Multivariable Ripple-Free Deadbeat Control

Transcription

1 Scholarly Jornal of Mathematics and Compter Science Vol. (), pp. 9-9, October Available online at ISS Scholarly-Jornals Fll Length Research aper Mltivariable Ripple-Free Deadbeat Control Hatem ELAYDI and Fadi ALBASH Electrical Engineering Department, Islamic University of Gaza,.O.BOX 8, Gaza, alestine Compter echnology and Indstrial rofessions Department, University College of Applied Sciences.O.BOX:45, Gaza, alestine Accepted 3 March he design of mltivariable ripple-free deadbeat controllers is a complex task. One approach which has shown promise for solving mltivariable ripple-free deadbeat control (MRFDC) problems is the se of Diophantine eqation parameters. he problem of solving robst mltivariable ripple free deadbeat with time delays has not been solved. his paper proposes a hybrid two degree of freedom controller tilizing the parameterization of Diophantine eqation to bild a mltivariable ripple-free deadbeat control (MRFDC). he important featre of the proposed approach is that it combines the concept of mltivariable inpt with robst ripple- free deadbeat control. Simlation reslts show that the otpt signal tracks the inpt sinsoidal signal in short settling time. Keywords: Single rate, Mltivariable Ripple Free Deadbeat Control (MRFDC), Diophantine eqations. IRODUCIO he stdy of deadbeat control started in 95 s. Deadbeat control makes the otpt of the systems tracks the reference inpt signal in a finite period of time. Deadbeat control achieves exact settling after a finite nmber of discrete sampling instants; however, there may exist ndesirable intersample ripple in the continos response. o obtain a ripple free deadbeat design, a continos internal model with no plant zeros cancellation is a reqirement (Franklm and Emaminaeini, 986 ). Deadbeat control plays a big role in tracking systems since the controller takes the system from any initial state to the origin in a minimm nmber of time steps. he attractive featre for the deadbeat response is that the tracking error goes down to exactly zero in a finite nmber of control steps. However, ripples may appear in the plant otpt between sampling instants. he ripple sorce comes from cancellation of plant zeros by controller poles reslts in controller modes that may be excited by the reference signal (Yamada and Fnahashi, 998). On the other hand, a deadbeat control has three disadvantages in practical field: the robstness isse, the distrbance sorce in the continos plant, and the Corresponding Athor helaydi@igaza.ed.ps efficiency of the tracking performance in the transient response (Sirisena, 985). Soltions to mltivariable and ripple-free deadbeat control had been presented by several researchers. Elaydi and az, 998, presented an Optimal Ripple-Free Deadbeat Controllers for Systems with ime Delays. A ripple free deadbeat controller for a system with time delays was proposed. Matrix parameterization of the Diophantine eqation was the approach sed to solve this problem. However, they didn't tackle the mltivariable problem. Ito,, improved performance of deadbeat servomechanism by means of mltirate inpt control. A state-space approach was designed to solve a mltirate inpt control problem. Mltirate inpt mechanism achieved shorter settling time than single rate control sing the same freqency of sampling. However, mltirate control often exhibited intersample ripple and this problem was solved here. Frthermore, the proposed method garanteed robstness against continos-time model ncertainty and distrbance. However, they didn't deal with varios reference inpt signals and the inpt signal was a step signal only. az, 6. proposed a ripple-free tracking approach with robstness. A hybrid two-degree-of freedom (DOF) controller considering optimization problem and robstness was proposed. his controller was given in terms of the soltion of two Diophantine eqations.

2 Elaydi and Albatsh However, he didn't tackle the mltivariable problem. Salgado and Oyarzn, 7, proposed two objective optimal mltivariable ripple-free deadbeat control. A parameterization of all stabilizing ripple-free deadbeat controller was given. Also, optimized control signal and tracking performance were kept minimal sing qadratic index. However, they didn't tackle the time delay problem and they didn't deal with varios reference inpt signals. Moreover, the inpt signal was jst a step signal and the paper never dealt with the robstness isse. his paper presents new methodology for designing mltivariable ripple free deadbeat controller to solve the tracking of an arbitrary reference signal and deals with the robstness isse. Ripple free deadbeat tracking is formlated based on the soltion of the Diophantine eqation. he ripple free deadbeat tracking formlation is based on az reslts which delt with robstness (az, 6) his paper also takes advantage of Salgado and Oyarzn approach which solved the mltivariable problem (Salgado and Oyarzn, 7). he mltivariable ripple free deadbeat control, based on the soltion of the parametrization of the Diophantine eqation, is proposed. he approach presented in this paper can handle systems with time delays, where the time delay is not an integer mltiple of the sampling time. A discretizing form of the system with the time delay is obtained and sed in the controller design. he inpt signal is not restricted to step bt it can be any arbitrary reference signal. his paper is organized as follows: section discsses the Diophantine Eqation parameterization, section 3 presents the soltion of the ripple-free deadbeat control problem, section 4 introdces the mltivariable ripple free deadbeat control, section 5 shows the proposed approach inclding MALAB simlation, and the last section concldes this paper. Diophantine Eqations arameterization he Diophantine eqation plays an important role in the design and synthesis of controllers in the freqency domain. he Diophantine eqation has an infinite nmber of soltions that all provide an internally stabilizing controller. However, the Diophantine eqation in a polynomial form masks its design freedom. A parameterization of the Diophantine eqation is obtained, allowing simple access to the degrees of freedom. olynomial mltiplication and division is given as matrix mltiplication. he parameterization of the Diophantine eqation is based on obtaining soltion in a matrix form (Elaydi, 998). Methods for solving the Diophantine eqation Given the two polynomials, A(q) and B(q), with B q b q b q b and m n m m ( ) m, and given two more polynomials Q n (q), Q d (q) and C(q) is defined as C ( q) A( q) Q ( q) B ( q) Q ( q) n he polynomial eqation (3) is called the Diophantine eqation. his eqation is linear in terms of the polynomials Q n (q) and Q d (q) for the known polynomials A(q), B(q) and C(q) where A(q) and B(q) are coprime. he coprimeness of A(q) and B(q) garantees the existence of a soltion to the above eqation for any arbitrary C(q). here are several methods for solving the Diophantine eqation sch as: Eclidean algorithm, Sylvester s resltant, Bezot s resltant, and MacDffee s resltant (Elaydi, 998, Kcera, 98). he soltion of the Diophantine eqation, sing the resolving matrix is not compact for optimization prposes. he former parameterization does not lend itself well characterizing control constraint conditions that arise in practice. Since the Diophantine eqation has two polynomial prodcts, the vectorization is a convenient method to express the eqation. Matrix Representation of olynomial rodcts Sppose the polynomials A(q) and B(q) are given sch that ((Elaydi, 998, Fan and Chang, 99) A q a a q a q a q m ( )... m B q b b q b q b q n ( )... n d () (3) he vectorized form of the polynomials is defined as a b r r A M, B M a m b n Indeed, any polynomial may be vectorized this way. his vectorization may be expressed as an operator. he expanded matrix form is also defined as r r A B Ap O, B O r r A B Lemma : m p p n p p p (6) (4) (5) (7) A q a q aq a and a n n ( ) n, () AB For vector cab be expressed as:

3 Scholarly J. Math. Comp. Sci. Figre : Hybrid DOF control configration. r r AB A B B A r n m (8) (where q=z - ) and z is the Z- transform variable along with a tracking filter M(s) (az, 8). roof of Lemma can be fond in (Elaydi, 998). Lemma illstrates the fact that the vectorization of a polynomial prodct may be written in terms of a matrix prodct. In a comparable way, a polynomial division may also be written as a matrix eqation. Assming now that B()=, the polynomial division can be considered now Aq ( ) C q c c q c q Bq ( ) ( )... oting that eqation (9) may also be written as A( q) B ( q) C ( q) (9) () oting that the left hand side of eqation () has at most m+ nonzero term; ths, the right hand side mst also have the same nmber of term. hs, restricting attention to a finite version of the seqence, the approach now considers the first coefficients of C. he trncated version of this can be written as: C ( q) c c q... c q Solving Ripple-Free deadbeat Control roblem System demonstration () Consider the DOF hybrid system as shown in Figre with the control system showing the continos-time plant G(s) and the DOF discrete time controller C(q) he ripple-free deadbeat control problem definition he ripple-free control problem has several goals (Sirisena, 985). he closed-loop system is internally stable. he error of the system e(t)=r(t)-y(t)= for all t s, where s is the nmber of steps to settle. he control signal (K) settles down after a finite nmber of steps. It may be necessary to impose constraints on the transient response that arise from practical implementation isses. o find the control signal for the DOF system, we discretize the reference signal R(s), the plant (s) and the filter M(s) then rearrange the block diagram for the system as shown in Figre. he controller may be realized sing the configration shown in Figre. We note that (q) and (q) may be implemented as FIR filters. A polynomial in q corresponds to a rational fnction in z with all the poles at the origin, where (q=z - ). ow we can find the DOF control law as the following form U ( q) V ( q) D ( q) c V ( q) ( q) R( q) ( q) Y ( q) () From () and (3), we get the transfer fnction of the control signal U(q) as follow: Uq ( ) ( q) R ( q) ( q) Y ( q) (3) (4) Dc ( q)

4 Elaydi and Albatsh Figre : Implementation of the Deadbeat controller. Ripple free deadbeat control problem (RFC)Soltion he controller polynomials are obtained by soltions of the Diophantine eqations (6) ( q) ( q) D ( q) Q ( q) (5) p r ( q) ( q) D ( q) D ( q) (6) p p c Where Q is a polynomial. he soltion of the RFC reqires the soltion of two Diophantine eqations. Any, and D c that satisfy the above Diophantine eqations give a soltion to the RFC. Since the Diophantine eqation has an infinite nmber of soltions, we will seek specific soltions that provide desired response. Or two Diophantine eqations have soltions as follow ( q) ( q) D ( q) ( q) (7) min r Q ( q) Q ( q) ( q) ( q) (8) min p ( q) ( q) D ( q) ( q) (9) min p D ( q) D ( q) ( q) ( q) () c c min p While v and v are arbitrary polynomials of degrees that define the degree of freedom in the design or free parameters. If we applied the controller in (8), we obtain the transfer fnction Y ( q) D c D Rq ( ) D D D C D C by (4) we also have the error signal E ( q) R ( q) Y ( q) ( ) R ( q) ( D Q ) Q r r r Dr () () Which is a polynomial implying that e(k)= for K s =deg( r Q ) Using (6) when p (q) (q)+d p (q)d c (q)=, we have the control signal transfer fnction D C D C D C D U( q) R ( q) D D Mltivariable ripple-free deadbeat control D (3) Salgado and Oyarzn, 7, presented a new method to demonstrate a mltivariable ripple-free deadbeat control. heir stdy was applied to discrete-time, stable, linear and time invariant plant model. A simple parameterization of all stabilizing ripple-free deadbeat controllers of a given order was considered. he free parameter was then optimized in the sense that a qadratic index is kept minimal. he optimality criterion had the advantage of acconting for both tracking performance and magnitde of the control effort. A control strategy leads to settle the tracking error seqence to zero in a minimm nmber of time steps based on pole zero cancellations between controller and plant model. he basic idea behind this approach is in order to avoid any intersample ripple after the settling time; the control seqence mst also reach its steady state in, at most, the same nmber of samples. Dealing with a combined optimality criterion and ensring a MIMO ripple-free deadbeat response were presented in this approach. Assmptions and definitions We consider the plant model G(q) to be a stable p p discrete-time transfer matrix. We will assme that G(q) is represented in right coprime polynomial matrix fraction description as G ( q) ( ) ( ) B q A q (4)

5 Scholarly J. Math. Comp. Sci. 3 Figre 3: Mltivariable Sampled data control loop Figre 4: Continos-time deadbeat response with ripple. where A(q) and B(q) are right coprime polynomial matrices of dimension p p. Coprime polynomial factorizations of transfer matrices can be implemented sing many methods sch as in (Wang and Davison, 973, atel, 98). We assme that G(q) has no zeros on q=, i.e. B() is non-singlar and we frther assme that B()=I. he non-singlarity of B() is a standard condition necessary for being able to track constant reference signals. Given a proper transfer matrix M(q), we define the right degree interactor (RDI) of M(q) as a polynomial matrix E(q) sch that the prodct M(q)E(q) is biproper, i.e. (Silva and Salgado, 5) lim M ( q) E ( q) D q where < det{d} < System demonstration Consider a continos-time plant with transfer fnction Gc(s) which is digitally controlled throgh a zero order sample and hold device with transfer fnction Gho(s), by a linear discrete-time feedback controller with transfer fnction, C(z). We can present sampled data control loop system in the block diagram as shown in Figre 3. ow, we will find the transfer fnction of the sampled data open loop system as follow: G ( q) B ( q) A( q) (6) where B(q) and A(q) are right coprime polynomial matrices. Given that the reference vector signal is p assmed to be step fnction, i.e. r(k)= νµ(k), ν then having a ripple-free deadbeat control loop means that y c (t) satisfies y ( t ), t c (7) Where is called the deadbeat horizon of the control system. A controller will be designed to achieve a ripplefree deadbeat response which provides perfect steady state tracking at D.C. and makes the otpt of the plant to settle in a finite nmber of samples, while avoiding any intersample ripple beyond the deadbeat horizon. Ripple in a deadbeat response arises when the controller cancels the minimm phase zeros of G(q). hose cancelled zeros appear as closed loop poles and generate the intersample response of the continostime otpt. he response of the system can be shown in Figre 4, where it is noticeable that, althogh the sampled otpt settles in one sample, the continostime otpt exhibits considerable ripple. he discrete-

6 Elaydi and Albatsh 4 time model sally contains sampling zeros located in the negative real axis; therefore its cancellation leads to oscillatory modes in the deadbeat response. o avoid the intersample ripple a sfficient condition mst be applied to the control seqence which settles in s samples. his condition is eqivalent to ( k ), k ss where the constant vector the control seqence MIMO ripple-free deadbeat controllers (8) is the steady state vale of he ripple-free condition eqation (8) implies that the control sensitivity mst have the form S ( q) Kqq ( ) (9) where the deadbeat horizon and K(q) is a polynomial matrix with degree sch that S (q) is proper. On the other hand, the tracking error signal mst also settle in a finite horizon, so that the complementary sensitivity fnction q G q S q B q A q K q q ( ) ( ) ( ) ( ) ( ) ( ) (3) mst have all its poles at q - =. his means that K(q) mst be factored as K(q)= A(q)V(q) with V(q) being a polynomial matrix. hs S ( q) A( q) V ( q) q (3) For convenience, we set =n+, where n is the degree of A(q) and Let also V(q) be written as V(q) = E(q)W(q), with E(q) being a RDI of A(q) q, hence making A(q)E(q) q biproper. hese definitions lead to S ( q) A( q) q n E ( q) W ( q) q (3) from where we have that the properness of S (q) depends on the properness of W ( q) q. It is worth noting that the degree of A(q) is always eqal to that of A(q)E(q). he form of (q), S (q) and the mltivariable deadbeat controller C(q) can then be obtained simply as ( q) B ( q) E ( q) W ( q) q S ( q) A( q) E ( q) W ( q) q C ( q) q S ( q) q I ( q) ss (33) (34) (35) C ( q) q S ( q) q I ( q) n (36) Since we also need perfect steady state tracking of constant references, we mst force ()=I, which, sing (33) and the fact that B()=I, implies that W()=E() -. he controller given in (36) is then a general form of a MIMO deadbeat controller for stable plants and constant reference signals. We need to bild the RDI E(q) which will be sed in (36). Using this formlation we have the advantage to provide a nitary E(q) that also satisfies E()=I, which certainly simplifies the condition imposed on W(q) to W()=I. In the seqel, we will always assme that E(q) is a nitary RDI (Silva and Salgado, 5). Moreover, from (34) it is clear that since, then the minimm deadbeat horizon is min = n, that is, the degree of A(q). his is the mltivariate version of the fact that for SISO systems, the minimm deadbeat horizon is given by the plant order. ext lemma gives a characterization of all polynomial matrices W(q) that yield the minimm horizon ripple-free deadbeat controller. Lemma : Consider a stable transfer matrix G(q) and the MIMO deadbeat controller of (34). hen the minimm horizon deadbeat controller is given by: C min A( q) E ( q) ( q) I B ( q) E ( q) (37) W ( q) q I, (36). and it is achieved by choosing in roof of Lemma can be fond in (Salgado and Oyarzn, 7). roposed methodology In this section, a mltivariable ripple free deadbeat control is introdced. A combination between the robst ripple-free deadbeat approach proposed by (az, 6) with the mltivariable ripple-free deadbeat approach proposed by (Salgado and Oyarzn, 7) is proposed for simlation of single rate ripple-free deadbeat control. Aftere that a MALAB code and a block diagram in Simlink to simlate the mltivariable ripple-free deadbeat control were developed. he code for mltivariable consists of several fnctions; one of them is the compting of the Diophantine eqation parameters. o illstrate the proposed controller design procedre in this paper, consider the continos time mltivariable plant model [7] of nd order sch as: Gc ( s) s s s s s s 3 s 7s 6 (38)

7 Scholarly J. Math. Comp. Sci. 5 Figre 5: Overall system sing Simlink. he mltivariable system has two inpts and two otpts with a transfer fnction matrix of x. he state eqation of an LI system in state-variable form is (Soliman and Srinath, 998) 3 4 (4) x ( t ) A x ( t ) B ( t ) y ( t ) C x ( t ) D ( t ) (39) (4) Where s 4 6s 5 (4) An overall system is presented sing MALAB Simlink as shown in Figre 5. he goal of the controller is to track the two inpt sinsodial signals. In Figre 5, we can see how we represent the two inpts by applying the concept in (39 and 4). Also, we divided the for transfer fnction of the plant by applying the concept in (39 and 4). Assme the transfer fnction of the plant is in the form of s 3 8 s 3 (43) 6 s s 3 (44)

8 Elaydi and Albatsh 6 Figre 6: Mltivariable system with two inpts and two otpts. s 4 3 7s 6 (45) he following steps mst be followed to obtain the controller. First step is finding the discrete-time reference signal. he second step is compting the discrete-time model for every plant. We choose a sampling time of. second inline with Salgado and Oyarzn, 7. For mltivariable plant in (38), the discrete-time model for all plants is compted as follow:.988q.8q.3775q.6q.5q.549q.56q.333q Gq ( ).5663q.39q.9q.95q.56q.333q.56q.333q (46) A MALAB code is sed to compte the Diophantine eqation parameters. (q), (q) and D c (q) are sed for finding the controller as shown in (4). hen, the ripplefree deadbeat control for the first plant in the system is obtained. he other ripple- free deadbeat control for the rest of other plant components is obtained in similar manners. Figre 6. Shows the mltivariable system with two inpts and two otpts giving a plant consisting of 4 transfer fnction matrix of size. he inpt consisting of = sin(wt) and = sin(wt). he otpt of the system consists of two parts: the first part combines the otpt of the plant and the otpt of the plant while the second part combines the otpt of the plant 3 and the otpt of the plant 4. From Figre (7) to (), we can see that the otpt signals track the inpt sinsoidal signals in minimm settling time (abot.7 second). Also, we can see that

9 Scholarly J. Math. Comp. Sci. 7 Figre 7: ime response for plant # Figre 8: ime response for plant #

10 Elaydi and Albatsh 8 Figre 9: ime response for plant #3 Figre : ime response for plant #4

11 Scholarly J. Math. Comp. Sci. 9 able : System preformance lant lant lant 3 lant 4 Overshoot 74 % 58 % 57 % 7 % Settling time.79 s.7 s.69 s.7 s Rise time.3 s.34 s.35 s.36 s Steady state error the error signal, e(t), and the control signals, (t), for every plant settles down in finite nmber of steps. Figre 7- show the time response for the plants, - 4. We note that the otpt signal tracks the inpt sinsoidal signal in short settling time. It's clear that the response is a periodic and the time domain performance for the otpt signal is illstrated below: able smmarizes the system performance in terms of the overshoot, settling time, rise time, and steady state error. hs, the reslts show that the otpt was able to track the sinsoidal inpts in a finite nmber of steps while driving the tracking error to zero. he overshoot is at an acceptable level and the settling and rising times are fine. hese reslts show improvement to the reslts obtained by both az, 8, Salgado and Oyarzn, 7. COCLUSIO A new approach shows promise for solving robst mltirate ripple-free deadbeat control (MRFDC) problems sing Diophantine eqation parameterization. his approach is based on the combining both the concept of mltivariabl and robst single rate. his paper proposed a hybrid two degree of freedom controller for the fixedorder constrained optimization problem addressing performance and robstness specifications tilizing the parameters of Diophantine eqation to bild a robst mltirate ripple-free deadbeat control. Simlation reslts showed that the otpt signal tracked the inpt sinsoidal signal in short settling time either in single rate or mltivariable setting. Also the ripple problem which cased by intersample was solved. he time domain specification for the otpt signal, control signal, error signal and the otpt of the filter signal were compted and satisfied the specifications. REFERECES Elaydi, H (998). Complex Controller Design Using Linear Matrix Ineqalities, h.d hesis, ew Mexico State University, USA,. Elaydi, H, az, R (998). Optimal Ripple-Free Deadbeat Controllers for Systems with ime Delays, hiladelphia, ennsylvania, roceedings of the American Control Conference, 7(6):87-4, Fan, C, Chang, F (99). A ovel Approach for Solving Diophantine Eqations, IEEE ransactions on circits and systems, 3(). Franklm, G, Emami-naeini, A (986). Design of Ripple-Free Mltivariable Robst Servomechanisms, IEEE ransactions on atomatic control, 3(7). Ito, H (). Improving performance of deadbeat servomechanism by means of mltirate inpt control, Department of Control Engineering and Science, Kysh Institte of echnology, Japan, Access on //8, /R/pdf/csse pdf. p.4. Kcera, V (98). A Deadbeat Servo roblem, Int. J. Contr., 36: 7-3. atel, R (98). Comptation of Matrix Fraction Descriptions of Linear ime- invariant Systems, IEEE ransactions on atomatic control, 6(). az, R (6). Ripple-free tracking with robstness, Int. J. contr., 79 (6). az, R (8). Compter Controlled Systems, Lectrer notes, ew Mexico state niversity. Salgado, M, Oyarzn, D (7). wo objective optimal mltivariable ripple- free deadbeat control, Department of Electronic Engineering, University ecnica Federico Santa Maria Valparaiso, Chile, Access on /4/9, Silva, E, Salgado, M (5). erformance bonds for feedback control of non- minimm phase MIMO systems with arbitrary delay strctre, IEEE roceedings-control heory and Applications, 5(): -9. Sirisena, H (985). Ripple-Free Deadbeat Control of SISO Discrete Systems, IEEE ransactions on atomatic control, 3(). Soliman S, Srinath M (998). Continos and Discrete Signals and Systems, ew Jersey, rentice-hall Inc,. Wang, S, Davison, E (973). A Minimization Algorithm for the Design of Linear Mltivariable Systems, IEEE ransactions on atomatic control, 8(3). Yamada, M, Fnahashi, Y (998). Mltivariable Deadbeat racking Controllers, roceedings of the 37th IEEE Conference on Decision and Control, ampa, Florida USA.