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Linear Algebra

  • 1. Introduction to Linear Algebra
  • 2. Matrices
  • 3. Matrix factorizations
    • 3.3. Schur decomposition
    • 3.4. Cholesky decomposition
    • 3.5. Jordan normal form
    • 3.6. Singular Value Decomposition
  • 4. Linear Systems
  • 5. Spectral decomposition
    • 5.1. Spectral decomposition of symmetric matrices
    • 5.2. Sensitivity of spectral decomposition
  • 6. Cayley-Hamilton theorem

Multivariable Calculus

  • 7. Introduction to multi-variable calculus

Differential Geometry

  • 8. Introduction to Differential Geometry

Vector and Tensor Algebra and Calculus

  • 9. Tensor Algebra
  • 10. Tensor Calculus in Euclidean Spaces
    • 10.5. Tensor Calculus in Euclidean Spaces - Cartesian coordinates in \(E^3\)
    • 10.6. Tensor Calculus in Euclidean Spaces - cylindrical coordinates in \(E^3\)
    • 10.7. Tensor Calculus in Euclidean Spaces - Spehrical coordinates in \(E^3\)
  • 11. Tensor Invariants
  • 12. Unitary and rotation tensors
  • 13. Isotropic Tensors
  • 14. Time derivative of integrals over moving domains
  • 15. Calculus identities

Functional Analysis

  • 16. Introduction to Functional Analysis
  • 17. Dirac’s delta

Complex Calculus

  • 18. Complex Algebra
  • 19. Complex Analysis
  • 20. Laplace Transform
    • 20.1. Definition and Properties
    • 20.2. Applications of Laplace Transform
  • 21. Fourier Transforms
    • 21.1. Fourier Series
    • 21.2. Fourier Transform
    • 21.3. Relations between Fourier transforms
    • 21.4. Practice on Fourier analysis on finite-time discrete-time signal
  • 22. Z-transform

Approximation of Functions

  • 23. Introduction to function approximation
  • 24. Lagrange interpolation

Calculus of Variations

  • 25. Introduction to Calculus of Variations

Ordinary Differential Equations

  • 26. Introduction to Ordinary Differential Equations
  • 27. Linear equations
  • 28. Linear systems
  • 29. Integration schemes for Cauchy problems
    • 29.1. Foundations
    • 29.2. Families of integration methods
    • 29.3. Stability of integration schemes
    • 29.4. Simplectic integrators
    • 29.5. Energy conservation of integration schemes
    • 29.6. Time-integration of a mechanical system

System Theory

  • 30. Introduction to System Theory
  • 31. Linear Time-Invariant Systems
  • 32. Coordinate transformation
  • 33. LTI system response
  • 34. LTI: stability and feedback
  • 35. Observability and Detectability
  • 36. Reachability and Controllability
  • 37. State-space realizations
  • 38. Kalman decomposition
  • 39. Lyapunov equation
  • 40. Order reduction of dynamical systems
    • 40.1. Balanced truncation
  • 41. Shape filter
  • 42. Padé approximation of time delay

Control Theory

  • 43. Introduction to control methods
  • 44. Closed-loop control: requirements and performance
    • 44.1. Stability (SISO)
    • 44.2. Reference tracking (SISO)
  • 45. Frequency domain control
  • 46. Optimal control
    • 46.1. Full-state feedback
      • 46.1.1. Generic ODE without exogenous inputs
      • 46.1.2. Linear system without exogenous inputs
      • 46.1.3. Linear system without exogenous inputs - infinite time horizon
    • 46.2. Full-state feedback (OLD)
    • 46.3. Phase and gain margin of optimal control
    • 46.4. Optimal observer for stochastic disturbance
    • 46.5. Optimal observer for deterministic disturbances
    • 46.6. Sub-optimal control for output feedback
    • 46.7. Combination of controller and observer
    • 46.8. Optimal control for reference tracking
      • 46.8.3. Example of optimal control for tracking reference input
    • 46.9. Kalman filter
    • 46.10. Hamilton Bellman Jacobi equation
  • 47. Pole placement
  • 48. Examples
    • 48.1. First order system - Reference tracking
    • 48.2. First order system w/ time delay - Reference tracking
    • 48.3. Inverted pendulum
    • 48.4. Inverted pendulum on a cart
    • 48.5. Inverted pendulum on a cart - only force actuation

Partial Differential Equations

  • 49. Introduction to Partial Differential Equations
  • 50. Elliptic equations
  • 51. Parabolic equations
  • 52. Hyperbolic equations
    • 52.1. Hyperbolic problems and conservation laws
    • 52.2. Hyperbolic problems - dimensions
    • 52.3. Scalar linear equation
    • 52.4. Linear vector equation
    • 52.5. Scalar non-linear equation
    • 52.6. Method of characteristics
    • 52.7. Hyperbolic problems in multi-dimensional domains
      • 52.7.1. General hyperbolic problem
      • 52.7.2. P-system
      • 52.7.3. Euler equations
      • 52.7.4. Shallow water
    • 52.8. Riemann problems
    • 52.9. Physical solution in hyperbolic problems
    • 52.10. Convexity in hyperbolic problems
    • 52.11. Non-convex hyperbolic problems
  • 53. Navier-Cauchy equations
  • 54. Navier-Stokes equations
  • 55. Arbitrary Lagrangian-Eulerian description

Numerical Methods for PDEs

  • 56. Introduction to numerical methods for PDEs
  • 57. Finite Element Method
    • 57.1. 1-dimensional Poisson equation
  • 58. Finite Volume Method
    • 58.1. 1-dimensional Poisson equation
    • 58.2. FVM for hyperbolic problems
    • 58.3. Boundary conditions in hyperbolic problems
    • 58.4. Examples of FVM for hyperbolic problems
      • 58.4.1. 1-dimensional P-system
      • 58.4.2. 1-dimensional Euler equations for Perfect Ideal Gas
      • 58.4.3. Quasi 1-dimensional Euler equations for Perfect Ideal Gas
      • 58.4.4. 1-dimensional Euler equations for Perfect Ideal Gas on moving mesh (ALE)
  • 59. Boundary Element Method

Boundary Methods for PDEs

  • 60. Green’s function method

Optimization

  • 61. Optimization

Reinforcement Learning

  • 62. Introduction to Reinforcement Learning
  • 63. Markov Processes
  • 64. Methods of solution of MPD: DP and LP
  • 65. Methods of solution of MPD: RL
  • 66. Large or Continuous MDPs
  • Repository
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  • .md

Examples

48. Examples#

List of examples of scripts

Frequency domain.

Optimal control.

  • First-order stable system with optimal control for reference tracking

  • First-order stable system with time delay optimal control for reference tracking

  • Mass-damper-spring system. Second-order stable system with optimal control for reference tracking

  • Inverted pendulum

  • Inverted pendulum on cart, with force and torque actuation

  • Inverted pendulum on cart, with only force actuation

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47. Pole placement

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48.1. First order system - Reference tracking

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