Chapter 2 Flashcards
What is the difference between a model and a simulation?
- A model represents a physical system
- A simulation is the repeated solving of a model in order to produce the behavior of the modeled system
Why do we simulate?
- Identify and compare feasibility of design systems
- Obtain insight into behavior of a system
- Simulator development
- Understand how a system will behave in different situations
What is control development and what are some examples?
It determines what inputs are needed to obtain a desired output
- Gain Tuning
- Preliminary Testing
- Model-based control
What are some simulation verification Techniques?
- Model in the Loop
- Simulation in the Loop
- Process in the Loop
- Hardware in the Loop
- Physical System Tests
What is model in the loop testing?
It tests if the controller logic works
- Testing on simulated system model
- Record and save IO behavior of model
What is Process in the Loop testing?
- Put controller code on embedded processor and run closed loop simulation on simulation plant
- Determine if processor is able to run the developed controller logic
What is Hardware in the Loop testing?
- For testing behavior that cannot be captured in simulation
- Run simulation model on real time system with real physical connections to embedded processor
- Check for problems in IO interfaces
What is a time invariant system?
Response signal does not change with time
Does not matter when in time an input is applied
What is an ordinary differential Equation?
- Depends on just one independent variable
What is a Partial Differential Equation?
Depends on more than one variables
What is the formula for the Laplace Transform?
F(s) = \int_{0}^{\infty} f(t)e^{-st} dt
What is the transfer function?
Ratio of Laplace transform of output to Laplace transform of input when initial conditions are assumed to be zero
- Relates input and output with an algebraic expression
What are the limitations of the transfer function?
- Only works for linear time-invariant system
- Only single input output systems
What is mathematical representation of transfer function?
Y(s)/ U(s) = (2s+3)/(s+1)(s+4)
What are the poles of a transfer function?
Values of s for which G(s) approaches infinity
What are zeros of a transfer function?
Values of s for which G(s) approaches 0
What is the gain of a transfer function?
Steady state value of G(s), s=0
n
What is a state space system?
- A representation of the dynamics of an nth order system as a set of first-order differential state equations
- State-space models describe temporal change (first-order time derivative) of the state variables as a function of current state variables and current inputs
What is the notation used for the state space system ?
/dot{x} = f(x,u), y = h(x,u)
x: state vector, describes current state completely. Components of x are state variables.
u: input vector, contains the input variables
y: output vector
What is the notation for Linear state space model?
\dot{x} = Ax + Bu, x(0) = x_0
y = Cx + Du
A: system matrix, represents system dynamics
B: input matrix, describes how inputs influence system
C: output matrix, defines how states are combined to yield desired output
D: feed-through matrix, ratio of output to input under steady state conditions
What is Linearity?
f(x+y) = f(x) + f(y)
What is homogeneity?
f(ax) = af(x)
When is a system linear?
If homogeneity and superposition hold
What do the eigenvalues of a system tell us?
How the system responds to disturbances over time
What do the eigenvectors of a system tell us?
To what extent each state is involved in each mode
How can we assess LTI system stability using eigenvalues
Left is stable right is unstable
What are static and dynamic stability?
Static stability is the tendency of a body to return to its original position when disturbed
Dynamic stability is the response of a body to a disturbance over time
What are global models?
Models that are valid across a whole envelope
What is the difference between a continuous and discrete signals?
Continuous signals have uncountable number of states and discrete signals have finite number of states.
What is an analog to digital converter?
Converts continuous signal to discrete signal using sampling
What does bandwidth describe?
Difference between highest and lowest frequency
What is the 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.
What do finite difference methods do?
Approximate the derivative of a signal at a specific point via a linear combination of values of that signal at a neighboring point
What are three examples of FD methods and what do they do?
Forward euler: uses linear combination of values after f
Central Scheme: Uses linear combination of values before and after f
Backward euler: uses linear combination of values before point
What is the order of accuracy of a FD scheme?
The power of delta x to which the truncation error is proportional.
What are the truncation errors of forward euler, central and backward euler?
O(\delta x), O(\delta x^2) ,O(\delta x)
What are the numerical methods for solving ODEs?
Single Step:
- x_{n+1} computed from x_n
- Eg. Forward euler, Rung Kutta
- Variable step size possible
Multi Step:
- E_{n+1} computed from multiple previous points
- Eg, Explicit midpoint method
- Variable step size difficult
What is the formula for forward / explicit euler?
x_{n+1} = x_n + hf(t_n,x_n)
What is the formula for backward/implicit euler?
x_{n+1} = x_n + hf(t_{n+1}, x_{n+1})
What is a stiff ODE?
An ODE is stiff if two or more significantly different time scales occur in the system
When is a numerical problem well-posed?
- If a solution u exists
- the solution is unique
- solution u is stable
