Mechano Informatics - Chapter 5 - Dynamics of Linear Systems (Part 2)

In this talk

1. 0:01

   State-space models and system behavior

   The lecture recaps the model ẋ = Ax + Bu and output y = Cx + Du, where the state collects variables such as a mass’s position and velocity. The choice of state variables shapes the resulting matrices and representation. A damped spring pendulum illustrates how a trajectory through position-and-velocity space describes the system over time.

2. 9:10

   Laplace transforms and basic signals

   The lecture introduces the Laplace transform as a way to analyze continuous-time, linear time-invariant systems in the complex domain. Examples include the transforms of a constant, a unit step, a ramp, an exponential, and a Dirac impulse. It also derives how differentiation maps to sF(s) minus the initial value, while integration maps to F(s)/s.

3. 35:03

   Solving the state equation and finding transfer functions

   Applying the transform to ẋ = Ax + Bu gives X(s) = (sI − A)⁻¹x(0) + (sI − A)⁻¹BU(s). The matrix inverse (sI − A)⁻¹ is the Laplace-domain form of the state-transition matrix, and inverse transformation recovers the time-domain response. For output y = Cx + Du, the input-output transfer function is C(sI − A)⁻¹B + D; in time, the response involves convolution.

4. 55:51

   Applications and changing state variables

   Examples show the method applied to a loaded DC motor and an inverted pendulum, with matrix inversion and inverse Laplace transforms used to obtain system behavior. The lecture then describes changing coordinates through a linear transformation, producing new A, B, and C matrices while preserving the system’s form. For two spring-coupled masses, center-of-mass position and the distance between the masses offer an alternative set of states.

5. 1:09:48

   Continuous and discrete system perspectives

   Continuous signals can be sampled for digital processing, where discrete-time dynamics use difference equations and the Z-transform. The lecture also points to Fourier methods in signal and image processing. It closes by connecting state-space and Laplace approaches as equivalent ways to analyze system dynamics and compute responses.