1.jpeg

## Chapter 1.1: Dynamic Systems in SS Representation * Differential equation order = number of states (denominator) ($n \to n_{eq}$). * Advantages: Robust, linear algebra, complex order systems, nonlinear, time varying systems, ($\phi$ $\neq$ 0). * Disadvantages: Mathematical model of system, compu...

## Chapter 1.1: Dynamic Systems in SS Representation * Differential equation order = number of states (denominator) ($n \to n_{eq}$). * Advantages: Robust, linear algebra, complex order systems, nonlinear, time varying systems, ($\phi$ $\neq$ 0). * Disadvantages: Mathematical model of system, computational science, matrix = arrays, engineering, control matrix mapping x to y, 1 row per output (y), $y = \sum Ax$. * Numerator > denominator -> No direct feed through, no jump at output. * Numerator = denominator -> Direct feed through, jump at output. * $P_T$, $P_T^2$ - All numerator is constant. * X (state vector) = ($x_1$, $x_2$) (state variables) (has all data required to describe the system). * State variables = n (den) -> min realization. * We need min 2n initial conditions. * Stable SS systems end in (0,0) -> A gives direction & strength. * SISO -> MIMO; same A. ## Chapter 1.2: Solving SS Equations * $x(t) = e^{At}x_0 + \int_{0}^{t} e^{A(t-\tau)} b(u(\tau))d\tau$. * $SI = (\sigma s)$ * $Anxn \to$ n eigenvalues, n eigenvectors * Eigenvalues of A * System is stable when all poles (a) of G(s) are negative; it only depends on A (dynamic part). * A (Eigenvalue) only scales down output (no rotation) * Negative A shrinks towards origin.