1

 general framework
 analysis and synthesis methods for unstructured uncertainty
 stability with structured uncertainties
 unstructured perturbation
 structured perturbation
 definition of SSV
 examples
 tighter bounds
 tightness of bounds
 computing bounds using Matlab
 structured robust stability
 structured robust stability theorem
 robust performance
 extension to nonlinear time varying uncertainties
 HIMAT example
 skewed problem
 overview on msynthesis

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 Every problem can be put in the general framework with
 For analysis, the controller can be absorbed into the system:

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4

 Assume D(s)=diag[d_{1}I_{r1},
…, d_{s}I_{rs},
D_{1},
…, D_{F}]
Î RH_{¥} with d_{i}_{¥}<1 and D_{j}_{¥}<1.
 Robust Stability Û The interconnection is stable.
 Stability Conditions:
 (1) (sufficient condition) M_{11}_{¥}_{ }£1
 Conservative: ignoring
structure of D(s).
 (2) (necessary conditions ) Test for each d_{i}_{ }(D_{j})
individually (assuming no uncertainty in other channels): (M_{11 })_{ii}_{
}_{¥}_{ }£1
 Optimistic: ignoring interaction between the d_{i}_{ }(D_{j}).

5

 Problem: Given M Î
C ^{p}^{´}^{ q} , find a smallest D Î C ^{p}^{´}^{ q}
(no structure imposed on D) in the sense of such that det(IMD)=0.
 It is easy to see that
 with a smallest “destabilizing”
 So the largest singular value of M can be defined as

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 Problem: Given M Î
C ^{p}^{´}^{ q} , find a smallest D Î D
 D={diag[d_{1}I_{r1},
…, d_{s}I_{rs},
D_{1},
…, D_{F}]
Î C ^{p}^{´}^{ q}
: d_{i}
Î C, D_{j }Î C ^{m}^{j}^{´}^{ m}^{j}
}
 such that det(IMD)=0.
 We shall call 1/a_{min}
as structured singular value (SSV).

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 For M Î C ^{n}^{´}^{ n}
, m_{D}(M) is defined as
 unless no D Î D makes IMD singular, in which
case m_{D}(M) :=0.
 If D={dI_{n} : d Î C} (S=1, F=0, r_{1}=n),
then m_{D}(M)=r(M), the spectral radius of M.
 If D={D Î C ^{n}^{´}^{n }} (S=0, F=1, m_{1}=n),
then m_{D}(M) =
.
 Thus we have the following bounds:
 r(M) £ m_{D}(M) £

8

 Let D={diag[d_{1}, d_{2}] Î C ^{2}^{´}^{ 2}
: d_{i}
Î C}. Consider
two matrices:
 Thus neither r(M)
nor provides useful
bounds even in these simple cases

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 To obtain tighter bounds, define U={U Î D : UU*=I_{n}}

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11

 Example: Let M be a 13x13 matrix and suppose D is given by
 where the size of D
is specified by the matrix blk.
 >> [bounds,rowd] = mu(M,blk)
 >> [D_{l},D_{r}] = unwrapd(rowd,blk)

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 We are interested in the following question: How large D (in the sense of D_{¥}) can be
without destabilizing the feedback system?
 Since the closedloop poles are
given by det(IG(s)D)=0
the feedback system becomes unstable if det(IG(s)D(s))=0 for some s in
the closed right half plane. Now
let a>0
be a sufficiently small number such that the closedloop system is
stable for all stable D_{¥}< a. Next
increase a
until a_{max}
so that the closedloop system becomes unstable. So a_{max} is the robust
stability margin.
 Consider a general feedback interconnection where D is structured uncertainty block and G(s) is the
interconnection matrix.

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 Define
 D :={D(·) Î RH_{¥} : D(s_{0}) Î D for all s_{0 }Î }
 Theorem: Let b>0.
The interconnected system is wellposed and internally stable for all D(·) Î D with D_{¥}< 1/b if and only if

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 Let
 Theorem: Let b>0.
For all D(s)
Î D with D_{¥}< 1/b , the system is wellposed, internally stable,
and F_{u}(G_{p},
D) _{¥}_{ }£ b if and only if

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 Suppose D Î D_{N}_{ } is a structured Nonlinear
(Timevarying) Uncertainty and suppose D is constant scaling matrix such
that DDD^{1}
Î D_{N}_{ }. (Note that we do not require DD=DD.)
 Then a sufficient stability condition for all D Î D_{N} with D_{¥}< 1, is (by small gain theorem)
 D^{1}G(s)D_{¥}_{ }£ 1

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 We shall first find a H_{¥} controller using Matlab (details will be
discussed in later chapters. Just follow the steps here.), which gives g=1.8612= G_{p}_{¥} , a
stabilizing controller K, and a closed loop transfer matrix G_{p}:

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 Now generate the singular value frequency responses of G_{p}:
 >> w = logspace(3,3,300);
 >> Gpf = frsp(G_{p},w);
% Gpf is the frequency response of G_{p};
 >> [u,s,v] = vsvd(Gpf);
 >> vplot(‘liv, m’, s)
 The singular value frequency responses of G_{p} are shown in
Figure 10.6

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 To test the robust stability, we need to compute G_{p11}_{¥} ,
 >> G_{p11} = sel(G_{p}, 1:2, 1:2);
 >> norm_of_G_{p11} = hinfnorm(G_{p11},0.001);
 which gives G_{p11}_{¥} =0.933<1. So the system is robustly stable.
 To check the robust performance, we shall compute the for each frequency with
 >> blk = [2,2;4,2];
 >> [bnds,dvec,sens,pvec] = mu(Gpf,blk);
 >> vplot(‘liv,m’,vnorm(Gpf),bnds)
 >> title(‘Maximum Singular Value and mu’)
 >> xlabel(‘frequency(rad/sec)’)
 >> text(0.01,1.7,’maximum singular value’)
 >> text(0.5,0.8,’mu bounds’)

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 The structured singular value
are shown in Figure 10.7. It is clear that the robust performance
is not satisfied. Note that
 Using a bisection algorithm, we can also find the worst performance:

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 Recall the skewed problem in Chapter 8. It can be shown that robust
performance problem is equivalent to a 2 block structure singular value
with the following interconnection matrix
 So the robust performance condition is
 for all w³0.

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 Now assume W_{e}=w_{s}I, W_{d}=I, W_{1}=I,
W_{2}=w_{t}I and P is stable and has a stable inverse
(I.e., minimum phase) and K(s)=P^{1}(s)l(s) such that K(s) is
proper and the closedloop is stable. Then
 S_{o }=_{ }S_{i }=1/(1+l(s))I=e(s)I, T_{o }=_{ }T_{i }=l(s)/(1+l(s))I=t(s)I
 and
 Let the SVD of P(jw) be
 P(jw)=USV*, S = diag(s_{1}, s_{2} , …, s_{m})
 with and where m is the
dimension of P.
 Then

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 Note that
 where P_{1} and P_{2} are permutation matrices.
 Hence

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 The maximum is achieved at
 and
 Conclusion: The structure singular value of the skewed problem is
(approximately) proportional to the square root of the condition number
of the plant.

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 Consider a general feedback system with interconnection G. Find a
stabilizing controller K so that the following mnorm is minimized:
 This problem is called msynthesis.
 The msynthesis is
not yet fully solved. But a reasonable approach is to “solve”
 by iteratively solving for K and D, i.e., first minimizing over K with D
fixed, then minimizing pointwise over D with K fixed, then again over K,
and again over D, etc. This is the socalled DK Iteration.

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 Fix D
 is a standard H_{¥}_{ } optimization problem.
 Fix K
 is a standard convex
optimization problem and it can be solved pointwise in the frequency
domain:
 Note that when S = 0, (no scalar blocks)

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 Details of DK Iterations:
 (i) Fix an initial estimate of the scaling matrix D_{w} Î D pointwise across frequency.
 (ii) Find scalar transfer functions d_{i}(s), d_{i}^{1}(s)
Î RH_{¥} for i =1,…,
(F1) such that d_{i}(jw)» d_{i}^{w} .
 (iii) Let D(s)=diag(d_{1}(s)I, …, d_{F1}(s)I, I).
Construct a state space model for system
 (iv) Solve an H_{¥} optimization problem to minimize
 over all stabilizing K’s. Denote the minimizing controller by

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 (v) Minimize
 over D_{w}
Î D pointwise across frequency. The
minimization itself produces a new scaling function .
 (vi) Compare with the previous estimate D_{w}_{ },
Stop if they are close, otherwise, replace D_{w} with
and return to step (ii).
 The joint optimization of D and K is not convex and the global
convergence is not guaranteed, many designs have shown that this
approach works well.
