Miscellaneous

→ Linear Quadratic Problem

references:

Linear–quadratic regulator
Quadratic programming

Linear quadratic problem, whose system dynamics are described by a set of linear differential equations (linear dynamic system), and cost function is described by a quadratic function ^[lq1].

EXAMPLE 1: finite-horizon, continuous-time LQR

continuous-time linear system, defined on $t\in [t_0,t_1]$ , described by

$\dot{\mathit{x}}=\mathit{A}\mathit{x}+\mathit{B}\mathit{u}$ (lq.1)

with a quadratic cost function defined as

$\mathit{J}={\mathit{x}}^T\left(t_1\right)\mathit{F}\left(t_1\right)\mathit{x}\left(t_1\right)+{\int }_{t_0}^{t_1}({\mathit{x}}^T\mathit{Q}\mathit{x}+{\mathit{u}}^T\mathit{R}\mathit{u}+2{\mathit{x}}^T\mathit{N}\mathit{u})dt$ (lq.2)

where $\mathit{F}\left(t_1\right)\ge \mathbf{0},\mathit{Q}\ge \mathbf{0},\mathit{R}>\mathbf{0},\mathit{N}>\mathbf{0}$

→ diff MPC LQR

LQR is continous infinite horizon solved by Riccati and Kalman, while MPC is discrete and finite horizon

→ matrix, vector, derivative, calculus, quadratic forms, ${\mathit{x}}^T\mathit{P}\mathit{x}$ , ${\mathit{A}}^T\mathit{P}+\mathit{P}\mathit{A}$

1. vector derivation
references:

Vector derivation of ${\mathit{x}}^{\mathit{T}}\mathit{x}$ ^[mvdc1]

→ first order differential function

1. reasoning process of (kf.6)
references:

Linear Differential Equations - Paul's Online Math Notes

process:
For LTI system in state space like

  $\frac{d\mathit{x}\left(t\right)}{dt}=\mathit{A}\mathit{x}\left(t\right)+\mathit{B}\mathit{u}\left(t\right)$ (fodf.1)

This is a special case of first order differential functions (see part 2 of this section).
solution see [fodf1] [fodf2]

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2. DE, ODE, PDE, category of differential equations
Differential equations (DE) is mathematical equations that relate some function with its derivatives ^[fodf3] .
DE can be categorized as several types: Ordinary/Partial, Linear/Non-linear, and Homogeneous/Inhomogeneous ^[fodf4] .

An ordinary differential equation (ODE) is a differential equation containing one or more functions of one independent variable and its derivatives, while partial differential equation may be respect to more than one independent variable ^[fodf5] .
The form of ordinary differential equation of order $n$ is ^[fodf6] :

  $F(x,y,\frac{df\left(x\right)}{dx},\dots ,\frac{d^{n}f\left(x\right)}{d{x}^n})=0$ (fodf.2)

If (fodf.2) not depending on $x$ is called autonomous ^[fodf7] .
If $F$ is a linear function ^{[fodf8] [fodf9]} of $f\left(x\right)$ and its derivatives, then the ODE is called linear, and its form may be rewritten as:

  $y^{\left(n\right)}=\sum _{i=0}^{n-1}{a}_i\left(x\right)y^{\left(i\right)}+r\left(x\right)$ (fodf.3)

where $a_i\left(x\right)$ and $r\left(x\right)$ are continuous functions in $x$ ^[fodf10] , notice that $a_i\left(x\right)$ could be function in $x$ other than constant number . If $r\left(x\right)=0$ , then linear ODE (fodf.3) is homogeneous with trivial solution ^[fodf11] $y=0$ . If $r\left(x\right)\ne 0$ , then linear ODE (fodf.3) is inhomogeneous. Inhomogeneous ODE can be solved if the general solution to the homogenous version is known ^[fodf12] .
If $n$ in (fodf.2) is 1, then such ODE is called first-order ODE ^[fodf13] . Thus, state function of an LTI system ^{[fodf14] [fodf15] [fodf16] [fodf17]} in form

  $\left\{\begin{array}{l}\dot{\mathit{x}}\left(t\right)=\mathit{A}\mathit{x}\left(t\right)+\mathit{B}\mathit{u}\left(t\right)\\ \mathit{y}\left(t\right)=\mathit{C}\mathit{x}\left(t\right)+\mathit{D}\mathit{u}\left(t\right)\end{array}\right.$ (fodf.4)

is a first-order ODE, linearity of (fodf.4) see [fodf18] , if $\mathit{u}\left(t\right)=0$ , then it is homogeneous and autonomous, the solution of which see [fodf19].

Whereas a typical partial differential equation ^[fodf20] of wave equation ^[fodf21] of three-dimension may be written in ^[fodf22]

  \begin{aligned} \nabla^2{\Psi(x,y,z;t)}:&=\frac{\partial^2\Psi(x,y,z;t)}{\partial x^2}+\frac{\partial^2{\Psi(x,y,z;t)}}{\partial y^2}+\frac{\partial^2{\Psi(x,y,z;t)}}{\partial z^2}\\&=\frac1{v^2}\frac{\partial^2{\Psi(x,y,z;t)}}{\partial t^2} \end{aligned} (fodf.5)

The universal form of a PDE is ^[fodf23] :

  $G(x_1,\dots ,x_n,g(x_1,\dots ,x_n),\frac{\partial g(x_1,\dots ,x_n)}{\partial x_1},\dots ,\frac{\partial g(x_1,\dots ,x_n)}{\partial x_n},\frac{{\partial }^{2}g(x_1,\dots ,x_n)}{\partial x_1\partial x_1},\dots ,\frac{{\partial }^{2}g(x_1,\dots ,x_n)}{\partial x_1\partial x_n},\dots )=0$ (fodf.6)

Name	Solvability	Form	Method	Tools
ODE	closed-form solution [fodf24] for linear ones [fodf25] only graphical and numerical for nonlinear ones	(fodf.2)
PDE	linear ones more difficult than ODE, but analytical solution exists, due to separation of variables. [fodf26]	(fodf.6)
Linear	see ODE and PDE
Non-linear	see ODE and PDE	Riccati equations (quadratic)
Homogeneous
Inhomogeneous
First-order	catogory see [fodf27]
High-order	reduction to first-order system [fodf28] [fodf29]