Backstepping Control Design and Its Applications to Vehicle Lateral. Control in Automated Highway Systems. Chieh Chen

Transcription

1 ackstepping ontrol Design and Its Applications to Vehicle Lateral ontrol in Automated Highway Systems by hieh hen.eng. (National Taiwan University, R.O.) 99 M.Eng. (University of alifornia, erkeley) 995 A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Engineering-Mechanical Engineering in the GRADUATE DIVISION of the UNIVERSITY of ALIFORNIA at ERKELEY ommittee in charge: Professor Masayoshi Tomizuka, hair Professor Karl Hedrick Professor Pravin Varaiya 996

2 The dissertation of hieh hen is approved: hair Date Date Date University of alifornia at erkeley 996

3 ackstepping ontrol Design and Its Applications to Vehicle Lateral ontrol in Automated Highway Systems opyright 996 by hieh hen

4 Abstract ackstepping ontrol Design and Its Applications to Vehicle Lateral ontrol in Automated Highway Systems by hieh hen Doctor of Philosophy in Engineering-Mechanical Engineering University of alifornia at erkeley Professor Masayoshi Tomizuka, hair In this dissertation the eort is to explore new aspects of recursive backstepping design methodology from both theoretical and application point of view. Three main topics are investigated in this dissertation from a backstepping perspective: control of multivariable nonlinear systems whose vector relative degrees are not well dened, steering control of light passenger vehicles on automated highways, and coordinated steering and braking control of commercial heavy vehicles on automated highways. For a class of ane multivariable nonlinear systems with an equal number of inputs and outputs, if the decoupling matrix is singular, the vector relative degree is not well dened. If the mutivariable nonlinear system is strongly invertible and strongly accessible, the vector relative degree of the system can be achieved by adding chains

5 2 of integrators to the input channels. Several versions of dynamic extension algorithms have been proposed to identify the input channels where dynamic compensators (or integrators) are needed to achieve the nonsingularity of the decoupling matrix. Once the vector relative degree is well dened by adding dynamic compensators in the input channel, the multivariable nonlinear system can be decoupled in the inputoutput sense. In this dissertation, instead of decoupling the nonlinear system by adding chains of integrators in the input channels, we modify the dynamic extension algorithm by incorporating backstepping design methods to partially close the loop in each design step. The resulting control law by this new approach is a static state feedback law. Although the nal closed loop form of the nonlinear system is not decoupled, each output is controlled to the desired value asymptotically. ackstepping design methodology is utilized for lateral control of light passenger vehicles and commercial heavy vehicles in Automated Highway Systems (AHS). The steering control algorithm for light passenger vehicles is designed by utilizing the robust backstepping technique, whereas the coordinated steering and braking control algorithm for commercial heavy vehicles is designed by applying the backstepping technique for multivariable nonlinear systems without vector relative degrees. For lateral control of light passenger vehicles, the lateral tracking error is aected by the relative yaw angle of the vehicle with respect to the road centerline. Then the relative yaw angle is controlled by the front wheel steering command. Intuitively, this backstepping control procedure resembles the human driver behavior. Mathe-

6 3 matically, there is no internal dynamics in this design; i.e., both the lateral and the yaw dynamics are under control. Another advantage of the backstepping controller is that the road disturbance, which does not satisfy the matching condition, can be attenuated eectively. Thus the backstepping design eectively utilizes the feedforward information of the road curvature to generate the feedforward part of the steering command. To satisfy both the ride comfort and safety requirements, we introduce a nonlinear spring term (nonlinear position feedback) which exhibits lower gains at small tracking errors and higher gains at larger tracking errors. Furthermore, to cope with nonsmoothness of the road disturbance, robust backstepping control methodology will be utilized. For lateral control of commercial heavy vehicles, a control oriented dynamic modeling approach for articulated vehicles is proposed. A generalized coordinate system is introduced to describe the kinematics of the vehicle. Equations of motion of a tractor-semitrailer vehicle are derived based on the Lagrange mechanics. Experimental studies are conducted to validate the eectiveness of this modeling approach. Two lateral control algorithms are designed for a tractor-semitrailer vehicle. The baseline steering control algorithm is designed utilizing input-output linearization, whereas the coordinated steering and braking control algorithm is designed based on the multivariable backstepping technique.

7 Professor Masayoshi Tomizuka Dissertation ommittee hair 4

8 iii To My Loving Parents,Wife and Daughter

9 iv ontents List of Figures List of Tables vii ix Introduction. Motivations, previous work, and objectives of this dissertation....2 ontributions of this Dissertation....3 Dissertation Outline ackstepping 5 2. Integrator ackstepping ackstepping for Strict-feedback Systems Adaptive ackstepping Robust ackstepping Sliding ontrol via ackstepping onclusions ackstepping ontrol Design of a lass of Multivariable Nonlinear Systems without Vector Relative Degrees Introduction Dynamic Extension Algorithm ombined Dynamic Extension and ackstepping Algorithm Design example : Planar Vehicle Decoupling ontrol by Dynamic Extension ontrol by ackstepping Design onclusions Lateral ontrol of Light Passenger Vehicles in Automated Highway Systems Introduction... 65

10 v 4.2 Vehicle Dynamics and ontrol Model Lateral ontrol of Light Passenger Vehicles Road Model ontroller Design Simulation Results onclusions Dynamic Modeling of Tractor-Semitrailer Vehicles for Automated Highway Systems Introduction Denition of oordinate System oordinate System Reference Frame Transformation between the inertial reference frame and the unsprung mass reference frame Vehicle Kinematics Tractor Kinematics Trailer Kinematics Kinetic Energy and Potential Energy Equations of Motion Generalized Forces Subsystems : Tire Model and Suspension Model Tire Model Suspension Model Model Verication: Simulation and Experimental Results onclusions Lateral ontrol of Tractor-SemitrailerVehicles on Automated Highways Introduction Road reference frame Steering ontrol Model (SIM) Model Simplication ontrol Model with respect to the Road Reference Frame Linear Analysis of the ontrol Model Steering ontrol of Tractor-Semitrailer Vehicles ontroller Design Simulation Results Steering and raking ontrol Model (SIM2) oordinated Steering and Independent raking ontrol ontroller Design Simulation Results... 6

11 vi 6.7 onclusions onclusions and Future Research Summary Suggested Future Research ibliography 7

12 vii List of Figures 3. lock diagram of the MIMO nonlinear system after recursive static state feedback control Dynamic extensions of the MIMO nonlinear system by adding chains of integrators to the appropriate input channel First Step of the ackstepping Designs ackstepping designs of the MIMO nonlinear system Planar Vehicle Ideal mass distribution for passenger car lock diagram of the lateral dynamics Denition of the desired yaw rate A smooth road model Simulation scenario Simulation results of backstepping controller at longitudinal velocity = 3 MPH, solid line : lateral position at.g., dashdot line : lateral position at rear Simulation results of backstepping controller at longitudinal velocity = 6 MPH, solid line : lateral position at.g., dashdot line : lateral position at rear oordinate System to Describe the Vehicle Motion Three Reference oordinates Inertial and Unsprung Mass Reference Frames Schematic Diagram of omplex Vehicle Model Denition of Tire Force in the artesian oordinate Tire Force Model omprehensive Tire Model (araket and Fancher) Suspension Model Step input response with the longitudinal vehicle speed 3 MPH, Step input response with the longitudinal vehicle speed 35 MPH... 3

13 viii 5. Step input response with the longitudinal vehicle speed 4 MPH Step input response with the longitudinal vehicle speed 46 MPH Unsprung Mass and Road Reference oordinates Simulation Scenario Input/Output Linearization ontrol Wheel Dynamics Input/Output Linearization ontrol with Trailer Independent raking Input/Output Linearization ontrol with Trailer Independent raking omparison of Input/Output Linearization ontrol with (solid line) and without (dashdot line) Trailer Independent raking

14 ix List of Tables 4. Notations of the Simplied ontrol Model Parameters of a Passenger ar Parameters of omplex Vehicle Model Parameters for a Tractor-Semitrailer Vehicle Suspension Parameters Tire and Wheel Parameters Nomenclature of ontrol Models

15 x Acknowledgements Iwould like to express my sincere thank to my advisor, Professor Masayoshi Tomizuka, for his timely guidance, encouragement and support. His judicious suggestions made the successful completion of this dissertation possible. He has been the best possible mentor in every sense of the word. I would like to thank Professor Kameshwar Poolla and Professor Wei Ren for their invaluable comments as members of my dissertation committee. Special thanks to my colleagues and friends with whom I had the greatest time: Dr. Tsu-hih hiu, Dr. Liang-Jung Huang, Dr. Tom Hessburg, Dr. Tony Phillips, Dr. in Yao, Victor hu, hieh hen, Weiguang Niu, Li Yi, arlos Osorio, Matt White, raig Smith, Mohammed Al-Majed, Wonshik hee, Hyeoncheol Lee, Pushkar Hingwe and Sujit Saraf. I give my special appreciation to Dr. Ho Seong Lee and Mohammed Al-Majed for their help in setting up the simulations and experiments and Rob ickel, raig Smith and Matt White for careful proof-reading of my dissertation. Iwould like to thank all my family members. My parents You Ying and Yuen Hua deserve special recognition for their continuous encouragement and seless support. My wife Eva should be specially honored for her constant encouragement and sacrice. Iamvery grateful to my daughter Stephanie. Her cheerful smile and lovely voice made my hard moments bearable. Lastly, and most importantly, I thank my Lord, Jesus hrist, for his grace and love.

16 hapter Introduction. Motivations, previous work, and objectives of this dissertation In this dissertation the eort is to explore new aspects, from both theoretical and application point of view, in the design of the backstepping control systems [35] with applications to vehicle lateral control in Automated Highway System (AHS). Three main topics are investigated in this dissertation from a backstepping perspective: control of multivariable nonlinear systems whose vector relative degrees are not well dened, steering control of light passenger vehicles on automated highways, and coordinated steering and braking control of commercial heavy vehicles on automated highways.

17 2 Vehicle Lateral ontrol in Automated Highway Systems. AHS technologies have attracted growing attention among researchers throughout the world in the past several years [3, 6, 7, 23, 32, 5, 58, 62, 67]. The principal motivation for an AHS is to increase highway capacity. Potential benets for AHS include: a substantial increase in lane capacity and therefore trac throughput improvement in driving safety on highways a decrease in travel time and therefore reduction in air pollution Due to the complexities of AHS, a hierarchy of system structure is proposed in [6, 62]. According to spatial and temporal scale, control tasks in this AHS architecture are organized into ve layers: network layer, link layer, coordination layer, regulation layer, and physical layer. ontrol tasks in the network layer and the link layer are executed in roadside systems, whereas the control tasks in the coordination layer and the regulation layer are performed in vehicle on-board computers. The physical layer represents the vehicle dynamics which is controlled by the regulation layer. y this classication, vehicle lateral control sits in the regulation layer and is one of the critical components in the framework of AHS. Lateral control in AHS consists of two maneuvers: lane following and lane change. The objective of lateral motion control for lane following is to achieve accurate tracking of a reference lane while maintaining an acceptable level of passenger comfort in the presence of disturbances and over a wide range of operating conditions. In this

18 3 dissertation, lateral motion control for lane following maneuvers is investigated for both light passenger cars and commercial heavy vehicles. In the area of lateral motion control of light passenger vehicles, previous studies have been conducted by utilizing both the linear control theory and the nonlinear control theory. Among the linear control strategies, Fenton et al. [7] designed a feedback steering controller by using the lead/lag compensator and root locus theory, where no preview information on the road curvature is used in calculating the steering command. Peng and Tomizuka [5] synthesized a Frequency Shaped Linear Quadratic (FSLQ) controller with preview. The main appeal of the linear FSLQ controller is that it provides a quantitative description of the trade-o between passenger ride-comfort and tracking performance. The FSLQ controller also possesses moderate modeling robustness properties. However, the longitudinal velocity is assumed to be constant in the linear control approach, and hence gain scheduling with respect to the longitudinal velocity is required to cover the full operating range of the vehicle. Among the nonlinear control strategies, Guldner et al. [22] and Pham et al. [52] developed steering controllers independently based on Sliding Mode ontrol (SM) theory. Hingwe and Tomizuka further pointed out that the major dierence between these two SM controllers is the location of an integrator within the feedback loop[26]. Simulations and experimental studies conducted in the alifornia PATH program have shown that the SM controllers possess superior lateral tracking performance [27]. Since SM explicitly takes into account parameter uncertainties in the design, the controller is

19 4 robust with respect to parametric uncertainties. Furthermore, the longitudinal velocity is regarded as a known time-varying parameter in the SM approach, thus gain scheduling with respect to the longitudinal velocity is not necessary. The drawback of this approach is that the ride quality is not explicitly considered in the design and the stability ofvehicle yaw dynamics (internal dynamics) is not guaranteed. Lane change maneuvers have been studied using magnetic referencing system in [8], where a sliding mode controller using ltered errors has been proposed as the tracking control algorithm and a reduced order Kalman lter is designed as the state estimator. Preliminary experimental results conducted in the alifornia PATH program showed the eectiveness of this controller at speeds up to 32 km/hr. In contrast to light passenger vehicles, less attention has been paid to control issues of commercial heavy vehicles for automated highway systems. The study of heavy vehicles for AHS applications has gained interest only recently [4,, 6, 32, 67, 75]. The study of lateral guidance of heavy-duty vehicles is important for several reasons. In 993, the share of the highway miles accounted for by truck trac was around 28% [25]. This is a signicant percentage of the total highway miles traveled by all the vehicles in US. According to Motor Vehicles Facts and Figures [42], the total number of registered trucks (light, commercial and truck-trailer combinations) formed approximately % of the national gures in 99 and 3.9% of the highway taxes came from heavy vehicles. Also, due to several economic and policy issues, heavy vehicles have the potential of becoming the main beneciaries of automated guidance

20 5 [32]. The main reasons are: On average, a truck travels six times the miles as compared to a passenger vehicle. Possible reduction in the number of drivers will reduce the operating cost substantially. Relative equipment cost for automating heavy vehicles is far less than for passenger vehicles. Automation of heavy vehicles will have a signicant impact on the overall safety of the automated guidance system. Trucking is a tedious job and automation will contribute positively to reducing driving stress and thereby increase safety. Thus commercial heavy vehicles will gain signicant benet from Advenced Vehicle ontrol Systems (AVS), and may actually become automated earlier than passenger vehicles due to economical considerations. ackstepping. ackstepping [3, 35, 34] is a recursive procedure which breaks a design problem for the full system into a sequence of design problems for lower order systems. The idea of breaking a dynamic system into subsystems is not unusual in the design of nonlinear controllers. Sliding control [6, 6] is such an example. The design of a sliding controller involves two steps: ) design a stable sliding surface to achieve the control objective, and 2) make the sliding surface attractive by pushing system states toward the surface. To facilitate the synthesis of a sliding controller,

21 6 the sliding surface is designed in such away that the relative degree from the control input to the sliding surface variable is one; i.e., a rst order system for the sliding variable. Therefore, it is easier to control system states toward the sliding surface than it is to control the original dynamic system, even in the face of plant nonlinearity and modeling uncertainty. However, robust sliding control for uncertain nonlinear systems requires that the matching condition be satised; that is, the uncertain terms enter the state equation at the same point as the control input. One of the advantages of backstepping design is that the matching condition can be relaxed for a class of nonlinear systems satisfying the so called strict feedback form. ontrol design for this class of nonlinear systems can be achieved by recursive designs of scalar (rst order) subsystems. y exploiting the extra exibility that exists with the scalar systems, the matching condition is not required. Another feature of backstepping designs is that they do not force the designed system to appear linear, which can avoid cancellations of useful nonlinearities. Furthermore, additional nonlinear damping terms can be introduced in the feedbackloop to enhance robustness. Depending on the structure of the lower order subsystems, this recursive design procedure can be categorized as integrator backstepping, backstepping, and block backstepping, which will be studied in this dissertation as design tools for vehicle lateral control and for control of multivariable nonlinear systems without vector relative degrees.

22 7 Steering ontrol of Light Passenger Vehicles via ackstepping. In this dissertation, we will apply the backstepping technique to the design of the steering control algorithm for light passenger vehicles on automated highways. The challenges in designing the vehicle lateral controller include: The lateral controller requires good road tracking performance as well as passenger ride quality and system safety. Small tracking errors are tolerable especially if passenger comfort can be achieved by avoiding unnecessarily small and high frequency steering command adjustments for small tracking errors when the lateral tracking error is under a safety range. On the other hand, lateral tracking errors can not be too large because: ). larger tracking errors may cause avehicle to collide with adjacent vehicles, and 2). larger tracking errors may cause the automated vehicle to run out of the lateral sensor range and become uncontrolled. Lateral dynamics strongly depends on the longitudinal velocity. It is known that the damping of lateral dynamics is inversely proportional to the longitudinal velocity. On the other hand, the desired yaw rate (the tracking signal for the car) is proportional to the longitudinal velocity for the same road curvature. Thus, it is more dicult to control a car negotiating curved sections at high speeds than at low speeds.

23 8 In the formulation of the vehicle lateral control problem, the desired yaw rate, _ d, which depends on the road curvature and the longitudinal speed, appears as a disturbance input for both the lateral dynamics and the yaw error dynamics. We shall call the d term the road disturbance. In the automated highway scenario, the road curvature is previewable, thus the road disturbance is known. However, the matching condition for the road disturbance is not satised since there is only one control input for front wheel steered vehicles. This imposes diculties in designing nonlinear controllers. Vehicle parameters, especially the vehicle mass and the tire cornering stiness, exhibit large uncertainties. In this dissertation, we propose to use the backstepping technique to takeinto account these challenges in the design of vehicle lateral controllers. In this approach, the lateral tracking error is aected by the relative yaw angle of the vehicle with respect to the road centerline. Then the relative yaw angle is controlled by the front wheel steering command. Intuitively, this backstepping control procedure resembles the human driver behavior. Mathematically, there is no internal dynamics in this design; i.e., both the lateral and the yaw dynamics are under control. Another advantage of the backstepping controller is that the road disturbance, which does not satisfy the matching condition, can be attenuated eectively. Thus the backstepping design eectively utilizes the feedforward information of the road curvature to generate the feedforward part of the steering command. To satisfy both the ride comfort and safety

24 9 requirements, we introduce a nonlinear spring term (nonlinear position feedback) which exhibits lower gains at small tracking errors and higher gains at larger tracking errors. The use of such nonlinear action has been applied to the active suspension controller by Lin and Kanellakopoulos [37] to optimize ride quality and suspension travel. A velocity-dependent nonlinear damping term can be easily incorporated in this nonlinear controller to cover the full envelope of operations. Furthermore, to cope with nonsmoothness of the road disturbance, robust backstepping control methodology [68, 7] will be utilized. oordinated Steering and Independent raking ontrol of ommercial Heavy Vehicles via ackstepping. For the lateral control of articulated heavy vehicles, two kinds of control inputs will be used: the steering angle and braking forces of the wheels. We will primarily rely on the front wheel steering angle. However, the braking on trailer units will also be investigated to enhance the stability of lateral motion. Specically, braking forces can be independently distributed over the inner and outer tires of the trailer so that the relative yaw errors between the tractor and the trailer are reduced. In designing the coordinated steering and braking control algorithm, we observe that the so called decoupling matrix for this system is singular; in other words, the vector relative degree is not well dened [3, 46]. Specically, when dierentiating the outputs the steering input appears \earlier" than the braking torque input. To overcome this diculty, we use the braking force generated at the tire/ground inter-

25 face as a virtual control input and then backstep to determine the real braking torque applied at the wheel. ontrol of Multivariable Nonlinear Systems without Vector Relative Degrees via ackstepping. Motivated by the steering and braking control via backstepping, control of a class of multivariable nonlinear systems without vector relative degrees is investigated from the backstepping perspective. A popular control approach for multivariable systems is to make one input control one output independent from other inputs and outputs, i.e., decoupling control or noninteraction control in the input/output sense. Decoupling of multivariable systems has been an active research subject in the past two or three decades. Morgan [4] gave a sucient condition for decoupling of multivariable linear systems in 964. Falb and Wolovich [5] introduced the decoupling matrix which is used to characterize necessary and sucient conditions for decoupling of multivariable systems by static state feedback. Gilbert [2] further classied three types of coupling for multivariable linear systems, that is, strong inherent coupling, no inherent coupling and weak inherent coupling. For a strong inherent coupling multivariable system, no control law can eect decoupling; for a no inherent coupling system, it can be decoupled by static state feedback; and for a weak inherent coupling system, it can not be decoupled by static state feedback, yet decoupling can be achieved by dynamic state feedback. Wang [65] developed a recursive algorithm to design a precompensator for the weak inherent coupling multivariable system. Static and dynamic state feedback control has also been applied

26 to achieve decoupling of nonlinear systems [3, 46] and several versions of dynamic extension algorithms [, 3, 45, 46, 73] have been proposed to identify the input channels where dynamic compensators (or integrators) are needed to achieve the nonsingularity of the decoupling matrix for the extended system. Once the vector relative degree is well dened by adding the integrators in the input channel, the extended multivariable nonlinear system can be decoupled in the input-output sense. In this dissertation, control of multivariable nonlinear systems will be studied via the block backstepping approach..2 ontributions of this Dissertation The contributions of this dissertation are summarized as follows. ackstepping control design of a class of multivariable nonlinear systems. A recursive algorithm is developed to control a class of square multivariable nonlinear systems whose decoupling matrices are singular. Past research on control of this class of systems emphasizes the decoupling or noninteraction control by adding integrators to the appropriate input channels; i.e., decoupling control by dynamic extension. We provide an alternative approach from the backstepping perspective to control this class of nonlinear systems. Specically, this new control procedure is developed based on the dynamic extension algorithm. Instead of adding integrators in input channels, we incorporate backstepping design methods to partially close the loop in each ofthe

27 2 design steps. The resulting control law obtained by this new approach is static state feedback. Steering ontrol of Light Passenger Vehicles. A backstepping controller is designed for lateral guidance of the passenger car in automated highway systems. In this design, the vehicle lateral displacement is aected by the relative yaw angle of the car with respect to the road centerline, and the relative yaw angle is controlled by the vehicle's front wheel steering angle. The main features of this nonlinear design are that the stability of both lateral and yaw error dynamics is ensured, and closed loop performance can be specied simultaneously. Furthermore, nonlinear position feedback, which acts as low gain control at small tracking errors and high gain control at larger tracking errors, is introduced as a trade-o between passenger ride comfort and tracking accuracy. Dynamic Modeling of Articulated ommercial Vehicles. In the literature, various mathematical models of heavy-dutyvehicles have been proposed for computer simulations in [38, 57, 64], where the goal is to develop a tool for predicting and evaluating the longitudinal and directional response of heavy-duty vehicles. Most of the mathematical models published in the literature adopt the Newtonian mechanics approach to describe the body dynamics of heavy-duty vehicles. In this dissertation, a control oriented dynamic modeling approach is proposed for articulated vehicles. A generalized coordinate system is dened in this approach to precisely describe the

28 3 kinematics of a vehicle. Equations of motion are derived based on the Lagrange mechanics. This modeling approach is validated by comparing eld test data of a class 8 tractor-semitrailer type articulated vehicle and the simulation results of the computer model. oordinated Steering and Independent raking ontrol of Tractor-Trailer Vehicles. Independent braking control has been investigated [44, 53, 72] as a safety augmented system for light passenger vehicles. However, independent braking control of articulated vehicles has not been seriously studied. In this dissertation, a steering control algorithm for tractor-semitrailer vehicles is designed as a baseline controller for lane following maneuver in AHS. To enhance safety, a coordinated steering and braking control algorithm is designed..3 Dissertation Outline The outline of the remainder of this dissertation is as follows. In chapter 2, the control system design based on backstepping is reviewed. hapter 3 presents a recursive control algorithm for a class of multivariable nonlinear systems whose vector relative degrees are not well dened. The controller design is based on both the dynamic extension algorithm and the backsteppping control algorithm.

29 4 A backstepping procedure for lateral control of passenger vehicles is developed in chapter 4. In chapter 5 a control oriented dynamic modeling approach for articulated vehicles is proposed. A generalized coordinate system is introduced to describe the kinematics of the vehicle. Equations of motion of a tractor-semitrailer vehicle are derived based on the Lagrange mechanics. Experimental studies are conducted to validate the eectiveness of this modeling approach. hapter 6 presents two lateral control algorithms for a tractor-semitrailer vehicle. The baseline steering control algorithm is designed utilizing input-output linearization, whereas the coordinated steering and braking control algorithm is designed based on the multivariable backstepping technique presented in chapter 3. In chapter 7 the main results of this dissertation are summarized and recommendations for future research are provided.

30 5 hapter 2 ackstepping In this chapter, backstepping design [9, 3, 34, 35, 36, 7] of nonlinear systems is reviewed. The recursive backstepping design methodology is originally introduced in adaptive control theory to systematically construct the feedback control law, the parameter adaptation law and the associated Lyapunov function for a class of nonlinear systems satisfying certain structured properties. In this chapter, various backstepping design techniques, including integrator backstepping, backstepping for strict-feedback systems, adaptive backstepping and robust backstepping, will be reviewed. For a more complete presentation of the adaptive backstepping, refer to a recent book by Krstic, Kanellakopoulos and Kokotovic [35]. Robust backstepping can be found in [7]. As we mentioned in chapter, backstepping is a recursive procedure which breaks a design problem for the full system into a sequence of design problems for lower order systems, and sliding control is such an example. We will give aninterpretation of

31 6 sliding control from the backstepping perspective. The organization of this chapter is as follows. In section 2., integrator backstepping is presented. ackstepping for a more general class of nonlinear systems is given in section 2.2. Adaptive and robust versions of backstepping designs are presented in sections 2.3 and 2.4, respectively. Sliding control is interpreted in section 2.5 from a backstepping perspective. onclusions of this chapter are drawn in the last section. 2. Integrator ackstepping Let us start the integrator backstepping by considering the second order system _x = x 2, x 3 + _ = u (2.) The design objective is that x(t)! ast!. The control law can be synthesized in two steps. We regard as a real control rst. y choosing the Lyapunov function candidate V = 2 x2 and the control law des =,x 2, k x (x) the control objective will be achieved. Nevertheless, is a state and can not be set to des.we dene the variable z =, des

32 7 as the deviation of from its desired value des. With the denition of the error variable, we have _z = _, _ des = u, (2x + k )(k x + x 3, z) Now the Lyapunov function candidate can be augmented as V 2 = V + 2 z2 It's time derivative is _V 2 = x(,x 3, k x + z)+z(u, (2x + k )(k x + x 3, z)) To make _ V 2 negative denite, we choose the control law u =,x +(2x + k )(k x + x 3, z), k 2 z Then we obtain _V 2 =,x 4, k x 2, k 2 z 2 which is negative denite. This implies that x! and! des asymptotically. In this example, is called a virtual control, and its desired value (x) is called a stabilizing function. We notice that the second order system (2.) can also be stabilized by a linearizing control law u =,(2x, 3x 2 )_x, k _x, k 2 x (2.2) However, the,x 3 term, which helps stabilizing Eq. (2.), is canceled by the linearizing control law (2.2). ackstepping design can avoid cancellation of useful nonlinearities.

33 8 The result of integrator backstepping is summarized in the following lemma. Lemma (Integrator ackstepping) onsider the system _x = f(x)+g(x) _ = u (2.3) where f() =. If there exists a stabilizing function = (x) and a positive denite, radially unbounded function V : R n! R such that then the control V (f(x)+g(x)(x)) < ; x u =,c(, (x)) + V (f(x)+g(x)), g(x); c> (2.4) x x asymptotically stabilizes the equilibrium point of (2.3). Proof: This can be easily veried by computing the derivative of the Lyapunov function candidate V a = V + (, (x))2 2 along the system trajectory (2.3) using the control law (2.4).

34 9 2.2 ackstepping for Strict-feedback Systems y recursively applying the integrator backstepping technique, a systematic design can be obtained for the strict-feedback system: _x = f(x)+g(x) _ = f (x; )+g (x; ) 2 _ 2 = f 2 (x; ; 2 )+g 2 (x; ; 2 ) 3 (2.5). _ k = f k (x; ; ; k )+g k (x; ; ; k )u where x 2 R n and ; ; k 2 R. The Lyapunov function and the control law will be constructed in a recursive manner. Step Design a continuously dierentiable stabilizing function = (x) for the x subsystem; i.e., construct a positive denite, radially unbounded function V (x) such that, with this control law, its time derivative where W (x) is positive denite. V (f(x)+g(x)(x)) <,W (x) x Step We start our backstepping procedure by considering the following subsystem _x = f(x)+g(x) _ = f (x; )+g (x; ) 2 (2.6)

35 2 In step, we assume is a virtual control and the control law = (x) stabilizes the x subsystem. To take into account the deviation of the state variable from the stabilizing function (x), we dene the error variable z =, (x) Then _z = _, (x) _x x = f (x; )+g (x; ) 2, (x) x (f(x)+g(x)((x)+z )) (2.7) We proceed in the same way as in integrator backstepping by augmenting the Lyapunov function V = V (x)+ 2 z2 We want to design a stabilizing function 2 = (x; z ) such that the time derivative of the Lyapunov function V is negative denite. _V = _ V (x)+z _z = V (x) x (f(x)+g(x)((x)+z )) + z _z (2.8) <,W (x)+ Substituting _z in (2.7) into (2.8), we obtain V (x) x g(x)z + z _z _V <,W (x)+ V (x) x g(x)z +z ff (x; )+g (x; ) 2, (x) (f(x)+g(x)((x)+z x ))g (2.9)

36 2 It is clear that, if g (x; ) 6=,bychoosing the stabilizing function for the virtual control 2 as 2 = (x; z ) n =,k g z, V (x)g(x), f x (x; )+ (x) (f(x)+g(x)((x)+z x )) o the derivative of the Lyapunov function in (2.9) becomes _V <,W (x), k z 2 (2.) Step 2 In this step, we will consider the subsystem _x = f(x)+g(x) _ = f (x; )+g (x; ) 2 (2.) _ 2 = f 2 (x; ; 2 )+g 2 (x; ; 2 ) 3 We observe that this subsystem can be written as _ X = F (X )+G (X ) 2 _ 2 = f 2 (X ; 2 )+g 2 (X ; 2 ) 3 (2.2) x where X = A, F f(x)+g(x) (X )= A, and G (X )= A.In f (x; ) g (x; ) this notation, the structure of the subsystem (2.2) is identical to that of step (2.6). Similarly, we dene the error variable z 2 = 2, (X )

37 22 We proceed in the same way as in step by augmenting the Lyapunov function V 2 = V (X )+ 2 z2 2 We can design a stabilizing function 3 = 2 (X ;z 2 ) such that the time derivative of the Lyapunov function V 2 is negative denite. This recursive procedure will terminate at the k,th step, where the actual control law for u will be designed. 2.3 Adaptive ackstepping In the previous two sections, we consider backstepping designs for nonlinear systems satisfying certain structured properties. In this section, we will present the idea of adaptive backstepping design procedure for a class of nonlinear systems with unknown parameters. The design procedure will be illustrated by an example. onsider _x = x 2 + (x ) _x 2 = u (2.3) where is an known constant parameter. Step We regard x 2 as a control input rst. Denote ^ as the estimated value for the parameter and the estimation error, ^ as ~. hoose the Lyapunov function candidate V (x ; ~ )= 2 x2 + 2 ~ 2 (2.4)

38 23 It is easy to see that with the control law x 2 =,k x, ^(x ) (x ; ^) (2.5) and the adaptation law _^ = (x )x the derivative of the Lyapunov function candidate (2.4) is (2.6) _V =,k x 2 (2.7) The function in (2.5) is called a stabilizing function for x 2, and in (2.6) is called a tuning function. Step 2 Since x 2 is not the control, we dene the deviation of x 2 from the desired stabilizing function as z = x 2, (x ; ^) With this new error variable z, the system (2.3) can be rewritten as _x =,k x + (x ~ )+z _z = _x 2, _ = u, _ (2.8) and the derivative of the Lyapunov function V is _V = x _x + ~ ~ _ =,k x 2 + x z + ( ~, _^) (2.9)

39 24 Further, the dynamics of the error variable is _z = _x 2, _ = u, x (x 2 + (x )), _^ ^ = u, x x 2, _^, ^ x (x ) (2.2) = u, x x 2, _^, ^ ^ x (x ), ~ x (x ) Augment the Lyapunov function by adding the error variable V 2 (x ;z; ~ )=V (x ; ~ )+ 2 z2 (2.2) y noting (2.9) and (2.2), the derivative ofv 2 can be computed as _V 2 = _ V + z _z =,k x 2 + x z + ( ~, _^) (2.22) Grouping similar terms, we obtain +z(u, x x 2, _^, ^ ^ x (x ), ~ x (x )) _V 2 =,k x 2 + ( ~, x (x )z, _^) +z(u + x, x x 2, _^, ^ ^ x (x )) (2.23) To make _ V2 in (2.23) nonpositive, we can choose the control law and the parameter adaptation law u =,k 2 z, x + x 2 + _^ x ^ + ^ (x ) (2.24) x _^ = (, x (x )z) = ((x )z, x (x )z) (2.25)

40 25 Then the derivative ofv 2 becomes _V 2 =,k x 2, k 2 z 2 (2.26) This implies that x! and z! asymptotically. 2.4 Robust ackstepping In this section, robust backstepping design [7] is illustrated by cnsidering the second order system _x = _x 2 = f(x )+x 2 + ~ (x ;t) u (2.27) where (x ~ ;t) is an unknown nonlinear function bounded by h (x ;t), i.e., j (x ~ ;t)j < h (x ;t). We observe that the uncertainty term ~(x ;t) in (2.27) enters the system dynamics one integrator prior than the control input u does. Since the stabilizing function i in each backstepping step is required to be continuously dierentiable, we can not use a discontinuous sign function for a stabilizing function. Therefore, a smooth approximation is used in the development of the control law. The following lemma quanties the approximation error of a sgn() function by ahyperbolic tanh() function. Lemma 2 Given any >, the following inequality holds h xsgn(x), h x tanh( hx )

41 26 where =:2785 and h is any positive number. The proof of this lemma can be found in [54] or [7]. The control algorithm for the system (2.27) is designed in two steps. Step We regard x 2 as a control input in this step. hoosing the Lyapunov function candidate V = 2 x2 (2.28) and the stabilizing function x 2 =,k x, f(x ), h (x ;t) sgn(x ) (2.29) then we have _V =,k x 2 + x ~ (x ;t), x h (x ;t) sgn(x ) < (2.3) This implies that the x dynamics will be stabilized asymptotically, even in the face of uncertain nonlinearity ~ (x ;t). To avoid using discontinuous stabilizing function, a smooth approximation for (2.29) is x 2 =,k x, f(x ), h (x ;t) tanh( h (x ;t)x ) (x ;t) (2.3) where in the argument oftanh() is a free design parameter. With this modied

42 27 stabilizing function (2.3), the derivative ofv can be recalculated as _V =,k x 2 + x ~ (x ;t), x h (x ;t) tanh( h (x ;t)x ),k x 2 + x h (x ;t)sgn(x ), x h (x ;t) tanh( h (x ;t)x ) (2.32),k x 2 + where the last inequality is obtained by lemma 2. Thus the state variable x will converge to a ball whose size depends on the freely adjusted parameter. Step 2 Since x 2 is not an actual control input, we dene an error variable z = x 2, (x ;t) Then the dynamic equation (2.27) becomes and the derivative ofv is _x + k x + h tanh( h x )=z + ~ (x; t) (2.33) _V,k x x z (2.34) Further, the dynamics of the error variable is _z = _x 2, _ (x ;t) = u, ( x _x + t ) (2.35) = u, x (,k x, h tanh( h x )+z + ~(x; t)), t

43 28 Augment the Lyapunov function V by including the error variable V 2 = V + 2 z2 Then _V 2 = _ V + z _z,k x x z + z _z =,k x x z + z(u, x (,k x, h tanh( h x )+z + ~(x ;t)), t ) Assume there exists a smooth function h 2 (x ;t) such that (2.36) It is clear that by choosing j x ~ (x ;t)j h 2 (x ;t) (2.37) u =,k 2 z, x + x (,k x, h tanh( h x )+z)+ t, h 2 tanh( h 2z 2 ) (2.38) The inequality (2.36) becomes _V 2,k x 2, k 2z 2 +, z x ~(x ;t), zh 2 tanh( h 2z 2 ) (2.39),k x 2, k 2 z which implies that the control law (2.38) renders x(t) globally uniformly bounded. 2.5 Sliding ontrol via ackstepping onsider an n, th order nonlinear system x (n) = f(x)+u (2.4)

44 29 where u is the control input, x is the output of interest, x =[x; _x; ;x n, ] T is the state vector, and the dynamics f(x) is not exactly known, but estimated as ^f(x). The estimation error on f(x) is assumed to be bounded by some known function F (x), that is, j ^f, fj F The design of a sliding controller involves two steps: ) design a stable sliding surface to achieve the control objective, and 2) make the sliding surface attractive by pushing system states toward the surface. These two steps can be interpreted as a backstepping procedure. Let ~x = x, x d be the tracking error. For simplicity, the sliding surface for the system (2.4) can be chosen as s =( d dt + )n, ~x (2.4) It is easy to see that s implies ~x!, or x! x d asymptotically. Step Assume s is the control input of the equation: ( d dt + )n, ~x = s (2.42) Eq. (2.42) can be written in state space form as d ~x = A~x + s (2.43) dt

45 3 where ~x =[~x; _~x; ; ~x (n,2) ] T, A =., n,,n n,2,n, 2 R (n,)(n,) A and =. 2 R (n,) A Since matrix A is stable, given any positive denite matrix Q 2 R (n,)(n,), there exists a positive matrix P 2 R (n,)(n,) satisfying the Lyapunov equation A T P + PA =,Q hoose the Lyapunov function candidate V =~x T P ~x y the denition of the sliding surface, the stabilizing function for s can simply chosen as, i.e., s = (~x) (2.44) which will achieve the control objective. With this stabilizing function, the time derivative ofv is _V =,~x T Q~x

46 3 Step 2 Since s is not the real control, s can not be set to all the time. Yet s can be adjusted by the real control input u. We augment the Lyapunov function V as V 2 = V + s 2 =~x T P ~x + 2 s2 (2.45) Its time derivative can be calculated as _V 2 =,~x T Q~x +2 T P ~xs + s _s (2.46) Furthermore, the derivative of s is _s = ~x n + ~x n, + + n _~x = x (n), x (n) d + ~x n, + + n _~x (2.47) We choose the control law = f(x)+u, x (n) d + ~x n, + + n _~x u =,k s, ^f(x)+x (n) d, ~x n,,, n _~x, F (x)sgn(s), 2 T P ~x (2.48) Then _V 2 =,~x T Q~x, k s 2 + s (f(x), ^f(x)), F (x)sgn(s),~x T Q~x, k s 2 < This implies that ~x and s converges to asymptotically. (2.49) Furthermore, to avoid chattering due to the switching function sgn(s) in (2.48), a smooth approximation is u =,k s, ^f(x)+x (n) d, ~x n,,, n _~x, F (x) tanh( F (x)s ), 2 T P ~x (2.5)

47 32 where in the argument of the tanh() term is a free design parameter. With this smooth control law and lemma 2, _ V2 becomes _V 2 =,~x T Q~x, k s 2 + s (f(x), ^f(x)) F (x)s, F (x) tanh( ),~x T Q~x, k s 2 + (2.5) Thus ~x will converge to a ball whose size depends on the freely adjusted parameter ; i.e., global uniform boundedness is achieved. For the traditional sliding control law, once the sliding surface, s, is designed, the control objective becomes s! in nite time. However, during the transition phase that the state have not reached the sliding surface, the behavior of the state x(t) is not guaranteed. We observe that the control law in (2.5) feedbacks one more term,,2 T P ~x, than the traditional sliding control law. This term will ensure that the tracking error ~x is still decreasing even during the transition phase of the sliding control. 2.6 onclusions ackstepping design of nonlinear system was reviewed in this chapter. backstepping is a recursive procedure which breaks a design problem for the full system into a sequence of design problems for lower order systems. Various backstepping design techniques, including integrator backstepping, backstepping for strict-feedback systems, adaptive backstepping and robust backstepping, were presented. Sliding control was illustrated from the backstepping perspective.

48 33 hapter 3 ackstepping ontrol Design of a lass of Multivariable Nonlinear Systems without Vector Relative Degrees 3. Introduction This chapter is concerned with the control of a class of ane multivariable nonlinear systems with an equal number of inputs and outputs. When the decoupling matrix is singular, the vector relative degree is not well dened [3, 46]. Furthermore, if the multivariable nonlinear system is strongly invertible [28] and strongly

49 34 accessible [46], the vector relative degree of the system can be achieved by adding chains of integrators to the input channels []. Several versions of dynamic extension algorithms [, 45, 65, 73] have been proposed to identify the input channels where dynamic compensators (or integrators) are needed to achieve the nonsingularity of the decoupling matrix. Once the vector relative degree is well dened by adding dynamic compensators in the input channel, the multivariable nonlinear system can be decoupled in the input-output sense. In this chapter, we are concerned with the class of strongly invertible and strongly accessible multivariable nonlinear systems whose decoupling matrix is singular. Since the decoupling matrix is singular, no static state feedback control law can cause the decoupling. However, it has been shown that decoupling can always be achieved by dynamic compensation for the strongly invertible and strongly accessible multivariable nonlinear system [], where the invertibility ofmultivariable nonlinear systems is given by Hirschorn [28]. Instead of attempting to decouple the nonlinear system by adding chains of integrators in the input channels, we provide an alternative approach to control the multivariable nonlinear systems. ased on the dynamic extension algorithm in [45], we utilize the backstepping design methodology [3, 34, 35] to partially close the loop in each design step. In this study, we shall use the dynamic extension algorithm to identify the input channels where controls appear \too early" in the input-output sense, and then we apply the backstepping control algorithm to partially close those loops. The resulting control law proposed in this chapter is static

50 35 state feedback. Even though the nal closed loop form of the nonlinear system is not decoupled, each output is controlled to the desired value asymptotically. The organization of this chapter is as follows. The dynamic extension algorithm proposed in [45] is reviewed in section 3.2. The combined dynamic extension and backstepping design algorithm for multivariable nonlinear systems is presented in section 3.3. ontrols of a planar vehicle, which illustrate and contrast both approaches, are designed in section 3.4. onclusions of this chapter are given in the last section. 3.2 Dynamic Extension Algorithm In this section, we will summarize some denitions and results from the geometric nonlinear control theory [3, 46] and review the dynamic extension algorithm presented in [45]. Denition onsider an ane nonlinear system having m inputs and m outputs _x = f(x)+ P m i= g i(x)u i (3.) and y = h (x) (3.2) y m = h m (x) where x 2 R n, f(x) and g i (x) are n, vectors. The nonlinear system (3.) and (3.2) is called input-output decoupled if, after a possible relabeling of the inputs, the following

51 36 two properties hold.. For each i, i m, the output y i is invariant under the inputs u j, j 6= i. 2. The output y i is not invariant with respect to the input u i, i m. Denition 2 (Lie Derivative) Let h : R n! R be a smooth scalar function, and f : R n! R n be a smooth vector eld on R n, then the Lie derivative of h with respect to f is a scalar function dened by L f h = h x f: Repeated Lie derivatives can be dened recursively L fh = h L i fh = L f (L i, f h) for i =; 2; ::: Similarly, ifg is another vector eld, then the scalar function L g L f h(x) is L g L f h = L g (L f h) With the notation of Lie derivative, the vector relative degree for a multivariable system is dened as follows. Denition 3 The system (3.) and (3.2) is said to have a vector relative degree fr ; :::; r m g atapoint x if. for each i, i m, (L g L k f h i(x); ;L gm L k f h i(x)) = (; ; ) for all k<r i,, and for all x in a neighborhood ofx,