Pulse Response Circulant Matrix and its Singular Value Decomposition to MIMO Control and Identification Kwang Soon Lee and Wangyun Won Abstract: Properties and potential applications of e bloc pulse response circulant matrix (PRCM) and its singular value decomposition (SVD) are investigated in relation to MIMO control and identification. he SVD of e PRCM is found to provide complete directional as well as frequency decomposition of a MIMO system in a real matrix form. hree examples were considered: design of MIMO FIR controller, design of robust reduced-order model predictive controller, and input design for MIMO identification. he examples manifested e effectiveness and usefulness of e PRCM in e design of MIMO control and identification. irculant matrix, SVD, MIMO control, identification. Keywords: Circulant matrix, identification, MIMO control, SVD.. IRODUCIO In recent years, e state space model-based approaches have dominated e controller design techniques. he transfer function-based meods are still important, however, in at e transfer function represents e complete frequency characteristics of a linear system, enabling us to view and analyze e system from a different angle. Despite its advantage, usefulness of e frequency domain approach is confined to SISO problems in many cases and is not easily extended to MIMO problems. It is primarily because a MIMO transfer function is represented by a matrix of complex functions and e techniques to analyze and interpret complex matrices are not as convenient as ose for real matrices. Additionally, e transfer function itself does not provide e directional information alough bo e frequency and directional characteristics are inherent in MIMO systems. As an alternative representation of e MIMO transfer function at can overcome e problems mentioned above, e bloc pulse response circulant matrix (PRCM) [,2] and its properties are introduced and investigated. he PRCM is a matrix at maps an input sequence to an output sequence, bo of which are periodic over an interval. It is composed of e Manuscript received October, 26; accepted June 25, 27. Recommended by Editorial Board member Jietae Lee under e direction of Editor ae-woong Yoon. he auors would lie to acnowledge e financial support from e Korea Research Foundation rough e Second Stage Brain Korea Program. Kwang Soon Lee and Wangyun Won are wi e Department of Chemical and Biomolecular Engineering, Sogang University, -Shinsoo-dong, Mapo-gu, Seoul 2-742, Korea ( s: {slee, wwy83}@sogang.ac.r). finite impulse response (FIR) coefficients and has one-to-one correspondence to an FIR model. he PRCM has been used in e filter design in signal processing [3,4]. he related concepts have also been introduced to controller design [5,6]. he applications have been restricted to SISO problems and MIMO extensions have not yet been made as far as e auors have surveyed. his paper shows at e singular value decomposition (SVD) of e bloc PRCM gives e directional and frequency decomposition of a MIMO system in a real matrix form, and is property can be utilized in control and identification of MIMO systems. As potential applications of bloc PRCM and its SVD, ree design problems are considered: MIMO feedbac control, reduced-order robust model predictive control (MPC), and input excitation signal for MIMO identification. Effectiveness of e PRCM is demonstrated rough numerical examples for e respective applications. In e subsequent sections, R,C, and e superscript * will be used to represent e spaces of real and complex numbers and e conjugate transpose of a complex matrix, respectively. 2. PULSE RESPOSE CIRCULA MARIX 2.. Definition and basic properties Consider an n u -input/n y -output asymptotically stable MIMO discrete-time system described by an impulse response model ˆ yz ( ) = Hz ( ) uz ( ), () ˆ ( ) =, ny nu. (2) = H z h z h R

2 Applications of Bloc Pulse Response Circulant Matrix and its Singular Value Decomposition to MIMO Control 59 Let e SVD of ˆ ( ) He be ˆ ˆ ˆ ˆ He ( ) = W() D() V () (3) n n n n wi ˆ y y (), ˆ n () u nu, ˆ y u W C V C D() R. he singular value matrix D ˆ () has nonnegative real elements in e principal diagonal, which are e directional gains of e system at frequency. he input and output singular matrices, V ˆ() and W ˆ (), are unitary. he above SVD provides e directional decomposition of a frequency response model, but e analytic SVD of ˆ ( j He ) is hard to obtain. What we can do best is e numerical decomposition at each. For asymptotically stable H ˆ, (2) can be approximated by e following finite impulse response (FIR) model for some : H( z ) = h z. (4) = Associated wi (4), we consider e following bloc PRCM H at represents e map between a sequence of -periodic input and a sequence of - periodic output: Y = HU, (5) Y y ( ) y ( + ), U u ( ) u ( + ), (6) H h h h h h h 2 2 h h h R ny nu In e followings, MIMO bloc PRCM will be simply called PRCM ver ere is no confusion. Consider e discrete Fourier transform (DF) matrix: m F d d d d d d d d d m 2π C. m m, (7) d e I ; ; I m denotes e m identity matrix. he DF matrix is symmetric and unitary, i.e., F = F and F F = FF = I. hen e following holds for e PRCM: Lemma : For a SISO PRCM, e eigenvalues and e normalized eigenvectors are He ( ) and e column vectors of F for,,, respectively. For a bloc PRCM, e same holds bloc-wise. he lemma is well nown for e SISO case [,2]. Proof for e MIMO case is given in e appendix. According to e lemma, H can be blocdiagonalized by e similarity transform wi F such at Y = HU, (8) Y FY, U FU, H FHF = bd H( e j ), (9) and bd denotes e bloc-diagonal matrix. It can be seen at H corresponds to e FIR model H at e frequencies,,. he following is essential in e analysis and synesis of a MIMO controller using e PRCM: Lemma 2: Any series, parallel, and/or feedbac interconnection of PRCM s is a PRCM. Conversely, if a series, parallel, and/or feedbac interconnection of some matrix A and PRCM s is a PRCM, A is also a PRCM. Proof is given in e appendix SVD of PRCM Let e SVD of PRCM H be represented as H = WDV, () ny nu ny ny nu nu W R, V R, and D R denote orogonal output and input singular matrices, and e singular value matrix, respectively. SISO Case: Let n = n =. o investigate e u properties of e matrices in (), we tae e DF on bo sides of (). y H = ( FWD ) ( FV) () Bo F W and F V are unitary and D is real and diagonal, and (7) represents e SVD of H. From e fact at H is diagonal wi complex elements, D = diag He ( j ), i.e., e diagonal elements of D are e amplitude ratios of e FIR model at each rearranged in order of descending magnitude according to e definition of e SVD. More details on e SVD are summarized in e following lemma: Lemma 3: Consider e SVD of a SISO PRCM in (). It is assumed at e SVD is rearranged in order of increasing for notational simplicity. Let w

3 5 Kwang Soon Lee and Wangyun Won and v be e column vectors of W and V, respectively, corresponding to He ( j ). hen D = diag He ( ),,, cos( + φ, ) 2 w cos( ( ) + φ, ) sin( + φ, ) 2 w (2) sin( ( ) + φ, ) cos( + φ2, ) 2 v cos( ( ) + φ2, ) sin( + φ2, ) 2 v sin( ( ) + φ2, ) φ arg ( He ) = φ, φ2, represents e phase angle. he SVD is not unique since φ, and φ 2, can be arbitrary wiin φ = φ, φ2,. Proof is given in e appendix. he above tells at e SVD of a SISO PRCM provides e complete information on e amplitude ratio and phase angle in a real matrix form. MIMO Case: he result of e SISO case can be extended to e MIMO case wi additional complexity. He ( ) for a MIMO system is an ny n u complex matrix and H is a bloc-diagonal matrix. Let e SVD of He ( ) be He ( ) = W( ) D( ) V( ), (3) ny ny nu nu W( ) C, V( ) C, and D ( ) ny nu. R he above represents e directional decomposition of He ( ). From (9), (), and (3), we have H = WDV = F HF = F bd[ W( )]bd[ D( )]bd[ V( )] F. (4) = W = D = V In e above, mbd consists of e directionally decomposed frequency gains. Expansion of (4) and rearrangement leads to e following lemma: Lemma 4: Consider e SVD of a MIMO PRCM as in (). he SVD is assumed to be rearranged in order of increasing. Let w and v be e bloc column vectors of W and V, respectively, and d be e corresponding diagonal bloc of D. hen using e SVD in (3), we have [ D ] e W( ) D= bd ( ) w d v + w d v 2 = Re ( ) e W( ) ( ) D( ) e V( ) e V( ). (5) Proof is given in e appendix. In bo e SISO and MIMO cases, column vectors of e input and output singular matrices are sampled sinusoids of different frequencies. herefore, e frequency associated wi each singular value can be identified by inspection. 3. APPLICAIOS 3.. Design of FIR controller Consider a MIMO discrete-time stable process yz ( ) = Gzuz ( ) ( ). (6) We want to design a feedbac regulator uz ( ) = Kz ( )( rz ( ) yz ( )), (7) such at e closed-loop transfer function is Gcl ( z ), i.e., yz ( ) = ( I+ GzKz ( ) ( )) GzKzrz ( ) ( ) ( ) (8) = G ( z) r( z). cl It is assumed at Gcl ( z ) is given so at K( z ) is asymptotically stable. Let G and G cl be e PRCM representations of Gz ( ) and Gcl ( z ), respectively, and K be e matrix representation of yet undetermined K( z ). he problem in (8) can be converted to ( I+ GK) GK = G. (9) cl Under e assumption at I G cl is invertible, K is obtained as + cl cl K = G G ( I G ), (2) + denotes a pseudo-inverse. From Lemma 2, K is a PRCM, and FIR controller can be constructed wi e first column bloc of K Design of robust MPC When a system is identified experimentally, e

4 Applications of Bloc Pulse Response Circulant Matrix and its Singular Value Decomposition to MIMO Control 5 high gain modes are easily excited and precisely identified as e low gain modes are not. Consequently, large error is usually incurred in e estimates of e low gain modes, and a model-based controller may result in poor performance or instability of e closed loop. One way to avoid is problem is to bloc e signals associated wi e low gain modes. For is, it is necessary to now e directions and frequency range over which e process gains are significantly small. We apply e above idea to e design of robust MPC. Let Δ U() t denote e future input moves determined at t. In e standard MPC meod, e dimension of Δ U() t is high and e required computation is heavy [7,8]. We address e problem of computation reduction togeer wi robustness enhancement against model uncertainty, and e following input blocing is proposed on e basis of e SVD of e PRCM. Let H be e PRCM of e process model. Assume at e SVD of H is given as Fig.. Magnitude ratios of diagonal elements of process and nominal model transfer function matrices. D H = W W V V, D D. (2) D2 In e above, e partition associated wi D 2 may include large model error. If e control input is restricted to e span of V, e low gain part will not be excited and e potential instability of e closed-loop can be avoided. he input movement is defined as Δ U() t = Vc(). t (22) (a) Regular MPC. MPC determines c ( t), whose dimension is smaller an at of Δ U (). t Example : he process and e nominal model are zero-order hold equivalents of e following transfer function matrices wi sampling interval of 2: ( )( ) 3 G pro s s+ s + s+ + () s =, s 2 + (s+ )(5 s + s+ ) nom s+ 3s+ G () s = s+ s+ (23) Bo e prediction and control horizons were chosen to be 9, and bo input and output weighting matrices were given as I. From Fig., it can be seen at significant error exists in e nominal model beyond =.rad/sec. Because of is, regular MPC (b) Reduced-order MPC. Fig. 2. Responses of regular and reduced-order MPC (solid line: y, dashed line: y 2 ). renders e closed-loop unstable as shown in Fig. 2(a). o improve e robustness, e SVD was conducted on e 8 8 PRCM formed wi ninety 2 2 pulse response coefficient matrices of e nominal model, and e first 2 column vectors of V was

5 52 Kwang Soon Lee and Wangyun Won selected as V, which roughly covers e frequency up to.2 rad/sec. he performance of e robust reduced-order MPC by input blocing is shown in Fig. 2(b). It can be seen at e low gain modes are effectively suppressed and e instability is removed leading to smooly converging responses to e set point changes Input design for MIMO identification he input design to excite all latent system modes wi prescribed S/ ratios has been a constant research subject in system identification [9,]. We propose a novel approach to is, e SVD of e PRCM is repeatedly applied to e estimated model. Consider e following system wi a PRCM representation: Y = HU+ E= WDV U+ E, (24) E is assumed to be a zero-mean output noise wi uniform variance over all modes. In order to estimate all e principal gains wi uniform accuracy, it is necessary to design e input so at e S/ ratio of e output for each principal direction is uniformly large. his can be achieved by e following input signal: nu α v = d U = W E (25) for some large α > d s and v s denote e singular values and e input singular vectors, respectively. In reality, accurate H is not available in advance, hence U may not be designed as intended. Instead, we can tae e following iterative steps for sequential improvement of e input signal: Step : Apply a uniformly distributed random signal to all inputs and identify e system. Let n =. Step 2: Let e PRCM of e estimated model be H ˆ n. ae e SVD of H ˆ n. Step 3: Construct e input signal as in (25) using e SVD of Ĥn for e next identification experiment. Step 4: Repeat steps 2 and 3 wi increasing n until e estimates of d s converge. In e first run, only e high gain modes will be identified accurately. he input directions for e low gains will be estimated inaccurately as a result, but e span of e directions will be correct because it is given to be orogonal to e high gain input directions. he low gain modes can be appropriately excited and be identified in e next run. Repeated application of (25) wi v and d, which are replaced by e estimates from e previous run, will progressively achieve balanced excitation of all modes. Fig. 3. Singular values of PRCM s. α can be given differently wi if it is desired to concentrate e input energy over a certain frequency range and/or input directions. Koung and MacGregor [] have proposed a similar idea to e above, but e meod was limited to e identification of e steady state gain. Example 2: he process to identify is e zeroorder hold equivalent of G pro ( s ) in (23) wi sampling period of 2. It is assumed at independent zero-mean Gaussian noises wi variance of.5 are imposed on each output. In e first experiment, independent PRBS s wi amplitude.5 was applied to each input for,5 sampling times and a model was identified using a subspace technique called 4SID [2]. In e subsequent experiments, e proposed meod was applied wi α = 2 using PRCM s wi = 5. In Fig. 3, e singular values of e PRCMs for e true process and e estimated model are compared. One can see e singular value estimates converge to e true values in just two runs. 4. COCLUSIOS It has been shown at e bloc PRCM is a useful alternative representation of a MIMO dynamic system and at its SVD can play as a powerful instrument in e design of MIMO control and identification. he PRCM s can replace e asymptotically stable transfer function matrices wi one-to-one correspondence. his property enables us to construct a MIMO FIR controller rough a simple matrix algebra. It was also proved at e SVD of a PRCM provides e complete directional and frequency decomposition of e PRCM in a real matrix form. Meods to design a robust controller and input signals for MIMO identification were considered as exemplary applications of is property.

6 Applications of Bloc Pulse Response Circulant Matrix and its Singular Value Decomposition to MIMO Control 53 APPEDIX A In e subsequent proofs, e ordinal numbers are counted from zero. Proof of Lemma : Let f be e column bloc of F. ( ) f = I e I e I (A.) What we have to show is at H f = f ( ), He (A.2) denotes e complex conjugate. Since m ( m e e ) e p bloc of H f is obtained as p e Hf ( = h p + he ( p ( p+ ) ( ) + h e + hp+ e p = e h + he ( ) + h e p m e I he i m= p = = e H( e ), (A.3) e p denotes a matrix at pics e p bloc. he above tells at f is e bloc eigenvector of H and e corresponding bloc eigenvalue is He ( j ). Proof of Lemma 2: We prove e lemma using an exemplary matrix equation Q= ( I+ HK) HK. First, let H and K be PRCM s. DF of e matrix equation yields FQF = F( I+ HK) HKF = ( I+ FHF FKF ) FHF FKF = ( I + HK) HK. (A.4) Since H and K are bloc-diagonal, Q= FQF is also bloc-diagonal, which implies at Q is a PRCM. Conversely, if Q and H are PRCM s, DF shows at FKF is bloc-diagonal, which means K is a PRCM. Proof of Lemmas 3 and 4: Let f be e column bloc of F. It holds at f = and f He ( ) = He ( ) for. Wiout loss of generality, is assumed to be an odd number. From (9)-() and (3), we have F HF H ( e H = WDV = = f ) f ( ) / 2 = = H() + H( e ) f + f = j ( ) / 2 ( f f ) ( f f ) He ( ) = H() + H( e ) f + f H( e ) = ( ) / 2 = ( ) / 2 = = H() + 2 Re f H( e ) f = H() + 2 Re f W( ) D( ) V( ) f. (A.5) For e SISO case, ( j He ) = re φ and D ( ) j = r. e φ can be split between W ( ) and V ( ) wi any ratio. Let jφ, jφ2, W( ) = e, V( ) = e (A.6) wi φ = φ φ2. hen,, f W D V f r j( nr+ φ, ) j( nc+ φ2, ) Re col row e e r cos(( nr ) ( nc 2 )) φ, φ, r cos( nr φ)cos( nc φ2 ),, + sin( nr + φ, )sin( nc + φ2, ) Re ( ) ( ) ( ) = = + + = + + cos( + φ, ) r = cos( ( ) φ ) +, cos( + φ2, ) cos( ( ) + φ2, ) sin( + φ, ) r + sin( ( ) φ, ) + sin( + φ2, ) sin( ( ) + φ2, ), (A.7) nr, nc,, and [ ] represents a matrix. If we define W =, w w V = v v, D= diag[ d] (A.8)

7 54 Kwang Soon Lee and Wangyun Won en we have H = d w v. (A.9) = Comparing (A.9) wi (A.5) after inserting (A.7) gives (2). For e MIMO case, we rearrange (A.5) using e following expression e W( ) Re f W( ) D( ) V( ) f = Re D ( ) ( ) e W( ) ( ) e V( ) e V( ) (A.) and e fact at e and e column blocs of W and V are associated wi f ( ) W and f ( ), V respectively. his results in (5) and proves Lemma 4. REFERECES [] R. M. Gray, oeplitz and circulant matrices: A review, available at /gray/toeplitz.pdf, 25. [2] R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, ew Yor, 985. [3] V. C. Liu and P. P. Vaidyanaan, Circulant and sew-circulant matrices as new normal-form realization of IIR digital filter, IEEE rans. on Circuits and Systems, vol. 35, no. 6, pp , 988. [4] J. Bae, J. Chun,. Jeong, B. Gu, and S.. Kim, Design of FIR/IIR lattice filter using e circulant matrix factorization, Proc. of Oceans MS/IEEE Conf. and Exhibition, vol., pp , 2. [5] L. Wang and W. R. Cluett, Frequency sampling filters: An improved model structure for step response identification, Automatica, vol. 33, no. 5, pp , 997. [6] L. Wang and W. R. Cluett, From Plant Data to Process Control: Ideas for Process Identification and PID Design, aylor and Francis, London, Research Monograph, 2. [7] M. Morari and J. H. Lee, Model predictive control - Past, present and future, Computers and Chemical Engineering, vol. 23, pp , 999. [8] S. J. Qin and. A. Badgwell, A survey of industrial model predictive control technology, Control Engineering Practice, vol., no. 7, pp , 23. [9] G. C. Goodwin and R. L. Payne, Dynamic System Identification: Experimental Design and Data Analysis, Academic Press, ew Yor, 977. [] L. Ljung, System Identification: heory for e User, 2nd ed., Prentice Hall, Upper Saddle River, 999. [] C. W. Koung and J. F. MacGregor, Robust identification for multivariable control: he design of experiments, Automatica, vol. 3, no., pp , 994. [2] P. V. Overschee and B. D. Moor, 4SID: Subspace algorims for e identification of combined deterministic-stochastic system, Automatica, vol. 3, no., pp , 994. Kwang Soon Lee was born in Seoul, Korea in 955. He obtained e B.S. degree in Chemical Engineering from Seoul ational University in 977, and e Ph.D. degree in Chemical Engineering from KAIS in 983 in e area of process control. Since 983, he has been wi e Department of Chemical and Biomolecular Engineering at Sogang University, he is currently a Professor. He has held visiting appointment at e University of Waterloo in Ontario, Canada, in 986 and also at Auburn University in Alabama, USA, in 995. His research has covered e meodologies of batch and repetitive process control, model predictive control, and system identification, and eir applications to various chemical processes, semiconductor manufacturing processes, and power plants. Wangyun Won was born in Seoul, Korea in 983. He obtained e B.S. and M.S. degrees in Chemical Engineering from Sogang University, Seoul, Korea in 25 and 27, respectively. His research interests are in e areas of system identification, model predictive control, and process optimization.