J. Cent. South Univ. Technol. (2010) 17: 572-579

DOI: 10.1007/s11771-010-0525-1                                                                    

Adaptive output feedback control for uncertain nonholonomic chained systems

YUAN Zhan-ping(袁占平)^{1, 2}, WANG Zhu-ping(王祝萍)^{1, 2}, CHEN Qi-jun(陈启军)^{1, 2}

1. Department of Electronics and Information Engineering, Tongji University, Shanghai 201804, China;

2. Key Laboratory of Embedded System and Service Computing, Ministry of Education, Tongji University,

Shanghai 201804, China

? Central South University Press and Springer-Verlag Berlin Heidelberg 2010

                                                                                                

Abstract: An adaptive output feedback control was proposed to deal with a class of nonholonomic systems in chained form with strong nonlinear disturbances and drift terms. The objective was to design adaptive nonlinear output feedback laws such that the closed-loop systems were globally asymptotically stable, while the estimated parameters remained bounded. The proposed systematic strategy combined input-state-scaling with backstepping technique. The adaptive output feedback controller was designed for a general case of uncertain chained system. Furthermore, one special case was considered. Simulation results demonstrate the effectiveness of the proposed controllers.

Key words: adaptive output feedback; input-state scaling; backstepping; nonholonomic system

                                                                                                           

1 Introduction

Over the last few years, controlling nonholonomic systems have received considerable attention. It is shown in BROCKETT’s work [1] that the origin of a simple nonholonomic system with restricted mobility would not be stabilized at the origin by any static continuous state feedback though it is open loop controllable. A number of approaches have been proposed for the problem, which can be classified as discontinuous time-invariant stabilization [2-3], time-varying stabilization [4-5] and hybrid stabilization [6-7]. More details can be found in Ref.[8].

One commonly used approach for controller design of nonholonomic systems is to convert, with appropriate state and input transformations, the original systems into some canonical forms for which controller design can be carried out more easily [9-12]. As illustrated in Refs.[8-9, 13], many nonlinear mechanical systems with nonholonomic constraints on velocities can be transformed, either locally or globally, to a system of canonical form via appropriate state and input transformation. The typical examples include tricycle type mobile robots, cars towing several trailers, a vertical rolling wheel, and a rigid spacecraft with two torque actuators.

Using the special algebra structure of the chained forms, some feedback strategies were proposed to stabilize nonholonomic systems. Adaptive controllers were proposed to stabilize nonholonomic systems with uncertainty in Refs.[14-15]. A hybrid feedback algorithm was presented to make an uncertain wheeled mobile robot globally asymptotically stable in Ref.[16]. To obtain practical point stabilization for a nonholonomic system, a neural network control was used in Refs.[17-18]. An adaptive state feedback control using input-to-state scaling technique was presented in Ref.[19]. Output feedback issue was addressed for the asymptotic and exponential stability properties with uncertainties [20-21], where the first subsystem is assumed to be linear. Recently, robust output-feedback exponential stabilization of uncertain chained systems was presented in Ref.[22], which relies on the known of the first disturbance and other bounded disturbances. In this work, output feedback controllers were designed using input-state scaling and backstepping technique to solve the problem of stabilization of a class of uncertain nonholonomic systems. The contributions of this paper are as follows. Firstly, an adaptive output feedback controller was proposed to stabilize the considered system, including the system which has uncertainties in x₀-subsystem. Secondly, the control design was developed without the priori knowledge on the bounds of the nonlinear drifts and only the system outputs were measurable. At last, an adaptive control based switching scheme was proposed to handle the uncontrollability of the first subsystem, which prevented the possible finite escape of system states, and at the same time guaranteed the bound of all the signals.

2 Problem formulation

A class of perturbed canonical nonholonomic systems in chained form were considered as follows:

1≤i＜n           (1)

  where u₀ and u₁ are control inputs; [x₀, x^T]^T  [x₀, x₁, …, x_n]^T∈Rⁿ⁺¹ are system states;  and (1≤i≤n) are vectors of smooth nonlinear functions of  x₀ and u₀, respectively,  is a vector of unknown bounded constant parameters; and d_i denotes the disturbed virtual control directions.

When we consider the output feedback control problem, system (1) can be rewritten as

1≤i＜n            (2)

  where an

  ×

×

×2≤i＜n    (3)

The control objective is to construct adaptive nonlinear control laws of the form

                          (4)

such that all solutions x₀(t) and x(t) of the closed-loop systems (1), (2) and (4) are bounded, and the adaptively adjusted parameter  and μ are bounded. Furthermore, x₀(t) and x(t) converge to zero when t→∞ and all other signals in the closed-loop system are bounded.

Through the work, the following assumption is proposed which is usually used in output feedback design [20, 22].

Assumption 1: The signs of d₀ and d are known. Without losing generality, we assume d₀ and d are positive constants.

Assumption 2: For  there is a smooth function vector  such that   where θ₀ is the first vector of θ.

For each 1≤i≤n, there is a known smooth function vectorsuch that≤ which is a function of u₀, x₀ and

Assumption 2 implies that the origin is an equilibrium point of system (1).

3 Output feedback control

In this section, an adaptive output feedback controller was designed using input-state scaling and backstepping techniques for the general case of uncertain chained system. Furthermore, one special case was considered. For clarity, the general case was divided into two categories: x₀(t₀)≠0 and x₀(t₀)=0.

3.1 Controller design when x₀(t₀)≠0

When x₀(t₀)≠0, the inherently triangular structure of system (1) suggests that we should design the control inputs u₀ and u₁ in two separate stages.

3.1.1 Control design for u₀

Control u₀ is designed to guarantee that x₀ converges to zero but never crosses zero. So we can choose u₀ as a modified form of SONTAG formula [23], which is also used in Refs.[19, 21].

Consider the control

                     (5)

      (6)

where k₀＞0 and  is the first estimate of

It can be seen that g₀(x₀)≠0 is guaranteed regardless of the values of x₀.

Choose  as

＞0      (7)

Considering the following Lyapunov function

                        (8)

with

The time derivative of V₀ is ≤-d₀k₀x₀². Accordingly, we can conclude that  is bounded, x₀→0 as t→∞ by using LaSalle’s Invariant Theorem. Using Eq.(5), the closed-loop dynamics of the x₀-subsystem is

              (9)

Since x₀(t) and  are bounded, the solution of Eq.(9) is

Here, we assume x₀(t₀)≠0. So it is concluded that u₀ can guarantee that x₀ does not cross zero for all  From the above analysis and Eq.(3), we can see that the x₀-state in Eqs.(1) and (2) can be globally regulated to zero via u₀ in Eq.(5) as t→∞. This phenomenon causes serious trouble in designing a global adaptive stabilization control input u₁ because, in the limit (u₀=0), the x-subsystem is uncontrollable. The following global input-to-state scaling transformation [19] will solve the problem:

1≤i≤n                           (10)

Under the new z-coordinates, with the choice of u₀ in Eq.(5), the x-system is transformed into

 

1≤i≤n-1

                   (11)

 

where

 (12)

A stable adaptive control scheme for the transformed system (11) is obtained by applying the backstepping technique such that all states asymptotically converge to zero and all signals in the closed-loop are bounded.

If u₀≠0 for every t≥0, the discontinuous state transformation (10) is applicable. Then system (11) has the following form:

                   (13)

where

     (14)

3.1.2 Control design of

Lemma 1: There exists a smooth function matrix  such that

                        (15)

Proof: The proof is straightforward from Assumption 2.

Denote C=[1, …, 0] and consider y=Cz as the output of the transformed system (13). It is easy to verify that the pair (C, A) is observable and the pair (A, b) is controllable. In the following, an observer is to be designed to estimate the unmeasurable states (z₂, …, z_n).

Since  is a continuous function, we can choose the matrix P(t) as the one used in output feedback design [22]. Then the following full-order observer can be designed:

 (16)

whereis defined by the following n×l matrix differential equation:

               (17)

This kind of filtering and parameter estimation methods can be found in Ref.[24].

Denote the observation error as  and the parameter estimation error asthe error dynamics is given by

            (18)

Note that  multiplying the right side of Eq.(17) by  and rearranging it, we obtain

           (19)

Definecombining with Eq.(19), Eq.(18) becomes

                      (20)

From Eqs.(13), (16) and (20), the overall system to be controlled is given by

         (21)

where l_i=C_iPC^TC and C_i (i=1, 2, …, n) are n-dimensional row vectors with zero elements except the ith element being 1.

In the following, the output feedback controller design is presented based on backstepping technique. It consists of n steps.

Step 1: Define  where  is viewed as the first virtual control input.

Using  to stabilize the  z₁)-subsystem of Eq.(21), Lyapunov function candidate is chosen as follows:

                (22)

The time derivative of Eq.(22) is given by

                               (23)

By completing the squares, we have

≤

By choosing  and the tuning function τ_i for θ as

            (24)

＞0   (25)

where k₁＞0.

It is noted that  is a smooth function and satisfies

Then the time derivative of V₁ is changed into the following form:

≤  (26)

where the coupling term will be cancelled at the next step.

Step i (2≤i≤n): At step (i-1), we have   and define  where  is referred to the ith virtual control.

              

Lyapunov function candidate is chosen as follows:

By completing the squares in Step 1, there exist a smooth nonnegative function  and a smooth function  such that

≤

Define h₁=0 and the following computable variables

                           

 

which are computable for 2≤i≤n.

Choose the ith tuning function τ_i and the ith virtual control(2≤i≤n-1) as

  

and actual control law u₁, parameter update law as

  (27)

Then at the last step, we have

≤.

Note that andare smooth functions and satisfy and   0×R^l.

So far, output feedback controller was completed for x₀(t₀)≠0.

3.2 Switching control when x₀(t₀) =0

When x₀(t₀)=0, an adaptive control based switching strategy is presented to solve the problem of finite escape for the class of systems under study, in which, the uncertainties do not satisfy the Lipschitz condition in general. So we can choose the following u₀ as in Refs.[19, 21].

                        (28)

where g₀ is given by Eq.(6) and  is updated by Eq.(7).

Choosing the same Lyapunov function Eq.(8), its time derivative is given by

≤                       (29)

which leads to the bound of x₀, and consequently the bound of  as well.

Using the control law Eq.(28), the closed-loop dynamics of the x₀-subsystem is

whereConsequ-ently, x₀ does not escape and x₀(t_s)≠0 for the control design can be carried out.

During the time period [0, t_s], using u₀ defined in Eq.(28), backstepping-based feedback  and a new update law can be obtained for system (2). Then, we conclude that x-state of Eq.(2) will not blow up during [0, t_s]. By Eq.(3), we can also obtain x-sate of Eq.(1) will not blow up during the time period [0, t_s]. Since x₀(t_s)≠0 at t_s, the control inputs u₀ and u₁ can be switched to Eqs.(5) and (27), respectively.

Theorem 1: Under Assumptions 1 and 2, if the output feedback laws Eqs.(5) and (27) are applied to Eqs.(2) and (1) with adaptation law  along with the above switching strategy, systems (2) and (1) are globally regulated at the origin. Furthermore, the estimated parameters are bounded.

Proof: Choose the Lyapunov function

where P is chosen according to Eq.(16). Then, the time derivative of V is given by

≤

which means  and  Since θ is a constant vector,  are bounded. Hence, as  and P(t) are bounded, we conclude that ζ is bounded.

From LaSalle’s Invariant Theorem, it implies that E(t)→0 as t→∞. We have ψ→0 as t→∞. Hence, it can be concluded that ζ→0 as t→∞ from Eq.(18). From the definition of  and the fact of  is bounded,  ζ→0 as t→∞. So e→0 as t→∞.

From the control design procedure for u₁, we can see that all the virtual controls and the actual control u₁ are smooth functions. So we have  (1≤i＜n) and

These properties along with E(t)→0 as t→∞ imply that →0 as t→∞. Since  e→0 as t→∞, which implies that (x₀(t), z(t))→0 as t→∞, we have→0 as t→∞. Accordingly, from the above we conclude that (x₀(t), x(t))→0 as t→∞.

3.3 One special case

In this case, in addition to Assumption 2 we further assume that x₀-subsystem in system (1) owns a special structure, i.e., , where c₀ is a known constant, and lower bound of d₀ is by d₀≥d₀₀＞0.

As in Section 3.1, consider x₀(t₀)≠0 first. Using the knowledge of c₀ and Assumption 1, control law u₀ can be chosen as

 k₀＞                        (30)

with respect to the following Lyapunov function candidate V₀=(1/2)x₀², whose time derivative is given by ≤0. As a consequence, we guarantee x₀→∞ as t→∞. Through transformation (10), x-system is transformed into

                          (31)

where

Denote C=[1, …, 0] and consider y=Cz=z₁ as the output of the transformed system (31). It is easy to verify that the system is observable and controllable. In the following, an observer is to be designed to estimate the unmeasurable states [z₂, …, z_n].

From the analysis in Section 3.1, the overall system to be controlled is given by

         (32)

In the following, the output feedback control is presented like Section 3.1.2.

Step 1: Define  where  is viewed as the first virtual control input. Choose the

Lyapunov function candidate as , where P is given by

                (33)

Since ≤, we choose the virtual

control  and tuning function τ₁ as

        (34)

 

The time derivative of V₁ is given by

≤

where ξ₁ξ₂ will be cancelled at the next step.

Step i (2≤i≤n): Consider the Lyapunov function

candidate  By completing the squares

as in Step 1, there exist a smooth nonnegative function p_i and a smooth function vector g­ such that

≤

Let h₁=0, and define

Choose virtual control and tuning function τ_i   (2≤i≤n-1) as

Then, actual control u₁, update law for  as

        (35)

Then, the time derivative of V_n is given as

≤

So far controller u₁ is obtained. When x₀(t₀)=0, the switching control strategy in Section 3.3 can be applied only when g₀=-k₀.

4 Simulation results

To verify our proposed controller, we consider the following low-dimensional system with parametric uncertainty:

            (36)

where unknown constant parameter a is assumed to be bounded; and d₀, d₁, d₂ are positive constants. The control objective is to design u₀ and u₁ such that (x₀(t₀), x₁(t), x₂(t))→0 as t→∞.

From the change form Eq.(3), system (36) can be turned into

From the analysis in Section 3, and for any k₀＞0, the control law (30) can be applied. After performing the input-state scaling (10), we have θ=a and

Then, the control law u₁ and update law are given by Eq.(34).

In the simulation, the design parameters are chosen as d₀=2, d₁=d₂=1, k₀=k₁=k₂=1, l₁=l₂=1 to place both poles as (A-LC) at -1. The system parameter is chosen as a=0.5 and the initial conditions are   and

As seen from Fig.1, all signals (x₀, x₁, d₁x₂) globally converge to zero. Fig.2 shows the control inputs. From Fig.2, it can be seen that the control efforts using the controller derived in this work are mild. Fig.3 shows that the estimation of the unknown constant is bounded but does not approach to its true value.

5 Conclusions

(1) An output feedback controller is proposed for a class of uncertain nonholonomic chained systems with

Fig.1 States of simulated system

Fig.2 Control of simulated system

Fig.3  of simulated system

drift nonlinearity and parameter uncertainties.

(2) An output feedback controller is addressed without imposing any restriction on the parameter uncertainty and the growth of the drift nonlinearities.

(3) The proposed design method does not require any priori knowledge on the bounds of the nonlinear drifts.

(4) To facilitate the adaptive controller design, an adaptive observer is constructed to handle the technical problem due to the presence of unavailable states in the regressor matrix.

(5) All the system states are proven to converge to the origin by choosing the design parameters and the estimated parameters maintain bounded. Simulation results demonstrate the effectiveness of the proposed control approach.

References

[1] BROCKETT R. Asymptotic stability and feedback stabilization [M]. Differential Geometry Control Theory. Basel: Birkhauser, 1983: 181-208.

[2] ASTOLFI A. Discontinuous control of nonholonomic systems [J]. Systems and Control Letters, 1996, 27: 37-45.

[3] MARCHAND N, ALAMIR M. Discontinuous exponential stabilization of chained form systems [J]. Automatica, 2003, 39: 343-348.

[4] WALSH G C, BUSHNELL L G. Stabilization of multiple input chained form control systems [J]. System and Control Letters, 1995, 25: 227-234.

[5] JIANG Zhong-ping. Iterative design of time-varying stabilizers for multi-input systems in chained form [J]. Systems and Control Letters, 1996, 28: 255-262.

[6] SORDALEN O J, de WIT C C. Exponential stabilization of nonholonomic chained systems [J]. IEEE Trans Automatic Control, 1995, 40(1): 35-49.

[7] KOLMANOVSKY I, MCCLAMROCH N H. Hybrid feedback laws for a class of cascaded nonlinear control systems [J]. IEEE Trans Automat Control, 1996, 41: 1271-1282.

[8] KOLMA NOVSKY I, MCCLAMROCH N H. Development in nonholonomic control problems [J]. IEEE Control System Magazine, 1995, 15(6): 20-36.

[9] MURRAY R, SASTRY S. Nonholonomic motion planning: Steering using sinusoids [J]. IEEE Trans Automat Control, 1993, 38: 700- 716.

[10] M'CLOSKEY R, MURRAY R. Convergence rate for nonholonomic systems in power form [C]// Proceedings of the American Control Conference. Chicago, 1992: 2489-2493.

[11] HUO W, GE S S. Exponential stabilization of nonholonomic systems: An eni approach [J]. International Journal of Control, 2001, 74(15): 1492-1500.

[12] OYA M, SU C Y, KATOH R. Robust adaptive motion/force tracking control of uncertain nonholonomic mechanical systems [J]. IEEE Transaction on Robotics and Automation, 2003, 19(1): 175-181.

[13] MURRAY R, SASTRY S. Steering nonholonomic systems in chained forms [C]// Proceedings of the 30th IEEE Conference on Decision and Control. Brightom, 1991: 1121-1126.

[14] COLBAUGH R, BARANY R, GLASS K. Adaptive control of nonholonomic mechanical systems [C]// Proceedings of the 35th IEEE Conference on Decision and Control. Kobe, 1996: 1428-1434.

[15] GE S S, WANG J, LEE T H, ZHOU G Y. Adaptive robust stabilization of dynamic nonholonomic chained systems [J]. Journal of Robotic Systems, 2001, 18(3): 119-133.

[16] HESPANHA J P, LIBERZON S, MORSE A S. Towards the supervisory control of uncertain nonholonomic systems [C]// Proceedings of the 1999 American Control Conference. San Diego, 1999: 3520-3524.

[17] FIERRO R, LEWIS F L. Control of a nonholonomic mobile robot: Backstepping kinematics into dynamics [C]// Proceedings of the 34th IEEE Conference on Decision and Control. New Orleans, 1995: 1722-1727.

[18] WANG Z P, GE S S, LEE T H. Robust adaptive neural network control of uncertain nonholonomic systems with strong nonlinear drifts [J]. IEEE Transaction on Systems, Man, and Cybernetics—Part B: Cybernetics, 2004, 34(5): 2048-2059.

[19] DO K D, PAN J. Adaptive global stabilization of nonholonomic systems with strong nonlinear drifts [J]. Systems and Control Letters, 2002, 46: 195-205.

[20] JIANG Zhong-ping. Robust exponential regulation of nonholonomic systems with uncertainties [J]. Automatica, 2000, 36: 189-209.

[21] GE S S, WANG Z P, LEE T H. Adaptive stabilization of uncertain nonholonomic systems by state and output feedback [J]. Automatica, 2003, 39: 1451-460.

[22] XI Zai-rong, FENG Gang, JIANG Zhong-ping, CHENG Dai-zhan. Output feedback exponential stabilization of uncertain chained systems [J]. Journal of The Franklin Institute, 2007, 344: 36-57.

[23] SONTAG E D. Smooth stabilization implies coprime factorization [J]. IEEE Trans Automat Control, 1989, 34(4): 435-443.

[24] KRSTIC M, KANELLAKOPOULOS I, KOKOTOVIC P V. Nonlinear and adaptive control design [M]. New York: Wiley, 1995.

                    

Foundation item: Project(60704005) supported by the National Natural Science Foundation of China; Project(07ZR14119) supported by Natural Science Foundation of Shanghai Science and Technology Commission; Project(2009AA04Z213) supported by the National High-Tech Research and Development Program of China

Received date: 2009-09-16; Accepted date: 2009-11-30

Corresponding author: CHEN Qi-jun, PhD, Professor; Tel: +86-21-69589378; E-mail: qjchen@tongji.edu.cn

(Edited by YANG You-ping)