- Email: [email protected]
From LUST to Brainwave: Fifteen Years of Persistent Excitation
Copyright @ IFAC Adaptive Systems in Control and Signal Processing, Glasgow, Scotland, UK, 1998
FROM LUST TO BRAINWAVE: FIFTEEN YEARS OF PERSISTENT EXCITATION Guy A. Dumont·
• Department of Electrical and Computer Engineering clo Pulp and Paper Centre, 2385 East Mall University of British Columbia, Vancouver, BC V6T 1Z4 Canada email: guyd
Abstract: Adaptive predictive controllers based on truncated orthonormal series representation of the process dynamics were first proposed in the mid 1980's. Those schemes have since been extended to stochastic, multivariable and even nonlinear systems, using various predictive or robust control techniques. Several advantages have been claimed to result from the use of an orthonormal series representation of process dynamics, particularly in an adaptive control framework. The Laguerre model is an output-error structure, is linear in the parameters, preserves convexity for the identification problem. It eliminates the parameter drift due to the influence of unmodelled dynamics on the nominal model, is stable, and robustly stabilizable as long as the unmodelled dynamics are stable. This effectively solves the so-called admissibility problem and makes the Laguerre structure particularly suitable for adaptive control applications in the process industries. Various adaptive controllers based on the Laguerre representation have been developed, and used in industrial applications, in diverse application such as pulping and bleaching, Ti02 pigment production, lime production, glass furnaces, gas desulfuration, fibre glass production, and beer brewing. In particular, considerable experience has been gained through the development of two commercial "general-purpose" adaptive controllers. This experience will be presented and discussed. The validity of the advantages claimed for the Laguerre methodology will be scrutinized. Finally, the ingredients necessary to the design of a robust adaptive process controller will then be reviewed and iterative control will be discussed in this framework. Copyright @ 1998 1FAC
1. INTRODUCTION
automatic tuning of PID controllers. The use of Laguerre functions in adaptive control was first proposed in Dumont and Zervos (1986) . This was followed by a significant increase in activity on the use of Laguerre functions to approximate infinite dimensional systems, e.g. Wahlberg (1991) and Makilii (1990). More recently, approximation of dynamic systems by generalized orthonormal basis functions has been studied by several research groups, e.g. Heuberger, Van den Hof and Bosgra (1995). The thrust of this recent work has been the development of orthonormal functions specifically tailored to the underlying dynamics of the system to be represented, using balanced realizations.
Over the last 15 years, there has been considerable interest in the use of orthonormal functions in system identification and adaptive control. Interest in the orthonormal series representation of signals goes back to the classical work of Wiener and Lee during the 1930's on network synthesis using Laguerre functions, Lee (1960). In the 1970's, there was a renewed interest in Laguerre functions, but as a tool for data reduction in system identification. The identification of a closed-loop system in terms of a truncated Laguerre series was considered in Zervos, Belanger and Dumont (1985) for
445
Discrete Laguerre Functions Most industrial applications of control use a discrete time model of the plant. When digitizing a continuous Laguerre model, one key requirement is that the orthonormality should be kept . Many standard methods, such as zero-orderholder method, bilinear transformation method, or backward difference method, can not be used. Those methods will not preserve the digitized Laguerre function orthonormal unless a sufficiently fast sampling rate is used (theoretically impossible). One way around this problem is the use of a fictitious hold that approximates a continuous signal by a straight line between the sampling points, and described by;
The objective is to capture the essential dynamics of the system in a series expansion involving as few terms as possible. The present paper is not an attempt to review the previously mentioned work, but rather to describe in an admittedly subjective way the use that has been made over the last 15 years of Laguerre filters to develop practical adaptive predictive control schemes that have met with some amount of success in industry.
2. LAGUERRE MODELLING 2.1
Laguerre Functions
G h (s )
Continuous Laguerre Functions The set of Laguerre functions is particularly appealing to describe stable dynamic systems because it is simple to represent and is similar to transient signals. It also closely resembles Pade approximants. The continuous Laguerre functions, a complete orthonormal set in L2[0, 00), can be represented by the simple and convenient ladder network shown in Figure 1 and can be described by:
=
J2P (s+p')"
l(t
+ 1) =
Al(t)
i=l, .. ,N(l)
+ bu(t)
y(t) = cTl(t)
+ 1) =
Al(t) + bu(t)
y(t) = cTl(t)
(5)
(6)
Alternatively, it is possible to define a set of ztransfer functions that are orthonormal and have a structure similar to the continuous Laguerre filters;
(2)
Li{z)
(3)
= v'f=l.i2 z-a
[h(t) l2(t) . . . IN(t) ]T, and cT = h C2·· · CN] . The l;'s are the outputs from each block in Fig. 1, and u(t), y(t) are the plant input and output respectively. A is a lower triangular N x N matrix where the same elements are found respectively across the diagonal or every subdiagonal , b is the input vector, and c is the Laguerre spectrum vector. The vector c gives the projection of the plant output onto the linear space whose basis is the orthonormal set of Laguerre functions. Some of the advantages of using the above series representation are that,(a) because of its resemblance to the Pade approximants timedelays can be very well represented as part of the plant dynamics, (b) theoretically the model order N does not affect the coefficients c;, and (c) extensions to multivariable schemes do not require the use of interactor matrices (Zervos and Dumont, 1988b) . with IT(t)
(4)
where, A is a lower triangular (N x N) matrix and b an (N x 1) vector, that depend solely on p and T. This state-space system is stable (p > 0), observable, and controllable. The output of the process to be modelled is then approximated by the weighted sum of the outputs of the Laguerre filters,
where i is the order of the function (i = 1, ..N), and p > 0 is the time-scale. Based on the continuous network compensation method, the Laguerre ladder network of Fig. 1 can be expressed in a stable, observable and controllable state-space form as,
l(t
Ts2
When using such a fictitious hold, the discretized Laguerre network is described by;
(s _ p)i-l
Fi(S)
= exp(sT) - 2 + exp( -sT)
=
(1z-a -
az)i-l
(7)
The Laguerre functions described by equation (7) have been used for instance in Wahlberg (1991). They can also be put in a state-space from as in (5 ).
2.2
Identification of Linear Systems
The classical method is to compute the Laguerre spectrum of a signal using the correlation method, i.e.
J b
Ci
=
h(t)li(t)dt
(8)
a
where h(t) is the plant impulse response and li(t) is the ith Laguerre function. In practice, it is more
446
U(s)
::LJi
L1 (s)
S+p
s-p S+p
I Cl I Y(s)
I I
L2(S)
I
C2
LN(S)
S-p S+p
-;--+- ••• -
-
I
I CN I
Summing Circuit
I
Fig. 1. Representation of plant dynamics using a truncated continuous Laguerre ladder network. realistic to use a least-squares type algorithm. Consider the real plant described by N
y(t) =
00
L e;Li(q) + L i=l
e;Li(q)
+ w(t)
thus reducing the detrimental effect of disturbances at high frequencies.
(9)
i=N+l
where w(t) is a disturbance. It is obvious that this model has an output-error structure, is linear in the parameters, and gives a convex identification problem. Because of that, and of the orthonormality of the Laguerre filter outputs (obtained if the plant is perturbed by a white noise or a PRBS), it is trivial to show that:
2.3 Extension to Nonlinear Systems The above methodology can be extended to nonlinear systems, following the work of Schetzen (1980) . The nonlinear Laguerre model is a special case of a Wiener model, where the linear dynamic part represented by a series of Laguerre filters is followed by a memory less nonlinear mapping. Such a nonlinear model can be derived from the Volterra series input-output representation, where the Volterra kernels are expanded via truncated Laguerre functions. A finite-time observable nonlinear system can be approximated as a truncated Wiener-Volterra series:
• Even if w(t) it colored and non-zero mean, simple least-squares provide consistent estimates of the e; 'so • The estimate of the nominal plant, i.e. of e;, for i = 1" , . ,N is unaffected by the presence of the unmodelled dynamics represented by e;, for i = N + 1, ... , 00. Wahlberg (1991) shows that the the mapping (1+ aeiw)(e iw + a) improves the condition number of the least-squares covariance matrix. Furthermore, the implicit assumption that the system is lowpass in nature reduces the asymptotic covariance of the estimate at high frequencies. For recursive least-squares, Gunnarsson and Wahlberg( 1991) show that the mean square error of the transfer function estimate can be approximated by
y(t) = ho(t)
+
L J... Jhn(n. , ·· ·, Tn) IT u(t - T;)dTi N
n
n= l
i=l
For instance, truncating the series after the second-order kernel:
J 00
y(t) = ho(t)
+
hI (Tr}U(t - Tr}dTl
+
(10)
o
JJ 00 00
h2(Tl, T2)U(t - Tr}U(t - T2) dTl dT2
Note that the case a = 0 corresponds to a Fm model. The MSE is seen to be proportional to the number of parameters. Compared with a FIR model, a representation in terms of an orthonormal series representation will be more efficient, will require less parameters and thus will give a smaller MSE. Furthermore, the disturbance spectrum is scaled by
o
0
Assuming that the Volterra kernels are in L 2[, 00) , they can be expanded and approximated as: N
h1(Tr}
= 2:>k4>k(Tl) k=l
447
(11)
N
h2(TI, T2 ) =
N
L L Cnm cPn(Tt}cPn(T2)
plant in 1988 (Dumont, Zervos and Pageau, 1990). This led to the development by Universal Dynamics Ltd of the Universal Adaptive Controller in 1989-1990. The original version was implemented as a stand-alone controller, later ported to a PC platform under OS2. The current version, called Brainwave is an OPC-compliant WindowsNT package. To this date, more than 200 hundred UAC/BrainWave applications have been performed in a variety of industries, for instance:
(12)
n=lm=l
Using Laguerre functions , this second-order nonlinear system can be expressed as the nonlinear state-space model:
= Al(t) + bu(t) yet) = Co + cTl(t) + IT(t)Dl(t) i(t)
(13) (14)
where c = {cd and D = {cnm } . Note that since the Volterra kernels are symmetric, Cnm = Cmn and thus D is symmetric. A discrete model can be derived in a similar form . Note that this model is linear in the parameters, and can thus be easily identified.
• Cement clinker grate cooler pressure control • Sulphur recovery unit tail gas ratio and O 2 control • Bleach plant compensated brightness and final brightness control • Glass forehearth temperature control • Chlor-alkali plant wastewater pH control • Beer brew-kettle boil level control • Kamyr digester effective alkali control
3. LAGUERRE-BASED CONTROL
Figure 2 below shows the engineering graphical user interface (G UI) for Brainwave. The user can provide initial estimates for gains, delays and time constants, but those are not critical. Various learning modes are available. Learning can be either continuous, or occur only at set point changes. The user has the choice between three preset rates of learning, and three preset closedloop responses: slow, normal and fast, thus making the setting of parameters relatively easy. For each loop, there is the possibility to have three feedforward compensators. In addition, future versions will offer the capability to do gain scheduling, using multiple models that can be learned automatically. This last feature is very useful on processes where the dynamics change significantly with the operating conditions, such as power boilers. From the operator's point of view, Brainwave looks just like a regular controller template, thus minimal training is necessary.
Once a system is represented a truncated by an orthonormal series representation, and put in a state-space form , virtually any control design could be used. However, in keeping with the objective of developing an adaptive controller suitable for the process industries, the use of a model-based predictive control technique is a logical choice. Zervos and Dumont (1988) used an extended-horizon control law, later extended to a full-fledged generalized-predictive control law (Elshafei, Dumont and Elnaggar, 1994). The use of a quadratic dynamic matrix control law is described in Finn, Wahlberg and Ydstie (1993). Multivariable and stochastic extensions are described respectively in Zervos and Dumont (1988b) and Zervos and Dumont (1988c) . Nonlinear control laws are described in Dumont and Fu (1993) and in Dumont, Fu and Lu (1994) .
4. INDUSTRlAL APPLICATIONS
4.2 Rotary Kiln Control Many industrial applications of Laguerre-based adaptive controllers have taken place over the last decade. Although a few of those applications have been developed as special purpose adaptive control systems for very specific and demanding applications, many applications have been in the form of a commercial general-purpose adaptive control package.
This methodology has been in use for several years on four Ti0 2 rotary kilns in Europe, (Dumont, 1992) in two plants with no staff formally trained in advanced control methods. An advantage of Laguerre-based representation of process dynamics is very useful when controlling variable flow systems, common in processes involving material flows , e.g. sheetmaking processes, band conveyors, flows in pipes, and rotary kilns. In many of those systems, the dynamics are inversely proportional to variables such as flows or speeds. On a rotary kiln , it is easy to compensate for changes in the rotational speed by making the Laguerre parameter p proportional to the rotational speed w. This would be more difficult to do with a transfer function model. In both plants, the control system has
4.1 From LUST to Broinwave The original adaptive controller based on a discretized continuous Laguerre network and on a simple extended-horizon control law, and called the Laguerre Unstructured Self-Thner (LUST) was successfully tested on an industrial bleach
448
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PV~
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PV
I"~·"U
SP
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.'
~ cv
::');·::'.7t;',ttf'
ff1De~
10. 00
Ff1
FF2De~
lu un
Ff2
Ff3De=--
la. 0 11
Ff3
Learning Se1UP
I I J:; Ir Ir Ir I J:;
i
_L..-ning~
IS
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-~ iiiiit';§M
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-
.
1[1) -NeW SP ~equest
Fig. 2. The Brainwave engineering graphical user interface, showing the various menus and options (Courtesy Universal Dynamics Technologies Inc.)
contributed to increased capacity and is crucial in running the calciners through frequent feed interruptions and feedrate changes while maintaining product quality.
require initialization when switching, as long as the models only differ by their c vector. Theoretical advantages Several theoretical advantages result from the use of an orthonormal series representation of pr
5. DISCUSSION
5.1 So What 's the Big Deal? There are both practical and theoretical advantages to the use of orthonormal functions in adaptive control. Practical advantages From the user 's point of view, the main practical advantage of this methodology is the minimal amount of prior knowledge required to commission a loop, essentially a rough estimate of the time delay and the dominant time constant. This greatly reduces the setup and commissioning time. Furthermore, because the time delay is implicitely described by the Laguerre network representation, it is easy to track variations of the time delay. When used in a multimodel system, the use of a Laguerre network with fixed pole greatly smoothes the transition between models, as the state is the same for all models, and thus does not
5.2 Nothing Comes For Free .. . As for any technique, there are some downsides to using orthonormal series in adaptive control. The
449
first, and most obvious one is the loss of physical insight. Indeed, poles and within some limits, zeros, are usually easy to interpret, and certainly are well known to control engineers. Looking at a Laguerre spectrum, does not on the other hand provide much in terms of physical interpretation. Another and more subtle problem comes from the use of an unstructured model, and arises when the frequency content of the excitation used during identification is incompatible with the choice of the Laguerre pole and the process dynamics. If the impulse response of the highest-order Laguerre has not vanished within the longest duration of the PRBS pulse, then it is very likely that an artifact in the identified response will result . This problem would not occur with a low-order transfer function model. The choice of the optimal Laguerre time scale is discussed e.g. in Fu and Dumont (1993). If the process to be modelled as oscillatory modes, then Laguerre filters are inadequate. In such cases, Kautz filters or tailored generalized orthogonal polynomials can be used, at the cost of increased prior knowledge requirements.
By means of this set of values, the C; and
Ci=I/2(m~ef'+~nef');
~ei
are:
i=I, ... ,n
and
~Ci = 1/2 (m~ef' - m~nef');
i = 1, .. . ,n
A second technique assumes that {h(k)} is defined by a set of the impulse responses, as:
where h(k) is the median value of {h(k)}; ~h(k) is the absolute value of the maximum deviation of {h(k)}, and ch(k) represents the uncertainty. Thus, the set of series coefficients is: 00
{Ci} = L{h(k)}i(q-l)O(k);
i = 1, .. . ,n
k=O
Developing this equation, the parameters C; and ~Ci can be obtained as: 00
C; = Lh(k)i(q-l)O(k) k=O
6. CURRENT WORK
00
~Ci = L
6.1 Robust Control
~h(k)i(q-l)O(k)
k=O
Recently, robust predictive control based on truncated orthonormal series representation was proposed (Oliveira et al.; 1996, 1998). Assuming uncertainty bounds in the Laguerre coefficients, a model with structured uncertainties is:
A third, and promising technique is based on the UBBE 1 approach, which assumes that the error between the model output and the measured process output satisfies:
le(k)1
n
y(k) = L Ci(ci)li(k)
= IYm(k) - y(k)1 ::; emax(k);
k E [1,K]
where Ym(k) is the measured output, K is the number of measurements, and emax (k) is known. Hence, the pair of input/output signals is assumed to be modeled as:
i=1
where liCk) is the output of the i - th function i(q-l) and Ci represents the uncertainty in each coefficient. Each Ci(ci) coefficient is represented by means of a median value and an absolute value of the maximum deviation ~Ci with respect to its median value, that is:
l(k+l) Ym(k)
= =
Al(k)+bu(k) l(k) + e(k)
eT
All parameters e that are consistent with the model structure, the prior error bounds and the measurement Ym (k ), belong to the so-called parameters membership set defined as:
with Ic; I ::; 1. A first method for the estimation of uncertain parameters assumes a set of M impulse responses:
K
S(k) =
n (Hi(k)nH!(k) ) k=1
n
hm(k) = Lef'i(q-l)O(k) ;
m = 1, . . . ,M
where
;=1
HICk) = { e I eT l(k) = h(k) - emax(k) }
and the coefficients ef' can be obtained as:
H2(k)
= { e I eT l(k) = h(k) + emax(k)
00
ef' = L
hm(k)i(q-l)o(k);
m = 1, ... , M 1
k=O
450
United But Bounded Error
}
6.3 Multivariable Systems
The membership set S has a poly topic shape, and its determination can become very complex when the number of measurements K increases. The polytope S is usually approximated by a simpleshaped set containing S, for instance an ellipsoid or an orthotope. Two methods to determine an outer bound orthotope approximation of S:
The use of Laguerre functions for describing and controlling multivariable systems was briefly and superficially explored in Zervos and Dumont (1988b) . Recent work (Ninness, G6mez and Weller, 1995; Fisher and Medvedev, 1998) shows that the use of orthonormal basis functions gives computationally simple and efficient identification schems for MIMO systems. In particular, Fisher and Medvedev show that using subspace identification methods in the Laguerre domain yields more accurate results, likely due to improved separation of plant and noise dynamics. Such techniques could potentially be used in conjunction with MIMO model-based predictive control.
• The non-recursive method of Milanese and Belfort (1982) , based on the solution of 2n linear programming problems, each one having 2K constraints. • The recursive method proposed by Messaoud and Favier (1994) , based on simple geometric considerations Once a model with uncertainties has been identified, a robust predictive controller can be designed. Solving the robust control problem as a min-max optimization, closed-loop stability can be guaranteed by sufficient conditions for the selection of the controller prediction horizon, Oliveira et al (1996). Like the FIR-based robust predictive controller of Campo and Morari (1987) , this representation gives a convex problem, and does not require assumptions about the order and time delay of the system. However, compared with a FIR model, it requires a smaller number of parameters to describe the model and its uncertainty.
7. CONCLUSIONS This paper has shown that the use of orthonormal basis functions, and in particular of Laguerre functions, for describing process dynamics can be beneficial in adaptive control. In particular, it has been shown that not only there are theroetical advantages in doing so, but perhaps most importantly, practical and industrially accepted adaptive controllers can be built around that methodology. Finally, the potential of orthonormal functions in designing robust adaptive controllers, performance enhancement controllers and multivariable adaptive controllers was discussed and shown to warrant further work.
6.2 Performance Enhancement Control
8. REFERENCES
In Dumont and Belanger (1978) , an adaptive controller was implemented around a closed-loop system under fixed minimum-variance control. It was shown that if the inner fixed MV controller was indeed providing minimum variance control, then the outer self-tuning controller would take no action. In case of non-optimality of the inner controller, the self-tuner would then adjust its parameters to enhance the performance of the closedloop system. This configuration was shown to provide an efficient and robust adaptive control loop. Recently, a related concept has been the object of rigorous research and seen the development of supporting theory based on the Youla-Kucera parameterization of all stabilizing controllers (Tay, Mareels and Moore, 1998) and iterative control design. The drawback of this methodology when using transfer function representations, is that each iteration increases the controller order, and thus controller order reduction is generally required . When using an orthonormal series representation, the controller order need not increase, and if it does, it is a fairly trivial matter to add a few filters to the orthonormal filter network. We are currently exploring this idea.
[] Campo, P.J . and M. Morari (1987), "Robust Model Predictive Control" , 1987 ACC, Minneapolis, June 1987, pp. 1021-1026. [] Dumont, G .A. (1992) , "Fifteen Years in the Life of an Adaptive Controller" , Plenary Talk, 4th IFAC Symp. on Adaptive Systems in Control and Signal Processing, July 1-3, 1992, Grenoble, France, pp. 109-120. [] Dumont, G.A. and P.R. Belanger (1978), "SelfTuning Control of a Titanium Dioxide Kiln", IEEE Transactions on Automatic Control, v. 23, pp. 532- 538. [] Dumont, G.A., A. Elnaggar, and A.L. Elshafei (1993) , "Adaptive Predictive Control of Systems with Time-Varying Time Delay", Int. J. Adaptive Control and Signal Processing, v . 7, no. 2, pp. 91-1Ol. [] Dumont, G .A. and Y . Fu (1993) , "Nonlinear Adaptive Control via Laguerre Expansion of Volterra Kernels", Int. J. Adaptive Control and Signal Processing, v. 7, no. 5, pp.367-382. [] Dumont, G .A., Y. Fu, and G . Lu (1994) , "Nonlinear Adaptive GPC and Applications", Advances in Model-Based Predictive Control, D.W .
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FROM LUST TO BRAINWAVE: FIFTEEN YEARS OF PERSISTENT EXCITATION Guy A. Dumont·
• Department of Electrical and Computer Engineering clo Pulp and Paper Centre, 2385 East Mall University of British Columbia, Vancouver, BC V6T 1Z4 Canada email: guyd
Abstract: Adaptive predictive controllers based on truncated orthonormal series representation of the process dynamics were first proposed in the mid 1980's. Those schemes have since been extended to stochastic, multivariable and even nonlinear systems, using various predictive or robust control techniques. Several advantages have been claimed to result from the use of an orthonormal series representation of process dynamics, particularly in an adaptive control framework. The Laguerre model is an output-error structure, is linear in the parameters, preserves convexity for the identification problem. It eliminates the parameter drift due to the influence of unmodelled dynamics on the nominal model, is stable, and robustly stabilizable as long as the unmodelled dynamics are stable. This effectively solves the so-called admissibility problem and makes the Laguerre structure particularly suitable for adaptive control applications in the process industries. Various adaptive controllers based on the Laguerre representation have been developed, and used in industrial applications, in diverse application such as pulping and bleaching, Ti02 pigment production, lime production, glass furnaces, gas desulfuration, fibre glass production, and beer brewing. In particular, considerable experience has been gained through the development of two commercial "general-purpose" adaptive controllers. This experience will be presented and discussed. The validity of the advantages claimed for the Laguerre methodology will be scrutinized. Finally, the ingredients necessary to the design of a robust adaptive process controller will then be reviewed and iterative control will be discussed in this framework. Copyright @ 1998 1FAC
1. INTRODUCTION
automatic tuning of PID controllers. The use of Laguerre functions in adaptive control was first proposed in Dumont and Zervos (1986) . This was followed by a significant increase in activity on the use of Laguerre functions to approximate infinite dimensional systems, e.g. Wahlberg (1991) and Makilii (1990). More recently, approximation of dynamic systems by generalized orthonormal basis functions has been studied by several research groups, e.g. Heuberger, Van den Hof and Bosgra (1995). The thrust of this recent work has been the development of orthonormal functions specifically tailored to the underlying dynamics of the system to be represented, using balanced realizations.
Over the last 15 years, there has been considerable interest in the use of orthonormal functions in system identification and adaptive control. Interest in the orthonormal series representation of signals goes back to the classical work of Wiener and Lee during the 1930's on network synthesis using Laguerre functions, Lee (1960). In the 1970's, there was a renewed interest in Laguerre functions, but as a tool for data reduction in system identification. The identification of a closed-loop system in terms of a truncated Laguerre series was considered in Zervos, Belanger and Dumont (1985) for
445
Discrete Laguerre Functions Most industrial applications of control use a discrete time model of the plant. When digitizing a continuous Laguerre model, one key requirement is that the orthonormality should be kept . Many standard methods, such as zero-orderholder method, bilinear transformation method, or backward difference method, can not be used. Those methods will not preserve the digitized Laguerre function orthonormal unless a sufficiently fast sampling rate is used (theoretically impossible). One way around this problem is the use of a fictitious hold that approximates a continuous signal by a straight line between the sampling points, and described by;
The objective is to capture the essential dynamics of the system in a series expansion involving as few terms as possible. The present paper is not an attempt to review the previously mentioned work, but rather to describe in an admittedly subjective way the use that has been made over the last 15 years of Laguerre filters to develop practical adaptive predictive control schemes that have met with some amount of success in industry.
2. LAGUERRE MODELLING 2.1
Laguerre Functions
G h (s )
Continuous Laguerre Functions The set of Laguerre functions is particularly appealing to describe stable dynamic systems because it is simple to represent and is similar to transient signals. It also closely resembles Pade approximants. The continuous Laguerre functions, a complete orthonormal set in L2[0, 00), can be represented by the simple and convenient ladder network shown in Figure 1 and can be described by:
=
J2P (s+p')"
l(t
+ 1) =
Al(t)
i=l, .. ,N(l)
+ bu(t)
y(t) = cTl(t)
+ 1) =
Al(t) + bu(t)
y(t) = cTl(t)
(5)
(6)
Alternatively, it is possible to define a set of ztransfer functions that are orthonormal and have a structure similar to the continuous Laguerre filters;
(2)
Li{z)
(3)
= v'f=l.i2 z-a
[h(t) l2(t) . . . IN(t) ]T, and cT = h C2·· · CN] . The l;'s are the outputs from each block in Fig. 1, and u(t), y(t) are the plant input and output respectively. A is a lower triangular N x N matrix where the same elements are found respectively across the diagonal or every subdiagonal , b is the input vector, and c is the Laguerre spectrum vector. The vector c gives the projection of the plant output onto the linear space whose basis is the orthonormal set of Laguerre functions. Some of the advantages of using the above series representation are that,(a) because of its resemblance to the Pade approximants timedelays can be very well represented as part of the plant dynamics, (b) theoretically the model order N does not affect the coefficients c;, and (c) extensions to multivariable schemes do not require the use of interactor matrices (Zervos and Dumont, 1988b) . with IT(t)
(4)
where, A is a lower triangular (N x N) matrix and b an (N x 1) vector, that depend solely on p and T. This state-space system is stable (p > 0), observable, and controllable. The output of the process to be modelled is then approximated by the weighted sum of the outputs of the Laguerre filters,
where i is the order of the function (i = 1, ..N), and p > 0 is the time-scale. Based on the continuous network compensation method, the Laguerre ladder network of Fig. 1 can be expressed in a stable, observable and controllable state-space form as,
l(t
Ts2
When using such a fictitious hold, the discretized Laguerre network is described by;
(s _ p)i-l
Fi(S)
= exp(sT) - 2 + exp( -sT)
=
(1z-a -
az)i-l
(7)
The Laguerre functions described by equation (7) have been used for instance in Wahlberg (1991). They can also be put in a state-space from as in (5 ).
2.2
Identification of Linear Systems
The classical method is to compute the Laguerre spectrum of a signal using the correlation method, i.e.
J b
Ci
=
h(t)li(t)dt
(8)
a
where h(t) is the plant impulse response and li(t) is the ith Laguerre function. In practice, it is more
446
U(s)
::LJi
L1 (s)
S+p
s-p S+p
I Cl I Y(s)
I I
L2(S)
I
C2
LN(S)
S-p S+p
-;--+- ••• -
-
I
I CN I
Summing Circuit
I
Fig. 1. Representation of plant dynamics using a truncated continuous Laguerre ladder network. realistic to use a least-squares type algorithm. Consider the real plant described by N
y(t) =
00
L e;Li(q) + L i=l
e;Li(q)
+ w(t)
thus reducing the detrimental effect of disturbances at high frequencies.
(9)
i=N+l
where w(t) is a disturbance. It is obvious that this model has an output-error structure, is linear in the parameters, and gives a convex identification problem. Because of that, and of the orthonormality of the Laguerre filter outputs (obtained if the plant is perturbed by a white noise or a PRBS), it is trivial to show that:
2.3 Extension to Nonlinear Systems The above methodology can be extended to nonlinear systems, following the work of Schetzen (1980) . The nonlinear Laguerre model is a special case of a Wiener model, where the linear dynamic part represented by a series of Laguerre filters is followed by a memory less nonlinear mapping. Such a nonlinear model can be derived from the Volterra series input-output representation, where the Volterra kernels are expanded via truncated Laguerre functions. A finite-time observable nonlinear system can be approximated as a truncated Wiener-Volterra series:
• Even if w(t) it colored and non-zero mean, simple least-squares provide consistent estimates of the e; 'so • The estimate of the nominal plant, i.e. of e;, for i = 1" , . ,N is unaffected by the presence of the unmodelled dynamics represented by e;, for i = N + 1, ... , 00. Wahlberg (1991) shows that the the mapping (1+ aeiw)(e iw + a) improves the condition number of the least-squares covariance matrix. Furthermore, the implicit assumption that the system is lowpass in nature reduces the asymptotic covariance of the estimate at high frequencies. For recursive least-squares, Gunnarsson and Wahlberg( 1991) show that the mean square error of the transfer function estimate can be approximated by
y(t) = ho(t)
+
L J... Jhn(n. , ·· ·, Tn) IT u(t - T;)dTi N
n
n= l
i=l
For instance, truncating the series after the second-order kernel:
J 00
y(t) = ho(t)
+
hI (Tr}U(t - Tr}dTl
+
(10)
o
JJ 00 00
h2(Tl, T2)U(t - Tr}U(t - T2) dTl dT2
Note that the case a = 0 corresponds to a Fm model. The MSE is seen to be proportional to the number of parameters. Compared with a FIR model, a representation in terms of an orthonormal series representation will be more efficient, will require less parameters and thus will give a smaller MSE. Furthermore, the disturbance spectrum is scaled by
o
0
Assuming that the Volterra kernels are in L 2[, 00) , they can be expanded and approximated as: N
h1(Tr}
= 2:>k4>k(Tl) k=l
447
(11)
N
h2(TI, T2 ) =
N
L L Cnm cPn(Tt}cPn(T2)
plant in 1988 (Dumont, Zervos and Pageau, 1990). This led to the development by Universal Dynamics Ltd of the Universal Adaptive Controller in 1989-1990. The original version was implemented as a stand-alone controller, later ported to a PC platform under OS2. The current version, called Brainwave is an OPC-compliant WindowsNT package. To this date, more than 200 hundred UAC/BrainWave applications have been performed in a variety of industries, for instance:
(12)
n=lm=l
Using Laguerre functions , this second-order nonlinear system can be expressed as the nonlinear state-space model:
= Al(t) + bu(t) yet) = Co + cTl(t) + IT(t)Dl(t) i(t)
(13) (14)
where c = {cd and D = {cnm } . Note that since the Volterra kernels are symmetric, Cnm = Cmn and thus D is symmetric. A discrete model can be derived in a similar form . Note that this model is linear in the parameters, and can thus be easily identified.
• Cement clinker grate cooler pressure control • Sulphur recovery unit tail gas ratio and O 2 control • Bleach plant compensated brightness and final brightness control • Glass forehearth temperature control • Chlor-alkali plant wastewater pH control • Beer brew-kettle boil level control • Kamyr digester effective alkali control
3. LAGUERRE-BASED CONTROL
Figure 2 below shows the engineering graphical user interface (G UI) for Brainwave. The user can provide initial estimates for gains, delays and time constants, but those are not critical. Various learning modes are available. Learning can be either continuous, or occur only at set point changes. The user has the choice between three preset rates of learning, and three preset closedloop responses: slow, normal and fast, thus making the setting of parameters relatively easy. For each loop, there is the possibility to have three feedforward compensators. In addition, future versions will offer the capability to do gain scheduling, using multiple models that can be learned automatically. This last feature is very useful on processes where the dynamics change significantly with the operating conditions, such as power boilers. From the operator's point of view, Brainwave looks just like a regular controller template, thus minimal training is necessary.
Once a system is represented a truncated by an orthonormal series representation, and put in a state-space form , virtually any control design could be used. However, in keeping with the objective of developing an adaptive controller suitable for the process industries, the use of a model-based predictive control technique is a logical choice. Zervos and Dumont (1988) used an extended-horizon control law, later extended to a full-fledged generalized-predictive control law (Elshafei, Dumont and Elnaggar, 1994). The use of a quadratic dynamic matrix control law is described in Finn, Wahlberg and Ydstie (1993). Multivariable and stochastic extensions are described respectively in Zervos and Dumont (1988b) and Zervos and Dumont (1988c) . Nonlinear control laws are described in Dumont and Fu (1993) and in Dumont, Fu and Lu (1994) .
4. INDUSTRlAL APPLICATIONS
4.2 Rotary Kiln Control Many industrial applications of Laguerre-based adaptive controllers have taken place over the last decade. Although a few of those applications have been developed as special purpose adaptive control systems for very specific and demanding applications, many applications have been in the form of a commercial general-purpose adaptive control package.
This methodology has been in use for several years on four Ti0 2 rotary kilns in Europe, (Dumont, 1992) in two plants with no staff formally trained in advanced control methods. An advantage of Laguerre-based representation of process dynamics is very useful when controlling variable flow systems, common in processes involving material flows , e.g. sheetmaking processes, band conveyors, flows in pipes, and rotary kilns. In many of those systems, the dynamics are inversely proportional to variables such as flows or speeds. On a rotary kiln , it is easy to compensate for changes in the rotational speed by making the Laguerre parameter p proportional to the rotational speed w. This would be more difficult to do with a transfer function model. In both plants, the control system has
4.1 From LUST to Broinwave The original adaptive controller based on a discretized continuous Laguerre network and on a simple extended-horizon control law, and called the Laguerre Unstructured Self-Thner (LUST) was successfully tested on an industrial bleach
448
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i
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Fig. 2. The Brainwave engineering graphical user interface, showing the various menus and options (Courtesy Universal Dynamics Technologies Inc.)
contributed to increased capacity and is crucial in running the calciners through frequent feed interruptions and feedrate changes while maintaining product quality.
require initialization when switching, as long as the models only differ by their c vector. Theoretical advantages Several theoretical advantages result from the use of an orthonormal series representation of pr
5. DISCUSSION
5.1 So What 's the Big Deal? There are both practical and theoretical advantages to the use of orthonormal functions in adaptive control. Practical advantages From the user 's point of view, the main practical advantage of this methodology is the minimal amount of prior knowledge required to commission a loop, essentially a rough estimate of the time delay and the dominant time constant. This greatly reduces the setup and commissioning time. Furthermore, because the time delay is implicitely described by the Laguerre network representation, it is easy to track variations of the time delay. When used in a multimodel system, the use of a Laguerre network with fixed pole greatly smoothes the transition between models, as the state is the same for all models, and thus does not
5.2 Nothing Comes For Free .. . As for any technique, there are some downsides to using orthonormal series in adaptive control. The
449
first, and most obvious one is the loss of physical insight. Indeed, poles and within some limits, zeros, are usually easy to interpret, and certainly are well known to control engineers. Looking at a Laguerre spectrum, does not on the other hand provide much in terms of physical interpretation. Another and more subtle problem comes from the use of an unstructured model, and arises when the frequency content of the excitation used during identification is incompatible with the choice of the Laguerre pole and the process dynamics. If the impulse response of the highest-order Laguerre has not vanished within the longest duration of the PRBS pulse, then it is very likely that an artifact in the identified response will result . This problem would not occur with a low-order transfer function model. The choice of the optimal Laguerre time scale is discussed e.g. in Fu and Dumont (1993). If the process to be modelled as oscillatory modes, then Laguerre filters are inadequate. In such cases, Kautz filters or tailored generalized orthogonal polynomials can be used, at the cost of increased prior knowledge requirements.
By means of this set of values, the C; and
Ci=I/2(m~ef'+~nef');
~ei
are:
i=I, ... ,n
and
~Ci = 1/2 (m~ef' - m~nef');
i = 1, .. . ,n
A second technique assumes that {h(k)} is defined by a set of the impulse responses, as:
where h(k) is the median value of {h(k)}; ~h(k) is the absolute value of the maximum deviation of {h(k)}, and ch(k) represents the uncertainty. Thus, the set of series coefficients is: 00
{Ci} = L{h(k)}i(q-l)O(k);
i = 1, .. . ,n
k=O
Developing this equation, the parameters C; and ~Ci can be obtained as: 00
C; = Lh(k)i(q-l)O(k) k=O
6. CURRENT WORK
00
~Ci = L
6.1 Robust Control
~h(k)i(q-l)O(k)
k=O
Recently, robust predictive control based on truncated orthonormal series representation was proposed (Oliveira et al.; 1996, 1998). Assuming uncertainty bounds in the Laguerre coefficients, a model with structured uncertainties is:
A third, and promising technique is based on the UBBE 1 approach, which assumes that the error between the model output and the measured process output satisfies:
le(k)1
n
y(k) = L Ci(ci)li(k)
= IYm(k) - y(k)1 ::; emax(k);
k E [1,K]
where Ym(k) is the measured output, K is the number of measurements, and emax (k) is known. Hence, the pair of input/output signals is assumed to be modeled as:
i=1
where liCk) is the output of the i - th function i(q-l) and Ci represents the uncertainty in each coefficient. Each Ci(ci) coefficient is represented by means of a median value and an absolute value of the maximum deviation ~Ci with respect to its median value, that is:
l(k+l) Ym(k)
= =
Al(k)+bu(k) l(k) + e(k)
eT
All parameters e that are consistent with the model structure, the prior error bounds and the measurement Ym (k ), belong to the so-called parameters membership set defined as:
with Ic; I ::; 1. A first method for the estimation of uncertain parameters assumes a set of M impulse responses:
K
S(k) =
n (Hi(k)nH!(k) ) k=1
n
hm(k) = Lef'i(q-l)O(k) ;
m = 1, . . . ,M
where
;=1
HICk) = { e I eT l(k) = h(k) - emax(k) }
and the coefficients ef' can be obtained as:
H2(k)
= { e I eT l(k) = h(k) + emax(k)
00
ef' = L
hm(k)i(q-l)o(k);
m = 1, ... , M 1
k=O
450
United But Bounded Error
}
6.3 Multivariable Systems
The membership set S has a poly topic shape, and its determination can become very complex when the number of measurements K increases. The polytope S is usually approximated by a simpleshaped set containing S, for instance an ellipsoid or an orthotope. Two methods to determine an outer bound orthotope approximation of S:
The use of Laguerre functions for describing and controlling multivariable systems was briefly and superficially explored in Zervos and Dumont (1988b) . Recent work (Ninness, G6mez and Weller, 1995; Fisher and Medvedev, 1998) shows that the use of orthonormal basis functions gives computationally simple and efficient identification schems for MIMO systems. In particular, Fisher and Medvedev show that using subspace identification methods in the Laguerre domain yields more accurate results, likely due to improved separation of plant and noise dynamics. Such techniques could potentially be used in conjunction with MIMO model-based predictive control.
• The non-recursive method of Milanese and Belfort (1982) , based on the solution of 2n linear programming problems, each one having 2K constraints. • The recursive method proposed by Messaoud and Favier (1994) , based on simple geometric considerations Once a model with uncertainties has been identified, a robust predictive controller can be designed. Solving the robust control problem as a min-max optimization, closed-loop stability can be guaranteed by sufficient conditions for the selection of the controller prediction horizon, Oliveira et al (1996). Like the FIR-based robust predictive controller of Campo and Morari (1987) , this representation gives a convex problem, and does not require assumptions about the order and time delay of the system. However, compared with a FIR model, it requires a smaller number of parameters to describe the model and its uncertainty.
7. CONCLUSIONS This paper has shown that the use of orthonormal basis functions, and in particular of Laguerre functions, for describing process dynamics can be beneficial in adaptive control. In particular, it has been shown that not only there are theroetical advantages in doing so, but perhaps most importantly, practical and industrially accepted adaptive controllers can be built around that methodology. Finally, the potential of orthonormal functions in designing robust adaptive controllers, performance enhancement controllers and multivariable adaptive controllers was discussed and shown to warrant further work.
6.2 Performance Enhancement Control
8. REFERENCES
In Dumont and Belanger (1978) , an adaptive controller was implemented around a closed-loop system under fixed minimum-variance control. It was shown that if the inner fixed MV controller was indeed providing minimum variance control, then the outer self-tuning controller would take no action. In case of non-optimality of the inner controller, the self-tuner would then adjust its parameters to enhance the performance of the closedloop system. This configuration was shown to provide an efficient and robust adaptive control loop. Recently, a related concept has been the object of rigorous research and seen the development of supporting theory based on the Youla-Kucera parameterization of all stabilizing controllers (Tay, Mareels and Moore, 1998) and iterative control design. The drawback of this methodology when using transfer function representations, is that each iteration increases the controller order, and thus controller order reduction is generally required . When using an orthonormal series representation, the controller order need not increase, and if it does, it is a fairly trivial matter to add a few filters to the orthonormal filter network. We are currently exploring this idea.
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"Mimo System Identification Using Orthonormal Basis Functions", 34th IEEE CDC, New Orleans, LA, December 1995, pp. 703-708. Oliveira, G.H.C., G. Favier, G.A. Dumont, and W.C. Amaral (1996), "Robust Predictive Control Based on Laguerre Filters Modeling" , 13th IFAC World Congress, San Francisco, June 30July 5, 1996, pp. G375-380. Oliveira, G.H.C., G. Favier, G.A. Dumont, and W .C. Amaral (1998) , "Uncertainties Identification using Laguerre Models with Application to Paper Machine Headbox", CESA '98, NabeulHammamet, Tunisia, April 1998, pp. 329-334. Schetzen, M. (1980), The Volterra and Wiener Theory of Nonlinear Systems,Wiley, New York Tay, T-T, I. Mareels, and J.B. Moore (1998) , High Performance Control, Birkhiiuser, Boston. Van den Hof, P.M., P.S. Heuberger and J . Bokor (1995) , "System Identification with Generalized Orthonormal Basis Functions", Automatica, v. 31 , no. 12, pp. 1821-1834. Wahlberg, B. (1991) , "System Identification Using Laguerre Models", IEEE Transactions on Automatic Control, v. 36, no. 5, pp. 551-562. Wahlberg, B. and P.M. Miikilii (1996), "On Approximation of Stable Linear Dynamical Systems using Laguerre and Kautz Functions", Automatica, v. 32, no. 5, pp. 693-708. Ydstie, B.E. (1996), "Certainty Equivalence Adaptive Control: What's New in the Gap", Chemical Process Control-V, Tahoe City, CA, January 1996, pp. 9-23. Zervos, C. and G. Dumont (1988a), "Deterministic Adaptive Control Based on Laguerre Series Representation", International Journal of Control, v. 48, no. 6, pp. 2333-2359. Zervos, C. and G.Dumont (1988b), "Multivariable Self-Tuning Control Based on Laguerre Series Representation", Workshop on Adaptive Control Strategies for Industrial Use, Kananaskis, Alberta, June 20-221988, pp. 184196. Zervos, C. and G. Dumont (1988c) , "Laguerre Functions in Stochastic Self-Tuning Control" , 1988 Workshop on Robust Adaptive Control, Aug. 22-24 1988, Newcastle, Australia, pp. 102107. Zervos, C., G. Dumont and G. Pageau (1990) , "Laguerre-Based Adaptive Control of pH in an Industrial Bleach Plant Extraction Stage" , Automatica, v. 26, no. 4, pp. 781-787.