Control EngineeringControl Engineering

Lecture #2Lecture #2 99thth Sep,2009 Sep,2009

Models of Physical SystemsModels of Physical Systems

Two types of methods used in system modeling:Two types of methods used in system modeling: (i) Experimental method(i) Experimental method (ii) Mathematical method(ii) Mathematical method Design of engineering systems by trying and error Design of engineering systems by trying and error

versus design by using mathematical models.versus design by using mathematical models. Mathematical model gives the mathematical Mathematical model gives the mathematical

relationships relating the output of a system to its relationships relating the output of a system to its input.input.

Models of Electrical CircuitsModels of Electrical Circuits Resistance circuit: Resistance circuit: v(t) = i(t) Rv(t) = i(t) R

Inductance circuit: Inductance circuit:

Models of Electrical CircuitsModels of Electrical Circuits Capacitance circuit: Capacitance circuit:

Models of Electrical CircuitsModels of Electrical Circuits

Kirchhoff’ s voltage law:Kirchhoff’ s voltage law:

The algebraic sum of voltages around any The algebraic sum of voltages around any closed loop in an electrical circuit is zero.closed loop in an electrical circuit is zero.

Kirchhoff’ s current law:Kirchhoff’ s current law:

The algebraic sum of currents into any The algebraic sum of currents into any junction in an electrical circuit is zero.junction in an electrical circuit is zero.

Models of Electrical CircuitsModels of Electrical Circuits Example: Example:

Transfer FunctionTransfer Function

Suppose we have a constant-coefficient Suppose we have a constant-coefficient linear differential equation with input linear differential equation with input f(t)f(t) and and output output x(t).x(t).

After Laplace transform we have After Laplace transform we have X(s)=G(s)F(s)X(s)=G(s)F(s)

We call We call G(s)G(s) the the transfer functiontransfer function..

An ExampleAn Example

Linear differential equationLinear differential equation

The Laplace transform is:The Laplace transform is:

An ExampleAn Example Differential equation:Differential equation:

Characteristic EquationCharacteristic Equation

Block Diagram and Signal Flow Block Diagram and Signal Flow GraphsGraphs

Block diagram:Block diagram:

Signal flow graph is used to denote graphically the transfer Signal flow graph is used to denote graphically the transfer function relationship:function relationship:

System interconnectionsSystem interconnections Series interconnectionSeries interconnection

Y(s)=H(s)U(s)Y(s)=H(s)U(s) where where H(s)=HH(s)=H11(s)H(s)H22(s).(s).

Parallel interconnectionParallel interconnection

Y(s)=H(s)U(s)Y(s)=H(s)U(s) where where H(s)=HH(s)=H11(s)+H(s)+H22(s).(s).

Feedback interconnectionFeedback interconnection

An ExampleAn Example

Parallel interconnection:Parallel interconnection:

Another exampleAnother example: :

Mason’s Gain FormulaMason’s Gain Formula

Motivation:Motivation:

How to obtain the equivalent Transfer Function?How to obtain the equivalent Transfer Function?

Ans: Mason’s formulaAns: Mason’s formula

Mason’s Gain FormulaMason’s Gain Formula

This gives a procedure that allows us to find the This gives a procedure that allows us to find the transfer function, by inspection of either a block transfer function, by inspection of either a block diagram or a signal flow graph.diagram or a signal flow graph.

Source NodeSource Node: signals flow away from the node.: signals flow away from the node. Sink nodeSink node: signals flow only toward the node.: signals flow only toward the node. PathPath: continuous connection of branches from one : continuous connection of branches from one

node to another with all arrows in the same direction.node to another with all arrows in the same direction. Forward pathForward path: is a path that connects a source to a : is a path that connects a source to a

sink in which no node is encountered more than sink in which no node is encountered more than once.once.

LoopLoop: a closed path in which no node is : a closed path in which no node is encountered more than once. Source node encountered more than once. Source node cannot be part of a loop.cannot be part of a loop.

Path gainPath gain: product of the transfer functions of : product of the transfer functions of all branches that form the path.all branches that form the path.

Loop gainLoop gain: products of the transfer functions : products of the transfer functions of all branches that form the loop.of all branches that form the loop.

NontouchingNontouching: two loops are non-touching if : two loops are non-touching if these loops have no nodes in common.these loops have no nodes in common.

An ExampleAn Example

Loop 1 Loop 1 (-G(-G22HH11) and loop 2 (-) and loop 2 (-GG44HH22) are not ) are not

touching.touching. Two forward paths:Two forward paths:

More Examples:More Examples:

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1

43211

34321143232

343213

1432

2321

43211

)(

1

1

GGGGsG

GGGGM

HGGGGHGGHGG

HGGGGL

HGGL

HGGL

GGGGP

Another Example:Another Example: