Dual-Layer Adaptive Sliding Mode Control

Transcription

1 Dual-Layer Adaptive Sliding Mode Control Christopher Edwards 1, and Yuri Shtessel 2 Abstract This paper proposes new and novel equivalent control-based adaptive schemes for both conventional and super-twisting sliding mode control algorithms. The approach is based on a dual layer nested adaptive scheme which is quite different to the existing schemes proposed in the sliding mode literature. The new adaptive schemes do not require knowledge of the minimum and maximum allowable values of the adaptive gains, and more importantly do not require information about the upper-bound of the disturbances and their derivatives. The underlying idea can be applied to a variety of sliding mode control structures and relies more on the availability of an online approximation of the equivalent control, than the precise control law used to induce and maintain sliding. In this paper, these ideas are incorporated within classical relay structures, unit vector control schemes and super-twisting controllers. The simulations presented in the paper confirm the effectiveness and simplicity of the proposed scheme. I. INTRODUCTION Sliding mode control is an established and increasingly popular methodology for controlling uncertain systems. In addition to creating closed-loop insensitivity to a certain class of uncertainty so-called matched uncertainty sliding mode controllers provide finite time convergence guarantees [1]. To create these attractive properties, sliding mode schemes are invariably nonlinear and contain discontinuities at some location within the control structure, even if the overall control signal is continuous. Another feature is that the controllers typically contain, within their structure, bounds of some kind on the uncertainty. These bounds can take different forms. In the simplest case upper bounds are required on the magnitude of the uncertainty. This is usually the case for conventional sliding mode controllers [1], [2]. In other cases, a bound on the derivative of the uncertainty is required [3]. Another popular constraint involves the Lipschitz gain of some appropriate higher derivative [4]. Whilst in principle conservative (gross) upper-bounds can be employed to guarantee sliding takes place, in practise such an approach has the disadvantage of increasing the likelihood of so-called chattering. This has motivated the need for what broadly speaking could be termed adaptive sliding mode control, whereby (usually) the scalar gains representing upper-bounds on the uncertainty or its derivatives are allowed to varying to ensure the enforcement and maintenance of a sliding mode. There is a rich history of literature in this sub-branch of sliding mode control theory largely because initially it was considered as a mechanism to try to mitigate the occurrence of chattering which has been a long standing 1 College of Engineering, Mathematics and Physical Sciences, University of Exeter, Exeter UK EX4 4QF, UK; c.edwards@exeter.ac.uk 2 Department of Electrical Engineering, University of Alabama, Huntsville, Alabama, USA; shtessel@ece.uah.edu problem. Some of the key papers in this area are [5] [9] and a recent overview of this field is given in [10]. Such a standpoint has also been adopted within fault tolerant control frameworks [11] [13] where post-fault adaptation is required to maintain sliding. In the last few years the creation of new Lyapunov functions for previously well established sliding mode control structures particularly for 2nd order sliding controllers (2-SM) such as the so-called twisting and super-twisting controllers has renewed interest in this area [14]. The literature expanding these Lyapunov ideas into the realm of adaptive sliding mode control is developing rapidly [15] [20]. Whilst it is intuitively clear that when sliding begins to deteriorate the controller gains must be increased, devising an effective way of lowering unnecessarily large gains once sliding is achieved, has proved more difficult. This paper follows the approach of [21], [22] in the sense that the adaption scheme relies on the availability, in real time, of the equivalent control signal. The equivalent control signal may be viewed as the average value the discontinuous signal must take in order to preserve/maintain a sliding motion [23]. Although by inception the equivalent control signal was regarded as an abstraction to help analyze the reduced order motion associated with sliding, an arbitrarily close approximation can be obtained by low-pass filtering the high (ideally infinite) frequency switching which is obtained during sliding. The use of the equivalent control plays a vital role in the adaptive scheme proposed in [21] and is also central to this paper. The main contribution of this paper is to propose new and novel equivalent control-based adaptive schemes for both conventional and super-twisting sliding mode control algorithms. The approach is based on a dual layer nested adaptive scheme which is quite different to the existing schemes proposed in the sliding mode literature [10], [15]. The new adaptive schemes do not require knowledge of the minimum and maximum allowed values of the adaptive gain and also do not require information about the upperbound of the disturbances and their derivatives that are necessary in [21]. The underlying idea can be applied to a variety of sliding mode control structures and relies more on the availability of the equivalent control than the precise control-law used to induce and maintain sliding. In this is paper, it is incorporated within classical relay structures, unit vector control schemes and super-twisting controllers. The simulations presented in the paper confirm the effectiveness and simplicity of the proposed schemes.

2 II. A SINGLE INPUT SINGLE OUTPUT FORMULATION Consider initially the proto-typical first-order sliding mode equation σ(t) = a(t)+u(t) (1) where σ(t) R represents the switching function to be forced to zero in finite time,u(t) represents the scalar control input to be manipulated and a(t) is a disturbance. Here it will be assumed that a(t) is not known but its magnitude and its derivative ȧ(t) are bounded so that a(t) < a 0 and ȧ(t) < a 1 where both a 0 and a 1 are finite. Consider the control law u(t) = (k(t) + η) sign(σ(t)) (2) where η is a small positive design constant and k(t) is a varying scalar term which satisfies an adaptive scheme, which will be explicitly defined later in the paper. The objective is to select k(t) as small as possible to ensure sliding takes place in (1). Note that a sufficient condition to enforce a sliding motion in (1) is that k(t) > a(t) (3) in which case the so-called η-reachability condition is satisfied [1], [2]. During the sliding motion σ(t) 0 and the so-called equivalent control u eq (t), which is the value the switched signal must take on average [23] to maintain sliding, must satisfy u eq (t) = a(t) i.e it must take the value to cancel exactly the imprecisely unknown uncertainty/disturbance. Consequently during sliding u eq (t) = a(t). Although the equivalent control was conceived as an abstraction to allow the analysis of the reduced order sliding motion, an arbitrarily close approximation to the equivalent control can be obtained by low pass filtering of the switching signal k(t) sign(σ(t)) [23]. In this way if ū eq (t) satisfies ū eq (t) = 1 τ (k(t) sign(σ(t)) ū eq(t)) (4) where τ > 0 is the (small) time constant, ū eq (t) u eq (t) can be arbitrary small (at least in theory) for small enough τ. As in [21], [22] the equivalent control will be exploited when constructing the adaptive algorithm for k(t). Define δ(t) = k(t) 1 u eq(t) ǫ (5) where 0 < < 1 is a design scalar and ǫ is a small positive scalar. Note that if δ = 0 then k(t) > u eq (t) = a(t). As argued above the inequality k(t) > a(t) is precisely the condition needed to maintain sliding in equation (1). Note that provided it is accepted that a good approximation to u eq (t) is available in real time through the filtering operation in (4), then the quantity δ(t) is known and can be used as part of the (adaptive) control algorithm. The adaptive control element k(t) in (2) will now be described. Specifically define k(t) = ρ(t)sign(δ(t)) (6) where ρ(t) is a varying scalar which has a physical interpretation: namely it represents an upper-bound on the rate of change of the disturbance. In this paper it is assumed that ρ(t) has the structure ρ(t) = r 0 +r(t) (7) where r 0 is a fixed positive scalar and the evolution of r(t) will also satisfy a differential equation (i.e an adaptive law) that will be described shortly. This dual layer adaptation is the crux of this paper. Precise details regarding the adaptation law for r(t) proposed in this paper will depend on the assumptions which are made with respect to the knowledge about a 1 (the upper-bound on the magnitude of the derivative of a(t)). In particular two layers of adaptation occur. One layer is concerned with the magnitude of the switching signal in the control law k(t). The rate at which k(t) can change is depends on the time-varying parameter r(t) which itself adapts in a way to ensure r(t)+r 0 > ȧ(t). This is the second adaptive layer in the scheme. This second layer obviates the need to know a-priori the bound a 1. Two situations will now be considered: firstly the situation when a 1 is known, and secondly the situation when it is unknown. A. Formulation when the bound a 1 is known In this section it is assumed that a 0 is unknown (but bounded), but a 1 is available: i.e the worst case rate of change of the disturbance a(t) is known. Define e(t) = a 1 / r(t) (8) Since (in this subsection) it is assumed that a 1 is known then e(t) is known and can be exploited in the adaptation scheme. Define ṙ(t) = γ δ(t) +r 0 γ sign(e(t)) (9) where γ is positive design scalar and δ(t) is defined in (5). Note that to realize this adaptive scheme, a 1 must be known in order to evaluate e(t) in (8). The dynamical system associated with the variables δ(t) and e(t) in (5) and (8) will now be analyzed using the Lyapunov function V = 1 2 δ γ e2 (10) Taking the derivative with respect to time Arguing as in [21], from equation (5) V = δ δ + 1 eė (11) γ δ = k(t) 1 d dt u eq(t) = k(t) 1 d a(t) (12) dt

3 Therefore from (6) it follows that δ δ δ(t) k(t)+ δ(t) a 1 = ρ(t) δ(t) + δ(t) a 1 = r 0 δ(t) r(t) δ(t) + δ(t) a 1 = r 0 δ(t) +e(t) δ(t) (13) from the definition of e(t) in (8). Therefore V r 0 δ(t) + δ(t) e(t) 1 γ e(t)ṙ(t) Since = r 0 δ(t) + δ(t) e(t) δ(t) e(t) r 0 1 γ e(t) 1 = r 0 δ(t) r 0 γ (e(t) = ( ) 1 2r 0 2 δ(t) + 1 2γ (e(t) r 0 2V (14) ( δ + e ) ( ) δ 2 1/2 2 2γ 2 + e 2 (15) 2γ for all δ and e. Inequality (14) implies V = 0 in finite time and therefore both δ(t) and e(t) become zero in some finite time t 0 > 0. Consequently from the definition of δ in (5) k(t) = u eq (t) + (1 ) u eq (t) +ǫ = a(t) + (1 ) a(t) +ǫ > a(t) (16) for all time t > t 0. This is exactly the condition necessary to maintain sliding. Also note that since e(t) is bounded, from the definition of e(t) in (8), r(t) and hence ρ(t) remains bounded. B. Formulation when a 1 is unknown In this subsection it is assumed that both a 0 and a 1 are unknown (but bounded). Since a 1 is not known the variable e(t) is unknown and the adaptive scheme in (9) can not be used. Instead define ṙ(t) = γ δ(t) (17) The new adaptive scheme now comprises (6), (7) and (17). The Lyapunov function V = 1 2 δ γ e2 from (10) will be employed to analyze the error variables e(t) and δ(t) from (5) and (8). As before, from (13) it follows that δ δ r 0 δ(t) +e(t) δ(t) (18) Now from the definition of e(t) in equation (8) and of r(t) from (17), it follows that ė(t) = ṙ(t) = γ δ(t). Therefore V r 0 δ(t) + δ(t) e(t) 1 γ eṙ(t) = r 0 δ(t) + δ(t) e(t) δ(t) e(t) = r 0 δ(t) (19) Since V 0 it follows that both e(t) and δ(t) remain bounded. Furthermore from LaSalle s invariance principal [24], δ(t) 0 as t. Consequently there exists a finite time t 0 such that δ(t) ǫ/2 for all time t > t 0 (where ǫ is the user defined small scalar which is involved in the definition of δ in (5)). Consequently from the definition of δ(t) in (5), for all t > t 0 Thus and therefore δ(t) = k(t) 1 u eq(t) ǫ < ǫ/2 k(t) 1 u eq(t) ǫ > ǫ/2 k(t) > u eq (t) + (1 ) u eq (t) + ǫ 2 (20) > a(t) (21) for all time t > t 0. From (21) it follows that the condition for maintaining a sliding motion on σ = 0 is guaranteed. Remark In equality (20), the right hand side establishes a bounding cone around the equivalent control u eq (t) involving both multiplicative (1 ) and fixed ǫ 2 components. This introduces robustness into the adaptive scheme since the value of u eq (t) can only be estimated by ū eq (t) through the low-pass filtering process in (4). The amount of allowable uncertainty is a function of the parameters and ǫ which are to be selected by the designer (subject to > 1 and ǫ > 0). In this way the designer can introduce his/her own preferred level of safety into the algorithm by widening the cone through making smaller and ǫ bigger although, of course, this must be traded-off against how much larger k(t) becomes with respect to a(t). Note since both e(t) and δ(t) remain bounded, from (8), the adaptive gain r(t) satisfies r(t) < a 1 + e(t) and so r(t) and hence ρ(t) remains bounded. Also from the definition of δ(t) in (5) it follows k(t) < δ(t) + 1 u eq(t) +ǫ < δ(t) + a 0 +ǫ (22) and so k(t) remains bounded. Note that prior to sliding taking place, the adaptive scheme for k(t) as described above is also applicable. Suppose no sliding takes place in the interval [ 0 t 0 ]. During this time sign(σ(t)) = 1 almost everywhere and therefore formally u eq (t) = k(t) almost everywhere. Substituting this value in (5), it follows that δ(t) = ( 1) k(t) ǫ < 0 almost everywhere since 0 < < 1. Consequently sign(δ(t)) = 1 almost everywhere and from (6) it follows k(t) = ρ(t) > ρ 0

4 and therefore the gain k(t) continues to grow at the rate dictated by ρ(t) which is always greater than r 0. Increasing k(t) is precisely what is need to induce a sliding motion. Remark Note that whilst certain parallels exist between the adaptive schemes in (6), (9) and (17) and those proposed in [21], the precise details are quite different. In this paper, compared to [21], no knowledge about the upper-bound a 0 is required. Furthermore in the scheme presented in Section II.B, no knowledge of the upper-bound on a 1 is required in the adaptive scheme. In [21] knowledge of both a 0 and a 1 are required (as is the polarity of the disturbance signal which is not allowed to change). Also the dual-layer nested adaptive architecture involving k(t) and r(t) is clearly quite different from the schemes proposed in [15]. III. A MULTI-INPUT MULTI-OUTPUT FORMULATION Now consider a multi-variable version of the first-order scalar equation in (1) given by σ(t) = a(t)+u(t) (23) where σ(t) R m represents the switching function to be forced to zero in finite time,u(t) R m represents the control input to be manipulated and a(t) R m is a disturbance which is imprecisely known. Here it will be assumed that a(t) is not known but its magnitude and its derivative ȧ(t) are bounded so that a(t) < a 0 and ȧ(t) < a 1 where both a 0 and a 1 are finite but unknown. Consider the unit vector control law [2], [25] given by u(t) = (k(t)+η) σ(t) (σ(t)) (24) where η is a positive design constant and k(t) is a varying term which satisfies an adaptive law. A sufficient condition to ensure sliding takes place on σ 0 in finite time is k(t) > a(t) (25) If (25) is satisfied then the multivariable η-reachability condition [1], [2] of the form σ T σ η σ is satisfied and sliding is guaranteed. In this multivariable situation the concept of equivalent control is equally applicable and so once sliding is achieved u eq (t) = a(t) (26) and u eq (t) = a(t). Now define a scalar δ(t) = k(t) 1 u eq(t) ǫ (27) where, as in Section II, the scalar parameters 0 < < 1 and ǫ > 0. As in Section II.B the gain k(t) adapts according to the law k(t) = ρ(t)sign(δ(t)) (28) where ρ(t) = r(t)+r 0 and ṙ(t) = γ δ(t) (29) with γ a positive user defined scalar. Although the control problem is a multivariable one, the approach in Section II can be employed here because δ(t) is once again a scalar. Define e(t) = a 1 / r(t) (30) then using the Lyapunov function V = 1 2 δ γ e2 and repeating the LaSalle argument in Section II, once again it can be demonstrated that in finite time k(t) > a(t) and the η-reachability condition σ T σ < η σ is satisfied and sliding is maintained. IV. ADAPTIVE CONTINUOUS HIGHER ORDER SLIDING MODE CONTROL: SUPER-TWISTING CASE Consider once again the scalar sliding variable dynamics in (1) under the assumption that ȧ a 1 where a 1 > 0. Now suppose that in instead of the relay structure in (2), u(t) is given by the so-called super-twisting control u(t) = λ σ(t) 1/2 sign(σ(t))+u 1 (t) (31) u 1 (t) = k sign(σ(t)) (32) which drives both σ, σ 0 in finite time if and λ > k > a 1 (33) 2 (k +a 1 )(1+q) k a 1 1 q (34) where a(t) < qu m, 0 < q < 1 and u(t) U m [26]. Assuming that the gain λ can be selected large enough so that equation (34) holds, the aim is to adapt the gain k(t) in equation (32) so that k(t) is close to a 1 whilst satisfying condition (34). This reduces the amplitude of the high frequency part of the super-twisting control in equation (32) which helps reduced chattering. Remark In the exposition above the original notation of [26] has been retained 1. However it is clear that qu m is essentially the bound a 0 on the magnitude of the disturbance a(t) employed earlier in Section II. Equations (1), (31) and (32) can be written in the form σ(t) = λ σ(t) 1/2 sign(σ(t))+v(t) (35) v(t) = ϕ(t) w(t) (36) where ϕ(t) = ȧ(t), and w(t) = k sign(σ(t)) (37) During a 2-SM σ = v = 0 (which is exactly equivalent to the condition σ = σ = 0) and the equivalent control w eq (t) = ϕ(t). Suppose d dt ϕ(t) a2. Again two cases will be considered: firstly the case when a 2 is known; and secondly the case when a 2 is unknown. 1 For further details, see the appendix.

5 A. Formulation when a 2 is known Assuming that w eq (t) is available (by filtering similar to the extraction process for ū eq (t) in (4)). Introduce a new variable δ(t) = k(t) 1 w eq(t) ǫ (38) where 0 < < 1 and ǫ > 0 is a small real number. The first layer of the proposed double-layer control gain adaptation algorithm is defined as k(t) = (r 0 +r(t)) sign(δ) (39) where r 0 > 0 is a small scalar, and the evolution of r(t) in the second layer of the adaptation algorithm is given by where ṙ(t) = γ δ(t) +r 0 γsign(e(t)) (40) e = a 2 / r(t) (41) and γ is a positive scalar. Note that in order to realize (40), the bound a 2 must be known. It follows from the definition of δ(t) in (38) that δ(t) = k(t) 1 and therefore d dt w eq(t) = k(t) 1 d ϕ(t) (42) dt δ δ δ k(t)+ δ a 2 r 0 δ r(t) δ + δ a 2 = ( r 0 +e) δ (43) Now the stability of the dynamics of δ(t) and e(t) from (38) and (41) will be studied. Using the Lyapunov function candidate V = 1 2 δ γ e2 (44) its derivative satisfies V ( r 0 +e) δ 1 γ eṙ(t) 1 = ( r 0 +e) δ e δ r 0 γ e ( ) = r δ +r 0 r 0 2V 1/2 2γ e (45) from inequality (15). The inequality V r 0 2V 1/2 from (45) guarantees that in finite time V 0, which means the finite time convergence of δ,e 0. This in turns guarantees the boundedness of k(t) and r(t) from (39) and (40). In the 2-SM (δ = e = 0 σ = v = 0), using (38), the following inequality holds k(t) = w eq (t) + 1 w eq(t) +ε = ϕ(t) + 1 ϕ(t) +ε > ϕ(t) (46) This means that equation (33) holds, and provided that λ in (34) is sufficiently large, the 2-SM is maintained. B. A formulation when a 2 is unknown If a 2 is unknown the formulation used in Section III.A is not applicable. However instead of the adaptive law in (40), consider (39) together with ṙ(t) = γ δ(t) (47) where γ is a positive scalar. The stability of the dynamics δ(t) and e(t) from (38) and (41) together with the dual stage adaptive laws (39) and (47) will now be investigated. Note this formulation does not require knowledge of a 2. Using the Lyapunov function candidate it can be shown that V = 1 2 δ γ e2 (48) V r 0 δ(t) and then arguing as in Section II.B, from LaSalle s principal, in finite time t 0 the inequality δ(t) < ǫ/2 is guaranteed and from (38) it follows k(t) > ϕ(t) (49) holds for all t > t 0. This means that equation (33) holds, and provided that λ from (34) is sufficiently large, the 2-SM is maintained. V. SIMULATION RESULTS Consider the scalar situation in (1) when the disturbance is given by a(t) = 2+sin(t). Suppose initially that a bound a 1 on the rate of change of the disturbance a(t) is known so that the adaptive scheme (2), (6) and (9) from Section II.A can be employed. In the simulations which follow a 1 = 1, = 0.99, r 0 = 0.5, ǫ = 0.01 and γ = 4. Figures 1-3 show the results when σ(0). Figure 1 shows that sliding is maintain throughout the simulation in the presence of the disturbance. Figure 2 shows that k(t) varies and attempts to track the disturbance. Since in this case = , very little margin of error between k(t) and a(t) is being implicity specified. In this situation, because it is assumed that a 1 is known, the gain r(t) tracks the upper-bound of ȧ(t) (in this case unity) in finite time. Fig. 1. Evolution of the switching function

6 Fig. 2. Evolution of the adaptive gain k(t) Fig. 4. Evolution of the switching function Fig. 3. Evolution of the gain r(t) and the control signal Fig. 5. Evolution of the adaptive gain k(t) Again consider the scalar situation in (1) when the disturbance is given by a(t) = 2 + sin(t). Now suppose that both a 0 and a 1 are not known. Now the adaptive scheme (2), (6) and (17) from Section II.B will be employed where = 0.99, r 0 = 0.5, ǫ = 0.01 and γ = 4. Figures 4-6 show the results when σ(0). Figure 4 shows that sliding is maintained throughout the simulation in the presence of the disturbance. Figure 5 shows that k(t) varies and attempts to track the disturbance since in this case = which means very little margin of error between k(t) and a(t) is being implicity specified. Compared to the case when a 1 is known, now it takes longer for k(t) to become close to a(t). Also it can be seen from Figure 6 that r(t) no longer converges to the true upper-bound on ȧ (which in this case is unity) but converges to a value which is greater than one. This does not contradict the theory, since there are no guarantees on the value to which r(t) converges because of the use of LaSalle s principal. The remainder of the results pertain to the super-twisting case. Again the scalar situation in (1) when the disturbance is given by a(t) = 2+sin(t) is considered. As before = 0.99, r 0 = 0.5, ǫ = 0.01 and γ = 4. Figures 7-9 show the results when the upper-bounda 2 is assumed to be known. Here nonzero initial conditions are taken for σ and σ. Figure 7 shows a 2-SM is obtained in finite time. Figure 8 shows that k(t) varies and attempts to track the disturbance. Figure 9 shows that r(t) accurately tracks the least upper-bound on ä. Figures show the results when the upper-bound a 2 Fig. 6. Evolution of the gain r(t) and the control signal is not known. Again non-zero initial conditions are taken for σ and σ. Figure 10 shows a 2-SM is obtained in finite time (although it takes longer to achieve compared to Figure 7). Figure 11 shows that k(t) varies and attempts to track the disturbance. Figure 12 shows that r(t) does not track the least upper-bound on ä (which in this case is unity) but converges to a value greater than one. This is in accordance with the theory. VI. CONCLUSION This paper has proposed a novel equivalent control-based adaptive schemes for both conventional and super-twisting sliding mode control algorithms. The assumptions on the

7 Fig. 7. Evolution of σ and σ in the super-twisting case Fig. 11. Evolution of the adaptive gain k(t) Fig. 8. Evolution of the adaptive gain k(t) Fig. 12. Evolution of r(t) and the switching component w(t) Fig. 9. Evolution of r(t) and the switching component w(t) particular one of the proposed schemes does not require information about the upper-bound of the disturbance and its derivative. Furthermore the underlying ideas can be applied to different sliding mode control structures and relies more on the availability of an online approximation of the equivalent control than the precise control-law used to induce and maintain sliding. The schemes can also be made robust to imperfections in the estimates of the equivalent control by choice of two specific scalar parameters. These trade-off robustness, against the degree of conservatism over-bounding the uncertainty. In this paper, these ideas were incorporated within classical relay structures, unit vector control schemes and super-twisting controllers. The simulations presented in the paper confirm the effectiveness and simplicity of the proposed schemes. REFERENCES Fig. 10. Evolution of σ and σ in the super-twisting case uncertainty and the knowledge about the bounds on the magnitude and the derivative of the uncertainty are weaker than those imposed in the comparable existing literature. In [1] Y. Shtessel, C. Edwards, L. Fridman, and A. Levant, Sliding Mode Control and Observation. Birkhauser, [2] C. Edwards and S. K. Spurgeon, Sliding Mode Control: Theory and Applications. Taylor & Francis, [3] A. Levant, Robust exact differentiation via sliding mode technique, Automatica, vol. 34, no. 3, pp , [4], Higher-order sliding modes, differentiation and outputfeedback control, International Journal of Control, vol. 76, no. 9-10, pp , [5] E. Dubrovskii, Application of the adaption principle for control systems with variable structure, in Proceedings of the National Conference on Control, Bulgaria Varna, [6] G. Ambrosino, G. Celentano, and F. Garofalo, Variable structure model reference adaptive control systems, International Journal of Control, vol. 39, pp , 1983.

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