OPTIMAL POLICIES AND THE STABILITY OF SYSTEM: AN APPROACH FROM TRANSIENT RESPONSE

Abstract

There are three purposes in this paper. The first is that we discuss the three different types of policy, namely, proportional, derivative and integral policy, in relation to the stability of the system. The second is that we discuss both the length of policy lag and the strength of policy in relation to the stability of the system. The third is that we derive the optimal policy to achieve the system performance.
The desired performance of a system may not be specified in terms of some performance criterion, but in terms of the transient response of a unit-step-function input. The system to be analysed is as follows.
C=c(Y-T) 0<c<1 I=(2μ/D+2μ)2νDY μ>0, ν>0 G=G X=X M=mY 0<m<1 T=tY 0<t<1 Y=C+I+G+X-M R=∫t0(X-M)dτ where C=consumption, I=investment, G=government expenditure, X=export, M= import, T=tax receipts, Y=national income, R=foreign currency reserves (measured by cumulated trade balance), D=diffential operator d/dt.
The correcting action taken is such that the government expenditure is made with a time lag to adjust the gap between optimal level of the foreign currency reserves and actual level of that. For example, proportional policy is shown by g=a1δ/D+δ(R-R*) where 1/δ is the time constant of the lag.
We shall show several conclusions.
(1) The derivative policy has the effect of stabilising the system, but the steady-state error still remains.
(2) The proportinal and integral policy have the effect of reducing the system stability, but the steady-state error does not remain.
(3) The existence of policy lag in proportinal policy makes the system unstable, but in derivative and integral policy policy, lag has only a little effect with respect to the system stability.
(4) The optimal policy for the unstable system is shown by derivative plus proportional policy, that is, g=δ/D+δ(b1+b2D)(R-R*).