2.4 MODEL DEVELOPMENT 19 number of basic models is small.But object-orientation is also becoming increas- ingly prevalent in digital design using hardware description languages,although in i text it should be regarded more in the context of an increase in in the development of text-based,software-like models,see for example Ecker and Mrva [93]. In mechanics object-orientation has only recently been implemented inorder to make modelling easier,whereby the work of Otter [308]and Kecskemethy [185] in particular.are worth mentioning.One explanation for this is the fact that the number of basic elements and the associated variation in mechanics greater than is the case in electronics.Furthermore,the classic modelling methods of mechanical engineering often lead to descriptions in the form of generalised coordinates,2 which are again incompatible with object-oriented modelling.The advantage of the generalised coordinates is that the resulting equation system has a minimum number of equations and.furthermore.the constraints can be disregarded for holonomous systems.This is attractive from a numerical point of view. How ever,generalised coordinates can only be specified by drawing upon knowledge of the entire system and not from the mole-hill perspective of a component. Resulting equations In this section we will investigate the equations that result from the various mod- elling forms.From a mathematical point of view,a digital gate or the setting of a digital signal in a hardware description language gives an instruction,which is executed after the passage of a predetermined time period.This period corresponds with the time delay of the described block.If the block is defined without a delay thena virtual period of time still passes,the delta time,in which although the sim- ulation time does not proceed,a check is made to ensure that the right-hand sides of all assignments have already been evaluated before the new value of the assign ment under consideration becomes effective.Otherwise the parallel processing of instructions would not be possible. In the case of an analogue circuit,the modified node voltage analysis is generally used,see Vlach and Singhal [410]for a good overview.This establishes differ- ential equations for capacitances and inductances.Transistor models can include one or more parasitic capacitances.Otherwise the heart of transistor models,like diode models,is made up of a parallel circuit consisting of a resistor and a current source,the parameters of which have to be set for each new time interval.This corresponds with an arbitrary linear characteristic that can be placed as a tangent at the current working point on the nonlinear characteristic of the transistor.Volt age and current sources each correspond with constraints that are formulated in algebraic equations.Resistors are also expressed in algebraic equations.Overall a differential-algebraic equation system is established that is also known as DAE 2 See Section 6.2
2.4 MODEL DEVELOPMENT 19 number of basic models is small. But object-orientation is also becoming increasingly prevalent in digital design using hardware description languages, although in this context it should be regarded more in the context of an increase in efficiency in the development of text-based, software-like models, see for example Ecker and Mrva [93]. In mechanics object-orientation has only recently been implemented in order to make modelling easier, whereby the work of Otter [308] and Kecskemethy [185] ´ in particular, are worth mentioning. One explanation for this is the fact that the number of basic elements and the associated variation in mechanics is significantly greater than is the case in electronics. Furthermore, the classic modelling methods of mechanical engineering often lead to descriptions in the form of generalised coordinates,2 which are again incompatible with object-oriented modelling. The advantage of the generalised coordinates is that the resulting equation system has a minimum number of equations and, furthermore, the constraints can be disregarded for holonomous systems. This is attractive from a numerical point of view. However, generalised coordinates can only be specified by drawing upon knowledge of the entire system and not from the mole-hill perspective of a component. Resulting equations In this section we will investigate the equations that result from the various modelling forms. From a mathematical point of view, a digital gate or the setting of a digital signal in a hardware description language gives an instruction, which is executed after the passage of a predetermined time period. This period corresponds with the time delay of the described block. If the block is defined without a delay, then a virtual period of time still passes, the delta time, in which although the simulation time does not proceed, a check is made to ensure that the right-hand sides of all assignments have already been evaluated before the new value of the assignment under consideration becomes effective. Otherwise the parallel processing of instructions would not be possible. In the case of an analogue circuit, the modified node voltage analysis is generally used, see Vlach and Singhal [410] for a good overview. This establishes differential equations for capacitances and inductances. Transistor models can include one or more parasitic capacitances. Otherwise the heart of transistor models, like diode models, is made up of a parallel circuit consisting of a resistor and a current source, the parameters of which have to be set for each new time interval. This corresponds with an arbitrary linear characteristic that can be placed as a tangent at the current working point on the nonlinear characteristic of the transistor. Voltage and current sources each correspond with constraints that are formulated in algebraic equations. Resistors are also expressed in algebraic equations. Overall a differential-algebraic equation system is established that is also known as DAE 2 See Section 6.2
20 2 PRINCIPLES OF MODELLING AND SIMULATION (differential-algebraic equation set).The number of equations depends upon the circuit and is very high,typically significantly above the number of degrees of freedom.The resulting system matrices are however only sparsely occupied. For multibody mechanics,the equations of motion are normally derived by means of the application of a classical principle,e.g.that of Lagrange or D'Alembert.When drawing up the equations it is possible to choose between two extremes.In one case the generalised coordinates,which fully describe the state of a system and which can also be regarded as degrees of freedom,are first determined.For n generalised coordinates (at least for holonomous systems)n equations can be drawn up.The constraints fall away,leaving a system of ordinary differential equations.However, these may turn out to be very complex.Alternatively,it is possible -as in electron- -to permit more unknowns and thereby obtain a system of differential equations for the motion of bodies and algebraic equations for the constraints,which may,for example,be caused by joints.This establishes asystem of DAEs,which can be solved using similar methods to those used in the circuit simulation,see for example Orlan- dea et al.[304].In both cases the number of degrees of freedom is relatively small in comparisonto those.The number of objects under consideration.such as bodies,joints,springs,shock absorbers,etc.is generally significantly below one hundred.However,the numerical problems caused by transitions between static and sliding friction,mechanical impacts,three-dimensional coordinate transformations and other effects,cannot be disregarded. In the representation of continuum mechanics by means of finite elements the number of degrees of freedom is significantly higher than those in multibody mechanics.The associated system matrices normally have a band shape,which the simulation exploits by suitably customised numerical procedures.Overall,this normally establishes a system of ordinary differential equations,the parameters of which,i.e.the inputs into the mass,damping and stiffness matrix,may however have to be recalculated at runtime. 2.4.4 Experimental modelling Introduction Experimental modelling consists of the development of mathematical models of dynamic systems on the basis of measured data or at least providing existing models with parameters.So neither the underlying physics nor the internal life of the system need necessarily play a role in model generation.In contrast to physical modelling there are procedures for experimental modelling in which the modelling can be wholly or partially automated. Table model The simplest method of incorporating measured data is by the formulation of table models that lead to a stepped or piece-wise linear characteristic.The problem with
20 2 PRINCIPLES OF MODELLING AND SIMULATION (differential-algebraic equation set). The number of equations depends upon the circuit and is very high, typically significantly above the number of degrees of freedom. The resulting system matrices are however only sparsely occupied. For multibody mechanics, the equations of motion are normally derived by means of the application of a classical principle, e.g. that of Lagrange or D’Alembert. When drawing up the equations it is possible to choose between two extremes. In one case the generalised coordinates, which fully describe the state of a system and which can also be regarded as degrees of freedom, are first determined. For n generalised coordinates (at least for holonomous systems) n equations can be drawn up. The constraints fall away, leaving a system of ordinary differential equations. However, these may turn out to be very complex. Alternatively, it is possible — as in electronics— to permit more unknowns and thereby obtain a system of differential equations for the motion of bodies and algebraic equations for the constraints, which may, for example, be caused by joints. This establishes a system of DAEs, which can be solved using similar methods to those used in the circuit simulation, see for example Orlandea et al. [304]. In both cases the number of degrees of freedom is relatively small in comparison to those in electronics. The number of objects under consideration, such as bodies, joints, springs, shock absorbers, etc. is generally significantly below one hundred. However, the numerical problems caused by transitions between static and sliding friction, mechanical impacts, three-dimensional coordinate transformations and other effects, cannot be disregarded. In the representation of continuum mechanics by means of finite elements the number of degrees of freedom is significantly higher than those in multibody mechanics. The associated system matrices normally have a band shape, which the simulation exploits by suitably customised numerical procedures. Overall, this normally establishes a system of ordinary differential equations, the parameters of which, i.e. the inputs into the mass, damping and stiffness matrix, may however have to be recalculated at runtime. 2.4.4 Experimental modelling Introduction Experimental modelling consists of the development of mathematical models of dynamic systems on the basis of measured data or at least providing existing models with parameters. So neither the underlying physics nor the internal life of the system need necessarily play a role in model generation. In contrast to physical modelling there are procedures for experimental modelling in which the modelling can be wholly or partially automated. Table model The simplest method of incorporating measured data is by the formulation of table models that lead to a stepped or piece-wise linear characteristic. The problem with
2.4 MODEL DEVELOPMENT 21 the trivial conversion of a table model is the abrupt changes or kinks that are caused by the fact that only a finite number of values are available.The difficulties are erical in nature since numerical oscillations may occur at abrupt change and kinks.These are caused by the fact that-as a result of feedback-different sections of the characteristic are approached alternately and this may impair or even prevent the convergence of the simulation.A possible solution is offered by procedures that smooth the characteristic,such as the Chebychev or Spline approximations. Parameter estimation and system identification In this connection we can differentiate between two aspects:Parameter estimation and system identification.Parameter estimation requires a model and considers the parameters that belong to it.Some parameters,such as mass or spring constants are generally accessible without parameter estimation,whereas other parameters e.g.coefficients of friction,can often only be determined within the framework of parameter estimation.The identified parameters then ensure the best possible correspondence between simulation and measurement. In system identification,on the other hand,a model for the system is created on this basis or selected from a group of candidates.This is generally efficient and numerically unproblematic.The quality criterion here is the degree of corre- spondence that can be achieved using parameter estimation.The two significant disadvantages of parameter estimation and system identification are that,firstly,a measured result must be available in advance,which means that the system can only be considered after its development and manufacture.Secondly,the results are often not transferable.or at least not in a straightforward manner.to variations of the system or of components There are typically four stages to a system identification,see for example, Kramer and Neculau [206]or Unbehauen [405]and Figure 2.5. Signal analysis Selection of a quality criterion Calculat Figure 2.5 System identification sequence
2.4 MODEL DEVELOPMENT 21 the trivial conversion of a table model is the abrupt changes or kinks that are caused by the fact that only a finite number of values are available. The difficulties are numerical in nature since numerical oscillations may occur at abrupt changes and kinks. These are caused by the fact that — as a result of feedback — different sections of the characteristic are approached alternately and this may impair or even prevent the convergence of the simulation. A possible solution is offered by procedures that smooth the characteristic, such as the Chebychev or Spline approximations. Parameter estimation and system identification In this connection we can differentiate between two aspects: Parameter estimation and system identification. Parameter estimation requires a model and considers the parameters that belong to it. Some parameters, such as mass or spring constants are generally accessible without parameter estimation, whereas other parameters, e.g. coefficients of friction, can often only be determined within the framework of parameter estimation. The identified parameters then ensure the best possible correspondence between simulation and measurement. In system identification, on the other hand, a model for the system is created on this basis or selected from a group of candidates. This is generally efficient and numerically unproblematic. The quality criterion here is the degree of correspondence that can be achieved using parameter estimation. The two significant disadvantages of parameter estimation and system identification are that, firstly, a measured result must be available in advance, which means that the system can only be considered after its development and manufacture. Secondly, the results are often not transferable, or at least not in a straightforward manner, to variations of the system or of components. There are typically four stages to a system identification, see for example, Kramer and Neculau [206] or Unbehauen [405] and Figure 2.5. Signal analysis Specification of the modelling method Selection of a quality criterion Calculation of the parameters Figure 2.5 System identification sequence
22 2 PRINCIPLES OF MODELLING AND SIMULATION The first stage of signal analysis is the establishment of a suitable test signal, which is triggered by the system.Possibilities here are step functions,rectan- gular pulses,triangular pulses and many more.An inspection in the frequency range facilitates investigations into whether the system to be identified is suffi- ciently excited over the spectrum of interest.Measurements are generally only made at discrete time points,so that a sampling interval must also be determined. Furthermore,a measurement time must be specified,the lower limit of which is determined by the point at which sufficient data is available for identification.The progressive nature of a real system imposes an upper limit on the measurement time.Then,signal processing procedures may also be used,such as averaging. root mean square calculation,or Fourier analysis,correlation analysis,and spec tral analysis.So,for example,statistically dispersive noise signal components can be disposed of by averaging similar measurements which,however,multiplies the measurement time In stage two,determining the model approach,we can choose between prefabri- cated and customised structures,see for example Ljung [234].The former may,for example,consist of canonical models in the state space and lead to a'black-box parameterisation,i.e.model structure and parameters have no physical significance, but rather serve merely as a vehicle for reflecting the observed behaviour.Cus- tomised equation system structures,on the other hand,are based upon a physical modelling of the system,so that the identified parameters also possess a physi- cal significance.In any case,however,all available information about the system should be fed into this.This applies in particular to the faults that are virtually always present,which in most cases rule out an exact solution. The identification typically rests upon minimising the discrepancy between mea- surement and simulated behaviour or a functional of this.Various quality criteria are used for this,one of which is selected in the third stage.Criteria are particularly frequently selected that assess a quadratic function of the measurement error. To conclude the identification,numerical procedures are used in order to min imise the quality criteria selected in the third stage.These procedures are performed for all model structures proposed in the second stage.so that not ony are the param- eters in question determined in this stage,the quality of the individual structures in relation to one another are also established.This facilitates a selection of the model structure. In the simplest case we can,as in Kramer and Neculau [206],quote the following equation for the system under investigation: (2.1) where xk denotes an input quantity,yk an output quantity,nk a disturbance vari- able in relation to the mea ement and a is the parameter to be estimated.This relationship should be modelled on the basis of the following approach: (2.2)
22 2 PRINCIPLES OF MODELLING AND SIMULATION The first stage of signal analysis is the establishment of a suitable test signal, which is triggered by the system. Possibilities here are step functions, rectangular pulses, triangular pulses and many more. An inspection in the frequency range facilitates investigations into whether the system to be identified is suffi- ciently excited over the spectrum of interest. Measurements are generally only made at discrete time points, so that a sampling interval must also be determined. Furthermore, a measurement time must be specified, the lower limit of which is determined by the point at which sufficient data is available for identification. The progressive nature of a real system imposes an upper limit on the measurement time. Then, signal processing procedures may also be used, such as averaging, root mean square calculation, or Fourier analysis, correlation analysis, and spectral analysis. So, for example, statistically dispersive noise signal components can be disposed of by averaging similar measurements which, however, multiplies the measurement time. In stage two, determining the model approach, we can choose between prefabricated and customised structures, see for example Ljung [234]. The former may, for example, consist of canonical models in the state space and lead to a ‘black-box’ parameterisation, i.e. model structure and parameters have no physical significance, but rather serve merely as a vehicle for reflecting the observed behaviour. Customised equation system structures, on the other hand, are based upon a physical modelling of the system, so that the identified parameters also possess a physical significance. In any case, however, all available information about the system should be fed into this. This applies in particular to the faults that are virtually always present, which in most cases rule out an exact solution. The identification typically rests upon minimising the discrepancy between measurement and simulated behaviour or a functional of this. Various quality criteria are used for this, one of which is selected in the third stage. Criteria are particularly frequently selected that assess a quadratic function of the measurement error. To conclude the identification, numerical procedures are used in order to minimise the quality criteria selected in the third stage. These procedures are performed for all model structures proposed in the second stage, so that not only are the parameters in question determined in this stage, the quality of the individual structures in relation to one another are also established. This facilitates a selection of the model structure. In the simplest case we can, as in Kramer and Neculau [206], quote the following equation for the system under investigation: yk = a · xk + nk (2.1) where xk denotes an input quantity, yk an output quantity, nk a disturbance variable in relation to the measurement and a is the parameter to be estimated. This relationship should be modelled on the basis of the following approach: yˆk = aˆ · xk (2.2)
2.4 MODEL DEVELOPMENT 23 X Real 一mi Figure 2.6 Comparison between real system and model for parameter estimation This can also be graphically represented as shown in Figure 2.6.The aim of this is to minimise the quality function Q.so that the estimated parameter a is optimised in relation to o. A common approach for the quality function Q is to find an expression that is proportional to the quadratic average of the error signal ex: Q=∑c=∑-2=∑-ax)月 (2.3) k=1 k=1 k=l wheren is the number of measurements.For a compact representation the signals should henceforth be regarded in the form of n-dimensional vectors: xT =[x1x2.xn] y=yy2.yal T=12.9l (2.4 eT=[el e2.en] Thus the quality function can be described in vector notation as follows: Q=eTe=(y-ax)T.(y-aix)=yTy-2ayTx +axx (2.5) Now Q should be minimised in relation to a.For this to be achieved the partial derivative of Q in relation to a must become zero,i.e.: =-2y5x+2x'x=0 aQ (2.6) Solving this with respect to a finally gives: (2.7) Equation (2.7)is also called a regression and represents the solution for the method of least squares [206].The inclusion of information on the interference process
2.4 MODEL DEVELOPMENT 23 Model â·xk xk nk − ek Q min â yk yk Real system a·xk ˆ Figure 2.6 Comparison between real system and model for parameter estimation This can also be graphically represented as shown in Figure 2.6. The aim of this is to minimise the quality function Q, so that the estimated parameter a is optimised ˆ in relation to Q. A common approach for the quality function Q is to find an expression that is proportional to the quadratic average of the error signal ek: Q = n k=1 e2 k = n k=1 (yk − ˆyk) 2 = n k=1 (yk − ˆa · xk) 2 (2.3) where n is the number of measurements. For a compact representation the signals should henceforth be regarded in the form of n-dimensional vectors: xT = [x1 x2 . xn] yT = [y1 y2 . yn] ˆyT = [ˆy1 ˆy2 . ˆyn] eT = [e1 e2 . en] (2.4) Thus the quality function can be described in vector notation as follows: Q = eTe = (y − ˆax) T · (y − ˆax) = yTy − 2ˆayTx + ˆa 2 xTx (2.5) Now Q should be minimised in relation to a. For this to be achieved the partial ˆ derivative of Q in relation to a must become zero, i.e.: ˆ ∂Q ∂ ˆa = −2yTx + 2ˆaxTx = 0 (2.6) Solving this with respect to a finally gives: ˆ ˆa = yTx xTx (2.7) Equation (2.7) is also called a regression and represents the solution for the method of least squares [206]. The inclusion of information on the interference process