Question

Objectives:

Model a feedback position control system with Multisim.

Study the behavior of the system as parameters are adjusted for different types of response.

Design Requirements:

A closed-loop control system is designed to allow an operator to set a reference angle and the inaccessible controlled element should turn by the same angle. A block diagram of the model is shown below. The gain in the feedback path is −1, and this forces to reach a final steady-state value equal to . All parameters except the gain adjustment in the forward path are fixed but that gain is adjustable.

Input R: 100 degrees          Load:      Integrator:      

Forward Gain To be adjusted to different settings in the steps that follow.

Feedback Gain . Run each plot below for the time range indicated or as needed.

Due to the time frames involved, set Maximum time step to 0.01 seconds.

Pg. 1       This page with data added as requested should be Page 1 of the report.

Pg. 2       Set and obtain a plot of until steady-state is reached. (Overdamped)

               

Pg. 3       Set and obtain a plot of until steady-state is reached. (Underdamped)

                                                               

Pg. 4       You are assigned the value of overshoot indicated below. Change your assigned value to boldface.

Overshoot Assignments:      I. 10%        II. 20%        III. 30%        IV. 40%        V. 50%

Adjust for the assigned overshoot. Obtain a plot of until steady-state is reached. List

                                      

time at which output first reaches 100. time at which the assigned overshoot is reached.

Pg. 5       For the value of in the previous step, obtain a plot also of angular velocity in the same interval.

Answer 1

A Closed-loop Control System, also known as a feedback control system is a control system which uses the concept of an open loop system as its forward path but has one or more feedback loops (hence its name) or paths between its output and its input. The reference to “feedback”, simply means that some portion of the output is returned “back” to the input to form part of the systems excitation.

Closed-loop systems are designed to automatically achieve and maintain the desired output condition by comparing it with the actual condition. It does this by generating an error signal which is the difference between the output and the reference input. In other words, a “closed-loop system” is a fully automatic control system in which its control action being dependent on the output in some way.

So for example, consider our electric clothes dryer from the previous open-loop tutorial. Suppose we used a sensor or transducer (input device) to continually monitor the temperature or dryness of the clothes and feed a signal relating to the dryness back to the controller as shown below.

Closed-loop Control

This sensor would monitor the actual dryness of the clothes and compare it with (or subtract it from) the input reference. The error signal (error = required dryness – actual dryness) is amplified by the controller, and the controller output makes the necessary correction to the heating system to reduce any error. For example if the clothes are too wet the controller may increase the temperature or drying time. Likewise, if the clothes are nearly dry it may reduce the temperature or stop the process so as not to overheat or burn the clothes, etc.

Then the closed-loop configuration is characterised by the feedback signal, derived from the sensor in our clothes drying system. The magnitude and polarity of the resulting error signal, would be directly related to the difference between the required dryness and actual dryness of the clothes.

Also, because a closed-loop system has some knowledge of the output condition, (via the sensor) it is better equipped to handle any system disturbances or changes in the conditions which may reduce its ability to complete the desired task.

For example, as before, the dryer door opens and heat is lost. This time the deviation in temperature is detected by the feedback sensor and the controller self-corrects the error to maintain a constant temperature within the limits of the preset value. Or possibly stops the process and activates an alarm to inform the operator.

As we can see, in a closed-loop control system the error signal, which is the difference between the input signal and the feedback signal (which may be the output signal itself or a function of the output signal), is fed to the controller so as to reduce the systems error and bring the output of the system back to a desired value. In our case the dryness of the clothes. Clearly, when the error is zero the clothes are dry.

The term Closed-loop control always implies the use of a feedback control action in order to reduce any errors within the system, and its “feedback” which distinguishes the main differences between an open-loop and a closed-loop system.The accuracy of the output thus depends on the feedback path, which in general can be made very accurate and within electronic control systems and circuits, feedback control is more commonly used than open-loop or feed forward control.

Closed-loop systems have many advantages over open-loop systems. The primary advantage of a closed-loop feedback control system is its ability to reduce a system’s sensitivity to external disturbances, for example opening of the dryer door, giving the system a more robust control as any changes in the feedback signal will result in compensation by the controller.

Then we can define the main characteristics of Closed-loop Control as being:

• To reduce errors by automatically adjusting the systems input.
• To improve stability of an unstable system.
• To increase or reduce the systems sensitivity.
• To enhance robustness against external disturbances to the process.
• To produce a reliable and repeatable performance.

Whilst a good closed-loop system can have many advantages over an open-loop control system, its main disadvantage is that in order to provide the required amount of control, a closed-loop system must be more complex by having one or more feedback paths. Also, if the gain of the controller is too sensitive to changes in its input commands or signals it can become unstable and start to oscillate as the controller tries to over-correct itself, and eventually something would break. So we need to “tell” the system how we want it to behave within some pre-defined limits.

Closed-loop Summing Points

For a closed-loop feedback system to regulate any control signal, it must first determine the error between the actual output and the desired output. This is achieved using a summing point, also referred to as a comparison element, between the feedback loop and the systems input. These summing points compare a systems set point to the actual value and produce a positive or negative error signal which the controller responds too. where: Error = Set point – Actual

The symbol used to represent a summing point in closed-loop systems block-diagram is that of a circle with two crossed lines as shown. The summing point can either add signals together in which a Plus ( + ) symbol is used showing the device to be a “summer” (used for positive feedback), or it can subtract signals from each other in which case a Minus ( − ) symbol is used showing that the device is a “comparator” (used for negative feedback) as shown.

Summing Point Types

Note that summing points can have more than one signal as inputs either adding or subtracting but only one output which is the algebraic sum of the inputs. Also the arrows indicate the direction of the signals. Summing points can be cascaded together to allow for more input variables to be summed at a given point.

Closed-loop System Transfer Function

The Transfer Function of any electrical or electronic control system is the mathematical relationship between the systems input and its output, and hence describes the behaviour of the system. Note also that the ratio of the output of a particular device to its input represents its gain. Then we can correctly say that the output is always the transfer function of the system times the input. Consider the closed-loop system below.

Typical Closed-loop System Representation

Where: block G represents the open-loop gains of the controller or system and is the forward path, and block H represents the gain of the sensor, transducer or measurement system in the feedback path.

To find the transfer function of the closed-loop system above, we must first calculate the output signal θo in terms of the input signal θi. To do so, we can easily write the equations of the given block-diagram as follows.

The output from the system is equal to:    Output = G x Error

Note that the error signal, θe is also the input to the feed-forward block:  G

The output from the summing point is equal to:    Error = Input - H x Output

If  H = 1 (unity feedback) then:

The output from the summing point will be:    Error (θe) = Input - Output

Eliminating the error term, then:

The output is equal to:    Output = G x (Input - H x Output)

Therefore:    G x Input = Output + G x H x Output

Rearranging the above gives us the closed-loop transfer function of:

The above equation for the transfer function of a closed-loop system shows a Plus ( + ) sign in the denominator representing negative feedback. With a positive feedback system, the denominator will have a Minus ( − ) sign and the equation becomes:  1 - GH.

We can see that when  H = 1 (unity feedback) and G is very large, the transfer function approaches unity as:

Also, as the systems steady state gain G decreases, the expression of:  G/(1 + G) decreases much more slowly. In other words, the system is fairly insensitive to variations in the systems gain represented by G, and which is one of the main advantages of a closed-loop system.

Multi-loop Closed-loop System

Whilst our example above is of a single input, single output closed-loop system, the basic transfer function still applies to more complex multi-loop systems. Most practical feedback circuits have some form of multiple loop control, and for a multi-loop configuration the transfer function between a controlled and a manipulated variable depends on whether the other feedback control loops are open or closed.

Consider the multi-loop system below.

Any cascaded blocks such as G1 and G2 can be reduced, as well as the transfer function of the inner loop as shown.

After further reduction of the blocks we end up with a final block diagram which resembles that of the previous single-loop closed-loop system.

And the transfer function of this multi-loop system becomes:

Then we can see that even complex multi-block or multi-loop block diagrams can be reduced to give one single block diagram with one common system transfer function.

Closed-loop Motor Control

So how can we use Closed-loop Systems in Electronics. Well consider our DC motor controller from the previous open-loop tutorial. If we connected a speed measuring transducer, such as a tachometer to the shaft of the DC motor, we could detect its speed and send a signal proportional to the motor speed back to the amplifier. A tachometer, also known as a tacho-generator is simply a permanent-magnet DC generator which gives a DC output voltage proportional to the speed of the motor.

Then the position of the potentiometers slider represents the input, θi which is amplified by the amplifier (controller) to drive the DC motor at a set speed N representing the output, θo of the system, and the tachometer T would be the closed-loop back to the controller. The difference between the input voltage setting and the feedback voltage level gives the error signal as shown.

Closed-loop Motor Control

Any external disturbances to the closed-loop motor control system such as the motors load increasing would create a difference in the actual motor speed and the potentiometer input set point.

This difference would produce an error signal which the controller would automatically respond too adjusting the motors speed. Then the controller works to minimize the error signal, with zero error indicating actual speed which equals set point.

Electronically, we could implement such a simple closed-loop tachometer-feedback motor control circuit using an operational amplifier (op-amp) for the controller as shown.

Closed-loop Motor Controller Circuit

This simple closed-loop motor controller can be represented as a block diagram as shown.

Block Diagram for the Feedback Controller

A closed-loop motor controller is a common means of maintaining a desired motor speed under varying load conditions by changing the average voltage applied to the input from the controller. The tachometer could be replaced by an optical encoder or Hall-effect type positional or rotary sensor.