Opened loop control of switching linear systems

Opened Loop Control of Switching Linear Systerns by v. ABADIE Laboratoire

and G. DAUPHIN-TANGUY

d’Automatique

BP 48 CitP ScientiJique,

et d ‘Informatique 59651

Villeneuve

Industrielle

d ‘Ascq Cedex,

de Lille, France

ABSTRACT : The purpose of this paper is to elaborate an opened loop control law for linear systems with Boolean inputs and to show a way to$nd a convergence domain of those systems. First, we determine the convergence domain ,for the state vector by expressing it with respect to the time using linear systems theory. Afterwards, the algorithm of control is determined. We apply those results to a two-dimensional system and a geometric approach is then presented. Finally, we present simulation results as an example.

I. Introduction Control of linear systems is a well-known problem, assuming the inputs are continuous ones. But, some systems are controlled by switches, especially power electronic systems. Those systems are usually controlled by the P.W.M. method (1) which consists of controlling the process by mean values. In this paper, we will use a global mode1 including Boolean inputs to elaborate upon an algorithm for the control of switching linear systems. The paper is outlined as follows. The first section describes the studied system. The second section presents the resolution of a linear state equation. In Section III, the convergence domain and the control law are pointed out. In the last part, we apply the proposed method to a two-dimensional linear system, and give a geometric approach to finding the convergence domain.

ZZ.Problem Statement Consider x = (x ,, . . . , x<,)~E RY the state vector, u = (U ,, . . . , ~4,)~E R” the input vector, and y = (y,, . . . , y,JT~R” the output vector. A, B and C are constant matrices with appropriate dimensions. The system mode1 is expressed as i = Ax+Bu

(1)

y = cx where the U, components have Boolean values. The aim of the study is to determine the sequence reach a desired point denoted by 2.

Q The Franklin Institute00164432/93

$6.OO+O.W

of u-vectors

allowing

x to

799

V. A&die

and G. Dauphin-Tunyuy

III. Resolution of Equation 3.1. General case According to linear systems theory

(2) we can express -u(r) as

x(t) = e *(~-~dx(~,,)

+

s’

e*“~~“‘Btr(B) d0

‘0

assuming t,, = 0 is the time origin. In the case where II, components are Booleans, the vector I* can have 2’” values which belong to the set denoted config (u) (3), as

=

config

1IIIIII I 0

1

0

1

0 .

0 .

1 .

1

;

0

;

0

; .“;

0

0

0

Between two commutations of the system, the input u is a constant vector denoted [config (u)]~. If we denote tk the time of the kth commutation and tk+, the time for the (k+ 1)th one, we can express x(t) for r belonging to [tl;, tk+,] as

s z

x(t)

= e A(i- ‘~‘_x(t~)+

e”” “‘Btl(0) d0 IA

(2)

which leads to X(T) = e *cr -QX(tk) + ([:

eA(rP’i)d0)B[configu

(u)]k.

(3)

3.2. Stability hypothesis We will suppose in what follows that the model expressed by Eq. (1) is asymptotically stable. Then, the matrix A is non-singular (4). We can calculate x(t) by using expression (3) : i.e. X(T) = e A’r~‘~~x(t,)-A-‘(I-eA”+‘~‘)B[config(~)]k with

r E [L

tk+

,I,

which leads, if B. [config (u)]~ is denoted

(4) Bn. to

x(r) = eA(r~‘,),~(t~)-A-‘(I-eA(T~‘h))Bk.

IV. Convergence

(5)

domain

Proposition 1 When we impose the same configuration [config (u)],, iE { 1,. . . , Y} of the set config (u) as input vector, in all the time domain [0, + co[, then ,‘i~mx(t) = x,,, where xix = --A- ‘Bi. Proof: If we impose only [config (u)],, for t E [0, a[, then tk = 0, tk+ , = co and Journal

800

of the Franklin Lnstu~te Pergamon Press Ltd

Opened Loop Control Law z = t. Then, we is asymptotically lim (x(t)) = -A,- E

have x(t) = e*@)x(O)-Astable, which involves

'[I-e*']B,. The system model jit (e*‘) = 0, which leads to

‘B, = x,, which demonstrates

n

the proposition.

Proposition 2 Consider 9 the convex n-dimensional polyhedra made by the set of points , ,Y~,~+}.9 represents the state vector convergence domain of the opened { X,r,... loop switching linear system expressed by Eq. (1). Proof: Consider X belonging to 9. X can be considered as the barycentra of the points {xIX , . , ,x~,~~,),with the loads {a,, . . . , a2”$}.Then, we have 2”’

X =

C a+,, ,=I

’

y ,g, a, = l witha,E[O,

l],fori~

l,...,

2”‘).

a

All the x,, can be determined by using proposition 1, and Eq. (6) allows us to determine the values of a, by resolving the system of n+ 1 equations and 2”’ unknown variables. The first step is to separate the time-axis into intervals with a constant period denoted Tmod. Each interval is decomposed into subintervals [CQI”,,J, 1 d k d 2’“- 1. Then, we apply on each interval [nT,,,,,, (n+ l)T,,J the same sequence for the vector u : the value [config (u)]~ is applied to the system during a time domain interval tk+ , - tk = QT,,,~~. This can be schematized as in Fig. 1. The problem is to calculate cli, ie { 1, . ,2’,} for having : ,Q x(t) = 2. According

to Eq. (5), we have ~(7) = eA(rP’A)X(tk)-A-

for z E [tk, tk+ J, where u is a constant

U

1‘

[

config (u)1, I config (u)1,

‘(I-~“(“L))B/,

vector, equal to [config (u)]~.

. . . [ config (u) 1,. . . [ conffg (u) ]2”

I FIG. 1. Representation Vol. 330, No 5, pp. 799-813, 1993 Printed m Great Britain

I (n+l)Tmod

t’

of input during a time-period [nT,,,,,, (n+ I)T,,,J.

801

V. Ahadie and G. Dauphin-Tanguy In the time domain x(c(, Tl,,,,J = -A-

x(0) = 0 :

’ [I-eA”lT-~~d]B,

~((a,+g~)T,~~~)

x(Tmod)

[0, Tmod], we have, assuming

= _eAzzT,“OdA-‘[I_eAa,~~,,,]B,_A~‘[I_eA12Tmud]B2

._eA~~Trn~ti . . eA~zTmodA_ ‘[I -eAz~Tmod]B,

=

_eA~z’JsTlnod. . eA~>TmmodA-’ [I_eAz2Tmod]B2

In the time domain

[7’,,,od,2T,,,,],

...

‘[I_eA~~~‘Tmmod]B~ ,,,.

-A-

we have :

x(T,,,,,, + c(, T,,,ad) = eAz~Tnl~dx(T,,,od)-A-

’ [I -eAxl q~lod]B,

x( Tmod+ (a, + a2) Tmod) = eAZZTm~~“[eAZl Tmod.x( Tlnod) -A-‘[I-~A”I~“,~,~]B,]_A-‘[I_~~~z~,~~~]B~

x(2T,,,,)

= eAz2”‘Tlnod . . . eAa1Tnwd~(Tmod) _ eA’,f,2Tmod . . eA~zTmxlA- ’ [I _ eAz, Trnm~]B, . .

Then, in the domain

_

’ [I _ eAaYTmod]Bz ,,,.

A

it leads to :

[IzT,,,,,~, (n+ I)T,,J, *m

C

x((n+

\

l)T,,,,,) Y

I

= eATlnodx(nT,,,,,J c

J

I,, + I

eA*z,j,Tmmt . . . e%,+

~lTnod~-

’ [I_eAa~TmmOd]B,

-,=, L

J,,

J

Y K

(7) with

which is equivalent

to : %,I

= eATlnodx,+K,

for n = 0, 1, . . .

(8)

If we choose

we have, using (Appendix

A) :

which leads to :

because we have set x,, = 0. Journal

802

of the Franklin Institute Pergamon Press Ltd

Opened Loop Control Law

Then x,

=

(I-(eATm~d)“)([T-eATmod]-‘K)

stable, then lim x(t) = lim X, = [I -eATmc,d]--‘K. If this limit is ,+ 3c n-x xX, using Eq. (7), we have

A is asymptotically denoted

x,

2”’ . . . e%, +I,*~ocIA~’ [I _ eAa,rmoct]Bi = (I_ e*LC~) 1 _ 1 eAfi,~~~rno, 1 ( i= I

We have to calculate

tl,, iE (1, . . ,2”‘> verifying

which leads to, (using Appendix

X, = f, which leads to

B) :

*tu

*m

c i=

By identifying

e XY ATme . . . &+

~AT~~~,d(T-ea,AT~~~~)BI = z, (I-ee”Tmc,d)4.

(10)

I

each term, it results in

By using the series-expansion

which can be written

of the exponential

terms, we have

as

with

with

fx= If we assume

&,I

+,.

. +a,,..

/IAT,,,,, I/<< 1, we can assimiliate

4 to its first-order

term,

which leads to Vol. 330, No. 5, PP. 799-8’3, Printed in Great Britain

‘993

803

V. Ahadie and G. Dauphin-Tanguq

where (Appendix

C) :

(a,- 4 ~~ 2

+...+LX~~~,) < a,.

-a,(r,+,

Then we have

llATnm
approximation

a, = a, iE(l,...,Y}’ Remark. It is well-known that IjL(A)j,,,,, d //A/J. Then, the assumption /IA I/ C-C1/T,nod is equivalent to Ilk(A) I,,,:,rCC1/T,,,od, which is a usual hypothesis for sampled systems where the sampling period has to be chosen according to the dynamics of the system.

V. Examples

and Simulations

Results

5. I. Example 1 Consider the two-dimensional

linear switching

system defined as

with I = (x,, .xJ’~ and u = (u,, uJT. Then, we have, by using the results of Section

x2-r

=

-A-%(;)=

x3,

=

-A--

x 4x,

=

-A--‘B

IV,

(-;)

‘B

; 0

=

; 0 8

The convergence domain can be represented as in Fig. 2. We want the system to converge, for example, to R = (b, i)T, which involves resolution of the equation set : 804

Journal

the

of the Franklin lnst~tule Pergamon Press Ltd

Opened Loop Control Law

*x l--

2

200 X

X

> x,

1

‘loo

-1 --

FIG. 2. Convergence domain of a two-dimensional

switching system.

composed of three equations and four unknown parameters. Then, one term is undetermined and it can be specified by using other criteria. For example, we can notice that X belongs to the triangular domain made by ix,_*, x2X, xdz}, which leads us to choose a3 = 0. It follows then that a, = : 5 u2 = n a3 =O, 0-4 = + If we define the norm of the matrix A as (5) : IJAIl = ~y,~W(Lj)l,j where A(i, ,j), i, j = I, 2 represent Then, it results in

the coefficients

(12)

= 13, of matrix A.

/I’AIj = 4. According to Section which leads to

IV results, we have to choose

T,,,“,, such that

I/AT,,,od// cc I,

Tmud << :. Remark. The eigenvalues of A, &(A), i =1,2 that Eq. (13) is equivalent to Tmod -cc 1/lI(A)I,,,. Vol. 330. No. 5, pp. 799-813. IYY3 Printed in Great Britain

(13) are -2,

-4.

Then,

we can see

805

V. Ahudiie uncl G. Duupllin-Tunyu~.

0

4T

QT

12 mod

1

12 mod

T

4T

nbxl

3’“Od

2T

mod

l-.-EL-

0

4T lErnod

FIG. 3. Representation

si 12 “lud

T mod

4T 3 “‘“d

2T

of the inputs of the two-dimensional

mod

switching

linear system.

The components of the input u can be represented with respect to t as in Fig. 3. The response of the model, simulated on a computer 386-AT using the simulation software A.C.S.L., with T ,,,,,<,= 10 ’ s is presented in Fig. 4. 5.2. E.~~tnpl~~ 2 Consider the electrical network defined as in Fig. 5. S,. i = I. 2, 3 denote switching components, with controllable

ON and OFF

R d

5: d

8 d

0.00

1.00

FIG. 4. Time-response

806

2.00

3.00

of the state variables

t 4.00

of example

I

FIG. 5. Representation

of an electrical network with switches.

states. The corresponding bond-graph is presented in Fig. 6, where components are modelled by TF-elements with Boolean moduli depending on the switch OFF or ON state) and R-elements associated resistances R,, (6). We can deduce the state equation by using bond-graph theory (7)

2 =

6 04

=

- &IL i - l/L

0

1/c ~ ~/R,,C-CI$/R,,C /

1 xct- i E,IR,,

with .x = (I?, q) r where p is the magnetic flux of the inductance of the condenser C. If we choose u = (IYZ:,FH:)~, we have the state equation :

the switching (M, = 0 or I with internal :

0

??I?

&IR,, I

m; 1

L and q the charge

i = A(u,)x+Bu with

RR -Fd

RR

RR

FIG. 6. Bond-graph Vol 33U.No. 5. pp.7YY-813, Pnntcd in Great Britam

1993

of an electrical

d

network

with switches.

807

V. Ahadie und G. Dauphin-Tcrngu~

1/c

-&IL

Mu,) =

-l/L

Some topologies

>

- l/R,,C-~rn’/R,,C

are not allowed

and

in the electrical

B =

network

0

0

E, IR,,

EJR
: i.e.

-S, and S3 cannot be ON simultaneously, neither Sz and Sjr or S, and S2 ; - moreover, the configuration where S,, S2 and S, are simultaneously OFF can be eliminated. Then,

it results that c nz,’ = I and u = (m:, rni)’ cannot

(I, I) ‘. With the previous by

-

&IL - l/L

.u= ‘I Consider

the electrical

assumptions,

we have a switching

I/C - I/R,,C-

I/R,,C

1( -‘+

0

E,iR,,

take the configuration linear model expressed

0 Ed& ! ”

(14)

A parameters

defined as R,, = 2Q L = 200 mH C=

1OmF

R,, = 40 R,, = 40 E, = +l2V E2 = -12V. Then Eq. (14) becomes

We can apply Section V results : i.e.

808

Journal

olthe

Frankhn Institute Pergamon Press Ltd

Opened Loop Control Law

FIG. 7. Convergence domain of the state vector of an electrical network.

the convergence domain (Fig. 7) Because xX, = -.Y?,_, the polygon representing is the segment [xj *, ~~~1. Th’IS is due to the structure of the electrical network where E, and Ez have a symmetrical function. Consider x E 9 such as, for example,

-y -(= 1 0.1

0.01

’

which corresponds to a current in the inductance L: i = p/L = 500 mA, and the voltage of the condenser C: u, = y/C = 1 V. According to Eq. (6), we can choose uj = 0, because f belongs to the segment [0, .uz,_]. Moreover, a4 = 0, because II = (I, 1)’ is not allowed. The equation set to be solved is : 0.3u, -0.3111 = 0.1 0.030 ( -o.o3a, : a,+a,

= 0.01

= 1

which leads to

By using definition

(12), we have )(Ajl = 100

which leads one to choose

Vol. 330, No. 5. pp. 799-813. 1993 Prmted in Great Britam

Tmod,such as

809

FIG. 8. Representation

of inputs

for a controlled

switching

electrical

system

We can represent the components of u in Fig. 8. The response of the model simulated with A.C.S.L,. on a 3%AT-computer, T 111 od = 10~ 4 s is shown in Fig. 9.

with

VI. Conclusion An opened loop control law and the determination of the convergence domain for linear systems with Boolean inputs have been discussed. The first step is to separate the time-axis into intervals with a constant period. Then, we apply on each interval the same sequence for the vector II. allowing one to reach a desired point .U. This sequence is calculated by considering the barycentra components of .? with respect to the tops of the convergence domain. Two examples are used to illustrate the control scheme, and simulations show that the state vector I reaches the desired convergence point -7. The possibility to allow the state vector to follow a desired trajectory X(t) in the convergence domain is presented in (8).

x, = 0.1

FIG. 9. Time response

of the state variables

of the electrical

switching

system.

Opened Loop Control Lou*

References (I) W. Leonard, “Control of Electrical Drives”, Springer, Berlin, 1985. (2) P. Borne, G. Dauphin-Tanguy, J. P. Richard, F. Rotella and 1. Zambettakis, “Modelisation et identification des processus,” Tome I, ‘Collection : Methodes et pratiques de I’ingcnieur”, Editions Technip, Paris, 1992. “Elaboration of a control law for switching (3) V. Abadic and G. Dauphin-Tanguy, nonlinear systems”, J. F~~uzkli~~Insf., Vol. 330, No. 4, pp. 6855693, 1993. (4) E. I. Jurv, “Inner and Stability of Dynamic Systems”. John Wiley, New York, 1974. (5) J. H. Wilkinson, “The Algebraic Eigenvalues Problem”, Oxford University Press, London, 1965. (6) J. P. Ducreux. A. Castelain, G. Dauphin-Tanguy and C. Rombaut, “Power electronics and electrical machines modelling using bond-graphs”, IMACS Transactions on “Bond-Graphs for Engineers”, (eds G. Dauphin-Tanguy and P. Breedveld), Elscvier, New York, 1992. to Physical System Dynamics”, (7) D. C. Karnopp and R. C. Rosenberg, “Introduction McGraw-Hill, New York. 1983. (8) V. Abaolie and G. Dauphin-Tanguy, “Control of switching continuous systems”, IEEESMC Conference, le Touquet (France), October 1993. Appendix A Firstly, we have to justify the inversibility of I- ch7~~~~,~~ : consider (A_,.i.,, , A,) the set of the A-eigcnvalucs. WC can write A = P ‘A’P where P is an inversible matrix of C”*” and A’ a triangular matrix, such as :

u>,,(l)

iz

.,_

I 0

Q:,.,(l) where rri,,. for to c. Then :

iE (2,.

, (I) and ,ie { I,

..’

u;.<, ,(I)

i.,, J

. (I ~ 1) are the coefficients

of A’ which can belong

Or, we have 0 n>,,(k)

(&)i

“.

f o

a:,,<,- ,(k)

(A,)“

(A’)” =

Then : r e” I7’,,,.>,, 0

where .yr,,, iE(2,...,4’

, , ,jE ( I,

Vo1.330.No.5.pp.799~813,1993 Prmtedin Great Briram

0

7

, q - I ) are complex coefficients.

811

V. Ahndiir and G. Duuplzin-Tm,qu~~ Then : 0 e’l’,‘?<,d detP

’

Then

det (I-c”‘“‘,“‘) = det

which leads to dct (1 _ ,,“‘,,,w) = fi (1 _e%L,,,,)~ 1.mI This determinant is equal to zero if and only if there exists & such as & = 0 which is impossible following the hypothesis of asymptotic stability of A. Then 1-e”7,~~,~~is an inversiblc matrix. Then, we can define r,, as :

where s,, is defined by

+ K, -y,,+ , = e* ‘m.,,,_y,, Then, it follows that z,,+ , + (I _ e” L ) ‘K = e”‘,,,,>,i(z,, +(I_,$“,,,,>,,)- ‘K)+K which leads to

-rr+I = c,’’ 111,>
(9) is expressed

A

e-?“,,,.m,’

as

i

then

812

Journal

or the Franklin

lnstitutc Pergamon Press Lid

Opened Loop Control Law

We have (I-eA7”~,~d)- ‘A- ’ = (A(I_eATm~d))m ’ = (A-_Ae*Tn~~~~)m ’ =

(A _ eaTnx,, A)-’ each term by A, we can demonstrate

Then, by multiplying

= A-‘(I_eATrn
Eq. (IO).

Appendix C We have : a, -

~ 2

ct,?

a,-a,Z

< 2 -(a,+, +~..+cc,rjt)a,

because (a,, , +~~~+~,~~~)a, > 0 due to the a,-definition. We have :

Then, we can conclude

that a,-a,2 2

Received : 30 December Accepted : 2.5 February

Vol. 330. No. 5, pp. 799-813. Prmted ,n Great Britain

1993

(a ,+,+”

.+xz,~~)a,
1992 1993

813