Practical Stabilization for Nonholonomic Chained Systems with Fast ...
Proceedings of the 2002 IEEE International Conference on Robotics & Automation Washington, DC • May 2002
Practical Stabilization for Nonholonomic Chained Systems with Fast Convergence, Pole-Placement and Robustness Ti-Chung Lee Department of Electrical Engineering, Ming Hsin Institute of Technology Hsinchu, Taiwan 304, R.O.C., E-mail: [email protected] Abstract
In this paper, instead of using an exponential stabilizer, a practical stabilization problem that can achieve a fast convergence like the exponential convergence will be studied. A novel stabilizer for the chained system (1) will be proposed using the switching controllers approach. The subsystem (1b) will be studied first and the closed-loop system will be shown to be exponentially stable under some extra conditions imposed on the controller u 0 . Due to this result, there will be several possibilities for the choice of u 0 such that the extra conditions are satisfied and simultaneously, solutions of the closed-loop system of subsystem (1a) are still in a pre-assigned domain. The proposed practical stability has the same effect as the usual exponential stability on achieving the fast convergence, and the controllers will be very simple and thus easily implemented. Furthermore, a bound relating to the measurement of the transient behavior of the closed-loop system will be given explicitly. A numerical method can be invoked to minimize the bound and thus, improve the transient response of the closed-loop system. This is achieved in present literature yet. Moreover, we will show that the proposed controller is robust w.r.t. some uncertainties in the model. When apply to the example given in [3], it will be shown that a practical stability with fast convergence result can be also guaranteed under the parametric uncertainties based on our approach.
In this paper, issues of practical stabilization problem for nonholonomic chained systems are discussed. A novel controller that can guarantee a fast convergence like the exponential convergence is proposed based on the switching controllers approach. The steady-state error can be specified in advance. Moreover, the convergent rate can be pre-assigned and the transient response can be improved using the pole-placement method. The proposed controller is shown to be robust w. r. t. some uncertainties in the model. Simulation results for a unicycle-modeled mobile robot that has a small bias in orientation, an unknown radius in the rear wheels and an unknown distance in between them are presented to validate the effectiveness of our approach. Keywords: Pole-placement, practical stabilization, nonholonomic chained systems, robust stability.
1. Introduction Many mechanical systems such as tractor-trailer systems and wheeled mobile robots etc., can be modeled by the chained system described in the following x& 0 = u 0 (1a) x& = u 0 Ax + Bu 1 , (1b) and where x = [ x1 , x 2 , L , x n ]T , B = [0,0,L,1]T ( A, B ) is in the controllable canonical form (CCF) [8]. In present literature, to achieve a fast convergence, several stabilizers to guarantee the exponential convergence were proposed recently [1], [10]. On the other hand, the robustness issue for the chained system (1) was also attracted much attention recently [3], [5], [7]. A mobile robot of unicycle type with a small bias in orientation was proposed in [5] as a counterexample to show that the controller from the homogeneous feedback is not robust w.r.t. a small parametric uncertainty. In [3], an interesting exponentially convergent robust controller was proposed using the discontinuous feedback. Quite recently, a different approach was proposed in [7] using hybrid feedback. However, to the best of our knowledge, a globally exponential stability for a mobile robot of unicycle type with a small bias in orientation is still unsolved in present literature. 0-7803-7272-7/02/$17.00 © 2002 IEEE
∀v = (v1 , v 2 , L, v n ) ∈ ℜ
{
,
v = v12 + v 22 + L + v n2
Notations: n
,
A = sup Av
,
v =1
}
1 , ∀x > 0 . B m = x ∈ ℜ x ≤ m and sign( x) = − 1 , ∀x < 0
2. Simultaneous stabilization for controllable systems 2.1 Stability and pole-placement Consider the following controllable time-invariant systems: x& = σ Ax + Bu , σ ∈ {- 1,1 } ,
linear (2)
where x ∈ ℜ , u ∈ ℜ and ( A, B) is a controllable pair. Let {λ1 , λ 2 , L , λ n } ⊆ (- ∞ , 0 ) be any pre-assigned set of eigenvalues. Then, there exist a matrix K 0 and a n
3534
m
unique positive definite matrix P such that the eigenvalues of A + BK 0 are λ1 , λ 2 , L , λ n and the following Ricatti equation holds [2] (3) P ( A + BK 0 ) + ( A + BK 0 ) T P + 2 PBB T P = 0 . Now, for each σ , a stabilizing controller can be given in the following u = σ ( K 0 + B T P) x − B T Px . (4) With the choice of controller u, the closed-loop system can be written into the following system: x& = Aσ x (5)
M 0 = max || V ⋅ diag (e λ t , e λ t , L , e λ t )V −1 ||≥ 1 . (11) 1
Then it can be seen that for every solution x(t ) of the system (5), the following inequality holds:
max | x (t ) |≤ M 0 | x (t 0 ) |, ∀t 0 ≥ 0 . (12) t0 ≤t We summarize the previous discussion into the following lemma. Lemma 2: Consider the system (5) where B = [0,0, L ,1]T and ( A, B ) is in the CCF. Then every solution x(t ) satisfies the inequalities (10) and (12).■
where Aσ = σ ( A + BK 0 + BB T P) − BB T P . Then, it can be seen that Aσ has the same eigenvalues with A + BK 0 . Thus, the following result can be given. Lemma 1: Consider the linear time-invariant systems (2). With the controller u given in (4), the origin of the closed-loop system (5) is simultaneously exponentially stable and λ 1 , λ 2 , L , λ n are poles of the system. ■
3. Main results 3.1 The exponential stability for partial state In the remainder of this paper, it is said that sign(u 0 ) can be defined on an open interval ( a, b) if there exists a finite subset {c1 , c 2 , L, c n } of ( a, b) such that u 0 (t ) ≠ 0 and sign(u 0 (t )) is a constant function for all t ∈ (a, b) − {c1 , c 2 , L , c n } . Let λ1 , λ 2 , L , λ n be any pre-assigned eigenvalues contained in (−∞,0) , K 0 is the matrix defined in (6) where a 0 , a1 ,L , a n −1 are the coefficients of the
2.2 Convergent rate and transient behavior for systems in CCF In this subsection, the exact solution of the closed-loop system (5) will be studied when ( A, B ) is in the controllable canonical form (CCF). Let us assume that B = [0,0,L,1]T and ( A, B ) is in the CCF. Let K 0 be given by (6) K 0 = [− a 0 ,− a1 ,− a 2 ,L ,−a n −1 ] . Then, for σ = 1, the system matrix Aσ = A + BK 0 can be written into the following companion form 0 0 A + BK 0 = M 0 − a0
1
0
0 M 0 − a1
1 M 0 − a2
0 L 0 O M . L 1 L a n −1
n
characteristic polynomial ∏(λ − λ i ) and P is the i =1
positive definite matrix solving the Ricatti equation (3). Choose the controller u1 as u1 = u0 ( K 0 + B T P ) x − | u0 | B T Px .
1 λ2 M λ n2 − 1
L
(7)
1 λ n . M λ nn − 1
L L O L
( a, b) . Then, with the controller u1 chosen as in the equation (13), every solution x(t ) satisfies the following inequalities: (i) | x(t) |≤ V ⋅ diag(eλ α(t ) , eλ α(t ) ,L, eλ α (t ) ) ⋅V −1 x(a) , (14) 1
(8)
(ii) | x(t ) |≤ M 0 | x(a) |,
∀a ≤ t ≤ b . (15) ■ In the following, let us define a function M (t ) by
(9)
1≤ i ≤ n
M (t ) = V ⋅ diag (e λ1t , e λ2t ,L , e λnt ) ⋅ V −1 , ∀t ≥ 0.
Then, every solution of x& = ( A + BK 0 ) x can be estimated by | x(t ) |≤ V ⋅ diag(e λ (t −t ) , e λ (t −t ) , L, e λ (t −t ) ) ⋅V −1 x(t 0 ) (10) ≤ e −λ (t −t ) || V || || V −1 || | x(t 0 ) |, ∀t ≥ t 0 ≥ 0. Similarly, every solution of x& = A x also 0
2
0
n
n
t
λ 0 = min λ i .
0
2
∀a ≤ t ≤ b , where α (t ) = ∫a| u 0 ( s ) | ds .
Let
1
(13)
Then, the following result can be derived in view of Lemmas 1 and 2. The proof is omitted due to a limited space. Proposition 1: Consider the subsystem (1b). Suppose u 0 (t ) is a continuous bounded function defined on an open interval (a, b) and sign(u 0 ) can be defined on
Consider the following Vandermode matrix 1 λ V (λ 1 , λ 2 , L , λ n ) = 1 M n −1 λ 1
n
2
t ≥0
(16)
and make the following condition for the controller u 0 .
0
(C1)
Let
t0 = 0
and
t
β (t , s) = ∫s u 0 (τ ) dτ
,
∀0 ≤ s ≤ t . Suppose there exist a positive constant γ < 1 , a positive constant T and a strictly increasing t n = ∞ and sequence {t1 , t 2 , L, t n , L} ⊆ [0, ∞) with lim n →∞
0
−1
satisfies the inequality (10). Thus, every solution of x& = Aσ x , with σ ∈ {− 1,1} , satisfies the inequality (10).
t i +1 − t i ≤ T , ∀i ∈ ℵ , such that for each i ∈ ℵ , u 0 (t )
Define a constant M 0 which is relating to the transient behavior of the closed-loop system (5) by
is a continuous bounded function defined on (t i −1 , t i ) ,
3535
with t1 = T t1 ( x 0 (0)) and t i = (i − 1)T + t1 , ∀i ∈ ℵ .
sign(u 0 ) (t i −1 , t i ) and can be defined on M ( β (t i +1 , t i )) ≤ γ . Using condition (C1) and Proposition 1, the following exponential convergence result for the partial state x can be given. Proposition 2: Consider the subsystem (1b). Suppose the controller u 0 has been chosen such that condition (C1) holds. Then, with the controller u1 chosen as in
t n = ∞ and t i +1 − t i = T , ∀i ∈ ℵ . Let Note that lim n →∞
µ 0 be the positive constant such that µ 0 = δ 1 h . 0
Now, the controller u 0 can be chosen as the following − σ 0 h(t ) x 0 , if x0 (0) > ε and 0 ≤ t < t1 u0 = ( 21) ti µ 0 (−1) T h(t ), if t i ≤ t < t i +1 , for some i ∈ ℵ. Then, it can be derived that
the equation (13), every solution x(t ) of the closed-loop system of (1b) satisfies the following inequalities: ∀t ≥ 0 , (i) | x(t ) |≤ k1 e -σ 1 t | x(0) |, (17) where σ 1 =
t i +1 i
definition of µ 0 . Thus, we have M ( β (t i +1 , t i )) = M (δ 1 ) ≤ γ , ∀i ∈ ℵ , by Lemma 3 and the choice of λ1 , λ 2 ,L , λ n . Note that by
M2 − ln(γ ) and k1 = 0 e σ 1t1 . γ T
(ii) | x(t ) |≤ M 02 | x(0) |,
∀t ≥ 0 .
(18) ■
the
x&0 (t ) = −σ 0 h(t ) x0 (t ) ,
(22)
when x 0 (0) > ε and 0 ≤ t < t 1 in view of the equation
3.2 The practical stability for full state: fast convergence and transient response In the following, a technique lemma that is relating to an estimation of the function M (t ) is given. Lemma 3: Let δ 1 , γ be two given positive constants, and {d 2 , d 3 , L, d n } ⊆ [1, ∞) be a given sequence of real number. For any λ > 0 , let the pre-assigned eigenvalue λ1 = −λ , and recursively define the other pre-assigned eigenvalues as λi = λi −1 − d i , ∀2 ≤ i ≤ n . Then, there is
(21). From the choice of controller u 0 , it is easy to see that condition (C1) holds. In the following, let us study the closed-loop system of (1a). Let x 0 (t ) be any solution of the closed-loop system of (1a). Then, it can be described into the following equation: −σ ∫ h(t ) dt x0 (0), if x0 (0) > ε , 0 ≤ t < t1 e (23) x0 (t ) = t t −t x0 (ti ) + (−1) T µ0 h ( s ) ds , if t t t , i . ≤ < ∈ ℵ + i i 1 ∫0 From the equation (23) and the definition of t1 , it can t
0
0
i
a large enough constant λ0 such that the following inequality holds: M (δ 1 ) ≤ γ , ∀λ ≥ λ 0 . (19) ■ Let us construct the controller u 0 . Let γ < 1 , m , ε < m and δ 1 < m − ε be four given positive constants. Then, there exists a set of pre-assigned eigenvalues λ1 , λ 2 , L , λ n such that M (δ 1 ) ≤ γ by Lemma 3. Let T be any given positive constant and h(t ) be any continuous periodic function defined on [0, ∞) with period T , h(0) = 0 and h(t ) ≠ 0 , ∀t ∈ (0, T ) . Choose a positive constant ln(γ ) σ0 = − that is equal to the constant σ 1 defined in T
i
be checked that x 0 (t1 ) = e − t1 ( x0 ( 0 )) h0σ 0 x 0 (0) ≤ ε ,
x0 (t 2 i −1 ) = x0 (t1 ) and x0 (t 2i ) = x0 (t1 ) ± δ 1 , ∀i ∈ ℵ . This results in x 0 (t ) ≤ x 0 (t1 ) + δ 1 ≤ ε + δ 1 ≤ m, ∀t ≥ t1 . Thus, we have the following estimation for any solution x 0 (t ) of the closed-loop system of (1a): −σ ∫ e x0 (t ) ≤ m, ε , 0
t 0
h ( t ) dt
x0 (0) , if x0 (0) > ε and 0 ≤ t < t1 if t ≥ t1
(24)
if t = t2 i −1 for some i ∈ℵ.
In particular, the following proposition can be given in view of Proposition 2. Proposition 3: Consider the system (1). Let γ < 1 , m ,
T
Proposition 2. Define a positive constant h0 = ∫ 0 h(t ) dt and a continuous function t1 : ℜ → [0, ∞) following 1 x0 ln , if x 0 > ε ε t1 ( x 0 ) = h0σ 0 0 , if x 0 ≤ ε .
T
β (t i +1 , t i ) = µ 0 ∫ t h(t ) dt =µ 0 ∫ 0 h(t ) dt =δ 1 , ∀i ∈ ℵ ,
as the
ε < m , δ 1 < m − ε and T be five given positive constants. With the controllers u1 and u 0 chosen as in the equation (13) and (21), every solution ( x 0 (t ), x(t )) of the closed-loop system satisfies the ■ inequalities (17)-(18) and (24). In the remainder of this section, let us show that the exponential practical stability can be guaranteed based on Proposition 3. For simplicity, the function h(t ) is
(20)
Let y denote the least integer lager than or equal to a real number y . For any initial state x 0 (0) , let {t1 , t 2 ,L, t n ,L} be a strictly increasing time sequence
3536
chosen as
h(t ) = 1 − cos(
2πt ) . Then, T
h0 = T
∆B(0,0) = 0 . Without lose of generality, let us assume that η 0 ∈ (−1,1) . This implies that 0 < 1 + η 0 < 2 . Choose the controller u 0 as in the first equation in (26). Then,
and
δ1 x(t ) = 0 . . By the equation (17), we have lim t →∞ T Thus, for any positive constant ε 1 , the constant µ0 =
{
}
t f = min t 2i −1 x(t 2i −1 ) ≤ ε 1 , ∀i ∈ ℵ
for x 0 (0) > ε , an inequality like the first inequality in
(25)
− ln γ (1 + η 0 ) in T view of the first equation in (29). Then, we can modify the constant t1 as in the following
(28) holds with σ 0 replaced by σ 0 =
is well-defined. We modify the controller u 0 in (21) by 2πt ln(γ ) T (1 − cos( T )) x0 , if x0 (0) > ε and 0 ≤ t < t1 ti 2πt δ u 0 = 1 ( −1) T (1 − cos( )), if ti ≤ t < ti + 1 , i ∈ ℵ with ti < t f T T if t ≥ t f . . 0,
Note that we also have
x 0 (t f ) ≤ ε
{
(26)
by the third
δ 1 (1 + η 0* ) < m − ε . Then, it can be directly checked that the second inequality in (28) also holds for any η 0 ∈ (−η 0∗ ,η 0∗ ) . Choose the controller u1 as in (13). Let us show that the first inequality in (29) holds for small values of η 0 and η1 . Again using the time-scaling method,
x(t ) = x(t f ) , ∀t ≥ t f , in this case. Now, the following theorem is readable from Proposition 3. Theorem 1: Consider the chained system (1). Let γ < 1 , m , ε < m , δ 1 < m − ε , T and ε 1 be six given positive constants. Let λ1 , λ 2 , L , λ n be a set of pre-designed eigenvalues such that M (δ 1 ) ≤ γ . Choosing the controllers u1 and u 0 as in equations (13) and (26), the closed-loop system is exponentially practically stable in the following sense. With M2 − ln(γ ) σ1 = σ 0 = and k1 = 0 e σ 1t1 , every solution γ T ( x 0 (t ), x(t )) of the closed-loop system satisfies the following inequalities: k 1 e −σ 1t x(0) , ∀0 ≤ t < t f (i) | x(t ) |≤ (27) ∀t ≥ t f , ε 1 ,
define
i −1
~ x& i = Aσ ~ xi + ∆Aσ (η 0 ,η1 ) ~ xi ,
− sign( x 0 (0)), if i = 1 and x 0 (0) > ε and σ = (−1) t i −1 / T , otherwise. Note that ∆Aσ is also a smooth function of η 0 and η1 with ∆Aσ (0,0) = 0 . Let x (τ ) be a solution of x (0) . Then, x (τ ) satisfies the x& = A x with x (0) = ~ σ
i
inequalities (10) and (12) by Lemma 2. Let e(τ ) = ~ xi (τ ) − x (τ ) . Then, e(0) = 0 and e(τ ) satisfies the following equation: e& = ( Aσ + ∆Aσ )e + ∆Aσ x . (32) A Since σ is a stable matrix, there is a positive definite Lyapunov matrix Qσ [2] such that
Qσ Aσ + AσT Qσ = − I . (33) Let λ0 is the constant defined in (9) and λ m (Qσ ) denote the minimum eigenvalue of Qσ . Then, for any positive constant r0 , there exist two small constants η 0∗
η1 ∈ ℜ is a unknown vector; ∆B(η 0 ,η1 ) and are both smooth n0
∆A(0,0) = 0
T
∆Aσ = σ (∆A + ∆B( K 0 + B T P)) − ∆BB T P
where ( A, B ) is in the CCF; η 0 is a unknown
with
(31)
where Aσ = σ ( A + BK 0 + BB P ) − BB P , T
3.3 A robust stability result In this subsection, we want to show that the controller proposed in previous subsection has the robust property w.r.t. some model uncertainties. Consider the following chained system with parametric uncertainties: x& 0 = u 0 + η 0 u 0 (29) x& = u 0 Ax + Bu1 + u 0 ∆A(η 0 ,η1 ) x + ∆B (η 0 ,η1 )u1 ,
functions
all
ti
■
∆A(η0 ,η1 )
for
i −1
τ ∈ [0, bi ] with bi = β (t i , t i −1 ) = ∫ t u 0 ( s) ds. Then,
t
matrix-valued
t
β i (t ) = β (t , t i −1 ) = ∫t u 0 ( s) ds
t i −1 ≤ t ≤ t i and all i ∈ ℵ . Let ~ xi (τ ) = x( β i−1 (τ )), for all
−σ 0 ∫ 0 1−cos(2Tπt ) dt x0 (0) , if x0 (0) > ε and 0 ≤ t < t1 e (ii) x0 (t ) ≤ m, (28) if t1 ≤ t < t f if t ≥ t f . ε ,
and
(30)
Define the time sequence {t i } as before, i.e., t i +1 = t i + T for all i ∈ ℵ and choose the controller u 0 as in the first and second equations in (26). Furthermore, let us choose a small positive constant η 0∗ < 1 such that
inequality in (24). In view of equation (13), u 0 = 0 implies that u1 = 0 . Thus, x 0 (t ) = x 0 (t f ) and
constant
}
t1 = T min i ∈ ℵ ∪ {0} x 0 (iT ) ≤ ε .
and η1∗ such that
Qσ ∆Aσ (η0 ,η1 ) < min(1 4 , r0 λ0 λm (Qσ ) ( V V −1 ) ) (34)
and
3537
4. Simulations and case study
for all η0 < η0∗ and η1 < η1∗ in view of ∆Aσ (0,0) = 0 .
Let us apply Theorem 2 to study the parking problem of the following mobile robot model [3]: x& = pv cos(θ + r ) (39) y& = pv sin(θ + r ) θ& = qw,
Let Vσ = e T Qσ e . Then, we have V&σ ≤ 2 Qσ ∆Aσ
2
2
x , ∀η0 < η0∗ , ∀η1 < η1∗ .
(35)
Note that e(0) = xi (0) − x (0) = 0 by definition. Integrating the two side of (35) and using the inequalities (10) and (34), we have 2
λm(Qσ ) e(τ) ≤ e (τ)Qσe(τ) ≤ e (0)Qσe(0) + 2 Qσ ∆Aσ ≤2
T
T
r02 λ 0 λ m (Qσ ) V V
2
V
2
V −1
x (0)
∫
o
where p and q are both unknown parameters relating to the unknown radius of the rear wheels and the distance between them and r denotes a small bias in orientation. We assume that r ≤ ∆r with ∆r being a
2
x(s) ds
2
(1 − e − 2 λ0τ )
2λ 0
2
−1
2
2 τ
known bound, and
2
≤ r02 λ m (Qσ ) x (0) , ∀τ ∈ [0, bi ] .
interval
(36)
Let p 0 =
e(τ ) ≤ r0 x (0) This implies that and ~ x i (τ ) ≤ e(τ ) + x (τ ) ≤ r0 x (0) + x (τ ) . Note that by xi (0) = x(t i −1 ). In view of the definitions, x (0) = ~ inequalities (11) and (13), the following inequalities x(t ) ≤ (r0 + M 0 ) x (0) ≤ (r0 + M 0 ) x(t i −1 ) (37) (38) and x(t ) = ~ x (b ) ≤ (r + M (b )) x(t ) i
i
i
0
hold for all t i −1 ≤ t ≤ t i , η 0 < η M (t )
is
the
function
∗ 0
and η1 < η
∗ 1
defined
ti
in
(16)
i >1 ,
bi = δ 1
of
hence
i −1
by
the
choice
x(t i ) ≤ (γ + r0 ) x(t i −1 )
u0
and
q max − q min q . Define η0 = −1∈[− ∆q0 , ∆q0 ] , q0 q min + q max and
η1 ≤ (∆r ) 2 + (∆p0 ) 2 .
Then,
η1 = (r , p )
with
q = q0 (1 + η 0 )
and
x 2 cosθ sin θ 0 x x = sin θ − cosθ 0 y , u0 q0 w u = p v − x q w .(40) 1 1 0 x 0 0 0 1 θ 1 0 In the new coordinate and by the definitions of η 0 and η1 , the mobile robot system can be transformed into a system of the form (29) where the matrix-valued functions ∆A(η 0 ,η1 ) and ∆B (η 0 ,η1 ) can be given in the following η0 − (1 + p) sin r − (1 + p) sin(r ) . (41) ∆A = , ∆B = p r r η − + cos − 1 + cos 0 p cos r − (1 − cos r ) 0
in view the inequality (37)
under the condition M (δ 1 ) ≤ γ . Choose a small positive constant r0 such that γ + r0 < 1 . Then it can be checked that Proposition 2 also holds for the system (29) where M 0 is replaced by M 0 + r0 and γ is replaced by γ + r0 . Now, Choosing the time constant t f as in the equation (25), Theorem 1
Thus, Theorem 2 can be applied to guarantee a globally exponential practical robust stability for small values of ∆q 0 , ∆p 0 , ∆r . This is achieved in present literature yet. Note that a two-wheeled mobile robot (see Fig. 1) can be modeled as a system of the form (39). Indeed, let w l and wr denote the angular velocities of the left wheel and the right wheel, respectively. Let R and L denote the unknown radius of the rear wheels and the distance between them, respectively. Define wr − wl wr + wl w= and v = . Then, it can be 2 2 modeled as a system of the form (39) with p = R and 2R q= [4]. For simulations, suppose L p = R = 0.11 (m) and L = 0.5 (m) . Then, q = 0.44 . Let ( pmin, pmax ) = (0.05,0.25) and (qmin , qmax ) = (0.2,0.5) . Then, p 0 = 0.15 and q 0 = 0.35. Using the numerical
also holds. Let us summarize the previous discussion into the following theorem. Theorem 2: Consider the chained system (29) with unknown parameters η 0 and η1 . Let γ < 1 , m , ε < m , δ 1 < m − ε , T and ε 1 be six given positive constants. Let r0 < 1 − γ be a positive constant and λ1 , λ 2 ,L, λ n be a set of pre-designed eigenvalues such that M (δ 1 ) ≤ γ . Let η0∗ and η1∗ be two positive constants such that δ 1 (1 + η 0∗ ) < m − ε , η 0∗ < 1 and the inequality (34) holds. Choose the controllers u1 and u 0 as in the equations (13) and (26). Then, every solution ( x 0 (t ), x(t )) of the closed-loop system satisfies the inequalities (27) and (28) with − ln(γ + r0 ) ( ) + M r − ln(γ ) σ1 = ,σ 0 = (1+η0 ) and k1 = 0 0 eσ1t1 γ + r0 T T for all η 0 < η 0∗ and η1 < η1∗ .
p min + p max p −p q + q max , q 0 = min , ∆p0 = max min 2 2 pmin + pmax
p = p 0 (1 + p ) . Define the new coordinates by [4]
and
Note that for
, p max )
p − 1 ∈ [− ∆p 0 , ∆p 0 ] p0
p=
where
bi = β (t i , t i −1 ) = ∫ t u 0 ( s ) ds.
min
and ∆q 0 =
i −1
i
(p
p and q belong to a known and (q min , q max ) , respectively.
■
3538
method, we find λ0 = 1.3 and δ 1 = 1 that determine an approximation optimal value M 0 = 1.3141 . Table 1 The parameters used in the simulation. m
T
γ
ε = ε1
[8] R. M. Murray and S. S. Sastry, “Nonholonomic motion planning: steering using sinusoids,” IEEE Trans. Automat. Contr., Vol. 38, pp. 700-716, 1993. [9] C. Samson, “Control of chained systems application to path following and time-varying point-stabilization of mobile robots,” IEEE Trans. Automat. Contr., Vol. 40, pp. 64-77, 1995. [10] O. J. Sordalen and O. Egeland, “Exponential stabilization of nonholonomic chained systems,” IEEE Trans. Automat. Contr., Vol. 40, pp. 35-49, 1995.
d2
2 5 0.5 0.003 11.3 For simulation, let the initial condition be x (0) = 0(m) , y (0) = 1(m) , θ (0) = 0(rad ) . Simulation results for the case r = 0 (having no bias in orientation) and the case r = π / 6 (having a small bias in orientation) are shown in Figs. 2-4. In each case, a satisfied result is achieved. It can be seen that the case of r = π / 6 has a more bad transient behavior than the case of r = 0 . This is due to the error in orientation.
Y v
r
θ +r
θ ( x, y )
5. Conclusion
2R L
A novel practical stabilizer was proposed for chained systems. The pole-placement method was used to improve the convergence rate and the transient response. From simulation study, the proposed exponential practical stabilizer has the same fast convergence behavior as the usual exponential stabilizer. Moreover, it was shown that the proposed controller is robust w.r.t. some uncertainties in the model. Acknowledgement: This work was supported by the National Science Council, Taiwan, R.O.C., under contracts NSC-90-2213-E-159-003.
X Fig. 1. A two-wheeled mobile robot . 2
1
x(m ) & y(m )
x(m ) & y(m )
1.5
0.5 0 -0.5
0
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40 Tim e(s ec)
60
0.5 0 -0.5
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40 Tim e(s ec)
60
80
40 60 Tim e(s ec) (b)
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1.5
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theta(rad)
theta(rad)
References
0
-1
80
1.5
[1] A. Astolfi, and W. Schaufelberger, “State and output feedback stabilization of multiple chained systems with discontinuous control,” in Proc. IEEE 35th Conf. Decision Contr., Kobe, 1996, pp.1443-1448. [2] C. T. Chen, Introduction to linear system theory. New York: Holt, Rinehart and Winston, 1986. [3] Z. P. Jiang, “Robust exponential regulation of nonholonomic systems with uncertainties,” Automatica, Vol. 36, pp. 189-209, 2000. [4] T. C. Lee, K. T. Song, C. H. Lee and C. C. Teng, “Tracking control of unicycle-modeled mobile robots using a saturation feedback controller,” IEEE Trans. Contr. Syst. Technology, Vol. 9, pp. 305-318, 2001. [5] D. A. Lizarraga, P. Morin and C. Samson, “Non-robustness of continuous homogeneous stabilizers for affine control systems,” in Proc IEEE 38th Conf. Decision Contr., Phoenix, AZ, 1999, pp.855-860. [6] R. T. M’Closkey, and R. M. Murray, “Exponential stabilization of driftless control systems using homogeneous feedback,” IEEE Trans. Automat. Contr., Vol. 42, pp. 614-628, 1997. [7] P. Morin and C. Samson, “Robust stabilization of driftless systems with hybrid open-loop/feedback control,” in Proc. ACC 2000, Chicago, IL, 2000, pp.3929-3933.
1
0
20
40 Tim e(s ec) (a)
60
1 0.5 0 -0.5
80
Fig. 2. Time history of position and angle ( x : solid line; y : dashed line). (a) r = 0 . (b) r = π / 6 . a n g u la r ve lo c it ie s (rp m ) 40 20 0 -2 0 -4 0
0
10
20
30
40
50
60
70 80 Tim e (s e c )
50
60
70 80 Tim e (s e c )
(a )
a n g u la r ve lo c it ie s (rp m ) 100 50 0 -5 0 -1 0 0
0
10
20
30
40 (b )
Fig. 3. Applied angular velocities (solid line: wr , dashed line: wl ). (a) r = 0 . (b) r = π / 6 . Y (m ) 1.5 1 0.5 0 -0 . 5 -0 . 4
-0 . 2
0
0.2
0.4
0.6
0.8
1 X(m )
0.6
0.8
1
1 . 2 X(m )
(a )
Y (m ) 2
1
0
-1 -0 . 2
0
0.2
0.4 (b )
Fig. 4. Moving trajectory of mobile robot. (a) r = 0 . (b) r =π /6 .
3539
Practical Stabilization for Nonholonomic Chained Systems with Fast Convergence, Pole-Placement and Robustness Ti-Chung Lee Department of Electrical Engineering, Ming Hsin Institute of Technology Hsinchu, Taiwan 304, R.O.C., E-mail: [email protected] Abstract
In this paper, instead of using an exponential stabilizer, a practical stabilization problem that can achieve a fast convergence like the exponential convergence will be studied. A novel stabilizer for the chained system (1) will be proposed using the switching controllers approach. The subsystem (1b) will be studied first and the closed-loop system will be shown to be exponentially stable under some extra conditions imposed on the controller u 0 . Due to this result, there will be several possibilities for the choice of u 0 such that the extra conditions are satisfied and simultaneously, solutions of the closed-loop system of subsystem (1a) are still in a pre-assigned domain. The proposed practical stability has the same effect as the usual exponential stability on achieving the fast convergence, and the controllers will be very simple and thus easily implemented. Furthermore, a bound relating to the measurement of the transient behavior of the closed-loop system will be given explicitly. A numerical method can be invoked to minimize the bound and thus, improve the transient response of the closed-loop system. This is achieved in present literature yet. Moreover, we will show that the proposed controller is robust w.r.t. some uncertainties in the model. When apply to the example given in [3], it will be shown that a practical stability with fast convergence result can be also guaranteed under the parametric uncertainties based on our approach.
In this paper, issues of practical stabilization problem for nonholonomic chained systems are discussed. A novel controller that can guarantee a fast convergence like the exponential convergence is proposed based on the switching controllers approach. The steady-state error can be specified in advance. Moreover, the convergent rate can be pre-assigned and the transient response can be improved using the pole-placement method. The proposed controller is shown to be robust w. r. t. some uncertainties in the model. Simulation results for a unicycle-modeled mobile robot that has a small bias in orientation, an unknown radius in the rear wheels and an unknown distance in between them are presented to validate the effectiveness of our approach. Keywords: Pole-placement, practical stabilization, nonholonomic chained systems, robust stability.
1. Introduction Many mechanical systems such as tractor-trailer systems and wheeled mobile robots etc., can be modeled by the chained system described in the following x& 0 = u 0 (1a) x& = u 0 Ax + Bu 1 , (1b) and where x = [ x1 , x 2 , L , x n ]T , B = [0,0,L,1]T ( A, B ) is in the controllable canonical form (CCF) [8]. In present literature, to achieve a fast convergence, several stabilizers to guarantee the exponential convergence were proposed recently [1], [10]. On the other hand, the robustness issue for the chained system (1) was also attracted much attention recently [3], [5], [7]. A mobile robot of unicycle type with a small bias in orientation was proposed in [5] as a counterexample to show that the controller from the homogeneous feedback is not robust w.r.t. a small parametric uncertainty. In [3], an interesting exponentially convergent robust controller was proposed using the discontinuous feedback. Quite recently, a different approach was proposed in [7] using hybrid feedback. However, to the best of our knowledge, a globally exponential stability for a mobile robot of unicycle type with a small bias in orientation is still unsolved in present literature. 0-7803-7272-7/02/$17.00 © 2002 IEEE
∀v = (v1 , v 2 , L, v n ) ∈ ℜ
{
,
v = v12 + v 22 + L + v n2
Notations: n
,
A = sup Av
,
v =1
}
1 , ∀x > 0 . B m = x ∈ ℜ x ≤ m and sign( x) = − 1 , ∀x < 0
2. Simultaneous stabilization for controllable systems 2.1 Stability and pole-placement Consider the following controllable time-invariant systems: x& = σ Ax + Bu , σ ∈ {- 1,1 } ,
linear (2)
where x ∈ ℜ , u ∈ ℜ and ( A, B) is a controllable pair. Let {λ1 , λ 2 , L , λ n } ⊆ (- ∞ , 0 ) be any pre-assigned set of eigenvalues. Then, there exist a matrix K 0 and a n
3534
m
unique positive definite matrix P such that the eigenvalues of A + BK 0 are λ1 , λ 2 , L , λ n and the following Ricatti equation holds [2] (3) P ( A + BK 0 ) + ( A + BK 0 ) T P + 2 PBB T P = 0 . Now, for each σ , a stabilizing controller can be given in the following u = σ ( K 0 + B T P) x − B T Px . (4) With the choice of controller u, the closed-loop system can be written into the following system: x& = Aσ x (5)
M 0 = max || V ⋅ diag (e λ t , e λ t , L , e λ t )V −1 ||≥ 1 . (11) 1
Then it can be seen that for every solution x(t ) of the system (5), the following inequality holds:
max | x (t ) |≤ M 0 | x (t 0 ) |, ∀t 0 ≥ 0 . (12) t0 ≤t We summarize the previous discussion into the following lemma. Lemma 2: Consider the system (5) where B = [0,0, L ,1]T and ( A, B ) is in the CCF. Then every solution x(t ) satisfies the inequalities (10) and (12).■
where Aσ = σ ( A + BK 0 + BB T P) − BB T P . Then, it can be seen that Aσ has the same eigenvalues with A + BK 0 . Thus, the following result can be given. Lemma 1: Consider the linear time-invariant systems (2). With the controller u given in (4), the origin of the closed-loop system (5) is simultaneously exponentially stable and λ 1 , λ 2 , L , λ n are poles of the system. ■
3. Main results 3.1 The exponential stability for partial state In the remainder of this paper, it is said that sign(u 0 ) can be defined on an open interval ( a, b) if there exists a finite subset {c1 , c 2 , L, c n } of ( a, b) such that u 0 (t ) ≠ 0 and sign(u 0 (t )) is a constant function for all t ∈ (a, b) − {c1 , c 2 , L , c n } . Let λ1 , λ 2 , L , λ n be any pre-assigned eigenvalues contained in (−∞,0) , K 0 is the matrix defined in (6) where a 0 , a1 ,L , a n −1 are the coefficients of the
2.2 Convergent rate and transient behavior for systems in CCF In this subsection, the exact solution of the closed-loop system (5) will be studied when ( A, B ) is in the controllable canonical form (CCF). Let us assume that B = [0,0,L,1]T and ( A, B ) is in the CCF. Let K 0 be given by (6) K 0 = [− a 0 ,− a1 ,− a 2 ,L ,−a n −1 ] . Then, for σ = 1, the system matrix Aσ = A + BK 0 can be written into the following companion form 0 0 A + BK 0 = M 0 − a0
1
0
0 M 0 − a1
1 M 0 − a2
0 L 0 O M . L 1 L a n −1
n
characteristic polynomial ∏(λ − λ i ) and P is the i =1
positive definite matrix solving the Ricatti equation (3). Choose the controller u1 as u1 = u0 ( K 0 + B T P ) x − | u0 | B T Px .
1 λ2 M λ n2 − 1
L
(7)
1 λ n . M λ nn − 1
L L O L
( a, b) . Then, with the controller u1 chosen as in the equation (13), every solution x(t ) satisfies the following inequalities: (i) | x(t) |≤ V ⋅ diag(eλ α(t ) , eλ α(t ) ,L, eλ α (t ) ) ⋅V −1 x(a) , (14) 1
(8)
(ii) | x(t ) |≤ M 0 | x(a) |,
∀a ≤ t ≤ b . (15) ■ In the following, let us define a function M (t ) by
(9)
1≤ i ≤ n
M (t ) = V ⋅ diag (e λ1t , e λ2t ,L , e λnt ) ⋅ V −1 , ∀t ≥ 0.
Then, every solution of x& = ( A + BK 0 ) x can be estimated by | x(t ) |≤ V ⋅ diag(e λ (t −t ) , e λ (t −t ) , L, e λ (t −t ) ) ⋅V −1 x(t 0 ) (10) ≤ e −λ (t −t ) || V || || V −1 || | x(t 0 ) |, ∀t ≥ t 0 ≥ 0. Similarly, every solution of x& = A x also 0
2
0
n
n
t
λ 0 = min λ i .
0
2
∀a ≤ t ≤ b , where α (t ) = ∫a| u 0 ( s ) | ds .
Let
1
(13)
Then, the following result can be derived in view of Lemmas 1 and 2. The proof is omitted due to a limited space. Proposition 1: Consider the subsystem (1b). Suppose u 0 (t ) is a continuous bounded function defined on an open interval (a, b) and sign(u 0 ) can be defined on
Consider the following Vandermode matrix 1 λ V (λ 1 , λ 2 , L , λ n ) = 1 M n −1 λ 1
n
2
t ≥0
(16)
and make the following condition for the controller u 0 .
0
(C1)
Let
t0 = 0
and
t
β (t , s) = ∫s u 0 (τ ) dτ
,
∀0 ≤ s ≤ t . Suppose there exist a positive constant γ < 1 , a positive constant T and a strictly increasing t n = ∞ and sequence {t1 , t 2 , L, t n , L} ⊆ [0, ∞) with lim n →∞
0
−1
satisfies the inequality (10). Thus, every solution of x& = Aσ x , with σ ∈ {− 1,1} , satisfies the inequality (10).
t i +1 − t i ≤ T , ∀i ∈ ℵ , such that for each i ∈ ℵ , u 0 (t )
Define a constant M 0 which is relating to the transient behavior of the closed-loop system (5) by
is a continuous bounded function defined on (t i −1 , t i ) ,
3535
with t1 = T t1 ( x 0 (0)) and t i = (i − 1)T + t1 , ∀i ∈ ℵ .
sign(u 0 ) (t i −1 , t i ) and can be defined on M ( β (t i +1 , t i )) ≤ γ . Using condition (C1) and Proposition 1, the following exponential convergence result for the partial state x can be given. Proposition 2: Consider the subsystem (1b). Suppose the controller u 0 has been chosen such that condition (C1) holds. Then, with the controller u1 chosen as in
t n = ∞ and t i +1 − t i = T , ∀i ∈ ℵ . Let Note that lim n →∞
µ 0 be the positive constant such that µ 0 = δ 1 h . 0
Now, the controller u 0 can be chosen as the following − σ 0 h(t ) x 0 , if x0 (0) > ε and 0 ≤ t < t1 u0 = ( 21) ti µ 0 (−1) T h(t ), if t i ≤ t < t i +1 , for some i ∈ ℵ. Then, it can be derived that
the equation (13), every solution x(t ) of the closed-loop system of (1b) satisfies the following inequalities: ∀t ≥ 0 , (i) | x(t ) |≤ k1 e -σ 1 t | x(0) |, (17) where σ 1 =
t i +1 i
definition of µ 0 . Thus, we have M ( β (t i +1 , t i )) = M (δ 1 ) ≤ γ , ∀i ∈ ℵ , by Lemma 3 and the choice of λ1 , λ 2 ,L , λ n . Note that by
M2 − ln(γ ) and k1 = 0 e σ 1t1 . γ T
(ii) | x(t ) |≤ M 02 | x(0) |,
∀t ≥ 0 .
(18) ■
the
x&0 (t ) = −σ 0 h(t ) x0 (t ) ,
(22)
when x 0 (0) > ε and 0 ≤ t < t 1 in view of the equation
3.2 The practical stability for full state: fast convergence and transient response In the following, a technique lemma that is relating to an estimation of the function M (t ) is given. Lemma 3: Let δ 1 , γ be two given positive constants, and {d 2 , d 3 , L, d n } ⊆ [1, ∞) be a given sequence of real number. For any λ > 0 , let the pre-assigned eigenvalue λ1 = −λ , and recursively define the other pre-assigned eigenvalues as λi = λi −1 − d i , ∀2 ≤ i ≤ n . Then, there is
(21). From the choice of controller u 0 , it is easy to see that condition (C1) holds. In the following, let us study the closed-loop system of (1a). Let x 0 (t ) be any solution of the closed-loop system of (1a). Then, it can be described into the following equation: −σ ∫ h(t ) dt x0 (0), if x0 (0) > ε , 0 ≤ t < t1 e (23) x0 (t ) = t t −t x0 (ti ) + (−1) T µ0 h ( s ) ds , if t t t , i . ≤ < ∈ ℵ + i i 1 ∫0 From the equation (23) and the definition of t1 , it can t
0
0
i
a large enough constant λ0 such that the following inequality holds: M (δ 1 ) ≤ γ , ∀λ ≥ λ 0 . (19) ■ Let us construct the controller u 0 . Let γ < 1 , m , ε < m and δ 1 < m − ε be four given positive constants. Then, there exists a set of pre-assigned eigenvalues λ1 , λ 2 , L , λ n such that M (δ 1 ) ≤ γ by Lemma 3. Let T be any given positive constant and h(t ) be any continuous periodic function defined on [0, ∞) with period T , h(0) = 0 and h(t ) ≠ 0 , ∀t ∈ (0, T ) . Choose a positive constant ln(γ ) σ0 = − that is equal to the constant σ 1 defined in T
i
be checked that x 0 (t1 ) = e − t1 ( x0 ( 0 )) h0σ 0 x 0 (0) ≤ ε ,
x0 (t 2 i −1 ) = x0 (t1 ) and x0 (t 2i ) = x0 (t1 ) ± δ 1 , ∀i ∈ ℵ . This results in x 0 (t ) ≤ x 0 (t1 ) + δ 1 ≤ ε + δ 1 ≤ m, ∀t ≥ t1 . Thus, we have the following estimation for any solution x 0 (t ) of the closed-loop system of (1a): −σ ∫ e x0 (t ) ≤ m, ε , 0
t 0
h ( t ) dt
x0 (0) , if x0 (0) > ε and 0 ≤ t < t1 if t ≥ t1
(24)
if t = t2 i −1 for some i ∈ℵ.
In particular, the following proposition can be given in view of Proposition 2. Proposition 3: Consider the system (1). Let γ < 1 , m ,
T
Proposition 2. Define a positive constant h0 = ∫ 0 h(t ) dt and a continuous function t1 : ℜ → [0, ∞) following 1 x0 ln , if x 0 > ε ε t1 ( x 0 ) = h0σ 0 0 , if x 0 ≤ ε .
T
β (t i +1 , t i ) = µ 0 ∫ t h(t ) dt =µ 0 ∫ 0 h(t ) dt =δ 1 , ∀i ∈ ℵ ,
as the
ε < m , δ 1 < m − ε and T be five given positive constants. With the controllers u1 and u 0 chosen as in the equation (13) and (21), every solution ( x 0 (t ), x(t )) of the closed-loop system satisfies the ■ inequalities (17)-(18) and (24). In the remainder of this section, let us show that the exponential practical stability can be guaranteed based on Proposition 3. For simplicity, the function h(t ) is
(20)
Let y denote the least integer lager than or equal to a real number y . For any initial state x 0 (0) , let {t1 , t 2 ,L, t n ,L} be a strictly increasing time sequence
3536
chosen as
h(t ) = 1 − cos(
2πt ) . Then, T
h0 = T
∆B(0,0) = 0 . Without lose of generality, let us assume that η 0 ∈ (−1,1) . This implies that 0 < 1 + η 0 < 2 . Choose the controller u 0 as in the first equation in (26). Then,
and
δ1 x(t ) = 0 . . By the equation (17), we have lim t →∞ T Thus, for any positive constant ε 1 , the constant µ0 =
{
}
t f = min t 2i −1 x(t 2i −1 ) ≤ ε 1 , ∀i ∈ ℵ
for x 0 (0) > ε , an inequality like the first inequality in
(25)
− ln γ (1 + η 0 ) in T view of the first equation in (29). Then, we can modify the constant t1 as in the following
(28) holds with σ 0 replaced by σ 0 =
is well-defined. We modify the controller u 0 in (21) by 2πt ln(γ ) T (1 − cos( T )) x0 , if x0 (0) > ε and 0 ≤ t < t1 ti 2πt δ u 0 = 1 ( −1) T (1 − cos( )), if ti ≤ t < ti + 1 , i ∈ ℵ with ti < t f T T if t ≥ t f . . 0,
Note that we also have
x 0 (t f ) ≤ ε
{
(26)
by the third
δ 1 (1 + η 0* ) < m − ε . Then, it can be directly checked that the second inequality in (28) also holds for any η 0 ∈ (−η 0∗ ,η 0∗ ) . Choose the controller u1 as in (13). Let us show that the first inequality in (29) holds for small values of η 0 and η1 . Again using the time-scaling method,
x(t ) = x(t f ) , ∀t ≥ t f , in this case. Now, the following theorem is readable from Proposition 3. Theorem 1: Consider the chained system (1). Let γ < 1 , m , ε < m , δ 1 < m − ε , T and ε 1 be six given positive constants. Let λ1 , λ 2 , L , λ n be a set of pre-designed eigenvalues such that M (δ 1 ) ≤ γ . Choosing the controllers u1 and u 0 as in equations (13) and (26), the closed-loop system is exponentially practically stable in the following sense. With M2 − ln(γ ) σ1 = σ 0 = and k1 = 0 e σ 1t1 , every solution γ T ( x 0 (t ), x(t )) of the closed-loop system satisfies the following inequalities: k 1 e −σ 1t x(0) , ∀0 ≤ t < t f (i) | x(t ) |≤ (27) ∀t ≥ t f , ε 1 ,
define
i −1
~ x& i = Aσ ~ xi + ∆Aσ (η 0 ,η1 ) ~ xi ,
− sign( x 0 (0)), if i = 1 and x 0 (0) > ε and σ = (−1) t i −1 / T , otherwise. Note that ∆Aσ is also a smooth function of η 0 and η1 with ∆Aσ (0,0) = 0 . Let x (τ ) be a solution of x (0) . Then, x (τ ) satisfies the x& = A x with x (0) = ~ σ
i
inequalities (10) and (12) by Lemma 2. Let e(τ ) = ~ xi (τ ) − x (τ ) . Then, e(0) = 0 and e(τ ) satisfies the following equation: e& = ( Aσ + ∆Aσ )e + ∆Aσ x . (32) A Since σ is a stable matrix, there is a positive definite Lyapunov matrix Qσ [2] such that
Qσ Aσ + AσT Qσ = − I . (33) Let λ0 is the constant defined in (9) and λ m (Qσ ) denote the minimum eigenvalue of Qσ . Then, for any positive constant r0 , there exist two small constants η 0∗
η1 ∈ ℜ is a unknown vector; ∆B(η 0 ,η1 ) and are both smooth n0
∆A(0,0) = 0
T
∆Aσ = σ (∆A + ∆B( K 0 + B T P)) − ∆BB T P
where ( A, B ) is in the CCF; η 0 is a unknown
with
(31)
where Aσ = σ ( A + BK 0 + BB P ) − BB P , T
3.3 A robust stability result In this subsection, we want to show that the controller proposed in previous subsection has the robust property w.r.t. some model uncertainties. Consider the following chained system with parametric uncertainties: x& 0 = u 0 + η 0 u 0 (29) x& = u 0 Ax + Bu1 + u 0 ∆A(η 0 ,η1 ) x + ∆B (η 0 ,η1 )u1 ,
functions
all
ti
■
∆A(η0 ,η1 )
for
i −1
τ ∈ [0, bi ] with bi = β (t i , t i −1 ) = ∫ t u 0 ( s) ds. Then,
t
matrix-valued
t
β i (t ) = β (t , t i −1 ) = ∫t u 0 ( s) ds
t i −1 ≤ t ≤ t i and all i ∈ ℵ . Let ~ xi (τ ) = x( β i−1 (τ )), for all
−σ 0 ∫ 0 1−cos(2Tπt ) dt x0 (0) , if x0 (0) > ε and 0 ≤ t < t1 e (ii) x0 (t ) ≤ m, (28) if t1 ≤ t < t f if t ≥ t f . ε ,
and
(30)
Define the time sequence {t i } as before, i.e., t i +1 = t i + T for all i ∈ ℵ and choose the controller u 0 as in the first and second equations in (26). Furthermore, let us choose a small positive constant η 0∗ < 1 such that
inequality in (24). In view of equation (13), u 0 = 0 implies that u1 = 0 . Thus, x 0 (t ) = x 0 (t f ) and
constant
}
t1 = T min i ∈ ℵ ∪ {0} x 0 (iT ) ≤ ε .
and η1∗ such that
Qσ ∆Aσ (η0 ,η1 ) < min(1 4 , r0 λ0 λm (Qσ ) ( V V −1 ) ) (34)
and
3537
4. Simulations and case study
for all η0 < η0∗ and η1 < η1∗ in view of ∆Aσ (0,0) = 0 .
Let us apply Theorem 2 to study the parking problem of the following mobile robot model [3]: x& = pv cos(θ + r ) (39) y& = pv sin(θ + r ) θ& = qw,
Let Vσ = e T Qσ e . Then, we have V&σ ≤ 2 Qσ ∆Aσ
2
2
x , ∀η0 < η0∗ , ∀η1 < η1∗ .
(35)
Note that e(0) = xi (0) − x (0) = 0 by definition. Integrating the two side of (35) and using the inequalities (10) and (34), we have 2
λm(Qσ ) e(τ) ≤ e (τ)Qσe(τ) ≤ e (0)Qσe(0) + 2 Qσ ∆Aσ ≤2
T
T
r02 λ 0 λ m (Qσ ) V V
2
V
2
V −1
x (0)
∫
o
where p and q are both unknown parameters relating to the unknown radius of the rear wheels and the distance between them and r denotes a small bias in orientation. We assume that r ≤ ∆r with ∆r being a
2
x(s) ds
2
(1 − e − 2 λ0τ )
2λ 0
2
−1
2
2 τ
known bound, and
2
≤ r02 λ m (Qσ ) x (0) , ∀τ ∈ [0, bi ] .
interval
(36)
Let p 0 =
e(τ ) ≤ r0 x (0) This implies that and ~ x i (τ ) ≤ e(τ ) + x (τ ) ≤ r0 x (0) + x (τ ) . Note that by xi (0) = x(t i −1 ). In view of the definitions, x (0) = ~ inequalities (11) and (13), the following inequalities x(t ) ≤ (r0 + M 0 ) x (0) ≤ (r0 + M 0 ) x(t i −1 ) (37) (38) and x(t ) = ~ x (b ) ≤ (r + M (b )) x(t ) i
i
i
0
hold for all t i −1 ≤ t ≤ t i , η 0 < η M (t )
is
the
function
∗ 0
and η1 < η
∗ 1
defined
ti
in
(16)
i >1 ,
bi = δ 1
of
hence
i −1
by
the
choice
x(t i ) ≤ (γ + r0 ) x(t i −1 )
u0
and
q max − q min q . Define η0 = −1∈[− ∆q0 , ∆q0 ] , q0 q min + q max and
η1 ≤ (∆r ) 2 + (∆p0 ) 2 .
Then,
η1 = (r , p )
with
q = q0 (1 + η 0 )
and
x 2 cosθ sin θ 0 x x = sin θ − cosθ 0 y , u0 q0 w u = p v − x q w .(40) 1 1 0 x 0 0 0 1 θ 1 0 In the new coordinate and by the definitions of η 0 and η1 , the mobile robot system can be transformed into a system of the form (29) where the matrix-valued functions ∆A(η 0 ,η1 ) and ∆B (η 0 ,η1 ) can be given in the following η0 − (1 + p) sin r − (1 + p) sin(r ) . (41) ∆A = , ∆B = p r r η − + cos − 1 + cos 0 p cos r − (1 − cos r ) 0
in view the inequality (37)
under the condition M (δ 1 ) ≤ γ . Choose a small positive constant r0 such that γ + r0 < 1 . Then it can be checked that Proposition 2 also holds for the system (29) where M 0 is replaced by M 0 + r0 and γ is replaced by γ + r0 . Now, Choosing the time constant t f as in the equation (25), Theorem 1
Thus, Theorem 2 can be applied to guarantee a globally exponential practical robust stability for small values of ∆q 0 , ∆p 0 , ∆r . This is achieved in present literature yet. Note that a two-wheeled mobile robot (see Fig. 1) can be modeled as a system of the form (39). Indeed, let w l and wr denote the angular velocities of the left wheel and the right wheel, respectively. Let R and L denote the unknown radius of the rear wheels and the distance between them, respectively. Define wr − wl wr + wl w= and v = . Then, it can be 2 2 modeled as a system of the form (39) with p = R and 2R q= [4]. For simulations, suppose L p = R = 0.11 (m) and L = 0.5 (m) . Then, q = 0.44 . Let ( pmin, pmax ) = (0.05,0.25) and (qmin , qmax ) = (0.2,0.5) . Then, p 0 = 0.15 and q 0 = 0.35. Using the numerical
also holds. Let us summarize the previous discussion into the following theorem. Theorem 2: Consider the chained system (29) with unknown parameters η 0 and η1 . Let γ < 1 , m , ε < m , δ 1 < m − ε , T and ε 1 be six given positive constants. Let r0 < 1 − γ be a positive constant and λ1 , λ 2 ,L, λ n be a set of pre-designed eigenvalues such that M (δ 1 ) ≤ γ . Let η0∗ and η1∗ be two positive constants such that δ 1 (1 + η 0∗ ) < m − ε , η 0∗ < 1 and the inequality (34) holds. Choose the controllers u1 and u 0 as in the equations (13) and (26). Then, every solution ( x 0 (t ), x(t )) of the closed-loop system satisfies the inequalities (27) and (28) with − ln(γ + r0 ) ( ) + M r − ln(γ ) σ1 = ,σ 0 = (1+η0 ) and k1 = 0 0 eσ1t1 γ + r0 T T for all η 0 < η 0∗ and η1 < η1∗ .
p min + p max p −p q + q max , q 0 = min , ∆p0 = max min 2 2 pmin + pmax
p = p 0 (1 + p ) . Define the new coordinates by [4]
and
Note that for
, p max )
p − 1 ∈ [− ∆p 0 , ∆p 0 ] p0
p=
where
bi = β (t i , t i −1 ) = ∫ t u 0 ( s ) ds.
min
and ∆q 0 =
i −1
i
(p
p and q belong to a known and (q min , q max ) , respectively.
■
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method, we find λ0 = 1.3 and δ 1 = 1 that determine an approximation optimal value M 0 = 1.3141 . Table 1 The parameters used in the simulation. m
T
γ
ε = ε1
[8] R. M. Murray and S. S. Sastry, “Nonholonomic motion planning: steering using sinusoids,” IEEE Trans. Automat. Contr., Vol. 38, pp. 700-716, 1993. [9] C. Samson, “Control of chained systems application to path following and time-varying point-stabilization of mobile robots,” IEEE Trans. Automat. Contr., Vol. 40, pp. 64-77, 1995. [10] O. J. Sordalen and O. Egeland, “Exponential stabilization of nonholonomic chained systems,” IEEE Trans. Automat. Contr., Vol. 40, pp. 35-49, 1995.
d2
2 5 0.5 0.003 11.3 For simulation, let the initial condition be x (0) = 0(m) , y (0) = 1(m) , θ (0) = 0(rad ) . Simulation results for the case r = 0 (having no bias in orientation) and the case r = π / 6 (having a small bias in orientation) are shown in Figs. 2-4. In each case, a satisfied result is achieved. It can be seen that the case of r = π / 6 has a more bad transient behavior than the case of r = 0 . This is due to the error in orientation.
Y v
r
θ +r
θ ( x, y )
5. Conclusion
2R L
A novel practical stabilizer was proposed for chained systems. The pole-placement method was used to improve the convergence rate and the transient response. From simulation study, the proposed exponential practical stabilizer has the same fast convergence behavior as the usual exponential stabilizer. Moreover, it was shown that the proposed controller is robust w.r.t. some uncertainties in the model. Acknowledgement: This work was supported by the National Science Council, Taiwan, R.O.C., under contracts NSC-90-2213-E-159-003.
X Fig. 1. A two-wheeled mobile robot . 2
1
x(m ) & y(m )
x(m ) & y(m )
1.5
0.5 0 -0.5
0
20
40 Tim e(s ec)
60
0.5 0 -0.5
0
20
0
20
40 Tim e(s ec)
60
80
40 60 Tim e(s ec) (b)
80
1.5
1
theta(rad)
theta(rad)
References
0
-1
80
1.5
[1] A. Astolfi, and W. Schaufelberger, “State and output feedback stabilization of multiple chained systems with discontinuous control,” in Proc. IEEE 35th Conf. Decision Contr., Kobe, 1996, pp.1443-1448. [2] C. T. Chen, Introduction to linear system theory. New York: Holt, Rinehart and Winston, 1986. [3] Z. P. Jiang, “Robust exponential regulation of nonholonomic systems with uncertainties,” Automatica, Vol. 36, pp. 189-209, 2000. [4] T. C. Lee, K. T. Song, C. H. Lee and C. C. Teng, “Tracking control of unicycle-modeled mobile robots using a saturation feedback controller,” IEEE Trans. Contr. Syst. Technology, Vol. 9, pp. 305-318, 2001. [5] D. A. Lizarraga, P. Morin and C. Samson, “Non-robustness of continuous homogeneous stabilizers for affine control systems,” in Proc IEEE 38th Conf. Decision Contr., Phoenix, AZ, 1999, pp.855-860. [6] R. T. M’Closkey, and R. M. Murray, “Exponential stabilization of driftless control systems using homogeneous feedback,” IEEE Trans. Automat. Contr., Vol. 42, pp. 614-628, 1997. [7] P. Morin and C. Samson, “Robust stabilization of driftless systems with hybrid open-loop/feedback control,” in Proc. ACC 2000, Chicago, IL, 2000, pp.3929-3933.
1
0
20
40 Tim e(s ec) (a)
60
1 0.5 0 -0.5
80
Fig. 2. Time history of position and angle ( x : solid line; y : dashed line). (a) r = 0 . (b) r = π / 6 . a n g u la r ve lo c it ie s (rp m ) 40 20 0 -2 0 -4 0
0
10
20
30
40
50
60
70 80 Tim e (s e c )
50
60
70 80 Tim e (s e c )
(a )
a n g u la r ve lo c it ie s (rp m ) 100 50 0 -5 0 -1 0 0
0
10
20
30
40 (b )
Fig. 3. Applied angular velocities (solid line: wr , dashed line: wl ). (a) r = 0 . (b) r = π / 6 . Y (m ) 1.5 1 0.5 0 -0 . 5 -0 . 4
-0 . 2
0
0.2
0.4
0.6
0.8
1 X(m )
0.6
0.8
1
1 . 2 X(m )
(a )
Y (m ) 2
1
0
-1 -0 . 2
0
0.2
0.4 (b )
Fig. 4. Moving trajectory of mobile robot. (a) r = 0 . (b) r =π /6 .
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