Adaptive robust control with limited knowledge on systems dynamics, 1st ed , spandan roy, indra narayan kar, 2020 1292

Studies in Systems, Decision and Control 257 Spandan Roy Indra Narayan Kar Adaptive-Robust Control with Limited Knowledge on Systems Dynamics An Artificial Input Delay Approach and Beyond Studies in Systems, Decision and Control Volume 257 Series Editor Janusz Kacprzyk, Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland The series “Studies in Systems, Decision and Control” (SSDC) covers both new developments and advances, as well as the state of the art, in the various areas of broadly perceived systems, decision making and control–quickly, up to date and with a high quality The intent is to cover the theory, applications, and perspectives on the state of the art and future developments relevant to systems, decision making, control, complex processes and related areas, as embedded in the fields of engineering, computer science, physics, economics, social and life sciences, as well as the paradigms and methodologies behind them The series contains monographs, textbooks, lecture notes and edited volumes in systems, decision making and control spanning the areas of Cyber-Physical Systems, Autonomous Systems, Sensor Networks, Control Systems, Energy Systems, Automotive Systems, Biological Systems, Vehicular Networking and Connected Vehicles, Aerospace Systems, Automation, Manufacturing, Smart Grids, Nonlinear Systems, Power Systems, Robotics, Social Systems, Economic Systems and other Of particular value to both the contributors and the readership are the short publication timeframe and the world-wide distribution and exposure which enable both a wide and rapid dissemination of research output ** Indexing: The books of this series are submitted to ISI, SCOPUS, DBLP, Ulrichs, MathSciNet, Current Mathematical Publications, Mathematical Reviews, Zentralblatt Math: MetaPress and Springerlink More information about this series at http://www.springer.com/series/13304 Spandan Roy Indra Narayan Kar • Adaptive-Robust Control with Limited Knowledge on Systems Dynamics An Artificial Input Delay Approach and Beyond 123 Spandan Roy Robotics Research Center International Institute of Information Technology Hyderabad, Telangana, India Indra Narayan Kar Department of Electrical Engineering Indian Institute of Technology Delhi New Delhi, Delhi, India ISSN 2198-4182 ISSN 2198-4190 (electronic) Studies in Systems, Decision and Control ISBN 978-981-15-0639-0 ISBN 978-981-15-0640-6 (eBook) https://doi.org/10.1007/978-981-15-0640-6 © Springer Nature Singapore Pte Ltd 2020 This work is subject to copyright All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed The use of general descriptive names, registered names, trademarks, service marks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore Dedicated to my parents, sister and grandparents Spandan Roy Dedicated to my former students Indra Narayan Kar Preface In the quest to negotiate the inevitable effects of parametric and non-parametric uncertainties in a system during a control task, researchers have broadly applied two different classes of control strategies, namely, adaptive control and robust control However, while a conventional adaptive control requires structural knowledge of the system, a conventional robust control requires predefined bound on the uncertainties It is often difficult to satisfy either of these constraints in practice Under such circumstances, the recent research direction has reoriented toward developing adaptive-robust control (ARC) law that can address the issues while retaining the individual advantages of the adaptive and robust controllers Unfortunately, to date, the state-of-the-art ARC designs have not been able to fully achieve the objective: they either depend on significant a priori knowledge of system model or impose restrictions on the state evolutions via various assumptions To reduce dependency on the accuracy of system modelling while designing a controller, researchers have also applied black-box techniques like neural-network (NN) However, such schemes require expertise knowledge apart from being computationally expensive In such scenario, an alternate literature has grown where time delay is invoked intentionally/artificially into a delay-free system to approximate the unknown/unmodelled system dynamics Such approximation method is typically called time-delayed estimation (TDE) or artificial delay-based approximation, and the resulting control law is termed in the literature as time-delayed control (TDC) Compared to an NN-based controller, TDC is found to be significantly easier to implement as well as it does not require any expertise knowledge In view of the capability of a TDE method in reducing dependency on system model, attempts have been made to reap this benefit to design ARC Nonetheless, there exist unsolved design issues of TDC which, apart from restricting its flexibility in practical applications, even may cause serious impediment to system stability Moreover, the state-of-the-art switching law based ARCs suffer from the over- and under-estimation problems of switching gain, where the first issue demands vii viii Preface unnecessary high control input and the second one compromises tracking accuracy These prevailing issues foster the aim of this book as to develop an ARC framework for a class of uncertain systems with minimal knowledge of system dynamics model, which can alleviate the over- and under-estimation problems of switching gain For control design purpose, this book particularly concentrates on a class of Euler-Lagrange (EL) systems which encompasses a large number of real-world systems ranging from simple robotic manipulator, mobile robot, pneumatic muscles to complex systems such as humanoids, ship dynamics, aircraft systems, etc EL systems have found enormous applications over the years in multitude of domains such as industry automation, planetary mission, surveillance, etc., to name a few Therefore, achieving autonomy in these systems while accomplishing a specified task has always attracted the control systems research community Based on the detailed discussions regarding the issues of conventional TDE-based controllers and of the ARCs in Chap 1, this book brings out five major research outcomes spanning across Chaps 2–6 and they are briefly enumerated below: • A new stability analysis, based on the Lyapunov-Razumikhin theorem, is carried out in Chap to solve various design issues of the state-of-the-art TDE-based control designs Accordingly, the study establishes a relation between controller gains and time delay; provides analytical measure to the impact of the selection of time delay on system stability and allows the continuous-time system to assimilate discrete-time feedback used for TDE process • A TDE-based ARC framework is proposed in Chap to solve the long standing challenge of alleviating the over- and under-estimation problems in the adaptive evaluation of switching gains • To reduce the effect of measurement error in the absence of state-derivatives, a new TDE-based controller as well as its ARC framework are derived in Chap via only position feedback The proposed controllers utilize only past position data (for EL systems) to estimate the state-derivative terms • The Lyapunov-Krasovskii based stability notions for TDE-based designs, treated as alternate/parallel study to the Lyapunov-Razumikhin based analysis in Chaps 2–4, are provided in Chap • A new ARC strategy is derived in Chap which, in contrast to a TDE-based law, does not require any knowledge of system dynamics terms Furthermore, the proposed ARC avoids any separate module for uncertainty approximation and hence, simplifies the controller structure To validate the effectiveness of all the controllers developed in this book, suitable experimental results are provided using a wheeled mobile robot which serves as an appropriate exemplar of EL systems Preface ix This book is intended for graduate students who wish to work in the fields of artificial delay-based design TDC and ARC Apart from being the first book on TDE that ensembles its motivation, origin, issues and corresponding novel solutions, this endeavour also extensively highlights how an ARC and a TDE method can benefit from each other when applied simultaneously Hyderabad, India New Delhi, India August 2019 Spandan Roy Indra Narayan Kar Acknowledgements Research is not a singular journey, but an outcome of plurality Besides the unconditional support of my family, I would like to acknowledge those who have helped me to grow as a researcher I would like to thank my colleagues Drs Sayan Basu Roy, Abhishek Dhar and Abhilash Patel for the various discussions leading to polishing and strengthening my existing knowledge base I would like to express my gratitude to my Master’s supervisor Prof Shambhunath Nandy (CSIR-CMERI, India) and my Postdoc supervisor Prof Simone Baldi (TU Delft, The Netherlands) whose dedication to work has been a great source of inspiration For a Control Systems researcher, negative feedback is more important With that spirit, I would like to thank those people whose (unjustified/unrealistic) comments/suggestions and attitude have taught me what I should never become as a researcher Hyderabad, India August 2019 Spandan Roy xi 130 Adaptive-Robust Control for Systems with State-Dependent … Let us define a scalar z as < z < λmin (G) Then using (6.41) and (6.45), (6.44) is modified as V˙1 ≤ −λmin ( )||e||2 − {λmin (G) − z}||e f ||2 − z||e f ||2 + ≤ − V1 − z||e f || + where 1ζ + ϑ ϑ, (6.46) min{(λmin (G) − z), λmin ( )} M Hence, V˙1 ≤ − V1 is guaranteed for this case when ||e f || = || ξ || ≥ ( 1ζ + ϑ)/z (6.47) Considering the stability results of all the cases, it can be realized that V˙1 ≤ − { , } V1 is guaranteed when ||ξ || ≥ max ι, ( ζ + ϑ)/z || || Therefore, it can be concluded that the closed-loop system is UUB Remark 6.6 It is noteworthy that the condition (6.23) is necessary for stability of the system Moreover, high values of ς helps to reduce ι which consequently can improve controller accuracy However, one needs to be careful that too high value of ς may excite the condition γ ≤ β leading to the increment in all the gains γ , θˆi , i = 0, 1, Further, the scalar terms z, ϑ, ψ, μ2 , ζ, θ¯i and γ¯ are only used for the purpose of analysis and not used to design control law Remark 6.7 The importance of the auxiliary gain γ can be realized from Theorem 6.1 and Theorem 6.2 It can be observed from (6.24) that t1 gets reduced due to the presence of α3 (contributed when γ˙ > as in (6.18)) which leads to faster adaptation Moreover, the negative fourth degree term ‘−ςβ||ξ ||4 ’ in f p (||ξ ||) (contributed when γ˙ < as in (6.18)) ensures system stability for Case (2) by making f p (||ξ ||) ≤ for ||ξ || ≥ ι This also indicates the reason for selecting β > while lower bounds of other gains θˆi , i = 0, 1, are selected as zero Special Case: The quadratic term ||ξ ||2 in the uncertainty bound (6.12) is con˙ (through Property in (6.10)) EL systems such tributed by the Coriolis term C(q, q) ˙ as robotic manipulator, underwater vehicles, ship dynamics etc includes C(q, q) However, there also exist EL systems (e.g reduced order WMR system, mass˙ For such systems, spring-damper system) which does not have the term C(q, q) 6.4 Stability Analysis of ASRC 131 the uncertainty bound (6.12) would turn out to posses the following LIP structure: ||σ || ≤ θ0∗ + θ1∗ ||ξ || Y (ξ )T ∗ , (6.48) where Y(ξ ) = [1 ||ξ ||]T and ∗ = [θ0∗ θ1∗ ]T Hence, following the switching gain laws (6.16)–(6.19), the control laws for uncertainty structure (6.48) are modified as ρˆ = θˆ0 + θˆ1 ||ξ || + γ YT ˆ + γ , (6.49) (i) for ||e f || ≥ θ˙ˆi = γ˙ = αi ||ξ ||i ||e f || if {eT e˙ > 0} ∨ { ˆ i=0 θi ≤ 0} ∨ {γ ≤ β} −αi ||ξ || ||e f || otherwise, i if {eT e˙ > 0} ∨ { α3 ||e f || −ς α3 ||ξ || ˆ i=0 θi ≤ 0} ∨ {γ ≤ β} (6.50) (6.51) otherwise, (ii) for ||e f || < θ˙ˆi = 0, γ˙ = 0, with θˆi (t0 ) > 0, i = 0, 1, γ (t0 ) > β (6.52) (6.53) Closed-loop system stability employing (6.49)–(6.52) can be analysed exactly like Theorem 6.2 using the following Lyapunov function: V1 = V + i=0 2 θ˜ + γ 2αi i 2α3 (6.54) One can verify that, the cubic polynomial 2|| || θ0∗ + θ1∗ ||ξ || + θ2∗ ||ξ ||2 ||ξ || in f p (||ξ ||) of Case (2) would be modified as a quadratic polynomial 2|| || θ0∗ + θ1∗ ||ξ || ||ξ || using (6.48) and (6.54) Hence, following the argument in Remark 6.7, it can be noticed that a cubic term −ς α3 ||ξ ||3 is selected in the adaptive law (6.52) for closed-loop system stability Thus, with EL system (6.6), two generally probable structures for ||σ || are: • Y(ξ ) = [1 ||ξ || ||ξ ||2 ]T , ∗ = [θ0∗ θ1∗ θ2∗ ]T and • Y(ξ ) = [1 ||ξ ||]T , ∗ = [θ0∗ θ1∗ ]T Both the aforementioned situations are covered here For better inference, the ASRC algorithm is summarized in Table 6.1 for various system structures 132 Adaptive-Robust Control for Systems with State-Dependent … Table 6.1 ASRC Algorithm for various system structures System structure LIP structure of ||σ || ˙ =0 (6.6) C(q, q) ˙ =0 C(q, q) (6.12) (6.48) Control law (6.15)–(6.20) (6.15), (6.49)–(6.53) 6.5 Comparison with Various Adaptive-Robust Law and Some Design Aspects It was mentioned in Sect 3.2 of Chap that two ARC laws were proposed in [28], i.e., Eqs (1.17) and (1.18) The ARC law (1.18) requires the knowledge of nominal system model while (1.17) does not However for both of these laws, as mentioned earlier, the ASMC in [28] assume that the uncertainties are upper bounded by an unknown constant, which is conservative for EL systems To gain further insight into the advantage of the proposed adaptive law, the ARC law (1.17) for switching gain K is rewritten as follow: K˙ = K¯ ||s||sgn(||s|| − ¯ ), K if K > K if K ≤ K , (6.55) where ¯ , K¯ , K ∈ R+ are user-defined scalars and s is the sliding surface It can be observed from (6.55) that when ||s|| ≥ ¯ the switching gain K increases monotonically even if error trajectories move close to ||s|| = This gives rise to the overestimation problem of switching gain Again, even if K is sufficient to keep ||s|| within ¯ , it decreases monotonically when ||s|| < ¯ So, at certain time, K would become insufficient and error will increase again However, K will not increase (rather it keeps on decreasing) until ||s|| > ¯ , which creates the underestimation problem Low (resp High) value of ¯ may force K to increase (resp decrease) for longer duration when ||s|| ≥ ¯ (resp ||s|| < ¯ ) resulting in escalation of the overestimation (resp underestimation) problem of ASMC Whereas, ASRC allows its gains to decrease when error trajectories move towards ||e|| = and ||e f || ≥ (overcoming overestimation problem) and keeps the gains unchanged when they are sufficient to keep the error within the ball B (overcoming underestimation problem) Nevertheless, the overall switching gain ρˆ will still increase or decrease if ||ξ || increases or decreases, respectively when ||e f || < as ρˆ is an explicit function of ||ξ ||, apart from θˆi ’s and γ It is interesting to note from (6.27) of Theorem 6.1 that a lower bound of ||e f || is required for the gains θˆi and γ For ||e f || < , this lower bound is and hence finite time decrement of the gains cannot be guaranteed Hence, the gains θˆi and γ are not updated when ||e f || < This, however, may lead to potential overestimation particularly for this case as gains not decrease even if the error decreases For ARTDC in Chap usage of acceleration feedback ensured that sT s˙ could have been 6.5 Comparison with Various Adaptive-Robust Law and Some Design Aspects 133 used in the ARC law and ||s|| > was considered Similarly, the ASMC work [34] can completely overcome the over- and under-estimation issue as it does not involve any threshold value in the adaptive law as in (6.17)–(6.19) Therefore, we say ASRC can alleviate the over- and under-estimation problem, rather overcoming like [34] On the other hand, ASRC does not use acceleration information and only uses position and velocity feedback More importantly, the major advantage of ASRC over ARTDC and ASMC [34] is that ASRC does not require any knowledge of the system dynamics at all, while the other two need the knowledge of the mass/inertia matrix However, as a trade-off, the complexity of ASRC is larger than that of ARTDC, as ASRC needs to adapt for three unknown parameters (θ0∗ , θ1∗ , θ2∗ ) compared to only one (the unknown upper bound of the TDE error c) for ARTDC Additionally, ASRC requires a stabilizing gain γ Hence, it is beneficial to reduce so that θˆi ’s and γ can be updated over a long range This can be accomplished by simply increasing η, as can be noted from (6.21) Moreover, it is noteworthy that increment of η does not necessarily demand increased control input This argument can be explained as follow: the control input τ can be written in the following way from (6.16)–(6.18) for ||e f || ≥ : τ= i=0 t tin ηαi ||ξ ||i ||e f ||dt + ηα3 t ||e f ||dt ||e f ||ε tin ef , ||e f || when gains increase and, τ= i=0 t tin (−ηαi ||ξ ||i ||e f ||)dt − ης α3 t ||ξ ||4 dt ||e f ||ε tin ef , ||e f || when gains decrease It is obvious that ρˆ would be non-negative by virtue of the ARC law (6.17)–(6.18) The important aspect to observe here is that η effectively governs the rate of change of θˆi ’s and γ along with αi ’s As a matter of fact, a designer can always split a desired value among η and αi For example, if a designer wants the gains to increase at a rate of 10 for θˆ0 , then it can be taken as η = 10 and α0 = giving a combined value of ηα0 = 10 The benefit of taking higher value of η is that, one can reduce for a fixed ε and can increase the adaptation range for the gains 6.6 Experimental Validation: Application to a WMR In this section, performance of the proposed ASRC is evaluated experimentally using the PIONEER-3 WMR in comparsion with ASMC [28] ASRC does not require any knowledge of the system dynamics/parameters Therefore, for fair comparison, we have selected the ASMC with adaptive law (6.55) which also does not require knowledge of the system dynamics for controller design However, ASMC assumes 134 Adaptive-Robust Control for Systems with State-Dependent … that the uncertainty is upper bounded by a constant and effects of such assumption on the controller performance would be studied consequently The reduced order dynamics (as mentioned in Chap 2) for the WMR is given below: ⎡ rw b ⎢ rw ⎢b q˙ = ⎢ ⎢ M R qă R + C R q R = τ , b b cos(ϕ) − d sin(ϕ) sin(ϕ) + d cos(ϕ) rw /b rw b rw b b b cos(ϕ) + d sin(ϕ) sin(ϕ) − d cos(ϕ) −rw /b ⎤ ⎥ ⎥ ⎥ q˙ R , ⎥ ⎦ (6.56) (6.57) S(q) where M R = ST MS = k1 k2 k2 , k1 (6.58) k1 = Iw + { I¯ + m(b2 /4 − d )}(rw /b2 ), k2 = {m(b2 /4 + d ) − I¯}(rw /b2 ), C R = ST (MS˙ + CS) = 0 , q R = [θr θl ]T , (6.59) where q ∈ R5 = {xc , yc , ϕ, θr , θl }, q R = [θr θl ]T ; τ = [τr , τl ]T is the control input vector; (xc , yc ) are the coordinates of the center of mass (CM) of the system and ϕ is the heading angle; (θr , θl ) and (τr , τl ) are rotation and torque inputs of the right and left wheels, respectively; m, rw and b represent the system mass, wheel radius and robot width, respectively; d is the distance to the CM from the center of the line joining the two wheel axis As WMR moves on ground, the gravity vector g(q) and the potential function would certainly be zero which implies that M R , C R satisfies the Properties and ˙ − 2C)e f = and [44] The main implication of system Property is to hold eTf (M this can be easily verified from (6.58)–(6.59) The WMR dynamics (6.56) is based on rolling without slipping condition and hence the term F(q˙ R ) is omitted However, in practical circumstances a WMR is always subjected to uncertainties like friction, slip, skid, external disturbance etc Hence, incorporating (6.59), the system dynamics (6.56) is modified as M R qă R + F(q˙ R ) + ds = τ , (6.60) where F(q˙ R ) and ds are considered to be the unmodelled dynamics and disturbance respectively The ASRC framework does not require any knowledge of M R , F and ds Further, unlike [11], it avoids any prior knowledge of the upper bound of uncertainties as they are estimated online by its adaptive law Hence, ASRC is insensitive towards the parametric variations and characteristics of uncertainty Since Coriolis component 6.6 Experimental Validation: Application to a WMR 135 is zero, the ASRC algorithm applied to the WMR is based on the control laws (6.15), (6.49)–(6.53) It is to be noted that S(q) is only used for coordinate transformation and WMR pose (xc , yc , ϕ) representation and, not for control law design 6.6.1 Experimental Scenario The WMR is directed to follow a circular path using the following desired trajectories: θrd = (4t + π/10) rad, θld = (3t + π/10) rad PIONEER uses two quadrature incremental encoders (500 ppr) and always starts from θr (0) = θl (0) = and the initial wheel position error (π/10, π/10) rad helps to realize the error convergence ability of the controllers The desired WMR pose (xcd , ycd , ϕ d ) and actual WMR pose (xc , yc , ϕ) can be determined from (2.36) using (θ˙rd , θ˙ld ) and (θ˙r , θ˙l ) (obtained from encoder) respectively with rw = 0.097 m, b = 0.381 m, d = 0.02 m (supplied by the manufacturer) The control laws for both ASRC and ASMC are written in VC++ environment Considering the hardware response time, the sampling interval is selected as 20 ms for all the controllers Further, to create a dynamic payload variation, a 3.5 kg payload is added (kept for s) and removed (for s) periodically on the robotic platform at different places for ASRC are selected as: G = = I, ε = 0.5, η = √ √ The controller parameters 2, αi = α3 = 10/ ∀i = 0, 1, β = 0.1, ς = 10 To properly judge the performance of ASRC, two different sets of initial gain conditions are selected: θˆi (0) = γ (0) = 20 and θˆi (0) = γ (0) = 10 ∀i = 0, For better clarity, we have denoted the ASRC with θˆi (0) = γ (0) = 20 as ASRC1 and, the ASRC with θˆi (0) = γ (0) = 10 as ASRC2 The reason for choosing ASRC with two different initial gain conditions would be clarified subsequently Further, the controller parameters for ASMC are selected as s = e f , K¯ = 10, K (0) = 35, ¯ = 0.5 6.6.2 Experimental Results and Comparison The path tracking performance of ASRC1 is depicted in Fig 6.2 while following the desired circular path The tracking performance comparison of ASRC1 and ASRC2 with ASMC is illustrated in Fig 6.3 in terms of E p (defined by the Euclidean distance in tracking error of xc and yc position) ASMC framework is built on the assumption that uncertainties are upper bounded by an unknown constant (i.e θ1∗ = θ2∗ = for general EL systems and θ1∗ = for WMR as C R = 0) This assumption is restrictive in nature for EL systems and the switching gain is thus insufficient to provide the necessary robustness As a matter of fact, both ASRC1 and ASRC2 provide better tracking accuracy over ASMC 136 Adaptive-Robust Control for Systems with State-Dependent … 3.5 2.5 c y (m) Desired path Path tracked with ASRC1 1.5 0.5 −0.5 −2 −1.5 −1 −0.5 0.5 x (m) 1.5 c Fig 6.2 Circular path tracking performance of ASRC1 0.25 ASRC1 ASMC ASRC2 0.15 p E (m) 0.2 0.1 0.05 10 15 20 25 time (sec) Fig 6.3 Tracking performance comparison of ASRC1, ASRC2 and ASMC 30 6.6 Experimental Validation: Application to a WMR 137 36 34 32 30 28 10 15 20 25 30 35 40 25 30 35 40 time (sec) 0 10 15 20 time (sec) Fig 6.4 The response of the switching gain K of ASMC To evaluate the benefit of the proposed adaptive-robust law, the evaluation of switching gain for ASMC and ASRC1 are provided in Figs 6.4 and 6.5 respectively Figure 6.4 reveals that K , the switching gain of ASMC, increases even when ||s|| approaches towards ||s|| = during the time t = 0−1.2 s This is due to the fact that K does not decrease unless ||s|| < ¯ and invites the overestimation problem On the other hand for ASRC1, it can be seen from Fig 6.5 that all the gains γ , θˆ0 , θˆ1 decrease when ||e R || (e R q R − qdR ) decreases during t = 0−1 s when ||e f R || ≥ (||e f R || e˙ R + e R ) Therefore, ASRC1 overcomes the overestimation problem which is encountered in ASMC Further, K decreases monotonically during t=1.2−38.5 s, when ||s|| < ¯ This monotonic decrement makes K insufficient to tackle uncertainties after certain time as a consequence of underestimation problem Therefore, ||s|| starts increasing again for t > 38.5 s leading to poor tracking accuracy and K increases again when ||s|| ≥ ¯ Gains of ASRC1, on the contrary, stay unchanged for t > s when the gains are sufficient to keep ||e f R || < avoiding any underestimation problem It can be noticed from Fig 6.5 that initial gains of ASRC1 are high enough such that ||e R || decreases from the beginning and so the gains γ , θˆi Hence, it would be prudent to verify the capability of ASRC in alleviating the overestimation- 138 Adaptive-Robust Control for Systems with State-Dependent … 20 18 16 14 12 10 15 20 25 30 35 40 25 30 35 40 time (sec) 2.5 1.5 0.5 10 15 20 time (sec) Fig 6.5 The response of the switching gains of ASRC1 underestimation problem while starting with relatively low gains Hence, the same experiment for ASRC is repeated with much lower initial value of the gains compared to ASRC1, which is termed as ASRC2 The tracking performance and evaluation of the switching gain for ASRC2 is shown in Figs 6.2 and 6.6, respectively It can be noticed that initially the tracking error is high for ASRC2 compared to ASRC1 and ASMC (initial gain K (0) = 35) due to low initial gains However, at t ≥ s tracking accuracy of ASRC2 begins to improve as the gains became sufficient to negotiate the uncertainties and eventually the tracking performance of ASRC2 is found to be similar to ASRC1 from t ≥ 12 s and much improved compared to ASMC This proves that the proposed adaptive law can perform satisfactorily even with low initial conditions of the gains Another important aspect to verify is whether ASRC2 can alleviate the overand under-estimation issues similar to ASRC1 It can observed from Fig 6.6 that when ||e f R || > , the gains follow the pattern of ||e R || Due to the initial low values, ||e R || increases and so the gains; similarly at t ≥ s the gains decrease as ||e R || decreases Further, at t ≥ s the gains remain unchanged when they were sufficient to keep the filtered tracking error ||e f R || ≤ , thus overcoming the underestimation problem Moreover, the gains not increase during t = 3.78–8 s when 6.6 Experimental Validation: Application to a WMR 139 25 20 15 10 10 15 20 25 30 35 25 30 35 time (sec) 0 10 15 20 time (sec) Fig 6.6 The response of the switching gains of ASRC2 ||e R || decrease sand thus avoids the overestimation problem Hence, low initial gain conditions not affect the capability ASRC2 in alleviating the over- and under-estimation problems 6.7 Summary This chapter introduces a new ARC architecture, ASRC, for a class of uncertain EL systems where the upper bound of the uncertainty satisfies a LIP structure, even though the uncertainties itself can be LIP or NLIP Unlike the conventional ARC strategies, ASRC neither requires the structural knowledge of the system nor presume the overall uncertainties or its time derivative to be upper bounded by a constant The switching control law of ASRC negotiates the uncertainties by exploiting the structure of the upper bound of the uncertainty without any knowledge of the system dynamics parameters Moreover, the unique polynomial LIP structure of the upper bound of uncertainty is exploited to derive the UUB stability of the closed-loop system The 140 Adaptive-Robust Control for Systems with State-Dependent … adaptive switching law of ASRC is able to alleviate the over- and under-estimation problems of switching gain Experimental results of ASRC using a wheeled mobile robot noted improved control performance in comparison to the existing adaptive sliding mode control 6.8 Notes The control framework in this chapter, like other chapters throughout this book, is dedicatedly designed for a class of EL systems [45] Nevertheless, interested readers can see [38], where, in line with the similar concept outlined in this chapter, we have designed an ARC for a quite a general class of nonlinear systems However, due to this generalization, the approach in [38], when applied to an EL system as a special case, requires some nominal knowledge of mass matrix and the knowledge of its upper bound References Hsia, T., Gao, L.: Robot manipulator control using decentralized linear time-invariant timedelayed joint controllers In: Proceedings IEEE International Conference on Robotics and Automation, pp 2070–2075 IEEE (1990) Hsia, T.S., Lasky, T., Guo, Z.: Robust independent joint controller design for industrial robot manipulators IEEE Trans Ind 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solutions of TDC, 31 E Euler-Lagrange Systems, I Issues of conventional TDC, Issues with conventional ARC, 44 L Linear-in-Parameters (LIP), 119 LK method for ARPOTDC, 102 M Monod kinetic, 119 O Over-estimation of switching gain, 10 P PIONEER, 33 Position-Only Time-Delayed Control (POTDC), 71 R Razumikhin condition, 14 S Sampled data stability, 32 State-dependent uncertainty, 118 State-dependent uncertainty in EL systems, 121 © Springer Nature Singapore Pte Ltd 2020 S Roy and I N Kar, Adaptive-Robust Control with Limited Knowledge on Systems Dynamics, Studies in Systems, Decision and Control 257, https://doi.org/10.1007/978-981-15-0640-6 143 144 State-derivative estimation, 71 T TDE error, 26 Time-Delayed Control (TDC), 24 Time-Delayed Estimation (TDE), 24 Author Index U Under-estimation of switching gain, 10 W Wheeled Mobile Robots (WMR), 33 ... parameters Lyapunov-Krasovskii Nonlinear in parameters Position Only Time-Delayed Control Robust Outer-loop Control Sliding Mode Control Time-Delayed Control Time-Delayed Estimation Uniformly Ultimately... Adaptive-Robust Control with Limited Knowledge on Systems Dynamics, Studies in Systems, Decision and Control 257, https://doi.org/10.1007/97 8-9 8 1-1 5-0 64 0-6 _1 Introduction simple electro-mechanical systems However,... interests include nonlinear systems theory, contraction analysis, adaptive robust control, time-delay system, robust-optimal control theory, cyber-physical systems, control of multi-agent systems, and