Unit 1 - Practice Quiz

In an open-loop system, there is no feedback path. The controller's action is pre-determined and is not influenced by the system's actual output.

The thermostat measures the room temperature (output) and compares it to the desired temperature (setpoint), adjusting the cooling action accordingly. This measurement and correction is the key feature of a closed-loop system.

The feedback element, such as a sensor, is responsible for measuring the actual output of the process and feeding that information back to the controller to be compared with the reference input.

By definition, the transfer function describes the system's dynamic behavior assuming that the system starts from rest, which means all initial conditions are zero.

The relationship between voltage and current for a capacitor is . Taking the Laplace transform gives , so the impedance .

A damper, or dashpot, produces a resistive force that is proportional to the velocity of the motion. This force, known as viscous friction, opposes the motion.

A linear system must satisfy the superposition principle, which includes both additivity and homogeneity. This means the response to a sum of inputs is the sum of the responses to each individual input.

Negative feedback allows the system to correct for errors caused by external disturbances or changes in its components, thus making the system more robust and its performance more reliable.

The traffic light's operation is based on a pre-set timer and does not change based on the actual traffic volume. Since it doesn't measure the output (traffic flow), it is an open-loop system.

The transfer function is an intrinsic property of the system, determined by its physical components (like resistors, masses, springs) and their configuration. It does not change with the input signal.

In the force-voltage analogy, Force is analogous to Voltage. The force in a spring is proportional to the integral of velocity (), while the voltage across a capacitor is proportional to the integral of current (). Thus, the spring constant 'k' is analogous to the reciprocal of capacitance, '1/C'.

The setpoint is the target value that the control system aims to maintain for the controlled variable. For example, the desired temperature in a furnace is the setpoint.

The error signal represents the deviation of the actual output from the desired output. The controller uses this signal to determine the necessary corrective action.

Using the voltage divider rule in the s-domain, the output voltage across the resistor is . Therefore, the transfer function is .

Newton's second law of motion is fundamentally expressed as F = ma, where F is the net force, m is the mass, and a is the acceleration of the body.

A time-invariant system is one whose characteristics do not change with time. Its response to an input depends only on the input itself, not on the time it is applied.

While negative feedback is used to stabilize systems, if the gain is excessively high or there are significant phase shifts in the loop, it can lead to instability and oscillations.

Poles are the roots of the denominator polynomial of the transfer function. They are critical because they determine the stability and dynamic behavior of the system.

The actuator is the device that takes the signal from the controller and produces a physical action. The thermostat (controller) sends a signal to turn the furnace (actuator) on or off.

Since open-loop systems do not have a feedback mechanism, they cannot sense or compensate for external disturbances or changes in the system itself, leading to inaccurate results.

Using voltage divider rule in the Laplace domain: . Substituting , , and , the expression simplifies to .

The closed-loop gain is . With , . If G becomes 110, the new gain is . The percentage change is minuscule, demonstrating that negative feedback greatly reduces sensitivity to changes in forward path gain.

Taking the Laplace Transform of the differential equation with zero initial conditions gives . Factoring out gives . The transfer function is the ratio , which is .

In the force-voltage analogy, mechanical and electrical differential equations are matched. This leads to the analogies: Force Voltage, Velocity Current, Mass Inductance, Damper Resistance, and Spring Constant Reciprocal of Capacitance (1/C).

Poles are the roots of the denominator polynomial (the characteristic equation). We need to solve . Factoring the polynomial gives , so the poles are at and .

Initially, varying the voltage without measuring the output speed is an open-loop system. Adding a tachometer provides feedback (measurement of the output speed), which is used to correct the input. This creates a closed-loop system, which is more robust to disturbances like changing loads.

The formula for sensitivity is . If the magnitude of the loop gain, , is much greater than 1, the denominator becomes very large. Therefore, the sensitivity approaches zero, meaning the closed-loop system is insensitive to changes in .

A linear system must satisfy the superposition principle. The equation fits this, as it contains only terms of y and its derivatives raised to the power of one, with constant coefficients. The term and are non-linear, and the coefficient makes the system time-variant.

In the force-current analogy (also known as the direct analogy), the roles of inductor and capacitor are swapped compared to the force-voltage analogy. The analogies are: Force Current, Mass Capacitor, Damper Conductor (1/R), and Spring Inductor.

For an inverting op-amp configuration, the transfer function is . Here, the input impedance and the feedback impedance is . Therefore, the transfer function is .

While negative feedback reduces the overall gain, its main advantage is making the system more robust. It decreases the effect of variations in component values (e.g., due to temperature or aging) and minimizes the impact of external disturbances on the output.

This is Newton's second law for rotation (). The term represents the torque required to overcome the moment of inertia (J) of the rotating body, where is the angular acceleration.

Negative feedback generally increases the bandwidth of a system, meaning it can respond to a wider range of input frequencies. This is associated with a smaller time constant, which results in a faster transient response (the system settles to its final value more quickly).

Because a closed-loop system measures the output and compares it to the desired input, it can take corrective action when external disturbances affect the output. Open-loop systems cannot do this, making them highly susceptible to such disturbances.

The error signal is the difference between the setpoint (desired output) and the measured actual output. Here, the setpoint is 60 mph and the measured output is 55 mph. The error is mph, which the controller acts upon.

The very definition of a transfer function relies on analyzing the system's response to an input without any influence from pre-existing energy or states. Therefore, all initial conditions (e.g., initial charge on a capacitor, initial velocity of a mass) are assumed to be zero.

The order of a mechanical system is determined by the number of independent energy storage elements. Each mass stores kinetic energy () and each spring stores potential energy (). Since there are two masses and two springs, there are four independent energy storage states (), leading to a fourth-order system described by four first-order differential equations or two second-order differential equations.

Using the voltage divider rule in the Laplace domain, the voltage across the inductor is . Substituting the impedances and , the transfer function becomes . This is a high-pass filter.

In positive feedback, the feedback signal is added to the reference input, which tends to drive the output further in the same direction, often leading to instability. In contrast, negative feedback subtracts the feedback signal from the reference input to create a corrective error signal.

A toaster is a classic open-loop system because its control action (heating time) is pre-set and independent of the actual output (the color or 'doneness' of the toast). It doesn't have a sensor to provide feedback on whether the desired brownness has been achieved.

The sensitivity of a closed-loop system to a parameter in the forward path is given by . With unity feedback and , the formula becomes . To have , we set . Solving for gives , which results in .

The transfer function has a zero in the right-half of the s-plane (a 'non-minimum phase' zero) at . Such zeros cause an initial undershoot in the step response, where the output initially moves in the opposite direction of the final value before correcting itself.

The system requires two equations of motion, one for each mass. The equation for will be , and for it is . Each equation is a second-order differential equation. When these two coupled equations are solved in the s-domain to find the transfer function , the characteristic polynomial in the denominator will be of order 4 (involving an term).

A lever relates forces and velocities by a ratio () and conserves power (). An ideal transformer relates voltages and currents by its turns ratio () and conserves power (). Therefore, a lever is analogous to a transformer, not a gearbox which is a rotational system component.

The open-loop DC gain is . Its sensitivity is . The closed-loop transfer function is , with DC gain . Its sensitivity is . Since and , . Thus, the closed-loop system is less sensitive to variations in 'a'.

Substituting into the expression for yields . The new pole, which defines the closed-loop bandwidth, is at . Therefore, the closed-loop bandwidth is increased by a factor of , which is the amount of feedback.

Linearity is composed of two properties: additivity (response to sum of inputs is sum of responses) and homogeneity (response to scaled input is scaled response). The system satisfies additivity and time-invariance, but it violates homogeneity because scaling the input by 2 does not scale the output by 2. A system must satisfy both additivity and homogeneity to be linear.

The closed-loop transfer function is . When the loop gain , we can approximate the denominator as . This simplifies the transfer function to . This means the overall system behavior is dictated by the characteristics of the feedback network , which is often built with precise and stable components, making the system performance robust against variations or uncertainties in the plant .

The simple multiplication of transfer functions in a cascade configuration assumes that the second system does not 'load' the first system. This assumption holds if the output impedance of the first stage is much smaller than the input impedance of the second stage. If loading occurs, the dynamics of the first system are altered by the connection of the second, and the overall transfer function must be re-derived for the combined system.

When reflecting mechanical impedances from the load side to the motor side through a gearbox, the inertia and damping are divided by the square of the gear ratio . Therefore, the total equivalent inertia seen by the motor is its own inertia plus the reflected load inertia: . Similarly, the equivalent damping is .

The standard second-order form is . By comparing denominators, we find and . Solving for gives . Let the new parameters be and . The new damping ratio is .

The transfer function for a positive feedback system is . The system becomes unstable when the denominator goes to zero, which means , or . This is the Barkhausen criterion for oscillation. For sustained oscillations at a frequency , the loop gain must be exactly 1 at that frequency.

In the Force-Current (F-I) analogy (also known as the force-current/mobility analogy), the analogous quantities are: Force (F) Current (I), and Velocity (v) Voltage (V). The constitutive relations are: Damper: , so (or conductance). Mass: , so . Spring: , so . Thus, a damper is analogous to a resistor/conductance and a mass to a capacitor.

In a unity feedback loop, the plant input is , where is the error signal from the reference. The output is assuming unity feedback where . Setting the reference to analyze the disturbance effect, we get . Rearranging gives , so the transfer function from disturbance to output is . For large , this value becomes small, demonstrating disturbance rejection.

We use the Laplace transform pair . For the given impulse response, and . The transfer function is . The poles are the roots of the denominator, found by setting . This gives , so , which means the poles are located at and .

The mass balance equation for the tank is . The term makes the differential equation non-linear. Linear control techniques and transfer functions are based on linear differential equations. Therefore, the system must be linearized (e.g., using a Taylor series expansion) around a specific operating height to create a linear model that is valid for small deviations from that point.

There is no 'free lunch' with feedback. The benefits come at a cost. The gain is reduced by the factor , which is the same factor that reduces sensitivity and increases bandwidth (gain-bandwidth product is often constant). Adding sensors, comparators, and controllers for feedback increases complexity and cost. Furthermore, feedback loops can introduce phase shifts that may lead to positive feedback at certain frequencies, causing oscillations and instability if not carefully designed.

A notch filter is designed to completely attenuate a specific frequency. In the s-plane, this corresponds to having a pair of zeros on the imaginary axis () at the notch frequency. The bridged-T network is an R-C circuit capable of creating these complex zeros without using inductors. Its analysis results in a second-order transfer function where the numerator becomes zero at a specific frequency, creating the notch.

The term 'Bs' in the characteristic equation corresponds to a damping force or torque that is proportional to velocity. In a rotational system, this is angular velocity, , whose Laplace transform is . A viscous damper or dashpot provides precisely this kind of damping torque (). Adding this component to the system's equation of motion () introduces the desired 'Bs' term in the denominator of the transfer function.

The system is non-linear due to the term. Let's check the properties. Homogeneity: If input is , output should be . The equation becomes . This is not . So, homogeneity is violated. Additivity: If input is , the term becomes , which is not the sum of the responses to and individually (). Therefore, it violates both principles of linearity.