Lecture 2 Flashcards
model vs simulation
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
goals of simulation
- identify and compare system designs and their performance
- obtain insight into the behaviour of a system or certain aspects of it
- simulator development
- understand how a system will behave in different situations and why
- control development - determine what inputs are needed to yield a desired behaviour
- optimization - predict the system’s behaviour for each possible variation of design choices
control development
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
optimization
predict the system’s behaviour for each possible variation of design choices
determine trade-off winners
simulation verification techniques
- model-in-the-loop (MIL) (most abstract)
- simulation-in-the-loop (SIL)
- processor-in-the-loop (PIL)
- hardware-in-the-loop (HIP)
- physical system tests (most physical)
model-in-the-loop (MIL)
does the controller logic work?
testing on simulated system model, recording and saving the IO-behaviour of the model response
simulation-in-the-loop (SIL)
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
processor-in-the-loop (PIL)
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
hardware-in-the-loop
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
physical system test
test controller directly on the system
logic controller
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
algebraic system vs dynamic system
algebraic system: current output depends on current input only
dynamic system: current output depends on current and previous input values
Laplace transformation
tool for analysing linear dynamic system
tells which frequencies and exponentials are present in a function
can simplify ODEs into algebraic equations
transfer function
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
elements of transfer function
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)
stable system
all poles of the transfer function are in the left half-plane
state-space model
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
equations of motion (EOM)
x’(t) = f(x,u)
states x: position, attitude, linear velocities, angular velocities
inputs u
LTI system
both linear (homogenity and superposition holds) and time-invariant system
has A, B, C, D matrices
linearisation
usign e.g. Taylor series
allows us to obtain linear approximations of nonlinear models around specific conditions
eigenvalues and eigenvectors
eigenvalues show how the system responds to a disturbance over time
eigenvectors show to what extent is each state involved in each mode
stability assessment of LTI
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
static stability
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
dynamic stability
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
global vs linear model
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
discretisation
contuniuous -> discrete
analog-to-digital converters (ADC)
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
bandwidth
difference between highest and lowest frequencies
Nyquist theorem
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
finite-difference (FD) methods
- forward Euler: linear combination of values f after the considered point
- central scheme: linear combination of values f before and after the considered point
- backward Euler: linear combination of values f before the considered point
numerical methods for solving linear ODEs
- single step method
- multi step method
- forward / explicit Euler
- backward / implicit Euler
- 2nd-order Runge-Kutta
- 4th-order Runge-Kutta
properties of numerical methods
- consistency error of single step approaches
- convergence
- numerical stability
stiffness
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
well-posed numerical problem
- solution u exists
- solution u is unique
- solution u is stable
step size
smaller step size: more accurate, slower
bigger step size: faster, less accurate, if too big then solution might become unstable
computational step solvers
variable step solvers: can modify step size during simulation depending on the integration error
fixed step solvers: maintain the same step size throughout simulation
simulation solvers
Simulink, Gazebo, Ansys, Microsoft Flight Simulator
