Learning Space/Optimization

Left & Right Eigenvectors and Eigenvalues

Eigenvalue \( \chi (\lambda) = \textbf{det}(\lambda\textit{I}-A)=0 \) 에서, \( \lambda \in \mathbb{C} \) : \( A \in \mathbb{C}^{n \times n} \) 의 eigenvalue (Right) eigenvector \( \exists\ nonzero \ v \in \mathbb{C}^n \) s.t. \(( \lambda I-A)v = 0 \), i.e., \(Av = \lambda v \) \(v \) : A의 eigenvector (Left) eigenvector \( \exists\ nonzero \ w \in \mathbb{C}^n \) s.t. \( w^{T}(\lambda I-A) = 0 \), i..

Reachability & Controllability

\( \dot{x}(t)=Ax(t) + Bu(t) \) \( y(t) = Cx(t) + Du(t) \) time interval \( [t_i, t_f] \) 동안에 대해 성립한다고 가정하자. State transfer : input \( u : [t_i, t_f] \to \mathbb{R}^m \) 으로 \(x(t_i) \to x(t_f) \) 로 transfer하는 것 Reachability 주어진 initial state \( x(t_0) \) 로부터 모든 final state \( x(t_1) \) 까지 state transfer이 가능할 때 \(x(t)\) is reachable (in t seconds or epochs) CT(Continuous Time) system : \( \dot{x}(..

Linear Quadratic Regulator(LQR)

목적 \( J(U) \) 를 최소화하는 \( u_{0}^{lqr}, \cdots, u_{N-1}^{lqr} \) 찾기 Discrete-time finite horizon discrete-time system : \( x_{t+1}=Ax_{t}+Bu_{t}, x_0=x^{init} \) \( x_{0}, x_{1}, \cdots \) 작은 경우, good regulation or control \( u_{0}, u_{1}, \cdots \) 작은 경우, small input effor or actuator authority Quadratic cost function을 정의하자. \( J(U) = \sum_{\tau =0}^{N-1} (x_\tau^{T}Qx_{\tau}+u_\tau^{T}Ru_{\tau})..