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STABILITY ANALYSIS FOR SANDWICH SYSTEMS WITH BACKLASH: AN LMI APPROACH
ROCOND'06, Toulouse, France, July 5-7, 2006
STABILITY ANALYSIS FOR SANDWICH SYSTEMS WITH BACKLASH: AN LMI APPROACH S. Tarbouriech ∗ and C. Prieur ∗
∗
LAAS-CNRS, 7 Avenue du Colonel Roche, 31077 Toulouse cedex 4, France. E-mails: [email protected], [email protected]
Abstract: This paper addresses the problem of stability analysis for a certain class of nonlinear systems resulting from a sandwiched backlash operator in the connection of a plant and an actuator device. Constructive conditions based on LMIs to ensure the stability of the sandwich system are proposed by using some suitable Lyapunov functions and generalized sector conditions. The associated set of all the admissible equilibrium points is precisely defined. Extensions to the case where some parameters of the backlash nonlinearity are uncertain are provided. c 2006 IFAC. Copyright Keywords: Stability analysis, backlash, generalized sector conditions, equilibrium points, LMI
1. INTRODUCTION This paper focuses on the class of nonlinear systems resulting from a sandwiched non-smooth nonlinear operator in the connection of a plant and an actuator device. Such a class of systems includes a wide variety of practical systems and devices, like servo system, flexible systems, ... Hence, non-smooth nonlinearities, as hysteresis, backlash or dead-zone often occur in real control process, due to physical, technological or safety constraints or imperfections, even inherent characteristic of considered controlled systems. The nonlinearity under consideration in this paper is a backlash. Such a nonlinearity is present in mechanical systems and its negligence during the control design or the stability analysis can lead to an important degradation of closed-loop performance or even to the lost of the stability (see, in particular, (Nordin et al., 2002)). Several approaches have been developed in the context of this type of non-smooth nonlinearity: see, for
example (Corradini and Orlando, 2002) and references therein. In particular, one can cite the works relative to sandwich systems (Taware and Tao, 2003). Moreover, some solutions consisting in applying inverse nonlinearity have been proposed (Taware and Tao, 2003), (Taware et al., 2002). The current paper proposes an approach based on quadratic Lyapunov function and generalized sector conditions using the knowledge on the nonlinearity. At the contrary of results developed in (Par´e et al., 2001) and (Park et al., 1998), the conditions do not need to verify some particular assumption on the system matrix at s = 0, where s is the Laplace variable. The conditions proposed are under an LMI form and allow to guarantee the stability of the closed-loop system and the convergence of its trajectories to a described set of associated equilibrium points. The conditions are extended to the case where some uncertainties affect the parameters of the nonlinearity. Moreover, the case where the nonlinearity is a dead-
Preprints of the 5th IFAC Symposium on Robust Control Design
ROCOND'06, Toulouse, France, July 5-7, 2006
zone nonlinearity is addressed (Tarbouriech et al., 2004).
where (`1 , . . . , `m ) is given in
Notations. For any vector x ∈
cl(i) 0 cr(i) .
the components of x, denoted x(i) , are nonnegative. For two vectors x, y of
2. PROBLEM FORMULATION This paper focuses on the following class of nonlinear systems resulting from a sandwiched backlash operator in the connection of a plant and an actuator system as shown in Fig. 1. actuator plant (x ) A
w
backlash nonlinearity (F)
system (x)
y
The plant is described by: (1)
where x ∈
Thus, Φ is a time-invariant nonlinearity with slope restriction, as in (Park et al., 1998). Note however that it is a memory-based operator, since to compute it, we need to have information of the past values of v (this is not the case in (Park et al., 1998)). Throughout the paper (Nordin et al., 2002; Corradini and Orlando, 2002), we consider that the nonlinearity is active, that is, we are especially interested by all initial conditions xA (t = 0) = xA (0) satisfying L(CA xA (0)−cl ) Φ[CA xA ](0) L(CA xA (0)−cr ) (5) Thus, the problem we intend to solve by exploiting some properties of sector nonlinearities can be summarized as follows. Problem 1. Determine the associated set of equilibrium points toward which the system (1), (2) and (3) converges when initialized as in (5).
3. PRELIMINARIES
Fig. 1. Complete system
x˙ = Ax + BΦ[w] y = Cx
(4)
(2)
where xA ∈
0 and (Φ[v](t))(i) = `(i) (v(i) (t) − cr(i) ) `(i) v˙ (i) (t) if v˙ (i) (t) < 0 and (Φ[v](t))(i) = `(i) (v(i) (t) − cl(i) ) 0 if ` (v(i) (t) − cl(i) ) ≥ (Φ[v](t))(i) (i) ≥ `(i) (v(i) (t) − cr(i) ).
Let us define the matrix L = diag(`1 , . . . , `m ). The following properties with respect to the nonlinear operator Φ can be stated. Lemma 1. For any diagonal positive definite matrix N1 and N2 in
(6) (7)
Proof. Let v ∈ C 1 ([0, +∞];
−(`(i) )2 v˙ (i) (t)N1(i,i) cr(i) < 0
(3)
since cr(i) > 0. • Assume now that (Φ[v](t))(i) = `(i) (v(i) (t) − cl(i) ) and v˙ (i) (t) < 0, then it follows:
Preprints of the 5th IFAC Symposium on Robust Control Design
ROCOND'06, Toulouse, France, July 5-7, 2006
! z}|{ ˙ Φ[v](t)0 N1 (Φ[v](t) − Lv(t))
Proposition 1. If there exist two symmetric positive definite matrices P ∈ <(n+p)×(n+p) , M ∈
=
(i)
−(`(i) )2 v˙ (i) (t)N1(i,i) cl(i) < 0 since cl(i) < 0. • Assume now that we are not in one of the z}|{ ˙ previous cases. Then (Φ[v](t))(i) = 0 and thus ! z}|{ ˙ 0 = 0. Φ[v](t) N1 (Φ[v](t) − Lv(t)) (i)
This proves (6). Let us prove (7). Let i ∈ {1, . . . , m}. Only two cases may occur z}|{ ˙ • either (Φ[v](t))(i) = 0, z}|{ ˙ • or (Φ[v](t) − Lv(t)) ˙ (i) = 0.
N1 +
(H + H 0 ) −M >0 2
A0 P + P A P B + C0 LN2 + (A−1 )0 C0 LN1 ? −2N2 <0
(10)
(11) where H = N1 LCA−1 B, then the system (9) is asymptotically stable, for all initial conditions xA (t = 0) = xA (0) satisfying (5). Proof. For conciseness, in this proof, we denote z }| ˙ { Φ˙ instead of Φ[w], and Φ instead of Φ[w]. Let us consider the following candidate of Lyapunov function 1
Therefore ! z}|{ z}|{ ˙ ˙ 0 Φ[v](t) N2 (Φ[v](t) − Lv(t))
V (X, w, t) = X(t)0 P X(t) + Q Z t ˙ 0 N1 (Φ(s) − Lw(s))ds −2 Φ(s) 0 Z t ˙ 0 N2 (Φ(s) ˙ −2 Φ(s) − LCX(s))ds
= 0,
(i)
and thus (7) follows. 2
(12)
0
+Φ(t)0 M Φ(t) 4. MAIN RESULTS 4.1 Stability analysis Since the backlash operator is defined in terms of its time derivative, it is particularly interesting to study the time-derivative version of systems (1) and (2): z }| ˙ { X˙ = AX + B Φ[w] Y = CX (8) ˙ XA = AA XA + BA Y W = CA XA where X = x, ˙ XA = x˙ A , Y = y, ˙ and W = w. ˙
where P = P 0 > 0, M = M 0 > 0, N1 and N2 are diagonal positive definite matrices in
Hence, by considering the augmented state vector X X= XA the system under consideration reads z }| ˙ { X˙ = AX + B Φ[w] W = CX
(9)
with A=
A 0 B A C AA
B=
B 0
C = 0 CA
The study of this system allows us to derive properties of the system (1)-(2), and therefore to solve Problem 1. The following proposition relative to system (9) can be first given.
(13)
By noting that one gets: x X=A + BΦ xA 1
Note that V is an operator with functions as arguments.
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one can replace in the above expression of V˙ x the term w = CA xA = C by C(A−1 X − xA A−1 BΦ); Note that since matrices A and AA are Hurwitz matrix A is non-singular. Hence, the time-derivative of V reads: V˙ = X0 (A0 P + P A)X + 2X0 P BΦ˙ −2Φ˙ 0 N2 (Φ˙ − LCX) −2Φ˙ 0 [N1 Φ − N1 LCA−1 X +N1 LCA−1 BΦ − M Φ] + Q˙ (14) = X0 (A0 P + P A)X + 2X0 P BΦ˙ −2Φ˙ 0 N2 (Φ˙ − LCX) +2Φ˙ 0 N1 LCA−1 X −2Φ˙ 0 [N1 + N1 LCA−1 B − M ]Φ + Q˙ Thus, by choosing Q given as follows: Q = Φ0 (N1 + N1 LCA−1 B − M )Φ
(15)
one can remove the cross-terms between Φ˙ and Φ in the expression of V˙ . Moreover one has to verify that this given Q is positive definite. For this, one wants to prove that: Φ0 (N1 + N1 LCA−1 B − M )Φ > 0, ∀Φ 6= 0 By noting that the part N1 LCA−1 B, denoted by H to simplify, is non-symmetric and satisfies H = 0 ) (H+H 0 ) + (H−H , one has to verify the following: 2 2 Q = Φ0 (N1 + (
(H + H 0 ) (H − H 0 ) + ) − M )Φ > 0 2 2 ∀Φ 6= 0 0
) Thus, since Φ0 (H−H Φ = 0, this implies, with 2 (10), Q > 0. Therefore the time-derivative of the Lyapunov function writes:
V˙ = X0 (A0 P + P A)X + 2X0 P BΦ˙ −2Φ˙ 0 N2 (Φ˙ − LCX) +2Φ˙ 0 N LCA−1 X 0 1 X X M ˙ = Φ˙ Φ
(16)
where the matrix M is the left term of relation (11). Hence, the satisfaction of relations (10) and (11) allows to guarantee that one V˙ < 0 along the trajectories of system (9). In summary, due to the definition of the backlash nonlinearity, the proof of Proposition is complete. 2 Remark 1. Relations (10) and (11) of Proposition 1 are linear in the decision variables P , M , Ni , i = 1, 2. Thus the test for their feasibility can be done by using efficient LMI solvers. Due to the linearity of conditions of Proposition 1, we do not need to suppose that the matrix CA−1 B is symmetric as in (Par´e et al., 2001). By the same way we do not need to verify an equality like N CA−1 B = B0 (A−1 )0 C0 N as in (Park et al., 1998).
4.2 Computation of the equilibrium-set Proposition 1 ensures the convergence to the origin of vector X. By using such a proposition, we can give a convergence result for the system (1) and (2). To do this, one has to exhibit the set of admissible equilibrium points (see also (Par´e et al., 2001)). −1 Proposition 2. Assume that I−LCA A−1 B A BA CA is non-singular, then by denoting −1 R = (I − LCA A−1 B)−1 A BA CA
(17)
S = (−A−1 B)+ (−R+ Lcl + R− Lcr ) −(−A−1 B)− (−R+ Lcr + R− Lcl )
(18)
T = (−A−1 B)+ (−R+ Lcr + R− Lcl ) −(−A−1 B)− (−R+ Lcl + R− Lcr )
(19)
then the set of equilibrium points for system (1)(2) is defined by: e x e e n p T E = (x , xA ) ∈ < × < , S xeA (20) with S S= −1 (−A−1 A BA C)+ S − (−AA BA C)− T (21) T T= −1 (−A−1 A BA C)+ T − (−AA BA C)− S (22) If cr = −cl , then by letting c = cr > 0, this set rewrites as: e x e e n p E = (x , xA ) ∈ < × < , Ss Ts xeA (23) with |A−1 B| |R| Lc Ss = (24) −1 |A−1 B| |R| Lc A BA C| |A −|A−1 B| |R| Lc Ts = (25) −1 |A−1 B| |R| Lc A BA C| |A
Proof. By setting xeA (t = 0) = xeA (0), let us note first that, if we have L(CA xeA (0) − cl ) Φ(t = 0) L(CA xeA (0) − cr ) then, due to (3), we have L(CA xeA (t)−cl ) Φ(t) L(CA xeA (t)−cr ), ∀t ≥ 0 (26) Consider now (xe , xeA ) an equilibrium point of (1)(2). Then Axe + BΦ[CA xeA ] = 0 and AA xeA + −1 BA Cxe = 0. Therefore xeA = A−1 BΦ, A BA CA and with (26), we get −1 −Lcl (I −LCA A−1 B)Φ −Lcr (27) A BA CA −1 Let us define R = (I −LCA A−1 B)−1 and A BA CA rewrite R = R+ − R− where R+ and R− are two
Preprints of the 5th IFAC Symposium on Robust Control Design
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5. EXTENSIONS
matrices with non-negative terms. Due to (27), we have −1 −R+ Lcl R+ (I − LCA A−1 B)Φ A BA CA −R+ Lcr (28) −1 −R− Lcl R− (I − LCA A−1 B)Φ A BA CA −R− Lcr
We get −R+ Lcl + R− Lcr Φ −R+ Lcr + R− Lcl
5.1 Stability in presence of unknown parameters in the nonlinearity In this section, we assume that the coefficients cr and cl needed to define the nonlinearity Φ are constant numbers but unknown, as considered in (Parlangeli and Corradini, 2005). More precisely, we assume that there exist ρl , ρr , ρl and ρr in
and, with xe = −A−1 BΦ, it follows: −1
(−A B)+ (−R+ Lcl + R− Lcr ) −(−A−1 B)− (−R+ Lcr + R− Lcl ) xe (−A−1 B)+ (−R+ Lcr + R+ Lcl ) −(−A−1 B)− (−R+ Lcl + R− Lcr )
(29)
(30)
e Thus, from (30) and xeA = −A−1 A BA Cx , we get −1 e (−A−1 A BA C)+ S − (−AA BA C)− T xA −1 −1 (−AA BA C)+ T − (−AA BA C)− S
(31)
By combining (30) and (31), we get (20). If moreover cr = −cl , then by letting c = cr > 0, S and T become respectively S = |A−1 B| |R| Lc ;
T = −|A−1 B| |R| Lc
Thus we get (23). 2 −1 Remark 2. If the matrix I − LCA A−1 B A BA CA is singular, then it is not possible to describe the lower and upper bounds on xe and xeA as in Proposition 2. In this case, the set of equilibrium points is defined as follows: e
−1
x ∈ (−A )BΩ(Φ) −1 xeA ∈ A−1 BΩ(Φ) A BA CA with Ω(Φ) = {Φ ∈
(33)
ρl Lcl ρl 0
(34)
and
By defining S = (−A−1 B)+ (−R+ Lcl + R− Lcr ) − (−A−1 B)− (−R+ Lcr +R− Lcl ) and T = (−A−1 B)+ (−R+ Lcr +R+ Lcl )−(−A−1 B)− (−R+ Lcl +R− Lcr ), we get S xe T
0 ρr Lcr ρr
(32)
By combining Proposition 1 and 2, we can provide a solution to Problem 1 as stated below. Theorem 1. If there exist two symmetric positive definite matrices P ∈ <(n+p)×(n+p) , M ∈
We can state a convergence result for the system (1), (2) in the presence of these unknown parameters in the nonlinearity as follows Theorem 2. If there exist two symmetric positive definite matrices P ∈ <(n+p)×(n+p) , M ∈
(35)
the solution of (1)-(2) converge to a point of B defined by e ˜ xe T ˜ B = (xe , xeA ) ∈
(39)
T˜ = (−A−1 B)+ (−R+ ρr + R− ρl ) −(−A−1 B)− (−R+ ρl + R− ρr )
(40)
If ρl = −ρr , and ρl = −ρr , then B rewrites e x e e n p ˜ ˜ B = (x , xA ) ∈ < × < , Ss Ts xeA (41) with |A−1 B| |R| ρr ˜ Ss = (42) −1 |A−1 B| |R| ρr A BA C| |A −|A−1 B| |R| ρr ˜ Ts = (43) −1 −|A−1 B| |R| ρr A BA C| |A
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Proof. To prove this theorem, we have to note that • Due to (33) and (34), (5) is guaranteed with the unknown parameters cr and cr , as soon as (35) is satisfied. • The converging set is defined by (20) where cl and cr are two unknown parameters. Due to (33) and (34), S and T can be estimated as follows: S (−A−1 B)+ (−R+ ρl + R− ρr ) −(−A−1 B)− (−R+ ρr + R− ρl ) T (−A−1 B)+ (−R+ ρr + R− ρl ) −(−A−1 B)− (−R+ ρl + R− ρr ) Thus the equilibrium set is bounded by B defined by (36). This concludes the proof of Theorem 2. 2
5.2 Dead-zone nonlinearity case The results developed can be extended by considering a dead-zone nonlinearity Φ defined by: `(i) (v(i) − br(i) ) if v(i) > br(i) (44) Φ[v](i) = `(i) (v(i) − bl(i) ) if v(i) < bl(i) 0 if bl(i) ≤ v(i) ≤ br(i) where (`1 , . . . , `m ) is given in
(45)
Thus, by using a modified version Lemma 1 to deal with the dead-zone nonlinearity defined in (44) as done in (Tarbouriech et al., 2004), one obtains the following proposition to prove the stability of the closed-loop system: ˜˙ = AX ˜ + BΦ[w] X ˜ w = CX ˜ = x0 x0A 0 . with X
(46)
Proposition 3. If there exist a symmetric positive definite matrix P ∈ <(n+p)×(n+p) and a diagonal positive definite matrix N ∈
REFERENCES Corradini, M. L. and G. Orlando (2002). Robust stabilization of nonlinear uncertain plants with backlash or dead zone in the actuator. IEEE Transactions on Control Systems Technology 10(1), 158 – 166. Macki, J. W., P. Nistri and P. Zecca (1993). Mathematical models for hysteresis. SIAM Review 35(1), 94–123. Nordin, M., X. Ma and P.O. Gutman (2002). Controlling mechanical systems with backlash: a survey. Automatica 38, 1633–1649. Par´e, T., A. Hassibi and J. How (2001). A kyp lemma and invariance principle for systems with multiple hysteresis non-linearities. Int. J. of Control 74(11), 1140–1157. Park, P.G., D. Banjerdpongchai and T. Kailath (1998). The asymptotic stability of nonlinear (lur’e) systems with multiple slope restrictions. IEEE Transactions on Automatic Control 43(7), 979–982. Parlangeli, G. and M. L. Corradini (2005). Variable structure control of systems with sandwiched backlash. In: 13th Mediterranean conf. On Control and Automation. Limassol, Cyprus. Tarbouriech, S., C. Prieur and J.M. Gomes da Silva Jr. (2004). Stability analysis and satbilization of systems presenting nested saturations. In: 43rd IEEE Conference on Decision and Control (CDC). Atlantis, Bahamas. pp. 5493–5498. Taware, A. and G. Tao (2003). Control of sandwich nonlinear systems. Lecture Notes in Control and Information Sciences, vol.288, Springer-Verlag, Berlin. Taware, A., G. Tao and C. Teolis (2002). Design and analysis of a hybrid control scheme for sandwich nonsmooth nonlinear systems. IEEE Transactions on Automatic Control 47(1), 145–150.