ch8

x1 (t ) a11 x1 (t ) a12 x2 (t ) a1N x N (t ) b1v(t ) x2 (t ) a21 x1 (t ) a 22 x2 (t ) a 2 N x N (t ) b2 v(t ) x N (t ) a N 1 x1 (t ) a N 2 x2 (t ) a NN x N (t ) bN v(t )
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Ch8. State Model Representation
Example 8.1 Consider the circuit in following figure. compute the currents in L1 and L2, and the voltage on C.
L1 x1 x3 R1 x1 e1 Let x1 be the current in L1, KVL : L2 x2 R2 x2 x3 e2 x2 be the current in L2, x3 be the voltage on C, KCL : Cx3 x1 x2 y R2 x2 e2
y(t ) f y(t ), v(t ), t
Defining the state x(t) of the system to be equal to y(t) results in the state model:
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Ch8. State Model Representation
The state equations describes the state response,while output equation gives the output response. The two parts correspond to cascade decomposition of the system as illustrated in the Figure below.
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Ch8. State Model Representation 3. Construction of State Models
Consider a single-input single-output continuous-time system given by the first-order input/output differential equation:
(1) (2)
In (1), A(t) is a N×N matrix, B(t) is an N-element column vector. In (2), C(t) is an N-element row vector and D(t) is a real-valued function of time. The number N of state variables is called the dimension of the state model (or system). If the system is time invariant, then A(t), B(t), C(t) and D(t) are constant.
x1 (t ) x (t ) 2 x (t ) x N (t )
The components x1(t),x2(t)….xN(t) are called the state variables of the system.
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Ch8. State Model Representation
If we get x1, x2, x3,we can get all the information about the system. So they are necessary and enough.
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Ch8. State Model Representation
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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Ch8. State Model Representation
Matrix form representaiton:
1 R1 1 0 0 L L L1 x1 1 1 1 x R2 1 1 e1 x2 0 x2 0 L L2 L2 e2 2 x3 x3 0 1 1 0 0 C C x1 e1 y 0 R2 0 x2 0 1 e2 x3
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Ch8. State Model Representation
1. State model
For a single-input single-output causal continuoustime system, its input : v(t) output: y(t) Consider the question: At a value t1 of the time variable t , is it possible to compute the output response y(t) from only the knowledge of the input v(t) for t t1 ? Obviously it is not. The reason is that the application of the input v(t) for t t1 may put energy into the system that affects the output response for t t1 .
2. State Equations
For a single-input single-output N-dimensional continuous-time system with state x(t) given by :
x1 (t ) x (t ) x (t ) 2 x N (t )
It can be modeled by the state equation given by :
derivative of the state vector
x(t ) f x(t ), v(t ),t equation
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Ch8. State Model Representation
For any time point t1, the state x(t) of the system at time t t1 is defined to be that portion of the past history t t1 of the system required to determine the output response y(t) for all t given the input v(t) t1 xat )time indicates the (t1 for A t1 t . t1nonzero state presence of energy in the system at time . t1 If the system is zero at t1, y(t) can be computed from v(t) fort t1 . If the system is not zero at t1, knowledge of the state is necessary to be able to compute the output y(t).
Ch8. State Model Representation
INTRODUCTION
There are two types of mathematical models of systems: input/output representation and statevariable 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.
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Ch8. State Model Representation
Rewrite the former equations, respectively, as
R1 1 1 x1 x1 x3 e1 L1 L1 L1 R2 1 1 2 x x2 x3 e2 L2 L2 L2 1 1 x3 x1 x2 C C y R2 x 2 e 2
Cascade structure corresponding to state model
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Ch8. State Model Representation
If f and g are linear, the state equations can be written in the form:
x(t ) A(t ) x(t ) B(t )v(t ) y(t ) C (t ) x(t ) D(t )v(t )
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Ch8. State Model Representation
In this case, the state model is given by: x(t ) Ax(t ) Bv(t )
(3) (4)
y(t ) Cx(t ) Dv(t )
With aij equal to the ij entry of A and bi equal to the ith component of B, (3) can be written in the expanded form :
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Ch8. State Model Representation
With C [c1 c2 cN ] , the expanded form of (4) is:
y(t ) c1x1 (t ) c2 x2 (t ) cN xN (t ) Dv(t )
From the expanded form of the state equations, it is seen that the derivative xi (t ) of the ith state variable and the output y(t) are equal to linear combinations of all the state variables and the input.