Design of optimal sampled-data tracking systems with preview

Copyright © IFAC Time Delay Systems,

ELSEVIER

Rocquencourt, France, 2003

IFAC PUBLICATIONS www.elsevier.com/localelifac

DESIGN OF OPTIMAL SAMPLED-DATA TRACKING SYSTEMS WITH PREVIEW K.Yu. Polyakov*,l E.N. Rosenwasser* H.P. Lampe**

• St. Petersburg State University of Ocean Technology Lotsmanskaya ul. 3, 190008 St. Petersburg, Russia ** Fachbereich Elektrotechnik, Universitiit Rostock D-18051 Rostock, Germany [email protected]

Abstract: The problem of optimal sampled-data (SD) tracking system design is considered for the case when the reference signal is known in advance over an interval T. A rigorous frequency-domain solution is given on basis of the Laplace transform in continuous time. Formulas determining the order of the optimal controller have been obtained. Limiting properties of the optimal systems as T -+ 00 have been investigated. Copyright © 2003 IFAC Keywords: Sampled-data systems, Tracking systems, Control system design, Preview

of the 'Hoc-norm of the associated operator. These papers made clear that serious problems arise in constructing equivalent state-space discrete-time models for SD systems with arbitrary pure delays or previews.

1. INTRODUCTION

In manv applications. for instance. in terrain following for planes or other vehicles, and control of robots, the desired motion trajectory is known in advance over a finite time interval T. In this case information about "future" values of the input (which is called "preview") can be used for decreasing the tracking error.

The problem of preview tracking contains elements of the form eST. Therefore, the system considered in continuous time is infinite-dimensional, and this feature causes significant problems for analysis and design. Such well-known methods of direct SD systems design as "lifting" (Chen and Francis, 1995) and "FR-operator" (Hagiwara and Araki. 1995) fail in this case, because so far they cannot handle time-delayed systems in a systematic way. The recent approach of (Nlirkin and Tadmor, 2002) is computationally very involved and applicable only to delays which are integer multiples of the sampling period.

Previously the preview control problem has been considered mostly for continuous- and discretetime LTI systems (see (Tomizuka. 1975; Grimhie. 1990: Halpern. 1994) and references therein). Le::>s attention was paid for sampled-data (SD) ::>y::>tellls that are t ime- periodic and. therefore, more complicated. A similar problem of timedelayed signal reconstruction for open-loop SD systems was investigated in (Khargonekar and Yalllamoto. 1996: Yamamoto and Hara. 1998). when' the t niCking error was estimated in terms

1

This paper presents a solution of the design problem for optimal digital preview control on basis of the frequency-domain theory of SD systems (Rosenwasser and Lampe. 2000), which takes into

Supported b~' the German science foundation (DFG)

327

account networks of the form eST without any approximations (Rosenwasser et al., 1997). As distinct from (Khargonekar and Yamamoto, 1996; Yamamoto and Hara, 1998), the optimization criterion is the L 2-norm of the continuous-time error signal, i.e., the integrated quadratic error between a desired signal and the output of the SD system. The proposed solution takes into account restrictions on the control power, and is valid for any T rather than for integer multiples of the sampling period.

delays in continuous-time elements and the computational delay. The transfer function of the hold circuit (not shown in Fig. 1) will be denoted by Ch(s). The solution presented here does not exploits the fact that this is a zero-order hold and is valid for any form of the modulated pulse. Let the reference signal r(t) be given as r( t) _ {

-

where ro(t) is some function, and let it have the Laplace transform R(s). Assume that r(t) is known in advance over an interval T and the input of the control system is excited by "future" values of the signal r(t) with preview T, i.e., r(t + T).

2. NOTATION Let T be the sampling period,

W

~ 27r IT the

sampling frequency and ( ~ the unit delay operator. The asterisk denotes the Hermitian conjugate function such that 2

e- sT

F*(s) ~ F(-s),

The aim of the system is to reconstruct, as precise as possible, some linear transformation y(t) of the input signal r(t). This desired transformation is given as a network with transfer function Q(s). In order to restrict the control power, let us introduce the ideal control signal u(t) given as a result of transformation of the reference signal r( t) by a linear network with stable transfer function Qu(s) (Rosenwasser and Lampe, 2000; Polyakov, 2001). Then, we can construct the cost function incorporating a weighted sum of the integral squared output and control errors:

C*(() = C(c- J ).

Real rational functions in s or ( will be called stable if they are analytic in ~s :::: 0 or j( I SI, respectively. Polynomials in ( are called stable if they are free of zeros in the closed unit disk. For any transfer function F(s), we define the discrete Laplace transform as (Rosenwasser and Lampe, 2000):

DF(S, t)

~~

f:

e(s+kjw)t

k=-oo

J 00

J =

F(s + kjw)

0 for t < 0 ro(t) for t :::: 0

[y(t) - y(t)]2

+ p2 [u(t) - u(t)]2 dt

(2)

-00

(1)

where p2 is a nonnegative weighting coefficient, and it is required to find the transfer function of an optimal stabilizing controller C(() which ensures the minimal value of J.

Denote the monic denominator of the function DP((. t) by dd(), and J(F) = degd F .

In (Rosenwasser and Lampe, 2000) a constructive

For any polynomial f(() free from zeros on the unit circle the superscript "+" denotes a stable factor, and "-" the strictly antistable cofactor such that f(() = f+ f-· Hereinafter, the tilde denotes the inverse polynomial in ( having the reverse order of coefficients:

method was developed for analysis and optimal design of SD systems on basis of Laplace transforms in continuous time. Recall that the system at hand includes a non-causal block which realizes preview. Therefore, even for zero initial energy of the system, there are nonzero signals for - T S t < 0, and the ordinary Laplace transform is seemingly inapplicable. Nevertheless, after changing the time zero point it is possible to construct an equivalent causal system which gives the same value of the cost function as the initial system.

By a quasi polynomial we mean a rational function in ( with poles (if any) only at ( = o.

Indeed, let

3 STATEMENT OF THE PROBLEM

i = t + aT for T

The block-diagram of the SD tracking system is shown in Fig. 1. It consists of a plant with transfer function F(s), actuator H(s), dynamic feedback C(s) and digital controller C(() with prefilter Fo(s). The blocks e- ST1 and e- ST2 simulate pure

=

aT - ()

(3)

where a is an integer and 0 S () < T. Then, for i < 0 all signals in the system are zero, and the criterion (2) takes the form

J 00

J = for tire scalar case

(y(i) - y(i)]2 + p2 [u(i) - u(i)]2 di. (4)

o

32X

sampled-data system

r{t

+ 7)

i

y{t)

Fig.!. Sampled-data tracking system Then, i is taken as new independent variable. Since the time zero point is shifted by an integer multiple of the sampling period, and the system is T-periodic, such operation causes no phase shift of the sampling unit in the new system.

The Laplace transforms of the desired signals are calculated in a classical way:

Y{s) = QRe- suT , Represent

Due to (3) the block Q is excited by the signal r{i-aT), while the input of the SD system is acted upon by r{i - B). These signals can be represented a<; results of transformations of r(i) by pure delay networks e- suT and e- slJ , respectively. Now, all networks are causal and all signals are zero for i < O. For brevity, we will write t instead of i.

and

7

7 71

71

in the form

= 1T + 'l/J = aT -

()

= 11T + 'l/JI = alT - BI

where 1, 11, a and at are integers such that 0 S BI < T, 0 S 'l/J < T, 0 S 'l/Jl < T. Then, the following equalities hold:

o S B < T,

Thus, the problem of optimal preview tracking system design consists in determining the transfer function of an stabilizing digital controller C{(), which ensures the minimal value of the cost function J in (4) for the equivalent system.

a=1+J.1., with

_ J.1.-

{O ,() = 0 I,B#O

= 11 + J.1.1

(7)

J.1.1 = { 0 , BI = 0 1 , B1 # o.

(8)

al

Hence, 4. EQUIVALENT DISCRETE PROBLEM

DFoR{S, -B) = DFoR{S, h - a)T + 'l/J) =e(r--u)T DFoR{S,'l/J)

Using Parseval's formula, we can write the cost function (4) in the form 3 and joo

J =

J IY -

YI 2 + p21U - UI 2 ds

YY' =es(r--ullT FHe slJI R'Q'ChDFoR{S,'l/J)C

(5)

UU' = es(r--u,)T H e slJI R'Q~ChDFoR{S,'l/J)C.

-joo

Using these relations, after discretization and passage to the variable ( gives (Rosenwasser and Lampe,2000)

where Y{s), U{s). Y{s) and U{s) denote the Laplace transforms of the corresponding signals. If necessary. frequency-dependent weighting functions can be used in (5) {see (Polyakov, 2001)).

J = _1. 2~J

Using the technique of (Rosenwasser and Lampe, 2UOO). we can write the Laplace images for y(t) and u(t) as

f

(ACC' - BC - WC'

+ E)

r

d( (

(9)

r

the unit circle passed anticlockwise, and

Y{s) = F H e- STI ChDFuR(S, -B)C

A{()

= DAuG"G;' {(, O)DFoR{(, 1/J)DFuR{C I , 1l')

U{s) = H e- ST1 ChDFuR{S, -B)C

B{() = (U I -,. D BoGh ((, Br)DFoR((, 1/1)

with

E{() = DEu{(,O)

with

+ p2)H H' (FQ' + p2Q~)H R*

Ao{s) = (FP" Bo{s) =

Eo{s) = {QQ* + p2QuQ~)RR' . Generally, Qu may have poles which are not poles of Q. Let dQ {() and dQu {() be the denominators of

Hereinafter function arguments are omitted for brevity if no confusion is possible.

:l

329

Theorem 1. Let AI-A4 hold. Then,

DQ(T, (, 0) and D Qu (T, (, 0), respectively. Denote by dQ , (() the least common multiple of dQ (() and dQ ,,((). Let also 0) = degd QJ .

i) the optimal digital stabilizing controller which minimizes (9) has the form C(() = ne de

5. CONTROLLER DESIGN All deviations hold under the following assumptions on the transfer functions:

d-g*(" P C

Calculating the discrete Laplace transforms DAuC"Ci,(T,(,O), DFuR(T,('"l/J), DBoc,,(T,(,Bd and DEo(T, (, 0), the functions A((), E(() and E(() can be written in the form (Rosenwasser and Lampe, 2000)

A(~)= d

02

02

d* ' - - ' - FH FH d FaR d FaR (3(UJ -"'( 02 E(()= d d* d FH QJR FoR

(10)

E(()=

(12)

c"

dQJRdQ1R

+ 7Td QJR

-- d-{3*O"("'(-Ul+" 1 2 (19)

such that deg 7T < deg d 1 + v; ii) the minimal value of the cost function equals

Notice that it follows from A3 and A4 that A o is proper and Eo and Eo are strictly proper.

a)

(18)

where the polynomials {P, D, 7T} are a solution, with 7T(() of minimal degree, of the following system of polynomial equations:

A!. R, Q and Qu are free of poles in ~ 8 > O. A2. F, H, Fo, and G are free of poles at the imaginary axis R 8 = O. 4 A3. FoR, QR and QuR are strictly proper. A4. Hand F H are at least proper.

.

d+ P FHFaC D

=

J

= opt

f r

(7T7T*

d l (d l )*gg*

+

E _ El Ej) d( (20) Al

(

where Al (()

= D AaChC~ ((, 0)

El (() = D BaCh ((, Bd·

iii) the polynomials de and ne satisfy (11)

degn e

~ X

degde

~ X

(21)

+ (1.,.

(22)

X = o(FHFoGR) + 01 - 1 + max(O,,-,d; iv) the characteristic polynomial of the closedloop equals, up to a constant factor, to

where 02(() is a polynomial, while a) ((), (3(() and E() are quasipolynomials Since A O(8) and E O(8) are Hermitian self-conjugate, we have

(23)

Let

•

The proof is omitted due to space.

Then, find a stable polynomial g(() as a result of the factorization with respect to the unit circle (13)

6. PROPERTIES OF JOPT(T)

(14)

In this section some properties of the optimal cost in dependence of the preview, i.e. of the function Jopt(T), are considered. In (Polyakov et al., 2003) it is shown that the following inequality holds for the similar open-loop problem for any TO:

where aT is an integer and 0 ~ BT < T. Construct the discrete transfer function of the closed-loop

JOpt(TO

where n(() and d(() = dFHFoC are polynomials such that n(O) = O. Denote d)() = dd c = dFHFoCC

(16)

a(() = gg"d cQ - nS"o;C- u ,.

(17)

+ kT)

~ Jopt(TO),

k

=

1,2,... (24)

It is well known (Rosenwasser and Lampe, 2000) that if there is an unstable block in the loop, not all stable solutions C correspond to stabilizing controllers. Hence, as distinct from the case with a stable plant, the optimization has to be performed on a subset of the set of stable functions C. This means that the optimal values of the cost function can either increase or decrease as k increases, and (24) does not hold. Indeed, let Co and Co be the optimal controller and the associated function (6) for some T = TO. Then, for the preview interval TO + kT (with integer k > 0) the same value of the

Let l/ be the minimal nonnegative integer such that3" n iC- U1 +" and 9"(" are polynomials in C.

In principle, there are cases when some of these functions may have neutral poles. but this question will not be exposed here. 4

330

..

cost fUllction can be ensured by the choice C = (kCo, i.e., extra information is simply ignored. Nevertheless, the corresponding controller may well not be stabilizing.

12

Next, the continuity of lopt(7) is investigated. Since 7 = -yT + 'I/; the dependence lopt(7) can be represented as a function lopth, '1/;) in the variables -y (an integer) and 'I/; (0 :S 'I/; < T). As follows from the formulas for the coefficients A and B in the functional (9), these functions are continuous with respect to 'I/; for 0 :S 'I/; < T. Then. due to the continuity of the solution of the optimization problem, the function lopth, '1/;) is also continuous for -y = const and 0 :S 'I/; < T, and jumps are only for 'I/; = O.

,,

"'"'0.8

.,"Po 6

06

, ,,, , ,

04

02

0

8

0

.

10

T1.me.

t

12

14

16

18

20

Fig. 2. Desired and actual processes for Ex. 1. Remark 1. If all blocks inside the loop are stable, we obtain 10 = O. Since the functions E, AJ, and B I are independent of '1/;, the value loo is also independent of '1/;. In this case the problem at hand is equivalent to the corresponding open-loop problem (see (Polyakov et al., 2003)), and (24) holds, because any stable solution C is admissible, i.e., is associated with a stabilizing controller.

If the pole excess of FoR is 1, the function DFoR((, '1/;) has a jump at 'I/; = 0 (Rosenwasser and Lampe, 2000). Physically, this means that the sampling unit is acted upon by a discontinuous signal. Therefore, the coefficients A and B in the integrand of (9) can have discontinuities. In this case the problem becomes non-robust in the sense that the cost function depends on the fact whether the sampling unit fixes the input signal at the moments kT + 0 or kT - O.

A peculiar feature of the problem under consideration is that the input signal passes through the sampling unit. Therefore, the output of the SD system depends only on the discrete values at tk = kT. This means that information is lost, and therefore lopt does not tend to zero as 7 -+ 00.

Limiting properties of the system as 7 -+ 00 are of special interest. It appears that the limiting value of lopt as 7 -+ 00 depends, in the general, on '1/;. Let us perform some preliminary transformations. Assume that there exists the factorization

7. NUMERICAL EXAMPLES Example 1. Consider the system in Fig. 1 with where the function K (() is stable together with its inverse. Denote F(s)

(25)

H(s)

= Fo(s) = G(s) = 1

1 1 R(s) = s2+ s ' Q(s) = O.ls+l TI = 1.5, 72 = 0, 7 = 2, T = 1.

where JT2 is a polynomial and {.} _ denotes the strictly proper strictly antistable part. Note that none of the functions aD, d, and K depends on T' Therefore, 1r2 is also independent of f. Nevertheless, K(() does depend on 1/.', hence so does JT2.

rl =

O.

Using the above formulas leads to the optimal controller

C(()= 7.567 - 6.771( + 0.4711(2 1 + 0.2674( - 0.7201(2 - 0.5968(3 + 0.04954(4

Theorem 2. Let lopt (r, 1p) be the minimal value of the cost function (9) for some known T and 'I/J. Then. for any 11.' there exists the limit loo('I/;) liIll,_a<: lopth,";:), which is equal to

~(

1

= 5s + l'

providing for 1 = 0.05173. For numerical calculations the DirectSD tool box for MATLAB was used (Polyakov et al., 1999). Desired and actual output processes in the system are shown in Fig. 2. The curve lopt(T) for this system is shown in Fig. 3. If the preview interval 7 increases, the curve tends asymptotically to the constant value loo = 0.007769 calculated by (26) (dashed line in Fig. 3). Circles denote the values of lopt(T) for integer T'S. Note that the cost function is nonzero for any 7.

(26)

where

331

even with the infinite preview interval it is impossible to obtain zero integral tracking error, because a portion of information is lost due to time quantization. The results can be applied in signal processing and predictive control.

REFERENCES Chen, T. and RA. Francis (1995). Optimal sampled-data control systems. SpringerVerlag. Berlin, Heidelberg, New York. Grimble, M.J. (1990). LQG-predictive optimal control for adaptive applications. Automatica 26(6), 949-96l. Hagiwara, T. and M. Araki (1995). FR-operator approach to the 'H2-analysis and synthesis of sampled-data systems. IEEE Trans. Autom. Contr AC-40(8), 1411-1421. Halpern, M.E. (1994). Preview tracking for discrete-time SISO systems. IEEE Trans. Autom. Contr AC-39(3), 589-592. Khargonekar, P.P. and Y. Yamamoto (1996). Delayed signal reconstruction using sampleddata control. In: Proc. 35th IEEE Con/. Decision Contr. Kyoto. pp. 1259-1263. Mirkin, L. and G. Tadmor (2002). Yet another 'Hoo-discretization. Tech. Report ETR-200203. Israel Institute of Technology. Polyakov, K.Yu. (2001). Polynomial design of optimal sampled-data tracking systems. 1: Quadratic optimization. Avtomatika i Telemechanika (2), 149-162. (in Russian). Polyakov, K.Yu., RN. Rosenwasser and B.P. Lampe (1999). DirectSD - a toolbox for direct design of sampled-data systems. In: Proc. IEEE Intern. Symp. CACSD'99. Kohala Coast, Island of Hawai'i, Hawai'i, USA. pp. 357-362. Polyakov, K.Yu., E.N. Rosenwasser and RP. Lampe (2003). Optimal open-loop tracking using sampled-data systems with preview. In: Proc. 11th IEEE Mediterr. Control Con/. Rhode, Greece. pp. IV04-D3. Rosenwasser, E.N. and B.P. Lampe (2000). Computer Controlled Systems - Analysis and Design with Process-orientated models. Springer-Verlag. London Berlin Heidelberg. Rosenwasser, RN., K.Yu. Polyakov and RP. Lampe (1997). Frequency domain method for 'H2 -optimization of time-delayed sampleddata systems. Automatica 33(7), 1387-1392. Tomizuka. M. (1975). Optimal continuous finite preview problem. IEEE Trans. Autom. Contr AC-20, 362-365. Yamamoto, Y. and S. Hara (1998). Performance lower bound for a sampled-data signal reconstruction. In: Open Problems in Mathematical Systems and Control Theory (V. Blondel, E. Sontag, M. Vidyasagar and J. Willems, Eds.). pp. 277-279. Springer. London.

Fig. 3. Curve Jopt(T) for Example 1. 25'r--~--------------' 0. ,..,':: 2'0,

c

,'\ ,

'

,. '.,

o

t

,. \,'

,/\ '\.'

\,

'

't

I

\,"

I'

I

1\

1\

\,1' "\," '\,,' \\1 I'' ,,' '\,"

.~

u U C

"

~ u

~

10

U

. .

00

.

Preview interval. T

"

Fig. 4. Curve Jopt(T) for Example 2.

Example 2. For the same system but with unstable plant 1 F(s) = - 55 - 1 the optimal digital controller

C()

=

14.75 - 17.28( + 3.573(2 1 + 0.67( - 0.9515(2 - 1.059(3 + 0.3401(4 gives J = 2.7072. The curve Jopt(T) for this system is shown in Fig. 4, where the dashed line is the periodic limiting curve Joo(T) from (26). As is evident from the graphs, the cost increases as T increases. Moreover, there is an "optimal" T, for which the error is minimal.

8. CONCLCSIONS

The optimal design of SD tracking systems with preview of the reference signal is formulated, and a rigorous solution is given in the frequency domain. Formulas are derived for calculating the order of the optimal controller based directly on initial data. It is shown that the optimal cost function approaches the T-periodic function (26) as T -+ 00. If all blocks in the loop are stable, this function is constant, precisely as for openloop systems. As distinct from the stationary case,

332