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Asymptotic Stabilization for a Class of Output-Constrained Nonholonomic Systems*
10th IFAC IFAC Symposium Symposium on on Nonlinear Nonlinear Control Control Systems Systems 10th 10th Symposium on Control Systems August 23-25, 2016. Monterey, California, USA 10th IFAC IFAC Symposium on Nonlinear Nonlinear Control Systems August 23-25, 2016. Monterey, California, USA 10th IFAC Symposium on Nonlinear Control Systems August USA Available August 23-25, 23-25, 2016. 2016. Monterey, Monterey, California, California, USA online at www.sciencedirect.com August 23-25, 2016. Monterey, California, USA
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Asymptotic Stabilization for a Class of Asymptotic Stabilization for Asymptotic Stabilization for a a Class Class of of Output-Constrained Nonholonomic Output-Constrained Nonholonomic ⋆ Output-Constrained Nonholonomic Systems ⋆⋆ Systems Systems ∗,∗∗ ∗∗∗ ∗∗
Fangzheng Yuqing,Wu Fushun Fangzheng Gao Gao ∗,∗∗ Yuqing,Wu ∗∗∗ Fushun Yuan Yuan ∗∗ ∗,∗∗ ∗∗∗ ∗∗ ∗,∗∗ Yuqing,Wu ∗∗∗ Fushun ∗∗ ∗∗ ∗∗∗∗ Fangzheng Gao ∗∗ Yanling Shang ∗∗∗∗ Yuan Fangzheng Gao Yuqing,Wu Yuan Hejun Yao ∗,∗∗ ∗∗∗ Fushun ∗∗ Hejun Yao Yanling Shang Fangzheng Gao Yuqing,Wu Fushun Yuan ∗∗ ∗∗∗∗ ∗∗ Yanling Shang ∗∗∗∗ Hejun Yao Hejun Yao ∗∗ Yanling Shang ∗∗∗∗ Hejun Yao Yanling Shang ∗ ∗ School of of Automation, Automation, Southeast Southeast University, University, Nanjing Nanjing 210096, 210096, School ∗ ∗ School of Automation, Southeast University, Nanjing 210096, University, Nanjing 210096, P. (e-mail: ∗ School of Automation, P. R. R. China China Southeast (e-mail: [email protected]). [email protected]). School of Automation, Southeast University, Nanjing 210096, ∗∗ P. R. China (e-mail: [email protected]). ∗∗ School of Mathematics P. R. China (e-mail: [email protected]). and Statistics, Anyang Normal University, School of Mathematics and Statistics, Anyang Normal University, P. R. China (e-mail: [email protected]). ∗∗ ∗∗ School of Mathematics and Statistics, Anyang Normal University, of and Statistics, Anyang Normal Anyang 455000, P. P. R. R. China (e-mail: [email protected]) ∗∗ School Anyang 455000, (e-mail: [email protected]) of Mathematics Mathematics andChina Statistics, Anyang Normal University, University, ∗∗∗School Anyang 455000, P. R. China (e-mail: [email protected]) ∗∗∗ Anyang 455000, P. R. China (e-mail: [email protected]) Institute of Automation, Qufu Normal University, Qufu 273165, 273165, Institute of Automation, Qufu Normal University, Qufu ∗∗∗ Anyang 455000, P. R. China (e-mail: [email protected]) ∗∗∗ of Automation, Qufu Normal University, Qufu 273165, Institute of Qufu Normal University, Qufu 273165, P. R. [email protected]) ∗∗∗ Institute P.Automation, R. China China (e-mail: (e-mail: [email protected]) Institute of Automation, Qufu Normal University, Qufu 273165, ∗∗∗∗ P. R. China (e-mail: [email protected]) ∗∗∗∗ School of P. R. China (e-mail: [email protected]) Software Engineering, Anyang Normal University, School of Software Engineering, Anyang Normal University, P. R. China (e-mail: [email protected]) ∗∗∗∗ ∗∗∗∗ School of Software Engineering, Anyang Normal University, School Software Anyang Normal Anyang 455000, P. Engineering, R. China China (e-mail: (e-mail: [email protected]) ∗∗∗∗Anyang 455000, P. R. [email protected]) School of of Software Engineering, Anyang Normal University, University, Anyang 455000, P. R. China (e-mail: [email protected]) Anyang 455000, P. R. China (e-mail: [email protected]) Anyang 455000, P. R. China (e-mail: [email protected]) Abstract: This This paper paper addresses addresses the the problem problem of of asymptotic asymptotic stabilization stabilization for for aa class class of of Abstract: Abstract: This paper addresses the problem of asymptotic stabilization for aa class of Abstract: This paper the problem of asymptotic for class of nonholonomic systems in chained with output constraint. nonlinear is nonholonomic systems inaddresses chained form form with output constraint. A Astabilization nonlinear mapping mapping is first first Abstract: This paper addresses the problem of asymptotic stabilization for a class of nonholonomic systems in chained form with output constraint. A nonlinear mapping is first nonholonomic systems in chained form with output constraint. A nonlinear mapping is first introduced to transform the output-constrained system into a new unconstrained one. Then, by introduced to transform thechained output-constrained systemconstraint. into a newAunconstrained one. Then, by nonholonomic systems in form with output nonlinear mapping is first introduced to transform the output-constrained system into aa new unconstrained one. Then, by introduced to the output-constrained system into one. Then, employing the backstepping technique and switching switching control strategy, state feedback feedback controller employing the backstepping and control aa state introduced to transform transform the technique output-constrained system into strategy, a new new unconstrained unconstrained one. controller Then, by by employing the backstepping technique and switching control strategy, a state feedback controller employing the backstepping switching control strategy, aa state feedback controller is successfully constructed totechnique guaranteeand that the states states of closed-loop closed-loop system are asymptotically asymptotically is successfully constructed to guarantee that the of system are employing the backstepping technique and switching control strategy, state feedback controller is successfully successfully constructed toviolation guarantee that the states of of closed-loop closed-loop system are asymptotically asymptotically is constructed guarantee states are regulated to without of the constraint. simulation example is regulated to zero zero withoutto of that the the constraint. Aclosed-loop simulationsystem example is provided provided to to is successfully constructed toviolation guarantee that the states ofA system are asymptotically regulated to zero without violation of the constraint. A simulation example is provided to regulated to zero without violation of the constraint. A simulation example is to demonstrate the effectiveness of the proposed method. demonstrate the effectiveness of the proposed method. A simulation example is provided regulated to zero without violation of the constraint. provided to demonstrate the effectiveness of the proposed method. demonstrate the effectiveness of the proposed method. demonstrate the effectiveness of the proposed method. © 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Nonholonomic Nonholonomic systems; systems; output output constraint; constraint; nonlinear nonlinear mapping; mapping; backstepping backstepping Keywords: Keywords: Nonholonomic systems; output constraint; nonlinear mapping; backstepping Keywords: Nonholonomic systems; output constraint; nonlinear mapping; backstepping Keywords: Nonholonomic systems; output constraint; nonlinear mapping; backstepping 1. INTRODUCTION INTRODUCTION state variables, variables, which which may may appear appear in in the the form form of of physical physical 1. state 1. state variables, which may appear in the form of physical 1. INTRODUCTION INTRODUCTION state variables, which may appear in the form of physical stoppages, saturation, or and safety specstoppages, saturation, or performance performance and safety spec1. INTRODUCTION state variables, which may appear in the form of physical stoppages, saturation, or performance and safety specstoppages, saturation, or performance and safety specifications. Violation of the constraints during operation During the the past past decades, decades, the the control control of of nonholonomic nonholonomic stoppages, ifications. Violation of or theperformance constraints during operation During saturation, and safety specifications. Violation of the constraints during operation During the past decades, the control of nonholonomic ifications. Violation of the constraints during operation may result in performance degradation, hazards or system During the past decades, the control of nonholonomic systems has attracted a great deal of attention because may result in performance degradation, hazards or system systems has past attracted a great deal of attention because ifications. Violation of the constraints during operation During the decades, the control of nonholonomic result in performance hazards systems attracted great of may result in performance degradation, hazards or system damage (Do, 2010; Tee, Ge Gedegradation, and Tay, Tay, 2009). 2009). On or thesystem other systems has attracted great deal deal of attention attention because they canhas model many a practical systems, such as asbecause mobile may damage (Do, Tee, and On the other they can model many systems, such mobile may result in 2010; performance degradation, hazards or system systems has attracted aapractical great deal of attention because damage (Do, 2010; Tee, Ge and Tay, 2009). On the other they can model many practical systems, such as mobile damage (Do, 2010; Tee, Ge and Tay, 2009). On the other hand, rigorous constraints bring more difficulties in control they can model many practical systems, such as mobile robots, car-like vehicle, under-actuated satellites and so hand, rigorous constraints bring more difficulties in control robots, car-like vehicle, under-actuated satellites and so damage (Do, 2010; Tee, Ge and Tay, 2009). On the other they cancar-like model vehicle, many practical systems,satellites such as and mobile hand, rigorous constraints bring more difficulties in control robots, under-actuated so hand, rigorous constraints bring more difficulties in control design, since its existence can cause severe deterioration robots, car-like vehicle, under-actuated satellites and so on. However, from a famous theorem due to Brockett design,rigorous since its existence bring can cause severe deterioration on. However, from a famous theorem due to Brockett hand, constraints more difficulties in control robots, car-like vehicle, under-actuated satellites and so since itsclosed-loop existence can cause severe on. from aa famous due to design, since existence cause deterioration of the nominal nominal performance. In deterioration recent years, years, on. However, However, from famous theorem dueeven to Brockett Brockett (1983), it is is well well known that no notheorem smooth (or (or even continu- design, of the performance. In recent (1983), it known that smooth continudesign, since its itsclosed-loop existence can can cause severe severe deterioration on. However, from a famous theorem due to Brockett of the nominal closed-loop performance. In recent years, (1983), it is well known that no smooth (or even continuof the nominal closed-loop performance. In recent driven by practical needs and theoretical challenges, the (1983), it is well known that no smooth (or even continuous) time-invariant static state feedback exists for the stadriven by practical needs and theoretical challenges, the ous) time-invariant static state feedback exists for the staof the nominal closed-loop performance. Inchallenges, recent years, years, (1983), it is well known that nofeedback smooth exists (or even continudriven by practical needs and theoretical the ous) time-invariant static state for the stadriven by practical needs and theoretical challenges, the control design for constrained nonlinear systems has beous) time-invariant static state feedback exists for the stabilization of nonholonomic systems. To give this difficulty a control design for constrained nonlinear systems has bebilization of nonholonomic systems. To give this difficulty a driven by practical needs and theoretical challenges, ous) time-invariant static state feedback exists for the sta-a control design for constrained nonlinear systems has the bebilization of nonholonomic systems. To give this difficulty control design for constrained nonlinear systems has become an important research topic and some interesting bilization of nonholonomic systems. To give this difficulty a solution, a number of challenging control approaches have come andesign important research topic and some interesting solution, aofnumber of challenging control approaches have control for constrained nonlinear systems has bebilization nonholonomic systems. To give this difficulty a come an important research topic and some interesting solution, a number of challenging control approaches have come an important research topic and some interesting results have been proposed (Guo, Wang and Wang, 2013; solution, a number of challenging control approaches have been proposed, which mainly includes discontinuous timeresults have been proposed (Guo, Wang and Wang, 2013; been proposed, which mainly includes discontinuous timecome an important research topic and some interesting solution, a number of challenging control approaches have results have been Niu proposed (Guo, Wang and Wang, 2013; been which includes discontinuous timebeen proposed Wang and Wang, 2013; Meng ethave al., 2016; 2016; et al., al., (Guo, 2015; Tee, Tee, Ren and Ge, 2011). 2011). been proposed, proposed, which mainly mainly includes discontinuous time- results invariant stabilization, smooth time-varying stabilization Meng al., et 2015; Ren and Ge, invariant stabilization, smooth time-varying stabilization resultset have been Niu proposed (Guo, Wang and Wang, 2013; been proposed, which mainly includes discontinuous timeMeng et al., 2016; Niu et al., 2015; Tee, Ren and Ge, 2011). invariant stabilization, smooth time-varying stabilization Meng et al., 2016; Niu et al., 2015; Tee, Ren and Ge, 2011). Nevertheless, it is difficult to generalize the exiting methinvariant stabilization, smooth time-varying stabilization and hybrid stabilizationsmooth (Astolfi, 1996; Jiang, Jiang, 1996; Xu Xu Meng Nevertheless, it is Niu difficult generalize theand exiting methand hybrid stabilization (Astolfi, 1996; 1996; et al., 2016; et al.,to 2015; Tee, Ren Ge, 2011). invariant stabilization, time-varying stabilization Nevertheless, it is difficult to generalize the exiting methand hybrid stabilization Jiang, Xu Nevertheless, it to methnonholonomic systems due to hybrid2000; stabilization (Astolfi, 1996; Jiang, 1996; Xu ods and Huo, Huo, 2000; Tian and and(Astolfi, Li, 2002; 2002;1996; Walsh and 1996; Bushnell, ods to to constrained constrained nonholonomic systemsthe dueexiting to its its special special and Tian Li, Walsh and Bushnell, Nevertheless, it is is difficult difficult to generalize generalize the exiting methhybrid stabilization (Astolfi, 1996; Jiang, 1996; Xu ods to constrained nonholonomic systems due to its special and Huo, 2000; Tian and Li, 2002; Walsh and Bushnell, ods to constrained nonholonomic systems due to its special structure, and hence, it is essential to develop techniques and Huo, 2000; Tian and Li, 2002; Walsh and Bushnell, 1995; Yuan and Qu, 2010). Mainly thanks to these valid structure, and hence, it is essential to develop techniques 1995; Yuan2000; and Tian Qu, 2010). Mainly thanks to these valid ods to constrained nonholonomic systems due to its special and Huo, and Li, 2002; Walsh and Bushnell, and hence, it with is essential to develop techniques 1995; Qu, Mainly to structure, hence, is to techniques for stabilization constraint of 1995; Yuan Yuan and and Qu, 2010). 2010). Mainly thanks thanks to these these valid approaches, number of interesting interesting results have valid been structure, for robust robust and stabilization constraint of nonholonomic nonholonomic approaches, aa number of results have been structure, and hence, it it with is essential essential to develop develop techniques 1995; Yuan and Qu, 2010). Mainly thanks to these valid for robust stabilization with constraint of nonholonomic approaches, a number of interesting results have been for robust stabilization with constraint of nonholonomic systems. However, to the best of our knowledge, there approaches, a number of interesting results have been established over the last years, see, e.g., Astolfi (1996); systems. However, to the best of our knowledge, established over the last years, see, e.g., Astolfi (1996); for robust stabilization with constraint of nonholonomic approaches, a number of years, interesting results have(1996); been systems. However, to the best of our knowledge, there there established over the last see, e.g., Astolfi systems. However, to the best of our knowledge, there has been no results addressing the issue of state/outputestablished over the last years, see, e.g., Astolfi (1996); Jiang (1996); Xu and Huo (2000); Tian and Li (2002); has been However, no results to addressing of state/outputJiang (1996); Xuthe andlast Huo (2000); Li (1996); (2002); systems. the best the of issue our knowledge, there established over years, see, Tian e.g., and Astolfi has been no results addressing the issue of state/outputJiang (1996); Xu and Huo (2000); Tian and Li (2002); has been no results addressing the issue of state/outputconstrained nonholonomic systems up to now. Jiang (1996); Xu and Huo (2000); Tian and Li (2002); Walsh and Bushnell (1995); Yuan and Qu (2010) and the constrained nonholonomic systems up to now. Walsh and Bushnell (1995); Yuan and Qu and (2010) and the has been no results addressing the issue of state/outputJiang (1996); Xu and Huo (2000); Tian Li (2002); nonholonomic up to Walsh Bushnell (1995); Yuan and and constrained nonholonomic systems systems to now. now. Walsh and andtherein. Bushnell (1995); the Yuan andof Qu (2010) and the the references therein. However, the effect ofQu the(2010) constraints is constrained references However, effect the constraints is constrained systems up up now. the Walsh and Bushnell (1995); Yuan and Qu (2010) and the In this this paper, paper,nonholonomic we focus focus our our attention attention onto solving the asympasympIn we on solving references therein. However, the effect of the constraints is references therein. However, the effect of the constraints is omitted in the above-mentioned results. In this paper, we focus our attention on solving the asympomitted in the above-mentioned results. references therein. However, the effect of the constraints is totic In this paper, we focus our attention on solving the totic stabilization for a class of output-constrained nonstabilization for a class of output-constrained nonomitted in the above-mentioned results. In thisstabilization paper, we focus our attention on solving the asympasympomitted in in the the above-mentioned above-mentioned results. results. totic for a class of output-constrained nonomitted totic stabilization for a class of output-constrained nonholonomic systems using state feedback. The contributions As aa matter matter of of fact, fact, many many practical practical systems systems are are subject subject totic holonomic systems for using state feedback. The contributions As stabilization a class of output-constrained nonholonomic systems using state feedback. The contributions As a matter of fact, many practical systems are subject holonomic systems using state feedback. The contributions is highlighted as follows. (i) The stabilization problem As a matter of fact, many practical systems are subject to constraints constraints onfact, their manipulated inputs, outputs and holonomic is highlighted as follows. (i) The stabilization problem of of to on their manipulated inputs, outputs and systems using state feedback. The contributions As a matter of many practical systems are subject highlighted as follows. (i) The stabilization of to is highlighted as follows. (i) The stabilization problem of nonholonomic systems with output constraintproblem is studied studied to constraints constraints on on their their manipulated manipulated inputs, inputs, outputs outputs and and is nonholonomic systems with output constraint is is highlighted as follows. (i) The stabilization problem of to constraints on their manipulated inputs, outputs and ⋆ This work is partially supported by National Nature Science Founnonholonomic systems with output constraint is studied ⋆ nonholonomic systems with output constraint is studied for the first time. (ii) A nonlinear mapping is introduced, This work is partially supported by National Nature Science Founfor the first time. (ii) A nonlinear mapping is introduced, nonholonomic systems with output constraint is studied ⋆ This work is partially supported by National Nature Science Foun⋆ for the the which first time. time. (ii) A nonlinear nonlinear mapping is introduced, introduced, dation of China China under supported Grants 61273091, 61273091, 61403003, the Project of This work is partially by National Naturethe Science Founfor first A mapping is dation of under Grants 61403003, Project of under the constrained interval is to ⋆ under the (ii) constrained interval is mapped mapped to the the This work is partially supported by National Naturethe Science Founfor the which first time. (ii) A nonlinear mapping is introduced, dation of China under Grants 61273091, 61403003, of Taishan Shandong of China, the PhD Programs under which the constrained interval is mapped to the dation ofScholar China of under GrantsProvince 61273091, 61403003, the Project Project of Taishan Scholar of Shandong Province of China, the PhD Programs under which the constrained interval is mapped to the whole real number space, and then the conventional nondation ofScholar China of under GrantsProvince 61273091, 61403003, the Project of whole real number space, and then the conventional nonunder which the constrained interval is mapped to the Taishan Shandong of China, the PhD Programs Foundation of Ministry Ministry of Education Education China, the thePhD Key Programs Program Taishan Scholar of Shandong Province of China, whole real number space, and then the conventional nonFoundation of of China, the Key Program whole real number space, and then the conventional nonlinear control technique can be directly used to design Taishan Scholar of Shandong Province of China, the PhD Programs linear control technique can be directly used to design Foundation of Ministry Ministry of Education Education of China, China, the Key Key of Program whole real number space, and then the conventional nonof Science Technology Research of Education Department Henan Foundation of of of the Program of Science Technology Research of Education Department Henan linear control technique can be directly used to design Foundation of Ministry of Education of China, the Key of Program linear control technique can directly used to controller for the the transformed system without considering controller for transformed system without of Science Technology of of linear control technique can be be directly usedconsidering to design design Province under Grants Research 13A120016, 14A520003 Department and the the Scientific Scientific and of Scienceunder Technology Research of Education Education Department of Henan Henan Province Grants 13A120016, 14A520003 and and controller for the transformed system without considering of Science Technology Research of Education Department of Henan controller for the transformed system without considering the initial values range. (iii) Based on a switching stratProvince under Grants 13A120016, 14A520003 and Scientific the initial for values range. (iii) Based a switching stratTechnological Anyang under 2015310. controller the transformed systemon without considering Province underProject Grantsof 13A120016, 14A520003 and the the Scientific and and Technological Project of Anyang City City under Grant Grant 2015310. the initial values range. (iii) Based on a switching stratProvince under Grants 13A120016, 14A520003 and the Scientific and the initial values range. (iii) Based on a switching stratTechnological Project of Anyang City under Grant 2015310. Technological Project of Anyang City under Grant 2015310. the initial values range. (iii) Based on a switching stratTechnological Project of Anyang City under Grant 2015310. Copyright © 2016 766 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2016, 2016 IFAC IFAC 766Hosting by Elsevier Ltd. All rights reserved. Copyright 2016 IFAC 766 Peer review© of International Federation of Automatic Copyright ©under 2016 responsibility IFAC 766Control. Copyright © 2016 IFAC 766 10.1016/j.ifacol.2016.10.256
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egy to eliminate the phenomenon of uncontrollability of u0 = 0, and by skillfully using the backstepping technique, a systematic state feedback control design procedure is proposed to render the states of closed-loop system to zero while the output constraint is not violated. Notations. Throughout this paper, the following notations are adopted. R+ denotes the set of all nonnegative real numbers and Rn denotes the real n-dimensional space. For any vector x = (x1 , · · · , xn )T ∈ Rn denote 1 x ¯i = (x1 , · · · , xi )T ∈ Ri , i = 1, · · · , n, |x| = ( ni=1 x2i ) 2 . Besides, the arguments of the functions will be omitted or simplified, whenever no confusion can arise from the context. For instance, we sometimes denote a function f (x(t)) by simply f (x), f (·) or f . 2. PROBLEM STATEMENT AND NONLINEAR MAPPING 2.1 Problem statement Since many nonlinear mechanical systems with nonholonomic constraints can be transformed to a canonical chained form representation(Murray and Sastry, 1993). In this paper, we consider the following chained nonholonomic system: x˙ 0 = u0 x˙ i = xi+1 u0 , i = 1, · · · , n − 1 (1) x˙ n = u1 where (x0 , x)T = (x0 , x1 , · · · , xn )T ∈ Rn+1 , u = (u0 , u1 )T ∈ R2 , y = (x0 , x1 )T ∈ R2 are the system state, control input and system output, respectively. The output y is required to remain in the set Ωy = {−ki < xi < ki , i = 0, 1} (2) where ki ’s are positive constants. The objective of this paper is to present a state feedback control design strategy which stabilizes the system (1) with the output constraint being not violated. Remark 1. The control problem proposed in this paper can be briefly verified by the daily experience of parking a car. A car is a 4th order nonholonomic system and the output y = (x0 , x1 )T is the displacement from the parking position (see Section 5). When the car’s initial position is far away from the parking position, one usually can drive directly to the parking position. The car’s body angle can be aligned without difficulties and no more maneuvers are needed. However, when the car’s initial position is close to the parking position, it might not be feasible to get to the parking position while aligning the car’s body angle at the same time. Therefore it is very necessary to develop control techniques for such outputconstrained nonholonomic systems for giving this difficulty a straightforward solution. 2.2 Nonlinear mapping To prevent the output y from violating the constraint, we define a nonlinear mapping which will be used to develop the control design and the main results. Define a one-to-one nonlinear mapping H : (x0 , x) → (η0 , η) as follows: 767
Fig. 1. Schematic illustration of the nonlinear mapping H0 . k + x 0 0 η0 = H0 (x0 ) = ln k − x 0 k + x0 1 1 η1 = H1 (x1 ) = ln k1 − x1 (3) η2 = H2 (x2 ) = x2 .. . ηn = Hn (xn ) = xn where H0 is shown in Fig.1. It is clear that function H0 is a continuous elementary function. From (3), we have 2 (4) x0 = H0−1 = k0 1 − η0 e +1 then the derivative of x0 is given by 2k0 eη0 η˙ 0 x˙ 0 = η (5) (e 0 + 1)2 Substituting (5) into the first equation of (1), we have 1 η0 (e + e−η0 + 2)u0 η˙ 0 = (6) 2k0 Similarly, we can obtain 1 η1 η˙1 = (e + e−η1 + 2)x2 (7) 2k1 By noting that η˙ i = x˙ i , i = 2, · · · , n, we can rewrite the system (1) as η˙ 0 = d0 (η0 )u0 η˙ 1 = d1 (η1 )u0 η2 (8) η˙ i = u0 ηi+1 , i = 2, · · · , n − 1 η˙ n = u1 where 1 η0 d0 (η0 ) = (e + e−η0 + 2) 2k0 (9) 1 η1 d1 (η1 ) = (e + e−η1 + 2) 2k1 Remark 2. Based on the nonlinear mapping H , we know that the state η0 (η1 ) is an unconstrained variable and is defined in the whole space R. In addition, no matter which number η0 (η1 ) is taking, x0 (x1 ) will stay in the constraint interval |x0 | < k0 (|x1 | < k1 ). That is, the control design for the unconstraint transformed system (8) is equivalent to the control design for the constraint initial system (1). 3. ROBUST CONTROL DESIGN In this section, we focus on designing robust controller for system (8) provided that η0 (t0 ) �= 0, while the case where
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the initial condition η0 (t0 ) = 0 will be treated in Section 4. The inherently triangular structure of system (8) suggests that we should design the control inputs u0 and u1 in two separate stages. 3.1 Design u0 for η0 -subsystem
3.3 Backstepping Design for u1 In this subsection, we shall construct a continuous state feedback controller u1 which is addressed in a step-bystep manner. For the consistency of the following inductive steps, we denote d˜i = 1 for i = 2, · · · , n.
For η0 -subsystem, we take the following control law 1 u0 = − λ0 η0 (10) d0 where λ0 is a positive design parameter. Under the control law (10), the following lemma can be established.
Step 1. Introduce the Lyapunov function V1 = z12 /2. With the help of (14) and (16), it can be verified that (17) V˙ 1 = d˜1 z1 z2 + z1 f˜1 ≤ d˜1 z1 z2 + z 2 ϕ˜1
Lemma 1. For any initial t0 ≥ 0 and any initial condition η0 (t0 ) ∈ R/{0}, the corresponding solution η0 (t) exists and satisfies limt→∞ η0 (t) = 0. Furthermore, the control u0 given by (10) also exists; does not cross zero and satisfies limt→∞ u0 (t) = 0.
(18)
Proof. Choosing the Lyapunov function V0 = η02 /2, a simple computation gives (11) V˙ 0 ≤ −λ0 η02 which implies (12) |η0 (t)| ≤ |η0 (t0 )|e−λ0 (t−t0 ) Consequently, η0 is globally exponentially convergent and does not cross zero for all t ∈ (t0 , ∞) provided that η0 (t0 ) �= 0. Furthermore, from equation (10), we can conclude that the u0 exists, does not cross zero for all t ∈ (t0 , ∞) independent of the η-subsystem and satisfies limt→∞ u0 (t) = 0.
Obviously, the first virtual controller 1 z2∗ = − (n + ϕ¯1 )z2 := −α1 (η0 , z1 )z2 ˜ d1 where α1 > 0 is a smooth function, leads to V˙ 1 ≤ −nz12 + d˜1 z1 (z2 − z2∗ )
(19)
Step i (i = 2, · · · , n). Suppose at step i − 1, there is a positive-definite and proper Lyapunov functional Vi−1 , and a set of smooth virtual controllers z1∗ , · · · , zi∗ defined by z1∗ = 0 ξ1 = z1 − z1∗ ∗ z2 = −α1 (η0 , z¯1 )ξ1 ξ2 = z2 − z2∗ (20) .. .. . . zi∗ = −αi−1 (η0 , z¯i−1 )ξi−1 ξi = zi − zi∗ with α1 (η0 , z1 ) > 0, · · · , αi−1 (η0 , z¯i−1 ) > 0 being smooth, such that i−1 V˙ i−1 ≤ −(n − i + 2) ξj2 + d˜i−1 ξi−1 (zi − zi∗ ) (21) j=1
3.2 Input-state-scaling transformation Since it has already proven that η0 can be globally exponentially regulated to zero as t → ∞. However, it is troublesome in controlling the η-subsystem via the control input u1 (t) because, in the limit (i.e. u0 (t) = 0), the ηsubsystem is uncontrollable. To avoid the phenomenon, the following discontinuous input-state-scaling transformation is employed. ηi zi = n−i , i = 1, · · · , n (13) u0 under which, the η-subsystem is transformed into z˙1 = d˜1 (η0 , z1 )z2 + f˜2 (t, η0 , z1 , u0 ) (14) z˙i = zi+1 + f˜i (t, η0 , zi , u0 ), i = 2, · · · , n − 1 z˙n = u1 + f˜n (t, η0 , zn , u0 ) where
d˜1 (η0 , z1 ) = d1 (η1 ) u˙ 0 f˜i (t, η0 , zi , u0 ) = −(n − i)zi u0
(15)
The following lemma gives the estimation of nonlinear function f¯i . Lemma 2. For i = 1, · · · , n, there exist nonnegative smooth functions ϕ˜i such that (16) |f˜i (t, η0 , zi , u0 )| ≤ ϕ˜i (η0 )|zi | where i = 1, · · · , n. Proof. The estimation can be easily obtained from (10), (15) and the transformation (13). The detailed proof is omitted here. 768
We intend to establish a similar property for z¯i -subsystem. Consider the following Lyapunov function candidate 1 (22) Vi (¯ zi ) = Vi−1 (¯ zi−1 ) + ξi2 2 Clearly i−1 ξj2 + d˜i−1 ξi−1 (zi − zi∗ ) V˙ i ≤ −(n − i + 2) j=1
+d˜i ξi zi+1 + ξi f˜i + ξi
i−1 ∂z ∗ i
j=1
∂zj
(d˜j zj+1 + f˜j )
(23)
∂z ∗ +ξi i d0 u0 ∂η0 ∗ , an appropriate To design the virtual controller zi+1 bounding estimate should be given for each term on the right-hand side of (23). To begin with, from (20), we have 1 2 (24) d˜i−1 ξi−1 (zi − zi∗ ) ≤ |ξi−1 ξi | ≤ ξi−1 + li1 ξi2 4 where li1 > 0 is a constant.
Then, according to (16) and (20), it follows that 1 2 ξi f˜i ≤ |ξi |ϕ¯i (|ξi | + αi−1 |ξi−1 |) ≤ ξi−1 + li2 ξi2 4 where li2 ≥ 0 is a smooth function.
(25)
Similar to (25), with the aid of (16) and (20), we get i−1 i−1 1 2 ∂zi∗ ˜ ξi (dj zj+1 + f˜j ) ≤ ξj + li3 ξi2 (26) ∂z 4 j j=1 j=1 for a smooth function li3 ≥ 0.
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Since zi∗ is a smooth function and satisfies ∂zi∗ zi∗ (η0 , 0) = 0, (η0 , 0) = 0 (27) ∂η0 Based on the completion of squares, it is deduced that there is a smooth function li4 ≥ 0 such that i−1 1 2 ∂z ∗ ξ + li4 ξi2 ξi i d0 u0 ≤ (28) ∂η0 4 j=1 j Substituting (24)–(26) and (28) into (23) yields i−1 V˙ i ≤ −(n − i + 2) ξj2 + d˜i ξi zi+1 j=1
(29)
+ξi (li1 + li2 + li3 + li4 ) Now, it easy to see that the smooth virtual controller 1 ∗ zi+1 n − i + 1 + li1 + li2 + li3 + li4 ξi =− (30) d˜i := −αi (η0 , z¯i )ξi renders i ∗ ˙ Vi ≤ −(n − i + 1) ξj2 + d˜i ξi (zi+1 − zi+1 ) (31) j=1
As i = n, the last step, we can construct explicitly a change of coordinates (ξ1 , · · · , ξn ), a positive-definite and proper Lyapunov function Vn (¯ zn ) and a smooth state feedback ∗ controller zn+1 of form (30) such that n ∗ ξj2 + ξn (u1 − zn+1 ) V˙ n ≤ − (32) j=1
Therefore, by choosing the actual control u1 as ∗ = −αn (η0 , z¯n )ξn u1 = zn+1 we get n V˙ n ≤ − ξj2
(33) (34)
j=1
We have thus far completed the controller design procedure for η0 (t0 ) �= 0. Without loss of generality, we can assume that t0 = 0. 4. SWITCHING CONTROL DESIGN AND MAIN RESULTS In the preceding section, we have given controller design for η0 (0) �= 0. Now, we discuss how to select the control laws u0 and u1 when η0 (0) = 0. In the absence of disturbances, the most commonly used control strategy is using constant control u0 = u∗0 �= 0 in time interval [0, ts ). However, for system (8) with non-Lipschitz nonlinearities, the choice of constant feedbacks may lead to the solution of the η0 -subsystem blow up before the given switching time ts . In order to prevent this finite escape phenomenon from happening, we give the switching control strategy for control input u0 by the use of state measurement of the η0 -subsystem in (8) instead of frequently-used time measurement. When η0 (0) = 0, we choose u0 as follow: u0 = u∗0 , u∗0 > 0 At η0 (0) = 0, we know that η˙0 (0) = d0 (0)u0 (0) = d0 (0)u∗0 > 0. Thus for a small positive constant δ, there 769
757
exists a small neighborhood Ω of η0 (0) = 0 such that |d0 η0 | ≤ δ. Suppose that η0∗ satisfies |η0∗ | = δ. In Ω, η0 is increasing until |η0 | > δ. Now, we define the switching control law u0 as u0 = u∗0 , u∗0 > 0, |η0 | ≤ |η0∗ | < δ
(35)
During the time period satisfying |η0 | ≤ using u0 ∗ defined in (35) and new u1 = zn+1 (η0 , z) obtained by the similar control design method as (33), it is concluded that the η-state of (8) cannot blow up for |η0 | ≤ |η0∗ |. At this time, η0 (ts ) is not zero (|η0 (ts )| > |η0∗ |), then, we switch to the control inputs u0 and u1 into (10) and (33), respectively. Thus, the following results are obtained. |η0∗ |,
Lemma 3. Under Assumption 1, if the proposed control design procedure together with the above switching control strategy is applied to system (8), then, for any initial conditions in the state space (η0 , η)T ∈ Rn+1 , uncertain system (8) globally asymptotic-regulated at origin. Proof. According to the above analysis, it suffices to prove the statement in the case where η0 (0) �= 0. Since we have already proven that limt→∞ η0 (t) = 0 in Lemma 1, we just need to show that limt→∞ η(t) = 0. In this case, noting that Vn is positive definite and radially unbounded, by (20) and (34), we get limt→∞ z(t) = 0. Furthermore, from the input-state-scaling transformation (13), we conclude that limt→∞ η(t) = 0. Thereby, the proof of Lemma 3 is completed. With the help of Lemma 3, we are ready to state the main results of this paper. Theorem 1. Under Assumption 1, if the proposed control design procedure together with the above switching control strategy is applied to system (1), then, for any initial conditions (x0 (0), x(0)) ∈ Θ = {(x0 , x)T ∈ Rn+1 | − ki < xi < ki , i = 0, 1}, the following properties hold. (i) The output y remains in the set Ωy = {−ki < xi (t) < ki , i = 0, 1}, ∀t ≥ 0, i.e., the output constraint is never violated. (ii) All the states of closed-loop system are asymptotically regulated to zero. Proof. From Lemma 3, we can easily see that the states ηi (t), i = 0, 1, · · · , n are bounded, and limt→∞ ηi (t) = 0. The bounded states ηi (t), i = 0, 1 together with the nonlinear mapping (3) lead to 2 (36) |x0 (t)| = k0 1 − η (t) < k0 e 0 +1 and 2 (37) |x1 (t)| = k1 1 − η (t) < k1 e 1 +1 that is, the output y = (x0 , x1 )T will remains in the set Ωy and never violates the constraint. Furthermore, limt→∞ ηi (t) = 0, i = 0, 1, · · · , n and (3) imply that implies that limt→∞ xi (t) = 0, i = 2, · · · , n and 2 lim x0 = lim k0 1 − η (t) t→∞ t→∞ e 0 +1 (38) 2 = 0 = k0 1 − lim e t→∞ η0 (t) + 1
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2 lim x1 = lim k1 1 − η (t) t→∞ t→∞ e 1 +1 2 =0 = k1 1 − lim η1 (t) + 1 t→∞ e Thus, the proof is completed.
(39)
5. APPLICATION TO PARK A CAR Consider a car driven by front wheels and containing two passive rear wheels (Mobayen, 2015), the kinematic equations of which can be represented by x˙ c = vcosθ y˙ c = vsinθ v (40) θ˙c = tanφ L φ˙ = w where (xc , yc ) are the coordinates of point P located between of the rear wheels, θ is the orientation angle of the car with respect to the X-axis, φ is the steering angle of the front wheels with respect to the car body, w is the steering velocity of the front wheels, v is the forward velocity of the rear wheels and L is the distance between the center of the front and rear wheels. Introducing the following change of coordinates x0 = xc , x1 = yc , x2 = tanθ 1 x3 = sec3 θtanφ, u0 = vcosθ (41) L 1 3 2 2 3 w = − u0 sin φsinθ + u1 cos θcos φ L L system (40) is transformed into the chained form as x˙ 0 = u0 , x˙ 1 = u0 x2 , x˙ 2 = u0 x3 , x˙ 3 = u1 (42) Clearly, system (42) is in the form of system (1) and y = (xc , yc ) = (x0 , x1 ). Obviously, when the car’s initial position is close to the parking position, how to park this car becomes the stabilization problem of system (42) with output constraint. To verify our proposed controller, we suppose that the output constraint is |x0 | < 1 and |x1 | < 1. Introduce a one-to-one nonlinear mapping H : (x0 , x) → (η0 , η) as follows: 1 + x 0 η0 = H0 (x0 ) = ln 1 − x 1 + x0 1 η1 = H1 (x1 ) = ln (43) 1 − x1 η2 = H2 (x2 ) = x2 η3 = H3 (x3 ) = x3 under which, system (42) can be rewritten as 1 η˙ 0 = (eη0 + e−η0 + 2)u0 2 1 η1 η˙ 1 = (e + e−η1 + 2)u0 η2 (44) 2 η˙ 2 = u0 η3 η˙ 3 = u1 If η0 (0) = 0, controls u0 and u1 are set as in Section 4 in interval [0, ts ), such that η0 (ts ) �= 0, then we can adopt the controls developed below. Therefore, without loss of generality, we assume that η0 (0) �= 0. For the η0 -subsystem, we can choose the control law u0 = −2λ0 η0 /(eη0 + e−η0 + 2). By introducing the input-state-scaling transformation 770
z1 = η1 /u20 , z2 = η2 /u0 , z3 = x3 , the η-subsystem of (44) is transformed into 2 2 1 u˙ 0 z˙1 = (ez1 u0 + e−z1 u0 + 2)z2 − z1 2 u0 u˙ 0 (45) z˙2 = z3 − z2 u0 z˙3 = u1 According to the design procedure shown in Section 3, we can explicitly construct a state feedback controller u1 = −b3 (z3 + b2 z2 + b1 z1 ) (46) with appropriate nonnegative smooth functions b1 , b2 , and b3 to globally asymptotically stabilize z-subsystem (45). In the simulation, by choosing initial value (x0 (0), x1 (0), x2 (0), x3 (0)) = (0.8, −0.98, −2, 1) and the gains for the control laws as k0 = 3, λ0 = 0.3, b1 = 4/(ez1 + e−z1 + 2), b2 = 1.5, b3 = 10, Fig. 2 is obtained to exhibit the responses of the closed-loop system. From the figure, it can be seen that all the closed-loop system states are asymptotically regulated to zero and the output constraint is never violated, which accords with the main results established in Theorem 1 and also demonstrates the effectiveness of the control method proposed in this paper. 6. CONCLUSION This paper has studied the problem of asymptotic stabilization by state feedback for a class of output-constrained nonholonomic systems for the first time. Based on the nonlinear mapping, and by using backstepping technique, a constructive design procedure for state feedback control is given. Together with a novel switching control strategy, the designed controller can guarantee that the closed-loop system states are asymptotically regulated to zero while the output constraint is not violated. In this direction, there are still remaining problems to be investigated. For example, an interesting research problem is how to design an output feedback stabilizing controller for the constrained nonholonomic systems studied in the paper. REFERENCES Astolfi, A.(1996). Discontinuous control of nonholonomic systems. Systems and Control Letters, 27, 37–45. Brockett, R.W. (1983). Asymptotic stability and feedback stabilization. in Brockett, R.W., Millman, R.S., Sussmann, H.J., Eds. Differential geometric control theory, 181–195, Birkhauser, Boston. Do, K.D.(2010). Control of nonlinear systems with output tracking error constraints and its application to magnetic bearings. International Journal of Control, 83, 1199– 1216. Gao, F.Z., Yuan, F.Y., and Wu, Y.Q.(2012). Statefeedback stabilisation for stochastic non-holonomic systems with time-varying delays. IET Control Theory and Applications, 6, 2593–2600. Ge, S.S., Wang, Z.P., and Lee, T.H.(2003). Adaptive stabilization of uncertain nonholonomic systems by state and output feedback. Automatica, 39, 1451–1460. Guo, T., Wang, D.L., and Wang, A.M.(2013). Adaptive backstepping control for constrained systems using nonlinear mapping. Acta Automatica Sinica, 39, 1558–1563.
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Jiang, Z.P. (1996). Iterative design of time-varying stabilizers for multi-input systems in chained form. Systems and Control Letters, 28, 255–262. Jiang, Z.P.(2000). Robust exponential regulation of nonholonomic systems with uncertainties. Automatica, 36, 189–209. Liu, Y.G., and Zhang, J.F. (2005). Output feedback adaptive stabilization control design for nonholonomic systems with strong nonlinear drifts. International Journal of Control, 78, 474–490. Meng, W.C., Yang, Q. M., Si, J.N, and Sun, Y.X.(2016). Adaptive neural control of a class of output-constrained nonaffine systems. IEEE Transactions on Cybernetics, 2016, 46, 85–95 Mobayen, S (2015). Fast terminal sliding mode tracking of non-holonomic systems with exponential decay rate. IET Control Theory and Applications, 9, 1294–1301. Murray, R.R., and Sastry, S.S.(1993). Nonholonomic motion planning: steering using sinusoids. IEEE Transactions on Automatic Control, 38, 700–716. Niu, B., Zhao X.D., Fan, X.D., and Cheng, Y.(2015). A new control method for state-constrained nonlinear switched systems with application to chemical process. International Journal of Control, 88, 1693–1701. Xu, W.L., and Huo, W. (2000). Variable structure exponential stabilization of chained systems based on the extended nonholonomic integrator. Systems and Control Letters, 41, 225–235. Tee, K.P., Ge, S.S., and Tay, E.H.(2009). Adaptive control of electrostatic microactuators with bidirectional drive. IEEE Transactions on Control Systems Technology, 17, 340–352. Tee, K.P., Ren, B., and Ge, S.S. (2011). Control of nonlinear systems with time-varying output constraints. Automatica, 47, 2511–2516. Tian, Y.P., and Li, S.H. (2002). Exponential stabilization of nonholonomic dynamic systems by smooth timevarying control. Automatica, 38, 1139–1146. Walsh, G.C., and Bushnell, L.G.(1995). Stabilization of multiple input chained form control systems. Systems and Control Letters, 25, 227–234. Wu, Y.Q., Zhao, Y., and Yu, J.B.(2013). Global asymptotic stability controller of uncertain nonholonomic systems. Journal of the Franklin Institute, 350, 1248–1263. Xi, Z.R., Feng, G., Jiang, Z.P., and Cheng, D.Z.(2007). Output feedback exponential stabilization of uncertain chained systems. Journal of the Franklin Institute, 344, 36–57. Yuan, H., and Qu, Z.H. (2010). Smooth time-varying pure feedback control for chained nonholonomic systems with exponential convergent rate. IET Control Theory and Applications, 4, 1235–1244.
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Asymptotic Stabilization for a Class of Asymptotic Stabilization for Asymptotic Stabilization for a a Class Class of of Output-Constrained Nonholonomic Output-Constrained Nonholonomic ⋆ Output-Constrained Nonholonomic Systems ⋆⋆ Systems Systems ∗,∗∗ ∗∗∗ ∗∗
Fangzheng Yuqing,Wu Fushun Fangzheng Gao Gao ∗,∗∗ Yuqing,Wu ∗∗∗ Fushun Yuan Yuan ∗∗ ∗,∗∗ ∗∗∗ ∗∗ ∗,∗∗ Yuqing,Wu ∗∗∗ Fushun ∗∗ ∗∗ ∗∗∗∗ Fangzheng Gao ∗∗ Yanling Shang ∗∗∗∗ Yuan Fangzheng Gao Yuqing,Wu Yuan Hejun Yao ∗,∗∗ ∗∗∗ Fushun ∗∗ Hejun Yao Yanling Shang Fangzheng Gao Yuqing,Wu Fushun Yuan ∗∗ ∗∗∗∗ ∗∗ Yanling Shang ∗∗∗∗ Hejun Yao Hejun Yao ∗∗ Yanling Shang ∗∗∗∗ Hejun Yao Yanling Shang ∗ ∗ School of of Automation, Automation, Southeast Southeast University, University, Nanjing Nanjing 210096, 210096, School ∗ ∗ School of Automation, Southeast University, Nanjing 210096, University, Nanjing 210096, P. (e-mail: ∗ School of Automation, P. R. R. China China Southeast (e-mail: [email protected]). [email protected]). School of Automation, Southeast University, Nanjing 210096, ∗∗ P. R. China (e-mail: [email protected]). ∗∗ School of Mathematics P. R. China (e-mail: [email protected]). and Statistics, Anyang Normal University, School of Mathematics and Statistics, Anyang Normal University, P. R. China (e-mail: [email protected]). ∗∗ ∗∗ School of Mathematics and Statistics, Anyang Normal University, of and Statistics, Anyang Normal Anyang 455000, P. P. R. R. China (e-mail: [email protected]) ∗∗ School Anyang 455000, (e-mail: [email protected]) of Mathematics Mathematics andChina Statistics, Anyang Normal University, University, ∗∗∗School Anyang 455000, P. R. China (e-mail: [email protected]) ∗∗∗ Anyang 455000, P. R. China (e-mail: [email protected]) Institute of Automation, Qufu Normal University, Qufu 273165, 273165, Institute of Automation, Qufu Normal University, Qufu ∗∗∗ Anyang 455000, P. R. China (e-mail: [email protected]) ∗∗∗ of Automation, Qufu Normal University, Qufu 273165, Institute of Qufu Normal University, Qufu 273165, P. R. [email protected]) ∗∗∗ Institute P.Automation, R. China China (e-mail: (e-mail: [email protected]) Institute of Automation, Qufu Normal University, Qufu 273165, ∗∗∗∗ P. R. China (e-mail: [email protected]) ∗∗∗∗ School of P. R. China (e-mail: [email protected]) Software Engineering, Anyang Normal University, School of Software Engineering, Anyang Normal University, P. R. China (e-mail: [email protected]) ∗∗∗∗ ∗∗∗∗ School of Software Engineering, Anyang Normal University, School Software Anyang Normal Anyang 455000, P. Engineering, R. China China (e-mail: (e-mail: [email protected]) ∗∗∗∗Anyang 455000, P. R. [email protected]) School of of Software Engineering, Anyang Normal University, University, Anyang 455000, P. R. China (e-mail: [email protected]) Anyang 455000, P. R. China (e-mail: [email protected]) Anyang 455000, P. R. China (e-mail: [email protected]) Abstract: This This paper paper addresses addresses the the problem problem of of asymptotic asymptotic stabilization stabilization for for aa class class of of Abstract: Abstract: This paper addresses the problem of asymptotic stabilization for aa class of Abstract: This paper the problem of asymptotic for class of nonholonomic systems in chained with output constraint. nonlinear is nonholonomic systems inaddresses chained form form with output constraint. A Astabilization nonlinear mapping mapping is first first Abstract: This paper addresses the problem of asymptotic stabilization for a class of nonholonomic systems in chained form with output constraint. A nonlinear mapping is first nonholonomic systems in chained form with output constraint. A nonlinear mapping is first introduced to transform the output-constrained system into a new unconstrained one. Then, by introduced to transform thechained output-constrained systemconstraint. into a newAunconstrained one. Then, by nonholonomic systems in form with output nonlinear mapping is first introduced to transform the output-constrained system into aa new unconstrained one. Then, by introduced to the output-constrained system into one. Then, employing the backstepping technique and switching switching control strategy, state feedback feedback controller employing the backstepping and control aa state introduced to transform transform the technique output-constrained system into strategy, a new new unconstrained unconstrained one. controller Then, by by employing the backstepping technique and switching control strategy, a state feedback controller employing the backstepping switching control strategy, aa state feedback controller is successfully constructed totechnique guaranteeand that the states states of closed-loop closed-loop system are asymptotically asymptotically is successfully constructed to guarantee that the of system are employing the backstepping technique and switching control strategy, state feedback controller is successfully successfully constructed toviolation guarantee that the states of of closed-loop closed-loop system are asymptotically asymptotically is constructed guarantee states are regulated to without of the constraint. simulation example is regulated to zero zero withoutto of that the the constraint. Aclosed-loop simulationsystem example is provided provided to to is successfully constructed toviolation guarantee that the states ofA system are asymptotically regulated to zero without violation of the constraint. A simulation example is provided to regulated to zero without violation of the constraint. A simulation example is to demonstrate the effectiveness of the proposed method. demonstrate the effectiveness of the proposed method. A simulation example is provided regulated to zero without violation of the constraint. provided to demonstrate the effectiveness of the proposed method. demonstrate the effectiveness of the proposed method. demonstrate the effectiveness of the proposed method. © 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Nonholonomic Nonholonomic systems; systems; output output constraint; constraint; nonlinear nonlinear mapping; mapping; backstepping backstepping Keywords: Keywords: Nonholonomic systems; output constraint; nonlinear mapping; backstepping Keywords: Nonholonomic systems; output constraint; nonlinear mapping; backstepping Keywords: Nonholonomic systems; output constraint; nonlinear mapping; backstepping 1. INTRODUCTION INTRODUCTION state variables, variables, which which may may appear appear in in the the form form of of physical physical 1. state 1. state variables, which may appear in the form of physical 1. INTRODUCTION INTRODUCTION state variables, which may appear in the form of physical stoppages, saturation, or and safety specstoppages, saturation, or performance performance and safety spec1. INTRODUCTION state variables, which may appear in the form of physical stoppages, saturation, or performance and safety specstoppages, saturation, or performance and safety specifications. Violation of the constraints during operation During the the past past decades, decades, the the control control of of nonholonomic nonholonomic stoppages, ifications. Violation of or theperformance constraints during operation During saturation, and safety specifications. Violation of the constraints during operation During the past decades, the control of nonholonomic ifications. Violation of the constraints during operation may result in performance degradation, hazards or system During the past decades, the control of nonholonomic systems has attracted a great deal of attention because may result in performance degradation, hazards or system systems has past attracted a great deal of attention because ifications. Violation of the constraints during operation During the decades, the control of nonholonomic result in performance hazards systems attracted great of may result in performance degradation, hazards or system damage (Do, 2010; Tee, Ge Gedegradation, and Tay, Tay, 2009). 2009). On or thesystem other systems has attracted great deal deal of attention attention because they canhas model many a practical systems, such as asbecause mobile may damage (Do, Tee, and On the other they can model many systems, such mobile may result in 2010; performance degradation, hazards or system systems has attracted aapractical great deal of attention because damage (Do, 2010; Tee, Ge and Tay, 2009). On the other they can model many practical systems, such as mobile damage (Do, 2010; Tee, Ge and Tay, 2009). On the other hand, rigorous constraints bring more difficulties in control they can model many practical systems, such as mobile robots, car-like vehicle, under-actuated satellites and so hand, rigorous constraints bring more difficulties in control robots, car-like vehicle, under-actuated satellites and so damage (Do, 2010; Tee, Ge and Tay, 2009). On the other they cancar-like model vehicle, many practical systems,satellites such as and mobile hand, rigorous constraints bring more difficulties in control robots, under-actuated so hand, rigorous constraints bring more difficulties in control design, since its existence can cause severe deterioration robots, car-like vehicle, under-actuated satellites and so on. However, from a famous theorem due to Brockett design,rigorous since its existence bring can cause severe deterioration on. However, from a famous theorem due to Brockett hand, constraints more difficulties in control robots, car-like vehicle, under-actuated satellites and so since itsclosed-loop existence can cause severe on. from aa famous due to design, since existence cause deterioration of the nominal nominal performance. In deterioration recent years, years, on. However, However, from famous theorem dueeven to Brockett Brockett (1983), it is is well well known that no notheorem smooth (or (or even continu- design, of the performance. In recent (1983), it known that smooth continudesign, since its itsclosed-loop existence can can cause severe severe deterioration on. However, from a famous theorem due to Brockett of the nominal closed-loop performance. In recent years, (1983), it is well known that no smooth (or even continuof the nominal closed-loop performance. In recent driven by practical needs and theoretical challenges, the (1983), it is well known that no smooth (or even continuous) time-invariant static state feedback exists for the stadriven by practical needs and theoretical challenges, the ous) time-invariant static state feedback exists for the staof the nominal closed-loop performance. Inchallenges, recent years, years, (1983), it is well known that nofeedback smooth exists (or even continudriven by practical needs and theoretical the ous) time-invariant static state for the stadriven by practical needs and theoretical challenges, the control design for constrained nonlinear systems has beous) time-invariant static state feedback exists for the stabilization of nonholonomic systems. To give this difficulty a control design for constrained nonlinear systems has bebilization of nonholonomic systems. To give this difficulty a driven by practical needs and theoretical challenges, ous) time-invariant static state feedback exists for the sta-a control design for constrained nonlinear systems has the bebilization of nonholonomic systems. To give this difficulty control design for constrained nonlinear systems has become an important research topic and some interesting bilization of nonholonomic systems. To give this difficulty a solution, a number of challenging control approaches have come andesign important research topic and some interesting solution, aofnumber of challenging control approaches have control for constrained nonlinear systems has bebilization nonholonomic systems. To give this difficulty a come an important research topic and some interesting solution, a number of challenging control approaches have come an important research topic and some interesting results have been proposed (Guo, Wang and Wang, 2013; solution, a number of challenging control approaches have been proposed, which mainly includes discontinuous timeresults have been proposed (Guo, Wang and Wang, 2013; been proposed, which mainly includes discontinuous timecome an important research topic and some interesting solution, a number of challenging control approaches have results have been Niu proposed (Guo, Wang and Wang, 2013; been which includes discontinuous timebeen proposed Wang and Wang, 2013; Meng ethave al., 2016; 2016; et al., al., (Guo, 2015; Tee, Tee, Ren and Ge, 2011). 2011). been proposed, proposed, which mainly mainly includes discontinuous time- results invariant stabilization, smooth time-varying stabilization Meng al., et 2015; Ren and Ge, invariant stabilization, smooth time-varying stabilization resultset have been Niu proposed (Guo, Wang and Wang, 2013; been proposed, which mainly includes discontinuous timeMeng et al., 2016; Niu et al., 2015; Tee, Ren and Ge, 2011). invariant stabilization, smooth time-varying stabilization Meng et al., 2016; Niu et al., 2015; Tee, Ren and Ge, 2011). Nevertheless, it is difficult to generalize the exiting methinvariant stabilization, smooth time-varying stabilization and hybrid stabilizationsmooth (Astolfi, 1996; Jiang, Jiang, 1996; Xu Xu Meng Nevertheless, it is Niu difficult generalize theand exiting methand hybrid stabilization (Astolfi, 1996; 1996; et al., 2016; et al.,to 2015; Tee, Ren Ge, 2011). invariant stabilization, time-varying stabilization Nevertheless, it is difficult to generalize the exiting methand hybrid stabilization Jiang, Xu Nevertheless, it to methnonholonomic systems due to hybrid2000; stabilization (Astolfi, 1996; Jiang, 1996; Xu ods and Huo, Huo, 2000; Tian and and(Astolfi, Li, 2002; 2002;1996; Walsh and 1996; Bushnell, ods to to constrained constrained nonholonomic systemsthe dueexiting to its its special special and Tian Li, Walsh and Bushnell, Nevertheless, it is is difficult difficult to generalize generalize the exiting methhybrid stabilization (Astolfi, 1996; Jiang, 1996; Xu ods to constrained nonholonomic systems due to its special and Huo, 2000; Tian and Li, 2002; Walsh and Bushnell, ods to constrained nonholonomic systems due to its special structure, and hence, it is essential to develop techniques and Huo, 2000; Tian and Li, 2002; Walsh and Bushnell, 1995; Yuan and Qu, 2010). Mainly thanks to these valid structure, and hence, it is essential to develop techniques 1995; Yuan2000; and Tian Qu, 2010). Mainly thanks to these valid ods to constrained nonholonomic systems due to its special and Huo, and Li, 2002; Walsh and Bushnell, and hence, it with is essential to develop techniques 1995; Qu, Mainly to structure, hence, is to techniques for stabilization constraint of 1995; Yuan Yuan and and Qu, 2010). 2010). Mainly thanks thanks to these these valid approaches, number of interesting interesting results have valid been structure, for robust robust and stabilization constraint of nonholonomic nonholonomic approaches, aa number of results have been structure, and hence, it it with is essential essential to develop develop techniques 1995; Yuan and Qu, 2010). Mainly thanks to these valid for robust stabilization with constraint of nonholonomic approaches, a number of interesting results have been for robust stabilization with constraint of nonholonomic systems. However, to the best of our knowledge, there approaches, a number of interesting results have been established over the last years, see, e.g., Astolfi (1996); systems. However, to the best of our knowledge, established over the last years, see, e.g., Astolfi (1996); for robust stabilization with constraint of nonholonomic approaches, a number of years, interesting results have(1996); been systems. However, to the best of our knowledge, there there established over the last see, e.g., Astolfi systems. However, to the best of our knowledge, there has been no results addressing the issue of state/outputestablished over the last years, see, e.g., Astolfi (1996); Jiang (1996); Xu and Huo (2000); Tian and Li (2002); has been However, no results to addressing of state/outputJiang (1996); Xuthe andlast Huo (2000); Li (1996); (2002); systems. the best the of issue our knowledge, there established over years, see, Tian e.g., and Astolfi has been no results addressing the issue of state/outputJiang (1996); Xu and Huo (2000); Tian and Li (2002); has been no results addressing the issue of state/outputconstrained nonholonomic systems up to now. Jiang (1996); Xu and Huo (2000); Tian and Li (2002); Walsh and Bushnell (1995); Yuan and Qu (2010) and the constrained nonholonomic systems up to now. Walsh and Bushnell (1995); Yuan and Qu and (2010) and the has been no results addressing the issue of state/outputJiang (1996); Xu and Huo (2000); Tian Li (2002); nonholonomic up to Walsh Bushnell (1995); Yuan and and constrained nonholonomic systems systems to now. now. Walsh and andtherein. Bushnell (1995); the Yuan andof Qu (2010) and the the references therein. However, the effect ofQu the(2010) constraints is constrained references However, effect the constraints is constrained systems up up now. the Walsh and Bushnell (1995); Yuan and Qu (2010) and the In this this paper, paper,nonholonomic we focus focus our our attention attention onto solving the asympasympIn we on solving references therein. However, the effect of the constraints is references therein. However, the effect of the constraints is omitted in the above-mentioned results. In this paper, we focus our attention on solving the asympomitted in the above-mentioned results. references therein. However, the effect of the constraints is totic In this paper, we focus our attention on solving the totic stabilization for a class of output-constrained nonstabilization for a class of output-constrained nonomitted in the above-mentioned results. In thisstabilization paper, we focus our attention on solving the asympasympomitted in in the the above-mentioned above-mentioned results. results. totic for a class of output-constrained nonomitted totic stabilization for a class of output-constrained nonholonomic systems using state feedback. The contributions As aa matter matter of of fact, fact, many many practical practical systems systems are are subject subject totic holonomic systems for using state feedback. The contributions As stabilization a class of output-constrained nonholonomic systems using state feedback. The contributions As a matter of fact, many practical systems are subject holonomic systems using state feedback. The contributions is highlighted as follows. (i) The stabilization problem As a matter of fact, many practical systems are subject to constraints constraints onfact, their manipulated inputs, outputs and holonomic is highlighted as follows. (i) The stabilization problem of of to on their manipulated inputs, outputs and systems using state feedback. The contributions As a matter of many practical systems are subject highlighted as follows. (i) The stabilization of to is highlighted as follows. (i) The stabilization problem of nonholonomic systems with output constraintproblem is studied studied to constraints constraints on on their their manipulated manipulated inputs, inputs, outputs outputs and and is nonholonomic systems with output constraint is is highlighted as follows. (i) The stabilization problem of to constraints on their manipulated inputs, outputs and ⋆ This work is partially supported by National Nature Science Founnonholonomic systems with output constraint is studied ⋆ nonholonomic systems with output constraint is studied for the first time. (ii) A nonlinear mapping is introduced, This work is partially supported by National Nature Science Founfor the first time. (ii) A nonlinear mapping is introduced, nonholonomic systems with output constraint is studied ⋆ This work is partially supported by National Nature Science Foun⋆ for the the which first time. time. (ii) A nonlinear nonlinear mapping is introduced, introduced, dation of China China under supported Grants 61273091, 61273091, 61403003, the Project of This work is partially by National Naturethe Science Founfor first A mapping is dation of under Grants 61403003, Project of under the constrained interval is to ⋆ under the (ii) constrained interval is mapped mapped to the the This work is partially supported by National Naturethe Science Founfor the which first time. (ii) A nonlinear mapping is introduced, dation of China under Grants 61273091, 61403003, of Taishan Shandong of China, the PhD Programs under which the constrained interval is mapped to the dation ofScholar China of under GrantsProvince 61273091, 61403003, the Project Project of Taishan Scholar of Shandong Province of China, the PhD Programs under which the constrained interval is mapped to the whole real number space, and then the conventional nondation ofScholar China of under GrantsProvince 61273091, 61403003, the Project of whole real number space, and then the conventional nonunder which the constrained interval is mapped to the Taishan Shandong of China, the PhD Programs Foundation of Ministry Ministry of Education Education China, the thePhD Key Programs Program Taishan Scholar of Shandong Province of China, whole real number space, and then the conventional nonFoundation of of China, the Key Program whole real number space, and then the conventional nonlinear control technique can be directly used to design Taishan Scholar of Shandong Province of China, the PhD Programs linear control technique can be directly used to design Foundation of Ministry Ministry of Education Education of China, China, the Key Key of Program whole real number space, and then the conventional nonof Science Technology Research of Education Department Henan Foundation of of of the Program of Science Technology Research of Education Department Henan linear control technique can be directly used to design Foundation of Ministry of Education of China, the Key of Program linear control technique can directly used to controller for the the transformed system without considering controller for transformed system without of Science Technology of of linear control technique can be be directly usedconsidering to design design Province under Grants Research 13A120016, 14A520003 Department and the the Scientific Scientific and of Scienceunder Technology Research of Education Education Department of Henan Henan Province Grants 13A120016, 14A520003 and and controller for the transformed system without considering of Science Technology Research of Education Department of Henan controller for the transformed system without considering the initial values range. (iii) Based on a switching stratProvince under Grants 13A120016, 14A520003 and Scientific the initial for values range. (iii) Based a switching stratTechnological Anyang under 2015310. controller the transformed systemon without considering Province underProject Grantsof 13A120016, 14A520003 and the the Scientific and and Technological Project of Anyang City City under Grant Grant 2015310. the initial values range. (iii) Based on a switching stratProvince under Grants 13A120016, 14A520003 and the Scientific and the initial values range. (iii) Based on a switching stratTechnological Project of Anyang City under Grant 2015310. Technological Project of Anyang City under Grant 2015310. the initial values range. (iii) Based on a switching stratTechnological Project of Anyang City under Grant 2015310. Copyright © 2016 766 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2016, 2016 IFAC IFAC 766Hosting by Elsevier Ltd. All rights reserved. Copyright 2016 IFAC 766 Peer review© of International Federation of Automatic Copyright ©under 2016 responsibility IFAC 766Control. Copyright © 2016 IFAC 766 10.1016/j.ifacol.2016.10.256
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egy to eliminate the phenomenon of uncontrollability of u0 = 0, and by skillfully using the backstepping technique, a systematic state feedback control design procedure is proposed to render the states of closed-loop system to zero while the output constraint is not violated. Notations. Throughout this paper, the following notations are adopted. R+ denotes the set of all nonnegative real numbers and Rn denotes the real n-dimensional space. For any vector x = (x1 , · · · , xn )T ∈ Rn denote 1 x ¯i = (x1 , · · · , xi )T ∈ Ri , i = 1, · · · , n, |x| = ( ni=1 x2i ) 2 . Besides, the arguments of the functions will be omitted or simplified, whenever no confusion can arise from the context. For instance, we sometimes denote a function f (x(t)) by simply f (x), f (·) or f . 2. PROBLEM STATEMENT AND NONLINEAR MAPPING 2.1 Problem statement Since many nonlinear mechanical systems with nonholonomic constraints can be transformed to a canonical chained form representation(Murray and Sastry, 1993). In this paper, we consider the following chained nonholonomic system: x˙ 0 = u0 x˙ i = xi+1 u0 , i = 1, · · · , n − 1 (1) x˙ n = u1 where (x0 , x)T = (x0 , x1 , · · · , xn )T ∈ Rn+1 , u = (u0 , u1 )T ∈ R2 , y = (x0 , x1 )T ∈ R2 are the system state, control input and system output, respectively. The output y is required to remain in the set Ωy = {−ki < xi < ki , i = 0, 1} (2) where ki ’s are positive constants. The objective of this paper is to present a state feedback control design strategy which stabilizes the system (1) with the output constraint being not violated. Remark 1. The control problem proposed in this paper can be briefly verified by the daily experience of parking a car. A car is a 4th order nonholonomic system and the output y = (x0 , x1 )T is the displacement from the parking position (see Section 5). When the car’s initial position is far away from the parking position, one usually can drive directly to the parking position. The car’s body angle can be aligned without difficulties and no more maneuvers are needed. However, when the car’s initial position is close to the parking position, it might not be feasible to get to the parking position while aligning the car’s body angle at the same time. Therefore it is very necessary to develop control techniques for such outputconstrained nonholonomic systems for giving this difficulty a straightforward solution. 2.2 Nonlinear mapping To prevent the output y from violating the constraint, we define a nonlinear mapping which will be used to develop the control design and the main results. Define a one-to-one nonlinear mapping H : (x0 , x) → (η0 , η) as follows: 767
Fig. 1. Schematic illustration of the nonlinear mapping H0 . k + x 0 0 η0 = H0 (x0 ) = ln k − x 0 k + x0 1 1 η1 = H1 (x1 ) = ln k1 − x1 (3) η2 = H2 (x2 ) = x2 .. . ηn = Hn (xn ) = xn where H0 is shown in Fig.1. It is clear that function H0 is a continuous elementary function. From (3), we have 2 (4) x0 = H0−1 = k0 1 − η0 e +1 then the derivative of x0 is given by 2k0 eη0 η˙ 0 x˙ 0 = η (5) (e 0 + 1)2 Substituting (5) into the first equation of (1), we have 1 η0 (e + e−η0 + 2)u0 η˙ 0 = (6) 2k0 Similarly, we can obtain 1 η1 η˙1 = (e + e−η1 + 2)x2 (7) 2k1 By noting that η˙ i = x˙ i , i = 2, · · · , n, we can rewrite the system (1) as η˙ 0 = d0 (η0 )u0 η˙ 1 = d1 (η1 )u0 η2 (8) η˙ i = u0 ηi+1 , i = 2, · · · , n − 1 η˙ n = u1 where 1 η0 d0 (η0 ) = (e + e−η0 + 2) 2k0 (9) 1 η1 d1 (η1 ) = (e + e−η1 + 2) 2k1 Remark 2. Based on the nonlinear mapping H , we know that the state η0 (η1 ) is an unconstrained variable and is defined in the whole space R. In addition, no matter which number η0 (η1 ) is taking, x0 (x1 ) will stay in the constraint interval |x0 | < k0 (|x1 | < k1 ). That is, the control design for the unconstraint transformed system (8) is equivalent to the control design for the constraint initial system (1). 3. ROBUST CONTROL DESIGN In this section, we focus on designing robust controller for system (8) provided that η0 (t0 ) �= 0, while the case where
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the initial condition η0 (t0 ) = 0 will be treated in Section 4. The inherently triangular structure of system (8) suggests that we should design the control inputs u0 and u1 in two separate stages. 3.1 Design u0 for η0 -subsystem
3.3 Backstepping Design for u1 In this subsection, we shall construct a continuous state feedback controller u1 which is addressed in a step-bystep manner. For the consistency of the following inductive steps, we denote d˜i = 1 for i = 2, · · · , n.
For η0 -subsystem, we take the following control law 1 u0 = − λ0 η0 (10) d0 where λ0 is a positive design parameter. Under the control law (10), the following lemma can be established.
Step 1. Introduce the Lyapunov function V1 = z12 /2. With the help of (14) and (16), it can be verified that (17) V˙ 1 = d˜1 z1 z2 + z1 f˜1 ≤ d˜1 z1 z2 + z 2 ϕ˜1
Lemma 1. For any initial t0 ≥ 0 and any initial condition η0 (t0 ) ∈ R/{0}, the corresponding solution η0 (t) exists and satisfies limt→∞ η0 (t) = 0. Furthermore, the control u0 given by (10) also exists; does not cross zero and satisfies limt→∞ u0 (t) = 0.
(18)
Proof. Choosing the Lyapunov function V0 = η02 /2, a simple computation gives (11) V˙ 0 ≤ −λ0 η02 which implies (12) |η0 (t)| ≤ |η0 (t0 )|e−λ0 (t−t0 ) Consequently, η0 is globally exponentially convergent and does not cross zero for all t ∈ (t0 , ∞) provided that η0 (t0 ) �= 0. Furthermore, from equation (10), we can conclude that the u0 exists, does not cross zero for all t ∈ (t0 , ∞) independent of the η-subsystem and satisfies limt→∞ u0 (t) = 0.
Obviously, the first virtual controller 1 z2∗ = − (n + ϕ¯1 )z2 := −α1 (η0 , z1 )z2 ˜ d1 where α1 > 0 is a smooth function, leads to V˙ 1 ≤ −nz12 + d˜1 z1 (z2 − z2∗ )
(19)
Step i (i = 2, · · · , n). Suppose at step i − 1, there is a positive-definite and proper Lyapunov functional Vi−1 , and a set of smooth virtual controllers z1∗ , · · · , zi∗ defined by z1∗ = 0 ξ1 = z1 − z1∗ ∗ z2 = −α1 (η0 , z¯1 )ξ1 ξ2 = z2 − z2∗ (20) .. .. . . zi∗ = −αi−1 (η0 , z¯i−1 )ξi−1 ξi = zi − zi∗ with α1 (η0 , z1 ) > 0, · · · , αi−1 (η0 , z¯i−1 ) > 0 being smooth, such that i−1 V˙ i−1 ≤ −(n − i + 2) ξj2 + d˜i−1 ξi−1 (zi − zi∗ ) (21) j=1
3.2 Input-state-scaling transformation Since it has already proven that η0 can be globally exponentially regulated to zero as t → ∞. However, it is troublesome in controlling the η-subsystem via the control input u1 (t) because, in the limit (i.e. u0 (t) = 0), the ηsubsystem is uncontrollable. To avoid the phenomenon, the following discontinuous input-state-scaling transformation is employed. ηi zi = n−i , i = 1, · · · , n (13) u0 under which, the η-subsystem is transformed into z˙1 = d˜1 (η0 , z1 )z2 + f˜2 (t, η0 , z1 , u0 ) (14) z˙i = zi+1 + f˜i (t, η0 , zi , u0 ), i = 2, · · · , n − 1 z˙n = u1 + f˜n (t, η0 , zn , u0 ) where
d˜1 (η0 , z1 ) = d1 (η1 ) u˙ 0 f˜i (t, η0 , zi , u0 ) = −(n − i)zi u0
(15)
The following lemma gives the estimation of nonlinear function f¯i . Lemma 2. For i = 1, · · · , n, there exist nonnegative smooth functions ϕ˜i such that (16) |f˜i (t, η0 , zi , u0 )| ≤ ϕ˜i (η0 )|zi | where i = 1, · · · , n. Proof. The estimation can be easily obtained from (10), (15) and the transformation (13). The detailed proof is omitted here. 768
We intend to establish a similar property for z¯i -subsystem. Consider the following Lyapunov function candidate 1 (22) Vi (¯ zi ) = Vi−1 (¯ zi−1 ) + ξi2 2 Clearly i−1 ξj2 + d˜i−1 ξi−1 (zi − zi∗ ) V˙ i ≤ −(n − i + 2) j=1
+d˜i ξi zi+1 + ξi f˜i + ξi
i−1 ∂z ∗ i
j=1
∂zj
(d˜j zj+1 + f˜j )
(23)
∂z ∗ +ξi i d0 u0 ∂η0 ∗ , an appropriate To design the virtual controller zi+1 bounding estimate should be given for each term on the right-hand side of (23). To begin with, from (20), we have 1 2 (24) d˜i−1 ξi−1 (zi − zi∗ ) ≤ |ξi−1 ξi | ≤ ξi−1 + li1 ξi2 4 where li1 > 0 is a constant.
Then, according to (16) and (20), it follows that 1 2 ξi f˜i ≤ |ξi |ϕ¯i (|ξi | + αi−1 |ξi−1 |) ≤ ξi−1 + li2 ξi2 4 where li2 ≥ 0 is a smooth function.
(25)
Similar to (25), with the aid of (16) and (20), we get i−1 i−1 1 2 ∂zi∗ ˜ ξi (dj zj+1 + f˜j ) ≤ ξj + li3 ξi2 (26) ∂z 4 j j=1 j=1 for a smooth function li3 ≥ 0.
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Since zi∗ is a smooth function and satisfies ∂zi∗ zi∗ (η0 , 0) = 0, (η0 , 0) = 0 (27) ∂η0 Based on the completion of squares, it is deduced that there is a smooth function li4 ≥ 0 such that i−1 1 2 ∂z ∗ ξ + li4 ξi2 ξi i d0 u0 ≤ (28) ∂η0 4 j=1 j Substituting (24)–(26) and (28) into (23) yields i−1 V˙ i ≤ −(n − i + 2) ξj2 + d˜i ξi zi+1 j=1
(29)
+ξi (li1 + li2 + li3 + li4 ) Now, it easy to see that the smooth virtual controller 1 ∗ zi+1 n − i + 1 + li1 + li2 + li3 + li4 ξi =− (30) d˜i := −αi (η0 , z¯i )ξi renders i ∗ ˙ Vi ≤ −(n − i + 1) ξj2 + d˜i ξi (zi+1 − zi+1 ) (31) j=1
As i = n, the last step, we can construct explicitly a change of coordinates (ξ1 , · · · , ξn ), a positive-definite and proper Lyapunov function Vn (¯ zn ) and a smooth state feedback ∗ controller zn+1 of form (30) such that n ∗ ξj2 + ξn (u1 − zn+1 ) V˙ n ≤ − (32) j=1
Therefore, by choosing the actual control u1 as ∗ = −αn (η0 , z¯n )ξn u1 = zn+1 we get n V˙ n ≤ − ξj2
(33) (34)
j=1
We have thus far completed the controller design procedure for η0 (t0 ) �= 0. Without loss of generality, we can assume that t0 = 0. 4. SWITCHING CONTROL DESIGN AND MAIN RESULTS In the preceding section, we have given controller design for η0 (0) �= 0. Now, we discuss how to select the control laws u0 and u1 when η0 (0) = 0. In the absence of disturbances, the most commonly used control strategy is using constant control u0 = u∗0 �= 0 in time interval [0, ts ). However, for system (8) with non-Lipschitz nonlinearities, the choice of constant feedbacks may lead to the solution of the η0 -subsystem blow up before the given switching time ts . In order to prevent this finite escape phenomenon from happening, we give the switching control strategy for control input u0 by the use of state measurement of the η0 -subsystem in (8) instead of frequently-used time measurement. When η0 (0) = 0, we choose u0 as follow: u0 = u∗0 , u∗0 > 0 At η0 (0) = 0, we know that η˙0 (0) = d0 (0)u0 (0) = d0 (0)u∗0 > 0. Thus for a small positive constant δ, there 769
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exists a small neighborhood Ω of η0 (0) = 0 such that |d0 η0 | ≤ δ. Suppose that η0∗ satisfies |η0∗ | = δ. In Ω, η0 is increasing until |η0 | > δ. Now, we define the switching control law u0 as u0 = u∗0 , u∗0 > 0, |η0 | ≤ |η0∗ | < δ
(35)
During the time period satisfying |η0 | ≤ using u0 ∗ defined in (35) and new u1 = zn+1 (η0 , z) obtained by the similar control design method as (33), it is concluded that the η-state of (8) cannot blow up for |η0 | ≤ |η0∗ |. At this time, η0 (ts ) is not zero (|η0 (ts )| > |η0∗ |), then, we switch to the control inputs u0 and u1 into (10) and (33), respectively. Thus, the following results are obtained. |η0∗ |,
Lemma 3. Under Assumption 1, if the proposed control design procedure together with the above switching control strategy is applied to system (8), then, for any initial conditions in the state space (η0 , η)T ∈ Rn+1 , uncertain system (8) globally asymptotic-regulated at origin. Proof. According to the above analysis, it suffices to prove the statement in the case where η0 (0) �= 0. Since we have already proven that limt→∞ η0 (t) = 0 in Lemma 1, we just need to show that limt→∞ η(t) = 0. In this case, noting that Vn is positive definite and radially unbounded, by (20) and (34), we get limt→∞ z(t) = 0. Furthermore, from the input-state-scaling transformation (13), we conclude that limt→∞ η(t) = 0. Thereby, the proof of Lemma 3 is completed. With the help of Lemma 3, we are ready to state the main results of this paper. Theorem 1. Under Assumption 1, if the proposed control design procedure together with the above switching control strategy is applied to system (1), then, for any initial conditions (x0 (0), x(0)) ∈ Θ = {(x0 , x)T ∈ Rn+1 | − ki < xi < ki , i = 0, 1}, the following properties hold. (i) The output y remains in the set Ωy = {−ki < xi (t) < ki , i = 0, 1}, ∀t ≥ 0, i.e., the output constraint is never violated. (ii) All the states of closed-loop system are asymptotically regulated to zero. Proof. From Lemma 3, we can easily see that the states ηi (t), i = 0, 1, · · · , n are bounded, and limt→∞ ηi (t) = 0. The bounded states ηi (t), i = 0, 1 together with the nonlinear mapping (3) lead to 2 (36) |x0 (t)| = k0 1 − η (t) < k0 e 0 +1 and 2 (37) |x1 (t)| = k1 1 − η (t) < k1 e 1 +1 that is, the output y = (x0 , x1 )T will remains in the set Ωy and never violates the constraint. Furthermore, limt→∞ ηi (t) = 0, i = 0, 1, · · · , n and (3) imply that implies that limt→∞ xi (t) = 0, i = 2, · · · , n and 2 lim x0 = lim k0 1 − η (t) t→∞ t→∞ e 0 +1 (38) 2 = 0 = k0 1 − lim e t→∞ η0 (t) + 1
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2 lim x1 = lim k1 1 − η (t) t→∞ t→∞ e 1 +1 2 =0 = k1 1 − lim η1 (t) + 1 t→∞ e Thus, the proof is completed.
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5. APPLICATION TO PARK A CAR Consider a car driven by front wheels and containing two passive rear wheels (Mobayen, 2015), the kinematic equations of which can be represented by x˙ c = vcosθ y˙ c = vsinθ v (40) θ˙c = tanφ L φ˙ = w where (xc , yc ) are the coordinates of point P located between of the rear wheels, θ is the orientation angle of the car with respect to the X-axis, φ is the steering angle of the front wheels with respect to the car body, w is the steering velocity of the front wheels, v is the forward velocity of the rear wheels and L is the distance between the center of the front and rear wheels. Introducing the following change of coordinates x0 = xc , x1 = yc , x2 = tanθ 1 x3 = sec3 θtanφ, u0 = vcosθ (41) L 1 3 2 2 3 w = − u0 sin φsinθ + u1 cos θcos φ L L system (40) is transformed into the chained form as x˙ 0 = u0 , x˙ 1 = u0 x2 , x˙ 2 = u0 x3 , x˙ 3 = u1 (42) Clearly, system (42) is in the form of system (1) and y = (xc , yc ) = (x0 , x1 ). Obviously, when the car’s initial position is close to the parking position, how to park this car becomes the stabilization problem of system (42) with output constraint. To verify our proposed controller, we suppose that the output constraint is |x0 | < 1 and |x1 | < 1. Introduce a one-to-one nonlinear mapping H : (x0 , x) → (η0 , η) as follows: 1 + x 0 η0 = H0 (x0 ) = ln 1 − x 1 + x0 1 η1 = H1 (x1 ) = ln (43) 1 − x1 η2 = H2 (x2 ) = x2 η3 = H3 (x3 ) = x3 under which, system (42) can be rewritten as 1 η˙ 0 = (eη0 + e−η0 + 2)u0 2 1 η1 η˙ 1 = (e + e−η1 + 2)u0 η2 (44) 2 η˙ 2 = u0 η3 η˙ 3 = u1 If η0 (0) = 0, controls u0 and u1 are set as in Section 4 in interval [0, ts ), such that η0 (ts ) �= 0, then we can adopt the controls developed below. Therefore, without loss of generality, we assume that η0 (0) �= 0. For the η0 -subsystem, we can choose the control law u0 = −2λ0 η0 /(eη0 + e−η0 + 2). By introducing the input-state-scaling transformation 770
z1 = η1 /u20 , z2 = η2 /u0 , z3 = x3 , the η-subsystem of (44) is transformed into 2 2 1 u˙ 0 z˙1 = (ez1 u0 + e−z1 u0 + 2)z2 − z1 2 u0 u˙ 0 (45) z˙2 = z3 − z2 u0 z˙3 = u1 According to the design procedure shown in Section 3, we can explicitly construct a state feedback controller u1 = −b3 (z3 + b2 z2 + b1 z1 ) (46) with appropriate nonnegative smooth functions b1 , b2 , and b3 to globally asymptotically stabilize z-subsystem (45). In the simulation, by choosing initial value (x0 (0), x1 (0), x2 (0), x3 (0)) = (0.8, −0.98, −2, 1) and the gains for the control laws as k0 = 3, λ0 = 0.3, b1 = 4/(ez1 + e−z1 + 2), b2 = 1.5, b3 = 10, Fig. 2 is obtained to exhibit the responses of the closed-loop system. From the figure, it can be seen that all the closed-loop system states are asymptotically regulated to zero and the output constraint is never violated, which accords with the main results established in Theorem 1 and also demonstrates the effectiveness of the control method proposed in this paper. 6. CONCLUSION This paper has studied the problem of asymptotic stabilization by state feedback for a class of output-constrained nonholonomic systems for the first time. Based on the nonlinear mapping, and by using backstepping technique, a constructive design procedure for state feedback control is given. Together with a novel switching control strategy, the designed controller can guarantee that the closed-loop system states are asymptotically regulated to zero while the output constraint is not violated. In this direction, there are still remaining problems to be investigated. For example, an interesting research problem is how to design an output feedback stabilizing controller for the constrained nonholonomic systems studied in the paper. REFERENCES Astolfi, A.(1996). Discontinuous control of nonholonomic systems. Systems and Control Letters, 27, 37–45. Brockett, R.W. (1983). Asymptotic stability and feedback stabilization. in Brockett, R.W., Millman, R.S., Sussmann, H.J., Eds. Differential geometric control theory, 181–195, Birkhauser, Boston. Do, K.D.(2010). Control of nonlinear systems with output tracking error constraints and its application to magnetic bearings. International Journal of Control, 83, 1199– 1216. Gao, F.Z., Yuan, F.Y., and Wu, Y.Q.(2012). Statefeedback stabilisation for stochastic non-holonomic systems with time-varying delays. IET Control Theory and Applications, 6, 2593–2600. Ge, S.S., Wang, Z.P., and Lee, T.H.(2003). Adaptive stabilization of uncertain nonholonomic systems by state and output feedback. Automatica, 39, 1451–1460. Guo, T., Wang, D.L., and Wang, A.M.(2013). Adaptive backstepping control for constrained systems using nonlinear mapping. Acta Automatica Sinica, 39, 1558–1563.
IFAC NOLCOS 2016 August 23-25, 2016. Monterey, California, USA Fangzheng Gao et al. / IFAC-PapersOnLine 49-18 (2016) 754–759
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