Convergence Analysis of Terminal ILC in the z Domain

Transcription

1 25 American Control Conference June 8-, 25 Portlan, OR, USA WeA63 Convergence Analysis of erminal LC in the Domain Guy Gauthier, an Benoit Boulet, Member, EEE Abstract his aer shows how we can aly -transform theory to analye the convergence of a terminal LC algorithm his aroach uses an equivalent system viewe in the cycle omain an analyes it with a -transform hen, conventional iscrete time control is alie to the equivalent system his control is viewe by the real system as a cycle-tocycle control herefore, the stability analysis of the controlle equivalent system corresons to convergence analysis use in LC Furthermore, a ea beat convergence is feasible an corresons to the fastest convergence rate of the LC algorithm NRODUCON HE iea of iterative learning control (LC) is to use the nowlege of revious outut error measurement to uate the inut so as to reuce the error Many aers have been written about LC; see the survey aer by Moore [] he terminal LC control (also calle oint-tooint LC) is an aroach whose goal is to reuce the error at the en of the cycle n rai thermal rocessing, terminal LC hels to reuce thicness error [2-6] n our roject we want to aly terminal LC to the reheat hase of the thermoforming rocess o now more about thermoforming, refer to [7] n most wors [2-5] the behavior of the terminal LC is analye via a classic convergence analysis, in the sense of the evolution of the norm of the error n this aer, we will use a new aroach base on an equivalent system built in the cycle omain, from the system in the time omain A close-loo analysis is one in the -omain with a controller connecte to the equivalent system his control aears for the system in the time omain as a cycle-to-cycle control he stability analysis one on the close-loo equivalent system corresons to a convergence analysis one with the corresoning cycle-to-cycle control n Section, we efine the system analye with our Manuscrit submitte for review on Setember 3 24 Guy Gauthier is with École e echnologie Suérieure, Montréal, Québec, Canaa (corresoning author to rovie hone: ; fax: ; guygauthier@ etsmtlca) Benoit Boulet is with McGill University, Montréal, Québec, Canaa ( boulet@cimmcgillca) aroach Section consiers terminal LC control for the SSO version of the system efine in Section Section V oes the same for the MMO case Section V gives simulation results obtaine with terminal LC esigne using the analysis one in Sections an V Finally, Section V conclues an suggests some irections for future wor with this aroach DEFNNG SYSEM O CONROL WH ERMNAL LC n this aer, we aly terminal LC control to a linear iscretie system reresente by: x( th) Ax() t Bu () t () y () t Cx () t where t, h, 2 h,, N h, h is the samling erio, N+ is the number of samles er cycle, an the subscrit is the cycle number Matrices A, B, an C are time n invariant he state vector is x () t, the inut vector is m u () t, an the outut vector is y () t Here, we assume that the number of inuts is the same as the number of oututs, so m he control tas is to uate the control inut u () t after cycle - such that the terminal outut y ( Nh) converges to a given terminal value y at time Nh From linear system theory, one can write the solution of () at t Nh : N N N j j x ( Nh) A x () A Bu ( j) (2) From this terminal state we calculate the corresoning terminal outut as: N N N j y ( Nh) CA x () C A Bu ( j) (3) j For the articular case of a thermoforming rocess, we ee the control inut (heater temerature setoint) constant uring a cycle So in this case, we can efine /5/$25 25 AACC 84

2 u( ) u( t) : u( ),, Nh an rewrite (3) as: y ( ) x ( ) u( ), (4) where y( ) y( Nh), x ( ) x () an the constant matrices an are efine as: N CA : (5) an: N N j : C A B j he change in notation is to emhasie the fact that for the cycle-to-cycle control cycle, can be looe at as a iscrete ste so that the system (4) is equivalent to iscrete time system hen, we will aly in cycle control on the terminal outut (4) that will aear lie cycle-to-cycle control to the system () he following assumtions are mae for this aer: A) Reetition of the initial state is satisfie hen x () x ( ) must be at the same value for all cycles A2) here exists a unique inut u () t u( ) such that the system exhibits the outut y( ) y( Nh) his forces the matrix to be of full ran Hence, the system must be comletely observable an controllable LC CONROL OF AN SSO SYSEM he terminal outut (4) can be controlle by any in cycle control n this section, we assume that the corresoning system () is SSO We analye this SSO system controlle by roortional cycle-to-cycle control an by integral cycle-to-cycle control A Proortional cycle-to-cycle control of an SSO system he roortional cycle-to-cycle controller for () is efine as: u ( ) e ( ) y y( ), (7) where is the roortional gain of the controller We can use -transform theory to analye the behavior of the controlle system hen, the -transform of (4) can be written as: yˆ ˆ ( ) u( ) xˆ (8) an the -transform of (7) is: u ˆ( ) yˆ ( ) yˆ ( ) where u ˆ( ), yˆ ( ), ˆ y( ) an xˆ ( ) are the -transforms of u ( ), y( ), y( ) an x ( ), resectively (6) (9) he close-loo transfer function is obtaine next: yˆ ( ) yˆ ( ) yˆ ( ) x ( ) () ˆ Rearranging the terms in (), one can write: yˆ y ( ) x ( ) ˆ () Proosition : Assume a constant initial state vector x ( ) an let the esire terminal value be y With the constraint that, the roortional cycle-tocycle control will converge to a terminal value of: y ( ) (2) as iff: (3) Proof: he -transform of x( ) an y are: xˆ ( ), (4) an: yˆ ( ) (5) By alying the final value theorem, one can write: y ( ) lim ( ) lim ˆ y y( ) (6) he stability of the close-loo system (corresoning to the convergence of the cycle-to-cycle control as given in (2)) eens on the root of the characteristic equation: (7) From -transform theory, we now that the close-loo system is stable iff the root : P is strictly insie the unit circle Since, we fin that for, we must have B ntegral cycle-to-cycle control of an SSO system Suose we try to control the terminal outut (4) with an integral control law exresse in the -omain as: u ˆ( ) ˆ y( ) yˆ ( ), (8) where is the integral gain of the controller 85

3 f we exress this control law in the cycle omain, we can write it as: u ( ) u ( ) y y( ) (9) u ( ) e ( ) One can see in (9), the usual LC control law, name integral tye LC (-LC) On some aers about terminal LC [2-5], a convergence analysis is erforme Here, we analye the stability in the -omain he two analyses are equivalent (when we consier LC control of orer one in [2-5]), therefore stability in the -omain imlies convergence of the terminal -LC algorithm Combining (8) an (8), we can get the close-loo transfer function: yˆ ( ) ˆ y y( ) xˆ ( ) (2) Rearranging all the terms, we have: yˆ ˆ ( ) y ( ) (2) xˆ ( ) Proosition 2: Assume a constant initial state vector x ( ) an the esire terminal value y he terminal -LC control will converge to the esire terminal value: ( ) (22) as iff: y 2 (23) Proof: With the -transform of x( ) an y efine in (4) an (5) an by alying the final value theorem, we can write: y ( ) lim ( ) lim ˆ y y( ) (24) Again, the stability of the close-loo system eens on the root of this characteristic equation: (25) hen, the close-loo root : must be strictly insie the unit circle to ensure stability (convergence) So to have, the gain must satisfy 2 outut to the esire one if the integral gain is selecte roerly When the gain is set to, we have the root of the characteristic equation equal to his is nown as the ea beat resonse in iscrete time control For that articular gain, the convergence of the terminal -LC algorithm is obtaine in only one cycle hat is the fastest rate of convergence that the -LC algorithm can achieve Proosition 3: Assume a constant initial state vector x ( ) an the esire terminal value y Assume also a gain he terminal -LC control will converge to the esire terminal value in one cycle Proof: With the selecte gain we can rewrite (2) as: yˆ ( ) ˆ ( ) y xˆ ( ) (26) Using the inverse -transform, we can write: y ( ) y ( ) x ( ) x ( ) (27) So at the secon cycle, we have irectly the esire terminal outut as y() y() an we stay on it for all subsequent cycle n ractice, the nowlege of is aroximate an then, using an LC algorithm is useful herefore, it is ifficult to obtain the ea beat resonse f nowlege of is erfect (an for a nown constant initial state vector x ( ) ), one can calculate irectly the inut without using LC, so it becomes useless V EXENSON O MMO SYSEMS n the revious section, we analye the close-loo behavior of (4) using roortional an integral control Since the integral control was shown as the more effective aroach in the SSO case, we will use only this control aroach on MMO systems Here, we assume that the number of inuts is equal to the number of oututs he integral control law for MMO systems can be efine as: y ˆ( ) G ˆ ˆ LC( ) y( ) y( ) (28) K ˆ LC y y( ) where K LC is a ositive efinite iagonal matrix an: As one can see, we have convergence of the terminal 86

4 G LC ( ) K LC (29) From (8) an (28), one can write this MMO close-loo equation: yˆ ( ) ˆ KLC y y( ) xˆ ( ) (3) which we can simlify to: yˆ ( ) ˆ K LC K LC y ( ) (3) K xˆ ( ) LC Proosition 4: Assume a constant initial state vector x ( ) an the esire terminal outut vector y he MMO terminal -LC control will converge to a terminal value of: y ( ) (32) as if all roots of et KLC are such that: j, j,, m (33) Proof: Following the roof of Proosition 2, by alying the final value theorem, we can write: y ( ) lim ( ) lim ˆ y y( ) (34) Now, the stability of the close-loo system eens on the root of the characteristic equation obtaine by calculating: et KLC (35) Because we assume that the number of inuts equals the number of oututs, the orer of the characteristic equation will be equal to the number of inuts an oututs For stability (convergence), the roots, 2,, m must lie strictly insie the unit circle, then j, j,, m Case : iagonal hat imlies the comlete ecouling of each inut/outut ynamic n that case, we have: m et K LC jjj (36) where, j,, m K LC j j are the elements of the iagonal of hen finally each gain j must be such that: j 2 jj (37) to ensure stability (convergence) Case 2: triangular n that case, we have the same result as case, since the eterminant of a iagonal matrix is the rouct of all element of the iagonal So we can fin the gains using (37) Case 3: is neither iagonal nor triangular n that case no simlification can be mae an we must calculate the roots of: et K (38) j LC From (38) we can fin gains that ensure that all, j,, m Another way to state the roots conition of (35) is to say that the eigenvalues of K LC are strictly insie a unit circle centere at (,) An that is equivalent to say that: K (39) LC his in of norm inequality aears in the LC literature [-5] he ea beat convergence for MMO systems is equivalent to the ea beat convergence in the SSO case if the matrix is iagonal when the gain matrix K LC is also iagonal For the case of non-iagonal matrix, ea beat convergence is not achievable with a iagonal gain matrix K LC A variant of terminal -LC control can be efine by relaxing the iagonal ositive efinite constraint on K LC An then, a goo choice for the matrix gain is K LC With this choice of matrix gain we can achieve ea beat control an have the MMO system converge in only one cycle Proosition 5: Assume a constant initial state vector x ( ) an the esire terminal outut vector y Assume also we have efine K LC he MMO terminal -LC control will converge to the esire terminal value in only one cycle Proof: Using an aroach similar to the roof of roosition 3, we can rewrite (3) as: yˆ ( ) yˆ ( ) xˆ ( ) (4) since K LC 87

5 Using the inverse -transform, one can write: y ( ) y ( ) x ( ) x ( ) (4) So at the secon cycle (=), we have irectly the esire terminal value at each outut an we stay on it for all subsequent cycles lie in the SSO case On the next section, simulation results will show the effectiveness of the -LC algorithm on MMO system V SMULAON RESULS o show the effectiveness of the control, we will tae as examle the following MMO system obtaine by iscretiing a continuous time system with a samling erio h = s: x( th) x( t) u (42) y() t x() t he initial state of the system is ( ) an the initial inut alie is u We want to reach the esire terminal value y 2 3 at t = s x () o esign a terminal -LC control, we nee to calculate: (43) Case : Design with a iagonal gain matrix K LC n this case, the terminal -LC control will be (in the - omain): u ˆ( ) G ˆ ˆ LC ( ) y ( ) y ( ) (44) yˆ y( ) 2 he gains are ajuste to have the roots of the characteristic equation: 2 ( ) (45) strictly insie the unit circle hen the gains must be in the following ranges 42 an for stability (convergence) Here, we select 25 an 2 33 to have a characteristic equation with two oles at Figure show the convergence of the inuts base on cycle simulation results Convergence is achieve in two cycles an is not ea beat since K LC is iagonal but is not Alie inuts teration Figure : Alie inut from cycle to cycle for Case Case 2: Design with a gain matrix K LC his is the MMO ea beat convergence esign Now the terminal -LC control is: u ˆ( ) G ˆ ˆ LC ( ) y ( ) y ( ) (46) yˆ y( ) herefore, we have: K LC 33 u () u 2 () (47) Figure 2 shows the ea beat convergence of the inuts base on cycle simulation results As one can see, the inuts converge in only one ste Note that, since we have erfect nowlege of the system, we can calculate irectly the otimal inut with: 88

6 * u y x (48) With the value efine earlier one can obtain this otimal * inut vector u As we can see from Figures an 2, the alie inuts have converge to this otimal inut vector SSO an MMO systems his novel aroach reuces convergence analysis of terminal LC to stability analysis on the -omain transfer function built from an equivalent system in cycle omain he simulation results show how effective the controller can be Alie inuts 4 2 u () u 2 () Alie inuts 4 2 u () u 2 () teration Figure 2: Alie inut from cycle to cycle for Case 2 Case 3: Effect of error in the evaluation of We will reeat Case 2, but we assume we have a wrong estimate of Suose we have evaluate: 6 2, (49) 5 but the real system matrix stays as show in (43) hen, the gain matrix K LC will be: K LC 2 (5) an we use this matrix for the cycle-to-cycle control on the real system Figure 3 shows the effect of the error in estimation of he convergence is slower than the two other cases because our value of is not exact he robustness of our aroach has to be evaluate in further wor But certainly if the error on is too large, it is ossible to have a non converging cycle-to-cycle control V CONCLUSON We use an aroach ifferent from the usual one to analye the convergence of terminal -LC algorithm of teration Figure 3: Alie inut from cycle to cycle for Case 3 Future wor will aress the robustness of the roose aroach to the uncertainty in an to changes in initial conitions We will also conuct the same in of analysis with robust control theory alie in the cycle omain to imrove cycle-to-cycle control in the time omain REFERENCES [] KL Moore, terative Learning Control: An Exository Overview, in Alie an Comutational Controls, Signal Processing, an Circuits, vol, no, 998, 5-24 [2] Y Chen, C Wen, High-Orer erminal terative Learning Control with an Alication to a Rai hermal Process for Chemical Vaor Deosition, in terative Learning Control, Convergence, Robustness an Alications, Lectures notes in control an information sciences, vol 248, Sringler-Verlag, 999, 95-4 [3] J-X Xu, Y Chen, H Lee, S Yamamoto, erminal iterative learning control with an alication to RPCVD thicness control, in Automatica, vol 35, 999, [4] Y Chen, J-X Xu, C Wen, A High-orer erminal terative Control Scheme, in Proc of the 36th EEE Conference on Decision an Control, San Diego, CA, December 997, [5] Y Chen, J-X Xu, H Lee, S Yamamoto, An terative Learning Control n Rai hermal Processing, in Proc he ASED nt Conf on Moeling, Simulation an Otimiation (MSO 97), Singaore, August 997, [6] D e Roover, A Emami-Naeini, J L Ebert, R L Kosut, Comman Shaing for MMO Nonlinear Systems using terative Learning Control with Alication to an RP System,, in Proc of the ASME Dynamic Systems an Control Divison-2, November 2, 53-6 [7] J L hrone, Unerstaning thermoforming, Hanser Garner Publications,