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Motor Modeling and Position Control LabWeek 3: Closed Loop Control1. ReviewIn the rst week of motor modeling lab, a mathematical model of a DC motor from rst principles wasderived to obtain a rst order system. The open and closed loop (proportional-derivative) control was imple-mented specically for this motor model. In the second week, a physical DC motor (Quanser SRV-02) wasused for open-loop control implementation and the rst order transient characteristics were observed. Basedon the model response, DC motor parameters (time constant) were estimated both by hand-calculations aswell as using MATLAB. You should have also observed in the open loop control of actual DC motor thatthe motor positions start to drift over time indicating continuous accumulation of error within the system.Another observation that should have been made is that there is no way to enforce the output of the motorto track the input voltage in the absence of any feedback loop.In the nal week of this lab, you will try to address some of these issues by realizing the benets ofclosed-loop control of DC motor. In particular, you will:1. study transient characteristics of a typical second order systemand evaluate model or systemresponsesusing these specications.2. extend the closed loop control implemented in the rst week of this lab to the actual DC motor3. analyze the eects of proportional-, derivative- and integral- control individually and in combinationon the closed loop response of motor4. solve a position control problem by calculating PD controller gains analytically and validate the con-trol by monitoring the motor response for dierent desired trajectories5. design a PID controller for the actual DC motor using Ziegler-Nichols method and compare theperformance with that of the PD controller2. DC Motor ModelWe derived the mathematical model of DC motor earlier and obtained the following rst order transferfunction that relates the motor velocity (rad/s) to input voltage (V) as:l(s)Vm(s) =Ks + 1. (1)where is the mechanical time constant of the system, and K is the steady state gain(also known as DCgain).Since, angular position can be obtained by integration of angular velocity, the open loop transfer functionbetween angular position (rad) and input voltage (V) can be obtained from (1) as in (2):l(s)Vm(s) =Ks (s + 1) =Ks2+ s =as2+ bs l (s) =1sl (s) (2)1The open loop control of the DC servo motor is given in Fig.11. You should remember that simulationof these two forms of mathematical motor models were performed in the rst week of motor lab as wellas the experiments with actual DC motor in the last week to estimate model parameters. Next, let us studyabout transient response characteristics of typical 2nd order systems.Figure 1: Open Loop Control of DC Servo Motor3. Transient Characteristics of Second Order SystemsSystems that store energy cannot respond instantaneously, so they exhibit a transient response when they aresubjected to inputs or disturbances. Consequently, the transient response characteristics constitute one of themost important factors in system design. In many practical cases, the desired performance characteristics ofcontrol systems can be given in terms of transientresponse specications and step inputs are commonly usedinput for this purpose, since such an input is easy to generate and is suciently drastic. Mathematically,if the response of a linear system to a step input is known, by principle of superposition and linear theoryassumptions, it is possible to compute the systems response to any input. However, the transient responseof a system to a unit stepinput depends on initial conditions. For convenience in comparing the transientresponses of various systems, it is thus a common practice to use standard initial conditions: the system isinitially considered to be at rest, with its output and all time derivatives thereof zero to facilitate comparisonof dierent transient responses (if you had not realized this earlier). By ensuring these protocols, it will thenbe possible to compare transient responses for dierent controller parameters and design our controllers forgiven transient response specications.In general, transfer function of a 2nd order system with input, u(t) and output, y(t) can be expressed asin (3),G(s) =Y(s)U(s) =k 2ns2+ 2ns + 2n(3)U(s) and Y(s) are Laplace transforms of u(t) and y(t) respectivelyThe response for such a systemis given in Fig.2 marked with the transient response specications that aredened below and any or all of the following parameters can be used to specify such responses completely:1. Rise time (tr): Time required for a signal to change from a specied low value to a specied highvalue. Typically, these values are 10% and 90% of the input step size.2. Maximum Overshoot (Mp): It is measured as a ratio of value from the maximum peak of the responsecurve measured to the desired response of the system and nal steady state value. Typically, it is1Note: When designing or analyzing a system, often it is useful to model the system graphically. Block Diagrams are a usefuland simple method for analyzing a system graphically. You will see a lot more of these in this lab manual2measured for step inputs in which case, the percentage overshoot (PO) is the maximum value minusthe input step size divided by the step size % of ratio of value.3. Settling time (ts): Time elapsed from the application of an ideal instantaneous step input to the timeat which the output has entered and remained within a specied error band (typically within 2 % or5% within the nal value).4. Delay time (td): Time required for the response to reach half the nal value the very rst time.5. Peak time (tp): Time required for the response to reach the rst peak of the overshoot.6. Steady-state error (ess): Dierence between the desired nal output_yssdes_ and the actual responsewhen the system reaches a steady state (yss), when its behavior may be expected to continue if thesystem is undisturbed.Figure 2: Typical 2nd Order Motor Response and Transient CharacteristicsGiven a 2nd order system response, these parameters can be manually estimated and responses fordierent inputs can be compared. At the same time, by following rigorous derivations, these parameters canbe expressed as a function of n and and without delving into full details, the nal results are summarizedin the table 1. We will discuss how you can use MATLAB to trivially calculate these values later.So once we give the values of tr, ts, td, tp and Mp, then the transient response from Fig.2 can be com-pletely specied. Nevertheless, in most real applications, desired values of these parameters would be givenand the objective will be to design controllers that can meet the requirements. Some desirable characteristics3Parameter Symbol (unit) FormulaRise Time tr(s) d , where = tan1___1 2__d = _1 2Maximum Overshoot Mp(%)e_1 2 100Delay Time td(s) 1 + 0.7nSettling Time ts(s)4n (2 % setting time)3n (5 % settling time)Peak Time tp(s) dSteady State Error ess _limt y(t)_ yssdesTable 1: Transient System Specicationsin addition of requiring a dynamic system to be stable, i.e., its response does not increase unbounded withtime (a condition that is satised for a second order system provided that 0) are the system shouldpossess: faster and instantaneous response minimal overshoot above the desired value (i.e., relatively stable) and ability to reach and remain close to the desired reference value in the minimum time possible.We will use these parameters to analyze the DC motor system under dierent form of controls andoptimize the controller gains to achieve desired performance by end of this lab session.4. Closed Loop Control SystemsClosed loop control system uses feedback to determine the actual input to the system. Feedback meansthat the information about the current states(e.g. position, velocity) of the system is used by the controllerto continuously correct the actual input to the system in order to reach the desired states. Typically, suchsystems are expressed using block representation and the resulting transfer function as in Fig. 3. Here(G(s)) refers to transfer function of the actual plant (or physical system) being controlled, [Gc (s)] refersto the Laplace transform of the controller, (H (s)) refers to output feedback transfer function and (E (s))is Laplace transform of the error signal term, _E (s) = L_u(t) y(t)__ that is input to the controller. Thecontroller output, Up (s), is given by, Up (s) = Gc (s) E (s) which is input to the controlled plant.In our experiments, we only use unity feedback systems, meaning, H(s) = 1.4Figure 3: Closed Loop Control of Plant (G(s))In control systems application, the input signal, U(s) for the system in Fig.3 or motor voltage, Vm (s)for the DC motor plant in (2) as well as in Fig. 1, is actually the desired trajectory or a set point that wewould like our system to execute or reach respectively over time. The controller, Gc (s), can be any type ofcontroller with transducer depending on the application. Then, the plant input, Up (s) is simply the actualcontrolled signal input to the plant to ensure it reaches the desired state. For example, in the case of DCmotor plant, Vm, will be desired angular trajectory, d (t) or set-point position, d for constant position andVi, will be actual controlled input signal in volts generated by the controller, Gc (s) fed to the DC motor.Most commonly used type of controllers are the ones discussed below, that we will also study in this lab.4.1 Proportional Control (Kp)A proportional controller (P) consists of only a constant gain block as shown in the block diagram Fig.4. The output of the controller will be a constant multiple of the dierence between the input signaland current state. For example, by adding a proportional gain blo

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