IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY 1

IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY 1 Robust Control With Exact Uncertainties Compensation: With or Without Chattering? Alejandra Ferreira, Member, IEEE, Francisco Javier Bejarano, and Leonid M Fridman, Member, IEEE Abstract The problem of robust exact output control for linear systems with smooth bounded matched unknown inputs is considered The higher order sliding mode observers provide both theoretically exact observation and unknown input identification In this paper, a methodology is proposed to select the most adequate output control strategy for matched perturbations compensation The aim of this paper is to investigate the possibility for exact uncertainties compensation using signals identified by high order sliding mode observers Towards this aim, we modify the hierarchical super-twisting observer in order to have the best observation and identification accuracy possible Then, two controllers are compared The first one is an integral sliding mode controller based on the observed values of the state variables The other strategy is based on the direct compensation of matched perturbations using their identified values The performance of both controllers is estimated in terms of the deterministic noise upper bounds, sampling step and execution time Based on these estimations, the designer may select the proper controller for the system Experimental results are given for an inverted rotary pendulum system Index Terms Observers, output robust control, sliding mode control, uncertainties identification I INTRODUCTION Motivation: CONTROL under heavy uncertainties is one of the main problems of modern control theory One of the most prospering control strategies for exact compensation of uncertainties is sliding mode control (see, eg, [1] [3]) Nevertheless, in many practical situations, the robustness properties of sliding mode controllers are tarnished by the so-called chattering effect (see, eg, [1], [4]) Recently, finite time exact observers based on higher order sliding modes were suggested [5] [12] These kind of observers provide both theoretically exact observation and unknown inputs identification Moreover, it was shown in [13] and [14] that the finite time theoretically exact observation and unknown inputs identification for linear time invariant systems with unknown inputs (LTISUI) is realizable if and only if the LTISUI is strongly observable In the design of an observer-based output robust control for LTISUI the next question arises: Manuscript received August 10, 2009; revised February 23, 2010 and May 17, 2010; accepted May 24, 2010 Manuscript received in final form July 29, 2010 Recommended by Associate Editor C Lagoa The authors are with the Department of Control, Engineering Faculty, Universidad Nacional Autónoma de México (UNAM), CP 04510, Mexico City, México (e-mail: da_ferreira@yahoocom) Digital Object Identifier 101109/TCST20102064168 Which type of uncertainties compensation is more reasonable, a sliding mode controller based on the observed state values, or a strategy based on direct compensation of matched perturbations using their identified values? Paper Contribution: In this paper, we propose a methodology to select the most adequate control strategy to compensate matched perturbations Towards this aim, the hierarchical super-twisting observer [14] is modified in order to use arbitrary order sliding mode robust exact differentiation [15], thereby achieving better precision in comparison with [13] and [14] The following two strategies are compared: compensation via a discontinuous integral sliding mode control based on the observed states which could present the chattering effect; compensation of matched perturbations via a continuous output feedback control that includes the identified perturbations The precision of both control strategies is estimated in terms of deterministic noise bounds, observer sampling step, and controller execution time Based on these estimations, a methodology is proposed which allows the designer to make an adequate selection of the control strategy The proposed methodology is experimentally validated in an inverted rotary pendulum system Paper Structure: In the next section, the problem statement is presented In Section III, we present a modification of the hierarchical observer Observation and identification precisions are discussed in Section IV In Section V, a continuous compensator for unmatched perturbations will be designed An integral sliding mode control strategy based on the state estimations provided by the higher order sliding mode observer is introduced in Section VI A methodology for the proper controller selection based on the hardware characteristics is proposed in Section VII Section VIII deals with experimental results illustrating the control schemes given in the manuscript II PROBLEM STATEMENT Consider a LTISUI where,,, are the state vector, the control, and the output of the system, respectively The disturbances and system uncertainties are represented by the unknown inputs function vector Furthermore, and The following conditions are assumed to be fulfilled henceforth A1) The pair is controllable (1) 1063-6536/$2600 2010 IEEE

2 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY A2) The system is strongly observable [16], (equivalently, the triple has no invariant zeros) A3) is absolutely continuous, there is a constant such that The goal of this paper is to design a robust output control for system (1) of the form (2) Now, we form the extended vector (5) where, which may be designed following any control strategy, is the nominal control for the nominal system (ie, ); is the identification-based compensator of the unknown input vector III OBSERVER DESIGN Now, we present a modification of the hierarchical high order sliding mode observer (see [14]) This scheme allows using a high order differentiator instead of a step by step differentiator in order to have a better observation and identification accuracy The high order sliding mode (HOSM) observer design consists of two stages: first, a Luenberger observer is used to maintain the norm of the estimation error bounded; then, by means of a differentiation scheme, the state vector is reconstructed Before introducing the observer, let us define the following notation For any matrix having, represents one of the matrices fulfilling and Let be a vector function, represents the th antidifferentiator of, ie,, A Hierarchical Super-Twisting Observer Modification Stage 1: In order to realize the observation process we need to be sure that the observation error will be bounded First, design an auxiliary dynamic system where and must be designed such that the matrix is Hurwitz Let, whose dynamic equations are Thus, has a bounded norm, ie, there exist a known constant and a finite time, such that (3) for all (4) Stage 2: This part of the state reconstruction is based on an algorithm that allows decoupling the unknown inputs from the successive derivatives of the output of the linear estimation error system Then, by moving the differentiation operator outside the parenthesis and defining, the following equation is obtained: The advantage of representing as in (6) rather than as in (5) can be better seen in the next step of the algorithm 2) Derive a linear combination of, ensuring that the derivative of this combination is unaffected by uncertainties, ie, Then form the extended vector Moving the differentiation operator outside the parenthesis from (7) we have that Using the previous equation, we can represent as a second-order derivative of a known function Then, from the above expression and from (6), and by moving the differentiation operator outside the parenthesis, we obtain where Thus, if needs to be reconstructed, it can be done from (8) using only a second order differentiator, as opposed to using (5) and (7) and differentiating twice j) A general step can be summarized as follows Derive Then, from the identity (6) (7) (8) (9) 0) Define 1) Derive a linear combination of the output, ensuring that the derivative of this combination is unaffected by the uncertainties, ie, we obtain the expression (10)

FERREIRA et al: ROBUST CONTROL WITH EXACT UNCERTAINTIES COMPENSATION 3 where Under A2) there exists a matrix, generated recursively by (10), that satisfies the condition (see, eg, [17]) This means that the algebraic equation has a unique solution for Such solution may be found by premultiplying both sides of the previous equation by That is (11) where Remark 1: From the above expression, it is clear that can be reconstructed not in an iterative manner, but in just one step using a high order differentiation; the only matrices that should be obtained in an iterative manner are and, which can be obtained using (9) with From (11), the reconstruction of is equivalent to the reconstruction of, which can be carried out by a linear combination of the output and its th derivatives Hence, a real time high order sliding mode differentiator will be used in order to provide the theoretically exact observation and unknown inputs identification Before introducing the HOSM differentiator, let us assume that we are dealing with smooth perturbations, ie: A4) has successive derivatives up to the order bounded by the same constant, ie,, for all ( is a known constant greater than zero) The previous statement allows realizing a th-order sliding mode differentiator, which is the highest order we can construct for this case Beforehand, let us define That is The HOSM differentiator is given by (12) (13) where is the differentiator order and,,, Consider,, the function vector is defined as The values of the s can be calculated as is shown in [15], is a Lipschitz constant of, which for our case can be calculated in the following way: starting from the fact that remains bounded by (4), the next derivative will be also bounded In general can be represented as a linear combination of and it can be verified that (14) B State Variables Observation In [15] there was shown that with the proper choice of the constants, there is a finite time such that the identity is achieved for every The vector can be reconstructed from the th-order sliding dynamics Thus, we achieve the identity, and consequently for all (15) where represents the estimated value of Therefore, the identity, for all is achieved C Uncertainties Identification Now, consider the system error dynamics (16) We can recover from the HOSM differentiator (13) in finite time, the equality is achieved for all and the next equation holds IV PRECISION OF THE OBSERVATION AND IDENTIFICATION PROCESSES (17) Suppose that we would like to realize observation with a sampling step while considering that a deterministic noise signal (a Lebesgue measurable function of time with a maximal magnitude ) is presented in the system output Let (18) where is the signal measured online As it follows from [15, Th 7], the error caused by the sampling time in the absence of noise for an th-order HOSM differentiator is for (19) Now, from [15, Th 6], the HOSM differentiator error provoked by a deterministic upper bounded noise will be for (20) Here, we are dealing with an th-order HOSM differentiator To recover the estimated state, differenti-

4 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY TABLE I PRECISION DUE TO SAMPLING STEP AND BOUNDED NOISE ations are needed From expressions (19) and (20), it follows that the observation error provoked by the sampling time is, while the observation error caused by a deterministic upper bounded noise is From (17), it follows that differentiations are needed in order to recover the estimated unknown input Therefore, from (19) the sampling step identification error will be, and the deterministic noise identification error (20) will be To analyze the total effect of both sampling step and deterministic noise errors, the next proposition is given Proposition 1: Let us assume, and with,, some positive constants and Then after a finite-time, the HOSM observation and identification error will be and, respectively Proof: It follows from [18, Th 31] Indeed, if is the signal to be differentiated, and it is affected by some noise (18), then according to the cited theorem, using the differentiator (13) of order, we achieve the following estimations after some finite time: Thus, from (12) and (15) we have that vector Thus, we need to differentiate times After the first applications of the STD, from (19) the observation error caused by the sampling time will be This error will be reflected as an output noise in the next application of the STD This means that after the second application of the STD, from (20) we will have an error of order So, applying recursively (20), after applications of the STD we will have a state estimation error For the unknown input identification it is necessary to differentiate one more time, ie, the STD is applied times successively, and the accuracy error (20) will increase to For the step-by-step observer given in [11] the error is the same as in the previously discussed scheme Now we consider the observer proposed in [13] For this observer two differentiation stages are needed to recover vector First, an th-order HOSM differentiator is applied, where is the biggest coordinate of the relative degree vector; it follows from (19) that the error will be Afterwards, a step-by-step super-twisting differentiator is used to achieve the full state space observation (20) and in this case the total observation error is Remark 5: The Euler integration method was used to realize the integration of the differentiator discontinuous (13) To avoid error amplification during the following stages another integration method should be used When no integration of discontinuous differential equations is needed, it is preferable to use the Runge Kutta integration method This way, the differentiator accuracy will be preserved throughout the entire process V EXACT OUTPUT FEEDBACK STABILIZATION (EOFS) Here, a compensation control law is designed based on the estimated states and the unknown input identification Consider the nominal system (21) As for the identification error, we obtain from (17) that is a stabilization control for the nom- The control signal inal system where the gain can be designed by any control strategy Let us design the second part of the control input (2) as Remark 2: Table I summarizes the observation and identification errors when the unknown input satisfies Assumption A4 Then, an th-order differentiator is used to improve precision Remark 3: The proposed observer here provides the best possible accuracy order with respect to the sampling step and bounded deterministic noises one can achieve on state observation and unknown inputs identification (see [19]) Remark 4: We would like to point out that the observer suggested here achieves a better precision than the recently published HOSM observers [11], [14], and [13] In the case of the hierarchical super-twisting observer [14], a step-by-step super-twisting differentiator (STD) scheme [20] (first-order HOSM differentiator) is used to recover the rows of where is the unknown input estimated in Section III-C Theoretically, assuming exact observation and identification, the equalities and hold after a finite time The EOFS control law is given by (22) applied to system (1) yields the following dynamic equation: Theoretically the continuous control exactly compensates the matched perturbations and the solutions for systems (1) and (21) coincide EOFS Realization Error: Theoretically, perturbations are exactly compensated in finite time Nevertheless, in the previous section we discussed how the discretization and deterministic

FERREIRA et al: ROBUST CONTROL WITH EXACT UNCERTAINTIES COMPENSATION 5 output noise, present in the observation and identification processes, affect the compensation accuracy Furthermore, an additional error, due to the actuator time constant, will cause an error of order (see [1] and [4]) Now, the EOFS controller stabilization error may be estimated by (23) VI OUTPUT INTEGRAL SLIDING MODE CONTROL (OISMC) In this section, we propose applying the ISM (see for example [2]) using the estimated states obtained by the HOSM observer; we will call it output integral sliding mode control (OISMC) Consider a control input of the form (2), where is the nominal control for the system without uncertainties (21) Let the nominal control be The compensator should be designed to reject the disturbance in the sliding mode on the manifold [21], ie, the equivalent control [1] The switching function is defined as where,,, and and the integral part is selected such that for all In other words, from the system state belongs to the sliding surface, where the equivalent sliding mode control should compensate the unknown input, that is, To achieve this purpose, is determined from the equation with form The switching surface takes the where The compensator is designed as a discontinuous unitary control Thus, the sliding mode manifold is attractive from if Finally, the control law (2) is designed as follows: VII EOFS AND OISMC COMPARISON Now, we analyze the accuracy of the HOSM observer and the identification procedure, combined with both control methodologies, EOFS (23) and OISMC (25) Recall that we are using a th-order HOSM differentiator and that we need the th and th derivatives for the state observation and unknown inputs identification, respectively Consider the following cases 1), ie, the controller execution error is greater than the identification process error In such a case, it would be suitable to use the EOFS strategy to avoid chattering 2), ie, the error related to the actuator time constant is less than the identification process error Thus, the error in the EOFS control strategy is mainly determined by the identification error In this case, OISMC strategy could be a better solution for systems tolerant to chattering with oscillation frequencies of order [7] 3), ie, the error provoked by the actuator time constant is less than the observation error Once again, the precision of the EOFS controller is determined by the precision of the identification process, and the precision of the OISMC controller is determined by the accuracy of the observation process However, it is necessary to remark that in this case the use of the OISMC controller could amplify the observer noise (see, eg, [7]) VIII EXPERIMENTAL RESULTS:INVERTED ROTARY PENDULUM Here, we present the experimental results of the inverted rotary pendulum system shown in Fig 1 An L-shaped arm, or hub, is connected to the dc motor shaft and pivots between 180 At the end of the arm, there is a suspended pendulum attached The experimental setup includes a PC equipped with a real time dspace acquisition platform with a minimum sample and integration time equal to 20 s The pendulum and hub angles are measured by two encoders with a resolution of 1024 pulses per rate and the motor driver bandwidth is 83 s The system state equations with, and, linearized along the point are (24) Again, in the ideal case, system (1) with given by (24) takes the form of (21) OISMC Realization Error: As we have seen, when the observation, identification and control processes are free from nonidealities, both controllers, EOFS and OISMC, give identical results However, in the practical case, the errors appearing in the complete control process should be taken into account In the case of the OISMC the stabilization error is the sum of the observation error plus the control error, ie, (25) A second-order HOSM differentiator is designed (ie, ) with gain The state is recovered using the first derivative (ie, ) The nominal control gain is given by First, an experiment was done to show the compensation effect using the EOFS control with 20 s and 400 s Recall that the state vector and the disturbances are reconstructed using the HOSM observer The pendulum position is shown in Fig 2 Specifically, from to 5 s, the control signal is acting without compensation ; from 5 s, a sinusoidal signal is added to the motor voltage acting as an

6 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY Fig 3 Comparison of the pendulum position x (t) [rad] between EOFS and OISM for three pairs of (; ) TABLE II PERFORMANCE OF THE SYSTEM FOR THREE SAMPLE AND EXECUTION TIMES Fig 1 Inverted rotary pendulum the previous section Table II summarizes the performance of the system with respect to the control design and sample and execution times Finally an experiment is done considering a sampling step 500 s and an execution time 80 s To verify the effects on the precision when different HOSM differentiator order is used Fig 4(a) shows the pendulum position when an OISMC based on the state estimation via a first order HOSM differentiator is used; on the other hand Fig 4(b) shows the stabilization of via a OISMC based on a second-order HOSM differentiator The precision is clearly improved by increasing the HOSM differentiator order Fig 2 Pendulum position x (t) [rad] and the estimated disturbance ^w(t) (dashed line) with the exogenous sinusoidal disturbance (solid line) exogenous matched perturbation; then in 10 s, the compensation control signal is added which considerably improves the performance of the pendulum around the equilibrium point It can be seen that, thanks to the control compensation, the system maintains robustness properties wrt the matched smooth perturbations Furthermore, three different experiments were done considering a different pair of values for and corresponding to the three main cases considered in Section VII Fig 3 shows the pendulum position using both methodologies In the left column we have the angular position using EOFS control for the considered cases; the right column shows the trajectories using OISMC Each row shows the results for the cases discussed in IX CONCLUSION Higher order sliding mode observers providing theoretically finite time exact state observation and estimation of absolutely continuous unknown inputs were presented These observers lead us to two possible approaches for absolutely continuous matched uncertainties rejection One is the compensation of such uncertainties using their identified values Another one is the sliding mode control design based on the estimated states values carried out by the observers In this paper: the hierarchical observation scheme is modified in order to use arbitrary order sliding mode robust exact differentiators [15] providing a better precision with respect to [14] and [13]; a theoretical accuracy analysis of the proposed higher order sliding mode observation and identification algorithms in terms of sampling time and deterministic noise upper bounds was presented;

FERREIRA et al: ROBUST CONTROL WITH EXACT UNCERTAINTIES COMPENSATION 7 Fig 4 Precision of x (t) [rad] using OISMC applying a (left) first-order HOSM differentiator and a (right) second-order HOSM differentiator two robust output feedback control strategies were compared: continuous compensation control based on the estimated states and the compensation of identified unknown inputs (EOFS); output integral sliding mode control based on estimated states (OISMC) a methodology is suggested for the selection of an appropriate controller based on the comparison of both control strategies considering the accuracy of observation and identification algorithms as well as the actuator time constant; the proposed methodology is experimentally validated in an inverted rotary pendulum system REFERENCES [1] V I Utkin, Sliding Modes in Control and Optimization Berlin, Germany: Springer Verlag, 1992 [2] V I Utkin, J Guldner, and J Shi, Sliding Modes in Electromechanical Systems London, UK: Taylor and Francis, 1999 [3] S Laghrouche, F Plestan, and A Glumineau, Higher order sliding mode control based on integral sliding mode, Automatica, vol 43, pp 531 537, 2007 [4] L Fridman, Singularly perturbed analysis of chattering in relay control systems, IEEE Trans Autom Control, vol 47, no 12, pp 2079 2084, Dec 2002 [5] J Alvarez, Y Orlov, and L Acho, An invariance principle for discontinuous dynamic systems with application to a coulomb friction oscillator, J Dyn Syst, Measure Control, vol 122, pp 123 126, 2000 [6] J Barbot, M Djemai, and T Boukhobza, Sliding mode observers, in Sliding Mode Control in Engineering, W Perruquetti and J Barbot, Eds New York: Marcel Dekker, 2002, pp 103 130 [7] I Boiko, Discontinuous Control Systems: Frequency-Domain Analysis and Design Boston, MA: Birkhäuser, 2009 [8] C Edwards and S Spurgeon, Sliding Mode Control London, UK: Taylor and Francis, 1998 [9] Y B Shtessel, I Shkolnikov, and A Levant, Smooth second-order sliding modes: Missile guidance application, 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with unknown inputs based on hierarchical super-twisting algorithm, Int J Rob Nonlinear Control, vol 17, pp 1734 1753, 2007 [15] A Levant, Higher-order sliding modes, differentiation and outputfeedback control, Int J Control, vol 76, pp 924 941, 2003 [16] M L J Hautus, Strong detectability and observerss, Linear Algebra and Its Applications, vol 50, pp 353 368, 1983 [17] B P Molinari, A strong controllability and observability in linear multivariable control, IEEE Trans Autom Control, vol 21, no 5, pp 761 764, Oct 1976 [18] M Angulo and A Levant, On robust output based finite-time control of LT I systems using HOSMs, in Proc IFAC Conf Anal Des Hybrid Syst (AHDS), 2009, pp 222 227 [19] A N Kolmogorov, On inequalities between upper bounds of consecutive derivatives of an arbitrary function defined on an infinite interval, (in Russian) Amer Math Soc Transl 2, pp 233 242, 1962 [20] A Levant, Robust exact differentiation via sliding mode technique, Int J Control, vol 34, pp 379 384, 1998 [21] B Drazenovic, The invariance conditions in variable structure systems, Automatica, vol 5, pp 287 295, 1969 Alejandra Ferreira was born in Mexico in 1976 She received the BSc degree from National Autonomous University of Mexico (UNAM), Mexico City, Mexico, in 2004, where she is pursuing the PhD degree in automatic control In 2000 2005, she was with the Instrumentation Department, Institute of Astronomy, UNAM Her professional interests include electronics design, observation, and identification of linear systems, sliding mode control, and its applications Francisco Javier Bejarano received the Master and Doctor degrees in automatic control from the CINVESTAV-IPN, Mexico City, Mexico, in 2003 and 2006, under the direction of Prof A Poznyak and Dr L Fridman He stayed one year at the ENSEA, France, and two years at UNAM, Mexico with respective posdoctoral positions He has published nine papers in international journals Leonid M Fridman (M 98) received the MS degree in mathematics from Kuibyshev State University, Samara, Russia, in 1976, the PhD degree in applied mathematics from the Institute of Control Science, Moscow, Russia, in 1988, and the DrSci degree in control science from Moscow State University of Mathematics and Electronics, Moscow, Russia, in 1998 From 1976 to 1999, he was with the Department of Mathematics, Samara State Architecture and Civil Engineering Academy From 2000 to 2002, he was with the Department of Postgraduate Study and Investigations at the Chihuahua Institute of Technology, Chihuahua, Mexico In 2002, he joined the Department of Control, Division of Electrical Engineering of Engineering Faculty, National Autonomous University of Mexico (UNAM), México He is an Editor of three books and five special issues on sliding mode control He has published over 200 technical papers His research interests include variable structure systems and singular perturbations Dr Fridman is an Associate Editor of the International Journal of System Science and Conference Editorial Board of IEEE Control Systems Society, Member of TC on Variable Structure Systems and Sliding mode control of IEEE Control Systems Society