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Linear systems Previous Next Contents

1.ii. Linear systems

(For a nice survey of linear systems and the pole placement problem, and methods developed to study it, see [By].)

Consider the following problem:
We wish to understand and control the behaviour of a (mechanical) linkage driven by torques ui applied to each joint with measured angular displacements yi:
Setting vi := i th angular velocity, this evolves according to the linearized Newton equation:
dvi /dt   =   Ii ui
dyi /dt   =   vi

   More generally, consider a physical system with m inputs and p outputs, which we model as vectors u in Rm and y in Rp. If this system is linear, or is near equilibrium, then there are n internal states x, which we consider as a vector in Rn, such that the system is governed by a first order linear evolution equation:
dx/dt  =   Ax + Bu
        y  =   Cx

We represent this schematically:
The Fourier transform of the first equation gives sx = Ax + Bu.  If we solve this for x and substitute into the second equation, we obtain:
y = C(sI - A)-1Bu.
The multiplier C(sI - A)-1B is called the transfer function. This p by m-matrix of rational functions determines the response of the measured quantities y in terms of the inputs u, in the frequency domain.
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