Controller Parameters Dependence on Model Information through ...
Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference Shanghai, P.R. China, December 16-18, 2009
WeBIn4.9
Controller Parameters Dependence on Model Information Through Dimensional Analysis P. Balaguer, A. Ibeas, C. Pedret and S. Alc´antara Abstract— In this article we make use of dimensional analysis in order to investigate the controller parameters dependence on model information (i.e. model parameters). The objective is to relate the influence that each model parameter has on each controller parameter. In order to accomplish this goal the model transfer function is first analyzed using dimensional analysis and characterized by means of of dimensionless numbers. Secondly each controller parameter is related with the model parameters. As a result it is derived the underlaying structure that any homogeneous tuning rule must follow. The general results are particularized for PID control of first order and second order systems. The results can be applied to PID tuning and PID tuning rules comparison.
I.
INTRODUCTION
Dimensional analysis [1] is a well known theory applied to physical problems which permits a variety of important results such as i) to understand the physical phenomenon, not only the depending quantities but also their relations, ii) to represent results in a more compact way by means of dimensionless numbers and iii) to generalize experimental results, thus reducing the experimental requirements. Control theory can beneficiate from dimensional analysis in a wide variety of ways as can be seen from the bibliography. In [2] dimensional analysis is explicitly presented as a theory to be used in control theory related problems. In particular dimensional analysis is used to define systems equivalence and to relate system sensitivity to dimensional concepts. From a practical point of view dimensionless models of vehicle dynamics are derived and used for controller synthesis. In [6] the Buckingham Π theorem is used in order to represent the PID controller parameters of first order plus dead time (FOPDT) model by means of dimensionless numbers, with the aim of performing a numerical optimization of the tuning parameters. Practical advances in control design are proposed in [3]. In particular it is shown that by means of dimensional analysis it is possible to reduce the scheduling parameters of gain scheduling controllers, thus reducing the analysis and design complexity. Dimensional analysis is also used to dump uncertainty representation in dimensionless parameters, what may lead to less conservative results. The financial support received from the Spanish CICYT programme under grants DPI2007-64570 is greatly recognized. P. Balaguer is with the Department of Industrial Systems Engineering and Design, Universitat Jaume I de Castell´o, Castell´o, Spain
[email protected] A. Ibeas, C. Pedret and S. Alc´antara are with the Department of Telecommunication and Systems Engineering, Autonomous University of Barcelona, Barcelona, Spain
978-1-4244-3872-3/09/$25.00 ©2009 IEEE
Recently, in [5] it is studied the physical dimensions of matrices in state-space models by means of dimensional analysis. Moreover, throughout the control bibliography, it is interesting to note that dimensional analysis results are used, although not stated explicitly, in order to tackle some control issues. For example in [4] dimensionless parameters of FOPDT model are used to compare distinct autotuning methods. From the previous review we can see that dimensional analysis permits to tackle distinct control theory issues. Although it has been mainly applied to identification and modeling of systems and systems uncertainty, the potential results of dimensional analysis related to controller design and controller comparison seems to have received less attention. In this article we apply dimensional analysis on the transfer function framework in order to determine the structural relation among model parameters and controller parameters. It is shown that an underlaying structure in fact exists. The main result is obtained by firstly representing the transfer function by means of dimensionless numbers, in such a way that the transfer function behavior is characterized using a reduced set of dimensionless parameters. The following questions are answered: i) Which are the physical dimensions of the transfer function parameters?, ii) Is it possible to reduce the number of transfer function parameters by means of dimensionless numbers? If the answer is affirmative, how many parameters can be reduced?, iii) What is a proper system of units for the transfer function parameters? Next, on the basis of the preceding results, the dependence of controller parameters on the transfer function dimensionless numbers is established. In particular the following questions are answered: i) What is the relationship among the controller parameters and the plant model parameters?, ii) Do all the model parameters affect in the same way all the controller parameters? and iii) Is it possible to state these relationships in a more compact way by means of dimensionless numbers? Finally the general results obtained are particularized for PID control of FOPDT and second order models in order to show the practical benefits of the results obtained. In particular the results obtained are useful for comparing PID tuning methods as the characterization of controllers and models by means of dimensionless numbers allows the reduction of parameters in the comparison procedure (i.e. the dimensionless number τ /T ). The contributions of the article are organized as follows.
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WeBIn4.9 In Section II the basic concepts of dimensional analysis are reviewed. Next it is shown in Section III that transfer functions can be represented by means of a reduced set of dimensionless numbers. In Section IV it is analyzed the controller parameters dependence on transfer function dimensionless numbers. Finally the theoretical results obtained in the preceding Sections are applied to PID control of first and second order systems. II.
DIMENSIONAL ANALYSIS
Dimensional Analysis [1] is a technique used extensively in physic problems (e.g. fluid dynamics). The idea on which dimensional analysis is based is that physical laws are dimensionally homogeneous (i.e. do not depend on the choice of any basic units of measurement), that is the law is valid in any system of dimensions. This leads to the fact that functions expressing physical laws have a fundamental property called generalized homogeneity or symmetry. This property allows the number of function arguments to be reduced, therefore making the dependence simpler. Following we briefly review the fundamental concepts of dimensional analysis and state the fundamental Buckingham Π Theorem.
have independent dimensions if none of these quantities have dimensions which can be represented in terms of a product of powers of the dimensions of the remaining quantities. For example, density ([ρ]=M L−3), velocity ([v]=LT −1) and force ([f ]=M LT −2) have independent dimensions. In fact there are no x and y that accomplish, for example, the following equation: [ρ] = [v]x [f ]y
(2)
II-B. Buckingham Π Theorem The Buckingham Π Theorem [1] states that if some governed parameter a is a function of n governing parameters, that is: a = f (a1 , . . . , ak , ak+1 , . . . , an )
(3)
in such a way that only the set of k parameters given by (a1 , . . . , ak ) have independent dimensions, then there exists n−k independent dimensionless parameters Πj j ∈ [1, n−k] such that the following relation can be established: Π = Φ(Π1 , . . . , Πn−k )
(4)
Each one of the dimensionless numbers related with the dependent governing parameters is given by:
II-A. Physical quantities, Units and Dimensions When we refer to the fact that the length of a cable is 5 meters we are stating the magnitude of a physical quantity. However, we could also say that the cable length is 5000 millimeters. The only difference being in the units used to describe the physical quantity. We then have that: P hysical quantity = (numerical value) × (U nit) (1) Thus in both cases we are referring to the same physical quantity (i.e. the cable length is constant), being the only difference the numerical value due to the difference in units (e.g. meters Vs millimeters). On the other hand, in both cases we are describing a length. The dimension is defined as a qualitative description of a sensory perception of a physical entity. It is clear that in both cases we are dealing with the same dimensions. We write the dimensions of a quantity φ between square brackets, that is [φ]. For example, the quantity lc that describes the length of a cable has length dimensions (i.e. L), that is [lc] = L. On the other hand, the velocity dimension is [v] = LT −1 (where T is time dimension). The systems of units is the set of fundamental units sufficient for measuring the properties of the class of phenomena under consideration. For example in order to study the dynamics of mechanical systems, one system of units could be mass, length and time (i.e. M,L and T) and a different system of units could be length, force and time (i.e. L,F and T). However both systems of units are equally valid to study the mechanical system. It is necessary to stress that the dimensions are defined by the systems of units employed. Finally the concept of independent dimensions is introduced. Given a set of quantities, these quantities are said to
Πj =
ak+j p r a1j . . . akj
(5)
where the exponents pj . . . rj are chosen such that the parameter Πj is dimensionless. The dimensionless number related with the governed parameter is: Π=
a ap1 . . . ark
(6)
Note that there are n − k dimensionless numbers related with each one of the n − k dependent governing parameters (i.e. Πj j ∈ [1, n − k]), plus one dimensionless number related with the governed parameter a (i.e. Π). As a result the function f (equivalently the governed parameter as a = f (a1 , . . . , an )) can be written in terms of a function Φ of smaller number of variables in the following form: f (a1 , . . . , an ) = ap1 . . . ark Φ(Π1 , . . . , Πn−k ) III.
(7)
TRANSFER FUNCTION DIMENSIONAL ANALYSIS
In this section we characterize the transfer function by means of dimensional analysis theory. The objective is to express a given transfer function by means of dimensionless numbers. Thus we are able to represent the transfer function information in the most compact way. In order to express a transfer function by means of dimensionless numbers, first we find the dimensions of the transfer function parameters. Secondly a suitable system of units for transfer function parameters is defined.
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WeBIn4.9 III-A. Analysis of Transfer Function Parameters Dimension The transfer function of a system described by a linear time invariant differential equation is defined as the quotient between the Laplace transform of the output and the Laplace transform of the input under the assumption of zero initial conditions. In order to apply the Buckingham Π Theorem to a transfer function, it is necessary to know the dimensions of the transfer function parameters. This is shown by the following lemma: Lemma 1: Consider a generic transfer function G(s) in pole-zero form G(s) = Ksc
gz (s) −τ s e gp (s)
(8)
with c ∈ Z and gz (s) gp (s)
= =
z Y
(Tzj s + 1)
j=1 p Y
(Tpi s + 1)
i=1
with the following transfer function parameters K, Tzj , Tpi and τ satisfying p − z − c ≥ 0. Note that multiple complex conjugate poles and zeros are also allowed by appropriate complex values of the terms Tpi and Tzj . Then the sets {Tpi } and {Tzj } must be closed under complex conjugation. The dimensions of the transfer function parameters are: [K]
= Y U −1 T c
[Tpi ] = [Tzj ] = T [τ ] = T where Y is the output dimension, U the input dimension and T the time dimension. Proof: The result of the lemma 1 follows from the following facts. First note that [gz (s)] = 1, that is, dimensionless. In fact the dimensions of Tzj are time dimensions (i.e. they are time constants), thus [Tzj ] = T . On the other hand, the dimensions of the s operator are [s] = T −1 . It can be seen either in the time domain where the s operator is transformed to operator p defined as p = d/dt or in the frequency domain where s is transformed to jω. Thus each product Tzl s yields a dimensionless number. The same reasoning applies to gp (s) and e−τ s . Furthermore the dimensions of the transfer function are [G(s)] = Y U −1 . As a conclusion of the above analysis it results that [G(s)] = [Ksc ], or equivalently [K] = [G(s)][sc ]−1 which yields: [K] = Y U −1 T c
(9)
As a result, the system of units required to study the inputoutput relation given by a transfer function is Y U T (i.e.
output dimension, input dimension and time dimension). In fact any variable or parameter dimensions can be obtained by a combination of the Y U T dimensions. III-B. Transfer Function Dimensionless Numbers In this section we make use of the Buckingham Π theorem presented in Section II in order to characterize the transfer function by means of dimensionless numbers. As a result we obtain the minimum number of dimensionless numbers required to characterize a transfer function. The number of dimensionless numbers is equal to n − k where n is the number of governing parameters and k is the number of parameters which have independent dimensions. We can consider the following situations regarding the number of physical quantities n considered in the relationship and the number of independent dimensions k: n > k: In this case the number of quantities n appearing in the relating equation is greater than the independent dimensions k. As a result we can find n−k dimensionless numbers than equivalently relate the above relation. n = k: In this case the number of physical quantities equals the number of independent dimensions then n−k = 0, so there are no dimensionless numbers. n < k: In this case the number of quantities n is less than the number of independent dimensions k. This case is not physically possible as the resulting equation is no longer homogenous. We now study the number of dimensionless numbers that can be derived from a general transfer function presented in (8). In order to accomplish this goal, it is necessary to calculate the number of governing variables n and the number of variables with independent dimensions k. First of all we divide the parameters in three groups: 1. Signals: the output y and the input u. 2. Transfer Function parameters: K, Tzj , Tpi , and τ . 3. Frequency variable: s. If we consider the output as the governed variable, we have: y = f (K, τ, Tp1 , . . . , Tpp , Tz1 , . . . , Tzz , s, u)
(10)
The number of parameters can be calculated as follows: y: The output of a transfer consists of only one parameter, that is #y=1 (i.e. # refers to cardinality). u: The input also consists of one parameter then #u=1; K: The system gain is again another single parameter #K=1. τ : The system delay quantity #τ =1 if there is time delay. In case there is not time delay, #τ =0. p: The system number of poles with associated parameter is given by p. In fact, if c < 0 there are c poles at the origin but with no associated parameter. z: The system number of zeros with associated parameter is given by z. If c > 0 there are c zeros at the origin but with no associated parameter. s: The complex variable is also a parameter governing the output, then #s=1.
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WeBIn4.9 Then given the relation Y (s) = G(s)U (s) we find that the number of governing quantities (we consider y the governed quantity) is given by:
τ¯ = Π1 =
τ , T
s¯ = Π2 = T s
(19)
yielding finally n = #u + #K + #τ + p + z + #s
(11)
Considering a SISO system we have #u=#K=1. Moreover #s=1. Then it follows that n=3+#τ +p+z. On the other hand as derived in the preceding section, we have that the fundamental system units YUT. The number of governing parameters with independent dimensions is k=3. This is easily seen by considering the input [u] = U , the gain [K] = Y U −1 T c and any other governing parameter, as the rest of them have time dimensions T . As a result, given any relation described by any transfer function G(s) we can expect the following number of dimensionless numbers: n − k = 3 + #τ + m + n − 3 = #τ + p + z
(12)
1 Y = e−¯τ s¯ KU (¯ s + 1)
Example 2: Consider now a second order system without time delay, that is: Y (s) =
Example 1: Consider a first order system plus time delay: Y (s) =
Ke−τ s U (s) (T s + 1)
(13)
Y = f (U, K, T1 , T2 , s)
(22)
Π1 =
T2 , Π2 = s¯ = T1 s T1
(23)
The dimensionless number related with the governed variable is the same as the example before, so we have: T2 Y = KU Φ , T1 s (24) T1 with function Φ given by
(14)
The number of dimensionless numbers is n − k = #τ + p+z. In the FOPTD case we have that #τ =1, #p=1 and #z=0, thus n-k=2. As a result we have two dimensionless number: τ Π1 = , Π2 = T s (15) T Note that the dimensionless numbers are not unique as T /τ and τ s are also valid. There is always an extra dimensionless number related with the governed variable, in this case: y Π = K −1 (16) u Then we have the following relationship Π = Φ(Π1 , Π2 ), which can also be written as: τ Y = KU Φ ,Ts (17) T with the function Φ given by: Y 1 = e−Π1 Π2 KU (Π2 + 1)
(21)
The number of dimensionless numbers is n−k = #τ +p+ z. Now we have that #τ =0, #p=2 and #z=0, thus the number of dimensionless numbers is again n-k=2. The dimensionless numbers are:
Firstly we state the governed parameter and the governing parameters. These are: Y = f (U, K, τ, T, s)
K U (s) (T1 s + 1)(T2 s + 1)
The relations among quantities considered is:
Thus, we can see that the number of dimensionless numbers is a function of the time parameters, that is the delay and the number of poles and zeros of the transfer function. III-C. Examples
(20)
(18)
Y 1 = KU (¯ s + 1)(Π1 s¯ + 1)
(25)
Example 3: Consider now a second order system with time delay, that is: Y (s) =
Ke−τ s U (s) (T1 s + 1)(T2 s + 1)
(26)
The relations among quantities considered is: Y = f (U, K, τ, T1 , T2 , s)
(27)
The number of dimensionless numbers is n − k = #τ + p + z. In this case we have that #τ =1, #p=2 and #z=0, thus the number of dimensionless numbers is now n-k=3. The dimensionless number are: Π1 =
T2 τ , Π2 = , Π3 = T1 s T1 T1
(28)
The dimensionless number related with the governed variable is the same as the example before, so we have: τ T2 Y = KU Φ , , T1 s (29) T1 T1 with function Φ given by
what can be easily shown by direct substitution. In the rest of the article we use the following more meaningful notation to refer to each dimensionless number:
1917
Y e−Π1 s¯ = KU (¯ s + 1)(Π2 s¯ + 1)
(30)
WeBIn4.9 IV.
CONTROLLER PARAMETERS DEPENDENCE ON DIMENSIONLESS NUMBERS
In this section we apply the dimensional analysis theory in order to analyze the controller parameters dependence on the model transfer function parameters. We then consider a generic controller C(s) given by: C(s) =
g c (s) Kc sq zc gp (s)
(31)
with p ∈ Z and gzc (s) gpc (s)
= =
zc Y
c (Tzj s + 1)
j=1 pc Y
c (Tpi s + 1)
i=1
c with the following transfer function parameters Kc , Tzj and c Tpi . The relative order of the system is pc− zc− q ≥ 0. Note that complex poles and zeros are also allowed by appropriate c c complex values of the terms Tpi and Tzj . In this case, the sets c c {Tpi } and {Tzj } must be closed under complex conjugation. Then we are faced with the problem of finding the controller parameters dependence on the model parameters, that is:
= f1 (K, τ, Tp1 , . . . , Tpp , Tz1 , . . . , Tzz ) = f2 (K, τ, Tp1 , . . . , Tpp , Tz1 , . . . , Tzz )
a function of n − k − 1 dimensionless numbers. These dimensionless numbers (i.e. (Π1 , . . . , Πn−k−1 )) are the ones generated from all transfer function time parameters (i.e. τ, Tp1 , . . . , Tpp , Tz1 , . . . , Tzz ). As a result we can write KKc = Φ1 (Π1 , . . . , Πn−k−1 ) c Tp1 = Φ2 (Π1 , . . . , Πn−k−1 ) τ .. . c Tppc = Φpc+1 (Π1 , . . . , Πn−k−1 ) τ c Tz1 = Φpc+2 (Π1 , . . . , Πn−k−1 ) τ .. . c Tzzc = Φpc+zc+1 (Π1 , . . . , Πn−k−1 ) τ Note that each one of the controller time constants are divided by τ . This is so because in order to form the dimensionless numbers Π1 , . . . , Πn−k−1 , the governing parameter τ was chosen to be the one forming the base of parameters with independent dimension. However this selection is arbitrary. In the following we particularize this general result for PID controllers of FOPTD models and SOPTD models. V.
APPLICATION TO PID CONTROL
Kc c Tp1 .. . c Tppc
= fpc+1 (K, τ, Tp1 , . . . , Tpp , Tz1 , . . . , Tzz ) (34)
V-A. First order plus time delay systems
c Tz1
= fpc+2 (K, τ, Tp1 , . . . , Tpp , Tz1 , . . . , Tzz ) (35)
In this section we tackle the problem of the PID controller parameter dependence on a FOPTD model, with transfer function G(s) given by:
(32) (33)
.. . c Tzzc
In what follows we apply the general results stated in Section IV to two distinct model plants to be controlled with PID control.
= fpc+zc+1 (K, τ, Tp1 , . . . , Tpp , Tz1 , . . . , Tzz ) (36)
Remark 1. Note that the governing parameters of each one of the controller parameters is a subset of the governing parameters of the system output as can be seen in equation (10). In fact, the controller parameters do not depend on the input u and on the complex variable s. Moreover note that in equation (10) the parameters K and u, due to their dimensions, only affect the governed variable y. Remark 2. The governing parameters on equations (32)(36) only have two distinct dimensions, the gain dimension K and the time dimension T . Moreover there is just one parameter with gain dimension, the transfer function gain K. It then follows, due to dimensional homogeneity, that the gain can not appear in the rest of the governing variables (all with time dimension) in order to make them dimensionless. In fact the only parameter that can be made dimensionless by means of the model gain K is the controller gain Kc . From the above remarks it can be seen that each one of the controller parameters (i.e. governed variables) are
G(s) =
Ke−τ s (T s + 1)
(37)
Recall from example 1 that in this case there is just one dimensionless number related with the transfer function time parameters, which is Tτ . As a result we have that KKc = Φ1
τ
T τ Ti = Φ2 T T τ Td = Φ3 T T Thus, the controller gain depends in an inverse way on the process gain and on an arbitrary function Φ (which defines the tuning rule) of the dimensionless parameter τ /T . The integral time and the derivative time do not depend on the model gain K but are an arbitrary function of the dimensionless parameter τ /T .
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WeBIn4.9
In this section we study the PID controller parameters dependence on the second order model plus delay given by: G(s) =
Ke−τ s (T1 s + 1)(T2 s + 1)
(38)
Taking the first time constant T1 as the independent dimension variable we have find the following dimensionless numbers of the transfer function time parameters TT21 and Tτ1 . Then the controller parameters are obtained as: T2 τ −1 , K c = K Φ1 T1 T1 T2 τ T i = T 1 Φ2 , T T 1 1 T2 τ T d = T 1 Φ3 , T1 T1
s¯ is the normalized laplace variable defined as s¯ = T s, with T the FOPDT time constant. τ¯ is the dimensionless time delay defined as τ¯ = τ /T It can be seen that the normalized error transfer function is a function of just one parameter, the dimensionless time delay τ¯. As a result the H∞ norm of normalized error ¯ s)||∞ ) can be calculated for distinct transfer function (||E(¯ Pade approximation orders as it is shown in Figure 1.
0.5
0.3
0.1
0 0
In this section we study the PID controller parameters dependence on the third order system given by: K (T1 s + 1)(T2 s + 1)(T3 s + 1)
0.4
0.2
Third order systems
G(s) =
1st Order Pade Approximation 2nd Order Pade Approximation 3rd Order Pade Approximation 4th Order Pade Approximation
0.6
As a result, an equivalent relation as the one obtained in the preceding case is obtained. However now the functions Φi i ∈ [1, 2, 3] defining each one of the controller parameters is a function of two dimensionless variables. V-C.
¯ s)||∞ ||E(¯
0.7
¯ s)||∞ ||E(¯
V-B. Second order plus time delay systems
0.2
0.4
0.6
0.8
1 τ/T
1.2
1.4
1.6
1.8
2
Fig. 1. Comparison of Pade Approximations of several Orders.
(39) VII.
CONCLUSIONS
Taking the first time constant T1 as the independent dimension variable we have find the following dimensionless numbers of the transfer function time parameters TT21 and TT31 . Then the controller parameters are obtained as: T2 T3 , Kc = K −1 Φ1 T1 T1 T2 T3 T i = T 1 Φ2 , T T 1 1 T2 T3 T d = T 1 Φ3 , T1 T1
In this article we have used dimensional analysis to investigate two important aspects of control theory. On the one hand it has been established that given a general transfer function, it is possible to characterize it by means of dimensionless numbers. The number of dimensionless numbers obtained are equal to the transfer function time parameters. On the other hand, dimensional analysis has been used to determine the controller parameters dependence on the model information (i.e. model parameters). These results are general and of interest in order to characterize plant behaviour for analysis purposes as well as to derive PID tuning rules.
We can see that the controller parameters are again a function of two dimensionless parameters.
R EFERENCES
VI.
APPLICATION EXAMPLE
In this section we compare Pade approximations of time delay in FOPDT models as the one presented by equation (20). By straightforward calculation the transfer function ¯ s) = G(¯ of the dimensionless approximation error E(¯ s) − ˆ G(¯ s) for a first order Pade approximation is given by τ¯ ¯− 1 1 −¯ τs 2s ¯ E(¯ s) = e − τ¯ (40) (¯ s + 1) ¯+ 1 2s where ¯ s) is the normalized error transfer function defined E(¯ s)−Yˆ (¯ s) ˆ s) the as Y (¯ KU(¯ s) , with Y (s) the FOPDT output, Y (¯ output of the approximate model, U (¯ s) the model input and K the FOPDT gain.
[1] G. I. Barenblatt. Dimensional Analysis. Gordon and Breach Science Publishers, 1987. [2] Sean Brennan. On Size and Control: The Use of Dimensional Analysis in Controller Design. PhD thesis, University of Illinois at UrbanaChampaign, 2002. [3] Haftay Hailu. Dimensional Transformation: A Novel Method for Gain Scheduling and Robust Control. PhD thesis, The Pennsylvania State University, 2006. [4] A. Leva. Comparative study of model-based PI(D) autotuning methods. In American Control Conference, 2007. [5] H. J. Palanthandalam-Madapusi, D. S. Berstein, and R. Venugopal. Dimensional analysis of matrices. IEEE Control Systems Magazine, December:100–109, 2007. [6] S. Tavakoli and M. Tavakoli. Optimal tuning of PID controllers for first order plus time delay models using dimensional analysis. In International Conference on Control and Automation, 2003.
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WeBIn4.9
Controller Parameters Dependence on Model Information Through Dimensional Analysis P. Balaguer, A. Ibeas, C. Pedret and S. Alc´antara Abstract— In this article we make use of dimensional analysis in order to investigate the controller parameters dependence on model information (i.e. model parameters). The objective is to relate the influence that each model parameter has on each controller parameter. In order to accomplish this goal the model transfer function is first analyzed using dimensional analysis and characterized by means of of dimensionless numbers. Secondly each controller parameter is related with the model parameters. As a result it is derived the underlaying structure that any homogeneous tuning rule must follow. The general results are particularized for PID control of first order and second order systems. The results can be applied to PID tuning and PID tuning rules comparison.
I.
INTRODUCTION
Dimensional analysis [1] is a well known theory applied to physical problems which permits a variety of important results such as i) to understand the physical phenomenon, not only the depending quantities but also their relations, ii) to represent results in a more compact way by means of dimensionless numbers and iii) to generalize experimental results, thus reducing the experimental requirements. Control theory can beneficiate from dimensional analysis in a wide variety of ways as can be seen from the bibliography. In [2] dimensional analysis is explicitly presented as a theory to be used in control theory related problems. In particular dimensional analysis is used to define systems equivalence and to relate system sensitivity to dimensional concepts. From a practical point of view dimensionless models of vehicle dynamics are derived and used for controller synthesis. In [6] the Buckingham Π theorem is used in order to represent the PID controller parameters of first order plus dead time (FOPDT) model by means of dimensionless numbers, with the aim of performing a numerical optimization of the tuning parameters. Practical advances in control design are proposed in [3]. In particular it is shown that by means of dimensional analysis it is possible to reduce the scheduling parameters of gain scheduling controllers, thus reducing the analysis and design complexity. Dimensional analysis is also used to dump uncertainty representation in dimensionless parameters, what may lead to less conservative results. The financial support received from the Spanish CICYT programme under grants DPI2007-64570 is greatly recognized. P. Balaguer is with the Department of Industrial Systems Engineering and Design, Universitat Jaume I de Castell´o, Castell´o, Spain
[email protected] A. Ibeas, C. Pedret and S. Alc´antara are with the Department of Telecommunication and Systems Engineering, Autonomous University of Barcelona, Barcelona, Spain
978-1-4244-3872-3/09/$25.00 ©2009 IEEE
Recently, in [5] it is studied the physical dimensions of matrices in state-space models by means of dimensional analysis. Moreover, throughout the control bibliography, it is interesting to note that dimensional analysis results are used, although not stated explicitly, in order to tackle some control issues. For example in [4] dimensionless parameters of FOPDT model are used to compare distinct autotuning methods. From the previous review we can see that dimensional analysis permits to tackle distinct control theory issues. Although it has been mainly applied to identification and modeling of systems and systems uncertainty, the potential results of dimensional analysis related to controller design and controller comparison seems to have received less attention. In this article we apply dimensional analysis on the transfer function framework in order to determine the structural relation among model parameters and controller parameters. It is shown that an underlaying structure in fact exists. The main result is obtained by firstly representing the transfer function by means of dimensionless numbers, in such a way that the transfer function behavior is characterized using a reduced set of dimensionless parameters. The following questions are answered: i) Which are the physical dimensions of the transfer function parameters?, ii) Is it possible to reduce the number of transfer function parameters by means of dimensionless numbers? If the answer is affirmative, how many parameters can be reduced?, iii) What is a proper system of units for the transfer function parameters? Next, on the basis of the preceding results, the dependence of controller parameters on the transfer function dimensionless numbers is established. In particular the following questions are answered: i) What is the relationship among the controller parameters and the plant model parameters?, ii) Do all the model parameters affect in the same way all the controller parameters? and iii) Is it possible to state these relationships in a more compact way by means of dimensionless numbers? Finally the general results obtained are particularized for PID control of FOPDT and second order models in order to show the practical benefits of the results obtained. In particular the results obtained are useful for comparing PID tuning methods as the characterization of controllers and models by means of dimensionless numbers allows the reduction of parameters in the comparison procedure (i.e. the dimensionless number τ /T ). The contributions of the article are organized as follows.
1914
WeBIn4.9 In Section II the basic concepts of dimensional analysis are reviewed. Next it is shown in Section III that transfer functions can be represented by means of a reduced set of dimensionless numbers. In Section IV it is analyzed the controller parameters dependence on transfer function dimensionless numbers. Finally the theoretical results obtained in the preceding Sections are applied to PID control of first and second order systems. II.
DIMENSIONAL ANALYSIS
Dimensional Analysis [1] is a technique used extensively in physic problems (e.g. fluid dynamics). The idea on which dimensional analysis is based is that physical laws are dimensionally homogeneous (i.e. do not depend on the choice of any basic units of measurement), that is the law is valid in any system of dimensions. This leads to the fact that functions expressing physical laws have a fundamental property called generalized homogeneity or symmetry. This property allows the number of function arguments to be reduced, therefore making the dependence simpler. Following we briefly review the fundamental concepts of dimensional analysis and state the fundamental Buckingham Π Theorem.
have independent dimensions if none of these quantities have dimensions which can be represented in terms of a product of powers of the dimensions of the remaining quantities. For example, density ([ρ]=M L−3), velocity ([v]=LT −1) and force ([f ]=M LT −2) have independent dimensions. In fact there are no x and y that accomplish, for example, the following equation: [ρ] = [v]x [f ]y
(2)
II-B. Buckingham Π Theorem The Buckingham Π Theorem [1] states that if some governed parameter a is a function of n governing parameters, that is: a = f (a1 , . . . , ak , ak+1 , . . . , an )
(3)
in such a way that only the set of k parameters given by (a1 , . . . , ak ) have independent dimensions, then there exists n−k independent dimensionless parameters Πj j ∈ [1, n−k] such that the following relation can be established: Π = Φ(Π1 , . . . , Πn−k )
(4)
Each one of the dimensionless numbers related with the dependent governing parameters is given by:
II-A. Physical quantities, Units and Dimensions When we refer to the fact that the length of a cable is 5 meters we are stating the magnitude of a physical quantity. However, we could also say that the cable length is 5000 millimeters. The only difference being in the units used to describe the physical quantity. We then have that: P hysical quantity = (numerical value) × (U nit) (1) Thus in both cases we are referring to the same physical quantity (i.e. the cable length is constant), being the only difference the numerical value due to the difference in units (e.g. meters Vs millimeters). On the other hand, in both cases we are describing a length. The dimension is defined as a qualitative description of a sensory perception of a physical entity. It is clear that in both cases we are dealing with the same dimensions. We write the dimensions of a quantity φ between square brackets, that is [φ]. For example, the quantity lc that describes the length of a cable has length dimensions (i.e. L), that is [lc] = L. On the other hand, the velocity dimension is [v] = LT −1 (where T is time dimension). The systems of units is the set of fundamental units sufficient for measuring the properties of the class of phenomena under consideration. For example in order to study the dynamics of mechanical systems, one system of units could be mass, length and time (i.e. M,L and T) and a different system of units could be length, force and time (i.e. L,F and T). However both systems of units are equally valid to study the mechanical system. It is necessary to stress that the dimensions are defined by the systems of units employed. Finally the concept of independent dimensions is introduced. Given a set of quantities, these quantities are said to
Πj =
ak+j p r a1j . . . akj
(5)
where the exponents pj . . . rj are chosen such that the parameter Πj is dimensionless. The dimensionless number related with the governed parameter is: Π=
a ap1 . . . ark
(6)
Note that there are n − k dimensionless numbers related with each one of the n − k dependent governing parameters (i.e. Πj j ∈ [1, n − k]), plus one dimensionless number related with the governed parameter a (i.e. Π). As a result the function f (equivalently the governed parameter as a = f (a1 , . . . , an )) can be written in terms of a function Φ of smaller number of variables in the following form: f (a1 , . . . , an ) = ap1 . . . ark Φ(Π1 , . . . , Πn−k ) III.
(7)
TRANSFER FUNCTION DIMENSIONAL ANALYSIS
In this section we characterize the transfer function by means of dimensional analysis theory. The objective is to express a given transfer function by means of dimensionless numbers. Thus we are able to represent the transfer function information in the most compact way. In order to express a transfer function by means of dimensionless numbers, first we find the dimensions of the transfer function parameters. Secondly a suitable system of units for transfer function parameters is defined.
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WeBIn4.9 III-A. Analysis of Transfer Function Parameters Dimension The transfer function of a system described by a linear time invariant differential equation is defined as the quotient between the Laplace transform of the output and the Laplace transform of the input under the assumption of zero initial conditions. In order to apply the Buckingham Π Theorem to a transfer function, it is necessary to know the dimensions of the transfer function parameters. This is shown by the following lemma: Lemma 1: Consider a generic transfer function G(s) in pole-zero form G(s) = Ksc
gz (s) −τ s e gp (s)
(8)
with c ∈ Z and gz (s) gp (s)
= =
z Y
(Tzj s + 1)
j=1 p Y
(Tpi s + 1)
i=1
with the following transfer function parameters K, Tzj , Tpi and τ satisfying p − z − c ≥ 0. Note that multiple complex conjugate poles and zeros are also allowed by appropriate complex values of the terms Tpi and Tzj . Then the sets {Tpi } and {Tzj } must be closed under complex conjugation. The dimensions of the transfer function parameters are: [K]
= Y U −1 T c
[Tpi ] = [Tzj ] = T [τ ] = T where Y is the output dimension, U the input dimension and T the time dimension. Proof: The result of the lemma 1 follows from the following facts. First note that [gz (s)] = 1, that is, dimensionless. In fact the dimensions of Tzj are time dimensions (i.e. they are time constants), thus [Tzj ] = T . On the other hand, the dimensions of the s operator are [s] = T −1 . It can be seen either in the time domain where the s operator is transformed to operator p defined as p = d/dt or in the frequency domain where s is transformed to jω. Thus each product Tzl s yields a dimensionless number. The same reasoning applies to gp (s) and e−τ s . Furthermore the dimensions of the transfer function are [G(s)] = Y U −1 . As a conclusion of the above analysis it results that [G(s)] = [Ksc ], or equivalently [K] = [G(s)][sc ]−1 which yields: [K] = Y U −1 T c
(9)
As a result, the system of units required to study the inputoutput relation given by a transfer function is Y U T (i.e.
output dimension, input dimension and time dimension). In fact any variable or parameter dimensions can be obtained by a combination of the Y U T dimensions. III-B. Transfer Function Dimensionless Numbers In this section we make use of the Buckingham Π theorem presented in Section II in order to characterize the transfer function by means of dimensionless numbers. As a result we obtain the minimum number of dimensionless numbers required to characterize a transfer function. The number of dimensionless numbers is equal to n − k where n is the number of governing parameters and k is the number of parameters which have independent dimensions. We can consider the following situations regarding the number of physical quantities n considered in the relationship and the number of independent dimensions k: n > k: In this case the number of quantities n appearing in the relating equation is greater than the independent dimensions k. As a result we can find n−k dimensionless numbers than equivalently relate the above relation. n = k: In this case the number of physical quantities equals the number of independent dimensions then n−k = 0, so there are no dimensionless numbers. n < k: In this case the number of quantities n is less than the number of independent dimensions k. This case is not physically possible as the resulting equation is no longer homogenous. We now study the number of dimensionless numbers that can be derived from a general transfer function presented in (8). In order to accomplish this goal, it is necessary to calculate the number of governing variables n and the number of variables with independent dimensions k. First of all we divide the parameters in three groups: 1. Signals: the output y and the input u. 2. Transfer Function parameters: K, Tzj , Tpi , and τ . 3. Frequency variable: s. If we consider the output as the governed variable, we have: y = f (K, τ, Tp1 , . . . , Tpp , Tz1 , . . . , Tzz , s, u)
(10)
The number of parameters can be calculated as follows: y: The output of a transfer consists of only one parameter, that is #y=1 (i.e. # refers to cardinality). u: The input also consists of one parameter then #u=1; K: The system gain is again another single parameter #K=1. τ : The system delay quantity #τ =1 if there is time delay. In case there is not time delay, #τ =0. p: The system number of poles with associated parameter is given by p. In fact, if c < 0 there are c poles at the origin but with no associated parameter. z: The system number of zeros with associated parameter is given by z. If c > 0 there are c zeros at the origin but with no associated parameter. s: The complex variable is also a parameter governing the output, then #s=1.
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WeBIn4.9 Then given the relation Y (s) = G(s)U (s) we find that the number of governing quantities (we consider y the governed quantity) is given by:
τ¯ = Π1 =
τ , T
s¯ = Π2 = T s
(19)
yielding finally n = #u + #K + #τ + p + z + #s
(11)
Considering a SISO system we have #u=#K=1. Moreover #s=1. Then it follows that n=3+#τ +p+z. On the other hand as derived in the preceding section, we have that the fundamental system units YUT. The number of governing parameters with independent dimensions is k=3. This is easily seen by considering the input [u] = U , the gain [K] = Y U −1 T c and any other governing parameter, as the rest of them have time dimensions T . As a result, given any relation described by any transfer function G(s) we can expect the following number of dimensionless numbers: n − k = 3 + #τ + m + n − 3 = #τ + p + z
(12)
1 Y = e−¯τ s¯ KU (¯ s + 1)
Example 2: Consider now a second order system without time delay, that is: Y (s) =
Example 1: Consider a first order system plus time delay: Y (s) =
Ke−τ s U (s) (T s + 1)
(13)
Y = f (U, K, T1 , T2 , s)
(22)
Π1 =
T2 , Π2 = s¯ = T1 s T1
(23)
The dimensionless number related with the governed variable is the same as the example before, so we have: T2 Y = KU Φ , T1 s (24) T1 with function Φ given by
(14)
The number of dimensionless numbers is n − k = #τ + p+z. In the FOPTD case we have that #τ =1, #p=1 and #z=0, thus n-k=2. As a result we have two dimensionless number: τ Π1 = , Π2 = T s (15) T Note that the dimensionless numbers are not unique as T /τ and τ s are also valid. There is always an extra dimensionless number related with the governed variable, in this case: y Π = K −1 (16) u Then we have the following relationship Π = Φ(Π1 , Π2 ), which can also be written as: τ Y = KU Φ ,Ts (17) T with the function Φ given by: Y 1 = e−Π1 Π2 KU (Π2 + 1)
(21)
The number of dimensionless numbers is n−k = #τ +p+ z. Now we have that #τ =0, #p=2 and #z=0, thus the number of dimensionless numbers is again n-k=2. The dimensionless numbers are:
Firstly we state the governed parameter and the governing parameters. These are: Y = f (U, K, τ, T, s)
K U (s) (T1 s + 1)(T2 s + 1)
The relations among quantities considered is:
Thus, we can see that the number of dimensionless numbers is a function of the time parameters, that is the delay and the number of poles and zeros of the transfer function. III-C. Examples
(20)
(18)
Y 1 = KU (¯ s + 1)(Π1 s¯ + 1)
(25)
Example 3: Consider now a second order system with time delay, that is: Y (s) =
Ke−τ s U (s) (T1 s + 1)(T2 s + 1)
(26)
The relations among quantities considered is: Y = f (U, K, τ, T1 , T2 , s)
(27)
The number of dimensionless numbers is n − k = #τ + p + z. In this case we have that #τ =1, #p=2 and #z=0, thus the number of dimensionless numbers is now n-k=3. The dimensionless number are: Π1 =
T2 τ , Π2 = , Π3 = T1 s T1 T1
(28)
The dimensionless number related with the governed variable is the same as the example before, so we have: τ T2 Y = KU Φ , , T1 s (29) T1 T1 with function Φ given by
what can be easily shown by direct substitution. In the rest of the article we use the following more meaningful notation to refer to each dimensionless number:
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Y e−Π1 s¯ = KU (¯ s + 1)(Π2 s¯ + 1)
(30)
WeBIn4.9 IV.
CONTROLLER PARAMETERS DEPENDENCE ON DIMENSIONLESS NUMBERS
In this section we apply the dimensional analysis theory in order to analyze the controller parameters dependence on the model transfer function parameters. We then consider a generic controller C(s) given by: C(s) =
g c (s) Kc sq zc gp (s)
(31)
with p ∈ Z and gzc (s) gpc (s)
= =
zc Y
c (Tzj s + 1)
j=1 pc Y
c (Tpi s + 1)
i=1
c with the following transfer function parameters Kc , Tzj and c Tpi . The relative order of the system is pc− zc− q ≥ 0. Note that complex poles and zeros are also allowed by appropriate c c complex values of the terms Tpi and Tzj . In this case, the sets c c {Tpi } and {Tzj } must be closed under complex conjugation. Then we are faced with the problem of finding the controller parameters dependence on the model parameters, that is:
= f1 (K, τ, Tp1 , . . . , Tpp , Tz1 , . . . , Tzz ) = f2 (K, τ, Tp1 , . . . , Tpp , Tz1 , . . . , Tzz )
a function of n − k − 1 dimensionless numbers. These dimensionless numbers (i.e. (Π1 , . . . , Πn−k−1 )) are the ones generated from all transfer function time parameters (i.e. τ, Tp1 , . . . , Tpp , Tz1 , . . . , Tzz ). As a result we can write KKc = Φ1 (Π1 , . . . , Πn−k−1 ) c Tp1 = Φ2 (Π1 , . . . , Πn−k−1 ) τ .. . c Tppc = Φpc+1 (Π1 , . . . , Πn−k−1 ) τ c Tz1 = Φpc+2 (Π1 , . . . , Πn−k−1 ) τ .. . c Tzzc = Φpc+zc+1 (Π1 , . . . , Πn−k−1 ) τ Note that each one of the controller time constants are divided by τ . This is so because in order to form the dimensionless numbers Π1 , . . . , Πn−k−1 , the governing parameter τ was chosen to be the one forming the base of parameters with independent dimension. However this selection is arbitrary. In the following we particularize this general result for PID controllers of FOPTD models and SOPTD models. V.
APPLICATION TO PID CONTROL
Kc c Tp1 .. . c Tppc
= fpc+1 (K, τ, Tp1 , . . . , Tpp , Tz1 , . . . , Tzz ) (34)
V-A. First order plus time delay systems
c Tz1
= fpc+2 (K, τ, Tp1 , . . . , Tpp , Tz1 , . . . , Tzz ) (35)
In this section we tackle the problem of the PID controller parameter dependence on a FOPTD model, with transfer function G(s) given by:
(32) (33)
.. . c Tzzc
In what follows we apply the general results stated in Section IV to two distinct model plants to be controlled with PID control.
= fpc+zc+1 (K, τ, Tp1 , . . . , Tpp , Tz1 , . . . , Tzz ) (36)
Remark 1. Note that the governing parameters of each one of the controller parameters is a subset of the governing parameters of the system output as can be seen in equation (10). In fact, the controller parameters do not depend on the input u and on the complex variable s. Moreover note that in equation (10) the parameters K and u, due to their dimensions, only affect the governed variable y. Remark 2. The governing parameters on equations (32)(36) only have two distinct dimensions, the gain dimension K and the time dimension T . Moreover there is just one parameter with gain dimension, the transfer function gain K. It then follows, due to dimensional homogeneity, that the gain can not appear in the rest of the governing variables (all with time dimension) in order to make them dimensionless. In fact the only parameter that can be made dimensionless by means of the model gain K is the controller gain Kc . From the above remarks it can be seen that each one of the controller parameters (i.e. governed variables) are
G(s) =
Ke−τ s (T s + 1)
(37)
Recall from example 1 that in this case there is just one dimensionless number related with the transfer function time parameters, which is Tτ . As a result we have that KKc = Φ1
τ
T τ Ti = Φ2 T T τ Td = Φ3 T T Thus, the controller gain depends in an inverse way on the process gain and on an arbitrary function Φ (which defines the tuning rule) of the dimensionless parameter τ /T . The integral time and the derivative time do not depend on the model gain K but are an arbitrary function of the dimensionless parameter τ /T .
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WeBIn4.9
In this section we study the PID controller parameters dependence on the second order model plus delay given by: G(s) =
Ke−τ s (T1 s + 1)(T2 s + 1)
(38)
Taking the first time constant T1 as the independent dimension variable we have find the following dimensionless numbers of the transfer function time parameters TT21 and Tτ1 . Then the controller parameters are obtained as: T2 τ −1 , K c = K Φ1 T1 T1 T2 τ T i = T 1 Φ2 , T T 1 1 T2 τ T d = T 1 Φ3 , T1 T1
s¯ is the normalized laplace variable defined as s¯ = T s, with T the FOPDT time constant. τ¯ is the dimensionless time delay defined as τ¯ = τ /T It can be seen that the normalized error transfer function is a function of just one parameter, the dimensionless time delay τ¯. As a result the H∞ norm of normalized error ¯ s)||∞ ) can be calculated for distinct transfer function (||E(¯ Pade approximation orders as it is shown in Figure 1.
0.5
0.3
0.1
0 0
In this section we study the PID controller parameters dependence on the third order system given by: K (T1 s + 1)(T2 s + 1)(T3 s + 1)
0.4
0.2
Third order systems
G(s) =
1st Order Pade Approximation 2nd Order Pade Approximation 3rd Order Pade Approximation 4th Order Pade Approximation
0.6
As a result, an equivalent relation as the one obtained in the preceding case is obtained. However now the functions Φi i ∈ [1, 2, 3] defining each one of the controller parameters is a function of two dimensionless variables. V-C.
¯ s)||∞ ||E(¯
0.7
¯ s)||∞ ||E(¯
V-B. Second order plus time delay systems
0.2
0.4
0.6
0.8
1 τ/T
1.2
1.4
1.6
1.8
2
Fig. 1. Comparison of Pade Approximations of several Orders.
(39) VII.
CONCLUSIONS
Taking the first time constant T1 as the independent dimension variable we have find the following dimensionless numbers of the transfer function time parameters TT21 and TT31 . Then the controller parameters are obtained as: T2 T3 , Kc = K −1 Φ1 T1 T1 T2 T3 T i = T 1 Φ2 , T T 1 1 T2 T3 T d = T 1 Φ3 , T1 T1
In this article we have used dimensional analysis to investigate two important aspects of control theory. On the one hand it has been established that given a general transfer function, it is possible to characterize it by means of dimensionless numbers. The number of dimensionless numbers obtained are equal to the transfer function time parameters. On the other hand, dimensional analysis has been used to determine the controller parameters dependence on the model information (i.e. model parameters). These results are general and of interest in order to characterize plant behaviour for analysis purposes as well as to derive PID tuning rules.
We can see that the controller parameters are again a function of two dimensionless parameters.
R EFERENCES
VI.
APPLICATION EXAMPLE
In this section we compare Pade approximations of time delay in FOPDT models as the one presented by equation (20). By straightforward calculation the transfer function ¯ s) = G(¯ of the dimensionless approximation error E(¯ s) − ˆ G(¯ s) for a first order Pade approximation is given by τ¯ ¯− 1 1 −¯ τs 2s ¯ E(¯ s) = e − τ¯ (40) (¯ s + 1) ¯+ 1 2s where ¯ s) is the normalized error transfer function defined E(¯ s)−Yˆ (¯ s) ˆ s) the as Y (¯ KU(¯ s) , with Y (s) the FOPDT output, Y (¯ output of the approximate model, U (¯ s) the model input and K the FOPDT gain.
[1] G. I. Barenblatt. Dimensional Analysis. Gordon and Breach Science Publishers, 1987. [2] Sean Brennan. On Size and Control: The Use of Dimensional Analysis in Controller Design. PhD thesis, University of Illinois at UrbanaChampaign, 2002. [3] Haftay Hailu. Dimensional Transformation: A Novel Method for Gain Scheduling and Robust Control. PhD thesis, The Pennsylvania State University, 2006. [4] A. Leva. Comparative study of model-based PI(D) autotuning methods. In American Control Conference, 2007. [5] H. J. Palanthandalam-Madapusi, D. S. Berstein, and R. Venugopal. Dimensional analysis of matrices. IEEE Control Systems Magazine, December:100–109, 2007. [6] S. Tavakoli and M. Tavakoli. Optimal tuning of PID controllers for first order plus time delay models using dimensional analysis. In International Conference on Control and Automation, 2003.
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