Engineering Methods and Software Support for ...

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Engineering Methods and Software Support for Control of Distributed Parameter Systems G. Hulkó, C. Belavý, P. Buček, K. Ondrejkovič, P. Zajíček University Center for Control of Distributed Parameter Systems Institute of Automation, Measurement and Applied Informatics of STU Nám. Slobody 17, 812 31 Bratislava , Slovak Republic, [email protected]

Abstract - In this paper advanced engineering methods are presented for modeling and discrete-time control of distributed parameter systems described for control purposes by numerical structures on complex-shape definition domains over 3D along with the software package Distributed Parameter Systems Blockset for MATLAB & Simulink Third-Party MathWorks product, inspired by the boom of numerical analysis of dynamics of machines and processes within engineering practice. The techniques offered are completed by a user-friendly web service tool, suitable for interactive solution of modeling and control problems of distributed parameter systems via the Internet. I. INTRODUCTION

T

he first monographs on distributed parameter systems (DPS) was published in the 1960ties of the last century by P. K. C. Wang „Control of distributed parameter systems“ (1964) and A. G. Butkovskij „Optimal control of distributed parameter systems“ (1965) and later by J. L. Lions „Optimal control of systems governed by partial differential equations“ (1971). In further decades, a good number of significant results, based mainly on analytical solutions of partial differential equations (PDE), were published. From quite a number of valuable publications let us note at least one three-volume book from series of Encyclopedia of Mathematics and Its Applications, Cambridge University Press: I. Lasiecka and R. Triggiani „Control Theory for Partial Differential Equations I.-III.“ (2000). Nowadays dramatic advances in information technologies open new horizons for numerical solutions of PDE, integral equations and integro-partial differential equations given on complex-shape 3D definition domains. In last decade these advances iniciate in engineering practice a boom of numerical analysis of dynamics (NAD) of diverse continuum-type processes as dynamical systems defined on complex-shape 3D domains. This boom opens new possibilities also for determination of dynamical characteristics of DPS described for control purposes by numerical structures over complex-shape 3D definition domains as distributed step, impulse and frequency responses. The goal of this paper is to present some advanced engineering methods and software tools for the solution of

several actual problems in engineering practice as decomposition of dynamics of DPS described for control purposes by numerical structures on complex-shape 3D domains into time- and space-components as well as decomposition of control synthesis into space- and timeproblems [7, 9-12]. Further in the paper, software support for modeling, control and design of DPS in the MATLAB & Simulink environment based on Distributed Parameter Systems Blockset for MATLAB & Simulink (DPS Blockset) a Third-Party MathWorks product www.mathworks.com/products/connections/ , [11] will be presented as well as the Interactive Service of the web portal www.dpscontrol.sk , [9] which enables interactive formulation and solution of model control problems for DPS via the Internet.

II. BOOM OF NUMERICAL ANALYSIS OF DYNAMICS - BOOM OF CONTROL OF DISTRIBUTED PARAMETER SYSTEMS In the beginning of the 1940ties the development of numerical methods for solution of PDE was initiated by the publications of A. Hrennikoff „Solution of Problems in Elasticity by the Framework Method“ (1941) and R. Courant „Variational Methods for Solution of Equilibrium and Vibration“ (1943). Thanks to the dramatic advances of information technologies a variety of numerical methods for the solution of PDE, integral equations and integro-partial differential equations given on complex-shape 3D definition domains were offered. A new branch of software engineering has emerged, where results in continuum-type systems modeling of all scientific and technical disciplines are offered in form of numerical dynamic models given on complex-shape 3D definition domains - used for the solution of diverse practical problems - like: Suite of CAE (Computer Aided Engineering) simulation solutions (ANSYS, FLUENT, STAR-CD,... – www.ansys.com , www.fluent.com , www.cd-adapco.com ); Scientific modeling environments (COMSOL – www.comsol.com ); Virtual try-out spaces (ESI Group – www.esi-group.com ); 3D PLM application software (Product Lifecycle Management solutions Dassault Systemes – www.3ds.com ); Design Analysis Solutions for Plastics (Moldflow Corporation - www.modflow.com ); Environmental and water resources software packages (MODFLOW,

Preprint submitted to 7th Asian Control Conference. Received February 24, 2009.

CONFIDENTIAL. Limited circulation. For review only. MODPATH, MT3D,..– www.modflow.com), etc. This boom also opens qualitatively new possibilities for determination of dynamical characteristics of controlled DPS - for example as discrete lumped-input and distributedoutput systems (LDS) with hold units (HLDS) on complex shape 3D definition domains - in all engineering disciplines. Hence appropriate conditions are met for the initiation of the similar boom of DPS control applications in engineering practice.

UK ( x,t ) – contrast material concentration in the input of excretion process YK ( x,t ) – contrast material concentration in the output of excretion process

{GK i }i – anatomic parts, subsystems of liver complex at blood and contrast material distribution, distributed input variable generators x = {x, y,z}

The anatomic structures, as controlled systems of these real distributed parameter control loops in internal organs of human body, are in fact lumped-input and distributedoutput systems , Fig. 2.

III. LUMPED-INPUT AND DISTRIBUTED-OUTPUT SYSTEMS The processes of metabolism in internal organs of the human body are realized at molecular level as interactions of fields of variables. Internal organs: liver, lungs, kidneys,... are in fact real distributed parameter systems of control given on complex-shape 3D definition domains [6], Fig. 1. Fig. 2 Lumped-input and distributed-output system – LDS.

{U ( t )} i

i

– lumped input variables

Y ( x,t ) = Y ( x, y, z, t ) – distributed output variable

A similar structure is generally obtained when distributedinput and distributed-output systems (DDS) in combination with different types of actuators form the distributed parameter controlled systems as depicted in Fig. 3.

Fig. 1 Process of liver excretion. H – liver LC – liver cellular structure BC – bile canaliculus S – sinusoid VP – portal vein VF – gall – bladder DCH – choledochus AH - arteria hepatica VH – hepatic vein {UK i ( t )}i – contrast material flows, lumped input variables

Fig. 3 Controlled DPS as LDS at control of metal body heating.

{SAi (s)}i , {SGi (s)}i , {Ti (x, y, z)}i

– models of actuators

DDS – distributed-input and distributed-output system {Ui }i – actuating variables Ω – definition domain – actuation subdomains

{Ωi }i

Preprint submitted to 7th Asian Control Conference. Received February 24, 2009.

CONFIDENTIAL. Limited circulation. For review only. LDS usually consist of DDS and different actuators:

Fig. 4 General systems.

units – HLDS – defined only on the interval 0, L nevertheless, the following results are valid in general for DPS given on 3D definition domains [7, 9-12]:

structure of lumped-input and distributed-output

LDS – lumped-input and distributed-output system {SAi }i – actuating members of lumped input variables

{GUi }i

– generators of distributed input variables

DDS – distributed-input and distributed-output system U(t) = {U i (t)}i – vector of lumped input variables of LDS

{UAi (t)}i {Ui (ξ, t)}i

Fig. 5 The i-th discrete-time impulse response of HLDS.

– output variables of lumped parameter actuators

GH i ( x i , k ) – partial discrete-time impulse response of HLDS in time

– distributed output variables of generators

- t - relation to the i-th input, in the point x i , where the

U(ξ, t) - overall distributed input variable for DDS Y(x, t) = Y(x, y, z, t) – distributed parameter output variable

response exhibits maximal amplitude {GHi ( x, k )}k – set of partial discrete impulse responses of HLDS to the i-th input in space - x - relation

Input-output dynamics of these DPS can be described by n

n

i =1

i =1

Y(x, t) = ∑ Yi (x, t) = ∑ Gi (x, t) ⊗ U i (t)

(1)

{GHR ( x, k )} i

k

– set of reduced partial discrete impulse responses of HLDS in space - x relation

or in discrete form n

n

i =1

i =1

Y(x, k) = ∑ Yi (x, k) = ∑ GH i (x, k) ⊕ U i (k)

(2)

where ⊗ marks convolution product and ⊕ marks convolution sum, Gi(x,t) – distributed impulse response of LDS to the i-th input, GHi(x,k) – discrete-time distributed impulse response of LDS with zero-order hold units – HLDS to the i-th input. Yi(x,t) - distributed output variable of LDS as a response to the i-th input, Yi(x,k) – discrete-time distributed output variable of LDS with zero-order hold units – HLDS as a response to the i-th input. Whereby discrete-time distributed parameter step responses { HH i ( x,k )}i of controlled DPS can be computed by common methods and software products of NAD. Then, discrete-time distributed parameter impulse responses are {GHi ( x,k ) = H Hi ( x,k ) − H Hi ( x,k-1)}i .

IV. DECOMPOSITION OF DYNAMICS At the decomposition of dynamics of DPS described for control purposes by numerical structures on complex-shape definition domains over 3D discrete-time distributed step and impulse responses will be used. But for an illustration, procedure of decomposition of dynamics and control synthesis will be shown on the LDS with zero-order hold

If the reduced characteristics are defined as ⎧ GH i (x, k) ⎫ ⎨GHR i (x, k) = ⎬ (3) GH i (x i , k) ⎭k ⎩ then the i-th discrete-time distributed output in (2) can be rewritten by means of the reduced characteristics as follows Yi ( x, k ) = GH i ( x i , k ) GHR i ( x, k ) ⊕ U i ( k )

(4)

At fixed xi the partial discrete-time distributed output variable in time direction Yi ( x i , k ) is given as the sum GH i ( x i , k ) ⊕ U i ( k ) when in the

convolution

chosen point xi {GHR i ( x i , q ) = 1}q = 0,k . At fixed k the partial discrete-time distributed output variable in space direction Yi ( x, k ) is given as a linear combination of elements

{GHR ( x, q )} i

q = 0,k

where

reduced

discrete

partial

distributed characteristics {GHR i ( x, q )}q = 0,k are multiplied by corresponding elements {GHi ( x i , q ) Ui ( k − q )}q =0,k . Fig. 6.

of

Preprint submitted to 7th Asian Control Conference. Received February 24, 2009.

the

set

CONFIDENTIAL. Limited circulation. For review only. V. DISTRIBUTED PARAMETER DISCRETE-TIME SYSTEM OF CONTROL Decomposition of HLDS dynamics into time and space components enables to decompose the control synthesis into space and time problems. In space domain it is the solution of the approximation problem on the set of reduced distributed discrete impulse responses {GHR i ( x,k )}i,k or on the set of reduced distributed steady-state transient responses {HHR i ( x,∞ )}i . In time domain the control synthesis is solved by discrete-time control synthesis methods for lumped parameter systems. Fig. 7. Fig. 6 Partial distributed output variables in time and space direction. HLDS – lumped-input and distributed-output system with zero-order hold units defined on the interval 0, L U i - i-th discrete-time lumped input variable Yi ( x i , k ) - i-th partial discrete-time distributed output variable in

time direction at chosen point x i on the interval 0, L Yi ( x, k ) - i-th partial distributed output variable in space direction at

point k of the time axis

This decomposition is valid for all given lumped input {Ui }i =1,n and correspondin output {Yi ( x, k )}i variables - so we obtain time and space components of LDS dynamics: Time components of dynamics {GHi ( x i ,k )}i – for given i and chosen xi - variable k Space components of dynamics {GHR i ( x,k )}i,k – for given i and chosen k – variable x

When reduced steady-state transient responses {HHR i ( x,∞ )}i are introduced and discrete transfer functions {SH i ( x i ,z )}i are assigned to partial transient responses with maximal amplitudes in points {x i }i on the interval 0, L we obtain time and space components of LDS dynamics for steady-state: Time components of dynamics {SHi ( x i ,z )}i - for given i and chosen xi - variable z Space components of dynamics {HHR i ( x,∞ )}i for given i in ∞ – variable x

In time the discretization is chosen by standard methods, in space it is given by distribution of the computational net.

Fig. 7 Distributed parameter discrete-time system of control. HLDS – controlled LDS with zero-order hold units CS – control synthesis TS – time control synthesis SS- space control synthesis K – time/space discretization Y ( x, t ) - distributed controlled variable W ( x, k ) , V ( x, t ) - reference and disturbance variables  E ( k ) – vector of control errors U ( k ) – vector of discrete-time control variables E ( x, k ) - distributed parameter control error

VI. DISTRIBUTED PARAMETER SYSTEMS BLOCKSET As software support for distributed parameter systems modeling, control and design problems in MATLAB & Simulink the Distributed Parameter Systems Blockset for MATLAB & Simulink – a Third-Party MathWorks product (DPS www.mathworks.com/products/connections/ Blockset) has been developed in program CONNECTIONS of The MathWorks, [11]. Fig. 8 shows the library of Distributed Parameter Systems Blockset.

Preprint submitted to 7th Asian Control Conference. Received February 24, 2009.

CONFIDENTIAL. Limited circulation. For review only. was modeled in the MODFLOW). The block Demos contains examples oriented to methodology of modeling and control synthesis. The DPS Wizard gives an automatized guide for arrangement and setting distributed parameter control loops in step-by-step operation.

Fig. 9 Distributed parameter control loop for discrete-time PID control of heating of complex-shape metal body in DPS Blockset environment. Fig. 8 The library of Distributed Parameter Systems Blockset for MATLAB & Simulink – Third-Party MathWorks product.

The HLDS block models controlled distributed parameter systems as lumped-input and distributed-output systems with zero-order hold units. DPS Control Synthesis provides feedback to distributed parameter controlled systems in control loops with blocks for discrete-time PID, Algebraic, State-Space and Robust control. The block DPS Input generates distributed quantities which can be used as distributed control variables 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. The block Tutorial presents methodological framework for formulation and solution of DPS control problems, based on [7, 9-11]. The block Show contains motivation examples: Control of temperature field of 3D metal body (the controlled system was modeled in the ANSYS), Fig. 9-13; Control of 3D beam of „smart“ structure (the controlled system was modeled in the ANSYS); Adaptive control of glass furnace (the controlled system was modeled in the COMSOL Multiphysics) and Groundwater remediation control (the controlled system

VII. INTERACTIVE CONTROL VIA INTERNET For interactive formulation and solution of DPS control problems via the Internet an Interactive Service was started on the web portal www.dpscontrol.sk [9]. In frame of problem formulation first the computational geometry and mesh are chosen on the complex-shape 3D definition domain, afterwards distributed transient responses are computed by means of numerical analysis of dynamics methods using appropriate software products to each input of the LDS. Finally, the distributed reference quantity is specified – see Fig. 10. As a solution to the interested person animated results of actuating quantities, quadratic norm of control error and distributed controlled quantity is sent in the form of DPS Blockset outputs – see Fig. 11-13. VIII. CONCLUSIONS In the paper advanced engineering methods are presented for modeling and discrete-time control of DPS, inspired by the boom of the numerical analysis of dynamics of machines and processes in the engineering practice, with respect to problems of discrete-time control of DPS described for control

Preprint submitted to 7th Asian Control Conference. Received February 24, 2009.

CONFIDENTIAL. Limited circulation. For review only.

Fig. 10 W ( x,y,z,∞ ) – distributed reference quantity of metal body

Fig. 11 {U i ( k )}i=1,4 – discrete-time lumped parameter actuating

heating given over the numerical net.

variables.

Fig. 12 Y ( x,y,z,t ) – distributed controlled quantity of metal body

Fig. 13 Quadratic norm of distributed control error E ( x,y,z,t ) .

heating over the numerical net. .

Preprint submitted to 7th Asian Control Conference. Received February 24, 2009.

CONFIDENTIAL. Limited circulation. For review only. purposes by numerical structures on complex-shape definition domains over 3D, or problems of discrete-time control of systems which dynamics are given by measured distributed parameter characteristics on 3D (obtained at validation of large-scale numerical models), etc. For solution of these problems the software support in the MATLAB & Simulink environment is offered together with interactive service for solution of model control problems via the Internet. ACKNOWLEDGMENT This work has been financially supported by the Slovak Scientific Grant Agency VEGA, project „Advanced Methods of Control of Distributed Parameter Systems“ (grant 1/0036/08) and the Slovak State Agency for Science and Technology for projects “ Predictive control of mechatronic systems with fast dynamics and constraints” (grant APVV-0280-06) and „Advanced Methods for Modeling, Control and Design of Mechatronical Systems as Lumped-input and Distributed-output Systems” (grant APVV-0160-07).

REFERENCES [1]

A. Hrennikoff, Solution of Problems in Elasticity by the Framework Method. ASME J. Appl. Mech. 8, A619–A715, 1941. [2] R. Courant, Variational Methods for Solution of Equilibrium and Vibration. Bull. Amer. Math. Soc. Volume 49, Number 1, 1-23, 1943. [3] P. K. C. Wang, Control of distributed parameter systems. In: Advances in Control Systems: Theory and Applications, 1, Academic Press, New York, 1964. [4] A. G. Butkovskij, Optimal control of distributed parameter systems. Nauka, Moscow, 1965. (in Russian) [5] J. L. Lions, Optimal control of systems governed by partial differential equations. Springer-Verlag, Berlin – Heidelberg - New York, 1971. [6] G. Hulkó, M. Mikulecký, Distrtibuted parameter model of liver dye excretion. Proceedings of the 1-st International Symposium on Mathematical Modelling of Liver Dye Excretion, SAV-UK, Bratislava – Smolenice, 1984. [7] G. Hulkó, et al., Modeling, Control and Design of Distributed Parameter Systems with Demonstrations in MATLAB. Publishing House of STU, Bratislava, www.mathworks.com/support/books/ 1998. [8] I. Lasiecka, R. Triggiani, Control Theory for Partial Differential Equations: Continuous and approximation theories I. Abstract parabolic systems, II. Abstract hyperbolic-like systems over a finite time horizon, ( III. – in preparation). Encyclopedia of Mathematics and Its Applications 74, Cambridge University Press, Cambridge UK, 2000. [9] G. Hulkó, et al., Distributed Parameter Systems. Web portal [Online]. Available: www.dpscontrol.sk , 2003-2007. [10] G. Hulkó, et al., MONOGRAPH on the web portal. [Online]. Available: www.dpscontrol.sk , 2003-2007. [11] G. Hulkó, et al., Distributed Parameter Systems Blockset for MATLAB & Simulink. Third-Party MathWorks product www.mathworks.com/products/connections/ , 2003-2007. [12] G. Hulkó, et al., Engineering Methods and Software Support for Modeling and Design of Discrete-Time Control of Distributed Parameter Systems. The European Control Conference 2009, Minitutorial, (Accepted for presentation).

Preprint submitted to 7th Asian Control Conference. Received February 24, 2009.