System Theory

CHAPTER 3

3.1 CONCEPTS AND TERMINOLOGY

The branch of modem engineering analysis known as system theory is highly relevant to the study of the flight of vehicles in the atmosphere and in space. The word system has long been current in such applications as “control system,” “navigation system,” and “hydraulic system.” In our present context we identify the vehicle itself as a system, of which the above examples are subsidiary systems, or associated systems.

We do not attempt to offer here a precise definitionf of a system—suffice it to say that it is an element, or an interconnected set of elements that is clearly identifiable and that has a state defined by the values of a set of vari­ables that characterize its instantaneous condition. The elements may be physical objects or devices, or they may be purely mathematical, i. e. equations expressing relationships among the variables. In the case of a physical system, the governing equations may or may not be known. A set of equations that constitutes a mathematical model of a physical system, is a mathematical system that is a more or less faithful image of the physical system, depending on the assumptions and approximation contained therein. The set of n variables that defines the state of the system is the state vector,

f See for example ref. 3.1, Sec. 1.10.

and the corresponding те-dimensional space is the state space. Some or all of the state variables, or quantities derived from them, are arbitrarily termed, according to the circumstances of the experiment or analysis, as outputs. The exact specification of a system is usually arbitrary, as will be seen in the following example; the “boundary” of the system under consideration in any given circumstance is chosen by the analyst or experimenter to suit his purpose.

In addition to the state variables, there is usually associated with a system a second set of variables called inputs. These are actions upon the system the physical origins of which are outside the system. Some of these are independent of the state of the system, being determined by processes entirely external to it; these are the nonautonomous inputs. Others, the autonomous inputs, have values fixed by those of the state variables them­selves, owing to internal interconnections or feedbacks, or a as result of environmental fields (e. g. gravity, aerodynamic, or electromagnetic) that produce reactions that are functions of the state variables. An output of one system may be an input to another, or to itself if there is a simple feedback. The state variables are unique functions of the nonautonomous inputs and of the initial conditions of the system. A system with only autonomous inputs is an autonomous system.

Every system has, as well as its state variables and inputs, a set of system parameters that characterize the properties of its elements—e. g. areas masses, and inductances. When these are constant, or nearly so, it is con­venient to consider them as a separate set. On the other hand, if some of them vary substantially in a manner that depends on the state variables, they may usefully be transferred to the latter set. The problem of system design, after the general configuration has been established, is primarily one of optimization in the system parameter space. Still another set of parameters is that associated with the environment—e. g. atmospheric density, gravi­tational field, and radiation field. In adaptive systems, some system parameters are made to be functions of the state variables and/or environmental param­eters in order to achieve acceptable performance over a wider range of operating conditions than would otherwise be possible.

The following example will serve to illustrate some of the above concepts.

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