A Comparison of Frequency-Domain and Optimal Control Methods in Macroeconomic Policy Design

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A COMPARISON OF FREQUENCY-DOMAIN AND OPTIMAL CONTROL METHODS IN MACROECONOMIC POLICY DESIGN P. F. Westaway* and

J.

M. Maciejowski**

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Abstract. This paper addresses the question of which of two approaches is most appropriate for economic policy design. It does this by comparing two alternative designs for a policy, one of which was obtained by frequency-domain methods, and the other by the use of linear optimal control theory. The conclusion reached is that both approaches can be used to design realistic economic control policies and, although each has its own advantages, on balance there is little to choose between them. However, it is seen to be very useful to supplement the optimal control design methodolo gy with the use of the frequency domain for specifying design objectives, and for analysing the resultin g policy des ign. This restores some s ym met r y tot het wo a p pro a c h e s, sin c e i t is qui t e us u a 1 to use the time domain to help specification and analysis when ' Nyquist -like' design methods are used. ~eywor~.

Economics;

feedback;

frequency-response;

optimal

control. 1.INTRODUCTION It is well known

that there is a dichotomy between the practice of industrial control system design, which, in so far as it makes use of theory at all, relies almost exclusively on 'classicaf frequency domain methods, and the academic pursuit of control studies, which centres on the time-domain solution of optimisation problems. The reason for this difference of approach is not merely the existence of a delay in the transfer of concepts and methods from academe to industry. Ra ther it is that the optimal control problem appears, at first sight, to bear little relation to most real control problems . One of the first economists to use control theory, (Phillips, 1954), used the 'classical' apparatus. But since then almost all applications of control techniques to economics have been based on optimal control. It is important to ques tion whe the r this choice is appropria te , since the economist who is interested in the formulation of realistic and effective policy finds himself in a position much more similar to that of the control system design engineer 165

than that of the academic control theorist. He has little or no idea of what might constitute an appropriate objective function for optimisation, he require s a simple policy , he needs to understand why the policy is as it is, he needs to be confiden t tha t the policy will work even if his model 0 f the economy is i naccu ra te, and he must take account of administrative and political constraints on the policy . In this work, the policy to b e des igned is to be ' New Keynesian', in the sense defined by Meade(1981). That is, individual policy instrunents are to be assi gned to the control of individual output variables and this assignment is the reverse of the ' Orthodox Keynesian' assignment. Tax rates are to be used to re gulate to tal money expendi tures on the produc ts 0 f industry, and the wage rate is to be used to regulate the level of employment. The exchange rate is to be used to regulate the balance of payments on current account. The model used as the basis of the investigation is a discrete-time (quarterly), non-linear model of the UK economy, full details of which can be found in Vines ~~. ,(1983) . A linear,

166

P.F. Westaway and J.M. Maciejowski

time-invariant model, which retained (appro~ imately) the small perturbation input-output properties of the non-linear model, was obtained by using an algori thm due to Kung (1979), and described in Maciejowski and Vines (1982). The design of the policies analysed in this paper was performed on the basis of that linear monel.

2.2 Linear Op timal Control By con tras t wi th the frequency-domain approach, 'optimal control' techniques are usually based in the time-domain. The feedback configuration is still as shown below in Fig.1.

REF, /

2. RECENT DEVELOR-1ENTS IN DESIGN TECHNIQUES. /

2.1 Frequency-domain Very successful nesign techniques for single variable control systems were evolved between 1 940 and 19')5. Th ese mad e use 0 f frequencydomain descriptions of signals (i.e. variables) and systems. Their success and populari ty wi th design engineers was due largely to the insights they gave into the problems of closedloop stability, performance and robustness in the face of uncertainty, ~nn to the relations between these aspects of feedback control sys terns. Th ey provided the n es igne r wi th concepts which fitten in well with his understanding of the feedback design problem and which aiden the development of his intuition. However, these 'classical' techniques were limited to the design of single-input singleoutput systems. This limitation precluned the use of these techniques to any non-trivial economic studies since these were essentially of a multivariable nature. In the 1970's, important work by Rosenbrock (1970,1974) and M.acFarlane (1q79) lerl to the generalisation of the well-tried 'classical' methods of stability assessment and performance prediction to the mul tivariable case. Coupled with the availability of powerful interactive computing facilities, this breakthrough allowed multivariable design problems to be handlerl. Consequently, the way was opened for the application of these techniques to the field of macroeconomic policy design. The essential problem tackled by the frequencydomain methods is the design of a control law K(z) (a matrix of z-transform transfer functions, if the problem is formulated in discrete-time) which, when coupled with the system that is to be controlled, G(z), in the negative feedback configuration shown in Fig. 1, gives 'satisfactory' characteristics of closed-loop objects such as the transfer function T(z)=G(z)K(z)(I+G(z)K(z))-1 and the sensitivity function, I-T(z) (Horowitz,1963, Doyle and Stein,1981). The design process is one of manipulation of K( z) so as to modi fy the properties of T(z) and I-T(z), particularly on the unit circle z=exp(i ). In the multivariable case, this manipulation is performed by the use of appropriate software.

Fig.

The system to be controlled must now be described by a state-space model, such as x(k+1) y( k)

Ax(k) + Bu(k) Cx(k) + Du(k).

A. control law is determined which minlmlses some pre-assigned performance cri terion such as limA(T), where A(T) is a quadratic objective T+oo function, typically '1'

A(T) k~6

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1

Here, u(k) is the (vector) control inputs, y(k) is the output to be controlled (to zero in this formulation), k is the time index and Q and R a re we igh ting ma trices. Tn e so lu tion to this problem is now well established (Kwakernaak and Sivan, 1972) and yields a control law in the form of a dynamic system as described bystatespace equations. If it is assumed that stochastic disturbances act upon the state and output of the system and that the true state x(k) cannot be directly observed, it is necessary to obtain an estimate of the state by the use of a Kalman filter. The details of this filter depend on the covariance matrices (Wand V, say) of the disturbances which are assumed to act on the state and the output. The optimal control law then consists of a series connection of the Kalman filter with a state feedbac k regula to r. Despite the apparent attractiveness of the 'optimality' of the resulting design, there are a number of major obstacles to using the optimal control approach successfully. In most design problems, no objective function is apparent and frequently the characteristics of the stochastic disturbances are not known accurately. The solution of the optimisation then becomes li tUe more than a device for narrowing down the range of possible designs. However, in these circumstances, the designer has the daunting task 0 f choosing the four ma t rices Q, R, Wand V in such a way that significant closed-loop properties are 'sa tis fac tory'. At hans( 1 971) ,

Frequency-Domain and Optimal Control Methods Kwakernaak and Si vane 1 972) and Anderson and Moore(1972), among others, have addressed themselves to the problem of choosing these four matrices, but major difficulties remain. 2.3 More Recent Developnents Spurred on by the difficulties of applying optimal control theory, a trend has developed amongst control theorists towards the incorporation of frequency-domain indicators within optimal control design methods. In particular, a powerful result by Safonov and Athans (1977) showed that 'linear optimal' state-feedback d esigns exhibited impressive stability margins and performance characteristics. However, it was emphasised by Doyl e and St ein( 1979) that these characteristics are not guaranteed if the state is not directly observed, but has to be estimated by an observer such as a Kalman filter. This led to a revival of the ideas first developed by Kwakernaak(1969) which indicated that desirable frequency-response characteristics could be attained by means of the outpu tfeedback optimal control techniques if an 'asymptotic recovery' design procedure was employed. These ideas were taken up by Doyle and Stein(1981) who used them to combine aspects of the frequency-domain approach and the optimal control approach. Their method is especially interesting and useful to the designer faced wi th a complex problem such as a mul tivariable economic system because the number of matrices to be adjusted is reduced from 4 to 2 • Furthermore, considerable guidance is given as to how they should be adjusted to give' good' performance, however that may be judged. A significant step forward is taken wi th this design technique because no pretence is made that a real cost function can be found which must be optimised or that the nature of the disturbances affecting the system is exactly known. Instead, these unknown parameters are themselves used as design parameters which are adjusted until 'desirable' stability and performance characteristics are secured. The developnent by Doyle and Stein(1981) is restricted to a continuous-time setting. A parallel developnent for the discrete- time case is given by Maciejowski and Westaway(1983). Significant differences exist between the two. The implementation of this design technique is d esc ri bed in Se c tion 3 and its re suI ts a re compared with the 'pure' frequency-domain approach in Section 4. 3. DETAILS OF DESIGN PROCEDURES 3.0 Se t-up of the general design problem Having pu t the design techniques tha t are to be compared into a general context, we are now in

167

a position to briefly describe our experiences with the two rival techniques. The linear model to be controlled is in state-space form and is of 21st order with 4 inputs (the rate of indirect tax, the real exchange rate, the wage rate and the rate of employee's national insurance contributions) and 3 outputs (nominal gdp, the balance of payments on current account and the level of unemployment). The struc tural constraint imposed upon the problem by the 'New Keynesian' instrument-targe t assignment is that the feedback policy, when expressed as a transfer function, should have the form shown below as K(z):

3.1 Frequency-domain design Although the model was specified in state-space form, a frequency-domain representation of it was available when necessa r y , using the relation; G(z)

=

C(zI-A)-1 B + D

The main idea behind the frequency-domain methods is the manipulation of frequency-domain properties such as the loop bandwid th and sta bilitymargins such that 'desirab le' performance is achieved. These frequency-response speci fica tions we re deduced from the time- response speci fica tions, wh ich themselves rela ted to the speed of recovery to exogenous disturbances and to permi t ted ins trumen t va ria tions. Howeve r, it was found that a considerable amount of trial and error was involved to match the 'desired' c harac teris tics to the achievable ones. Fuller details of the specifications and of the design process are available in Maciejowski and Vines (1981) . In fact, the method used by Maciejowski and Vines(1981) was a mixture of two techniques. Firstly, a one-loop-at-a-time d esi gn strategy was adopted whereby the control laws for each target-instrument assignment were designed separately. The loops were designed in order of speed of attainable speed of response (that is, first the demand loop, secondly the balance of payments loop and thirdly, the unemployment loop) and for the second and third loops, the control laws of the earlier loops were taken as being in operation. It was necessary to use this procedure because 0 f the spa rse s truc ture imposed on the controller. Consequently, this technique involved solving a series of singleinput single-output design problems so the well understood' classical' methods could be applied in order to 'bend' the Nyquist loci to force them to have the required characteristics.

168

P.F. Westaway and J.M. Maciejowski

The design was then improved by using a frequency-domain optimisation algorithm to tune the controller parameters so that the closed-loop transfer function approached the specified desired closed-loop transfer function more closely. It is this improved design which will be compared wi th the optimal control design in Section 4. 3.2 Op tima 1 con trol design Before starting the design, the model was augmented wi th artificial integrator states in order to force the controller to have integral action (see Anderson and l-bore(1972) or Kwakernaak and Sivan(1972)). Tne addition of these states to the 21st order model made the overall model order too large to work with convenien tly. A model reduc tion technique based on the theory of balanced realisations (Moore, 1978 ) was used to reduce the overall model order while preserving the input-output properties. The standard solution to the optimal control problem yields a cross-coupled control law, namely one in which the 'New Keynesian' structure is not preserved. In order to force the controller to have this structure, it was necessary to adopt a one-loop-a t-a-time desi g n strategy just as was done for part of the frequency-domain design. For each loop, the following two-stage procedure was carried ou t. Firstly, by manipulation of the covariance ma t rices 0 f the s ta tedisturbances and output-disturbances, Wand V respectively, the open-loop gain/frequency characteristics of the Kalman fil ter were manipulated until the zero-crossover frequency was near the required closed-loop bandwid th. We set W=BBt+P (where P is a diagonal positivedefinite ma trix) and V= ;, 1. Fo r a fixed P, thi s allowed the zero-crossover frequency to be ad jus ted by the variation 0 f the scala r (decreasing I' g~v~ng a higher crossove'r frequency). Prior ideas of what constituted a desirable bandwidth were deduced from the results of the frequency-domain design, which was the first one to be carried out. The second stage of the design strategy involved using the 'asymptotic recovery' procedure so that the open-loop gain/frequency characteristics of the Kalman filter were forced to be exhi bi ted by the series combination of the control law and the economic model. This was achieved by setting R=rI and Q=CtC and reducing r to zero. This forces the open-loop transfer func tion 0 f the fil ter to be 'recovered' approximately at the output of the compensated system. The technical details behind the theory of this design technique are given in Doyle and Stein, (1981) and Maciejowski and Westaway, (1983).

The final step in the overall implementation arises from consideration of the fact that the control law so derived is too complex to be poli tically and administratively feasible since it is of the same dynamic order as the reduced model. By using the same reduction algorithm as was used to reduce the original model, it was possible to derive a control scheme whose performance was similar to that derived by the design technique but which was only of 2nd order. As stated above, this design procedure was carried ou t for each loop in turn. When all the loops had been designed, the fi rs t lOop's control law was re-designed with the other two loops closed, in order to compensate for the deterioration in performance which had occurred as a result of the closing of the second and third loops. However, very little improvement was gained as a resul t 0 f this procedure. Th is contrasted with the frequency-domain optimisation method which yielded significant improvements over the one-loop-at-a-time results. 4. COMPARISON OF POLICY DESIGNS There are a number of ways to compare 2 sets of resul ts derived from the design techniques described in section 3. In keeping with the rest of this paper, the resul ts wi 11 be analysed and compared from the perspective of both the timedomain and the frequency-domain. 4.1 Time-simulation comparison. In the time-domain, it is useful to look at the simulated step response of each loop. Figures 2-4 show this for the mul tivariable frequencydomain design and Figures 5-7 for the optimal control design. These figures show the results of attempting to achieve a 1% sustained increase for each of the output variables in turn. Figures 2 and 5 show the resul to f a step demand on nominal gdp, Figures 3 and 6 the e ffec t 0 f a step on the balance 0 f paym en ts and Figures 4 and 7 the effects of a step demand on unemployment. In all these cases, Figures (a) and (b) show the movements of the policy targets and instruments, respectively. Nominal gdp loop: Comparing the two designs, the optimal control policy attains a rise time for money gdp of 4 quarters wi th a slight overshoot of the target, target interaction of 50% on the balance of payments and instrument variations of at most 1.5% per quarter. In contrast, the 'multivariable' design has a much faster rise time of two quarters, but this is paid for in terms of larger target interaction (70% for the balance of payments) and larger more oscillatory movements in the tax rate and exchange rate.

Frequency-Domain and Optimal Control Methods Balance of payments loop: feature of the compa rison be tween Figures 3 and 6 is tha t the instrument variations are almost identical for each case, both exhibiting the sharp5% initial devaluation in the exchange rate to initiate the corrective action . Similarly, both designs show the familiar 'J-curve' effect (that is, the initial deterioration of the balance of payments before an improvement is obtained) of roughly the same magni tude and duration. A more surprising similarity is the tendency for the balance of payments to fall away slightly (to a differing degree) in its rise to the target before finally tracking the target at the steady-state. The judgement on the comparison between the two designs is not clear cut. The frequencyresponse design displays a settling time of 20 quarters, with moderate target interaction (40% on nominal gdp ) while the optimal control design shows a slightly quicker settling time with similar target interaction although a small error remains. An immediately obvious

Unemployment loop: By far the largest discrepancy between the two approaches to policy design appears in the unemployment loop. While the frequencyresponse based design secures a rise time for unemployment of 20 quarters wi th ini tial variations in the wage rate of approximately 1% per quarter, the optimal control design failed to attain a rise time any quicker than 40 quarters, in spite of larger initial wage rate varia tions. However, the frequency-res ponse design does incur very high levels of target interaction. Also, it is true to say that the slow response of unemployment may not be too serious because exogenous changes in unemployment can only be caused by technical progress and labour supply changes , both of which are very gradual phenomena . 4.2. Frequency-domain Comparison. It is also useful to examine the frequencyresponse properties 0 f the two sets 0 f designs. Figure 8 (for the mul tivariable design) and Figure 9 (for the optimal control design) show the magni tudes of the diagonal elements of the closed loop t rans fer- func tion, Le. it shows graphs of 'l'ii(e j e, ) against e. fori=1,2,3,both axes having logari thmic scales. These show the bandwid ths of the three feedback 'loops' (money gdp, balance of payments and unemployment). For the frequency domain design, these are 1.74, 1.00 and 0 . 54 rad/quarter respectively and for the optimal control design these are 0.62, 1.06 and 0 . 095, respectively. These bandwidths are consistent with the observations of the time simulations because the slower recovery times of the nominal gdp and unemployment loop with the optimal control design are reflected in

169

correspondingly lower bandwidths. It is useful to compare the bandwidths of the closed-loop transfer function with Figures 10 and 11 which show the magni t udes 0 f the diagona 1 e lemen ts 0 f the sensi tivi ty func tion, Le. graphs of I-'l'ii (e L ) for the frequencyresponse and optimal control designs, respec t ively. They show that exogenous disturbances appearing independently on each of the three controlled outputs are attenuated at frequencies below 0.46, 0 .1 7 and 0.21 rad/quarter, respectively, for the frequency-domain design and at frequencies below 0.40, 0 . 16 and 0.12 rad/quarter, respectively, for the optimal con trol design, and are magni fied at highe r frequencies . From the point of view of regula ting agains t freq uen t random dis turbances, it is possible to determine how' efficient' a loop design is by comparing the frequency at which instruments cease to have any beneficial effect (i . e . the frequencies indicated by the sensi tivity function in Figures 10 and 11) with the frequency up to which policy instruments react to measurement errors (indicated by the closedloop bandwid th in figure s 8 and 9). An asses sment along these lines for the frequency-domain design indicates that the control of money gdp and the balance of payments is 'inefficient' in the sense that the policy instruments are reacting to measurement noise at f~equencies 4 - 6 times higher than those at which they are effective. Control of unemployment is 'more' efficient in this respect, since the ratio of the two frequencies is only 2. 6 in this case . On the other hand, the optimal control design yields slightly better resul ts by these criteria. While the 'efficiency' of the balance of payments loop is approximately the same, the lowe r bandwid th 0 f the money gdp loop has caused the control to be more' efficient' in the sense defined above, the ratio of frequencies being 1 .5. Similarly, the s lowe r unemploym en t loop has led to greater 'efficiency' with a freq uency ratio of 1.2. A further observation in favour of the frequency-response properties of the optimal control design is that the magnitude characteristics show less tendency to peak which implies more accurate tracking of the targets at the relevant frequencies. While these frequency-domain considerations prove useful for purposes of comparison, it should also be borne in mind that, if the rej ection 0 f more realis tic dis turbances is considered, the designs may not appear so 'inefficient'. For more discussion of this point, see Vines ~ a1.(1983). (It should be no ted tha t neither 0 f the policies analysed in this paper is the policy proposed in Vines et a1. (1983» .

5 . DISCUSSION. The main conclusion which emerges from the above comparison is that the two design

170

P.F. Westaway and J.M. Maciejowski

approaches produce macroeconomic policy rules which are very similar in their effect, with many of the contrasts in the time and frequencyresponses being attributable to the differences in the speeds of the loops eventually achieved rather than to the differences in methods used to attain those designs. However, a number of more general points are worth making which are not made apparent purely by observing the two sets of results. It is a very important addi tional point that the frequency-domain based study (Maciejowski and Vines,1982) was carried out first. It is beyond dispute that this made the LQG-based design attempt al together easier. The trial and error initially involved in gauging suitable bandwidths, feasible instrument variations, target-instrument trade-offs was virtually eliminated in the second study as a result of the 'experience' gained in the first. On the other hand, the 'optimal control' approach allowed a more systematic implementation of the design strategy, even though the one-loop-at-a-time approach was used. In particular, the straightforward manipulation of 2 scalar parameters involved in this approach can be compared wi th the complicated experimentation with alternative polezero placements involved in finding an 'ideal' transfer function. Another useful by-produc t of this comparison is that a number of features are apparent which when observed in an individual design may seem to be quirks of the particular control law used, but when seeing them repeated using different design techniques, may be ins tead id en tified as 'real' constraints in policy design which prompt the search for economic causes. Our experience in this paper, and indeed in more recent work, has shown that it is possible to adopt a syn thesis of the approaches which allows these sophistica ted techniques to design an approxima tely 'correc t' controller which can then be fine-tuned to the desired specifi ca tions by using much simpler ~ hoc procedures. In conclusion then, it appears that the 'frequency- domain' and the 'optimal control' approaches are of approximately equal utility for the design of macroeconomic policies. Each has its own advantages and drawbacks, but perhaps the deciding fac tors in determining which to use should be the experience of the personnel involved and the availability of appropriate software. Acknowledgement We are grateful to the Science and Ehgineering Research Council for the facili ties to use the Cambridge Linear Analysis and Design Package. Peter Westaway would also like to thank them for the financial support received during the course of this work.

REFERENCES Anderson,B.D.O. and Moore,J. B. (1971). Linear Optimal Control. Englewood Cliffs, N.J.: Prentice Hall. Athans,M. (1971). 'The role and use of the stochastic linear-quadratic-Gaussian problem in control system design. IEEE Transactions on Automatic Control, AC-16, 529-552. Doyle,J.C. and Stein-;G:'l(979). Robustness with observers. IEEE Transactions on Automatic Control, AC-24. Doyle,J. C. and Stein,G. (1981). Mul tivariable feedback design; concepts for a classical/modern syn thesis. IEEE Transac tions on Automatic Control, AC-26, 4-16. Kwakernaak,H. (1969~timal low-sensitivity linear feedback systems. Automatica, Vol.5. Kwakernaak,H. and Sivan,R. (1972). Linear Optimal Control Systems. New York: WileyIn te rscience. Kung,S. (1979). A new low-order approximation algorithm via singular value decompositions. Proceedings of the Control and Decision Conference, December. MacFarlane, A.G.J. (1979). Complex Variable Methods for Linear Mul tivariable Feedback Systems. Taylor and Francis, London. Maciejowski,J.M. and Vines,D.A. (1981). Decoupled control of a macroeconomic model usin g frequency-domain me thods. Su bmi t ted to the Journal of Economic Dynamics and Control. Maciejowski,J.M. and Vines,D.A. (1982). The design and performance of a mul tivariable macroeconomic feedback regula tion policy. Pro ceedings of the IEEE Conference on Adaptiv~ Multivariable Systems, Hull, U.K., July. Maciejowski,J. M. and Westaway,P. F. (1983). Asymptotic recovery for discrete-time systems. Proceedings of the American Control Conference, San Francisco, June. Meade,J. E. (1982). Stagflation, Volume 1: Wagefixing. London: Allen and Unwin. Moore,B.C. (198 1). Principal components analysis in linear systems: controllability, observability and model reduction. IEF; E Transactions on Automatic Control, AC-26, 4-16. Phillips,A.W. (1954). Stabilisation policy in a closed economy. Economic Journal, Volume 64 , 290-323. Rosenbrock,H.H. (1970). State-space and mul tivariable theory. London: Nelson. Rosenbrock,H.H. (1974). Compute!'-aided control system design. New York: Academic Press. Safonov,M.G. and Athans,M. (1977). Gain and phase margins of mul t i-loop LQG regula tors. IEEE Transactions on Automatic Co ntrol, AC-22. Vines,D.A., Maciejowski,J.M. and Meade,J.E. (1983). Stagflation, Volume 2: Demand Management. London: Allen and Unwin.

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