Control of Distributed Parameter Systems - Engineering Methods and Software Support in the MATLAB & Simulink Programming Environment

Distributed parameter systems (DPS) are systems with state/output quantities X(x,t) /Y(x,t) – parameters which are defined as quantity fields or infinite dimensional quantities distributed through geometric space, where x – in general is a vector of the three dimensional Euclidean space. Thanks to the development of information technology and numerical methods, engineering practice is lately modelling a wide range of phenomena and processes in virtual software environments for numerical dynamical analysis purposes such as ANSYS www.ansys.com, FLUENT (ANSYS Polyflow) www.fluent.com , ProCAST www.esi-group.com/products/casting/, COMPUPLAST – www.compuplast.com, SYSWELD – www.esi-group.com/products/welding, COMSOL Multiphysics www.comsol.com, MODFLOW, MODPATH,... www.modflow.com , STAR-CD www.cd-adapco.com, MOLDFLOW www.moldflow.com, ... Based on the numerical solution of the underlying partial differential equations (PDE) these virtual software environments offer colorful, animated results in 3D. Numerical dynamic analysis problems are solved both for technical and non-technical disciplines given by numerical models defined in complex 3D shapes. From the viewpoint of systems and control theory these dynamical models represent DPS. A new challenge emerges for the control engineering practice, which is the objective to formulate control problems for dynamical systems defined as DPS through numerical structures over complex spatial structures in 3D. The main emphasis of this chapter is to present a philosophy of the engineering approach for the control of DPS given by numerical structures, which opens a wide space for novel applications of the toolboxes and blocksets of the MATLAB & Simulink software environment presented here.

Input-output dynamics of these DPS can be described, from zero initial conditions, by ( ) () or in discrete form

GG
(2) where ⊗ marks convolution product and ⊕ marks convolution sum, G i (x,t) -distributed parameter impulse response of LDS to the i-th input, GH i (x,k) -discrete time (DT) distributed parameter impulse response of LDS with zero-order hold units H -HLDS to the i-th input, Y i (x,t) -distributed output quantity of LDS to the i-th input, Y i (x,k) -DT distributed output quantity of HLDS to the i-th input.For simplicity in this chapter distributed quantities are considered mostly as continuous scalar quantity fields with unit sampling interval in time domain.Whereas DT distributed parameter step responses

H
of HLDS can be computed by common analytical or numerical methods then DT distributed parameter impulse responses can be obtained as (3)

Decomposition of dynamics
The process of dynamics decomposition shall be started from DT distributed parameter step and impulse responses of the analysed LDS.For an illustration, procedure of decomposition of dynamics and control synthesis will be shown on the LDS with zero-order hold units H -HLDS -distributed only on the interval [ ] 0,L , with output quantity discretised in time relation and continuous in space relation on this interval.Nevertheless the following results are valid in general both for continous or discrete distributed quantities in space relation given on compex-shape definition domains over 3D as well.
, then the i-th DT distributed output quantiy in (2) can be rewritten by the means of the reduced characteristics as follows At fixed i x the partial DT distributed output quantity in time direction ( ) ii Yx , k i s g i v e n a s the convolution sum holds at the fixed point i x .At fixed k, the partial discrete distributed output quantity in space direction , where the reduced discrete partial distributed characteristics are multiplied by corresponding elements of the set Hx , q Uk q = − G ., see Fig. 5.
Yx , k 0 ≠ can be considered as time components and then in the steady-state When distributed quantities are used in discrete form as finite sequences of quantities, the discretization in space domain is usually considered by the computational nodes of the numerical model of the controlled DPS over the compex-shape definition domain in 3D.

Distributed parameter systems of control
Based on decomposition of HLDS dynamics into time and space components, possibilities for control synthesis are also suggested by an analogous approach.In this section a methodical framework for the decomposition of control synthesis into space and time problems will be presented by select demonstration control problems.In the space domain control synthesis will be solved as a sequence of approximation tasks on the set of space components of controlled system dynamics, where distributed parameter quantities in particular sampling times are considered as continuous functions on the interval [ ] 0,L as elements of strictly convex normed linear space X with quadratic norm.It is necessary to note as above that the following results are valid in general for DPS given on compexshape definition domains in 3D both for continous or discrete distributed quantities, in the space relation as well.In the time domain, the control synthesis solutions are based on synthesis methods of DT lumped parameter systems of control.

Open-loop control
Assume the open-loop control of a distributed parameter system, where dynamic characteristics give an ideal representation of controlled system dynamics and ( ) = , that is with zero initial steady-state, in which all variables involved are equal to zero -see see Fig. 6 for reference.Let us consider a step change of distributed reference quantity - =∞ , see Fig. 7.For simplicity let the goal of the control synthesis is to generate a sequence of control inputs

()
Uk in such fashion that in the steady-state, for k →∞, the control error ( )( ) ( ) will approach its minimal value ( ) First, an approximation problem will be solved in the space synthesis (SS) block: Fig. 6.Distributed parameter open-loop system of control LDS -lumped-input and distributed-parameter-output system H -zero-order hold units HLDS -controlled system with zero-order hold units CS -control synthesis TS -time part of control synthesis SS -space part of control synthesis Wx , ∞  -vector of lumped reference quantities ()

Uk
-vector of lumped control quantities From approximation theory in this relation is known the theorem: Let Fn be a finite-dimensional subspace of a strictly convex normed linear space X.Then, for each f ∈ X, there exists a unique element of best approximation.(Shadrin, 2005).So the solution of the approximation problem ( 10) is guaranteed as a unique best approximation is the vector of optimal approximation parameters.Hence: Rz , a r e to be chosen such that for k →∞ the following relation holds: Wk / Uk  -vector of lumped reference/control quantities When the individual components of the vector are input in the particular control loops: SH x ,z ,R z , the control processes take place.The applied control laws result in the sequences of control inputs: U k , and respectively the output quantities, for k →∞ converging to reference quantities Values of these lumped controlled quantities in new steady-state will be further denoted as , which implies that the overall distributed output quantity at the time k →∞: ( ) gives the unique best approximation of the distributed reference variable: ( ) Therefore the control error has a unique form as well, with minimal quadratic norm Thus the control task, defined at equation ( 9), is accomplished with Ex , 0 ∞=  -see Fig. 10 . for reference.As the conclusion of this section we may state that the control synthesis in open-loop control system is realized as: Time Tasks of Control Synthesis -in lumped parameter control loops Space Tasks of Control Synthesis -as approximation task.
When mathematical models cannot provide an ideal representation of controlled DPS dynamics and disturbances are present with an overall effect on the output in steady-state expressed by Finally at the design stage of a control system, for a given desired quality of control δ in space domain, it is necessary to choose appropriate number and layout of actuators for the fulfillment of this requirement ( ) ( )  12).In the k-th step in block SS2 at approximation of ( ) time components of partial output quantities Wx , ∞  are computed.Then on the output of the algebraic block is . These sequences . Thus the control task, defined by equation ( 9) is accomplished as given by relation ( 16).In case of the uncertainty of the control process relations similar to (17-18) are also valid.
Let´s consider now the approximation of ( ) -this actually means that the control task defined in equation ( 9), is accomplished as given by relation ( 16).
Finally we may state as a summary, that in closed-loop control with RHLDS the control synthesis is realized as: Time Tasks of Control Synthesis -on the level of lumped parameter control loops Space Tasks of Control Synthesis -as approximation tasks.
At the same time the solution of the approximation problem in block SS1 on the approximation set where ( )

Closed-loop control
Let us now consider a distributed parameter feedback loop as featured in Fig. 14. with initial conditions as above, where in timestep k an approximation problem is solved and as a result in timestep k a vector () The problem solution is obtained by approximation.
Fig. 14.Distributed parameter closed-loop system of control HLDS -LDS with zero-order hold units is valid, so the above given control task ( 9) is accomplished -whereas in the steady-state . By concluding the above presented discussion, the control synthesis in closed-loop control is realized as: Time Tasks of Control Synthesis -on the level of lumped parameter control loops Space Tasks of Control Synthesis -as approximation tasks.
When mathematical models cannot provide an ideal description of controlled DPS dynamics and disturbances are present with an overall effect on the output in steady-state, expressed by ( ) EY x,∞ then the realtions similar to (17-18) are also valid here.In practice mostly only reduced distributed parameter transient responses in steady-state are considered for the solution of the approximation tasks in the block SS of the scheme in Fig. 14. along with robustification of controllers Rz .
For simplicity problems of DPS control have been formulated here for the distributed desired quantity ( ) W x,k is assumed, the control synthesis is realized similarly: -In Space Domain -as problem of approximation in particular sampling intervals -In Time Domain -as control synthesis in lumped parameter control loops, closed throughout structures of the distributed parameter control loop.The solution of the presented problems of control synthesis an assumption is used, that in the framework of the chosen control systems the prescribed control quality can be reached both in the in space and time domain.However in the design of actual control systems for the given distributed parameter systems, usually the • optimization of the number and layout of actuators • optimization of dynamical characteristics of lumped/distributed parameter actuators • optimization of dynamical characteristics of lumped parameter control loops is required and necessary.(Hulkó et al., 2003(Hulkó et al., -2010)).Fig. 16.shows The library of DPS Blockset.The HLDS and RHLDS blocks model controlled DPS dynamics described by numerical structures as LDS with zero-order hold units -H.DPS Control Synthesis provides feedback to distributed parameter controlled systems in control loops with blocks for discrete-time PID, Algebraic, State- Space and Robust Synthesis.The block DPS Input generates distributed quantities, which can be used as distributed reference quantities or distributed disturbances, etc. DPS Display presents distributed quantities with many options including export to AVI files.The block DPS Space Synthesis performs space synthesis as an approximation problem.As a demonstration, some results of the discrete-time PID control of complex-shape metal body heating by the DPS Blockset are shown in , where the heating process was modelled by finite element method in the COMSOL Multiphysics virtual software environment -www.comsol.com.

Distributed Parameter Systems Blockset for MATLAB & Simulink
The block Tutorial presents methodological framework for formulation and solution of control problems for DPS.The block Show contains motivation examples such as: Control of temperature field of 3D metal body (the controlled system was modelled in the virtual software environment COMSOL Multiphysics); Control of 3D beam of "smart" structure (the controlled system was modelled in the virtual software environment ANSYS); Adaptive control of glass furnace (the controlled system was modelled by Partial Differential Equations Toolbox of the MATLAB ), and Groundwater remediation control (the controlled system was modelled in the virtual software environment MODFLOW).The block Demos contains examples oriented at the methodology of modelling and control synthesis.The DPS Wizard gives an automatized guide for arrangement and setting distributed parameter control loops in step-by-step operation.

Interactive control via the Internet
For the interactive formulation and solution of DPS demonstration control problems via the Internet, an Interactive Control service has been started on the web portal Distributed Parameter Systems Controlwww.dpscontrol.sk of the IAMAI-MEF-STU (Hulkó, 2003(Hulkó, -2010) ) -see Fig. 20. for a screenshot of the site.In the framework of the problem formulation, first the computational geometry and mesh are chosen in the complex 3D shape definition domain, then DT distributed transient responses are computed in virtual software environments for numerical dynamical analysis of machines and processes.Finally, the distributed reference quantity is specified in points of the computational mesh -Fig.18. Representing the solution to those interested animated results of actuating quantities, quadratic norm of control error, distributed reference and controlled quantity are sent in the form of DPS Blockset outputs -see Fig. 17

Conclusion
The aim of this chapter is to present a philosophy of the engineering approach for the control of DPS -given by numerical structures, which opens a wide space for novel applications of the toolboxes and blocksets of the MATLAB & Simulink software environment.This approach is based on the general decomposition into time and space components of controlled DPS dynamics represented by numerically computed distributed parameter transient and impulse characteristics, given on complex shape definition domains in 3D.Starting out from this dynamics decomposition a methodical framework is presented for the analogous decomposition of control synthesis into the space and time subtasks.In space domain approximation problems are solved, while in the time domain control synthesis is realized by lumped parameter SISO control loops (Hulkó et al., 1981(Hulkó et al., -2010)).Based on these decomposition a software product named Distributed Parameter Systems Blockset for MATLAB & Simulink -a Third-Party software product of The MathWorkswww.mathworks.com/products/connections/ has been developed within the program CONNECTIONS of The MathWorks Corporation, (Hulkó et al., 2003(Hulkó et al., -2010)), where time domain toolboxes and blocksets of software environment MATLAB & Simulink as Control Systems Toolbox, Simulink Control Design, System Identification Toolbox,... are made use of.In the space domain approximation problems are solved as optimization problems by means of the Optimization Toolbox.For the further support of research in this area a web portal named Distributed Parameter Systems Control -www.dpscontrol.skwas realized (Hulkó et al., 2003(Hulkó et al., -2010)), see Fig. 20. for an illustration.On the above mentioned web portal, the online version of the monograph titled Modeling, Control and Design of Distributed Parameter Systems with Demonstrations in MATLAB -www.mathworks.com/support/books/(Hulkó et al., 1998), is presented along with application examples from different disciplines such as: control of technological and production processes, control and design of mechatronic structures, groundwater remediation control, etc.This web portal also offers for those interested the download of the demo version of the Distributed Parameter Systems Blockset for MATLAB & Simulink with Tutorial , Show , Demos and DPS Wizard.This portal also offers the Interactive Control service for interactive solution of model control problems of DPS via the Internet.

Acknowledgment
This work was supported by the Slovak Scientific Grant Agency VEGA under the contract No. 1/0138/11 for project "Control of dynamical systems given by numerical structures as distributed parameter systems" and the Slovak Research and Development Agency under the contract No. APVV-0160-07 for project "Advanced Methods for Modeling, Control and Design of Mechatronical Systems as Lumped-input and Distributed-output Systems" also the project No. APVV-0131-10 "High-tech solutions for technological processes and mechatronic components as controlled distributed parameter systems".

Fig. 4
Fig. 4. i-th discrete distributed parameter impulse response of HLDS ( ) ii Hx , k G -partial DT impulse response in time, t -relation to the i-th input considered as response with maximal amplitude in point Fig. 5. Partial distributed output quantities in time and space direction -dimensional subspace of approximation functions Fn in the strictly convex normed linear space of distributed parameter quantities X on [ ] 0,L with quadratic norm, where the approximation problem is to be solved, see Fig.8. for reference.

Fig. 9 .
Fig. 9. SISO lumped parameter control loops in the block TS TS -time part of control synthesis ( ) { } ii i SH x ,z -time components of HLDS dynamics () { } i i R z -lumped parameter controllers Fig. 10.Quantities of distributed parameter open-loop control in new steady-state HLDS -controlled system with zero-order hold units { } i i U -lumped control quantities ( ) ( ) { } ii i ii i Yx , / W Wx , ∞= ∞   -controlled/reference quantities in new steady-state ( ) { } ii i Ex , ∞  -lumped control errors ( ) Yx , ∞  -controlled distributed quantity in new steady-state z .-This analysis of control synthesis process shows that synthesis in time domain is realized on the level of one parameter

Fig. 11 .
Fig. 11.Distributed parameter closed-loop system of control with reduced space components of output quantity Fig. 13.Lumped parameter SISO control loops -i-th control loop ( ) ii SH x ,z -i-th time component of HLDS dynamics As a software support for DPS modelling, control and design of problems in MATLAB & Simulink the programming environment Distributed Parameter Systems Blockset for MATLAB & Simulink (DPS Blockset) -a Third-Party Product of The MathWorks www.mathworks.com/products/connections/-Fig.15., has been developed within the program CONNECTIONS of The MathWorks Corporation by the Institute of Automation, Measurement and Applied Informatics of Mechanical Engineering Faculty, Slovak University of Technology in Bratislava (IAMAI-MEF-STU)

Fig. 15 .
Fig. 15.Distributed Parameter Systems Blockset on the web portal of The MathWorks Corporation Fig. 20.Web portal Distributed Parameter Systems Control with monograph Modeling, Control and Design of Distributed Parameter Systems with Demonstrations in MATLAB and service Interactive Control This decomposition is valid for all given lumped input { }