1-s2.0-000510989500093C-main

Auromoticn, Vol. 31, No. 11, pp. 1637-1647. 1995

Pergamon ooo5-109q95)ooo93-3 Copyright 0 1995 Elsevier Science Ltd

Printed m Great Britain. All rights reserved ocw1098/95 $9.50 + 0.M)

Model Validation for Robust Control: an Experimental Process Control Application*

ROY S. SMITH?

Robust control (discrete frequency-domain) model validation theory is discussed and contrasted with validation for standard probabilistic models. A process control experiment illustrates the application of the theory to the

engineering assessment of model adequacy.

Key Words-Model validation; uncertain dynamic systems; identification; robust control; complex perturbations; fault detection.

Abstract-The practical application of robust control design methodologies depends on the ability to develop suitable models of physical systems. Model validation provides a means of assessing the applicability of a given robust control model (nominal model with linear fractional norm bounded perturbations and norm bounded unknown inputs) with respect to an input-output experiment.

This paper describes the practical application of the model validation theory (for X-/p framework models) to a laboratory process control problem. A discrete-time fre- quency domain approach is used. Two candidate robust control models are postulated for the system. An experiment is performed and the theory is used to quantitatively assess the applicability of each of the candidate models.

1. INTRODUCTION

The motivating problem is the design of a control system for a physical plant. The robust control modeling framework includes norm- bounded perturbations, in addition to additive noise, to describe the uncertainty in the relationship between the nominal plant model and the physical system. Standard identification techniques have emphasized probabilistic de- scriptions of this uncertainty, typically using a probabilistic noise description for the purpose. The mismatch in these frameworks has hindered the practical application of robust control and has led to an increased interest in the problem of suitably modeling the uncertainty between nominal models and physical systems.

Robust control models, for the X-/p design methodology, are described by linear fractional transformations upon a stable, linear time-

* Received 7 September 1993; revised 1 November 1994; received in final form 9 June 1995. This paper was not presented at any IFAC meeting. This paper was recom- mended for publication in revised form by Associate Editor Bo Wahlberg under the direction of Editor Torsten Sijderstrom. Corresponding author Dr Roy S. Smith. Tel. +l 805 893 2967: Fax +l 805 893 3262: E-mail [email protected].

t Electrical and Computer Engineering Department, University of California, Santa Barbara, CA 93106, U.S.A.

invariant (LTI), norm-bounded perturbation. Unknown signals, such as noise or disturbances, are assumed to be of unity bounded energy. A robust control model is in fact a model set generated by all possible perturbations and additive signals satisfying the norm bounds.

The greater richness of this modeling frame- work carries with it greater difficulties in developing models. A designer must now distinguish between, and provide bounds for, the perturbations and the additive disturbance/noise signals, in addition to estimating a nominal model. Standard identification techniques cur- rently do not include such norm bounds on the perturbations. Recent work has considered modeling systems with perturbations in addition to additive noise. See for example, the work of Goodwin et al. (1990) and Hjalmarsson and Ljung (1992).

An alternative approach to developing robust control models is the area of identification in Xm, developed by Parker and Bitmead (1987) and Helmicki et al. (1991) with additional work by Makila and Partington (1991) and Gu and Khargonekar (1991). A priori Xm perturbation bounds are assumed, and data is taken and analyzed in order to provide a compatible model, meeting the bounds with a small amount of additive noise. The full generality of the robust control of framework cannot yet be handled, and the perturbation bound depends only upon the a priori assumptions, rather than the experimental data.

Krause et al. (1990) and Kosut et al. (1990) have also studied the problem of parameter identification in the presence of unmodeled dynamics. A similar formulation is used by Zhou and Kimura (1994). In their work, the poles of the system are prescribed and the search for parameters in the presence of bounded noise and

1637

1638 R. S. Smith

bounded perturbations is a convex optimization problem.

Assumptions on the physical system are usually applied in the theoretical study of identification. This is a natural thing to do-if an algorithm will not work on an idealized system then one has little confidence of it working on a physical system.

The model validation methods described here approach the uncertainty modeling problem from a different point of view. No assumptions are made about the nature of the physical system. Rather, measurements are taken, and the assumption that the model describes the system is directly tested. Simply stated, the model validation problem is Could the model have generated the observed datum? In a robust control context, this is equivalent to asking whether or not there exists an unknown disturbance/noise signal and an unknown per- turbation, within the assumed bounded sets, accounting for the observed input-output datum. Model validation gives a means of studying the relationship between an uncertainty model and experimental data. In this sense, it is complementary to any uncertainty identification methodology.

Formally stated, model validation can deter- mine whether or not a single experimental input-output datum could have been generated by the model. A model can only be invalidated by this procedure-model invalidation is a more appropriate term, although the subsequent discussion uses the more common term: model validation. Repeated experiments where every datum can be accounted for by the model does, however, lead to confidence about the suitability of the model. This paper will illustrate that useful engineering conclusions can be drawn from a validation analysis.

Model validation was studied in detail for Xm/p robust control models using frequency domain input-output data, by Smith and Doyle (1989,1992). Krause et al. (1989) studied a similarly motivated problem: the implications of test data on determining stability margins.

Poolla et al. (1992) have studied model validation in the time domain, using a discrete- time 5% framework. The use of the time domain is more appealing in its closer connection with experimental data. However, their approach requires structural constraints on the model, including a restriction on how the perturbation can enter the system. Zhou and Kimura (1994) have also studied this problem using a similar approach, and they include unknown real-valued parameters in their formulation. Recently, Smith and Dullerud (1994) and Rangan and Poolla

(1995) have developed model validation methods for sampled-data systems. Both the discrete-time and sampled-data formulations involve large computational problems. This work focuses on the more computationally tractable frequency- domain case.

It should be borne in mind that the closed-loop iterative identification approaches of Schrama (1992) and Schrama and Van den Hof (1992) in the Xm case, and Zang et al. (1991) in the linear-quadratic case, have shown that a good open-loop model is not necessary for a good closed-loop design. Using an open-loop model/system matching criterion may not be appropriate for the purposes of closed-loop design. Fortunately, the use of the linear fractional framework allows model validation to be performed for closed-loop experiments.

Experimental studies are invaluable in deter- mining the value of any theoretical development, particularly in the area of system modeling. This is the main contribution of this paper, and to this end significant detail on the experimental aspects has been included. The relevant theoretical results are only outlined, since more formal statements are available elsewhere (Smith and Doyle, 1989, 1992). A process control experi- ment-the Caltech two-tank experiment-is used for the study. This will be used to determine which of two structurally different models is best suited to modeling the given system.?

1.1. Organization of the paper A more formal discussion of model validation

is given in Section 2. The probabilistic model setting is considered initially illustrate how the model framework influences the nature of the validation results. %-/p model structures are then considered and stated as an optimization problem. The issues that arise from considering a finite amount of discrete-time data in the frequency domain are also covered.

Section 3 introduces the two-tank system used for the experimental study. A first-principles model is developed and compared to experimen- tally estimated transfer functions. Also included is a discussion on how the perturbation models were obtained for this particular system.

Two robust control models are proposed for the system in Section 4. They differ in the choice of the nominal model. An input-output experiment is performed and the model valida- tion theory is applied to quantify the relative

t An abbreviated description of this work was presented at the 12th IFAC World Congress (Smith, 1993).

A process control model validation application 1639

applicability of the models. Section 5 concludes with a summary of the work presented here.

1.2. Notation A matrix-valued transfer function will be

denoted by P(s) in the continuous-time domain, P(z) or P(e) in the discrete-time domain, or, where there is no ambiguity, simpy P. omax(A) is the maximum singular value of a matrix A. Unless otherwise stated, the signal norm used will be the two-norm;

Ilw 02 = [ Irn W)12 dt]. -3c

If w is vector-valued then the spatial norm is the Euclidean norm. The set of bounded energy signals is denoted by .& The norm of a system P will be the Xm norm (induced 2-norm).

If P is matrix-valued then the maximum singular value is taken as the spatial norm,

II p II m = sip flmax[W41.

2. MODEL VALIDATION THEORY

2.1. Vaiidation of probabilistic models Ljung (1987) gives an overview of model

validation for probabilistic models. This area is discussed here, since it illustrates the approach is a more common framework.

Take, for example, an output error model in discrete time:

Y(k) = P(z)@) + E(k),

where e(k) E N(0, A). The notation P(z)u(k) denotes the z-domain transfer function model, P(z) operating on the input sequence u(k). Accounting for the input-output discrepancy via

E(k), as a realization of a sequence of independent random variables with a specified probability density function, is a part of the chosen modeling framework.

Model validation is the test of this model with respect to a given experimental datum, denoted here by (f(k), a(k)). This involves testing the assumption that E(k) E N(0, h). Ideally, the datum (P(k), C(k)) h as been taken for the purpose of validation and was not previously used for the model development. The use of fresh data is referred to as cross-validation.

The first step is to calculate the observed residuals,

E(k)=p(k)-P(z)tZ(k), k=O ,... ,N-1.

There are several features of the probabilistic model that can be tested. The first is the hypothesis that the actual residuals i(k) are independently distributed. This can be done by examining an estimate of the covariance

8,(r) = $8(k)i(k + 7). k 1

If the residuals are independent then

5 N,M

:=-E- -$ g,(7)* R,(O) += *

is asymptotically X*(M)-distributed, and this can be posed as a hypothesis test.

Another feature of this form of probabilistic model is that the residuals e(k) are uncorrelated with the input u(k). This model assumption can be tested by examining the sample covariances for the observed residuals

&,,(z) = i 2 Z(k)li(k - T). k T

If 8(k) and li (k) are independent then fi fi,,( 7) will be asymptotically normally distributed. This can also be formally posed as a hypothesis test.

Note that the framework used to develop the model-in this case a probabilistic framework- is also applied for the validation test. Model validation results are therefore stated as statistical hypothesis tests. Note also that the independence of the input and the unknown signal E(k) is an assumption of the model. The following section will illustrate that for deter- ministic (norm-bounded) perturbation-based models, deterministic tests are available.

2.2. Validation of T/p models Uncertainty is modeled by norm-bounded

perturbations and norm-bounded disturbance/ noise signals in the robust control Xm/,u framework. Model validation therefore amounts to the question of whether or not there exists a perturbation, and a disturbance/noise signal, accounting for the datum. The deterministic nature of the model assumptions leads to a deterministic model validation test.

The linear fractional model framework, modified for the purposes of identification and validation, is shown in Fig. 1. The input-output relationship is given by

Y = i&W - fnA)-[p,, Pnl

+ P22 .31,[;]

=:W', A,[;], (1)

1640 R. S. Smith

Fig. 1. The generic structure for model validation and identification problems.

where A is an unknown, structured, norm- bounded perturbation. The case addressed in this work is the most common, and assumes that both P and A are LTI. Multiple perturbations are included in the model by assuming that A belongs to prespecihed block diagonal structure, denoted by A. As P(s) can be scaled, it is assumed, without loss of generality, that A E BA (i.e. A E A and ~~A~~m~ 1).

This modeling framework includes multiple perturbations, and allows the modeling of interconnected uncertain systems. Note that it can be applied to model uncertain closed-loop systems. These model assumptions, and this framework, exactly match those used for X design and p analysis and synthesis.

In the model validation problem, the system input is partitioned into measured or known inputs u and unknown inputs W, which are assumed (as a part of the modeling framework) to belong to BL$ (unit ball of 5!Q. Model validation considers the case where the model P and block structure A are given, perhaps by ad hoc identification methods, or first-principles modeling. Furthermore, an experiment has been performed to give the datum (u, y). Referring to the model structure of Fig. 1, the model validation problem can be formally stated as follows.

Problem 1. (Model validation.) Given a robustly stable transfer function model P, a perturbation block structure A and an experimental datum (u, y), does there exist (w, A), w E BL!$;, A E BA such that

Y = Fu(Pt A)[ ;].

Robustly stable means that the model set F,(P, A) is stable for all A E BA. The computa- tional approach presented here is based on the following constant-matrix problem.

Problem 2. (Constant matrix, minimum w.) Given vectors y and u, a matrix P and a specified block structure A, calculate,

subject to

(i) A E A and c,,,,,(A) I 1;

(ii) Y = F,V, A)[ z]. This is a linearly constrained structured

singular-value problem, and an upper bound on 9 can be calculated via a convex LMI optimization. For three or fewer perturbation blocks in A, the convex optimization actually gives 9. Details of the solution of this problem are given by Smith and Doyle (1992).

The Xp/p model validation problem is solved by optimization, and therefore involves a deterministic test. Note also that the resulting minimum norm w is a function of the input signal U, and therefore, unlike the probabilistic model case, is correlated with U. This is a function of the worst-case modeling framework, rather than the model validation theory. This characteristic also arises in the X design theory (Doyle et al., 1989). In model validation, the (w, A) obtained from the above optimization are, in a sense, best case. Accounting for the datum with w E BL$ and )/A/J,5 1 gives no hard guarantee of applicability of these norm bounds for the true system. As stated before, in a formal sense it is only possible to invalidate models.

2.3. Frequency-domain validation issues The datum is almost invariably obtained by

sampling, and this section illustrates how Problem 2 can be used to solve the model validation problem for discrete-time models on a frequency by frequency basis.

The datum consists of N input-output samples (u(k), y(k)), k = 0, . . . , N - 1. The model is given in the discrete domain

m

P(z) = 2 &Wk, k=O

where p(k) are the pulse response coefficients. A discrete Fourier transform (DFT) of the data is taken via

Y(n) := -$N~1y(k)e--j2X*niN (3) k 0

The expression of the input-output constraint (2) in terms of Y(n) and U(n) poses a potential difficulty. It is straightforward to show (see

A process control model validation application 1641

Theorem 2.1 Ljung, 1987) that,

Y(n) = P(en)U(n) + R(n),

where &On := &ZlmN, and R(n) satisfies

(4)

IRWI 5 & (max W)l)( 2 k Ip(k)l). I k=O

Under certain conditions, R(n) = 0 for n = 0 , * . . , N - 1. These conditions are

(i) u(k) is periodic and N is an integer multiple of the period; or

(ii) p(k) = 0 for k 5 1 (P is a static system).

As the amount of data taken increases (N increases), the size of jR(n)J relative to IIu(n)))2 decreases. By taking sufficient data, (R(n)1 can usually be made small enough that the approximation R(n) = 0 yields useful results. In the experimental example presented here, the model and experimental configuration actually give R(n) = 0.

To pose the discrete-time frequency-domain model validation, assume that R(n) = 0, n = 0 * * , N - 1. The model is available in the discrete-time, P(e), n = 0, . . . , N - 1, and the continuous-time, unsmoothed, datum (u(k), y(k)) is transformed into discrete time via the DPT given in (3). Assume the block structure A to consist of m perturbation blocks. For each value of n, n = 0, . . . , N - 1, solve the following matrix problem.

Problem 3.

P = WC&i& II W(n) II * II ) subject to

(i) II WQMl 5

i=l,...,m; (ii) Y(n) V(n) = [P*,(e++) P*,(e) P&+)]

[ 1 W(n) ; U(n) where (*)i denotes the components correspond- ing to the inputs or outputs of the ith block of the block structure A.

Note that this is a more detailed formulation of Problem 2. There is no W(n), with IIW(nh< 9, that, together with a A(n), accounts for the datum (condition (ii) in Problem 3), and satisfies A(n) E BA (condition (i) in Problem 3). Clearly then, if 9 > 1, the datum invalidates the model. Note also that the

magnitude of 9 gives a useful measure of how close the model is to being invalidated by the particular datum, and this can be useful in making engineering decisions about the suitabi- lity of the model.

3. OUTLINE OF THE PHYSICAL SYSTEM AND MODELS

Two robust control models are analyzed with respect to an experimental datum. Although each model includes a structured perturbation and weighted unknown disturbance input, in this case they differ only in the nominal model. This section describes the physical system and the two models, and gives details on how the perturba- tion descriptions were obtained.

3.1. A description of the two-tank experiment The Caltech two-tank experiment is depicted

in Fig. 2 and illustrated schematically in Fig. 3. This experiment has been used for several other robust control studies, and is documented in more detail elsewhere (Smith et al., 1987; Smith and Doyle 1988).

The work presented here will concentrate only on the top tank (tank 1). The input hot-water flow fh and cold-water flow fc are the control inputs. Measurements are made of the water level height hl and the water temperature tl at the tank base. The height of the tank is large compared with the diameter, and, although it is

Fig. 2. The two-tank system. The inlet pressure regulators and computer controlled valves are in the upper right of the picture. Tank 1 and the flow metering tubes are in the upper left. The tank 1 temperature and pressure (height) sensors are located at the bottom of the tank. The tube to tank 2

exits from the lower left of tank 1.

1642 R. S. Smith

hl

I -

-

TANK1

-L fl 1 supply at temp td

I fd

1 I

fl + fd

Fig. 3. Schematic diagram of the two-tank system. Only tank 1 is considered for this experiment.

stirred, mixing dynamics are evident in the t1 response. A pipe outlet at the base results in a very linear height response. The time constant of the temperature response is a nonlinear function of the height.

3.2. A nonlinear first-principles model A first-principles modeling approach is used to

develop an initial model. In order to proceed, it is necessary to make some typical, and unfortunately inaccurate, assumptions: no ther- mal losses in the system, perfect mixing, the flow out of the tank is related only to hr, and there are no thermal or flow delays.

The relationship between the outlet flow fi and h1 is modeled as hi = cyfi - p, where a, p > 0 and fr 2 P/a. Conservation of mass yields the following model for the height response:

h = afi - P, where Al is the cross-sectional area of the tank.

To obtain the tr response model, it is useful to define a variable, El : = hItI, which can loosely be thought of as the energy in tank 1. Conservation of energy leads to the following model:

+_1 &a

t1+,. 1

The system units are scaled such that all inputs and output lie between 0 and 1, and in this set of units Al = 91.4. The above theoretical model is only a good approximation over a range of h, : 0.15 5 h, I 0.75. Static measurements have been performed in order to estimate a and p, giving a = 1.34 and p = 0.6.

The pneumatic actuators have limited band- width, and this is modeled by

fh = Pactfhc

l f =1+0.05s hc

where fhc is the commanded identical model applies to fc

hot flow. An and &. The

objective here is to quantitatively analyze the adequacy of two linear models at the operating point: h, = 0.47 and tl = 0.5.

3.3. Open-loop experiments Open-loop identification experiments were

performed to assess the quality of the above model and give some indication of the frequency range over which the first principles model is a accurate. In the tank height case the empirical transfer function from the total input flow fh +fc to h, was compared to the linearized model. The agreement with the theoretical model is excellent over four decades of frequency. For brevity, this data is not illustrated here. The open-loop identification of the t1 response is of more interest. Linearized models were generated at a

A process control model validation application 1643

i

10-2 10-l

frequency: Hz

- i : : : j ;

i :

3

1:51 .._ : : : j :

i : : : : i : .:--:- : :

.i..i. : : : :

.i..j. : :

L

10'

-

, . .

.-

.1

I..A

. ..L

. ..A

..I

Y

10

frequency: Hz

Fig. 4. Transfer function between fhC - f, and f,: experimental data and models (solid lines). h, = 0.15 and h, = 0.75. The models shown here also include the effect of the anti-aliasing filters.

series of fixed heights for comparison purposes. The input waveforms were generated such that fhC = -f& ther e b y maintaining a constant hr. For illustrative purposes, Fig. 4 shows a smoothed estimate of the empirical transfer function from fhC -fcC (= 2fhC) to tI estimated from the experimental data and calculated from the model. To give some idea of the variation in response for different hI, the two cases are shown: h, = 0.15 and hI = 0.75.

There are obvious discrepancies between the theoretical model and the identified transfer function estimate. Note in particular the extra phase, particularly in the h, = 0.75 case. This is

not altogether unexpected, since the tank is tall and thin and poorly stirred, in contrast to the assumptions made in the development of the theoretical model.

3.4. Outline of the perturbation model Figure 5 illustrates the structure of the

uncertain model. Output multiplicative pertur- bations are used on both the h, and tl outputs. Sensor noise is modeled by the unknown signals w1 and w2. P,,, is a nominal linear time- invariant model.

The two nominal models to be considered are as follows.

1644 R. S. Smith

whn Y

Fig. 5. Schematic diagram of the top-tank robust control model.

Model A.

hl = 1 +1&s (.ti +.a

1.25 tl =

1+ 53.8sfh.

Model B.

hl =

E, = 1 + & 5s (1.34& - O-6&),

tl = & (E, - 0.5hl).

Model B ,is a linearization of the nonlinear model given in Section 3.2, and is a MIMO model. Model A is a further simplification, obtained by using the assumption that hl is constant in linearizing the model for tl. There are several engineering motivations for prefer- ring the simpler Model A. It can be statically decoupled, allowing the use of independent SISO designs for the control of h, and tl. Independent control of each loop allows independent start-up and tuning, and gives better performance in the event of a failure of one actuator or one sensor.

In Model A each loop contains a perturbation, which must account for the unmodeled effects of the cross-coupling between height and tempera- ture, in addition to the unmodeled dynamics evident in Fig. 4. The issue here is whether or not the perturbation description given is sufficient to cover such affects, and model validation will be used to answer this question.

Models A and B have the same perturbation and noise weights, described below. The

L

-0.5

Fig. 6. Nyquist plot of the nominal t, response (h, held fixed) with the discs described by A2 W,, P,,,, iA21 = 1, shown

superimposed.

perturbation weights are chosen as

w,, = 0.5

1 + 0.25s

w t1

= 0.1 + 16.5s

1+0.2.Y

and the noise weights are chosen as W,, = 0.01 and W,, = 0.03. The perturbation weight W,, is illustrated graphically in Fig. 6. The effect of the weight is shown superimposed on a Nyquist plot of the nominal response. For brevity, only the temperature perturbation weight is shown; W,, is so small that it is barely discernible on a Nyquist plot. The appropriate interpretation is that at each frequency the response of all models in the set is described by a disk in the complex plane. This is only valid when considering either of hl or t1 with the other held fixed.

A process control model validation application 1645

The perturbation weights have been obtained by several ad hoc methods. These include

0)

(ii)

estimating the discrepancy between the nominal model and experimental empirical transfer function estimates (see Fig. 4);

using relay feedback and a series of phase lead controllers to induce limit cycles in each loop-this can be used to give some idea of the frequencies where nonlinear and/or nonrepeatable behavior dominate the closed-loop response (further details are are given in Smith and Doyle, 1993).

The noise weights were determined by comparing the relative sizes of the noise and typical output signals during open-loop opera- tion. There is currently no comprehensive theory behind weight selection, and this is an area of active research. Model validation does give a means of assessing the quality of a weighted perturbation model obtained by such ad hoc methods.

The perturbation associated with the tempera-

r4 r 0 01 0 cl 1 pi1

ture is large (refer to Fig. 6), and it is this that tempts one to use a simplified nominal model (Model A). The hope is that the discrepancies in the behavior due to the effects of changing h, will be captured in the A2 perturbation model.

It will be seen that the hI output perturbation weight W,, is in fact adequate, even though it is extremely small. While the temperature pertur- bation weight (W,,) appears to be generous, the following model validation analysis will illustrate that the decoupled model (Model A) does not adequately describe the observed temperature response.

4. MODEL VALIDATION FOR THE TANK SYSTEM

4.1. The frequency-domain problem set-up The continuous-time models described above

were converted into discrete-time models by zero-order hold equivalence at the same frequency used in the physical system.

At each frequency, the interconnection struc- ture P is therefore

rv1 i

00 0

1 0 K 0 1 0 W,

where P is shown divided up into the partitions & illustrated in Fig. 1. In the model validation application P,,, will be a discrete-time zero- order equivalent of either Model A or Model B. The perturbation block structure A consists of two SISO blocks.

With this particular model structure, the experiment can be configured such that the discrete-time model can be tested in the frequency domain without any approximation error (i.e., referring to (4), R(n) = 0). To achieve this, periodic signals will be used for the input. Note also that the unknown signals affect the output only via static weights, which means that R(n) is also zero for those components of the model.

4.2. The experimental datum A periodic input signal was applied to the flow

actuators, and allowed to run for at least five periods to remove any initial transient, before input-output data was recorded. The signal was generated by low-pass filtering (with roll-off just beyond the expected system roll-off) the output of a random noise generator. Figure 7 shows a window of output data taken from this

experiment. Only the output signals are shown, since these will be compared with the size of the unknown signals obtained from the model validation analysis.

The sampling period was 0.1 s, and 4096 sample points were recorded during the experi- ment. The period of the input signal was 204.8s, and the record therefore contains two periods. The initial and final points in the record are at the same value, and the record contains an integer number of periods. As the models represent a linearization about an operating point, a constant bias was subtracted from both the input and output signals.

4.3. Model validation results Because there are only two perturbations, the

model validation optimization problem is con- vex. Problem 3 is solved, at each frequency, by the NPSOL optimization package (Gill et al., 1986), which uses a sequential quadratic programming method.

4.3.1. Model A. For Model A-the simpler, decoupled, model-the minimum (1 W(n) II2 was 1.32. As this is larger than one, there is no element in the model set (which includes the

1646 R. S. Smith

0 0 150 300 409.5

Tim(scnmds)

Fig. 7. Model validation experiment: output singals t, and h,.

assumption that 11 W(n) (I2 5 1) that can account for the experimental datum. A time-domain representation (calculated via the inverse DET function) of the resulting minimum-norm signals is given in Fig. 8. The signals are scaled to represent their contribution to the output tI (i.e. u2 and W,,w, are plotted). The signals associated with h, are not plotted, since they are both less than 0.01 and do not contribute significantly to the norm of the required noise signal.

The noise on the temperature output w2 is almost a sinusoid of the same period as the input signal. Recall that the nominal model for the temperature response is minimum-phase, even though it is known from the previous input- output experiments that considerable delay is expected (see Section 3.3 and Fig. 4). The intent was to select the perturbation weight W,, large enough to account for the nonminimum-phase behavior. This has not been achieved, since the

0.10

0.05

0 .5

-0.05

-0.10 -

4.15 !

Fig. 8. Minimum-jlwll time-domain signals for Model A. Fig. 9. Minimum-ljw 11 time-domain signals for Model B. Only the scaled temperature signals (u2 and W,,w,) are Only the scaled temperature signals (u2 and W,,w,) are

shown. shown.

amount of noise required to also account for the residual y - Pz3u was larger than modeled.

This model validation experiment indicates that Model A cannot account for the observed behavior (I( W(n)ll, 2 1.32). There are several immediate possibilities for modifying the model so that this datum may be accounted for. One is to simply scale the w input, Applying an additional scaling factor of 0.75 is sufficient. The alternative is to modify the weights on the perturbation blocks. The discussion in the previous paragraph suggests that modifying W,, in order to increase its value at least at the dominant frequency of the response may lower the norm of the w required to account for the datum. Either of these model changes will result in a more conservative controller design.

Note that the size of Wmw2 is of the order of the deviation in tI itself, suggesting that something more fundamental is wrong with the model. An examination of the nominal behavior shows that, in fact, the nominal model does a very poor job of approximating the observed datum. The model validation analysis given here indicates that the perturbation description is inadequate to account for the discrepancy.

4.3.2. Model B. Using Model B as the nominal yields a minimum-norm W(n) such that 1) W(n)l12 = 0.53. Figure 9 again gives the results of the model validation optimization for the unknown temperature-related signals. This da- tum can be accounted for by the model with

IIWtn)ll2

A process control model validation application 1647

5. CONCLUSIONS

In any identification/design problem the true system cannot be described by a nominal model. Robust control models include perturbations and unknown signals; however, it is still a matter of judgement whether or not the model is adequate to describe the system. The model validation procedure gives a means of quantitatively addressing this question on an experiment-by- experiment basis.

In the experimental system considered here, Model B is clearly more sophisticated than Model A, and it would be expected to be a better description of the physical system. The issue illustrated here is that model validation gives a quantitative assessment of how good each model is on a particular experiment, and, perhaps more importantly, how much better is Model B than Model A.

Model validation techniques using sampled time-domain data have recently been developed. These result in a single optimization, approxi- mately N times as large as those given here, leading to a significantly increased computational effort. In addition, these methods cannot yet directly deal with the full generality of LFI perturbation models.

A discrete-time frequency-domain approach was applied in this case, giving $N independent optimization problems to be solved. The model structure and experimental configuration meant that this could be done without introducing error in mapping the finite-time input-output re- lationship to the frequency domain. In general, this will not be the case. An exact quantification of the effects of R(n) ZO would allow an intelligent choice between an exact sampled-data calculation and a faster, less accurate, discrete- time frequency-domain calculation. This is left as an area for future research.

Acknowledgements-This work has been supported by the California Institute of Technology via the Program in Advanced Technology, the National Science Foundation, the Jet Propulsion Laboratory, and the National Aeronautics and Space Administration. The author would also like to thank John Doyle for guidance and discussions on this work. Robert Kosut also contributed some useful thoughts on the discrete-time frequency-domain setting.

REFERENCES

Doyle, J. C., K. Glover, P. Khargonekar and B. Francis (1989). State-space solutions to standard Hz and H, control problems. IEEE Trans. Autom. Control, AC-34, 831-847.

Gill, P. E., W. Murray, M. A. Saunders and M. H. Wright (1986). Users guide for NPSOL (version 4.0): a Fortran package for nonlinear programming. Technical Report SOL 86-2, Systems Optimization Laboratory, Department of Operations Research, Stanford University.

Goodwin, G. C., B. Ninness and M. E. Salgado (1990). Quantification of uncertainty in estimation. In Proc.

American Control Conf, pp. 2400-2405. Gu, G. and P. P. Khargonekar (1991). Linear and nonlinear

algorithms for identification in H with error bounds. In Proc. American Control Conf, Boston, MA, pp. 6469.

Helmicki, A. J., C. A. Jacobson and C. N. Nett (1991). Control oriented system identification: a worst- case/deterministic approach in H,. ZEEE Trans. Autom. Control, AC-Xi, 1163-1176.

Hjalmarsson, H. and L Ljung (1992). Estimating model variance in the case of undermodeling. IEEE Trans. Autom. Control, AC-37, 10041008.

Kosut, R. L. M. Lau and S. Boyd (1990). Parameter set identification of systems with uncertain nonparametric dynamics and distrubances. In Proc. 29th IEEE Conf on Decision and Control, Honolulu, HI, pp. 3162-3167.

Krause, J. M. (1989). Stability margins with real parameter uncertainty: test data implications. In Proc. American Control Conf, Pittsburgh, PA, pp. 1441-1445.

Krause, J. M., P. P. Khargonekar and G. Stein (1990). Robust parameter adjustment with nonparametric weighted-ball-in-H uncertainty. IEEE Trans. Autom. Control, AC-35,225-229.

Ljung, L. (1987). System Identification, Theory for the User. Series. Prentice-Hall, Englewood Cliffs, NJ.

Makila, P. M. and J. R. Partington (1991). Robust approximation and identification in H. In Proc. American Control Conf, Boston, MA, pp. 70-76.

Parker, P. J. and R. R. Bitmead (1987). Adaptive frequency response identification. In Proc. 26th IEEE Conf on Decision and Control, Los Angeles, CA, pp. 348-353.

Poolla, K., P. Khargonekar, A. Tikku, J. Krause and K. Nagpal (1992). A time-domain approach to model validation. In Proc. American Control Conf, pp. 313-317.

Rangan, S. and K. Poolla (1995). Time-domain sampled-data model validation. In Proc. American Control Conf, to appear.

Schrama, R. J. P. (1992). Accurate identification for control: the necessity of an iterative scheme. IEEE Trans. Autom. Control, AC-37, 991-994.

Schrama, R. J. P. and P. M. J. Van den Hof (1992). An iterative scheme for identification and control design based on coprime factorizations. In Proc. American Control Conf, Chicago, IL, pp. 2842-2846.

Smith, R. S. (1993). Model validation for robust control: an experimental process control application. In Proc. 12th ZFAC World Congress, Sydney, Vol. 9, pp. 61-64.

Smith, R. S. and Doyle, J. C. (1988). The two tank experiment: a benchmark control problem. In Proc. American Control Conf, Atlanta, GA, Vol. 3, pp. 403-415.

Smith, R. S. and J. C. Doyle (1989). Model invalidation-a connection between robust control and identification. In Proc. American Control Conf, Pittsburgh, PA, pp. 1435-1440.

Smith, R. S. and J. C. Doyle (1992). Model validation: a connection between robust control and identification. IEEE Trans. Autom. Control, AC-37,942-952.

Smith, R. S. and J. C. Doyle (1993). Closed loop relay estimation of uncertainty bounds for robust control models. In Proc. 12th ZFAC World Congress, Sydney, Vol. 9, pp. 57-60.

Smith, R. S. and G. Dullerud (1995). Validation of continuous-time control models by finite experimental data. IEEE Trans. Autom. Control, to appear.

Smith, R. S.. J. C. Doyle, M. Morari and A. Skjellum (1987). A case study using CL: laboratory process control problem. In Proc. ZFAC World Congress, Munich, Vol. 8, pp. 403-415.

Zang, Z., R. R. Bitmead and M. Gevers (1991). H, iterative model refinement and control robustness enhancement. In Proc. 30th IEEE Conf on Decision and Control, Brighton, U.K., pp. 279-284.

Zhou, T. and H. Kimura (1994). Input-output extrapolation-minimization theorem and its application to model validation and robust identification. In R. Smith and M. Dahleh (Eds), The Modeling of Uncertainty in Control: Proceedings of the 1992 Santa Barbara Workshop, pp. 127-137. Springer-Verlag, New York.