Lecture #9 Control Engineering REVIEW SLIDES Reference: Textbook by Phillips and Habor

Mathematical Modeling

Models of Electrical Systems R-L-C series circuit, impulse voltage source: R-L-C series circuit, impulse voltage source:

Model of an RLC parallel circuit: Model of an RLC parallel circuit:

Kirchhoff’ s voltage law: 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: 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 Mechanical Systems Mechanical translational systems. Mechanical translational systems. Newton’s second law: Newton’s second law: Device with friction (shock absorber): Device with friction (shock absorber): B is damping coefficient. B is damping coefficient. Translational system to be defined is a spring (Hooke’s law): Translational system to be defined is a spring (Hooke’s law): K is spring coefficient K is spring coefficient

Model of a mass-spring-damper system: Model of a mass-spring-damper system: Note that linear physical systems are modeled by linear differential equations for which linear components can be added together. See example of a mass-spring-damper system. Note that linear physical systems are modeled by linear differential equations for which linear components can be added together. See example of a mass-spring-damper system.

Simplified automobile suspension system: Simplified automobile suspension system:

Mechanical rotational systems. Mechanical rotational systems. Moment of inertia: Moment of inertia: Viscous friction: Viscous friction: Torsion: Torsion:

Model of a torsional pendulum (pendulum in clocks inside Model of a torsional pendulum (pendulum in clocks inside glass dome); glass dome); Moment of inertia of pendulum bob denoted by J Moment of inertia of pendulum bob denoted by J Friction between the bob and air by B Friction between the bob and air by B Elastance of the brass suspension strip by K Elastance of the brass suspension strip by K

Differential equations as mathematical models of physical systems: similarity between mathematical models of electrical circuits and models of simple mechanical systems (see model of an RCL circuit and model of the mass-spring-damper system). Differential equations as mathematical models of physical systems: similarity between mathematical models of electrical circuits and models of simple mechanical systems (see model of an RCL circuit and model of the mass-spring-damper system).

Laplace Transform

Name Time function f(t) Laplace Transform Unit Impulse  (t) 1 Unit Step u(t)1/s Unit ramp t 1/s 2 nth-Order ramp t n n!/s n+1 Exponential e -at 1/(s+a) nth-Order exponential t n e -at n!/(s+a) n+1 Sine sin(bt) b/(s 2 +b 2 ) Cosine cos(bt) s/(s 2 +b 2 ) Damped sine e -at sin(bt) b/((s+a) 2 +b 2 ) Damped cosine e -at cos(bt) (s+a)/((s+a) 2 +b 2 ) Diverging sine t sin(bt) 2bs/(s 2 +b 2 ) 2 Diverging cosine t cos(bt) (s 2 -b 2 ) /(s 2 +b 2 ) 2

Find the inverse Laplace transform of F(s)=5/(s 2 +3s+2). Solution: Solution:

Find inverse Laplace Transform of

Find the inverse Laplace transform of F(s)=(2s+3)/(s 3 +2s 2 +s). Solution: Solution:

Laplace Transform Theorems

Transfer Function

After Laplace transform we have X(s)=G(s)F(s) After Laplace transform we have X(s)=G(s)F(s) We call G(s) the transfer function. We call G(s) the transfer function.

System interconnections System interconnections Series interconnection 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 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 Feedback interconnection

Transfer function of a servo motor: Transfer function of a servo motor:

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. 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. Source Node: signals flow away from the node. Sink node: signals flow only toward 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. Path: continuous connection of branches from one node to another with all arrows in the same direction.

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

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

State Variable System:

Solutions of state equations: Solutions of state equations:

Responses

System Responses (Time Domain) First order systems: First order systems: Transient response Transient response Steady state response Steady state response Step response Step response Ramp response Ramp response Impulse response Impulse response Second order systems Second order systems Transient response Transient response Steady state response Steady state response Step response Step response Ramp response Ramp response Impulse response Impulse response

Time Responses of first order systems The T.F. for first order system:

is called the time constant is called the time constant Ex. Position control of the pen of a plotter for a digital computer: is too slow, is faster. Ex. Position control of the pen of a plotter for a digital computer: is too slow, is faster.

System DC Gain System DC Gain In general: In general: