H_{2} control synthesis for LTI plant
[K,CL,GAM,INFO] = H2SYN(P,NMEAS,NCON)
[K,CL,GAM,INFO] = H2SYN(P,NMEAS,NCON)
computes a stabilizing
H_{2} optimal controller K
for
a partitioned LTI plant P
:
H_{2} control
system CL = lft(P,K)
:
The LTI system P
is partitioned such that inputs to
B_{1} are the disturbances, inputs to
B_{2} are the control inputs, output of
C_{1} are the errors to be kept small, and
outputs of C_{2} are the output measurements
provided to the controller. B_{2} has column size
(NCON
) and C_{2} has row size
(NMEAS
). The controller, K
, is a state-space
(ss
) model and has the same number of states as
P
.
If P
is constructed with mktito
, you can omit NMEAS
and NCON
from
the arguments.
The closed-loop system is returned in CL
and
the achieved H_{2} cost γ in GAM
. INFO
is
a struct
array that returns additional information
about the design.
Output Arguments | Description |
---|---|
K | LTI controller |
CL= lft(P,K) | LTI closed-loop system $$T{y}_{1}{u}_{1}$$ |
GAM = norm(CL) | H_{2} optimal cost γ = $$\begin{array}{l}\Vert {T}_{{y}_{1}{u}_{1}}\Vert 2\\ \end{array}$$ |
INFO | Additional output information |
Additional output — structure array INFO
containing
possible additional information depending on METHOD
)
INFO.NORMS | Norms of four different quantities, full information control cost (FI), output estimation cost (OEF), direct feedback cost (DFL) and full control cost (FC). NORMS = [FI OEF DFL FC]; |
INFO.KFI | Full-information gain matrix (constant feedback) $${u}_{2}(t)={K}_{FI}x(t)$$ |
INFO.GFI | Full-information closed-loop system |
INFO.HAMX | X Hamiltonian matrix (state-feedback) |
INFO.HAMY | Y Hamiltonian matrix (Kalman filter) |
h2syn
computes the H_{2}
optimal controller. The standard implicit assumptions for the existence of an optimal
solution are:
(A, B_{2}, C_{2}) are stabilizable and detectable.
P_{12} and P_{21} have no zeros on the imaginary axis (continuous time) or unit circle (discrete time).
D_{12} has full column rank and D_{21} has full row rank. In other words, P_{12} and P_{21} have no zeros at infinity.
If the second or third of these conditions are violated, then h2syn
makes slight adjustments to the plant data to enforce these conditions. This process is
called regularization. It allows h2syn
to
compute the optimal controller for a nearby problem even when the implicit assumptions of
H_{2} synthesis are not all satisfied.
[1] Safonov, M.G., A.J. Laub, and G. Hartmann, “Feedback Properties of Multivariable Systems: The Role and Use of Return Difference Matrix,” IEEE Trans. of Automat. Contr., AC-26, pp. 47-65, 1981.