2.1.1 Standard stabilization problem

The optimal stabilization problem with deterministic coefficients is presented first. This initial form, which does not fit to the application of the control theory, is transformed to a more convenient form. In the control form of the system, the constraints and the objective functions are rewritten. Following Turnovsky, let a system be described by the following matrix equation

The system (1) consists in q1 target variables in instantaneous and delayed vectors Y and q2 policy instruments in instantaneous and delayed vectors U . The maximum delays are m and n for Y and U respectively. The squared q1×q1 matrices A are associated to the targets, and the q1×q2 matrices B are associated to the instruments. All elements of these matrices are subject to stochastic shocks. Suppose that the objective of the policy maker is to stabilize the system close to the long-run equilibrium, a quadratic objective function will be

where M is a strictly positive definite costs matrix associated to the targets and N a positive definite matrix associated to the instruments. According to (1), the two sets Y¯ and U¯ of long-run objectives are required to satisfy

Letting the deviations be Yt−Y¯=yt and Ut−U¯=ut , the optimal problem is

2.1.2 State-space form of the system

The constraint (2) is transformed into an equivalent first order system [Preston & Pagan, 1982]

where xt=(ytyt−1yt−2,...,yt−m+1utut−1,...,ut−n+1) is the g×1 state vector with g=mq1+nq2 . The control vector is vt=ut . The block matrix A and the vector B are defined by

Any stabilization of a linear system requires that the system be dynamically controllable over some time period [Turnovsky, 1977]. The condition for the full controllability of the system states that it is possible to move the system from any state to any other.

Theorem 2.1.2 (Dynamic controllability condition). A necessary and sufficient condition for a system to be dynamically controllable over some time period

Proof. In [Turnovsky, 1977], pp. 333-334.

The objective function (3) may be also written as

where θ includes past y ’s and u ’s before t=1 . Letting M˜=M/m and N˜=N/n , the block diagonal matrix M* is defined by

The stabilization problem, (2) is transformed to the control form

Since the matrices M* and N are strictly positive, the optimal policy exits and is unique.

2.1.3 Backward recursive resolution method

Let a formal stabilization problem be expressed with a discrete-time deterministic system

In the quadratic cost function of the problem, the n state vector y and the m control vector x are deviations from long-run desired values, the positive semi-definite matrices Mn×n and Nm×m are costs with having values away from the desired objectives. The constraint of the problem is a first order dynamic system [1] - with matrices of coefficients An×n and Bn×m . The objective of the policy maker is to stabilize the system close to its long-run equilibrium. To find a sequence of control variables such that the state variables y _t can move from any initial y0 to any other state yT , the dynamically controllable condition is given by a rank of a concatenate matrix equal to n

The solution is a linear feedback control given by

where we have

The optimal policy is then determined according a backward recursive procedure from terminal step T to the initial conditions, such as

2.1.4 The stochastic control problem

Uncorrelated multiplicative and additive shocks: The dynamic system is now subject to stochastic disturbances with random coefficients and random additive terms to each equation. The two sets of random deviation variables are supposed to be uncorrelated. [2] -

The constant matrices An×n and Bn×m are the deterministic part of the coefficients. The random components of the coefficients are represented by the matrices Φn×n and Ψm×m . Moreover, we have the stochastic assumptions : the elements ϕijtψijt and εit are identically and independently distributed (i.i.d.) over time with zero mean and finite variances and covariances. The elements of Φt are correlated with those of Ψt , the matrices Φt and Ψt are uncorrelated with εt . The solution is a linear feedback control given by

[3] -

Correlated multiplicative and additive shocks: The assumption of non correlation in the original levels equation, will necessarily imply correlations in the deviations equation. Let the initial system be defined in levels by the first order stochastic equation

and the stationary equation

By subtracting these two matrix equations and letting yt≡Yt−Y* and xt≡Xt−X* , we have

where the additive composite disturbance ε' denotes a correlation between the stochastic component of the coefficients and the additive disturbance. The solution to the stabilization problem takes a similar expression as in the uncorrelated case. We have the solution

where

The optimal policy then consists of a feedback component R together to a fixed component p . The system will oscillate about the desired targets.

2.2.1 Basic stochastic multiplier-accelerator model

Structural model: The discrete time model consists in two equations, one is the final form of output equation issued from a multiplier-accelerator model with additive disturbances, the other is a stabilization rule [Howrey, 1967; Turnovsky, 1977]

where Y denotes the total output, G the stabilization oriented government expenditures, B¯ a time independent term to characterize a full-employment policy [Howrey, 1967] and ε random disturbances (serially independent with zero mean, constant variance) from decisions only. The policy parameters are g1g2 and Y¯ is a long run equilibrium level [4] - .

Time path of output: Combining the two equations, we obtain a second order linear stochastic difference equation (SDE)

where B¯ is a residual expression. Provided the system is stable [5] - , the solution is given by

where C1C2 are arbitrary constants given the initial conditions and r1r2 the roots of the characteristic equation: r1r2=(−b±b2−4c)/2 The time path of output is the sum of three terms, expressing a particular solution, a transient response and a random response respectively.

2.2.2 Stabilization of the model

Iso-variance and iso-frequencies loci: Let the problem be simplified to [Howrey, 1967]

Figure 1 shows the iso-variance and the iso-frequencies contours together with the stochastic response to changes in the parameters b and c . Attempts to stabilize the system may increase its variance ratio σy2/σε2 . As coefficient bc being held constant, the peak is shifted to a higher frequency.

Figure 1.

Iso-variance (a) and iso-frequencies (b) contours

Asymptotic variance of output: Provided the stability conditions are satisfied (the characteristic roots lie within the unit circle in the complex plane), the transient component will tend to zero. The system will fluctuate about the stationary equilibrium rather than converge to it. The asymptotic variance of output is

Speed of convergence: The transfer function (TF) of the autoregressive process (4) is given by

We then have the asymptotic spectrum

The time-dependent spectra are defined by

In this application, the parameters take the values b=−1.1,c=.5,σε2=1 as in [Howrey, 1967]. Figure 2 shows how rapid is the convergence of the first ten log-spectra to the asymptotic log- spectrum. [Nerlove et al., 1979].

Figure 2.

Convergence to the asymptotic log- spectrum

Optimal policy: Policies which minimize the asymptotic variance are such

The output will then fluctuate about Y¯ with variance σε2 .

3.1.1. Structural form of the Phillips’ model

The Phillips’model [Phillips, 1954; Allen, 1955; Phillips, 1957; Turnovsky, 1974; Gandolfo, 1980; Shone, 2002] is described by the continuous-time system

where I˙ and Y˙ denote the first derivatives w.r.t. time of the continuous-time variables I(t) and Y(t) respectively. All yearly variables are continuous twice-differentiable functions of time and all measured in deviations from the initial equilibrium value. The aggregate demand Z consists in consumption C , investment I and autonomous expenditures of government G in equation (5). Consumption C depends on income Y without delay and is disturbed by a spontaneous change u at time t=0 in equation (6). The variable u(t) is then defined by the step function u(t)=0 , for t0 and u(t)=1 for t≥1 . The coefficient c is the marginal propensity to consume. Equation (7) is the linear accelerator of investment, where investment is related to the variation in demand. The coefficient v is the acceleration coefficient and β denotes the speed of response of investment to changes in production, the time constant of the acceleration lag being β−1 years. Equation (8) describes a continuous gradual production adjustment to demand. The rate of change of production Y at any time is proportional to the difference between demand and production at that time. The coefficient α is the speed of response of production to changes in demand. Simple exponential time lags are then used in this model [7] - .

3.1.2. Block-diagram of the Phillips’ model

The block-diagram of the whole input-output system (without PID tuning) is shown in Figure 3 with simulation results. Figure 4. shows the block-diagram of the linear multiplier-accelerator subsystem. The multiplier-accelerator subsystem shows two distinct feedbacks : the multiplier and the accelerator feedbacks.

Figure 3.

Block-diagram of the system and simulation results

3.1.3. System analysis of the Phillips’ model

The Laplace transform of X(t) is defined by

Omitting the disturbance u(t) , the model (5-8) is transformed to

The TF of the system is

Taking a unit investment time-lag with β=1 together with α=4,c=34 and v=35 , we have

The constant of the TF is then 4, the zero is at s=−1 and poles are at the complex conjugates s=.2±j . The TF of system is also represented by H(jω) . The Bode magnitude and phase, expressed in decibels 20log10 are plotted with a log-frequency axis. The Bode diagram in Figure 5 shows a low frequency asymptote, a resonant peak and a decreasing high frequency asymptote. The cross-over frequency is 4 (rad/sec). To know how much a frequency will be phase-shifted, the phase (in degrees) is plotted with a log-frequency axis. The phase cross over is near 1 (rad/sec). When ω varies, the TF of the system is represented in Figure 5 by the Nyquist diagram on the complex plane.

3.1.4 PID control of the Phillips’ model

The block-diagram of the closed-loop system with PID tuning is shown in Figure 6. The PID controller in Figure 7 invokes three coefficients. The proportional gain Kpe(t) determines the reaction to the current error. The integral gain

bases the reaction on sum of past errors. The derivative Gain Kde˙ determines the reaction to the rate of change of error. The PID controller is a weighted sum of the three actions. A larger Kp will induce a faster response and the process will oscillate and be unstable for an excessive gain. A larger Ki eliminates steady states errors. A larger Kd decreases overshoot [Braae & Rutherford, 1978 [8] - A PID controller is also described by the following TF in the continuous s-domain [Cominos & Nunro, 2002]

The block-diagram of the PID controller is shown in Figure 7.

Figure 6.

Block diagram of the closed-looped system

3.2.1. Structural form of the Goodwin’s model

The extended model of Goodwin [Goodwin, 1951; Allen, 1955; Gabisch & Lorenz, 1989] is a multiplier-accelerator with a nonlinear accelerator. The system is described by the continuous-time system

The aggregate demand Z in equation (13) is the sum of consumption C and total investment I [9] - . The consumption function in equation (14) is not lagged on income Y . The investment (expenditures and deliveries) is determined in two stages: at the first stage, investment I in equation (15) depends on the amount of the investment decision B with an exponential lag; at the second stage the decision to invest B in equation (16) depends non linearly by Φ on the rate of change of the production Y . Equation (17) describes a continuous gradual production adjustment to demand. The rate of change of supply Y is proportional to the difference between demand and production at that time (with speed of response α ). The nonlinear accelerator Φ is defined by

where M is the scrapping rate of capital equipment and L the net capacity of the capital-goods trades. It is also subject to the restrictions

The graph of this function is shown in Figure 8.

3.2.2. Block-diagrams of the Goodwin’s model

The block-diagrams of the nonlinear multiplier-accelerator are described in Figure 9.

3.2.3 Dynamics of the Goodwin’s model

The simulation results show strong and regular oscillations in Figure 10. The Figure 11 shows how a sinusoidal input is transformed by the nonlinearities. The amplitude is strongly amplified, and the phase is shifted.

Figure 10.

Simulation on the nonlinear accelerator

3.2.4 PID control of the Goodwin’s model

Figure 12 shows the block-diagram of the closed-loop system. It consists of a PID controller and of the subsystem of Figure 9. The simulation results which have the objective to maintain the system at a desired level equal to 2.5. This objective is reached with oscillations within a time-period of three years. Thereafter, the system is completely stabilized.

4.1.1 Fuzzy logic controller

A fuzzy logic controller (FLC) acts as an artificial decision maker that operates in a closed-loop system in real time [Passino & Yurkovitch, 1998]. Figure 13 shows a simple control problem, keeping a desired value of a single variable. There are two conditions: the error and the derivative of the error. This controller has four components: (a) a fuzzification interface to convert crisp input data into fuzzy values, (b) a static set of "If-Then" control rules which represents the quantification of the expert’s linguistic evaluation of how to achieve a good control, (c) a dynamic inference mechanism to evaluate which control rules are relevant, and (d) the defuzzification interface that converts the fuzzy conclusions into crisp inputs of the process [10] - These are the actions taken by the FLC. The process consists of three main stages: at the input stage 1 the inputs are mapped to appropriate functions, at the processing stage 2 appropriate rules are used and the results are combined, and at the output stage 3 the combined results are converted to a crisp value input for the process.

4.1.2 Fuzzyfication and fuzzy rules

Simple control example: Let us consider a simple control example of TISO (Two Inputs Single Output) Mamdani fuzzy controller. The fuzzy controller uses identical input fuzzy sets, namely "Negative", "Zero" and "Positive" MFs. The system output is supposed to follow

as in Figure 14. The error is defined by e(t)=r(t)−x(t) where r(t) is the reference input, supposed to be constant (a set point) [11] - . Then we have de(t)dt≡e˙=−x˙ .

Figure 14.

System output, fuzzy rules and phase trajectory

Fuzzification: Membership functions. A membership function (MF) assigns to each element x of the universe of discourse

The triangular MF of Figure 15 is defined by μ(x)=max{min{x−ab−ac−xc−b},0} where abc . A fuzzy set A˜ is then defined as a set of ordered pairs A˜={xμA˜(x)|x∈X}

According to the Zadeh operators, we have

The overlapping MFs of the two inputs error and change-in-error and the MF of the output control-action show the most common triangular form in Figure 15. The linguistic label of these MFs are Negative", "Zero" and "Positive" over the range [−100,100] for the two inputs and over the range [−1,1] for the output.

Figure 15.

Membership functions of the two inputs and one output

Fuzzy rules: Fuzzy rules are coming from expert knowledge and consist in "If-Then" statements. An antecedent block is placed between "If" and "Then" and a consequent block is following "Then"[12] - . Let the continuous differentiable variables

These FAM (Fuzzy Associative Memory)-rules [13] - are those of the Figure 16. These nine rules will cover all the possible situation. According to rule (PL,NL;ZE), the system output is below the set point (positive error) and is increasing at this point. The controller output should then be unchanged. On the contrary, according to rule (NL,NL;NL), the system output is above the set point (negative error) and is increasing at this point. The controller output should then decrease the overshoot. The commonly linguistic states of the TISO model are denoted by the simple linguistic set A={NL,ZE;PL}. The binary input-output FAM-rules are then triples such as (NL,NL;NL): "If" input e is Negative Large and e˙ is Negative Large "Then" control action v is Negative Large. The antecedent (input) fuzzy sets are implicitly combined with conjunction "And".

4.1.3 Fuzzy inference and control action

Fuzzy inference: In Figure 17, the system combines logically input crisp values with minimum, since the conjunction "And" is used. Figure 18 produces the output set, combining all the rules of the simple control example, given crisp input values of the pair

Figure 17.

FAM influence procedure with crisp input measurement

Figure 18.

Output fuzzy set from crisp input measurements

Defuzzyfication: The fuzzy output for all rules are aggregated to a fuzzy set as in Figure 18. Several methods can be used to convert the output fuzzy set into a crisp value for the control-action variable v. The centroid method (or center of gravity (COG) method) is the center of mass of the area under the graph of the MF of the output set in Figure 18. The COG corresponds to the expected value

In this example, vc=−.124 for the pair of crisp inputs (ee˙)=(−55,20)