Discretization of Continuous Linear Anisochronic Models Milan Hofreiter Czech echnical University in Prague Faculty of Mechanical Engineering echnicka 4, 66 7 Prague 6 CZECH REPUBLIC E-mail: hofreite@fsid.cvut.cz Abstract: Much work has been done on analysis and control of continuous or discrete time-delay systems. However, the discretization of continuous time-delay systems has not been extensively studied. his paper introduces an original approach to the discrete approximation of continuous-time linear anisochronic models by discrete models for the purposes of numerical modelling, simulation, identification and control. Depending on the required numerical accuracy of the approximation, this method employing the aylor series enables us to determine the order and the resulting structure of a discrete linear model appropriate for numerical processing. he technique is simultaneously applied, more in greater detail, to a continuous anisochronic model of the nd order which is often used in applications including various forms of delays that are very closely related to the existence of continuously distributed parameters. In addition, this work includes derived, original formulae which enable us to convert directly an original continuous linear anisochronic model of the nd order into the corresponding discrete approximation. Finally, an example is given of the application of the method, and a measure of agreement is shown between the original continuous anisochronic model representing the system and a discrete approximation of it. Keywords: discretization, delay, anisochronic model, internal point delay Milan Hofreiter Born st March 95 in Prague, Associate Professor, Institute of Instrumentation of Control echnology, Section of Automatic Control and Engineering, CU Prague, Fac. of Mech. Eng., Education: Graduated 975 from CU Prague, Fac. of Mech. Eng., CSc degree awarded in 98 with a dissertation on: Choice of inputs and order of the regression model of a stochastic system. Associate Professor since 998. Fields of research: System Identification, Computer Modelling, Adaptive Control. Practice: Research engineer at the Institute SVÚS Prague (975-993), Assistant Professor, Department of Automatic Control, CU Prague, Fac. of Mech. Eng., (993-998), Associate Professor, Institute of Instrumentation and Control Engineering, Section of Automatic Control and Engineering, CU Prague, Fac. of Mech. Eng.,( 998 till present). List of publications contains : 3 textbooks, research reports, 45 papers, 4 patents. Introduction he rapid development of digital technology has affected the fields of modelling, simulation, identification and control of so-called hereditary systems, i.e. systems where time delays and latencies are an essential feature of process dynamics. ime delays and latencies are closely connected with mass, energy and information transport. Considerable attention has therefore been paid to these systems in recent years. o describe such models, so-called anisochronic models (as opposed to isochronic models) are nowadays often applied. Anisochronic models enable multi-parameter systems (multi-dimensional systems) to be described including not only transport delays of these input variables but also internal delays of their state variables. For more detailed information about anisochronic models, see e.g. [7] and [9]. he linear models represented by the matrix differential equation () and the matrix equation () are anisochronic models which have been often used. m xt & () = Axt ( ϑ ) + B ut ( τ j ) () i i j i= j= n yt () = C xt () () where, x& () t is the first derivative of state vector x at time t, xt ( ϑ i ) is the state vector at time t ϑ i, ut ( τ j ) is its input vector at time t τ j, Studies in Informatics and Control, Vol., No., March 3 69
A, i =,, L, m and B, j =,, L, n and C i j are particular constant matrices, ϑ =, ϑ i >, i=,, L, m, and τ =, τ j >, j =,, L, n are the delays of the state variables and of the input variables respectively. If anisochronic model () is to be used for digital modelling, simulation, identification or control, it is necessary to find the corresponding discrete approximation of this linear model. In the case of discrete approximation of continuous linear models where the influences of transport delays are not considered, the field of theory of alternative discrete model design has been well covered, see, e.g. [3]. Although much work has been done on analysis and control of continuous time-delay systems, there are only a few published methods of discrete approximation of linear anisochronic models for cases which respect internal and external point delays. he basic reason is that the characteristic equations of anisochronic models are not algebraic but transcendental. Such equations admit an arbitrarily large number of roots in the complex domain, and each of the roots tends to influence the resulting response of the system even in the case of an unlimited set of roots. For this reason not all the roots of the characteristic equation can be determined, as required in common methods of classic discrete approximation of continuous models. his problem has been solved by searching for dominant roots [4], [6], [8], using FIR filters [5] or with the help of recursive solution of the state vector []. In this paper the proposed discretization method does not require determination of the roots of the characteristic equation, and allows us to design the discrete model with respect to the required accuracy.. Discrete Approximation of a Linear Anisochronic Model In order to avoid looking for the roots of the characteristic equation of an anisochronic model, the aylor series expansion (3) evaluated in the neighbourhood of the operating point x() t is used to obtain its discrete approximation. 3 & && &&& L (3) xt ( + ) = xt () + xt () + xt () + xt () +! 3! where is a step of discrete approximation he derivatives of the state vector can be obtained using equation (); e.g. it holds for m && xt () = Axt & ( ϑ ) + B ut & ( τ j ). (4) i i j i= j= n && x() t he goal is to obtain a model suitable for digital control where the input vector changes only at discrete moments. Assumption (5) can be accepted for time instants t = k, k =,,, L [ ] ut τ &( ) = for j =,, L,n (5) j [ ] where is the vector of zeros. Applying formula () once again, formula (4) yields m m n && xt () = Ai Aixt ( ϑi ϑ i) + Bjut ( ϑi τ j) = i= i= j= 7 Studies in Informatics and Control, Vol., No., March 3
m m m n = A A x( t ϑ ϑ ) + A B u( t ϑ τ ) i i i i i j i j i= i= i= j= m n xt ( + ) = xt ( ) + Aixt ( ϑ i) + Bjut ( τj) + i= j=. (6) Each state vector derivative occurring in the aylor series expansion (3) can be evaluated by application of the method introduced above, and the resulting formula (7) is obtained after substitution into those derivatives. m m m n + A A x( t ϑ ϑ ) + A B u( t ϑ τ +! 3 3! ) i i i i i j i j i= i= i= j= m m m + AiAiAi3x( t ϑi ϑi ϑ i3) + i= i= i3= m m n + AiAiBj u( t ϑi ϑi τj i= i= j= ) + L (7) he number of elements of the aylor series expansion is optional and should be chosen with respect to the required accuracy of the approximation. his results from the fact that the approximation error of th aylor s polynomial of the l order at time t + is vector et+ ( ), l+ ( l+ ) et ( + ) = x ( ξ)! ( l + ) where ξ (, tt+). he particular value of the ( l + ) th state vector derivative at time ξ, i.e. ( l ) x + ( ξ), can again be calculated according to formula (). If the time arguments of the state vectors or the input vectors are not commonly integer multiples of the discrete approximation time step, it is necessary to apply a method that uses known vectors obtained at time instants k, where k =,,, L and which thus enables the state vectors to be estimated. Various interpolation methods [4] can be utilized for this purpose. So-called linear interpolation is one of the simplest methods and is satisfactory in most cases. Interpolation using spline functions is a more sophisticated method that takes advantage of the application of formula (). 3. Discrete Approximation of a Anisochronic and Order Model In this section, the described common method of discrete approximation will be applied to a linear anisochronic model of the nd order with the s-operator transcendental transfer function (8) Ke Gs () = ϑ s ( s+ ) ( s+ e ) τ s (8) where K is the static gain, τ is the transport delay of the input,, are time constants, and ϑ is the feedback delay (which is determined by the location of the inflex point of the transient response, see Fig.) ransfer function (8) corresponds to differential equation (9) && yt () + yt & () + yt & ( ϑ ) + yt ( ϑ ) = Kut ( τ) (9) Studies in Informatics and Control, Vol., No., March 3 7
Because of its considerable universality (its structure remains the same regardless of the plant which is to be approximated), this model has been used increasingly in recent years for describing real processes in practice, e.g. [],[6],[7],[9]. Using the standard method of successive integration, model (9) will be transformed into the state-space representation. x ( t ϑ) = Ku t ϑ x() t x( t ϑ) () = () x () t () x& () t ( ) x& t x t yt =. () hen, the set of equations () can be modified as given by (), which corresponds to the matrix notation () xt &() = A xt () + Axt ( ϑ ) + But ( τ). () where x () t xt () = x() t and A = (3) A = K B =. (4) (5) By applying the technique introduced in section, the elements of the aylor expansion series (3) can be determined with the use of formula (). As it holds A A =, (6) the state-space model representation below can be derived () xt ( +=Φ ) ( ) xt ( ) + AΦ( ) xt ( ϑ+ ) A Φ( ) xt ( ϑ+ ) 3 3 L + A Φ ( ) x( t 3 ϑ ) + +Φ ( ) B u( t τ ) + 3 + AΦ ( ) But ( τ ϑ ) + A Φ ( ) But ( τ ϑ ) + L where Φ ( ) is the fundamental matrix which is defined by the infinity series 3 (7) A 3 ( ) e I A A A 3! Φ = = + + + +L (8) where I is the unit matrix, and it holds for Φ ( ), i =,,3, L that i( ) i ( x) dx i Φ = Φ. (9) he fundamental matrix Φ ( ) may easily be derived by the inverse Laplace transform 7 Studies in Informatics and Control, Vol., No., March 3
{ [ ] } L s I A Φ ( ) = () t L = where denotes the inverse Laplace transform, and s is a complex variable which is related to the Laplace transform. It follows from formulae (3) and () for Φ ( ) ( ) Φ = ( e ) e. () By recurrent computation using (9) and (), Φ ( ), i =,,3, L can be evaluated. hus, for example, it holds for Φ ( ), Φ ( ), Φ ( ) that 3 Φ( ) = + e e () Φ( ) = (3) e + + e 3 3! Φ3( ) = 3 3 3. (4) 3! e + + e + If the formulae introduced above are utilized to obtain the discrete approximation of the anisochronic model represented by the operator transfer function (9), i.e. by differential equation () with parameters = s,= s, K =, ϑ=.3 s, τ =.45 s, then the resulting anisochronic model discrete approximation with the discrete time step =.s is xk ( + ) = P xk ( ) + Pxk ( 3.) + P xk ( 6.4) + Pxk ( 9.6) + L 3 L+ U u( k 4.5) + U u( k 7.7) + U u( k.9) + L yk ( ) = Cxk ( ) where P ( ).49 3.48 =Φ =, P.95.95 = AΦ( ) =.49 3,.48 4.63 5.9 3 P = A Φ( ) = 4.63 5.9 3, C = [.5], 3 5.6 7.3 5. P3 = A Φ3( ) = 5.6 7.3 5, U =Φ( ) B = 3 4,837, 8.9 5. 6 U = AΦ( ) B =, U A 3( ) B 5 8.9 = Φ =. 6, i (5) Studies in Informatics and Control, Vol., No., March 3 73
and it holds for the discrete time k that t = k. he decreasing influence of the former state space variables and also of the inputs can be observed on the previous formulae, and consequently the order of the discrete model may easily be set with respect to the required accuracy. Simultaneously, it is easily seen that the columns of particular matrices Pi i, =,, L and of each vector U j, j =,, L contain the same values. For the case of considering only matrices P, P, P and vector U for a discrete model, the transient responses (Fig.) and frequency responses (Fig.3) of the two compared models are shown to demonstrate agreement between the anisochronic model (9) and its discrete approximation (5). he influence of former values of the state variables and of the inputs are neglected (elements of matrices 4 Pi= i, 3,4, L and vectors U j, j =,, L are less than ). 4. Conclusion Discrete approximation of continuous systems is a necessary step when their models are to be numerically implemented. he original method of discrete approximation of a linear anisochronic model introduced here can be applied to the field of numerical modelling, simulation, identification and control of systems with external and internal delays. he method does not require knowledge of the roots of a characteristic transcendental equation, and allows the structure and the order of the discrete model to be estimated in relation to the required accuracy of the approximation. he method also enables derivation of expressions (7), (8) and (9), which are applicable to the wide class of hereditary systems described by the anisochronic model. Further research will be aimed toward further generalization and seeking for explicit formulae applicable in the field of technical cybernetics. Acknowledgement: his research has been supported by GACR grant No. GP 65 74 Studies in Informatics and Control, Vol., No., March 3
τ ϑ time Figure. y(t).9.8.7.6.5 anisochronic model.4.3. discrete approximation. 5 5 time [s] Figure. Studies in Informatics and Control, Vol., No., March 3 75
Im.. discrete approximatio.4.6 anisochronic model (9).8.4...4.6.8 Re Figure 3. REFERENCES. HOFREIER, M., Parameter Identification of Anisochronic Models, In: Annals of DAAAM for & Proceedings of the th International DAAAM Symposium, Jena, pp.85-86, 4-7th October.. JUGO, J., Discretization of Continuous ime-delay Systems, 5th IFAC World Congress on Automatic Control, IFAC, Barcelona, Spain,. 3. OGAA, K., Discrete-ime Control Systems, Prentice Hall Inc., London, 995. 4. PEROVA, R., Discrete Approximation of Anisochronic Models of a Delay System in a ransform, CU Prague, Ph.D. hesis, Prague (in Czech), CZ.,. 5. DE LA SEN, M. and LUO, N., Discretization and FIR Filtering of Continuous Linear Systems with Internal and External Point Delays, Int. J. Control, 6, n. 6, pp. 3-46, 994 6. VYHLIDAL,. and ZIEK, P., Control System Design Based on a Universal First Order Model with ime Delays, In: : Acta Polytechnica, vol. 4, no. 4-5, pp. 49-53,. 7. ZIEK, P., ime Delay Control System Design Using Functional State Models, CU Reports No./998, Prague, 998. 8. ZIEK, P. and PEROVA, R., Discrete Approximation of Anisochronic Models of Systems with Delays, In: Intelligent Systems for Practice, Ostrava, CZ., pp. 9-6,. 9. ZIEK, P. and HLAVA, J., Anisochronic Internal Model of ime-delay Systems, Control Engineering Practice, Vol. 9, No. 5, pp. 5-56,. 76 Studies in Informatics and Control, Vol., No., March 3