ME 433 – STATE SPACE CONTROLeus204/teaching/ME433/... · 5 ME 433 - State Space Control 39 Let us...
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31 ME 433 - State Space Control
ME 433 – STATE SPACE CONTROL
Lecture 3
32 ME 433 - State Space Control
MECHANICAL SYSTEM: Newton’s law
Which are the equilibrium points when Tc=0?
At equilibrium:
damping coefficient
angular velocity
angular acceleration
moment of inertia
Stable
Unstable
Dynamic Model
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33 ME 433 - State Space Control
What happens around θ=0?
By Taylor Expansion:
Linearized Equation:
y
sin(
y)
Linearization
34 ME 433 - State Space Control
Reduce to first order equations:
State Variable Representation
Is this state-space representation unique?
State-variable Representation
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35 ME 433 - State Space Control
What happens around θ=π?
By Taylor Expansion:
Linearized Equation:
sin(
x)
Linearization
36 ME 433 - State Space Control
Reduce to first order equations:
State Variable Representation
State-variable Representation
Is this state-space representation unique?
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37 ME 433 - State Space Control
We consider the linear, time-invariant system
We define the state transformation
Then we can write
to obtain
State Transformation
The state-space representation is NOT unique!
38 ME 433 - State Space Control
We consider the linear, time-invariant, homogeneous system
Time-invariant Dynamics:
where A is a constant n×n matrix. The solution can be written as
Solution of State Equation
where
We can note that
Then,
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39 ME 433 - State Space Control
Let us assume that x(τ) is known. Then,
Solution of State Equation
The homogeneous solution can be finally written as
We consider now the linear, time-invariant, non-homogeneous system
We assume a “particular” solution of the form
Then,
and
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The overall solution can be written as
Solution of State Equation
At t=τ
We finally can write the solution to the state equation as
and the system output as
Note that B, C and D can be functions of time.
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41 ME 433 - State Space Control
We consider the linear, time-variant, homogeneous system
Time-variant Dynamics:
where A is a time-variant n×n matrix. The solution can be written as
Solution of State Equation
where Φ(t,τ) is known as the “state-transition” matrix. For time-invariant systems, the state transition matrix is only function of t-τ,
We can write the state and output as
42 ME 433 - State Space Control
Solution of State Equation Properties of the state transition matrix:
For time-invariant systems:
NOTE: only if .
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43 ME 433 - State Space Control
We consider the linear, time-invariant system
And we solve to obtain
Solution by the Laplace Transform:
We Laplace transform the state equation to obtain
Solution of State Equation
44 ME 433 - State Space Control
We inverse Laplace transform to obtain
where we have used that
Solution of State Equation
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45 ME 433 - State Space Control
The transfer function does NOT depend on the state choice because it represents the input-output relationship.
Proof: In class
State Transformation
We Laplace transform the output equation to obtain
Assuming x(0)=0 and combining state and output equations we obtain the transfer function
Given two state space representations
46 ME 433 - State Space Control
Scalar Differential Equation
Transfer Function
State Variable Representation
Transfer Function
Model Representation
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47 ME 433 - State Space Control
Example:
Find: - Transfer function between q(t) and u(t) - Scalar ODE for q(t)
Scalar Differential Equation
State Variable Representation
Transfer Function
Transfer Function
Model Representation
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Example:
Find: - State variable representation
Scalar Differential Equation
State Variable Representation
Model Representation
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49 ME 433 - State Space Control
Example:
Find: - State variable representation
Transfer function
State Variable Representation
Model Representation
50 ME 433 - State Space Control
Example: Find: - State variable representation
Transfer function
State Variable Representation
Model Representation
Scalar Differential Equation
State Variable Representation
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51 ME 433 - State Space Control
Choosing
Model Representation
Controller Form
52 ME 433 - State Space Control
Choosing
Model Representation
Observer Form
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53 ME 433 - State Space Control
Choosing
Model Representation
Modal Form