Main Content

Mixed
*H*_{2}/*H*_{∞}
synthesis with regional pole placement constraints

`[`

employs LMI techniques to compute an output-feedback control law `K`

,`CL`

,`normz`

,`info`

]
= h2hinfsyn(`P`

,`Nmeas`

,`Ncon`

,`Nz2`

,`Wz`

,`Name,Value`

)*u* =
*K*(*s*)*y* for the control problem of the following illustration.

The LTI plant `P`

has partitioned state-space form given
by

$$\begin{array}{c}\dot{x}=Ax+{B}_{1}w+{B}_{2}u,\\ {z}_{\infty}={C}_{1}x+{D}_{11}w+{D}_{12}u,\\ {z}_{2}={C}_{2}x+{D}_{21}w+{D}_{22}u,\\ y={C}_{y}x+{D}_{y1}w+{D}_{y2}u.\end{array}$$

The resulting controller `K`

:

Keeps the

*H*_{∞}norm*G*of the transfer function from*w*to*z*_{∞}below the value you specify using the`Name,Value`

argument`'HINFMAX'`

.Keeps the

*H*_{2}norm*H*of the transfer function from*w*to*z*_{2}below the value you specify using the`Name,Value`

argument`'H2MAX'`

.Minimizes a trade-off criterion of the form

$${W}_{1}{G}^{2}+{W}_{2}{H}^{2},$$

where

*W*_{1}and*W*_{2}are the first and second entries in the vector`Wz`

.Places the closed-loop poles in the LMI region that you specify using the

`Name,Value`

argument`'REGION'`

.

Use the input arguments `Nmeas`

, `Ncon`

, and
`Nz2`

to specify the number of signals in
*y*, *u*, and
*z*_{2}, respectively. You can use
additional `Name,Value`

pairs to specify additional options for
the computation.

Do not choose weighting functions with poles very close to

*s*= 0 (*z*= 1 for discrete-time systems). For instance, although it might seem sensible to choose*W*= 1/*s*to enforce zero steady-state error, doing so introduces an unstable pole that cannot be stabilized, causing synthesis to fail. Instead, choose*W*= 1/(*s*+*δ*). The value*δ*must be small but not very small compared to system dynamics. For instance, for best numeric results, if your target crossover frequency is around 1 rad/s, choose*δ*= 0.0001 or 0.001. Similarly, in discrete time, choose sample times such that system and weighting dynamics are not more than a decade or two below the Nyquist frequency.

[1] Chilali, M., and P. Gahinet, “H_{∞} Design
with Pole Placement Constraints: An LMI Approach,” *IEEE Trans. Aut.
Contr*., 41 (1995), pp. 358–367.

[2] Scherer, C., “Mixed H2/H-infinity Control,” *Trends in
Control: A European Perspective*, Springer-Verlag (1995),
pp.173–216.