STATE DEPENDENT INPUT DELAY

A STATE FEEDBACK CONTROL METHODOLOGY FOR NONLINEAR SYSTEMS WITH TIME/STATE DEPENDENT INPUT DELAY Gildas Besan¸ con ∗ Didier Georges ∗ Zohra Benayache ∗

∗

Control Systems Department1 GIPSA-lab UMR 5216, BP 46 38402 Saint-Martin d’H`eres, France

Abstract: This paper proposes a methodology for state feedback stabilization of nonlinear systems with time-varying input delay, and more particularly when the delay varies w.r.t. the state variables. The control design approach is based on state prediction computation, and the numerical issue resulting from the actual implementation of such a control law is also discussed. The overall method is in particular illustrated on an example of water flow control in open-channel systems. Copyright © 2007 IFAC

Keywords: Nonlinear control, nonlinear time-delay systems, time-varying delay, state-dependent delay, state feedback, state predictor.

1. INTRODUCTION Time-delay systems belong to the class of infinite-dimensional systems. In the linear case, a time-delay system has in general an infinite number of eigenvalues. Control laws have been proposed to assign a finite number of eigenvalues in closed loop (Manitius and Olbrot [1979]). This approach is called the finite spectrum assignment problem. Solutions to this problem are obtained in terms of delay-distributed control laws. However, the implementation of delay-distributed control laws is difficult due to the integral term which cannot be computed explicitly. In (Manitius and Olbrot [1979]), it is suggested to approximate the integral by a sum of point-wise delays by using a quadrature rule. However, this approach may fail due to the occurrence of unstable poles introduced by the discretization procedure (Assche et al. [1999]). The use of block-pulse functions has also been proposed in (Fattouh et al. [2001]). 1

Former LAG, Laboratoire d’Automatique de Grenoble.

More recently, a safe implementation of delaydistributed control laws has been proposed by using a low-pass filter in the control loop (Mondie and Michiels [2003]). In (Maza-Casas et al. [2000]) a passivity-based control scheme is proposed for the stabilization of SISO nonlinear systems with input delay. However, distributed-delay control laws for nonlinear systems has not yet been extensively studied. Other approaches have been proposed for special cases (Mazenc et al. [2003], Mazenc and Bliman [2006], Zhang et al. [2006]). In the present paper, the purpose is to extend the so-called finite spectrum assignment approach already available for linear input-delayed systems, to the case of nonlinear input-delayed systems, with even state-dependent input delays. Notice that the stability problem for such systems with state-dependent delays was previously considered in Verriest [2002], but for linear systems, and that the work we propose here comes as a continuation of Georges et al. [2007] where the problem was already dedicated to nonlinear systems, but basi-

cally with constant input delays. The paper in organized as follows: section 2 first presents the formal statement of the proposed approach. Section 3 then discusses numerical issues for practical implementation, while section 4 proposes a possible use in water flow control as an illustrative example of application. Section 5 finally gives some conclusions. 2. A STABILIZATION SCHEME FOR NONLINEAR SYSTEMS WITH VARYING INPUT DELAY

u(t) = Φ(x(t)) ensuring the closed-loop stability of the non-delayed system x(t) ˙ = F (x(t), u(t)). The main issue remains the computation at time t of the prediction of the state at time t + δ(t), denoted by xp (t, t + δ(t)), which is given by: xp (t, t + δ(t)) = x(t)+ t+δ(t) Z F (xp (t, θ), u(θ − τ (θ, xp (t, θ))))dθ. + t

The predicted state xp (t, t + δ(t)) may also be defined in terms of an operator Ψ defined as follows:

Let us consider nonlinear systems with a varying input delay of the following general form:

xp (t, t + δ(t)) = Ψ(x(t), {u(θ)}θ∈[t−τ,t] )

(3)

Finally, the control law is given by x(t) ˙ = F (x(t), u(t − τ (t, x(t))))

(1)

where x(t) ∈IRn is the state vector, u(t) ∈IRm is the control input, and τ (t, x(t)) is a varying delay with known smooth evolution w.r.t. its arguments. The origin of the system is supposed to be an equilibrium point (F (0, 0) = 0). Clearly, for causality, τ (t, x(t)) should remain larger than 0 for any time and realizable trajectory, and in particular around the origin x = 0. The purpose here is to design a state feedback law in order to stabilize the origin of the system in closed loop. To that end, the idea is to extend the so-called finite spectrum assignment approach already available for linear input-delayed systems (Mondie and Michiels [2003]). This approach is based on the following principle: firstly a prediction of the state at an appropriate prediction time δ, denoted by xp (t, t + δ), is computed from the available state x(t) at time t and input controls u(θ), θ ∈ [t − δ, t]. Then the predicted state is used to compute the control law. The prediction time is chosen so that the effect of the delay vanishes and the closed-loop system is no more a time-delay system. In the case of constant delay, the prediction time is just given by the delay τ itself (e.g. as in Georges et al. [2007]). When the delay might be varying τ (t), the prediction horizon cannot be τ anymore but a time-varying prediction horizon δ(t) chosen such that δ(t) = τ (t + δ(t)). This condition is used to ensure that a control input u(t) can be computed since in this case one has u(t − τ (t + δ(t)) + δ(t)) = u(t)(Witrant et al. [2004]). The same obviously occurs when the delay further depends on the state: δ is to be chosen such that δ(t) = τ (t + δ(t), x(t + δ(t))). The stability of the closed-loop system x(t ˙ + δ(t)) = F (x(t + δ(t)), Φ(x(t + δ(t)))) (2) expressed in the time coordinate t + δ(t), can then be guaranteed if there exists a smooth feedback

u(t) = Φ(Ψ(x(t), {u(θ)}θ∈[t−τ,t] ))

(4)

From now on, in order to simplify a little bit the notations, but still keeping the specificity of a state-dependent varying delay, we will limit the presentation to the case when τ = τ (x(t)) (without restriction). The main result in that respect can then be stated as follows: Theorem 1. If there exists a smooth state feedback Φ(x) making the origin of x(t) ˙ = F (x(t), Φ(x(t)) locally exponentially stable - in the sense that (Khalil [1996]): there exists D ⊂ Rn containing and a C 1 positive definite function V : D → R such that ∀x ∈ D: (i) c1 kxk2 ≤ V (x) ≤ c2 kxk2 (ii)

∂V (x) F (x, Φ(x)) ≤ −c3 kxk2 ∂x

for positive constants c1 , c2 and c3 , then the control law (4) makes the origin of (1) locally asymptotically stable. Proof: First of all, define f (x) := F (x, Φ(x)) and z(t) := x(t + δ(t)). Then the closed-loop system (1)-(4) can be re-written w.r.t. z and f as: ˙ z(t) ˙ = (1 + δ(t))f (z(t)).

(5)

Let us then consider V (z) as a candidate Lyapunov function for this system. Clearly: ∂V ˙ V˙ = (1 + δ(t)) f (z(t)). ∂x Now notice that from the definition of δ we have: ∂τ ∂τ [1 − (z)f (z)]δ˙ = (z)f (z) (6) ∂x ∂x ∂τ where ∂x (z)f (z) vanishes when z goes to zero. ¯ ⊂IRn of states Hence there exists a domain D z of D and containing the origin, such that 1 − ∂τ ˙ ∂x (z)f (z) > 0, and consequently such that δ >

¯ −1. Then using condition (ii), we get on D:

which can be written as:

˙ 3 kzk2 . V˙ ≤ −(1 + δ)c

X(t) = H(X(t), x(t)).

Now using assumption (i) and integrating the above inequality from 0 to t, we obtain:

The fixed point computation can then be performed e.g. by using a Newton-Raphson method. This technique being time consuming - and therefore not appropriate for on-line computation - let us instead propose another approach based on ”dynamic inversion”.

V (t) ≤ V (0)e−φ(t) , where φ(t) =

c3 c1

Zt

˙ (1 + δ(θ))dθ

0

c3 φ(t) = (t + δ(t) − δ(0)) c1 Again using assumption (i), the following inequality holds: c2 kz(t)k2 ≤ kz(0)k2 e−φ(t) . c1 Clearly δ ≥ 0 since τ ≥ 0 (at least locally), and thus φ(t) → +∞. This yields that for any initial condition in some neighborhood of 0,

Suppose that we seek for the solution of G(x, t) = 0, where G is a nonlinear C 1 -function :IRn × [0, +∞(→IRn and the Jacobian matrix ∂G ∂x is supposed to be invertible. The main idea is how to compute the solution of the differential equation G˙ + ΛG = 0

∂G ∂G x˙ + + ΛG(x, t) = 0. ∂x ∂t

t−→∞

•

Notice that for a system without any finite escape time, if conditions of theorem 1 hold on D =IRn with V being radially unbounded and for any x: ∂τ (x)F (x, Φ(x)) < 1 ∂x τ (x) ≥ 0

(7) (8)

we further get a global convergence result. Obviously conditions on τ are rather strong, but they are necessary for some well-posedness of the model (since (7) is necessary for t > τ (x(t)) to be satisfied at any time in closed loop, while (8) simply means causality).

3. AN IMPLEMENTATION SCHEME FOR THE PROPOSED FEEDBACK LAW In this section, a method for practical implementation of the previously proposed control law is discussed. It is based on an approximate computation of the predicted state xp (t, t + δ(t)). Using one step of the backward Euler method (the implicit Euler method), we can indeed obtain an approximation of the predicted state x ˆp (t, t+δ(t)) as follows: x ˆp (t, t + δ(t)) = x(t) + δ(t)F (ˆ xp (t, t + δ(t)), u(t)).

(11)

where Λ is any positive definite matrix ensuring the asymptotic stability of this equation. In the coordinates x, the equation (11) is equivalent to

lim kx(t + δ(t))k = 0

and finally limt−→∞ kx(t)k = 0.

(10)

Since

(12)

∂G ∂x

has full rank, (12) is equivalent to  −1   ∂G ∂G x˙ = − + ΛG(x, t) . (13) ∂x ∂t

The motivation may be found in the fact that if the initial state x0 is a solution of G(x, t = 0) = 0, then the trajectory x(t) of (11), is a solution of G(x(t), t) = 0, ∀t > 0. Since (11) is asymptotically stable, even when the initial state is not a solution of G(x, t = 0) = 0, x(t) will reach asymptotically the manifold G(x, t) = 0, since the solution of (11) is G(x, t) = e−Λt G(x(0), 0) and lim G(x(t), t) = t→+∞

0 for all Λ > 0. The coefficient Λ can be used to control the speed of convergence. The application of this approach to (10) leads to the state-prediction-based control law given by: ∂H −1 X˙ = (Id − ) [F (x(t), u(t − τ (x(t)))) ∂X −Λ(X − H(X, x(t)))] (14) u(t) = Φ(X(t)) with

(15)

∂H ∂τ ∂F = F +τ ∂X ∂x ∂x ∂F ∂Φ +τ ∂x ∂x

An important issue now is to verify that the control law (14)-(15) can still stabilize the closed loop. To that end, it can be noticed that this practical realization of the control law introduces an additive perturbation term depending on the

The problem is now to compute on-line the fixed point X(t) := x ˆTp (t, t + δ(t)) solution of x ˆp (t, t + δ(t)) = x(t) + τ (ˆ xp (t, t + δ(t)))× F (ˆ xp (t, t + δ(t)), Φ(ˆ xp (t, t + δ(t))))

(9)

We could also consider several steps of any implicit integration schemes. In that case, the fixed point vector would include additional discretization states.

state x(t) for the closed-loop dynamics expressed at time t + δ(t). From the fixed point problem (10) indeed it is first clear that there exists a function ψ such that x ˆp (t, t + δ) = ψ(x(t)) (using the implicit function theorem). Then, with the same notation z(t) = x(t+δ(t)) as in the proof of theorem 1, the closed-loop system can be expressed as ˙ (z(t), Φ(ψ(x(t)))) z(t) ˙ = (1 + δ)F ˙ (z(t), Φ(z(t))) = (1 + δ)F ˙ (z(t), Φ(ψ(z(t − δ))))−F (z(t), Φ(z(t)))] +(1 + δ)[F ˙ = (1 + δ){F (z(t), Φ(z(t)))+P (z(t), z(t − δ))} (16) Notice that from a Taylor series expansion of z(t) at time t − δ, we get an expression of z(t) w.r.t. z(t−δ) and clearly P vanishes when its arguments are zero. Consequently, on some neighborhood of the origin small enough, we can get: kP (z(t), z(t− δ))k ≤ γkz(t−δ)k for some γ. Considering first the control law obtained by on-line solving of the fixed point equation (10), this motivates the following statement: Theorem 2. If the following conditions hold: (1) There exist a smooth control law Φ and a Lyapunov function V (x) such that the following assumptions hold: ∀x ∈ D with D ⊂ Rn a domain containing the origin, • c1 kxk2 ≤ V (x) ≤ c2 kxk2 ∂V (x) F (x, Φ(x)) ≤ −c3 kxk2 ∂x ∂V (x) • k k ≤ c4 kxk ∂x where c1 , c2 , c3 and c4 are some positives scalar numbers (roughly Φ is a stabilizing control law for the non-delayed system Khalil [1996]). (2) Equation (10) admits a fixed point (roughly τ (.)F (., Φ(.)) is a contracting map). (3) The perturbation term P (z(t), z(t−δ)) satisfies kP (z(t), z(t − δ))k ≤ γkz(t − δ)k, ∀z(t) ∈ c3 D and γ ’small enough’, e.g. γ ≤ 4c , 4 •

then the closed-loop system is locally asymptotically stable. Proof: We introduce the Lyapunov-Krasovskii function candidate W (zt ) = V (z) + Zt µ kz(θ)k2 dθ, where zt = z(t + θ), θ ∈ [−δ, 0] t−δ

as usual in Lyapunov-Krasovskii formalism (see e.g. Hale and Lunel [1993]), and µ > 0 is to be specified later on. The time derivative of W is given by:

˙ ∂V (z) F (z, Φ(z)) ˙ = (1 + δ) (17) W ∂z ˙ ∂V (z) P (z(t), z(t − τ )) + (1 + δ) ∂z ˙ + µ[kz(t)k2 − (1 − δ(t))kz(t − δ)k2 ]. (18) Using again the expression (6), we that: 1 − ∂τ have 1−ε δ˙ ≥ ε whenever ∂x F (z, Φ(z)) ≤ 2−ε for any 0 < ε < 1. Moreover, this also guarantees that 2−ε 1 + δ˙ > ε˜ with ε˜ = 3−2ε > 0. Finally, it is also clear that in this case, 1 + δ˙ < 2. Hence, given such an ε, and using conditions 1) and 2) of the theorem, we can obtain on some small enough neighborhood of the origin: ˙ ≤ −c3 ε˜kz(t)k2 + 2c4 γkz(t)kkz(t − δ)k W + µ[kz(t)k2 − εkz(t − δ)k2 ]

(19)


˙ ≤ − kx(t)k kx(t − δ)k i.e. W    c3 ε˜ − µ −c4 γ kx(t)k × . −c4 γ εµ kx(t − δ)k (20) Hence choosing µ < c3 ε˜, we get that for: p (c3 ε˜ − µ)εµ γ< c4

(21)

the right-hand side of the above inequality is ˙ ≤ −ρkz(t)k for negative definite, and thus W some ρ > 0 which gives the local asymptotical stability of z = 0 by the Lyapunov-Krasovskii stability result, and finally that of x = 0 for the system in time t. Notice that choosing e.g. µ = c32ε˜ , condition (21) √ ε˜ ε . Since the right-hand side can becomes γ < c32c4 c3 be made arbitrarily close to 2c by choosing ε 4 close enough to 1 (by lower values), this can in c3 turn make(21) to be satisfied whenever γ ≤ 4c , 4 which ends the proof. Now if we consider the control law with the ’dynamic inversion’ for the fixed point resolution, then the stability can still be guaranteed by invoking Tikhonov’s theorem (Khalil [1996]) in a similar way as in Georges et al. [2007] for constant input delays.

4. AN APPLICATION EXAMPLE TO OPEN CHANNEL WATER FLOW CONTROL In order to demonstrate the effectiveness of the here-proposed approach, let us consider the example of water flow control in open-channel systems: water flow dynamics in such systems can classically be described by a couple of nonlinear partial differential equations known as Saint-Venant equations (see e.g. Chow [1988]).

In the performed simulations, the numerical values were also taken from the case study of Litrico and Pomet [2003], namely a 10km-long and 8m-wide channel, with a 0.04% slope. The control in that case was obtained on the basis of some feedback linearizing control law for the non-delayed system, with the purpose of changing the output value from y = 0.93m3 /s to y = 1.24m3 /s within about 5 hours (following the performances of Litrico and Pomet [2003]).

1.4

1.3

in m3/s

1.2

1.1

1

0.9

0.8

0

2

4

6 Time in seconds

8

10

12 4

x 10

Fig. 1. Downstream water flow rate step response. a simple example, we can consider the case of a pendulum to be controlled via some delayed input. The dynamics of the system in that case are given by: x˙ 1 (t) = x2 (t) x˙ 2 (t) = −x2 (t) − sin(x1 (t)) + u(t − τ (t)) (24) y = x1 Let us for instance assume that here: τ (t) = 1 + 0.5 cos(t). The stabilization results obtained when following the methodology proposed in this paper for instance to achieve the stabilization at xd1 = π3 are illustrated by 2 below. prediction horizon & delay

x(t) ˙ = A(x2 (t))x(t) + B(x2 (t))u(t − τ (x2 (t))) y(t) = x2 (t) (23)
T where x = x1 x2 is the state vector roughly defined by Q (for x2 ) and its time derivative (for x1 ), and A, B, τ are nonlinear functions explicitly given in Litrico and Pomet [2003].

Downstream water flow rate 1.5

2 δ(t) τ(t) 1.5

1 π/3 0.5

0

0

2

4

6

8

0

2

4

6

8

10

12

14

16

18

20

10

12

14

16

18

20

pendulum angle (rad)

In a simplified approach, they can be reduced to a single diffusive wave equation of the following form: ∂Q ∂Q ∂2Q (22) + ν(Q) − ∆(Q) 2 = 0 ∂t ∂x ∂x where Q(x, t) is the water flow at point x and time t, ν is the celerity coefficient and ∆ the diffusion one. The boundary conditions are here given by: Q(0, t) = u(t) where u is the control input, and limx→∞ ∂Q ∂x = 0. The output to be controlled will here be y(t) = Q(L, t) where L is the length of the considered channel. Clearly in that case, the partial differential equation (22) carries some delay effect from the control u(t) to the output y(t). In (Litrico and Pomet [2003]), the authors have proposed some nonlinear ODE model with statedependent delayed output to approximate the dynamics described by (22). It can easily be checked that a similar model can be obtained with delayed input, yielding a representation of the following form:

π/3

time(s)

The resulting control law was tested in simulation on a complete Saint-Venant model, using the classical finite-difference Preissman scheme for numerical description of such dynamics Georges and Litrico [Eds], and the corresponding stabilization results are presented on figure 1: it can indeed be seen that the control purpose is here achieved fairly well, with only a very small static error (less than 1%) likely to be due to numerical approximations. Notice that the methodology given in details for the case of state-dependent input delays can also be used for systems with time-varying delays: as

Fig. 2. Pendulum angle response

5. CONCLUSIONS In this paper, a control methodology for nonlinear systems with state-dependent input delay has been proposed: it is formally based on an appropriate state prediction, and an approximate implementation scheme. The method is illustrated on a control problem of open channel water flow dynamics, relying on a recently proposed model for such dynamics. It can also work for timevarying delays under exogenous laws.

REFERENCES V. Van Assche, M. Dambrine, and J.F. Lafay. Some problems arising in the implementation of distributed-delay control law. In 38 th IEEE Conference on Decision and Control, 1999. V.T. Chow. Open-Channel Hydraulics. McGrawHill, 1988. A. Fattouh, O. Sename, and J.M. Dion. Pulse controller design for linear systems with delayed state and control. In IFAC Symposium on System Structure and Control, 2001. D. Georges and X. Litrico(Eds). Automatique pour la gestion des ressources en eau (in French). Herm`es, 2002. D. Georges, G. Besan¸con, Z. Benayache, and E. Witrant. A nonlinear state feedback design for nonlinear systems with input delay. In European Control Conf. ECC’07 (To appear), 2007. J.K. Hale and S.M. Verduyn Lunel. Introduction to functional differential equations. Springer Verlag, New York, 1993. H.K. Khalil. Nonlinear systems. Prentice-Hall, Inc, 1996. X. Litrico and JB. Pomet. Nonlinear modelling and control of a long river stretch. In European Control Conf, Cambridge, GB, 2003. A.Z. Manitius and A.W. Olbrot. Finite spectrum assignement problem for systems with delays. IEEE Transactions Automatic Control, 24(4), 1979. L. Maza-Casas, M. Velasco-Villa, and J.A. Alvarez-Gallegos. Compensation of input time delay for a class of nonlinear systems. In 39 th IEEE Conference on Decision and Control, 2000. F. Mazenc and P.A. Bliman. Backstepping design for time-delay nonlinear systems. IEEE Trans. Automatic Control, 51(1):149–154, 2006. F. Mazenc, S. Mondie, and S.-I. Niculescu. Global asymptotic stabilization for chains of integrators with a delay in the input. IEEE Transactions Automatic Control, 48:57–63, 2003. S. Mondie and W. Michiels. Finite spectrum assignement of unstable time-delay systems with a safe implementation. IEEE Transactions Automatic Control, 48(12), 2003. E.I. Verriest. Stability of systems with state dependent and random delays. IMA J. Math. Control and Info., 19:103–114, 2002. E. Witrant, C. Canudas de Wit, D. Georges, and M. Alamir. Remote stabilization via timevarying communication delays: Application to tcp networks. In IEEE Conference on Control Applications, 2004. X. Zhang, H. Gao, and Zhang. Global asymptotic stabilization of feedforward nonlinear systems with a delay in the input. Int. Journal of Systems Science, 37(3):141 – 148, 2006.