MIMO systems Systems with more than one input and output A system - - PowerPoint PPT Presentation

MIMO systems

Systems with more than one input and output

A system with Multiple Inputs and Multiple Outputs is called a MIMO system. A system with m inputs and p outputs can be represented as    Y1(s) . . . Yp(s)   

• =Y (s)

=    G11(s) . . . G1m(s) . . . . . . Gp1(s) . . . Gpm(s)   

• =G(s)

   U1(s) . . . Um(s)   

• =U(s)

If m = p we say that the system is square.

1 / 4 hans.rosth@it.uu.se Discrete-time

MIMO systems

Be careful!

Fr G Fy

• r

+ u y − The closed loop system: Y (s) = G(s)(Fr(s)R(s) − Fy(s)Y (s)) (I + G(s)Fy(s))Y (s) = G(s)Fr(s)R(s) Y (s) = (I + G(s)Fy(s))−1G(s)Fr(s)

• =Gc(s)

R(s) Sensitivity function: S(s) = (I + G(s)Fy(s))−1 Complementary sensitivity function: T(s) = I−S(s) = S(s)(S(s)−1−I) = (I+G(s)Fy(s))−1G(s)Fy(s)

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MIMO systems

Straightforward with state space representation

Almost all formulas and results look the same for SISO and MIMO systems:

• ˙

x = Ax + Bu, y = Cx + Du G(s) = C(sI − A)−1B + D Poles: 0 = det(sI − A) x(t) = eA(t−t0)x(t0) + t

t0

eA(t−τ)Bu(τ)dτ S =

• B

AB . . . An−1B

• ,

O =      C CA . . . CAn−1      Controllability and observability ⇔ S and O of full rank.

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MIMO systems

How to put up a state space representation

Not obvious how to set up a state space model for a given MIMO G(s). In general the controller and observer canonical forms do not work. Two exceptions:

◮ SIMO — one input: Controller canonical form works ◮ MISO — one output: Observer canonical form works

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