Design of an Industrial Distributed Controller Near Spatial Domain Boundaries

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1 Design of an Inustrial Distribute Controller Near Spatial Domain Bounaries Stevo Mijanovic, Greg E. Stewart, Guy A. Dumont, an Michael S. Davies ABSTRACT This paper proposes moifications to an inustrial paper machine cross-irectional (CD) controller, initially esigne assuming process spatial invariance, near the sheet eges where the assumption of spatial invariance is clearly violate. The resulting esign problem is equivalent to a blockecentralize static output feeback problem. The propose approach sequentially synthesizes each block, resulting in an internally stable loop with nominal performance an robustness criteria satisfie. I. INTRODUCTION The assumption of spatial-invariance is necessary for the application of the many new tools [], [], [5], [] evelope for the analysis an controller synthesis of spatially istribute control systems. With this assumption, controller synthesis can be significantly simplifie. However, when the controllers are implemente on real systems with a spatial omain of finite with, the actual spatial bounary conitions (BC) are invoke [3]. Because of the iealize BC assume in the esign process, there is no guarantee of performance or even stability aroun the bounaries. A class of boune, spatially istribute systems with its bounary conitions, for which stability an performance are guarantee after implementing a controller esigne assuming spatial invariance of the process is presente in [7]. An inustrial example of a spatially istribute, large scale, multivariable process is the paper machine crossirectional (CD) control system. A paper machine prouces a sheet of paper up to m wie. The objective of CD control is to reuce the variations of the paper sheet properties of interest (which inclue weight, moisture content, an thickness) in the cross-irection (the irection perpenicular to sheet travel) as efficiently as possible [3], [4], []. Most often, each one of these properties is controlle with a separate controller using 3 3 ientically constructe an evenly istribute actuators in the crossirection. The scanning sensor is mounte ownstream from the actuator array an measures the sheet properties at up to locations across the sheet. The CD component of S. Mijanovic, G.A. Dumont, an M.S. Davies are with the Department of Electrical an Computer Engineering, University of British Columbia, 356 Main Mall, Vancouver, B.C., Canaa, V6T Z4 stevom,guy,mike@ece.ubc.ca G.E. Stewart is with Honeywell Process Solutions, 5 brooksbank Avenue, North Vancouver, B.C., Canaa, V7J 3S4 greg.stewart@honeywell.com the measurement profile is subsequently spatially low-pass filtere an ownsample to the actuator resolution. Inustrial CD controllers, of interest in this paper, are essentially D (spatio-temporal) filters, causal in time an noncausal in the spatial omain [5], [9], []. In particular, we will consier the case in which these systems have been tune using a two-imensional loop shaping technique [], []. The spatial invariance approximation is central to this technique. The two-imensional loop shaping technique has been successfully introuce to the inustry, an to ate has been implemente on more than paper machines worlwie. However, as the process characteristics are ifferent aroun the eges from those in the rest of the sheet, the initially compute spatially-invariant CD controllers are moifie near the sheet eges before implementation on a paper machine [9]. The main contribution of this paper is a novel approach to the reesign of CD controllers to accommoate spatial omain bounaries, where the initial controller has been compute assuming iealize perioic bounary conitions. The objective is state in terms of a static output feeback (SOF) esign problem via appropriately efine linear fractional transformations (LFTs). The subsequent esign approach is base on a novel low-banwith SOF controller synthesis algorithm. The algorithm is sequentially implemente on two constant matrix components in the existing inustrial controller, moifying the CD control law near spatial omain bounaries by irectly taking into account the relevant control engineering criteria: closeloop stability, performance, an robustness. The objective of this work was to prepare for an inustrial trial, that has subsequently been complete [8]. The main characteristics of the paper machine CD control systems an the objective of this work are given in Section II. The new approach to CD control near spatial omain bounaries is etaile in Section III. An example is presente in Section IV an conclusions in Section V. II. PROBLEM STATEMENT A. Paper Machine CD Control The stanar moel of a paper machine CD control system, subject to process output isturbances, is given by: y(z) = G(z) u(z) + (z), u(z) = K(z) y(z) () This is one for convenience as we will implement the propose esign on a controller tune in this way. The propose esign only requires the original controller to be stabilizing.

2 where y(z),u(z) C n (C - set of complex numbers) are the Z-transforms of the output (measurement) profile an the input (actuator) profile respectively, an (z) C n is the Z-transform of the process output isturbance. The objective of the CD controller K(z) C n n is rejection of isturbances (z). The number of actuators 3 n 3 as state above. The process moel s transfer matrix G(z) C n n can be written as: G(z) = [I Az ] Bz () where the constant matrices A an B R n n have a Toeplitz symmetric structure : A = toeplitz{a,,,...,} B = toeplitz{b,b,...,b lb,,...,} (3) The coefficients moelling the spatial response [b,,b lb ], the iscrete-time pole a, an the process moel eatime are ientifie from input-output paper machine ata, e.g. using commercial software escribe in [6]. This structure an the use of a ban-iagonal Toeplitz matrix B in () is stanar in the moelling of CD processes [4]. The structure of the inustrial CD controller of interest in this paper, K(z) in (), is given by: K(z) = [I Dz ] D C c(z) (4) with real matrices C an D R n n, an scalar transfer function c(z). The scalar transfer function c(z) is a stable eatime compensator known as the Dahlin controller in the process inustries []. The two-imensional loop shaping controller tuning technique, presente in [], is base on approximating the process moel as spatially-invariant by imposing spatially perioic bounary conitions. This amounts to approximating the process Toeplitz symmetric matrices A an B in () (3), with the corresponing circulant symmetric matrices A cs,b cs R n n : A cs = toeplitz{a,,,...,} B cs = toeplitz{b,b,...,b lb,,...,,b lb,...,b } (5) Such an approximation, physically speaking, is equivalent to assuming that the paper machine prouces a tube rather than a sheet of paper. As a result of the assume process spatial invariance, an optimal controller will also be spatiallyinvariant []: C cs = toeplitz{c,c,...,c lc,,...,,c lc,...,c } D cs = toeplitz{,,..., l,,...,, l,..., } (6) where the coefficients [c,c,...,c lc ] an [,,..., l ] are etermine by the controller tuning, an specify the number of off-iagonal elements of the matrices C an D respectively. Denote using MATLAB notation. However, the physical reality is that the controller must be implemente on a process that is more accurately moelle by the Toeplitz matrices given in (3). As a result, the initially compute spatially-invariant controller K(z), given with (4) an (6), is moifie near the spatial omain bounaries before it is implemente in the inustrial control system. The current inustrial practice esigns these moifications base on signal processing techniques for extening finite with signals: zero, average, an reflection paing [9]. Subsequently, the same control law is implemente near the bounaries as in the mile of the sheet. Unfortunately, such an a hoc moification of the CD control law near the eges oes not take into account close-loop characteristics of the resulting control system, an can lea to seriously egrae performance (even instability) near spatial bounaries [9]. B. Objective of this work Let us efine the moifications to the existing controller matrices in (4) in terms of aitive matrix perturbations δc,δd R n n in Fig.. The elements of δc an δd are given by, δc ij, i n C an n n C + i n [δc] ij = j n C an n n C + j n, otherwise (7) δ ij, i n D an n n D + i n [δd] ij = j n D an n n D + j n, otherwise (8) with l c n C n/, l n D n/, an n C,n D n. However, most often n C,n C,n D,n D n, resulting in only the upperleft an lower-right corners of the matrices δc an δd being ifferent from zero. The parameters l c an l are the respective withs of the matrix bans of C an D in (6). y C δc C u C c(z) y D δd D z - u D u G (z) P(z) Fig.. CD control system with control law moifications, δc an δd, near the sheet eges. By factoring out controller perturbations δc an δd as shown in Fig., a lower linear fractional transformation (LFT) as illustrate in Fig., can be efine. The generalize plant P(z) in Fig. consists of the close-loop transfer functions that can be obtaine, after some straightforwar algebra, from the system shown in Fig.. As pointe out in Section II-A, the moifications δc an δd currently use in inustrial CD control systems o not take into account relevant control engineering criteria y

3 Fig.. (LFT). u C u D P(z) δd δc E y D y C w=[y u] T Problem reformulate in terms of linear fractional transformation an can lea to very poor control near spatial omain bounaries. The objective of this work is to fin a block-iagonal compensator E in Fig., such that: () δd,δc R n n are static matrices with n C,n C,n D,n D n in (7) an (8). () The resulting close-loop system is stable. (3) The close-loop performance of the system, as measure by the -norm of the process output vector at low frequencies, is improve: y(e jω,e) < y(e jω,), ω < ω b, (9) for some ω b >. (4) The gain M(e jω,e) : u is limite across all the frequencies, i.e. for a given weight W(e jω ) > : σ(m(e jω,e)) < W(e jω ), ω, () where σ( ) enotes the maximum singular value. The first requirement above is the consequence of the main objective of this work: esigning a localize moification of the existing control law near the sheet eges without changing the controller structure. The nee for the secon requirement is obvious. The thir requirement is in accorance with the main objective of CD control: the reuction of process output variations as measure by its vector -norm. The fourth requirement is a result of the esire preservation of the system robustness characteristics 3. III. SEQUENTIAL DESIGN OF SOF COMPENSATORS In this work the esign of δd an δc in Fig. will be performe sequentially as it is very ifficult to esign a static output feeback (SOF) compensator E with an aitional block-iagonal structure constraint 4. A novel low-banwith SOF controller esign algorithm, outline in Section III-A, is use for computing CD controller moifications in Section III-B. 3 In the two-imensional loop shaping proceure, the process moel uncertainty is moelle as aitive uncertainty, an subsequently K[I GK] is a measure of system robustness. 4 A wie variety of SOF problems are analyze in [3] an references therein. A. SOF controller synthesis In this work, we are concerne with a stable, finiteimensional generalize plant N(z): N (z) N (z) N(z) = N (z) N (z), () N 3 (z) N 3 (z) Fig. 3. illustrate in Fig. 3. The signal (z) represents the ex u N(z) K y w a w b Diagram of the lower linear fractional transformation F l (N, K). ogenous input, w a (z) an w b (z) represent the exogenous outputs, y(z) represents the feeback signal, an u(z) represents the control signal. Let the generalize plants N a (z) an N b (z) be efine as 5, [ ] [ ] N N N a = N N, N N 3 N b =, () 3 N 3 N 3 then the input-output transfer functions, (z) w a (z) an (z) w b (z), are given by lower linear fractional transformations 5 : F l (N a,k) = N + N K(I N 3 K) N 3 F l (N b,k) = N + N K(I N 3 K) N 3 (3) The objective is to esign a compensator such that: the controller K(z) = K is a static matrix, the feeback system with K(z) = K is stable, (c) the compensator improves the close-loop performance as measure by the Frobenius norm, F l (N a (e jω ),K ) F < F l (N a (e jω ),) F, (4) ω < ω b, for some ω b >, () the performance at higher frequencies is not overly egrae. In other wors, a constraint, F l (N b (e jω ),K ) <, ω (5) is satisfie. It can be seen that the above requirements completely correspon to the requirements () () in Section II-B. Also, the requirement (c) above is closely linke to the requirement (3), as the Frobenius norm is relate to the sum of all singular values [], an y irectly epens on the singular values of the corresponing close-loop transfer function y. Since N(z) in () is stable, the internal stability of the close-loop system in Fig. 3 is equivalent to the inputoutput stability of K(z)(I N 3 (z)k(z)) in (3). We can 5 Transfer functions argument (z) is omitte for brevity.

4 then write own the familiar parametrization of stabilizing controllers K(z) for the feeback system in Fig. 3, K(z) = Q(z)(I + N 3 (z)q(z)) (6) for stable Q(z) (see for example [4]), leaing to the convenient form of the LFTs in (3), F l (N a (z),k(z)) = N (z) + N (z)q(z)n 3 (z) (7) F l (N b (z),k(z)) = N (z) + N (z)q(z)n 3 (z) (8) Consier the low-frequency requirement on the Frobenius norm in (4). Using (7) we can write the LFT at steaystate (ω = ), F l (N a (e j ),K(e j )) = N (e j )+N (e j ) Q(e j ) N 3 (e j ) (9) Now consier the following optimization problem motivate by (9), Q = arg min Q R J(N a(e j ),ρ,q) J = N (e j ) + N (e j ) Q N 3 (e j ) F + ρ Q F () A close-form solution to this optimization problem for a real static matrix Q is given in [8]. Subsequently, the resulting Q from () is use to form the static controller K (requirement above), K = Q (I + N 3 (e j )Q ) () The first term in optimization () is intene to aress the above specifie performance requirement (4), while the secon term is intene to limit the magnitue of the synthesize matrix Q. The conitions on ρ an generalize plant N(z), such that the stability conition an the ynamical conition () above are satisfie, are etermine by Theorem below. The close-loop performance improvement (4), with the CD controller esigne using the above outline algorithm is guarantee by Theorem. Theorem : (Stability an Full Banwith Performance Limit) If N(z) in () is stable, σ(f l (N b (e jω ),)) < for all ω, an ρ > β in () where β = r r 3 σ(n (e j )) σ(n 3 (e j )) σ(n (e j )) { N3(z) N 3(e j ) + N } (z) N 3 (z) N (z) with the integers r an r 3 enoting the rank of N (e j ) an N 3 (e j ), respectively. Then K synthesize from () () stabilizes the feeback system in Fig. 3 an σ(f l (N b (e jω ),K )) <, for all ω (3) where σ( ) enotes the maximum singular value. Proof: Given in [8]. Theorem : (Low Frequency Performance Improvement) If N(z) in () is stable, then for any K constructe from ()-() that stabilizes the system in Fig. 3, there exists a frequency ω b > such that Fl (N a (e jω ),K ) F < Fl (N a (e jω ),) F (4) () for all ω < ω b. Proof: Given in [8]. The value for the weight ρ in () is etermine through bisection on ρ to prouce a K in () such that the requirements () are successfully trae off, an is initialize with the value compute base on Theorem. Subsequently, the bisection continues as long as the stability an full banwith requirements are satisfie, an until the ifference between the two consecutive values of the weight ρ in () is smaller than some specifie value of the tolerance. B. Computing CD controller moifications The first step in the propose CD controller moification is a replacement of the initially compute spatiallyinvariant controller matrices C cs an D cs in (6) with their corresponing Toeplitz symmetric matrices. Next, the final controller moifications are compute using the SOF algorithm in turn on the δd an δc matrices in Fig. an, presente in Section III-A. Since the SOF algorithm requires stable systems, the close-loop stability of the system with Toeplitz symmetric process an controller moels, is assume. Base on numerous inustrial ata, as well as simulation stuies, this is not a restrictive assumption an is only violate in certain pathological examples. However, if the system with Toeplitz symmetric process an controller moels is not stable, then a stabilization proceure is require. Such a stabilization proceure is presente in [8]. Fig. 4. W u P(z) W 6 y W u c W 7 y c W 3 y u W 4 W 5 P e (z) Isolating system inputs/outputs near one ege. Since the two eges of the paper machine are moelle to be ientical, it is enough to retune the controller at one ege only. Subsequently, the corresponing moifications at the other ege can easily be foun by symmetry arguments. Factoring out the control system inputs an outputs near one ege, base on the close-loop system in Fig., is performe with rectangular weights W i, i =,,...,7, as illustrate in Fig. 4. The weighting matrices W i are efine

5 as: W = [I nd nd (n n }{{ D)], W 3 = [I nc nc (n n } C)], }{{} n D n n C n W 4 = [I ny ny (n n y)], }{{} W 5 = [I nu nu (n n u)], }{{} W = n y n n [ ] [ u n ] I n I, W 6 = nd, (n n ) n }{{ (n nd) n }}{{ D } W 7 = n n n n [ ] D I nc (n nc) n }{{ C } n n C (5) The matrices W,W 3,W 6,W 7 are use to convert the matrix sub-block esign into a full-block esign problem (D e = W T 6 δd W T, C e = W T 7 δd W T 3 ), i.e., [D e ] ij =δ ij, i n D an j n D, [C e ] ij =δc ij, i n C an j n C, (6) where δc ij an δ ij are the same as those elements in (7) (8). In other wors, C e R nc nc an D e R nd nd are the non-zero, upper-left elements of the matrices δc an δd in (7) (8), respectively. The matrices W,W 4,W 5, on the other han, are use to isolate the sheet eges for consieration in the performance esign. The orer of the transfer matrix P e (z) in Fig. 4 can easily reach into the thousans for practical systems. However, most of these states can be eliminate using the orer reuction proceure base on Hankel singular values as they are mainly relate to the inputs/outputs locate in the mile of the sheet an the other machine ege. Base on the iagram given in Fig. 4, the linear fractional transformations (LFT s) for computing moifications C e an D e in (6) can be efine. They are presente in Fig. 5a an 5b. The esign of C e an D e is performe by alternately synthesizing one matrix component while holing the other fixe. The synthesis proceure is the SOF metho of Section III-A. P e (z) De P C (z) P e (z) Ce P D (z) computer), an a harware-in-the-loop simulator (resiing on another computer). The CD control system, presente below, escribes an array of n = 36 slice lip actuators being use to control the paper sheet basis weight profile 6. The parameters of the process moel in () (3) were ientifie using software escribe in [6] with the size of the matrix B in (3) ban l b = 6 an {b,...,b 6 } = 3 {.65,.44,.789,.38,.69,.9,.} a =.855, = (7) The feeback controller K(z) in (4) was esigne using the stanar two-imensional loop shaping technique [], []. The obtaine controller parameters in (6) ha matrix ban sizes l c = 4,l =, with: {c,,c 4 }={.89,.9,.487,.856,.48} {, }={.9878,.6} (8) The parameters of the Dahlin controller c(z) in (4) are also prouce by the two-imensional loop shaping esign, but as not being central to this work, they are omitte here for brevity. Subsequently, the initially compute circulantsymmetric controller matrices are replace with the corresponing Toeplitz symmetric matrices. After confirming stability of the system with the process an controller Toeplitz symmetric matrices, the proceure for moifying CD control near the eges, presente in Section III, can be implemente. The close-loop simulations have been performe with the steay-state process output isturbance, (z) in (), as shown in Fig. 6. Near one ege (left sie), the isturbance has a significant high spatial frequency content, an near the other (right sie), has a smoother appearance. The first type of isturbance very often leas to system instabilities in real life CD systems as the actuator array attempts to remove the uncontrollable (high frequency) moes of the process output isturbance. The secon type of isturbance (introuce near the right ege) is usually successfully attenuate by the control systems. The close-loop simulation results [gsm] Ce De Cross irection Fig. 5. Linear fractional transformations for sequentially computing C e an D e moifications. IV. EXAMPLE Simulation stuies presente in this section were carrie out using the real inustrial ientification an controller tuning software etaile in [6], [] (resiing on one Fig. 6. Process output isturbance (at zero temporal frequency ω = ). are shown in Fig. 7, an summarize in Table I. Fig. 7 illustrates the close-loop steay-state process output an actuator profile using reflection paing (one of the techniques currently use in inustry). Simulation results, with the controller tune in turn conservatively, balance, 6 More etails about the system can be foun in [8].

6 an aggressively using the new approach from Section III, are given in Fig. 8. In all three cases, the sizes of the rectangular weights W i,i =,...,7 in (5) were chosen as, n C = 5,n C = 8,n D = 3,n D = 8,n u = 8,n = 8,n y = 8, an the output vectors w a (z) an w b (z) in Fig. 3 as, w a (z) = [k P y(z) u(z)] T, w b (z) = +k R u(z), where y(z) an u(z) are the process output an control signal respectively, an k P an k R are the tuning variables. For the CD process moel given with (7), σ(g(e j )) = 3.56, where σ( ) enotes the maximum singular value. The tuning variables, in the case of conservative tuning were chosen as: k P = 3, k R =., balance tuning: k P = 6, k R =., an in the case of aggressive tuning: k P = 4, k R = Cross irection Fig. 8. Steay-state process output an control signal, using the new technique - conservative tuning Cross irection Fig. 7. Steay-state process output an control signal, using the current inustrial technique - reflection paing Cross irection Fig. 9. Steay-state process output an control signal, using the new technique - balance tuning. It can be seen from Fig. 8 an Table I that, by using the approach presente in Section III, a successful trae-off between performance (a flat CD profile) an the corresponing control signal magnitue, can be achieve. From Table I, it can be notice that in case of all three tunings base on the new approach, the output profile has been improve in comparison to the result achieve with the existing technique. Also, in the cases of conservative an balance tunings, such a result has been achieve with less actuator usage. Current metho New approach (three ifferent tunings) Conservative Balance Aggressive y u TABLE I -NORM OF THE PROCESS OUTPUT y AND CONTROL SIGNAL u STEADY-STATE PROFILES SHOWN IN FIG. 7. V. CONCLUSIONS In this paper, a new approach to paper machine CD control near spatial omain bounaries has been presente. The new technique irectly takes into account relevant control Cross irection Fig.. Steay-state process output an control signal, using the new technique - aggressive tuning. engineering criteria: close-loop stability, performance, an robustness. Cross-irectional controllers are often esigne uner the assumption of spatial invariance, which is clearly violate near spatial omain bounaries (paper machine eges). As a result, such controllers often result in an excessively large control signal originating from the spatial bounaries that can even lea to control system instability. The new

7 approach moifies the CD control law near the paper sheet eges by sequentially applying a low-banwith static output feeback esign on two matrix components of the existing inustrial controller, without changing the controller structure or complexity. An example was presente of poor control near the eges, using the currently implemente technique in inustry. It was illustrate that by implementing the newly evelope technique, the results were substantially improve. With the new technique, it is possible to achieve the esire level of aggressiveness of CD controller near spatial omain bounaries. The new technique has also been successfully teste on a paper machine in a working paper mill [8]. ACKNOWLEDGEMENTS The financial support of the first author s work by the Natural Sciences an Engineering Research Council of Canaa, the Science Council of British Columbia, an Honeywell Process Solutions - North Vancouver is gratefully acknowlege. REFERENCES [] B. Bamieh, F. Paganini, an M. Dahleh. Distribute control of spatially invariant systems. IEEE Trans. on Automatic Control, 47(7):9 7, July. [] R. D Anrea an G.E. Dulleru. Distribute control esign for spatially interconnecte systems. IEEE Trans. on Automatic Control, 48(9): , September 3. [3] S.R. Duncan. The cross-irectional control of web forming processes. PhD thesis, University of Lonon, UK, 989. [4] A.P. Featherstone, J.G. VanAntwerp, an R.D. Braatz. Ientification an Control of Sheet an Film Processes. Springer-Verlag, Lonon,. [5] D.M. Gorinevsky, S. Boy, an G. Stein. Optimization-base tuning of low-banwith control in spatially istribute systems. In Proc. of American Control Conference, pages , Denver, Colorao, USA, June 3. [6] D.M. Gorinevsky an C. Gheorghe. Ientification tool for crossirectional processes. IEEE Trans. on Control Systems Technology, (5):69 64, September 3. [7] C. Langbort an R. D Anrea. Imposing bounary conitions for a class of spatially-interconnecte systems. In Proc. of American Control Conference, pages 7, Denver, Colorao, USA, June 3. [8] S. Mijanovic. Paper Machine Cross-Directional Control Near Spatial Domain Bounaries. PhD thesis, University of British Columbia, BC, Canaa, 4 ( [9] S. Mijanovic, G.E. Stewart, G.A. Dumont, an M.S. Davies. Stability-preserving moification of paper machine cross-irectional control near spatial omain bounaries. In Proc. of IEEE Conference on Decision an Control, pages 43 49, Las Vegas, Nevaa, USA, December. [] S. Skogesta an I. Postlethwaite. Multivariable feeback control: analysis an esign. Wiley, New York, 996. [] G.E. Stewart, D.M. Gorinevsky, an G.A. Dumont. Feeback controller esign for a spatially-istribute system: The paper machine problem. IEEE Trans. on Control Systems Technology, (5):6 68, September 3. [] G.E. Stewart, D.M. Gorinevsky, an G.A. Dumont. Two-imensional loop shaping. Automatica, 39(5):779 79, May 3. [3] V.L. Syrmos, C.T. Aballah, P. Dorato, an K. Grigoriais. Static output feeback - a survey. Automatica, 33():5 37, February 997. [4] K. Zhou, J.C. Doyle, an K. Glover. Robust an Optimal Control. Prentice Hall, New Jersey, 996.