1
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- H2 optimal control
- stability margins of H2 controllers
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2
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- Consider a general LFT system
- Assumptions:
- (i) (A,B2) is stabilizable and (C2,A) is
detectable;
- (ii) D12 has full
column rank with [D12 D^] unitary, and
- D21 has
full row rank with
unitary;
- (iii) has
full column rank for all w;
- (iv) has
full row rank for all w.
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3
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- H2 Problem: find a stabilizing controller K that minimizes ||Tzw||2
- Let X2 and Y2 be stabilizing solutions to
- There exists a unique optimal controller
- Moreover,
- min ||Tzw||22
= ||GcB1||22 +||F2Gf
||22 =
||GcL2||22 +||C1Gf||22
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4
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- Proof: Note that
- ||Tzw||22 = ||GcB1||22
+||F2Gf -QV ||22 = ||GcB1||22
+||F2Gf ||22 +||Q||22
- And Q = 0 gives the unique optimal control: K=Fl(M2,0).
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5
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- LQR margin: ³ 60o
phase margin and ³ 6dB gain margin.
- LQG or H2 Controller:
No guaranteed margin.
- Example:
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6
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- Suppose the controller implemented in the system (or plant G22)
is actually
- K=kKopt ,
- with a nominal value k=1. Then the closed-loop system A-matrix becomes
- The characteristic polynomial has the form
- With a1= a+b-4+2(k-1)ab, a0=1+(1-k)ab
- - necessary for stability: a0>0
and a1>0.
- -a>>1 and b>>1 and k¹1 Þ a0 » (1-k)ab, and a1 » 2(k-1)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.
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7
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- 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.
- H2 (LQG) controllers have no global system-independent 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 H2)
control law will approximate the loop properties of the regular LQR
control.
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