Systems & Control Letters 57 (2008) 105–111www.elsevier.com/locate/sysconle

Shift-invariant signal norms for fault detection and control

Andreas Johansson∗

Control Engineering Group, Lulea University of Technology, 971 87 Lulea, Sweden

Received 13 November 2006; received in revised form 12 July 2007; accepted 17 July 2007Available online 10 September 2007

Abstract

Two desirable properties of a signal norm to be used for the purpose of optimal control or fault detection are highlighted. These propertiesare shift-invariance and the permitting of persistent signals. A class of norms, called the window norms, is analyzed. Among norms consideredfor control purposes, they appear to be the only ones, save the L∞-norm, that satisfy both properties. The signal spaces defined by the windownorms, however, include some interesting signals that are not in the L∞-space. The window norms have been found suitable for application infault detection and are here also considered for optimal control. It is shown that they can be taken as a support for the concept of L1-controlbut may also suggest a new class of optimal controllers.© 2007 Elsevier B.V. All rights reserved.

Keywords: Signal norms; Window norms; Optimal control; L1-control; H∞-control; Fault detection; Thresholds

1. Introduction

The ability to quantify the size of a signal, such as a distur-bance or an error signal, is fundamental in both fault detectionand feedback control. The signals involved in these applicationsare e.g. control errors, control signals, residuals, disturbances,etc. and they are often modelled as unit steps, sinusoids, orrealizations of white or colored noise. All of these examplesare persistent signals, i.e. they do not vanish with time.

In this context, it is obvious to demand that the norm usedfor quantifying the signal size permits persistent signals i.e. thatthese are bounded in the sense of the norm in question. The basisfor the popular H∞-controller [20] is the well-known L2 signalnorm, since the H∞-norm is induced from this. Unfortunately,persistent signals are unbounded in the sense of the L2-norm,a fact which has motivated some work aiming to justify theH∞-controller with a signal norm that does allow persistentsignals. To this end, both bounded power norms [12,13] andsome seminorms [13,20] have been proposed.

Another demand which is quite natural to place on a norm isthat the value should not change if the signal is shifted in time,as pointed out in [11]. Intuitively, two disturbance signals orcontrol error signals, that are distinguished only by time-shift

∗ Tel.: +46 920 49 2334; fax: +46 920 49 2043.E-mail address: Andreas.Johansson@csee.ltu.se.

0167-6911/$ - see front matter © 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.sysconle.2007.07.002

should be considered equal in size. Notwithstanding this, to thebest of our knowledge, the only shift-invariant signal norm thatpermits persistent signals and has been considered for optimalcontrol is the L∞ signal norm. The induced norm is then theL1-norm of the system impulse response, thus leading to theconcept of L1-control [3,18]. Shift-invariant signal norms hasrecently appeared as a topic of current research in the field offunctional analysis. The specific problems that are investigatedconcern primarily the uniqueness of norms [5,19]. In this paper,we will justify in detail why shift-invariance is important inboth control and fault detection. We will also analyze a class ofshift-invariant signal norms called window norms in terms oftheir application to optimal control and fault detection. Thesenorms allow persistent signals and, compared to the L∞-norm,they define a larger signal space while being less conservativein the modelling of disturbances.

2. Preliminaries

In the following, a signal is a causal (or one-sided) functionof time t, i.e. a function defined on R+ := {t ∈ R|t �0} ex-cept possibly a set of measure zero. For t < 0, the value canbe assumed to be 0. All signals are assumed to be integrable(in the Lebesgue sense) on any closed interval, i.e. belong tothe space L of locally integrable functions.

106 A. Johansson / Systems & Control Letters 57 (2008) 105–111

Functions and operations applied to signals u, v, such as ab-solute value |u|, power up, or multiplication uv is to be inter-preted pointwise, so that e.g. |u|(t) := |u(t)| and (uv)(t) :=u(t)v(t). Equalities and inequalities between functions are to beinterpreted pointwise and almost everywhere, i.e. u�v meansu(t)�v(t) for all t ∈ R+ except possibly a set of measure zero.A function which is zero for all t �0 is denoted 0. The unitstep function is denoted � and the unit pulse function of width� is denoted ��, i.e.

�(t) ={

1 t �0,

0 otherwise,

��(t) ={

1 0� t < �,

0 otherwise.

The truncation operator P� is defined as (P�u)(t) = ��(t)u(t)

and the right shift operator S� for ��0 is

(S�u)(t) ={

u(t − �) t ��,

0 otherwise.

A norm is a functional, i.e. a mapping, denoted ‖ · ‖, from asignal into R+ that satisfies the following [8]:

Property 1.

(a) ‖u‖ = 0 ⇒ u = 0.(b) ‖�u‖ = |�|‖u‖.(c) ‖u + v‖�‖u‖ + ‖v‖.

A seminorm is a functional that satisfies Property 1except (a).

A star between two functions g, u ∈ L denotes convo-lution, i.e.

(g ∗ u)(t) :=∫ t

0g(t − �)u(�) d�.

A linear operator defined by convolution by a weighting func-tion is denoted by the symbol of the weighting function writtenin capital bold-face font. Thus e.g. Gu := g ∗ u. The identityoperator is denoted I, i.e. Iu := u. An induced operator normis denoted by the same symbol as the signal norm from whichit is induced, i.e.

‖G‖ := sup‖u‖�=0

‖Gu‖‖u‖ .

2.1. Signal spaces

Define the Lp-norm, 1�p < ∞ and the L∞-norm as

‖u‖pp :=

∫ ∞

0|u(t)|p dt ,

‖u‖∞ := ess supt �0

|u(t)|,

where ess supt∈� denotes supremum over all t ∈ � exceptpossibly a set of measure zero. The Lp-space, 1�p�∞ isthen the set of functions u ∈ L such that ‖u‖p < ∞ while

Lep, 1�p�∞ denotes the extended Lp-space, i.e. the set of

functions u ∈ L such that ‖P�u‖p < ∞ for all ��0. It isstraightforward to verify (e.g. using Lemma 2) that Le

q ⊂ Lep

if q > p.Obviously, the Lp spaces do not include persistent signals,

i.e. signals that do not, in some sense, vanish as t → ∞, suchas a step function or a sinusoid. Instead, one may use

‖u‖pAp

:= limT →∞

1

T

∫ T

0|u(t)|p dt (1)

which, in the case of p=2 is known as average power. Clearly,this quantity does not satisfy Property 1(a) since signals in Lp

will have ‖u‖pAp

= 0. In addition, the signal space defined byAp = {u ∈ L|‖u‖Ap < ∞} is not linear. As shown in [14], itmay occur that u, v ∈ Ap but ‖u + v‖Ap is undefined.

Since the above is only a seminorm, the following two normshave also been proposed (see [13] for the case of p = 2 withdouble-sided time axis)

‖u‖pSp

:= supT >0

1

T

∫ ∞

0|u(t)|p dt ,

‖u‖p

S′p

:= supT >1

1

T

∫ ∞

0|u(t)|p dt .

The corresponding signal spaces are denoted Sp and S′p,

respectively. In discrete time, the difference between Sp andS′

p becomes insignificant and this case is treated in [17] for ageneral p�1. In addition, several seminorms have been pro-posed, similar to (1) [13] and based on spectral density [20].

For the L1-norm and the L∞-norms, the induced norm isknown to be the L1-norm of the impulse response, i.e. ‖G‖1 =‖G‖∞ = ‖g‖1, while for the L2-norm it is the H∞-norm, i.e.‖G‖2 =‖G‖∞ where G is the Fourier transform of the impulseresponse g. The H∞-norm is also induced from the boundedpower norms ‖ · ‖S2 and ‖ · ‖S′

2(when considering a double-

sided time axis) [12,13] as well as several seminorms [13,20]including the average power ‖ · ‖A2 (with double-sided timeaxis) [10]. Unfortunately, nonpersistent signals are zero in thesense of these seminorms and can thus not be handled in thesame framework as the persistent signals.

The space AP of almost periodic functions, endowed with thenorm L∞, has also been considered in the context of optimalcontrol [16]. The elements in this space belong to the linearspan of the functions e�(t) := ei�t where � ∈ R. Although afunction in AP is not necessarily periodic, it is restricted to becontinuous, which excludes many interesting functions used inthe analysis of control systems.

The norm

supt �0

∫ t

t−1|u(�)| d� (2)

allows persistent signals and is also shift-invariant. It is used in[15] when analyzing stability of linear differential equations. Inthe sequel we will analyze a generalization of it in the contextof fault detection and optimal control.

A. Johansson / Systems & Control Letters 57 (2008) 105–111 107

2.2. Some inequalities

Three well-known inequalities will be required for the proofsin the sequel. For convenience we state them here in the par-ticular form to used.

Let f ∈ Lep and g ∈ Le

q with p, q �1 and 1/p + 1/q = 1.Then, the Hölder inequality is

∫ b

a

|f (�)g(�)| d��(∫ b

a

|f (�)|p d�

)1/p(∫ b

a

|g(�)|q d�

)1/q

.

Let f, g ∈ Lep with p�1. Then, the Minkowski inequality is

(∫ b

a

|f (�) + g(�)|p d�

)1/p

�(∫ b

a

|f (�)|p d�

)1/p

+(∫ b

a

|g(�)|p d�

)1/p

.

Let be a convex function and let f, g, ◦ f ∈ L, g�0 and∫ b

ag(�) d� > 0. Then, the Jensen inequality [9] is

(∫ b

af (�)g(�) d�∫ b

ag(�) d�

)�∫ b

a(f (�))g(�) d�∫ b

ag(�) d�

.

The following two lemmas are used to improve readability ofthe proofs in the sequel. Lemma 1 is proved1 in [6] in thegeneral case of matrix-valued signals and Lemma 2 is a con-sequence of the Hölder inequality.

Lemma 1. Let g, u, v ∈ L, w ∈ Le∞, and define w(t) :=sup

�∈[0,t]|w(�)|. Then

(a) If g�0 and v�u then g ∗ v�g ∗ u.(b) |g ∗ u|� |g| ∗ |u|.(c) If g�0 then g ∗ |uw|�(g ∗ |u|)w.

Lemma 2. Let q �p�1 and u ∈ Leq . Then, for b�a,

∫ b

a

|u(�)|p d��(b − a)1−p/q

(∫ b

a

|u(�)|q d�

)p/q

.

Proof. Let s = q/p�1 and r = q/(q − p)�1 so that 1/s +1/r = 1. From the Hölder inequality it then follows that

∫ b

a

|u(�)|p d��(∫ b

a

d�

)1/r(∫ b

a

|u(t)|ps d�

)1/s

= (b − a)1−p/q

(∫ b

a

|u(�)|q d�

)p/q

. �

3. Shift-invariant norms

Definition 1. A signal norm ‖ · ‖ is said to be shift-invariant if‖u‖ = ‖S�u‖ for any ��0.

1 The proof is actually for g ∈ Lep and u, v ∈ Le

q with 1/p + 1/q = 1but will, for the formulation below, be exactly the same.

From the perspective of optimal control, it seems quite nat-ural that two disturbance signals that are distinguished only bya time-shift should be considered equal in size from a controlpoint of view. Similarly, two control error signals that are onlydistinguished by a time-shift should also be considered equalin size. Indeed, if a control error would be considered smallerwhen shifted in time, then a control design algorithm couldproduce a “better” controller by simply delaying the controlerror. This is in fact a conceivable scenario as indicated by thefollowing example:

Example 1. Consider the scalar, causal process G and the twocontrollers F1 and F2 =S(I + F1G(I −S))−1F1 where S isa time-shift. The controller F2 may be implemented as a time-shift in series with a feedback loop with F1 in the forward pathand G(I −S) in the feedback. Thus F2 is causal provided thatF1 is. Furthermore, straightforward calculations reveal that theclosed-loop systems T1 = (I + GF1)

−1GF1 and T2 = (I +GF2)

−1GF2 are related as T2 =ST1 and thus the control errorof the two closed-loop systems differ only by a time-shift.

Fault detection essentially amounts to online validation ofa model for the unfaulty process. A fault detection algorithmtypically consists of two parts, residual generation and residualevaluation. The residual generator is e.g. a Luenberger observeror some other filter that, based upon the measured signals,generates an output which is supposed to be nonzero in the caseof fault and ideally zero otherwise [4]. If the process model isperfectly known, then the residual will be possible to expressas r =Ge+Hf where e is some disturbance, f is the fault whileG and H represent known dynamics. The basic problem in thecontext of residual generation is to minimize the sensitivity ofthe residual with respect to the disturbance while maintaininga high sensitivity to the fault. This can be formulated as anoptimal control problem and the justification for using shift-invariant signal norms is therefore evident in this context also.

Since the residual is always affected by disturbances, onemust be able to decide whether it is large enough to infer thepresence of a fault. In its simplest form, online residual eval-uation can be formulated as comparing some functional of thetruncated residual Pt r to a threshold, for each time t. If thethreshold is exceeded at time t, then a fault is declared. Moregenerally, define a set of admissible disturbances as the ballE = {e ∈ L|‖e‖�} for some signal norm ‖ · ‖. Then, given atruncated residual Pt r , determine whether there exists an e ∈E such that Pt r =PtGe. If the answer is no, then a fault is de-clared. Now consider two disturbance signals e1 and e2 =Se1only distinguished by a time-shift S and resulting in the tworesiduals r1 and r2 =Sr1. Provided time-invariant conditions,it seems reasonable that these two disturbances should give riseto the same fault decision, albeit time-shifted. If the norm usedin the definition of E is not shift-invariant, however, this maynot be the case, for then it may occur that e1 ∈ E and e2 /∈ Eor vice versa.

From the definition of shift-invariance, it is clear that Lp andL∞ are shift-invariant norms while Sp and S′

p are not. Of thesenorms, the L∞-norm is thus the only one that is shift-invariant

108 A. Johansson / Systems & Control Letters 57 (2008) 105–111

and permits persistent signals. The main drawback of theL∞-norm is that it provides no averaging of the signal, i.e. thatnarrow peaks have an excessive influence. This is the reasonfor introducing the Window norms.

4. The Window norms

Definition 2. A window function is a piecewise continuoussignal w which is not identically zero (a.e.) and satisfies0�w(t)�Ce−at for all t �0 and for some positive C and a.

Note that a function that satisfies this property belongs toany Lp space for p�1 and also L∞. Given a window functionw, and a signal u ∈ L, a generalization of the norm (2), herenamed window norm, may be defined as

‖u‖pw,p := sup

t �0

∫ t

0w(t − �)|u(�)|p d�

= ‖w ∗ |u|p‖∞. (3)

Remark 1. It is straightforward to verify that u ∈ Lep (or

equivalently |u|p ∈ Le1) is necessary for ‖u‖w,p < ∞. This, in

combination with the fact that w is a bounded function yieldsthat w ∗ |u|p is continuous (see e.g. [2, Exercise 5.3.7]) so thatsup can be used in the definition above instead of the moregeneral ess sup.

Theorem 1. Let w be a window function. Then, for p�1, thefunctional ‖·‖w,p defined by (3) is a shift-invariant signal norm.

Proof. First it will be shown that (3) constitutes a norm, i.e.satisfies Property 1, and then that it satisfies Definition 1. Thecondition of piecewise continuity implies that w can be boundedfrom below by a pulse function, i.e. there exist positive, finite D,�, and � such that w(t)�D��(t −�). Thus, due to Lemma 1(a),‖u‖p

w,p = ‖w ∗ |u|p‖∞ �‖D�� ∗ |u|p‖∞ �D∫ t

t−� |u(�)|p d�for all t �0. In consequence, ‖u‖w,p > 0 if ‖u‖p > 0 whichshows Property 1(a). Property 1(b) follows from ‖�u‖p

w,p =‖w ∗ |�u|p‖∞ = ‖w ∗ (|�|p|u|p)‖∞ = ‖|�|p(w ∗ |u|p)‖∞ =|�|p‖w ∗ |u|p‖∞ =|�|p‖u‖p

w,p. To show the triangle inequality(Property 1(c)), first note that the Minkowski inequality leadsto(∫ t

0w(t − �)|u(�) + v(�)|p d�

)1/p

=(∫ t

0|w(t − �)1/pu(�) + w(t − �)1/pv(�)|p d�

)1/p

�(∫ t

0w(t−�)|u(�)|p d�

)1/p

+(∫ t

0w(t−�)|v(�)|p d�

)1/p

i.e. (w ∗ |u + v|p)1/p �(w ∗ |u|p)1/p + (w ∗ |v|p)1/p. Thus

‖u + v‖w,p = ‖(w ∗ |u + v|p)1/p‖∞�‖(w ∗ |u|p)1/p + (w ∗ |v|p)1/p‖∞�‖(w ∗ |u|p)1/p‖∞ + ‖(w ∗ |v|p)1/p‖∞= ‖u‖w,p + ‖v‖w,p.

Finally, shift-invariance follows from ‖Su‖pw,p = ‖w ∗

|Su|p‖∞ = ‖w ∗ (S|u|p)‖∞ = ‖S(w ∗ |u|p)‖∞ = ‖w ∗|u|p‖∞ = ‖u‖p

w,p for an arbitrary time-shift S. �

The window norms were introduced in [7] for the case p =1 and later generalized to p�1. Intuitively, it considers theLp-norm of the signal over a window whose shape is givenby w. The window is then slid over all t �0 and the windownorm is given as the largest of these time-shifted Lp-norms.For p�1, a signal space based upon the window norms maybe defined as

Wp = {u ∈ L|‖u‖�1,p < ∞}.Although there are many potential choices of w, all the normsfor a fixed p are equivalent, as shown in the following theorem.In consequence, the signal space {u ∈ L|‖u‖w,p < ∞} remainsequal to Wp regardless of the choice of the window function w.

Theorem 2. Let p�1. Then, for any window functions w

and w0, the corresponding norms ‖u‖w,p and ‖u‖w0,p areequivalent in the sense that there exist a, b > 0 such thata‖u‖w0,p �‖u‖w,p �b‖u‖w0,p.

Proof. Recall that there exist positive, finite D0, �0, and �0such that w0(t)�D0��0(t −�0). Furthermore, the condition ofexponential boundedness implies that w can be bounded by

w(t)�Ce−at �C

∞∑n=0

��0(t − n�0)e−an�0

� C

D0

∞∑n=0

w0(t + �0 − n�0)e−an�0 .

Thus, for an arbitrary u ∈ Lep,∫ t

0w(t − �)|u(�)|p d�

�∫ t

0

C

D0

∞∑n=0

w0(t − � + �0 − n�0)e−an�0 |u(�)|p d�

� C

D0

∞∑n=0

e−an�0

∫ t

0w0(t − � + �0 − n�0)|u(�)|p d�

� C

D0

∞∑n=0

e−an�0‖u‖pw0,p = C/D0

1 − e−a�0‖u‖p

w0,p.

Thus ‖u‖w,p �b‖u‖w0,p for a finite nonzero b. Switching w

and w0 and repeating the procedure yields the result. �

As mentioned before, it is straightforward to verify thatWp ⊂ Le

p. The following theorem clarifies the relation to someother important signal spaces. Please note that ⊂ denotes strictinclusion.

Theorem 3. Let q > p�1. Then

(a)⋃

p�n�∞Ln ⊂ Wp.(b) Wq ⊂ Wp.(c) Wp ⊂ S′

p.

A. Johansson / Systems & Control Letters 57 (2008) 105–111 109

Proof. For Part (a), consider an arbitrary u ∈ Ln, n�p. SinceLn ⊂ Le

n ⊂ Lep, we may calculate

(�1 ∗ |u|p)(t)=∫ t

t−1|u(�)|p d��

(∫ t

t−1|u(�)|n d�

)p/n

�‖u‖pn ,

where the first inequality was due to Lemma 2. Thus‖u‖�1,p �‖u‖n and consequently Ln ⊆ Wp. For the case ofu ∈ Le∞,

(�1 ∗ |u|p)(t) =∫ t

t−1|u(�)|p d�� sup

��0|u(�)|p �‖u‖p∞

and consequently⋃

p�n�∞ Ln ⊆ Wp. To see that the inclu-sion is strict, consider the periodic function

ur(t) = 1

(t − t�)1/r, (4)

where ·� denotes integer part. Clearly, ur belongs to neitherLp, p�1 nor L∞ regardless of the choice of r > 0. However,it is easily verified that

‖ur‖p�1,p =

∫ 1

0|ur(t)|p dt

=∫ 1

0t−p/r dt =

{ r

r − p, r > p,

∞, r �p

and thus ur ∈ Wp for r > p.For Part (b), take an arbitrary u ∈ Wq ⊂ Le

q ⊂ Lep. Then, by

Lemma 2,

(�1 ∗ |u|p)(t) =∫ t

t−1|u(�)|p d�

�(∫ t

t−1|u(�)|q d�

)q/n

�‖u‖p�1,q

i.e. Wq ⊆ Wp. To see that the inclusion is strict, consider onceagain the signal (4) (or, equivalently, just one period of it) withr = q so that ur ∈ Wp but ur /∈ Wq .

For Part (c), take an arbitrary u ∈ Wp ⊂ Lep and divide the

interval [0, T ] into T � + 1 subintervals of width 1. Then,

∫ T

0|u(�)|p d��

T �∑n=0

∫ n+1

n

|u(�)|p d�

= T �∑n=0

∫ T

0�1(n + 1 − �)|u(�)|p d�

�(T + 1)‖u‖p�1,p.

Thus

‖u‖p

S′,p = supT >1

1

T

∫ T

0|u(�)|p d�� sup

T >1(1 + 1/T )‖u‖p

�1,p

= 2‖u‖p�1,p

and consequently Wp ⊆ S′p. Finally, to see that this inclu-

sion is also strict, consider a signal consisting of pulses of

exponentially increasing amplitude and interval

up(t) = 21/p�2(t) +∞∑

n=1

2n/p�1(t − 2n). (5)

This signal is clearly unbounded in the sense of thenorm on Wp. However, it is straightforward to verify that∫ T

0 |up(�)|p d��2T with equality for e.g. T �3 and thus‖up‖p

S′p

= 2. �

4.1. Applications of the window norms

Clearly, the proposed signal spaces Wp have a number ofadvantages. They are quite large since they contain all spacesLq for q �p and also persistent signals, including some signalsnot in the L∞-space (such as (4)). Yet they exclude some patho-logical bounded power signals (such as (5)). Furthermore, thecorresponding window norms ‖ · ‖w,p are shift-invariant andalso quite intuitive. Compared to norms and seminorms definedusing the autocorrelation function or in the frequency domain,it is often quite easy to judge the size of a signal in a windownorm from an ordinary time plot.

The introduction of the norm (3) was originally motivated byits application to the residual evaluation problem in fault de-tection. A fault is declared to have occurred if some evaluationfunction E of the residual r exceeds a threshold �. Provided thatthe disturbance e is known to be bounded in the sense ‖e‖�,the threshold should then satisfy �(t)�sup‖e‖� (Er)(t) in or-der to avoid false alarms. In practice, the L2-norm over a slid-ing window is often used as evaluation function [4] or moregenerally [6,7]

(Ew,pr)(t) :=(∫ t

0w(t − �)|r(�)|p d�

)1/p

= (w ∗ |r|p)1/p(t), (6)where w is a window function. The connection to the windownorm becomes apparent from the relation ‖u‖w,p =‖Ew,pu‖∞.

The residual is, in general, affected by disturbances as well asknown signals, the latter due to model uncertainty. Calculating adetection threshold for this general case with model uncertaintyis treated in [1,6]. For simplicity, however, consider the casewhen the residual can be expressed as the output of a linearsystem with impulse response g and input e. If the disturbancee is assumed to be bounded in the sense of the norm (3) and thewindow function w and the power p is the same in the norm(3) as in the evaluation function (6), then calculating the faultdetection threshold is particularly easy. This fact can be statedas follows:

Proposition 1. Let u ∈ Wp and g ∈ L1. Then Ew,p(g ∗u)�‖u‖w,p(|g| ∗ �).

Proof. The Jensen inequality, with a = 0, b = t , (·) = | · |p,f (�) = |u(t − �)|, and g substituted by |g| gives, for all t �0,(∫ t

0 |u(t − �)||g(�)| d�∫ t

0 |g(�)| d�

)p

�∫ t

0 |u(t − �)|p|g(�)| d�∫ t

0 |g(�)| d�. (7)

110 A. Johansson / Systems & Control Letters 57 (2008) 105–111

Thus

(Ew,p(g ∗ u))p = w ∗ |g ∗ u|p�w ∗ (|g| ∗ |u|)p

= w ∗(( |g| ∗ |u|

|g| ∗ �

)p

(|g| ∗ �)p)

�w ∗( |g| ∗ |u|p

|g| ∗ �(|g| ∗ �)p

)

= w ∗ ((|g| ∗ |u|p)(|g| ∗ �)p−1)

�(w ∗ (|g| ∗ |u|p))(|g| ∗ �)p−1

= (|g| ∗ (w ∗ |u|p))(|g| ∗ �)p−1

�(|g| ∗ (�‖u‖pw,p))(|g| ∗ �)p−1

= ‖u‖pw,p(|g| ∗ �)p,

where the first inequality is due to Lemma 1(b), the secondinequality is from (7) and the third inequality follows fromLemma 1(c) and the fact that |g| ∗ � is nondecreasing. In ad-dition, Lemma 1(a) was used in the first, second and fourthinequality. �

Clearly, one may obtain very different norms depending onthe choice of the window function. Consider letting w approacha unit step function which results in the Wp-norm approachingthe Lp-norm. On the other hand, letting w approach the Diracdelta function yields the L∞-norm. Note, however, that neitherthe unit step function nor the Dirac delta function satisfies theconditions of a window function. Instead, a typical choice ofw could be a pulse function whose duration covers the rele-vant time constants. For implementation purposes, a favorablechoice of w is a decreasing exponential function, since cal-culation of Ew,pr (and consequently ‖r‖w,p) then essentiallyamounts to filtering through a first-order filter. This is desir-able in the context of fault detection, where the calculation isto be performed in real time. Using a decreasing exponentialfunction can then also be interpreted as applying a forget-ting factor in order to attach more weight to the most recentdata.

Apart from its application in fault detection, the benefits ofthe Wp-norms makes it conceivable for basing controller op-timization on. This, of course, requires the ability to calculatethe operator norm induced from the Wp-norms. As mentionedbefore, both the Lp-norm and the L∞-norm can be consideredas extreme cases of the Wp-norms. The corresponding inducedoperator norms are then the H∞-norm (for p = 2) and theL1-norm, respectively. Thus, the operator norms of the win-dow norms will, in general, depend on both w and p. Exactexpressions for these operator norms are nontrivial to find andis left as an open problem but the following theorem gives theH∞-norm and the L1 norm as bounds for all these norms.

Theorem 4. Let w be a window function and g ∈ L1 an arbi-trary impulse response with G as the corresponding operator.Then, for p�1, ‖G‖∞ �‖G‖w,p �‖g‖1.

Proof. For the first inequality, we assume that the supremumin the H∞-norm is attained, i.e. there exists an such that|G( )| = ‖G‖∞. If this is not the case, then there exists asequence n such that |G( n)| → ‖G‖∞ and the followingproof is valid for each point in this sequence. The response tothe input u(t) = sin( t) is

y(t) =∫ t

0g(�)u(t − �) d�

=∫ ∞

0g(�) sin( (t − �)) d� −

∫ ∞

t

g(�) sin( (t − �)) d�

= |G( )| sin( t + � G( )) − �(t),

where �(t) = ∫∞t

g(�) sin( (t − �)) d�. Let tn = (2n� −� G( ))/ for all integers n such that tn �0 and de-fine a sequence of truncated and time-shifted outputs asyn = StnPtny. Due to shift-invariance, it is clear that‖yn‖w,p =‖StnPtny‖w,p =‖Ptny‖w,p �‖y‖w,p. Furthermore,yn(t)=|G( )| sin( t)−�n(t) where �n=StnPtn� and then thetriangle inequality gives ‖yn‖w,p � |G( )|‖u‖w,p − ‖�n‖w,p.Thus it remains to show that ‖�n‖w,p → 0. Now,

|�(t)| =∣∣∣∣∫ ∞

t

g(�) sin( (t − �))d�

∣∣∣∣ �∫ ∞

t

|g(�)| d� := �(t)

and thus ‖�‖w,p �‖�‖w,p and ‖�n‖w,p �‖�n‖w,p where�n =StnPtn �. Since all the window norms for a fixed p�1 areequivalent by Theorem 2, it suffices to show that ‖�n‖w,p → 0for one particular window function, e.g. w = �1. With thischoice, ‖�n‖p

w,p = ∫ 10 (�n(t))

p dt since �n(t) is nonincreasing.Also, limt→∞ �(t) = 0 and thus limn→∞ �n(t) = 0 point-wise on any interval. This, in combination with the fact that|�n(t)|�‖g‖1 enables the use of Arzelà’s theorem to show that∫ 1

0 (�n(t))p dt → 0 so that ‖�n‖w,p → 0 and the result follows.

To show the second inequality, consider an arbitrary u ∈ Wp.Then ‖Gu‖w,p = ‖g ∗ u‖w,p = ‖Ew,p(g ∗ u)‖∞ �‖u‖w,p‖g‖1where the inequality is from Proposition 1. �

Theorem 4 shows that the L1-norm is an upper bound forthe operator norms induced from the window norms. Sincethe Wp-space is larger than L∞-space, this can be taken asan improved support for the concept of L1-control which isnormally motivated by the L∞-norm. Another interpretation ofTheorem 4 is that the window norms (primarily with p=2) withthe window w as a degree of freedom could potentially leadto a class of optimal controllers in between the two extremesH∞-control and L1-control.

Remark 2. Theorem 4 is likely to be valid in a much moregeneral setting of shift-invariant norms as is straightforwardto show in discrete time. Assume that Property 1(c) can beextended to countable sums, i.e.∥∥∥∥∥

∞∑k=0

uk

∥∥∥∥∥ �∞∑

k=0

‖uk‖. (8)

A. Johansson / Systems & Control Letters 57 (2008) 105–111 111

Then, since discrete time convolution between an in-put u and a pulse response g ∈ �1 can be expressed asy =∑∞

j=0 g(j)(Sj u), the output is bounded by

‖y‖ =∥∥∥∥∥∥

∞∑j=0

g(j)Sj u

∥∥∥∥∥∥ �∞∑

j=0

‖g(j)Sj u‖

=∞∑

j=0

|g(j)|‖Sj u‖ = ‖u‖∞∑

j=0

|g(j)| = ‖u‖‖g‖1,

where (8) was used in the inequality, Property 1(b) in the secondequality, and shift-invariance in the third equality. Thus, the�1-norm is an upper bound for the induced norm of all shift-invariant norms in discrete time that satisfy (8).

5. Conclusions and open problems

Shift-invariance and the permitting of persistent signals havebeen highlighted as two desirable properties of a signal norm tobe used for the purpose of optimal control or fault detection. Aclass of norms, the window norms, that satisfy both properties isanalyzed. These norms have been found suitable for applicationin fault detection and are here also shown to be conceivable forformulating optimal control. They can be taken as a support forthe concept of L1-control but have possibly the potential to bethe basis for a new class of optimal controllers. Open problemsin this context are determining completeness of the signal spacedefined by the window norms and to find exact expressions fortheir induced operator norms.

Acknowledgments

The author wishes to thank the Hjalmar Lundbohm researchcenter funded by LKAB for financing this research. The veryinsightful and helpful comments by the anonymous reviewersare gratefully acknowledged.

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