quasi-continuous high-order sliding-mode controllers - Semantic Scholar
Proceedings of the 42nd IEEE Conference on Decision and Control Maui, Hawaii USA, December 2003
ThP14-6
QUASI-CONTINUOUS HIGH-ORDER SLIDING-MODE CONTROLLERS Arie Levant School of Mathematical Sciences, Tel-Aviv University, Ramat-Aviv, 69978 Tel-Aviv, Israel E-mail: [email protected] . Tel.: 972-3-6408812, Fax: 972-3-6407543 discrete sampling and, properly used, totally remove the chattering effect. In order to remove the chattering, the control derivative is to be treated as a new control. The discontinuity set of controllers [8, 10] is a stratified union of manifolds with codimension varying in the range from 1 to r. Unfortunately, the complicated structure of the controller discontinuity set causes certain redundant transient chattering. To avoid it one needs an additional artificial increase of the relative degree [8]. A sliding-mode controller of a new type is proposed in this paper, providing for the control being a function of s, (r-1) && , ..., s , continuous everywhere except the s& , s manifold. defined by the r-sliding mode equations
Abstract. A universal finite-time-convergent controller is developed capable to control the output s of any uncertain SISO system of a known permanent relative degree r. The mode s º 0 (r-sliding mode) is established by means of a (r-1) control dependent only on s, s& , ..., s and continuous (r-1) & everywhere except the set s = s = ... = s = 0. An output-feedback controller version is also developed. With s being the tracking deviation, the exact finite-timeconvergent output tracking is provided in the absence of output noises, otherwise the tracking accuracy is proportional to the magnitude of the noise. In the latter case s º 0 is not attained producing an unswitched control continuous in time. The resulting performance is significantly improved compared with known r-sliding controllers.
(r-1)
&& = ... = s s = s& = s
(1)
The mode s º 0 is established after a finite-time transient. In the presence of errors in evaluation of the output and its derivatives, a motion in some vicinity of (1) takes place. As a result the control is practically a continuous function of time, for the trajectory does not hit manifold (1). The lacking derivatives are produced in their turn by recently proposed robust exact differentiators with finitetime convergence [7, 10]. The resulting output-feedback controller provides for exact tracking s = 0 after a finitetime transient when the measurements of the deviation s are exact, and for the tracking error proportional to the maximal measurement error otherwise [9, 10]. Its transient is smoother and the tracking accuracy is higher than those of the known r-sliding controllers [8, 10]. The simulation demonstrates the practical applicability of the new controller.
1. Introduction Control under heavy uncertainty conditions remains one of the main subjects of the modern control theory. While a number of advanced methods like adaptation, absolute stability methods or the back-stepping procedure are based on relatively detailed knowledge of the controlled system, the sliding-mode control approach requirements are more moderate. The idea is to react immediately to any deviation of the system from some properly chosen constraint steering it back by a sufficiently energetic effort. Slidingmode implementation is based on its insensitivity to external and internal disturbances and high accuracy [13,14]. The main drawback of the standard sliding modes is mostly related to the so-called chattering effect caused by the high-frequency control switching [12, 4]. Let the constraint be given by the equation s = s - w(t) = 0, where s is some available output variable of an uncertain single-input-single-output (SISO) dynamic system and w(t) is an unknown-in-advance smooth input to be tracked in real time. Then the standard sliding-mode control u = - k sign s may be considered as a universal output controller applicable if the relative degree is 1, i.e. if & u > 0. Highers& explicitly depends on the control u and s¢ order sliding mode [6, 1] is applicable for controlling SISO uncertain systems with arbitrary relative degree. The correspondent finite-time-convergent controllers (r-sliding controllers) [8, 10] require actually only the knowledge of the system relative degree. The produced control is a discontinuous function of the tracking deviation s and of && , ..., its real-time-calculated successive derivatives s, s& , s (r-1) s . The controllers provide also for higher accuracy with
0-7803-7924-1/03/$17.00 ©2003 IEEE
= 0.
2.
Preliminaries and the problem statement
Consider a smooth dynamic system with smooth output function s, and let the system be closed by some possiblydynamical discontinuous feedback and understood in the Filippov sense [3]. Then, provided that successive total (r-1) time derivatives s, s& , ..., s are continuous functions of the closed-system state-space variables; and set (1) is a (r-1) non-empty integral set, the motion on the set s = ... = s = 0 is called r-sliding (rth order sliding) mode [6, 1, 10]. The standard sliding mode used in the most variable structure systems (VSS), is of the first order (s is continuous, and s& is discontinuous). Consider a dynamic system of the form x& = a(t,x) + b(t,x)u,
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s = s(t, x),
(2)
n
n+1
continuous everywhere (i.e. it can be redefined by (i-1) continuity) except the point s = s& = ... = s = 0. Theorem 1. Let the parameters b1,..., br-1, a be chosen sufficiently large in the list order, then the controller
Here x Î R , a, b and s: R ® R are unknown smooth functions, u Î R, n is also uncertain. The relative degree r of the system is assumed to be constant and known. The task is to provide in finite time for exact keeping of s º 0. Extend system (2) by introduction of a fictitious t t variable xn+1 = t, x& n +1 = 1 . Denote ae = (a,1) , be = (b,0) , where the last component corresponds to xn+1. The equality of the relative degree to r means that the Lie derivatives r-2 Lbes, LbeLaes, ..., LbeLae s equal zero identically in a
(r-1)
u = - aYr-1,r (s, s& , ..., s
r-1
(r) r
(3)
r-1
where h(t,x) = Lae s, g(t,x) = LbeLae s. Obviously, in order to provide for s º 0, any continuous control u = (r-1) U(s, s& , ..., s ) has to satisfy the equality r-1
r
U(0,0, ..., 0) = - Lae s / LbeLae s , whenever (1) holds. That is obviously impossible due to the problem uncertainty. Thus, the control is to be discontinuous at least on set (1), and the problem is actually to establish the r-sliding mode s = 0. It is supposed that r-1
r
0 < Km £ LbeLae s £ KM, | Lae s | £ C r-1
for some Km, KM, C > 0. Since LbeLae s =
¶ ¶u
r £ 4: 1. u = - a sign s, 1/2 -1 1/2 2. u = - a(| s& |+ |s| ) ( s& + |s| sign s), 2/3 -1/2 2/3 -1 && |+ 2 (| s& |+ |s| ) | s& + |s| sign s | ] 3. u = - a [| s 2/3 -1/2 2/3 && + 2 (| s& |+ |s| ) ( s& + |s| sign s ) ], [s 4. j3,4 = 3/4 -1/3 3/4 -1/2 &&& && |+(| s& |+0.5|s| ) | s& +0.5 |s| sign s|] s + 3[| s 3/4 -1/3 3/4 && +(| s& |+0.5|s| ) ( s& +0.5 |s| sign s)], [s N3,4 = 3/4 -1/3 3/4 -1/2 && |+(| s& |+0.5|s| ) | s& +0.5 |s| sign s|] | &&& s |+ 3[| s 3/4 -1/3 3/4 && +(| s& |+0.5|s| ) ( s& +0.5 |s| sign s)|, |s u = - a j3,4 / N3,4 .
(4) (r)
r
s , Lae s =
(r)
s |u=0 , conditions (4) are reformulated in terms of inputoutput relations. It is also assumed that trajectories of (2) are infinitely extendible in time for any Lebesguemeasurable bounded control u(t, x). In practice it means that the system be weakly minimum phase. 3. Construction of the r-sliding controller Let i = 1,..., r-1. Denote j0,r = s, N0,r = |s|, Y0,r = j0,r /N0,r = sign s, (i)
(5)
provides for the appearance of the r-sliding mode s º 0 attracting trajectories in finite time. It follows from Proposition 1 that control (5) is continuous everywhere except the r-sliding mode s = s& = (r-1) ... = s = 0. Each time some finite-time transient process is mentioned in this paper it means that the transient time is a locally bounded function of initial conditions. Each choice of the parameters b1,..., br-1determines a controller family applicable to all systems (2) of the relative degree r. Parameter a > 0 is to be chosen specifically for any fixed C, Km, KM, most conveniently by computer simulation in order to avoid the use of redundantly large estimations of C, Km, KM. The proposed controller is easily generalized: coefficients of Ni,r, ji,r may be any positive numbers etc. Obviously, a is to be negative (r) ¶ s < 0. Following are controllers with tested bi for with ¶u
vicinity of a given point and LbeLae s is not zero at the point [5]. It is easy to check [5] that the control first time appears explicitly in the rth total time derivative of s and s = h(t,x) + g(t,x)u,
).
The identity s/|s| = sign s, s ¹ 0, is used here. Note that the sign function does not introduce here more discontinuity than in the identity y = |y| sign y. The control is here a continuous function of time everywhere except the r-sliding set. Nevertheless, it may change very fast, when its derivative turns into infinity. Consider now the discrete sampling case, when the control value is updated at the moments ti, ti+1 - ti = t > 0, t Î [ti, ti+1). Theorem 2. With discrete measurements controller (5) provides in finite time for keeping the inequalities
(i)
ji,r = s +bi N i--11/(, rr -i +1) ji-1,r, Ni,r= |s |+bi N i--11/(, rr -i +1) |ji-1,r|, Yi,r = ji,r / Ni,r where b1,..., br-1 are positive numbers. Recall that according to the Filippov definition values of the feedback function on any set of the zero Lebesgue measure do not matter. The following proposition is easily proved by induction using the obvious identity (i) ji,r = s +bi N i(-r1-,ir) /( r -i +1) Yi-1,r .
r
r-1
(r-1)
|s| < a0t , | s& | < a1t , ..., |s
Proposition 1. Let i = 0, ..., r -1. Ni,r is positive definite, (i) i.e. Ni,r = 0 iff s = s& = ... = s = 0. The inequality |Yi,r| £ 1 (i-1) holds whenever Ni,r > 0. The function Yi,r(s, s& , ..., s ) is
| < ar-1t
with some positive constants a0, a1, ..., ar-1. That is the best possible accuracy attainable with (r) discontinuous s [6]. The following result shows robustness of controller (5) with respect to measurement errors.
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(i)
(r-i)/r
2. if | f(t) - f0(t)| £ e, then for some positive constants mi, ni depending exclusively on the parameters of differentiator (6)
Theorem 3. Let s be measured with accuracy nie for some fixed ni > 0, i = 1, ..., r-1. Then there are such positive constants mi that for any e > 0 the following inequalities are established in finite time: (i)
|s | £ mi e
(r - i)/r
(i)
, i = 0, ..., r - 1.
3. if f(t) = f0(t), but f(t) is sampled with the constant step t > 0, then for some mi, ni
4. Robust exact differentiation Controller (5) requires real-time exact calculation or direct (r-1) (r) (r) measurement of s, s& , ..., s . The identity s = u ¶¶u s r
(i)
(r)
0
0
0
Parameters l0i being tuned for L = 1, the parameters are easily recalculated for any value of L by formula 1/(n-i+1) li = l0i L . Following are some computer-tested parameters of the 5th-order differentiator tuned with L = 1: l0 = 8, l1 = 5, l2 = 3, l3 = 2, l4 = 1.5, l5 = 1.1. 5. Universal output-feedback SISO controller Consider uncertain system (2), (4). Combining controller (5) and differentiator (6) achieve
0
Lipschitz constant L > 0. Then the nth-order differentiator i has the outputs zi = Dn , i = 0, 1, ..., n, defined recursively as follows: z&0 = v , v = -l | z0 - f(t)|
n/(n + 1)
u = - a Yr(z0, z1, ..., zr-1), 1/r
z&0 = v0, v0 = - l0,0 L
sign(z0 - f(t)) + z1,
0
z1 = Dn-1 (v(×), L), ..., zn = Dn-1
n-1
Here D0(f(×), L) is a simple nonlinear filter l > L.
n/(n + 1)
z& 0 = v0, v0 = -l0 | z0 - f(t)| sign(z0 - f(t)) + z1, (n-1)/n z& 1 = v1, v1 = -l1 | z1 - v0| sign(z1 - v0) + z2, ... (6) 1/2 z&n -1 = vn-1, vn-1 = -ln-1| zn-1 - vn-2| sign(zn-1- vn-2)+ zn, z& n = -ln sign(zn - vn-1), where li > 0 are chosen sufficiently large in the reverse order. Note that it contains actually all the lower-order differentiators and each recursive step requires tuning one parameter only. Remark. It is easy to check that differentiator (5) may be rewritten in the non-recursive form (7)
where i = 0,..., n-1, z&n = - lˆ n sign(z0 - f(t)). lˆ i are calculated on the basis of l , ..., l , lˆ = l . 0
n
n
n
(i)
|s | £ mi e
Following relations are established in finite time with properly chosen parameters [10]: 1. if f(t) = f0(t) then (i) z0 = f0(t); zi = vi-1 = f0 (t),
s|
sign(z0 - s) + z1,
(r-2)/ (r-1)
where parameters li of differentiator (9) are chosen (r) according to the condition |s | £ L, L ³ C + aKM. 1/(r- i) Relations li = l0i L are used, where l0i are chosen in advance for L = 1. Hence, in case when C and Km , KM are known, only one parameter a is really needed to be tuned, otherwise both L and a might be found in computer simulation. Due to the recursive form of differentiator (6) the abovementioned computer-tested values of parameters can be taken for any r: l0, r-1 = 1.1, l0, r-2 = 1.5, l0, r-3 = 2, l0, r-4 = 3, l0, r-5 = 5, l0, r-6 = 8. These values are sufficient for up to the 5th order differentiation and r £ 6. The lacking values need to be tuned in the unlikely case of r > 6. Theorem 4. Let s be measured with a Lebesguemeasurable noise h, |h| £ e. Then with properly chosen parameters of controller (8), (9) the following inequalities are established in finite time in the closed system (2), (8), (9) for some positive mi:
Thus, the nth-order differentiator [10] has the form
(n - i)/(n + 1) z&i = - lˆ i |z0 - f(t)| sign(z0 - f(t)) + zi+1,
| z0 1/(r-1)
(8) (r-1)/r
z&1 = v1, v1 = - l0,1 L | z1 - v0| sign(z1 - v0) + z2, ... (9) 1/2 1/ 2 z&r - 2 = vr-2, vr-2 = - l0, r-2 L | zr-2 - vr-3| sign(zr-2- vr-3)+ zr-1, z&r -1 = - l0, r-1 L sign(zr-1 - vr-2),
(v(×), L).
D0: z& = -l sign(z - f(t)),
n-i+1
|zi - f0 (t)| £ mi t , i = 0, ..., n, (i+1) n-i |vi - f0 (t)| £ ni t , i = 0, ..., n - 1.
+ La s implies |s | £ C + aKM , which allows implementation of robust exact (r-1)th-order differentiators [7, 10]. Let input signal f(t) be a function defined on [0, ¥), consisting of a bounded Lebesgue-measurable noise with unknown features and an unknown base signal f0(t) with the nth derivative having a known Lipschitz constant L > 0. Denote by Dn-1(f(×),L) the (n-1)th-order differentiator i producing outputs Dn-1 , i = 0, 1, ..., n-1, being estimations (n-1) (n-1) of f , f& , &f& , ..., f for any input f(t) with f having 0
(n - i +1)/(n + 1)
|zi - f0 (t)| £ mi e , i = 0, ..., n, (i+1) (n - i)/(n + 1) |vi - f0 (t)| £ ni e , i = 0, ..., n-1;
(r - i)/r
, i = 0, ..., r - 1.
Theorem 4 means that with exact measurements (i.e. e = 0) an r-sliding mode s º 0 is established in the closed system globally attracting trajectories in finite time.
i = 1, ..., n;
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Theorem 5. Let t > 0 be the constant input sampling interval and the noises be absent. Then the following inequalities are established in finite time for some positive constants mi: (i)
|s | £ mi t 6.
The condition |Yi,r| ³ x is assured outside of W2. Prove now that W2 is invariant and attracts the trajectories with large bi+1. The “upper” boundary of W2 is given by the (i) equation p+ = s - F+ = 0. Suppose that at the initial moment p+ > 0 and, therefore, Yi,r ³ x. Taking into account & £ & is homogeneous and, according to Lemma 1, F that F
r-i
, i = 0, ..., r - 1.
+ ( r - i -1) /( r - i ) k N i,r
The proofs
As follows from (3), (4) (r) s Î [-C, C] + [Km, KM] u . (10) The differential inclusion is understood here in the Filippov sense [3], which means that the right-hand vector set is enlarged in a special way in order to satisfy certain convexity and semicontinuity conditions. This inclusion does not “remember” anything on system (2) except the constants r, C, Km, KM. Thus, providing for the (i) convergence of s to zero or to some small vicinity of the origin, the tracking problem is simultaneously solved for all systems (2) satisfying (4). All the proofs are based on homogeneity reasoning. Proof of Theorem 1 is based on a few Lemmas. Only the main proof points are listed below. Assign the weight (i) (homogeneity degree [11]) r - i to s , i = 0, ..., r - 1 and the weight 1 (minus system homogeneity degree) to t. Lemma 1. The weight of Ni,r equals r - i, i = 0, ..., r - 1. Each homogeneous locally-bounded function w(s, s& , ..., (i) s ) of the weight r - i satisfies the inequality | w | £ c Ni,r for some c > 0. Lemma 2. Let 1 £ i £ r-2, then for any positive bi, gi with sufficiently large bi+1 > 0 and sufficiently small gi+1> 0 the inequality (i+1) |s + b i+1 N i(,rr-i -1) /( r -i ) Yi,r| £ gi+1 N i(,rr-i -1) /( r -i )
+
and |p+| £ k1 Ni,r for some k, k1 > 0, achieve
differentiating that & p& + £ (-bi+1 x + gi+1) N i(,rr-i -1) /( r -i ) - F + £ (-bi+1 x + gi+1 + k) N i(,rr-i -1) /( r -i ) . Hence p+ vanishes in finite time. Thus, the trajectory inevitably enters the region W2 in finite time. Similarly, the trajectory enters W2 if the initial value of p+ is negative and, therefore, Yi,r £ - x. Obviously, W2 is invariant. Choosing F- and F+ sufficiently close to f- and f+ on the homogeneous sphere and bi+1 large enough, achieve due to Lemma 1 the statement of Lemma 2.n Since N0,r = |s|, j0,r = s, Lemma 2 is replaced by the next simple Lemma with i = 0. (r-1}/r (r-1}/r sign s| £ g1|s| Lemma 3. The inequality | s& + b1|s| provides with 0 £ g1< b1 for the establishment in finite time and keeping of the identity s º 0. The proof of the Theorem is now finished by the similar proof that for any small g > 0 with sufficiently large a the (r-1) inequality |s + br-1 N r1-/ 22, r Yr-2,r | £ g Nr-2,r is established in finite time and kept afterwards. n Other proofs are standard, follow the proofs of the analogous theorems from [8, 9, 10]. and are based on the homogeneity reasoning. Consider, for example, Theorem 4. (i) Denote xi = zi - s . Taking into account h Î [-e, e] and form (7) of the differentiator achieve
provides for the finite-time establishment and keeping of the inequality (i) |s + b i N i(-r1-,ir) /( r -i +1) Yi-1,r | £ gi N i(-r1-,ir) /( r -i +1) .
(r-1)
u = - a sign(Yr-1,r(s+x0, s& +x1, ...,s
(i)
Proof. Consider the point set W = {(s, s& , ..., s )| |Yi,r| £ x} for some fixed x > 0. Simple calculations show that W Ì W1 with small x, where W1 is defined by the inequality
+ xr-1)),
(11)
(r-1)/r x& 0 Î - lˆ 0 | x0 + [-e, e] | sign(x0 +[-e, e]) + x1, (r-2)/r x& Î - lˆ | x + [-e, e]| sign(x + [-e, e]) + x ,
1
(i)
|s + b i N i(-r1-,ir) /( r -i +1) Yi-1,r | £ 4x N i(-r1-,ir) /( r -i +1) .
1
0
0
2
...
(i)
(12)
1/r x& r - 2 Î - lˆ r - 2 | x0 + [-e, e]| sign(x0 + [-e, e])+ xr-1, x& Î - lˆ sign(x + [-e, e]).
That implies in its turn f- £ s £ f+ , where f-, f+ are (i) homogeneous functions of s, s& , ..., s of the weight r - i. p/r Restricting f- and f+ to the homogeneous sphere s + p/(r-1) (i) p/(r-i) s& + ...+ (s ) = 1, where p is the least multiple of 1, 2, ..., r - i, achieve some continuous on the sphere functions f1- and f1+. Functions f1- and f1+ can be approximated on the sphere by some smooth functions f2and f2+ from beneath and from above respectively. Functions f2- and f2+ are extended by homogeneity to the (i) smooth homogeneous functions F- and F+ of s, s& , ..., s (i) of the weight r - i, so that W Ì W2 = {(s, s& , ..., s )| F- £ (i) s £ F+}.
r -1
r -1
0
Dynamics (12) of the differentiator is independent of the system dynamics (10), (11). Thus, with e = 0 derivative deviations xi vanish in finite time [10]. Hence, trajectories of (10) - (12) converge to 0 in finite time with e = 0 (Theorem 1). It is easy to see that with e > 0 the transformation (i)
(t, s , xi )
r-i (i)
r-i
a ( kt, k s , k xi), i = 0, ..., r-1
(13)
transfers trajectories of (10) - (12) into trajectories of (10) r (12) but with the changed noise magnitude k e. This
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deviations, steering angle q and its derivative u are shown in Fig. 2a, b, c, d respectively. It is seen from Fig. 3c that the control u remains continuous until the entrance into the 3-sliding mode. The steering angle q remains rather smooth and is quite feasible.
homogeneity implies the desired asymptotics of the residual attracting set [9, 10]. n 7. Simulation example: car control Consider a simple kinematic model of car control x& = v cos j, y& = v sin j, j& = v/l tan q, q& = u, where x and y are Cartesian coordinates of the rear-axle middle point, j is the orientation angle, v is the longitudinal velocity, l is the length between the two axles and q is the steering angle (Fig. 1). The task is to steer the car from a given initial position to the trajectory y = g(x), while g(x) and y are assumed to be measured in real time.
Fig. 2: 3-sliding car control
Fig. 1: Kinematic car model Define s = y - g(x). Let v = const = 10 m/s, l = 5 m, g(x) = 10 sin(0.05x) + 5, x = y = j = q = 0 at t = 0. The relative degree of the system is 3 and 3-sliding controller N°3 can be applied here. It was taken a = 0.5, L = 100. The resulting output-feedback controller (7), (8) is 2/3 -1/2
2/3
Fig. 3: Differentiator convergence -3
-1
u = - 0.5 [|z2|+ 2 (|z1|+ | z0| ) | z1+ | z0| sign z0 | ] 2/3 -1/2 2/3 [z2+ 2 (|z1|+ | z0| ) (z1+ | z0| sign z0 )], 2/3 z&0 = v0, v0 = - 9 | z0 - s| sign(z0 - s) + z1, z&1 = v1, v1 = - 15 | z1 - v0| z& 2 = - 110 sign(z2 - v1).
1/2
-5
With t = 10 the tracking accuracies |s| £ 2.5×10 , | s& | -3 && | £ 0.11 were attained, which corresponds to £ 2.4×10 , | s the asymptotics stated in Theorem 5. Convergence of the differentiator outputs to the directly calculated derivatives of s is demonstrated in Fig. 3. In the presence of output noise with the magnitude && | £ 0.01m the tracking accuracies |s| £ 0.04, | s& | £ 0.2, | s 1.8 were obtained. With the measurement noise of the magnitude 0.1 the accuracies changed to |s| £ 0.4, | s& | £ && | £ 3.2 which corresponds to the asymptotics stated 0.9, | s by Theorem 4. The performance of the controller with the measurement error magnitude 0.1m is shown in Fig. 4. It is seen from Fig. 4c that the control u is continuous function of t. The steering angle vibrations have magnitude of about 12 degrees and frequency 1 which is also quite feasible. The performance does not change when the frequency of the noise varies in the range 100 - 100000. The performance of the standard 3-sliding controller [810] in the absence of noises is demonstrated in Fig. 5. The advantages of the new controller are obvious (compare Figs. 2d, 5b). Simulation shows that the standard controller is also much more sensitive to the parameter choice.
sign(z1 - v0) + z2,
Mark that the differentiator parameter L = 100 is deliberately enlarged in order to provide for faster convergence and higher robustness with respect to sharp commands caused by possible measurement errors (otherwise L = 22 would be sufficient in the absence of sampling noises). The control was applied only from t = 0.5 in order to provide some time for the differentiator convergence. The integration was carried out according to the Euler method (the only reliable integration method with discontinuous dynamics), the sampling step being equal to -4 the integration step t = 10 . In the absence of noises the -8 -4 && | £ tracking accuracies |s| £ 3.12×10 , | s& | £ 1.4×10 , | s 0.011 were attained. The car trajectory, 3-sliding tracking
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While the sensitivity of the higher-order differentiation to input noises grows severely with the differentiation order [7], the sensitivity of the proposed controller to the higherorder derivative estimation errors decreases exactly in the same proportion. That results in the acceptable performance of the proposed output-feedback controller. Both the proposed controller and its output-feedback version are very robust with respect to measurement noises. The simulation shows that it is probably the first practically applicable output-feedback r-sliding controller with r > 2. The direct measurements of successive output derivatives can be avoided. Only boundedness of the measurement noise is needed, no frequency considerations are relevant. References [1] G. Bartolini, A. Ferrara, A. Levant, E. Usai. On second order sliding mode controllers. Variable Structure Systems, Sliding Mode and Nonlinear Control (Lecture Notes in Contr. and Inf. Science, 247, (K.D. Young and U. Ozguner (Ed)), pp. 329-350, Springer-Verlag, London, 1999. [2] C. Edwards, S. K. Spurgeon. Sliding Mode Control: Theory and Applications, Taylor & Francis, 1998 [3] A.F. Filippov. Differential Equations with Discontinuous Right-Hand Side, Kluwer, Dordrecht, the Netherlands, 1988. [4] L. Fridman, Singularly perturbed analysis of chattering in relay control systems, IEEE Trans. on Automatic Control, vol. 47(12), pp. 2079-2084, 2002 [5] A. Isidori. Nonlinear Control Systems, second edition, Springer Verlag, New York, 1989. [6] A. Levant (L.V. Levantovsky). Sliding order and sliding accuracy in sliding mode control, International Journal of Control, 58(6), pp.1247-1263, 1993. [7] A. Levant. Robust exact differentiation via sliding mode technique, Automatica, 34(3), pp. 379-384, 1998. [8] A. Levant. Universal SISO sliding-mode controllers with finite-time convergence, IEEE Trans. on Automatic Control, vol. 46(9), pp. 1447-1451, 2001. [9] A. Levant. Universal output-feedback SISO controller, Proc. of the 15th IFAC Congress, July 2002, Barcelona, 2002. [10] A. Levant, Higher-order sliding modes, differentiation and output-feedback control, International Journal of Control, 76 (9/10), pp. 924-941, 2003. [11] L. Rosier, Homogeneous Lyapunov function for homogeneous continuous vector field. System and Control Letters, 19, pp. 467–473, 1992. [12] J.-J. E. Slotine, and W. Li, Applied Nonlinear Control (London: Prentice-Hall, Inc.) , 1991 [13] V.I. Utkin. Sliding Modes in Optimization and Control Problems, Springer Verlag, New York, 1992. [14] A.S.I Zinober, (Ed.). Variable Structure and Lyapunov Control, Springer-Verlag, Berlin, 1994.
Fig. 4: Performance with the input noise magnitude 0.1m
a: Car trajectory
b: Steering angle
Fig. 5: Performance of the standard 3-sliding controller [8] 8. Conclusions A new arbitrary-order sliding mode controller is proposed. It is actually only the second known family of such controllers. It is also a sliding-mode controller of a new type, because it provides for sliding motion on a manifold of codimension higher than 1 by means of control continuous everywhere except this manifold. As a result the chattering effect of such a controller is significantly reduced. The real-time exact differentiator [10] of the appropriate order is combined with the proposed controller providing for the full SISO control based on the input measurements only, when the only information on the controlled uncertain process is actually its relative degree. The obtained controller is locally applicable to generalcase weakly-minimum-phase SISO systems; it is also globally applicable if the relative degree is constant and few boundedness restrictions hold globally. In the absence r of noises the resulting accuracy is proportional to t , t being a sampling period and r being the relative degree. That is the best possible accuracy with discrete sampling and discontinuous control [6]. Artificially increasing the relative degree, arbitrarily smooth control may be produced, which totally removes the chattering effect.
4610
ThP14-6
QUASI-CONTINUOUS HIGH-ORDER SLIDING-MODE CONTROLLERS Arie Levant School of Mathematical Sciences, Tel-Aviv University, Ramat-Aviv, 69978 Tel-Aviv, Israel E-mail: [email protected] . Tel.: 972-3-6408812, Fax: 972-3-6407543 discrete sampling and, properly used, totally remove the chattering effect. In order to remove the chattering, the control derivative is to be treated as a new control. The discontinuity set of controllers [8, 10] is a stratified union of manifolds with codimension varying in the range from 1 to r. Unfortunately, the complicated structure of the controller discontinuity set causes certain redundant transient chattering. To avoid it one needs an additional artificial increase of the relative degree [8]. A sliding-mode controller of a new type is proposed in this paper, providing for the control being a function of s, (r-1) && , ..., s , continuous everywhere except the s& , s manifold. defined by the r-sliding mode equations
Abstract. A universal finite-time-convergent controller is developed capable to control the output s of any uncertain SISO system of a known permanent relative degree r. The mode s º 0 (r-sliding mode) is established by means of a (r-1) control dependent only on s, s& , ..., s and continuous (r-1) & everywhere except the set s = s = ... = s = 0. An output-feedback controller version is also developed. With s being the tracking deviation, the exact finite-timeconvergent output tracking is provided in the absence of output noises, otherwise the tracking accuracy is proportional to the magnitude of the noise. In the latter case s º 0 is not attained producing an unswitched control continuous in time. The resulting performance is significantly improved compared with known r-sliding controllers.
(r-1)
&& = ... = s s = s& = s
(1)
The mode s º 0 is established after a finite-time transient. In the presence of errors in evaluation of the output and its derivatives, a motion in some vicinity of (1) takes place. As a result the control is practically a continuous function of time, for the trajectory does not hit manifold (1). The lacking derivatives are produced in their turn by recently proposed robust exact differentiators with finitetime convergence [7, 10]. The resulting output-feedback controller provides for exact tracking s = 0 after a finitetime transient when the measurements of the deviation s are exact, and for the tracking error proportional to the maximal measurement error otherwise [9, 10]. Its transient is smoother and the tracking accuracy is higher than those of the known r-sliding controllers [8, 10]. The simulation demonstrates the practical applicability of the new controller.
1. Introduction Control under heavy uncertainty conditions remains one of the main subjects of the modern control theory. While a number of advanced methods like adaptation, absolute stability methods or the back-stepping procedure are based on relatively detailed knowledge of the controlled system, the sliding-mode control approach requirements are more moderate. The idea is to react immediately to any deviation of the system from some properly chosen constraint steering it back by a sufficiently energetic effort. Slidingmode implementation is based on its insensitivity to external and internal disturbances and high accuracy [13,14]. The main drawback of the standard sliding modes is mostly related to the so-called chattering effect caused by the high-frequency control switching [12, 4]. Let the constraint be given by the equation s = s - w(t) = 0, where s is some available output variable of an uncertain single-input-single-output (SISO) dynamic system and w(t) is an unknown-in-advance smooth input to be tracked in real time. Then the standard sliding-mode control u = - k sign s may be considered as a universal output controller applicable if the relative degree is 1, i.e. if & u > 0. Highers& explicitly depends on the control u and s¢ order sliding mode [6, 1] is applicable for controlling SISO uncertain systems with arbitrary relative degree. The correspondent finite-time-convergent controllers (r-sliding controllers) [8, 10] require actually only the knowledge of the system relative degree. The produced control is a discontinuous function of the tracking deviation s and of && , ..., its real-time-calculated successive derivatives s, s& , s (r-1) s . The controllers provide also for higher accuracy with
0-7803-7924-1/03/$17.00 ©2003 IEEE
= 0.
2.
Preliminaries and the problem statement
Consider a smooth dynamic system with smooth output function s, and let the system be closed by some possiblydynamical discontinuous feedback and understood in the Filippov sense [3]. Then, provided that successive total (r-1) time derivatives s, s& , ..., s are continuous functions of the closed-system state-space variables; and set (1) is a (r-1) non-empty integral set, the motion on the set s = ... = s = 0 is called r-sliding (rth order sliding) mode [6, 1, 10]. The standard sliding mode used in the most variable structure systems (VSS), is of the first order (s is continuous, and s& is discontinuous). Consider a dynamic system of the form x& = a(t,x) + b(t,x)u,
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s = s(t, x),
(2)
n
n+1
continuous everywhere (i.e. it can be redefined by (i-1) continuity) except the point s = s& = ... = s = 0. Theorem 1. Let the parameters b1,..., br-1, a be chosen sufficiently large in the list order, then the controller
Here x Î R , a, b and s: R ® R are unknown smooth functions, u Î R, n is also uncertain. The relative degree r of the system is assumed to be constant and known. The task is to provide in finite time for exact keeping of s º 0. Extend system (2) by introduction of a fictitious t t variable xn+1 = t, x& n +1 = 1 . Denote ae = (a,1) , be = (b,0) , where the last component corresponds to xn+1. The equality of the relative degree to r means that the Lie derivatives r-2 Lbes, LbeLaes, ..., LbeLae s equal zero identically in a
(r-1)
u = - aYr-1,r (s, s& , ..., s
r-1
(r) r
(3)
r-1
where h(t,x) = Lae s, g(t,x) = LbeLae s. Obviously, in order to provide for s º 0, any continuous control u = (r-1) U(s, s& , ..., s ) has to satisfy the equality r-1
r
U(0,0, ..., 0) = - Lae s / LbeLae s , whenever (1) holds. That is obviously impossible due to the problem uncertainty. Thus, the control is to be discontinuous at least on set (1), and the problem is actually to establish the r-sliding mode s = 0. It is supposed that r-1
r
0 < Km £ LbeLae s £ KM, | Lae s | £ C r-1
for some Km, KM, C > 0. Since LbeLae s =
¶ ¶u
r £ 4: 1. u = - a sign s, 1/2 -1 1/2 2. u = - a(| s& |+ |s| ) ( s& + |s| sign s), 2/3 -1/2 2/3 -1 && |+ 2 (| s& |+ |s| ) | s& + |s| sign s | ] 3. u = - a [| s 2/3 -1/2 2/3 && + 2 (| s& |+ |s| ) ( s& + |s| sign s ) ], [s 4. j3,4 = 3/4 -1/3 3/4 -1/2 &&& && |+(| s& |+0.5|s| ) | s& +0.5 |s| sign s|] s + 3[| s 3/4 -1/3 3/4 && +(| s& |+0.5|s| ) ( s& +0.5 |s| sign s)], [s N3,4 = 3/4 -1/3 3/4 -1/2 && |+(| s& |+0.5|s| ) | s& +0.5 |s| sign s|] | &&& s |+ 3[| s 3/4 -1/3 3/4 && +(| s& |+0.5|s| ) ( s& +0.5 |s| sign s)|, |s u = - a j3,4 / N3,4 .
(4) (r)
r
s , Lae s =
(r)
s |u=0 , conditions (4) are reformulated in terms of inputoutput relations. It is also assumed that trajectories of (2) are infinitely extendible in time for any Lebesguemeasurable bounded control u(t, x). In practice it means that the system be weakly minimum phase. 3. Construction of the r-sliding controller Let i = 1,..., r-1. Denote j0,r = s, N0,r = |s|, Y0,r = j0,r /N0,r = sign s, (i)
(5)
provides for the appearance of the r-sliding mode s º 0 attracting trajectories in finite time. It follows from Proposition 1 that control (5) is continuous everywhere except the r-sliding mode s = s& = (r-1) ... = s = 0. Each time some finite-time transient process is mentioned in this paper it means that the transient time is a locally bounded function of initial conditions. Each choice of the parameters b1,..., br-1determines a controller family applicable to all systems (2) of the relative degree r. Parameter a > 0 is to be chosen specifically for any fixed C, Km, KM, most conveniently by computer simulation in order to avoid the use of redundantly large estimations of C, Km, KM. The proposed controller is easily generalized: coefficients of Ni,r, ji,r may be any positive numbers etc. Obviously, a is to be negative (r) ¶ s < 0. Following are controllers with tested bi for with ¶u
vicinity of a given point and LbeLae s is not zero at the point [5]. It is easy to check [5] that the control first time appears explicitly in the rth total time derivative of s and s = h(t,x) + g(t,x)u,
).
The identity s/|s| = sign s, s ¹ 0, is used here. Note that the sign function does not introduce here more discontinuity than in the identity y = |y| sign y. The control is here a continuous function of time everywhere except the r-sliding set. Nevertheless, it may change very fast, when its derivative turns into infinity. Consider now the discrete sampling case, when the control value is updated at the moments ti, ti+1 - ti = t > 0, t Î [ti, ti+1). Theorem 2. With discrete measurements controller (5) provides in finite time for keeping the inequalities
(i)
ji,r = s +bi N i--11/(, rr -i +1) ji-1,r, Ni,r= |s |+bi N i--11/(, rr -i +1) |ji-1,r|, Yi,r = ji,r / Ni,r where b1,..., br-1 are positive numbers. Recall that according to the Filippov definition values of the feedback function on any set of the zero Lebesgue measure do not matter. The following proposition is easily proved by induction using the obvious identity (i) ji,r = s +bi N i(-r1-,ir) /( r -i +1) Yi-1,r .
r
r-1
(r-1)
|s| < a0t , | s& | < a1t , ..., |s
Proposition 1. Let i = 0, ..., r -1. Ni,r is positive definite, (i) i.e. Ni,r = 0 iff s = s& = ... = s = 0. The inequality |Yi,r| £ 1 (i-1) holds whenever Ni,r > 0. The function Yi,r(s, s& , ..., s ) is
| < ar-1t
with some positive constants a0, a1, ..., ar-1. That is the best possible accuracy attainable with (r) discontinuous s [6]. The following result shows robustness of controller (5) with respect to measurement errors.
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(i)
(r-i)/r
2. if | f(t) - f0(t)| £ e, then for some positive constants mi, ni depending exclusively on the parameters of differentiator (6)
Theorem 3. Let s be measured with accuracy nie for some fixed ni > 0, i = 1, ..., r-1. Then there are such positive constants mi that for any e > 0 the following inequalities are established in finite time: (i)
|s | £ mi e
(r - i)/r
(i)
, i = 0, ..., r - 1.
3. if f(t) = f0(t), but f(t) is sampled with the constant step t > 0, then for some mi, ni
4. Robust exact differentiation Controller (5) requires real-time exact calculation or direct (r-1) (r) (r) measurement of s, s& , ..., s . The identity s = u ¶¶u s r
(i)
(r)
0
0
0
Parameters l0i being tuned for L = 1, the parameters are easily recalculated for any value of L by formula 1/(n-i+1) li = l0i L . Following are some computer-tested parameters of the 5th-order differentiator tuned with L = 1: l0 = 8, l1 = 5, l2 = 3, l3 = 2, l4 = 1.5, l5 = 1.1. 5. Universal output-feedback SISO controller Consider uncertain system (2), (4). Combining controller (5) and differentiator (6) achieve
0
Lipschitz constant L > 0. Then the nth-order differentiator i has the outputs zi = Dn , i = 0, 1, ..., n, defined recursively as follows: z&0 = v , v = -l | z0 - f(t)|
n/(n + 1)
u = - a Yr(z0, z1, ..., zr-1), 1/r
z&0 = v0, v0 = - l0,0 L
sign(z0 - f(t)) + z1,
0
z1 = Dn-1 (v(×), L), ..., zn = Dn-1
n-1
Here D0(f(×), L) is a simple nonlinear filter l > L.
n/(n + 1)
z& 0 = v0, v0 = -l0 | z0 - f(t)| sign(z0 - f(t)) + z1, (n-1)/n z& 1 = v1, v1 = -l1 | z1 - v0| sign(z1 - v0) + z2, ... (6) 1/2 z&n -1 = vn-1, vn-1 = -ln-1| zn-1 - vn-2| sign(zn-1- vn-2)+ zn, z& n = -ln sign(zn - vn-1), where li > 0 are chosen sufficiently large in the reverse order. Note that it contains actually all the lower-order differentiators and each recursive step requires tuning one parameter only. Remark. It is easy to check that differentiator (5) may be rewritten in the non-recursive form (7)
where i = 0,..., n-1, z&n = - lˆ n sign(z0 - f(t)). lˆ i are calculated on the basis of l , ..., l , lˆ = l . 0
n
n
n
(i)
|s | £ mi e
Following relations are established in finite time with properly chosen parameters [10]: 1. if f(t) = f0(t) then (i) z0 = f0(t); zi = vi-1 = f0 (t),
s|
sign(z0 - s) + z1,
(r-2)/ (r-1)
where parameters li of differentiator (9) are chosen (r) according to the condition |s | £ L, L ³ C + aKM. 1/(r- i) Relations li = l0i L are used, where l0i are chosen in advance for L = 1. Hence, in case when C and Km , KM are known, only one parameter a is really needed to be tuned, otherwise both L and a might be found in computer simulation. Due to the recursive form of differentiator (6) the abovementioned computer-tested values of parameters can be taken for any r: l0, r-1 = 1.1, l0, r-2 = 1.5, l0, r-3 = 2, l0, r-4 = 3, l0, r-5 = 5, l0, r-6 = 8. These values are sufficient for up to the 5th order differentiation and r £ 6. The lacking values need to be tuned in the unlikely case of r > 6. Theorem 4. Let s be measured with a Lebesguemeasurable noise h, |h| £ e. Then with properly chosen parameters of controller (8), (9) the following inequalities are established in finite time in the closed system (2), (8), (9) for some positive mi:
Thus, the nth-order differentiator [10] has the form
(n - i)/(n + 1) z&i = - lˆ i |z0 - f(t)| sign(z0 - f(t)) + zi+1,
| z0 1/(r-1)
(8) (r-1)/r
z&1 = v1, v1 = - l0,1 L | z1 - v0| sign(z1 - v0) + z2, ... (9) 1/2 1/ 2 z&r - 2 = vr-2, vr-2 = - l0, r-2 L | zr-2 - vr-3| sign(zr-2- vr-3)+ zr-1, z&r -1 = - l0, r-1 L sign(zr-1 - vr-2),
(v(×), L).
D0: z& = -l sign(z - f(t)),
n-i+1
|zi - f0 (t)| £ mi t , i = 0, ..., n, (i+1) n-i |vi - f0 (t)| £ ni t , i = 0, ..., n - 1.
+ La s implies |s | £ C + aKM , which allows implementation of robust exact (r-1)th-order differentiators [7, 10]. Let input signal f(t) be a function defined on [0, ¥), consisting of a bounded Lebesgue-measurable noise with unknown features and an unknown base signal f0(t) with the nth derivative having a known Lipschitz constant L > 0. Denote by Dn-1(f(×),L) the (n-1)th-order differentiator i producing outputs Dn-1 , i = 0, 1, ..., n-1, being estimations (n-1) (n-1) of f , f& , &f& , ..., f for any input f(t) with f having 0
(n - i +1)/(n + 1)
|zi - f0 (t)| £ mi e , i = 0, ..., n, (i+1) (n - i)/(n + 1) |vi - f0 (t)| £ ni e , i = 0, ..., n-1;
(r - i)/r
, i = 0, ..., r - 1.
Theorem 4 means that with exact measurements (i.e. e = 0) an r-sliding mode s º 0 is established in the closed system globally attracting trajectories in finite time.
i = 1, ..., n;
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Theorem 5. Let t > 0 be the constant input sampling interval and the noises be absent. Then the following inequalities are established in finite time for some positive constants mi: (i)
|s | £ mi t 6.
The condition |Yi,r| ³ x is assured outside of W2. Prove now that W2 is invariant and attracts the trajectories with large bi+1. The “upper” boundary of W2 is given by the (i) equation p+ = s - F+ = 0. Suppose that at the initial moment p+ > 0 and, therefore, Yi,r ³ x. Taking into account & £ & is homogeneous and, according to Lemma 1, F that F
r-i
, i = 0, ..., r - 1.
+ ( r - i -1) /( r - i ) k N i,r
The proofs
As follows from (3), (4) (r) s Î [-C, C] + [Km, KM] u . (10) The differential inclusion is understood here in the Filippov sense [3], which means that the right-hand vector set is enlarged in a special way in order to satisfy certain convexity and semicontinuity conditions. This inclusion does not “remember” anything on system (2) except the constants r, C, Km, KM. Thus, providing for the (i) convergence of s to zero or to some small vicinity of the origin, the tracking problem is simultaneously solved for all systems (2) satisfying (4). All the proofs are based on homogeneity reasoning. Proof of Theorem 1 is based on a few Lemmas. Only the main proof points are listed below. Assign the weight (i) (homogeneity degree [11]) r - i to s , i = 0, ..., r - 1 and the weight 1 (minus system homogeneity degree) to t. Lemma 1. The weight of Ni,r equals r - i, i = 0, ..., r - 1. Each homogeneous locally-bounded function w(s, s& , ..., (i) s ) of the weight r - i satisfies the inequality | w | £ c Ni,r for some c > 0. Lemma 2. Let 1 £ i £ r-2, then for any positive bi, gi with sufficiently large bi+1 > 0 and sufficiently small gi+1> 0 the inequality (i+1) |s + b i+1 N i(,rr-i -1) /( r -i ) Yi,r| £ gi+1 N i(,rr-i -1) /( r -i )
+
and |p+| £ k1 Ni,r for some k, k1 > 0, achieve
differentiating that & p& + £ (-bi+1 x + gi+1) N i(,rr-i -1) /( r -i ) - F + £ (-bi+1 x + gi+1 + k) N i(,rr-i -1) /( r -i ) . Hence p+ vanishes in finite time. Thus, the trajectory inevitably enters the region W2 in finite time. Similarly, the trajectory enters W2 if the initial value of p+ is negative and, therefore, Yi,r £ - x. Obviously, W2 is invariant. Choosing F- and F+ sufficiently close to f- and f+ on the homogeneous sphere and bi+1 large enough, achieve due to Lemma 1 the statement of Lemma 2.n Since N0,r = |s|, j0,r = s, Lemma 2 is replaced by the next simple Lemma with i = 0. (r-1}/r (r-1}/r sign s| £ g1|s| Lemma 3. The inequality | s& + b1|s| provides with 0 £ g1< b1 for the establishment in finite time and keeping of the identity s º 0. The proof of the Theorem is now finished by the similar proof that for any small g > 0 with sufficiently large a the (r-1) inequality |s + br-1 N r1-/ 22, r Yr-2,r | £ g Nr-2,r is established in finite time and kept afterwards. n Other proofs are standard, follow the proofs of the analogous theorems from [8, 9, 10]. and are based on the homogeneity reasoning. Consider, for example, Theorem 4. (i) Denote xi = zi - s . Taking into account h Î [-e, e] and form (7) of the differentiator achieve
provides for the finite-time establishment and keeping of the inequality (i) |s + b i N i(-r1-,ir) /( r -i +1) Yi-1,r | £ gi N i(-r1-,ir) /( r -i +1) .
(r-1)
u = - a sign(Yr-1,r(s+x0, s& +x1, ...,s
(i)
Proof. Consider the point set W = {(s, s& , ..., s )| |Yi,r| £ x} for some fixed x > 0. Simple calculations show that W Ì W1 with small x, where W1 is defined by the inequality
+ xr-1)),
(11)
(r-1)/r x& 0 Î - lˆ 0 | x0 + [-e, e] | sign(x0 +[-e, e]) + x1, (r-2)/r x& Î - lˆ | x + [-e, e]| sign(x + [-e, e]) + x ,
1
(i)
|s + b i N i(-r1-,ir) /( r -i +1) Yi-1,r | £ 4x N i(-r1-,ir) /( r -i +1) .
1
0
0
2
...
(i)
(12)
1/r x& r - 2 Î - lˆ r - 2 | x0 + [-e, e]| sign(x0 + [-e, e])+ xr-1, x& Î - lˆ sign(x + [-e, e]).
That implies in its turn f- £ s £ f+ , where f-, f+ are (i) homogeneous functions of s, s& , ..., s of the weight r - i. p/r Restricting f- and f+ to the homogeneous sphere s + p/(r-1) (i) p/(r-i) s& + ...+ (s ) = 1, where p is the least multiple of 1, 2, ..., r - i, achieve some continuous on the sphere functions f1- and f1+. Functions f1- and f1+ can be approximated on the sphere by some smooth functions f2and f2+ from beneath and from above respectively. Functions f2- and f2+ are extended by homogeneity to the (i) smooth homogeneous functions F- and F+ of s, s& , ..., s (i) of the weight r - i, so that W Ì W2 = {(s, s& , ..., s )| F- £ (i) s £ F+}.
r -1
r -1
0
Dynamics (12) of the differentiator is independent of the system dynamics (10), (11). Thus, with e = 0 derivative deviations xi vanish in finite time [10]. Hence, trajectories of (10) - (12) converge to 0 in finite time with e = 0 (Theorem 1). It is easy to see that with e > 0 the transformation (i)
(t, s , xi )
r-i (i)
r-i
a ( kt, k s , k xi), i = 0, ..., r-1
(13)
transfers trajectories of (10) - (12) into trajectories of (10) r (12) but with the changed noise magnitude k e. This
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deviations, steering angle q and its derivative u are shown in Fig. 2a, b, c, d respectively. It is seen from Fig. 3c that the control u remains continuous until the entrance into the 3-sliding mode. The steering angle q remains rather smooth and is quite feasible.
homogeneity implies the desired asymptotics of the residual attracting set [9, 10]. n 7. Simulation example: car control Consider a simple kinematic model of car control x& = v cos j, y& = v sin j, j& = v/l tan q, q& = u, where x and y are Cartesian coordinates of the rear-axle middle point, j is the orientation angle, v is the longitudinal velocity, l is the length between the two axles and q is the steering angle (Fig. 1). The task is to steer the car from a given initial position to the trajectory y = g(x), while g(x) and y are assumed to be measured in real time.
Fig. 2: 3-sliding car control
Fig. 1: Kinematic car model Define s = y - g(x). Let v = const = 10 m/s, l = 5 m, g(x) = 10 sin(0.05x) + 5, x = y = j = q = 0 at t = 0. The relative degree of the system is 3 and 3-sliding controller N°3 can be applied here. It was taken a = 0.5, L = 100. The resulting output-feedback controller (7), (8) is 2/3 -1/2
2/3
Fig. 3: Differentiator convergence -3
-1
u = - 0.5 [|z2|+ 2 (|z1|+ | z0| ) | z1+ | z0| sign z0 | ] 2/3 -1/2 2/3 [z2+ 2 (|z1|+ | z0| ) (z1+ | z0| sign z0 )], 2/3 z&0 = v0, v0 = - 9 | z0 - s| sign(z0 - s) + z1, z&1 = v1, v1 = - 15 | z1 - v0| z& 2 = - 110 sign(z2 - v1).
1/2
-5
With t = 10 the tracking accuracies |s| £ 2.5×10 , | s& | -3 && | £ 0.11 were attained, which corresponds to £ 2.4×10 , | s the asymptotics stated in Theorem 5. Convergence of the differentiator outputs to the directly calculated derivatives of s is demonstrated in Fig. 3. In the presence of output noise with the magnitude && | £ 0.01m the tracking accuracies |s| £ 0.04, | s& | £ 0.2, | s 1.8 were obtained. With the measurement noise of the magnitude 0.1 the accuracies changed to |s| £ 0.4, | s& | £ && | £ 3.2 which corresponds to the asymptotics stated 0.9, | s by Theorem 4. The performance of the controller with the measurement error magnitude 0.1m is shown in Fig. 4. It is seen from Fig. 4c that the control u is continuous function of t. The steering angle vibrations have magnitude of about 12 degrees and frequency 1 which is also quite feasible. The performance does not change when the frequency of the noise varies in the range 100 - 100000. The performance of the standard 3-sliding controller [810] in the absence of noises is demonstrated in Fig. 5. The advantages of the new controller are obvious (compare Figs. 2d, 5b). Simulation shows that the standard controller is also much more sensitive to the parameter choice.
sign(z1 - v0) + z2,
Mark that the differentiator parameter L = 100 is deliberately enlarged in order to provide for faster convergence and higher robustness with respect to sharp commands caused by possible measurement errors (otherwise L = 22 would be sufficient in the absence of sampling noises). The control was applied only from t = 0.5 in order to provide some time for the differentiator convergence. The integration was carried out according to the Euler method (the only reliable integration method with discontinuous dynamics), the sampling step being equal to -4 the integration step t = 10 . In the absence of noises the -8 -4 && | £ tracking accuracies |s| £ 3.12×10 , | s& | £ 1.4×10 , | s 0.011 were attained. The car trajectory, 3-sliding tracking
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While the sensitivity of the higher-order differentiation to input noises grows severely with the differentiation order [7], the sensitivity of the proposed controller to the higherorder derivative estimation errors decreases exactly in the same proportion. That results in the acceptable performance of the proposed output-feedback controller. Both the proposed controller and its output-feedback version are very robust with respect to measurement noises. The simulation shows that it is probably the first practically applicable output-feedback r-sliding controller with r > 2. The direct measurements of successive output derivatives can be avoided. Only boundedness of the measurement noise is needed, no frequency considerations are relevant. References [1] G. Bartolini, A. Ferrara, A. Levant, E. Usai. On second order sliding mode controllers. Variable Structure Systems, Sliding Mode and Nonlinear Control (Lecture Notes in Contr. and Inf. Science, 247, (K.D. Young and U. Ozguner (Ed)), pp. 329-350, Springer-Verlag, London, 1999. [2] C. Edwards, S. K. Spurgeon. Sliding Mode Control: Theory and Applications, Taylor & Francis, 1998 [3] A.F. Filippov. Differential Equations with Discontinuous Right-Hand Side, Kluwer, Dordrecht, the Netherlands, 1988. [4] L. Fridman, Singularly perturbed analysis of chattering in relay control systems, IEEE Trans. on Automatic Control, vol. 47(12), pp. 2079-2084, 2002 [5] A. Isidori. Nonlinear Control Systems, second edition, Springer Verlag, New York, 1989. [6] A. Levant (L.V. Levantovsky). Sliding order and sliding accuracy in sliding mode control, International Journal of Control, 58(6), pp.1247-1263, 1993. [7] A. Levant. Robust exact differentiation via sliding mode technique, Automatica, 34(3), pp. 379-384, 1998. [8] A. Levant. Universal SISO sliding-mode controllers with finite-time convergence, IEEE Trans. on Automatic Control, vol. 46(9), pp. 1447-1451, 2001. [9] A. Levant. Universal output-feedback SISO controller, Proc. of the 15th IFAC Congress, July 2002, Barcelona, 2002. [10] A. Levant, Higher-order sliding modes, differentiation and output-feedback control, International Journal of Control, 76 (9/10), pp. 924-941, 2003. [11] L. Rosier, Homogeneous Lyapunov function for homogeneous continuous vector field. System and Control Letters, 19, pp. 467–473, 1992. [12] J.-J. E. Slotine, and W. Li, Applied Nonlinear Control (London: Prentice-Hall, Inc.) , 1991 [13] V.I. Utkin. Sliding Modes in Optimization and Control Problems, Springer Verlag, New York, 1992. [14] A.S.I Zinober, (Ed.). Variable Structure and Lyapunov Control, Springer-Verlag, Berlin, 1994.
Fig. 4: Performance with the input noise magnitude 0.1m
a: Car trajectory
b: Steering angle
Fig. 5: Performance of the standard 3-sliding controller [8] 8. Conclusions A new arbitrary-order sliding mode controller is proposed. It is actually only the second known family of such controllers. It is also a sliding-mode controller of a new type, because it provides for sliding motion on a manifold of codimension higher than 1 by means of control continuous everywhere except this manifold. As a result the chattering effect of such a controller is significantly reduced. The real-time exact differentiator [10] of the appropriate order is combined with the proposed controller providing for the full SISO control based on the input measurements only, when the only information on the controlled uncertain process is actually its relative degree. The obtained controller is locally applicable to generalcase weakly-minimum-phase SISO systems; it is also globally applicable if the relative degree is constant and few boundedness restrictions hold globally. In the absence r of noises the resulting accuracy is proportional to t , t being a sampling period and r being the relative degree. That is the best possible accuracy with discrete sampling and discontinuous control [6]. Artificially increasing the relative degree, arbitrarily smooth control may be produced, which totally removes the chattering effect.
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