Globally Stable Output-Feedback Sliding Mode Control with Asymptotic Exact Tracking Eduardo V. L. Nunes, Liu Hsu and Fernando Lizarralde Abstract— An output-feedback sliding mode controller is proposed for uncertain plants with relative degree higher than one in order to achieve asymptotic exact tracking of a reference model. To compensate the relative degree, a lead filter scheme is proposed such that global stability and asymptotic exact tracking are obtained. The scheme is based on a convex combination of a linear lead filter with a robust exact differentiator, based on second order sliding modes. keywords: Sliding Mode Control, Uncertain Systems, Tracking Control, Model Reference, Exact Differentiator. I. I NTRODUCTION Robustness and adaptation are the main trends to cope with systems with poor modeling or large uncertainties, including parameter variations, unmodeled dynamics and external disturbances. An important technique to control systems under large uncertainties, effective in several practical applications in engineering, is variable structure control based on sliding modes, or, for short, sliding mode control (SMC). Recently, a growing number of research papers about the subject, both on theoretical and application grounds can be observed. The power of the SMC to deal with nonlinear plants together with newly introduced concepts like terminal sliding mode control [3], higher order sliding modes [11], [2] and also the progress in output feedback SMC [10], [9], [1], [14], have significantly widened the range of applicability of SMC. In the recent papers [13], [14], interesting output feedback SMCs based on higher order sliding were proposed for plants of arbitrary relative degree. The main idea that allowed the completion of the feedback control scheme was the so called robust exact differentiator introduced in [12]. The class of controllers, based on exact differentiators, may lead to exact output tracking but, so far, stability or convergence has been proved only locally [14]. On the other hand, an earlier output feedback SMC scheme, named, VS-MRAC (Variable structure Model Reference Adaptive Control), introduced in [7], [10] has the capability of guaranteeing global exponential stability. However, for plants of relative degree higher than one, the tracking error becomes arbitrary small but not necessarily zero. This paper represents a preliminary attempt to achieve global stability and asymptotic exact tracking controllers using exact differentiators. To this end, we have restricted the detailed theoretical development to SISO (single-input/single-output) uncertain linear plants of relative degree two. Extension to higher relative degree seems quite immediate using the differentiators introduced in [2], [14], while extension to nonlinear plants could be done using the recent extensions of the VS-MRAC to MIMO (multi-input/multioutput) and nonlinear plants [9], [8]. This work was supported in part by FAPERJ, CNPq and CAPES. E. V. L. Nunes and L. Hsu are with Dept. of Electrical Eng./COPPE, Federal University of Rio de Janeiro, Rio de Janeiro, Brazil. eduardo@coep.ufrj.br, liu@coep.ufrj.br F. Lizarralde is with the Dept of Electronic Eng./EE, Federal University of Rio de Janeiro, Rio de Janeiro, Brazil. fernando@coep.ufrj.br II. P RELIMINARIES Prior to developing the theoretical content of the paper, it is important to clarify some notation which might otherwise lead to some confusion. Here, similarly to adaptive control literature, a dual time-domain/frequency domain notation is often adopted. Rigorously, one should use “s” for the Laplace variable (frequency-domain) and “p” for the differential operator “d/dt”. However, for the sake of simplicity, the symbol “s” will here represent either the Laplace variable or the differential operator (d/dt), according to the context. Some further notation is also introduced, according to [10]: a) In what follows, all K’s denote positive constants, operator norms (||H||) are L∞ induced norms, π(t) is an exponentially decaying function (i.e. |π(t)| ≤ Re−λt , for some positive scalars λ, R and ∀t). b) Operators and convolution operators: Refer to ([10], [15]) for precise meaning of mixed time domain (state-space) and Laplace transform domain (operator) representations. III. P ROBLEM S TATEMENT Consider an uncertain SISO LTI plant with known relative degree n∗ and transfer function Gp (s) = Kp Np (s)/Dp (s), where deg(Dp ) = n, with input u and output yp . The reference model, having input r and output ym , also has relative degree n∗ and is given by M (s) = Km /Dm (s), where deg(Dm ) = n∗ , Np , Dp , Dm are monic polynomials and Dm (s) is a Hurwitz polynomial. The main objective is to find a control law u(t) such that the output error e0 := yp − ym tends asymptotically or in finite time to zero, for arbitrary initial conditions and uniformly bounded arbitrary piecewise continuous reference signals r(t). The control input u can be parameterized as u(t) = θ T ω(t), where θ ∈ IR2n is the parameter vector and ω ∈ IR2n is the regressor vector obtained from input and output state variable filters [6]. Considering the usual MRAC design assumptions, the error equations for a plant under the action of an input disturbance de , is of the form (see [6][10] for details) State-Space form: ė = Ac e + k∗ bc (u + Ū ) (1) e0 I/O form: e0 = = hTc e; k∗ M (s)[u + Ū ] (2) where k∗ = Kp /Km , Ū = −u∗ + Wd de , u = u∗ = θ∗T ω is the ideal control signal which matches the plant to the model (with de = 0), Wd (s) = [k∗ M (s)]−1 W̄d is proper and stable and W̄d (s) is the closed-loop transfer function from the input disturbance de to e0 with u = u∗ (see [10] for details). The input disturbance is assumed to be piecewise continuous or locally integrable and uniformly bounded. It is also assumed that an instantaneous upper bound d¯e (t) of de (t) is known, satisfying d¯e (t) ≥ |de (t)| (∀t). From the control parameterization u(t) = θ T ω(t), we now make the following assumption on the class of admissible control laws. The control signal satisfies the inequality sup |u(t)| ≤ Kω sup ||ω(t)|| + Kδ ; t t ∀t (3) where Kω , Kδ are positive constants. This assumption guarantees that no finite time escape occurs in the system signals. Indeed, in this case the system signals will be regular and therefore can grow at most exponentially [15]. This bound guarantees that all systems signals are in L∞e . Model r M ym γ de u + yp + Gp + f − e0 D ē0 + Plant ẽ0 + βα (t) IV. VARIABLE S TRUCTURE M ODEL R EFERENCE A DAPTIVE C ONTROL (VS-MRAC) −1 When the relative degree of the plant is n∗ = 1, the main idea of the VS-MRAC is to close the error loop with an appropriate modulated relay, i.e. u = −f (t)sign(e0 ). In this case, the reference-model M (s) can be chosen strictly positive real (SPR), so that an ideal sliding loop (ISL) [5] is formed around the relay function. Figure 1 presents the block diagram of the VS-MRAC for n∗ = 1. The control signal u can be generated with a modulation function satisfying f (t) ≥ |u∗ | + |Wd de | + δ (4) where δ is an arbitrary positive constant. This modulation function guarantees that the above scheme is globally exponentially stable and the output error e0 becomes identically zero after some finite time, according to Lemma 1 in [10]. Model ym M r f de u + Gp + yp − e0 + Plant Fig. 2. VS-MRAC using a linear lead fi lter for relative degree compensation, with an uniformly bounded output measurement error. A. Stability Analysis In what follows the stability analysis of the LF/VS-MRAC will be presented. It should be stressed that Filippov’s definition of solution for differential equations with discontinuous right-hand sides is assumed [4]. Note that u or de are not necessarily functions of t in the usual sense when sliding modes take place. In order to avoid clutter, we will denote by u(t) and de (t) the locally integrable functions which are equivalent to u and de , respectively, in the sense of equivalent control, along any given Filippov solution of the closed-loop system. It should be stressed that Filippov solution is, by definition, absolutely continuous. Then, along any such solution, u or de can be replaced by u(t) or de (t) respectively, in the right-hand side of the governing differential equations. Although the equivalent control u(t) = ueq (t) is not directly available, filtering u with any strictly proper filter G(s) gives G(s)u = G(s)u(t) = G(s)ueq (t). From Fig. 2, one has ē0 = (γ + D)e0 Relay VS-MRAC for (6) which, from (2), can be rewritten as (in fact, immersed) −1 Fig. 1. Relay n∗ = 1. where T Consider the parameter uncertainty upper bound vector θ̄ defined as θ̄T = [θ̄1 · · · θ̄2n ], with θ̄i > |θi∗ |. In order to satisfy (4) the modulation function f (t) can be implemented as follows: f (t) = θ̄T |ω(t)| + dˆe (t) + δ (5) where |ω(t)|T = [|ω1 (t)| , · · · , |ω2n (t)|], dˆe (t) is an upper bound for |Wd de (t)| and is obtained from d¯e ≥ |de (t)| filtered by a first ke ¯ order filter i.e. dˆe (t) = p+γ de where γ = mink |Re(pk )|, with pk being the poles of Wd (for details see Lemma 3 in [10]). For the case of plants with relative degree n∗ > 1 the reference model transfer function cannot be chosen SPR. For simplicity consider only the case n∗ = 2. To overcome the relative degree problem we propose the scheme of Figure 2, named LF/VSMRAC, where D(s) = s/F (τ s), with F (τ s) being a Hurwitz polynomial in τ s, deg(F ) = l and F (0) = 1. As τ tends to zero the transfer function from e0 to ē0 , namely, La (s) = D(s) + γ approximates the polynomial operator L(s) = s + γ. Therefore the scheme depicted in Figure 2 approximately compensates the excess of relative degree. Moreover, for convenience, it is assumed that M L(s) = Km /(s + am ). An additional signal βα (t) has also been introduced in the control loop and, for the time being, it will be regarded as a bounded disturbance. Later on, we will synthesize the signal βα (t) in order to achieve asymptotic exact tracking. ē0 = k∗ M L[u + Ū ] + βŪ + βu (7) βŪ = − [k∗ M (F − 1) D] Ū (8) ∗ βu = − [k M (F − 1) D] u (9) Note that the transfer function M (s) [F (τ s) − 1] D(s) is BIBO stable and strictly proper. The auxiliary error ẽ0 is given by ẽ0 = ē0 + βα (10) where |βα (t)| is a bounded disturbance. From now on, let z denote the full error state vector of the system (1)(7)-(9). In order to fully account for the initial conditions, it is convenient to partition z as z T = [(z 0 )T , zeT ] zeT = [eT , ēT ] where ēT = [ē1 , . . . , ēl ] correspond to the state vector of the lead filter, and similarly as in [10], z 0 denotes the transient state corresponding to operators Wd and M (s) [F (τ s) − 1] D(s). In what follows, EXP and EXP 0 denote any term of the form K ||z(0)|| e−at and K z 0 (0) e−at , respectively, where a is some (generic) positive constant [5]. The following proposition characterizes the convergence properties of the error ē0 (t). Proposition 1: Consider the error equation (7), with u = −f (t)sign(ẽ0 ). If the relay modulation function f (t) is defined as in (5), M L(s) is of the form M L(s) = Km /(s+am ) (Km , am > 0) and |βα (t)| ≤ τ KR then, |ē0 (t)| ≤ τ Kē0 C(t) + EXP, ∀t ≥ 0 (11) where Kē0 > 0 is a constant, τ is the time constant of F −1 and C1 (t) = sup ||ω(t)|| ; C(t) = Kθ C1 (t) + Kβ (12) t for some constants Kθ , Kβ > 0. (Proof: see Appendix.) The stability result can be stated in the following theorem: Theorem 1: Consider the system (1)(6)(10), with u = −f (t)sign(ẽ0 ). Assume that (4) is satisfied. If M L(s) = Km /(s + am ) (Km , am > 0). Then, for sufficiently small τ > 0, the complete error system, with state z, is globally exponentially stable with respect to a residual set of order τ , i.e., there exist positive constants Kz and a such that ∀z(0), ∀t ≥ 0, ||z(t)|| ≤ Kz e−at ||z(0)|| + O(τ ). (Proof: see [10].) The following corollaries will be useful in the theoretical analysis presented in section VI. Corollary 1: For all R > 0, ∃ τ > 0 sufficiently small such that for some finite T , ||z(t)|| < R, ∀t ≥ T VI. VS-MRAC BASED ON A G LOBAL ROBUST E XACT D IFFERENTIATOR (GRED) In sections IV and V two solutions for derivative estimation were discussed. The lead filter, proposed in Section IV, leads to global stability, but cannot provide exact derivative. On the other hand the RED, proposed in section V, can give the exact derivative. However, when used in a feedback loop, only local convergence properties can be guaranteed since the boundedness condition (15) may not be valid for any initial conditions. The idea is to combine both estimators in order to accomplish the following tasks: • To globally drive the system trajectories into an invariant compact set DR in some finite time. • To drive the full error state asymptotically to zero Here we propose a control scheme, named GRED/VS-MRAC (see Fig. 3), based on a weighted switching scheme in order to achieve global asymptotic convergence of the full error state to zero. In this scheme the derivative of the output error e0 can be estimated using either the lead filter or the RED. (13) Corollary 2: The signal ë0 (t) is bounded, i.e., there exists a positive constant Ka such that Model r γ ym M |ë0 (t)| ≤ Ka , ∀ t ≥ 0 u (Proof: see Appendix) The drawback of this approach is that the system only guarantees error convergence to a residual set of order τ and thus the chattering phenomena may arise. Gp yp − − e0 + where e0 (t) is a measurable locally bounded input signal, α, λ > 0 and v(t) is the output of the differentiator. Let e0 (t) be a signal having derivative with Lipschitz constant C2 . If the following sufficient conditions α + C2 α − C2 ẽrl modulate êg + RED êr ẽ0 + 1−α Plant + Π Relay −1 To circumvent the above problem we will consider the following differentiator based on second-order sliding-mode, proposed in [12] ẋ = v (14) v = u1 − λ|x − e0 (t)|1/2 sign(x − e0 (t)) u̇1 = −αsign(x − e0 (t)) λ2 ≥ 4C2 f + + V. ROBUST E XACT D IFFERENTIATOR (RED) α > C2 , GRED Π α + + êl D de (15) are satisfied, then the output v(t) converges to ė0 (t) in a finite time. This result is formally stated in the following Theorem Theorem 2: Consider system (14). Let α and λ be such that inequality (15) is satisfied. Then, provided e0 (t) has a derivative with Lipschitz’s constant C2 or bounded second derivative, the equality v(t) = ė0 (t) is fulfilled identically after a finite-time transient process. (Proof: see [12]) This differentiator can provide, in absence of noise, the exact derivative. In the presence of noise the RED has accuracy proportional to the square root of the noise magnitude. Another important aspect that must be pointed out is the fact that the state of the RED cannot escape in a finite time, provided the input signal has bounded second derivative, even if (15) does not hold. This result is stated in the following Lemma. Lemma 1: Consider system (14). If |ë0 (t)| ≤ Ka ∀t, for some positive constant Ka , then the system state cannot diverge in finite time (Proof: see appendix) Fig. 3. GRED/VS-MRAC: VS-MRAC using a GRED for relative degree compensation. The block composed by the lead filter and the RED, denoted by Global Robust Exact Differentiator (GRED), can be seen as a single differentiator with input e0 and output êg given by the convex combination êg = α(ẽrl )êl (t) + [1 − α(ẽrl )] êr (16) where êl and êr are estimations of ė0 provided by the lead filter and the RED respectively and ẽrl = êr − êl . The switching function α(ẽrl ) is a continuous, state dependent modulation which allows the controller to smoothly change between both estimators. This function assumes values in the set [0, 1] and it will be defined later on. The estimate given by the lead filter and the RED can be written as follows êl (t) êr (t) = = ė0 (t) + ǫl (t) ė0 (t) + ǫr (t) (17) where ǫl (t) and ǫr (t) are estimation errors of the lead filter and the RED respectively. From (17), equation (16) can be rewritten as êg (t) = ė0 (t) + ǫ(t) (18) ǫ(t) = α(ẽrl )ǫl (t) + [1 − α(ẽrl )] ǫr (t) (19) where Thus, the error ẽ0 (see Fig. 3) can be written as ẽ0 (t) = ė0 + γe0 + ǫ(t) (20) The estimation error ǫ(t) can be considered as an output measurement error. Thus we can define the following auxiliary error ê0 := ė0 + γe0 (21) From (21) and considering system (1) (with relative degree two), we can describe the GRED/VS-MRAC as follows. State-Space form: ė = Ac e + k∗ bc (u + Ū ) (22) ê0 I/O form: ê0 u = = = ĥT e k∗ M (s)L(s)[u + Ū ] −f (t)sign(ê0 + ǫ) (23) (24) Since, by assumption, M (s)L(s) = Km /(s + am ), the system {Ac , bc , ĥT } is SPR. We now propose a switching law for α(.) in order to guarantee global stability and to ensure that the full error state converges to zero. To this end, the lead filter must be fully activated, when the system state is far from the equilibrium, so as to drive the system close to the origin. Then, after a finite time transient the RED takes over, providing exact estimation. In order to avoid “nested” discontinuities (outside the scope of Filippov’s theory), we choose the following (continuous) weighted switching law for α: α(ẽrl ) = ( 0, (|ẽrl |−ǫM +c)/c, 1, for |ẽrl | < ǫM − c for ǫM −c ≤ |ẽrl | < ǫM for |ẽrl | ≥ ǫM (25) where 0 < c < ǫM and ǫM = τ KR (26) where KR is an appropriate positive constant. Proposition 2: Consider the estimation error ǫ(t) defined in (19). Using the above weighted switching function α, ǫ(t) can be rewritten as ǫ(t) = ǫl (t) + βα (ẽrl (t)) (27) where βα (ẽrl (t)) is absolutely continuous in t and uniformly bounded by |βα (ẽrl (t))| < ǫM , ∀t ≥ 0 (28) (Proof: see Appendix.) According to Proposition 2, for the switching function (25), the GRED can be seen as a lead filter with transfer function D(s) plus an output measurement error βα (ẽrl ). Hence, the system can be represented as in Fig. 2 and, consequently, Theorem 1 holds if all signals in the system are defined for all t, that is, belong to L∞e . In order to show that the latter condition is true for the system with the GRED block, we only have to show that the signals in the block RED are in L∞e . This argument can be proved by contradiction as follows. Suppose that the maximal interval of finiteness of the signals in the RED is [0, TM ). During this interval, all conditions of Theorem 1 hold and thus all signals of the remaining subsystems of the GRED/VS-MRAC are bounded by a constant, and in particular |ë0 (t)|, from Corollary 2. This leads to a contradiction with Lemma 1 whereby, the signals in RED could not diverge unboundedly as t → TM . As a consequence of the continuation theorem for differential equations (in Filippov’s theory), TM must be ∞, which means that all signals are defined ∀t > 0. Therefore, according to Theorem 1 the full error system with state z is globally exponentially stable with respect to a residual set of order τ . Now, we will analyze the convergence of the RED. In order to apply Theorem 2 we have to find an upper bound to the signal ë0 (t). According to Corollary 1 the full error state is steered to an invariant compact set DR := {z : ||z(t)|| < R} in some finite time T1 ≥ 0. After the error state enters the set DR the signal ë0 (t) can be bounded according to the following Proposition. Proposition 3: Consider the control scheme of the GRED/VSMRAC, represented by (22)(24)(19), with α(ẽrl ) defined in (25). The modulation function f (t) is defined as in (5). If ||e(t)|| < R, ∀t > T1 then, sup |ë0 (t)| ≤ C2 (29) t≥T1 (Proof: see Appendix.) Since the RED is time invariant its initial conditions can be considered in t = T1 . According to Lemma 1 the initial conditions are finite. If the parameters α and λ were adjusted, satisfying condition (15), then from Theorem 2 the estimation error ǫr (t) converges to zero in a finite time T2 . This convergence result is formally stated in the following Lemma. Lemma 2: Consider the system (22) (24) (19), with the switching function defined in (25). The modulation function f (t) is defined as in (5). If condition (15) is satisfied, for C2 given by proposition 3, then êr (t) = ė0 (t) after some finite time T2 . From Lemma 2 the estimation error ǫr (t) becomes zero after some finite time. Thereafter, the RED will remain active if the threshold ǫM is chosen larger than the upper bound of the residual estimation error of the lead filter. One suitable way to do this is to choose ǫM such that ǫM > ǭl +c, where ǭl is the upper bound of the lead filter estimation error ǫl (t) when the error state is within the invariant compact set DR . This upper bound is characterized in the following proposition. Proposition 4: Consider the system (22) (24) (19), with the switching function defined in (25). The modulation function f (t) is defined as in (5). The lead filter estimation error ǫl (t) can be bounded for t ≥ T1 by lim sup |ǫl (ts)| < ǭl t→∞ ts≥t (30) where ǭl = τ Kl C2 , Kl is a positive constant and C2 is defined in Proposition 3. (Proof: see Appendix.) If ǫM is chosen appropriately, then the weighted switching function α(ẽrl ) = 0, ∀t ≥ T2 , which implies that ǫ(t) = 0, ∀t ≥ T2 . In this case an ideal sliding loop is formed and applying Lemma 1 in [10] to system (22) (24), with f (t) defined as in (5), one can conclude that the error state e will converge exponentially to zero and the output error ê0 becomes identically zero after some finite time. Since F (τ s) is a Hurwitz polynomial and the tracking error e0 (t) converges exponentially to zero, one can conclude that the lead filter state vector ē will also converges exponentially to zero, which implies that after some finite time the full error state z converges exponentially to zero. The convergence properties of the proposed system concluded above can be formalized in the following Theorem. Theorem 3: (Main Result) Consider the error system of the GRED/VS-MRAC, depicted in Fig. 3, with switching function α defined in (25) and modulation function f (t) defined in (5). If KR is such that ǫM = τ KR satisfies ǫM > ǭl + c, (31) then, for sufficiently small τ > 0, the full error system with state z is globally exponentially stable with respect to a residual set of (a) 0.5 1.2 1 0 -0.5 0 1 2 3 4 5 6 7 8 8 α(ẽrl ) 0.8 0.6 10 0.4 (b) 0.5 0.2 0 0 -0.5 0 1 2 3 4 5 6 time 7 8 8 -0.2 0 10 Fig. 4. (a) Tracking error e0 (t) for ǫM = 0 and c = 0 (lead fi lter only); (b) Tracking error e0 (t) for ǫM = 20τ and c = 5τ (ǫM and c from (25)). 1 2 3 4 5 time 6 7 8 8 10 Fig. 5. Time behavior of switching function α(ẽrl ) for ǫM = 20τ and c = 5τ (see Fig. 4 (b) ). 2 1.5 1 0.5 0 -0.5 -10 1.2 1 (a) 0.8 1 2 3 4 5 6 7 8 8 α(ẽrl ) VII. S IMULATION R ESULTS This section presents some illustrative simulation examples which highlights the performance of the proposed control scheme. Case 1: Uncertain plant with relative degree (n∗ = 2) The plant is considered unknown and is given by Gp (s) = 2 2 . The reference model is chosen to be M (s) = (s+2) 2. (s+1)(s−2) We consider de (t) = sqw(5t), where sqw denotes a unit square wave and r(t) = sin(0.5t). The modulation function is given by f (t) = θ̄T |ω| + fo , where θ̄T = [6, 10, 2, 2] and fo = 1.5. Other design parameters are: L(s) = s + 2; RED : α = 1.1C2 ; 1/2 λ = 0.5C2 ; C2 = 30; lead filter: F (τ s) = (τ s+1)2 ; τ = 0.02; plant initial conditions: yp (0) = 10; ẏp (0) = 5 As shown in Fig 4, if the velocity is estimated using only the lead filter, i.e. α(ẽrl ) = 1, the output tracking error do not converges to zero. As was expected using the GRED very precise tracking is achieved even in the presence of large disturbance de (t). In this case an ideal sliding loop is obtained. to that obtained using only the lead filter, which demonstrates the robustness of the proposed scheme. This result motivates further research to investigate the influence of unmodeled dynamics on the proposed controller. 0.6 10 0.4 2 1.5 1 0.5 0 -0.5 -10 0.2 (b) order τ . Moreover after some finite time the derivative estimation becomes exact and only given by the RED (α(ẽrl ) = 0), and the full error state z, as well as the output tracking error e0 (t) tend exponentially to zero. 0 1 2 3 4 5 6 time 7 8 8 -0.2 0 10 Fig. 7. (a) Tracking error e0 (t) for ǫM = 0 and c = 0; (b) Tracking error e0 (t) for ǫM = 60τ and c = 40τ . 1 2 3 4 5 time 6 7 8 8 10 Fig. 8. Time behavior of switching function α(ẽrl ) for ǫM = 60τ and c = 40τ (see Fig. 7 (b) ) VIII. CONCLUSIONS In this paper, an output feedback sliding mode controller for uncertain plants with relative degree higher than one is proposed. The presented controller uses a convex combination of a linear lead filter with a robust exact differentiator in order to achieve global stability and asymptotic exact tracking of a model reference. One key element that has allowed the solution of the problem with only output feedback was the error ẽrl , i.e., the difference between the velocity estimates provided by the RED and the lead filter. The detailed theoretical analysis has been restricted to uncertain plants of relative degree two. The extension to arbitrary relative degree will be presented in a future work. Simulation results are presented to validate the analysis and to illustrate the robustness of the proposed scheme to external disturbances and unmodeled dynamics. A PPENDIX Proof of Proposition 1: From (7) and (10), one has For the above parameters and conditions, if only the RED is used for velocity estimation (α(ẽrl ) = 0) the system becomes unstable (see Fig. 6) 2000 1500 ẽ0 = k ∗ M L[−f (t)sign(ẽ0 ) + Ū ] + βŪ + βu + βα where βŪ and βu are defi ned in (8) and (9) respectively, and |βα | ≤ τ KR . According to assumption (3), one can also choose the constants K θ , Kβ such that supt Ū (t) ≤ C(t). Then, from (8), one has 0 (t) ≤ ||k ∗ M (F − 1) D|| C(t) = τ KβŪ C(t) sup βŪ (t)−βŪ 1000 | t 500 e0 0 -500 From (9) and (3) one has -1000 -2000 -2500 0 {z O(τ ) | t 1 2 3 4 5 time 6 7 8 8 10 Fig. 6. System instability when only the RED is used for velocity estimation. {z O(τ ) It is straightforward to conclude that (33) } 0 (t) ≤ ||k ∗ M (F − 1) D|| C(t) = τ Kβu C(t) sup βu (t)−βu -1500 (32) (34) } sup |βŪ (t)| ≤ τ KβŪ C(t) + EXP 0 (35) sup |βu (t)| ≤ τ Kβu C(t) + EXP 0 (36) t Case 2: Uncertain plant (n∗ = 2) with unmodeled dynamics In this case we consider the same example of case 1 except for the planth which includes an unmodeled dynamic, i.e. i 1 2 Gp (s) = (µs+1) , where µ = 0.1 was chosen for (s+1)(s−2) simulation purposes. The same i control design (for the nominal h 2 is considered. The plant initial plant G0p (s) = (s+1)(s−2) conditions are: yp (0) = 1 and ẏp (0) = 5. As shown in Fig. 7, the tracking performance of the control scheme using the GRED for velocity estimation is clearly superior t Since |βα | ≤ τ KR , for appropriate constants Kθ and Kβ , one has sup |βα (t)| ≤ τ Kβα C(t) (37) t Using the results obtained in (35), (36) and (37), if f (t) ≥ Ū , applying Lemma 2 in [10] to (32), one has |ẽ0 | ≤ τ Kẽ0 C(t) + EXP , which implies, from (10), that |ē0 | ≤ τ Kē0 C(t) + EXP (38) Proof of Proposition 2: Consider the switching function proposed in (25) there are three possible cases: Case 1: (|ẽrl | ≥ ǫM ) In this case α(ẽrl ) = 1, then, from (19), one has ǫ(t) = ǫl (t). Thus βα (ẽrl ) = 0, satisfying condition (28) Case 2: (ǫM − c ≤ |ẽrl | < ǫM ) In this case the following statement can be made |ẽrl | = ǫM − δ1 (ẽrl ) (39) 0 < δ1 (ẽrl ) ≤ c (40) where Substituting (39) in (19), using (25), one can rewrite ǫ(t) = ǫl + βα (ẽrl ) where: βα (ẽrl ) = ± |ẽrl | = ǫM − c − δ2 (ẽrl ) (41) 0 < δ2 (ẽrl ) ≤ ǫM − c (42) where From (41) and (19), one has ǫ(t) = ǫl + βα (ẽrl ) where βα (ẽrl ) = ± [ǫM − c − δ2 (ẽrl )]. Then, from (42), condition (28) is also satisfi ed for this case. Finally, βα (ẽrl (t)) is absolutely continuous in t since α(ẽrl ) is Lipschitz continuous and êr (t) and êl (t) are absolutely continuous since they are Filippov Solutions. T 2 Proof of Proposition   3: From (1) it follows that ë0 =T hc2 Ac e + k ∗ hT c Ac bc u + Ū . Thus ë0 can be bounded by |ë0 | ≤ hc Ac ||e|| + k ∗ hT c Ac bc 2f (t), which, can be rewritten, from (5), as (43) Using the relation ω = ωm + Ωe and the fact that ||e|| ≤ R, one has (44) t≥T1 Proof of Proposition 4: The lead fi lter estimation of the output derivative is given by êl = F −1 (s)ė0 + π. Thus, from (17), one has ǫl = h 1 − F (τ s) i F (τ s) ė0 + π (45) Equation (45) can be written as follows ǫl = −τ Q(τ s) ë0 + π F (τ s) where Q(τ s) = [F (τ s) − 1] /τ s Substituting (44) into (46), it follows that sup |ǫl (t)| ≤ τ Kl̄ C2 + π t≥T1 where Kl̄ is a positive constant and C2 is defi ned in Proposition 3 Then it is straightforward to see that lim sup |ǫl (ts)| ≤ τ Kl C2 t→∞ ts≥t where Kl > Kl̄ . = = ζ − λF (ε) −αsign(ε) − ë0 (47) where F (ε) = |ε|1/2 sign(ε). Lyapunov-like function: V (ε, ζ) = α |ε| + ζ 2 /2 which has from (47) the following time derivative V̇ = −λα |ε|1/2 − ζ ë0 ≤ −ζ ë0 ≤ |ζ| |ë0 | √ Since |ë0 | < Ka and |ζ| ≤ 2V 1/2 , one has √ V̇ ≤ 2Ka V 1/2 we know that if Vc (0) = V (0), then Using (40) condition (28) can be easily verifi ed Case 3: (|ẽrl | < ǫM − c) In this case α(ẽrl ) = 0, which implies, from (19), that ǫ(t) = ǫr (t). For this case the following statement can be made sup |ë0 (t)| ≤ C2 ε̇ ζ̇ Using the comparison equation √ 1/2 V̇c = 2Ka Vc δ1 (ẽrl ) [ǫM − δ1 (ẽrl )] c |ë0 | ≤ K1 ||e|| + K2 ||ω|| + K3 Proof of Lemma 1: Using the following variable transformations ε := x − e0 and ζ := u1 − ė0 system (14) can be rewritten as (46) V (t) ≤ Vc (t); ∀t ≥ 0 ρ2 = Vc , one obtains √ 2ρρ̇ = 2Ka ρ √ √ For ρ(0) 6= 0 → ρ̇ = 2Ka /2 → ρ(t) √ = 2Ka t/2 + ρ(0) For ρ(0) = 0 → ρ(t) ≡ 0 or ρ(t) = √ 2Ka t/2 1/2 2 2Ka t/2 + V (0) . 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