Solving Inverse Problems

Numerical Methods

Algorithms for approximating solutions to mathematical problems that are difficult/impossible to solve analytically

Inverse Problems

Known output + model and unknown input OR known input + output and unknown model

Time Invariant

System properties do not change with time

Linear

System response satisfies superposition principle

Linear and Time-Invariant

LTI System (closed loop)

Closed-Loop Poles

Determine the stability and performance of a closed-loop system

Bisection Method

Iterative method for finding roots predicated on the Intermediate Value Theorem

Bisection Method

Root lies between m and b

If f(m) > 0 for the bisection method

Root lies between a and m

If f(m) < 0 for the bisection method

Root found

If f(m) = 0 within tolerance for the bisection method

Linear

Convergence rate of bisection method

Newton-Raphson Method

Uses the rate of change of f(x) to speed up bisection method (also: 1st order Taylor approximation)

Newton-Raphson Method Formula

Newton-Raphson Method

Quadratic

Convergence rate of Newton-Raphson method

Secant Method

Uses Newton-Raphson method formula but approximated f’(x) when unable to find analytically

Backward Finite Difference

Approximates f’(x) for secant method

Secant Method Formula

Secant Method

Draw a line between f(p₀) and f(p₁), x-int of line is p₂ → repeat w/ p₁ and p₂

Superlinear

Convergence rate of secant method

a and b

Initial guesses for bisection method

p₀

Initial guess for Newton-Raphson method

p₀ and p₁

Initial guesses for secant method

Yes

Convergence guarantee for bisection method?

No

Convergence guarantee for Newton-Raphson method?

No

Convergence guarantee for secant method?

Lambda Function

Anonymous functions, simple enough to be defined in one line and only used once

Lamba Function Syntax

lambda x: [func]

Import Callable

from collections.abc import Callable

Callable

Used to define the input and output types of a function

Callable Syntax

def function(param_1: Callable[[type], type], param_2: type) -> type

LIFO

Last in, first out

Call Stack

Memory area where items are stored and retrieved on a LIFO basis → from functions calling each other

Stack Frame

When function is called, frame is created to hold all info needed to execute function call

Stack Overflow

Stack storage is filled (recursion cannot go arbitrarily deep)

RecursionError

Max depth exceeded

Returns current value of recursion limit

sys.getrecursionlimit

Sets recursion limit

sys.setrecursionlimit

Recursive Limitation

Can return a RecursionError → memory fills up quickly and can only calculate smaller numbers/recursions