The *H*_{2} optimal control
theory has its roots in the frequency domain interpretation the cost
function associated with time-domain state-space LQG control theory [1]. The equations
and corresponding nomenclature used here are taken from the Doyle *et
al.*, 1989 [2]-[3].

`h2syn`

solves the *H*_{2} optimal
control problem by observing that it is equivalent to a conventional
Linear-Quadratic Gaussian (LQG) optimal control problem. For simplicity, we shall
describe the details of algorithm only for the continuous-time case,
in which case the cost function *J*_{LQG} satisfies

$$\begin{array}{c}{J}_{LQG}=\underset{T\to \infty}{\mathrm{lim}}E\left\{\frac{1}{T}{\displaystyle {\int}_{0}^{T}{y}_{1}^{T}{y}_{1}dt}\right\}\\ =\underset{T\to \infty}{\mathrm{lim}}E\left\{\frac{1}{T}{\displaystyle {\int}_{0}^{T}\left[{x}^{T}{u}_{2}^{T}\right]\left[\begin{array}{cc}Q& {N}_{c}\\ {N}_{c}^{T}& R\end{array}\right]\left[\begin{array}{c}x\\ {u}_{2}\end{array}\right]dt}\right\}\\ =\underset{T\to \infty}{\mathrm{lim}}E\left\{\frac{1}{T}{\displaystyle {\int}_{0}^{T}\left[{x}^{T}{u}_{2}^{T}\right]\left[\begin{array}{c}{C}_{1}^{T}\\ {D}_{12}^{T}\end{array}\right]\left[\begin{array}{cc}{C}_{1}& {D}_{12}\end{array}\right]\left[\begin{array}{c}x\\ {u}_{2}\end{array}\right]dt}\right\}\end{array}$$

with plant noise *u*_{1} channel
of intensity I, passing through the matrix [B1;0;D12] to produce equivalent
white correlated with plant ξ and white measurement noise θ
having joint correlation function

$$\begin{array}{c}E\left\{\left[\begin{array}{c}\xi (t)\\ \theta (t)\end{array}\right]{\left[\begin{array}{cc}\xi (\tau )& \theta (\tau )\end{array}\right]}^{T}\right\}=\left[\begin{array}{cc}\Xi & {N}_{f}\\ {N}_{f}^{T}& \Theta \end{array}\right]\delta (t-\tau )\\ =\left[\begin{array}{c}{B}_{1}\\ {D}_{21}\end{array}\right]\left[\begin{array}{cc}{B}_{1}^{T}& {D}_{21}^{T}\end{array}\right]\delta (t-\tau )\end{array}$$

The *H*_{2} optimal controller *K*(*s*)
is thus realizable in the usual LQG manner as a full-state feedback *K*_{FI} and
a Kalman filter with residual gain matrix *K*_{FC}.

**Kalman Filter**

$$\begin{array}{l}\dot{\widehat{x}}=A\widehat{x}+{B}_{2}{u}_{2}+{K}_{FC}({y}_{2}-{C}_{2\widehat{x}}-{D}_{22}{u}_{2})\\ {K}_{FC}=(Y{C}_{2}^{T}+{N}_{f}){\Theta}^{-1}=(Y{C}_{2}^{T}+{B}_{1}{D}_{21}^{T}){(}^{{D}_{21}}\end{array}$$

where *Y* = *Y*^{T}≥0
solves the Kalman filter Riccati equation

$$Y{A}^{T}+AY-(Y{C}_{2}^{T}+{N}_{f}){\Theta}^{-1}({C}_{2}Y+{N}_{f}^{T})+\Xi =0$$

**Full-State Feedback**

$$\begin{array}{l}{u}_{2}={K}_{FI}\widehat{x}\\ {K}_{FI}={R}^{-1}({B}_{2}^{T}X+{N}_{c}^{T})={D}_{12}^{T}{D}_{12}{)}^{-1}({B}_{2}^{T}X+{D}_{12}^{T}{C}_{1})\end{array}$$

where *X* = *X*^{T}≥0
solves the state-feedback Riccati equation

$${A}^{T}X+XA-(X{B}_{2}+{N}_{c}){R}^{-1}({B}_{2}^{T}X+{N}_{c}^{T})+Q=0$$

The final *positive*-feedback *H*_{2}^{ }optimal
controller $${u}_{2}=K(s){y}_{2}$$ has a familiar closed-form

$$K(s):=\left[\begin{array}{cc}A-{K}_{FC}{C}_{2}-{B}_{2}{K}_{FI}+{K}_{FC}{D}_{22}{K}_{FI}& {K}_{f}\\ -{K}_{FI}& 0\end{array}\right]$$

`h2syn`

implements the continuous optimal *H*_{2} control
design computations using the formulae described in the Doyle, *et
al. *[2]; for discrete-time plants, `h2syn`

uses
the same controller formula, except that the corresponding discrete
time Riccati solutions (dare) are substituted for *X* and *Y*.
A Hamiltonian is formed and solved via a Riccati equation. In the
continuous-time case, the optimal *H*_{2}-norm
is infinite when the plant *D*_{11} matrix
associated with the input disturbances and output errors is *non-*zero;
in this case, the optimal *H*_{2} controller
returned by `h2syn`

is computed by first setting *D**11* to
zero.

**Optimal Cost GAM**

The full information (FI) cost is given by the equation $${\left(\text{trace}({{B}^{\prime}}_{1}{X}_{2}{B}_{1})\right)}^{{}^{\frac{1}{2}}}$$. The output estimation cost
(OEF) is given by $${\left(\text{trace}({F}_{2}{Y}_{2}{{F}^{\prime}}_{2})\right)}^{{}^{\frac{1}{2}}}$$, where
. The disturbance feedforward cost
(DFL) is $${\left(\text{trace}({{L}^{\prime}}_{2}{X}_{2}{L}_{2})\right)}^{{}^{\frac{1}{2}}}$$, where *L*_{2} is
defined by $$-({Y}_{2}{{C}^{\prime}}_{2}+{B}_{1}{{D}^{\prime}}_{21})$$ and the full control cost (FC)
is given by $${\left(\text{trace}({C}_{1}{Y}_{2}{{C}^{\prime}}_{1})\right)}^{{}^{\frac{1}{2}}}$$. *X*_{2} and *Y*_{2} are
the solutions to the *X* and *Y* Riccati
equations, respectively. For for continuous-time plants with zero
feedthrough term (`D11`

= 0), and for all discrete-time
plants, the optimal *H*_{2} cost *γ* =
is

GAM =sqrt(FI^2 + OEF^2+ trace(D11*D11'));

otherwise, `GAM`

= `Inf`

.