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 Robust stabilization of Coprime factors
 Robust Stabilization of Normalized Coprime Factors
 H_{¥} Loop
Shaping Design
 Weighted H_{¥} Control Interpretation
 Optimal Stability Margin
 Further Guidelines for Loop Shaping

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 Given nominal model P(s).
 (1) Loop Shaping: Obtain a desired openloop shape (singular values) by
using a precompensator W_{1} and/or a postcompensator W_{2},
 P_{s}=W_{2}PW_{1}
 Assume that W_{1} and W_{2} are such that P_{s}
contains no hidden modes.
 (2) (a) Calculate robust stability margin b_{opt}(P_{s}).
If b_{opt}(P_{s}) <<1, return to (1) and adjust W_{1}
and W_{2 }. (b) Select e £ b_{opt}(P_{s})
, then synthesize a stabilizing controller K_{¥} which satisfies
 (3) The final controller K=W_{1}K_{¥}W_{2}

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 A typical design works as follows: the designer inspects the openloop
singular values of the nominal plant, and shapes these by pre and/or
postcompensation until nominal performance (and possibly robust
stability) specifications are met. (Recall that the openloop shape is
related to closedloop objectives.) A feedback controller K_{¥} with
associated stability margin (for the shaped plant) e £ b_{opt}(P_{s})
is then synthesized. If b_{opt}(P_{s}) is small, then
the specified loop shape is incompatible with robust stability
requirements, and should be adjusted accordingly, then K_{¥} is
reevaluated.
 Note that the final controller is K=W_{1}K_{¥}W_{2 }, so it is necessary to check if
the loop properties are significantly changed. It is helpful to choose W_{1}
and W_{2} with small condition numbers.
 Only W_{1} or W_{2} is needed if P is SISO.

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 >> b_{p,k} = emargin (P,K); % given P and K, compute b_{P,K}
 >> [K_{opt}, b_{p,k} ]=ncfsyn(P,1); % find the
optimal controller K_{opt}.
 >> [K_{sub}, b_{p,k} ] = ncfsyn(P,2); % find a
suboptimal controller K_{sub}.

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 Let P = NM^{1} be normalized right coprime factorization. Then
 small l(P) Þ small b_{opt}(P)
 Let the open righthalf plane zeros and poles of P be:
 z_{1}, z_{2},…,
z_{m}, p_{1}, p_{2},…,
p_{k}
 Define
 Then P(s)=P_{0}(s)N_{z}(s)/N_{p}(s)
 where P_{0}(s) has no open righthalf plane zeros and poles.

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 Let N_{0}(s) and M_{0}(s) be stable and minimum phase
spectral factors:
 Then P_{0}= N_{0} / M_{0} is a normalized coprime factorization
and (N_{0} N_{z}) and (M_{0} N_{p}) form
a pair of normalized coprime factorizations of P.
 Thus
 where r>0, p/2<q< p/2, and

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 K_{q}(w/r) large near w=r: N_{0}(re^{j}^{q}) will be
small if P(jw) is
small near w=r
and M_{0}(re^{j}^{q}) will be small if P(jw) is large near w=r.
 Large q: K_{q}(w/r) very large near w=r and small otherwise. Hence N_{0}(re^{j}^{q}) and M_{0}(re^{j}^{q}) will
essentially be determined by P(jw) in a very narrow frequency range near w=r when q is large.
 On the other hand, when q is small, a larger range of frequency response P(jw) around w=r will have affect on
the value N_{0}(re^{j}^{q}) and M_{0}(re^{j}^{q})(This, in
fact, will imply that a rightplane zero (pole) with a much larger real
part than the imaginary part will have much worse effect on the
performance than a rightplane zero (pole) with a much larger imaginary
part than the real part).

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 Recall (b_{opt}(P))
^{2} £N_{0}(s)N_{z}(s)^{2}
+M_{0}(s)N_{p}(s)^{2} " Re(s)>0
 Let s=re^{j}^{q} and note that N_{z}(z_{i})=0 and
N_{p}(p_{j})=0. Then the bound can be small if
 N_{z}(s) and N_{p}(s) are both small for some s.
That is, N_{z}(s)»0 (i.e., s is close to a righthalf plane zero of
P) and N_{p}(s)»0 (i.e., s is close to a righthalf plane pole of
P).
 This is possible if P(s) has a righthalf plane zero near a righthalf
plane pole. (See Example 16.1.)
 N_{z}(s) and M_{0}(s) are both small for some s.
That is, N_{z}(s)»0 (i.e., s
is close to a righthalf plane zero of P) and M_{0}(s)»0 (i.e., P(jw) is large around w=s=r ).
 This is possible if P(jw) is large around w=r where r is the modulus of a righthalf plane
zero of P (See Example16.2)

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 N_{p}(s) and N_{0}(s) are both small for some s.
That is, N_{p}(s)»0 (i.e., s is close to a righthalf plane pole of
P) and N_{0}(s)»0 (i.e., P(jw) is small around w=s=r).
 This is possible if P(jw) is small around w=r where r is the modulus of a righthalf plane
pole of P (See Example 16.3)
 N_{0}(s) and M_{0}(s) are both small for some s.
That is, N_{0}(s)»0 (i.e., P(jw) is small around w=s=r) and M_{0}(s)»0 (i.e., P(jw) is large around w=s=r).
 The only way in which P(jw) can be both small and large at frequencies
near w=r is
that P(jw)
is approximately equal to 1 and the absolute value of the slope of P(jw) is large. (See
Example 16.4)

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 Example 16.1
 b_{opt}(P_{1})
will be very small for all K whenever r is close to 1 (i.e.,
whenever there is an unstable pole close to an unstable zero).

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 Example 16.2
 b_{opt}(P_{2}) will be small if P_{2}(jw) is
large around w =1, the modulus of the righthalf plane zero.
 Note that b_{opt}(L/s) = 0.707 for any L and b_{opt}(P_{2})® 0.707 as K ® 0. This
is because P_{2}(jw) around the frequency of the righthalf plane
zero is very small as K ® 0.

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 b_{opt}(P_{3}) will be small if P_{3}(jw) is large around the
frequency of w=1 (the modulus of the righthalf plane zero).
 For zeros with the same modulus, b_{opt}(P_{3})
will be smaller for a plant with relatively larger real part zeros than
for a plant with relatively larger imaginary part zeros (i.e., a pair of
real righthalf plane zeros has a much worse effect on the performance
than a pair of almost pure imaginary axis righthalf plane zeros of the
same modulus).

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 Example 16.3
 b_{opt}(P_{4}) will be small if P_{4}(jw) is small around the
frequency of w=1 (the modulus of the righthalf plane pole).
 Note that b_{opt}(P_{4})® 0.707
as K ® ¥. This is because P_{4}(jw) is very large around
the the frequency of the modulus of the righthalf plane pole as K ® ¥.

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 b_{opt}(P_{5}) will be small if P_{5}(jw) is small around the
frequency of the modulus of the righthalf plane pole.
 For poles with the same modulus, b_{opt}(P_{5}) will be
smaller for a plant with relatively larger real part poles than for a
plant with relatively larger imaginary part poles(i.e., a pair of real
righthalf plane poles has a much worse effect on the performance than a
pair of almost pure imaginary axis righthalf plane poles of the same
modulus).

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 Example 16.4
 K = 10^{5} : slope near
crossover is not too large Þ b_{opt}(P_{6})
not too small.
 K = 10^{4} : Similar.
 K = 0.1: slope near crossover is
quite large Þ b_{opt}(P_{6}) quite small.

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 Based on the preceding discussion, we can give some guidelines for the
loopshaping design.
 The loop transfer function should be shaped in such a way that it has
low gain around the frequency of the modulus of any righthalf plane
zero z. Typically, it requires that the crossover frequency be much
smaller than the modulus of the righthalf plane zero; say, w_{c}<z/2
for any real zero and w_{c} <z for any complex zero with a
much larger imaginary part than the real part (see Figure 16.6).
 The loop transfer function should have a large gain around the frequency
of the modulus of any righthalf plane pole.
 The loop transfer function should not have a large slope near the crossover
frequencies.
 These guidelines are consistent with the rules used in classical
control theory (see Bode[1945] and Horowitz[1963]).
