— Function File: hinfdemo ()

H-infinity design demos for continuous SISO and MIMO systems and a discrete system. The SISO system is difficult to control because it is non-minimum-phase and unstable. The second design example controls the jet707 plant, the linearized state space model of a Boeing 707-321 aircraft at v=80 m/s (M = 0.26, Ga0 = -3 deg, alpha0 = 4 deg, kappa = 50 deg). Inputs: (1) thrust and (2) elevator angle Outputs: (1) airspeed and (2) pitch angle. The discrete system is a stable and second order.

SISO plant:

               s - 2
               G(s) = --------------
               (s + 2)(s - 1)

               
               +----+
               -------------------->| W1 |---> v1
               z   |                    +----+
               ----|-------------+
               |             |
               |    +---+    v   y  +----+
               u *--->| G |--->O--*-->| W2 |---> v2
               |    +---+       |   +----+
               |                |
               |    +---+       |
               -----| K |<-------
               +---+

               min || T   ||
               vz   infty

W1 und W2 are the robustness and performance weighting functions.

MIMO plant:

The optimal controller minimizes the H-infinity norm of the augmented plant P (mixed-sensitivity problem):

               w
               1 -----------+
               |                   +----+
               +---------------------->| W1 |----> z1
               w         |   |                   +----+
               2 ------------------------+
               |   |            |
               |   v   +----+   v      +----+
               +--*-->o-->| G  |-->o--*-->| W2 |---> z2
               |          +----+      |   +----+
               |                      |
               ^                      v
               u                       y (to K)
               (from controller K)

               +    +           +    +
               | z  |           | w  |
               |  1 |           |  1 |
               | z  | = [ P ] * | w  |
               |  2 |           |  2 |
               | y  |           | u  |
               +    +           +    +

Discrete system:

This is not a true discrete design. The design is carried out in continuous time while the effect of sampling is described by a bilinear transformation of the sampled system. This method works quite well if the sampling period is “small” compared to the plant time constants.

The continuous plant:

               1
               G (s) = --------------
               k      (s + 2)(s + 1)

is discretised with a ZOH (Sampling period = Ts = 1 second):

               
               0.199788z + 0.073498
               G(z) = --------------------------
               (z - 0.36788)(z - 0.13534)

               
               +----+
               -------------------->| W1 |---> v1
               z   |                    +----+
               ----|-------------+
               |             |
               |    +---+    v      +----+
               *--->| G |--->O--*-->| W2 |---> v2
               |    +---+       |   +----+
               |                |
               |    +---+       |
               -----| K |<-------
               +---+

               min || T   ||
               vz   infty

W1 and W2 are the robustness and performance weighting functions.