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Discrete variable-structure integral controllers*
PII: S0005–1098(97)00197–0
Automatica, Vol. 34, No. 3, pp. 355—361, 1998 ( 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0005-1098/98 $19.00#0.00
Brief Paper
Discrete Variable-Structure Integral Controllers* C. BONIVENTO,- M. SANDRI‡ and R. ZANASIKey Words—Robust control; variable-structure systems; discontinuous control; integral control; discretetime sytems.
(1994) and in Bonivento and Zanasi (1994), which uses a nonlinear integral action for estimating the external disturbance and keeping small the discontinuous term in the control law and the subsequent chattering effect. The original DIC design was given for the continuous-time case. In this paper, starting from some previous partial results presented in Bonivento and Zanasi (1996) and in Bonivento et al. (1996), a discretised version of the DIC algorithm is properly modified in order to take into account the time-derivative of the controlled variable. The so-obtained discrete-time controller shows to be very effective in the disturbance estimation and in the chattering reduction especially when the system parameters are known and constant. If this is not the case, a new ‘‘adaptive’’ control structure which also estimates the ‘‘inertia’’ parameter of the system is then proposed. This new control structure is analysed and its asymptotic stability is proved in the case of constant disturbances. The paper is organised as follows. In Section 2 a brief review of the DIC technique is presented together with a first modified discrete version of the DIC control algorithm. In Section 3, the basic properties of this discrete-time controller are summarised. In Section 4, an adaeptive version of the proposed control law is introduced and analysed. Finally, in Section 5, some simulation results on the velocity control of a DC motor confirm the effectiveness of the proposed control structure.
Abstract—The subject of this paper is the analysis and the design of a class of robust discrete-time variable-structure controllers of integral type. We refer to the discontinuous integral control (DIC) technique, already developed by the authors for the time-continuous case, where a nonlinear integral action is used for estimating the external disturbance and keeping small the discontinuous term in the control law and the subsequent chattering effect. A discretised version of the DIC algorithm, which is properly modified in order to take into account the time-derivative information on the controlled variable, is introduced and discussed. The so-obtained discrete-time controller shows to be very effective in reducing the chattering when the system parameters are known. Moreover, to cope with a specific case of unknown parameters, a new adaptive control structure is proposed and analysed. The local asymptotic stability of the resulting closed-loop system is proved in the case of constant external disturbances. Finally, simulation results on the velocity control of a DC motor end the paper. ( 1998 Elsevier Science Ltd. All rights reserved. 1. INTRODUCTION
While variable-structure control (VSC) was originally introduced in the continuous-time case, see Utkin (1977), a deeper investigation of the discrete case is necessary for a satisfactory implementation in computer-controlled systems. In this paper we introduce a discrete-time sliding mode algorithm. The definition of the discrete-time sliding mode is not uniform in the literature, see, for example, Drakunov and Utkin (1992) and Yu (1993). Here the expression ‘‘discrete sliding mode’’ will be used for referring to system trajectories forced to stay in a proper neighbourhood of a given sliding manifold. The framework of our considerations is the discontinuous integral control (DIC) technique, introduced in Nersisian and Zanasi (1993) and in Zanasi (1993) and further developed in Bonivento et al.
2. DISCRETE VS CONTROLLERS
Let us consider the following uncertain system: yR (t)"t(t)#u(t),
*Received 30 August 1996; revised 19 May 1997; received in final form 17 October 1997. A preliminary version of this paper was presented at the ‘‘13th IFAC World Congress’’, San Francisco, USA, 30 June—5 July 1996. This paper was recommended for publication in revised form by Associate Editor Hassan Khalil under the direction of Editor Tamer Bas,ar. Corresponding author Dr. R. Zanasi. Tel. #39 51 6443034; Fax #39 51 6443073; E-mail [email protected]. -DEIS, Department of Electronics, Computer and System Science University of Bologna, V.le Risorgimento 2, 40136 Bologna, Italy. ‡EDUE ITALIA S.p.A., Modena, Italy.
(1)
where y3R is the controlled output, u3R the input, and t(t) an external unknown disturbance. A significant class of MIMO uncertain systems can be reduced to the parallel of first-order systems like (1) by using a proper state space transformation, see Zanasi (1993). In the literature it is well-known that when the disturbance is bounded (Dt(t)D(* ), the 0 sliding mode surface y"0 is exactly reached and 355
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maintained by using the simple discontinuous control law u(t)"!k@ sgn y(t) with k@'* . The slid0 ing mode surface y"0 is reached ‘‘exactly’’ only in the continuous-time control case when the switching frequency is infinite. On the contrary, in the discrete-time control case when the control law u(t) can vary only at the sampling instants t"n¹ (¹ 4 4 is the sampling period) only the d-boundedness of the output can be obtained: DyD(d. Due to the presence of the switching term k@ sgn y(t) the output y is affected by an oscillation whose minimum amplitude is k@ ¹ and in any case is proportional to 4 k@. Since k@'* , when a large disturbance t(t) acts 0 on the system, the amplitude of the output oscillation is also large: the system is affected by a large amount of chattering. For the continuous-time case, in Nersisian and Zanasi (1993) and in Zanasi (1993), a control structure (DIC) was proposed which allows the reduction of k@ by introducing in the control law a non-linear integral term tI (t) which estimates the external disturbance t(t), as follows: yR (t)"t(t)#u(t), u(t)"!k sgn y(t)!tI (t),
(2)
tRI (t)"h sgn y(t).
y(n#1)"y(n)!k¹ sgn y(n) s
P
(n#1)¹
4
#
In fact, the interesting benefit of (2) is that k;k@. When the controlled system is in sliding mode we have yR "y"0 and ‘‘k sgn y"t!tI ’’: the ‘‘equivalent control’’ corresponding to the switching term k sgn y is equal, in each instant, to the estimation error e(t)"t(t)!tI . By integrating the term h sgn y, in each instant tI moves in the direction of reducing the error e(t), that is, it estimates the disturbance t(t). In Nersisian and Zanasi (1993) and in Zanasi (1993) it was shown that, if the derivative of the disturbance is bounded (DtQ D(* ), 1 with a proper choice of parameter h (h'* ) 1 control structure (2) ensures the sliding mode on y"0 even if parameter k is arbitrarily small (k'0). Our purpose is now to derive a similar control solution in the discrete-time case. First, we can simply start considering a discrete version of control law (2) by using the Euler formula, which results in the following overall controlled system: yR (t)"t(t)#u(n), u(n)"!k sgn y(n)!tI (n),
of the discrete control law (3) with respect to the continuous one, is that estimation tI (n) is now ‘‘discontinuous’’, that is, at each sampling time, estimation tI (n) varies up or down of the quantity h¹ . s When the disturbance is zero (or constant), the estimation tI (n) switches between the two values tI and tI #h¹ and produces on the controlled 4 system an effect similar to the one produced by the switching term k sgn y, that is, an output oscillation. If we want to track a disturbance t with a large derivative (* 'DtQ D), parameter h must be larger 1 than * (h'* ), and so the oscillation h¹ gener1 1 4 ated by the estimator will be large too. This oscillation is a major drawback to be avoided. In order to reduce this oscillation, generated as explained above by the simple estimation mechanism present in (3), we proposed a proper modification, see Bonivento et al. (1995). The basic properties of this new control structure will be now reconsidered and extended to the case when the system inertia is not known and to be estimated. By integrating the first equation in system (3) in the time interval [n¹ , (n#1)¹ ], we obtain s 4
(3)
tI (n)"tI (n!1)#h ¹ sgn y(n). s In (3), a mixed notation has been used: yR (t) and t(t) are continuous-time variables, while u(n), y(n) and tI (n) are variables which change values only at sampling instants t"n¹ . An important difference 4
n¹ 4
t(t) dt!tI (n) ¹ . s
(4)
If we denote by tM (n#1) the mean value of the external disturbance t (t) in the time interval [n¹ , (n#1) ¹ ] 4 4 1 tM (n#1)" ¹ 4
P
(n#1) ¹
n¹
4
t(t) dt,
4
equation (4) can be written as *y(n#1) #k sgn y(n)"tM (n#1)!tI (n), (5) ¹ s where *y(n#1)"y(n#1)!y(n). The term on the left-hand side of equation (5) has the physical meaning of mean disturbance estimation error in the sampling interval [n¹ , (n#1) ¹ ] and there4 s fore seems to be a suitable function to use in the discrete disturbance estimator in the place of function sgn y. With this modification the controlled system (3) becomes yR (t)"t(t)!k sgn y(n)!tI (n), tI (n)"tI (n!1)#k
C
D
*y(n) #k sgn y(n!1) e ¹ 4
(6)
where k is a new design parameter. The *y(n)/¹ e 4 is the discrete derivative of output y. When e(n)"*y(n)/¹ #k sgn y(n!1) is zero, the es4 timator tI (n) is constant and therefore no additional oscillations are introduced in the system.
Brief Papers 3. BASIC PROPERTIES OF THE PROPOSED CONTROL LAW
At the sampling instants, system (6) becomes y(n#1)"y(n)#¹ [!k sgn y(n) s #tM (n#1)!tI (n)],
C
(7)
D
*y(n) #k sgn y(n!1) ¹ s and by using the ‘‘disturbance estimation error’’ t3(n)"tI (n!1)#k e
*y(n) #k sgn y(n!1), e(n)" ¹ 4 system (7) can be rewritten as follows:
(8)
e(n#1)"tM (n#1)!tI (n), (9) tI (n)"tI (n!1)#k e(n) . e System (9) is now linear with respect to the new variables e(n) and tI (n), and tM (n) is the new system input. From (8) we obtain y(n#1)"y(n)!k ¹ sgn y(n)#e(n#1) ¹ , (10) 4 4 that is, the output dynamics is completely determined by the estimation error e(n). From (10) it is clear that in the vicinity of the sliding surface y"0, a discrete sliding mode starts only if the disturbance estimation error e(n) is smaller than the amplitude k of the switching term: De(n)D(k.
(11)
When the discrete sliding mode is reached, the output y is forced to stay within the bounded region DyD(2 k¹ . The equation which describes the 4 dynamics of the estimation error e(n) as a function of the external disturbance tM (n#1) can be obtained from (9) by direct elimination of the variable tI (n) e(n#1)#(k !1) e(n)"*tM (n#1), (12) e where *tM (n#1)"tM (n#1)!tM (n). From (12) it follows that e(z) z!1 " . (13) tM (z) z#k !1 e System (12) is stable if and only if the pole of the Z transfer function (13) belongs to the open unit disk, that is if D1!k D(1 % 0(k (2. It is evie e dent that the best choice for parameter k is k "1 e e because in this case the dynamics of system (12) is of deadbeat type. In the following, we will consider only the case k "1. e Let e(R) denote the steady-state value of the disturbance estimation error. In Bonivento et al. (1995) it has been shown that for system (12) the following results hold: (a) If disturbance t(t) is constant, the disturbance estimation error e(R) is zero
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t(t)"c"e(R)"0. In this case, tI (t) exactly estimates the disturbance. (b) If disturbance t(t) is a ramp, the estimation error e(R) is constant and proportional to the slope of the ramp: t(t)"r t"e (R)" 0 r ¹ . The controlled system (7) reaches the 0 4 discrete sliding mode y"0 only if k'Dr D ¹ . 0 4 (c) If disturbance t(t) is a sinusoid, the estimation error e(R) is a sinusoid: t(t)"sin ut" e(R)"A(u) sin (ut#u(u)), where A(u)"2(1!cos u ¹ )/(u ¹ ). Note 4 4 that when u ¹ P0 the estimation error 4 tends to zero (limu¹ P0 A(u)"0), that is the 4 estimation error is small when the sampling frequency is much higher than the disturbance frequency: u
K
K
tM (n)!tM (n!1) (* , t ¹ 4 the estimation error e(R) is also bounded: e(R)"* ¹ . When the sampling time t 4 ¹ P0, the estimation error e(n) tends to 4 zero. The above results show that in steady-state condition the estimation error e(n) is bounded (De(n)D(* ¹ ) when the discrete derivative of the t 4 external disturbance is bounded. So, if the chosen value of k satisfies k'* ¹ we are sure that in t 4 a finite time a discrete sliding mode will be present in the system according to (11). The resulting closed-loop structure (6), which can be rewritten as yR (t)"t(t)#u(n), u(n)"!k sgn y(n)!tI (n), tI (n)"tI (n!1)#e(n),
(14)
*y(n) e(n)" #k sgn y(n!1), ¹ s is very simple, easily implementable and very effective in rejecting external disturbances with bounded derivative. More or less it has the same control capabilities of the simple discretised controller (3) with the main advantage of a reduced chattering, especially when the disturbance is constant. 4. ESTIMATION OF THE SYSTEM ‘‘INERTIA’’
The control structures considered in the previous sections refer to the simple controlled system (1), that is an integrator. Let us now consider the following system JyR "t(t)#u(t) ,
(15)
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where J is a parameter (here called ‘‘inertia’’) which is supposed to be unknown, constant or slowly time-varying. This case is especially relevant for mechanical systems. Let J (n) denote the ‘‘estima% tion’’ of the inertia J at the sampling instant t"n¹ . Applying to system (15) the analogous 4 control law defined in (14) and using J instead of J, % the following discrete closed-loop structure is obtained: ¹ y(n#1)"y(n)# 4 [tM (n#1)#u(n)], J u(n)"!k sgn y(n)!tI (n), tI (n)"tI (n!1)#eN (n),
(16)
J (n!1) *y(n) #k sgn y(n!1) , eN (n)" e ¹ 4 where eN (n) plays the role of ‘‘disturbance estimation pseudo-error’’. The ‘‘real’’ disturbance estimation error e(n) is J *y(n) #k sgn y(n!1). (17) e(n)" ¹ 4 The ‘‘pseudo-error’’ eN (n) differs from the real one e(n) because it uses the inertia J (n!1), estimated % at the previous instant t"(n!1) ¹ , instead of 4 using the real inertia J. The errors e(n) and eN (n) are coupled by the equation eN (n)!k sgn y(n!1) J (n!1) " e . e(n)!k sgn y(n!1) J
ties of the controlled system, let us consider the following updating function J (n)"J (n!1)#a *eN (n) % %
where *eN (n)"eN (n)!eN (n!1), * sgn y(n!1)" sgn y(n!1)! sgn y(n!2) and a is a proper design parameter. In (21) the term [* sgn y(n!1)]/2 has the function to allow the updating of inertia J (n) only when the controlled system is in the % neighbourhood of the sliding mode surface y"0, that is when y(n) changes the sign: sgn y(n!2)" !sgn y(n!1). The following result holds. ¹heorem 1. Let us consider the controlled system (16), (21) when the external disturbance t(t) and the unknown inertia J are constant. The overall controlled system is asymptotically stable in a neighbourhood of the equilibrium point e(n)"0 and J (n)"J if and only if the positive parameter % a satisfies the inequality J a( . 3k
C
J (n!1) J (n!1) *eN (n)" % e(n)#k 1! % J J
D
C
D
J (n!1) "*tM (n#1)!k 1! % sgn y(n!1). J (19) Note that when J "J, equation (19) reduces to % equation (12). When J is constant with J OJ and % % the disturbance has bounded time-derivative, system (19) is linear with bounded input, so its stability is equivalent to the condition
K
K
J %!1 (1"J (2J. % J
(20)
Inequality (20) suggests that, when J is not known, for the stability of the controlled system it is better to ‘‘underestimate’’ the initial value of J . Due to % the presence of the last term on the right-hand side of equation (19), the estimation error e(n) is not null even when the disturbance tM (n#1) is constant (*tM (n#1)"0). To ensure in (16) both asymptotic estimation of the real value J and stability proper-
D
J (n!2) ]sgn y(n!1)! % e(n!1) J
If J PJ then eN (n)Pe(n). From (16) and (17) one % obtains the dynamics of the estimation error e(n):
C
(22)
Proof. From (18) it follows that
(18)
J (n!1) e(n#1)# e !1 e(n) J
* sgn y(n!1) , (21) 2
C
D
J (n!2) sgn y(n!2). !k 1! % J By using the sliding mode condition sgn y(n!2) sgn y(n!1)"!1, the updating function (21) transforms as follows: J (n)"J (n!1)#a*eN (n) sgn y(n!1) % %
C
J (n!1)#J (n!2) e "J (n!1)#2ka 1! % % 2J
C
#a
D
D
J (n!1) J (n!2) % e(n)! % e(n!1) J J
]sgn y(n!1) .
(23)
If we introduce the ‘‘inertia estimation error’’ J (n)"J (n)!J as new variable substituting J (n), z % % equations (19) and (23) are transformed as follows: J (n!1) e(n#1)"! z e(n)#*tM (n#1) J J (n!1) sgn y(n!1), #k z J
Brief Papers k J (n)"J (n!1)!a [J (n!1)#J (n!2)] z z z J z
C C
D D
J (n!1) e(n) sgn y(n!1) #a 1# z J J (n!2) e(n!1) sgn y(n!1). !a 1# z J (24)
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a "a(1#x )[b sgn y(n!1)!x ] 23 2 1 ]sgn y(n)!ba!ax sgn y(n). 1 When the system is in sliding mode, that is when sgn y(n!1)"!sgn y(n), the Jacobian matrix A(x(n), n) in the ‘‘zero’’ equilibrium point x(n)"0 is 0
0
!b sgn y(n)
Let us consider the case when the external disturbance t(t) is constant: *tM (n#1)"0. If we introduce the state space variables
A(0, n)" !a sgn y(n) (1!ba)
!2ab
0
0
e(n) x (n)" , 1 J
Since this linearised system is a periodically timevarying system with period n"2, it is well known, see, for example, Bittanti (1986), that its asymptotic stability is equivalent to the condition that the eigenvalues of related transition matrix
J (n) x (n)" z , 2 J
J (n!1) x (n)" z 3 J
and the auxiliary constant b"k/J, system (24) becomes x (n#1)"x [b sgn y(n!1)!x ], 1 3 1 x (n#1)"aM(1#x ) x [b sgn y(n!1)!x ] 2 2 3 1 !(1#x ) x N sgn y(n) 3 1 #(1!ba) x !bax , 2 3 x (n#1)"x . (25) 3 2 For the sake of brevity, on the right-hand side of equations (25), the state variables x (n), x (n) and 1 2 x (n) have been denoted as x , x and x , respec3 1 2 3 tively. System (25) can also be represented in the compact form x(n#1)"F(x(n), n), where x(n)"[x (n) x (n) x (n)]T and F(x(n), n) is a 1 2 3 proper nonlinear function of the state x(n), depending also on the time-varying ‘‘parameter’’ sgn y(n). One can directly verify that x "0, x "0 and 1 2 x "0 is an equilibrium point for system (25). 3 Equivalently, e(n)"0 and J (n)"0 is an equilib% rium point for system (24). Since in the equilibrium point we are in sliding mode condition, the timevarying parameter sgn y(n) is periodic with period n"2 and then sgn y(n!1) sgn y(n)"!1 and sgn 2y(n)"1. The stability of the discrete nonlinear system (25) in a neighbourhood of the ‘‘zero’’ equilibrium point can be proved by using the first Lyapunov theorem. The Jacobian matrix of the system is a 0 a 11 13 LF(x(n), n) A(x(n), n)" " a a a 21 22 23 Lx(n) 0 1 0 where a "!x 11 3 a "b sgn y(n!1)!x 13 1 a "!a[(1#x ) x #(1#x )] sgn y(n), 21 2 3 3 a "(1!ba)#ax [b sgn y(n!1)!x ] sgn y(n), 22 3 1
1
.
A(0, n#1) A(0, n)" 0
b sgn y(n)
0
a(ab!1) sgn y(n) 1!4ba#b2a2 !3ab#2a2b2 !a sgn y(n)
1!ab
!2ab
belong to the open unit disc in the complex plane. The characteristic equation of matrix A(0, n#1) A(0, n) is the following: z3#(!1#6c!c2) z2#2c(1#c) z!c2"0, (26) where c"ab. The coefficients of equation (26) are nonlinear functions of c. The root locus of equation (26) numerically computed when c varies from 0 to 1 is reported in Fig. 1. The small circles here shown denote the position of the three roots when c"1/3. By using the Jury’s test, one can easily find that the three roots of the characteristic equation (26) belong to the open unit disk for 0(c(1 , that is 3 when 0(a(J/(3k). Correspondingly, for the first Lyapunov theorem the nonlinear discrete system (25) is asymptotically stable in the equilibrium point x"0. Theorem 1 is proved. K Remark 1. When disturbance t(t) and inertia J are constant and unknown, inequality (22) is a necessary and sufficient condition for the asymptotic estimation of the real values: tI (n)Pt and J (n)PJ. Since inertia J is supposed to be un% known, in the synthesis of the parameter a we have to use a lower-bound J of the inertia J, that is .*/ a(J /(3k). .*/ Remark 2. Note that relation (17) can also be rewritten as follows ¹ y(n)"y(n!1)# s [!k sgn y(n!1)#e(n)]. J
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Fig. 1. The global behavior of the root locus when c"ba3[0,1].
The dynamics of the controlled variable y(n) is completely determined by the ‘‘real’’ disturbance estimation error e(n). As for the time-invariant case J"1, also in this case it is clear that a discrete sliding mode starts in the system when De(n)D(k. In Theorem 1 it is shown that for constant disturbances both the ‘‘real’’ the ‘‘estimation’’ errors e(n) and eN (n) tend to zero after a transient. So, we have proved that in a finite time a discrete sliding mode starts in the system. Remark 3. Note that Theorem 1 could also be extended: when the disturbance t(t), instead of being constant, is a ramp (*tM "r ), it can be 0 argued that the new equilibrium point e(n)"r , 0 J (n)"0 is asymptotically stable if a is ‘‘sufficiently z small’’.
Fig. 2. Block scheme used in simulation.
5. SIMULATION RESULTS
Let us consider the block scheme of Fig. 2 representing the velocity control of a DC motor where J is the inertia of the motor, b the linear friction, u(t) the input torque, t (t) an external disturbance, r(n) q the reference signal, y(n)"r(n)!u(n) the tracking error and H (s) the zero-order hold. The electrical 0 part of the motor is supposed to be much faster than the mechanical one and for this reason it has been disregarded. The moment of inertia J of the motor is supposed to be unknown. The aim of the controller is to keep the tracking error y(n) as small as possible even when the inertia J of the motor is time-varying. In Fig. 3 are reported the simulation results obtained when inertia J varies as follows: J(t)"0.04#0.01 sgn [sin (15 t)]
Fig. 3. Simulation results when r(n)"10, t(t)"!7.5 and J is time-varying.
and when r(n)"10 and t (t)"!7.5. Note that q the total external disturbance t(t) acting on the system is now constituted by the contributions of
Brief Papers
361 6. CONCLUSIONS
The discontinuous integral control technique has been discretised and suitable modifications have been proposed and analysed. The first introduced modification improves the discrete-time estimation of the external disturbance t(t). The second modification allows also the estimation of the unknown ‘‘inertia’’ J of the system. In the latter adaptive case, the local asymptotic stability of the proposed control structure has been proved. The obtained simulations support the theoretical results and show good parameter convergence also when inertia J is time-varying. Fig. 4. Simulation results obtained when J, r(n) and t (t) are time-varying.
both the reference signal r(n) and the load disturbance t (t). In the upper part of Fig. 3 the conq trolled variable u(n) is reported together with the reference signal r(n), while the middle and low parts of the figure show the estimated disturbance tI (n) and the estimated inertia J (n), respectively. The used % parameters are ¹ "0.005 s, k"0.8, a"0.005, 4 b"0.002 and J (0)"0.001. The estimation tI (n) % and the input u(n) have been saturated to the maximum available torque [!15, 15]. Note that in steady-state conditions the estimates tI (n) and J (n) % tend asymptotically to the real values t(t) e J, despite of the fact that a small oscillation is present on the controlled variable u(t). In Fig. 4 are reported the results obtained in the same conditions of the previous simulation when the reference signal r(n) and the disturbance t (t) are time-varying: q r(n)"10 cos(10 n¹ ), t (t)"!7 sin (8 t). 4 q The obtained simulation results show the good performance of the proposed control structure even when the disturbance is time-varying.
REFERENCES Bittanti, S. (1986). Deterministic and stochastic linear periodic systems. ¹ime Series and ¸inear Systems. Springer, Berlin, pp. 141—182. Bonivento, C., A. Nersisian, A. Tonielli and R. Zanasi (1994). A cascade structure for robust control design. IEEE ¹rans. Autom. Control, 39, 846—849. Bonivento, C. and R. Zanasi (1994). Discontinuous integral control. Proc. of »S¸¹’94, Benevento, Italy, pp. 86—92. Bonivento, C., M. Sandri and R. Zanasi (1995). Discrete LowChattering Variable Structure Controllers. Proc. ECC’95, Roma, Italy, pp. 1455—1460. Bonivento, C. and R. Zanasi (1996). Advances in variable structure control. In Colloquium on Automatic Control, C. Bonivento, G. Marro and R. Zanasi eds, LNCIS series, Vol. 215. Springer, Berlin, pp. 177—226. Bonivento, C., M. Sandri and R. Zanasi (1996). Discrete variable structure integral controllers. Proc. 13th IFAC ¼orld Congress, San Francisco, U.S.A., Vol. H, pp. 333—338. Drakunov, S. V. and V. I. Utkin (1992). Sliding mode control in dynamic systems. Int. J. Control, 55, 1029—1037. Fridman, L. and A. Levant (1994). Higher order sliding modes as a natural phenomenon in control theory. Proc. »S¸¹’94, Benevento, Italy, pp. 302—309. Nersisian, A. and R. Zanasi (1993). A modified variable structure control algorithm for stabilization of uncertain dynamical systems. Int. J. Robust Nonlinear Control, 3, 199—209. Utkin, V. I. (1977). Variable structure systems with sliding modes. IEEE ¹rans. Autom. Control, 22, 212—222. Zanasi, R. (1993). Sliding mode using discontinuous control algorithms of integral type. Int. J. Control (Special Issue on Sliding Mode Control), 57, 1079—1099. Yu, X. H. (1993). Discrete variable structure control system. Int. J. Systems Sci. 24, 373—386.
Automatica, Vol. 34, No. 3, pp. 355—361, 1998 ( 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0005-1098/98 $19.00#0.00
Brief Paper
Discrete Variable-Structure Integral Controllers* C. BONIVENTO,- M. SANDRI‡ and R. ZANASIKey Words—Robust control; variable-structure systems; discontinuous control; integral control; discretetime sytems.
(1994) and in Bonivento and Zanasi (1994), which uses a nonlinear integral action for estimating the external disturbance and keeping small the discontinuous term in the control law and the subsequent chattering effect. The original DIC design was given for the continuous-time case. In this paper, starting from some previous partial results presented in Bonivento and Zanasi (1996) and in Bonivento et al. (1996), a discretised version of the DIC algorithm is properly modified in order to take into account the time-derivative of the controlled variable. The so-obtained discrete-time controller shows to be very effective in the disturbance estimation and in the chattering reduction especially when the system parameters are known and constant. If this is not the case, a new ‘‘adaptive’’ control structure which also estimates the ‘‘inertia’’ parameter of the system is then proposed. This new control structure is analysed and its asymptotic stability is proved in the case of constant disturbances. The paper is organised as follows. In Section 2 a brief review of the DIC technique is presented together with a first modified discrete version of the DIC control algorithm. In Section 3, the basic properties of this discrete-time controller are summarised. In Section 4, an adaeptive version of the proposed control law is introduced and analysed. Finally, in Section 5, some simulation results on the velocity control of a DC motor confirm the effectiveness of the proposed control structure.
Abstract—The subject of this paper is the analysis and the design of a class of robust discrete-time variable-structure controllers of integral type. We refer to the discontinuous integral control (DIC) technique, already developed by the authors for the time-continuous case, where a nonlinear integral action is used for estimating the external disturbance and keeping small the discontinuous term in the control law and the subsequent chattering effect. A discretised version of the DIC algorithm, which is properly modified in order to take into account the time-derivative information on the controlled variable, is introduced and discussed. The so-obtained discrete-time controller shows to be very effective in reducing the chattering when the system parameters are known. Moreover, to cope with a specific case of unknown parameters, a new adaptive control structure is proposed and analysed. The local asymptotic stability of the resulting closed-loop system is proved in the case of constant external disturbances. Finally, simulation results on the velocity control of a DC motor end the paper. ( 1998 Elsevier Science Ltd. All rights reserved. 1. INTRODUCTION
While variable-structure control (VSC) was originally introduced in the continuous-time case, see Utkin (1977), a deeper investigation of the discrete case is necessary for a satisfactory implementation in computer-controlled systems. In this paper we introduce a discrete-time sliding mode algorithm. The definition of the discrete-time sliding mode is not uniform in the literature, see, for example, Drakunov and Utkin (1992) and Yu (1993). Here the expression ‘‘discrete sliding mode’’ will be used for referring to system trajectories forced to stay in a proper neighbourhood of a given sliding manifold. The framework of our considerations is the discontinuous integral control (DIC) technique, introduced in Nersisian and Zanasi (1993) and in Zanasi (1993) and further developed in Bonivento et al.
2. DISCRETE VS CONTROLLERS
Let us consider the following uncertain system: yR (t)"t(t)#u(t),
*Received 30 August 1996; revised 19 May 1997; received in final form 17 October 1997. A preliminary version of this paper was presented at the ‘‘13th IFAC World Congress’’, San Francisco, USA, 30 June—5 July 1996. This paper was recommended for publication in revised form by Associate Editor Hassan Khalil under the direction of Editor Tamer Bas,ar. Corresponding author Dr. R. Zanasi. Tel. #39 51 6443034; Fax #39 51 6443073; E-mail [email protected]. -DEIS, Department of Electronics, Computer and System Science University of Bologna, V.le Risorgimento 2, 40136 Bologna, Italy. ‡EDUE ITALIA S.p.A., Modena, Italy.
(1)
where y3R is the controlled output, u3R the input, and t(t) an external unknown disturbance. A significant class of MIMO uncertain systems can be reduced to the parallel of first-order systems like (1) by using a proper state space transformation, see Zanasi (1993). In the literature it is well-known that when the disturbance is bounded (Dt(t)D(* ), the 0 sliding mode surface y"0 is exactly reached and 355
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maintained by using the simple discontinuous control law u(t)"!k@ sgn y(t) with k@'* . The slid0 ing mode surface y"0 is reached ‘‘exactly’’ only in the continuous-time control case when the switching frequency is infinite. On the contrary, in the discrete-time control case when the control law u(t) can vary only at the sampling instants t"n¹ (¹ 4 4 is the sampling period) only the d-boundedness of the output can be obtained: DyD(d. Due to the presence of the switching term k@ sgn y(t) the output y is affected by an oscillation whose minimum amplitude is k@ ¹ and in any case is proportional to 4 k@. Since k@'* , when a large disturbance t(t) acts 0 on the system, the amplitude of the output oscillation is also large: the system is affected by a large amount of chattering. For the continuous-time case, in Nersisian and Zanasi (1993) and in Zanasi (1993), a control structure (DIC) was proposed which allows the reduction of k@ by introducing in the control law a non-linear integral term tI (t) which estimates the external disturbance t(t), as follows: yR (t)"t(t)#u(t), u(t)"!k sgn y(t)!tI (t),
(2)
tRI (t)"h sgn y(t).
y(n#1)"y(n)!k¹ sgn y(n) s
P
(n#1)¹
4
#
In fact, the interesting benefit of (2) is that k;k@. When the controlled system is in sliding mode we have yR "y"0 and ‘‘k sgn y"t!tI ’’: the ‘‘equivalent control’’ corresponding to the switching term k sgn y is equal, in each instant, to the estimation error e(t)"t(t)!tI . By integrating the term h sgn y, in each instant tI moves in the direction of reducing the error e(t), that is, it estimates the disturbance t(t). In Nersisian and Zanasi (1993) and in Zanasi (1993) it was shown that, if the derivative of the disturbance is bounded (DtQ D(* ), 1 with a proper choice of parameter h (h'* ) 1 control structure (2) ensures the sliding mode on y"0 even if parameter k is arbitrarily small (k'0). Our purpose is now to derive a similar control solution in the discrete-time case. First, we can simply start considering a discrete version of control law (2) by using the Euler formula, which results in the following overall controlled system: yR (t)"t(t)#u(n), u(n)"!k sgn y(n)!tI (n),
of the discrete control law (3) with respect to the continuous one, is that estimation tI (n) is now ‘‘discontinuous’’, that is, at each sampling time, estimation tI (n) varies up or down of the quantity h¹ . s When the disturbance is zero (or constant), the estimation tI (n) switches between the two values tI and tI #h¹ and produces on the controlled 4 system an effect similar to the one produced by the switching term k sgn y, that is, an output oscillation. If we want to track a disturbance t with a large derivative (* 'DtQ D), parameter h must be larger 1 than * (h'* ), and so the oscillation h¹ gener1 1 4 ated by the estimator will be large too. This oscillation is a major drawback to be avoided. In order to reduce this oscillation, generated as explained above by the simple estimation mechanism present in (3), we proposed a proper modification, see Bonivento et al. (1995). The basic properties of this new control structure will be now reconsidered and extended to the case when the system inertia is not known and to be estimated. By integrating the first equation in system (3) in the time interval [n¹ , (n#1)¹ ], we obtain s 4
(3)
tI (n)"tI (n!1)#h ¹ sgn y(n). s In (3), a mixed notation has been used: yR (t) and t(t) are continuous-time variables, while u(n), y(n) and tI (n) are variables which change values only at sampling instants t"n¹ . An important difference 4
n¹ 4
t(t) dt!tI (n) ¹ . s
(4)
If we denote by tM (n#1) the mean value of the external disturbance t (t) in the time interval [n¹ , (n#1) ¹ ] 4 4 1 tM (n#1)" ¹ 4
P
(n#1) ¹
n¹
4
t(t) dt,
4
equation (4) can be written as *y(n#1) #k sgn y(n)"tM (n#1)!tI (n), (5) ¹ s where *y(n#1)"y(n#1)!y(n). The term on the left-hand side of equation (5) has the physical meaning of mean disturbance estimation error in the sampling interval [n¹ , (n#1) ¹ ] and there4 s fore seems to be a suitable function to use in the discrete disturbance estimator in the place of function sgn y. With this modification the controlled system (3) becomes yR (t)"t(t)!k sgn y(n)!tI (n), tI (n)"tI (n!1)#k
C
D
*y(n) #k sgn y(n!1) e ¹ 4
(6)
where k is a new design parameter. The *y(n)/¹ e 4 is the discrete derivative of output y. When e(n)"*y(n)/¹ #k sgn y(n!1) is zero, the es4 timator tI (n) is constant and therefore no additional oscillations are introduced in the system.
Brief Papers 3. BASIC PROPERTIES OF THE PROPOSED CONTROL LAW
At the sampling instants, system (6) becomes y(n#1)"y(n)#¹ [!k sgn y(n) s #tM (n#1)!tI (n)],
C
(7)
D
*y(n) #k sgn y(n!1) ¹ s and by using the ‘‘disturbance estimation error’’ t3(n)"tI (n!1)#k e
*y(n) #k sgn y(n!1), e(n)" ¹ 4 system (7) can be rewritten as follows:
(8)
e(n#1)"tM (n#1)!tI (n), (9) tI (n)"tI (n!1)#k e(n) . e System (9) is now linear with respect to the new variables e(n) and tI (n), and tM (n) is the new system input. From (8) we obtain y(n#1)"y(n)!k ¹ sgn y(n)#e(n#1) ¹ , (10) 4 4 that is, the output dynamics is completely determined by the estimation error e(n). From (10) it is clear that in the vicinity of the sliding surface y"0, a discrete sliding mode starts only if the disturbance estimation error e(n) is smaller than the amplitude k of the switching term: De(n)D(k.
(11)
When the discrete sliding mode is reached, the output y is forced to stay within the bounded region DyD(2 k¹ . The equation which describes the 4 dynamics of the estimation error e(n) as a function of the external disturbance tM (n#1) can be obtained from (9) by direct elimination of the variable tI (n) e(n#1)#(k !1) e(n)"*tM (n#1), (12) e where *tM (n#1)"tM (n#1)!tM (n). From (12) it follows that e(z) z!1 " . (13) tM (z) z#k !1 e System (12) is stable if and only if the pole of the Z transfer function (13) belongs to the open unit disk, that is if D1!k D(1 % 0(k (2. It is evie e dent that the best choice for parameter k is k "1 e e because in this case the dynamics of system (12) is of deadbeat type. In the following, we will consider only the case k "1. e Let e(R) denote the steady-state value of the disturbance estimation error. In Bonivento et al. (1995) it has been shown that for system (12) the following results hold: (a) If disturbance t(t) is constant, the disturbance estimation error e(R) is zero
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t(t)"c"e(R)"0. In this case, tI (t) exactly estimates the disturbance. (b) If disturbance t(t) is a ramp, the estimation error e(R) is constant and proportional to the slope of the ramp: t(t)"r t"e (R)" 0 r ¹ . The controlled system (7) reaches the 0 4 discrete sliding mode y"0 only if k'Dr D ¹ . 0 4 (c) If disturbance t(t) is a sinusoid, the estimation error e(R) is a sinusoid: t(t)"sin ut" e(R)"A(u) sin (ut#u(u)), where A(u)"2(1!cos u ¹ )/(u ¹ ). Note 4 4 that when u ¹ P0 the estimation error 4 tends to zero (limu¹ P0 A(u)"0), that is the 4 estimation error is small when the sampling frequency is much higher than the disturbance frequency: u
K
K
tM (n)!tM (n!1) (* , t ¹ 4 the estimation error e(R) is also bounded: e(R)"* ¹ . When the sampling time t 4 ¹ P0, the estimation error e(n) tends to 4 zero. The above results show that in steady-state condition the estimation error e(n) is bounded (De(n)D(* ¹ ) when the discrete derivative of the t 4 external disturbance is bounded. So, if the chosen value of k satisfies k'* ¹ we are sure that in t 4 a finite time a discrete sliding mode will be present in the system according to (11). The resulting closed-loop structure (6), which can be rewritten as yR (t)"t(t)#u(n), u(n)"!k sgn y(n)!tI (n), tI (n)"tI (n!1)#e(n),
(14)
*y(n) e(n)" #k sgn y(n!1), ¹ s is very simple, easily implementable and very effective in rejecting external disturbances with bounded derivative. More or less it has the same control capabilities of the simple discretised controller (3) with the main advantage of a reduced chattering, especially when the disturbance is constant. 4. ESTIMATION OF THE SYSTEM ‘‘INERTIA’’
The control structures considered in the previous sections refer to the simple controlled system (1), that is an integrator. Let us now consider the following system JyR "t(t)#u(t) ,
(15)
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where J is a parameter (here called ‘‘inertia’’) which is supposed to be unknown, constant or slowly time-varying. This case is especially relevant for mechanical systems. Let J (n) denote the ‘‘estima% tion’’ of the inertia J at the sampling instant t"n¹ . Applying to system (15) the analogous 4 control law defined in (14) and using J instead of J, % the following discrete closed-loop structure is obtained: ¹ y(n#1)"y(n)# 4 [tM (n#1)#u(n)], J u(n)"!k sgn y(n)!tI (n), tI (n)"tI (n!1)#eN (n),
(16)
J (n!1) *y(n) #k sgn y(n!1) , eN (n)" e ¹ 4 where eN (n) plays the role of ‘‘disturbance estimation pseudo-error’’. The ‘‘real’’ disturbance estimation error e(n) is J *y(n) #k sgn y(n!1). (17) e(n)" ¹ 4 The ‘‘pseudo-error’’ eN (n) differs from the real one e(n) because it uses the inertia J (n!1), estimated % at the previous instant t"(n!1) ¹ , instead of 4 using the real inertia J. The errors e(n) and eN (n) are coupled by the equation eN (n)!k sgn y(n!1) J (n!1) " e . e(n)!k sgn y(n!1) J
ties of the controlled system, let us consider the following updating function J (n)"J (n!1)#a *eN (n) % %
where *eN (n)"eN (n)!eN (n!1), * sgn y(n!1)" sgn y(n!1)! sgn y(n!2) and a is a proper design parameter. In (21) the term [* sgn y(n!1)]/2 has the function to allow the updating of inertia J (n) only when the controlled system is in the % neighbourhood of the sliding mode surface y"0, that is when y(n) changes the sign: sgn y(n!2)" !sgn y(n!1). The following result holds. ¹heorem 1. Let us consider the controlled system (16), (21) when the external disturbance t(t) and the unknown inertia J are constant. The overall controlled system is asymptotically stable in a neighbourhood of the equilibrium point e(n)"0 and J (n)"J if and only if the positive parameter % a satisfies the inequality J a( . 3k
C
J (n!1) J (n!1) *eN (n)" % e(n)#k 1! % J J
D
C
D
J (n!1) "*tM (n#1)!k 1! % sgn y(n!1). J (19) Note that when J "J, equation (19) reduces to % equation (12). When J is constant with J OJ and % % the disturbance has bounded time-derivative, system (19) is linear with bounded input, so its stability is equivalent to the condition
K
K
J %!1 (1"J (2J. % J
(20)
Inequality (20) suggests that, when J is not known, for the stability of the controlled system it is better to ‘‘underestimate’’ the initial value of J . Due to % the presence of the last term on the right-hand side of equation (19), the estimation error e(n) is not null even when the disturbance tM (n#1) is constant (*tM (n#1)"0). To ensure in (16) both asymptotic estimation of the real value J and stability proper-
D
J (n!2) ]sgn y(n!1)! % e(n!1) J
If J PJ then eN (n)Pe(n). From (16) and (17) one % obtains the dynamics of the estimation error e(n):
C
(22)
Proof. From (18) it follows that
(18)
J (n!1) e(n#1)# e !1 e(n) J
* sgn y(n!1) , (21) 2
C
D
J (n!2) sgn y(n!2). !k 1! % J By using the sliding mode condition sgn y(n!2) sgn y(n!1)"!1, the updating function (21) transforms as follows: J (n)"J (n!1)#a*eN (n) sgn y(n!1) % %
C
J (n!1)#J (n!2) e "J (n!1)#2ka 1! % % 2J
C
#a
D
D
J (n!1) J (n!2) % e(n)! % e(n!1) J J
]sgn y(n!1) .
(23)
If we introduce the ‘‘inertia estimation error’’ J (n)"J (n)!J as new variable substituting J (n), z % % equations (19) and (23) are transformed as follows: J (n!1) e(n#1)"! z e(n)#*tM (n#1) J J (n!1) sgn y(n!1), #k z J
Brief Papers k J (n)"J (n!1)!a [J (n!1)#J (n!2)] z z z J z
C C
D D
J (n!1) e(n) sgn y(n!1) #a 1# z J J (n!2) e(n!1) sgn y(n!1). !a 1# z J (24)
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a "a(1#x )[b sgn y(n!1)!x ] 23 2 1 ]sgn y(n)!ba!ax sgn y(n). 1 When the system is in sliding mode, that is when sgn y(n!1)"!sgn y(n), the Jacobian matrix A(x(n), n) in the ‘‘zero’’ equilibrium point x(n)"0 is 0
0
!b sgn y(n)
Let us consider the case when the external disturbance t(t) is constant: *tM (n#1)"0. If we introduce the state space variables
A(0, n)" !a sgn y(n) (1!ba)
!2ab
0
0
e(n) x (n)" , 1 J
Since this linearised system is a periodically timevarying system with period n"2, it is well known, see, for example, Bittanti (1986), that its asymptotic stability is equivalent to the condition that the eigenvalues of related transition matrix
J (n) x (n)" z , 2 J
J (n!1) x (n)" z 3 J
and the auxiliary constant b"k/J, system (24) becomes x (n#1)"x [b sgn y(n!1)!x ], 1 3 1 x (n#1)"aM(1#x ) x [b sgn y(n!1)!x ] 2 2 3 1 !(1#x ) x N sgn y(n) 3 1 #(1!ba) x !bax , 2 3 x (n#1)"x . (25) 3 2 For the sake of brevity, on the right-hand side of equations (25), the state variables x (n), x (n) and 1 2 x (n) have been denoted as x , x and x , respec3 1 2 3 tively. System (25) can also be represented in the compact form x(n#1)"F(x(n), n), where x(n)"[x (n) x (n) x (n)]T and F(x(n), n) is a 1 2 3 proper nonlinear function of the state x(n), depending also on the time-varying ‘‘parameter’’ sgn y(n). One can directly verify that x "0, x "0 and 1 2 x "0 is an equilibrium point for system (25). 3 Equivalently, e(n)"0 and J (n)"0 is an equilib% rium point for system (24). Since in the equilibrium point we are in sliding mode condition, the timevarying parameter sgn y(n) is periodic with period n"2 and then sgn y(n!1) sgn y(n)"!1 and sgn 2y(n)"1. The stability of the discrete nonlinear system (25) in a neighbourhood of the ‘‘zero’’ equilibrium point can be proved by using the first Lyapunov theorem. The Jacobian matrix of the system is a 0 a 11 13 LF(x(n), n) A(x(n), n)" " a a a 21 22 23 Lx(n) 0 1 0 where a "!x 11 3 a "b sgn y(n!1)!x 13 1 a "!a[(1#x ) x #(1#x )] sgn y(n), 21 2 3 3 a "(1!ba)#ax [b sgn y(n!1)!x ] sgn y(n), 22 3 1
1
.
A(0, n#1) A(0, n)" 0
b sgn y(n)
0
a(ab!1) sgn y(n) 1!4ba#b2a2 !3ab#2a2b2 !a sgn y(n)
1!ab
!2ab
belong to the open unit disc in the complex plane. The characteristic equation of matrix A(0, n#1) A(0, n) is the following: z3#(!1#6c!c2) z2#2c(1#c) z!c2"0, (26) where c"ab. The coefficients of equation (26) are nonlinear functions of c. The root locus of equation (26) numerically computed when c varies from 0 to 1 is reported in Fig. 1. The small circles here shown denote the position of the three roots when c"1/3. By using the Jury’s test, one can easily find that the three roots of the characteristic equation (26) belong to the open unit disk for 0(c(1 , that is 3 when 0(a(J/(3k). Correspondingly, for the first Lyapunov theorem the nonlinear discrete system (25) is asymptotically stable in the equilibrium point x"0. Theorem 1 is proved. K Remark 1. When disturbance t(t) and inertia J are constant and unknown, inequality (22) is a necessary and sufficient condition for the asymptotic estimation of the real values: tI (n)Pt and J (n)PJ. Since inertia J is supposed to be un% known, in the synthesis of the parameter a we have to use a lower-bound J of the inertia J, that is .*/ a(J /(3k). .*/ Remark 2. Note that relation (17) can also be rewritten as follows ¹ y(n)"y(n!1)# s [!k sgn y(n!1)#e(n)]. J
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Fig. 1. The global behavior of the root locus when c"ba3[0,1].
The dynamics of the controlled variable y(n) is completely determined by the ‘‘real’’ disturbance estimation error e(n). As for the time-invariant case J"1, also in this case it is clear that a discrete sliding mode starts in the system when De(n)D(k. In Theorem 1 it is shown that for constant disturbances both the ‘‘real’’ the ‘‘estimation’’ errors e(n) and eN (n) tend to zero after a transient. So, we have proved that in a finite time a discrete sliding mode starts in the system. Remark 3. Note that Theorem 1 could also be extended: when the disturbance t(t), instead of being constant, is a ramp (*tM "r ), it can be 0 argued that the new equilibrium point e(n)"r , 0 J (n)"0 is asymptotically stable if a is ‘‘sufficiently z small’’.
Fig. 2. Block scheme used in simulation.
5. SIMULATION RESULTS
Let us consider the block scheme of Fig. 2 representing the velocity control of a DC motor where J is the inertia of the motor, b the linear friction, u(t) the input torque, t (t) an external disturbance, r(n) q the reference signal, y(n)"r(n)!u(n) the tracking error and H (s) the zero-order hold. The electrical 0 part of the motor is supposed to be much faster than the mechanical one and for this reason it has been disregarded. The moment of inertia J of the motor is supposed to be unknown. The aim of the controller is to keep the tracking error y(n) as small as possible even when the inertia J of the motor is time-varying. In Fig. 3 are reported the simulation results obtained when inertia J varies as follows: J(t)"0.04#0.01 sgn [sin (15 t)]
Fig. 3. Simulation results when r(n)"10, t(t)"!7.5 and J is time-varying.
and when r(n)"10 and t (t)"!7.5. Note that q the total external disturbance t(t) acting on the system is now constituted by the contributions of
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361 6. CONCLUSIONS
The discontinuous integral control technique has been discretised and suitable modifications have been proposed and analysed. The first introduced modification improves the discrete-time estimation of the external disturbance t(t). The second modification allows also the estimation of the unknown ‘‘inertia’’ J of the system. In the latter adaptive case, the local asymptotic stability of the proposed control structure has been proved. The obtained simulations support the theoretical results and show good parameter convergence also when inertia J is time-varying. Fig. 4. Simulation results obtained when J, r(n) and t (t) are time-varying.
both the reference signal r(n) and the load disturbance t (t). In the upper part of Fig. 3 the conq trolled variable u(n) is reported together with the reference signal r(n), while the middle and low parts of the figure show the estimated disturbance tI (n) and the estimated inertia J (n), respectively. The used % parameters are ¹ "0.005 s, k"0.8, a"0.005, 4 b"0.002 and J (0)"0.001. The estimation tI (n) % and the input u(n) have been saturated to the maximum available torque [!15, 15]. Note that in steady-state conditions the estimates tI (n) and J (n) % tend asymptotically to the real values t(t) e J, despite of the fact that a small oscillation is present on the controlled variable u(t). In Fig. 4 are reported the results obtained in the same conditions of the previous simulation when the reference signal r(n) and the disturbance t (t) are time-varying: q r(n)"10 cos(10 n¹ ), t (t)"!7 sin (8 t). 4 q The obtained simulation results show the good performance of the proposed control structure even when the disturbance is time-varying.
REFERENCES Bittanti, S. (1986). Deterministic and stochastic linear periodic systems. ¹ime Series and ¸inear Systems. Springer, Berlin, pp. 141—182. Bonivento, C., A. Nersisian, A. Tonielli and R. Zanasi (1994). A cascade structure for robust control design. IEEE ¹rans. Autom. Control, 39, 846—849. Bonivento, C. and R. Zanasi (1994). Discontinuous integral control. Proc. of »S¸¹’94, Benevento, Italy, pp. 86—92. Bonivento, C., M. Sandri and R. Zanasi (1995). Discrete LowChattering Variable Structure Controllers. Proc. ECC’95, Roma, Italy, pp. 1455—1460. Bonivento, C. and R. Zanasi (1996). Advances in variable structure control. In Colloquium on Automatic Control, C. Bonivento, G. Marro and R. Zanasi eds, LNCIS series, Vol. 215. Springer, Berlin, pp. 177—226. Bonivento, C., M. Sandri and R. Zanasi (1996). Discrete variable structure integral controllers. Proc. 13th IFAC ¼orld Congress, San Francisco, U.S.A., Vol. H, pp. 333—338. Drakunov, S. V. and V. I. Utkin (1992). Sliding mode control in dynamic systems. Int. J. Control, 55, 1029—1037. Fridman, L. and A. Levant (1994). Higher order sliding modes as a natural phenomenon in control theory. Proc. »S¸¹’94, Benevento, Italy, pp. 302—309. Nersisian, A. and R. Zanasi (1993). A modified variable structure control algorithm for stabilization of uncertain dynamical systems. Int. J. Robust Nonlinear Control, 3, 199—209. Utkin, V. I. (1977). Variable structure systems with sliding modes. IEEE ¹rans. Autom. Control, 22, 212—222. Zanasi, R. (1993). Sliding mode using discontinuous control algorithms of integral type. Int. J. Control (Special Issue on Sliding Mode Control), 57, 1079—1099. Yu, X. H. (1993). Discrete variable structure control system. Int. J. Systems Sci. 24, 373—386.