51st IEEE Conference on Decision and Control

December 10-13, 2012. Maui, Hawaii, USA

Digital stabilization of delayed-input strict-feedforward dynamics

Salvatore Monaco, DorotMe Normand-Cyrot and Valentin Tanasa

Abstract- The paper deals with sampled-data stabilization of delayed input-affine dynamics. To remove the infinite dimen­sionality, the design is set in the sampled-data context and the delay is compensated with a discrete-time predictor. A simulated example illustrates the performances. Keyword Nonlinear Systems, Sampled-Data Systems, Delay Predictor

I. INTRODUCTION AND PROBLEM STATEMENT

Starting from [1] the interest in using compensation of long input delays in control systems is still at the attention of the scientific community and different approaches have been successfully developed in the literature. Looking at the problem of feedback stabilization, the reduction model approach introduced in [2], [3], [4] shows that a controller designed on the delay free dynamics solves the problem when it is computed on the predicted state. Unfortunately such a solution, the "extension" of the Smith predictor to linear unstable systems, brings to a controller which is im­plicitly defined due to the intrinsic infinite dimension of the continuous-time problem. Approximation techniques can be used to provide solutions which bring to adopt discretization schemes. In this context, the interest for looking a sampled-data solu­tion to the stabilization problem, where piecewise constant controls are computed from the data available at discrete-time instants, is clear. The dedicated literature mainly considers the case in which the digital controller is an emulation of the continuous-time one and studies the effect of the sampling length on the stability of the overall system. Recently, in [5], the case of linear input delayed dynamics is studied with reference to the solution provided through the reduction model approach. Pursuing a Lyapunov based analysis, it is shown that the stabilizing properties of a continuous feed­back solution can be preserved under digital implementation when limiting the sampling period to a bounded region. The extension of the Artstein's reduction method to the nonlinear context in [6], [7], [8] and [9] and the approach proposed in [10] to handle the sampled-data context are at the basis of our investigation. As in the cited references we consider the problem of stabilization in the presence

S.Monaco is with Dipartimento di Informatica e Sistemistica 'Antonio Ruberti', Universita di Roma "La Sapienza", via Ariosto 25, 00185 Roma, Italy. salvatore. monaco@dis. uniromal. it

D.Normand-Cyrot and V. Tanasa are with Laboratoire des Signaux et Systemes, CNRS-Supelec, Plateau de Moulon, 91190 Gif-sur-Y vette, France cyrot@lss.supelec.fr; tanasa@lss.supelec.fr

V. Tanasa is also with University Politehnica of Bucharest, Romania The work of V. Tanasa was supported in part by Romanian Ministry

of Education, Research, Youth and Sport through UEFISCDI, under grant TE23212010

of arbitrarily long delay on the input for a plant which is delay free smoothly stabilizable. In [10], assuming the measurements available at discrete-time instants only, the authors provide computable predictors. Two different cases are examined either assuming the existence for the delay free case of a discrete-time stabilizing controller or through emulation of a continuous-time stabilizer. The computational advantage of the approach is revealed when considering integrable dynamics as strict feedforward ones. In this paper, we consider input-affine dynamics with input delay and we assume the existence of a global stabilizer for the delay free case. We show that the sampled-data stabilizer introduced in [11] and computed from the output of a discrete-time predictor achieves global stabilization. For dynamics admitting exact sampled models as polyno­mial strict feed forward ones, the predictor can be explicitly described. The approach applies to dynamics which can be transformed under preliminary coordinates change and feedback into finitely discretizable dynamics. The controller is specified by means of its asymptotic expansions which makes easier the computation of the approximated solutions [11], [12]. The paper is organized as follows. The proposed strategy is developed in section II; the case of a finite sampled predictor is also discussed as the digital equivalent of an integrable predictor proposed in [10]. A simulated exampled illustrates the performances in section III. Some Notations. Maps and vector fields are assumed smooth (i.e. infinitely differentiable - C=), vector fields are forward complete to prevent from finite escape time under bounded inputs . The set %' (resp. %'d) of admissible inputs consists of all U-valued piecewise continuous (resp. piecewise constant) functions on R. Lf = L�1 fi(')!xt , denotes the Lie derivative

Li operator and eJ := 1 + L;2:1 if the Lie series exponential operator, associated with a vector field f (1 is the identity operator on RIl); "(x)" or ''Ix'' denotes the evaluation at a point x of a generic map. Given two vector fields on RIl, adJg = [f,g] = [LJ,Lg] = LfoLg - LgoLf indicates the Lie bracket of vector fields. Time dependency in discrete-time of a function A" is denoted A, (k) or A,k.

II. DIGITAL LYA PUNOV STABILIZATION

Let us consider the continuous-time single input affine dy­namics with delayed input

(1) Xe denotes the equilibrium point f(xe) = 0, supposed without loss of generality equal to zero; Xc indicates the continuous­time state behavior; the delay 'T is known. It is assumed that:

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A - The delay free system (1) is smoothly stabilizable; i. e. there exists a feedback u(t) = y(x(t)) with y(o) = ° and a Lyapunov function V> ° with V(O) = ° such that

In what follows, the sampling period 8 E (0, T*], given within a finite time interval, and the delay is chosen so that r = N 8 for a suitable integer N. (The more general case r = N 8 + i', i' E [0,8) can be studied through simple extension).

A. The reduction method in continuous time

The reduction method (see [8], [9]) provides a stabilizer of the form u(t) = nt) = y(zc(t)) with for all t � °

zc(t) =xc(t+r) =xc(t)+ J�1(Lf+U(S)Lg)(zc(S))dS (2)

and the predictor dynamics zc(t) = !(zc(t)) +u(t)g(zc(t)). The main difficulty stands in the computation of u(t) accord­ing to the implicit integral equation (2) with initial conditions verifying nt) = ro(t) for all t E [-r,O], where ro is any continuous R valued function defined on [-r,O] such that

ro(o) = y(x(O)) + 1°1: (Lt+ ro(s)Lg)y(zc(s))ds. (3)

and zc(O) = Xc (0) + J� 1:(Lt+ro(s)Lg)(zc(s))ds.

B. The reduction method in discrete time

In a discrete time context, the reduction method is equivalent to perform N dynamical extensions so greatly simplifying the design and the computation of the controller. Consider a discrete time dynamics described as a first order difference equation parameterized by the control variable with N input delays

(4)

Assuming as in A the existence in the delay free case of a smooth stabilizing feedback Uk = Y(Xk) with Y(O) = 0, then denoting by Zk the state predictor N steps ahead, i.e. Zk = Xk+N, it is a matter of computations to verify that the feedback Uk = Y(Zk) achieves asymptotic stabilization of the input-delayed dynamics (4) with predictor dynamics Zk+1 = !(Zk) + ukg(zd · Substituting Zk with xk+N one gets

which just corresponds to set vI(k) = Y(Xk) in the extended dynamics described below

The same approach applies to the sampled-data dynamics.

C. Stabilization under sampling To define the digital stabilizer in the delay free case, we reformulate the input-output matching under digital control proposed in [l3] in terms of the input-Lyapunov matching. Setting r = ° in (4) and the input is kept constant over successive intervals of length 8 > 0, u(t) = Ud (k) for t E [k8, (k + 1 )8), then the equivalent sampled-data dynamics is described by the map

xd(k+ 1) = e°(f+Ud(k)g)Xd(k). (6)

The following result is recalled from [l3].

Proposition 2.1: Given an input-affine dynamics (4) as­sumed forward complete and a smooth feedback u(t) = y(x(t)) satisfying A, then for all 8 E (0, T*] and LgV(x) # 0, there exists a digital feedback of the form

8i Ud = yO (x) = Yo(x) + � (i+ I)! y;(x)

with yO (0) = ° which assures stabilization with the same performances of the continuous controller at the sampling instants ( Lyapunov-matching at the sampling instants); i.e. for all k, by setting xd(k) = xc(t = k8) the equality

V(xd(k+ 1)) -V(xd(k)) = l(k+I)O (Lf + y(xc(s) )Lg) V (xc(s) )ds ko ' (7)

is satisfied by an appropriate choice of ud(k) = yO (xd(k)). The existence of yO (x) is ensured by the condition Lg V(x) # ° and by the implicit function Theorem. The solution is expressed by its series expansion in powers of 8 around Yo(x). The proof is constructive and works out by equating terms of the same power in 8 in the respective expansions with respect to 8 of both members of the equality (7) (see [14]). For the first terms one gets

Yo(x(k))

Y2(x(k))

Y(X(t))I,=ko; rt(x(k)) = y(x(t))I'=ko .. rt(x(k)) = y(x(t)) + 2Lg V (x(t)) ad1t,g] V (x(t)) I,=ko'

so recovering the continuous solution and its derivative for Yo and YI. It must be noted that the approximate controller Yo(x(k)) satisfies (7) with an error in 0(8); Yo(x(k)) + �rt(x(k)) with an error in 0(82); Yo(x(k)) + �rt(x(k)) + �� Y2 (x( k)) with an error in 0(83) and so on.

Remark - Yo(x(k)) and rt(x(k)) do not depend on V except through y(x) while from J'2(x(k)), the additional terms de­pend explicitly on the chosen Lyapunov function.

III. DIGITAL STABILIZATION WIT H DELAYED INPUT

The main result can now be stated. Two problems are solved, a digital stabilizer is built for the delay free case, a sampled-data predictor is computed by taking advantage of the structural properties of the dynamics (possible finite sampled model).

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Theorem 3.1: Consider an input-affine continuous-time dy­namics (4) with input delay r ?: 0, satisfying A. Assuming that r = N8 for an integer N with sampling period 8, then the origin is asymptotically stabilized by the sampled-data state feedback, piece-wise constant over 8

(8)

satisfying (7) with predictor dynamics

zd(k+ 1) = e°(f+'f (Zd(k))g)Zd (k) (9)

where zd(k) is computed at time k8 from the state measure­ment xd(k) = xc(t = k8) according to

zd(k) - eO (f+lld(k-N)g) eo(f+lld(k-l)g) I I (10) - o • • • 0 d Xd(k) with initial conditions Xd(O),Ud(i);i E [-N, ... ,-I] and Zd(O) = e°(f+lld(-N)g)o ... oe°(f+lld(-I)g)ldlxd(O).

Proof The proposed digital feedback strategy (8) with Zd (k) = Xd (k + N) is equivalent to solve the stabilizing problem for the delay free dynamics with N dynamical extensions

eo(f+VdJ (k)g)Xd(k) Vd2(k)

with initial conditions Xd(O), Vdi(O) = Ud( -N + i-I) for i E [-N,-I] and Ud(O) = eo(f+lld(-N)g)o ... oe°(f+lld(-l)g)yOlxd(O). To show asymptotic stability one considers the extended Lyapunov function

which gives

W(k+ 1) - W(k) = V(xd(k+ 1)) - V(xd(k)) 1

. .0 2 1 . .0 2

2(Vdl(k)-r (xd(k))) +2(Ud(k)-r (xd(k+N)))

V(xd(k+ 1)) - V(xd(k))

when setting ud(k) = yO (xd(k + N)) and thus Vdl (k) = yO(xd(k)). Moreover, by construction, the feedback strategy Vdl (k) guarantees strict negativity of V (xd(k + 1)) - V(xd(k)) since it satisfies

Remark - The Xd dynamics matches the Zd dynamics with a delay of N steps with the same closed loop performance so recalling the Smith predictor strategy of the linear context.

Remark - The digital controller (8) with zd(k) given by (10) rewrites as

so recovering for N = 0 that rO(xd(k),O) = yO (xd(k)). How­ever, the computation of the controller at time k according to the formulation (8-10) is more robust to measurement uncertainties since it takes into account the data measurement at each step. This aspect is made clear by the computational algorithm described below.

Remark - Theorem 3.1 holds true in the more general case of a fraction nary delay r as r = N 8 + r; r E [0,8 (. In such a case, one gets in place of (9) the delayed input xd-dynamics

xd(k+ 1) = e't(f+lld(k-N-l)g)oe(O-'t)(f+lld(k-N)g)Xd(k) (12)

so that one sets in place of (10)

zd(k) xd((k+N)8 + r) = e't(f+lld(k-N-l)g) 0 • • • 0e0(f+Ud(k-l)g)Xd(k).

It is a matter of computation to verify that zd(k) sat­isfies the predictor dynamics (9) with initial conditions Xd(O),Zd(O), ud(i) for all k E [-N -1, ... , -1] satisfying

Zd(O) = ei'(f+lld(-N-l)g)o ... oe°(f+lld(-l)g) (Xd(O)).

Remark - The case of systems with delayed measurements can be treated along the same lines. However, in that case, the given initial conditions are no more the control values but the state values xd(i) for i= [-N,-I].

Remark - In [10], the authors discuss the case of state delay measurements in a sampled-data stabilizing context under emulated controller.

IV. ABOUT A PPROXIMATE OR FINITE SOLUTIONS

The computation of the approximate or finite order solutions makes reference to the expansion of the solutions as series in 8. This concerns either the stabilizer functional yO ( -) or the predictor functional eO (f+Udg) 1d. Approximate solutions refer to truncations of these series expansions at a prefixed power of 8, while finite order refers to the nullity of the terms in O( 8P) from some integer P. To get finiteness in 8 of yO (.) is a difficult task. It depends on the dynamics itself and on the nonlinear function V to be matched. In practice, approximate solutions of low order (one or two) behave well as illustrated through various examples in [15] and the references therein. Finiteness of the predictor order refers to finite discretizability of the dynamics eO (f+lldg) ld itself under Ud constant. Such a property can be verified either directly or through preliminary coordinates change and feedback as studied in [16]. A typical example of finitely discretizable dynamics is that of strict feedforward dynamics discussed in the sequel.

Finite discretizability is verified when the sampled data equivalent dynamics to (4) is described by a finite number of terms; i.e. for Ud constant, there exists P ?: 1 such that

P 8i . e°(f+Udg)x = x + � -L' x k k L. ., f+Udg k·

i=l 1.

In such a case, the predictor dynamics (9) is described by a finite number of terms and thus is exactly computable.

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This occurs when the dynamics exhibits a triangular form with polynomial non Iinearities. The strict feedback form is described below

Xn-I (t) xn(t)

fn-I (xn) + gn-I (xn)u(t) u(t)

with R-valued mapping J;,gi;i E [1,n] polynomials in their arguments. It is a matter of computations to verify that one gets finiteness in the powers of 0 due to the triangular form. Example - In a comparative spirit, let us consider the system in strict feedforward form studied in [10]

X2+X� X3 +X3U(t - 'T) u(t - 'T).

In the delay free case 'T = 0, setting u(t) = Uk for t E [ko, (k+ 1)0), one gets the sampled equivalent dynamics in 03 below

Xlk+1 Xlk + o (X2k +X�k) 02 03

+ 2T(X3k + 3X3kUk) + 3(Uk + 3u�)

02 X2k+1 X2k + o (X3k +X3kUk) + 2T (Uk + u�)

X3k+1 X3k + OUk·

When 'T = No, the predicted state z(k) = x(k + N) is described through composition over N time steps. Iterated sampling over a finite number of time steps to get a predictor dynamics of order N is in line with integration over a finite time interval, namely the delay time, as proposed in [10] to define the integrated predictor. The main difference with our work is that the authors in [10] assumed the existence of a piecewise constant stabilizer in the delay free case while we build an ad hoc digital feedback under the assumption of existence of a continuous-time stabilizer. Then, in the most favorable cases, prediction is achieved either through integrability in [10] or finite sampling in the present case. The underlying property is the triangular form of the dynamics which makes finite the number of iterative integrals or time derivatives being integration and derivation two formal reverse operators.

V. THE VAN DER POL EXA MPLE

Let the strict feedforward system studied in [10] and repre­senting the Van der Pol oscillator with known delay 'T.

X2(t) -x�(t)u(t - 'T) u(t - 'T).

The sampled-data equivalent dynamics

(13) (14)

In the delay free case, the equivalent sampled dynamics is of order 3 in 0

b2 b3 XIk+1 Xlk +bX2k(1-X2kUk) + 2TUk(l +2X2kUk) + 3u� X2k+1 X2k + bUk'

Scenario Crit CC ECC SLMC Tr 5.69 5.27 5.46

8= 0.5,1"= 0 Eu 0.283 1.0850 0.6486 E(x) 0.0377 0.0361 0.0383 Tr 5.69 4.8100

8= I,N= 0 Eu 0.283 - 2.8649 E(x) 0.0377 0.0627

The delay free digital stabilizing feedback According to [17], the feedback Uc = -XI -2.x2 - � achieves stabilization with the Lyapunov function

1 X�2 1 2X�2 V = 2 (XI + 3) + 2 (X2 + 2xI + 3) .

In the delay free case, the digital controller Ud of the form

o 02 ud = UdO + '2Udl + 3TUd2 + ...

is built to satisfy at the sapling instants t = ko the series equality 1(k+1)8

V(x(k+ 1)) - V(x(k)) = V(xc(t))dt k8

where xc(t) indicates the closed-loop continuous-time state dynamics. One gets

udo(k)

The performance of the approximate solution are compared to the emulated solutions for various sampling periods, in the Figures 1, 2. The predictor based digital stabilizing feedback When the delay is equal to No, one computes accordingly the composed dynamics over N steps,

Zl (k) Xlk + 0 (NX2k -X�k � Uk-i)

z2(k)

02 (N (N )2)

+ 2 ,t; (2i -1 )Uk-i -2X2k ,t; uk-i

03 (fUk_i)3

3 i=1 N

X2k + 0 L Uk-i· i=1

A. Simulations: the delay free case Simulations are performed and the following performance parameters are compared in Tables I and II: Tr, the time necessary to ultimately bound the evolution to five percent of the Xo norm, E", the weighted integral over [0, Tr] of u2 (t)

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Be

�4

%�--

��--�---6�--�--�10 time (8)

(a) Lyapunov V

o:l-1

-2

-3

-��--�----�---

6�--���

time (81

(b) Control

10

(c) Xl

Fig. 1. Van der Pol, 0 = 0.50s, N = 0

Scenario Crit CC ECC SLMC Tr - 4.44 4.75

0= 0.5,N = 3 Eu - 1.4151 0.8595 E(x) - 0.047791 0.046624 Tr - - 8.09

0= I,N= 3 Eu - - 2.0926 E(x) - - 0.0363

and E, the weighted integral over [0, Tr] of the x (t ) norm. CC denotes the continuous-time controller; ECC its imple­mentation constant over the sampling interval (emulated con­troller); SLMC denotes the sampled-data Lyapunov-matching controller, approximated at the second order. Simulations from initial state Xo = (1, 1) T are performed for different values of 8. Until 8 = 0.9, all the control laws achieve stabilization, the parameters values are specified in Table V for 8 = 0.5. It results that for 8 small enough, both the controllers achieve stabilization with similar performances except for the control effort which is in favor of SLMC. In Figure 1 are depicted the Lyapunov function, control laws and Xl evolutions respectively for 8 = 0.5. The results depicted in Figure 2 show that for 8 = 1, ECC does not work anymore while SLMC still achieves stabilization.

-1

-2

6 time (8)

(a) Lyapunov V

10

-����----�---

6�--���

time {81

(b) Control

4 6 time (81

(c) Xl,

10

Fig. 2. Van der Pol, 0 = Is, N = 0

B. Simulations: the input-delayed case Simulations are performed for the same conditions but in presence of an input delay for different values of 8 and 'r. The results are depicted in Figure 3 for 8 = 0.5 and N = 3 showing that both the predictor-based control laws ECC (P ) and SLMC (P) achieve stabilization with values of the per­formance parameters specified in Table V. The obtained results compared with the delay free case ones show the effectiveness of the proposed discrete-time predictor based control law. Figure 4 shows that with 8 = 1 ECC (P ) does not work anymore while SLMC (P) still achieves stabilization according to the parameters specified in Table V. The result is in accordance with the delay free case; the comparison between Eu and E in Tables 1. and 2. shows the effectiveness of the proposed discrete-time predictor.

VI. CONCLUDING REMARKS

The problem of stabilization of delayed-input nonlinear systems has been investigated in the digital context. A predictor based extension of the digital stabilizing controller given in [II] has been proposed. For systems admitting computable sampled-data equivalent models, the proposed approach represents a direct way to solve the problem. The

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Be

�4

%���----�----�6-----------10 time (8)

(a) Lyapunov V

6 time {81

(b) Control

4 6 time {81

(c) Xl

10

Fig. 3. Van der Pol, 0 = 0.50s, N = 3

Van der Pol simulated example illustrates the performances. Further investigation will concern the use of approximated predictor dynamics and the case of dynamics admitting finitely computable models under preliminary feedback [16].

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-2

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(a) Lyapunov V

6 time {81

(b) Control

4 6 time {81

(c) Xl

I· SLMC(P)1

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I· SLMC(p)1

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