Proceedings of the 45th IEEE Conference on Decision & Control Manchester Grand Hyatt Hotel San Diego, CA, USA, December 13-15, 2006
FrA10.1
Multivariable Second Order Sliding Mode Control of Mechanical Systems Elisabetta Punta Abstract— The paper proposes an approach to second order sliding mode control for multi-input multi-output (MIMO) nonlinear uncertain systems. With respect to standard sliding mode control, the second order sliding mode techniques for singleinput single-output (SISO) systems show the same properties of robustness and precision, feature a higher order accuracy and can be exploited to eliminate the chattering effect. The extension of these results to the MIMO nonlinear systems is a challenging matter. In the present paper, the validity is extended to a quite large class of nonlinear processes affected by uncertainties of general nature; the control design is simple; the conditions of existence on the controllers are weak. The proposed procedure, which represents a general approach to the second order sliding mode control of MIMO systems, is applied to the control of mechanical systems.
I. INTRODUCTION In order to overcome the problem of chattering in classical sliding mode control, a new approach called “higher order sliding mode” has been recently proposed [1], [2], [3]. In £rst order sliding mode control the sliding variable is selected such that it has relative degree one with respect to the control. The discontinuous control signal acts on s, ˙ the £rst time derivative of the sliding output s, to enforce a sliding motion on s = 0. The concept of higher order sliding modes has been developed as the generalization of £rst order sliding modes: the control acts on the higher derivatives of s. For example, in the second order sliding modes, the control affects s¨, the second derivative of the sliding variable. The higher order sliding mode control provides a natural solution to avoid the chattering effect [4], shows robustness against various kinds of uncertainties such as external disturbances and measurement errors (as the standard sliding mode control [5]) and provides a higher order precision [1]. The main results concern the second order sliding mode control [1], [3], [6], even if sliding mode strategies of order higher than two have been proposed for SISO nonlinear systems [7], [8], [9], [10]. The extension of these results to the MIMO nonlinear systems is a challenging matter. Very few results on the higher order sliding mode control for MIMO nonlinear systems have been presented, mainly due to the non-applicability of Lyapunov’s direct method. Some existing results, [11], [12], [13], apply to second order sliding mode control systems with a “low” coupling between the inputs and the sliding variables. However, this condition is quite restrictive (for Work partially supported by PRAI-FESR Liguria E. Punta is with Institute of Intelligent Systems for Automation, National Research Council of Italy, ISSIA-CNR, Genoa, Italy
[email protected]
1-4244-0171-2/06/$20.00 ©2006 IEEE.
example it does not apply to general Lagrangian systems). For the class of Lagrangian systems solutions are given in [14] and [11]; they require the exploitation of observers, rely on the hierarchical sliding mode concept and ensure asymptotic convergence. The solution in [12] is based on regular form, while the one in [13] is based on quite restrictive conditions. In [15] a generalization of [10] is proposed by using the restrictive “low” coupling condition. The paper presents a MIMO second order sliding mode strategy [16], the validity of which is extended to a quite large class of nonlinear processes affected by uncertainties of general nature (for example, but not only, the Lagrangian systems). In Section II the “suboptimal” second order sliding mode algorithm for SISO control systems is brie¤y introduced. The considered MIMO second order control systems are described in the following section. In Section IV the second order sliding mode control strategy for multivariable nonlinear uncertain systems is presented in details. The control design is simple; the conditions of existence on the controllers are weak. The proposed procedure represents a general approach to the second order sliding mode control of MIMO systems. In Section V the MIMO control procedure is applied to the position tracking of a planar two link manipulator. The considered mechanical system is studied in two cases and simulation results are shown. II. SISO SECOND ORDER SLIDING MODE CONTROL SYSTEMS Consider an uncertain single-input nonlinear system x˙ = f (t, x, u),
t ≥ 0,
(1)
where x ∈ X ⊆ R is the measurable state vector, u ∈ U ⊂ R is the bounded control input and f (t, x, u) : [0, +∞) × Rn ×U → Rn is a suf£ciently smooth uncertain vector £eld. De£ne a proper sliding manifold in the state space n
s(t, x) = 0,
(2)
where s(t, x) : [0, +∞) × X → R is a known, single valued function. The control objective is that of steering to zero, possibly in £nite time, the measurable sliding output s(t, x). Assume that the sliding variable has a globally de£ned relative degree two [17], which implies that the second derivative of s can be expressed as s¨ = ϕ(t, x) + γ(t, x)u,
(3)
where the control gain function γ is always separated from zero. It can often be assumed that the sign of γ is known
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and that, under sensible conditions, the uncertainties of the system are bounded. The effective application of Lyapunov-based or classical £rst order sliding mode control design methods ([18], [19]) to system (1) with (2) is possible. The main drawbacks are when the relative degree p of the measurable regulated quantity s is higher than one; the control methods generally require the knowledge of the derivatives of s up to the (p−1)th order. As far as p = 2, the usually nonmeasurable s˙ must be estimated by means of some observer (e.g., “high-gain” observer [18] or sliding differentiator [20]). If the state trajectory of system (1) lies on the intersection of the two manifolds s = 0 and s˙ = 0 in the state space, then system (1) evolves featuring a second order sliding mode on the sliding manifold (2), [21]. A. Second Order Sliding Mode Control for SISO Systems The sliding variable s can be considered as a suitable output of the uncertain system (1); the control aim is that of steering this output to zero in £nite time. The second order sliding mode approach allows for the £nite time stabilization of both the output variable s and its time derivative s˙ by de£ning a suitable discontinuous control function which can be either the actual plant control or its time derivative, depending on the system relative degree [6]. The following standard notation will be used throughout the paper. A prime denotes transpose and | · | is the Euclidean norm or the induced matrix norm. Let us set y1 = s, it has been shown that, under sensible conditions, apart from a possible initialization phase, the second order sliding mode problem is equivalent to the £nite time stabilization problem for the following uncertain second order system [6] y˙ 1 = y2 , y˙ 2 = ϕ(t, y) + γ(t, y)v,
(4)
where y2 is not available, but its sign is known. The uncertain functions ϕ(t, y) and γ(t, y) are such that |ϕ(t, y)| < Φ, 0 < Γ1 < γ(t, y) < Γ2 ,
(y1 − βy1Mi ) y1Mi ≥ 0, (y1 − βy1Mi ) y1Mi < 0. 2Φ where V > max 2Φ , ∗ Γ1 (1+β)α Γ1+(β−1)Γ2 is the control 2 is the modumagnitude parameter, α∗ > max 1, (1−β)Γ (1+β)Γ1 lation parameter and β ∈ [0, 1) is the anticipation parameter; y1Mi is the last “singular point” of the sliding output y1 (i.e., the value of s at the most recent time instant at which s˙ is zero). By properly setting the controller parameters V , α∗ and β, it turns out that both s and s˙ converge to zero in £nite time. Moreover some stability and transient requirements can be obtained (this possibility can be exploited in many mechanical applications in order to avoid impacts and stickslip phenomena [23]). The discontinuous control law is given by (6), except in a possible initialization phase [3], [22]. The control problem is solved by (6) even when the bounds on the system are not constants and depend on the modulus of s [14], [23]. When not even the sign of y2 is available, the values y1Mi cannot be evaluated with ideal precision. The detection of the extremal values can be approximatively performed introducing an arbitrarily small time delay δ. Even in this case, the control law (6) has been proven, [3], to cause the convergence of the system’s trajectories to a δ-vicinity of the origin of the state plane. α(t) =
1 α∗
if if
III. MIMO SECOND ORDER SLIDING MODE CONTROL SYSTEMS Consider an uncertain multi-input nonlinear system x˙ = f (t, x, u),
t ≥ 0,
(7)
where x ∈ X ⊆ Rn is the measurable state vector, u ∈ U ⊂ Rk is the bounded vector of the control inputs and f (t, x, u) : [0, +∞)×Rn ×U → Rn is a suf£ciently smooth uncertain vector £eld. De£ne a proper sliding manifold in the state space s(t, x) = 0,
(8)
for all y ∈ Y ⊆ R2 , such that the uncertain dynamics (4) is bounded. If system (1) has relative degree p = 2 with respect ˙ to y1 = s, then v = u, while, if p = 1, v = u. The second order sliding mode approach [6] solves the stabilization problem for (4) by requiring the knowledge of y1 and just the sign of y2 . Some algorithms, which propose a solution to the above control problem, have been presented in the literature [1], [3], [22], [6]. In the present paper the focus is on the so-called “suboptimal” second order sliding mode algorithm which derives from a suboptimal feedback implementation of the classical time optimal control for a double integrator [3], [22]. The class of “suboptimal” second order sliding mode algorithms can be de£ned by the following control law
where s(t, x) : [0, +∞) × X → Rm is a known vector function such that for each t, s(t, · ) ∈ C 2 (X, Rm ), the m×n Jacobian matrix ∂s(t,x) has maximum rank m for almost ∂x every t, x ∈ X and m ≤ n. The control objective is that of steering to zero, possibly in £nite time, each component of the measurable sliding vector s(t, x). In this paper we consider the nominal system (7), (8) with a special structure, as follows. Let n be even and partition the state variable as n (9) x = (q , p ) , q ∈ Rj , j = . 2 We assume that j = k = m and p 0m×m f (t, x, u) = + u, (10) A(x) B(q)
v(t) = −α(t)V sign (y1 − βy1Mi ) ,
s(t, x) = Cx + D (t) ,
(5)
(6)
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where A(x) ∈ Rm , B(q) ∈ Rm×m , C is a constant m × 2m matrix and D(t) ∈ Rm is such that its second time derivative is bounded . For the sake of simplicity and without loss of generality we consider the case in which D (t) = 0. The special structure (9), (10), (11) is motivated by applications to mechanical systems for which q is the vector of the Lagrangian coordinates and p = q. ˙ We emphasize that the matrix B in (10) does not depend on p and is positive de£nite. In £nite time the measurable sliding output s(x) must be steered to zero by means of the control vector u. Let us choose C = [ Im×m 0m×m ], the second time derivative of s can be expressed as s¨ = A(s, s) ˙ + B(s)u.
(12)
The following conditions on the uncertain B(s) and A(s, s) ˙ hold (13) 0 < λ1 ≤ λmin ≤ λmax ≤ λ2 , |A(s, s)| ˙ ≤ A |s| ,
IV. SECOND ORDER SLIDING MODE CONTROL FOR MIMO SYSTEMS Let us consider the following scalar function 1 2 1 s s = |s| , (15) 2 2 by de£nition W (s) is positive de£nite; its £rst time derivative is given by ˙ (t) = s s˙ W (16) W (s) =
while the second time derivative is (17)
By posing y1 = W , we obtain y˙ 1 = y2 , ˙ + G(s)u, y˙ 2 = F (s, s)
(18)
˙ and where the uncertain terms are F (s, s) ˙ = s˙ s˙ + s A(s, s) G(s) = s B(s). The original MIMO control problem is solved if we are able to steer to zero y1 of system (18). De£nition 1: Consider system (18) at the time instants tMi : y2 (tMi ) = 0, i = 1, 2, . . .
(19)
[s1 (tMi ) , . . . , sm (tMi )] [s1Mi , . . . , smMi ] , i = 1, 2, . . . .
u = −Usign (s) ˙ ,
by (15) y1Mi ≥ 0, i = 1, 2, . . .. De£nition 2: At the time instants (19), the variable y1 assumes the values 1 2 s (tMi ) + . . . + s2m (tMi ) . (21) y1Mi = y1 (tMi ) = 2 1
(23)
A|s| λ1 ,
guarantees that an extremal point is reached with U > in £nite time. Proof. See [16]. Theorem 1: Consider system (18) at t = tM1 and let the control vector u, for t ∈ [tM1 , +∞), be designed according to the following |ul | = ρ(t)V,
l = 1, . . . , m,
(24)
if (y1 − βy1Mi ) y1Mi ≥ 0, if (y1 − βy1Mi ) y1Mi < 0,
(25)
sign (u) = −sign (y1 − βy1Mi ) sign (s) .
(26)
ρ(t) =
√1 m α∗ √ m
The control parameters are chosen
2A|sM1 | 2A|sM1 | V > 0, V > max λ1 +4(β−1)λ2 , (1+β)α∗ λ1 +5(β−1)λ2 , 2 , β ∈ [0, 1) . α∗ > max 1, 5(1−β)λ (1+β)λ1 (27) In £nite time, both y 1 and y2 are steered to zero; it turns out that both s and s˙ converge to zero in £nite time. Proof. See [16]. Remark 1: It can be noticed that, when |s| = 0, the control matrix G (s) of system (18) is zero, nevertheless the controllability of the system is not lost since the control matrix of (12) is B (s) > 0. Remark 2: The MIMO second order sliding mode control strategy expressed by (24)–(26) is robust with respect to the uncertainties, the norm of which can be upper-bounded. Even in the MIMO case, the knowledge of the bounds (13) and (14) on the uncertain terms results suf£cient to compute the control parameters according to (27). V. MECHANICAL SYSTEMS In this section we consider, as an example, the position tracking control of a two-link planar manipulator [5], [24]. In the Lagrangian formulation, the dynamics of a robot with j = 2 degrees of freedom is expressed as follows M (q)¨ q + C(q, q) ˙ q˙ + g(q) = τ
(20)
(22)
Nothing can be said on the sign and the value assumed by the extremal points slMi , l = 1, . . . , m, i = 1, 2, . . .. Proposition 1: Consider system (18) at t = 0 and assume that y1 (0) is not a singular point, i.e. y2 (0) = 0. The application of the control vector
and de£ne the singular points of the variable y 1 as y1Mi = y1 (tMi ) , i = 1, 2, . . . ,
= =
sMi
(14)
where λmin and λmax are, respectively, the minimum and maximum eigenvalue of B(s), while λ1 , λ2 , A are known positive constants.
¨ (t) = s˙ s˙ + s A(s, s) W ˙ + s B(s)u.
De£ne the extremal points of the sliding vector s as
(28)
where q ∈ Rj is the vector of the Lagrangian generalized position coordinates, τ ∈ Rj is the vector of the control generalized forces, C(q, q) ˙ q˙ is the vector of the Coriolis and centrifugal terms, g(q) is the vector of the gravitational effects, and M (q) is the positive de£nite matrix of the system. Equation (28) can be written in the form (10)
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FrA10.1 The designed control vector τ is discontinuous (Figure 4) and acts on the second time derivative of the sliding vector s, which is steered to zero in £nite time. 3
2.5
2
1.5
ee
1
y
with x = (q , q˙ ) ∈ Rn and n = 4. For the considered manipulator (Figure 1), the vector q contains the two angular positions of the system; the matrices and vector in (28) are expressed by M11 = 13 m1 l12 + m2 (l12 + 13 l22 + l1 l2 cos(q2 )), M12 = m2 ( 13 l22 + 12 l1 l2 cos(q2 )), M21 = M12 , M22 = 13 m2 l22 , C11 = −m2 l1 l2 sin(q2 )q˙2 , C12 = − 12 m2 l1 l2 sin(q2 )q˙2 , C21 = 12 m2 l1 l2 sin(q2 )q˙2 , C22 = 0, g1 = 12 m1 Gl1 cos(q1 ) + m2 G(l1 cos(q1 ) + 12 l2 cos(q1 + q2 )), g2 = 12 m2 Gl2 cos(q1 + q2 ); it is set m1 = m2 = 3 Kg, m l1 = l2 = 1 m and G = 9.8 sec 2.
0.5
5
0
4
−0.5
−1
3 0
q
1
2
2
y
3 x
4
5
6
ee
2
Fig. 2.
The trajectory of the end effector in the xee , yee space.
1 0.1
q1
0
0
−0.1
s1
−0.2 −0.3
−1 −1
0
1
2
3
4
5
−0.4
6
x
Fig. 1. mass.
−0.5
The planar manipulator with two links of the same length and
0
5
0
5
10
15
10
15
1
Case 1. First we consider the case in which the velocity vector q˙ is not available. Since the control aim is the position tracking control, let us choose s(t, x) ∈ Rm with m = 2 as s = [q − qd (t)]
0.8
s
2
0.6 0.4 0.2 0 t [sec]
(29)
where qd (t) is the desired pro£le for the position vector. The equations (28) and (29) satisfy the conditions posed in the previous sections in order to apply the proposed MIMO second order sliding mode control method. In fact the nominal system expressed by (28) and (29) is in accordance to (7), (8) with f and s as in (10) and (11) respectively, therefore it is possible to £nd a control vector u = τ (t, x) designed according to Theorem 1 and expressed by (24), (25), (26). The desired angular behaviours qd (t) are chosen such that the corresponding end-effector trajectory in the vertical Cartesian plane follows a circle. In Figure 2 it is reported the obtained end-effector behaviour. It can be noted that the initial conditions q1 (0) and q2 (0) are chosen so that, at the beginning, the end-effector position does not belong to the desired surface, nevertheless, under the control action τ , the desired sliding manifold is reached and maintained. Figure 3 shows the convergence to zero of the £rst and second component of the s vector.
Fig. 3.
In £nite time s 1 and s2 converge to zero.
Case 2. In this second case the velocity vector q˙ is assumed to be measurable. Let us derive equation (28) and obtain M (q) q (3) + Cε (q, q, ˙ τ ) = τ˙ , (30) ˙ g˙ depends ˙ τ ) = M˙ + C q¨+C˙ q+ where the vector Cε (q, q, on (q, q, ˙ τ ) since q¨, from (28), is a function of (q, q, ˙ τ ). The torque vector τ is assumed to be measurable. In order to perform the position tracking control, let us choose s (t) ∈ Rm with m = 2 as s = [q˙ − q˙d (t)] + Λ [q − qd (t)]
(31)
where qd (t), q˙d (t) are the desired pro£les for positions and velocities respectively and Λ is a diagonal matrix with positive elements λii > 0, i = 1, 2. De£ne (32) X = Q , Q˙ = (Q , P ) ,
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2000
1000
τ1
2.5 0
2 −1000
1.5 −2000
0
5
10
15
y
ee
1 2000
0.5
τ2
1000
0
0
−0.5
−1000
−2000
−1 0 0
5
10
1
2
15
Fig. 4.
3 x
4
5
6
ee
t [sec]
Fig. 5.
The two components τ1 and τ2 of the control vector τ .
where the vector Q ∈ Rm , m = 2, is de£ned as Q = (q˙ + Λq) .
0.5 0 −0.5
The equations (30) and (31) satisfy the conditions posed in the previous section in order to apply the proposed MIMO second order sliding mode control method. In fact the nominal system expressed by (30), (31) and (32) is in accordance to (7), (8) and (9) with f and s as in (10) and (11) respectively, therefore it is possible to £nd a control vector u = τ˙ (t, X ) designed according to Theorem 1 and expressed by (24), (25), (26).
s1
−1 −1.5 −2 −2.5 −3
0
5
0
5
10
15
10
15
5 4 3
s
2
The simulation results of the application of the previously introduced MIMO second order sliding mode control procedure to system (30) and (31) are presented.
2 1 0
Even in this case the desired angular behaviours qd (t) and q˙d (t) are chosen such that the corresponding end-effector trajectory in the vertical Cartesian plane follows a circle.
t [sec]
Fig. 6.
In Figure 5 it is reported the obtained end-effector behaviour. As in the previous case the initial conditions q1 (0) and q2 (0) are chosen so that, at the beginning, the endeffector position does not belong to the desired surface, nevertheless, under the control action τ˙ , the desired sliding manifold is reached and maintained.
In £nite time s 1 and s2 converge to zero.
1
500
τ
Figure 6 shows the convergence to zero of s1 and s2 , the two components of the sliding vector s. The designed control vector τ˙ is discontinuous (Figure 8) and acts on the second time derivative of the sliding vector s, which is steered to zero in £nite time.
0
−500
0
5
0
5
10
15
10
15
150
In this case the torque vector τ , which is computed by integrating the discontinuous vector τ˙ , results to be continuous (Figure 7). Remark 3: In both Case 1 and Case 2 the position tracking control problem for system (28) is solved by the proposed strategy directly, without introducing any auxiliary system and without relying on a hierarchy in the reaching of the sliding manifolds, how it is required by the MIMO second order sliding mode procedures in [11], [14], [23].
The trajectory of the end effector in the xee , yee space.
100
τ
2
50 0 −50 −100 −150
t [sec]
Fig. 7. The two components τ1 and τ2 of the continuous torque vector τ .
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4
4
x 10
1
dτ /dt
2
0
−2
−4
0
5
10
15
10
15
4
4
x 10
2
dτ /dt
2
0
−2
−4
0
5 t [sec]
Fig. 8.
The two components τ˙1 and τ˙2 of the control vector τ˙ .
VI. CONCLUSIONS The paper proposes an approach to second order sliding mode control for MIMO nonlinear uncertain systems. The extension of the SISO second order sliding mode techniques to the MIMO nonlinear systems is a challenging matter. In this paper, the presented method applies to a quite large class of nonlinear processes affected by uncertainties of general nature, the control design is simple and the conditions of existence on the controllers are weak. The presented procedure is applied to the position tracking control of a planar two link manipulator. The simulation results show that the proposed control method represents a general approach to the second order sliding mode control of MIMO systems.
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