Mechano Informatics - Exercise 3 - Dynamics of Linear Systems
A lecturer from HumanoidRobots explains how to model linear dynamical systems with state-space equations and Laplace transforms, using mechanical, robotic, and electrical examples.
In this talk
- 0:06
State-space models and a damped mass
Linear time-invariant systems can be represented with state and output equations using matrices A, B, C, and D. For a mass connected to a spring and damper, position and velocity are the states, and applied force is the input. Newton’s law yields the state equations; the output can be set equal to the state vector.
- 10:06
Coupled masses and an inverted pendulum
A two-mass system illustrates that state variables can be chosen in different ways, such as using one mass’s position and the distance between the masses. The examples then extend to an inverted pendulum, which can approximate Segway balance or aspects of humanoid walking. Its equations are nonlinear, so a small-angle approximation makes a linear state-space model possible.
- 29:32
An RLC circuit and Laplace methods
For a parallel RLC circuit, the inductor current and capacitor voltage serve as states, with incoming current as the input. The lecture derives their first-order equations from Kirchhoff’s current rule and the component relations. It then uses Laplace transforms and partial fractions to solve a third-order differential equation with a ramp input, explaining how initial conditions appear in the transformed equation.
- 49:10
Applications of the Laplace transform
Radioactive decay gives an exponential response, with atom count decreasing at a rate set by the decay constant. The transform also gives a capacitor’s impedance as 1/(Cs). For a resistor-capacitor circuit, the current decays exponentially, with a time constant determined by RC.
- 57:11
Using the models in practice
State-space models support analysis, controller design, and numerical simulation. The examples’ MATLAB code lets students vary masses, spring constants, damping, and inputs to inspect system behavior. The lecturer closes by noting that Laplace-domain methods can make circuit calculations easier to follow than working directly with time-domain integrals.
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