State Space Control | Control Systems - Electrical Engineering (EE) PDF Download

 Page 1


  
ME 433 – STATE SPACE CONTROL 
  
MECHANICAL SYSTEM: Newton’s law 
Which are the equilibrium points when T
c
=0? 
At equilibrium: 
damping coefficient 
angular velocity 
angular acceleration 
moment of inertia 
Stable 
Unstable 
Dynamic Model 
Page 2


  
ME 433 – STATE SPACE CONTROL 
  
MECHANICAL SYSTEM: Newton’s law 
Which are the equilibrium points when T
c
=0? 
At equilibrium: 
damping coefficient 
angular velocity 
angular acceleration 
moment of inertia 
Stable 
Unstable 
Dynamic Model 
  
What happens around ?=0? 
By Taylor Expansion: 
Linearized Equation: 
y 
sin(y) 
Linearization 
 
Reduce to first order equations: 
State Variable 
Representation 
Is this state-space representation unique?  
State-variable Representation 
Page 3


  
ME 433 – STATE SPACE CONTROL 
  
MECHANICAL SYSTEM: Newton’s law 
Which are the equilibrium points when T
c
=0? 
At equilibrium: 
damping coefficient 
angular velocity 
angular acceleration 
moment of inertia 
Stable 
Unstable 
Dynamic Model 
  
What happens around ?=0? 
By Taylor Expansion: 
Linearized Equation: 
y 
sin(y) 
Linearization 
 
Reduce to first order equations: 
State Variable 
Representation 
Is this state-space representation unique?  
State-variable Representation 
  
What happens around ?=p? 
By Taylor Expansion: 
Linearized Equation: 
sin(x) 
Linearization 
 
Reduce to first order equations: 
State Variable 
Representation 
State-variable Representation 
Is this state-space representation unique?  
Page 4


  
ME 433 – STATE SPACE CONTROL 
  
MECHANICAL SYSTEM: Newton’s law 
Which are the equilibrium points when T
c
=0? 
At equilibrium: 
damping coefficient 
angular velocity 
angular acceleration 
moment of inertia 
Stable 
Unstable 
Dynamic Model 
  
What happens around ?=0? 
By Taylor Expansion: 
Linearized Equation: 
y 
sin(y) 
Linearization 
 
Reduce to first order equations: 
State Variable 
Representation 
Is this state-space representation unique?  
State-variable Representation 
  
What happens around ?=p? 
By Taylor Expansion: 
Linearized Equation: 
sin(x) 
Linearization 
 
Reduce to first order equations: 
State Variable 
Representation 
State-variable Representation 
Is this state-space representation unique?  
  
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! 
 
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,  
Page 5


  
ME 433 – STATE SPACE CONTROL 
  
MECHANICAL SYSTEM: Newton’s law 
Which are the equilibrium points when T
c
=0? 
At equilibrium: 
damping coefficient 
angular velocity 
angular acceleration 
moment of inertia 
Stable 
Unstable 
Dynamic Model 
  
What happens around ?=0? 
By Taylor Expansion: 
Linearized Equation: 
y 
sin(y) 
Linearization 
 
Reduce to first order equations: 
State Variable 
Representation 
Is this state-space representation unique?  
State-variable Representation 
  
What happens around ?=p? 
By Taylor Expansion: 
Linearized Equation: 
sin(x) 
Linearization 
 
Reduce to first order equations: 
State Variable 
Representation 
State-variable Representation 
Is this state-space representation unique?  
  
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! 
 
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,  
 
Let us assume that x(t) 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 
 
The overall solution can be written as 
Solution of State Equation 
At t=t 
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.