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Transcript
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: Model of an RLC parallel circuit: 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. Models of Mechanical Systems Mechanical translational systems. Newton’s second law: Device with friction (shock absorber): B is damping coefficient. Translational system to be defined is a spring (Hooke’s law): K is spring coefficient 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. Simplified automobile suspension system: Mechanical rotational systems. Moment of inertia: Viscous friction: Torsion: Model of a torsional pendulum (pendulum in clocks inside glass dome); Moment of inertia of pendulum bob denoted by J Friction between the bob and air by B 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). Laplace Transform Name Unit Impulse Unit Step Unit ramp nth-Order ramp Exponential Time function f(t) (t) u(t) t tn e-at nth-Order exponential t n e-at Sine sin(bt) Cosine cos(bt) Damped sine e-at sin(bt) Damped cosine e-at cos(bt) Diverging sine t sin(bt) Diverging cosine t cos(bt) Laplace Transform 1 1/s 1/s2 n!/sn+1 1/(s+a) n!/(s+a)n+1 b/(s2+b2) s/(s2+b2) b/((s+a)2+b2) (s+a)/((s+a)2+b2) 2bs/(s2+b2)2 (s2-b2) /(s2+b2)2 Find the inverse Laplace transform of F(s)=5/(s2+3s+2). Solution: Find inverse Laplace Transform of Find the inverse Laplace transform of F(s)=(2s+3)/(s3+2s2+s). Solution: Laplace Transform Theorems Transfer Function Transfer Function After Laplace transform we have X(s)=G(s)F(s) We call G(s) the transfer function. 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). Feedback interconnection 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. 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. 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. 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 Loop 1 (-G2H1) and loop 2 (-G4H2) are not touching. Two forward paths: State Variable System: Solutions of state equations: Responses System Responses (Time Domain) First order systems: Transient response Steady state response Step response Ramp response Impulse response Second order systems Transient response Steady state response Step response Ramp response Impulse response Time Responses of first order systems The T.F. for first order system: b0 Y ( s) K G ( s) R( s ) s a0 s 1 a0 1 / b0 K / themeaningof and K will be clear below. In general, with initialcondition y0 : y0 (K / ) Y ( s) R( s) s (1 / ) s (1 / ) Unit step response: R(s) 1/ s (K / ) K K Y ( s) s[ s (1 / )] s s (1 / ) y (t ) K (1 e t / ), t 0 y (t ) K (1 e t / )u (t ) First term steady - stateresponse(from thepole of input R(s)). Second term natural(or transient)response(originatefrom thepole of theT .F.) is called the time constant Ex. Position control of the pen of a plotter for a digital computer: 1sec is too slow, 0.1sec is faster. lim t y (t ) lim s 0 sY ( s ) K K steady - state response and time constant for general first order T.F. System DC Gain In general: limt y (t ) lim s 0 sY ( s ) lim s 0 sG ( s ) R( s ) For unit step input : R(s) 1 / s limt y (t ) lim s 0 G ( s ) G (0). T hereforeG (0) is thesteady - stategain for constant (unit step)input.T hisis true,independent of theorder of thesystem.Hence G(0) is called theDC gain.