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## Automatic Control Theory

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**Automatic Control Theory**Taught by Prof. ZHANG Duanjin School of Information Engineering, Zhengzhou University, P R China Email: djzhang@zzu.edu.cn**Chapter2 System Modeling**• 2.1Mathematical models • 2.2Dynamics equation • 2.3State space and state equation • 2.4Linear differential equation • 2.5State equation • 2.6Transfer function(matrix) • 2.7Transfer function of closed-loop system • 2.8Fundamental parts • 2.9Signal flow graphs • 2.10Impulse response and step response • 2.11Summary Excises**2.1Mathematical Models**• The fundamental concept of mathematical • model • The fundamental forms of mathematical • models • The method of modeling • The steps of analyzing and studying a • dynamic system • Instructional objectives • Appendix : Property of linear system**1.Mathematical model**If the dynamic behavior of a physical system can be represented by an equation, or a set of equations, this is referred to as the mathematical model of the system. Model of system – the relationship between variables in system. Why do we must study the model of control system? Quantitative mathematical models must be obtained to understand and control complex dynamic systems. (analysis and design)**2.The fundamental forms of mathematical model**• Mathematical expression • Differential Equation :(time domain model)are used because the systems are dynamic in nature. • Impulse transfer function; • state-space equation. • Assumptions are needed because of the complexity of systems and the ignorance of all the relevant factors..**Transfer function(s domain model)**• Laplace transform is utilized to simplify the method of solution for linear equations. • Block diagram and Signal flow diagram • Response Curve (non parameter model) • frequency response curve; Bode diagram**3.The method of modeling**• Analytical method :It can be constructed from knowledge of the physical characteristicsof the system • Experimental method • Others. (NN) • Linear Time-invariant (constant) Parameter-lumped**4.The steps of analyzing and studying a dynamic system**• Define the system and its components. • Formulate the mathematic model and list the necessary assumptions. • Write the differential equations describing the model. • Solve the equations for the desired output variables. • Examine the solutions and the assumptions. • If necessary, reanalyze or redesign the system.**5.Instructional objectives**• Develop dynamic models of physical components. • Derivation of transfer functions. • Block diagram representation. • Block diagram rules and simplify the block diagram to determine the closed-loop transfer function.**6.Appendix : Property of linear system**The mathematical model of a system is linear, if it obeys the principle of superposition (or principle of homogeneity). This principle implies that if a system model has responses y1(t) and y2(t) to any two inputs x1(t) and x2(t) respectively, then the system response to the linear combination of these inputs: ax1(t) + bx2(t) is given by the linear combination of the individual outputs, i.e. ay1(t) + by2(t).**2.2Dynamics equation**2.2.1Derivation of the differential equations 2.2.2 Linear approximations of nonlinear equation 2.2.3 The differential equations of complex plants 2.2.4 Single variable differential equation’s derivation of original equations 2.2.5 Dynamics equation of discrete-time system Return**2.2.1 Derivation of the differential equations**In order to analyze the behavior of physical systems in time domain, we write the differential equations representing those systems: Electrical systems (KVL and/or KCL, a electric network can be modeled as a set of nodal equations using Kirchhoff’s current law or Kirchhoff’s voltage law).**Mechanical systems (Newton’s laws of motion)**• Hydraulic systems (Thermodynamic & Conservation of matter) • Thermal systems (Heat transfer laws, Conservation of energy) • The standard differential equation is: output input**The number of independent energy storage components in a**system determine the order of that system. • An nth order differential equation implies n independent energy storing components and you need n initial conditions. • wheren≥mfor a physically realizable system.**R**L + + uc(t) C u(t) • Example 2.2： Differential equation of a R-L-C network. • uc(t) is the output voltage and u(t)is the voltage source. i(t) Figure2.2 R-L-C network**The equation of the RLC network shown in Fig.2.2 is**obtained by writing the Kirchhoff voltage equation, yielding Therefore, solving Eq.(2) for i and substituting in Eq.(1), we have**Analogous variables and systems**Rewriting the following equations in terms of displacement and voltage respectively It is obvious that they are equivalent. Displacement x(t) and voltage uc(t) are equivalent variables, usually called analogous variables, and the systems are analogoussystems.**The concept of analogous systems is a very useful and**powerful technique for system modeling. • Analogous systems with similar solutions exist for electrical, mechanical, thermal, and fluid systems. • The analyst can extend the solution and the understanding of one system to all analogous systems with the same describing differential equations. • Analogous systems have the same form solution.**Conclusion:**• Different physical system can gain similar differential equations. • The differential equations reflect the essence characteristics of system.**Ia**MΩML J U(t) Ea Load Ra,La Φd Example 2.3 Differential equation of dc motor. dc motor wiring diagram**The armature current is related to the input voltage**applied to the armature as： • where Eais the back electromotive-force voltage proportional to the motor speed, U is input voltage, Ia is the armature current,，Lais the motor inductance, Ra is motor resistance.**M is the motor torque, ML is the load torque, J is the rotor**inertia, Ω is the (angular) velocity of the motor bearings. wherekd is defined as the motor constant.**There are two first-order differential equations and two**algebraic equations, and six variables U, Ea,,Ia, M, ML, J,Ω. Uand MLare the input variables,which lead to the motor’s movement. If we regard as Ω be the output variable, others be the middle variables. The differential equation of the motor-load combination is**is the field time constant of the**armature; • is the time constant of the motor armature; • Generally, • If Tais neglected, the equation is Return**2.2.2 Linear Approximations of nonlinear equation**A great majority of physical systems are linear within some range of the variables. However, all systems ultimately become nonlinear as the variables are increased without limit. A system is defined as linear in terms of the system excitation and response. Keywords:operating point; small-signal conditions; Taylor series; continuous; linear approximation**The relationship of two variables written as**, where f(x) indicates y is a function of x. The normal operating point is designated by x0. Because the curve (function) is continuous over the range of interest, a Taylor series expansion about the operating point may be utilized. Then we have**The slope at the operating point,**• is a good approximation to the curve over a small range of (x-x0)(small perturbation), the deviation from the operating point. Then, as a reasonable approximation, the equation can be rewritten as the linear equation • where m is the slope at the operating point.**Example 2.4 Nonlinear differential equation**• Operating point:（T0，u0） • Small change:**If the dependent variable y depends upon several**excitation variables, then the functional relationship is written as • The Taylor series expansion about the operating point x0(x10, x20,…, xn0) is useful for a linear approximation to the nonlinear function. When the higher-order terms are neglected, the linear approximation is written as Return**2.2.3 The differential equations of complex plants**• Generally steps: • (1)Confirm the I/O variables of system and every components; • (2)Write the dynamics equations of every components using physical laws; • (3)Check up the number of equations and decide if it is equal to the number of unknown variables or not. • (4)Expurgate the inside variables from the set of the differential equations, and change to the standard form.**Example 2.5 Servo-system**• Work principle: Two same changeable resistor are supplied by the same dc electrical source; The arm of resistor 1 can rotate by handle 3. Supposing ψ、φ represent the position of the arm of the two resistors respectively. If ψis not equal toφ, then the error signal upis formed, which be amplified by 4, finally the field current If of the dc dynamotor’s field winding is formed, which lead to the voltage’s change of dynamotor 5. So the dc motor 8 rotates, which lead to the load 10 rotates too by the gear 9. So the arm of resistor 2 starts to move, as far asψ=φ.**Write out dynamics equations of every components:**• (1)the set of changeable resistances： • Inputs:ψ,φ; output:up • （1） • Where kd is the coefficient of the resistance. • (2)amplifier： • Input:up , output:If; • or （2） • where ka is the magnified multiple of voltage; Rf is the sum resistance of output circuit; Lf is inductance of field winding 6; Tf=Lf /Rfis the time constant of field circuit.**(3)dynamotor-motor:**• Input:If;output: Ω. • From example 2.2: • (3) • (4)drive institution: • Input:Ω;output:φ. • (4) • where kt is the drive ratio of the drive institution.**Equation（1）~（4）are the mathematics models of the**servo-system. It consists of one second-order differential equation, two first-order differential equations and one algebraic equation. There are six variables such asψ、φ、up 、If、ML 、Ω.To the whole system, the inputs are ψ andML, others are the inside dependent variables, and the number is equal to the number of the equations. Return**2.2.4Scalar differential equation’s derivation of original**equations (1)Confirm the I/O variables of system and every components; (2)Write the dynamics equations of every components using physical laws; (3)Check up the number of equations and decide if it is equal to the number of unknown variables or not. (4)Expurgate the inside variables from the set of the differential equations, and change to the standard form. Return**2.2.5Dynamics equation of discrete-time system**• The discrete-time approximation is based on the division of the time axis into sufficiently small time increments. The values of the input/output are evaluated at the successive time intervals, that is, t=0,T,2T,…, where T is the increment of time: =T. If the time increment T is sufficiently small compared with the time constants of the system, the discrete-time series will be reasonably accurate.**Difference equation: reflect the relationship of the input**and output series at the every sample points of the discrete time system. Using T denotes the space between two sampling time. Difference equation describes the movement of these plants. Details in chapter nine.