Linearisation Proof

What is Taylor Series?

The Taylor series is a way to approximate a function around a point.

For a function $f(x)$, the Taylor series around the point $a$ is:

$f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f^{(3)}(a)}{3!}(x-a)^3 + \dots$

The approximation is valid around values $x=a$.

HLT Form

An equivalent form in HLT, where $x = a + h$, the point we linearise around plus a pertubation:

$f(a + h) = f(a) + h f'(a) + \frac{h^2}{2!} f''(a) + \frac{h^3}{3!} f^{(3)}(a) + \cdots$

This is a simple substitution and arguably makes the formula more intuitive. $h$ is the perturbation around the point $a$.

Extension to Multivariable Functions

If $f(x, u)$ is a function of multiple variables, then:

$f(x, u) \approx f(a, b) + \frac{\partial f}{\partial x}(a, b)(x - a) + \frac{\partial f}{\partial u}(a, b)(u - b) + \dots$

So you're still using values and derivatives at the point $(a, b)$, but now considering partial derivatives.

In HLT form, where $x = a + h_{x}$ and $u = a + h_{u}$:

$f(a+h_{x},b+h_{u})\approx f(a,b)+\frac{\partial f}{\partial x}(a,b)h_{x}+\frac{\partial f}{\partial u}(a,b)h_{u}+\dots$

Proof

Starting Point: Nonlinear System

$\dot{x}(t)=f(x(t),u(t)),\quad y(t)=h(x(t),u(t))$

Step 1: Nominal Trajectory and Perturbations

We want to linearise the system around the nominal trajectory or point $(x^\star(t), u^\star(t))$.

Perturbations around the nominal:

$x(t) = x^\star(t) + x_p(t), \quad u(t) = u^\star(t) + u_p(t)$

where $x_p(t), u_p(t)$ are small perturbations.

Step 2: Taylor Expansion of $f$

Taylor expansion allows us to model the system around our nominal point$(x^\star(t), u^\star(t))$

$\begin{aligned}\dot{x}(t) & =f(x(t),u(t))\\ & =f(x^{\star}(t),u^{\star}(t))\\ & \quad+\frac{\partial f}{\partial x}(x^{\star}(t),u^{\star}(t))x_{p}(t)+\frac{\partial f}{\partial u}(x^{\star}(t),u^{\star}(t))u_{p}(t)+\text{higher order terms}\end{aligned}$

Think of $x_{p}\left(t\right)$ as $h$ in the HLT form, but for multivaritate.

Step 3: Use Nominal Dynamics

By definition of nominal trajectory:

$\dot{x}^{\star}(t)=f(x^{\star}(t),u^{\star}(t))$

Substitute this back (replaces first term):

$\dot{x}(t)=\dot{x}^{\star}(t)+\frac{\partial f}{\partial x}(x^{\star}(t),u^{\star}(t))x_{p}(t)+\frac{\partial f}{\partial u}(x^{\star}(t),u^{\star}(t))u_{p}(t)+\text{higher order terms}$

Step 4: Express Perturbation Dynamics

Subtract $\dot{x}^\star(t)$ from both sides:

$\dot{x}(t) - \dot{x}^\star(t) = \frac{\partial f}{\partial x}(x^{\star}(t),u^{\star}(t))x_{p}(t)+\frac{\partial f}{\partial u}(x^{\star}(t),u^{\star}(t))u_{p}(t)+\text{higher order terms}$

Recall:

$\dot{x}_p(t) = \dot{x}(t) - \dot{x}^\star(t)$

Hence (variables hidden and neglect higher order terms):

$\dot{x}_{p}(t)=\frac{\partial f}{\partial x}x_{p}(t)+\frac{\partial f}{\partial u}u_{p}(t)$

This is a linear time-varying system describing the perturbation dynamics.

Summary:

$\boxed{\begin{aligned}\dot{x}_{p}(t) & =A(t)x_{p}(t)+B(t)u_{p}(t)\\ \text{where}\quad A(t) & =\frac{\partial f}{\partial x}(x^{\star}(t),u^{\star}(t)),\quad B(t)=\frac{\partial f}{\partial u}(x^{\star}(t),u^{\star}(t))\end{aligned}}$

Matrix Form

Both $f$ and $x$ are vectors (multiple functions and states), so we write the different state differentials horizontally and the function changes vertically.

$A(t) = \begin{bmatrix} \frac{\partial f_1}{\partial x_1}(x^\star(t), u^\star(t)) & \cdots & \frac{\partial f_1}{\partial x_n}(x^\star(t), u^\star(t)) \\ \vdots & \ddots & \vdots \\ \frac{\partial f_n}{\partial x_1}(x^\star(t), u^\star(t)) & \cdots & \frac{\partial f_n}{\partial x_n}(x^\star(t), u^\star(t)) \end{bmatrix}$

and

$B(t) = \begin{bmatrix} \frac{\partial f_1}{\partial u_1}(x^\star(t), u^\star(t)) & \cdots & \frac{\partial f_1}{\partial u_m}(x^\star(t), u^\star(t)) \\ \vdots & \ddots & \vdots \\ \frac{\partial f_n}{\partial u_1}(x^\star(t), u^\star(t)) & \cdots & \frac{\partial f_n}{\partial u_m}(x^\star(t), u^\star(t)) \end{bmatrix}$

Certainly! The matrices you provided are the Jacobian matrices of the function $f$ evaluated at the nominal trajectory $(x^\star(t), u^\star(t))$.

We commonly define:

$A(t) := \frac{\partial f}{\partial x}(x^\star(t), u^\star(t)) \in \mathbb{R}^{n \times n}$

and

$B(t) := \frac{\partial f}{\partial u}(x^\star(t), u^\star(t)) \in \mathbb{R}^{n \times m}$

Explicitly,Summary:

• $A(t)$ captures how the system dynamics change with respect to the state $x$ near the nominal trajectory.

• $B(t)$ captures how the system dynamics change with respect to the input $u$ near the nominal trajectory.

These matrices are crucial in the linearized system:

$\dot{x}_p(t) = A(t) x_p(t) + B(t) u_p(t)$

Let me know if you want me to write the corresponding matrices for the output function $h$ as well!