1

 model uncertainty
 small gain theorem
 additive uncertainty
 multiplicative uncertainty
 coprime factor uncertainty
 other tests
 robust performance
 skewed specifications
 example: siso vs mimo

2

 Suppose P is the nominal model
and K is a controller.
 Nominal Stability (NS): if K stabilizes the nominal P.
 Robust Stability (RS): if K stabilizes every plant in P.
 Nominal Performance (NP): if the performance objectives are satisfied
for the nominal plant P.
 Robust Performance(RP): if the performance objectives are satisfied for
every plant in P.

3

 Example 1: Consider a uncertain transfer function

4

 Another way to bound the frequency response is to treat and as norm
bounded uncertainities; that is,
 P(s,a,b)Î {P_{0}+W_{1
}D_{1}+W_{2}D_{2} 
D_{i}_{¥}_{ }£ 1}
 with P_{0}=P(s,0,0) and
 It is in fact easy to show that
 {P_{0}+W_{1}D_{1}+W_{2}D_{2} 
D_{i}_{¥}_{ }£ 1} = {P_{0}+WD 
D_{¥}_{ }£ 1}
 with W=W_{1}+W_{2}.

5

 Example 2: Consider a process control model
 Take the nominal model as
 Then for each frequency, all possible frequency responses are in a box,
as shown in Figure 8.16.
 D_{a}(jw)=G(jw)G_{0} (jw)
 To get an additive weighting W_{a}, use the following Matlab
procedure:

6

 >> mf=ginput(50) % pick 50
points: the first column of mf is the frequency points and the second
column of mf is the corresponding magnitude responses.
 >>magg=vpck(mf(:2),mf(:,1)); % pack them as a varying matrix.
 >> Wa=fitmag(magg); % choose the order of W_{a} online. A
thirdorder W_{a} is sufficient for this example.
 >> [A,B,C,D]=unpck(W_{a} ) %converting into statespace.
 >> [Z,P,K]=ss2zp(A,B,C,D) % converting into zero/pole/gain form.

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 We get
 And the frequency response of W_{a} is also plotted in Figure
8.17. Similarly, define the multiplicative uncertainty
 and a W_{m} can be found such that D_{m}(jw)£ W_{m}(jw) as
shown in Figure 8.18. A W_{m} is given by

8

 Small Gain Theorem: Suppose MÎ (RH_{¥})^{pxq} . Then the system is wellposed
and internally stable for all D ÎRH_{¥} with
 (a) D_{¥} £1/g if and only if M_{¥} < g ;
 (b) D_{¥} <1/g if and only if M_{¥} £ g .

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 Proof: Assume g = 1. System is stable iff det(IM D) has no zero in the
closed righthalf plane for all D ÎRH_{¥} and D_{¥} £1.
 (Ü ) det(IM D) ¹0 for all D ÎRH_{¥} and D_{¥} £1 since
 l(IM D)  ³ 1max l(M D)  ³ 1 M_{¥} >0
 (Þ ) Suppose M_{¥} ³1. There exists a D ÎRH_{¥} with D_{¥} £1 such that det(IM (s)D(s)) has a zero on the imaginary axis, so the
system is unstable. Suppose w Î R_{+}È {¥} is such that
s_{1}(M(jw_{0 })) ³ 1. Let M(jw_{0 })= U(jw_{0 })S(jw_{0 })V*(jw_{0 }) be a singular value decomposition
with
 U(jw_{0 })=[u_{1},u_{2},…,u_{p}],
V(jw_{0 })=[v_{1},v_{2},…,v_{p}],
 S(jw_{0 })=diag[s_{1}, s_{2},…]

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 We shall construct a D ÎRH_{¥} such that
D(jw_{0 })= (1/s_{1}) v_{1}u_{1}*
and D_{¥} £1. Indeed, for such D(s),
 det(IM (jw_{0
})D(jw_{0 }))=det(I
U(jw_{0 })S(jw_{0 })V(jw_{0 }) (1/s_{1}) v_{1}u_{1}*
)
 =1 u_{1}* U(jw_{0 })S(jw_{0 })V(jw_{0 })v_{1}
(1/s_{1})
=0
 and thus the closedloop system is either not wellposed (if w_{0} = ¥) or unstable (if w_{0} Î R_{+}). There
are two different cases:
 (1) w_{0}
= 0 or ¥ :
Then U and V are real matrices. Choose
 D= (1/s_{1}) v_{1}u_{1}*
Î R^{qxp}
 (2) 0<w_{0}
< ¥ :
write u_{1 }and v_{1}in the following form
 where u_{1i }, v_{1j }Î R are chosen so that q_{i}, f_{j }Î [p,0).

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 Choose b_{i }³0 and a_{j }³0 so that
 Let
 Then D_{¥} =1/s_{1} £1 and D(jw_{0 })= (1/s_{1}) v_{1}u_{1}* .

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 The small gain theorem still holds even if D and M are infinite dimensional. This is
summarized as the following corollary.
 Corollary: The following statements are equivalent:
 (i) The system is wellposed and internally stable for all D ÎH_{¥} with D_{¥} £1/g .
 (ii) The system is wellposed and internally stable for all D ÎRH_{¥} with D_{¥} £1/g .
 (iii) The system is wellposed and internally stable for all D ÎR^{qxp} with D £1/g .
 (iv) M_{¥} £ g
 It can be shown that the small gain
condition is sufficient to guarantee internal stability even if D is a nonlinear and
time varying “stable” operator with an appropriately defined stability
notion, see Desoer and Vidyasagar [1975].

13

 Define
 S_{o}=(I+PK)^{1}, T_{o}=PK(I+PK)^{1}
S_{i}=(I+KP)^{1}, T_{i}=KP(I+KP)^{1}
 Let P={P+W_{1 }DW_{2} : D_{ }Î_{ }RH_{¥}_{ }}
and let K stabilize P. Then the
closedloop system is wellposed and internally stable for all D_{¥} <1 if
and only if W_{2}KS_{o}W_{1 }_{¥} £ 1.

14

 Define
 S_{o}=(I+PK)^{1}, T_{o}=PK(I+PK)^{1}
S_{i}=(I+KP)^{1}, T_{i}=KP(I+KP)^{1}
 Let P={(I+W_{1 }DW_{2})P: D_{ }Î_{ }RH_{¥}_{ }}
and let K stabilize P. Then the
closedloop system is wellposed and internally stable for all D_{¥} <1 if
and only if W_{2}T_{o}W_{1 }_{¥} £ 1.

15

 Let be stable left
coprime factorization and let K stabilize P. Suppose
 with Then
the closedloop system is wellposed and internally stable for all D_{¥} <1 if
and only if

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18

 Consider a feedback system with uncertain plant P_{D}_{ },
the weighted sensitivity function is defined as the transfer function
from d to e:
 T_{ed }=W_{e}(I+ P_{D}_{ }K)^{1}
 Suppose our performance
objective is
 T_{ed}_{¥} =sup_{{d2 }_{£}_{1}} e_{2} £1
 Suppose P_{D}_{ }Î {(I+DW_{2})P: D_{ }Î_{ }RH_{¥}_{ },_{ }D_{¥} <1 } and K stabilizes P. Then the robust
performance is guaranteed if

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 Suppose P_{D}_{ }Î {P(I+wD): D_{ }Î_{ }RH_{¥}_{ }, _{ }D_{¥} <1 } and
K stabilizes P.
 robust stability:
 wT_{i}_{¥} £1
 nominal performance:
 W_{e}S_{o}_{¥} £1.
 robust performance is guaranteed if

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 Covering Uncertainty:
 since

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22

 Spinning body with constant velocity in the z direction.

23

 Each loop has the openloop transfer function as 1/s so each loop has
phase margin f_{max}=f_{min}=90^{o}
and gain margin k_{max}=0, k_{min}=¥
 Suppose one loop transfer function is
perturbed
 Denote z(s)/w(s)=T_{11}(s) = 1/(s+1). Then the maximum
allowable perturbation is d_{¥} <1/T_{11}(s)_{¥} =1, which
is independent of a.
 However, if both loops are perturbed at the same time, then the maximum
allowable perturbation is much smaller, as shown below
 The system is robustly stable for every such D iff

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25

 In particular, consider
 Then the closedloop system is stable for every such D iff
 has no zero in the closed righthalf plane.
 Hence the stability region is given by
 2+ d_{11}+_{
}d_{22 }>0, 1+d_{11}+d_{22}+(1+a^{2})d_{11}d_{22}>0
 The system is unstable with
 d_{11 }=d_{22 }= (1+a^{2})^{1/2}
