02 Mathematical Models of Systems Mathematical Models of Mechatronics Systems. Cesare Fantuzzi, Univeristà degli Studi di Modena e Reggio Emilia

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1 Chapter02 Modeling 02 Mathematical Models of Systems Mathematical Models of Mechatronics Systems Versione Stampabile della Lezione Controlli Automatici del 7 March 2012 Cesare Fantuzzi, Univeristà degli Studi di Modena e Reggio Emilia 02 Modeling 1 1 Content and Learning Objectives The learning objectives of this lesson and exercises 1 Understand quantitative mathematical models of physical systems to design and analyze control systems We will consider a wide range of systems, including mechanical, hydraulic, and electrical We will consider the general case in which dynamic behavior is described by ordinary differential equations Since most physical systems are nonlinear, we will discuss linearization approximations 2 We will discuss internal and external behavior: Internal behavior description requires the definition of system states, that take into consideration hidden system operation (eg system actions that are not measured with sensors) that might have or minght not have influence on system output External behavior describes the system operation visible from the analysis of the system input and output (eg monitored with sensors) 3 We will then proceed to obtain the input-output relationship for components and subsystems in the form of transfer functions 4 We will recall Laplace transform as a tool for analysis of mathematical model 5 The transfer function blocks can be organized into block diagrams or signal-flow graphs to graphically depict the interconnections 02 Modeling 2 Contents 1 Content and Learning Objectives 1 2 Mechatronic System Analysis and Design 1 21 The definition of Mechatronic Systems 1 22 System Modeling 3 3 Differential equations of physical systems 9 02 Modeling 3 2 Mechatronic System Analysis and Design 21 Mechatronic Systems The definition of Mechatronic Systems Mechatronics is a holistic approach to system design and manufacture that integrates deveral disciplines, such as: Mechanic Engineering, Electronic Engineering Computer Sciences, Software Engineering, 1

2 Control Engineering, Mechatronics is the combination of several branches of engineering as mechanical, electronic, computer, software and control in order to design and manufacture devices and machines which are supervised and controlled by an automatic system Mechatronics is a multidisciplinary field of engineering, that is to say it rejects splitting engineering into separate disciplines, and for this reason it s also sometimes named as System Engineering to outline the holistic approach to the complete system design and development French standard NF E gives the following definition: approach aiming at the synergistic integration of mechanics, electronics, control theory, and computer science within product design and manufacturing, in order to improve and/or optimize its functionality Thus we can say that Mechatronics is a science that defines the System Engineering Approach to modern system design and development: The System Engineering Approach following this approach the design of a system considers a global approach, in which all the aspects, also named system views: mechanics, physics, electronics, control, are considered all togheter at the real beginning of the desgin phase 02 Modeling 4 Disk Drive System A disk Drive System A disk drive system is composed by mechanical, electrical and control parts System engineering approach requires to desing the mechanical part considering at the same time the electronic part (eg size and position of the electrical motor) and the control part (eg a spring used to auto-park the system at home position during the shut down might be replaced by a proper control of the recording heads) System Engineering Design for Mechatronics Mechatronic System Design deals with the integrated and optimal design of a physical system, including sensors, actuators, and electronic components, and its embedded digital control system The integration is respect to both hardware components and information processing An optimal choice must be made with respect to the realization of the design specifications in the different domains Functionality can be achieved either by solutions in the physical domain or by information processing in electronics or software Computer control offers extreme flexibility 02 Modeling 5 02 Modeling 6 Mechatronic System Design Team Mechanical Engineers Electro- Mechanical Engineers Modeling & Simulation Electrical Engineers Sensors Actuators Computer System Engineers Embedded Control Control Engineers Mechatronic System Design 02 Modeling 7 2

3 The troubles of the Engineers The Solution Model Based design System modeling for System Engineering 02 Modeling 8 System modeling for System Engineering Develop models of systems is a important step in system engineering Before a new prototype design for an automobile braking system or a multimillion dollar aircraft is developed and tested in the field, it is common place to test drive the separate components and the overall system in a simulated environment based on some form of model A system model allows to understand and test systems in an economical and safe manner The benefit of having a model is to be able to explore the intrinsic behavior of a system in an economical and safe manner The physical system being modeled may be inaccessible or even nonexistent as in the case of a new design The system model is based on physical observation expressed using mathematics The behavior of dynamic systems can be explained by mathematical equations and formulae, which embody either scientific principles or empirical observations, or both, related to the system 02 Modeling 9 22 System Modeling Model Based Design In the evaluation of a system design: the design-and-test approach should be replaced by modeling-analysis-simulation-design-test process The new approach requires Multidisciplinary approach that crosses domanin boundaries, to obtain a physical and mathematical modeling Model-Based Design (MBD) is a method to address the problem of system engineering in the desing and devleopment of complex mechatronic devices and components In Model-based design, development is manifested in these steps: 1 Defining the specifications of the system under development, both technical (eg speed, processing time, etc) and non-technical (eg costs, availability of spare replacment parts, etc) 2 Modeling the system using physical model described by different kinds of mathematics (continuous time differential equations, discrete time equations, discrte events, stochastic systems, etc) 3 Analyzing the behaviour of the system and develop the control system to assure system specification fulfilment 4 Simulating the overall system, 5 if necessary modify the behavior of the system and/or of the controller, that is rerun the development process from second step (system behaviour analysis) or third step (controller developement) 02 Modeling 10 3

4 6 Deploy the system into the final artifact The model based design paradigm is significantly different from traditional design methodology Rather than design and develop mechanical parts, often without a clear understanding on system specification and with no consideration about control system, designers use MBD to define models with advanced functional characteristics using modular models described using different kind of matehmatical formulae These models can be collected into component libraries of continuous-time and discrete-time building blocks These built models used with simulation tools can lead to rapid prototyping, software testing, and verification Not only is the testing and verification process enhanced, but also, in some cases, hardwarein-the-loop simulation can be used with the new design paradigm to perform testing of dynamic effects on the system more quickly and much more efficiently than with traditional design methodology The main steps in MBD approach are: 1 Plant modeling Plant modeling can be data-driven or physical principles based Data-driven plant modeling uses techniques such as System Identification With system identification, the plant model is identified by acquiring and processing raw data from a real-world system and choosing a mathematical algorithm with which to identify a mathematical model Various kinds of analysis and simulations can be performed using the identified model before it is used to design a model-based controller First principles based modeling is based on creating a block diagram model that implements known differential-algebraic equations governing plant dynamics A type of first principles based modeling is physical modeling, where a model is created by connectings blocks that represent physical elements that the actual plant consists of 2 Controller analysis and synthesis The mathematical model conceived in step 1 is used to identify dynamic characteristics of the plant model A controller can be then be synthesized based on these characteristics 3 Offline simulation and real-time simulation The time response of the dynamic system to complex, time-varying inputs is investigated This is done by simulating a simple LTI (Linear Time Invariant) system or a non-linear model of the plant with the controller Simulation allows specification, requirements, and modeling errors to be found immediately, rather than later at real system development and test Real-time simulation can be done by automatically generating code for the controller developed in the previous step This code can be deployed to a special real-time protoyping computer that can run the code and control the operation of the plant If plant prototype is not available, or testing on the prototype is dangerous or expensive, code can be automatically generated from the plant model This code can be deployed to the special real-time computer that can be connected to the target processor with running controller code This way, controller can be tested in real-time against a real-time plant model 4 Deployment Ideally this is done via automatic code generation from the controller developed in step 3 It is unlikely that the controller will work on the actual system as well as it did in simulation, so an iterative debugging process is done by analyzing results on the actual target and updating the controller model Model based design tools allow all these iterative steps to be performed in a unified visual environment Some of the notable advantages MBD offers in comparison to the traditional approach are: MBD provides a common design environment, which facilitates general communication, data analysis, and system verification between development groups Engineers can locate and correct errors early in system design, when the time and financial impact of system modification are minimized Design reuse, for upgrades and for derivative systems with expanded capabilities, is facilitated Physical System Modeling Design Concepts A Spindle actuation system of a Lathe An example of mechatronic system is the electrical actuation system of the spindle in a metalworking lathe The physical system is formed by a rotating actuator (electrical motor) with a transmission that is contained in a flexible linear suspension 02 Modeling 11 4

5 Physical System Modeling Design Concepts The system model The system model represents a system consisting of a rotating actuator with ideal transmission that is contained in a flexible linear suspension 02 Modeling 12 Physical system, physical model and Mathematical Model system How to develop a model for a physical Identify the system (eg by defining system characteristics, behavior, boundaries, etc) which is the object of the study Isolate a physical model for each component of the system (eg define general characteristics of the system, identify input, output and state variables for each component) Determine physical laws (mathematical description) that describe the behaviour of each compoennt Identify parameters present in the above physical laws 02 Modeling 13 General Mathematical models An unified description of Mathematical Systems We are going to use quantitative mathematical models of physical systems The system dynamic behavior of continuous time systems is generally described by ordinary differential equations This approch is very general and it can be used for a wide range of physical systems moreover, both the system model (physical part of a mechatronic system) and the control system (implemented ina computer, that is a immaterial part of a mechatronic system) can be described with the same mathematics and graphical schemes In summary, we use quantitative mathematical models of physical systems to design and analyze control systems The dynamic behavior is generally described by ordinary differential equations We will consider a wide range of systems, including mechanical, hydraulic, and electrical Since most physical systems are nonlinear, we will discuss linearization approximations, which allow us to use linear methods such as Laplace transform methods We will then proceed to obtain the input output relationship for components and subsystems in the form of transfer functions The transfer function blocks can be organized into block diagrams or signalflow graphs to graphically depict the interconnections Block diagrams (and signal-flow graphs) are very convenient and natural tools for designing and analyzing complicated control systems 02 Modeling 14 Develop a model of a physical system A model is a representation of the process or a system existing in reality or planned for realization which expresses the essential attributes of a process or a system in a useful form Norbert Wiener, 1945 Mathematical models are needed when quantitative relationships are required, for example, to represent the detailed behavior of the output of a feedback system to a given input Development of mathematical models is usually based on principles from the physical, biological, social, or information sciences, depending on the control system application area, and the complexity of such models varies widely One class of models, commonly called linear systems, has found very broad application in control system science Techniques for solving linear system models are well established and documented in the literature of applied mathematics and engineering, 02 Modeling 15 5

6 The process of obtaining a model The process of modeling can be summarized as follows: 1 Identify the real system to be modeled 2 Identify components and describe necessary assumptions and simplifications 3 Write the differential equations describing the model 4 Simulate the model and, conceptually in parallel, execute the real system, and obtain a time history, data from sensor for real model, simulation data for the model 5 Examine if the difference between the two is within a specified accuracy 6 If necessary, renalyze or redesign the mathmatical model Physical System Physical laws Measurements Mathematical Model Coefficients (values) Parametric equations (structure) Real system execution Sensor acquisition Computer simulation Simulation data collect Change in: (1) coefficients or (2) equations Model Validation Model in the range of accuracy? Figure 1: The process of obtaining a model 02 Modeling 16 A model of an open Tank A model of a real think The model capture only relevant thinks capable to describe the system behavior that is of interests The figures represent respectively a real tank system and a description of the physical aspects that are of importance in the modeling step The primary input of the system is the liquid flow rate F 1 (t), an independent variable measured in appropriate units such as cubic feet per minute (volumetric flow rate) or pounds per hour (mass flow rate) Responding to changes in the input are dependent variables H(t) and F 0 (t) the fluid level, and flow rate from the tank, respectively Once the derivation is completed, we can use the model to predict the outflow and fluid level response to a specific input flow rate F 1 (t),t > 0 6

7 Note that we have restricted the set of possible inputs to F 1 (t) and in the process relegated the remaining independent variables, that is, other variables which affect F 0 (t) and H(t), to secondary importance (sometime referred as huger order effects), such as fluid evaporation It s important to note that those assumption should be validated by observation and actual verification that those terma are really negligible to the prediction of output variables 02 Modeling 17 Derivation of model The mathemaical model of the system is obtained from the observation of tank conditions at two discrete points in time, as if snapshots of the tank were available at times t and t + t, as shown in following figure: Notation: F 1 (t) : Input flow at time t,ft 3 /min H(t) : Liquid level at time t,ft, F 0 (t) : Output flow at time t,ft 3 /min A : Cross-sectional area of tank, ft 2 02 Modeling 18 Mathematical model At time t + t, from the physical law of conservation of volume, we obtain: V (t + t) = V (t) + V where V (t) is the volume of liquid in the tank at time t, V is the change in volume from time t to t + t The volume of liquid in the tank at times t and t + t is given by: V (t) = AH(t) ; and: V (t + t) = AH(t + t) Previous euqations assume constant cross-sectional area of the tank, that is, A is independent of H The change in volume from t to t + t is equal to the volume of liquid flowing in during the interval t to t + t minus the volume of liquid flowing out during the same period of time The liquid volumes are the areas under the input and output volume flow rates from t to t + t 02 Modeling 19 Expressing these areas in terms of integrals: t+ t t+ t V = V (t + t) V (t) = F 1 (t)dt F 0 (t)dt t t 02 Modeling 20 The integrals in previous Equation can be approximated by assuming F 1 (t) and F o (t) are constant over the interval t to t + t t+ t t+ t F 1 (t)dt F 1 t; F 0 (t)dt F 0 t t t Previous equations are reasonable approximations provided that t is small So that, we obtain: V F 1 t F 0 t 7

8 F 0 = 1 R H 02 Modeling 23 and then: and then, we obtain, finally: AH(t + t) AH(t) + [F 1 F 0 ] t AH(t + t) AH(t) [F 1 F 0 ] t where H is the change in liquid level over the interval (t,t + t) Note that H/ t is the average rate of change in the level H over the interval (t,t + t) It is the slope of the secant line from point A to point B in the following figure 02 Modeling 21 In the limit as t approaches zero, point B approaches point A, and the average rate of change in H over the interval (t,t + t) becomes the instantaneous rate of change in H at time t, that is: H lim = dh t 0 t dt where we call dh the first derivative of function H(t) dt From the graph, it can be seen that the firt orderd derivative is equal to the slope of the tangent line of the function H(t) at t (point A) Taking the limit as t approaches zero and using the definition of the derivative give: lim A H = lim t 0 t [F 1 F 0 ] A H = [F 1 F 0 ] t 0 t 02 Modeling 22 Since there are two dependent variables, a second equation or constraint relating F 0 and H is required in order to solve for either one given the input function F 1 (t) It is convenient at this point to assume that F 0 is proportional to H, that is, F 0 = ch, being c the constant of proportionality expressed as 1/R, where R is call ed the fluid resistance of the tank In this example, the model is a coupled set of equations One is a linear differential equation and the other is an algebraic equation, also linear The differential equation is first order since only the first derivative appears in the equation and the tank dynamics are said to be first order The outflow F 0 can be eliminated from the model equations obtaining: A dh + 1 dt R H = F 1 Before a particular solution to previous equation for some F 1 (t), t 0 can be obtained, the initial tank level H(0) must be known 02 Modeling 24 Assigment 21 Mathematical model of a tank with variable cross-sectional area 02 Modeling 25 8

9 h h = 0 Top view P 0 P(h) h = 1 P 1 Lateral view L P(h) = P 0 + (P 1 P 0 ) h, h [0,1] 3 Differential equations of physical systems Modeling using mathematics The differential equations describing the dynamic performance of a physical system are obtained by utilizing the physical laws of the proces This approach applies equally well to mechanical, electrical, fluid, and thermodynamic systems The physical laws define relationships between fundamental quantities and are usually represented by equations A differential equation is any algebraic or transcendental equality which involves either differentials or derivatives 02 Modeling 26 Example in mechanical domain The scalar version of Newton s second law states that, if a force of magnitude f is applied to a mass M units, the acceleration a of the mass is related to f by equation f = Ma f = M( dv dt ) The velocity v = v(t) and force f = f (t) are dependent variable, the time t is the independent variable 02 Modeling 27 Example in electric domani Ohm s law states that, if a voltage of magnitude v is applied across a resistor of R units, the current i through the resistor is related to v by the equation v = Ri; v = R( dq dt ) The charge q = q(t) and the voltage v = v(t) are dependent variable, the time t is the independent variable 02 Modeling 28 A linear equation A linear equation is an equation consisting of sum of linear terms A linear term is one which is first degree in dependent variables and their derivatives If any term of differential equation contains higher power, products, or transcendental functions of the dependent variables, it is nonlinear ( dy dt )3,u dy dt,sin(u) 02 Modeling 29 9

10 time variant - time variant equations A time-invariant equation is an equation in which none of the terms depends explicitly on the independent variable time A time-variable equation is an equation in which one or more terms depend explicitly on the independent variable time A system in which time is the independent variable is called causal if the output depends only on present and past values of the input That is, if y(t) is the output, then y(t) depends only on the x(t d ) for values of t d t 02 Modeling 30 Through and Across variables Consider a torsional spring-mass system with applied torque T a (t), assume the torsional spring element is massless Suppose we want to measure the torque T s (t) transmitted to the mass m Since the spring is massless, the sum of the torques acting on the spring itself must b e zero: T a (t) T s (t) = 0, which implies that T s (t) = T a (t) We see immediately that the external torque T a (t) applied at the end of the spring is transmitted through the torsional spring Because of this, we refer to the torque as a through-variable In a similar manner, the angular rate difference associated with the torsional spring element is ω(t) = ω s (t) ω a (t) Thus, the angular rate difference is measured across the torsional spring element and is referred to as an across-variable These same types of arguments can be made for most common physical variables (such as force, current, volume, flow rate, etc) These definitions allows a generalization of approach to system modeling that is considered as the moder approach to modeling (such as Matlab s SIMSCAPE, 20 Sim, Dymola, etc) 02 Modeling 31 An Example of Across variable Across-variables are defined by measuring a difference, or drop, across an element, that is between nodes on a graph (across one or more branches) These variables sum to zero around any closed loop on the graph Two across variable examples are (i) velocity drop in mechanical systems, and (ii) voltage drop in electrical systems Across variables satisfy the compatibility condition n v i = 0 i=1 around the n elements in any loop on a graph, which is clearly a generalization of Kirchoff s voltage law 02 Modeling 32 An Example of Through variable Through variables are measured through an element, that is considered as being transmitted through an element unchanged These variables sum to zero at the nodes on a graph, and are said to satisfy the continuity condition Example of through variables defined are (i) current in electrical systems, and (ii) force in mechanical systems 10

11 Figure 2: To measure voltage drop in an electrical circuit you must connect a voltmeter across an element Through variables satisfy the continuity condition n f i = 0 i=1 in the n elements connected to any node on a graph, which is clearly a generalization of Kirchoff s current law Figure 3: to measure force in a mechanical system, or current in an electrical system a sensor must be inserted in series with an element 02 Modeling 33 Summary of Through - and Across-Variables for Physical Systems System Variable Through Element Variable Across Element (difference) Electrical Current, i Voltage, V a V b Mechanical translational Force, F Velocity, v a v b Mechanical rotational Torque, T Angular velocity, ω a ω b Fluid Fluid volumetric rate of flow, Q Pressure, P a P b Thermal Heat flow rate, q Temperature, T a T b 02 Modeling 34 11

12 Generalized elements An unified view of elements that affects variables in physical systems Systems considered in their energy domains can be generalized in three primitive modeling elements: Energy storage element in which energy is a function of the through variable (Inductive storage) Energy storage element in which energy is a function of the across variable (Capacitive storage) Energy dissipative element 02 Modeling 35 Inductive Storage through-variable These are the energy storage elements in which the stored energy is a function of the Physical Element Electrical inductance Equation Energy L i V 2 V 1 V 2 V 1 = L dt di E = 1 2 Li2 Translational spring K F v 2 v 1 v 2 v 1 = K 1 df dt E = 1 2 F2 k Rotational spring K ω 2 Tω 1 Fluid inertia I Q P 2 P 1 ω 2 ω 1 = K 1 dt dt P 2 P 1 = I dq dt E = 1 2 T 2 K E = 1 2 IQ2 02 Modeling 36 Capacitive Storage across-variable These are the energy storage elements in which the stored energy is a function of the Physical Element Electrical capacitance Equation Energy C i i = C d(v 2 V 1 ) dt E = 1 2 V 2 V C(V 2 V 1 ) 2 1 Translational mass M F F = M dv 2 dt E = 2 1Mv2 2 v 2 v 1 = const Rotational mass T = J dω 2 T J E = 1 dt 2 J(ω 2) 2 ω 1 = const ω 2 02 Modeling 37 12

13 Energy dissipators the element) These are the dissipative elements (non-energy storage, the power flow is always into Physical Element Electrical resistance Equation Power R i i = R 1 (V 2 V 1 ) P = R 1 (V 2 V 1 ) 2 V 2 V 1 Translational mass b F F = bv 2 P = bv 2 2 v 2 v 1 = const Rotational mass b T T = bω 2 P = bω2 2 ω 2 ω 1 = const 02 Modeling 38 An example: An automobile shock absorber An automotive shock absorber A shock absorber is a mechanical device designed to smooth out or damp shock impulse, and dissipate kinetic energy 02 Modeling 39 Shock absorber model Conceptual model The system can be represented by a mass M with a spring K and a friction effect B Free body diagram We model the wall friction as a viscous damper, the friction force is linearly proportional to the velocity of the mass In reality the wall friction may behave as a Coulomb damper- a dry friction,which is a nonlinear function of mass velocity and possesses a discontinuity around zero velocity 13

14 v 1 = 0 F k F M F b v 2 r 02 Modeling 40 Deriving mathematical equations The mathematical model can be described as: a capacitive storage (a mass M), an inductive storage (a spring K) and a dissipative element (a dumper B) Summing the forces acting on M and utilizing Newton s second law yields: F + F k + F b = r F = M dv 2 dt ; F b = Bv 2 ; v 2 = 1 df k K dt ; v 2 = dy dt r = M d2 y dt 2 + B dy dt + Ky (1) 02 Modeling 41 A system with two carts connected by a flexible link Conceptual Model Free body diagram M 2 F b1 Fb2 F M2 v 2 F k F M1 M 1 v 1 u 02 Modeling 42 Mathematical model Three elements are present: Two capacitive storage elements: F M1 = M 1 x 1, and F M2 = M 2 ẍ 2, One inductive storage element: F k = (x 2 x 1 )k Two dissipative elements: F b1 = b 1 ẋ 1, and F b2 = b 2 ẋ 2 so that we can write: F M1 + F k + F b1 = u, so that M 1 ẍ 1 = k(x 1 x 2 ) b 1 ẋ 1 + u F M2 F k + F b2 = 0, so that M 2 ẍ 2 = +k(x 1 x 2 ) b 2 ẋ 2 02 Modeling 43 14

15 An example An Electrical Circuit RLC Circutit current driven L i L C i C V 1 V 2 R i R i R + i L + i C = i, i = V 2 V 1 R i +C d(v 2 V 1 ) + 1 dt L t (V 2 V 1 )dt 0 02 Modeling 44 Assigments 22 and 23 models Calculate the differential equation corresponding to the following conceputal 02 Modeling 45 Linear approximations of physical systems A non linear system 02 Modeling 46 15

16 Linearization Using linear equations to describe a non linear world Most, almost all, physical systems are nonlinear on the other hands the mathematical equations are tractable if they are linear therefore we will discuss the linearization process to obtain a mathematically tractable equation from a real model 02 Modeling 47 Superposition and homogeneity principles A great majority of physical systems are linear within some (sometime small) range of the process variables In general, however systems ultimately become always nonlinear as the variables are increased without limit For example, a spring-mass-damper system has a linear behaviour as long as the mass is subjected to small deflections However, if the force were continually increased, eventually the spring would be overextended and break This behavior applies in general to all the actuation systems For example an electrical motor has a range of working mode that corresponds to a maximum torque that it can actuate If a greater torque is requested by the control system, the motor gives a maximum torque, but not anymore This can of behaviour is called saturation of the actuator Therefore the question of linearity and the range of applicability must be considered for each system A system is defined as linear in terms of the system excitation, the system input: u(t), and response, the system output: y(t) Having these concepts defined, it can de introduced the: Superposition principle When the system at rest is subjected to an excitation u 1 (t) it provides a response y 1 (t) Furthermore, when the system is subjected to an excitation u 2 (t), it provides a corresponding response y 2 (t) For a linear system, it is necessary that the excitation u 1 (t) + u 2 (t) result in a response y 1 (t) + y 2 (t) This i s usually called the principle of superposition Homogeneity principle Furthermore, the magnitude scale factor must be preserved in a linear system Again, consider a system with an input u(t) that results in an output y(t) Then the response of a linear system to a constant multiple β of an input u(t) must be equal to the response to the input multiplied by the same constant so that the output is equal to βy(t) This is called the property of homogeneity In other words we define as linear a system which behavior can be described by a mathematical function: f : u U y Y so that given any two real numbers λ and γ and any two values u 1,u 2 U, the following equality holds: f (λu 1 + γu 2 ) = λ f (u 1 ) + γ f (u 2 ) 02 Modeling 48 System linearization The linearity of many mechanical and electrical elements can be assumed over a reasonably large range of the variables This is not usually the case for thermal and fluid elements, which are more frequently nonlinear in character Fortunately, however, one can often linearize nonlinear elements assuming small-signal conditions This is the normal approach used to obtain a linear equivalent circuit for electronic circuits and transistors Consider a general element with an excitation (through) variable u(t) and a response (across) variable y(t) The relationship of the two variables is written as y(t) = f (u(t)) where f (u(t)) indicates that output y(t) is a nonlinear function f () of input u(t) We designate the normal operating point of the system as u 0 If we assume that f () is contiunous arount operating point u 0, that is usually the case of physical system, expecially if the product of through and across variable is an energy, we can write the Taylor series of function f () around the operating point u 0 : y(t) = f (u 0 ) + d f (u(t)) du(t) (u(t) u 0 )+ u(t)=u0 d 2 f (u(t)) du(t) 2 (u(t) u 0 ) 2 + u(t)=u0 2! 02 Modeling 49 16

17 The slope at the operating point, d f (u(t)) du(t) u(t)=u0 is a good approximation to the curve over a small range of (u(t) u 0 ), the deviation from the operating point Then, as a reasonable approximation, we can take only the first term of the Taylor series: y(t) = f (u 0 ) + d f (u(t)) du(t) where m is the slope at the operating point (u(t) u 0 ) = m(u(t) u 0 ) u(t)=u0 Finally, it can be written a linear approximation of nonlinear system: 02 Modeling 50 or y(t) f (u 0 ) = m(u(t) u 0 ) y(t) = m u(t) 02 Modeling 51 An example Application to a non linear spring Consider the case of a mass M sitting on a nonlinear spring, which have the following nonlinear characteristics: f = y 2 The normal operating point is the equilibrium position that occurs when the spring force balances the gravitational force Mg, where g is the gravitational constant Thus, we obtain: Mg = f 0 = y 2 0 For the nonlinear spring with the equilibrium position is The linear model for small deviation is where Thus, y 0 = (Mg) f = m y m = d f dy y= (Mg) 02 Modeling 52 m = 2y 0 = 2 (Mg) A linear approximation is as accurate as the assumption of small signals is applicable to the specific problem 02 Modeling 53 17