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Chap. 6 State-Space Topics for Continuous
Systems
State-Space Model
If a LTI system with M inputs , K outputs and N states is described by
state-model, then we have
State Equation:
Output Equation:
where
: the state vector consists of N state
variables,
: the input vector consists of N input
variables, and
: the output vector consists of K
output variables.
:
matrix of constants.
:
matrix of constants.
:
matrix of constants.
:
matrix of constants.
Initial Condition Solution of the State Equation
Example:
Likewise, for any N, the general IC solution is
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Note:
Define
, such that
.
Problem 6.1:
.
A=[-1 0 ; 0 -2];
v0=[4;-2.5];
tspan=[0 1];
df=@(t,v) A*v;
[t vv]=ode45(df,tspan,v0);
plot(t,vv);
Complete Solution of State Equation
Multiply the state equation by
Therefore,
Integrate from 0 to t, we have
and rearrange, we have
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Problem 6.2:
a.
Find the complete solution.
b.
If
, find
.
Stability
The initial condition response of a stable causal system
approaches zero as
. That is,
.
Problem 6.3: Check the stability of the following A matrix
a.
b.
c.
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Numerical Evaluation of the State Equation
Let
be a small value,
. Then
Similarly,
This is call state difference equation.
Problem 6.4: Evaluate
and
of Problem 6.2.
Problem 6.5: For the state model of Problem 6.2, let
,
and
, find
. Compare the result
with analytical solution.
Solution of the State Equation by Laplace System
Take the Laplace transform of the state equation, we have
where
.
Taking inverse Laplace transform, we have
where
is called transition matrix.
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Problem 6.6: Find the transition matrix of Problem 6.2.
Characteristic Equation and Eigenvalues
Characteristic Equation: the determinant of the matrix
, that is
Or factor into its root as
where
are the eigenvalues.
For stable system,
.
Problem 6.7:
a.
b.
c.
Find the characteristic equation.
Find the eigenvalues. Use MATLAB function eig to verify. Is
the system stable?
Find
if
and
.
d.
Using residue to find
e.
Find
.
Matrix of Transfer Functions
If no initial condition, i. e.,
From
.
, then,
, we have
,
Therefore, the transfer function, now in matrix form, is
Problem 6.8: A band pass filter is described by the state equation
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.
a.
b.
Find the transfer function matrix.
Find the frequency response.
Transfer Function to State Equation
For Single-Input-Single-Output (SISO) LTI system,
Let
.
Likewise,
.
Let
.
Then,
Similarly, if
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If
Problem 6.11: Find the state-space representation of the transfer
function
Single-Input-Multiple-Output (SIMO) System
If there are more than one output, then
, where
.
.
Problem 6.12: Find the state and output equation of the following
transfer function.
Example 6.1: A causal LTI system is described by the state and
output equations
,
with
a.
b.
c.
d.
e.
f.
,
and
.
Find
due to the initial condition only.
Find the transfer function.
What are the system poles? Is the system stable?
Find
due to the input only.
Use MATLAB ode45 to check your analytical results.
If
, is the stability of the system affected?
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Example 6.2: Two tanks are connected as shown in Fig. E6.2a. The
flow rates of the inputs to the tanks are
,
and
. The
fluid levels are
is described by
and
. The follow rate between the two tanks
. The pertinent equations are
where is the area of both tanks.
a.
Find the state space equations with
b. Given that
,
and
c.
d.
e.
the only output.
, solve for
.
Use
, find
and use MATLAB to verify b.
For zero inputs, find the time when the two fluid levels become
the same. Compare your analytical result with MATLAB using
c.
If
and
, use MATLAB to find
.