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Introduction to Systems Theory and Control Systems

Introduction to Systems Theory and Control Systems 29
Introduction to Systems Theory and Control Systems Paula Raica Department of Automation Dorobantilor Str., room C21, tel: 0264 401267 Baritiu Str., room C14, tel: 0264 202368 email: Paula.Raicaaut.utcluj.ro http://rocon.utcluj.ro/st Technical University of ClujNapoca Introduction to Systems Theory and Control SystemsCourse organization Lectures: 2h/week, room D21 (Baritiu) Lab exercises: 4h / 2 weeks, 7173 Dorobatilor Str., yellow building on the right. Lab C01: ground floor Taught by: Paula Raica (lectures) Lab: Group When Where Who 30421 Friday 1216 C01 Paula Raica 30422 Thursday 1216 C01 Iulia Clitan 30423 Tuesday 1216 C01 Iulia Clitan Alexandru Codrean (odd) 30424 Thursday 812 C01 Zoltan Nagy (even) Paula Raica (odd) 30425 Tuesday 1620 C01 Zoltan Nagy (even) Introduction to Systems Theory and Control SystemsSystems Theory. Grading Point accumulation Exams (see ”Course calendar and grading”): lab tests including homework assignments (optional) midterm exam (optional) final exam Control Challenge Lab work policy Prerequisites: Differential equations, Linear algebra, Laplace transform, Complex numbers Introduction to Systems Theory and Control SystemsCourse objective The general objective of the course is to introduce the fundamental principles of linear system modeling, analysis and feedback control and to evaluate feedback control systems with desired behavior. Introduction to Systems Theory and Control SystemsSystems Theory System : A set or arrangement of entities so related or connected so as to form a unity or organic whole. (Iberall) Systems theory: interdisciplinary field which studies systems. Founded by Ludwig von Bertalanffy, William Ross Ashby and others between the 1940s and the 1970s on principles from physics, biology and engineering Grew into numerous fields including philosophy, sociology, organizational theory, management and economics among others. Cybernetics is a related field, sometimes considered as a part of systems theory. Introduction to Systems Theory and Control SystemsControl Systems Engineering Understanding systems Control of systems Modeling and control of modern, complex, interrelated systems traffic control systems, chemical processes, robotic systems industrial automation systems. Control Systems Engineering is based on the foundations of feedback theory and linear systems analysis. Introduction to Systems Theory and Control SystemsHistorical Background 300 B.C, Greece : development of the float regulator mechanisms, mid 1860s, J.C.Maxwell: the first formal study of the ”Theory of Control”, mid 1890s, E.J. Routh and A.M. Lyapunov: ”Routh Stability Test” and the ”Lyapunov Stability Criteria” 1930s, H. Nyquist (Bell Telephone Laboratories): applied frequency analysis to control systems design 1930, H.W.Bode: designed electronic amplifiers using the concepts of feedback control 1950s onwards: control theory evolved with new mathematical techniques applied and computer technology. Introduction to Systems Theory and Control SystemsDiscipline of Control Systems Multidisciplinary field. Covers Mechanical Engineering, Chemical Engineering, Electrical and Electronic Engineering, Environmental, Civil, Physics, Economics and Finance, Artificial Intelligence and Computer Science Taught in the main stream Engineering and Physics courses ComputerControlled Systems: complex field in control engineering. Concepts overlap with the branch of Physics and Electrical Engineering known as Digital Signal Processing (DSP) and Communication Systems Introduction to Systems Theory and Control SystemsDynamical system A dynamical system is a system whose behavior changes over time, often in response to external stimulation or forcing. Inputs (cause) = quantities that are acting on the system from the environment Outputs (effect) = the results of the input acting on the system. Inputs, outputs = signals Introduction to Systems Theory and Control SystemsExample: Helicopter Figure: Helicopter Outputs Inputs: the actual position (coordinates the power produced by x, y, z) the engines orientation (roll, pitch, yaw) the pilot control inputs velocity the wind = disturbance Introduction to Systems Theory and Control SystemsTerminology Control of an inverted pendulum Figure: Elements in a control system Block diagram. Input. Output. Plant (Process). Measurement. Signals. Setpoint (Reference value). Comparator. Compensator. Actuator. Disturbance. Openloop. Closedloop. Negative Feedback. Introduction to Systems Theory and Control SystemsClosedloop control system for HDD Figure: A hard disk control system Introduction to Systems Theory and Control SystemsExamples Oven temperature control (closedloop) Washing machine: (openloop) Central heating: (closedloop) vs. radiator( openloop) Automobile steering control system Desired Actual course of course of Error travel Steering travel Driver Automobile mechanism Desired direction of travel Actual direction Measurement, visual and tactile of travel Introduction to Systems Theory and Control SystemsApplications Figure: Traffic control Figure: Maglev Introduction to Systems Theory and Control SystemsApplications Introduction to Systems Theory and Control SystemsApplications Figure: Chemical industry, energy Introduction to Systems Theory and Control SystemsApplications Introduction to Systems Theory and Control SystemsCourse contents Mathematical models of linear time invariant systems systems. Transfer functions, statespace models, block diagram models Analysis of linear continuous systems. Characteristics and performance. Stability of linear continuous systems. System analysis using root locus. Frequency response. Bode diagrams. Controller design. Leadlag compensation. PID control. State feedback Sampleddata systems. Digital control systems. Introduction to Systems Theory and Control SystemsBibliography R.C.Dorf, R.Bishop, ”Modern Control Systems”, AddisonWesley, 2011; K.Ogata , ”Modern Control Engineering”, Prentice Hall, 1990. K.Dutton, S. Thompson, B. Barraclough, ”The Art of Control Engineering”, AddisonWesley, 1997 M. Ha˘nga˘nu¸t, Teoria sistemelor, UTCluj, 1996 T. Colo¸si, Elemente de teoria sistemelor si reglaj automat, UTCluj, 1981 Introduction to Systems Theory and Control SystemsIntroduction to Control System Modeling Paula Raica Department of Automation Dorobantilor Str., room C21, tel: 0264 401267 Baritiu Str., room C14, tel: 0264 202368 email: Paula.Raicaaut.utcluj.ro http://rocon.utcluj.ro/st Technical University of ClujNapoca Introduction to Control System ModelingIntroduction A mathematical model is an equation or set of equations which adequately describes the behavior of a system. Two approaches to finding the model: Lumpedparameter modeling: for each element a mathematical description is established from the physical laws. System identification: an experiment can be carried out and a mathematical model can be found from the results. The important relationship is that between the manipulated inputs and measurable outputs. y(t) u(t) Dynamic System output input Introduction to Control System ModelingLumpedparameter models The systems studied in this course are: Linear must obey the principle of superposition Stationary (or time invariant) the parameters inside the element must not vary with time. Deterministic The outputs of the system at any time can be determined from a knowledge of the system’s inputs up to that time. Examples. 1 The resistor: i(t) = v(t) R R di(t) 1 The inductor: i(t) = v(t)dt or v(t) = L L dt dv(t) The capacitor: i(t) = C dt Introduction to Control System ModelingExamples Springmassdamper system k Friction f Mass M y(t) Force r(t) 2 d y(t) dy(t) M +f +ky(t) = r(t) 2 dt dt where: f is the friction coefficient, M the mass, k the stiffness of the linear spring. Introduction to Control System Modeling displacementPrinciple of superposition A system is defined as linear in terms of the system excitation and response. Additivity x (t) → y (t) 1 1 x (t) → y (t) 2 2 x (t)+x (t) → y (t)+y (t) 1 2 1 2 Homogeneity x(t) → y(t) mx(t) → my(t) Introduction to Control System ModelingLinear Approximation Nonlinear system 2 y = x Nonlinear system y = mx +b Linear about an operating point x ,y for small changes Δx and 0 0 Δy. When x = x +Δx and y = y +Δy: 0 0 y +Δy = mx +mΔx +b 0 0 and therefore Δy = mΔx Introduction to Control System ModelingLinear Approximation If the dependent variable y depends upon several excitation variables x ,x ,...,x : 1 2 n y = g(x ,x ,...,x ). 1 2 n The Taylor series expansion about the operating point x ,x ,...,x (the higherorder terms are neglected): 10 20 n0 ∂g y = g(x ,x ,...,x )+ (x −x )+ 10 20 n0 x=x 1 10 0 ∂x 1 ∂g ∂g + (x −x )+ ...+ (x −x ) x=x 2 20 x=x n n0 0 0 ∂x ∂x 2 n where x is the operating point. 0 Introduction to Control System ModelingExample Pendulum oscillator The torque on the mass is: T = MgLsin(x) The equilibrium condition for the o mass is x = 0 . 0 ∂sinx ∼ T −T MgL (x−x ), = 0 x=x 0 0 ∂x where T = 0. 0 o o T = MgL(cos0 )(x−0 ) = MgLx The approximation is reasonably accurate for −π/4≤ x ≤ π/4. Introduction to Control System ModelingLinear Approximation Input x(t) and a response y(t): y(t) = g(x(t)) Taylor series expansion about the operating point x : 0 dg x−x 0 y = g(x) = g(x )+ +higher order terms 0 x=x 0 dx 1 The slope at the operating point, dg m = , x=x 0 dx dg y = g(x )+ (x−x ) = y +m(x−x ), 0 x=x 0 0 0 0 dx Finally, this equation can be rewritten as the linear equation (y−y ) = m(x−x ) or Δy = mΔx 0 0 Introduction to Control System ModelingExample. Magnetic levitation The system: an ironcore electromagnet and the steel ball levitated by the electromagnet. Electromagnetic force F : m 2 i (t) F = C m 2 z (t) Introduction to Control System ModelingExample. Magnetic levitation Input: the current through the coils of the electromagnet i(t) Output: the displacement of the ball z(t) The equation of motion for the ball: 2 i (t) mz¨(t) = mg−C 2 z (t) Nonlinear model Introduction to Control System ModelingExample. Magnetic levitation linearization Rewrite the equation: 2 i (t) g(z¨(t),z(t),i(t)) = mz¨(t)−mg +C = 0 2 z (t) Choose an operating point: (z¨ , z , i ) such that 0 0 0 2 i 0 mz¨ = mg −C 0 2 z 0 Write the truncated Taylor series around the operating point: ∂g 0 = g(z¨(t),z(t),i(t)) ≈ g(z¨ ,z ,i )+ (z¨(t)−z¨ )+ 0 0 0 0 (z¨ , z , i ) 0 0 0 ∂z¨ ∂g + (z(t)−z )+ (z¨ , z , i ) 0 0 0 0 ∂z ∂g + (i(t)−i ) 0 (z¨ , z , i ) 0 0 0 ∂i Introduction to Control System ModelingExample. Magnetic levitation linearization Compute the partial derivatives and evaluate them at the operating point. The Taylor series expansion is: 2 i i 0 0 0≈ 0+m·(z¨(t)−z¨ )−2C ·(z(t)−z )+2C ·(i(t)−i ) 0 0 0 3 2 z z 0 0 Denote the variations around the operating point by: Δz¨(t) = z¨(t)−z¨ , Δz(t)= z(t)−z and Δi(t) = i(t)−i 0 0 0 Linear differential equation in terms of Δz¨(t), Δz(t), Δi(t): 2 i i 0 0 mΔz¨(t) = 2C Δz(t)−2C Δi(t) 3 2 z z 0 0 Introduction to Control System ModelingTo do Review: Differential equations Linear algebra Laplace transform Check the course webpage: http://rocon.utcluj.ro/st Download the exercises (ControlEngineering.pdf) and detailed lecture notes Introduction to Control System Modeling
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