Lecture 2

Question 1

model vs simulation

Answer 1

model: simplified representation of a physical system

simulation: repeated solving (computation) of a model (equations) in order to reproduce the behaviour of the modeled system

Question 2

goals of simulation

Answer 2

1. identify and compare system designs and their performance
2. obtain insight into the behaviour of a system or certain aspects of it
3. simulator development
4. understand how a system will behave in different situations and why
5. control development - determine what inputs are needed to yield a desired behaviour
6. optimization - predict the system’s behaviour for each possible variation of design choices

Question 3

control development

Answer 3

determine what inputs are needed to yield a desired behaviour

gain tuning: most widely used controllers are linear (otherwise they’re expensive)

preliminary testing

model-based control

Question 4

optimization

Answer 4

predict the system’s behaviour for each possible variation of design choices

determine trade-off winners

Question 5

simulation verification techniques

Answer 5

1. model-in-the-loop (MIL) (most abstract)
2. simulation-in-the-loop (SIL)
3. processor-in-the-loop (PIL)
4. hardware-in-the-loop (HIP)
5. physical system tests (most physical)

Question 6

model-in-the-loop (MIL)

Answer 6

does the controller logic work?

testing on simulated system model, recording and saving the IO-behaviour of the model response

Question 7

simulation-in-the-loop (SIL)

Answer 7

generating code only from the controller model and replacing the controller block with this code

gives an idea whether the control logic (the controller model) can be converted to code and if its hardware implementable

Question 8

processor-in-the-loop (PIL)

Answer 8

put controller code on embedded processor and run closed-loop simulation again on simulation plant

determine if the processor is able to run the developed controller logic

Question 9

hardware-in-the-loop

Answer 9

for testing of behaviour which cannot be captured in simulation

run simulation model on real-time system with real physical connections to embedded processor

do problems arise due to communication and IO interfaces

Question 10

physical system test

Answer 10

test controller directly on the system

Question 11

logic controller

Answer 11

industrial computer that has been ruggedized (adapted to harsh environmental conditions) and adapted for the control of manufacturing processes, such as assembly lines, machines, robotic devices, or any activity that requires high reliability, ease of programming, and process fault diagnosis

Question 12

algebraic system vs dynamic system

Answer 12

algebraic system: current output depends on current input only

dynamic system: current output depends on current and previous input values

Question 13

Laplace transformation

Answer 13

tool for analysing linear dynamic system

tells which frequencies and exponentials are present in a function

can simplify ODEs into algebraic equations

Question 14

transfer function

Answer 14

ratio of Laplace transform of output and Laplace transform of input, when initial conditions are assumed to be zero

relates input and output with an algebraic expression

limitations: only linear time-invariant systems and only single-input single-output (SISO) systems

Question 15

elements of transfer function

Answer 15

poles: values of s for which G(s) goes to infinity

zaros: values of s for which G(s) goes to zero

gain: steady-state value of G(s)

Question 16

stable system

Answer 16

all poles of the transfer function are in the left half-plane

Question 17

state-space model

Answer 17

representation of the dynamics of an n-th order system as a set of first-order differential state equations (and algebraic output equations)

describes temporal change (first-order time derivative) of the state variable as a function of current state variables and current inputs

x - state vector
u - input vector
y - output vector

Question 18

equations of motion (EOM)

Answer 18

x’(t) = f(x,u)

states x: position, attitude, linear velocities, angular velocities

inputs u

Question 19

LTI system

Answer 19

both linear (homogenity and superposition holds) and time-invariant system

has A, B, C, D matrices

Question 20

linearisation

Answer 20

usign e.g. Taylor series

allows us to obtain linear approximations of nonlinear models around specific conditions

Question 21

eigenvalues and eigenvectors

Answer 21

eigenvalues show how the system responds to a disturbance over time

eigenvectors show to what extent is each state involved in each mode

Question 22

stability assessment of LTI

Answer 22

stable if real part of eigenvalue is below zero

unstable if real part of eigenvalue is over zero

undefined if real part of eigenvalue is zero

Question 23

static stability

Answer 23

initial tendency of a body to return to its original position when disturbed

a statically stable system can be dynamically unstable

a statically unstable system cannot be dynamically stable

Question 24

dynamic stability

Answer 24

response of a body to a disturbance over time (frequency, damping, etc.)

a statically stable system can be dynamically unstable

a statically unstable system cannot be dynamically stable

Question 25

global vs linear model

Answer 25

global: model that is valid across a whole envelope (all operating conditions)

linear: captures only the behaviour of a nonlinear system in a specific operating condition

Question 26

discretisation

Answer 26

contuniuous -> discrete

Question 27

analog-to-digital converters (ADC)

Answer 27

system that converts an analog signal, such as a sound picked up by a microphone or light entering a digital camera, into a digital signal

Question 28

bandwidth

Answer 28

difference between highest and lowest frequencies

Question 29

Nyquist theorem

Answer 29

to reproduce a signal without any distortion or loss of data, the sampling frequency must be greater than twice the maximum signal frequency or twice the bandwidth

Question 30

finite-difference (FD) methods

Answer 30

1. forward Euler: linear combination of values f after the considered point
2. central scheme: linear combination of values f before and after the considered point
3. backward Euler: linear combination of values f before the considered point

Question 31

numerical methods for solving linear ODEs

Answer 31

1. single step method
2. multi step method
3. forward / explicit Euler
4. backward / implicit Euler
5. 2nd-order Runge-Kutta
6. 4th-order Runge-Kutta

Question 32

properties of numerical methods

Answer 32

1. consistency error of single step approaches
2. convergence
3. numerical stability

Question 33

stiffness

Answer 33

ODE is stiff if two or more significantly different time scales occur in the system

differential equation for which certain numerical methods for solving the equation are numerically unstable, unless the step size is taken to be extremely small

it has proven difficult to formulate a precise definition of stiffness, but the main idea is that the equation includes some terms that can lead to rapid variation in the solution

Question 34

well-posed numerical problem

Answer 34

1. solution u exists
2. solution u is unique
3. solution u is stable

Question 35

step size

Answer 35

smaller step size: more accurate, slower

bigger step size: faster, less accurate, if too big then solution might become unstable

Question 36

computational step solvers

Answer 36

variable step solvers: can modify step size during simulation depending on the integration error

fixed step solvers: maintain the same step size throughout simulation

Question 37

simulation solvers

Answer 37

Simulink, Gazebo, Ansys, Microsoft Flight Simulator