# System models handout

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EE 3CL4, §2 / 97 Tim Davidson Modelling physical systems Trans Newton Mech Rot Newton Mech Linearization Laplace transforms EE3CL4: Introduction to Linear Control Systems Section 2: System Models Laplace in action Transfer function Tim Davidson Step response Transfer fn of DC motor Our first control system design Block diagram models Block dia transform McMaster University Winter 2016 EE 3CL4, §2 / 97 Outline Tim Davidson Modelling physical systems Trans Newton Mech Rot Newton Mech Linearization Laplace transforms Laplace in action Transfer function Modelling physical systems Translational Newtonian Mechanics Rotational Newtonian Mechanics Linearization Laplace transforms Laplace transforms in action Transfer function Step response Transfer fn of DC motor Our first control system design Block diagram models Block dia transform Step response Transfer function of DC motor Our first control system design Block diagram models Block diagram transformations EE 3CL4, §2 / 97 Tim Davidson Differential equation models Modelling physical systems Trans Newton Mech Rot Newton Mech Linearization Laplace transforms Laplace in action • Most of the systems that we will deal with are dynamic • Differential equations provide a powerful way to describe dynamic systems • Will form the basis of our models Transfer function Step response Transfer fn of DC motor Our first control system design Block diagram models Block dia transform • We saw differential equations for inductors and capacitors in 2CI, 2CJ • What about mechanical systems? both translational and rotational EE 3CL4, §2 / 97 Tim Davidson Translational Spring Modelling physical systems Trans Newton Mech Rot Newton Mech Linearization Laplace transforms F (t): resultant force in direction x Recall free body diagrams and “action and reaction” • Spring k : spring constant, Lr : relaxed length of spring Laplace in action Transfer function Step response Transfer fn of DC motor Our first control system design Block diagram models Block dia transform F (t) = k [x2 (t) − x1 (t)] − Lr EE 3CL4, §2 / 97 Translational Damper Tim Davidson Modelling physical systems Trans Newton Mech F (t): resultant force in direction x Rot Newton Mech Linearization Laplace transforms • Viscous damper b: viscous friction coefficient Laplace in action Transfer function Step response Transfer fn of DC motor Our first control system design Block diagram models Block dia transform F (t) = b dx2 (t) dx1 (t) − = b v2 (t) − v1 (t) dt dt EE 3CL4, §2 / 97 Mass Tim Davidson Modelling physical systems Trans Newton Mech F (t): resultant force in direction x Rot Newton Mech Linearization Laplace transforms • Mass: M Laplace in action Transfer function Step response Transfer fn of DC motor Our first control system design Block diagram models Block dia transform F (t) = M d xm (t) dvm (t) =M = Mam (t) dt dt EE 3CL4, §2 / 97 Tim Davidson Rotational spring Modelling physical systems Trans Newton Mech T (t): resultant torque in direction θ Rot Newton Mech Linearization Laplace transforms Laplace in action • Rotational spring k : rotational spring constant, φr : rotation of relaxed spring Transfer function Step response Transfer fn of DC motor Our first control system design Block diagram models Block dia transform T (t) = k [θ2 (t) − θ1 (t)] − φr EE 3CL4, §2 / 97 Rotational damper Tim Davidson Modelling physical systems T (t): resultant torque in direction θ Trans Newton Mech Rot Newton Mech Linearization Laplace transforms • Rotational viscous damper b: rotational viscous friction coefficient Laplace in action Transfer function Step response Transfer fn of DC motor Our first control system design Block diagram models Block dia transform T (t) = b dθ2 (t) dθ1 (t) − = b ω2 (t) − ω1 (t) dt dt EE 3CL4, §2 10 / 97 Rotational inertia Tim Davidson Modelling physical systems Trans Newton Mech T (t): resultant torque in direction θ Rot Newton Mech Linearization Laplace transforms • Rotational inertia: J Laplace in action Transfer function Step response Transfer fn of DC motor Our first control system design Block diagram models Block dia transform T (t) = J d θm (t) dωm (t) =J = Jαm (t) dt dt EE 3CL4, §2 11 / 97 Tim Davidson Modelling physical systems Example system (translational) Horizontal Origin for y : y = when spring relaxed Trans Newton Mech Rot Newton Mech Linearization Laplace transforms Laplace in action Transfer function Step response Transfer fn of DC motor Our first control system design • F = M dvdt(t) • v (t) = dydt(t) • F (t) = r (t) − b dydt(t) − ky (t) Block diagram models Block dia transform M d y (t) dy (t) +b + ky (t) = r (t) dt dt EE 3CL4, §2 81 / 97 Time constants Tim Davidson Modelling physical systems Trans Newton Mech Rot Newton Mech • Initial model Linearization Laplace transforms G(s) = 5000 s(s + 20)(s + 1000) Laplace in action Transfer function • Motor time constant = 1/20 = 50ms Step response • Armature time constant = 1/1000 = 1ms Transfer fn of DC motor • Hence Our first control system design Block diagram models Block dia transform ˆ G(s) ≈ G(s) = s(s + 20) EE 3CL4, §2 83 / 97 Tim Davidson Modelling physical systems Trans Newton Mech A simple feedback controller Now that we have a model, how to control? Simple idea: Apply voltage to motor that is proportional to error between where we are and where we want to be Rot Newton Mech Linearization Laplace transforms Laplace in action Transfer function Step response Transfer fn of DC motor Our first control system design Block diagram models Block dia transform Here, V (s) = Va (s) and Y (s) = θ(s) EE 3CL4, §2 84 / 97 Tim Davidson Simplified block diagram Modelling physical systems Trans Newton Mech Rot Newton Mech Linearization Laplace transforms Laplace in action Transfer function Step response Transfer fn of DC motor Our first control system design Block diagram models • What is the transfer function from command to position? Derive this yourself Ka G(s) Y (s) = R(s) + Ka G(s) ˆ • Using second-order approx G(s) ≈ G(s) = s(s+20) , Y (s) ≈ s2 5Ka R(s) + 20s + 5Ka Block dia transform • For < Ka < 20: overdamped; for Ka > 20: underdamped EE 3CL4, §2 85 / 97 Tim Davidson Modelling physical systems Trans Newton Mech Rot Newton Mech Response to r (t) = 0.1u(t); Ka = 10 Poles in s-plane Response Linearization Laplace transforms Laplace in action Transfer function Step response Transfer fn of DC motor Our first control system design Block diagram models Block dia transform Slow Slower than IBMs first drive from late 1950’s Disks in the 1970’s had 25ms seek times; now < 10ms Perhaps increase Ka ? That would result in a “bigger” input to the motor for a given error EE 3CL4, §2 86 / 97 Tim Davidson Modelling physical systems Response to r (t) = 0.1u(t); Ka = 10, 15 Trans Newton Mech Rot Newton Mech Linearization Poles in s-plane Response Laplace transforms Laplace in action Transfer function Step response Transfer fn of DC motor Our first control system design Block diagram models Block dia transform Changing Ka changes the position of the closed-loop poles Hence, step response changes EE 3CL4, §2 87 / 97 Tim Davidson Modelling physical systems Response to r (t) = 0.1u(t); Ka = 10, 15, 20 Trans Newton Mech Rot Newton Mech Linearization Poles in s-plane Response Laplace transforms Laplace in action Transfer function Step response Transfer fn of DC motor Our first control system design Block diagram models Block dia transform Changing Ka changes the position of the closed-loop poles Hence, step response changes (now critically damped) EE 3CL4, §2 88 / 97 Tim Davidson Modelling physical systems Response to r (t) = 0.1u(t); Ka = 10, 15, 20, 40 Trans Newton Mech Rot Newton Mech Linearization Poles in s-plane Response Laplace transforms Laplace in action Transfer function Step response Transfer fn of DC motor Our first control system design Block diagram models Block dia transform Changing Ka changes the position of the closed-loop poles Hence, step response changes (now underdamped) EE 3CL4, §2 89 / 97 Tim Davidson Modelling physical systems Response to r (t) = 0.1u(t); Ka = 10, 15, 20, 40, 60 Trans Newton Mech Rot Newton Mech Linearization Poles in s-plane Response Laplace transforms Laplace in action Transfer function Step response Transfer fn of DC motor Our first control system design Block diagram models Block dia transform Changing Ka changes the position of the closed-loop poles Hence, step response changes (now more underdamped) EE 3CL4, §2 90 / 97 Tim Davidson Modelling physical systems Response to r (t) = 0.1u(t); Ka = 10, 15, 20, 40, 60, 80 Trans Newton Mech Rot Newton Mech Linearization Poles in s-plane Response Laplace transforms Laplace in action Transfer function Step response Transfer fn of DC motor Our first control system design Block diagram models Block dia transform What is happening to the settling time of the underdamped cases? Only just beats IBM’s first drive What else could we with the controller? Prediction? EE 3CL4, §2 92 / 97 Tim Davidson Modelling physical systems Trans Newton Mech Rot Newton Mech Bock diagram models • As we have just seen, a convenient way to represent a transfer function is via a block diagram Linearization Laplace transforms Laplace in action Transfer function Step response Transfer fn of DC motor • In this case, U(s) = Gc (s)R(s) and Y (s) = G(s)U(s) • Hence, Y (s) = G(s)Gc (s)R(s) Our first control system design • Consistent with the engineering procedure of breaking Block diagram models things up into little bits, studying the little bits, and then put them together Block dia transform EE 3CL4, §2 93 / 97 Tim Davidson Simple example Modelling physical systems Trans Newton Mech Rot Newton Mech Linearization Laplace transforms Laplace in action Transfer function Step response Transfer fn of DC motor Our first control system design Block diagram models Block dia transform • Y1 (s) = G11 (s)R1 (s) + G12 (s)R2 (s) • Y2 (s) = G21 (s)R1 (s) + G22 (s)R2 (s) EE 3CL4, §2 94 / 97 Tim Davidson Example: Loop transfer function Modelling physical systems Trans Newton Mech Rot Newton Mech Linearization Laplace transforms Laplace in action Transfer function • Ea (s) = R(s) − B(s) = R(s) − H(s)Y (s) • Y (s) = G(s)U(s) = G(s)Ga (s)Z (s) Step response • Y (s) = G(s)Ga (s)Gc (s)Ea (s) Transfer fn of DC motor • Y (s) = G(s)Ga (s)Gc (s) R(s) − H(s)Y (s) Our first control system design Block diagram models Block dia transform Y (s) G(s)Ga (s)Gc (s) = R(s) + G(s)Ga (s)Gc (s)H(s) • Each transfer function is a ratio of polynomials in s • What is Ea (s)/R(s)? EE 3CL4, §2 95 / 97 Tim Davidson Modelling physical systems Trans Newton Mech Rot Newton Mech Linearization Laplace transforms Laplace in action Transfer function Step response Transfer fn of DC motor Our first control system design Block diagram models Block dia transform Block diagram transformations EE 3CL4, §2 96 / 97 Tim Davidson Modelling physical systems Trans Newton Mech Rot Newton Mech Linearization Laplace transforms Laplace in action Transfer function Step response Transfer fn of DC motor Our first control system design Block diagram models Block dia transform Using block diagram transformations EE 3CL4, §2 97 / 97 Tim Davidson Modelling physical systems Trans Newton Mech Rot Newton Mech Linearization Laplace transforms Laplace in action Transfer function Step response Transfer fn of DC motor Our first control system design Block diagram models Block dia transform Using block diagram transformations
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