API Reference

Systems: Core Classes

Abstract base classes and core system implementations

systems.base.core.SymbolicSystemBase	Abstract base class for symbolic systems (time-domain agnostic).
systems.base.core.ContinuousSystemBase	Abstract base class for all continuous-time dynamical systems.
systems.base.core.DiscreteSystemBase	Abstract base class for all discrete-time dynamical systems.
systems.base.core.ContinuousSymbolicSystem	Concrete symbolic continuous-time dynamical system.
systems.base.core.DiscreteSymbolicSystem	Concrete symbolic discrete-time dynamical system.
systems.base.core.ContinuousStochasticSystem	Concrete symbolic continuous-time stochastic dynamical system (SDE).
systems.base.core.DiscreteStochasticSystem	Concrete symbolic discrete-time stochastic dynamical system.
systems.base.core.DiscretizedSystem	Pure wrapper providing discrete interface to continuous systems.
systems.base.core.discretize	Convenience wrapper for creating a discretized system.

Systems: Numerical Integration

ODE and SDE integrators for multiple backends

systems.base.numerical_integration.is_sde_method	Check if integration method is for stochastic differential equations.
systems.base.numerical_integration.is_fixed_step	Check if integration method uses fixed time stepping.
systems.base.numerical_integration.normalize_method_name	Normalize method names across backends to canonical form.
systems.base.numerical_integration.get_available_methods	Get available integration methods for a backend.
systems.base.numerical_integration.validate_method	Validate method for backend and system type.
systems.base.numerical_integration.get_method_info	Get comprehensive information about a method.
systems.base.numerical_integration.list_all_methods	List all integration methods across all categories.
systems.base.numerical_integration.get_implementing_library	Get which library/package implements this method.
systems.base.numerical_integration.IntegratorFactory	Factory for creating numerical integrators.
systems.base.numerical_integration.IntegratorBase	Abstract base class for numerical integrators.
systems.base.numerical_integration.ScipyIntegrator	Adaptive integrator using scipy.integrate.solve_ivp.
systems.base.numerical_integration.DiffraxIntegrator	JAX-based ODE integrator using the Diffrax library.
systems.base.numerical_integration.TorchDiffEqIntegrator	PyTorch-based ODE integrator using the torchdiffeq library.
systems.base.numerical_integration.stochastic.SDEIntegratorFactory	Factory for creating SDE numerical integrators.
systems.base.numerical_integration.stochastic.SDEIntegratorBase	Abstract base class for SDE integrators.
systems.base.numerical_integration.stochastic.DiffEqPySDEIntegrator	Julia-based SDE integrator using DifferentialEquations.jl via diffeqpy.
systems.base.numerical_integration.stochastic.TorchSDEIntegrator	PyTorch-based SDE integrator using the torchsde library.
systems.base.numerical_integration.stochastic.DiffraxSDEIntegrator	JAX-based SDE integrator using the Diffrax library.
systems.base.numerical_integration.stochastic.CustomBrownianPath	Custom Brownian motion that uses provided dW increments.
systems.base.numerical_integration.stochastic.create_custom_or_random_brownian	Create either custom or random Brownian motion for Diffrax.

Systems: Utilities

Backend management, linearization, and code generation

systems.base.utils.BackendManager	Manages backend detection, conversion, and device placement.
systems.base.utils.CodeGenerator	Orchestrates code generation and caching for a dynamical system.
systems.base.utils.DynamicsEvaluator	Evaluates forward dynamics across backends.
systems.base.utils.EquilibriumHandler	Manages multiple equilibrium points for a dynamical system.
systems.base.utils.LinearizationEngine	Computes linearized dynamics across backends.
systems.base.utils.ObservationEngine	Evaluates observation/output functions across backends.
systems.base.utils.SymbolicValidator	Validates symbolic dynamical system definitions.
systems.base.utils.stochastic.NoiseCharacteristics	Container for noise structure analysis results.
systems.base.utils.stochastic.NoiseCharacterizer	Analyzes symbolic diffusion expressions to determine noise structure.
systems.base.utils.stochastic.analyze_noise_structure	Convenience function for analyzing noise structure.
systems.base.utils.stochastic.DiffusionHandler	Handles code generation and caching for diffusion terms.
systems.base.utils.stochastic.SDEValidator	Validates stochastic dynamical system definitions.

Control Design & Analysis

Control synthesis, system analysis, and classical control functions

control.ControlSynthesis	Control synthesis wrapper for system composition.
control.SystemAnalysis	System analysis wrapper for composition.
control.design_lqr	Design Linear Quadratic Regulator (LQR) controller.
control.design_lqg	Design Linear Quadratic Gaussian (LQG) controller.
control.design_kalman_filter	Design Kalman filter for optimal state estimation.
control.analyze_stability	Analyze system stability via eigenvalue analysis.
control.analyze_controllability	Test controllability of linear system (A, B).
control.analyze_observability	Test observability of linear system (A, C).

Visualization

Plotting and visualization tools for dynamical systems

visualization.TrajectoryPlotter	Time-domain trajectory visualization.
visualization.PhasePortraitPlotter	Phase space visualization for dynamical systems.
visualization.ControlPlotter	Control system analysis visualization.
visualization.ColorSchemes	Predefined color palettes for plotting.
visualization.PlotThemes	Complete plotting theme configurations.

Builtin: Deterministic Continuous

Continuous-time deterministic dynamical systems

systems.builtin.deterministic.continuous.LinearSystem	First-order linear system: dx/dt = -ax + bu
systems.builtin.deterministic.continuous.AutonomousLinearSystem	Autonomous first-order linear system: dx/dt = -a*x
systems.builtin.deterministic.continuous.LinearSystem2D	Two-dimensional linear system.
systems.builtin.deterministic.continuous.SymbolicPendulum	Simple inverted pendulum system - first-order state-space formulation.
systems.builtin.deterministic.continuous.SymbolicPendulum2ndOrder	Inverted pendulum - second-order formulation (returns ONLY acceleration).
systems.builtin.deterministic.continuous.CartPole	Cart-pole system (inverted pendulum on cart) - classic underactuated system.
systems.builtin.deterministic.continuous.Lorenz	Lorenz system - famous chaotic dynamical system from atmospheric convection.
systems.builtin.deterministic.continuous.ControlledLorenz	Lorenz system - famous chaotic dynamical system from atmospheric convection with a forcing term
systems.builtin.deterministic.continuous.VanDerPolOscillator	Van der Pol oscillator - self-excited nonlinear oscillator with limit cycle.
systems.builtin.deterministic.continuous.ControlledVanDerPolOscillator	Van der Pol oscillator - self-excited nonlinear oscillator with limit cycle and forcing term
systems.builtin.deterministic.continuous.DuffingOscillator	Duffing oscillator - nonlinear oscillator with cubic stiffness term.
systems.builtin.deterministic.continuous.CoupledOscillatorSystem	Coupled mass-spring-damper system with rotational coupling - first-order formulation.
systems.builtin.deterministic.continuous.NonlinearChainSystem	Chain of five coupled nonlinear oscillators - first-order formulation.
systems.builtin.deterministic.continuous.DubinsVehicle	Dubins vehicle - kinematic car model with unicycle dynamics.
systems.builtin.deterministic.continuous.PathTracking	Path tracking error dynamics for a vehicle following a circular reference path.
systems.builtin.deterministic.continuous.SymbolicQuadrotor2D	Planar quadrotor (quadcopter) - second-order formulation.
systems.builtin.deterministic.continuous.SymbolicQuadrotor2DLidar	Symbolic representation of a planar (2D) quadrotor with lidar-based partial observations.
systems.builtin.deterministic.continuous.PVTOL	Planar Vertical Take-Off and Landing (PVTOL) aircraft - second-order formulation.
systems.builtin.deterministic.continuous.Manipulator2Link	Two-link planar robotic manipulator - second-order formulation.
systems.builtin.deterministic.continuous.FifthOrderMechanicalSystem	Fifth-order mechanical system - extremely high-order dynamics.
systems.builtin.deterministic.continuous.ContinuousBatchReactor	Continuous-time chemical batch reactor with temperature control.
systems.builtin.deterministic.continuous.ContinuousCSTR	Continuous-time Continuous Stirred-Tank Reactor (CSTR) with cooling jacket.

Builtin: Deterministic Discrete

Discrete-time deterministic dynamical systems

systems.builtin.deterministic.discrete.DiscreteOscillator	Discrete-time damped harmonic oscillator with external forcing.
systems.builtin.deterministic.discrete.DiscretePendulum	Discrete-time simple pendulum with friction and optional control.
systems.builtin.deterministic.discrete.DiscreteDoubleIntegrator	Discrete-time double integrator (position-controlled point mass).
systems.builtin.deterministic.discrete.DiscreteDoubleIntegratorWithForce	Discrete double integrator with explicit force input and mass.
systems.builtin.deterministic.discrete.DiscreteRobotArm	Discrete-time single-link robot arm with gravity and friction.
systems.builtin.deterministic.discrete.DiscreteCartPole	Discrete-time cart-pole (inverted pendulum on cart) system.
systems.builtin.deterministic.discrete.DifferentialDriveRobot	Discrete-time differential drive mobile robot with nonholonomic constraints.
systems.builtin.deterministic.discrete.DiscreteBatchReactor	Discrete-time chemical batch reactor with temperature control.
systems.builtin.deterministic.discrete.DiscreteCSTR	Discrete-time Continuous Stirred-Tank Reactor (CSTR) with cooling jacket.
systems.builtin.deterministic.discrete.DiscreteSolowModel	Discrete-time Solow-Swan neoclassical growth model.
systems.builtin.deterministic.discrete.LogisticMap	The logistic map: x[k+1] = r·x[k]·(1 - x[k])
systems.builtin.deterministic.discrete.HenonMap	The Hénon Map: Paradigm of 2D dissipative chaos and strange attractors.
systems.builtin.deterministic.discrete.StandardMap	The Standard Map (Chirikov-Taylor Map): Paradigm of Hamiltonian chaos.

Builtin: Stochastic Continuous

Continuous-time stochastic dynamical systems (SDEs)

systems.builtin.stochastic.continuous.BrownianMotion	Standard Brownian motion (Wiener process) - pure diffusion process.
systems.builtin.stochastic.continuous.BrownianMotion2D	Two-dimensional Brownian motion with independent components.
systems.builtin.stochastic.continuous.BrownianMotionWithDrift	Alias for GeometricBrownianMotion.
systems.builtin.stochastic.continuous.GeometricBrownianMotion	Geometric Brownian motion with multiplicative (state-dependent) noise.
systems.builtin.stochastic.continuous.OrnsteinUhlenbeck	Ornstein-Uhlenbeck process with mean reversion and additive noise.
systems.builtin.stochastic.continuous.MultivariateOrnsteinUhlenbeck	Multivariate Ornstein-Uhlenbeck process with coupling and correlated noise.
systems.builtin.stochastic.continuous.CoxIngersollRoss	Cox-Ingersoll-Ross process with mean reversion and square-root diffusion.
systems.builtin.stochastic.continuous.LangevinDynamics	Langevin dynamics - particle in viscous fluid with thermal noise.
systems.builtin.stochastic.continuous.StochasticDoubleIntegrator	Stochastic double integrator - canonical LQG benchmark system.
systems.builtin.stochastic.continuous.ContinuousStochasticPendulum	Stochastic pendulum with random forcing - canonical nonlinear stochastic system.
systems.builtin.stochastic.continuous.StochasticCartPole	Stochastic inverted pendulum on cart with random disturbances.
systems.builtin.stochastic.continuous.StochasticLorenz	Stochastic Lorenz system - deterministic chaos with random perturbations.
systems.builtin.stochastic.continuous.StochasticSIR	Stochastic SIR epidemic model with demographic noise.
systems.builtin.stochastic.continuous.ContinuousStochasticBatchReactor	Continuous-time stochastic batch reactor with process noise.
systems.builtin.stochastic.continuous.ContinuousStochasticCSTR	Continuous-time stochastic CSTR with multiple equilibria and Brownian noise.

Builtin: Stochastic Discrete

Discrete-time stochastic dynamical systems

systems.builtin.stochastic.discrete.DiscreteAR1	First-order autoregressive process with additive noise.
systems.builtin.stochastic.discrete.DiscreteWhiteNoise	Pure white noise process - memoryless random sequence.
systems.builtin.stochastic.discrete.DiscreteRandomWalk	Random walk - non-stationary process with unit root.
systems.builtin.stochastic.discrete.DiscreteARMA11	ARMA(1,1) process - combines autoregressive and moving average.
systems.builtin.stochastic.discrete.DiscreteVAR1	Vector Autoregressive process of order 1 - multivariate time series model.
systems.builtin.stochastic.discrete.DiscreteGARCH11	GARCH(1,1) - time-varying volatility model (Nobel Prize 2003).
systems.builtin.stochastic.discrete.DiscreteStochasticDoubleIntegrator	Discrete-time stochastic double integrator - canonical digital LQG benchmark.
systems.builtin.stochastic.discrete.DiscreteStochasticPendulum	Discrete-time stochastic pendulum for digital control and RL.
systems.builtin.stochastic.discrete.DiscreteStochasticQueue	Discrete-time stochastic queue with random arrivals and service.
systems.builtin.stochastic.discrete.DiscreteStochasticBatchReactor	Discrete-time stochastic batch reactor for digital control and estimation.
systems.builtin.stochastic.discrete.DiscreteStochasticCSTR	Discrete-time stochastic CSTR with multiple steady states and process noise.
systems.builtin.stochastic.discrete.DiscreteStochasticLogisticMap	Discrete-time stochastic logistic map - chaos meets noise.

Types: Core

Fundamental vectors, matrices, dimensions, and function types

types.core.ScalarLike	Scalar value in any backend.
types.core.ArrayLike	Array-like type supporting multiple backends.
types.core.StateVector	State vector x ∈ ℝⁿˣ.
types.core.ControlVector	Control input vector u ∈ ℝⁿᵘ.
types.core.OutputVector	Output/observation/measurement vector y ∈ ℝⁿʸ.
types.core.NoiseVector	Noise/disturbance vector w ∈ ℝⁿʷ (stochastic systems only).
types.core.ParameterVector	Parameter vector θ ∈ ℝⁿᵖ.
types.core.StateMatrix	State matrix (nx, nx).
types.core.InputMatrix	Input matrix B (nx, nu).
types.core.OutputMatrix	Output/observation matrix (ny, nx).
types.core.DiffusionMatrix	Diffusion/noise gain matrix (nx, nw) - stochastic systems only.
types.core.FeedthroughMatrix	Feedthrough/direct transmission matrix (ny, nu).
types.core.GainMatrix	Gain matrix for control or estimation.
types.core.CovarianceMatrix	Covariance matrix (symmetric, positive semidefinite).
types.core.SystemDimensions	System dimensions as dictionary.
types.core.DynamicsFunction	Dynamics function f(x, u).
types.core.OutputFunction	Output/observation function h(x).
types.core.DiffusionFunction	Diffusion function g(x, u) for stochastic systems.
types.core.ControlPolicy	Control policy/controller π(x).
types.core.TimeVaryingControl	Time-varying control function u(t).
types.core.FeedbackController	Feedback controller with time awareness π(x, t).
types.core.ControlInput	Unified type for control inputs to integration methods.

Types: Backend & Configuration

Backend support and configuration types

types.backends.Backend	Backend identifier for numerical computation.
types.backends.Device	Device identifier for hardware acceleration.
types.backends.BackendConfig	Backend configuration dictionary.
types.backends.IntegrationMethod	Integration method for continuous-time systems (ODEs).
types.backends.SDEIntegrationMethod	SDE integration method for stochastic differential equations.
types.backends.DiscretizationMethod	Discretization method for continuous → discrete transformation.
types.backends.OptimizationMethod	Optimization method for control/estimation problems.
types.backends.NoiseType	Noise structure classification for stochastic systems.
types.backends.SDEType	SDE interpretation type.
types.backends.ConvergenceType	SDE convergence type for numerical integration.
types.backends.IntegratorConfig	Configuration for continuous-time integrators.
types.backends.SDEIntegratorConfig	Configuration for SDE integrators.
types.backends.DiscretizerConfig	Configuration for system discretization.
types.backends.SystemConfig	Complete system configuration dictionary.
types.backends.validate_backend	Validate and normalize backend string.
types.backends.validate_device	Validate device for given backend.
types.backends.get_backend_default_method	Get default integration method for backend.

Types: Trajectories & Simulation

Time series, trajectories, and simulation results

types.trajectories.TimePoints	Array of time points for simulation or evaluation.
types.trajectories.TimeSpan	Time interval for continuous integration: (t_start, t_end).
types.trajectories.StateTrajectory	State trajectory over time.
types.trajectories.ControlSequence	Control input sequence over time.
types.trajectories.OutputSequence	Output/measurement sequence over time.
types.trajectories.NoiseSequence	Noise/disturbance sequence for stochastic simulation.
types.trajectories.SimulationResult	Result from continuous-time system simulation.
types.trajectories.IntegrationResult	Result from continuous-time integration (ODE/SDE solver).
types.trajectories.TrajectoryStatistics	Statistical summary of trajectory.
types.trajectories.TrajectorySegment	Segment of trajectory between two time points.

Types: Control Results

Control design and analysis result types

types.control_classical.LQRResult	Linear Quadratic Regulator (LQR) design result.
types.control_classical.LQGResult	Linear Quadratic Gaussian (LQG) controller design result.
types.control_classical.KalmanFilterResult	Kalman Filter (optimal state estimator) design result.
types.control_classical.StabilityInfo	Stability analysis result dictionary.
types.control_classical.ControllabilityInfo	Controllability analysis result.
types.control_classical.ObservabilityInfo	Observability analysis result.
types.control_classical.PolePlacementResult	Pole placement (eigenvalue assignment) result.
types.control_classical.LuenbergerObserverResult	Luenberger observer (deterministic state estimator) design result.
types.control_advanced.MPCResult	Model Predictive Control (MPC) solution result.
types.control_advanced.MHEResult	Moving Horizon Estimation (MHE) result.
types.control_advanced.H2ControlResult	H₂ optimal control result.
types.control_advanced.HInfControlResult	H∞ robust control result.
types.control_advanced.LMIResult	Linear Matrix Inequality (LMI) solver result.
types.control_advanced.AdaptiveControlResult	Adaptive control result.
types.control_advanced.SlidingModeResult	Sliding Mode Control (SMC) result.

Types: Estimation

State estimation result types

types.estimation.EKFResult	Extended Kalman Filter (EKF) state and result.
types.estimation.UKFResult	Unscented Kalman Filter (UKF) result.
types.estimation.ParticleFilterResult	Particle Filter (Sequential Monte Carlo) result.

Types: Linearization

Linearization and Jacobian types

types.linearization.LinearizationResult	Flexible linearization result type.
types.linearization.DeterministicLinearization	Linearization result for deterministic systems: (A, B).
types.linearization.StochasticLinearization	Linearization result for stochastic systems: (A, B, G).
types.linearization.ContinuousLinearization	Alias for continuous-time deterministic linearization.
types.linearization.DiscreteLinearization	Alias for discrete-time deterministic linearization.
types.linearization.ContinuousStochasticLinearization	Alias for continuous-time stochastic linearization.
types.linearization.DiscreteStochasticLinearization	Alias for discrete-time stochastic linearization.
types.linearization.FullLinearization	Complete linearization including output: (A, B, C, D).
types.linearization.FullStochasticLinearization	Complete stochastic linearization: (A, B, G, C, D).
types.linearization.ObservationLinearization	Observation/output linearization: (C, D).
types.linearization.StateJacobian	State Jacobian ∂f/∂x.
types.linearization.ControlJacobian	Control Jacobian ∂f/∂u.
types.linearization.OutputJacobian	Output Jacobian ∂h/∂x.
types.linearization.DiffusionJacobian	Diffusion Jacobian ∂g/∂x (stochastic systems).
types.linearization.LinearizationCacheKey	Cache key for linearization results.

Types: Symbolic

Symbolic computation types

types.symbolic.SymbolicExpression	Single symbolic expression.
types.symbolic.SymbolicMatrix	Matrix of symbolic expressions.
types.symbolic.SymbolicStateEquations	Symbolic state equations: f(x, u, params).
types.symbolic.SymbolicJacobian	Jacobian matrix: ∂f/∂x.

Types: Advanced

Reachability, robustness, optimization, and learning types

types.reachability.ReachabilityResult	Reachability analysis result.
types.reachability.ROAResult	Region of Attraction (ROA) analysis result.
types.reachability.CBFResult	Control Barrier Function (CBF) result.
types.reachability.CLFResult	Control Lyapunov Function (CLF) result.
types.robustness.RobustStabilityResult	Robust stability analysis result.
types.robustness.TubeMPCResult	Tube-based MPC result.
types.optimization.OptimizationResult	General nonlinear optimization result.
types.optimization.TrajectoryOptimizationResult	Trajectory optimization result.
types.learning.TrainingResult	Neural network training result.
types.learning.RLTrainingResult	Reinforcement learning training result.
types.contraction.ContractionAnalysisResult	Contraction analysis result.
types.contraction.CCMResult	Control Contraction Metrics (CCM) result.
types.conformal.ConformalPredictionResult	Conformal prediction result for test points.

Types: Protocols

Structural subtyping protocols for system interfaces

types.protocols.ContinuousSystemProtocol	Minimal interface for continuous-time dynamical systems.
types.protocols.DiscreteSystemProtocol	Minimal interface for discrete-time dynamical systems.
types.protocols.StochasticSystemProtocol	System with stochastic dynamics (continuous or discrete).
types.protocols.LinearizableContinuousProtocol	Continuous system with linearization capability.
types.protocols.LinearizableDiscreteProtocol	Discrete system with linearization capability.

State Observers

State estimation and observer design

observers.LinearObserver	Linear state observer with constant gain.
observers.ExtendedKalmanFilter	Extended Kalman Filter (EKF) for nonlinear state estimation.

# API Reference {.doc .doc-index} ## Systems: Core Classes Abstract base classes and core system implementations | | | | --- | --- | | [systems.base.core.SymbolicSystemBase](systems.base.core.SymbolicSystemBase.qmd#cdesym.systems.base.core.SymbolicSystemBase) | Abstract base class for symbolic systems (time-domain agnostic). | | [systems.base.core.ContinuousSystemBase](systems.base.core.ContinuousSystemBase.qmd#cdesym.systems.base.core.ContinuousSystemBase) | Abstract base class for all continuous-time dynamical systems. | | [systems.base.core.DiscreteSystemBase](systems.base.core.DiscreteSystemBase.qmd#cdesym.systems.base.core.DiscreteSystemBase) | Abstract base class for all discrete-time dynamical systems. | | [systems.base.core.ContinuousSymbolicSystem](systems.base.core.ContinuousSymbolicSystem.qmd#cdesym.systems.base.core.ContinuousSymbolicSystem) | Concrete symbolic continuous-time dynamical system. | | [systems.base.core.DiscreteSymbolicSystem](systems.base.core.DiscreteSymbolicSystem.qmd#cdesym.systems.base.core.DiscreteSymbolicSystem) | Concrete symbolic discrete-time dynamical system. | | [systems.base.core.ContinuousStochasticSystem](systems.base.core.ContinuousStochasticSystem.qmd#cdesym.systems.base.core.ContinuousStochasticSystem) | Concrete symbolic continuous-time stochastic dynamical system (SDE). | | [systems.base.core.DiscreteStochasticSystem](systems.base.core.DiscreteStochasticSystem.qmd#cdesym.systems.base.core.DiscreteStochasticSystem) | Concrete symbolic discrete-time stochastic dynamical system. | | [systems.base.core.DiscretizedSystem](systems.base.core.DiscretizedSystem.qmd#cdesym.systems.base.core.DiscretizedSystem) | Pure wrapper providing discrete interface to continuous systems. | | [systems.base.core.discretize](systems.base.core.discretize.qmd#cdesym.systems.base.core.discretize) | Convenience wrapper for creating a discretized system. | ## Systems: Numerical Integration ODE and SDE integrators for multiple backends | | | | --- | --- | | [systems.base.numerical_integration.is_sde_method](systems.base.numerical_integration.is_sde_method.qmd#cdesym.systems.base.numerical_integration.is_sde_method) | Check if integration method is for stochastic differential equations. | | [systems.base.numerical_integration.is_fixed_step](systems.base.numerical_integration.is_fixed_step.qmd#cdesym.systems.base.numerical_integration.is_fixed_step) | Check if integration method uses fixed time stepping. | | [systems.base.numerical_integration.normalize_method_name](systems.base.numerical_integration.normalize_method_name.qmd#cdesym.systems.base.numerical_integration.normalize_method_name) | Normalize method names across backends to canonical form. | | [systems.base.numerical_integration.get_available_methods](systems.base.numerical_integration.get_available_methods.qmd#cdesym.systems.base.numerical_integration.get_available_methods) | Get available integration methods for a backend. | | [systems.base.numerical_integration.validate_method](systems.base.numerical_integration.validate_method.qmd#cdesym.systems.base.numerical_integration.validate_method) | Validate method for backend and system type. | | [systems.base.numerical_integration.get_method_info](systems.base.numerical_integration.get_method_info.qmd#cdesym.systems.base.numerical_integration.get_method_info) | Get comprehensive information about a method. | | [systems.base.numerical_integration.list_all_methods](systems.base.numerical_integration.list_all_methods.qmd#cdesym.systems.base.numerical_integration.list_all_methods) | List all integration methods across all categories. | | [systems.base.numerical_integration.get_implementing_library](systems.base.numerical_integration.get_implementing_library.qmd#cdesym.systems.base.numerical_integration.get_implementing_library) | Get which library/package implements this method. | | [systems.base.numerical_integration.IntegratorFactory](systems.base.numerical_integration.IntegratorFactory.qmd#cdesym.systems.base.numerical_integration.IntegratorFactory) | Factory for creating numerical integrators. | | [systems.base.numerical_integration.IntegratorBase](systems.base.numerical_integration.IntegratorBase.qmd#cdesym.systems.base.numerical_integration.IntegratorBase) | Abstract base class for numerical integrators. | | [systems.base.numerical_integration.ScipyIntegrator](systems.base.numerical_integration.ScipyIntegrator.qmd#cdesym.systems.base.numerical_integration.ScipyIntegrator) | Adaptive integrator using scipy.integrate.solve_ivp. | | [systems.base.numerical_integration.DiffraxIntegrator](systems.base.numerical_integration.DiffraxIntegrator.qmd#cdesym.systems.base.numerical_integration.DiffraxIntegrator) | JAX-based ODE integrator using the Diffrax library. | | [systems.base.numerical_integration.TorchDiffEqIntegrator](systems.base.numerical_integration.TorchDiffEqIntegrator.qmd#cdesym.systems.base.numerical_integration.TorchDiffEqIntegrator) | PyTorch-based ODE integrator using the torchdiffeq library. | | [systems.base.numerical_integration.stochastic.SDEIntegratorFactory](systems.base.numerical_integration.stochastic.SDEIntegratorFactory.qmd#cdesym.systems.base.numerical_integration.stochastic.SDEIntegratorFactory) | Factory for creating SDE numerical integrators. | | [systems.base.numerical_integration.stochastic.SDEIntegratorBase](systems.base.numerical_integration.stochastic.SDEIntegratorBase.qmd#cdesym.systems.base.numerical_integration.stochastic.SDEIntegratorBase) | Abstract base class for SDE integrators. | | [systems.base.numerical_integration.stochastic.DiffEqPySDEIntegrator](systems.base.numerical_integration.stochastic.DiffEqPySDEIntegrator.qmd#cdesym.systems.base.numerical_integration.stochastic.DiffEqPySDEIntegrator) | Julia-based SDE integrator using DifferentialEquations.jl via diffeqpy. | | [systems.base.numerical_integration.stochastic.TorchSDEIntegrator](systems.base.numerical_integration.stochastic.TorchSDEIntegrator.qmd#cdesym.systems.base.numerical_integration.stochastic.TorchSDEIntegrator) | PyTorch-based SDE integrator using the torchsde library. | | [systems.base.numerical_integration.stochastic.DiffraxSDEIntegrator](systems.base.numerical_integration.stochastic.DiffraxSDEIntegrator.qmd#cdesym.systems.base.numerical_integration.stochastic.DiffraxSDEIntegrator) | JAX-based SDE integrator using the Diffrax library. | | [systems.base.numerical_integration.stochastic.CustomBrownianPath](systems.base.numerical_integration.stochastic.CustomBrownianPath.qmd#cdesym.systems.base.numerical_integration.stochastic.CustomBrownianPath) | Custom Brownian motion that uses provided dW increments. | | [systems.base.numerical_integration.stochastic.create_custom_or_random_brownian](systems.base.numerical_integration.stochastic.create_custom_or_random_brownian.qmd#cdesym.systems.base.numerical_integration.stochastic.create_custom_or_random_brownian) | Create either custom or random Brownian motion for Diffrax. | ## Systems: Utilities Backend management, linearization, and code generation | | | | --- | --- | | [systems.base.utils.BackendManager](systems.base.utils.BackendManager.qmd#cdesym.systems.base.utils.BackendManager) | Manages backend detection, conversion, and device placement. | | [systems.base.utils.CodeGenerator](systems.base.utils.CodeGenerator.qmd#cdesym.systems.base.utils.CodeGenerator) | Orchestrates code generation and caching for a dynamical system. | | [systems.base.utils.DynamicsEvaluator](systems.base.utils.DynamicsEvaluator.qmd#cdesym.systems.base.utils.DynamicsEvaluator) | Evaluates forward dynamics across backends. | | [systems.base.utils.EquilibriumHandler](systems.base.utils.EquilibriumHandler.qmd#cdesym.systems.base.utils.EquilibriumHandler) | Manages multiple equilibrium points for a dynamical system. | | [systems.base.utils.LinearizationEngine](systems.base.utils.LinearizationEngine.qmd#cdesym.systems.base.utils.LinearizationEngine) | Computes linearized dynamics across backends. | | [systems.base.utils.ObservationEngine](systems.base.utils.ObservationEngine.qmd#cdesym.systems.base.utils.ObservationEngine) | Evaluates observation/output functions across backends. | | [systems.base.utils.SymbolicValidator](systems.base.utils.SymbolicValidator.qmd#cdesym.systems.base.utils.SymbolicValidator) | Validates symbolic dynamical system definitions. | | [systems.base.utils.stochastic.NoiseCharacteristics](systems.base.utils.stochastic.NoiseCharacteristics.qmd#cdesym.systems.base.utils.stochastic.NoiseCharacteristics) | Container for noise structure analysis results. | | [systems.base.utils.stochastic.NoiseCharacterizer](systems.base.utils.stochastic.NoiseCharacterizer.qmd#cdesym.systems.base.utils.stochastic.NoiseCharacterizer) | Analyzes symbolic diffusion expressions to determine noise structure. | | [systems.base.utils.stochastic.analyze_noise_structure](systems.base.utils.stochastic.analyze_noise_structure.qmd#cdesym.systems.base.utils.stochastic.analyze_noise_structure) | Convenience function for analyzing noise structure. | | [systems.base.utils.stochastic.DiffusionHandler](systems.base.utils.stochastic.DiffusionHandler.qmd#cdesym.systems.base.utils.stochastic.DiffusionHandler) | Handles code generation and caching for diffusion terms. | | [systems.base.utils.stochastic.SDEValidator](systems.base.utils.stochastic.SDEValidator.qmd#cdesym.systems.base.utils.stochastic.SDEValidator) | Validates stochastic dynamical system definitions. | ## Control Design & Analysis Control synthesis, system analysis, and classical control functions | | | | --- | --- | | [control.ControlSynthesis](control.ControlSynthesis.qmd#cdesym.control.ControlSynthesis) | Control synthesis wrapper for system composition. | | [control.SystemAnalysis](control.SystemAnalysis.qmd#cdesym.control.SystemAnalysis) | System analysis wrapper for composition. | | [control.design_lqr](control.design_lqr.qmd#cdesym.control.design_lqr) | Design Linear Quadratic Regulator (LQR) controller. | | [control.design_lqg](control.design_lqg.qmd#cdesym.control.design_lqg) | Design Linear Quadratic Gaussian (LQG) controller. | | [control.design_kalman_filter](control.design_kalman_filter.qmd#cdesym.control.design_kalman_filter) | Design Kalman filter for optimal state estimation. | | [control.analyze_stability](control.analyze_stability.qmd#cdesym.control.analyze_stability) | Analyze system stability via eigenvalue analysis. | | [control.analyze_controllability](control.analyze_controllability.qmd#cdesym.control.analyze_controllability) | Test controllability of linear system (A, B). | | [control.analyze_observability](control.analyze_observability.qmd#cdesym.control.analyze_observability) | Test observability of linear system (A, C). | ## Visualization Plotting and visualization tools for dynamical systems | | | | --- | --- | | [visualization.TrajectoryPlotter](visualization.TrajectoryPlotter.qmd#cdesym.visualization.TrajectoryPlotter) | Time-domain trajectory visualization. | | [visualization.PhasePortraitPlotter](visualization.PhasePortraitPlotter.qmd#cdesym.visualization.PhasePortraitPlotter) | Phase space visualization for dynamical systems. | | [visualization.ControlPlotter](visualization.ControlPlotter.qmd#cdesym.visualization.ControlPlotter) | Control system analysis visualization. | | [visualization.ColorSchemes](visualization.ColorSchemes.qmd#cdesym.visualization.ColorSchemes) | Predefined color palettes for plotting. | | [visualization.PlotThemes](visualization.PlotThemes.qmd#cdesym.visualization.PlotThemes) | Complete plotting theme configurations. | ## Builtin: Deterministic Continuous Continuous-time deterministic dynamical systems | | | | --- | --- | | [systems.builtin.deterministic.continuous.LinearSystem](systems.builtin.deterministic.continuous.LinearSystem.qmd#cdesym.systems.builtin.deterministic.continuous.LinearSystem) | First-order linear system: dx/dt = -a*x + b*u | | [systems.builtin.deterministic.continuous.AutonomousLinearSystem](systems.builtin.deterministic.continuous.AutonomousLinearSystem.qmd#cdesym.systems.builtin.deterministic.continuous.AutonomousLinearSystem) | Autonomous first-order linear system: dx/dt = -a*x | | [systems.builtin.deterministic.continuous.LinearSystem2D](systems.builtin.deterministic.continuous.LinearSystem2D.qmd#cdesym.systems.builtin.deterministic.continuous.LinearSystem2D) | Two-dimensional linear system. | | [systems.builtin.deterministic.continuous.SymbolicPendulum](systems.builtin.deterministic.continuous.SymbolicPendulum.qmd#cdesym.systems.builtin.deterministic.continuous.SymbolicPendulum) | Simple inverted pendulum system - first-order state-space formulation. | | [systems.builtin.deterministic.continuous.SymbolicPendulum2ndOrder](systems.builtin.deterministic.continuous.SymbolicPendulum2ndOrder.qmd#cdesym.systems.builtin.deterministic.continuous.SymbolicPendulum2ndOrder) | Inverted pendulum - second-order formulation (returns ONLY acceleration). | | [systems.builtin.deterministic.continuous.CartPole](systems.builtin.deterministic.continuous.CartPole.qmd#cdesym.systems.builtin.deterministic.continuous.CartPole) | Cart-pole system (inverted pendulum on cart) - classic underactuated system. | | [systems.builtin.deterministic.continuous.Lorenz](systems.builtin.deterministic.continuous.Lorenz.qmd#cdesym.systems.builtin.deterministic.continuous.Lorenz) | Lorenz system - famous chaotic dynamical system from atmospheric convection. | | [systems.builtin.deterministic.continuous.ControlledLorenz](systems.builtin.deterministic.continuous.ControlledLorenz.qmd#cdesym.systems.builtin.deterministic.continuous.ControlledLorenz) | Lorenz system - famous chaotic dynamical system from atmospheric convection with a forcing term | | [systems.builtin.deterministic.continuous.VanDerPolOscillator](systems.builtin.deterministic.continuous.VanDerPolOscillator.qmd#cdesym.systems.builtin.deterministic.continuous.VanDerPolOscillator) | Van der Pol oscillator - self-excited nonlinear oscillator with limit cycle. | | [systems.builtin.deterministic.continuous.ControlledVanDerPolOscillator](systems.builtin.deterministic.continuous.ControlledVanDerPolOscillator.qmd#cdesym.systems.builtin.deterministic.continuous.ControlledVanDerPolOscillator) | Van der Pol oscillator - self-excited nonlinear oscillator with limit cycle and forcing term | | [systems.builtin.deterministic.continuous.DuffingOscillator](systems.builtin.deterministic.continuous.DuffingOscillator.qmd#cdesym.systems.builtin.deterministic.continuous.DuffingOscillator) | Duffing oscillator - nonlinear oscillator with cubic stiffness term. | | [systems.builtin.deterministic.continuous.CoupledOscillatorSystem](systems.builtin.deterministic.continuous.CoupledOscillatorSystem.qmd#cdesym.systems.builtin.deterministic.continuous.CoupledOscillatorSystem) | Coupled mass-spring-damper system with rotational coupling - first-order formulation. | | [systems.builtin.deterministic.continuous.NonlinearChainSystem](systems.builtin.deterministic.continuous.NonlinearChainSystem.qmd#cdesym.systems.builtin.deterministic.continuous.NonlinearChainSystem) | Chain of five coupled nonlinear oscillators - first-order formulation. | | [systems.builtin.deterministic.continuous.DubinsVehicle](systems.builtin.deterministic.continuous.DubinsVehicle.qmd#cdesym.systems.builtin.deterministic.continuous.DubinsVehicle) | Dubins vehicle - kinematic car model with unicycle dynamics. | | [systems.builtin.deterministic.continuous.PathTracking](systems.builtin.deterministic.continuous.PathTracking.qmd#cdesym.systems.builtin.deterministic.continuous.PathTracking) | Path tracking error dynamics for a vehicle following a circular reference path. | | [systems.builtin.deterministic.continuous.SymbolicQuadrotor2D](systems.builtin.deterministic.continuous.SymbolicQuadrotor2D.qmd#cdesym.systems.builtin.deterministic.continuous.SymbolicQuadrotor2D) | Planar quadrotor (quadcopter) - second-order formulation. | | [systems.builtin.deterministic.continuous.SymbolicQuadrotor2DLidar](systems.builtin.deterministic.continuous.SymbolicQuadrotor2DLidar.qmd#cdesym.systems.builtin.deterministic.continuous.SymbolicQuadrotor2DLidar) | Symbolic representation of a planar (2D) quadrotor with lidar-based partial observations. | | [systems.builtin.deterministic.continuous.PVTOL](systems.builtin.deterministic.continuous.PVTOL.qmd#cdesym.systems.builtin.deterministic.continuous.PVTOL) | Planar Vertical Take-Off and Landing (PVTOL) aircraft - second-order formulation. | | [systems.builtin.deterministic.continuous.Manipulator2Link](systems.builtin.deterministic.continuous.Manipulator2Link.qmd#cdesym.systems.builtin.deterministic.continuous.Manipulator2Link) | Two-link planar robotic manipulator - second-order formulation. | | [systems.builtin.deterministic.continuous.FifthOrderMechanicalSystem](systems.builtin.deterministic.continuous.FifthOrderMechanicalSystem.qmd#cdesym.systems.builtin.deterministic.continuous.FifthOrderMechanicalSystem) | Fifth-order mechanical system - extremely high-order dynamics. | | [systems.builtin.deterministic.continuous.ContinuousBatchReactor](systems.builtin.deterministic.continuous.ContinuousBatchReactor.qmd#cdesym.systems.builtin.deterministic.continuous.ContinuousBatchReactor) | Continuous-time chemical batch reactor with temperature control. | | [systems.builtin.deterministic.continuous.ContinuousCSTR](systems.builtin.deterministic.continuous.ContinuousCSTR.qmd#cdesym.systems.builtin.deterministic.continuous.ContinuousCSTR) | Continuous-time Continuous Stirred-Tank Reactor (CSTR) with cooling jacket. | ## Builtin: Deterministic Discrete Discrete-time deterministic dynamical systems | | | | --- | --- | | [systems.builtin.deterministic.discrete.DiscreteOscillator](systems.builtin.deterministic.discrete.DiscreteOscillator.qmd#cdesym.systems.builtin.deterministic.discrete.DiscreteOscillator) | Discrete-time damped harmonic oscillator with external forcing. | | [systems.builtin.deterministic.discrete.DiscretePendulum](systems.builtin.deterministic.discrete.DiscretePendulum.qmd#cdesym.systems.builtin.deterministic.discrete.DiscretePendulum) | Discrete-time simple pendulum with friction and optional control. | | [systems.builtin.deterministic.discrete.DiscreteDoubleIntegrator](systems.builtin.deterministic.discrete.DiscreteDoubleIntegrator.qmd#cdesym.systems.builtin.deterministic.discrete.DiscreteDoubleIntegrator) | Discrete-time double integrator (position-controlled point mass). | | [systems.builtin.deterministic.discrete.DiscreteDoubleIntegratorWithForce](systems.builtin.deterministic.discrete.DiscreteDoubleIntegratorWithForce.qmd#cdesym.systems.builtin.deterministic.discrete.DiscreteDoubleIntegratorWithForce) | Discrete double integrator with explicit force input and mass. | | [systems.builtin.deterministic.discrete.DiscreteRobotArm](systems.builtin.deterministic.discrete.DiscreteRobotArm.qmd#cdesym.systems.builtin.deterministic.discrete.DiscreteRobotArm) | Discrete-time single-link robot arm with gravity and friction. | | [systems.builtin.deterministic.discrete.DiscreteCartPole](systems.builtin.deterministic.discrete.DiscreteCartPole.qmd#cdesym.systems.builtin.deterministic.discrete.DiscreteCartPole) | Discrete-time cart-pole (inverted pendulum on cart) system. | | [systems.builtin.deterministic.discrete.DifferentialDriveRobot](systems.builtin.deterministic.discrete.DifferentialDriveRobot.qmd#cdesym.systems.builtin.deterministic.discrete.DifferentialDriveRobot) | Discrete-time differential drive mobile robot with nonholonomic constraints. | | [systems.builtin.deterministic.discrete.DiscreteBatchReactor](systems.builtin.deterministic.discrete.DiscreteBatchReactor.qmd#cdesym.systems.builtin.deterministic.discrete.DiscreteBatchReactor) | Discrete-time chemical batch reactor with temperature control. | | [systems.builtin.deterministic.discrete.DiscreteCSTR](systems.builtin.deterministic.discrete.DiscreteCSTR.qmd#cdesym.systems.builtin.deterministic.discrete.DiscreteCSTR) | Discrete-time Continuous Stirred-Tank Reactor (CSTR) with cooling jacket. | | [systems.builtin.deterministic.discrete.DiscreteSolowModel](systems.builtin.deterministic.discrete.DiscreteSolowModel.qmd#cdesym.systems.builtin.deterministic.discrete.DiscreteSolowModel) | Discrete-time Solow-Swan neoclassical growth model. | | [systems.builtin.deterministic.discrete.LogisticMap](systems.builtin.deterministic.discrete.LogisticMap.qmd#cdesym.systems.builtin.deterministic.discrete.LogisticMap) | The logistic map: x[k+1] = r·x[k]·(1 - x[k]) | | [systems.builtin.deterministic.discrete.HenonMap](systems.builtin.deterministic.discrete.HenonMap.qmd#cdesym.systems.builtin.deterministic.discrete.HenonMap) | The Hénon Map: Paradigm of 2D dissipative chaos and strange attractors. | | [systems.builtin.deterministic.discrete.StandardMap](systems.builtin.deterministic.discrete.StandardMap.qmd#cdesym.systems.builtin.deterministic.discrete.StandardMap) | The Standard Map (Chirikov-Taylor Map): Paradigm of Hamiltonian chaos. | ## Builtin: Stochastic Continuous Continuous-time stochastic dynamical systems (SDEs) | | | | --- | --- | | [systems.builtin.stochastic.continuous.BrownianMotion](systems.builtin.stochastic.continuous.BrownianMotion.qmd#cdesym.systems.builtin.stochastic.continuous.BrownianMotion) | Standard Brownian motion (Wiener process) - pure diffusion process. | | [systems.builtin.stochastic.continuous.BrownianMotion2D](systems.builtin.stochastic.continuous.BrownianMotion2D.qmd#cdesym.systems.builtin.stochastic.continuous.BrownianMotion2D) | Two-dimensional Brownian motion with independent components. | | [systems.builtin.stochastic.continuous.BrownianMotionWithDrift](systems.builtin.stochastic.continuous.BrownianMotionWithDrift.qmd#cdesym.systems.builtin.stochastic.continuous.BrownianMotionWithDrift) | Alias for GeometricBrownianMotion. | | [systems.builtin.stochastic.continuous.GeometricBrownianMotion](systems.builtin.stochastic.continuous.GeometricBrownianMotion.qmd#cdesym.systems.builtin.stochastic.continuous.GeometricBrownianMotion) | Geometric Brownian motion with multiplicative (state-dependent) noise. | | [systems.builtin.stochastic.continuous.OrnsteinUhlenbeck](systems.builtin.stochastic.continuous.OrnsteinUhlenbeck.qmd#cdesym.systems.builtin.stochastic.continuous.OrnsteinUhlenbeck) | Ornstein-Uhlenbeck process with mean reversion and additive noise. | | [systems.builtin.stochastic.continuous.MultivariateOrnsteinUhlenbeck](systems.builtin.stochastic.continuous.MultivariateOrnsteinUhlenbeck.qmd#cdesym.systems.builtin.stochastic.continuous.MultivariateOrnsteinUhlenbeck) | Multivariate Ornstein-Uhlenbeck process with coupling and correlated noise. | | [systems.builtin.stochastic.continuous.CoxIngersollRoss](systems.builtin.stochastic.continuous.CoxIngersollRoss.qmd#cdesym.systems.builtin.stochastic.continuous.CoxIngersollRoss) | Cox-Ingersoll-Ross process with mean reversion and square-root diffusion. | | [systems.builtin.stochastic.continuous.LangevinDynamics](systems.builtin.stochastic.continuous.LangevinDynamics.qmd#cdesym.systems.builtin.stochastic.continuous.LangevinDynamics) | Langevin dynamics - particle in viscous fluid with thermal noise. | | [systems.builtin.stochastic.continuous.StochasticDoubleIntegrator](systems.builtin.stochastic.continuous.StochasticDoubleIntegrator.qmd#cdesym.systems.builtin.stochastic.continuous.StochasticDoubleIntegrator) | Stochastic double integrator - canonical LQG benchmark system. | | [systems.builtin.stochastic.continuous.ContinuousStochasticPendulum](systems.builtin.stochastic.continuous.ContinuousStochasticPendulum.qmd#cdesym.systems.builtin.stochastic.continuous.ContinuousStochasticPendulum) | Stochastic pendulum with random forcing - canonical nonlinear stochastic system. | | [systems.builtin.stochastic.continuous.StochasticCartPole](systems.builtin.stochastic.continuous.StochasticCartPole.qmd#cdesym.systems.builtin.stochastic.continuous.StochasticCartPole) | Stochastic inverted pendulum on cart with random disturbances. | | [systems.builtin.stochastic.continuous.StochasticLorenz](systems.builtin.stochastic.continuous.StochasticLorenz.qmd#cdesym.systems.builtin.stochastic.continuous.StochasticLorenz) | Stochastic Lorenz system - deterministic chaos with random perturbations. | | [systems.builtin.stochastic.continuous.StochasticSIR](systems.builtin.stochastic.continuous.StochasticSIR.qmd#cdesym.systems.builtin.stochastic.continuous.StochasticSIR) | Stochastic SIR epidemic model with demographic noise. | | [systems.builtin.stochastic.continuous.ContinuousStochasticBatchReactor](systems.builtin.stochastic.continuous.ContinuousStochasticBatchReactor.qmd#cdesym.systems.builtin.stochastic.continuous.ContinuousStochasticBatchReactor) | Continuous-time stochastic batch reactor with process noise. | | [systems.builtin.stochastic.continuous.ContinuousStochasticCSTR](systems.builtin.stochastic.continuous.ContinuousStochasticCSTR.qmd#cdesym.systems.builtin.stochastic.continuous.ContinuousStochasticCSTR) | Continuous-time stochastic CSTR with multiple equilibria and Brownian noise. | ## Builtin: Stochastic Discrete Discrete-time stochastic dynamical systems | | | | --- | --- | | [systems.builtin.stochastic.discrete.DiscreteAR1](systems.builtin.stochastic.discrete.DiscreteAR1.qmd#cdesym.systems.builtin.stochastic.discrete.DiscreteAR1) | First-order autoregressive process with additive noise. | | [systems.builtin.stochastic.discrete.DiscreteWhiteNoise](systems.builtin.stochastic.discrete.DiscreteWhiteNoise.qmd#cdesym.systems.builtin.stochastic.discrete.DiscreteWhiteNoise) | Pure white noise process - memoryless random sequence. | | [systems.builtin.stochastic.discrete.DiscreteRandomWalk](systems.builtin.stochastic.discrete.DiscreteRandomWalk.qmd#cdesym.systems.builtin.stochastic.discrete.DiscreteRandomWalk) | Random walk - non-stationary process with unit root. | | [systems.builtin.stochastic.discrete.DiscreteARMA11](systems.builtin.stochastic.discrete.DiscreteARMA11.qmd#cdesym.systems.builtin.stochastic.discrete.DiscreteARMA11) | ARMA(1,1) process - combines autoregressive and moving average. | | [systems.builtin.stochastic.discrete.DiscreteVAR1](systems.builtin.stochastic.discrete.DiscreteVAR1.qmd#cdesym.systems.builtin.stochastic.discrete.DiscreteVAR1) | Vector Autoregressive process of order 1 - multivariate time series model. | | [systems.builtin.stochastic.discrete.DiscreteGARCH11](systems.builtin.stochastic.discrete.DiscreteGARCH11.qmd#cdesym.systems.builtin.stochastic.discrete.DiscreteGARCH11) | GARCH(1,1) - time-varying volatility model (Nobel Prize 2003). | | [systems.builtin.stochastic.discrete.DiscreteStochasticDoubleIntegrator](systems.builtin.stochastic.discrete.DiscreteStochasticDoubleIntegrator.qmd#cdesym.systems.builtin.stochastic.discrete.DiscreteStochasticDoubleIntegrator) | Discrete-time stochastic double integrator - canonical digital LQG benchmark. | | [systems.builtin.stochastic.discrete.DiscreteStochasticPendulum](systems.builtin.stochastic.discrete.DiscreteStochasticPendulum.qmd#cdesym.systems.builtin.stochastic.discrete.DiscreteStochasticPendulum) | Discrete-time stochastic pendulum for digital control and RL. | | [systems.builtin.stochastic.discrete.DiscreteStochasticQueue](systems.builtin.stochastic.discrete.DiscreteStochasticQueue.qmd#cdesym.systems.builtin.stochastic.discrete.DiscreteStochasticQueue) | Discrete-time stochastic queue with random arrivals and service. | | [systems.builtin.stochastic.discrete.DiscreteStochasticBatchReactor](systems.builtin.stochastic.discrete.DiscreteStochasticBatchReactor.qmd#cdesym.systems.builtin.stochastic.discrete.DiscreteStochasticBatchReactor) | Discrete-time stochastic batch reactor for digital control and estimation. | | [systems.builtin.stochastic.discrete.DiscreteStochasticCSTR](systems.builtin.stochastic.discrete.DiscreteStochasticCSTR.qmd#cdesym.systems.builtin.stochastic.discrete.DiscreteStochasticCSTR) | Discrete-time stochastic CSTR with multiple steady states and process noise. | | [systems.builtin.stochastic.discrete.DiscreteStochasticLogisticMap](systems.builtin.stochastic.discrete.DiscreteStochasticLogisticMap.qmd#cdesym.systems.builtin.stochastic.discrete.DiscreteStochasticLogisticMap) | Discrete-time stochastic logistic map - chaos meets noise. | ## Types: Core Fundamental vectors, matrices, dimensions, and function types | | | | --- | --- | | [types.core.ScalarLike](types.core.ScalarLike.qmd#cdesym.types.core.ScalarLike) | Scalar value in any backend. | | [types.core.ArrayLike](types.core.ArrayLike.qmd#cdesym.types.core.ArrayLike) | Array-like type supporting multiple backends. | | [types.core.StateVector](types.core.StateVector.qmd#cdesym.types.core.StateVector) | State vector x ∈ ℝⁿˣ. | | [types.core.ControlVector](types.core.ControlVector.qmd#cdesym.types.core.ControlVector) | Control input vector u ∈ ℝⁿᵘ. | | [types.core.OutputVector](types.core.OutputVector.qmd#cdesym.types.core.OutputVector) | Output/observation/measurement vector y ∈ ℝⁿʸ. | | [types.core.NoiseVector](types.core.NoiseVector.qmd#cdesym.types.core.NoiseVector) | Noise/disturbance vector w ∈ ℝⁿʷ (stochastic systems only). | | [types.core.ParameterVector](types.core.ParameterVector.qmd#cdesym.types.core.ParameterVector) | Parameter vector θ ∈ ℝⁿᵖ. | | [types.core.StateMatrix](types.core.StateMatrix.qmd#cdesym.types.core.StateMatrix) | State matrix (nx, nx). | | [types.core.InputMatrix](types.core.InputMatrix.qmd#cdesym.types.core.InputMatrix) | Input matrix B (nx, nu). | | [types.core.OutputMatrix](types.core.OutputMatrix.qmd#cdesym.types.core.OutputMatrix) | Output/observation matrix (ny, nx). | | [types.core.DiffusionMatrix](types.core.DiffusionMatrix.qmd#cdesym.types.core.DiffusionMatrix) | Diffusion/noise gain matrix (nx, nw) - stochastic systems only. | | [types.core.FeedthroughMatrix](types.core.FeedthroughMatrix.qmd#cdesym.types.core.FeedthroughMatrix) | Feedthrough/direct transmission matrix (ny, nu). | | [types.core.GainMatrix](types.core.GainMatrix.qmd#cdesym.types.core.GainMatrix) | Gain matrix for control or estimation. | | [types.core.CovarianceMatrix](types.core.CovarianceMatrix.qmd#cdesym.types.core.CovarianceMatrix) | Covariance matrix (symmetric, positive semidefinite). | | [types.core.SystemDimensions](types.core.SystemDimensions.qmd#cdesym.types.core.SystemDimensions) | System dimensions as dictionary. | | [types.core.DynamicsFunction](types.core.DynamicsFunction.qmd#cdesym.types.core.DynamicsFunction) | Dynamics function f(x, u). | | [types.core.OutputFunction](types.core.OutputFunction.qmd#cdesym.types.core.OutputFunction) | Output/observation function h(x). | | [types.core.DiffusionFunction](types.core.DiffusionFunction.qmd#cdesym.types.core.DiffusionFunction) | Diffusion function g(x, u) for stochastic systems. | | [types.core.ControlPolicy](types.core.ControlPolicy.qmd#cdesym.types.core.ControlPolicy) | Control policy/controller π(x). | | [types.core.TimeVaryingControl](types.core.TimeVaryingControl.qmd#cdesym.types.core.TimeVaryingControl) | Time-varying control function u(t). | | [types.core.FeedbackController](types.core.FeedbackController.qmd#cdesym.types.core.FeedbackController) | Feedback controller with time awareness π(x, t). | | [types.core.ControlInput](types.core.ControlInput.qmd#cdesym.types.core.ControlInput) | Unified type for control inputs to integration methods. | ## Types: Backend & Configuration Backend support and configuration types | | | | --- | --- | | [types.backends.Backend](types.backends.Backend.qmd#cdesym.types.backends.Backend) | Backend identifier for numerical computation. | | [types.backends.Device](types.backends.Device.qmd#cdesym.types.backends.Device) | Device identifier for hardware acceleration. | | [types.backends.BackendConfig](types.backends.BackendConfig.qmd#cdesym.types.backends.BackendConfig) | Backend configuration dictionary. | | [types.backends.IntegrationMethod](types.backends.IntegrationMethod.qmd#cdesym.types.backends.IntegrationMethod) | Integration method for continuous-time systems (ODEs). | | [types.backends.SDEIntegrationMethod](types.backends.SDEIntegrationMethod.qmd#cdesym.types.backends.SDEIntegrationMethod) | SDE integration method for stochastic differential equations. | | [types.backends.DiscretizationMethod](types.backends.DiscretizationMethod.qmd#cdesym.types.backends.DiscretizationMethod) | Discretization method for continuous → discrete transformation. | | [types.backends.OptimizationMethod](types.backends.OptimizationMethod.qmd#cdesym.types.backends.OptimizationMethod) | Optimization method for control/estimation problems. | | [types.backends.NoiseType](types.backends.NoiseType.qmd#cdesym.types.backends.NoiseType) | Noise structure classification for stochastic systems. | | [types.backends.SDEType](types.backends.SDEType.qmd#cdesym.types.backends.SDEType) | SDE interpretation type. | | [types.backends.ConvergenceType](types.backends.ConvergenceType.qmd#cdesym.types.backends.ConvergenceType) | SDE convergence type for numerical integration. | | [types.backends.IntegratorConfig](types.backends.IntegratorConfig.qmd#cdesym.types.backends.IntegratorConfig) | Configuration for continuous-time integrators. | | [types.backends.SDEIntegratorConfig](types.backends.SDEIntegratorConfig.qmd#cdesym.types.backends.SDEIntegratorConfig) | Configuration for SDE integrators. | | [types.backends.DiscretizerConfig](types.backends.DiscretizerConfig.qmd#cdesym.types.backends.DiscretizerConfig) | Configuration for system discretization. | | [types.backends.SystemConfig](types.backends.SystemConfig.qmd#cdesym.types.backends.SystemConfig) | Complete system configuration dictionary. | | [types.backends.validate_backend](types.backends.validate_backend.qmd#cdesym.types.backends.validate_backend) | Validate and normalize backend string. | | [types.backends.validate_device](types.backends.validate_device.qmd#cdesym.types.backends.validate_device) | Validate device for given backend. | | [types.backends.get_backend_default_method](types.backends.get_backend_default_method.qmd#cdesym.types.backends.get_backend_default_method) | Get default integration method for backend. | ## Types: Trajectories & Simulation Time series, trajectories, and simulation results | | | | --- | --- | | [types.trajectories.TimePoints](types.trajectories.TimePoints.qmd#cdesym.types.trajectories.TimePoints) | Array of time points for simulation or evaluation. | | [types.trajectories.TimeSpan](types.trajectories.TimeSpan.qmd#cdesym.types.trajectories.TimeSpan) | Time interval for continuous integration: (t_start, t_end). | | [types.trajectories.StateTrajectory](types.trajectories.StateTrajectory.qmd#cdesym.types.trajectories.StateTrajectory) | State trajectory over time. | | [types.trajectories.ControlSequence](types.trajectories.ControlSequence.qmd#cdesym.types.trajectories.ControlSequence) | Control input sequence over time. | | [types.trajectories.OutputSequence](types.trajectories.OutputSequence.qmd#cdesym.types.trajectories.OutputSequence) | Output/measurement sequence over time. | | [types.trajectories.NoiseSequence](types.trajectories.NoiseSequence.qmd#cdesym.types.trajectories.NoiseSequence) | Noise/disturbance sequence for stochastic simulation. | | [types.trajectories.SimulationResult](types.trajectories.SimulationResult.qmd#cdesym.types.trajectories.SimulationResult) | Result from continuous-time system simulation. | | [types.trajectories.IntegrationResult](types.trajectories.IntegrationResult.qmd#cdesym.types.trajectories.IntegrationResult) | Result from continuous-time integration (ODE/SDE solver). | | [types.trajectories.TrajectoryStatistics](types.trajectories.TrajectoryStatistics.qmd#cdesym.types.trajectories.TrajectoryStatistics) | Statistical summary of trajectory. | | [types.trajectories.TrajectorySegment](types.trajectories.TrajectorySegment.qmd#cdesym.types.trajectories.TrajectorySegment) | Segment of trajectory between two time points. | ## Types: Control Results Control design and analysis result types | | | | --- | --- | | [types.control_classical.LQRResult](types.control_classical.LQRResult.qmd#cdesym.types.control_classical.LQRResult) | Linear Quadratic Regulator (LQR) design result. | | [types.control_classical.LQGResult](types.control_classical.LQGResult.qmd#cdesym.types.control_classical.LQGResult) | Linear Quadratic Gaussian (LQG) controller design result. | | [types.control_classical.KalmanFilterResult](types.control_classical.KalmanFilterResult.qmd#cdesym.types.control_classical.KalmanFilterResult) | Kalman Filter (optimal state estimator) design result. | | [types.control_classical.StabilityInfo](types.control_classical.StabilityInfo.qmd#cdesym.types.control_classical.StabilityInfo) | Stability analysis result dictionary. | | [types.control_classical.ControllabilityInfo](types.control_classical.ControllabilityInfo.qmd#cdesym.types.control_classical.ControllabilityInfo) | Controllability analysis result. | | [types.control_classical.ObservabilityInfo](types.control_classical.ObservabilityInfo.qmd#cdesym.types.control_classical.ObservabilityInfo) | Observability analysis result. | | [types.control_classical.PolePlacementResult](types.control_classical.PolePlacementResult.qmd#cdesym.types.control_classical.PolePlacementResult) | Pole placement (eigenvalue assignment) result. | | [types.control_classical.LuenbergerObserverResult](types.control_classical.LuenbergerObserverResult.qmd#cdesym.types.control_classical.LuenbergerObserverResult) | Luenberger observer (deterministic state estimator) design result. | | [types.control_advanced.MPCResult](types.control_advanced.MPCResult.qmd#cdesym.types.control_advanced.MPCResult) | Model Predictive Control (MPC) solution result. | | [types.control_advanced.MHEResult](types.control_advanced.MHEResult.qmd#cdesym.types.control_advanced.MHEResult) | Moving Horizon Estimation (MHE) result. | | [types.control_advanced.H2ControlResult](types.control_advanced.H2ControlResult.qmd#cdesym.types.control_advanced.H2ControlResult) | H₂ optimal control result. | | [types.control_advanced.HInfControlResult](types.control_advanced.HInfControlResult.qmd#cdesym.types.control_advanced.HInfControlResult) | H∞ robust control result. | | [types.control_advanced.LMIResult](types.control_advanced.LMIResult.qmd#cdesym.types.control_advanced.LMIResult) | Linear Matrix Inequality (LMI) solver result. | | [types.control_advanced.AdaptiveControlResult](types.control_advanced.AdaptiveControlResult.qmd#cdesym.types.control_advanced.AdaptiveControlResult) | Adaptive control result. | | [types.control_advanced.SlidingModeResult](types.control_advanced.SlidingModeResult.qmd#cdesym.types.control_advanced.SlidingModeResult) | Sliding Mode Control (SMC) result. | ## Types: Estimation State estimation result types | | | | --- | --- | | [types.estimation.EKFResult](types.estimation.EKFResult.qmd#cdesym.types.estimation.EKFResult) | Extended Kalman Filter (EKF) state and result. | | [types.estimation.UKFResult](types.estimation.UKFResult.qmd#cdesym.types.estimation.UKFResult) | Unscented Kalman Filter (UKF) result. | | [types.estimation.ParticleFilterResult](types.estimation.ParticleFilterResult.qmd#cdesym.types.estimation.ParticleFilterResult) | Particle Filter (Sequential Monte Carlo) result. | ## Types: Linearization Linearization and Jacobian types | | | | --- | --- | | [types.linearization.LinearizationResult](types.linearization.LinearizationResult.qmd#cdesym.types.linearization.LinearizationResult) | Flexible linearization result type. | | [types.linearization.DeterministicLinearization](types.linearization.DeterministicLinearization.qmd#cdesym.types.linearization.DeterministicLinearization) | Linearization result for deterministic systems: (A, B). | | [types.linearization.StochasticLinearization](types.linearization.StochasticLinearization.qmd#cdesym.types.linearization.StochasticLinearization) | Linearization result for stochastic systems: (A, B, G). | | [types.linearization.ContinuousLinearization](types.linearization.ContinuousLinearization.qmd#cdesym.types.linearization.ContinuousLinearization) | Alias for continuous-time deterministic linearization. | | [types.linearization.DiscreteLinearization](types.linearization.DiscreteLinearization.qmd#cdesym.types.linearization.DiscreteLinearization) | Alias for discrete-time deterministic linearization. | | [types.linearization.ContinuousStochasticLinearization](types.linearization.ContinuousStochasticLinearization.qmd#cdesym.types.linearization.ContinuousStochasticLinearization) | Alias for continuous-time stochastic linearization. | | [types.linearization.DiscreteStochasticLinearization](types.linearization.DiscreteStochasticLinearization.qmd#cdesym.types.linearization.DiscreteStochasticLinearization) | Alias for discrete-time stochastic linearization. | | [types.linearization.FullLinearization](types.linearization.FullLinearization.qmd#cdesym.types.linearization.FullLinearization) | Complete linearization including output: (A, B, C, D). | | [types.linearization.FullStochasticLinearization](types.linearization.FullStochasticLinearization.qmd#cdesym.types.linearization.FullStochasticLinearization) | Complete stochastic linearization: (A, B, G, C, D). | | [types.linearization.ObservationLinearization](types.linearization.ObservationLinearization.qmd#cdesym.types.linearization.ObservationLinearization) | Observation/output linearization: (C, D). | | [types.linearization.StateJacobian](types.linearization.StateJacobian.qmd#cdesym.types.linearization.StateJacobian) | State Jacobian ∂f/∂x. | | [types.linearization.ControlJacobian](types.linearization.ControlJacobian.qmd#cdesym.types.linearization.ControlJacobian) | Control Jacobian ∂f/∂u. | | [types.linearization.OutputJacobian](types.linearization.OutputJacobian.qmd#cdesym.types.linearization.OutputJacobian) | Output Jacobian ∂h/∂x. | | [types.linearization.DiffusionJacobian](types.linearization.DiffusionJacobian.qmd#cdesym.types.linearization.DiffusionJacobian) | Diffusion Jacobian ∂g/∂x (stochastic systems). | | [types.linearization.LinearizationCacheKey](types.linearization.LinearizationCacheKey.qmd#cdesym.types.linearization.LinearizationCacheKey) | Cache key for linearization results. | ## Types: Symbolic Symbolic computation types | | | | --- | --- | | [types.symbolic.SymbolicExpression](types.symbolic.SymbolicExpression.qmd#cdesym.types.symbolic.SymbolicExpression) | Single symbolic expression. | | [types.symbolic.SymbolicMatrix](types.symbolic.SymbolicMatrix.qmd#cdesym.types.symbolic.SymbolicMatrix) | Matrix of symbolic expressions. | | [types.symbolic.SymbolicStateEquations](types.symbolic.SymbolicStateEquations.qmd#cdesym.types.symbolic.SymbolicStateEquations) | Symbolic state equations: f(x, u, params). | | [types.symbolic.SymbolicJacobian](types.symbolic.SymbolicJacobian.qmd#cdesym.types.symbolic.SymbolicJacobian) | Jacobian matrix: ∂f/∂x. | ## Types: Advanced Reachability, robustness, optimization, and learning types | | | | --- | --- | | [types.reachability.ReachabilityResult](types.reachability.ReachabilityResult.qmd#cdesym.types.reachability.ReachabilityResult) | Reachability analysis result. | | [types.reachability.ROAResult](types.reachability.ROAResult.qmd#cdesym.types.reachability.ROAResult) | Region of Attraction (ROA) analysis result. | | [types.reachability.CBFResult](types.reachability.CBFResult.qmd#cdesym.types.reachability.CBFResult) | Control Barrier Function (CBF) result. | | [types.reachability.CLFResult](types.reachability.CLFResult.qmd#cdesym.types.reachability.CLFResult) | Control Lyapunov Function (CLF) result. | | [types.robustness.RobustStabilityResult](types.robustness.RobustStabilityResult.qmd#cdesym.types.robustness.RobustStabilityResult) | Robust stability analysis result. | | [types.robustness.TubeMPCResult](types.robustness.TubeMPCResult.qmd#cdesym.types.robustness.TubeMPCResult) | Tube-based MPC result. | | [types.optimization.OptimizationResult](types.optimization.OptimizationResult.qmd#cdesym.types.optimization.OptimizationResult) | General nonlinear optimization result. | | [types.optimization.TrajectoryOptimizationResult](types.optimization.TrajectoryOptimizationResult.qmd#cdesym.types.optimization.TrajectoryOptimizationResult) | Trajectory optimization result. | | [types.learning.TrainingResult](types.learning.TrainingResult.qmd#cdesym.types.learning.TrainingResult) | Neural network training result. | | [types.learning.RLTrainingResult](types.learning.RLTrainingResult.qmd#cdesym.types.learning.RLTrainingResult) | Reinforcement learning training result. | | [types.contraction.ContractionAnalysisResult](types.contraction.ContractionAnalysisResult.qmd#cdesym.types.contraction.ContractionAnalysisResult) | Contraction analysis result. | | [types.contraction.CCMResult](types.contraction.CCMResult.qmd#cdesym.types.contraction.CCMResult) | Control Contraction Metrics (CCM) result. | | [types.conformal.ConformalPredictionResult](types.conformal.ConformalPredictionResult.qmd#cdesym.types.conformal.ConformalPredictionResult) | Conformal prediction result for test points. | ## Types: Protocols Structural subtyping protocols for system interfaces | | | | --- | --- | | [types.protocols.ContinuousSystemProtocol](types.protocols.ContinuousSystemProtocol.qmd#cdesym.types.protocols.ContinuousSystemProtocol) | Minimal interface for continuous-time dynamical systems. | | [types.protocols.DiscreteSystemProtocol](types.protocols.DiscreteSystemProtocol.qmd#cdesym.types.protocols.DiscreteSystemProtocol) | Minimal interface for discrete-time dynamical systems. | | [types.protocols.StochasticSystemProtocol](types.protocols.StochasticSystemProtocol.qmd#cdesym.types.protocols.StochasticSystemProtocol) | System with stochastic dynamics (continuous or discrete). | | [types.protocols.LinearizableContinuousProtocol](types.protocols.LinearizableContinuousProtocol.qmd#cdesym.types.protocols.LinearizableContinuousProtocol) | Continuous system with linearization capability. | | [types.protocols.LinearizableDiscreteProtocol](types.protocols.LinearizableDiscreteProtocol.qmd#cdesym.types.protocols.LinearizableDiscreteProtocol) | Discrete system with linearization capability. | ## State Observers State estimation and observer design | | | | --- | --- | | [observers.LinearObserver](observers.LinearObserver.qmd#cdesym.observers.LinearObserver) | Linear state observer with constant gain. | | [observers.ExtendedKalmanFilter](observers.ExtendedKalmanFilter.qmd#cdesym.observers.ExtendedKalmanFilter) | Extended Kalman Filter (EKF) for nonlinear state estimation. |