Chap. 2. Lyapunov for Nonlinear System

Two Methods of Lyapunov for Nonlinear Systems

1. First/indirect method

Let $\boldsymbol{x}^*$ be an equilibrium of

$$\dot{\boldsymbol{x}} = \boldsymbol{f}(\boldsymbol{x}), \quad \boldsymbol{x} \in \mathcal{X} \subseteq \mathbb{R}^n$$

and $\boldsymbol{f} : \mathcal{X} \rightarrow \mathbb{R}^n$ twice continuously differentiable in an environment of $\boldsymbol{x}^*$. The linearization

$$\Delta \dot{\boldsymbol{x}} = \boldsymbol{A}\Delta \boldsymbol{x}, \quad \boldsymbol{A} = \left. \frac{\partial \boldsymbol{f}(\boldsymbol{x})}{\partial \boldsymbol{x}} \right|_{\boldsymbol{x}^*} \tag{2.16}$$

with $\Delta \boldsymbol{x} = \boldsymbol{x} - \boldsymbol{x}^*$, is a local approximation of the dynamics of the nonlinear system. $\sigma(\boldsymbol{A})$ denotes the spectrum of the matrix $\boldsymbol{A}$ (the entirety of all eigenvalues) and $\lambda_i(\boldsymbol{A})$ denotes a single eigenvalue. The equilibrium $\boldsymbol{x}^*$ is

• asymptotically stable if $\sigma(\boldsymbol{A}) \in \mathbb{C}^-$,
• unstable if at least one eigenvalue $\lambda_i(\boldsymbol{A}) \in \mathbb{C}^+$.
• If at least one eigenvalue lies on the imaginary axis (while the others are in $\mathbb{C}^-$), the equilibrium is called “non-hyperbolic” and no direct stability statement is possible.

• Example: Nonlinear System $\dot{x} = x(1-x)$ two Equilibrium Points: $x^*_1 = 0 \quad x^*_2 = 1$ let $f(x) = x - x^2$, find the deriviate

$$A = \frac{\partial f(x)}{\partial x} = 1 - 2x$$

1. Substitute $x^*_1 = 0$ in to the derivative formula. eigenvalue is 1 $\in \mathbb{C}^+$, so this equilibrium point is unstable.
2. Substitute $x^*_2 = 1$ in to the derivative formula. eigenvalue is $\lambda = -1<0$, so it is asymptotically stable.

if eigenvalue $\lambda = 0$ lies on the imaginary axis, the equilibrium is called “non-hyperbolic” and no direct stability statement is possible.

2. Second/direct method

Let $\boldsymbol{x}^*$ be an equilibrium of

$$\dot{\boldsymbol{x}} = \boldsymbol{f}(\boldsymbol{x}), \quad \boldsymbol{x} \in \mathcal{X} \subseteq \mathbb{R}^n.$$

Let $V : D \rightarrow \mathbb{R}$, $D \subseteq \mathcal{X}$, be a continuously differentiable function with $V(\boldsymbol{x}^*) = V^*$ such that

$$\begin{aligned} \text{(i)} \quad & V(\boldsymbol{x}) > V^* \quad \text{on} \quad D \setminus \{\boldsymbol{x}^*\},\\ \text{(ii)} \quad & \dot{V}(\boldsymbol{x}) = \frac{\partial V(\boldsymbol{x})}{\partial \boldsymbol{x}}\boldsymbol{f}(\boldsymbol{x}) \leq 0 \quad \text{on} \quad D. \end{aligned}$$

Then $V(\boldsymbol{x})$ is called a Lyapunov function and the equilibrium $\boldsymbol{x}^*$ is stable (in the sense of Lyapunov). If, in addition,

$$\dot{V}(\boldsymbol{x}) < 0 \quad \text{on} \quad D \setminus \{\boldsymbol{x}^*\}, \tag{2.19}$$

then $\boldsymbol{x}^*$ is asymptotically stable.

1. Condition (i) — $V(\boldsymbol{x})$ is Positive Definite

• Academic English Expression: > $V(\boldsymbol{x})$ is locally positive definite. (Or: $V(\boldsymbol{x})$ is strictly positive for all non-zero states in the neighborhood of $\boldsymbol{x}^*$).
• Standard Sentence for Papers: “The candidate Lyapunov function $V(\boldsymbol{x})$ must be zero at the equilibrium point $\boldsymbol{x}^*$ and strictly positive everywhere else in the domain $D \setminus \{\boldsymbol{x}^*\}$.“

2. Condition (ii) — $\dot{V}(\boldsymbol{x})$ is Negative Semi-Definite

• Academic English Expression: > The time derivative $\dot{V}(\boldsymbol{x})$ is negative semi-definite along the system trajectories.
• Standard Sentence for Papers: “The time derivative of $V(\boldsymbol{x})$ along the system trajectories is non-positive ($\dot{V}(\boldsymbol{x}) \leq 0$), which implies that the system’s total energy is non-increasing over time, guaranteeing Lyapunov stability.”

3. Additional Condition — $\dot{V}(\boldsymbol{x})$ is Negative Definite (for Asymptotic Stability)

• Academic English Expression: > The time derivative $\dot{V}(\boldsymbol{x})$ is strictly negative definite.
• Standard Sentence for Papers: “If, in addition, the time derivative $\dot{V}(\boldsymbol{x})$ is strictly negative ($\dot{V}(\boldsymbol{x}) < 0$) for all $\boldsymbol{x} \in D \setminus \{\boldsymbol{x}^*\}$, the system energy strictly dissipates, ensuring that the equilibrium $\boldsymbol{x}^*$ is asymptotically stable.”

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Summary of Core Terminology

Mathematical Property	Academic Terminology	Chinese Translation
$V(\boldsymbol{x}) > 0$	Positive definite	正定
$V(\boldsymbol{x}) \geq 0$	Positive semi-definite	半正定
$\dot{V}(\boldsymbol{x}) \leq 0$	Negative semi-definite	半负定
$\dot{V}(\boldsymbol{x}) < 0$	Negative definite	负定

Domain of attraction A Lyapunov function $V(\boldsymbol{x})$ provides an estimate of the domain of attraction of $\boldsymbol{x}^*$. Let

$$\mathcal{X}_V = \{\boldsymbol{x} \in \mathcal{X} \mid V(\boldsymbol{x}) > V^*\} \cup \boldsymbol{x}^* \tag{2.20}$$

and

$$\mathcal{X}_{\dot{V}} = \{\boldsymbol{x} \in \mathcal{X} \mid \dot{V}(\boldsymbol{x}) < 0\} \cup \boldsymbol{x}^* . \tag{2.21}$$

We consider the level sets of $V(\boldsymbol{x})$

$$\Omega_c := \{\boldsymbol{x} \in \mathcal{X} \mid V^* \leq V(\boldsymbol{x}) \leq c\}. \tag{2.22}$$

Let $\bar{c}$ be the value of $V(\boldsymbol{x})$ on the largest closed and bounded (i.e., compact) level set $\Omega_{\bar{c}}$, which is completely contained in $\mathcal{X}_V \cap \mathcal{X}_{\dot{V}}$. Then all trajectories that start in $\Omega_{\bar{c}}$ remain in $\Omega_{\bar{c}}$ and approach $\boldsymbol{x}^*$ asymptotically for $t \rightarrow \infty$. $\Omega_{\bar{c}}$ is an estimate of the domain of attraction of $\boldsymbol{x}^*$.

Explanation:

• $\boldsymbol{x}^*$ is an isolated equilibrium, there is no other point s.t. $\dot{\boldsymbol{x}} = \mathbf{0}$ in $\Omega_{\bar{c}}$.
• In $\Omega_{\bar{c}}$, except for $\boldsymbol{x}^*$, $\dot{V}(\boldsymbol{x}) = \frac{\partial V(\boldsymbol{x})}{\partial \boldsymbol{x}}\dot{\boldsymbol{x}} < 0$ holds, and hence $\frac{\partial V(\boldsymbol{x})}{\partial \boldsymbol{x}} \neq \mathbf{0}^T$. $\Rightarrow$ There are no further stationary points of $V(\boldsymbol{x})$.
• $\Rightarrow$ Every point in $\Omega_{\bar{c}}$ is “transient” ($\dot{\boldsymbol{x}} \neq \mathbf{0}$) and $V(\boldsymbol{x})$ decreases strictly: $\dot{V}(\boldsymbol{x}) < 0$.
• $\Rightarrow$ As $V(\boldsymbol{x})$ is bounded from below by $V^* = V(\boldsymbol{x}^*)$, every trajectory which starts in $\Omega_{\bar{c}}$ must end asymptotically in $\boldsymbol{x}^*$.

Remark 2.1. The quadratic Lyapunov function $V(\Delta \boldsymbol{x}) = \frac{1}{2}\Delta \boldsymbol{x}^T \boldsymbol{P} \Delta \boldsymbol{x}$, where $\boldsymbol{P} = \boldsymbol{P}^T > 0$ is the solution of a Lyapunov equation (2.14) and the state matrix $\boldsymbol{A}$ represents the linearization according to (2.16), serves also as a Lyapunov function for the nonlinear system $\dot{\boldsymbol{x}} = \boldsymbol{f}(\boldsymbol{x})$ –- in a region around the equilibrium $\boldsymbol{x}^*$, where the linearization is a sufficiently good approximation of the nonlinear system. This can be seen as follows. We consider w.l.o.g. $\boldsymbol{x}^* = \mathbf{0}$ and write the differential equation as

$$\dot{\boldsymbol{x}} = \underbrace{\boldsymbol{f}(\boldsymbol{x}^*)}_{=\mathbf{0}} + \boldsymbol{A}\boldsymbol{x} + \boldsymbol{r}(\boldsymbol{x}),$$

where $\boldsymbol{r}(\boldsymbol{x})$ represents a residual term of order $\mathcal{O}(\|\boldsymbol{x}\|^2)$. Express now the time derivative of the quadratic Lyapunov function:

$$\begin{aligned} \dot{V} &= \frac{1}{2}\boldsymbol{x}^T \boldsymbol{P} (\boldsymbol{A}\boldsymbol{x} + \boldsymbol{r}(\boldsymbol{x})) + \frac{1}{2}(\boldsymbol{x}^T \boldsymbol{A}^T + \boldsymbol{r}^T(\boldsymbol{x}))\boldsymbol{P}\boldsymbol{x} \\ &= \frac{1}{2}\boldsymbol{x}^T \underbrace{(\boldsymbol{P}\boldsymbol{A} + \boldsymbol{A}^T \boldsymbol{P})}_{=-\frac{1}{2}\boldsymbol{x}^T \boldsymbol{Q}\boldsymbol{x}, \quad \boldsymbol{Q}>0} \boldsymbol{x} + \underbrace{\boldsymbol{x}^T \boldsymbol{P}\boldsymbol{r}(\boldsymbol{x})}_{\text{perturbation}}. \end{aligned}$$

As long as the quadratic first term dominates the second non-quadratic perturbation term, $V(\boldsymbol{x})$ is a Lyapunov function for the nonlinear system.

Control Lyapunov Functions While Lyapunov functions were defined for autonomous systems, the concept of Control Lyapunov Functions (CLFs) for input-affine systems takes into account rendering the time derivative $\dot{V}(\boldsymbol{x})$ negative by means of the control input. For the discussion of CLFs, we assume that the input-affine system (2.23) has an equilibrium $(\boldsymbol{x}^*, \boldsymbol{u}^*) = (\mathbf{0}, \mathbf{0})$, which implies $\boldsymbol{f}(\mathbf{0}) = \mathbf{0}$. Note that this can always be achieved by a simple coordinate shift $(\boldsymbol{x}, \boldsymbol{u}) \mapsto (\boldsymbol{x} - \boldsymbol{x}^*, \boldsymbol{u} - \boldsymbol{u}^*)$.

Definition 2.3 (Control Lyapunov Function, CLF). A scalar function $V : \mathcal{X} \rightarrow \mathbb{R}$ is a Control Lyapunov Function (CLF) for the system (2.23) with equilibrium $(\boldsymbol{x}^*, \boldsymbol{u}^*) = (\mathbf{0}, \mathbf{0})$ if the following implication holds for $\boldsymbol{x} \neq \mathbf{0}$:

$$\frac{\partial V}{\partial \boldsymbol{x}}\boldsymbol{G}(\boldsymbol{x}) = \mathbf{0} \quad \Rightarrow \quad \frac{\partial V}{\partial \boldsymbol{x}}\boldsymbol{f}(\boldsymbol{x}) < 0. \tag{2.27}$$

In words: Whenever the decrease of $V(\boldsymbol{x})$ cannot be induced by the control input (through the vector field $\boldsymbol{G}\boldsymbol{u}$), $V(\boldsymbol{x})$ must decrease due to the drift term $\boldsymbol{f}(\boldsymbol{x})$.

Input Affine System $\dot{\boldsymbol{x}} = \boldsymbol{f}(\boldsymbol{x}) + \boldsymbol{G}(\boldsymbol{x})\boldsymbol{u}$

• $f(x)$ is the drift term. The inherent tendency of a system to move when left unctrolled and to its own devicces.
• $G(x)u$ is the control term. How the controller u apply weights to the system.

$$\dot{V} = \frac{\partial V}{\partial \boldsymbol{x}}\dot{\boldsymbol{x}} = \underbrace{\frac{\partial V}{\partial \boldsymbol{x}}\boldsymbol{f}(\boldsymbol{x})}_{a(\boldsymbol{x})} + \underbrace{\frac{\partial V}{\partial \boldsymbol{x}}\boldsymbol{G}(\boldsymbol{x})}_{\boldsymbol{b}^T(\boldsymbol{x})}\boldsymbol{u} = a(\boldsymbol{x}) + \boldsymbol{b}^T(\boldsymbol{x})\boldsymbol{u}$$

如果Controller失灵了，需要能够通过系统惯性降下来。可以通过设计信号u，把整体的 $\dot{V}$ 降下来，能够做到这一点的$V(x)$就叫CLF。如果找到了CLF，就需要算出来u。

Sontag’s Formula Once a CLF $V(\boldsymbol{x})$ according to the previous definition is found, an asymptotically stabilizing feedback control law can be constructed, the so-called Sontag’s formula. Define

$$a(\boldsymbol{x}) := \frac{\partial V(\boldsymbol{x})}{\partial \boldsymbol{x}}\boldsymbol{f}(\boldsymbol{x}) \quad \text{and} \quad \boldsymbol{b}(\boldsymbol{x}) := \left(\frac{\partial V(\boldsymbol{x})}{\partial \boldsymbol{x}}\boldsymbol{G}(\boldsymbol{x})\right)^T. \tag{2.28}$$

Recall that $a(\boldsymbol{x}) < 0$ must hold whenever $\boldsymbol{b}(\boldsymbol{x}) = \mathbf{0}$ for $\boldsymbol{x} \neq \mathbf{0}$. Sontag’s formula is then given by

$$\boldsymbol{u}_S(\boldsymbol{x}) = \begin{cases} -\boldsymbol{b}(\boldsymbol{x})\frac{a(\boldsymbol{x}) + \sqrt{a^2(\boldsymbol{x}) + (\boldsymbol{b}^T(\boldsymbol{x})\boldsymbol{b}(\boldsymbol{x}))^2}}{\boldsymbol{b}^T(\boldsymbol{x})\boldsymbol{b}(\boldsymbol{x})}, & \boldsymbol{b}(\boldsymbol{x}) \neq \mathbf{0}, \\ \mathbf{0}, & \boldsymbol{b}(\boldsymbol{x}) = \mathbf{0}. \end{cases} \tag{2.29}$$

The control law has an interpretation in terms of optimal control, as it minimizes a certain cost functional.

• 平滑性（Small Control Property）: 随着状态x越来越接近远点，算出来的控制量$u_s$会丝滑的减小到0，不会产生剧烈的震荡或突变。
• 当 $b(x) = 0$ 时，没有输入，输出为0,系统靠天然drift。
• 当 $\boldsymbol{b}(\boldsymbol{x}) \neq \mathbf{0}$ 时， 根号可以用最省力的方式抵消掉不稳定的趋势，让 $\dot{V}<0$。