MECH 466 - Lecture5-DCMotors-2009W

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Unformatted text preview: x’(t)=f(t) Example: Mx’’(t)=f(t) M(t)x’’(t)=f(t) For time-invariant systems: For time- Time shift Sys Linearization of water tank model (Laboratory 1) Mass flow rate equation Mass Time shift Q. How to linearize? linearize? This course deals with time-invariant systems. This time2008/09 MECH466 : Automatic Control 17 Linearization of water tank model (cont’d) 2008/09 MECH466 : Automatic Control 18 Linearization of water tank model (cont’d) Why is the system linear when α=1? Why α=1? 1. Select an operating point h0, around which you want to linearize the equation. 2. Consider a small perturbation from h0 System 3. Substitute this into the above equation: 3. 2008/09 MECH466 : Automatic Control 19 2008/09 MECH466 : Automatic Control 20 5 Linearization of water tank model (cont’d) Some remark Actually, this “linearized” model is not linear, but Actually, linearized” not affine, due to the constant term. RHS : However, if we define a new input However, Linearized model then the system is linear. You don’t need this fact in Laboratory #1. You don’ This model is accurate enough ONLY around h0 !!! !!! 2008/09 MECH466 : Automatic Control 21 Linearization of a pendulum model Motion of the pendulum Motion 2008/09 MECH466 : Automatic Control 22 Linearization of a pendulum model around general operating point Taylor series expansion of f(x) around x=xo Taylor f(x) Q. How to linearize? linearize? Pendulum example Pendulum For a small Linearized model Linearized Linearized model 2008/09 MECH466 : Automatic Control 23 2008/09 MECH466 : Automatic Control 24 6 Time delay examples (p.115) Time delay transfer function TF derivation TF Delay time (Memorize this!) The more time delay is, the more difficult to The control (Imagine that you are controlling the temperature of your shower with a very long hose. You will either get burned or frozen!) 2008/09 MECH466 : Automatic Control 25 2008/09 MECH466 : Automatic Control Summary and Exercises Main message until this point Modeling of, DC motor, nonlinear systems and Modeling systems with time delay Next Next 26 Many systems can be represented as transfer functions! Stability of linear control systems, one of the most Stability important topics in feedback control Exercises Exercises Read Sections 2.8, 2.10, 2.11 & Lab #1 manual Read “Review of the modeling of fluid-flow through a fluidconstriction” . Problems 2.45, 2.49, 2.50 in page 100. Problems 2008/09 MECH466 : Automatic Control Using the transfer functions, …. (to be continued) 27 2008/09 MECH466 : Automatic Control 28 7...
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