User:Egm6321.f10.team4.Yoon/Mtg14

EGM6321 - Principles of Engineering Analysis 1, Fall 2010

SC-L1-ODE-VC and SC-L1-ODE-CC

Mtg 14: Thu, 23 Sep 10

Page 14-1

Linear Time-Invariant system $\displaystyle \equiv$ SC-L1-ODE-CC
Linear Time-Variant system $\displaystyle \equiv$ SC-L1-ODE-VC

Particular case: When n=1 (L1-ODE-VC)

Eqn.(2) p.13-2: $\displaystyle \begin{align} \dot x(t) = a(t)x(t)+b(t)u(t) \end{align}$

(1)

Open loop: u(t) is prescribed.

HW: L1-ODE-CC $\displaystyle x(t) = [\exp a(t-t_0)] x(t_0) + \int^t_{t_0} [\exp a(t-\tau)]b (\tau) u(\tau) \, d \tau$ $\displaystyle \color{red}{\begin{align} (2) \end{align}}$

L1-ODE-VC: Varying coefficients

$\displaystyle x(t) = \left[\exp \int^t_{t_0} a(\tau) d\tau \right] x(t_0) + \int^t_{t_0} \left[ exp \int^t_{\tau} a(s) ds \right] b(\tau) u(\tau) \, d\tau$

(3)

SC-L1-ODE-CC: Constant coefficients

$\displaystyle \underbrace{\mathbf{x}(t)}_{\color{blue}{n \times 1}}=\left[ \exp [ \underbrace{\mathbf{A}}_{\color{blue}{n \times n}}(t-t_0) ] \right]_{\color{blue}{n \times n}} \underbrace{\mathbf{x}(t_0)}_{\color{blue}{n \times 1}} + \int^t_{t_0} \left[ \exp [ \underbrace{\mathbf{A}(t-\tau)}_{\color{blue}{n \times n}} ] \right]_{\color{blue}{n \times n}} \underbrace{\mathbf{B}}_{\color{blue}{n \times m}}\underbrace{\mathbf{u}(\tau)}_{\color{blue}{m \times 1}}d\tau$

(4)

Page 14-2

Recall:

$\displaystyle \begin{align} \exp x = 1+ \frac{x}{1!} + \frac{x^2}{2!} + \cdot \cdot \cdot = \sum^{\infty}_{k=0} \frac{x^k}{k!} \end{align}$

(1)

$\displaystyle \begin{align} \exp \underbrace{\mathbf{A}}_{n\times n} = \underbrace{\mathbf{I}}_{n\times n} + \frac{1}{1!}\underbrace{\mathbf{A}}_{n\times n} + \frac{1}{2!} \mathbf{A}^2 + \cdot \cdot \cdot = \sum^{\infty}_{k=0} \frac{1}{k!}\mathbf{A}^k \end{align}$

(2)

HW: Generalized Eqn.(4) p.14-1, 14-2 to SC-L1-ODE-C

Bryson&Ho 1975 p.450

$\displaystyle \begin{align} \mathbf{x}(t)= \mathbf{\Phi}(t,t_0) \mathbf{x}(t_0) + \int^t_{t_0} \mathbf{\Phi}(t,\tau)\mathbf{B}(\tau)\mathbf{u}(\tau)d\tau \end{align}$

(3)

Compare Eqn.(3) to Eqn.(4) p.14-1 (Constant Coefficients)

$\displaystyle \begin{align} \underbrace{\mathbf{\Phi}(t,t_0)}_{\color{blue}{n\times n}} = \exp \underbrace{\mathbf{A}(t-t_0)}_{\color{blue}{n\times n}} \end{align}$

(4)

Page 14-3

$\displaystyle \mathbf{\Phi}$ is related to Integrating Factor

Prop. of $\displaystyle \color{blue}{\mathbf{\Phi}(t, t_0)}$: (State transition matrix)

$\displaystyle \begin{align} \begin{cases} &\displaystyle \frac{d}{dt}\underbrace{\mathbf{\Phi}(t,t_0)}_{\color{blue}{n\times n}} = \underbrace{\mathbf{A}}_{\color{blue}{n\times n}} \underbrace{\mathbf{\Phi}(t,t_0)}_{\color{blue}{n\times n}}\\ &\mathbf{\Phi}(t_0,t_0) = \underbrace{\mathbf{I}}_{\color{blue}{n\times n}}\\ \end{cases} \end{align}$

(1)

Recall SC-L1-ODE-CC