Linear systems

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Linear systems

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 u_i applied to each joint with measured angular displacements y_i:

Setting v_i := i th angular velocity, this evolves according to the linearized Newton equation:

dv_i /dt = I_i u_i

dy_i /dt = v_i

More generally, consider a physical system with m inputs and p outputs, which we model as vectors u in R^m and y in R^p. 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.