Open-loop deadbeat control of multidimensional systems

Open-loop Deadbeat Control of hfultidimensional Systems by S. G. TZAFESTAS Control

Systems

Laboratory,

Electrical

Engineering Department,

University of

Patras, Patras, Greece and N. J. THEODOROU Hellenic Air Force Technology Research Center (KETA), Faliro, Athens, Greece

Delta

Falirou, PaIeo

For a given linear, shift-invariant, single-input single-output (SISO), m-D discrete : system, with real constant coeficients, an input sequence is speczjied such that the system achieves deadbeat behaviour. This means that the system output reaches a steady ualue after a minimum number of steps in the space domain and remains at that value thereafter. In the present paper this steady value is chosen to be zero. No feedback control is applied and the appropriate input sequence, leading to deadbeat response, is found by pure algebraic methodology. ABSTRACT

I. Zntmduction In the present paper linear, shift-invariant, SISO, m-dimensional (m-D), discrete systems with real constant coefficients are examined. Although some work has been done concerning the deadbeat control of 1-D systems, there are very few results for the case of multidimensional systems. Kaczorek (1) has examined a kind of deadbeat behaviour in 2-D systems, in the sense that the output deviation from a reference input and the input vanish after a minimal 2-D space interval. Theodorou and Tzafestas (2) developed a deadbeat controller using output (for m-D systems) and state feedback (for 2-D systems), which results in a steady output, after a minimal space interval, and for an appropriate input sequence. A comparison of the above multidimensional feedback control methods may be found in (2). The present work involves an open loop deadbeat control method for m-D systems which may be viewed as an extension of the work of Rao and Janakiraman (3) for 1-D systems. Here, the deadbeat behaviour of a given system has the following meaning : specify an input sequence, that vanishes after a minimum number of steps in the space domain, such that the system output also vanishes after a minimumbut possibly different from the previous-number of steps in the space domain. The paper is organized as follows : in Section II the minimum space interval of the unknown input sequence is specified (beyond this space interval the input is zero), in Section III an appropriate input sequence is found within the above mentioned space interval, such that the output reaches zero within a possibly different but minimum space interval and in Section IV an example illustrates the validity of the proposed method.

0

The Frankhn Institute 00164032/85$3.00+0.00

311

S. G. Tzafestas and N. J. Theodorou II. Minimum Znput Space Interval A linear, shift-invariant, ficients can be represented

SISO, m-D, discrete system, by its difference equation :

with real constant

coef-

(11 a,, . . . , O=

1,

(K, ,..., K,,Jn,) of minimum duration, that is u(n,, . . . , n,)

for

(0,. . . ,O) < (n,, . . . , n,) < (1i,. . . ,1,)

n,) = 0

@,,...,

for

(2)

(n,, . . . , n,) > (Z1,.. . ,l,)

such that the output y(n,,. . . ,n,) of this system reaches zero after a minimum number of steps in the space domain and remains zero thereafter. According to (l), the output y(n,, . . . , n,) depends on the previous output n,-ii,) which cover an m-D space n, i,) and inputs u(ni i,, . . . , Yh-k..., interval (N,, . . . , IV,). Hence, if one starts with an input sequence (2) and some initial conditions, for example -j,)

Y(-j i,.“,

for

(0,. . .,O) < (jl,. . .,j,)

< (IV,,. . ., N,)

(3)

then one may calculate ~(0, . . . , 0),y( LO,. . . , 0), etc., successively. In this way, all the initial conditions (3) are taken into account when the output reaches the point (N r,. . . , NJ, but not before that. Now, if the output y(n,, . . . , n,,,) is restricted to be zero in the following interval, i.e. y(ni,...

,n,) = 0

then all the next outputs Yh...,

n,)

for

(1i ,...,

I,) 0, and the conjecture that a solution of the type (13) exists has been proved. Special case 1: 2-D systems

Input sequence to be specified : uh,

f4

for

(O,O) d h,

f12)<

(4, 4).

(16)

Output sequence to be set equal to zero : An,, nz) = 0

for

(4,U

G (n,, n,) < WI + 4, N2 + 4).

(17)

Here A =(N,+1,+1)(N,+Z,+l)-(Z,+l)(&+l)

(18)

B=(l,+l)(&+l)-1.

(1%

Thus A < B implies (N, + I, + 1) (N, + 1, + 1) < 2(1, + l)(& + 1).

(20)

Special case 2 : 1-D systems

Input sequence u(n) for

0 Q n < 1.

(21)

l = X(z,, . . . ,z,)z;il

c

. . . z,‘m+ f i=l

l$kl,...,

k&l;

. . . . A,-i$m

jr,=1

..,ziim

(25)

of the sequence

x(. . . ,

x( . . . . -j, ,..., zi ,... )zj$...zk’z;il wherek=k,,...,ki,l=~,,...,l,_,i=l

,..., m,

x(. . . , -jk,. . . , z2,. . .) is the -jk,. . . , jl,. . .), i.e.

(m- i)-D 2

transform

X’(. . . . -j, ,..., j, ,...) = j, f _. . . . j, !_O x(. . . , -j,, . . . , j,, . . .)zi$l.. Im a-

. z;!+-L.

(26)

In the above notation k < A implies k, < 1, etc., otherwise, if k > 1, the notation should be changed to x(. . . , jL,. . . , -j,, . . .), X’(. . . ,zl,. . . , -jk,. . .), etc. Equation (25) is shown in the Appendix. Applying the 2 transform on both sides of(l), one obtains

c

X

lskl,..., ki,il,...,

x Y’(. .

I,-i