[10.9 SUMMARY](#page-15-0)

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10.9 SUMMARY

An Nth-order system can be described in terms of N key variables—the state variables of the system. The state variables are not unique; rather, they can be selected in a variety of ways. Every possible system output can be expressed as a linear combination of the state variables and the inputs. Therefore, the state variables describe the entire system, not merely the relationship between certain input(s) and output(s). For this reason, the state variable description is an internal description of the system. Such a description is therefore the most general system description, and it contains the information of the external descriptions, such as the impulse response and the transfer function. The state variable description can also be extended to time-varying parameter systems and nonlinear systems. An external description of a system may not characterize the system completely.

The state equations of a system can be written directly from knowledge of the system structure, from the system equations, or from the block diagram representation of the system. State equations consist of a set of N first-order differential equations and can be solved by time-domain or frequency-domain (transform) methods. Suitable procedures exist to transform one given set of state variables into another. Because a set of state variables is not unique, we can have an infinite variety of state-space descriptions of the same system. The use of an appropriate transformation allows us to see clearly which of the system states are controllable and which are observable.

REFERENCES

• 1. Kailath, Thomas. Linear Systems. Prentice-Hall, Englewood Cliffs, NJ, 1980.
• 2. Zadeh, L., and C. Desoer. Linear System Theory. McGraw-Hill, New York, 1963.

PROBLEMS

• 10.1-1 Convert each of the following second-order differential equations into a set of two first-order differential equations (state equations). State which of the sets represent nonlinear equations.
  • (a) y¨ +10y˙ +2y = x

1 ⁺ 1 H _

network in Fig. P10.2-2.

network in Fig. P10.2-3.

Figure P10.2-1

1 ² F

• (b) ^y¨ ⁺2eyy˙ ⁺log*^y* ^{= x}
• (c) y¨ +φ1(y)y˙ +φ2(y)y = x
• 10.2-1 Write the state equations for the RLC network in Fig. P10.2-1.

x 3

10.2-2 Write the state and output equations for the

10.2-3 Write the state and output equations for the

2

10.2-4 Write the state and output equations for the electrical network in Fig. P10.2-4.

Figure P10.2-4

Figure P10.2-2

10.2-5 Write the state and output equations for the network in Fig. P10.2-5.

10.2-6 Write the state and output equations of the system shown in Fig. P10.2-6.

Figure P10.2-6

10.2-7 Write the state and output equations of the system shown in Fig. P10.2-7.

Figure P10.2-7

10.2-8 For a system specified by the transfer function

$$H(s) = \frac{3s + 10}{s^2 + 7s + 12}$$

write sets of state equations for DFII and its transpose, cascade, and parallel forms. Also write the corresponding output equations.

10.2-9 Repeat Prob. 10.2-8 for

\n(a)

\n