STABILIZATION OF DISCRETE-TIME NONLINEAR SYSTEMS. APPLICATION TO A MANIPULATOR ROBOT K. E. Bouazza∗ , M. Boutayeb‡ and M. F. Khelfi∗ ∗
Computer Science Department, Faculty of Sciences, University of Oran Es-Senia.Algeria. ‡ LSIIT-CNRS, Universit´ e Louis Pasteur de Strasbourg, Pˆ ole API, Bv. S. Brant, 67400 Illkirch Cedex, France. e-mail :
[email protected]
Abstract- In this note, we investigate the problem of state feedback stabilization for affine nonlinear discrete-time systems. From a prescribed Lyapunov function, we introduce a modified Riccati equation so that the proposed state feedback law covers a large class of dynamical systems. In particular when the unforced dynamical model is not Lyapunov stable or when the system has unstabilizable first order linearization. Sufficient conditions for asymptotic stabilization are expressed in terms of matrix inequalities. Simplicity of implementation and performances of the proposed technique will be shown through a relevant application. In fact, the theory developed will be applied on a three axes direct-drive manipulator.
I.
Introduction
During the last four decades, one of the research area that was largely investigated in the literature is, undoubtedly, control design and stabilization of nonlinear dynamical systems, in particular for continuous time models, we refer the reader to some basic works [1], [2], [3], [4], [5], [6], [7] in this field. Later, with the growing developments in digital systems and microcomputer applications, few results based on feedback linearization [2], on the theory of passive systems [8], [9], or on the stability of the unforced dynamics [10], [11] have been established for a class of nonlinear discrete -time systems. Notice however that most of the important results were not reported to the discretetime case. Let us mention, for example, non smooth state feedback stabilization or systems with uncontrollable unstable linearization. The application of the passivity theory in discretetime nonlinear systems was inspired by stability of dissipative systems results [12], [13], [14], [15] and stabi-
lization results developed in [16], [17], [18] in continuoustime (i. e. choosing a Lyapunov function V , and an output y = (Lg V )T renders the composite system x˙ = f (x)+g(x)u and y = (Lg V (x))T passive, and then stabilizable by the output feedback u(x) = −y(x)). The transfer of this result to the discrete-time case is quite difficult. In fact, in discrete-time, to be passive, and under appropriate hypothesis, the system must be written in the form (cf. [10]) xk+1
=
f (xk ) + g(xk )uk
yk
= +
h(xk ) + J(xk )uk =
∂V ∂α
(1) !T g(xk ) α=f (xk )
! ∂ 2 V 1 T g (xk ) g(xk ) uk 2 ∂α α=f (xk )
In other words, there is no way to choose ! ∂V yk = h(xk ) or yk = g(xk ) ∂α α=f (xk )
(2)
(3)
such that a discrete-time system formed by (1) and y = h(xk ) is passive. This is a real drawback for this method. The second main approach used in the stabilization of discrete-time nonlinear system is also derived from the continuous-time (see for instance [19], [20], [5] and [3]). It deals with the stabilization of nonlinear systems with stable free dynamics, we quote in this field the nice works of Byrnes and Lin [10], [8], [11] in discrete-time. The basic idea is the following one, if there exist a Lyapunov function V : Rn → R which is positive definite and proper, such that the unforced dynamic system of (1), xk+1 = f (xk ) satisfies V (f (xk )) ≤ V (xk ) then stabilization of (1) is achieved through a state feedback control law which is constructed upon this Lyapunov function.
In many practical situations, in particular when complex large-scale systems are considered, it is well known that designing a Lyapunov function is a hard and fastidious task. This represents the main drawback of this second design approach. Inspired by the optimal quadratic control method, the aim of this contribution is to investigate the problem of stabilization of affine nonlinear discrete-time systems when the corresponding Lyapunov function is a standard well-known one. One of the main features is that we introduce a modified Riccati equation so that the proposed state feedback law works for a wide range of dynamical systems, in particular, when the unforced model is not Lyapunov stable or when the systems has unstabilizable first order linearization. Sufficient conditions for asymptotic stabilization are expressed in terms of matrix inequalities that depend closely on some weighting factors fixed by the user. In the last two sections, a relevant example will be given to show simplicity of implementation and high performances of the proposed approach. The theory developed will be applied, and simulation results of the proposed controller on a three-link manipulator are provided.
II.
Problem formulation and main result
In this paper we consider a class of affine nonlinear discrete-time systems of the form xk+1 = A(xk )xk + g(xk )uk
(4)
xk ∈ Rn and uk ∈ Rm denote the state and control vectors respectively at time instant k. The matrix A(.) and g(.) are nonlinear maps not necessarily smooth. Without loss of generality, we assume that zero is the equilibrium point of the system (4). For simplicity of notations, we replace A(xk ) and g(xk ) by Ak and gk in the rest of the paper. The purpose of this work is to design an explicit state feedback law so that x = 0 is an asymptotically stable equilibrium point of the closed loop system. The main motivation for investigating the stabilization problem is, in contrast to a wide range of methods, to consider a common Lyapunov function for all affine systems of the form of (4) to achieve the stability analysis. Inspired by the well known result on optimal control, we propose in the following theorem an explicit state feedback law to achieve stabilization of (4).
Theorem If there exists an integer N so that √
αk k Ak k< 1
(5)
and 1/2
Pk ΛTk ATk (Pk−1 + gk Rk−1 gkT )−1 Ak + Qk 1/2
× Λk P k
−1
≤ αk (1 − δ)In
(6)
where Pk is a symmetric positive definite matrix obtained from the following modified Riccati equation Pk+1 = αk (ΛTk Pk Λk + LTk Rk Lk + Qk )
(7)
Λk = Ak − gk Lk
(8)
with
then the system (4) is asymptotically stabilizable at the equilibrium point x = 0 by the state feedback law uk = −Lk xk
(9)
Lk = (Rk + gkT Pk gk )−1 gkT Pk Ak
(10)
with
The arbitrary positive real parameters {αk }k=1,.... and the positive definite matrices Rk and Qk are fixed by the user. Proof The sufficient conditions (5) and (6) for asymptotic stabilization follow from two independents parts. Let us first recall, that the idea behind this approach is to consider a common Lyapunov function for all systems of the form of (4), which is Vk = xTk Pk−1 xk
(11)
A strictly decreasing sequence {Vk }k=1,... means that there exists a positive scalar 0 < δ < 1 such that ∆V
=
Vk+1 − Vk ≤ −δVk
(12)
After matrix manipulation, (6) becomes 1/2
Pk ΛTk ATk Pk (I + gk Rk−1 gkT Pk )−1 Ak −1
+Qk )
1/2
Λk P k
≤ αk (1 − δ)In
(13)
We notice that I + gk Rk−1 gkT Pk
−1 −1 = I − gk I + Rk−1 gkT Pk gk × Rk−1 gkT Pk
(14)
Using (14) and (13), we obtain 1/2 Pk ΛTk ATk Pk I − gk (I + Rk−1 gkT Pk gk )−1 Rk−1 −1 1/2 Λk Pk ≤ αk (1 − δ)In (15) ×gkT Pk Ak + Qk
1/2
Λk P k
≤ αk (1 − δ)In
(17)
which under condition (5) and for a given large positive parameter λ, satisfies
which, can be rewritten as 1/2 Pk ΛTk ATk Pk Ak − ATk Pk gk (Rk + gkT Pk gk )−1 gkT ×Pk Ak + Qk ]
−1
1/2
Λk P k
≤ αk (1 − δ)In
(25)
(16)
1/2 Pk ΛTk ATk Pk I − gk (Rk + gkT Pk gk )−1 gkT Pk Ak −1
0 < λIn ≤ Pk ≤ λIn
It is easy to verify, from (7), that since αk Qk (Qk is fixed by the user) is positive definite we have λIn < Pk . The second inequality Pk ≤ λIn may be deduced from the sufficient condition (5). Indeed, the proof is straightforward if we consider the following auxiliary Riccati equation P k+1 = αk ATk P k Ak + Qk (26)
or
+Qk ]
Now, we will prove that the matrix Pk is bounded from above and below for all k, i. e. there exists λ and λ so that
then
P k ≤ λIn for all k
−1 1/2 Pk ΛTk ATk Pk Ak − ATk Pk gk Lk + Qk 1/2
× Λk P k
≤ αk (1 − δ)In
(18)
−1 ⇒ ΛTk ATk Pk Ak − ATk Pk gk Lk + Q × Λk ≤ αk (1 − δ)Pk−1
(19)
ΛTk ATk Pk Ak − ATk Pk gk Lk − LTk gkT Pk Ak −1 +LTk gkT Pk Ak + Qk Λk ≤ αk (1 − δ)Pk−1 (20) a multiplication by In , give ΛTk ATk Pk Ak −ATk Pk gk Lk −LTk gkT Pk Ak + LTk (Rk −1 Λk +gkT Pk gk )(Rk + gkT Pk gk )−1 gkT Pk Ak + Qk ≤ αk (1 −
On the other hand, when the arbitrary initial matrices are chosen to be P0 ≤ P 0 (< λIn )
(21)
Pk+1 = αk ATk Pk−Pk gk (Rk + gkT Pk gk )−1gkT Pk ×Ak + Qk ) ≤ αk ATk Pk Ak + Qk ≤ P k+1 = αk AT P k Ak + Qk ≤ λIn
ΛTk ATkPk Ak − ATkPk gk Lk − LTkgkTPk Ak + LTkRk Lk −1 +LTkgkTPk gk Lk + Qk Λk ≤ αk (1 − δ)Pk−1 (22) then ΛTk (Ak − gk Lk )T Pk (Ak − gk Lk ) + LTk RLk −1
Λk ≤ αk (1 − δ)Pk−1
(23)
which is nothing else than, −1 ΛTk Pk+1 Λk ≤ (1 − δ)Pk−1
So, the equation (12) is proved.
(24)
(29)
so the boundness of Pk is proved. Since Vk is a strictly decreasing sequence and Pk is bounded, it follows that 0 ≤ ϕkxk k ≤ Vk ≤ (1 − δ)k V0 0 ≤ ϕ lim kxk k ≤ lim Vk≤V0 lim (1−δ)k= 0 (30) k→∞
with
k→∞
k→∞
0 < ϕIn ≤ Pk−1
which is
+Qk )
(28)
we have, by the use of (8) and (10)
after adding and subtracting the same term, we obtain
δ)Pk−1
(27)
This ends the proof. Remarks i. Through condition (5), we notice that the main contribution of the proposed approach with respect to the existing results is that we introduce the weighting factor αk to control boundedness of Pk and, by the way, to relax the Lyapunov stability condition of the unforced dynamic system without preliminary coordinate transformations of the initial system. A simple choice of αk to ensure the condition (5) is αk =
1
¯ n for Pk < λI otherwise
1 (1+kA(xk )k)4
(31)
However, since αk is in the right hand side of the condition (6), the more αk is small, the more this condition is hard to verify in real time implementation, especially for some nonlinear systems. ii. Notice that the control law presented here, can also be applied to discrete-time nonlinear system of the form xk+1 = f (xk ) + g(xk )uk
(32)
1
In fact, if f (x) is of class C , and f (0) = 0, it is possible to write (32) in the equivalent form (4) with f (xk ) = A(xk )xk R 1 (txk ) where A(xk ) = 0 ∂f∂x dt k
III.
IV.
Simulation results
The three-link robot manipulator used in this study was constructed with two NSK motors and third arm which moves in the vertical plan, and which is equipped by an electro-pneumatic prehensile. The system is controlled in torque mode by a PC Pentium II 200 MHZ workstation via the Real-Time Workshop Simulink. The control algorithm was implemented on a Texas Instruments TMS320C31 Digital signal processing (DSP) floating 32-bits floating point processor.
Stabilization of manipulator robot
Control design theory of manipulator robots has grown up in the last twenty years, as we can see in the following different works [21], [22], [23], [24], [25], [26] and [27], and the references therein. The dynamics of a robot manipulator is described using the Euler-Lagrange equation of motion by ([24])
Figure 1: CRAN-Longwy testing bench : a manipulator robot.
Γ = H(q)¨ q + C(q, q) ˙ q˙ + G(q) + Fs sign(q) ˙ + Fv q˙ (33) Where q(t), q(t), ˙ q¨(t) ∈ Rn denote the link position, velocity and acceleration vectors, respectively, H(q) ∈ Rn×n is the symmetric and positive definite inertia matrix, C(q, q)∈ ˙ Rn×n is the centripetal-Coriolis n matrix, G(q)∈ R represents the gravity vector, Fs and Fv (∈ Rn×n ) are diagonal matrices representing the viscous and Coulomb friction coefficients, and Γ ∈ Rn is the torque input vector. From equation (33) the following state space model is obtained x˙ = Ab (x)x + gb (x)u (34) where
x =
Ab (x) gb (x)
q q˙
In −H −1 (q) (C(q, q) ˙ + Fv ) 0 = H −1 (q)
=
0 0
u = Γ − Fs sign(q) ˙ − G(q)
(35) (36) (37) (38)
The Coulomb friction and the gravity forces are added to the input torque Γ, to compensate the friction and gravity effects on the control. Now, we can apply the result developed so far on the testing bench of our laboratory, a three axis manipulator robot.
A global closed loop identification was done successfully for this robot [28], and gives the following model p1 + 2P3 c2 p2 + p3 c2 0 p2 0 q¨ Γ = p2 + p3 c2 0 0 p4 −2p3 s2 q˙2 −p3 s2 q˙2 0 0 0 0 q˙ + 0 + p3 s2 q˙1 0 0 0 −m3 g fs 1 0 0 + 0 fs 2 0 sign(q) ˙ 0 0 fs 3 fv 1 0 0 + 0 fv 2 0 q˙ (39) 0 0 fv 3 where c2 ans s2 denote cos(q2 ) and sin(q2 ) respectively. The parameters identified are given by p1 = 19.1337, p2 = 2.8522, p3 = 4.1214, p4 = 1.2568, fv1 = 2.1815, fv2 = 0.2347, fv3 = 37.8840, fs1 = 9.4226, fs2 = 1.9238, and fs3 = 6.8821. Writing this model in a state space form (34), and after discretization step, using the Euler method, we get the following state space system xk+1 = A(xk )xk + g(xk )uk
(40)
with
A(xk ) = I + T Ab (x)
(41)
g(xk ) = T gb (x)
(42)
and
where T is the sampling time. The system obtained is critical from the point of view of the stability, because on the linearized model, more than one eigenvalue are on the unit circle. Usually in the literature, in this case, the center manifold theory is applied, which is in general quite difficult, and only an approximated solution is used. In our case, there is no need to apply this theory, nor to look for a stable Lyapunov function for the free dynamics. Applying the control law (9), we ensure as shown in the figures below, that the system Σ is globally asymptotically stabilized (GAS). All simulations were performed at a sampling period of 7 ms. Some numerical results are depicted on Fig. 2-3. One can see that both positions and velocities converges to the equilibrium, with R = 1∗I3 , Q = 10∗I6 , P0 = 150∗I6 and x0 = [−π(rad) π(rad) 5(cm) 0 0 0]. 5
4
V.
Conclusion
In this contribution a simple and useful approach to deal with stabilization of affine nonlinear discrete-time systems is presented. We proposed an explicit state feedback law, based on a modified Riccati equation, that works even when the unforced dynamic model is not Lyapunov stable, which was an open problem. Finally , from a given Lyapunov function, sufficient conditions for asymptotic stabilization are established in terms of matrix inequalities. Numerical simulations was successfully applied to a bench of our laboratory, three-link manipulator robot, and show the high quality and the interesting proprieties of the control law, such us the simplicity of implementation and the rapidity of convergence.
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Figure 2: Position stabilization -joint1, joint2 (rad) and joint3 (cm)- with respect to time t(s).
0.5
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Figure 3: velocity stabilization -joint1, joint2(rad/s) and joint3(cm/s) - with respect to time t(s). Implementation and experimental results are under study and will be given in the near future.
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