Regulation of linear discrete time-varying systems

Regulation of linear discrete time-varying systems

Citation for published version (APA):

Engwerda, J. C. (1988). Regulation of linear discrete time-varying systems. Technische Universiteit Eindhoven.

https://doi.org/10.6100/IR289505

DOI:

10.6100/IR289505

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Published: 01/01/1988

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REGULATION OF LINEAR

DISCRETE

TIME-VARYING SYSTEMS

PROEFSCHRIFT

TER VERKRIJGING V AN DE GRAAD V AN DOCTOR AAN DE TECHNISCHE UNIVERSITEIT EINDHOVEN, OP GEZAG VAN DE RECTOR MAGNIFICUS, PROF. IR. M. TELS, VOOR EEN

COMMISSIE AANGEWEZEN DOOR HET COLLEGE VAN DEKANEN IN HET OPENBAAR TE VERDEDIGEN OP VRIJDAG

2 SEPTEMBER 1988 TE 16.00 UUR

DOOR

JACOB CHRISTIAAN ENGWERDA

Prof.Dr.Ir. M.L.J. Hautus de eo-promotor

Traditioneel wordt het voorwoord van een proefschrift meestal gebruikt om al die mensen te bedanken die hebben bijgedragen aan het tot stand kornen ervan. De vraag die hierbij onmiddellijk rijst is hoever je terug moet. gaan in de tijd en of je alle mensen moet bedanken die een duidelijke invloed hebben gehad op de keuzes die je hebt gemaakt en datgene wat je uiteindelijk hebt bereikt. Ikwil van dit voorwoord echter geen bibliografie maken. Daarvoor voel ik me nog te jong. Daarom, alle mensen die zich straks

eekort gedaan voelen omdat ik ze niet expliciet heb genoemd moeten de vol-gende zin nog maar eens goed lezen. Hardstikke bedankt voor alles wat je voor me gedaan hebt.

Ik zou dit voorwoord nu natuurlijk kunnen afsluiten. Maar ja, dan zou ik een aantal mensen groot onrecht aandoen. En dit is wel het laatste wat ik met dit proefschrift beoog.

Omdat ik toch ergens moet beginnen wil ik allereerst een viertal mensen bedanken die een belangrijke rol hebben gespeeld in mijn wiskunde opleiding voordat ik naar de universiteit ging. Dit zijn achtereenvolgens geweest de heren Plant, De Boer, Smit en Van der Berg. Verder hoort in dit rijtje eigenlijk mijn broer Chris ook thuis, want zonder hem was ik waarschijnlijk niet verder met de wiskunde gekomen dan de 4e klas van het atheneum.

Van de medewerkers van de Rijksuniversiteit te Groningen wil ik in het bijzonder de heren Klamer en Scheelbeek bedanken, die mij vooral tijdens de kandidaatsfase veel geholpen hebben met respectievelijk het vak metri-sche ruimten en de samenstelling van mijn studiepakket.

Ook Jan Willems wil ik bij deze nog eens bedanken, want zonder diens inspi-rerende werking had ik misschien geen specialisatie systeemtheorie gekozen. Verder ben ik mijn studiemaatjes Jan Schut, Rein Smedinga en Lourens Aalders nog steeds zeer erkentelijk voor onze samenwerking tijdens deze periode. Als laatste uit bet rijtje Groningen wil ik dan nog Pieter Otter noemen, die voor mij ook na de studie steeds een bron van inspiratie is gebleven.

Zo, een derde gedeelte van de bedankjes zit er nu ongeveer op. Het bedenken van mooie volzinnen begint me al steeds moeilijker te vallen. Alle zinnen beginnen onderhand op elkaar te lijken vind ik. Hopelijk blijkt een en

ander nog verteerbaar tot het eind. Anders bedank ik straks nog mensen in mijn proefschrift die het werk alweer dichtgeklapt hebben voordat zij het voor hen belangrijkste deel ervan hebben gelezen. Nog effetjes volhouden mensen, a.u.b.

Nu dan de periode Eindhoven.

Allereerst denk ik hierbij aan Kees Praagman die geeh:I!r<ende deze tijd erg veel tijd en energie heeft gestoken in mijn begeleiding. Het is fantastisch wat je gedaan hebt, Kees! Daarnaast ben ik ook Malo Hautus veel dank

ver-schuldigd voor het altijd klaar staan om oplossingen proberen te vinden voor vragen van mijn kant en voor het zeer gedegen lezen van het manuscript van dit proefschrift. Ook mijn oud-kamergenoot Jacob van der Woude moet ik in dit verband noemen. Hij heeft mij vaak geholpen bij het formuleren en oplossen van problemen. Verder wil ik de secretaresses Harma Koops en Caroline Verhulst bedanken voor het vele werk wat ze mij uit handen hebben genomen. Tot slot van deze periode wil ik nog bedanken de kerncommissie bestaande uit de beren Kwakernaak en Olsder, en mevrouw Baselmans-Weijers voor het typen van dit proefschrift.

Zo, en dan nu nog de mensen die met het directe werk misschien iets minder te maken hebben gehad, maar die zeker indirect een belangrijke rol hebben gespeeld bij de realisatie ervan. Sommigen daarvan waren zelfs onmisbaar. Tot de laatste categorie behoren zeker mijn ouders, Chris, opa en (wijlen) mijn oma. ((Jolk. Jlllijn vrienden Eddie, Gerrie, Wim en Bert waren altijd troll!Wle JN;,:m1il;genoten in de :s'txijd. Verder denk ik uit mijn Eindhoven-tij.d met tple-JZii:er terug aan met :m.mne Rik, iFJrans., Ingrid, Geert en Diny, bij wie ik vele

gezellige uren heb 'llll(l)jl)Jr~Jr~dlnit.

Tot slot noem ik jou, Carilllle, apart, omdat je ook :®ID"m lli>ij:~~:ondere plaats inneemt.

Voordat nu iedereen (al dan niet) teleu.Jrgest:ellil! llr.rett boekje dichtgooit, moet mij nog van het hart dat ik geprobeerd heb om in nog enigszins

begrijpe-lijke (wiskundige) taal in hoofdstuk 1.1 uit te leggen waarmee ik me nu in vredesnaam bezighoud in de rest van dit boekje. Voor mensen met een middel-bare opleiding hoop ik dat het te volgen is. Het zal misschien wel wat moeite kosten, maar ja, je krijgt in het leven nu eenmaal bijna niets voor niets. Succes bij het lezen, en nogmaals bedankt,

Voorwoord List of Symbols

I. Introduction

1.1. An illustrative economic example 1.2. Some basic notions

1.3. Mathematical preliminaries

11. Point Controllability Problems

11.1. Controllability and Reachability 11.2. Reconstructability

11.3. Stabilizability

11.4. Observability and Detectability

Ill. Target Path Problems

111.1. The general Target-Path Controllability problem 111.2. The time-invariant case

111.3. The Decoupled-Target-Path Controllability problem

IV. Characterization of Admissible Trajectories

i V 1 11 14 18 19 24 27 35 43 44 49 53 57

IV. 1. Strongly admissible target paths 58

IV.2. The approximately admissible target paths of level a 62

IV.3. The asymptotically admissible target paths 65

IV.4. Admissibility consequences of predescribed successful

control structure 67

V. Asymptotically Admissible Target Paths: Controller Design 71

V.1. The EQL-regulator: Introduction 72

V.2. The EQL-regulator: The finite planning horizon case 74

V.3. Convergence of the Recursive Riccati Equation 76

V.4. Convergence of the finite planning horizon EQL regulator 80 V.S. The EQL-regulator: The infinite planning horizon case 94

V.6. Design of a successful controller 98

V.7. The infinite horizon EQL problem: Some special cases 100

VI. Tracking Target Paths with only Short Run Information 106

VI.t. A minimum norm controller 107

VII. Characterization of Admissible Minimum Variance Controlled

Target Paths 119

VII.1. The Extended Minimum-Variance controller 120

VII.2. The Minimum Variance controller 126

VII.3. The construction of stabilizing weighting matrices

for time-invariant systems 130

VII.4. A simulation study 133

Conclusion 143

References 144

Subject index 149

Samenvatting 152

Symbol Contents

A, B, C, G basic system matrices

AT transposed of A A+ Moore-Penrose inverse of A A -1 inverse image of A A(k+i ,k)

c

CL-system d det A diag(A,B) Im A Ker A K. 1

complementary orthogonal matrix of A in IRn, i.e. Im A (t) Im Al. "" IRn

orthogonal projection onto Ker A

transition matrix: i > 0

i=O

A(k) mod Rk

A(k+i-1)

* ... *

A(k)

I

Algebraic .Riccati Equation

constmlption

Closed Loop system

exogenous input determinant of A Kronecker delta investment image of A kernel of A controllability index Chapter Page L2 12 I.2 13 I.3 14 I. 3 16 VII.1 122 I.2 13 II.3 30 V.7 102 I.1 3 V. 1 73 I.1 11 VII.4 134 I.1 3 VII.3 132

m MV n P(k,Q,R) JR r RRE, RRE1_, RRE" r(A) S[k,k-N] cri (A) a' (A)=a 1 (A) TPC(k₀;p,q) u(k) v[k,.t] V[k,k-N] W[k,k+i] eigenvalue of A

number of control (input) variables

Minimum Variance

impulse response from u[k

0+p+q-2,k0] to y [k

0+p+q-1,k0+p]

number of state variables

I -B (k} (R(k) + BT (k) Q(k+l )B(k)) B T (k) Q (k+l)

set of real numbers

number of output variables

Recursive Riccati Equations

spectral radius of A reachability subspace

dimension ~

Sufficient Control Existence conditions

reachability matrix

[B(k) !A(k+1 ,k)B(k-1)

l· ..

:A(k+1 ,k-N+1)B(k-N)] singular value of A

largest singular value of A

Target Path Controllable at k

0 with lead p control (input) at time k

unobservable subspace at k 0

unused reachability subspace in TPC(k₀;p,i-1)

reference value for variable v

T T T (v (k), ••• ,v (Q.)) [G(k)lA(k+1,k)G(k-1)!···:A(k+1,k-N+1)G(k-N)] [C(k)T: ••• ;{C(k+i)A(k+i,k)}T]T,

.

_.

i > 0 1.3 16 1.2 12 VII.2 126 III. 1 45 1.2 12 VII. 1 120 I. 2 12 I. 2 12 V 74, 79,100 I. 3 16 II. 1 20 II. 3 29 V.4 80 1.2 13 1.3 17 I. 3 17 III. 1 45 1.1 8 II.4 36 III.l 46 I.2 13 1.2 13 IV. 1 59 1.2 13

x(k,k

0,x0,u) state at time k resulting from initial state x

0 at k0 and u

xstab(ko) stabilizability subspace at ko x-(ko) potential stabilizability subspace

X-(A(•,k

0)) modal subspace y(k,k

0,x0,u) output at time k resulting from initial state x 0 at k0 and u y

zk

0 zko(k) ~ 3 V

:=

00 A< (;) II•IIE HI s llxllQ (!}

n

u

n

I

a. i=1 ~ 0 p,q B national income

zero-controllability subs pace at k 0 command error flow

almost equal to

there exists

for all

definition

infinity

B -A is positive (semi-) definite

Euclidean norm

spectral norm (also operator- or 2-norm)

T

x Qx direct sum

intersection union

zero matrix with p rows and q columns

1.2 13 II.3 28 II.3 30 II.3 28 1.2 13 1.1 3 II. 1 20 IV. 1 59 !.2 12 1.2 14 II.3 29

We start this thesis bij considering a macroeconomic example. It concerns the most elementary form of the dynamic multiplier-accelerator model

Samuelson (1939} and Hicks (1950) developed in order to give an explanation for the existence of trade cycles in economics.

This example serves purely illustrative purposes. The objective is to readers not familiar with linear systems theory an idea what this thesis is all about. The section ends with an overview of the main problems that are studied in the forthcoming chapters. Furthermore, a more realistic version of the economic example is then provided. This version is used in Chapter 7 for simulation experiments.

Section

i

contains introduction of the system considered in this thesis. Moreover, several notions which appear frequently throughout the thesis are discussed.

The last section, Section 3, contains a collection of elementary mathemat-ical results which are needed in the sequel. Most of the results concern matrix properties. Special attention is paid to the existence and geometric

interpretation of the Moore-Penrose inverse of a matrix.

I. 1. An illustrative economic example

In this section we briefly illustrate, by means of a macroec.onornic model, the main problems that are an~lyzed in this thesis.

This macroeconomic model is kept rather simple for didactical reasons. The model structure we obtain from it, however, is very general and captures a. very broad class of models studied in today's linear systems theory. In the remainder of this thesis we analyze this general class of models. For this reason, the results obtained in this thesis are not only of interest for economics but also for other applications which fit into this genera 1 theoretical framewr'rk.

The example originates from the early study of cyclical fluctuations in the level of economic activity which can be observed by examining annual changes over a long period of years.

Figure 1 provides an example. It shows the evolution of the percentual annual growth of the real national income for the Dutch economy over the past three decades (see De Roos (1985), pp. 185).

10

11150 '52 '54 '56 '56 '80 '82

·e.c

'68 '68 '70 '72 '7.. '78 '78 '80 '82

·e.c

....

_

_{10-years moving average}

Figure 1. Percentual growth of real national income.

Samuelson was the first to study this phenomenon in (1939). We shall pursue his treatment of this problem and we will give a more updated version of his theory later on.

First, however, we introduce some notation. Note that we can determine the real national income, denoted by Y, at each year. To distinguish the national incomes in different years, we shall add the year about which we

are talking between brackets behind the symbol Y. So, e.g., Y(195l) denotes the level of real national income in the year 1951.

Now, Samuelson assumed in his analysis that in a country for which the influence of foreign countries on its economy can be neglected (like, e.g. the U.S. and U.S.S.R. some years ago), the following relations hold:

(a) The national income (denoted by the symbol Y) is the sum of con-sumption (C), investment (I} and government expenditures (Ge) at any year. That is,

Y(k) C(k) + I(k) + Ge(k) at any year k. (1)

Here Y(k) denotes the value of the national income at year k, C(k) for consumption, etc.

(b) The consumption in the current year is a fraction of national income from the previous year. Thus, we can write

C(k) c(k) x Y(k-1) , (2)

where c(k) is a known factor, which may change in time. In economic literature, this factor is known as the marginal propensity to con-sume.

(c) The investment adjustes fully to lagged changes in income. Or, more explicitly, the induced investment is a fraction of the difference between the national income in the last year and national income in the year before that. In formula:

I(k) = v(k) x (Y(k-1) Y(k-2)) ,

where v(k) is a time-dependent factor, in literature known as the

accelerator.

(3)

From relation (c) we see that the investment depends on changes income, whereas changes in investment cause changes in income according to (a). Due to this modelling, cycles in income may occur like we observed in Figure 1. We illustrate this in an example.

Example: Assume that in the year 1950 for a fictitious country, consump-tion would be 400 (billions of dollars, or any other currency), investment 100, and the amount of government expenditures also 100. Moreover, assume that the factors c(k) and v(k) are equal to 0.8 and 1, respectively, at any year after 1950.

Then, according to (1), national income in the year 1950 would be:

Y(1950)

:=

400 + 100 + 100 600 •

Moreover, we have according to (2) that

C(1950)

=

0.8 x Y(1949) •

As C(1950) was 400, it follows that national income for the country in 1949 must have been 500.

Under the assumption that the government expenditures do not change in time, we can now calculate consumption, investment and national income for the year 1951. This \vorks as follows. According to (2), C(1951) will be equal to 0.8 x 600

=

480, and using (3) we see that Y(1951)

=

Y(1950) +

- Y(1949), which yields 100, using the above calculations. At last, we derive from (1) that national income in 1951 will be 480+ 100+ 100

=

680. We can continue these calculations for the year 1952 etc. The result is

plotted in Figure 2. The figure contains unmistakebly a number of cycles. c

TINE IN QUARTERS 1950 1957 1964 1971 1978 1985 766.2905 728.2905 690.2905 652.2905 {/)

""

614.2905 a:

_.

576-2905 "NAT.INCONE" ...J 0 c _638.2906 1.1.. 500.2906 0 {/) _462.2905 :z

)

0 424-2905 ...J ...J _386.2905 al 348-2905 310.2906 272-2905 234.2906 1960 1957 1964 1971 1976 1985 1 I ME IN OUART£R5 Figure 2.

So much for the example. We reconsider our model (a) upto (c). Note that, at this moment, it is unclear what the effect of investment will be on consumption and whether consumption depends on investment and government expenditures.

To get a better insight into these relations, we perform some manipulations. From {a), we have that national income at time k-1 equals the sum of con-sumption, investment and government expenditures, i.e.:

Y(k-1) C(k-1) + I{k-1) + (k-1) • (4)

Therefore, see {2), consumption at time k is given by .the following rela-tion:

C(k) c(k) x C(k-1) + c(k) x I(k-1) + c(k) x Ge(k-1)

Moreover, we obtain from (2} (provided c(k)

F

0) that

1

Y{k-1)

=

c(k) x C(k)

Consequently, national income at time k-2 is given by

1

c(k- 1) x C(k-1) •

Substitution of this last relation and (5) into the investment relation

(3) then yields

I(k) (1- c(k-l)) 1 x v(k) x C(k-1) + v(k) x I(k-1) + v(k) x Ge(k_.1).

(5)

(6)

From relations (5) and (6) we observe that consumption and investment in fact depend on the lagged values of these variables and lagged government expenditures.

To stress this direct relationship we rewrite these two relations in the following, more concise, way:

( C(k)) ( c(k) l(k) = ( 1 -

c(~-1))v(k)

+ (G (k-1)) • c(k)) (C(k-1·)) (c(k)) v(k) I(k-1) v(k) e (7)

Addendum on matrices and vectors

To enable readers, not faailiar with matrix and vector notation, to comprehend at least the first section of the thesis (which includes an overview of the rest of the thesis), we provide here some elementary properties.

To that end, we exaaine equation (7) in more detail. In this equation the rectangu-lar arrays ( c(k) (t - c

(~-I)

)v(k)

c(k})

and v(k)

(

c(k))

v(k)

are called matrices and the other arrays

(

C(k-1))

I(k-t) and

are called vectors. Now, the product between a matrix and vector can be defined in

a general way. ( ) ( )

To that extent • let A denote the matrix all a12 • and x denote the vector v 1 •

a21 a22 v2

(

all x vi+ al2x v2)

Then the product A x x equals by definition the vector •

a21 x vi+ a22x v2

Since it is rather cumbersome to write down always the multiplication sign, x, when-ever a multiplication is performed, we will drop it from now on. So,

Finally, we define the sum of two vectors, say

(~~)

and

(:~)·

as the vector from which each row consists of the sum of the corresponding rows of both vectors, i.e.

We conclude this addendum with.an example: Let

(

0.8

A a -0.25 Then Ax + Bu •

(

0.8

-0.25

(

400)

X • t

100

0.18) (410000)

+

(0.18) (100) - ( 320

+

80) ( 80) (480)

-100 +

100

+ 100 - 100 •

Readers not familiar with this notation are advised to read the addendum on matrices and vectors on the adjacent page.

Example (continued}: According to (7), we can calculate consumption and investment in 1951 from their previous values as follows:

(

c (

1 951)) (

o.

8 1(1951)

=

-0.25 + (G (1950)) 0.8) (C(1950).) (0.8) 1 I ( 1950) 1 e

Applying the multiplication yields

( C(1951)) = ( 0.8 C(1950) + 0.8 1(1950) + 0.8Ge(1950)) • 1(1951) -0.25C(1950) + 1(1950) + G (1950) e Since C(1950) = 400, 1(1950) immediately that

100 and Ge(1950) = 100, we obtain now

( C(1951)) =··r480) 1(1951) 100 Now, denote in (7) ( c(k) the matrix 1 (1-c(k-1))v(k) c(k)) v(k) ( c(k)) the matrix . · by B(k-1) , v(k) ( C(k-1)) the vector I(k-1) by x(k-1) and

the vector (G (k-.1)) by u(k-1) . e

by A(k-1) ,

·11'11!!em we recognize the &lll<0Wii.n-g :sltlnfclt:ure (systemS) in (7):

[J

S:

x{k)

=

A(k-l)x(k-1) + B(k-l)u(k-:-1) • x(k

where x

0 denotes the vector consisting of the initial values of consumption and investment at time k₀•

It may be clear that the values of the matrices A(k-1) and B(k-1) are rather arbitrary. Since, moreover, the multiplication and addition of matrices and vectors can be generalized straightforwardly to larger

dimen-sions, we note that this structure is very general. For this reason, special names are assigned to the vectors and matrices appearing in this relation. The matrices A(k) and B(k) are called the (structural) parameters

of the system. Since x(k-1) contains information concerning variables appearing in th~ system at time k-1, which must be remembered in order to be able to determine the values of these variables at time k, it is called

the state (vector) of the system. Finally, u(k-1) is called the control

(vector) of the system, since it contains all variables which can be used

to regulate the system.

Now, the primary reason to study the relations (a) to (c) was to give an explanation for the appearance of cycles in national income. National income, however, does not show up in the description of S. So, S does not completely describe our original three relations. But this incompleteness can be quickly remedied. From (a) we have that national income at time k is given by the relation

Y(k)

=

C(k) + I(k) + Ge(k) •

Using now the previously introduced notation (see (7)), we can rewrite this relation as Y(k) (1 ( C(k)) 1) . + (1)(G (k)) • I(k) e

Now, let C(k) denote the matrix (1 1) and D(k) the matrix (1). Then, with the notation introduced in (8) we see that national income satisfies the relation

Y(k) • C(k)x(k) + D(k)u(k) •

Since national income is induced by the systemS, we henceforth will call it the output of the system. Note that in general there will exist more than one output variable and that the matrices C(k) and O(k), which are

also called system parameters, will be time dependent. Recapitulating, we have the following system:

s :

_y x(k+1)

y(k)

A(k)x(k) + B(k)u(k) C(k)x(k) + D(k)u(k) ,

where x(k) is the state, u(k) the control and y(k) (= Y(k)) is the output of the system •

. For this system,_ Samuelson investigated how cycles in national income can

occur' provided the system

~aramete~~ a~~

.all

con~tant

in time and gavern..:. ment expenditures (control) do not fluctuate either. He showed that under

these conditions cycles in national income will occur if the system matrix A satisfies some additional requirements (imaginary eigenvalues).

The assumption that government expenditures are kept constant in time is, however, rather unrealistic. In reality, government expenditures are. used to realize political goals. It is from this point of view that we will be considering the system

S

_y here.

The first question one can ask in this setting is whether a specified political goal can be attained somewhere in time starting from ~ prede-scribed initial state. This question is considered in Chapter 2· .• In

general, however, the system cannot be forced to stay at this prespecif~ed

goal over a time-interval but will pass through this point. Not until some time has passed, the desired goal can be achieved again, etc. From the economic policy point of view, therefore, a more interesting question is whether the system is able to stay on any prespecified target patln over

some time-interval, irrespective of what tbe value of 'the state is now. This problem is dealt with in Chapter 3.

Quite another approach is taken in Chapter 4. There, the question is raised of how to characterize all time paths of political goals which can be tracked exactly, approximately and ultimately. In addition to answering

this question, we provide a way to check whether a prespecified target trajectory possesses any of the three properties mentioned above. Moreover, if a prespecified target trajectory satisfies any of these three require-ments, a control is constructed for the system Sy .which gives rise to an output trajectory which either tracks the prespecified trajectory '.exactly,

Chapters 5 and 6 elaborate on this subject. More specifically, Chapter 6 treats the possibility of designing a successful control if the future system parameters are unknown. Finally, Chapter 7 shows that under some conditions a certain elementary type of control is successful in tracking target paths exactly and ultimately. Unfortunately, it has som.e disadvan-tages too. A simulation study is performed to illustrate these. This study is performed on an extension of the economic model discussed above. This extended model is considered below.

We complete this section by considering a more realistic version of the economic example and discussing the consequences of this adaptation for the general system S •

y

In our example we assumed that consumption depends upon the income of the last year, while investment fully adjusts to lagged changes in income. However, it is commonly agreed on that as well for consumption as for investment, there exists a nominal path from which these variables to not deviate too much. If we incorporate this assumption in our model, then the consumption relation (2) alters into

C(k)

=

C(k) + c(k)Y(k-1) (2')

and the investment relation (3) into

I(k) I(k) + v(k)(Y(k-1)-Y(k-2)) (3')

where C(k) and I(k) are known time-varying quantities.

Performing operations on the updated model similar to the relations (1)~

(2') and (3•) then yields:

( C(k)) ( c(k) I(k). = (1-

c(~-1))v(k)

+ (G (k)) +

c(k)) (C(k-1)) (c(k))

Y (k) I (k-1) v (k) e +

(~

Y(k)

=

(1 1)

₍

C(k))

. + ( 1 )(G (k}) •

.

· I(k) · e

We recognize the following system in (9):

x(k)

=

A(k-1 )x(k-1) + B(k-1)u(k-1) + Ge (k-J)d(k-1)

syd=

y(k) C(k)x(k) + D(k)u(k) •

Here d(k) denotes the vector consisting of known coefficients:

[

C(k) )

I(k)+v(k)

~(k-

1

)

I (k-1)

and G(k) equals de matrix

(b

~)·

In the sequel, d(k) will be called the disturbance (vector) or exogenous input (vector).

Compared to the system S , we see that the only difference consists of the

y

disturbance vector d(k) which is entering now the system. By taking d(k) equal to zero we see that we get back. our initial system Sy. So, Syd con-tains S as a special case.

y

For this reason, the problems mentioned in the outline of the thesis are not studied for the systemS , but for the systemS d" However, since

y . y

including the disturbances only complicates the analysis of the problems considered in Chapters 2 and 3, and it has no essential influence on the basic results, we shall be dealing with S in those chapters.

y

I • 2. Some basic notions

In the previ~ section. we argued that the most general system we will be considering isSyd"

Now,

with notation as in Section

1,

define

x'(k)

:=

(x(k))

and u(k)

:q (k) ~- u(k+1) •

( A(k-1) x' (k)

=

0 B(k-1))

(0)

0 x'(k-1) + 1 v(k-1) + ( G(k-1)) + 0 d(k-1) y(k)

=

(C(k) D(k) )x' (k) •

So we see that the basic structure of the systems we consider is given by (forget the previous notation):

x(k+1) A(k)x(k) + B(k)u(k) + G(k)d(k) _{X '} y(k)

=

C(k)x(k) ,

where x(k) E IRn is the state, u(k) E IRm the applied control (c.q. input), d{k) E IRs the exogenous input (c.q. disturbance), y(k) E IRr the output and x the initial state of the system.

Here IRP, p ~ 0, is the p-dimensional Euclidean space. The system parameters are assumed to be bounded, that is, there exists a constant M such that the Euclidean norm of any of these matrices is smaller than M at any time. The

Euclidean nonn of a matrix A :"' (a .. )~ .

1 is defined as

~J ~.J=

Since this model description represents the same dynamics as

S

d' we use ·y

this model from now on. Moreover, two special cases of this system will frequently arise. First, there is the corresponding system forS • This

y

one, denoted by I:y' is obtained by taking d (k) equal to zero ~n I:yd. T.l)e second one, which will be mainly used for analytical purposes in Chapters 5 and 6, is obtained by taking C(k) equal to the identity matrix. So, in that case all states are outputs as well. The corresponding system is denoted by I:d.

Some rudimentary notation

In the sequel, the notation x(k,k

0,x,u) is used to denote the state of the system at time k resulting from the initial state x at time k

0 when the input u[k

0,k-1] is applied. The notation v[k,t] is used for the stacked vector sequence (vT(k), vT(k+1), ••• , vT(t))T if k is smaller than t and

T T T T T

(v (k), v (k-1), ••• , v (.Q,)) otherwise. Here v denotes the tran:spose of

v. Similarly, y(k,k

0,x,u) is defined as the output at time k resulting from x(k,k

0,x,u).

One of the questions in this thesis we will be dealing with is whether predescribed output variables can be tracked. Hence, reference values for various variables will appear throughout the text. Therefore, an unequivo-cal relation between the notation of any variable and that of its desired value is needed. We achieve this by marking predescribed variables with an

*

asterisk. So, y (k) denotes a reference value for the output at time k. Moreover, special names are attached to the error between output and its reference value as well as the error between input and its desired value. They are called the .command error and the control error, respectively. The evolution of the state when no control is applied to the system L ,

y x(k,k

0,x,O), plays an important role in the analysis of our systems. It is called the flow of the system.and it equals A(k+l,k₀)x. Here, A(k+1,k

0) is a shorthand notation for the matrices product A(k)•A(k-1)• ••• •A(k

0), which is defined to be the identity matrix if k+1 =

ko

and remains undefined whenever k+1 is smaller than k

0. In literature, this product is know as the transition matrix.

Two other matrices which play an important role in the analysis of these systems are the reachability matrix S[k

0+i.k0] and the observability matrix W{k

0,k0+i]. defined for positive integers L The image of the reachability matrix indicates which states can be reached by I:y at time k

0+i starting from the initial state zero at time k

0• Formally S[k0+i,k0) equals the composite matrix

The observability matrix, W[k

and it tells us which initial states cannot be observed at k

0 from the future input and output data upto time k

0+i, if the initial state of the system L is unknown. Especially in Chapter 2 the importance of these

y

matrices will become apparent.

Finally, we note that in addition to the Euclidean norm, several other matrix norms exist. In particular, we draw attention to the so-called spectral norm. The spectral norm of a matrix A is defined as

flAx tiE sup ,....--,, •

x#O IIXIIE

This norm will be used throughout the thesis too.

I. 3. Mathematical preliminaries

We proceed with giving a number of elementary lemmas which are used in the forthcoming chapters. The first lemma makes use of the Moore-Penrose inverse of a matrix, which is properly defined e.g. in Lancaster (1985), Section 12.8. We state in this section some important properties oJ it. But first we introduce it intuitively. To this extent, let A be a real n x m.matrix. Given this matrix, we are interested in the solution of the equation Ax

=

b. If this equation has no solution because b ! Im A, we take the best substitute, i.e. we minimize the distance IIAx- bilE. The minimization condition does not necessarily determine x uniquely, so we take of all possible solutions the one that minimizes the norm of x. In this way we obtain then for any b a unique solution of the equation, denoted by A+b. This, on its term, determines then uniquely the Moore-Penrose inverse, A+, of. matrix A. Penrose showed in (1955) that this in-verse can be formally defined as follows.

Definition 1: Let A E 1Rnxm (i.e. the set of all real n x m matrices). Then the Moore-Penrose inverse of A is the unique matrix A+ E 1Rmxn for which:

1) 3) AA+ A = A ; (AA+)T = AA+ 2) 4)

From Lancaster (1985), Chapter 12.9, we quote the following lemmas stating that the Moore-Penrose inverse always yields a best approximate solution to linear equations.

Lemma 2; Let A E lRnxm. Then the equation Ax = b is solvable iff rank [A : b]

=

rank A. In that case, a solution is provided by x

=

A+b.

I

Moreover, this solution is uniquely determined iff rank A= m. It then equals (AT A)-l ATb.

Lemma 3: m in !lAx - b liE

X

The next three lemmas are used in Chapter 3 to deduce an algorithm for target-path controllability. Since Lemma 6 is well-known, its proof is omitted.

Lellml'la 1.!: Let A E mnx:m

~

B € lRpxm. Then Im

(!)

i) Im A

=

lliLti!Jl and

(or .equivalently: i) Im B mP and ii) A Ker B

Pf'Oaf~

!From as well the condition Im

(~)

= lRn+p as both conditions i) and

ii} it is clear that always the inequality m ;;;; n+p holds .• So, we can make without loss of genei:ali ty a c!tecomposit:i.an of the form lRm

=

X f9 Ker A. With respect to a basis adapted to this

deoGmposi~ion,

matrix

(!)

has the

following structure:

:

..

)

,,

where Ker A' 0 .

Therefore, Im

(!)

=

IRn+p iff A' is inver.tible ana. [m B"

0

0

Corollary 5: Let A € lRnxm, B € lRnxq, C E lRpxq. Then Im (

~ ~)

lRn+p iff

i) lm C = mP ;

ii) Im A + B Ker C .. lRn [J

Lemma 6: Let S be a linear map from V ..,. W, and f. any linear subspace of V. Then

s-

1 _(S(£))

_·=

_{{v E V}

_I

_{S(v) € S(£)}}

f. + Ker S •

Corollary 7: Let A E lRnxm and let V be a linear subs pace of lRm. Then

AV

lRn is equivalent to i) Ker A + V .. lRm and 0 ii) Im A = lRn • o Lemma 8: Proof:

Ker C

n

A KerB • A Ker (:A) •

x € Ker C

n

A Ker B ~

there exists a vector b such that Cx

=

0, x = Ab and Bb there exists ab E Ker (:A) such that x =Ab*

x €

A

Ker

(:A)

In time-invariant systems the eigenvalues of the system matrix A, denoted

by Ai(A), i = 1, ••• ,s, play ;m important role. In particular in stability

questions the spectral radius, r(A) := mrx

!

\(A)

I'

is of special interest. In stability questions for time-varying systems this role is taken over in some sense (which will become apparent in Chapter VI) by the largest

singular value of this matrix. Formally, the singular values of a matrix are introduced as follows.

Let A be an n x m real matrix. Then the m x m matrix A1A is positive semi-definite. Therefore, see Lancaster (1985) Theorem 5.4.1, ATA has nonnega-tive eigenvalues. The square roots of these eigenvalues are then called the singular values and denoted by cr'(A) := cr

1(A) ~ ••• ~ crm(A).

Note that the singular values of A are sometimes defined as the square roots of the eigenvalues of the n x n matrix AAT. But, since the nonzero eigenvalues of ATA and AAT coincide, this difference in definition is not highly significant.

The following theorem is well known.

nxm Theorem 9: Let A E 1R , and o

1 (A) ••• ~ or(A) > 0 be the nonzero singular values of A.

Then there exist unitary matrices U E lRnxn and V E lRmxm such that

A UT ( D (1) 0 n-r,r

where D ~ diag(o.) is an r x r matrix and 0 denotes a zero matrix with

1 p,q

p rows and q columns. o

The representation (1) is referred to as a singular value decomposition of the matrix A.

The last item concerns the convergence of the solution of an inhomogeneous linear recurrence relation. A necessary condition (which is not sufficient) on the inhomogeneous term is derived,

Lemma 10: Let IIA(k)UE ~ c for all k, and let {e(k)} satisfy

e(k+1) = A(k)e(k) + v(k)

Then e(k) ~ 0 implies v(k) ~ 0.

In this chapter, several point controllability problems are analyzed for

~ • In all these problems, the central issue is whether the system can be y

controlled from a certain initial state, at some point in time, to another one. Although some of these problems were treated before (see e.g. Meditch (1969) Theorem 2.2 and Theorem 2.4, Ludyck (1981) Theorem 2.2 and Theorem 2.4, and Kwakernaak (1972) Theorem 6.7), the given solutions were incom-plete.

So, for completeness' sake, we give results for various controllability and reachability problems, together with elementary proofs. These problems are treated in the first section.

It sometimes is easier to perform calculations with input/output descrip-tions than with state-space descripdescrip-tions. Therefore, we investigate in Section 2 whether from a model given in state-space representation, an input/output (i/o) representation can be found. For time-invariant systems this, so-called, i/o-convertibility is always possible due to Cayley

Hamilton's theorem. For time-varying systems, however, this property ceases to hold. We will give a sufficient condition for the existence of such a relationship. This condition is called reconstructibility and it is, there-fore, defined in this section first. We also give a necessary and suffi-cient condition for reconstructibility.

In Section 3, we analyze the stabilizability problem in more detail. Based on the decomposition of the state-space at any point in time into a reach-able and unreachreach-able subspace, we discern various types of systems. One of them is the class of periodically smoothly exponentially stabilizable systems. This kind of systems will appear to play a similar role as the stabilizable systems in time-invariant quadratic minimization problems. This will become clear in Chapter 5. Another consequence of this state-space decomposition is that now easily a characterization of the stabiliz-ability subspace can be given.

The (dual) concepts for detectability are defined in the succeeding sec-tion. This section contains also some preliminary results for Chapter V concerning systems that are both periodically smoothly exponentially stabilizable and periodically smoothly exponentia1ly de,tectable.

11. 1. Controllability and Reachability

In this section, we treat the generalizations of the following target point c.ontrollability problems for time-invariant systems: the zero-controllability problem, the output-zero-controllability problem and the reachability problem. First, we introduce the definitions of these con-cepts. They are visualized in Figure 1-3.

output-controllable at k

₀

~ Figure 3 reachable at k

₀

~

---~x-+-~---·---~--~-.

time

ko

N output

*

y N N

Definition 1: The initial state x of the system I:Y is said to be

zero-controllable at

ko

i f there exists a control sequence u[k

0,N-1] with k

0 < N < oo, such that x(N,k0,x,u)

=

0.

The output y* is said to be output-controllable from zero at k₀ if there exists a control sequence u[k

0,N-1], with k0 < N < ®, such that

*

y(N,k

0,o,u)

=

y •

The state x* is said to be reachable at k₀ from zero if there exists a

*

control sequence u[N,k

0-1]. with -oo < N < k0, such that x(k0,N,O,u) x • Now, let I:. be a linear subspace of IRn (respectively IRr). Then I: is

y called L-zero-controllable at k

0 if all states x € I:. are zero-controllable

at k

0• Completely analogously one defines £-output-controllability and £-reachability of,!; • For each of these concepts, there is a maximal

sub-y

space £ having this property. The subspaces of all zero-controllable states at k

0 and reachable states from zero at k0 are denoted by zk0 and

Rko•

respectively. It is easily shown that all spaces defined here indeed are linear subspaces. In case the maximal subspaces equal IRn (respectively IRr) we talk about zero-controllability, output-controllability and

reachabil-ity of I:Y at k

0, respectively.

Remark 1:

In the last-defined system properties, the qualification 'from zero' is dropped for reachability and output-controllability. This is due to the fact that reachability and output-controllability from zero imply reachability and output-controllability, respectively, from any initial state. Moreover, note that zero-controllability does not imply controll-ability in the sense that in that case any state can be obtained from any initial state. This is easily seen by considering the system x(k)

=

0 for all k.

Remark 2: The concepts of zero-controllability and reachability are

time-dual. This property is used in the proof of a result about reachability.

Remark 3: In literature, one often encounters the concept of

output-controllability of the system at k

0 to a prespecified target y*. Usually, this is defined as the property of L that for any initial state x at k

0

y

*

there exists a finite control sequence u[k

With this concept then also output-controllability of Ey at

ko

can be defined. We prefer, however, to define output-controllability as the property that the output can be controlled towards any reference point, for initial state zero of

As was already noted in the introduction of this chapter, many proofs have been given in the past concerning results for zero-controllability and reachability which turned out to be wrong. For this reason, we will not only state here the results but also give elementary proofs of them. The first result is about zero-controllability.

Theorem 2: Ey is £-zero-controllable at k₀ iff there exists an integer M > k

0 such that

Proof: Clearly, A(M,k

0)£ c lrn S[M-1,k0] is equivalent to: for all t € £, 3 u[M-1 ,k

0] such that

From this, the sufficiency of the condition is clear. To prove the necessity of the condition, we let e

1, ••• ,ek be a basis for£.

Asst~e that u.[k

0,N.-1] steers to zero, and that M= max N •• If N. <M

l. l. i l. l.

for some index i, then we define a new extended control sequence as fol-lows:

Let 2 be any element of £, say 1(. <'k Li=t ai ei. Then the control sequence

steers ~. to zero at M. So, for R- € £ the equation

Note that as a special case of this theorem we obtain that ! is

zero-y

controllable iff there exists an integer M > k₀ such that Im A(M,ko) c c Im

s

[M-t,k

0].

Since the proof of the result concerning reachability is a complete dual-ization of the proof of Theorem 2, we treat this property now first. The next theorem states in particular that there exists a finite integer M such that Rko

=

_{Im S[k0-t,M].}

Theorem 3: ry is £-reachable from zero at k

0 iff there exists an integer M< k

0 such that£ c Im S[k0-1,M].

Proof: It is easily shown that the condition is sufficient.

To prove the necessity of the condition, let e

1, ••• ,ek be a basis for£ again. Then for each e., there exists an input sequence u.[N.,k

0-t] that

~ ~ l

steers the state from zero at Ni to at k 0•

Let M be the minimum of Ni, i

=

l, ... ,k. If Ni >M for some index i, then we define a new control sequence ui[M,k

0-1] as follows:

Now, let t be any element of£, say t

=

l~

1

~.e

.• Then the control l= l 1

sequence

l~=

1

_{ai ui[M,k0-1] steers the initial state of the system from} zero to t at k

0• In other words, there exists an integer M such that any t E £ can be reached at k

0 from zero at M. So for any t E £ there exists an input sequence u[M,k

0-t] such that the following equation holds:

This proves the theorem.

By taking

£

equal to IRn we obtain that ry is reachable at k

0 iff there exists an integer M< k

0 such that rank S[k0-t,M]

=

n.

The next item we discuss is output-controllability. A special cased con-cerning this subject was stated by Kwakernaak (1972) in Theorem 6.7. No proof was, however, provided. We give here a proof, though not construc-tive, of this generalization from his concept of complete controllability.

It seems to be difficult to give a constructive proof, since an output which can be obtained at time k is, in general, unobtainable at time k+1. So, a different method of analysis is required than in the proof of the previous theorem.

In order to obtain solvability conditions for this problem we first give a lemma. This lemma has a set-theoretic background. It tells us that IRn cannot be covered by a countable set ·of proper linear sQhspaces.

Lemma 4: Let I be an index set and £(i) and £ be linear subspaces of lRn with dimension £(i) < dimension £. Then U £(i) =

£

implies that I is

iEI

uncountable. o

The theorem now reads as fDllows.

Theorem 5: Ey is £-output-controllable from zero at

ko

iff there exists an integer M> k

0 such that£ c Im C(M)S[M-t,k0].

Proof: The sufficiency of the condition is again easily shown.

The necessity of the condition is proved by contradiction. Assume that for each integer M> k₀, £ ~ Im C(M)S[M-t,k₀]. Then for each M,

£

n

Im C(M}S[M-l,ko]

is a linear subspace of dimension smaller than the dimension of £. Since each output y € £ can be obtained from zero at k₀ we know that the collec-tion

u

£

n

Im C{i)S[i-1,k₀

J

i€1

covers £, where I = {k

0,k0+1 , ••• }. So, we have a countable collection of subspaces, all with lower dimension than the dimension of £, which cover £.

This clearly contradicts Lemma 4. IJ

A special case of this theorem is obtained again by taking £ equal to IRr. The theorem then states that

L

is output-controllable from zero iff there

y exists an integer M > k

Another interesting aspect is that the rank condition given in the theorem is equivalent to the following statement: there exists an integer M > k

0 such that for ally € £an input sequence u[k₀,M-1] exists which steers the output from the initial state x(k₀)

=

0 to y at M.

This is a result which, due to the linearity of r , also holds for £-zero-_y controllability and £-reachability. So,

r

_y has one of these properties iff there exists a uniform time M at which for each ~ E £ this property holds. We formulate this observation in a proposition.

Proposition 6: In the definitions of zero-controllability,

output-controll-ability and reachoutput-controll-ability the quantifier V £E£, 3 N such that etc. may be interchanged.

11.2. Reconstrcctibility

The concept of reconstructibility is defined for time-varying systems, e.g. by Ludyck in (1981), Chapter 2.4. In that chapter he also gives a necessary and sufficient condition for it. His proof, however, was incor-rect. Therefore, we provide a correct proof. Our proof is inspired by a proof that J.C. Willems gave in (1980) for the reconstructibility of

time-invariant systems. But first we give a definition of this concept.

Definition 7: ry is called reconstructible at k

0 i f there exists a time k₀- N such that x(k₀,k

0-N,x,u) is uniquely determined by u[k0-N,ko-1] and

y[k₀-N,k₀-1].

Theorem 8: ry is reconstructible at k₀ iff there exists a positive

in-teger N such that

Proof: First we note that

It follows from the definition that !:y is reconstructible at k₀ iff A(k

0,k0-N)x(k0-N) can be determined uniquely from :tihe past observations

This is not possible for all states at k₀ iff the'I'e t!xist two states

.x~:(~-'N) and x{k

0-N) such that

and

Or, equivalently,

and

In the remainder of this section we show that reconstructibility of !: at y

D

k

0 is a sufficient condition for obtaining an input/output (i/o) represen-tation. We call this property i/o-convertibility. Its definition reads as follows.

Definition 9: !:Y is called i/o-convertible at k

0 if there exists an N > 0 and matrices Pk(i), Qk(i) such that for all k ~

ko•

the dynamics of !:y can be described by the following input/output relation:

k-1

y

<k>

I

r

p

k

<

o

y

<

i

>

+ Qk

u)

u

<

o } .

i=~-N

It is clear fr{)JD the proof of Theorem

·s ttmtt,

.ii.Jf !: is reconstructible,

y

for some matrix

A direct consequence is

Proposition 10: Let ky be reconstructible at k₀• Then LY is i/o--convertible.

Moreover, the input/output relation is given by

where

v(i)

=

y(i)- C(i)S[i-1 ,k

0-N) u{i-1 ,k0-N] ,

Since this input/output relation depends severely on the initial state x(k₀) of the system, we give sufficient conditions under which a dynamic input/output relation (i.e. a relation which is independent of the initial state of the system) is obtained. This is the content of Theorem 11.

Theorem 11: Assume that there exists an N > 0 such that W[k,k+N] is full

row rank for any k. Then the following dynamic input/output relation holds for

:r :

y

y(k+N+l) = C(k+N+l){W[k,k+N} WT[k,k+N]}-f W[k,k+N] •

• (y

[k,k+N]

Here, M(k;l,N) is as defined in Section 111.1.

Proof: Obvious from the consideration that

IJ

y[k,N+k) = WT[k,N+k]x(k) + (

0

)

for any k.

T. T . · W[k+1,N+k+1]

w

Ik·H,N+k+1]

=

W[k,N+k]

w

[k,N+k] +

+ AT(k+N,k)GT('k-t;JD)<C(k+N)A(k+N,k) - AT(k)GT(k+1)C(k+1)A(k) •

From this equality-. ,a lr<&:ursion formula for the calculation of ·

{W[k,k+N] .WT[k,k+NJ}-1 ·can easily be obtained. This formula can be used to <calculate the inverse 'Wit:lh less difficulties.

llii~JI.. Stabilizability

Ln ~he theory of linear rtime-varying difference equations the concept of 'uniform asymptotic stability' plays an important role. This, since ac-cording to a theorem of Poii:ncare-Bendixson uniform asymptotic stability of the linearized system implies uniform asymptotic stability of the non-linear system. Now, J.L. Willems proved in (1970), Theorem 7.5.2, that the flow of E is uniformly asymptotic stable if and only if it is

exponential-y

ly stable.

Therefore, a natural question ~n the study of the system E is under which

y

conditions it will be exponentially stabilizable. That is, under which conditions does there exist a control sequence in the form of a state feedback, such that the resulting closed-loop system becomes exponentially stable.

For time-invariant systems it is well known under which conditions the system E is stabilizable.

y

For time-varying systems, however, this question is much more complicated, and a general theory about it is lacking. Hager (1976) and Anderson (1981)

took a lead with the introduction of definitions of detectability and stabilizability for discrete time-varying linear systems and some applica-tions to control and filtering problems. Their definiapplica-tions, however, do not give a clear insight into the basic underlying structural problems. To obtain a better insight, the state-space decomposition given by Ludyck in \1981) seems to be a more promising approach.

Therefore, we shall explore his ideas and extend them in this section and the next one. But first

we

introduce the elementary concepts of asymptotic stability and stabilizability together with some notation.

Definition l2: An initial state x of the system !. is said to be y

-asymptotically stable at k

0: if lim x(k,k_k-+<» 0,x,O)

=

0;

stabilizable at k

0: i f there exists a bounded control sequence u[k0, •]

such that lim x(k,k

0,x,u) = 0.

k-+<»

In the sequel, the subspace consisting of all asymptotically stable states at k

0 is denoted by X-(A(•,k0)). The stabilizabllity subspace at~· Le.

the subspace consisting of all initial states at k₀ which are stabilizable, is abbreviated by Xstab(kij). In case X-(A(•,k₀)) and Xstab(k₀) equal IRn we say that !.y is asymptotically stable and stabilizable at k₀, respectively. For time-invariant systems, we know that the stabilizability subspace is given by

(1)

where

Z

is the zero-controllability subspace.

Now, whenever the system is stabilizable at k₀, we know that in the time-invariant case it is always possible to construct for any unstable state at k₀ a control sequence which regulates this state to the stable subspace at time k₀+n. So, for a stabilizable time-invariant system there exists for any initial state a control u[k

0,k0+n-1] such that from time

ko

+ n on, no control is needed anymore to obtain convergence of the state towards

zero.

This property does not longer hold for time-varying systems. A simple example illustrates this phenomenon.

Example 13: Take

A(k)

=

(:

:)

B(k)

(:)

C(k) I • k "' 100,200,. •••

A(k) = (: :k) ; B(k)

(:)

_; C(k) I , k- 100,200, •••• This system is stabilizable. At any point 100k+1 in time we must, however, control the system in order to achieve this. Moreover, we see that

X-(A(•,k

0))

=

0 and zk0

=

(~).

So, property (1) ceases to hold.

In the rest of this section we embed the stabilizability subspace in a subspace from which one expects at a first glance that it equals .the

stabiliza~ility subspace. Though this is not the case in general, it turns

out that for a very broad class of systems, equality is.obtained. The advantage of this new introduced subspace, called :the potential stabiliz-ability subspace, is that it resembles a ststabiliz-ability subspace.

The introd1,1ction of this subspace requires a state-space decomposition. 'This state-space decomposition is based on the pr~ty that the time-:dependent reachability .subspace is A(k) invariaut ;(i.e~ A(k)Rk c: Rk+l).

This property is proved first now.

umma 14:

A(k)Rk + Im B(k1 = Rk+l

Proof: It is obvious :that if x is reachable at k, A(k)x is reachable at k+1 (take u(k) = 0). F1urthermore, any element in the image of B(k) is reachable at k+1. So one inclusion is clear.

To prove the other inclusion, let x be an element of Rk+l' By definition, there exist then a finite integer N, smaller than k+1, and a control sequence u[N,k] such that x = x(k+1 ,N,O,u).

Now, let i := x(k,N,O,u). Then obviously i E

Rk.

'Since, moreover, x

=

A(k)i + B(k)u(k), the other inclusion is clear with this, too.

The required state-space decomposition results immediately from this lemma. A proof of it can also be found in Ludyck (1981), Theorem 6.1. The

corol-lary uses the following notation: rk := dim ~ and op.q :·;

with p rows and q columns (see also Chapter I.3.9).

zero matrix

Corollary 15: Choose a subspace Xk such that IRn = Rk., Xk. With respect to

a basis adapted to this decomposition, ~ is described by the next recur-rence equation:

C'

Od}

1 ,

( x'

(k))

2

x2

(k.)

where the subsystem x1(k+l)

=

At₁(k)xl(k) + B1(k)u(k) is reachable at any time k, the state-space dimension of x;(k) is rk and that of xi(k) is n-rk.

Note that this state-space decomposition can be obtained by a state trans-formation x'(k)

=

T(k)x(k), with HT(k)D

2 1, and such that all the

trans-formed matrices remain bounded, if we take Xk

=

~

and the basis ortho-normal.

We now want to define the potential-stabilizability subspace at k 0• To this extent, we reconsider Ly for k > k

0, assuming that A(k) and B(k) are zero for all k < k₀. Furthermore, we introduce AR(k):= A(k) mod Rk. This is a mapping from m.n mod Rk-+ m.n mod Rk+l which, due to Lemma 14, is well defined. Note that AR(k

0

)

is defined on IRn (because

Rko

=

0).

Now, consider in the quotient-space, m.n mod Rk, the usual norm

lil

=

inf { llyll : y

e:

IRn, y-x E ~} , Then,

Definition 16: The potential-stabilizability subspace at k

0 is defined as follows:

To motivate the study of this potential-stabilizability subspace we first characterize this subspace in two examples. The first example we consider is the autonomous system, i.e. B(k)

=

0 for all k. In that case is Rk

=

0 for all k. Consequently, the potential-stabilizability subspace coincides with the stability and stabilizability subspaces.

The second example concerns time-invariant systems, i.e. A(k)

=

A and B(k) = B for all k. In that case the potential-stabilizability subspace coincides with the stabilizability subspace, as is shown later on in this section.

These two examples sugg~st that the potential-stabilizability subspace is closely related to the stabilizability subspace. The exact relation is re'V'""a.led in the next two theorems. The first theorem gives a justification of the ehosen name for 'the potential-stabilizability subspace.

~f: Let x be = t element of the stabilizabillii.try. >Subspace at k

0• lB;y cd.efiui:tion ·tbe>re ·exists then a control sequence

.ufk:o• •]

such that in

;x(t'k++-:1) .A!(k.)x,(k) + B{kl)u(:k), x(k

0)

=

x, lim k:+oo x'Ck,llL~lll'~ -'1!) = '0. But this is

,e.Qui:valent to the existence of a control sequence

u''iill)-0,. .. ] in a trans-ifoo:rmed system x' (k+ 1) = A' {k) x' (k) + B' (k) u' (k), x 1 _{(ko) = x, such that}

'lii.m x''(k~ko· ,x,u') = 0, if :the transformation matrices have e.g. a norm .one

ik._,

and 't'he corresponding system parameters remain bounded.

Take the transformation now conform to the one discussed immediately after Corollary 15. Then we observe that

x' (k ) '" ₀ x

S.o, ii.lt ii-s :e!l·e·ar that if x is stabilizable, then necessarily in the above

decomposed system x:2(k+1) =

A2

2(k)x:2(k), with x2(ko) = x,converges to zero when k tends to infinity. This implies that x is an element of the

poten-tial-stabilizabili:ty .subspace, which had to be proved.

In order to prove the converse statement we would have to construct a control sequence which steers incoming exogemous influences arising from

the second state c~:t (see Corollary AS) to zero in the first state

component. Systems which haw~ ~~is property will be called smoothly

con-trollable in the sequel. A 'PTO!per definition is given below.

However. even in case the whole secQnd state component of the system con-verges to zero, it is in general not possible to construct such a sequence which results in the convergence of x(k,ko,x,u) to zero. An example of

such a system is Example 13 with matrix B(k) replac~d by