Exercises: Block diagrams and simplification

Explain the elements of a control system block diagram: what blocks, arrows, summing junctions, and pick-off points represent, using a home heating system with a thermostat as an example.

Explain the difference between an open-loop and a closed-loop system, with an everyday example of each (for example, a timer toaster versus an oven with a thermostat), and explain which one corrects disturbances and why.

The figure shows the block diagram of a system with negative feedback. Identify each indicated element (reference signal, error signal, and output signal) and explain what the summing point computes at each instant.

Explain in your own words what the transfer function of a block is (the ratio between its output and its input) and why, when the blocks are constant gains, you can work with numbers instead of differential equations.

In a block diagram, distinguish between a signal and a physical variable: give three examples of signals in an oven control system (reference temperature in degrees, thermocouple voltage in millivoltios, heating element power in watts) and explain which block converts one into another.

A summing junction receives three signals: $a$ with sign $+$, $b$ with sign $-$, and $c$ with sign $-$. Write the expression for the output signal and calculate its value for $a = 10$, $b = 3$, and $c = 4$. What would it represent in a control system if a signal enters with a negative sign?

A pick-off point splits a signal towards two blocks. Explain whether the signal value is 'divided by two' or copied in full to both branches, and why in block diagrams the answer is the latter (the signal is information, not a current being split).

An amplifier has a gain of $10$ and a sensor has a gain of $0{,}01\ \text{V/}{}^\circ\text{C}$. State the units of the gain for each block and explain why the gain of the complete loop (amplifier, process, and sensor) must be dimensionless.

Two blocks in series have transfer functions $G_1 = 5$ and $G_2 = 2$. a) Calculate the equivalent transfer function of the combination. b) If an input signal of $3$ units is applied, calculate the output signal of each block. c) Explain the general rule for reducing blocks in series.

Two blocks in parallel receive the same input and their outputs are added: $G_1 = 4$ and $G_2 = 6$. a) Calculate the equivalent transfer function. b) Explain how the result would change if the summing junction subtracted the second output instead of adding it. c) Compare with the series case: why are they added in parallel and multiplied in series?

A negative feedback loop has $G = 10$ in the forward path and $H = 0{,}1$ in the feedback path. a) Calculate the closed-loop transfer function $M = \frac{G}{1 + G \cdot H}$. b) Compare the closed-loop gain with the open-loop gain and explain what is lost and gained by closing the loop (accuracy, disturbance rejection).

Three blocks in series have gains of $G_1 = 2$, $G_2 = 5$, and $G_3 = 0{,}5$. a) Calculate the equivalent gain. b) For an input of $4$ units, calculate the output of each intermediate block and the final output. c) Explain why the order of the blocks does not alter the total gain, but does alter the intermediate signals.

The figure shows a system with a block $G_1 = 3$ in series with the parallel sum of $G_2 = 2$ and $G_3 = 4$. a) Reduce the combination to a single equivalent block step by step. b) Calculate the total output for an input of $5$ units. c) Check the result by calculating the output of each branch separately.

A negative feedback loop has $G = 20$ and $H = 0{,}05$. a) Calculate the closed-loop gain $M = \frac{G}{1 + G \cdot H}$. b) Repeat the calculation for $G = 10$ and $H = 0{,}1$, and discover what both loops have in common. c) Reason what value $M$ tends towards when the product $G \cdot H$ is very large and what the system gain then depends on.

In a unity feedback loop ($H = 1$) with $G = 50$: a) calculate the transfer function $M$ and the output for a reference of $10$ units. b) Calculate the error $e = r - y$. c) Explain why increasing $G$ reduces the error but never eliminates it in a system without an integrator.

An open-loop heater with gain $5$ receives a setpoint of $8$ units; a disturbance subtracts $6$ units from the process output. a) Calculate the final output in open loop. b) Explain qualitatively what would happen with that same disturbance in closed loop: who 'detects' that the output has dropped and what do they do?

A vehicle equipped with cruise control is driving along a flat motorway at a constant speed of $120\text{ km/h}$. Suddenly, it begins to climb a long slope with a $6\,\%$ incline and a headwind of $40\text{ km/h}$.

1. a) Design the detailed block diagram of the closed-loop system explicitly indicating the exact point where these disturbances act and the physical quantities (with their units) of each signal flowing between blocks.
2. b) Justify analytically, from the block diagram equations, why an open-loop system would experience a drastic drop in speed while the closed-loop system is able to mitigate this drop.
3. c) If the controller is solely proportional ($P$) and the engine reaches its maximum allowable torque during the climb, predict with reasoning what will happen to the final speed of the car and the steady-state error. Does the closed loop succeed in maintaining the $120\text{ km/h}$ setpoint in this real scenario? Justify your answer.

A maintenance technician claims: "Since the feedback system in the figure has a high total forward gain ($G_1 \cdot G_2 = 10$, with $G_1 = 2$ and $G_2 = 5$, in series) and the reference is constant ($r = 10$), the output measured by the sensor, with gain $H = 0{,}2$ in the negative feedback, should exactly equal the reference, making the error signal at the actuator input zero if everything works properly".

1. a) Show analytically whether the technician's claim is correct or wrong: reduce the diagram to its equivalent block step by step (first the blocks in series and then the feedback loop), calculate the value of the output $y$ in steady state, and determine the exact numerical value of the actuating error signal $e = r - H \cdot y$.

1. b) Justify the paradox physically: why is it essential, in a loop with proportional action, that the error $e$ is NOT zero in order to maintain a nonzero steady output?

1. c) If the steady-state error is required to be reduced by half without changing the sensor gain ($H = 0{,}2$), predict what modification you would have to make to the forward gain and explain what negative consequences this measure could cause in a real physical actuator.

During the installation of a control loop with a forward path transfer function $G = 10$ and a feedback sensor with gain $H = 0{,}1$, a technician mistakenly reverses the polarity at the comparator, introducing positive feedback instead of negative feedback.

1. a) Derive the closed-loop transfer function $M = \frac{Y}{R}$ for this erroneous configuration and numerically calculate the overall gain $M$ if $H$ undergoes slight deviations due to thermal effects, taking the values $0{,}05$, $0{,}09$, and $0{,}099$.
2. b) Interpret and predict what will happen in the physical system when the product $G \cdot H$ approaches $1$. What behavior will the operator observe in the actuator in the event of a minimal disturbance, even with a zero setpoint ($r = 0$)?
3. c) Justify, by mathematically contrasting positive versus negative feedback, why the latter provides stability and robustness against manufacturing tolerances or thermal variations in the process components.

The angular positioning of a motor shaft is analyzed using a cascade control system. The motor plant ($G_2 = 4$) features an inner negative velocity feedback loop ($H_2 = 0{,}25$). Ahead of this inner loop, an adjustable preamplifier of gain $K$ is placed. The entire system is closed with an outer position loop featuring unity negative feedback ($H_1 = 1$).

1. a) Model the system's block diagram and obtain the overall equivalent transfer function $M = \frac{Y}{R}$ as a function of $K$.
2. b) If $K = 2$ is set, calculate the steady-state output $y$ for a step reference input $r = 10$ and compute the steady-state position error.
3. c) A student proposes simplifying the setup by eliminating the inner feedback ($H_2 = 0$) and adjusting a new value of $K$ to obtain exactly the same overall static transfer function. Determine this value of $K$ and refute their proposal by justifying what dynamic advantage and internal disturbance rejection in the motor would be lost by eliminating the minor loop.

In a thermal plant with gain $G = 9$ and negative feedback with sensor $H = 0{,}1$, a constant reference setpoint $r = 10$ is applied. While the system is in steady state, an unexpected heat leak occurs, modeled as an additive disturbance $p = -2$ right at the plant output.

1. a) From the block diagram and applying the principle of superposition, formally derive the expression for the output $y(r, p)$ without assuming preset formulas, and calculate the final temperature reached by the furnace.
2. b) Calculate the temperature that would have been reached if the system were operating in open loop, having previously calibrated its gain to yield the same nominal output for $r = 10$.
3. c) The specifications are redefined, requiring that the impact of any external disturbance on the output does not exceed $4\,\%$ of the magnitude of said disturbance ($|\Delta y| \le 0{,}04 \cdot |p|$). Determine the minimum allowable forward loop gain $G$ (keeping $H = 0{,}1$) and critically analyze what practical problems forcing that value would cause in reality.

For the angular guidance of a receiving antenna using a proportional control loop with unity feedback ($H = 1$), the steady-state relative tracking error for constant setpoints is required to be strictly less than $2\,\%$.

1. a) Derive the expression relating the relative error $\frac{e}{r}$ to the static gain $G$ of the forward path and calculate the minimum value required for this gain.
2. b) When testing the prototype with the calculated gain, it is observed that the motor exhibits sharp, continuous oscillations before stopping, and the amplifier even enters thermal protection mode. Justify technically and physically the causes of this behavior when greatly increasing $G$.
3. c) Without modifying the motor or the sensor, design a conceptual alternative in the controller structure (change of control type) that allows reducing the position error to exactly zero while maintaining a moderate proportional gain that does not compromise stability or saturate the actuator.

The output flow rate $q$ (in $\text{m}^3/\text{h}$) of a non-linear control valve follows the curve $q(u) = u^2$, where $u$ is the control signal in volts ($u \in [0, 5]\text{ V}$). The valve is integrated into a linear closed loop designed and tuned to operate around the nominal operating point $u_0 = 2\text{ V}$.

1. a) Linearize the valve behavior around $u_0 = 2\text{ V}$ using a first-order Taylor series expansion and calculate the incremental transfer function $G_v = \frac{\Delta q}{\Delta u}$ at that point.
2. b) Assume the closed loop is stable and behaves optimally at $u_0 = 2\text{ V}$. Predict and justify how the system response will change (speed, stability, and oscillations) if the operating point changes abruptly to full opening ($u = 4{,}5\text{ V}$) or to minimum flow rates near shutoff ($u = 0{,}2\text{ V}$).
3. c) Based on your findings, refute with quantitative arguments the following statement: “Since the implemented controller is linear and has fixed parameters, the loop will respond with the same speed and damping to flow variations regardless of the level at which the valve is operating.”

A cylindrical tank with a cross-sectional area $A = 2\text{ m}^2$ has an inflow pump with flow rate $q_e(t) = K_b \cdot u(t)$, with $K_b = 0{,}05\text{ m}^3/(\text{s}\cdot\text{V})$, controlled by voltage $u(t)$. The tank supplies a consumption network that draws an unexpected outflow $q_s(t)$ (disturbance). The actual level $h(t)$ is measured with a linear sensor with gain $K_s = 2\text{ V/m}$.

1. a) Derive the differential equation of the mass balance in the tank, express it in the Laplace domain ($s$), and draw the detailed block diagram of the closed-loop control system, clearly identifying the transfer functions of each block, the summing junction for the disturbance, and the physical units of each signal.
2. b) Analytically demonstrate why the tank plant acts as a natural integrator and indicate which 'system type' classification (type 0, 1, or 2) the open-loop combination acquires if a proportional controller $K_p$ is used.
3. c) For said proportional controller ($K_p = 4\text{ V/V}$), quantitatively predict the steady-state error that the water level will experience in response to a unit step change in the level setpoint ($h_{ref} = 1\text{ m}$ with $q_s = 0$) and compare it with the steady-state error caused by a continuous and constant outflow extraction $q_s = 0{,}02\text{ m}^3/\text{s}$.