DOCSLIB.ORG

Lecture 3: Transfer Function and Dynamic Response Analysis

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Lecture 3: Transfer Function and Dynamic Response Analysis Transfer function approach Dynamic response Summary Basics of Automation and Control I Lecture 3: Transfer function and dynamic response analysis Paweł Malczyk Division of Theory of Machines and Robots Institute of Aeronautics and Applied Mechanics Faculty of Power and Aeronautical Engineering Warsaw University of Technology October 17, 2019 © Paweł Malczyk. Basics of Automation and Control I Lecture 3: Transfer function and dynamic response analysis 1 / 31 Transfer function approach Dynamic response Summary Outline 1 Transfer function approach 2 Dynamic response 3 Summary © Paweł Malczyk. Basics of Automation and Control I Lecture 3: Transfer function and dynamic response analysis 2 / 31 Transfer function approach Dynamic response Summary Transfer function approach 1 Transfer function approach SISO system Definition Poles and zeros Transfer function for multivariable system Properties 2 Dynamic response 3 Summary © Paweł Malczyk. Basics of Automation and Control I Lecture 3: Transfer function and dynamic response analysis 3 / 31 Transfer function approach Dynamic response Summary SISO system Fig. 1: Block diagram of a single input single output (SISO) system Consider the continuous, linear time-invariant (LTI) system defined by linear constant coefficient ordinary differential equation (LCCODE): dny dn−1y + − + ··· + _ + = an n an 1 n−1 a1y a0y dt dt (1) dmu dm−1u = b + b − + ··· + b u_ + b u m dtm m 1 dtm−1 1 0 initial conditions y(0), y_(0),..., y(n−1)(0), and u(0),..., u(m−1)(0) given, u(t) – input signal, y(t) – output signal, ai – real constants for i = 1, ··· , n, and bj – real constants for j = 1, ··· , m. How do I find the LCCODE (1)? . How do I find the coefficients ai and bj? © Paweł Malczyk. Basics of Automation and Control I Lecture 3: Transfer function and dynamic response analysis 4 / 31 Transfer function approach Dynamic response Summary Definition Definition 1 (Transfer function) The transfer function G(s) of a linear, time- invariant differential equation system is defined as the ratio of the Laplace transform of the output Y(s) (response function) to the Laplace transform of the input U(s) (driving Fig. 2: Block diagram of a single input single output (SISO) system function) under the assumption that all initial conditions are zero. Let us transform Eq. (1): n n−1 a s + a − s + ··· + a s + a Y(s) = n n 1 1 0 m m−1 (2) = bms + bm−1s + ··· + b1s + b0 U(s) then P − m k ( ) m + − m 1 + ··· + + Y s bms bm 1s b1s b0 Pk=0 bks G(s) = = − = n (3) n − n 1 ··· k U(s) ans + an 1s + + a1s + a0 k=0 aks Comment 1 For causal systems m ≤ n (proper transfer functions). © Paweł Malczyk. Basics of Automation and Control I Lecture 3: Transfer function and dynamic response analysis 5 / 31 Transfer function approach Dynamic response Summary Poles and zeros Definition 2 (Characteristic equation) The characteristic equation of a system is defined as the equation obtained by setting the characteristic polynomial of a transfer function G(s) in Eq. (3) to zero, i.e.: n n−1 N(s) = ans + an−1s + ··· + a1s + a0 = 0 (4) Definition 3 (Pole) The roots of the characteristic equation N(s) = 0 are called poles of a transfer function G(s). Definition 4 (Zero) m m−1 The roots of the polynomial M(s) = bms +bm−1s +···+b1s+b0 are called zeros of a transfer function G(s). © Paweł Malczyk. Basics of Automation and Control I Lecture 3: Transfer function and dynamic response analysis 6 / 31 Transfer function approach Dynamic response Summary Transfer function for multivariable system Fig. 3: Block diagram of a multiple input multiple output (MIMO) system Definition 5 (Transfer function for MIMO systems) th th The transfer function between the j input uj(t) (j = 1, 2, ··· , p) and the i output yi(t) (i = 1, 2, ··· , q) is defined as Yi(s) Gij(s) = (5) Uj(s) with Uk(s) = 0 for k = 1, 2, ··· , p, k ≠ j under the assumption that all initial conditions are zero. When all the p inputs are in action, the ith output transform is written Yi(s) = Gi1(s)U1(s) + Gi2(s)U2(s) + ··· + Gip(s)Up(s) (6) © Paweł Malczyk. Basics of Automation and Control I Lecture 3: Transfer function and dynamic response analysis 7 / 31 Transfer function approach Dynamic response Summary Transfer matrix It is convenient to express Eq. (6) in matrix-vector form: Y(s) = G(s)U(s) (7) where U (s) Y (s) 1 1 U2(s) Y2(s) U(s) = ··· , Y(s) = ··· (8) Up(s) Yq(s) are the transformed p×1 input vector and the transformed q×1 output vector, whereas G (s) G (s) ··· G (s) 11 12 1p G21(s) G22(s) ··· G2p(s) G(s) = (9) . ··· . Gq1(s) Gq2(s) ··· Gqp(s) is the q × p transfer-function matrix. © Paweł Malczyk. Basics of Automation and Control I Lecture 3: Transfer function and dynamic response analysis 8 / 31 Transfer function approach Dynamic response Summary Transfer functions of physical systems Electrical network transfer functions V(s) Component Voltage-current Voltage-charge G(s) = I(s) Z t 1 1 1 v(t) = C i(τ)dτ v(t) = C q(t) Cs 0 dq(t) v(t) = Ri(t) v(t) = R dt R di(t) d2q(t) v(t) = L dt v(t) = L dt2 Ls Mechanical system transfer functions F(s) Component Force-velocity Force-displacement G(s) = X(s) Z t f(t) = k v(τ)dτ f(t) = kx(t) k 0 dx(t) f(t) = bv(t) f(t) = b dt bs 2 dv(t) d x(t) 2 f(t) = m dt f(t) = m dt2 ms © Paweł Malczyk. Basics of Automation and Control I Lecture 3: Transfer function and dynamic response analysis 9 / 31 Transfer function approach Dynamic response Summary Analogies between electrical and mechanical systems Electrical network transfer functions Mechanical system transfer functions V(s) F(s) Component Voltage-current Voltage-charge G(s) = I(s) Component Force-velocity Force-displacement G(s) = X(s) Z Z t t 1 1 1 v(t) = C i(τ)dτ v(t) = C q(t) Cs f(t) = k v(τ)dτ f(t) = kx(t) k 0 0 dq(t) dx(t) v(t) = Ri(t) v(t) = R dt R f(t) = bv(t) f(t) = b dt bs 2 2 di(t) d q(t) ( ) = dv(t) ( ) = d x(t) 2 v(t) = L dt v(t) = L dt2 Ls f t m dt f t m dt2 ms The equations of motion and the behavior of systems involving various physical media are found to be analogous. Force is analogous to voltage, and velocity to current. Velocity is analogous to voltage, and force to current. Different analogies exist. © Paweł Malczyk. Basics of Automation and Control I Lecture 3: Transfer function and dynamic response analysis 10 / 31 Transfer function approach Dynamic response Summary Example Example 1 Consider the mechanical system (Fig. 4). We assume that the system is linear. The external force u(t) is the input signal, and the displacement y(t) of the mass is the output. The displacement y(t) is measured from the equilibrium position in the absence of external force. Write the equations of motion and find the transfer function for the system. The equations of motion for the system are as follows m¨y + by_ + ky = u The Laplace transform of the eq. (y(0) = 0, y_(0) = 0): Fig. 4: Mechanical ms2Y(s) + bsY(s) + kY(s) = U(s) system Y(s) 1 G(s) = = Transfer function U(s) ms2 + bs + k Comment 2 The characteristic eq. N(s) = ms2 + bs + k has two poles. Comment 3 The transfer function conveys useful information: Y(s) ≈ 1 G(s) = U(s) k for small s (spring drive by a force) Y(s) ≈ 1 G(s) = U(s) ms2 for large s (mass driven by a force) © Paweł Malczyk. Basics of Automation and Control I Lecture 3: Transfer function and dynamic response analysis 11 / 31 Transfer function approach Dynamic response Summary Properties of a transfer function 1 The transfer function of a system is a mathematical model of that system, in that it is an operational method of expressing the differential equation that relates the output variable to the input variable. 2 The transfer function is a property of a system itself, unrelated to the magnitude and nature of the input or driving function. 3 The transfer function includes the units necessary to relate the input to the output; however, it does not provide any information concerning the physical structure of the system. 4 If the transfer function of a system is known, the output or response can be studied for various forms of inputs with a view toward understanding the nature of the system. 5 If the transfer function of a system is unknown, it may be established experimentally by introducing known inputs and studying the output of the system (identification). © Paweł Malczyk. Basics of Automation and Control I Lecture 3: Transfer function and dynamic response analysis 12 / 31 Transfer function approach Dynamic response Summary Dynamic response 1 Transfer function approach 2 Dynamic response Time response Unit impulse response Response by convolution Unit step response Transient and steady state response Total response – example Example of a heat-flow model 3 Summary © Paweł Malczyk. Basics of Automation and Control I Lecture 3: Transfer function and dynamic response analysis 13 / 31 Transfer function approach Dynamic response Summary Time response Definition 6 (Time response) The time response represents how the state and output of a dynamic system with nonzero initial conditions changes in time when subjected to a particular input.
Load more

Recommended publications
  • Input and Output Directions and Hankel Singular Values CEE 629
    Principal Input and Output Directions and Hankel Singular Values CEE 629. System Identification Duke University, Fall 2017 1 Continuous-time systems in the frequency domain In the frequency domain, the input-output relationship of a LTI system (with r inputs, m outputs, and n internal states) is represented by the m-by-r rational frequency response function matrix equation y(ω) = H(ω)u(ω) . At a frequency ω a set of inputs with amplitudes u(ω) generate steady-state outputs with amplitudes y(ω). (These amplitude vectors are, in general, complex-valued, indicating mag- nitude and phase.) The singular value decomposition of the transfer function matrix is H(ω) = Y (ω) Σ(ω) U ∗(ω) (1) where: U(ω) is the r by r orthonormal matrix of input amplitude vectors, U ∗U = I, and Y (ω) is the m by m orthonormal matrix of output amplitude vectors, Y ∗Y = I Σ(ω) is the m by r diagonal matrix of singular values, Σ(ω) = diag(σ1(ω), σ2(ω), ··· σn(ω)) At any frequency ω, the singular values are ordered as: σ1(ω) ≥ σ2(ω) ≥ · · · ≥ σn(ω) ≥ 0 Re-arranging the singular value decomposition of H(s), H(ω)U(ω) = Y (ω) Σ(ω) or H(ω) ui(ω) = σi(ω) yi(ω) where ui(ω) and yi(ω) are the i-th columns of U(ω) and Y (ω). Since ||ui(ω)||2 = 1 and ||yi(ω)||2 = 1, the singular value σi(ω) represents the scaling from inputs with complex am- plitudes ui(ω) to outputs with amplitudes yi(ω).
    [Show full text]
  • An Approximate Transfer Function Model for a Double-Pipe Counter-Flow Heat Exchanger
    energies Article An Approximate Transfer Function Model for a Double-Pipe Counter-Flow Heat Exchanger Krzysztof Bartecki Division of Control Science and Engineering, Opole University of Technology, ul. Prószkowska 76, 45-758 Opole, Poland; [email protected] Abstract: The transfer functions G(s) for different types of heat exchangers obtained from their par- tial differential equations usually contain some irrational components which reflect quite well their spatio-temporal dynamic properties. However, such a relatively complex mathematical representa- tion is often not suitable for various practical applications, and some kind of approximation of the original model would be more preferable. In this paper we discuss approximate rational transfer func- tions Gˆ(s) for a typical thick-walled double-pipe heat exchanger operating in the counter-flow mode. Using the semi-analytical method of lines, we transform the original partial differential equations into a set of ordinary differential equations representing N spatial sections of the exchanger, where each nth section can be described by a simple rational transfer function matrix Gn(s), n = 1, 2, ... , N. Their proper interconnection results in the overall approximation model expressed by a rational transfer function matrix Gˆ(s) of high order. As compared to the previously analyzed approximation model for the double-pipe parallel-flow heat exchanger which took the form of a simple, cascade interconnection of the sections, here we obtain a different connection structure which requires the use of the so-called linear fractional transformation with the Redheffer star product. Based on the resulting rational transfer function matrix Gˆ(s), the frequency and the steady-state responses of the approximate model are compared here with those obtained from the original irrational transfer Citation: Bartecki, K.
    [Show full text]
  • Control Theory
    Control theory S. Simrock DESY, Hamburg, Germany Abstract In engineering and mathematics, control theory deals with the behaviour of dynamical systems. The desired output of a system is called the reference. When one or more output variables of a system need to follow a certain ref- erence over time, a controller manipulates the inputs to a system to obtain the desired effect on the output of the system. Rapid advances in digital system technology have radically altered the control design options. It has become routinely practicable to design very complicated digital controllers and to carry out the extensive calculations required for their design. These advances in im- plementation and design capability can be obtained at low cost because of the widespread availability of inexpensive and powerful digital processing plat- forms and high-speed analog IO devices. 1 Introduction The emphasis of this tutorial on control theory is on the design of digital controls to achieve good dy- namic response and small errors while using signals that are sampled in time and quantized in amplitude. Both transform (classical control) and state-space (modern control) methods are described and applied to illustrative examples. The transform methods emphasized are the root-locus method of Evans and fre- quency response. The state-space methods developed are the technique of pole assignment augmented by an estimator (observer) and optimal quadratic-loss control. The optimal control problems use the steady-state constant gain solution. Other topics covered are system identification and non-linear control. System identification is a general term to describe mathematical tools and algorithms that build dynamical models from measured data.
    [Show full text]
  • Step Response of Series RLC Circuit ‐ Output Taken Across Capacitor
    ESE 271 / Spring 2013 / Lecture 23 Step response of series RLC circuit ‐ output taken across capacitor. What happens during transient period from initial steady state to final steady state? 1 ESE 271 / Spring 2013 / Lecture 23 Transfer function of series RLC ‐ output taken across capacitor. Poles: Case 1: ‐‐two differen t real poles Case 2: ‐ two identical real poles ‐ complex conjugate poles Case 3: 2 ESE 271 / Spring 2013 / Lecture 23 Case 1: two different real poles. Step response of series RLC ‐ output taken across capacitor. Overdamped case –the circuit demonstrates relatively slow transient response. 3 ESE 271 / Spring 2013 / Lecture 23 Case 1: two different real poles. Freqqyuency response of series RLC ‐ output taken across capacitor. Uncorrected Bode Gain Plot Overdamped case –the circuit demonstrates relatively limited bandwidth 4 ESE 271 / Spring 2013 / Lecture 23 Case 2: two identical real poles. Step response of series RLC ‐ output taken across capacitor. Critically damped case –the circuit demonstrates the shortest possible rise time without overshoot. 5 ESE 271 / Spring 2013 / Lecture 23 Case 2: two identical real poles. Freqqyuency response of series RLC ‐ output taken across capacitor. Critically damped case –the circuit demonstrates the widest bandwidth without apparent resonance. Uncorrected Bode Gain Plot 6 ESE 271 / Spring 2013 / Lecture 23 Case 3: two complex poles. Step response of series RLC ‐ output taken across capacitor. Underdamped case – the circuit oscillates. 7 ESE 271 / Spring 2013 / Lecture 23 Case 3: two complex poles. Freqqyuency response of series RLC ‐ output taken across capacitor. Corrected Bode GiGain Plot Underdamped case –the circuit can demonstrate apparent resonant behavior.
    [Show full text]
  • A Passive Synthesis for Time-Invariant Transfer Functions
    IEEE TRANSACTIONS ON CIRCUIT THEORY, VOL. CT-17, NO. 3, AUGUST 1970 333 A Passive Synthesis for Time-Invariant Transfer Functions PATRICK DEWILDE, STUDENT MEMBER, IEEE, LEONARD hiI. SILVERRJAN, MEMBER, IEEE, AND R. W. NEW-COMB, MEMBER, IEEE Absfroct-A passive transfer-function synthesis based upon state- [6], p. 307), in which a minimal-reactance scalar transfer- space techniques is presented. The method rests upon the formation function synthesis can be obtained; it provides a circuit of a coupling admittance that, when synthesized by. resistors and consideration of such concepts as stability, conkol- gyrators, is to be loaded by capacitors whose voltages form the state. By the use of a Lyapunov transformation, the coupling admittance lability, and observability. Background and the general is made positive real, while further transformations allow internal theory and position of state-space techniques in net- dissipation to be moved to the source or the load. A general class work theory can be found in [7]. of configurations applicable to integrated circuits and using only grounded gyrators, resistors, and a minimal number of capacitors II. PRELIMINARIES is obtained. The minimum number of resistors for the structure is also obtained. The technique illustrates how state-variable We first recall several facts pertinent to the intended theory can be used to obtain results not yet available through synthesis. other methods. Given a transfer function n X m matrix T(p) that is rational wit.h real coefficients (called real-rational) and I. INTRODUCTION that has T(m) = D, a finite constant matrix, there exist, real constant matrices A, B, C, such that (lk is the N ill, a procedure for time-v:$able minimum-re- k X lc identity, ,C[ ] is the Laplace transform) actance passive synthesis of a “stable” impulse response matrix was given based on a new-state- T(p) = D + C[pl, - A]-‘B, JXYI = T(~)=Wl equation technique for imbedding the impulse-response (14 matrix in a passive driving-point impulse-response matrix.
    [Show full text]
  • Frequency Response and Bode Plots
    1 Frequency Response and Bode Plots 1.1 Preliminaries The steady-state sinusoidal frequency-response of a circuit is described by the phasor transfer function Hj( ) . A Bode plot is a graph of the magnitude (in dB) or phase of the transfer function versus frequency. Of course we can easily program the transfer function into a computer to make such plots, and for very complicated transfer functions this may be our only recourse. But in many cases the key features of the plot can be quickly sketched by hand using some simple rules that identify the impact of the poles and zeroes in shaping the frequency response. The advantage of this approach is the insight it provides on how the circuit elements influence the frequency response. This is especially important in the design of frequency-selective circuits. We will first consider how to generate Bode plots for simple poles, and then discuss how to handle the general second-order response. Before doing this, however, it may be helpful to review some properties of transfer functions, the decibel scale, and properties of the log function. Poles, Zeroes, and Stability The s-domain transfer function is always a rational polynomial function of the form Ns() smm as12 a s m asa Hs() K K mm12 10 (1.1) nn12 n Ds() s bsnn12 b s bsb 10 As we have seen already, the polynomials in the numerator and denominator are factored to find the poles and zeroes; these are the values of s that make the numerator or denominator zero. If we write the zeroes as zz123,, zetc., and similarly write the poles as pp123,, p , then Hs( ) can be written in factored form as ()()()s zsz sz Hs() K 12 m (1.2) ()()()s psp12 sp n 1 © Bob York 2009 2 Frequency Response and Bode Plots The pole and zero locations can be real or complex.
    [Show full text]
  • Linear Time Invariant Systems
    UNIT III LINEAR TIME INVARIANT CONTINUOUS TIME SYSTEMS CT systems – Linear Time invariant Systems – Basic properties of continuous time systems – Linearity, Causality, Time invariance, Stability – Frequency response of LTI systems – Analysis and characterization of LTI systems using Laplace transform – Computation of impulse response and transfer function using Laplace transform – Differential equation – Impulse response – Convolution integral and Frequency response. System A system may be defined as a set of elements or functional blocks which are connected together and produces an output in response to an input signal. The response or output of the system depends upon transfer function of the system. Mathematically, the functional relationship between input and output may be written as y(t)=f[x(t)] Types of system Like signals, systems may also be of two types as under: 1. Continuous-time system 2. Discrete time system Continuous time System Continuous time system may be defined as those systems in which the associated signals are also continuous. This means that input and output of continuous – time system are both continuous time signals. For example: Audio, video amplifiers, power supplies etc., are continuous time systems. Discrete time systems Discrete time system may be defined as a system in which the associated signals are also discrete time signals. This means that in a discrete time system, the input and output are both discrete time signals. For example, microprocessors, semiconductor memories, shift registers etc., are discrete time signals. LTI system:- Systems are broadly classified as continuous time systems and discrete time systems. Continuous time systems deal with continuous time signals and discrete time systems deal with discrete time system.
    [Show full text]
  • State Space Models
    State Space Models MUS420 Equations of motion for any physical system may be Introduction to Linear State Space Models conveniently formulated in terms of its state x(t): Julius O. Smith III ([email protected]) Center for Computer Research in Music and Acoustics (CCRMA) Input Forces u(t) Department of Music, Stanford University Stanford, California 94305 ft Model State x(t) x˙(t) February 5, 2019 Outline R State Space Models x˙(t)= ft[x(t),u(t)] • where Linear State Space Formulation • x(t) = state of the system at time t Markov Parameters (Impulse Response) • u(t) = vector of external inputs (typically driving forces) Transfer Function • ft = general function mapping the current state x(t) and Difference Equations to State Space Models inputs u(t) to the state time-derivative x˙(t) • Similarity Transformations The function f may be time-varying, in general • • t Modal Representation (Diagonalization) This potentially nonlinear time-varying model is • • Matlab Examples extremely general (but causal) • Even the human brain can be modeled in this form • 1 2 State-Space History Key Property of State Vector The key property of the state vector x(t) in the state 1. Classic phase-space in physics (Gibbs 1901) space formulation is that it completely determines the System state = point in position-momentum space system at time t 2. Digital computer (1950s) 3. Finite State Machines (Mealy and Moore, 1960s) Future states depend only on the current state x(t) • and on any inputs u(t) at time t and beyond 4. Finite Automata All past states and the entire input history are 5.
    [Show full text]
  • MUS420 Introduction to Linear State Space Models
    State Space Models MUS420 Equations of motion for any physical system may be Introduction to Linear State Space Models conveniently formulated in terms of its state x(t): Julius O. Smith III ([email protected]) Center for Computer Research in Music and Acoustics (CCRMA) Input Forces u(t) Department of Music, Stanford University Stanford, California 94305 ft Model State x(t) x˙(t) February 5, 2019 Outline R State Space Models x˙(t)= ft[x(t),u(t)] • where Linear State Space Formulation • x(t) = state of the system at time t Markov Parameters (Impulse Response) • u(t) = vector of external inputs (typically driving forces) Transfer Function • ft = general function mapping the current state x(t) and Difference Equations to State Space Models inputs u(t) to the state time-derivative x˙(t) • Similarity Transformations The function f may be time-varying, in general • • t Modal Representation (Diagonalization) This potentially nonlinear time-varying model is • • Matlab Examples extremely general (but causal) • Even the human brain can be modeled in this form • 1 2 State-Space History Key Property of State Vector The key property of the state vector x(t) in the state 1. Classic phase-space in physics (Gibbs 1901) space formulation is that it completely determines the System state = point in position-momentum space system at time t 2. Digital computer (1950s) 3. Finite State Machines (Mealy and Moore, 1960s) Future states depend only on the current state x(t) • and on any inputs u(t) at time t and beyond 4. Finite Automata All past states and the entire input history are 5.
    [Show full text]
  • Simplified, Physically-Informed Models of Distortion and Overdrive Guitar Effects Pedals
    Proc. of the 10th Int. Conference on Digital Audio Effects (DAFx-07), Bordeaux, France, September 10-15, 2007 SIMPLIFIED, PHYSICALLY-INFORMED MODELS OF DISTORTION AND OVERDRIVE GUITAR EFFECTS PEDALS David T. Yeh, Jonathan S. Abel and Julius O. Smith Center for Computer Research in Music and Acoustics (CCRMA) Stanford University, Stanford, CA [dtyeh|abel|jos]@ccrma.stanford.edu ABSTRACT retained, however, because intermodulation due to mixing of sub- sonic components with audio frequency components is noticeable This paper explores a computationally efficient, physically in- in the audio band. formed approach to design algorithms for emulating guitar distor- tion circuits. Two iconic effects pedals are studied: the “Distor- Stages are partitioned at points in the circuit where an active tion” pedal and the “Tube Screamer” or “Overdrive” pedal. The element with low source impedance drives a high impedance load. primary distortion mechanism in both pedals is a diode clipper This approximation is also made with less accuracy where passive with an embedded low-pass filter, and is shown to follow a non- components feed into loads with higher impedance. Neglecting linear ordinary differential equation whose solution is computa- the interaction between the stages introduces magnitude error by a tionally expensive for real-time use. In the proposed method, a scalar factor and neglects higher order terms in the transfer func- simplified model, comprising the cascade of a conditioning filter, tion that are usually small in the audio band. memoryless nonlinearity and equalization filter, is chosen for its The nonlinearity may be evaluated as a nonlinear ordinary dif- computationally efficient, numerically robust properties.
    [Show full text]
  • Control System Design Methods
    Christiansen-Sec.19.qxd 06:08:2004 6:43 PM Page 19.1 The Electronics Engineers' Handbook, 5th Edition McGraw-Hill, Section 19, pp. 19.1-19.30, 2005. SECTION 19 CONTROL SYSTEMS Control is used to modify the behavior of a system so it behaves in a specific desirable way over time. For example, we may want the speed of a car on the highway to remain as close as possible to 60 miles per hour in spite of possible hills or adverse wind; or we may want an aircraft to follow a desired altitude, heading, and velocity profile independent of wind gusts; or we may want the temperature and pressure in a reactor vessel in a chemical process plant to be maintained at desired levels. All these are being accomplished today by control methods and the above are examples of what automatic control systems are designed to do, without human intervention. Control is used whenever quantities such as speed, altitude, temperature, or voltage must be made to behave in some desirable way over time. This section provides an introduction to control system design methods. P.A., Z.G. In This Section: CHAPTER 19.1 CONTROL SYSTEM DESIGN 19.3 INTRODUCTION 19.3 Proportional-Integral-Derivative Control 19.3 The Role of Control Theory 19.4 MATHEMATICAL DESCRIPTIONS 19.4 Linear Differential Equations 19.4 State Variable Descriptions 19.5 Transfer Functions 19.7 Frequency Response 19.9 ANALYSIS OF DYNAMICAL BEHAVIOR 19.10 System Response, Modes and Stability 19.10 Response of First and Second Order Systems 19.11 Transient Response Performance Specifications for a Second Order
    [Show full text]
  • Mathematical Modeling of Control Systems
    OGATA-CH02-013-062hr 7/14/09 1:51 PM Page 13 2 Mathematical Modeling of Control Systems 2–1 INTRODUCTION In studying control systems the reader must be able to model dynamic systems in math- ematical terms and analyze their dynamic characteristics.A mathematical model of a dy- namic system is defined as a set of equations that represents the dynamics of the system accurately, or at least fairly well. Note that a mathematical model is not unique to a given system.A system may be represented in many different ways and, therefore, may have many mathematical models, depending on one’s perspective. The dynamics of many systems, whether they are mechanical, electrical, thermal, economic, biological, and so on, may be described in terms of differential equations. Such differential equations may be obtained by using physical laws governing a partic- ular system—for example, Newton’s laws for mechanical systems and Kirchhoff’s laws for electrical systems. We must always keep in mind that deriving reasonable mathe- matical models is the most important part of the entire analysis of control systems. Throughout this book we assume that the principle of causality applies to the systems considered.This means that the current output of the system (the output at time t=0) depends on the past input (the input for t<0) but does not depend on the future input (the input for t>0). Mathematical Models. Mathematical models may assume many different forms. Depending on the particular system and the particular circumstances, one mathemati- cal model may be better suited than other models.
    [Show full text]