1

 H_{2} optimal control
 stability margins of H_{2} controllers

2

 Consider a general LFT system
 Assumptions:
 (i) (A,B_{2}) is stabilizable and (C_{2},A) is
detectable;
 (ii) D_{12 } has full
column rank with [D_{12 } D_{^}] unitary, and
 D_{21} has
full row rank with
unitary;
 (iii) has
full column rank for all w;
 (iv) has
full row rank for all w.

3

 H_{2} Problem: find a stabilizing controller K that minimizes T_{zw}_{2}
 Let X_{2} and Y_{2} be stabilizing solutions to
 There exists a unique optimal controller
 Moreover,
 min T_{zw}_{2}^{2
}=^{ }G_{c}B_{1}_{2}^{2 }+F_{2}G_{f
}_{2}^{2 }=^{
}G_{c}L_{2}_{2}^{2 }+C_{1}G_{f}_{2}^{2}

4

 Proof: Note that
 T_{zw}_{2}^{2 }=^{ }G_{c}B_{1}_{2}^{2
}+F_{2}G_{f }QV_{ }_{2}^{2 }=^{ }G_{c}B_{1}_{2}^{2
}+F_{2}G_{f }_{2}^{2 }+Q_{2}^{2}
 And Q = 0 gives the unique optimal control: K=F_{l}(M_{2},0).

5

 LQR margin: ³ 60^{o}
phase margin and ³ 6dB gain margin.
 LQG or H_{2} Controller:
No guaranteed margin.
 Example:

6

 Suppose the controller implemented in the system (or plant G_{22})
is actually
 K=kK_{opt} ,
 with a nominal value k=1. Then the closedloop system Amatrix becomes
 The characteristic polynomial has the form
 With a_{1}= a+b4+2(k1)ab, a_{0}=1+(1k)ab
  necessary for stability: a_{0}>0
and a_{1}>0.
 a>>1 and b>>1 and k¹1 Þ a_{0 }» (1k)ab, and a_{1 }» 2(k1)ab
 a>>1 and b>>1 (q and s), the system is unstable for arbitrarily small
perturbations in k in either direction. Thus, by choice of q and s the gain margins may
be made arbitrarily small.

7

 It is interesting to note that the margins deteriorate as control
weight (1/q) gets small (large q) and/or system driving noise gets large
(large s). In
modern control folklore, these have often been considered ad hoc means
of improving sensitivity.
 H_{2} (LQG) controllers have no global systemindependent guaranteed robustness properties.
 Improve the robustness of a given
design by relaxing the optimality of the filter with respect to error
properties. LQG loop transfer recovery (LQG/LTR) design technique. The
idea is to design a filtering gain in such way so that the LQG (or H_{2})
control law will approximate the loop properties of the regular LQR
control.
