控制理论基础-自控a_ch2(1)

1,CH2 . Mathematical Models of Systems,2.1 Introduction2.2 Differential Equations of physical systems2.3 Linear Approximations of systems 2.4 The Laplace transform2.5 The Transfer Function of Linear Systems2.6 Block Diagram Models2.7 Signal-Flow Graph Models,2,2.1 Introduction,The approach to dynamic system problem:,6. If necessary, reanalyze or redesign the system.,1. Define the system and its components.,2. Formulate the mathematical model and list the necessary assumptions.,3. Write differential equations describing the model.,4. Solve the equations for the desired output variables,5. Examine the solutions and the assumptions.,3,2.2 Differential Equations,The differential equations describing the dynamic performance of a physical system are obtained by utilizing the physical laws of the process. This approach applies equally well to mechanical, electrical, fluid, and thermodynamic systems.,（运用物理学定律列写微分方程）,机械学、电力电子学、流体力学、热力学系统,4,Table 2.1 Summary of Through- and Across-Variables for Physical Systems,Electrical: current i, charge q, voltage v, flux linkage (磁链) lMechanical translational: force F, translational momentum P, velocity v, displacement yMechanical rotational: torque T, angular momentum h, angular velocity w, angular displacement qFluid: fluid volumetric rate of flow Q, volume V, pressure P, pressure momentum gThermal: heat flow rate q, heat energy H, temperature T,5,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 Kirchhoffs current law or Kirchhoffs voltage law). Mechanical systems (Newtons laws of motion) Hydraulic systems (Thermodynamic & Conservation (守恒) of matter) Thermal systems (Heat transfer laws, Conservation of energy),Page 44, Table 2.2,6,平衡,7,8,9,动能,焦耳,10,势能,地心引力,11,11,(电动势),12,12,13,Example.2.1 Displacement system of spring-mass-damper,14,Example 2.2： Differential equation of a RLC network. uc(t) is the output voltage and u(t) is the voltage source.,Figure2.2 RLC network,15,The equation of the RLC network :,Therefore, we have,16,are the time constants of the network.,17,Analogous variables and systems,Displacement x(t) and voltage uc(t) are analogous variables, and the systems are analogous systems (相似系统).,18,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.,19,Conclusion: Different physical system can gain similar differential equations.The differential equations reflect the essence (本质) characteristics of system.,20,The standard differential equation:,output,input,21,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. where nm for a physically realizable system.,22,Example 2.3 Differential equation of dc motor.,dc motor wiring diagram,23,The armature(电枢) current is related to the input voltage applied to the armature as： Ea is the back electromotive-force voltage proportional（比例） to the motor speedU is input voltageIa is the armature current,La is the motor inductanceRa is motor resistance.,24,M is the motor torque, ML is the load torque, J is the rotor（转子） inertia, is the (angular) velocity of the motor bearings.,where kd is defined as the motor constant.,25,26,There are two first-order differential equations and two algebraic equations, and six variables U, Ea, Ia, M, ML, J,. U and ML are the input variables, which lead to the motors movement.,Regard as be the output variable:,27,is the field time constant of the armature (电磁时间常数); is the time constant of the motor armature (机电时间常数);Generally, If Ta is neglected, the equation is,28,2.3 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.,29,A linear system satisfies the properties of superposition and homogeneity.Taylor series expansion,30,Approximation: or,31,32,Example 2.4 Nonlinear differential equationOperating point:（T0，u0）Small change:,33,Linear approximation of differential equation,34,y depends upon several excitation variables: The approximation:,35,2.4 The Laplace Transform,1. Definition of the Laplace transform,拉普拉斯变换是 f(t)从时域到复频域F(s)的积分变换。,f(t)：原函数；F(s)：f(t)在s域中的象函数。,拉普拉斯反变换：Inverse of Lapalace transform,36,s-plane,37,2. Laplace transform theorems,Theorem 1 Linearity,Theorem 2 Complex differentiation,Theorem 3 Real integration,38,Theorem 4 Delay theorem,Theorem 5 Shifting theorem,Theorem 6 Final value theorem,Theorem 7 Initial value theorem,39,常见函数的拉氏变换,1、指数函数,2、单位阶跃,3、正弦函数,4、余弦函数,5、冲激函数,Table 2.3 at page 51,40,3. Partial-Fraction Expansion Theorems,Root of N(s)=0-zerosRoot of D(s)=0-poles D(s)=0-characteristic function,设法把F(s)分解成若干个较简单的、能够从表中查到的项的和，通过查表，可直接得到所求的原函数，这称为拉普拉斯反变换的部分分式法。,41,系数的确定：,(1) 不等实根,D(s)=0的根有三种情况：,42,(2) 共轭复数根,43,设F(s)在s1处有三重根，则：,(3) 重根,44,45,4. Application of the Laplace transform,电阻元件：,LU=L Ri，LU=U(s)，Li=I(s)U(s)=RI(s),46,电容元件:,47,UL(s)=sLIL(s)LiL(0),电感元件:,48,图(a),图(b),49,2.5 Transfer function,One of the most powerful tools for control system analysis and design is the transfer function.For a SISO system with input u(t) and output y(t), the transfer function is defined as (Zero initial conditions): Laplace transform of the input-output relation of a system.,50,Steps:Define the system and its components and identify the input and output of the system.Formulate the mathematical model and list the necessary assumptions.Write the differential equations describing the system.Linearize the differential equation.(*)Take Laplace transform with Zero initial conditions.Determine the ratio output/input which is called the Transfer Function.,51,Consider a linear time invariant system defined by:,52,The above transfer function can be written as:,53,n = l+k Order of the systemzi Zeros of the systempi Poles of the systemK = b0 / a0 Gain of the systeml Type of the system Poles and zeros are either real, or form complex conjugate pairs.,54,Zeros and Poles,55,Characters:A transfer function is an input-output description of the behavior of a system. 2. The transfer function does not include any information concerning the internal structure of the system and its behavior. 3. Transfer function is a property of system itself, independent of the magnitude and nature of the input or driving function.,56,4. Transfer function may be established experimentally by introducing known input and studying the output of the sys