ch8 State Model Rereferens.ppt

There are two types of mathematical models of systems: input/output representation and state-variable representation.
The input/output representation describes the input/output behavior of systems.
The state-variable representation describes the internal behavior of systems.
The objective of this chapter: define the state model and study the basic properties of this model for both continuous-time and discrete-time systems.
INTRODUCTION
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1. State model
For a single-input single-output causal continuous-time system, its
input : v(t)
output: y(t)
Obviously it is not. The reason is that the application of the input v(t) for may put energy into the system that affects the output response for .
Consider the question:
At a value t1 of the time variable t , is it possible pute the output response y(t) from only the knowledge of the input v(t) for ?
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If the system is zero at t1, y(t) can puted from v(t) for .
If the system is not zero at t1, knowledge of the state is necessary to be able pute the output y(t).
For any time point t1, the state x(t) of the system at time is defined to be that portion of the past history of the system required to determine the output response y(t) for all given the input v(t) for . A nonzero state at time indicates the presence of energy in the system at time .
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Example Consider the circuit in following figure. compute the currents in L1 and L2, and the voltage on C.
Let x1 be the current in L1, x2 be the current in L2,
x3 be the voltage on C,
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Rewrite the former equations, respectively, as
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If we get x1, x2, x3,we can get all the information about the system. So they are necessary and enough.
Matrix form representaiton:
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ponents x1(t),x2(t)….xN(t) are called the state variables of the system.
From the example, if the given system is finite dimensional, the state x(t) of the system at time t is an N-element column vector given by:
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2. State Equations
For a single-input single-output N-dimensional conti