Mechano Informatics - Chapter 5 - Dynamics of Linear Systems

In this talk

1. 2:22

   Why model dynamic systems

   Dynamic systems include processes that change over time, from pendulums and robot joints to weather and planetary motion. Mathematical models let engineers examine properties such as stability and robustness, and provide a basis for designing control systems.

2. 4:12

   Feedback control in practice

   A controller compares a measured output with a desired value and adjusts the input to reduce the difference, even when disturbances are not fully known. Examples include keeping a robot joint at a target speed despite changing loads, holding a room near 19 degrees despite an open window, and steering a car to stay in its lane.

3. 23:22

   State-space models and their solutions

   For continuous systems, the lecture uses differential equations and represents linear dynamics as a state equation with matrices A and B. For constant A, the state transition matrix is the matrix exponential; the resulting state depends on the initial condition and on accumulated inputs. The output can depend on the state alone or on both state and input.

4. 1:02:39

   Turning Newton’s law into a model

   For a mass acted on by force, the chosen state variables are position and velocity, while force divided by mass supplies the acceleration input. This gives a two-variable state-space model with A = [[0, 1], [0, 0]] and B = [[0], [1]]. The example shows how a second-order physical equation can be rewritten as a first-order system.

5. 1:14:13

   Properties of the transition matrix

   The transition matrix starts at the identity when the initial and target times match, and it composes across intervals: the transition from one time to another can be found by multiplying the matrices for intermediate steps. It remains invertible even when A is singular, allowing states to be related across time.