Control Engineering Lecture #2 Lecture #2 9 th Sep, th Sep,2009

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.

Models of Electrical Circuits  Resistance circuit: v(t) = i(t) R  Inductance circuit: 

Models of Electrical Circuits  Capacitance circuit:

Models of Electrical Circuits  Kirchhoff’ s voltage law: The algebraic sum of voltages around any closed loop in an electrical circuit is zero. 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. The algebraic sum of currents into any junction in an electrical circuit is zero.

Models of Electrical Circuits  Example:

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.

An Example An Example  Linear differential equation  The Laplace transform is:

An Example An Example  Differential equation:

Characteristic Equation Characteristic Equation

Block Diagram and Signal Flow Graphs  Block diagram:  Signal flow graph is used to denote graphically the transfer function relationship:

 System interconnections  Series interconnection Y(s)=H(s)U(s) where H(s)=H 1 (s)H 2 (s). Y(s)=H(s)U(s) where H(s)=H 1 (s)H 2 (s).  Parallel interconnection Y(s)=H(s)U(s) where H(s)=H 1 (s)+H 2 (s). Y(s)=H(s)U(s) where H(s)=H 1 (s)+H 2 (s).

 Feedback interconnection

An Example An Example

 Parallel interconnection:

Another example :

Mason’s Gain Formula  Motivation: How to obtain the equivalent Transfer Function? Ans: Mason’s formula

Mason’s Gain Formula Mason’s 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.

 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.

An Example An Example  Loop 1 (-G 2 H 1 ) and loop 2 (-G 4 H 2 ) are not touching.  Two forward paths:

More Examples:

Another Example: