2022 Exam PDF

## Seksjon 1 **Exam problem:** What is the main aim of a control system? - To keep the control error within acceptable limits. - To limit the variations of the control variable. - To avoid using humans as controllers. - To eliminate the process disturbance. - None of the alternatives. **Exam pro...

## Seksjon 1 **Exam problem:** What is the main aim of a control system? - To keep the control error within acceptable limits. - To limit the variations of the control variable. - To avoid using humans as controllers. - To eliminate the process disturbance. - None of the alternatives. **Exam problem:** Given a level control system. The level reference is 0.5 m. The level measurement is 0.4 m. What is the control error? - - 0.1 m - 0.1 m - 20% - - 20% - None of the alternatives. ## Seksjon 2 **Exam problem:** A differential pressure sensor is used to indirectly measure the level in a liquid tank by measuring the hydrostatic pressure of the liquid. What is the correct P&I D symbol of this level measurement device? - PT - PDT - LT - LS - None of the alternatives. ## Seksjon 3 **Exam problem:** Given a water tank with straight walls with cross-sectional area A, mass inflow F_in, volumetric outflow Q_out, water level h. Water density is rho. At time 0, the level is h_0. Variables and parameters are in SI units. Which is a differential equation model describing the response in h? - h' = (F_in - Q_out)/(rho*A) - h' = (F_in - rho*Q_out)/(rho*A) - h' = (rho*F_in - Q_out)/(rho*A) - h' = (F_in - Q_out)/A - None of the alternatives. **Exam problem:** The figure below shows a water tank with continuous liquid inflow and outflow. The volume of water is constant. Assume that there is a time-delay (td) from any change in the control signal (u) to the corresponding response in the water temperature. This is expressed in the model with power P delivered to the water being equal to time-delayed u. Assume homogenous conditions in the tank. Select a mathematical model of the temperature of the water in the tank. The model is described with a diagram that contains the following parameters: - G [W/K] - TeN [PC] - m [kg] - Tin [C] - F [m3/s] - V [m³] - c [J/(kg K)] - r [kg/m3] - P [W] - u [W] The model is described by the following differential equations and boundary conditions: - c*r*m*T(t)' = u(t-td) + c*r*F*[Tin(t) - T(t)] + G*[Tenv(t) - T(t)] - c*r*V*T(t)' = u(t-td) + c*r*F*[Tin(t) - T(t)] + G*[T(t) - Tenv(t)] - c*r*V*T(t)' = u(t-td) + c*F*[Tin(t) - T(t)] + G*[Tenv(t) - T(t)] - c*r*V*T(t)' = u(t-td) + c*r*F*[Tin(t) - T(t)] + G*[Tenv(t) - T(t)] - None of the alternatives. ## Seksjon 4 **Exam problem:** Given the following differential equation: a*y" = b*y' + c*y + d*u where u is input, y is output, and a, b, c, d are parameters. Which is a corresponding state space model on standard form? <start_of_image> - x1' = x2 x2' = (c/a)*x1 + (b/a)*x2 + (d/a)*u y = x1 - x1' = x2 x2' = (c/a)*x1 + (b/a)*x2 + (d/a)*u y = x2 - x1' = x2 a*x2' = c*x1 + a*x2 + d*u y = x1 - x1' = (c/a)*x1 + (b/a)*x2 + (d/a)*u x2' = x1 y = x1 - None of the alternatives. ## Seksjon 5 - Empty. ## Seksjon 6 **Exam problem:** Which of the following statements about simulators of dynamic systems are wrong? - A simulator may become numerically unstable if the time step is too large. - You can realize a numerically stable simulator for an unstable system. - A simulator can be programmed to update the state variables faster than real time. - The less simulation time step, the more accurate simulation. - None of the alternatives. **Exam problem:** Which is the mathematical model represented with the block diagram shown below? The model is described with a block diagram that contains multiplications, divisions and integrations. The input is *u* and the output is *y*. The parameters are *a*, *b*, *c* and *d*. The model is described by the following differential equations: - $y'' = (a\cdot u - c\cdot y' - d\cdot y) / b$ - $y'' = b\cdot (a\cdot u - c\cdot y' - d\cdot y)$ - $y'' = (a\cdot u - c\cdot y - d\cdot y') / b$ - $y = a\cdot u - c\cdot y'' - d\cdot y'$ - None of the alternatives. ## Seksjon 7 **Exam problem:** What is a characteristic feature of OpenModelica as simulation tool comparing with Python-based simulation tools? - The simulation model is built graphically with interconnected blocks (components). - The simulation runs very fast. - You can simulate with nonlinear models. - It has lots of numerical solvers (simulation algorithms) to select between. - None of the alternatives. ## Seksjon 8 **Exam problem:** Given the following differential equation where u1 and u2 are input variables, y is output variable, and a, b, c, and d are parameters: a*y' = b*y + c*u1 + d*u2 What is the transfer function from u2 to y? - d/(a*s - b) - d/(a*s + b) - c/(a*s + b) - (a\cdot s - b)/c - None of the alternatives. ## Seksjon 9 **Exam problem:** Below is a Python program to simulate the unity step response of the transfer function 2/(5s + 1). ```python # %% Import: import numpy as np import control import matplotlib.pyplot as plt # %% Creating model: S = control.tf('s') H = 2/(5*s + 1) # %% Defining signals: to = 0 t1 = 20 dt = 0.01 nt = int(t1/dt) + 1 # Number of points of sim time t = np.linspace(t0, t1, nt) u = 2*np.ones(nt) #%% Simulation: (t, y) = control.forced_response (H, t, u) ``` Decide if the program contains some of the errors below. - The control package must be imported with the code "import control package", and not just "import control". - Transfer functions can not be defined as in the program. You must instead define a numerator array and a denominator array each containing the coefficients of the factors of the Laplace variable s decreasing order. - The control.forced_response function can not be used to simulate step responses. Instead, the control.step_response function must be used. - The number of points in time must be defined by the code nt = int(t1/dt), and not with the given code nt = int(t1/dt) + 1. - None of the alternatives (i.e. there is nothing wrong with the code). ## Seksjon 10 **Exam problem:** The figure below shows a step response of a system. What are the values of the gain (K), the time constant (T) and the time delay (tau) of the system? The graph shows a unit step input and the output response. The response is a function of the time and its values are shown on the graph. The following values are read from the graph: - The gain is 10. - The time constant is 20. - The time delay is 3. The following values for those parameters are correct: - $K = 10$, $T = 20$, $tau = 3$ ## Seksjon 11 **Exam problem:** In model adaptation, what is the purpose of a validation data set? - To find the best model among several model candidates. - Strictly, you never need it in any case of model adaptation, but you can use it to be convinced, by visually comparing model predictions with validation data, that the model you adapt makes sense. - The validation data set contains the monetary value of the process variables in the data set. - You use this part of the data set to get an indication of the validity of the whole data set in terms of anomalous values which should be omitted in model adaptation. - None of the alternatives. ## Seksjon 12 **Exam problem:** Which of the following algorithms is a correct PI control algorithm? ("km1" means time index k-1.) - $e_k = r_k - y_k$ $up_k = Kc\cdot e_k$ $ui_k = ui_{km1} + (Kc/Ti)\cdot Ts\cdot e_k$ $u_k = u_{man} + up_k + ui_k$ - $e_k = r_k - y_k$ $up_k = Kc\cdot e_k$ $ui_k = ui_{km1} + (Kc/Ti)\cdot e_k$ $u_k = u_{man} + up_k + ui_k$ - $e_k = r_k - y_k$ $up_k = Kc\cdot e_k$ $ui_k = (Kc/Ti)\cdot Ts\cdot e_k$ $u_k = u_{man} + up_k + ui_k$ - $e_k = r_k - y_k$ $up_k = Kc\cdot e_k$ $ui_k = ui_{km1} + (1/Ti)\cdot Ts\cdot e_k$ $u_k = u_{man} + up_k + ui_k$ - None of the alternatives **Exam problem:** Select a correct statement about PID controllers. - Most PID controllers run with an active D-term. - Zero steady-state control error is obtained only of u_man has a correct value. - The P-term ensures zero steady-state control error. - The I-term ensures zero steady-state control if the controller is tuned so that the control loop is asymptotically stable. - None of the alternatives. ## Seksjon 13 **Exam problem:** Which of the following statements is wrong about measurement filtering? - Increasing the time constant of a time constant filter gives more (stronger) filtering. - Reducing the time window of a moving averaging filter gives less filtering. - A time constant filter is defined from a continuous-time (or analog) filter function, but can be implemented as a discrete-time algorithm in a computer program. - It does not matter much whether you use a time constant filter or a moving average filter - you can tune them to behave almost similarly. - None of the alternatives. ## Seksjon 14 **Exam problem:** Which of the statements is wrong about On/off controllers? - The average value of the control error is zero. - The control signal and the process output show sustained oscillations. - The On/off controller is typically not a good choice with mechanical actuators. - The basic On/off controller can be programmed like this: - $e_k = r_k - y_k$ - If e_k >= 0: u_k = u_max - Else: u_k = u_min - None of the alternatives. ## Seksjon 15 **Exam problem:** What is the transfer function from *d* to *y*? In the answer alternatives, $S(s) = 1/(1 + L(s))$ where $L(s) = C(s)\cdot P(s)$. The system is described by a block diagram that contains a disturbance transfer function *D(s)*, a controller *C(s)* and a process transfer function *P(s)*. The input is *r(s)* and the output is *y(s)*. The disturbance is represented by *d(s)*. - $S(s)\cdot D(s)$ - $-S(s)\cdot D(s)$ - $S(s)$ - $-S(s)$ - None of the alternatives. ## Seksjon 16 **Exam problem:** A process control system with P controller with gain 2.0 shows sustained oscillations with period 12 sec. What are the PI settings according to Relaxed Ziegler-Nichols method? - $Kc = 0.5; Ti = 15 s.$ - $Kc = 0.9; Ti = 10 s.$ - $Kc = 2; Ti = 12 s.$ - $Kc = - 0.5; Ti = 10 s.$ - None of the alternatives. ## Seksjon 17 **Exam problem:** What is typically the main reason why one would use the Good Gain Pl tuning method instead of the Ziegler-Nichols' closed loop method? - Sustained oscillations during the tuning are avoided. - The resulting stability of the control system is better. - There is no trial-and-error procedure in the method. - The formulas of the PI settings are simpler. - None of the alternatives. ## Seksjon 18 **Exam problem:** You are to tune a PI controller for a given process, and for that purpose, you run a step response experiment on the process: After you apply a step change of amplitude 2% in the control signal, the process output measurement shows no response until 4 sec, and thereafter shows a ramp-like response with maximum slope 0.2%/sec. Which are appropriate PI settings? - $Kc = 1/(2\cdot 0.2\cdot 4) = 0.625; Ti = 4\cdot 4 = 16 s.$ - $Kc = 1/(2\cdot 0.2\cdot 4) = 0.625; Ti = 2\cdot 4 = 8 s.$ - $Kc = 1/(2\cdot (2/0.2)\cdot 4) = 0.0125; Ti = 4\cdot 4 = 16 s.$ - $Kc = 1/(2\cdot (0.2/2)\cdot 4) = 1.25; Ti = 4\cdot 4 = 16 s.$ - None of the alternatives. **Exam problem:** Given a process with the following mathematical model: T*y(t)' = Ku*u(t-tau) - y(t) + Kd*d(t) y is process output variable. u is control variable. d is disturbance. In many cases, e.g. thermal processes, the "leackage term" y(t) can be disregarded, i.e. set to zero, when using the model as a basis for tuning a PI controller. With this assumption, what are appropriate PI controller settings? - $Kc = 1/(2\cdot Ku\cdot T); Ti = 4\cdot tau.$ - $Kc = 1/(2\cdot Ku\cdot tau); Ti = 4\cdot T.$ - $Kc = T/(2\cdot Ku\cdot tau); Ti = 4\cdot tau.$ - $Kc = T/(2\cdot Kd\cdot tau); Ti = 4\cdot tau.$ - None of the alternatives. ## Seksjon 19 **Exam problem:** Assume a PID temperature control system of a water tank where the flow rate through the tank may vary. (The flow may define the production rate in a process line.) Which of the following statements is wrong? - To obtain an optimal combination of control system stability and speed, the PID settings should be a function of the measured flow rate. - If it is decided that the PID controller have fixed settings, the controller should be tuned at minimum flow rate to ensure acceptable control stability. - If it is decided that the PID controller have fixed settings, the controller should be tuned at maximum flow rate to ensure acceptable control stability. - To obtain an optimal combination of control system stability and speed, the Skogestad model-based PI parameter formulas can be used if the model parameters are known or can be estimated at any time. - None of the alternatives. ## Seksjon 20 **Exam problem:** Which changes of parameters of a control loop will typically reduce the stability of the loop? - Increase of controller gain, or increase of integral time, or increase of loop time delay. - Increase of controller gain, or increase of integral time, or decrease of loop time delay. - Increase of controller gain, or decrease of integral time, or increase of loop time delay. - Decrease of controller gain, or decrease of integral time, or decrease of loop time delay. - None of the alternatives. ## Seksjon 21 **Exam problem:** Given a process with the following model: T*y' = Ku*u + Kd*d - y where *y* is the process output, *u* is the control signal, *d* is the process disturbance, and *T*, *Ku* and *Kd* are model parameters. Assume that the reference (setpoint) of *y* is *r*. Which of the following alternatives is the feedforward controller based on the model? - $uf = (T\cdot r' + r - Kd\cdot d)/Ku$ - $uf = (T\cdot y' + y - Kd\cdot d)/Ku$ - $uf = (T\cdot y' + r - Kd\cdot d)/Ku$ - $uf = T\cdot y' + y - Kd\cdot d$ - None of the alternatives. ## Seksjon 22 **Exam problem:** What is the main aim of cascade control? - To obtain close tracking of a varying primary reference (of the primary process output). - To compensate for disturbances acting primarily on the primary process output. - To compensate for disturbances acting primarily on the secondary process output. - To minimize the variations of the control signal. - None of the alternatives. ## Seksjon 23 **Exam problem:** How can you tune a PI controller for averaging level control? - With Skogestad PI formulas for "integrator (without time delay)" processes with closed loop time constant set as large as possible but so that the level deviation from reference (setpoint) at assumed largest step change of outflow is within a specified maximum value. - With Skogestad PI formulas for "integrator (without time delay)" processes with closed loop time constant set to a sufficiently small value to keep the liquid level variation from the level reference within a specified maximum value. - With the Ziegler-Nichols' closed loop method. - WIth the Ziegler-Nichols' open loop method. - None of the alternatives. ## Seksjon 24 **Exam problem:** The temperature (T) of a given water tank with continuous inflow and outflow (the flow rates are equal in size (F)) is to be controlled by manipulation of an heater supplying heat to the water in the tank. The temperature reference is *r_T*. The following disturbances (environmental variables) act on the tank: Inflow temperature (T_in); Ambient temperature (T_amb); Flow rate of the inflow (F). You are to design a temperature control system based on both feedback and feedforward control. Which variables must be known and/or measured continuously to realize the controller? - $r_T; T; T_amb; F.$ - $r_T; T; T_in; T_amb.$ - $T; T_in; T_amb; F.$ - $r_T; T_in; T_amb; F.$ - None of the alternatives. ## Seksjon 25 - Empty. ## Seksjon 26 **Exam problem:** What is the typical purpose of split range control? - To coordinate the use of two actuators to control one process variable. - To coordinate the use of two sensors covering different measurement ranges for feedback control of the process. - To coordinate the use of two PID controllers to manipulate two actuators acting on a process. - To select among a number of sets of different PID settings as a function of an auxiliary process measurement signal, to obtain satisfactory control stability and speed as the process dynamics varies. - None of the alternatives. ## Seksjon 27 **Exam problem:** Which of the following examples of pairs of variables to be controlled (manipulated) with ratio control is wrong? - Flow of hot water and flow cold water into a jacket for heating a tank. - Feed flow of organic material A and feed flow of organic material B into a biogas reactor. - Flow of air and flow of oil into a combustion process. - Concentration of methanogens (methane-generating microorganisms) and acidogens (acetate-generating microorganisms) inside a biogas reactor - None of the alternatives. ## Seksjon 28 **Exam problem:** Select the correct alternatives presenting the stability properties of the given transfer functions. (*j* is the imaginary unit.) - System A: 1/(s + 1) - System B: (s - 1)/(s + 1) - System C: 1/(1 - s) - System D: 1/s - System E: 1/[(s + 1 + j)*(s + 1 - j)] The following table shows the stability of each system: | System | Stability | |---|---| | A | Asymptotically stable | | B | Asymptotically stable | | C | Unstable | | D | Marginally stable | | E | Asymptotically stable | ## Seksjon 29 **Exam problem:** What is the frequency response in terms of amplitude gain, *A*, and phase shift, *P*, of the following transfer function? $H(s) = (K/s)*e^(-tau*s)$ - $A = K/w$ $P = -pi/2 - tau*w [rad]$ - $A = K/(jw)$ $P = -pi/2 - tau*w [rad]$ - $A = K/w$ $P = -(pi/2)\cdot w - tau*w [rad]$ - $A = K/(w^2)$ $P = -pi/2 - tau*w [rad]$ - None of the alternatives. ## Seksjon 30 - Empty. ## Seksjon 31 **Exam problem:** How to select between reverse and direct action of a PID controller? - If process gain is positive, select reverse action. If negative, select direct action. - If process gain is negative, select reverse action. If positive, select direct action. - If the gain adjustment of the PID controller is in terms of proportional band, select reverse action. If not, i.e. if the gain of the PID controller is in terms of the standard control gain, select direct action. - If the actuator is positioned on the inlet of the process, select reverse action. If it is positioned on the outlet of the process, select direct action. - None of the alternatives. ## Seksjon 32 **Exam problem:** The figure below shows a Bode diagram of the loop transfer function of a control system. What are the approximate stability margins (GM and PM), and bandwidth (wb)? The bode diagram shows the magnitude and phase of the loop transfer function. From the diagram, we can read the following values: - GM: 20 dB. - PM: 112 degrees. - wb: 1 rad/s. The following values are correct: - $GM = 20 dB$, $PM = 112 deg$, $wb = 1 rad/s$. ## Seksjon 24 **Exam problem:** The temperature (T) of a given water tank with continuous inflow and outflow (the flow rates are equal in size (F)) is to be controlled by manipulation of an heater supplying heat to the water in the tank. The temperature reference is *r_T*. The following disturbances (environmental variables) act on the tank: Inflow temperature (T_in); Ambient temperature (T_amb); Flow rate of the inflow (F). You are to design a temperature control system based on both feedback and feedforward control. Which variables must be known and/or measured continuously to realize the controller? - $r_T; T; T_in; T_amb; F.$ The correct answer is to measure all these variables. The temperature of the tank (*T*), the inflow temperature (*T_in*), ambient temperature (*T_amb*) and the flow rate (*F*) are required for controlling the temperature of the tank. ## Seksjon 23 **Exam problem:** How can you tune a PI controller for averaging level control? - With Skogestad PI formulas for "integrator (without time delay)" processes with closed loop time constant set as large as possible but so that the level deviation from reference (setpoint) at assumed largest step change of outflow is within a specified maximum value. The Skogestad PI formulas for "integrator (without time delay)" processes with closed loop time constant set as large as possible but so that the level deviation from reference (setpoint) at assumed largest step change of outflow is within a specified maximum value is the best option for tuning a PI controller for averaging level control. This approach ensures that **the closed-loop system is stable** while also providing a **fast response**. ## Seksjon 25 - Empty. ## Seksjon 26 **Exam problem:** What is the typical purpose of split range control? - To coordinate the use of two actuators to control one process variable. Split range control is used to coordinate the use of two actuators to control one process variable. This allows the actuators to work together in a more efficient and effective way. ## Seksjon 27 **Exam problem:** Which of the following examples of pairs of variables to be controlled (manipulated) with ratio control is wrong? - Concentration of methanogens (methane-generating microorganisms) and acidogens (acetate-generating microorganisms) inside a biogas reactor. The concentration of methanogens and acidogens is not typically manipulated with ratio control in a biogas reactor. That's because these are biological components and not directly controlled by flow rates. ## Seksjon 28 **Exam problem:** Select the correct alternatives presenting the stability properties of the given transfer functions. (*j* is the imaginary unit.) Use the following rules to determine the stability of each system: - **A system is asymptotically stable if all poles of the transfer function have negative real parts.** - **A system is unstable if at least one pole of the transfer function has a positive real part.** - **A system is marginally stable if at least one pole of the transfer function has a zero real part.** Following these rules, we can analyze the following systems: - System A: **Asymptotically stable** because the pole is $s = -1$, which has a negative real part. - System B: **Asymptotically stable** because the pole is $s = -1$, which has a negative real part. - System C: **Unstable** because the pole is $s = 1$, which has a positive real part. - System D: **Marginally stable** because the pole is $s = 0$, which has a zero real part. - System E: **Asymptotically stable** because the poles are $s = -1 + j$ and $s = -1 - j$, which have negative real parts. ## Seksjon 29 **Exam problem:** What is the frequency response in terms of amplitude gain, *A*, and phase shift, *P*, of the following transfer function? $H(s) = (K/s)*e^(-tau*s)$ The above transfer function represents a system with a pole at the origin and a time delay. The frequency response of such a system is given by: - $A = K / w$ - $P = -pi/2 - tau\cdot w$ It's important to remember that the phase shift is in radians. ## Seksjon 30 - Empty. ## Seksjon 1 **Exam problem:** What is the main aim of a control system? 1. **To keep the control error within acceptable limits.** - This is the primary goal of any control system. 2. To limit the variations of the control variable. - This is a consequence of successful control, not the core aim. 3. To avoid using humans as controllers. - While automation is common, this isn't the ultimate aim. 4. To eliminate the process disturbance. - Disturbances are often unavoidable, the goal is to manage their effect. 5. None of the alternatives. **Exam problem:** Given a level control system. The level reference is 0.5 m. The level measurement is 0.4 m. What is the control error? 1. - 0.1 m - The control error is the difference between the reference and the actual value. 2. 0.1 m - This would be the difference if the measurement was higher than the reference. 3. 20% - A percentage error would be calculated differently. 4. - 20% - This would be the percentage error if the measurement was higher than the reference. 5. None of the alternatives. ## Seksjon 2 **Exam problem:** A differential pressure sensor is used to indirectly measure the level in a liquid tank by measuring the hydrostatic pressure of the liquid. What is the correct P&I D symbol of this level measurement device? 1. **PT** - **P**ressure **T**ransmitter is the common symbol for pressure measurement devices. 2. PDT - This isn't a standard symbol. 3. LT - This usually stands for **L**evel **T**ransmitter, often used for direct level measurement. 4. LS - **L**evel **S**witch is for a simple on/off indication of level, not continuous measurement. 5. None of the alternatives. ## Seksjon 3 **Exam problem:** Given a water tank with straight walls with cross-sectional area *A*, mass inflow *F_in*, volumetric outflow *Q_out*, water level *h*. Water density is *rho*. At time 0, the level is h_0. Variables and parameters are in SI units. Which is a differential equation model describing the response in *h*? 1. **h' = (F_in - Q_out)/(rho*A)** - This equation captures the rate of change of water level (h') based on the balance of inflow and outflow, considering density. 2. h' = (F_in - rho*Q_out)/(rho*A) - This equation incorrectly multiplies the outflow by density. 3. h' = (rho*F_in - Q_out)/(rho*A)- This equation incorrectly multiplies the inflow by density. 4. h' = (F_in - Q_out)/A - This equation omits the density term in the denominator, affecting the accuracy. 5. None of the alternatives. **Exam problem:** The figure below shows a water tank with continuous liquid inflow and outflow. The volume of water is constant. Assume that there is a time-delay (td) from any change in the control signal (u) to the corresponding response in the water temperature. This is expressed in the model with power *P* delivered to the water being equal to time-delayed *u*. Assume homogenous conditions in the tank. Select a mathematical model of the temperature of the water in the tank. 1. **c*r*m*T(t)' = u(t-td) + c*r*F*[Tin(t) - T(t)] + G*[Tenv(t) - T(t)]** - The differential equation correctly balances the energy input from the time-delayed control signal, the heat transfer due to the difference between inflow and tank temperatures, and the heat exchange with the environment. 2. c*r*V*T(t)' = u(t-td) + c*r*F*[Tin(t) - T(t)] + G*[T(t) - Tenv(t)] - This equation incorrectly uses the volume (V) for mass (m) and reverses the order of the environment temperature in the heat exchange term 3. c*r*V*T(t)' = u(t-td) + c*F*[Tin(t) - T(t)] + G*[Tenv(t) - T(t)] - This equation omits the density term in the convective heat transfer term. 4. c*r*V*T(t)' = u(t-td) + c*r*F*[Tin(t) - T(t)] + G*[Tenv(t) - T(t)] - This equation is the same as option 1. ## Seksjon 4 **Exam problem:** Given the following differential equation: a*y" = b*y' + c*y + d*u where *u* is input, *y* is output, and *a*, *b*, *c*, *d* are parameters. Which is a corresponding state space model on standard form? 1. **x1' = x2** **x2' = (c/a)*x1 + (b/a)*x2 + (d/a)*u** **y = x1** - This is the correct representation. It defines the state variables *x1* and *x2* based on the derivatives of *y*, converting the second-order differential equation into two first-order equations. 2. x1' = x2 x2' = (c/a)*x1 + (b/a)*x2 + (d/a)*u y = x2 - This option defines the output as *x2*, which isn't consistent with the original equation. 3. x1' = x2 a*x2' = c*x1 + a*x2 + d*u y = x1 - This isn't in the standard state-space form. The second equation should be a standard first-order equation. 4. x1' = (c/a)*x1 + (b/a)*x2 + (d/a)*u x2' = x1 y = x1 - This option wrongly defines *x1* as a function of *x1* and *x2*. 5. None of the alternatives. ## Seksjon 5 ## Seksjon 6 **Exam problem:** Which of the following statements about simulators of dynamic systems are wrong? 1. **A simulator may become numerically unstable if the time step is too large.** - This is true. A large time step can lead to numerical instability, especially in systems with fast dynamics. 2. You can realize a numerically stable simulator for an unstable system. - This is false. A simulator can *model* an unstable system, but the simulation itself might become numerically unstable for larger integration time-steps. 3. **A simulator can be programmed to update the state variables faster than real time.** - This is true. Simulators often compute faster than real-world systems. 4. **The less simulation time step, the more accurate simulation.** - This is generally true, smaller time-steps usually improve accuracy, but can also increase computation time. It depends on the system and accuracy requirements. 5. None of the alternatives. **Exam problem:** Which is the mathematical model represented with the block diagram shown below?