Page 378 - Schaum's Outline of Theory and Problems of Signals and Systems

Chapter 7

State Space Analysis

7.1 INTRODUCTION
So far we have studied linear time-invariant systems based on their input-output
relationships, which are known as the external descriptions of the systems. In this chapter
we discuss the method of state space representations of systems, which are known as the
internal descriptions of the systems. The representation of systems in this form has many
advantages:

1. It provides an insight into the behavior of the system.
2. It allows us to handle systems with multiple inputs and outputs in a unified way.
3. It can be extended to nonlinear and time-varying systems.
Since the state space representation is given in terms of matrix equations, the reader
should have some familiarity with matrix or linear algebra. A brief review is given in App.
A.

7.2 THE CONCEPT OF STATE
A. Definition:

The state of a system at time to (or no) is defined as the minimal information that is
sufficient to determine the state and the output of the system for all times t 2 to (or
n 2 no) when the input to the system is also known for all times t 2 to (or n 2 no). The
variables that contain this information are called the state variables. Note that this
definition of the state of the system applies only to causal systems.
Consider a single-input single-output LTI electric network whose structure is known.
Then the complete knowledge of the input x(t) over the time interval -m to t is sufficient
to determine the output y(t) over the same time interval. However, if the input x(t) is
known over only the time interval to to t, then the current through the inductors and the
voltage across the capacitors at some time to must be known in order to determine the
output y(t) over the time interval to to t. These currents and voltages constitute
the "state" of the network at time to. In this sense, the state of the network is related to
the memory of the network.

B. Selection of State Variables:
Since the state variables of a system can be interpreted as the "memory elements" of
the system, for discrete-time systems which are formed by unit-delay elements, amplifiers,
and adders, we choose the outputs of the unit-delay elements as the state variables of the
system (Prob. 7.1). For continuous-time systems which are formed by integrators, ampli-
fiers, and adders, we choose the outputs of the integrators as the state variables of the
system (Prob. 7.3). For a continuous-time system containing physical energy-storing ele-