| A2. Control systems | 1998-10-22 |
| Basic theory | To contents |
The control system theory is used for analyzing dynamic systems. It does not have to be a control problem, all dynamic systems can be analyzed. The design of regulators and feed-back is only one part of the theory.
The same theory can be used to analyze the economic circuits. By using a more abstract description, it is avoided that a general view is hidden by the details.
The theory is described in a number of books, only one is cited below.
The definition of a state is given in the book Reglerteknik (Control systems) , ref. (1) chap 8.1, page 137:
"A dynamic system is characterized by the fact that the value of the output signal now, y(t), not only depends upon the value of the input signal now, u(t), but also upon (previous input signals) u(s), s<t. This means that we can not forecast the effect of the input signal that we choose, if we do not have additional information about the system at time t (e.g. u(s), s<t). … Information about the system at time t, such that it is possible to forecast the effect of an applied input signal u(t), t >= t is called the state of the system at time t. Obviously, {u(s), s<t}is such an information quantity. However it is awkward to handle, and we desire a more concise description of the state."
Italic = my addendum.
An explanatory example is given in ref. (1) chap 8.2, page 138:
"Study a system that is described by
| dy(t)/dt + a*y(t) = b*u(t). | Eq. (8.1) (equations numbered as in the book) |
Assume that we are confronted to the system at time t0, without knowing what happened before that time. From the expression
| dy(t)/dt = -a*y(t) + b*u(t). | Eq. (8.1´) one member moved. |
we see that if we know y(t0) and u(t), t >= t0
then y(t), t >= t0 can be computed. Thus y(t0)
is a state for the system at the time t0."
In the theory of differential equations, y(t0) is called
a starting condition. For this simple system it is necessary to
know one value = one state variable in order to calculated to
future development of the system.
Equation (8.1´) can be described graphically in this diagram. The connection lines and arrows mean signals, and not flows.
Figure A2:1 Feed-back system with one variable. (1/s) stands for integration.
If the picture is drawn a bit more schematic we get:
Figure A2:2 Simplified picture of feed-back system.
A more general system with a number of variables can be decsribed by the equation system:
| dy(t)/dt = A y(t) + B u(t) | Eq. (8.6) |
In this case u(t) and y(t) are vectors. A and B are matrixes. u(t) are the input signals to the system, y(t) is the state vector. (Simplified from ref (1) page 140). A diagram of this system looks like this:
Figure A2:3 Feed-back system with a number of variables.
| Right member | R.m = A Y(t) + B U(t) | eq. (A2:1) |
| Static equations | S(p) X(t) = R.m. | eq. (A2:2) |
| Correction to the state variables | DY(t) = C X(t) | eq. (A2:3) |
| Dynamic equations | Y(t+1) = Y(t) + DY(t) | eq. (A2:4) |
S(p) is the system matrix that determines the static values of all flows X(t) when the state variables Y(t) and the input signals U(t) are given. The matrix S(p) depends upon a number of parameters p(t) which also can vary from year to year. The input signals U(t) are extensive quantities, flows. The parameters p(t) are intensive quantities that do not depend upon the size of the system, they only determine relative distribution between flows.
The matrices A and B constitutes jointly the right member by making a column vector from the elements in Y(t) and U(t). They determine the size of the economy, e.g. the volume of the production and the number of employees.
The dynamic behavior of the system is due to the change of Y(t) until the next year. The extent of the change depends upon a number of the flows X(t) that are selected by the matrix C.
The interdependences are shown in the diagram:
Figure A2:4 Dynamic feed-back economic system.
The static variables X(t) are calculated by solving the equation
system S(p) X(t) = R.m. There is an
equivalent formulation in the diagram, where X(t) is expressed
explicitly by the help of the inverse S(p)-1
of the system matrix.
A more schematic diagram looks like this:
Figure A2:5 Simplified picture of the feed-back economic system.
At a given time instance U(t), Y(t) and p(t) may be regarded as given. Then the flows X(t) can be calculated as for a static system. At the end of the year some accounts (number of employees, bank accounts) have changed, which is expressed by the increments DY(t). The accounts of the next year have somewhat different values Y(t+1). During the year t+1, the input signals U(t+1) and the parameters p(t+1) may have new values which gives new flows X(t+1). To the start of the calculation, the states (endogenous variables) Y(t0) during the first year t0 have to be known. U(t) and p(t) are exogenous and give at all times.
The design of dynamic models that are continuos in time are described in ref (2) Model desing and simulation. Methods for parameter identification are also described.
Advanced methods for the analysis of multivariable systems are found in ref (3) Multivariable feedback design. Methods for determining the stability of the systems are presented and how to design optimal feed-back loops. The methods are not immediately necessary for the problems that we study, however the book gives a general understanding of feed-back systems.
Sampled systems are described in ref (4) Digital Signal Processing. The consequences of observing the system only at certain time instances, e.g. at each year turn or every three months. Systems with time delays (time lag) are treated. It is shown how to draw graphs for digital filters. All systems with time descrete input and output signals may be regarded as digital filters.
The literature is extensive, many similar books are easily found.
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