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Automation for Robotics by Luc Jaulin

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3

Linear Systems

The study of linear systems [BOU 06] is fundamental for the proper understanding of the concepts of stability and the design of linear controllers. Let us recall that linear systems are of the form:

images/ch3_image_1_3.jpg

for continuous-time systems and:

images/ch3_image_1_5.jpg

for discrete-time systems.

3.1. Stability

A linear system is stable (also called asymptotically stable in the literature) if, after a sufficiently long period of time, the state no longer depends on the initial conditions, no matter what they are. This means (see Exercises 3.1 and 3.2) that:

images/ch3_image_1_9.jpg

images/ch3_image_1_10.jpg

In this expression, we can see the concept of matrix exponential. The exponential of a square matrix M of dimension n can be defined through its integer series development:

images/ch3_image_2_1.jpg

where In is the identity matrix of dimension n. It is clear that eM is of the same dimension as M. Here are some of the important properties concerning the exponentials of matrices. If 0n is the zero matrix of n × n and if M and N are two matrices n × n, then:

CRITERION OF STABILITY.– There is a ...

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