As mentioned in [[- Robotics, Mechanics and Control -]], certain subfields may use slightly different notation based on the application. >[!brainwaves] Intuition >LTI systems are a **special case of a [[Static, Dynamic and Stochastic Systems|dynamical system]]** that is extensively studied in control theory. They are based on the assumption that the systems behavior is constant in time (governed by physical law) and that the system is linear (we can understand the whole by analyzing its parts and superposition). Mathematically, the functional is not explicitly dependent on the independent variable $t$. Additionally, we assume the [[ODEs - Ordinary Differential Equations|reformulation to a first order system]]. This yields the general formulation$\begin{align}\dot{\mathbf{x}}&=\mathbf{A}\mathbf{x}+\mathbf{Bu} \\ \mathbf{y}&=\mathbf{Cx}+\mathbf{Du}\end{align}$ In practice, controllers are implemented on a digital controller and therefore require a [[Linearization and Discretization of State Space Models|discretized version]], which for LTI system can be derived exactly$\begin{align}\mathbf{x}_{k+1}&=(\mathbb{1}+\Delta t\mathbf{A})\mathbf{x}_k+\Delta_t\mathbf{Bu}_k \\ \mathbf{y}_k&=\mathbf{Cx}_k+\mathbf{Du}_k\end{align}$Under these assumptions, we can conduct a vast amount of analysis and derive an abundance of properties. For details , see - [[Controllability and Stabilizability of LTI Systems]] - [[Observability and Detectability of LTI Systems]] - [[Lyapunov Stability for LTI Systems]] >[!success] Non-Uniqueness of State Representation >The input-output behavior $\mathbf{u}\mapsto \mathbf{y}$ is entirely defined by the initial condition $\mathbf{x}(0)$ and the control input sequence $\{\mathbf{u}_k\}_{k=0,1,...}$. **For LTI systems, the choice of a state representation is not unique**. >[!proof] Proof >Consider any [[Maps and Induced Structures - Functions, Pushforwards and Pullbacks|bijective]] [[Matrices|linear transformation]] $\mathbf{T},\quad \det (\mathbf{T}) \neq 0 \quad \tilde{\mathbf{x}}= \mathbf{Tx}$and insert it into our system$\begin{align} \mathbf{T}^{-1}\tilde{\mathbf{x}}_{k+1}&=(\mathbb{1}+\Delta t\mathbf{A})\mathbf{T}^{-1}\tilde{\mathbf{x}}_k+\Delta_t\mathbf{Bu}_k \\ \mathbf{y}_k&=\mathbf{C}\mathbf{T}^{-1}\tilde{\mathbf{x}}_k+\mathbf{Du}_k. \end{align}$Therefore, we can simply redefine all the matrices with the state $\mathbf{x}$ while leaving input and output unchanged. >[!brainwaves] Intuition >Since the transformation is bijective, we essentially just say that as long as the new state representation does not collapse dimensions / looses information, the basis in which we express the state does not matter.