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TRACKING OF NONHOLONOMIC CHAINED-FORM SYSTEMS WITHOUT PERSISTENT EXCITATION1
TRACKING OF NONHOLONOMIC CHAINED-FORM SYSTEMS WITHOUT PERSISTENT EXCITATION 1 Yu-Ping Tian 2 , Ke-Cai Cao
School of Automation, Southeast University, Nanjing, 210096, P.R.China
Abstract: Global K-exponential controllers are constructed for the tracking control problem of nonholonomic systems in chained-form whose reference targets are allowed to converge to a point exponentially. By using a novel transformation and the cascade-design method, the tracking control problem is converted into a stabilization problem of two simple subsystems. Then, a design approach based on linear matrix inequalities is developed for stabilizing the subsystems. The assumption on the reference signal is much more relaxed than those given in the c previous papers. Copyright ° 2007 IFAC Keywords: Nonlinear systems, Tracking, Stabilization, Mobile robots
1. INTRODUCTION As shown by Murray and Sastry (1993), through a suitable change of coordinates and state feedback many mechanical systems with nonholonomic constraints can be transformed into the following chained form x˙ 1 = u1 , x˙ 2 = u2 , x˙ 3 = x2 u1 , .. . x˙ n = xn−1 u1 ,
(1)
where u = (u1 , u2 )T is the input and x = (x1 , · · · , xn )T is the state. The system of this canonical form has been extensively studied by many researchers as a benchmark example in the area of control of nonholonomic systems. Efforts 1
This work was supported by National Natural Science Foundation of China (under grants 60425308 and 60673058), National “863” Programme of China (under grant 2006AA04Z263). 2 Corresponding author.([email protected])
have been devoted to both stabilization and tracking of the system (see, e.g., Kolmanovsky and McClamroch (1995); Astolfi (1996); Samson (1995); Tian and Li (2002); Hespanha and Morse (1999) for stabilization and Jiang (1999); Walsh et al. (1994); Lee et al. (2001); Lefeber et al. (2000) for tracking, respectively). The tracking problem for the chained-form system is to design appropriate controllers u1 and u2 so that the state trajectory of system (1) can follow a vector-valued reference signal xd (t) = (x1d (t), · · · , xnd (t))T which is produced by a virtual system of the same form as system (1) with inputs u1d , u2d . A common assumption in most of the papers on the tracking problem is that the inputs of the reference system satisfy certain kinds of conditions such as persistent excitation (PE) or not converging to zero. The fulfillment of these conditions implies that the reference target is always moving and never stops at any point. In the case of tracking control of a mobile robot, which are usually modeled as a three-dimensional chained-form system, this assumption implies that the line velocity or angular velocity of the target
must not converge to zero (Jiang and Nijmeijer, 1997). Concerning tracking an n-dimensional chained-form system, Jiang (1999) requires u1d must not converge to zero while Lefeber (2000) and Lefeber et al. (2000) impose a PE condition on u1d . These conditions mean that tracking of straight lines is also excluded besides stabilization. In other studies such as Aneke et al. (2003), the PE condition is also imposed on the first input of the second-order chained-form system. All of these restrictions make it impossible to treat the tracking problem and the stabilization problem simultaneously. Inspired by Samson’s time-varying stabilizer (Samson, 1995), Lee et al. developed a time-varying “universal” controller to achieve both stabilization and tracking of mobile robots (Lee et al., 2001). Universal controllers for mobile robots are also presented in Do et al. (2004a,b). However, as pointed in Lee et al. (2001), there is no straight way to extend these design methods to an ndimensional chained-from system. Moreover, since some PE signals are introduced into the virtual control in the design of these universal controllers, the convergence rate of the tracking error cannot be exponential. Recently, the authors show that the globally Kexponential stability can be achieved in the tracking problem for the chained-form system with target signals that may exponentially decay to zero (Tian and Cao, 2006). A linear time-varying tracking controller is derived in Tian and Cao (2006) under the following assumption on the reference signal u1d .
In this paper we study the global tracking problem for the chained-form system under the following assumption. Assumption 3. There exist constants λ ≥ 0, T > 0 and D 6= 0 (|D| > T > 0) such that d(t) = u1d (t)/ exp(−λ(t − t0 )) satisfies |d(t) − D| ≤ T. Obviously, this assumption is very close to Assumption 2. It does not requires d(t) approaches to any constant. Actually, in the case λ = 0 Assumption 3 reduces to the assumption on u1d given by Jiang (1999). However, compared to Jiang (1999), our assumption allows u1d to converge to zero exponentially and has no restrictions on u2d . The main idea of this paper can be sketched as follows. Using a modified a dilation technique for nonholonomic systems (Astolfi, 1996; Tian and Li, 2002), we introduce a time-varying coordinate transformation to extract the controllable part from the reference system. Then, a cascade-design method is adopted to transform the stabilization of the tracking-error system into two stabilization problems for subsystems. One of them contains parameter uncertainty due to the uncertainty in reference signal. So, a designed method based on a linear matrix inequality (LMI) is developed for obtaining tracking controllers.
2. PRELIMINARIES 2.1 K-exponential stability of cascaded systems Consider a time-varying system given by
Assumption 1. There exist a λ ≥ 0 and a D 6= 0 such that
z˙1 = f1 (t, z1 ) + g(t, z1 , z2 )z2 , z˙2 = f2 (t, z2 ).
d(t) = u1d (t)/ exp(−λ(t − t0 ))
We call system (3) a cascaded system because it can be viewed as the following system
is bounded, continuous and satisfies Z ∞ lim d(t) = D, kd(t) − Dkdt < +∞, t→∞
Σ1 : (2)
0
where t0 represents the initial time from which the control action starts. It should be pointed out that this assumption is still quite restrictive. It requires d(t) approaches to a constant. In the case of λ = 0, it cannot go back to the PE assumption considered in the aforementioned references. A more relaxed assumption was also considered in Tian and Cao (2006), which was given as follows. Assumption 2. There exists a λ ≥ 0 such that u1d / exp(−λ(t − t0 )) is persistently exciting. However, under this assumption, only the existence of time-varying controller for global Kexponential tracking was proved and no explicit form of the controller was given.
z˙1 = f1 (t, z1 )
(3)
(4)
perturbed by the output of the system Σ2 :
z˙2 = f2 (t, z2 ).
(5)
Lemma 1. (Lefeber, 2000; Panteley et al., 1998) The cascaded time-varying system (3) is globally K-exponentially stable if the following assumptions are satisfied: (1) the subsystems (4) is globally uniformly exponentially stable ( GUES ); (2) the function g(t, z1 , z2 ) satisfies the following condition for all t ≥ t0 kg(t, z1 , z2 )k ≤ θ1 (kz2 k) + θ2 (kz2 k)kz1 k , where θi (·) : R+ → R+ (i = 1, 2) are continuous functions; (3) the subsystem (5) is globally K-exponentially stable.
2.2 Quadratic stabilization of uncertain systems Consider the following system x˙ = (A + ∆A(t))x, n×n
(6)
n×n
where A ∈ R ,∆A(t) ∈ R and ∆A(t) = EΣ(t)F with E, F ∈ Rn×n and unknown Σ(t) belongs to the following sets: Ω = {Σ(t)|Σ(t)T Σ(t) ≤ I, ∀t}. Definition 1. (Petersen and Savkin, 1999) System (6) is quadratically stable if there exist a positive definite matrix P and a positive number a such that for all t xT (t)[AT P + P A]x(t) + 2x(t)∆AT (t)P x(t) ≤ −a||x(t)||2 is satisfied. From Definition 1 it can be obtained that if a system is quadratically stable then the system is exponentially stable for all ∆A(t)(Σ(t) ∈ Ω), and for any initial condition x(0) the following inequality holds for all t ≥ 0 a σmax (P ) ||x(0)||2 exp(− t), ||x(t)||2 ≤ σmin (P ) σmin (P ) where σmax (P ) and σmin (P ) represent the maximum and minimum eigenvalues of matrix P , respectively. Lemma 2. (Petersen and Savkin, 1999) System (6) is quadratically stable if and only if A is Hurwitz and ||F (sI − A)−1 E||∞ < 1. 3. DESIGN OF TRACKING CONTROLLERS 3.1 Cascaded system based on dilation Define the tracking error as ei = xi − xid and it is easy to obtain the tracking error dynamics of the chained-form system as follows e˙ 1 = u1 − u1d , e˙ 2 = u2 − u2d , e˙ 3 = u1d e2 + x2 (u1 − u1d ), .. . e˙ n = u1d e(n−1) + xn−1 (u1 − u1d ).
(7)
y˙ 1 = (n − 2)λy1 + (u1 − u1d ) exp((n − 2)λ(t − t0 )). (10) d(t)−D T , Denote Y = [y2 , y3 , · · · , yn ] and Σ(t) = T then the transformed system (9)(10) can be viewed as the following system Y˙ = (A + EΣ(t)F )Y + Bv (11) cascaded by the system (10), where v = u2 − u2d and A, B, E, F are defined as follows
0 0 ··· 0 0 1 . . D λ . 0 0 0 . , B = . .. .. A= , ... ... . . . . . . 0 0 0 · · · (n − 3)λ 0 0 0 0 ··· D (n − 2)λ 0 0 ··· 0 0 T 0 ··· 0 0 . . 1 0 . 0 0 0 T ... 0 0 E= ... ... . . . ... ... , F = ... ... . . . ... ... . 0 0 · · · 0 0 0 0 · · · T 0 0 0 ··· 1 0 0 0 ··· 0 T 3.2 Design of controller u1 Under the control law u1 = u1d − exp(−(n − 2)λ(t − t0 ))k1 y1 , k1 > (n − 2)λ,
(12)
the closed-loop subsystem of (10) becomes y˙ 1 = (−k1 + (n − 2)λ)y1 ,
Introduce a coordinate transformation for system (7) as y1 = e1 / exp(−(n − 2)λ(t − t0 )), y2 = e2 , y3 = e3 / exp(−λ(t − t0 )), .. . yn = en / exp(−(n − 2)λ(t − t0 )).
Then, the transformed system is given by y˙ 2 0 0 ··· 0 0 y2 y˙ 3 d(t) λ · · · y3 0 0 .. .. .. . . .. . . .. .. . = . . . . y˙ n−1 0 0 · · · (n − 3)λ yn−1 0 y˙ n 0 0 ··· d(t) (n − 2)λ yn 0 u1 − u1d 1 x2 exp(−λ(t − t )) 0 0 .. .. . + . (u2 −u2d )+ u1 − u1d 0 xn−2 exp(−(n − 3)λ(t − t0 )) 0 u1 − u1d xn−1 exp(−(n − 2)λ(t − t0 )) (9) and
(13)
which is globally exponentially stable. 3.3 Design of controller u2
(8)
Now, we design a state-feedback law v = KY so that the closed-loop system Y˙ = (A + BK + EΣ(t)F )Y (14) is quadratically stable.
By Lemma 2, we know that system (14) is quadratically stable if and only if A + BK is Hurwitz and ||F (sI − A − BK)−1 E||∞ < 1. Further, according to H ∞ control theory, A + BK is Hurwitz and ||F (sI − A − BK)−1 E||∞ < 1 if and only if there exists a positive definite matrix P such that (A+BK)T P +P (A+BK)+P E(P E)T +F T F < 0. (15) By Schur’s Complement Lemma the inequality (15) is equivalent to the following inequality (A + BK)T P + P (A + BK) P E F T ET P −I 0 < 0. F 0 −I (16) Pre-multiplying and post-multiplying (16) by diag{P −1 , I, I}, we get −1 P (A + BK)T +(A + BK)P −1 E (F P −1 )T ET −I 0 −1 FP 0 −I < 0. Letting X = P −1 , W = KX, the above inequality can also be rewritten as (AX + BW )T +(AX + BW ) E (F X)T ET −I 0 < 0. FX 0 −I (17) We say the above LMI is solvable if there exist W ? and a positively definite X ? such the inequality is satisfied.
3.4 Stability theorem Theorem 3. Consider the chained-form tracking error dynamics (7). Assume that u1d satisfies Assumption 3 and x2d , x3d , · · · , x(n−1)d are bounded. Then, the state feedback control u1 = u1d − exp(−(n − 2)λ(t − t0 ))k1 y1 (18) u2 = u2d + W ? (X ? )−1 Y
(19)
globally K-exponentially stabilizes the error system (7), where k1 > (n − 2)λ and W ? , X ? are solutions of (17). Proof. Since the system formed by (9) and (10) can be regarded as a system (11) cascaded by (10), we check three conditions of Lemma 1 for the cascaded system. • Since the quadratical stability implies the exponential stability, the GUES property of (11) is obtained from the controller designed in Section 3.3.
• Under the control law u1 , the interconnection term of system (11) and (10) can be rewritten as g(t, Y, Z)Z, where 0 y2 g(t, Y, Z) = −k1 e−(n−3)λ(t−t0 ) . . . yn−1 0 x2d e−(n−3)λ(t−t0 ) −k1 , .. . x(n−1)d Z = y1 , and Y is defined in Section 3.1. Since x2d , · · · , x(n−1)d are bounded, the second condition is satisfied. • The GUES property of system (10) follows immediately from the controller constructed in Section 3.2. Therefore, the third condition is also satisfied. Therefore, by Lemma 1 we conclude that the cascaded system formed by (9) and (10) is globally K-exponentially stable. According to the transformation (8) we know that under the control input (18) (19) the original chained form error dynamics (7) are also globally K-exponentially stable.
4. APPLICATION In this section we apply the control law proposed in Theorem 3 to the following unicycle mobile robot x˙ c = v cos θc , y˙ c = v sin θc , θ˙c = ω.
(20)
Although this system has been extensively studied by many researcher, the new assumption given in this paper (Assumption 3) allows us to discuss some interesting tracking tasks which has not been addressed before. The reference trajectory is generated by a virtual mobile robot which shares the same form as system (20) with reference input vd and ωd . In the following discussion we let t0 = 0 for simplicity. Let ωd = exp(−0.5t)(1 + 0.5 sin(t)). Then, it is easy to check that u1d = ωd = exp(−0.5t)(1 + 0.5 sin(t)), u2d = vd − (xd sin(θd ) − yd cos(θd ))ωd in chained form and λ = 0.5, T = 0.5, D = 1 in Assumption 3. Initial conditions of the reference system in chained-form and the error system are selected as [−2.9856 0.1570 1.3153] and [−0.080 0.8415 − 0.2579], respectively. Using the LMI toolbox in Matlab, we get
1.4
0.4
tracking errors (m)
(xd,yd) (x ,y ) c c
1.2
1
Y−direction (m)
−0.8
0
1
2
3
4
0.6
5
time (s)
6
7
8
9
10
0.1
tracking errors (rad)
0.4
0
−0.2 −1.1
−1
−0.9
−0.8
−0.7
X−direction (m)
−0.6
−0.5
−0.4
−0.3
Fig. 1. Moving path of mobile robots tracking errors (m)
(xc−xd) (yc−yd)
−0.2
−0.8
0
1
2
3
4
5
time (s)
6
7
θc−θd 0.05
0
−0.05
−0.1
0
1
2
3
4
5
time (s)
6
7
8
9
10
Fig. 4. Tracking errors
0.4
8
0.02
9
10
θ −θ c
tracking errors (rad)
−0.2
0.8
0.2
d
0
moving path of the mobile robot on the plane . The tracking errors are presented in Fig. 2. • vd = sin t In this case the target is in a periodic movement along a line when time goes to infinite. Fig. 3 shows the moving path of the mobile robot on the plane. The tracking errors are presented in Fig. 4.
−0.02 −0.04 −0.06 −0.08
0
1
2
3
4
5
time (s)
6
7
8
9
10
1.4 1.2
5. CONCLUSION Global K-exponential tracking controllers are constructed for the tracking control problem of nonholonomic systems in chained form. A coordinate transformation based on dilation is introduced and then the cascade-design method is adopted to transform the tracking problem into two simple stabilization problem. The obtained results show that the popular condition of persistent excitation or not converging to zero is not necessary for the tracking control of nonholonomic systems in chained form.
Fig. 2. Tracking errors (xd,yd) (xc,yc)
1 0.8
Y−direction (m)
(xc−xd) (yc−yd)
0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −2
−1.8
−1.6
−1.4
−1.2
−1
X−direction (m)
−0.8
−0.6
−0.4
−0.2
Fig. 3. Moving path of mobile robots · ¸ 5.2461 −1.9637 X= , −1.9637 1.2180 £ ¤ W = −7.7397 −1.0523 . Then, the control gain for the second subsystem is W (X)−1 = [−4.5362 − 8.1771]. For the first subsystem we can set k1 = 3 which ensures k1 > (n − 2)λ. This paper does not impose any restrictions on u2d . In simulation the following two cases are considered. • vd = exp(−0.5t) In this case the target asymptotically approaches a steady point. Fig. 1 shows the
REFERENCES Aneke, N.P. , H. Nijimeijer, A.G.de Jager (2003). Tracking Control of Second-order Chained Form Systems by Cascade Backstepping, International Jounal of Robust and Nonlinear Control, 13, 95-115. Astolfi, A. (1996). Discontinuous control of nonholonomic systems, System & Control Letters, . 27, 37-45. K.D. Do, Jiang, Z.-P., J. Pan (2004a) A global output-feedback controller for simultaneous tracking and stabilization of unicycle-type mobile robots, IEEE Transactions on Robotics and Automation, 20, 589-594. Do, K.D., Z.-P. Jiang, J. Pan (2004b). Simultaneous tracking and stabilization of mobile robots: an adaptive approach, IEEE Transactions on Automatic Control, 49, 1147-1152.
Hespanha, J.P., A.S. Morse (1999). Stabilization of nonholonomic integrators via logic-based switching, Automatica, 35, 385C393. Jiang, Z.-P., H. Nijmeijer (1997). Tracking control of mobile robots: a case study in backstepping, Automatica, 33, 1393-1399. Jiang, Z.-P. (1999). A recursive technique for tracking control of nonholonomic systems in the chained form, IEEE Transactions on Automatic Control, 44, 265-279. Kolmanovsky, I., N.H. McClamroch (1995). Developments in nonholonomic control systems, IEEE Control Systems Magazine, 15, 20-36. Lee, T.-C., K.-T. Song, C.-H. Lee, & C.-C. Teng (2001). Tracking control of unicycle modeled mobile robots using a saturation feedback controller, IEEE Transactions Control Systems Technology, 9, 305-318. Lefeber, E., A. Robertsson, H. Nijmeijer (2000). Linear controllers for exponential tracking of systems in chained-form, International Journal of Robust Nonlinear Control, 10, 243-263. Lefeber, E. (2000). Thesis: Tracking Control of Nonlinear Mechanical Systems, University of Twente, The Netherlands. Murray, R.M., S.S. Sastry (1993). Noholonomic motion planning: steering using sinusoids, IEEE Transactions on Automatic Control, 38, 700716. Panteley, E., E. Lefeber, A. Lor´ıa and H. Nijmeijer (1998). Exponential tracking control of a mobile car using a cascaded approach, Proceedings of the IFAC Workshop on Motion Control, Grenoble, France, pp. 221-226. Petersen, I.R., A.V. Savkin (1999). Robust Kalman Filtering for Signals and Systems with Large Uncertainties, Birkhauser, Boston, MA, USA. Samson, C. (1999). Control of chained system application to path following and timevarying point-stabilization of mobile robots, IEEE Transactions on Automatic Control, 40, 64-77. Tian, Y.-P., S. Li (2002). Exponential stabilization of nonholonomic dynamic systems by smooth time-varying control, Automatica, 38, 1139-1146. Tian, Y.-P., K.-C. Cao (2006). Time-varying linear controllers for exponential tracking of nonholonomic systems in chained form, International Journal of Nonlinear and Robust Control, in press. Walsh, G., D. Tilbury, R. Murry and J.P. Launond (1994). Stabilization of trajectories for systems with nonholonomic constraints, IEEE Transactions on Automatic Control, 39, 216-222.
School of Automation, Southeast University, Nanjing, 210096, P.R.China
Abstract: Global K-exponential controllers are constructed for the tracking control problem of nonholonomic systems in chained-form whose reference targets are allowed to converge to a point exponentially. By using a novel transformation and the cascade-design method, the tracking control problem is converted into a stabilization problem of two simple subsystems. Then, a design approach based on linear matrix inequalities is developed for stabilizing the subsystems. The assumption on the reference signal is much more relaxed than those given in the c previous papers. Copyright ° 2007 IFAC Keywords: Nonlinear systems, Tracking, Stabilization, Mobile robots
1. INTRODUCTION As shown by Murray and Sastry (1993), through a suitable change of coordinates and state feedback many mechanical systems with nonholonomic constraints can be transformed into the following chained form x˙ 1 = u1 , x˙ 2 = u2 , x˙ 3 = x2 u1 , .. . x˙ n = xn−1 u1 ,
(1)
where u = (u1 , u2 )T is the input and x = (x1 , · · · , xn )T is the state. The system of this canonical form has been extensively studied by many researchers as a benchmark example in the area of control of nonholonomic systems. Efforts 1
This work was supported by National Natural Science Foundation of China (under grants 60425308 and 60673058), National “863” Programme of China (under grant 2006AA04Z263). 2 Corresponding author.([email protected])
have been devoted to both stabilization and tracking of the system (see, e.g., Kolmanovsky and McClamroch (1995); Astolfi (1996); Samson (1995); Tian and Li (2002); Hespanha and Morse (1999) for stabilization and Jiang (1999); Walsh et al. (1994); Lee et al. (2001); Lefeber et al. (2000) for tracking, respectively). The tracking problem for the chained-form system is to design appropriate controllers u1 and u2 so that the state trajectory of system (1) can follow a vector-valued reference signal xd (t) = (x1d (t), · · · , xnd (t))T which is produced by a virtual system of the same form as system (1) with inputs u1d , u2d . A common assumption in most of the papers on the tracking problem is that the inputs of the reference system satisfy certain kinds of conditions such as persistent excitation (PE) or not converging to zero. The fulfillment of these conditions implies that the reference target is always moving and never stops at any point. In the case of tracking control of a mobile robot, which are usually modeled as a three-dimensional chained-form system, this assumption implies that the line velocity or angular velocity of the target
must not converge to zero (Jiang and Nijmeijer, 1997). Concerning tracking an n-dimensional chained-form system, Jiang (1999) requires u1d must not converge to zero while Lefeber (2000) and Lefeber et al. (2000) impose a PE condition on u1d . These conditions mean that tracking of straight lines is also excluded besides stabilization. In other studies such as Aneke et al. (2003), the PE condition is also imposed on the first input of the second-order chained-form system. All of these restrictions make it impossible to treat the tracking problem and the stabilization problem simultaneously. Inspired by Samson’s time-varying stabilizer (Samson, 1995), Lee et al. developed a time-varying “universal” controller to achieve both stabilization and tracking of mobile robots (Lee et al., 2001). Universal controllers for mobile robots are also presented in Do et al. (2004a,b). However, as pointed in Lee et al. (2001), there is no straight way to extend these design methods to an ndimensional chained-from system. Moreover, since some PE signals are introduced into the virtual control in the design of these universal controllers, the convergence rate of the tracking error cannot be exponential. Recently, the authors show that the globally Kexponential stability can be achieved in the tracking problem for the chained-form system with target signals that may exponentially decay to zero (Tian and Cao, 2006). A linear time-varying tracking controller is derived in Tian and Cao (2006) under the following assumption on the reference signal u1d .
In this paper we study the global tracking problem for the chained-form system under the following assumption. Assumption 3. There exist constants λ ≥ 0, T > 0 and D 6= 0 (|D| > T > 0) such that d(t) = u1d (t)/ exp(−λ(t − t0 )) satisfies |d(t) − D| ≤ T. Obviously, this assumption is very close to Assumption 2. It does not requires d(t) approaches to any constant. Actually, in the case λ = 0 Assumption 3 reduces to the assumption on u1d given by Jiang (1999). However, compared to Jiang (1999), our assumption allows u1d to converge to zero exponentially and has no restrictions on u2d . The main idea of this paper can be sketched as follows. Using a modified a dilation technique for nonholonomic systems (Astolfi, 1996; Tian and Li, 2002), we introduce a time-varying coordinate transformation to extract the controllable part from the reference system. Then, a cascade-design method is adopted to transform the stabilization of the tracking-error system into two stabilization problems for subsystems. One of them contains parameter uncertainty due to the uncertainty in reference signal. So, a designed method based on a linear matrix inequality (LMI) is developed for obtaining tracking controllers.
2. PRELIMINARIES 2.1 K-exponential stability of cascaded systems Consider a time-varying system given by
Assumption 1. There exist a λ ≥ 0 and a D 6= 0 such that
z˙1 = f1 (t, z1 ) + g(t, z1 , z2 )z2 , z˙2 = f2 (t, z2 ).
d(t) = u1d (t)/ exp(−λ(t − t0 ))
We call system (3) a cascaded system because it can be viewed as the following system
is bounded, continuous and satisfies Z ∞ lim d(t) = D, kd(t) − Dkdt < +∞, t→∞
Σ1 : (2)
0
where t0 represents the initial time from which the control action starts. It should be pointed out that this assumption is still quite restrictive. It requires d(t) approaches to a constant. In the case of λ = 0, it cannot go back to the PE assumption considered in the aforementioned references. A more relaxed assumption was also considered in Tian and Cao (2006), which was given as follows. Assumption 2. There exists a λ ≥ 0 such that u1d / exp(−λ(t − t0 )) is persistently exciting. However, under this assumption, only the existence of time-varying controller for global Kexponential tracking was proved and no explicit form of the controller was given.
z˙1 = f1 (t, z1 )
(3)
(4)
perturbed by the output of the system Σ2 :
z˙2 = f2 (t, z2 ).
(5)
Lemma 1. (Lefeber, 2000; Panteley et al., 1998) The cascaded time-varying system (3) is globally K-exponentially stable if the following assumptions are satisfied: (1) the subsystems (4) is globally uniformly exponentially stable ( GUES ); (2) the function g(t, z1 , z2 ) satisfies the following condition for all t ≥ t0 kg(t, z1 , z2 )k ≤ θ1 (kz2 k) + θ2 (kz2 k)kz1 k , where θi (·) : R+ → R+ (i = 1, 2) are continuous functions; (3) the subsystem (5) is globally K-exponentially stable.
2.2 Quadratic stabilization of uncertain systems Consider the following system x˙ = (A + ∆A(t))x, n×n
(6)
n×n
where A ∈ R ,∆A(t) ∈ R and ∆A(t) = EΣ(t)F with E, F ∈ Rn×n and unknown Σ(t) belongs to the following sets: Ω = {Σ(t)|Σ(t)T Σ(t) ≤ I, ∀t}. Definition 1. (Petersen and Savkin, 1999) System (6) is quadratically stable if there exist a positive definite matrix P and a positive number a such that for all t xT (t)[AT P + P A]x(t) + 2x(t)∆AT (t)P x(t) ≤ −a||x(t)||2 is satisfied. From Definition 1 it can be obtained that if a system is quadratically stable then the system is exponentially stable for all ∆A(t)(Σ(t) ∈ Ω), and for any initial condition x(0) the following inequality holds for all t ≥ 0 a σmax (P ) ||x(0)||2 exp(− t), ||x(t)||2 ≤ σmin (P ) σmin (P ) where σmax (P ) and σmin (P ) represent the maximum and minimum eigenvalues of matrix P , respectively. Lemma 2. (Petersen and Savkin, 1999) System (6) is quadratically stable if and only if A is Hurwitz and ||F (sI − A)−1 E||∞ < 1. 3. DESIGN OF TRACKING CONTROLLERS 3.1 Cascaded system based on dilation Define the tracking error as ei = xi − xid and it is easy to obtain the tracking error dynamics of the chained-form system as follows e˙ 1 = u1 − u1d , e˙ 2 = u2 − u2d , e˙ 3 = u1d e2 + x2 (u1 − u1d ), .. . e˙ n = u1d e(n−1) + xn−1 (u1 − u1d ).
(7)
y˙ 1 = (n − 2)λy1 + (u1 − u1d ) exp((n − 2)λ(t − t0 )). (10) d(t)−D T , Denote Y = [y2 , y3 , · · · , yn ] and Σ(t) = T then the transformed system (9)(10) can be viewed as the following system Y˙ = (A + EΣ(t)F )Y + Bv (11) cascaded by the system (10), where v = u2 − u2d and A, B, E, F are defined as follows
0 0 ··· 0 0 1 . . D λ . 0 0 0 . , B = . .. .. A= , ... ... . . . . . . 0 0 0 · · · (n − 3)λ 0 0 0 0 ··· D (n − 2)λ 0 0 ··· 0 0 T 0 ··· 0 0 . . 1 0 . 0 0 0 T ... 0 0 E= ... ... . . . ... ... , F = ... ... . . . ... ... . 0 0 · · · 0 0 0 0 · · · T 0 0 0 ··· 1 0 0 0 ··· 0 T 3.2 Design of controller u1 Under the control law u1 = u1d − exp(−(n − 2)λ(t − t0 ))k1 y1 , k1 > (n − 2)λ,
(12)
the closed-loop subsystem of (10) becomes y˙ 1 = (−k1 + (n − 2)λ)y1 ,
Introduce a coordinate transformation for system (7) as y1 = e1 / exp(−(n − 2)λ(t − t0 )), y2 = e2 , y3 = e3 / exp(−λ(t − t0 )), .. . yn = en / exp(−(n − 2)λ(t − t0 )).
Then, the transformed system is given by y˙ 2 0 0 ··· 0 0 y2 y˙ 3 d(t) λ · · · y3 0 0 .. .. .. . . .. . . .. .. . = . . . . y˙ n−1 0 0 · · · (n − 3)λ yn−1 0 y˙ n 0 0 ··· d(t) (n − 2)λ yn 0 u1 − u1d 1 x2 exp(−λ(t − t )) 0 0 .. .. . + . (u2 −u2d )+ u1 − u1d 0 xn−2 exp(−(n − 3)λ(t − t0 )) 0 u1 − u1d xn−1 exp(−(n − 2)λ(t − t0 )) (9) and
(13)
which is globally exponentially stable. 3.3 Design of controller u2
(8)
Now, we design a state-feedback law v = KY so that the closed-loop system Y˙ = (A + BK + EΣ(t)F )Y (14) is quadratically stable.
By Lemma 2, we know that system (14) is quadratically stable if and only if A + BK is Hurwitz and ||F (sI − A − BK)−1 E||∞ < 1. Further, according to H ∞ control theory, A + BK is Hurwitz and ||F (sI − A − BK)−1 E||∞ < 1 if and only if there exists a positive definite matrix P such that (A+BK)T P +P (A+BK)+P E(P E)T +F T F < 0. (15) By Schur’s Complement Lemma the inequality (15) is equivalent to the following inequality (A + BK)T P + P (A + BK) P E F T ET P −I 0 < 0. F 0 −I (16) Pre-multiplying and post-multiplying (16) by diag{P −1 , I, I}, we get −1 P (A + BK)T +(A + BK)P −1 E (F P −1 )T ET −I 0 −1 FP 0 −I < 0. Letting X = P −1 , W = KX, the above inequality can also be rewritten as (AX + BW )T +(AX + BW ) E (F X)T ET −I 0 < 0. FX 0 −I (17) We say the above LMI is solvable if there exist W ? and a positively definite X ? such the inequality is satisfied.
3.4 Stability theorem Theorem 3. Consider the chained-form tracking error dynamics (7). Assume that u1d satisfies Assumption 3 and x2d , x3d , · · · , x(n−1)d are bounded. Then, the state feedback control u1 = u1d − exp(−(n − 2)λ(t − t0 ))k1 y1 (18) u2 = u2d + W ? (X ? )−1 Y
(19)
globally K-exponentially stabilizes the error system (7), where k1 > (n − 2)λ and W ? , X ? are solutions of (17). Proof. Since the system formed by (9) and (10) can be regarded as a system (11) cascaded by (10), we check three conditions of Lemma 1 for the cascaded system. • Since the quadratical stability implies the exponential stability, the GUES property of (11) is obtained from the controller designed in Section 3.3.
• Under the control law u1 , the interconnection term of system (11) and (10) can be rewritten as g(t, Y, Z)Z, where 0 y2 g(t, Y, Z) = −k1 e−(n−3)λ(t−t0 ) . . . yn−1 0 x2d e−(n−3)λ(t−t0 ) −k1 , .. . x(n−1)d Z = y1 , and Y is defined in Section 3.1. Since x2d , · · · , x(n−1)d are bounded, the second condition is satisfied. • The GUES property of system (10) follows immediately from the controller constructed in Section 3.2. Therefore, the third condition is also satisfied. Therefore, by Lemma 1 we conclude that the cascaded system formed by (9) and (10) is globally K-exponentially stable. According to the transformation (8) we know that under the control input (18) (19) the original chained form error dynamics (7) are also globally K-exponentially stable.
4. APPLICATION In this section we apply the control law proposed in Theorem 3 to the following unicycle mobile robot x˙ c = v cos θc , y˙ c = v sin θc , θ˙c = ω.
(20)
Although this system has been extensively studied by many researcher, the new assumption given in this paper (Assumption 3) allows us to discuss some interesting tracking tasks which has not been addressed before. The reference trajectory is generated by a virtual mobile robot which shares the same form as system (20) with reference input vd and ωd . In the following discussion we let t0 = 0 for simplicity. Let ωd = exp(−0.5t)(1 + 0.5 sin(t)). Then, it is easy to check that u1d = ωd = exp(−0.5t)(1 + 0.5 sin(t)), u2d = vd − (xd sin(θd ) − yd cos(θd ))ωd in chained form and λ = 0.5, T = 0.5, D = 1 in Assumption 3. Initial conditions of the reference system in chained-form and the error system are selected as [−2.9856 0.1570 1.3153] and [−0.080 0.8415 − 0.2579], respectively. Using the LMI toolbox in Matlab, we get
1.4
0.4
tracking errors (m)
(xd,yd) (x ,y ) c c
1.2
1
Y−direction (m)
−0.8
0
1
2
3
4
0.6
5
time (s)
6
7
8
9
10
0.1
tracking errors (rad)
0.4
0
−0.2 −1.1
−1
−0.9
−0.8
−0.7
X−direction (m)
−0.6
−0.5
−0.4
−0.3
Fig. 1. Moving path of mobile robots tracking errors (m)
(xc−xd) (yc−yd)
−0.2
−0.8
0
1
2
3
4
5
time (s)
6
7
θc−θd 0.05
0
−0.05
−0.1
0
1
2
3
4
5
time (s)
6
7
8
9
10
Fig. 4. Tracking errors
0.4
8
0.02
9
10
θ −θ c
tracking errors (rad)
−0.2
0.8
0.2
d
0
moving path of the mobile robot on the plane . The tracking errors are presented in Fig. 2. • vd = sin t In this case the target is in a periodic movement along a line when time goes to infinite. Fig. 3 shows the moving path of the mobile robot on the plane. The tracking errors are presented in Fig. 4.
−0.02 −0.04 −0.06 −0.08
0
1
2
3
4
5
time (s)
6
7
8
9
10
1.4 1.2
5. CONCLUSION Global K-exponential tracking controllers are constructed for the tracking control problem of nonholonomic systems in chained form. A coordinate transformation based on dilation is introduced and then the cascade-design method is adopted to transform the tracking problem into two simple stabilization problem. The obtained results show that the popular condition of persistent excitation or not converging to zero is not necessary for the tracking control of nonholonomic systems in chained form.
Fig. 2. Tracking errors (xd,yd) (xc,yc)
1 0.8
Y−direction (m)
(xc−xd) (yc−yd)
0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −2
−1.8
−1.6
−1.4
−1.2
−1
X−direction (m)
−0.8
−0.6
−0.4
−0.2
Fig. 3. Moving path of mobile robots · ¸ 5.2461 −1.9637 X= , −1.9637 1.2180 £ ¤ W = −7.7397 −1.0523 . Then, the control gain for the second subsystem is W (X)−1 = [−4.5362 − 8.1771]. For the first subsystem we can set k1 = 3 which ensures k1 > (n − 2)λ. This paper does not impose any restrictions on u2d . In simulation the following two cases are considered. • vd = exp(−0.5t) In this case the target asymptotically approaches a steady point. Fig. 1 shows the
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