In this chapter various lumped and distributed models of LDS will be formulated in both, time and transform (L or Z) domains. We will start with basic dynamic characteristics of LDS, then we will show the decomposition of dynamics to time and space components.
As a conclusion of this chapter, the accuracy of models formulated using discretization in the space domain will be analysed.
Linear deterministic models
For illustration, first assume a linear timeinvariant LDS, distributed within the interval <0,L>, and timespatial variables in an axonometric coordinate system. Results achieved will be then generalized for common types of DPS.
Further assume, provided zero initial and boundary conditions are given, that an input variable U_{i}(t) is fed into LDS. The output of the system will be in the form of distributed variable, Y_{i}(x,t) – see Fig. 3.1.1.
Fig. 3.1.1 LDS response to individual action of ith input variable
LDS  – lumpedinput/distributedoutput system 
U_{i}(t)  – ith lumped input variable 
Y_{i}(x,t)  – ith distributed output variable 
Y_{i}(x_{i},t)  – ith partial distributed output variable at point x_{i} 
The relationship between these variables can be described by the convolution product:
(3.1.1) 
where is the LDS distributed impulse response function, see Fig. 3.1.2. When U_{i}(t) is a unitstep (Heaviside) function in the output of LDS the distributed transient response function, _{i}(x,t), can be obtained.
Fig. 3.1.2 LDS distributed impulse response function
LDS  – lumpedinput/distributedoutput system 
U_{i}(t) = (t)  – ith lumped input, unit impulse (Dirac) (t)function 
_{i}(x,t)  – ith distributed impulse response function 
_{i}(x_{i},t)  – ith partial distributed impulse response function at point x_{i} 
Similarly, it is possible to get relationships between the individual input variables and the corresponding distributed output variables .
The resulting distributed output variable can be then expresed by the equation
(3.1.2) 
Now, choose measurement points with coordinates x_{1},…,x_{i},…,x_{n} near to input points of the control variables, U_{1}(t),…,U_{i}(t),…,U_{n}(t) feeding. At these points, an ideal measurement process can be assumed, without any distortion of measured variables. At these points, on individual surfaces Y_{1}(x,t),…,Y_{i}(x,t),…,Y_{n}(x,t) partial distributed outputs – curves Y_{1}(x_{1},t),…,Y_{i}(x_{i},t),…,Y_{n}(x_{n},t) – are obtained – see Fig. 3.1.3.
Further, at points x_{1},…,x_{i},…,x_{n} and on surfaces of individual distributed impulse response functions, partial distributed impulse response functions ,…,,…,, Fig. 3.1.2, can be found.
Using relationship (3.1.1), it yields:
(3.1.3)  
(3.1.4)  
(3.1.5) 
See Fig. 3.1.3.
Fig. 3.1.3 Partial distributed output variables at measurement points
x_{1},.., x_{i},.., x_{n}
LDS  – lumpedinput/distributedoutput system 
U_{1}(t),…,U_{i}(t),…,U_{n}(t)  – lumped input variables 
Y_{1}(x_{1},t),…,Y_{i}(x_{i},t),…,Y_{n}(x_{n},t)  – partial distributed output variables at points x_{1},…,x_{i},…,x_{n} 
Assuming discrete input variables and zeroorder hold units, {H_{i}}_{i} . Discrete models can be expressed, similarly, in the form of the convolution summation:
(3.1.6) 
where is distributed impulse response function of LDS with zeroorder hold unit, H_{i}: HLDS.
For simplicity, assume further T = 1 and = 0. Hence eq. (3.1.6) gives:
(3.1.7) 
See Fig. 3.1.4. When zeroorder hold units, {H_{i }}_{i} are used, the distributed discrete impulse response functions for t = k can be obtained as follows
(3.1.8) 
Fig. 3.1.4 Influence of the U_{i}(k) sequence on LDS with hold unit H_{i}
LDS  – lumpedinput/distributedoutput system 
H_{i}  – zeroorder hold unit 
U_{i}(k)  – ith input sequence 
Y_{i}(x,k)  – ith distributed output variable 
Y_{i}(x_{i},k)  – ith partial distributed output variable at point x_{i} 
Then the overall distributed output variable of LDS and {H_{i }}_{i}: HLDS can be expressed in the form:
(3.1.9) 
Hereinafter, the lumped and distributed variables will be mostly assumed to be defined in discrete time instants. When reguired, distributed disturbances and output variable of HLDS block will be assumed as continuous signals, Figs. 3.1.5, 5.2.1, 5.2.2, etc.
Fig. 3.1.5 HLDS output variable
LDS  – lumpedinput/distributedoutput system 
H  – block of zeroorder hold units {H_{i }}_{i} 
Y(x,t)  – continuous distributed output variable 
– vector of lumped input variables 
For points x_{1},…,x_{i},…,x_{n} located in surroundings of lumped input variables feeding, partial distributed output variables, Y_{1}(x_{1},k),…,Y_{i}(x_{i},k),…,Y_{n}(x_{n},k) or partial distributed impulse response functions, H_{1}(x_{1},k),…,H_{i}(x_{i},k),…,H_{n}(x_{n},k) can be obtained.
Then the following convolution summations hold true:
(3.1.10)  
(3.1.11)  
(3.1.12) 
Operating in s or z domain, we get the relations:
(3.1.13)  
(3.1.14)  
(3.1.15)  
(3.1.16)  
(3.1.17) 
where {S_{i}(x,s)}_{i} and S_{1}(x_{1},s),…,S_{i}(x_{i},s),…,S_{n}(x_{n},s) are the respective transfer functions.
Further, it holds
(3.1.18)  
(3.1.19)  
(3.1.20)  
(3.1.21)  
(3.1.22) 
where {SH_{i}(x,z)}_{i}, resp. SH_{1}(x_{1},z),…,SH_{i}(x_{i},z),…,SH_{n}(x_{n},z) are transfer functions of the distributed system under consideration assuming zeroorder hold units.
This approach allows to associate LDS and zeroorder hold units {H_{i}}_{i}: HLDS with distributed and lumped blocks as shown in Figs. 3.1.6 – 8.
Fig. 3.1.6 Distributed systems associated with HLDS
Fig. 3.1.7 HLDS model
Fig. 3.1.8 Lumped systems associated with HLDS
When the vector of lumped variables, {U_{i}(k)}_{i} , operates as input of the block HLDS, the overall output variable, Y(x,k) is obtained. Let us associate lumped and distributed components with this distributed variable by means of introduced lumped and distributed systems.
When the vector of {U_{i}(k)}_{i }is entering at points x_{1},…,x_{i},…,x_{n} , in outputs of blocks SH_{1}(x_{1},z),…, SH_{i}(x_{i},z),…, SH_{n}(x_{n},z) partial distributed output variables Y_{1}(x_{1},k),…,Y_{i}(x_{i},k),…,Y_{n}(x_{n},k) are generated. Distributed output variables from blocks SH_{1}(x,z),…,SH_{i}(x,z),…,SH_{n}(x,z): Y_{1}(x,k),…, Y_{i}(x,k),…,Y_{n}(x,k) „slide„ on these trajectories – see Fig. 3.1.9.
Hence, the overall output variable is given by the relationship:
Fig. 3.1.9 Y_{i}(x,k) „ sliding“ on trajectory Y_{i}(x_{i},k)
HLDS  – lumpedinput/distributedoutput system with zeroorder hold units 
U_{i}(k)  – ith lumped input variable 
Y_{i}(x_{i},k)  – ith partial distributed output variable 
Y_{i}(x,k)  – ith distributed output variable 
– sliding direction 
When the discrete representation is considered, the distributed system dynamics is given by the set of discrete distributed impulse response functions, . Let us introduce on this set the reduced characteristics in the following form:
(3.1.23)  
(3.1.24)  
(3.1.25) 
Some reduced characteristics are shown in Fig. 3.1.10.
Fig. 3.1.10 Partial reduced discrete impulse responses
HLDS  – lumpedinput/distributedoutput system with zeroorder hold units 
– ith distributed impulse response function  
– ith partial distributed discrete impulse response function at point x_{i}, along time k  
,  – ith partial distributed discrete impulse response functions in the direction x 
,  – ith reduced partial distributed discrete impulse response functions in the direction x 
Using the reduced functions, eq. (3.1.7) yields:
(3.1.26)  
(3.1.27)  
(3.1.28) 
For x_{i} e.g. eq. (3.1.27) gives
(3.1.29) 
since HR_{i}(x_{i},k) = 1.
It means that H_{i}(x_{i},k) determines the partial distributed output variable in the time domain at point x_{i}. See Fig. 3.1.11.
Fig. 3.1.11 Distributed output variable in time and space domains
HLDS  – lumpedinput/distributedoutput system with zeroorder hold units 
U_{i}  – ith lumped input variable 
Y_{i}(x_{i},k)  – ith partial distributed output variable in the time domain 
Y_{i}(x,k)  – ith partial distributed output variable in the space domain 
The distributed output variable in the space domain at a point k is given again by linear combination of reduced dynamic characteristics, {HR_{i}(x,k)}_{i,k} according to eq. (3.1.2628).
Following this approach, the dynamics of the object under consideration is decomposed to
– time components
{H_{i}(x_{i},k)}_{i,k}  (3.1.30) 
for i, x_{i }chosen – variable k
– space components
{HR_{i}(x,k)}_{i,k}  (3.1.31) 
for i, k chosen – variable x.
Now let us analyse the decomposition of the distributed output variable and dynamic characteristics in steadystate.
The steadystate of distributed output variable, Y(x,), can be expressed using steadystate distributed transient response functions: , where are the transient responses of HLDS to unitstep functions, 1(k), at each input, , Fig. 3.1.12.
Fig. 3.1.12 ith distributed transient response of HLDS
HLDS  – lumpedinput/distributedoutput system with zeroorder hold units 
U_{i} = 1(k)  – ith lumped input variable, discrete unitstep function 
– ith distributed transient response  
– ith distributed transient response at a time q  
– ith partial distributed transient response at a point x_{i} 
Now, introduce reduced steadystate distributed transient responses at a time k – Fig. 3.1.13 – in the form:
(3.1.32) 
Then
(3.1.33)  
(3.1.34)  
(3.1.35) 
See Fig. 3.1.14, where
(3.1.36)  
(3.1.37) 
In this limit case, the dynamics of the object under consideration in steadystate is split into
– time components
(3.1.38) 
– space components
(3.1.39) 
Fig. 3.1.13 ith steadystate distributed transient response and reduced transient response
HLDS  – lumpedinput/distributedoutput system with zeroorder hold units 
– ith steadystate distributed transient response  
– ith steadystate distributed reduced transient response  
U_{i} = 1(k)  – ith lumped input variable, discrete unitstep function 
Fig. 3.1.14
HLDS  – lumpedinput/distributedoutput system with zeroorder hold units 
{U_{i}()}_{i}  – steadystate lumped input variables 
Y(x,)  – overall distributed output variable 
Y_{1}(x_{1},),…,Y_{i}(x_{i},),..,Y_{n}(x_{n},)  – lumped output variables, partial distributed output variables in points x_{1},…,x_{i},…,x_{n} respectively 
{HR_{i}(x, )}_{i=1,n}  – steadystates of the distributed reduced transient responses 
In control the further decomposition of the distributed system dynamics will be frequently used
– time components
(3.1.40) 
– space components
{HR_{i}(x, )}_{i}  (3.1.41) 
Model accuracy
In modeling of LDS, discretization in both, time and space domain, similarly as in numerical solutions of PDE or FEM, will be used . The distributed output variable will be calculated or measured at points appropriately chosen within an interval <0,L>, in the form of discrete values. In other points of this interval, values of distributed output variable will be fitted using appropriate interpolations methods.
Let us select the following points for spatial discretization from the above interval
{0=x_{1},..,x_{i},..,x_{j},…,x_{n},…,x_{m}=L}
Notice that the above points are, in general, different from the ones selected for measurement in the neighbourhood of actuating inputs.
Now let us carry out spatial discretization of the discretetime model, having lumpedinput distributedoutput – eq. (3.1.9). Then at points {x_{j}}_{j=1,m} chosen we have
(3.2.1) 
or in a vector form
(3.2.2) 
where {U_{i}(k)}_{i,k} are sequences of HLDS input variables and {H_{i}(x_{j},k)}_{i,j,k} represent the values of discrete distributed impulse response functions. Assume, that for particular i, j and r > q, the following relationship holds true: H_{i}(x_{j},r) = 0, Fig. 3.2.1. Where q denotes the LDS settling time.
Fig. 3.2.1 Values of ith distributed impulse response function at points {x_{j}}_{j=1,m}
HLDS  – lumpedinput/distributedoutput system with zeroorder hold units 
U_{i}  – ith lumped input variable 
– values of ith discrete distributed impulse response function after discretization in space domain 
Eq. (3.2.2) can be rewritten into the form:
(3.2.3) 
Here, particular matrices correspond to distributed discrete impulse response functions of the system under consideration, see Fig. 3.2.1.
Values of distributed output variable at other points of the interval <0,L> can be obtained by interpolation methods through spline functions. The approximation procedure is to be chosen with respect to the desired accuracy. Therefore in further no separate distinction will be made between approximates and approximated variables.
In further presented results will be exploited at optimal design of DPS. Therefore, the reader interested mainly in modeling and control area can skip the rest and continue with next chapter.
The accuracy of the model constructed is ensured by standard methods of numerical mathematics. As an illustration, let us now analyse the accuracy of the model in the neigbourhood of the steadystate distributed output variable, e.g. of Y(x,), see Fig. 3.1.14.
Differential properties of Y(x,) will be evaluated using modulus of continuity and higher order derivative norms in appropriate functional spaces [22, 53, 54, 75, 83]:
(3.2.4) 
for h L
(3.2.5)  
etc.  (3.2.6) 
In the interpolation using cubic spline functions, provided uniform displacement of approximation nodes is assumed, the deviations between distributed variable, Y(x,), and its approximation, Ya(x,), are less than maximal values presented in Tab. 3.1. These values were determined in the theory of spline functions, e.g. Y. S. Zavialov et. al. [83].
Tab. 3.1 Deviations between Y(x, ) and Ya(x,)
Classes of approximated functions Y(x,)  Maximal values of deviations between functions Y(x,) and Ya(x,) 
– net of approximating nodes in the interval <0,L> : 0 = x_{1} <… < x_{i} < x_{i+1} <… < x_{m} = L, length of an interval <x_{i}, x_{i+1>}: x_{i+1} – x_{i} = h 

– modulus of continuity  
C^{k}<0,L>  – class of functions with kth order continuous derivatives 
– class of functions f(x) in interval <0,L>, whose elements have n1 degree continuous derivatives and nth derivatives are from space L_{P}<0,L>, 1 p  
– n > k is a class of functions, f(x), where: f(x)C^{k}<0,L> and f(x)C^{n}<x_{i},x_{i+1}>, i=1,2,….,m1  
– n > k, 1 p is a class of functions, f(x), where: f(x)C^{k}<0,L> and f(x) , i=1,2,….,m1 
For a model accuracy specified, it is possible to determine backward the equidistant displacement of approximation nodes in an interval <0,L> for calculating and measuring the discrete values, {Y(x_{j},)}_{j }of the distributed output variable.
For example, from the second row of Tab. 3.1, for a desired model accuracy , , it comes:
(3.2.7) 
The greatest value of h, satisfying the above inequality is assumed, whose integer multiple is L. In such a way, Tab. 3.1 can be transformed into Tab. 3.2.
It is known, that in approximations using spline functions a significant reduction of the number of nodes can be achieved by their nonequidistant location. For given Y(x,) and accuracy, , specified, an adequate model can be fit by minimization of the deviations norm chosen with respect to nonequidistant distribution of the nodes. This model describes changes of the distributed output variable in neighbourhood of Y(x,) with the accuracy requested, assuming minimal number of discrete values, {Y(x_{j}, )}_{j }of the distributed output variable. Hence, the model of minimal complexity is fit.
Tab. 3.2 Maximal distances of approximation nodes for requested
Classes of approximated functions Y(x,)  Maximal distances of approximations nodes h 
The approximated distributed output variable using spline approximation is then got in the form:
(3.2.8) 
where SA stands for spline approximation.
For accuracy prescribed, , Fig. 3.2.2, the distributed output variable, Y(x,k), and its approximation, Ya(x,k), determine in a fully equivalent way the changes of the distributed output. Therefore, in accordance with conventions used in the domain of numerical solutions of PDE and FEM, no difference will be made, hereinafter, between approximates and approximated variables. Therefore Y(x,k) will stand for the model output in general.
Fig. 3.2.2 Equivalence of Y(x,) and Ya(x,) for given
HLDS  – lumpedinput/distributedoutput system with zeroorder hold units 
{U_{i}}_{i}  – lumped input variables 
Y(x,)/Ya(x,)  – distributed output variable and its approximation in steadystate 
– accuracy of the approximation 
<<< Previous  Next >>> 