Real-Time Implementation of Adaptive Optimal Controller Based on a Pseudo-Space Model into PLC

Transcription

1 Proceedings of the th WSEAS International Conference on SYSEMS, Agios Nikolaos, Crete Island, Greece, Jl 3-5, 7 3 Real-ime Implementation of Adaptive Optimal Controller Based on a Psedo-Space Model into PLC PER PIVOKA, VLASIMIL LORENC Department of Atomatic Control and Instrmentation Faclt of Electrical Engineering and Commnication, Brno Universit of echnolog Kolejni 4, 6 Brno CZECH REPUBLIC Abstract: - his paper describes the design and implementation of adaptive linear optimal controller (LQ) based on a psedo-space model into PLC. For identification of the controlled sstem an algorithm based on artificial neral netork is sed. he method of comfortable implementation of algorithms from the MALAB/Simlink environment into PLC is shon. In the first step of implementation the algorithm is designed in the simlation environment. he second step contains verification of the designed algorithm dring real process control, and finall is the algorithm moved from MALAB/Simlink into PLC. he basic condition for realization of the described implementation method is the existence of real-time commnication beteen MALAB/Simlink and PLC. his article shos ho real-time commnication can be created sing PLC B&R. he behavior of the described adaptive LQ algorithm is compared ith that of adaptive PSD controller dring control of the real sstem. Ke-Words: - Adaptive controller, Optimal controller, MALAB, Simlink, PLC, Levenberg-Marardt, Neral Netorks, Self-ning Introdction he prpose of adaptive controllers is to adapt control la parameters of control la to changes of the controlled sstem. Man tpes of adaptive controllers are knon. In this article the adaptive self-tning LQ controller is described. he scheme of this controller is separated into the to main parts: identification and controller. In this ork, identification based on neral netork approach is sed and the control algorithm is the linear adratic optimal controller. Figre shos the architectre of the self-tning LQ controller here denotes desired vale, denotes action vale and denotes otpt of the controlled sstem. On-line Identification he most important part of self-tning sstems is the on-line identification algorithm. Widel sed method for identification of sstems is recrsive least sares method (RLS). Instead of RLS, the identification method based on neral netork can be sed. A ver fast algorithm for training neral netorks is the Levenberg-Marardt (LM) algorithm. he main idea of on-line identification is that according to the measred inpt to the identified sstem (t) and the corresponding sstem otpt (t) e are able to find the vector of sstem parameters. ( Plant ( Specification of control la LQ controller Process parameters On-line NN identification Plant NN ( - + Fig.. Architectre of self-tning LQ controller. Fig.. he principle of identification of sstem sing neral netork.

2 Proceedings of the th WSEAS International Conference on SYSEMS, Agios Nikolaos, Crete Island, Greece, Jl 3-5, 7 4 For compting of the identified sstem otpt e can se the linear ARX model b z + b z + b z F M. () 3 3 ( z ) 3 + az + az + a3z he ARX model can be ritten in vector form as follos ( ϕ ( θ ( here [ ( k )... ( k m) ( k )... ( k n ] () ϕ ( k ) ) (3) is the vector of measred inpts and otpts and [ b (... b ( a (... a ( k ] θ k ) (4) ( ) m is the vector of estimated sstem parameters and k denotes discrete time. As it has been mentioned above, the Levenberg Marardt method can be sed for training of the neral netork. he ne vector of parameters is in each step given b next eation. ( JJ + λi ) J ε ( θ ( k + ) θ ( (5) here J is Jacobian matrix in the form ε ε... j J, (6) ε p ε p... j and is the vector of errors beteen the otpt of the sstem and the otpt of the model for all training patterns. he parameter p denotes the nmber of training patterns and j denotes the nmber of estimated parameters. he reason h identification based on neral netork approach as sed instead of RLS is that sing a neral netork e can set a shorter sampling period [7]. hen the controller is able to reject errors faster. 3 Linear LQ Controller he action vale of LQ controller is solved according to the identified model () and according to the minimization of adratic performance (7) in each step. Qadratic performance is defined b n k + k k+ ( ( ( ) + ( ( ( ) J x ( Qxk ( ) + (7) here ( denotes the desired vale, ( denotes sstem otpt and ( is action vale. Parameter ( is a signal eal to the desired vale, hich is sed for elimination of offset. Parameters ( ) denote eights for otpt (action) vale, k denotes the first step hile the minimization is sed and x (Qx( denotes the minimm in the last step k +. When e ork ith the psedo state matrix S [S,S x,s,s ] defined b [ b ] S [ ] S (8) [ ] S S x b bm a a (9) n and hen e introdce the psedo state vector z ( [x(, (,(] and x [(, x(k-),(,(] S(z(k-) e can rerite the adratic performance to the more sitable form [, ] k + k k + J z ( Qz(. () Using a nonstandard state vector e can bild niversal adratic performance. For example, for standard penalization according to eation (7) matrix Q is defined as follos Q. () he method for minimization in each step is knon [, ] and is described b next eations.

3 Proceedings of the th WSEAS International Conference on SYSEMS, Agios Nikolaos, Crete Island, Greece, Jl 3-5, 7 5 ( ) ( ) k + z k + S QSz ( k + ) z ( k + ) Hz( k + ) J () he minimm of () is given for derivation b (k +) eal to zero * J ( k z m + ) ( k + ) ( H H H H ) z ( k + ) ϕϕ ϕ here ( k + ) (k + -) and ϕ z m is z ( k + ) m H Hϕ H Hϕ Hϕϕ For next minimization step e can rite * J ( k + ) ( k + ) Qz( k + ) (3) ithot (4) * zm ( k + ) H zm( k + ) + (5) + z Where matrix H* H xx H x H - H x is defined at previos step. he algorithm given above is nsitable for practical calclations. he reason is that matrix H ceased to be positivel definite (semi-definite). his means that the algorithm can be nstable. he soltion is LD-FIL decomposition []. When e ork ith matrix G instead of H here HGDG and G is a loer trianglar matrix and D is a diagonal matrix, e can rerite adratic performance to the form G [ (, ( k ), (, ( ] ϕ (6) Minimization of eation (5) at step k is G ( + G ϕ ϕ( k ) (7) Where G and G are sb-matrices of G. Finall, the action vale is in each step compted as follos ( G G ϕ( k ) ϕ. o smmarize in the table belo, e can see the recrsive algorithm for compting of the LQ controller. Step Eation and Notes. H* H xx H x H - H x recrsivel solves lost fnction. GDG LD-FIL decomposition 3. ( G Gx x( k ) solves action vale able. Iteration algorithm of LQ controller. 4 Implementation of designed control algorithm into PLC 4. Introdction he described control algorithm as created in the MALAB/Simlink environment and then as implemented to the programmable logic controller (PLC) B&R. Design and implementation ere done in three steps. In the first step the algorithm as designed and tested onl sing MALAB/Simlink. In the second step MALAB/Simlink as interconnected ith the PLC. his step allos verification properties of the designed control algorithm dring control of the real sstem. It is important to notice that in this case e have the same possibilities for verification of the control algorithm behavior, and e can ver easil change its parameters and monitor the alit of real process reglation. In the third step the algorithm is moved from MALAB/Simlink to the PLC. For fast and comfortable accomplishment of this step it is necessar to se a langage hich can be sed in MALAB/Simlink and in the PLC environment as ell. In or case the langage ANSI C as sed. his implementation method is faster than in the case hen the control algorithm is developed directl in the environment of PLC. he reason is that MALAB/Simlink allos more comfortable methods and tools for design, debgging and testing of algorithms. Another major advantage is that the PLC is not necessar for the first step of implementation. 4. Connection beteen MALAB/Simlink and PLC B&R For realization of the described implementation scheme it is imperative to establish connection beteen MALAB/Simlink and PLC. his commnication as carried ot sing Process Visalization Interface (PVI). he PVI is a file of tools and libraries for the operating sstem MS Windos and is provided as a component of the

4 Proceedings of the th WSEAS International Conference on SYSEMS, Agios Nikolaos, Crete Island, Greece, Jl 3-5, 7 6 B&R AUOMAION NE softare. he PVI procres a commnication interface beteen PLC B&R and applications on PC ith MS Windos [6]. As a phsical netork laer Ethernet or RS-3 link can be sed. he most important part of PVI is the PVI Manager. his tilit provides netork connection beteen PC and PLC. Using PVI libraries and some standard development environment (e.g. MS Visal Stdio or Borland Delphi) it is possible to create ser application hich can, b force of PVI Manager, commnicate ith PLC. In or case, the commnication client as created as the s-fnction for MALAB/Simlink. he s-fnction represents a real controlled sstem. hat means that the s-fnction realizes transfer of the action vale compted in MALAB to the PLC, and from the PLC vales representing the otpt of the real controlled sstem are transferred. In the PLC, a task is rnning hich onl resends these vales into and from the real process sing D/A and A/D converters. Next figre shos the scheme of the described commnication. response ith the self-tning PSD controller based on the modified Ziegler Nichols method [, 4]. In both cases identification ith the neral netork approach and the same setp and initial conditions ere sed. he sampling period as set to s s. U [V] reference vale action vale sstem otpt time [s] Fig. 4. Real process control sing LQ controller (., ). 8 6 reference vale action vale sstem otpt Real controlled sstem U [V] 4 D/A Converter A/D Converter PLC PVI Manager Ethernet or RS-3 PC time [s] Fig. 5. Real process control sing self-tning PSD controller MALAB/Simlink on Control algorithm s-fnction PVI Client Fig. 3. he scheme of commnication beteen MALAB/Simlink and PLC B&R. 5 Real Process Control Reslts he described LQ controller as implemented into the PLC B&R. Used as the controlled sstem as the laborator model ith the transfer fnction F ( s). In the figre 3 e can (s + )( s + ) see the real process response ith the described LQ control algorithm. Figre 4 shos real the process In comparison of to previos figres e can see that sing LQ controller as reach better reslts notabl in case of changes of action vale. As it as shon above e can modif the criterion of LQ controller b changing of to parameters and. he parameter inflences the changes of action vale and is standard penalization of control error. In fact the criterion is dependent on relation of these to parameters. If e set, then e have onl one parameter for changing of criterion. he responses of LQ controller are optimal in regard of the criterion bt it doesn t mean that the are optimal from the ser s viepoint. he ser specifications can be ver easil inclded to the control la sing parameter.

5 Proceedings of the th WSEAS International Conference on SYSEMS, Agios Nikolaos, Crete Island, Greece, Jl 3-5, Conclsion It as described design of adaptive LQ controller. his algorithm solves one iteration in each step. hat means the time for compting is short and this algorithm is sitable for implementation in indstrial controller. As it as shon above the properties of control la can be changed sing one parameter. his algorithm as implemented to the indstrial controller appling of third-step implementation scheme, hich allos control of real processes directl from MALAB/Simlink environment. B sing this method can be saved time for development of the control algorithm, becase MALAB/Simlink is able to give more comfortable possibilities for design and debgging of algorithm in comparison ith the environment of PLC. he commnication beteen MALAB/Simlink and PLC B&R sing PVI as described. Acknoledgement: he paper has been prepared as a part of the soltion of Czech Science Fondation GAR project No. /6/3 Soft Compting in Control and b the Czech Ministr of Edcation in the frame of MSM MSM6359 Intelligent Sstems in Atomation. References: [] Švancara, K., Adaptive Optimal Controller ith Identification Based on Neral Netorks, PhD thesis, FEEC BU, Brno 4. [] Bobál V., Böhm J., Practical Aspects of Self- ning Controllers, Algorithms and Implementation, VUIUM, 999, in Czech, ISBN [3] Švancara K., Pivonka P., Closed Loop On-line Identification Based on Neral Netorks in Adaptive Optimal Controller, In Proceedings of th Zitta Fzz Colloim, Zitta (German),. [4] Koínek, V., Comnication MALAB-Ethernet. Diploma thesis in Czech, FEEC BU, Brno 3. [5] Lorenc V., Self-ning Controllers in MALAB - B&R Environment, Diploma thesis in Czech, FEEC BU, Brno 6. [6] B&R Docmentation. Atomation Net/PVI Online Help, AS4Ee, 997-7, [7] Veleba, V., he advantages of identification based on neral netork approach, In Proceedings of the th conference Stdent EEIC, BRNO 5.