Nonlinear and Adaptive Control of a Growth Model with Knowledge

Copyright © IFAC Modelling and Control of National Economies, Beijing, PRC, 1992

NONLINEAR AND ADAPTIVE CONTROL OF A GROWTH MODEL WITH KNOWLEDGE! E, Casares*, G, Garza** and R. Kelly***,2 *&oncmics Deparlmull, Queen Mary and West{u!ld College , University of London, Mile End Road, London, UK **Deparlamenlo de Econom{a, Universidad Aw6noma Melropolitana-Azcapolzalco, Ave. San Pablo 180 , Azcopo/zalco, 02200 Mexico, D .P., Mexico *"Divisi6n de Fisica Aplicada, CICESE, Carrelera Tijuana-Ensenada Km. 107, Apdo Poslal2732, Ensenada , B.C., 22800, Mexico

economy. An endogeneous technical progress la learning by doing is then being consider~. [Arrow (1962), Romer (19&6, 19&9)]. It is then aSSUMed that, due to the existence of measurement problems, the value of the rate of capital depreciation is an unknown, as Romer .uggest~ in [ROIIIer (19&9)]: "co\J1\tri •• with lar6e recent addition. of tn.ir capital .toc~ may haue less ~prec iat ion tha" the steady-state approximation would sU66est" Taking into account this economic enviroment and considerinQ the saving rate as the economic policy variable the tracking aim problems, aboved mentioned, are solved designing a saving policy rule (or controller). Two cases are considered, one with known parameters and other wlth unknown rate of capital depreciation. A formal proof is provided to show that the tracking aim is achieved. It is noted that the savings rate influences the level of net per-capita product in the nonendogenous growth case and the growth rate of output in the endogenous case. The paper is organized in the following way: Section 2 contains a description of the economic model; in Section 3 we state the control problem; section 4 contains a review of the state feedback linearization approach and the design of the nonlinear controller for the econa.ic IIIodel; with the Methodology in Section 4 and assuming a parameter ot the model as unknown, Section 5 contains the design of the adaptive controller; Section 6 offers brief conclusions of this paper.

a

Abst.ract. In this paper we consider the control of an aggregated growth model augmented by an externality, that measures the existing technical or scientific knowledge, in the production. This externality is assu.ed to be related to the aggregated capital-labor ratio. First, a nonlinear controller is proposed based on the state feedback linearization approach. Second, assuming parameter uncertainty in the model, an adaptive controller is proposed to solve tracking aim problems.

1

Int.roduct.ion

An economic policy problem can be formulated in different ways. ~t for example it is intended that the net product of an economy follows a certain desired trend, a tracking aim problem is faced. Additional difficulties rise, within this sa.e type of problem, when the economic model has unknown or uncertain nUMerical value of so~ parameters. Using control theory tecniques both problems can be solved. In the first case, with known parameters, nonlinear control techniques are suitable. With uncertain or unknown parameters the problem can be dealt with adaptive c ontrol techniques. In this paper both cases are considered for a growlng economy with en dogeneous technical progress and unknown rate of capital depreciation. It is assumed that the economy is ruled by a Solow growth model aug.ented by an externality (Sala-i-Martin (1990». This externality measureS the existing technical or scientific knowledge in the

2

The output trend of an econOMy, producing one single good using capital and labor as inputs, is assumed to be ruled by a Solow type growth model [Solow (1956), Burmeister and Debell (1970)] augmented by an externality in the production. The externality, that MeasureS the existing technical or scientific knowledge in the economy, is assumed to be related to the aggregated capital-Iabor

. . . xico

2

Author

• hou Ld

~

Le

",hem

all

The econo.uc lIIOdel

cor,. •• pond.nce

o.dclr-••• d

7

exogenously given rate n, we have:

ratio. This ~eans that an endongenous technical progress ~ la learning by doing is being considered. First it is assumed that a typical firm of this single-good producing economy has the following production function:

If Ol + e < 1, the equation describes the nonendogenous growth case, Ol + e = I the endogenous growth case. Given the initial condition of the capital-labor ratio, k(O), and numerical values for the parameters, the evolution of kIt) can be obtained. The capital-labor ratio will tend to an equilibrium point

(La) where V, 1S the net product obtained by fir.. " K. and L, are the amount of capi tal and labor used by f ir~ " Ol the distribution para~eter and A the technical progress. Constant returns to scale on capital and labor are assumed as well as a technical progress that hit all the production function. A technical progress A la learning by doing is considered where technological or scientific knowledge is assumed to be a function of the aggregated capital-labor ratio in the economy. Knowledge is considered as a public good. Therefore we have: (l.b) A = [KILl e where

k*, as t

k

y

is the degree of the externality. Substituting (l.b) in (La) and assuming that we have identical firms, the aggregated production function is:

=

KOl L"- Ol (K/L)e

where V is the national product, K the total capital stock and L the labor force in the economy. Note that K = NK, and L NL" where N is the number of firms in the economy. The resulting aggregate production function has constant returns to scale (increasing returns to scale are considered in the Arrow-Romer case) and it can be expressed in per capita terms as:

3

K

4

~/L

nonendogenous

s k Ol • e_(n +6) k kOl •

e

(2) (3)

The control problem

Nonlinear control

The design of a nonlinear controller that solves the control problem stated previously is based on the state feedback linearization approach developed by Sastry and Bodson (1989). This approach consists in obtaining a linear system when the dynamic model and the controller equations are put together. This ~ethodology has great advantages, since there is a well developed theory on linear differential systems that can be used.

(1.c)

where 6 is the rate of capital depreciation. The accounting economic system, which gives us the capital-labor trend, is then represented by: k/k = K/K -

the

From a control theory point of view the economic model, given by equations (2) and (3), represent a dynamic nonlinear system with an input variable, set), a state variable, kIt), and an output variable, yet). Given an arbitrarily chosen desired trend for the output variable, yd, the control problem can be stated as: to design a controller to compute the input variable or the savings rate, set), such that the output variable or product per unit of labor, yet), follows the arbitrarily chosen desired trend, yd(t). Two cases for this control proble~ are considered in this paper. In the first one, all the parameters of the economic model are assumed to be known and constant and the state feedbak linearization approach is used to obtain the nonlinear controller. In the second case, the rate of capital depreciation, 6, is assumed to be unknown and adapti v e control techniques are used to design the controller.

where y is the net product per unit of labor and k is the total capital stock per unit of labor. On a closed economy, net investment is equal to the amount of savings minus depreciation. Savings are assumed to be a proportion of the current income. Solow considered a constant and exogenously given proportion and given the value of the exogenous parameters, the equilibrium capital-Iabor ratio, and therefore the per-capita product, is determined. In this paper the savings rate, s, is endogenized by the design of the saving policy rule, this means that s adjusts in order to reach the desired output or capital-labor ratio trend. We exogenously choose a desired capital-labor ratio trend and obtain the saving rate that allows the system to follow it. The controller then determines, at any point in time, the proportion [S = sVl of the current income to be save, then this proportion may vary over time. Net investment is then defined as: 6

in

The stylized facts can be fulfilled in the nonendogenous growth case if L is redefined as the effective labor force (with exogenous technical progress) and n as (n + A), where A is the constant growth rate of the exogeneous augmenting labor force technical progress. All the per unit of labor variables in the model being redefined in terms of effective labor. We consider s as the economic policy variable or control variable, k the state variable and y the output variable. In the next sections the saving policy rule or controller will be obtained for the general case, when a + e < 1. The endogeneous growth is a particular case.

e

V

increases,

growth case or to a constant growth rate of the capital-Iabor, g, in the endogenous case. This constant growth rate is defined as g = [ s - (n + 6) 1. The overall economic model is then described by a nonlinear equation system given by:

( 1.d)

Considering (l.c) and (l.d) and assumming that the labor force grows at an

4.1

8

State feedback 11nearlzation approach

The basic concepts of state linearization

can

be

follows. Consider system written the form:

a

x

f(X) + g(X)

Y

hO)

feedback

summarized

dynaMic

k + (1n, d -

aB

equation

u

s

= (n+6) +

k)

k

When substituting the controller equation in the economic model we obtain the same linear differential equation given by (a) and the stability proof

(4)

remains.

where X is the state variable, u the input variable, and Y the output variable. Assume that:

* *

g(X)

>

for all X

0

economic model is an unknown

h(X) satisfies: Xl ~ X2 h -l ( X2 ) f or a 11 Xl, X2

.. E

h - l(X.) ~•

IR.

The input to state linearization obtained by choosing u as: [- fIX) + {Xd

u

+ (1(Xd -

X)}J

is

(5)

g(X) where (1 > 0 and Xd is the desired for the state variable. By substituting (5) in (4), a differential equation is obtained: (Xd -

=

X) + (1(Xd -X)

0

The nonlinear

(n s

linear

k

k

Y

(01.

6)

...

+ k ca

can be written in the same where: g(X)

hO) f(

(5)

X)

-

(n

k(Ot

...

(kd -

form

as

given

=

(10)

.

0

=2

function if a positive

V(k,8) with

r k k

respect

+ 2 8 8

to

(12)

Using equation (12) we can define 8 such that satisfies the Lyapunov condition for stability, that is when V(k,8)

(a)

8

=r ~

k k, 0

then,

(13)

From the definition 8 8 - 6, and considering that 6 is a constant, we have, 8

r

8

k k

(14)

.

Integrating 8 from (14), we obtain 8(t): •

8(t) k

the

k)]

which is a linear differential equation that eKponentially converges to zero. This means that [kd(t) - kIt») ~ 0 as t ~ m and kit) ~ kd(t) i.-plies y(t) ~ Yd(t) as t m. The control aim is then achieved. In the endogeneous growth case, a + 8 = 1, the economic model equations are , simplified as: s k -

on

V(k,8) can be defined, such that

Differentiating time, we have:

by

kd is the desired capital-effective labor ratio. Note that substituting (7) in the economic model equations, we obtain:

k

unknown

(9)

= k(8-6)

k)

which is a positive definite the adaptive gain, r, is constant.

wh~re

Y

the

(11 )

(])

=

(9)

(4),

€h

+ [ kd + (1 (kd -

-I<)

)

V(k,8) ~ O,then the stability is assured. Let us propose the following candidate Lyapunov function:

8. s

+ 6)k

+ (1(kd

k)

First, we choose 8(t) such that stability, on the Lyapunov sense, for the closed loop system is guaranteed. Define

y

I<)

(1 (kd -

estimate of Substituting

=

s

(kd -

+

+ (1(kd

k)

V(k,8) 6)k

kd

k kd - k and 8 8 - 6. The Lyapunov Method states that if a positive definite

The nonlinear controller, is then defined by: (n +

[

y

function +

+

=-

dynamic equation of the economic model k = sy - (n + 6)k, we obtain the closed-loop system equation:

The economic model equations: 6) k

8)k

+

where €I is the parameter, 6.

(6)

con~roller

(n +

.

trend

From the theory on linear systems we know that (6) has eKponentially decreasing to zero solutions provided that (1 > o. Hence [Xd (t) - X(t») ~ 0 as t ~ m. Moreover, considering the assuMption on h(X) with X. = X(t) and X2 = Xd(t), we conclude that X(t) ~ Xd(t) implies that Y(t) ~ Yd(t) as t ~ m. 4.2

con~rol

Let us assume that a parameter of the cons~ant, 6 for eKample. Using adaptive control techniques, an adaptive controller can be obtained such that guarantees the stability of the system, on a Lyapunov sense, and that verifies the control aim independently from the value of the unknown parameter. See Astrom (19a3) and Astrom (19a]). We propose the following control law:

~.

E

Adap~ive

5

=

l

Io

r kiT) kiT) d(T)

.

+ 8(0)

(n + 6) k

The error .odel equations are given by equat i'ons (10) and (14):

and the controller is defined as:

9

then

k

-

(1 k

19

y

le. k

considered as unknown a parameter in the model and an adaptive controler is proposed to solve the control proble•.

k 8

REFERENCES

whose origin le. , 19 [0, 0) is an equilibrium, which has been proven stable in the Lyapunov sense. To prove that the control aim is achieved, we use the following lemma shown in Oesoer and Vidyasagar (1975 ) . Co nsider a continuous function 0< . ~ ilL If: If:

LEMMA.

<

I f (t) 12 dt

Arrow, K. :l., (1962) "The econa.ic implications of learning by doing", Review of Economic Studies. Vol 29.

(2)

Astrom K. :l., B. Witternmark, "Adaptive Control " , Addison \&Iesley, 1989.

(3)

Burmeister E., R. Oobell, "l1athematical theories of econO
(4)

Oesoder y Vidyasagar, "Feedback systems: input-output properties", Academic Press, 1975.

f:

00

J

*

(1)

00

o

*

1

*

f I t)

Ht)

<

1

for all t

Cl>

E

IR ...

then, ~

as

0

t.

CI>.

(5)

00

Afterward s , we show that Cl>

and Ik (t ) 1

<

Io

for all t

00.

Ik(t)1 ~

2

<

dt

2

fl ' om 0 to

Cl> ,

V[k( .o ),e(oo») -

introduction to

l10dern Thomas

Nelson and sons, 1975.

O.

[6]

ROMer P. 11., (1986) "Increasing returns and long-run growth" in :lournal of Political Economy, vol 94.

[7]

Ro.er P. 11. , AccuOllUlation in

Inte g rating both sides of V(k,e) 2 y (1k

:lones, H.

5au

Theories of Economic Growth",

we have:

(1989), "Capital the Theory of

V[UO),e(O») Long-run Growth" in "Modern t:JUSlness Cycle Theory", Barro R. :l., editor, Basil Blackwell.

and, V

Now , ince V[k(t), e(t»)

e (t)

and

<

=y

00

k (t) 2

+

V(k(t), e(t») ~ 0, we have that

k and 19 are bounded functions. In addition, assuming kd bounded, we have also k bou nded. Using this, from error model equ a tion

k

= -

(1k -

Sastry, S. Bodson M., "Adaptive control. Stability. Convergence and Robustness, Prentice-Hall 1989.

(9]

Solow, R. 11., "A contribution to the theory of economic growth. Quarterly :lournal of Economics. Vol LXX. 1956. QQ 65-94.

[UO), 19 (0») 2 y (1

2

(8]

k 19

we conclude that k is a bounded

function

too, henc . Ikl < 00 for all t ~ O. Invo cing Lemma above, we can guarantee that kIt) • 0 as t • 00. Finally, this in t " rns implies that y(t) • yd(t) as

t •

00.

6

Concluding re . . rks

In this paper we have presented an applicati o n of the adaptive control theory to an aggregated nonendogeneous growth model. It is considered a tracking trend control a l m for the production per unit of labor. The main difficulties found to solve the control problem came fra. the nonlinear structure of the model. ' Two alternatives were considered in this paper. F I rst, it is assumed that the numerical value of all the parameters in the mode l are known and a nonlinear controlle r , based on the recently developed formalism of linearization by state feedback is proposed. Second, it is

10