Introduction to State Space Design

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Introduction to State Space Design
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Definitions State The state of a dynamic
system is the smallest set of variables (called
state variables) such that the knowledge of these
variables at t t0, together with the knowledge
of the input for t gt t0, completely determines
the behavior of the system for any time t gt
t0. The concept of a state is applicable to
mechanical systems and other systems such as
biological, economical, social systems.
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State variables The state variables of a dynamic
system are the variables making up the smallest
set of variables that determine the state of the
dynamic system. State Space The n-dimensional
space whose coordinate axes consist of the
x1-axis, x2-axis, , xn-axis, where x1, x2, , xn
are the state variables, is called a state space.
Any state can be represented by a point in this
state space.
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The concept of the state of a dynamic system

• State
• Cart position
• Cart velocity
• Angle of the pendulum
• Angular velocity of the pendulum

Inverted pendulum
Free joint
Cart
Force
Wheel
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The concept of the state vector
We write the coordinates of the tip of the vector
combining the origin of the state space and the
current state of the system
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State variable 3 x3
t 1
t 0
t 4
State vectors at different times
State trajectory
t 2
t 3
(0, 0, 0)
State variable 1 (x1)
Origin of the state space
State variable 2 x2
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State variable 3 x3
The concept of control in a state space point of
view
Desired state trajectory
t 1
t 0
t 4
State vectors at different times
State trajectory
t 2
t 3
(0, 0, 0)
State variable 1 (x1)
Origin of the state space
State variable 2 x2
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State space representation of a linear dynamical
system
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• Applying the state space representation
• Minimum number of state variables that describe
  the state of the system completely
• Components of the state vector should be linearly
  independent.
• How do we estimate?
• The order of the differential equation
• The number of independent energy storage elements
  of a system

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RL network
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RLC network
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Converting a transfer function to a state space
model
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State space model
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Phase variable form
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Conversion to controller canonical form
First renumber the phase variables in reverse
order
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Controller canonical form
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Observer canonical form
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Observer canonical form from the controller
canonical form
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a. State-spacerepresentationof a plantb.
plant with state-feedback
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Design a linear state-feedback controller to
yield 20 overshoot and a settling time of 2
seconds for a plant
Represent the system in phase variable form
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Check whether it is controllable or not using the
controlability matrix
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Controllability matrix
Is it controllable? Check the determinant
Then consider the desired properties and
calculate the damping ratio and undamped natural
frequency
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Add a pole at -6 to compensate for the zero and
calculate the desired characteristic polynomial
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Next calculate the characteristic polynomial of
the compensated system in the form
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The compensated system
The characteristic polynomial
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Compare the desired polynomial values with those
of the compensated system and find the gains
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Observer design
Why do we need observers? Because in practice we
can not measure all the states for state feedback
control. This is due to the unjustifiable cost
of some sensors especially used for measurement
of velocity and acceleration like Gyros. Another
reason is that the accuracy of these instruments
heavily depends on the alignment that can
deteriorate over time. Therefore, a better way
would be to use other states to Observe the
states that can not be measured.
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Conceptual state-space designconfiguration,
showingplant, observer, and controller
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Observability
A system is said to be completely observable if
every state x(t0) can be determined from the
observation of y(t) over a finite time interval
t0 lt t lt t1. The system is therefore completely
observable if every transition of the state
eventually affects every element of the output
vector.
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Observability matrix
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Comparison ofa. observable andb.
unobservablesystems
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State space representation of a linear dynamical
system Without observed state variables
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The observer model
Subtracting this model from the state space model
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Subtracting this model from the state space model
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Simulation showingresponse of observera.
closed-loopb. open-loop withobserver
gainsdisconnected
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Design an observer for the plant given bellow so
that the observer will respond 10 times faster
than the controller designed in the previous
exercise.
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Conceptual state-space designconfiguration,
showingplant, observer, and controller