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MSU ME 451 - Modeling and Experimental Validation of a First Order Plant Model

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Short Form Laboratory ReportReferences:1. Objective2. Background2.1. First-order systems2.2. DC Servo Motor SystemPre-Lab Sample Questions3. Description of Experimental Setup3.1. Hardware and software3.2. Basic setup (refer to circuit diagram Fig 5 given below)4. Experimental ProceduresPart A: Steady state responsePart B: Transient response: Effect of operating pointQuestions:Part C: Transient response: Effect of varying rotor inertia JDepartment of Mechanical EngineeringMichigan State UniversityEast Lansing, MI 48824-1226ME451 LaboratoryModeling and Experimental Validation of aFirst Order Plant Model: DC Servo Motor__________________ME451 Laboratory Manual Pages, Last Revised: 1/14/2010ME 451: Control Systems LaboratorySend comments to: Dr. Clark Radcliffe, ProfessorModeling and Experimental Validation of a First Order Plant Model: DC Servo Motor2ME 451: Control Systems LaboratoryReferences:C.L. Phillips and R.D. Harbor, Feedback Control Systems, Prentice Hall, 4th Ed.Section 2.7, pp. 38-43: Electromechanical SystemsSection 4.1, pp. 116-120: Time Response of First Order SystemsAppendix B, pp. 635-650: Laplace Transform (Particularly the “Final Value Theorem…)1. ObjectiveLinear time-invariant dynamical systems are categorized under first-order systems, second-order systems, and higher-order systems. The transfer function of all first-order systems has a standard form. This enables us to investigate the response of first-order systems collectively, for any specific input. The response of a first-order system depends on its DC gain, K, and time constant, . Both K and  are function of system parameters. The objective of this experiment is to model a first-order system and investigate the effect of system parameters on its response to a step input.We choose to experiment with an armature controlled DC servomotor, which behaves as a first-order system when the armature voltage is the input and the angular speed is the output. We obtain the transfer function of the motor and identify specific parameters of the system that affect system response. Specifically, we identify system parameters that individually affect the DC gain and the time constant and vary these parameters to experimentally verify the change in system response.2. Background2.1. First-order systemsThe standard form of transfer function of a first-order system is:)1()()()(sKsUsYsG (1)where Y(s) and U(s) are the Laplace transforms of the output and input variables, respectively, Kis the DC gain, and  is the time constant. For a unit step input /s U(s) 1, the response of the system is:)1(1)1()()()()(ssKssKsUsUsYsY(2)The inverse of the resulting Laplace transform can be easily found (see the Appendix in your text). Typically the inverse is available in standard tables. In this case,)1()1(1)1()(/11teKssLKssKLty(3)It is clear from (3)) that Ky  as t. The DC gain can therefore be interpreted as the final value of the output for a unit step input. The time constant is the time required for y(t) to reach 63.2% of its final value. Indeed, at =t , K. y(t) 6320 for a unit step input. For a unit step input, the change in input is one (1). In general, for a step input of magnitude A, at =t , KA. y(t) 6320. The response of the first-order system to a unit step input is shown in Fig.1a for two cases. For a system gain1K, the system’s output change is less than the input change applied. For a system gain1K. the system’s output change is more than the input change applied. The results plotted are for a system operating for small positive input and output deviations from zero (the origin).Modeling and Experimental Validation of a First Order Plant Model: DC Servo Motor3ME 451: Control Systems LaboratoryFigure 1a: First-Order System Step ResponseFigure 1b: Periodic First-Order System Step ResponseThe step response of the DC motor will be evaluated with a square wave input composed of a series of positive and negative steps. As shown in Figure 1b, these steps produce repeated positive and negative changes in a 1st order system’s output. Assuming the positive system’s response reaches steady-state for each positive and negative input, the gain and time constant parameters can be separately evaluated for both positive and negative input changes. Thespecific values for the gain and time constant parameters for the above systems are computed below. Notice that thesystem gains are equal but that the positive and negative change time constants are not. For both parameters, an average is typically used as a representative value. The variation from the average indicates the repeatability of the measurement.Table 1: Typical Gain and Time Constant Measurements and ComputationsInput ChangeInputOutput ChangeOutputTime to63.2% changeSystem GainOutput/InputTime ConstantRising 0.9-0.0 = 0.9 0.65 0.8-0.0 = 0.8 0.65/0.9 = 0.7 0.8Falling 0.0-0.9=-0.9 0.65 6.2-5.0 = 1.2 0.65/0.9 = 0.7 1.2Average ----------- -------------- -------------- 0.7 ± 0.0 1.0 ± 0.2Modeling and Experimental Validation of a First Order Plant Model: DC Servo Motor4ME 451: Control Systems LaboratoryFigure 2: The DC Servomotor (Phillips and Harbor)2.2. DC Servo Motor SystemA schematic diagram of an armature controlled DC servomotor is shown in Fig.2. The system variables include:ae: armature drive potential (volts). me: back emf potential (volts)ai: armature current (Amps)T: torque produced by motor (N-m): angular position of motor shaft (radians)dtd: angular velocity of motor shaft (rad/sec)The parameters of the system include:mR: armature resistance (Ohms)mL: armature inductance (Henry)J: moment of inertia of motor shaft (Kg-m2)B: coefficient of viscous friction (N-m-sec/rad)The system parameters not shown in Fig.2 include:TK: torque constant (N-m/Amp)bK: motor back emf constant (volt-sec/rad)The torque constant TK models (Phillips and Harbor, 4th Edition,, Section 2.7.2) the relationship between the electric current ai input and motor torque T output.)()( siKsTaT(4)Modeling and Experimental Validation of a First Order Plant Model: DC Servo MotorT,5ME 451: Control Systems LaboratoryThe back EMF constant bK models the relationship between the motor speed  input and the electrical back emf be produced by the DC motor, )()( sKsebm (5)The


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MSU ME 451 - Modeling and Experimental Validation of a First Order Plant Model

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