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Lecture7-Routh-HurwitzCriterion-2009W.pdf-MECH466 : Automatic Control 1 MECH466:

Problem Set # 6- Winter 2009 MECH 375 Problem Set # 6 Given: Wed., Feb. 11, 2009 Recommended Completion Date: Wed., Feb. 25, 2009 Do not submit for grading Problem 1: A plate of stainless steel ( ρ = 7817 kg/m 3 ; c = 460 J/kg- ° C; k = 19 W/m- ° C) has thickness of 3 cm and is initially at a uniform temperature of T i = 500 ° C. The plate is suddenly exposed to same convection c

MECH | 375 | Heat Transfer I | homework | Problem Set # 6- Winter 2009 M

MECH466: Exercises for Lecture 6 6.1 P ( s )= s 5 +3 s 4 +5 s 3 +4 s 2 + s +3 Routh array for P ( s ) is s 5 1 5 1 s 4 3 4 3 s 3 11 / 3 0 s 2 4 3 s 1 - 11 / 4 s 0 3 Since there are two sign changes (4 →- 11 / 4 → 3) in the ﬁrst column of the Routh array, P ( s ) has two RHP roots and three LHP roots. 6.9 The characteristic equation of the closed-loop system is 1+ 240 ( s + 1)( s + 2)( s + 3)( s + 4) | {z } G ( s ) =0 , i.e., P ( s ) := s 4 + 10 s 3 + 35 s 2 + 50 s + 264 = 0 Routh array for P ( s ) is s 4 1 35 264 s 3 10 50 s 2 30 264 s 1 - 38 s 0 264 Since there are two sign changes (30 →- 38 → 264) in the ﬁrst column of the Routh array, P ( s ) has two RHP roots. Therefore, the closed-loop system is unstable. 6.14 The characteristic equation of the closed-loop system is 1+ 1 2 s 4 +5 s 3 + s 2 +2 s | {z } G ( s ) =0 , i.e., P ( s ) := 2 s 4 +5 s 3 + s 2 +2 s +1=0 Routh array for P ( s ) is s 4 2 1 1 s 3 5 2 s 2 1 / 5 1 s 1 - 23 s 0 1 Since there are two sign changes (1 / 5 →- 23 → 1) in the ﬁrst column of the Routh array, P ( s ) has two RHP roots. Therefore, the closed-loop system is unstable. 1

MECH | 466 | Automatic Control | question bank | MECH466: Exercises for Lecture

4 2008/09 MECH466 : Automatic Control 13 Categories of Automatic Control Categories of Automatic Control   Classical control (contents in this course) Classical control (contents in this course)   Transfer function Transfer function   Frequency domain Frequency domain   Modern control (1960 Modern control (1960 - - ) )   State State - - space model space model   Time domain Time domain   Optimality Optimality   Post Post - - modern control (1980 modern control (1980 - - ) )   Robust control Robust control   Hybrid control, etc. Hybrid control, etc. 2008/09 MECH466 : Automatic Control 14 Weak points of classical control Weak points of classical control   Single Single - - input single input single - - output (SISO) output (SISO) (Multivariable (MIMO) control theory) (Multivariable (MIMO) control theory)   Linear time Linear time - - invariant invariant (State (State - - space, nonlinear control theory) space, nonlinear control theory)   No optimality in design No optimality in design (Optimal control theory) (Optimal control theory)   No robustness, except GM and PM No robustness, except GM and PM (Robust control theory) (Robust control theory) Much more control theory has been, Much more control theory has been, and is being, developed! and is being, developed! 2008/09 MECH466 : Automatic Control 15 Systematic controller design process Systematic controller design process Plant Plant Input Input Output Output Reference Reference Sensor Sensor Actuator Actuator Controller Controller Disturbance Disturbance 1. Modeling Mathematical model Mathematical model 2. Analysis Controller Controller 3. Design 4. Implemenation 2008/09 MECH466 : Automatic Control 16 Goals of this course Goals of this course To learn basics of feedback control systems To learn basics of feedback control systems   Modeling Modeling as a transfer function and a block diagram as a transfer function and a block diagram • • Laplace transform (Mathematics!) Laplace transform (Mathematics!) • • Mechanical, electrical, electromechanical systems Mechanical, electrical, electromechanical systems   Analysis Analysis • • Step response, frequency response Step response, frequency response • • Stability: Stability: Routh Routh - - Hurwitz criterion, Hurwitz criterion, Nyquist Nyquist criterion criterion   Design Design • • Root locus technique, frequency response technique, Root locus technique, frequency response technique, PID control, lead/lag compensator PID control, lead/lag compensator   Theory, (simulation with Theory, (simulation with Matlab Matlab ), practice in laboratories ), practice in laboratories

MECH | 466 | Automatic Control | lecture notes | 4 2008/09 MECH466 : Automatic 

MECH 366 Assignment #3 D) Submit a summary sheet of the 1 st order and 2 nd order systems using the following format: Model Input Initial conditions Time Domain Laplace Poles Notes First order  y’’ + y = u(t) Step A y(0) = y 0 y step = (y 0 – A)e -t/  + A = y 0 e -t/  +A(1-e -t/  ) = free response + forced response G(s) = 1/(  s+1) U(s) = 1/s s=-1/  Can also plot the poles here Ca

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