1.9 INTERNAL AND EXTERNAL [DESCRIPTIONS OF A](#page-8-0) SYSTEM
← Back to LINEAR SYSTEMS AND SIGNALS Overview
1.9 INTERNAL AND EXTERNAL DESCRIPTIONS OF A SYSTEM
The input–output relationship of a system is an external description of that system. We have found an external description (not the internal description) of systems in all the examples discussed so far. This may puzzle the reader because in each of these cases, we derived the input–output relationship by analyzing the internal structure of that system. Why is this not an internal description? What makes a description internal? Although it is true that we did find the input–output description by internal analysis of the system, we did so strictly for convenience. We could have obtained the input–output description by making observations at the external (input and output) terminals, for example, by measuring the output for certain inputs, such as an impulse or a sinusoid. A description that can be obtained from measurements at the external terminals (even when the rest of the system is sealed inside an inaccessible black box) is an external description. Clearly, the input–output description is an external description. What, then, is an internal description? An internal description is capable of providing complete information about all possible signals in the system. An external description may not give such complete information. An external description can always be found from an internal description, but the converse is not necessarily true. We shall now give an example to clarify the distinction between an external and an internal description.
Let the circuit in Fig. 1.41a with the input x(t) and the output y(t) be enclosed inside a “black box” with only the input and the output terminals accessible. To determine its external description, let us apply a known voltage x(t) at the input terminals and measure the resulting output voltage y(t).
Let us also assume that there is some initial charge Q0 present on the capacitor. The output voltage will generally depend on both, the input x(t) and the initial charge Q0. To compute the output resulting because of the charge Q0, assume the input x(t) = 0 (short across the input). In this case, the currents in the two 2 resistors in the upper and the lower branches at the output terminals are equal and opposite because of the balanced nature of the circuit. Clearly, the capacitor charge results in zero voltage at the output.†
† The output voltage y(t) resulting because of the capacitor charge [assuming x(t) = 0] is the zero-input response, which, as argued above, is zero. The output component due to the input x(t) (assuming zero initial capacitor charge) is the zero-state response. Complete analysis of this problem is given later in Ex. 1.21.
Figure 1.41 A system that cannot be described by external measurements.
Now, to compute the output y(t) resulting from the input voltage x(t), we assume zero initial capacitor charge (short across the capacitor terminals). The current i(t) (Fig. 1.41a), in this case, divides equally between the two parallel branches because the circuit is balanced. Thus, the voltage across the capacitor continues to remain zero. Therefore, for the purpose of computing the current i(t), the capacitor may be removed or replaced by a short. The resulting circuit is equivalent to that shown in Fig. 1.41b, which shows that the input x(t) sees a load of 5, and
Also, because y(t) = 2i(t),
This is the total response. Clearly, for the external description, the capacitor does not exist. No external measurement or external observation can detect the presence of the capacitor. Furthermore, if the circuit is enclosed inside a “black box” so that only the external terminals are accessible, it is impossible to determine the currents (or voltages) inside the circuit from external measurements or observations. An internal description, however, can provide every possible signal inside the system. In Ex. 1.21, we shall find the internal description of this system and show that it is capable of determining every possible signal in the system.
For most systems, the external and internal descriptions are equivalent, but there are a few exceptions, as in the present case, where the external description gives an inadequate picture of the system. This happens when the system is uncontrollable and/or unobservable.
Figure 1.42 shows structural representations of simple uncontrollable and unobservable systems. In Fig. 1.42a, we note that part of the system (subsystem S2) inside the box cannot be controlled by the input x(t). In Fig. 1.42b, some of the system outputs (those in subsystem S2) cannot be observed from the output terminals. If we try to describe either of these systems by applying an external input x(t) and then measuring the output y(t), the measurement will not characterize the complete system but only the part of the system (here S1) that is both controllable
Figure 1.42 Structures of uncontrollable and unobservable systems.
and observable (linked to both the input and output). Such systems are undesirable in practice and should be avoided in any system design. The system in Fig. 1.41a can be shown to be neither controllable nor observable. It can be represented structurally as a combination of the systems in Figs. 1.42a and 1.42b.