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Chapter 2Modeling of Uncertain Systems

As discussed in Chap. 1, it is well understood that uncertainties are unavoidablein a real control system. The uncertainty can be classied into two categories: dis-turbance signals and dynamic perturbations. The former includes input and outputdisturbance (such as a gust on an aircraft), sensor noise and actuator noise, etc.The latter represents the discrepancy between the mathematical model and the ac-tual dynamics of the system in operation. A mathematical model of any real systemis always just an approximation of the true, physical reality of the system dynam-ics. Typical sources of the discrepancy include unmodeled (usually high-frequency)

dynamics, neglected nonlinearities in the modeling, effects of deliberate reduced-order models, and system-parameter variations due to environmental changes andtorn-and-worn factors. These modeling errors may adversely affect the stability andperformance of a control system. In this chapter, we will discuss in detail how dy-namic perturbations are usually described so that they can be accounted for in sys-tem robustness analysis and design.

2.1 Unstructured Uncertainties

Many dynamic perturbations that may occur in different parts of a system can, how-ever, be lumped into one single perturbation block , for instance, some unmodeled,high-frequency dynamics. This uncertainty representation is referred to as unstruc-tured uncertainty. In the case of linear, time-invariant systems, the block maybe represented by an unknown transfer function matrix. The unstructured dynamicsuncertainty in a control system can be described in different ways, such as is listedin the following, where G p (s) denotes the actual, perturbed system dynamics and

G o (s) a nominal model description of the physical system.1. Additive perturbation (see Fig. 2.1):

G p (s) = G o (s) + (s) (2.1)

D.-W. Gu et al., Robust Control Design with MATLAB ,Advanced Textbooks in Control and Signal Processing,

13

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14 2 Modeling of Uncertain Systems

Fig. 2.1 Additiveperturbation conguration

Fig. 2.2 Inverse additiveperturbation conguration

Fig. 2.3 Input multiplicativeperturbation conguration

Fig. 2.4 Outputmultiplicative perturbationconguration

Fig. 2.5 Inverse inputmultiplicative perturbationconguration

2. Inverse additive perturbation (see Fig. 2.2 ):

G p (s) 1 = G o (s)

1 + (s) (2.2)

3. Input multiplicative perturbation (see Fig. 2.3):

G p (s) = G o (s) I + (s) (2.3)

4. Output multiplicative perturbation (see Fig. 2.4):

G p (s) = I + (s) G o (s) (2.4)5. Inverse input multiplicative perturbation (see Fig. 2.5):

G p (s) 1 = I + (s) G o (s)

1 (2.5)

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2.1 Unstructured Uncertainties 15

Fig. 2.6 Inverse outputmultiplicative perturbationconguration

Fig. 2.7 Left coprime factorperturbations conguration

Fig. 2.8 Right coprimefactor perturbationsconguration

6. Inverse output multiplicative perturbation (see Fig. 2.6):

G p (s) 1

= G o (s) 1

I + (s) (2.6)7. Left coprime factor perturbations (see Fig. 2.7):

G p (s) = ( M + M ) 1( N + N ) (2.7)

8. Right coprime factor perturbations (see Fig. 2.8):

G p (s) = (N + N )(M + M ) 1 (2.8)

The additive uncertainty representations give an account of absolute error be-tween the actual dynamics and the nominal model, while the multiplicative repre-sentations show relative errors.

In the last two representations, ( M, N) / (M,N) are left/right coprime factoriza-tions of the nominal system model G o (s) , respectively; and ( M , N ) / ( M , N )are the perturbations on the corresponding factors [111].

The block (or, ( M , N ) / ( M , N ) in the coprime factor perturbationscases) is uncertain, but usually is norm-bounded. It may be bounded by a knowntransfer function, say [(j) ] (j) , for all frequencies , where is a knownscalar function and [] denotes the largest singular value of a matrix. The uncer-tainty can thus be represented by a unit, norm-bounded block cascaded with ascalar transfer function (s) .

It should be noted that a successful robust control-system design would dependon, to a certain extent, an appropriate description of the perturbation considered,though theoretically most representations are interchangeable.

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Fig. 2.9 Absolute and relative errors in Example 2.1

Example 2.1 The dynamics of many control systems may include a slow part anda fast part, for instance in a dc motor. The actual dynamics of a scalar plant maybe

G p (s) = ggain G slow (s)G fast (s)

where ggain is constant, and

G slow (s) =1

1 + sT ; G fast (s) =

11 + sT

, 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 theadditive and multiplicative perturbations, respectively. It can easily be worked outthat

a (s) = G p ggain G slow = ggain G slow (G fast 1)

= ggain sT

(1 + sT )( 1 + sT)

m (s) =G p ggain G slow

ggain G slow= G fast 1 =

sT 1 + sT

The magnitude Bode plots of a and m can be seen in Fig. 2.9, where ggainis assumed to be 1. The difference between the two perturbation representations isobvious: though the magnitude of the absolute error may be small, the relative errorcan be large in the high-frequency range in comparison to that of the nominal plant.

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2.2 Parametric Uncertainty 17

2.2 Parametric Uncertainty

The unstructured uncertainty representations discussed in Sect. 2.1 are useful indescribing unmodeled or neglected system dynamics. These complex uncertainties

usually occur in the high-frequency range and may include unmodeled lags (timedelay), parasitic coupling, hysteresis, and other nonlinearities. However, dynamicperturbations in many industrial control systems may also be caused by inaccuratedescription of component characteristics, torn-and-worn effects on plant compo-nents, or shifting of operating points, etc. Such perturbations may be represented byvariations of certain system parameters over some possible value ranges (complex orreal). They affect the low-frequency range performance and are called parametricuncertainties.

Example 2.2 A massspringdamper system can be described by the following sec-ond order, ordinary differential equation:

md2x(t)

dt 2 + c

dx(t)dt

+ kx(t) = f( t )

where m is the mass, c the damping constant, k the spring stiffness, x(t) the dis-placement and f (t ) the external force. For imprecisely known parameter values, thedynamic behavior of such a system is actually described by

(m o + m )d2x(t)

dt 2 + (c o + c )

dx(t)

dt + (k o + k )x(t) = f( t ) (2.9)

where mo , co , and ko denote the nominal parameter values and m , c and k possiblevariations over certain ranges.

By dening the state variables x1 and x2 as the displacement variable and itsrst order derivative (velocity), the second order differential equation ( 2.9 ) may berewritten into a standard state-space form

x1 = x2

x2 =1

mo + m (ko + k )x 1 (c o + c )x 2 + f

y = x1

Further, the system can be represented by an analog block diagram as in Fig. 2.10 .Notice that 1mo + m can be rearranged as a feedback in terms of

1mo and m . Fig-

ure 2.10 can be redrawn as in Fig. 2.11 , by pulling out all the uncertain variations.Let z1 , z2 , and z3 be x2 , x2 , and x1 , respectively, considered as another, ctitious

output vector; and, d 1 , d 2 , and d 3 be the signals coming out from the perturbationblocks m , c , and k , as shown in Fig. 2.11 . The perturbed system can be arranged

in the following state-space model and represented as in Fig. 2.12 :

x1x2

= 0 1

komo como

x1x2

+ 0 0 0 1 1 1

d 1d 2d 3

+ 0

1mo

f

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Fig. 2.10 Analog block diagram of Example 2.2

Fig. 2.11 Structured uncertainties block diagram of Example 2.2

z1z2z3

= komo

como

0 11 0

x1x2

+ 1 1 10 0 00 0 0

d 1d 2d 3

+

1mo00

f (2.10)

y = 1 0x1x2

The state-space model of ( 2.10 ) describes the augmented, interconnection systemM of Fig. 2.12 . The perturbation block in Fig. 2.12 corresponds to parametervariations and is called parametric uncertainty. The uncertain block is not afull matrix but a diagonal one. It has certain structure, hence the terminology of structured uncertainty. More general cases will be discussed shortly in Sect. 2.4 .

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2.3 Linear Fractional Transformations 19

Fig. 2.12 Standardconguration of Example 2.2

Fig. 2.13 Standard M conguration

2.3 Linear Fractional Transformations

The block diagram in Fig. 2.12 can be generalized to be a standard congurationto represent how the uncertainty affects the input/output relationship of the controlsystem under study. This kind of representation rst appeared in the circuit analysisback in the 1950s [140, 141]. It was later adopted in the robust control study [145]for uncertainty modeling. The general framework is depicted in Fig. 2.13 .

The interconnection transfer function matrix M in Fig. 2.13 is partitioned as

M = M 11 M 12M 21 M 22

where the dimensions of M 11 conform with those of . By routine manipulations,it can be derived that

z = M 22 + M 21 (I M 11 ) 1M 12

if (I M 11 ) is invertible. When the inverse exists, we may dene

F(M,) = M 22 + M 21 (I M 11 ) 1M 12

F(M,) is called a linear fractional transformation (LFT) of M and . Be-cause the upper loop of M is closed by the block , this kind of linear fractionaltransformation is also called an upper linear fractional transformation (ULFT), anddenoted with a subscript u , i.e. F u (M,) , to show the way of connection. Similarly,

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Fig. 2.14 Lower LFTconguration

there are also lower linear fractional transformations (LLFT) that are usually usedto indicate the incorporation of a controller K into a system. Such a lower LFT canbe depicted as in Fig. 2.14 and dened by

F l (M,K) = M 11 + M 12 K(I M 22 K) 1M 21

With the introduction of linear fractional transformations, the unstructured un-certainty representations discussed in Sect. 2.1 may be uniformly described byFig. 2.13 , with appropriately dened interconnection matrices M s as listed below.

1. Additive perturbation:

M =0 I I G o

(2.11)

2. Inverse additive perturbation:

M = G o G o G o G o (2.12)

3. Input multiplicative perturbation:

M = 0 I G o G o

(2.13)

4. Output multiplicative perturbation:

M =0 G oI G o

(2.14)

5. Inverse input multiplicative perturbation:

M = I I G o G o

(2.15)

6. Inverse output multiplicative perturbation:

M = I G o I G o

(2.16)

7. Left coprime factor perturbations:

M = M 1G

0 G oI M 1G G o

(2.17)

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2.4 Structured Uncertainties 21

where G o = M 1G N G , a left coprime factorization of the nominal plant; and, the

perturbed plant is G p = ( M G + M ) 1( N G + N ) .

8. Right coprime factor perturbations:

M = M 1G 0 M

1G

G o I G o (2.18)

where G o = N G M G 1 , a right coprime factorization of the nominal plant; and,the perturbed plant is G p = (N G + N )(M G + M ) 1 .

In the above, it is assumed that [I M 11 ] is invertible. The perturbed systemis thus

G p (s) = F u (M,)

In the coprime factor perturbation representations, ( 2.17 ) and (2.18 ), =[ M N ] and =

M N

, respectively. The block in (2.11 )(2.18 ) is supposedto be a full matrix, i.e. it has no specic structure .

2.4 Structured Uncertainties

In many robust design problems, it is more likely that the uncertainty scenario is amixed case of those described in Sects. 2.1 and 2.2. The uncertainties under con-sideration would include unstructured uncertainties, such as unmodeled dynamics,as well as parameter variations. All these uncertain parts still can be taken out fromthe dynamics and the whole system can be rearranged in a standard congurationof (upper) linear fractional transformation F (M,) . The uncertain block wouldthen have the following general form:

= diag [1I r 1 , . . . , s I r s , 1 , . . . , f ], i C , j C m j m j (2.19)

where si = 1 r i +f j = 1 m j = n with n is the dimension of the block . We may

dene the set of such as . The total block thus has two types of uncertainblock: s repeated scalar blocks and f full blocks. The parameters i of the repeatedscalar blocks can be real numbers only, if further information of the uncertainties isavailable. However, in the case of real numbers, the analysis and design would beeven harder. The full blocks in ( 2.19 ) need not be square, but by restricting them assuch makes the notation much simpler.

When a perturbed system is described by an LFT with the uncertain block of (2.19 ), the considered has a certain structure. It is thus called structured un-certainty. Apparently, using a lumped, full block to model the uncertainty in suchcases, for instance in Example 2.2 , would lead to pessimistic analysis of the systembehavior and produce conservative designs.

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