· Control systems use some output state of a system and a desired state to make control decisions.
· In general we use negative feedback systems because,
- they typically become more stable
- they become less sensitive to variation in component values
- it makes systems more immune to noise
· Consider the system below, and how it is enhanced by the addition of a control system.
Figure 19.1 An example of a feedback controller
Figure 19.2 Rules for a feedback controller
· Some of the things we do naturally (like the rules above) can be done with mathematics
· The basic equation for a PID controller is shown below. This function will try to compensate for error in a controlled system (the difference between desired and actual output values).
Figure 19.3 The PID control equation
· The figure below shows a basic PID controller in block diagram form.
Figure 19.4 A block diagram of a feedback controller
· The PID controller is the most common controller on the market.
· Even though the transfer function uses the Laplace `s', it is still a ratio of input to output.
· Find an input in terms of the Laplace `s'
· The table below is for typical control system types,
· Consider the basic transform tables. A superficial examination will show that the denominator (bottom terms) are the main factor in determining the final form of the solution. To explore this further, consider that the roots of the denominator directly impact the partial fraction expansion and the following inverse Laplace transfer.
· When designing a controller with variable parameters (typically variable gain), we need to determine if any of the adjustable gains will lead to an unstable system.
· Root locus plots allow us to determine instabilities (poles on the right hand side of the plane), overdamped systems (negative real roots) and oscillations (complex roots).
· Note: this procedure can take some time to do, but the results are very important when designing a control system.
· Consider the example below,
· Consider the previous example, the transfer function for the whole system was found, but then only the denominator was used to determine stability. So in general we do not need to find the transfer function for the whole system.
· Consider the example,
· The basic procedure for creating root locus plots is,
1. write the characteristic equation. This includes writing the poles and zeros of the equation.
2. count the number of poles and zeros. The difference (n-m) will indicate how many root loci lines end at infinity (used later).
3. plot the root loci that lie on the real axis. Points will be on a root locus line if they have an odd number of poles and zeros to the right. Draw these lines in.
4. determine the asymptotes for the loci that go to infinity using the formula below. Next, determine where the asymptotes intersect the real axis using the second formula. Finally, draw the asymptotes on the graph.
5. the breakaway and breakin points are found next. Breakaway points exist between two poles on the real axis. Breakin points exist between zeros. to calculate these the following polynomial must be solved. The resulting roots are the breakin/breakout points.
6. Find the points where the loci lines intersect the imaginary axis. To do this substitute the phasor for the laplace variable, and solve for the frequencies. Plot the asymptotic curves to pass through the imaginary axis at this point.
· Consider the example in the previous section,
· Plot the root locus diagram for the function below,
·
3. Given the transfer function below, and the input `x(s)', find the output `y(t)' as a function of time.
8. Draw a detailed root locus diagram for the transfer function below. Be careful to specify angles of departure, ranges for breakout/breakin points, and gains and frequency at stability limits.
10. Draw the root locus diagram for the transfer function below,
11. Draw the root locus diagram for the transfer function below,
12. The block diagram below is for a motor position control system. The system has a proportional controller with a variable gain K.
a) Simplify the block diagram to a single transfer function.
b) Draw the Root-Locus diagram for the system (as K varies). Use either the approximate or exact techniques.
c) Select a K value that will result in an overall damping factor of 1. State if the Root-Locus diagram shows that the system is stable for the chosen K.
13. Draw a Bode Plot for either one of the two transfer functions below.
15. Given the system transfer function below.
a) Draw the root locus diagram and state what values of K are acceptable.
b) Select a gain value for K that has either a damping factor of 0.707 or a natural frequency of 3 rad/sec.
c) Given a gain of K=10 find the steady state response to an input step of 1 rad.
d) Given a gain of K=10 find the response of the system as
17. The equation below describes a dynamic system. The input is `F' and the output is `V'. It has the initial values specified. The following questions ask you to find the system response to a unit step input using various techniques.
a) Find the response using Laplace transforms.
b) Find the response using the homogenous and particular solutions.
c) Put the equation is state variable form, and solve it using your calculator. Sketch the result accurately below.
18. A feedback control system is shown below. The system incorporates a PID controller. The closed loop transfer function is given.
a) Verify the close loop controller function given.
b) Draw a root locus plot for the controller if Kp=1 and Ki=1. Identify any values of Kd that would leave the system unstable.
c) Draw a Bode plot for the feedback system if Kd=Kp=Ki=1.
d) Select controller values that will result in a natural frequency of 2 rad/sec and damping factor of 0.5. Verify that the controller will be stable.
e) For the parameters found in the last step find the initial and final values.
f) If the values of Kd=1 and Ki=Kd=0, find the response to a ramp input as a function of time.
19. The following system is a feedback controller for an elevator. It uses a desired heigh `d' provided by a user, and the actual height of the elevator `h'. The difference between these two is called the error `e'. The PID controller will examine the value `e' and then control the speed of the lift motor with a control voltage `c'. The elevator and controller are described with transfer functions, as shown below. all of these equations can be combined into a single system transfer equation as shown.
a) Find the response of the final equation to a step input. The system starts at rest on the ground floor, and the input (desired height) changes to 20 as a step input.
b) Write find the damping factor and natural frequency of the results in part a).
c) verify the solution using the initial and final value theorems.
1. a) The block diagram below is for an angular positioning system. The setpoint is a desired angle, which is converted to a desired voltage. This is compared to a feedback voltage from a potentiometer. A PID controller is used to generate an output voltage to drive a DC motor. Simplify the block diagram.
b) Given the transfer function below, select values for Kp, Ki and Kd that will result in a second order response that has a damping factor of 0.125 and a natural frequency of 10rad/s. (Hint: eliminate Ki).
c) The function below has a step input of magnitude 1.0. Find the output as a function of time using numerical methods. Give the results in a table OR graph.
d) The function below has a step input of magnitude 1. Find the output as a function of time by integrating the differential equation (i.e., using the homogeneous and particular solutions).
e) The function below has a step input of magnitude 1. Find the output as a function of time using Laplace transforms.
f) Given the transfer function below; a) apply a phasor/Fourier transform and express the gain and phase angle as a function of frequency, b) calculate a set of values and present them in a table, c) use the values calculated in step b) to develop a frequency response plot on semi-log paper, d) draw a straight line approximation of the Bode plot on semi-log paper.
2. Select a controller transfer function, Gc, that will reduce the system to a first order system with a time constant of 0.5s, as shown below.