Control Engineering

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Control Engineering

• Lecture 2
• 9th Sep,2009

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Models of Physical Systems

• Two types of methods used in system modeling
• (i) Experimental method
• (ii) Mathematical method
• Design of engineering systems by trying and error
  versus design by using mathematical models.
• Mathematical model gives the mathematical
  relationships relating the output of a system to
  its input.

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Models of Electrical Circuits

• Resistance circuit v(t) i(t) R
• Inductance circuit

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Models of Electrical Circuits

• Capacitance circuit

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Models of Electrical Circuits

• Kirchhoff s voltage law
• The algebraic sum of voltages around any
  closed loop in an electrical circuit is zero.
• Kirchhoff s current law
• The algebraic sum of currents into any
  junction in an electrical circuit is zero.

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Models of Electrical Circuits

• Example

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Transfer Function

• Suppose we have a constant-coefficient linear
  differential equation with input f(t) and output
  x(t).
• After Laplace transform we have X(s)G(s)F(s)
• We call G(s) the transfer function.

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An Example

• Linear differential equation
• The Laplace transform is

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An Example

• Differential equation

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Characteristic Equation

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Block Diagram and Signal Flow Graphs

• Block diagram
• Signal flow graph is used to denote graphically
  the transfer function relationship

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• System interconnections
• Series interconnection
• Y(s)H(s)U(s) where H(s)H1(s)H2(s).
• Parallel interconnection
• Y(s)H(s)U(s) where H(s)H1(s)H2(s).

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• Feedback interconnection

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An Example

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• Parallel interconnection

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Another example

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Masons Gain Formula

• Motivation
• How to obtain the equivalent Transfer Function?
• Ans Masons formula

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Masons Gain Formula

• This gives a procedure that allows us to find the
  transfer function, by inspection of either a
  block diagram or a signal flow graph.
• Source Node signals flow away from the node.
• Sink node signals flow only toward the node.
• Path continuous connection of branches from one
  node to another with all arrows in the same
  direction.
• Forward path is a path that connects a source to
  a sink in which no node is encountered more than
  once.

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• Loop a closed path in which no node is
  encountered more than once. Source node cannot be
  part of a loop.
• Path gain product of the transfer functions of
  all branches that form the path.
• Loop gain products of the transfer functions of
  all branches that form the loop.
• Nontouching two loops are non-touching if these
  loops have no nodes in common.

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An Example

• Loop 1 (-G2H1) and loop 2 (-G4H2) are not
  touching.
• Two forward paths

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More Examples
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Another Example