EE 3CL4, §2 1 / 97 Tim Davidson Modelling physical systems EE3CL4 C01: Trans. Newton. Mech. Rot. Newton. Mech. Introduction to Linear Control Systems Linearization Laplace Section 2: System Models transforms Laplace in action Transfer Tim Davidson function Step response McMaster University Transfer fn of DC motor Winter 2020 Our first model-based control system design Block diagram models Block dia. transform.
EE 3CL4, §2 2 / 97 Outline Tim Davidson Modelling physical systems 1 Modelling physical Translational Newtonian Mechanics systems Trans. Newton. Rotational Newtonian Mechanics Mech. Rot. Newton. Mech. Linearization Linearization 2 Laplace transforms Laplace transforms 3 Laplace in action Laplace transforms in action 4 Transfer function 5 Transfer function Step response 6 Step response Transfer fn of DC motor Transfer function of DC motor 7 Our first model-based control system Our first model-based control system design 8 design Block diagram Block diagram models 9 models Block dia. transform. Block diagram transformations
EE 3CL4, §2 4 / 97 Differential equation models Tim Davidson Modelling physical systems Trans. Newton. Mech. • Most of the systems that we will deal with are dynamic Rot. Newton. Mech. Linearization • Differential equations provide a powerful way to Laplace describe dynamic systems transforms Laplace in • Will form the basis of our models action Transfer function • We saw differential equations for inductors and Step response capacitors in 2CI, 2CJ Transfer fn of DC motor • What about mechanical systems? Our first both translational and rotational model-based control system design Block diagram models Block dia. transform.
EE 3CL4, §2 5 / 97 Translational Spring Tim Davidson Modelling physical systems Trans. Newton. F ( t ) : resultant force in direction x Mech. Rot. Newton. Mech. Recall free body diagrams and “action and reaction” Linearization Laplace • Spring. k : spring constant, L r : relaxed length of spring transforms Laplace in action Transfer function Step response Transfer fn of DC motor Our first � � F ( t ) = k [ x 2 ( t ) − x 1 ( t )] − L r model-based control system design Block diagram models Block dia. transform.
EE 3CL4, §2 6 / 97 Translational Damper Tim Davidson Modelling physical systems Trans. Newton. F ( t ) : resultant force in direction x Mech. Rot. Newton. Mech. Linearization Laplace • Viscous damper. b : viscous friction coefficient transforms Laplace in action Transfer function Step response Transfer fn of DC motor � dx 2 ( t ) − dx 1 ( t ) � � � F ( t ) = b = b v 2 ( t ) − v 1 ( t ) Our first model-based dt dt control system design Block diagram models Block dia. transform.
EE 3CL4, §2 7 / 97 Mass Tim Davidson Modelling physical systems F ( t ) : resultant force in direction x Trans. Newton. Mech. Rot. Newton. Mech. Linearization Laplace • Mass: M transforms Laplace in action Transfer function Step response Transfer fn of DC motor F ( t ) = M d 2 x m ( t ) = M dv m ( t ) Our first = Ma m ( t ) model-based dt 2 dt control system design Block diagram models Block dia. transform.
EE 3CL4, §2 8 / 97 Rotational spring Tim Davidson Modelling physical systems Trans. Newton. T ( t ) : resultant torque in direction θ Mech. Rot. Newton. Mech. Linearization Laplace • Rotational spring. k : rotational spring constant, transforms φ r : rotation of relaxed spring Laplace in action Transfer function Step response Transfer fn of DC motor Our first � � T ( t ) = k [ θ 2 ( t ) − θ 1 ( t )] − φ r model-based control system design Block diagram models Block dia. transform.
EE 3CL4, §2 9 / 97 Rotational damper Tim Davidson Modelling physical systems T ( t ) : resultant torque in direction θ Trans. Newton. Mech. Rot. Newton. Mech. Linearization • Rotational viscous damper. Laplace transforms b : rotational viscous friction coefficient Laplace in action Transfer function Step response Transfer fn of DC motor Our first � d θ 2 ( t ) − d θ 1 ( t ) model-based � � � T ( t ) = b = b ω 2 ( t ) − ω 1 ( t ) control system dt dt design Block diagram models Block dia. transform.
EE 3CL4, §2 10 / 97 Rotational inertia Tim Davidson Modelling physical systems Trans. Newton. T ( t ) : resultant torque in direction θ Mech. Rot. Newton. Mech. Linearization Laplace • Rotational inertia: J transforms Laplace in action Transfer function Step response Transfer fn of DC motor T ( t ) = J d 2 θ m ( t ) = J d ω m ( t ) Our first = J α m ( t ) model-based dt 2 dt control system design Block diagram models Block dia. transform.
EE 3CL4, §2 11 / 97 Example system (translational) Tim Davidson Horizontal. Origin for y : y = 0 when spring relaxed Modelling physical systems Trans. Newton. Mech. Rot. Newton. Mech. Linearization Laplace transforms Laplace in action Transfer function Step response • F = M dv ( t ) dt Transfer fn of DC motor • v ( t ) = dy ( t ) dt Our first model-based • F ( t ) = r ( t ) − b dy ( t ) − ky ( t ) control system dt design Block diagram models M d 2 y ( t ) + bdy ( t ) Block dia. transform. + ky ( t ) = r ( t ) dt dt
EE 3CL4, §2 12 / 97 Example, continued Tim Davidson Modelling physical systems Trans. Newton. Mech. Rot. Newton. Mech. Linearization Laplace transforms M d 2 y ( t ) + bdy ( t ) + ky ( t ) = r ( t ) Laplace in action dt dt Transfer function Resembles equation for parallel RLC circuit. Step response Transfer fn of DC motor Our first model-based control system design Block diagram models Block dia. transform.
EE 3CL4, §2 13 / 97 Example, continued Tim Davidson Modelling physical systems Trans. Newton. Mech. Rot. Newton. Mech. Linearization Laplace transforms Laplace in action Transfer function Step response Transfer fn of • Stretch the spring a little and hold. DC motor Our first • Assume an under-damped system. model-based control system • What happens when we let it go? design Block diagram models Block dia. transform.
EE 3CL4, §2 15 / 97 Taylor’s series Tim Davidson Modelling • Nature does not have many linear systems physical systems • However, many systems behave approximately linearly Trans. Newton. Mech. in the neighbourhood of a given point Rot. Newton. Mech. Linearization • Apply first-order Taylor’s Series at a given point Laplace transforms • Obtain a locally linear model Laplace in • Use this to obtain insight into behaviour of physical action Transfer system via Laplace Transforms, poles and zeros, etc function Step response • In this course we will focus on the case of a single Transfer fn of DC motor linearized differential equation model for the system, in Our first which the coefficients are constants model-based control system • e.g., in previous examples mass, viscosity and spring design Block diagram constant did not change with time, position, velocity, models temperature, etc Block dia. transform.
EE 3CL4, §2 16 / 97 Pendulum example Tim Davidson Modelling physical systems Trans. Newton. Mech. Rot. Newton. Mech. Linearization Laplace transforms Laplace in action Transfer • Assume shaft is light with respect to M , function and stiff with respect to gravitational forces Step response • Torque due to gravity: T ( θ ) = MgL sin( θ ) Transfer fn of DC motor • Apply Taylor’s series around θ = 0: Our first � � model-based θ − θ 3 / 3 ! + θ 5 / 5 ! − θ 7 / 7 ! + . . . T ( θ ) = MgL control system design • For small θ around θ = 0 we can build an approximate Block diagram model that is linear models Block dia. transform. T ( θ ) ≈ MgL θ
EE 3CL4, §2 18 / 97 Laplace transform Tim Davidson • Once we have a linearized differential equation (with Modelling constant coefficients) we can take Laplace Transforms physical systems to obtain the transfer function Trans. Newton. Mech. Rot. Newton. Mech. • We will consider the “one-sided” Laplace transform, for Linearization signals that are zero to the left of the origin. Laplace transforms � ∞ 0 − f ( t ) e − st dt Laplace in F ( s ) = action Transfer function Step response � T . � ∞ mean? lim T →∞ • What does Transfer fn of DC motor • Does this limit exist? Our first model-based control system • If | f ( t ) | < Me α t , then exists for all Re( s ) > α . design Includes all physically realizable signals Block diagram models Block dia. transform. • Note: When multiplying transfer function by Laplace of input, output is only valid for values of s in intersection of regions of convergence
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