2
Modelling of Uncertain Systems
As discussed in Chapter 1, it is well understood that uncertainties are unavoidable in a real control system. The uncertainty can be classified into two
categories: disturbance signals and dynamic perturbations. The former includes input and output disturbance (such as a gust on an aircraft), sensor
noise and actuator noise, etc. The latter represents the discrepancy between
the mathematical model and the actual dynamics of the system in operation.
A mathematical model of any real system is always just an approximation
of the true, physical reality of the system dynamics. Typical sources of the
discrepancy include unmodelled (usually high-frequency) dynamics, neglected
nonlinearities in the modelling, effects of deliberate reduced-order models, and
system-parameter variations due to environmental changes and torn-and-worn
factors. These modelling errors may adversely affect the stability and performance of a control system. In this chapter, we will discuss in detail how dynamic perturbations are usually described so that they can be well considered
in system robustness analysis and design.
2.1 Unstructured Uncertainties
Many dynamic perturbations that may occur in different parts of a system can,
however, be lumped into one single perturbation block ∆, for instance, some
unmodelled, high-frequency dynamics. This uncertainty representation is referred to as “unstructured” uncertainty. In the case of linear, time-invariant
systems, the block ∆ may be represented by an unknown transfer function
matrix. The unstructured dynamics uncertainty in a control system can be
described in different ways, such as is listed in the following, where Gp (s)
denotes the actual, perturbed system dynamics and Go (s) a nominal model
description of the physical system.
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2 Modelling of Uncertain Systems
1. Additive perturbation:
Fig. 2.1. Additive perturbation configuration
Gp (s) = Go (s) + ∆(s)
(2.1)
2. Inverse additive perturbation:
Fig. 2.2. Inverse additive perturbation configuration
(Gp (s))−1 = (Go (s))−1 + ∆(s)
3. Input multiplicative perturbation:
Fig. 2.3. Input multiplicative perturbation configuration
(2.2)
2.1 Unstructured Uncertainties
Gp (s) = Go (s)[I + ∆(s)]
15
(2.3)
4. Output multiplicative perturbation:
Fig. 2.4. Output multiplicative perturbation configuration
Gp (s) = [I + ∆(s)]Go (s)
(2.4)
5. Inverse input multiplicative perturbation:
Fig. 2.5. Inverse input multiplicative perturbation configuration
(Gp (s))−1 = [I + ∆(s)](Go (s))−1
(2.5)
6. Inverse output multiplicative perturbation:
(Gp (s))−1 = (Go (s))−1 [I + ∆(s)]
(2.6)
7. Left coprime factor perturbations:
Gp (s) = (M̃ + ∆M̃ )−1 (Ñ + ∆Ñ )
(2.7)
8. Right coprime factor perturbations:
Gp (s) = (N + ∆N )(M + ∆M )−1
(2.8)
16
2 Modelling of Uncertain Systems
Fig. 2.6. Inverse output multiplicative perturbation configuration
Fig. 2.7. Left coprime factor perturbations configuration
Fig. 2.8. Right coprime factor perturbations configuration
The additive uncertainty representations give an account of absolute error
between the actual dynamics and the nominal model, while the multiplicative
representations show relative errors.
In the last two representations, (M̃ , Ñ )/(M, N ) are left/right coprime factorizations of the nominal system model Go (s), respectively; and (∆M̃ , ∆Ñ )
/(∆M , ∆N ) are the perturbations on the corresponding factors [101].
The block ∆ (or, (∆M̃ , ∆Ñ ) /(∆M , ∆N ) in the coprime factor perturbations cases) is uncertain, but usually is norm-bounded. It may be bounded by
a known transfer function, say σ[∆(jω)]≤ δ(jω), for all frequencies ω, where
δ is a known scalar function and σ[·] denotes the largest singular value of a
matrix. The uncertainty can thus be represented by a unit, norm-bounded
block ∆ cascaded with a scalar transfer function δ(s).
2.2 Parametric Uncertainty
17
It should be noted that a successful robust control-system design would
depend on, to certain extent, an appropriate description of the perturbation
considered, though theoretically most representations are interchangeable.
Example 2.1
The dynamics of many control systems may include a “slow” part and a “fast”
part, for instance in a dc motor. The actual dynamics of a scalar plant may
be
Gp (s) = ggain Gslow (s)Gfast (s)
where, ggain is constant, and
Gslow (s) =
1
;
1 + sT
Gfast (s) =
1
;
1 + αsT
α ≤≤ 1.
In the design, it may be reasonable to concentrate on the slow response part
while treating the fast response dynamics as a perturbation. Let ∆a and ∆m
denote the additive and multiplicative perturbations, respectively. It can be
easily worked out that
∆a (s) = Gp − ggain Gslow = ggain Gslow (Gfast − 1)
−αsT
= ggain
(1 + sT )(1 + αsT )
Gp − ggain Gslow
−αsT
∆m (s) =
= Gfast − 1 =
ggain Gslow
1 + αsT
The magnitude Bode plots of ∆a and ∆m can be seen in Figure 2.9, where
ggain is assumed to be 1. The difference between the two perturbation representations is obvious: though the magnitude of the absolute error may be
small, the relative error can be large in the high-frequency range in comparison
to that of the nominal plant.
2.2 Parametric Uncertainty
The unstructured uncertainty representations discussed in Section 2.1 are
useful in describing unmodelled or neglected system dynamics. These complex uncertainties usually occur in the high-frequency range and may include
unmodelled lags (time delay), parasitic coupling, hysteresis and other nonlinearities. However, dynamic perturbations in many industrial control systems
may also be caused by inaccurate description of component characteristics,
torn-and-worn effects on plant components, or shifting of operating points,
etc. Such perturbations may be represented by variations of certain system
parameters over some possible value ranges (complex or real). They affect the
low-frequency range performance and are called “parametric uncertainties”.
18
2 Modelling of Uncertain Systems
Bode plot (Magnitude)
0
Solid line: absolute error
−20
Dashed line: relative error
Magnitude (dB)
−40
−60
−80
−100
−120
−140
−3
10
−2
10
−1
10
0
10
1
10
Frequency (rad/s)
2
10
3
10
4
10
5
10
Fig. 2.9. Absolute and relative errors in Example 2.1
Example 2.2
A mass-spring-damper system can be described by the following second-order,
ordinary differential equation
m
d2 x(t)
dx(t)
+c
+ kx(t) = f (t)
2
dt
dt
where, m is the mass, c the damping constant, k the spring stiffness, x(t) the
displacement and f (t) the external force. For imprecisely known parameter
values, the dynamic behaviour of such a system is actually described by
(mo + δm )
d2 x(t)
dx(t)
+ (co + δc )
+ (ko + δk )x(t) = f (t)
2
dt
dt
where, mo , co and ko denote the nominal parameter values and δm , δc and δk
possible variations over certain ranges.
By defining the state variables x1 and x2 as the displacement variable and
its first-order derivative (velocity), the 2nd-order differential equation (2.2)
may be rewritten into a standard state-space form
x˙1 = x2
1
[−(ko + δk )x1 − (co + δc )x2 + f ]
mo + δm
y = x1
x˙2 =
2.2 Parametric Uncertainty
19
Fig. 2.10. Analogue block diagram of Example 2.2
Further, the system can be represented by an analogue block diagram as
in Figure 2.10.
1
Notice that mo +δ
can be rearranged as a feedback in terms of m1o and δm .
m
Figure 2.10 can be redrawn as in Figure 2.11, by pulling out all the uncertain
variations.
Fig. 2.11. Structured uncertainties block diagram of Example 2.2
Let z1 , z2 and z3 be x˙2 , x2 and x1 , respectively, considered as another,
fictitious output vector; and, d1 , d2 and d3 be the signals coming out from the
perturbation blocks δm , δc and δk , as shown in Figure 2.11. The perturbed
20
2 Modelling of Uncertain Systems
system can be arranged in the following state-space model and represented as
in Figure 2.12.
 
¸ d1
·
¸
0
1
0
ẋ1
x1
0 0 0  
d
+ 1 f
=
+
2
ko
co
−m
−m
ẋ2
x2
−1 −1 −1
mo
o
o
d3


   ko





1
co
· ¸
− mo − m
z1
d1
−1 −1 −1
mo
o
x
1
 z2  =  0
+  0 0 0   d2  +  0  f
1 
x2
z3
d3
0 0 0
0
1
0
· ¸
£
¤ x1
y= 10
x2
·
¸
·
¸·
¸
·
(2.9)
Fig. 2.12. Standard configuration of Example 2.2
The state-space model of (2.9) describes the augmented, interconnection
system M of Figure 2.12. The perturbation block ∆ in Figure 2.12 corresponds
to parameter variations and is called “parametric uncertainty”. The uncertain
block ∆ is not a full matrix but a diagonal one. It has certain structure,
hence the terminology of “structured uncertainty”. More general cases will be
discussed shortly in Section 2.4.
2.3 Linear Fractional Transformations
The block diagram in Figure 2.12 can be generalised to be a standard configuration to represent how the uncertainty affects the input/output relationship
2.3 Linear Fractional Transformations
21
of the control system under study. This kind of representation first appeared
in the circuit analysis back in the 1950s ([128, 129]). It was later adopted in the
robust control study ([132]) for uncertainty modelling. The general framework
is depicted in Figure 2.13.
Fig. 2.13. Standard M -∆ configuration
The interconnection transfer function matrix M in Figure 2.13 is partitioned as
·
¸
M11 M12
M=
M21 M22
where the dimensions of M11 conform with those of ∆. By routine manipulations, it can be derived that
£
¤
z = M22 + M21 ∆(I − M11 ∆)−1 M12 w
if (I − M11 ∆) is invertible. When the inverse exists, we may define
F (M, ∆) = M22 + M21 ∆(I − M11 ∆)−1 M12
F (M, ∆) is called a linear fractional transformation(LFT) of M and ∆.
Because the “upper”loop of M is closed by the block ∆, this kind of linear
fractional transformation is also called an upper linear fractional transformation(ULFT), and denoted with a subscript u, i.e. Fu (M, ∆), to show the
way of connection. Similarly, there are also lower linear fractional transformations(LLFT) that are usually used to indicate the incorporation of a controller
K into a system. Such a lower LFT can be depicted as in Figure 2.14 and
defined by
Fl (M, K) = M11 + M12 K(I − M22 K)−1 M21
With the introduction of linear fractional transformations, the unstructured uncertainty representations discussed in Section 2.1 may be uniformly
described by Figure 2.13, with appropriately defined interconnection matrices
M s as listed below.
22
2 Modelling of Uncertain Systems
Fig. 2.14. Lower LFT configuration
1. Additive perturbation:
·
0I
M=
I Go
¸
(2.10)
2. Inverse additive perturbation:
·
−Go Go
M=
−Go Go
¸
(2.11)
3. Input multiplicative perturbation:
·
0 I
M=
Go Go
¸
(2.12)
4. Output multiplicative perturbation:
·
M=
0 Go
I Go
¸
(2.13)
5. Inverse input multiplicative perturbation:
·
M=
−I I
−Go Go
¸
(2.14)
6. Inverse output multiplicative perturbation:
·
M=
−I Go
−I Go
¸
(2.15)
2.4 Structured Uncertainties
23
7. Left coprime factor perturbations:
·
−1
−M̃G

0
M=
−1
M̃G
¸·
¸
−Go

I
Go
(2.16)
−1
where Go = M̃G
ÑG , a left coprime factorisation of the nominal plant;
and, the perturbed plant is Gp = (M̃G + ∆M̃ )−1 (ÑG + ∆Ñ ).
8. Right coprime factor perturbations:
·£
M=
¤ −1 ¸
−1
£−MG 0¤ MG
−Go I
Go
(2.17)
where Go = NG MG −1 , a right coprime factorisation of the nominal plant;
and, the perturbed plant is Gp = (NG + ∆N )(MG + ∆M )−1 .
In the above, it is assumed that [I − M11 ∆] is invertible. The perturbed
system is thus
Gp (s) = Fu (M, ∆)
In the coprime factor
representations, (2.16) and (2.17), ∆ =
· perturbation
¸
¤
∆M
∆M̃ ∆Ñ and ∆ =
, respectively. The block ∆ in (2.10)–(2.17) is
∆N
supposed to be a “full” matrix, i.e. it has no specific structure.
£
2.4 Structured Uncertainties
In many robust design problems, it is more likely that the uncertainty scenario
is a mixed case of those described in Sections 2.1 and 2.2. The uncertainties
under consideration would include unstructured uncertainties, such as unmodelled dynamics, as well as parameter variations. All these uncertain parts
still can be taken out from the dynamics and the whole system can be rearranged in a standard configuration of (upper) linear fractional transformation
F (M, ∆). The uncertain block ∆ would then be in the following general form
∆ = diag [δ1 Ir1 , · · · , δs Irs , ∆1 , · · · , ∆f ] : δi ∈ C, ∆j ∈ C mj ×mj
(2.18)
Pf
Ps
where i=1 ri + j=1 mj = n with n is the dimension of the block ∆. We
may define the set of such ∆ as ∆. The total block ∆ thus has two types of
uncertain blocks: s repeated scalar blocks and f full blocks. The parameters δi
of the repeated scalar blocks can be real numbers only, if further information
of the uncertainties is available. However, in the case of real numbers, the
analysis and design would be even harder. The full blocks in (2.18) need not
be square, but by restricting them as such makes the notation much simpler.
24
2 Modelling of Uncertain Systems
When a perturbed system is described by an LFT with the uncertain block
of (2.18), the ∆ considered has a certain structure. It is thus called “structured
uncertainty”. Apparently, using a lumped, full block to model the uncertainty
in such cases, for instance in Example 2.2, would lead to pessimistic analysis
of the system behaviour and produce conservative designs.
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