7
Introduction to LFTs
1. Notation and definition
2. parameters as LFTs
3. Inverses of LFTs
4. Interconnections of LFTs
5. Simple transfer function
6. State-Space variations as LFTs
7. LFT Small-Gain theorem
7.1
Representing Uncertainty
Procedure to represent Uncertainty in models:
• Use Linear Fractional Transformations (LFTs)
• Separate what is known from what is unknown in a feedback-like connection
• Bound the possible values of the unknown elements
This approach is a direct generalization of state-space realizations, where a
linear, dynamical system is written as a feedback connection of
• a constant matrix,
A B
, and
C D
• a dynamic element made up of a diagonal matrix of integrators (or
delays, in the discrete case)
This realization greatly facilitates manipulation and computation of linear
systems, and Linear Fractional Transformations (LFT) provide the same
capability for uncertain systems.
90
7.2
General Uncertainty Modeling: LFTs
Linear Fractional Transformations (LFT) are a powerful and flexible approach to represent uncertainty
• Complex matrix M, relating R and v
v
M
r
v = Mr
Partition into top and bottom
v1 M11 M12
r1
v1 = M11r1 + M12r2
v2 M21 M22
r2
v2 = M21r1 + M22r2
• Matrix ∆ relating v2 to r2, as
v2
-
- r2
∆
r2 = ∆v2
• The linear fractional transformation (LFT) of M by ∆
v1 M11 M12
M21 M22
-
r1
∆
Eliminate v2 and r2, leaving
v1 =
h
i
M11 + M12 ∆ (I − M22∆)−1 M21 r1
|
= FL (M, ∆) r1
91
{z
FL (M,∆)
}
7.3
Upper and Lower LFTs
v1 M11 M12
M21 M22
-
r1
v1 = FL (M, ∆) r1
∆
The notation FL indicates that the lower loop of M is closed with ∆.
If the upper loop of M is closed with Ω, then we have
-
v2 Ω
M11 M12
M21 M22
v2 = FU (M, Ω) r2
r2
where
h
FU (M, Ω) := M22 + M21 Ω (I − M11Ω)−1 M12
7.4
i
Uncertain Parameters as LFTs
How do we use LFTs to represent an uncertain parameter?
Suppose c is a parameter, and it is known to take on values
2.0 ≤ c ≤ 2.8
Write this as c = 2.4 + 0.4δc where δc ∈ [−1, 1]. This is a linear fractional
transformation!
2.4 0.4
, δc
c = FL
1 0
So, everywhere
c appears in a block diagram, simply replace it with
92
2.4
0.4
1
0
-
7.5
δc
Inverses of LFTs
If the gain c−1 also appears, the LFT representation can still be used, because
inverses of LFTs are LFTs (on the same δ). Note that
1
c−1 =
2.4 + 0.4δc
1
δc
− 6·2.4
1
=
+
2.4 1 − (− 61 )δc
=
So, everywhere
1 c
1
2.4
FL 1
2.4
, δc
appears in a block diagram, replace it with
1
2.4
1
2.4
-
7.6
− 16
− 16
− 16
− 16
δc
Inverses of LFTs, General Cases
The general case for inverses can be solved with the matrix inversion lemma.
Specifically, given a matrix H, there exists matrices HLI and HU I such that
for all ∆ and Ω
[FL (H, ∆)]−1 = FL (HLI , ∆)
,
93
[FU (H, Ω)]−1 = FU (HU I , Ω)
In fact, with H =
,
H21 H22
HLI =
and
the formulas for HLI and HU I are just
−1
H11
−1
−H11
H12
−1
H21H11
H22 −
−1
H21H11
H12
−1
H12 H22
−1
H22
−1
H11 − H12 H22 H21
−1
−H22
H21
HU I =
7.7
H11 H12
Uncertain 2nd order MCK system
• Single degree-of-freedom mass/damper/spring system with uncertain elements
mẍ + cẋ + kx = u
• Parametric uncertainty (nominal value and range of possible variation)
m = m̄ (1 + 0.5δm) ,
c = c̄ (1 + 0.3δc) ,
k = k̄ (1 + 0.4δk )
with −1 ≤ δm , δc, δk ≤ 1. This represents 50% uncertainty in m, 30%
uncertainty in c, 40% uncertainty in k.
• Block diagram is
-d
u
−
−6
I
@
-
@
@
@
ẍ
1
m
-
ẋ
R
c
-
x
R
-y
k
• Define matrices
Mmi :=
Now, replace
−0.5
−0.5
- 1
m
1
m̄
1
m̄
-
, Mc :=
,
c
94
c̄ 0.3c̄
1
0
and
, Mk :=
k
k̄ 0.4k̄
1
0
respectively with
δm -
Mmi
-
-
Mc
Mk
-
δc
δk
• Eventually we will separate what is known (Mmi, Mc , Mk and integrators) from what is unknown (δm, δc , δk ), so
– draw the block diagram without the δ’s, but
– label the signals which go to and from the δ’s.
wm
u
z
-m
-
Mmi
-e
−
−6
I
@
@
@
@
ẍ
-
ẋ
R
-
x
R
-y
Mc
zc wc
zk Mk
wk
• Let Gmck be the 4-input (wm, wc , wk , u), 4-output (zm , zc , zk , y), 2-state
system shown above and depicted below
zm zc zk y
Gmck wm
wc
wk
u
• Note that Gmck only depends on
– m̄, c̄, k̄, 0.5, 0.4 and 0.3 and
– the original differential equation which relates u to y.
Hence, Gmck is known.
• Uncertain behavior of the original system is characterized by an upper
95
LFT of Gmck with a diagonal uncertainty matrix
y=
FU Gmck ,
|
δm 0
δm 0
-
0 δc
0
0
{z
∆
0
0
u
δk
}
0
0 δc 0
0
0 δk
y
Gmck
u
• The unknown matrix ∆, (the perturbation) is “structured” as certain
elements are known to be zero.
7.8
Interconnections of LFTs
Extremely important property of LFTs – typical algebraic operations such
as
• frequency response,
• cascade connections,
• parallel connections, and
• feedback connections
preserve the LFT structure. Hence, typical interconnections of LFTs are still
in the form of an LFT.
For this reason, the LFT is an excellent choice for a general hierarchial
representation of uncertainty. Some additional examples follow.
96
7.9
Cascade Connection of LFTs
Cascade connection of FL (M, ∆) with FU (G, Ω), so that
y = FL (M, ∆) FU (G, Ω) u
-
y
M11 M12
M21 M22
-
Ω
G11 G12
G21 G22
u
∆
Draw a box around M and G, isolating them from ∆ and Ω, calling the
boxed items Q.
-
zΩ
Ω
6
y
M11 M12
M21 M22
?
z∆
-
6w
wΩ
?
G11 G12
G21 G22
u
Q
∆
∆
Q is made up of the elements of M and G, and relates the variables (u, w∆, wΩ)
to (y, z∆ , zΩ )
y
z∆ zΩ Q
97
u
w∆
wΩ
From the diagram, Q is easily calculated as
Since
∆ 0
y
z∆
zΩ
=
M11G22 M12 M11 G21
M21G22 M22
0
G12
{z
Q
|
is the matrix that relates
M21 G21
G11
}
z∆
→
u
w∆
w∆
wΩ
so cascade con0 Ω
zΩ
wΩ
nection of FL (M, ∆) and FU (G, Ω) is yet another LFT, namely
Q,
FL (M, ∆) FU (G, Ω) = FL
∆ 0
0 Ω
,
as shown below
y
Q
u
-∆
-Ω
Similar manipulations can be carried out for parallel connections, as well as
feedback connections, and arbitrary combinations of these.
7.10
Example, Feedback of LFTs
A complicated feedback connection with 3 LFTs
e
1
G1 -∆
d1
G2
u1
1
d3
-
-∆
G3
∆3 98
2
y3-
can be drawn as a single LFT on the diagonal matrix containing the 3 individual perturbations,
-
∆1
0
0
0
∆2
0
0
0
∆3
P
e1 y3 d1
d3
u1
Here, P depends only on G1 , G2, G3 and the interconnection diagram and is
easy to calculate with the interconnection program sysic.
Note: uncertainty matrix affecting P is structured, with a block-diagonal
structure. This is an extremely important observation.
General uncertainty at component level becomes structured uncertainty at the interconnection level
7.11
Parameter Uncertainty in Transfer Functions
SISO process with
• an uncertain gain,
• first-order lag with uncertain time constant, and
• uncertain delay (modeled with a Pade approximation).
The transfer function for the process is
1 −γs + 1
u(s)
τ s + 1 γs{z+ 1 }
gain | {z } |
y(s) = |{z}
K
lag
Assume that
99
Pade
• each term is uncertain, with
– K ∈ [1 3],
– γ ∈ [0.05 0.15] and
– τ ∈ [1 2].
• K and γ are linearly related, so that as K takes on values from 1 → 3,
γ simultaneously takes on values from 0.05 → 0.15.
Represent these variations with two uncertainties, δ1 and δ2, with
K = 2 + δ1 , γ = 0.1 + 0.05δ1 , τ = 1.5 + 0.5δ2
where −1 ≤ δ1 , δ2 ≤ 1.
A block diagram of
−γs+1
γs+1
-
2
is
- f - γ −1
−
6
Similarly, a block diagram of
1
τ s+1
-f −
6
-
-f
−
6
R
is
-
τ −1
-
R
Use upper-loop LFTs to model K, γ −1 and τ −1 , define matrices
MK :=
0 1
1 2
, MγI :=
1
−2
− 12
10
10
, Mτ I :=
1
−3
− 31
2
3
2
3
so that K = FU (MK , δ1), γ −1 = FU (MγI , δ1), τ −1 = FU (Mτ I , δ2).
The first-order Pade system is of the form FU (GP , δ1), and the first-order lag
is of the form FU (GL , δ2), where GP and GL are known, 2-input, 2-output,
1-state systems
100
zγ y1 GP
zτ y2 wγ
u1
GL
wτ
u2
GP is shown in detail below and can be built easily using sysic.
wγ
-
u1
-f −
6
2
zγ
-
-
MγI
-
-f
− y1
6
R
The internal structure of GL is even easier,
wτ
zτ
-
-
Mτ I
-e u2 6−
-
-
R
y2
The uncertain K is directly represented using FU (MK , δ1).
The final connection is simply a 4-input, 4-output, 2-state system Gproc
zγ zk zτ y
with internal structure
wγ
-
u
-
Gproc
zγ 6 wτ
?
-
-
wγ
wk
wτ
u
zτ 6 wk
?
6 -
GP
zk 6
?
6 -
GL
-
6
MK
y
The uncertain system’s behavior is an LFT,
Gproc ,
y = FU
101
δ1 I 2 0
0
δ2
- -
u
Note that the perturbation δ1 is repeated twice, due to the coupled variation
in K and γ.
7.12
Linear State-Space Uncertainty
In the special case of linear uncertainty in a state-space model, the uncertainty description can be built up even more easily.
Consider an uncertain state-space model,
ẋ(t)
y(t)
=
=
A0 +
C0 +
m
X
m
X
δ i A i B0 +
i=1
m
X
i=1
m
X
δiCi D0 +
i=1
A 0 B0
C0 D0
+
m
X
i
δ i Bi
δi Di
i=1
δi
A i Bi
Ci Di
x(t)
u(t)
x(t)
u(t)
where for each i = 1, 2, . . . , m
A i Bi
Ci Di
∈ R(n+nu )×(n+ny )
Let
ri := rank
A i Bi
Ci Di
and factor each matrix (using svd, for instance) as
where
Ei
Fi
A i Bi
Ci Di
(n+ny )×ri
∈R
=
,
Ei
Fi
Gi Hi
Gi Hi
∈ Rri ×(n+ny )
Now, define a linear system Gss , with extra inputs and outputs via the state
102
equations
ẋ
y
z1
..
.
zm
=
A0
B0 E 1 · · · E m
C0
D0 F1 · · ·
G1 H1
..
..
.
.
0 ···
.. . . .
.
Gm Hm
0 ···
Fm
0
..
.
0
x
u
w1
..
.
wm
as shown
y
z1 z2 ··
z ·
Gss
m
u
w1
w2
··
· w
m
The uncertain system is represented as an LFT around Gss , namely
y = FL (Gss , ∆) u
where ∆ maps z → w, and has the structure given as
∆ = {diag [δ1 Ir1 , . . . , δmIrm ] : δi ∈ R}
This approach, developed by Morton and McAfoos, has its roots in the
Gilbert realization.
7.13
Unmodeled Dynamics
Models of uncertainty are not limited to parametric uncertainty. Often,
• a low order, nominal model which suitably describes the low-mid frequency range behavior of the plant is available,
• high-frequency plant behavior is uncertain, and
• even the dynamic order of the actual plant is not known, and
• something richer than parametric uncertainty is needed to represent this
uncertainty.
103
Use a multiplicative, unmodeled dynamics uncertainty model.
This allows one to specify a frequency-dependent percentage uncertainty in
the actual plant behavior. To specify the uncertainty set, choose two things:
1. a nominal model, G(s),
2. a multiplicative uncertainty weighting function, Wu(s)
In the notes Robust Performance of Uncertain SISO Systems, we
defined the multiplicative uncertainty set as
M (G, Wu) := G̃ :
G̃(jω) − G(jω) G(jω)
≤ |Wu(jω)|
with the additional restriction that the number of right-half plane (RHP)
poles of G̃ be equal to the number of right-half plane poles of G.
Interpretations:
• At each frequency, |Wu (jω)| represents the maximum potential percentage difference between all of the plants represented by M (G, Wu) and
the nominal plant model G.
• In that sense, M (G, Wu) is a ball of possible plants, centered at G.
• On a Nyquist plot, a disk of radius |Wu(jω)G(jω)|, centered at G(jω) is
the set of possible values that G̃(jω) can take on, due to the uncertainty
description.
Here, we take an alternate approach, and model the uncertain values of G
with a 3-block system, namely
-
Wu
u
6
Behavior of
plants in M(G, Wu)
∆
+
h- ?
+
104
G
-
y
where ∆ ∈ S, and k∆k∞ ≤ 1.
Hence, ∆, G and Wu are now viewed as 3 separate physical devices which are
interconnected to form G̃. Note: this is an alternate approach to unmodelled
dynamics uncertainty modeling.
7.14
Multiplicative Uncertainty as LFT
As usual, we represent the uncertain plant set M (G, Wu) as an LFT by
separating the known elements (G and Wu) from the unknown elements
(∆).
-
∆
w
z
?
6
Wu
u
+
- ?
h-
6
-
+
Hmult
G
-
-
y
Known, 2-input,
2-output, linear
system
The uncertain component is now represented as y = FU (Hmult, ∆) u, with
Hmult having the value
z
y
=
|
0 Wu
G G
{z
Hmult
}
w
u
Hmult is easily calculated using sysic or simpler manipulations.
≫
≫
≫
≫
≫
≫
≫
Hmult= madd(mmult([1;0],wu,[0 1]),mmult([0;1],G,[1 1]));
% or
systemnames = ’wu G’;
sysoutname = ’Hmult’;
inputvar = ’[w;u]’;
input to wu = ’[u]’;
input to G = ’[w+u]’;
105
≫ outputvar = ’[wu;G]’;
≫ sysic
In summary, the multiplicative unmodeled dynamics uncertainty model
• captures a wide variety of plant variations;
• represents not only parameter variations, but also represents unmodelled
dynamics;
• in Nyquist plots, it represents disk-uncertainty at each frequency;
• is a coarse, yet simple, approach to putting uncertainty into models.
7.15
Additive unmodeled dynamics
Additive (as opposed to multiplicative) uncertainty may also be used.
Given a nominal model G and an additive uncertainty weighting function
Wu, the additive uncertainty set is defined by the block diagram
-W
u
u
-
-
G
Behavior of
plants in A(G, Wu)
∆
-?
g
-
It can be expressed as an LFT in the usual manner. Essentially, it accomplishes the same as multiplicative
7.16
Modeling Mixed Uncertainty
Uncertainty may be mixed, including both parametric uncertainty and and
unmodeled dynamics.
For example, uncertain 2nd order system with parametric uncertainty in the
parameters m, c and k and additional high-frequency unmodeled dynamics
using the multiplicative uncertainty model.
106
The block diagram is
δm 0
-
0
0 δc 0
0
0 δk
6
y
δ(s)
?
?
Gmck
6
Hmix
Wu
d?
6
u
The possible behavior is represented as an LFT
y = FU (Hmix, ∆)u
where the perturbation matrix ∆ has the structure
∆ = diag {δm , δc , δk , δ(s)}
and Hmix is the known 5-input, 5-output system inside the box.
7.17
Robustness of Uncertain Systems
We now know how to represent uncertain systems as LFTs on unknown, but
structured uncertainty matrices
Next question – how to analyze the robustness of these uncertain systems?
Answers are provided using the the structured singular value, µ
7.18
µ Analysis, Robust Stability
Every µ-analysis consists of the following steps:
1. Recast the problem into the feedback loop diagram (called the analysis
diagram) below,
107
-
∆
M
where
• M is a known linear system,
• ∆ is a structured perturbation
2. Calculate a frequency response of M
3. Describe the structure of the perturbations ∆.
4. Run the command mu on the frequency response
5. Plot the bounds obtained from the µ calculation
7.19
Analysis Diagram from System Block diagram
Consider the mass/damper/spring system, Hmix, with both parametric uncertainty and unmodeled dynamics, along with feedback controller K, so
u = Ky
-
∆
Hmix
-
K
Group Hmix and K together, namely M := FL (Hmix, K), to get
108
-
-
∆
Hmix
⇔
∆
Hmix
-
∆
⇔
M
-
7.20
K
-
K
M := FL (Hmix, K)
Structure of Uncertainty
The structure of the perturbation matrices must be passed to the mu command. Three attributes about each uncertainty block must be specified:
1. The type (real parameter .vs. unmodeled dynamics) of the perturbation
2. The dimension of the perturbation
3. The # of independent locations that the particular uncertainty occurs
(Gain/Pade/Lag example had δ1 affecting the system in 2 independent
locations)
Uncertainty structure information is stored as a n × 2 array (called the block
structure array) where
• n is the number of different perturbation elements in the uncertainty
matrix.
• The i’th row describes the i’th uncertain block, using conventions
– A scalar real parameter is denoted [-1 1] (or [-1 0]).
– A repeated (f times) real parameter is denoted [-f 0].
– A 1×1 (ie., scalar) unmodeled dynamics perturbation (later we refer
to this as complex) is denoted [1 1].
– A r × c (ie., full, rectangular) unmodeled dynamics block is denoted
[r c].
109
Ordering of the uncertainty elements is (and must be) consistent with the
ordering of the input/output channels of the known systems
Use the notation ∆ to represent the set of all perturbation matrices with
the appropriate structure.
δm 0
-
0
0 δc 0
0
0 δk
6
y
∆ :=
?
?
Gmck
δ(s)
δ1 0
0
0 δ2 0
0
0
0 δ3 0
0
0
0 δ4
6
Wu
?
d
0
u
6
Hmix
: δ1 ∈ R, δ2 ∈ R, δ3 ∈ R, δ4(s)
and is represented as
≫ deltaset = [-1 1;-1 1;-1 1;1 1];
% m/c/k/unmodeled
Now that the uncertainty structure has been represented, we can compute
the size of perturbations against which the system is robustly stable
7.21
Robust Stability Tests
We need to calculate a frequency response of M, and then compute the
structured singular value (µ) of M with respect to the uncertainty set ∆.
At each frequency, the matrix M(jω) is passed to the µ algorithm, and
µ (M(jω)) is computed and then plotted.
Use notation µ∆ (M(jω)), to emphasize the dependency of the function on
both
110
• M, and
• on the uncertainty set ∆
Let β denote the peak (across frequency ω) of µ∆ (M(jω))
max µ∆ (M(jω)) =: β
ω∈R
-
∆
M
µ calculation on known transfer function M gives
max µ∆ (M(jω)) =: β
ω∈R
which is interpreted as follows:
1. For all perturbation matrices ∆ with
• the appropriate structure (ie., any ∆ ∈ ∆),
• max
σ̄ [∆(jω)] < β1 ,
ω
the perturbed system is stable
2. Moreover, there is a particular perturbation matrix with
• ∆ ∈ ∆, and
• max
σ̄ [∆(jω)] =
ω
1
β
that causes instability.
Hence, we think of
1
max
µ∆ (M(jω))
ω
as a stability margin with respect to the structured uncertainty set ∆ affecting
M.
111
7.22
Robust Stability Tests, Upper and Lower Bounds
The software does not compute µ exactly, but bounds it from above and
below by several optimization steps. Let
• βu := peak (across frequency) of the upper bound for µ,
• βl := peak of the lower bound for µ.
Then
• For all perturbation matrices ∆ ∈ ∆ satisfying
max
σ̄ [∆(jω)] <
ω
1
,
βu
the perturbed system is stable;
• There is a particular perturbation matrix ∆ ∈ ∆ satisfying
max
σ̄ [∆(jω)] =
ω
1
βl
that causes instability.
The gap between the upper and lower bounds translates into gaps between
the conclusions “guaranteed robust stability” and “not robustly stable.”
The destabilizing perturbation matrix (of size
the µ calculation using the command dypert.
7.23
1
βl )
Robust Performance Test using µ
Manipulate into the feedback loop diagram below,
-
∆
e
M
112
d
can be constructed from
where
• M is a known linear system,
• ∆ is a structured perturbation from a problem-dependent allowable uncertainty set ∆, and
• d and e are the generalized disturbance and error that characterizes the
performance objective.
The transformation from a system block diagram to the M − ∆ analysis
diagram is as before, except that exogenous signals (disturbances and errors)
remain also.
7.24
MIMO Performance, H∞
The performance of MIMO control systems will be characterized using H∞
norms. We assume that good performance is equivalent to
kT k∞ := max σ̄ (T (jω)) ≤ 1
ω∈R
where T is the weighted, closed-loop transfer function matrix of interest.
For robust performance of uncertain systems, T is the uncertain transfer
function from d → e, so T = FU (M, ∆).
-
∆
M
e
d
The transfer function from d → e is a function of ∆, through the elements
of M, and the LFT formula FU .
Question: How big can the transfer function FU (M, ∆) get as ∆ takes on
its allowed values?
Precisely: Given M and ∆, the LFT
113
-
∆
e
M
d
is said to achieve Robust Performance if for all perturbations
• ∆ ∈ ∆ and
• max
σ̄ [∆(jω)] < 1,
ω
the LFT is stable, and has kFU (M, ∆)k∞ ≤ 1
7.25
Performance via Robustness
How can µ be used to assess robust performance?
Main idea – relate the size of a transfer function to a robust stability test.
Suppose that T is a given, stable system, with input dimension nd and
output dimension ne . By the Nyquist and small-gain theorem, we know that
kT k∞ ≤ 1 if and only if the feedback loop shown below is stable for every
stable ∆F (s) (of dimension nd × ne ) satisfying k∆F k∞ < 1.
T
⇔
-
∆F
T
≤1
∞
Stable for all k∆F k∞ < 1
Hence, a transfer function T is small, ie.,
kT k∞ ≤ 1
if and only if T can tolerate all possible stable feedback perturbations ∆F
(with k∆F k∞ < 1) without leading to instability.
114
The size of a transfer function can be determined using
a robust stability test
This ultimately allows us to pose the robust performance question as a robust
stability question.
For robust performance problems, transfer function in question is an LFT,
T = FU (M, ∆)
-
∆
M
e
d
Following the argument presented, it follows that
kFU (M, ∆)k∞ ≤ 1
for all pertubations ∆ ∈ ∆ satisfying max
σ̄ [∆(jω)] < 1 if and only if the
ω
LFT below
-
∆
M
-
∆F
is stable for all ∆ ∈ ∆ and all stable ∆F satisfying
max
σ̄ (∆(jω)) < 1
ω
and
max
σ̄ (∆F (jω)) < 1.
ω
But this is exactly a Robust Stability problem for M, subjected to perturbation matrices of the form
∆P =
∆
0
0 ∆F
115
where ∆F is a full unmodeled dynamics block.
Hence, we use robust stability techniques – on a larger problem, computing
µ∆ (M(jω)) P – to determine bounds on robust performance for our original
problem!
We use an additional (fictitious) uncertainty element, and determine the
robust stability of the extended system, and finally make conclusions about
the robust performance of the original uncertain system, FU (M, ∆).
IMPORTANT
Robust k·k∞ Performance is characterized by introducing a fictitious uncertainty block across the disturbance/error channels and carrying out a Robust Stability Analysis
7.26
Robust Performance Test
In summary, each robust performance µ-analysis consists of the following
steps:
1. Recast the problem into the feedback loop below
-
∆
M
e
d
where M is a known linear system, and ∆ ∈ ∆ is a structured perturbation, and d and e are the generalized disturbance and error that
characterize the performance objective.
2. Calculate a frequency response of M
3. Describe the structure of the perturbation set ∆.
116
4. Use the dimensions of the disturbance/error channels to define a fictitous uncertainty block, and augment this with the actual uncertainty
structure of ∆ giving an extended uncertainty set ∆P .
5. Compute µ∆ (M(jω)) P on the frequency response of M, using the augmented uncertainty set ∆P
6. Plot the bounds obtained from the µ calculation
-
∆
e
M
∆
0
d
Augmented Uncertainty Set
∆P :=
0 ∆F
where ∆F is a full, unmodeled dynamics block of dimension nd × ne .
Let β denote the peak of the µ-plot
max µ∆ (M(jω)) P =: β
ω∈R
1. For all perturbation matrices satisfying
• ∆ ∈ ∆ and
• max
σ̄ [∆(jω)] < β1 ,
ω
the perturbed system is stable and kFU (M, ∆)k∞ ≤ β
2. Moreover, there is a particular perturbation satisfying
• ∆ ∈ ∆ and
• max
σ̄ [∆(jω)] =
ω
1
β
that causes either kFU (M, ∆)k∞ = β, or instability.
Recall, exact computation of µ is not possible, so what implications are true
using the bounds...
117
7.27
Robust Performance Tests, Bounds
Let
• βu := peak (across frequency) of the upper bound for µ∆ (M(jω)) P
• βl := peak (across frequency) of the lower bound for µ∆ (M(jω)) P .
Then
• For all perturbation matrices satisfying
– ∆∈∆
– max
σ̄ [∆(jω)] <
ω
1
βu ,
the perturbed system is stable and kFU (M, ∆)k∞ ≤ βu.
• There is a particular perturbation matrix satisfying
– ∆ ∈ ∆ and
– max
σ̄ [∆(jω)] =
ω
1
βl
that causes either kFU (M, ∆)k∞ ≥ βl , or instability.
Hence the gap between the upper and lower bounds leads to gaps in the
inability to precisely determine the robust performance.
118
© Copyright 2025 Paperzz