Dynamical Systems

  "In short, a system is any process or entity that has one or more well-defined inputs and one or more well-defined outputs. Examples of systems include a simple physical object obeying Newtonian mechanics, and the US economy! Systems can be physical, or we may talk about a mathematical description of a system. The point of modeling is to capture in a mathematical representation the behavior of a physical system. As we will see, such representation lends itself to analysis and design, and certain restrictions such as linearity and time-invariance open a huge set of available tools." [hover_system_2022]

  Dynamical systems theory concerns the evolution of systems in time, where the state of a system can be described by a finite number of parameters. This framework applies to both continuous-time systems modeled by ordinary differential equations (ODEs) and discrete-time systems modeled by state machines or difference equations.

The focus of dynamical systems is to answer these central questions: What are the equilibrium or time-periodic solutions of the dynamics? Are these solutions stable controllable and/or observable? What is the long-time asymptotic behavior of the solutions?

1. Modeling Fundamentals

System State: $\mathbf{x}(t)\in {\mathbb{R}}^n$

• Minimal set of independent variables fully characterizing the system at time $t$.
• Given $\mathbf{x}(t_0)$ and the system dynamics, the future and past behavior $\mathbf{x}(t)$ can be predicted.
• The choice of state variables is not unique: multiple valid representations exist.
• Variables may be physical quantities measured directly (position, velocity, temperature) or abstract constructs that have no physical interpretation.

Ordinary Differential Equations

Linear ODEs

$$\dot{\mathbf{x}}=A\mathbf{x}$$

where $A\in {\mathbb{R}}^{n\times n}$ is the system matrix and $\mathbf{x}\in {\mathbb{R}}^n$ is the state vector.

Superposition: Let $f$ be a linear function, then $$f(x_1)+f(x_2)=f(x_1+x_2)$$ $$\alpha \cdot f(x)=f(\alpha \cdot x)$$

Representing Higher-Order ODEs as First-Order Systems

Any higher-order ODE can be reformulated as a system of first-order ODEs by introducing additional state variables. For the second-order equation: $\ddot{y}+5\dot{y}+6y=u(t)$ introduce state variables $x_1=y$ and $x_2=\dot{y}$, yielding: $$\left[\begin{array}{c}{\dot{x}}_1\\ \\ {\dot{x}}_2\end{array}\right]=\left[\begin{array}{cc}0& 1\\ & \\ -6& -5\end{array}\right]\left[\begin{array}{c}{x}_1\\ \\ x_2\end{array}\right]+\left[\begin{array}{c}0\\ \\ 1\end{array}\right]u(t)$$

Note that $x_2=\dot{x_1}$

Nonlinear ODEs

$$\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})$$

where $\mathbf{f}:{\mathbb{R}}^n\to {\mathbb{R}}^n$ is a nonlinear function.

Equilibrium Points: Static Operating Modes

$$\exists {\mathbf{x}}^{\star }\in {\mathbb{R}}^n$$ $$\dot{\mathbf{x}}=\mathbf{f}({\mathbf{x}}^{\star })=0$$

Linear systems have only the origin as equilibrium when $\det (A)\ne 0$. Nonlinear systems may have multiple equilibrium points.

Classification of Equilibrium Points

The stability of an equilibrium point can be determined by the eigenvalues of the Jacobian matrix at that point. The classification is:

TODO

Add time response vs pole zero map.

Phase Portraits: Visual Analysis of System Trajectories

A phase portrait is a geometric representation of all possible trajectories in the state space (phase space). For two-dimensional systems, the phase plane shows the complete qualitative behavior without solving the ODEs explicitly.

Vector Field and Integral Curves

At each point $\mathbf{x}$ in the phase space, the vector $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})$ indicates the instantaneous velocity of the trajectory. By plotting these velocity vectors at a grid of points, we obtain the vector field. The integral curves (solution trajectories) are curves that follow the vector field everywhere tangentially.

TODO2

add plot showing example phase portrait of a system.

2. Representations

State-Space Form (Time Domain)

The state-space representation is the standard form for modern control systems. It consists of two equations:

State Equation (Dynamics): $\dot{\mathbf{x}}=A\mathbf{x}+B\mathbf{u}$

Output Equation (Measurement): $\mathbf{y}=C\mathbf{x}+D\mathbf{u}$

• $\mathbf{x}\in {\mathbb{R}}^n$: state vector
• $\mathbf{u}\in {\mathbb{R}}^m$: input vector ($m$ inputs)
• $\mathbf{y}\in {\mathbb{R}}^p$: output vector ($p$ outputs)
• $A\in {\mathbb{R}}^{n\times n}$: system matrix (determines open-loop dynamics and stability)
• $B\in {\mathbb{R}}^{n\times m}$: input matrix (how controls influence each state)
• $C\in {\mathbb{R}}^{p\times n}$: output matrix (which states are measured)
• $D\in {\mathbb{R}}^{p\times m}$: feed-through matrix (direct coupling from input to output)

Advantages Over Transfer Functions

1. Initial Conditions: State-space naturally incorporates non-zero initial conditions $\mathbf{x}(0)=\mathbf{x}\_0$.
2. MIMO Systems: Handles multiple inputs and outputs directly; transfer functions require transfer function matrices.
3. State Feedback Control: Enables direct state feedback design: $\mathbf{u}=-K\mathbf{x}$.

Transfer Functions (Frequency Domain)

The transfer function relates a system's output to the input through the Laplace transform, assuming zero initial conditions:

$$\begin{gathered} G(s)=\frac{Y(s)}{U(s)}=\mathcal{L} \\ y(t) \\ /\mathcal{L} \\ u(t) \end{gathered}$$

where $s\in \mathbb{C}$ is the complex frequency variable.

Rational Function Form

Transfer functions are typically expressed as ratios of polynomials: $$G(s)=\frac{b_m{s}^m+b_{m-1}{s}^{m-1}+\cdots +b_{1}s+b_0}{a_n{s}^n+a_{n-1}{s}^{n-1}+\cdots +a_{1}s+a_0}$$

Note on "Proper": In general control theory, a system is considered Proper if $n\ge m$ (covering both the first two rows). This is the condition required for a system to have a valid State-Space representation.

Poles and Zeros: Frequency Domain Structure

$$G(s)=\frac{N(s)}{D(s)}=K\frac{{\prod }_{i=1}^m(s-z_i)}{{\prod }_{j=1}^n(s-p_j)}$$

$$\mathcal{Z}=\{s\in \mathbb{C}\mid N(s)=0\}⟹\forall z\in \mathcal{Z},G(z)=0$$

$$\mathcal{P}=\{s\in \mathbb{C}\mid D(s)=0\}⟹\forall p\in \mathcal{P},\underset{s\to p}{\lim }|G(s)|=\infty$$

Example

$$G(s)=\frac{2s+1}{(s+3)(s+2)}=\frac{2(s+0.5)}{(s+3)(s+2)}$$

$$\mathcal{P}=\{-3,-2\}$$

$$\mathcal{Z}=\{-0.5\}$$

Time Domain Response Mappings (${\mathcal{L}}^{-1}$)

1. Real Pole ($p_i=-\sigma ,\sigma >0$): $$G(s)=\frac{1}{s+\sigma }\stackrel{{\mathcal{L}}^{-1}}{\to }g(t)=e^{-\sigma t}u(t)$$ $$\tau =\frac{1}{\sigma }$$

2. Complex Conjugate Poles ($p_{1,2}=-\alpha \pm j{\omega }_d$): $$G(s)=\frac{{\omega }_d}{(s+\alpha {)}^2+{\omega }_d^2}\stackrel{{\mathcal{L}}^{-1}}{\to }g(t)=e^{-\alpha t}\sin ({\omega }_{d}t)u(t)$$

3. Zeros and Derivative Action: Let $H(s)=\frac{1}{D(s)}$ and $G(s)=(s-z)H(s)$. $${\mathcal{L}}^{-1}\{G(s)\}=\frac{d}{dt}h(t)-z\cdot h(t)$$

4. RHP Zero (Non-Minimum Phase) ($z>0$): Step response $Y(s)=\frac{1}{s}\frac{K(s-z)}{D(s)}$ $$\dot{y}(0^{+})=\lim \_s\to \infty s[sY(s)-y(0^{+})]\propto -K(\text{Initial undershoot})$$

Stability Criteria

$$\text{BIBO Stable}⟺\forall p_j\in \mathcal{P},\text{Re}(p_j)<0$$

$$\text{Unstable}⟺\exists p_j\in \mathcal{P},\text{Re}(p_j)\ge 0$$

Pole-Zero Plots

The pole-zero diagram plots poles (marked as ×) and zeros (marked as ○) on the complex $s$-plane. The real axis represents damping (leftward = more damping), and the imaginary axis represents oscillation frequency. This visualization immediately reveals stability (all poles must be in the left-half plane for BIBO stability) and transient characteristics.

Discrete-Time Systems and Z-Transform

For digital implementations, continuous-time systems must be converted to discrete-time systems. This requires sampling and the Z-transform.

Sampling and Difference Equations

A continuous-time signal $x(t)$ is sampled at rate $f_s=1/T$: $$x[k]=x(kT),k=0,1,2,\dots$$

where $T$ is the sampling period. Differential equations become difference equations: $$\dot{x}(t)\approx \frac{x[k]-x[k-1]}{T}$$

Z-Transform Definition

The Z-transform is the discrete-time counterpart of the Laplace transform: $$X(z)=\sum \_{k=0}^{\infty }x[k]z^{-k}$$

It converts difference equations into algebraic equations in the $z$-domain, enabling algebraic analysis of discrete systems.

Discrete Transfer Function

For a discrete-time LTI system, the transfer function is: $$H(z)=\frac{Y(z)}{U(z)}=\frac{b_0+b_1{z}^{-1}+\cdots +b_m{z}^{-m}}{1+a_1{z}^{-1}-\cdots +a_n{z}^{-n}}$$

The impulse response at samples is obtained by inverse Z-transforming $H(z)$.

Continuous-to-Discrete Conversion Methods

Several methods map continuous transfer functions $G(s)$ to discrete transfer functions $G(z)$:

The Tustin (bilinear) transform is the industry standard because it preserves the stability property: continuous-stable poles map to discrete-stable poles inside the unit circle.

3. System Analysis

Stability Criteria: Predicting Long-Term Behavior

Stability analysis determines whether the system remains bounded and whether it converges to equilibrium over time. Different criteria apply to continuous and discrete systems.

Continuous-Time Stability

For a linear continuous-time system $\dot{\mathbf{x}}=A\mathbf{x}$, stability is determined by the eigenvalues of the system matrix $A$:

• Asymptotically Stable (AS): All eigenvalues have negative real parts: $\text{Re}({\lambda }_i)<0$

  • Equivalently: all poles lie in the Left-Half Plane (LHP) of the $s$-plane
  • Behavior: every trajectory converges to the origin as $t\to \infty$
  • General solution: $\mathbf{x}(t)=e^{At}\mathbf{x}(0)\to 0$ as $t\to \infty$

• Marginally Stable (MS): Some eigenvalues on the imaginary axis ($\text{Re}(\lambda )=0$), rest in LHP

  • Example: undamped oscillator has poles at $s=\pm j\omega$
  • Behavior: bounded oscillation without decay; limit cycles

• Unstable: At least one eigenvalue with positive real part: $\text{Re}({\lambda }_i)>0$

  • Behavior: exponential divergence from equilibrium
  • Solutions grow unboundedly

Discrete-Time Stability

For discrete-time systems, stability is determined by pole locations in the $z$-plane:

• Stable: All poles satisfy $|z_i|<1$ (inside unit circle)

  • Behavior: state decays exponentially to zero
  • Time-domain decay: $|x[k]|\sim |z_i{|}^k|x[0]|$
  • System response bounded for bounded inputs

• Marginally Stable: Poles on unit circle $|z|=1$

  • Behavior: bounded oscillation (no decay)
  • Exception: multiple poles at the same location on unit circle cause instability

• Unstable: Poles outside unit circle $|z_i|>1$

  • Behavior: exponential growth of state

Lyapunov Stability Theory for Nonlinear Systems

For nonlinear systems $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})$, Lyapunov theory provides a powerful framework that does not require solving the ODEs explicitly.

A scalar function $V(\mathbf{x})$ is a Lyapunov function for an equilibrium \(\mathbf{x}^*\) if:

1. \(V(\mathbf{x}^*) = 0\) (the equilibrium has zero "energy")
2. $V(\mathbf{x})>0$ for all \(\mathbf{x} \neq \mathbf{x}^*\) (positive definite)
3. $\dot{V}(\mathbf{x})=\nabla V\cdot \mathbf{f}(\mathbf{x})<0$ along trajectories (energy dissipation)

If a Lyapunov function exists, then:

• The equilibrium is globally asymptotically stable (GAS)
• All trajectories converge to \(\mathbf{x}^*\) regardless of initial condition

[stanford-lyapunov],[washington-lyapunov]

The Lyapunov function represents generalized "energy" that is not necessarily mechanical energy. If energy always decreases (except at equilibrium), the system must dissipate and settle at the equilibrium. This is a fundamental principle: systems naturally evolve toward states of lowest energy.

Frequency Response: Magnitude and Phase vs. Frequency

Frequency response characterizes how a system responds to sinusoidal inputs of varying frequencies. This is essential for understanding bandwidth, resonance, and control design.

When a linear system is driven by a sinusoidal input $u(t)=U_0\sin (\omega t)$, the steady-state output is also sinusoidal: $$y(t)=Y_0\sin (\omega t-\varphi )$$

The frequency response describes:

• Magnitude (Gain): $|G(j\omega )|=Y_0/U_0$ (amplitude scaling)
• Phase Shift: $\varphi (\omega )=\angle G(j\omega )$ (time delay expressed as phase)

A Bode plot consists of two semi-logarithmic graphs:

1. Magnitude Plot:

   • Vertical axis: Gain in decibels: $\text{Gain (dB)}=20\log \_10|G(j\omega )|$
   • Horizontal axis: Frequency in log scale: $\log \_10(\omega )$ [rad/s]
   • Key frequency: -3 dB point marks the bandwidth

2. Phase Plot:

   • Vertical axis: Phase angle in degrees: $\varphi (\omega )=\angle G(j\omega )$
   • Horizontal axis: Frequency in log scale

The logarithmic scale makes it easy to see the system response over multiple decades of frequency, and the dB scale makes multiplying gains (cascading systems) equivalent to adding them on the plot.

Asymptotic Approximation

• Each zero at $s=-z_i$ contributes:

  • $+20$ dB/decade slope above the corner frequency $\omega =z_i$
  • $+90°$ phase shift

• Each pole at $s=-p_i$ contributes:

  • $-20$ dB/decade slope above the corner frequency $\omega =p_i$
  • $-90°$ phase shift

• A pole at the origin (integrator) contributes:

  • Initial slope of $-20$ dB/decade
  • Phase shift of $-90°$

This asymptotic method enables rapid sketching and visualization of frequency response without detailed computation.

Gain Margin and Phase Margin: Stability Robustness

Gain and phase margins quantify how much uncertainty or perturbation a system can tolerate before becoming unstable. They are critical for robust controller design.

Gain Margin (GM)

The maximum multiplicative increase in system gain before the closed-loop system becomes unstable.

Measured at the phase crossover frequency $\omega \ast pc$ (where phase = $-180°$): $$\text{GM (dB)}=0-|G(j\omega \ast pc)|\text{ (in dB)}=-20\log \ast 10|G(j\omega \ast pc)|$$

• GM > 0 dB: System is stable
• GM = 0 dB: System is marginally stable (critically damped)
• GM < 0 dB: System is unstable

A typical design specification is $\text{GM}>6$ dB, which allows the system to tolerate a gain increase by a factor of 2 before instability.

Phase Margin (PM)

Definition: The maximum phase lag increase before the closed-loop system becomes unstable.

Measured at the gain crossover frequency $\omega \ast gc$ (where $|G(j\omega )|=0$ dB, i.e., unity gain): $$\text{PM}=180°-|\angle G(j\omega \ast gc)|$$

or equivalently: $$\text{PM}=-180°-\angle G(j{\omega }_{gc})$$

when using the convention that the phase of the open-loop transfer function is negative.

• PM > 0°: System is stable
• PM = 0°: System is marginally stable
• PM < 0°: System is unstable

A typical design specification is $\text{PM}>45°$, which means the system can tolerate up to 45° of additional phase lag from unmodeled dynamics.

Nyquist Criterion

In the Nyquist plot, the critical point is $(-1,0j)$. Gain and phase margins are measures of distance from this critical point:

• Gain margin: how far the magnitude can increase before the plot passes through $-1$
• Phase margin: how much additional phase lag is tolerable before crossing $-1$

Controllability

Controllability determines whether the control input can drive the system to any desired state in finite time. This is essential for control system design.

LTI Continuous-Time Controllability

A system $\dot{\mathbf{x}}=A\mathbf{x}+B\mathbf{u}$ is controllable if, for any initial state $\mathbf{x}(0)=\mathbf{x}0$ and any desired final state $\mathbf{x}$f, there exists a finite time $t_f>0$ and control input $\mathbf{u}(t)$, $t\in [0,t_f]$, such that $\mathbf{x}(t_f)=\mathbf{x}\_f$.

Kalman Controllability Test

A system is controllable if and only if the controllability matrix has full rank: $$\text{rank}(\mathcal{C})=n$$

where the controllability matrix is: $$\mathcal{C}=\left[\begin{array}{ccccc}B& AB& A^{2}B& \cdots & A^{n-1}B\end{array}\right]\in {\mathbb{R}}^{n\times nm}$$

This is an $n\times nm$ matrix formed by stacking the $n\times m$ matrices. The rank must equal the system order $n$.

• $B$ column: direct influence of input on the states
• $AB$ column: influence of input through one state transition (dynamics)
• $A^{k-1}B$: influence through longer paths in the state space

If rank$(\mathcal{C})<n$, some state direction is unreachable: the control input cannot affect that mode, making it impossible to steer the system to arbitrary states.

Observability

Observability determines whether we can uniquely estimate the internal > state from measurements of the system output. > This is essential for observer design and state feedback implementation.

LTI Continuous-Time Observability

A system with state dynamics $\dot{\mathbf{x}}=A\mathbf{x}$ and measurement $\mathbf{y}=C\mathbf{x}$ is observable if, given the measurement $\mathbf{y}(t)$ over any finite time interval $[0,t_f]$, the initial state $\mathbf{x}(0)$ can be uniquely determined.

Kalman Observability Test

A system is observable if and only if the observability matrix has full rank: $$\text{rank}(\mathcal{O})=n$$

where the observability matrix is: $$\mathcal{O}=\left[\begin{array}{c}C\\ CA\\ C{A}^2\\ ⋮\\ C{A}^{n-1}\end{array}\right]\in {\mathbb{R}}^{pn\times n}$$

This is an $pn\times n$ matrix formed by stacking $p\times n$ matrices. The rank must equal the system order $n$.

• $C$ row: direct measurement of states
• $CA$ row: information about states after one state transition
• $C{A}^{k-1}$ row: information over longer time horizons

If rank$(\mathcal{O})<n$, some state component is unobservable: it affects the dynamics but never influences the measurements, making it impossible to determine that state from output alone.

4. Simulation

Initial Value Problems (IVP): Problem Formulation

An Initial Value Problem (IVP) is an ODE together with an initial condition. Numerical methods solve IVPs to simulate dynamical systems.

Standard Form

$$\dot{\mathbf{x}}=\mathbf{f}(t,\mathbf{x}),\mathbf{x}(t_0)={\mathbf{x}}_0$$

Given the function $\mathbf{f}$, the initial state $\mathbf{x}\_0$, and the initial time $t_0$, we seek the solution $\mathbf{x}(t)$ for $t\ge t_0$ \cite{frontiers-rk4}.

Existence and Uniqueness (Picard-Lindelöf Theorem)

A unique solution exists in a neighborhood of $t_0$ if:

1. $\mathbf{f}(t,\mathbf{x})$ is continuous in both arguments
2. $\mathbf{f}$ is Lipschitz continuous in $\mathbf{x}$: $\Vert \mathbf{f}(t,\mathbf{x}\_1)-\mathbf{f}(t,\mathbf{x}\_2)\Vert \le L\Vert \mathbf{x}\_1-\mathbf{x}\_2\Vert$ for some constant $L$

Most practical systems satisfy these conditions, ensuring that numerical solutions are meaningful.

Euler Integration: Forward Stepping

The Forward Euler method (also called RK1) is the simplest explicit numerical integrator. Given step size $h$:

$${\mathbf{x}}_{k+1}={\mathbf{x}}_k+h\cdot \mathbf{f}(t_k,{\mathbf{x}}_k)$$

where $t\_k+1=t_k+h$.

From the Taylor series expansion: $$\mathbf{x}(t+h)=\mathbf{x}(t)+h\dot{\mathbf{x}}(t)+\frac{h^2}{2}\ddot{\mathbf{x}}(\xi )\xi \in (t,t+h)$$

The Euler method truncates after the first-order term, replacing the derivative with its value at the current time: $$\mathbf{x}(t+h)\approx \mathbf{x}(t)-h\mathbf{f}(t,\mathbf{x}(t))$$

Error Analysis

• Local Truncation Error (LTE): $\mathcal{O}(h^2)$ per step (from truncating the Taylor series)
• Global Error: $\mathcal{O}(h)$ over a fixed time interval (accumulated over multiple steps)
• Stability Requirement: For numerical stability of linear systems, the step size must satisfy $|1+h\lambda |<1$, where $\lambda$ is an eigenvalue

The stability constraint is severe for stiff systems (systems with wide range of timescales), requiring tiny step sizes \cite{frontiers-rk4}.

Disadvantages

• Only first-order accurate (slow convergence with step size)
• Unstable for stiff systems without prohibitively small time steps
• Rarely used in practice for engineering applications

References

@book{hover_system_2022,
  title = {System {Design} for {Uncertainty}},
  url = {<https://eng.libretexts.org/Bookshelves/Mechanical_Engineering/System_Design_for_Uncertainty_(Hover_and_Triantafyllou)>},
  abstract = {This text covers the design, construction, and testing of field robotic systems, through team projects with each student responsible for a specific subsystem. Projects focus on electronics, instrumentation, and machine elements. Design for operation in uncertain conditions is a focus point, with ocean waves and marine structures as a central theme. Topics include basic statistics, linear systems, Fourier transforms, random processes, spectra, ethics in engineering practice, and extreme events with applications in design.},
  language = {en},
  urldate = {2025-12-24},
  publisher = {Engineering LibreTexts},
  author = {Hover, Franz S. and Triantafyllou, Michael S.},
  month = mar,
  year = {2022},
  note = {License: CC BY-NC-SA 4.0. Original source: MIT OpenCourseWare (Fall 2009)},
}


@article{aero_students_phase,
  title = {Phase {Portraits} and {Stability}},
  url = {http://www.aerostudents.com/courses/differential-equations/phasePortraitsAndStability.pdf},
  author = {{AeroStudents}},
}