# Multivariable control via Transfer Matrices?

• posted

reading my old copy of 'Modern Control Engineering' by Ogata, 1970, (ok, no longer 'modern'), pg 117 describes a method of multivariable control using a 'transfer matrix' of Laplace transfer functions where Gij(s) is the transfer function from ith input to jth output.

Would this be practical to actually implement? Assuming I could model each Gij(s) as a simple 2nd order function, and only need 10Hz response, and have 5 inputs (thermocouples) and 5 outputs (heaters), could one DSP physically handle the Z transform matrix inversions in

10Hz realtime?

Or, is there a simpler way?

tia!

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• posted

The problem with the transfer matrix approach is that it isn't clear if the states of two separate transfer functions are coupled or not. So a system that looks like

[ 1 ] x(n+1) = b x(n) + (1-b) u(n), y(n) = [ ] x(n) [ 1 ]

has the same transfer function matrix as

[b 0] [1-b] x(n+1) = [ ] x(n) + [ ] u(n), y(n) = x(n). [0 b] [1-b]

I much prefer a state-space approach to design here, because (a) it clears up such ambiguities and (b) it makes it easier to build in nonlinearities if necessary.

At a 10Hz sampling rate you don't need no stinking DSP. You should be able to implement 5 2nd-order thermal control loops with an 8-bit processor using all integer math. With a 16-bit DSP you can do this using high-level code and floating point.

• posted

Since all of the denominators would be identical, (the characteristic equation), then no, it would not be efficient to duplicate that for each of the numerators, unless you had a lot of pole/zero cancellations.

10Hz on a DSP? 5x5 matrix inversion? no problem.

By your description I'm assuming maybe the input is really five temperature errors calculated from 5 reference values and 5 thermocouples and that this is a control system. In a State Space format, the matrices are of size A (2x2), B (2x5), C(5x2), D(5x5). and it probably implements a PI type of control with feedforward if the D term is nonzero. I'm guessing at the details because you haven't provided them. But there isn't any reason why you couldn't implement this as a discrete state space controller. Depending upon the precison that you need, at 10 Hz, you could run this on an 4-10MHz

8-bit PIC. You don't need a DSP to implement this control.

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