NONLINEAR SYSTEMS — COMPLETE NOTES

ESSENTIAL MATRIX DERIVATIVE RULES

1. Derivative of a Transpose

Let $X = X(t)$.

\[\frac{d}{dt}(X^\top) = \left(\frac{dX}{dt}\right)^\top\]

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2. Matrix Product Rule

\[\frac{d}{dt}(XY) = \dot X Y + X \dot Y\]

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3. Quadratic Form

Let

\[V(x) = x^\top A x\]

Then

\[\nabla V(x) = (A + A^\top)x\]

If $A = A^\top$,

\[\nabla V(x) = 2Ax\]

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4. Chain Rule (Lyapunov Use)

For

\[\dot x = f(x)\] \[\dot V(x) = \nabla V(x)^\top f(x)\]

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PART I — LINEAR VS NONLINEAR SYSTEMS

1. Linear Systems

A linear time-invariant system:

System:

\[\dot{x} = A x\]

Solution:

\[x(t) = e^{At} x(0)\]

The stability of the system is determined by the eigenvalues $\lambda_i$ of $A$.

• $\operatorname{Re}(\lambda_i) < 0$ for all $i$ $\Rightarrow$ asymptotically stable

• $\operatorname{Re}(\lambda_i) > 0$ for some $i$ $\Rightarrow$ unstable

• $\operatorname{Re}(\lambda_i) = 0$ for some $i$

  • if the eigenvalues on the imaginary axis are simple $\Rightarrow$ Lyapunov stable
  • otherwise $\Rightarrow$ unstable

Properties of linear systems:

• Superposition principle holds
• The equilibrium is typically unique: $x = 0$
• No limit cycles exist
• The global behavior is completely determined by the spectrum of $A$

1.1 limit cycle

A limit cycle is an isolated periodic orbit of a nonlinear autonomous system

\[\dot{x} = f(x)\]

A trajectory $\gamma $ is a limit cycle if

• it is periodic
• there are no other periodic orbits arbitrarily close to it

Nearby trajectories may approach the orbit (stable), move away from it (unstable), or approach from one side only (semi-stable).

Limit cycles do not occur in linear systems.

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Example

Consider the system in polar coordinates

\[\dot R = -R(R^2-1), \qquad \dot \theta = 1\]

Radial equilibria satisfy

\[\dot R = 0 \Rightarrow R = 0,\; R = 1\]

For $0<R<1$, $\dot R > 0 $ so the radius increases.
For $R>1$, $ \dot R < 0 $ so the radius decreases.

Thus trajectories move toward R=1 while $\theta $ keeps rotating.

The circle

\[R = 1\]

is therefore a stable limit cycle. you can have unstable one too if you like

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2. Nonlinear Systems

General nonlinear autonomous system:

\[\dot{x} = f(x)\]

Equilibria satisfy:

\[f(x_e) = 0\]

Nonlinear systems may exhibit:

• Multiple equilibria
• Limit cycles
• Bifurcations
• Finite escape time
• Complex attractors

Superposition does not hold:

\[f(x_1 + x_2) \neq f(x_1) + f(x_2)\]

Behavior depends on geometry of the vector field.

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PART II — MATHEMATICAL FOUNDATIONS

3. Normed Spaces

A norm satisfies:

1. $\Vert x\Vert \ge 0$, and $\Vert x\Vert =0 \iff x=0$
2. $\vert\alpha x\vert = \vert\alpha\vert\vert x\vert$
3. $\Vert x+y\Vert \le \Vert x\Vert +\Vert y\Vert $

Common norms in $\mathbb{R}^n$:

\(\Vert x\Vert _2 = \sqrt{x^\top x}, \quad\) \(\Vert x\Vert _1 = \sum |x_i|, \quad\) \(\Vert x\Vert _\infty = \max_i |x_i|\)

All norms are equivalent in finite dimensions.

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4. Completeness

A sequence converges if:

\[\Vert x_n - x\Vert \to 0\]

(in strict definition: every ε>0, there exists an integer N such that for all n≥N the above <ε)

A sequence is Cauchy if:

\[\Vert x_n - x_m\Vert \to 0 \quad \text{as } n,m \to \infty\]

A space is complete if every Cauchy sequence converges in that space.

actually in $\mathbb{R}$ , every cauthy seq is converged.

A complete normed space is called a Banach space.

Completeness is required for fixed-point theorems.

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5. Contraction Mapping

A mapping $P$ is a contraction if:

\[\Vert P(x)-P(y)\Vert \le L \Vert x-y\Vert , \quad 0 \le L < 1\]

Banach Fixed-Point Theorem

If $P$ is a contraction on a complete space:

• A unique fixed point exists
• x is cauthy and iteration converges to fixed point $x^*$

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PART III — EXISTENCE & UNIQUENESS OF ODEs

6. Integral Form of ODE

Given:

\[\dot{x} = f(t,x), \quad x(t_0)=x_0\]

Integral form:

\[x(t)=x_0+\int_{t_0}^{t} f(s,x(s))\,ds\]

Define operator:

\[(Px)(t)=x_0+\int_{t_0}^{t} f(s,x(s))\,ds\]

Solving the ODE is equivalent to solving:

\[Px = x\]

Thus the ODE becomes a fixed-point problem.

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7. Lipschitz Continuity

Global Lipschitz:

\[\Vert f(x)-f(y)\Vert \le L \Vert x-y\Vert\]

Local Lipschitz guarantees local existence and uniqueness.(with p.w. continuity)

piecewise continuity looks like not continued

Global Lipschitz guarantees global existence (the condition for no finite escape time).

• you can actually take the derivative of the linear system, then the answer is the lipschitz constant
• lipschitz continuity $\rightarrow $ continuity

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8. Finite Escape Time

Example:

\[\dot{x} = 1 + x^2\]

Solution:

\[x(t)=\tan t\]

Blow-up occurs at:

\[t=\frac{\pi}{2}\]

Conclusion:

• Local Lipschitz ⇒ local solution
• Global growth control ⇒ global solution

linear system does not have a finite escape time because the derivatives always a constant, so it is globally lipschitz $\rightarrow$ no finite escape time

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9. Grönwall Inequality

If

\[u(t) \le C + \int_{t_0}^{t} a(s) u(s)\,ds\]

Then

\[u(t) \le C \exp\!\left(\int_{t_0}^{t} a(s)\,ds\right)\]

Used for:

• Continuous dependence on IC
• Growth bounds

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10. Continuous Dependence on IC (Initial Conditions)

four question for nonlinear system

1. solution exist?

2. solution unique?

3. have finite escape time?
4. continuous on IC?

Consider

\[\dot{x} = f(t,x), \quad x(t_0)=x_0\]

Assume:

• $f$ is continuous in $t$
• $f$ is locally Lipschitz in $x$

Then solutions depend continuously on the initial condition.

More precisely:

For $\forall$ $T>t_0$ and $\forall$ $\varepsilon>0$, there $\exists$ $\delta>0$ such that

\[\Vert x_0 - y_0\Vert < \delta \Rightarrow \sup_{t \in [t_0,T]} \Vert x(t,x_0) - x(t,y_0)\Vert < \varepsilon\]

This means small perturbations in the initial condition produce small changes in the entire trajectory over finite time intervals.

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PART IV — STABILITY THEORY

10. Equilibrium

\[f(x_e)=0\]

11. Compact

close + bounded = compact

[0,1]

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11. Lyapunov Stability

Stable if:

For $\forall$ $\varepsilon>0$, there $\exists$ $\delta>0$ such that

\[\Vert x(0)-x_e\Vert <\delta \Rightarrow \Vert x(t)-x_e\Vert <\varepsilon\]

that 𝑡 means any time 𝑡 ≥ 0 not 𝑡 → ∞

Asymptotically stable if additionally:

\[x(t)\to x_e\]

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PART V — LYAPUNOV DIRECT METHOD

12. Lyapunov Function

A scalar function $V(x)$ satisfies:

• $V(x)>0$ for $x\neq 0$
• $V(0)=0$

Derivative along trajectories:(basically $\dot x = f(x)$)

\[\dot{V}(x)=\nabla V(x)^\top f(x)\]

If

\[\dot{V}(x)\le 0\]

→ Stable.

If

\[\dot{V}(x)<0\]

→ Asymptotically stable.

intuition:

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13. Global AS and LaSalle Thm (one different condition for AS)

If:

• $V(x)>0$
• $\dot{V}(x)\le 0$
• $V(x)\to \infty$
  as $x \to \infty$ (radially unbounded)

Then we say it is globally AS (remember the picture Prof draw in class)

LaSalle Thm preset

• SISL
• let $\mathcal{S}’= {x\in D \mid \dot V(x)=0 } $, if the only solution of the system dynamics within S is x(t)=$x_e$ = 0

then $x_e$ is AS

S is invariance set, in set S the $\dot{V} =0$, the other region in D is $\dot{V} <0$

this basically says the point will fall downward and fix only to the origin

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14. Instability Theorem (Chetaev)

If:

• $V(0)=0$
• $V(x)>0$
• $\dot{V}(x)>0$ in a wedged region

or say:

in some point in the $B_\delta(X_e) $ $V(x)>0$,

and $\exists \varepsilon$ s.t. all $\dot{V}(x)$ >0 in this { $B_\epsilon(X_e)$ $\vert$ $V(x)>0$ }

Then equilibrium is unstable.

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PART VI — LINEARIZATION & INDIRECT METHOD

14. Lyapunov Equation

For the linearized system \(\dot{x} = A(x - x_e),\)

consider the quadratic Lyapunov function

\[V(x) = (x - x_e)^{\top} P (x - x_e),\]

where $P = P^{\top} > 0$.

The matrix $P$ satisfies the Lyapunov equation

\[A^{\top} P + P A = -Q,\]

where $Q = Q^{\top} > 0$. You can just assign $I$

If such a positive definite $P$ exists, then the equilibrium $x_e$ is locally asymptotically stable.

because the Lyapunov Equation makes

\[\dot{V} (x)= (x - x_e)^{\top} Q (x - x_e)<0\]

15. Linearization

Linearization matrix:

\[A = \frac{\partial f}{\partial x}\Big|_{x_e}\]

Approximation:

\(\dot{x} \approx A(x-x_e)\)

Linearization Example (Equilibrium Not at Origin)

Consider the nonlinear system

\[\begin{cases} \dot x_1 = x_2 \\ \dot x_2 = -x_1 + 1 - (x_1 - 1)^3 \end{cases}\]

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1. Equilibrium

Solve

\[x_2 = 0, \quad -x_1 + 1 - (x_1 - 1)^3 = 0\]

We obtain

\[x_e = (1,0).\]

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2. Jacobian

Let

\[f_1 = x_2, \quad f_2 = -x_1 + 1 - (x_1 - 1)^3.\]

Then

\[A(x) = \begin{bmatrix} 0 & 1 \\ -1 - 3(x_1 - 1)^2 & 0 \end{bmatrix}.\]

Evaluate at $x_e = (1,0) $:

\[J(x) = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}.\]

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3. Linearized System

Define shifted state $(x-x_e)$

\[\tilde x = \begin{bmatrix} x_1 - 1 \\ x_2 \end{bmatrix}.\]

Then the linear approximation is

\[\dot{\tilde x} = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} \tilde x.\]

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16. Lyapunov Indirect Method

after linearization,

If eigenvalues of $J(x)$:

• $\forall$ Re(λ)<0 → locally asymptotically stable
• $\exists$ Re(λ)>0 → unstable
• $\exists$ Re(λ)=0 → inconclusive (cannot say SISL) (if the linear system neither grows nor decays → higher-order nonlinear terms decide, so the test is inconclusive.)

Indirect method is local.

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PART VII — REGION OF ATTRACTION

17. Region of Attraction (ROA)

Defined as:

\[\mathcal{R} = \{x_0 : \lim_{t\to\infty} x(t,x_0)=0\}\]

Exact ROA is difficult to compute.

Using quadratic Lyapunov function:

\[V(x)=x^\top P x\]

Level set:

\[x^\top P x \le c\]

provides an inner estimate of ROA.

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PART IX — TIME-VARYING SYSTEMS

19. Time-Varying System

\[\dot{x}=f(t,x)\]

Uniform stability means $\delta$ does not depend on initial time.

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20. Class-$\mathcal{K}$ Functions

A function $\alpha:[0,a)\to[0,\infty)$ is class-$\mathcal{K}$ if:

• Continuous
• Strictly increasing
• $\alpha(0)=0$

Used in Lyapunov bounds:

\[\alpha_1(\Vert x\Vert ) \le V(t,x) \le \alpha_2(\Vert x\Vert )\]

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21. Exponential Stability

Equilibrium is exponentially stable if:

\[\Vert x(t)\Vert \le M e^{-\lambda t} \Vert x_0\Vert\]

for some $M>0$, $\lambda>0$.

Lyapunov condition:

If

\[\alpha_1\Vert x\Vert ^2 \le V(x) \le \alpha_2\Vert x\Vert ^2\]

and

\[\dot{V}(x) \le -\alpha_3 \Vert x\Vert ^2\]

Then system is exponentially stable.

Exponential stability implies asymptotic stability.

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PART VII — REGION OF ATTRACTION

17. Region of Attraction (ROA)

Defined as:

\[\mathcal{R} = \{x_0 : \lim_{t\to\infty} x(t,x_0)=0\}\]

Exact ROA is difficult to compute.

Using Lyapunov function:

\[V(x) > 0,\quad \dot{V}(x) < 0\]

Sublevel set:

\[\Omega_c = \{x : V(x) \le c\}\]

provides an inner estimate of ROA.

For quadratic Lyapunov function:

\[V(x)=x^\top P x\]

the level set

\[x^\top P x \le c\]

is an ellipse.

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PART VIII — TIME-VARYING SYSTEMS

18. Time-Varying System

\[\dot{x}=f(t,x)\]

Equilibrium satisfies:

\[f(t,x_e)=0,\quad \forall t\]

Region of attraction may depend on time and can shrink.

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19. Stability Types

• Stability may depend on initial time $t_0$
• Uniform stability does not depend on $t_0$

Uniform stability is stronger.

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PART IX — LYAPUNOV FOR TIME-VARYING SYSTEMS

20. Class-$\mathcal{K}$ Functions

A function $\alpha:[0,a)\to[0,\infty)$ is class-$\mathcal{K}$ if:

• Continuous
• Strictly increasing
• $\alpha(0)=0$

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21. Locally Positive Definite

\[V(t,x) \ge \alpha(\Vert x\Vert)\]

for some $\alpha \in \mathcal{K}$.

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22. Decrescent Function

\[V(t,x) \le \gamma(\Vert x\Vert)\]

for some $\gamma \in \mathcal{K}$.

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23. Equivalent Condition

Define:

\[W(x)=\inf_{t\ge0} V(t,x)\]

Then $W(x)$ is positive definite.

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24. Stability Theorem

If:

• $f$ is locally Lipschitz in $x$
• $V$ is locally positive definite
• $V$ is decrescent
• $\dot{V}(t,x) \le 0$

Then equilibrium is uniformly stable.

If:

\[\dot{V}(t,x) < 0\]

Then equilibrium is uniformly asymptotically stable.

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PART X — EXPONENTIAL STABILITY

25. Definition

Equilibrium is exponentially stable if:

\[\Vert x(t)\Vert \le M e^{-\lambda t} \Vert x_0\Vert\]

for some $M>0$, $\lambda>0$.

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26. Lyapunov Condition for ES

If:

\[c_1 \Vert x\Vert^2 \le V(x) \le c_2 \Vert x\Vert^2\]

and

\[\dot{V}(x) \le -c_3 V(x)\]

Then system is exponentially stable.

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27. Relation

\[\text{Exponential Stability} \Rightarrow \text{Asymptotic Stability} \Rightarrow \text{Stability}\]

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28. Example

\[\dot{x} = -x^3\]

System is asymptotically stable but not exponentially stable.

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PART XI — INDIRECT METHOD

29. Linearization

Linearize system at equilibrium:

\[A = \frac{\partial f}{\partial x}\Big|_{x=0}\]

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30. Result

• If all eigenvalues of $A$ have negative real parts → locally exponentially stable
• If any eigenvalue has positive real part → unstable
• If eigenvalues on imaginary axis → inconclusive

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PART XII — CONTROL LYAPUNOV FUNCTION (CLF)

31. Problem

Design control:

\[u = \alpha(x)\]

such that:

\[\dot{x} = f(x,u)\]

is stable.

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32. CLF Definition

A function $V(x)$ is a CLF if:

\[V(x) > 0,\quad V(0)=0\]

and

\[\inf_{u} \frac{\partial V}{\partial x} f(x,u) < 0,\quad \forall x \ne 0\]

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33. Interpretation

There always exists a control $u$ such that:

\[\dot{V}(x,u) < 0\]

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34. Control Design Idea

1. Choose Lyapunov function $V(x)$
2. Design $u(x)$ such that:

\[\dot{V}(x) < 0\]

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PART XIII — STABILIZATION

35. Full-State Feedback

Closed-loop system:

\[\dot{x} = f(x,\alpha(x))\]

Goal: make equilibrium globally asymptotically stable.

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36. Key Idea

Control enforces energy decrease:

\[\dot{V}(x) < 0\]

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37. Outcome

• Stability achieved
• Possibly exponential convergence