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Chapter 1 deals with plant process characterization and the PID control of the plant using computer simulation software. The software being use is LabVIEW. The report will also present an introduction to a plant setup, wiring and proper operation. Theory There are many situations that require some type of servo-control system. This section reviews the fundamentals of PID controllers. Proportional-Integral-Derivative (PID) Control PID controllers are commonly used to regulate the time-domain behavior of many different types of dynamic plants. These controllers are extremely popular because they can usually provide good closed-loop response characteristics, can be tuned using relatively simple design rules, and are easy to construct using either analog or digital components. Consider the feedback system architecture that is shown in Figure 1.1, where it can be assumed that the plant is a DC motor whose shaft position must be accurately regulated. r(t) y(t) e(t) Figure 1.1: PID control The PID controller K(s) is placed in the forward path, so that its output becomes the voltage applied to the motor’s armature. The feedback signal is either an angular shaft position or velocity, measured by a potentiometer or a tachometer, respectively. In the block diagram, these transducer dynamics are in the feedback path. The output position signal y(t), or velocity signal, is summed with a reference signal r (t), or command signal, K(S) DC motor Sensor Dynamics Σ CHAPTER 1: Plant Process Characterization and PID 2 to form the error signal e (t). Finally, the error signal is the input to the PID controller. The concept for this closed-loop system is simple. The sensor, which is attached to the motor shaft, provides a voltage that is compared to the reference voltage r (t) at the summer. When this error signal is non-zero, there will be an input to the controller, and hence some action taken by the DC motor. Once the sensor signal is equal to the reference signal, there is no input to the controller and no voltage applied to the motor, causing the motor to stop. However, this simple explanation does not provide us a with a method whereby the motor position can be brought to an exact position (with absolute accuracy) and does not tell us how to make the motor perform this positioning task as quickly as possible. Before examining the input-output relationships and design methods for the PID controller, it is helpful to review typical characteristics observed for the velocity response of a DC motor to a step voltage input. Different characteristics of the motor response (steady-state error, peak overshoot, rise time, etc.) are controlled by selection of the three gains that modify the PID controller dynamics. This is discussed in detail below. The PID controller is defined by the following relationship between the controller input e(t) and the controller output v(t) that is applied to the motor armature: Taking the Laplace transform of this equation gives the transfer function K(s): This transfer function clearly illustrates the proportional, integral, and derivative gains that make up the PID compensation. Select new definitions for the gain terms according to: dt tde KdeKtEKtV diP )( )()()( 0 ∫ ++= τ ττ )( )( )( )( SK S K K sE sV sK d i p ++== KK p = i i T K K = dd KTK = 3 Then, the transfer function can be expressed to easily show that the PID controller leads to a pole at the origin of the Laplace plane and design freedom over two zeros: The Vi used for this laboratory contains the capability of varying the control inputs (Kp, Ki and Kd). It also can make the system an open loop control or close loop control. A close loop control system is a system that can control an output with feedback, thus giving the system a real time control over the response of the system, as shown in figure 1.2. Figure 1.2: PID controller using LabVIEW (closed loop) An open loop control system is a system that has no type of real time control for the plant’s output, due to there being no feedback in the system. Figure 1.3 graphical represents the open loop system. ) 11 ()( 2 did d TT S T S S KT sK ++== 4 Figure 1.3: PID controller using LabVIEW (open loop) The Vi can also vary the damping ratio and natural frequency. The damping ratio is defined as the ratio of the damping of the system over the critic damping, where critical damping is the point were you have the fastest response without overshoot and ringing. The natural frequency is the frequency of the system or better known as its resonance frequency. When the damping ratio is greater than one, we can see the output with the slowest response, known as over damping. When the damping ration is one, we can see the output with the fastest response without overshooting and ringing, known as critical damping. When the damping ratio is between one and zero, we can see a fast response with overshooting and ringing, this is called under damping. The transfer function of the system is used as the basic characteristic of the plant, as shown in figure 4. The transfer function is used to understand the plant in used. Figure 1.4: Transfer function of the plant Discussion In the initial tasks of this lab we simulated a basic transfer function with the PID LabVIEW Vi. We then observed the response of the system in an open loop configuration. Our next few tasks consisted of making a closed loop system, and then by manipulating the PID variables of the system while keeping the Transfer function 5 constant, we acquired our results. We observed that this system setup allowed the response to be conformed to an ideal response without having to vary the transfer function. For the lab we were first given a graph with the response curve of some unknown transfer function. We initially made sure to set the P=1, I=0, and D=0. This was to insure that there would be no PID control influence upon the response curve. Figure 1.5 shows the open loop response that we were trying to conform our system to. 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.5 0.0 0.1 0.2 0.3 0.4 y(t)y(t) u(t)u(t) e(t)e(t) r(t)r(t) Figure 1.5: Open loop response By doing this we could then determine the transfer function of the unknown system. Once the PID variables were set we then varied only the damping ratio number till our 6 response resembled what is seen in figure 1.5. Graphically we achieved a fairly accurate representation of the unknown curve. Our response is shown in figure 1.6. Figure 1.6: Our response Once this response was displayed on the graph we then considered the transfer function in figure 1.4 to be the transfer function of the unknown response. If you carefully observe the responses in figures 1.5 and 1.6 you can see that both of there rise time resemble one another. One problem with the results is the fact that the resolution between the desired response and our response is considerably different. This variance unfortunately will bring some percentage of error into our system's result, therefore the Zn= 38.00 are not completely accurate. Once the desired response was achieved we then closed the system loop and observed that our response had changed. The response’s rise time had decreased and introduced a fairly large overshoot in to the system, yet the 7 overall amplitude of the response curve had been impeded. This result came from the introduction of the force feedback into the system, also known as closed loop control. Figure 1.7 is a display of the closed loop influence on the response curve. Figure 1.7: Closed loop response Finally by randomly manipulating the PID variables, of the closed loop system, we achieved our desired curve. Figure 1.6, which is the given response, and figure 1.7, which is our created response, and is an acceptable representation of one another. Again there remains a resolution discrepancies between the two therefore we can not say are results are 100% accurate, but our results are within acceptable tolerances 8 . 1.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.30.0 0.1 0.2 y(t)y(t) u(t)u(t) e(t)e(t) r(t)r(t) Figure 1.8: The given response Figure 1.9: The response 9 Figure 1.10: RPM Vs. Vout of system Figure 1.11 Vin ramp Vs. Vout response Conclusion In conclusion we observed that both the transfer function and the PID control could have a large influence upon the response of the system. More specifically the transfer functions damping ratio can be adjusted so in crease of decrease the response curves rise RPM vs. Vout of system y = 822.57x + 41.566 0 500 1000 1500 2000 2500 3000 0 1 2 3 4 Vout of system PRM of system RPM Linear (RPM) Vin ramp Vs. Vout response y = 0.0899x - 0.0642 y = 0.3415x - 7E-14 -2 0 2 4 6 8 10 12 0 10 20 30 40 time sec voltage OUTPUT INPUT Linear (OUTPUT) Linear (INPUT) 10 time. From our observation was concluded that as the Zn increases the rise time decreases and as the Zn decreases to one the rise time increase to a point. Once in that critically damped state the PID controls can be used to fine-tune the response. The P: proportional directly multiplies the amplitude to the response, while the integral and differential work against on another to fine-tune the shape of the curve. Over all we considered the lab to be very informative. Our only suggestion is that the lab needs to address the resolution difference so that we can increase the accuracy of our results.