Chap. 4. Controllability and Feedback Linearization

we start with discussion of controllability and different related notions for nonlinear input-affine systems. Based thereon, we present a condition, under which a nonlinear system can be made completely linear by state feedback and a state transformation. The condition for input-state-linearizability being quite restrictive, we end the chapter with the presentation of input-output linearization and the definition of internal/zero dynamics.

使用state feedback 和state transformation来将nonlinear 系统转化为completely linear， condition比较有限制性，最后会讲到i/o linearizaiton 和 internal/zero dynamiccs。

4.1 Controllability Definition and Driftless Systems

Linear vs. Nonlinear Controllability

• For linear systems $\dot{x} = Ax + Bu$, controllability is a global property easily verified via the Kalman rank condition.
• For nonlinear systems, local linearization can be highly misleading. Crucial system capabilities often get lost under linearization (e.g., a linearized unicycle model incorrectly suggests the vehicle cannot move sideways). Thus, we must employ differential geometric tools.

Definition 4.1 (Controllability)

The generic nonlinear system: $\dot{x} = f(x) + \sum_{i=1}^m g_i(x)u_i \quad \text{(4.1)}$ with $x \in \mathcal{X} \subset \mathbb{R}^n$ and $u \in \mathcal{U} \subset \mathbb{R}^m$ is controllable if for any two states $x_1, x_2 \in \mathcal{X}$, there exists a finite time control input $u : [0, T] \to \mathcal{U}$ ($T < \infty$) such that the trajectory steers from $x(0) = x_1$ to $x(T) = x_2$.

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Driftless Systems

A system is called driftless if the drift vector field $f(x) = 0$. This implies that if no control input is applied ($u=0$), the system immediately stops moving and preserves its state. It is formulated as:

$\dot{x} = g_1(x)u_1 + \dots + g_m(x)u_m \quad \text{(4.2)}$

Geometric Interpretation of Flow

Let $\Delta = \text{span}\{g_1, \dots, g_m\}$ be the distribution of control vector fields with $\text{dim}(\Delta) = m$. Let $\bar{\Delta} = \text{inv}(\Delta)$ be its involutive closure with $\text{dim}(\bar{\Delta}) = l$.

• If $l < n$, the flow generated by alternating the control inputs $u_1, \dots, u_m$ is tightly confined to an $l$-dimensional subspace (manifold). The state space $\mathcal{X}$ is foliated into $l$-dimensional submanifolds called leaves. The system is permanently trapped on the leaf dictated by its initial condition $x(0)$.
• If $l = n$, the flow is not confined to a lower-dimensional subspace. This full-rank expansion is a necessary and sufficient condition for the system to break free from geometric traps and roam the entire state space.

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The Chow-Rashevskii Theorem

Proven independently by Chow (China, 1930s) and Rashevskii (USSR, 1930s), this theorem provides the mathematical framework for the controllability of driftless nonholonomic systems.

Theorem 4.1 (Controllability of Driftless Systems)

The driftless system given by Equation (4.2) is controllable if and only if its involutive closure spans the entire tangent space at each point:

$\text{dim }\bar{\Delta}(x) = n \quad \forall x \in \mathcal{X}$