- Email: [email protected]
Novel Anti-windup PID Controller Design under Holonomic Endpoint Constraints for Euler-Lagrange Systems with Actuator Saturation
Proceedings of the 19th World Congress The International Federation of Automatic Control Cape Town, South Africa. August 24-29, 2014
Novel Anti-windup PID Controller Design under Holonomic Endpoint Constraints for Euler-Lagrange Systems with Actuator Saturation Mitsuru Kanamori* Keigo Iwagami* *Maizuru National College of Technology, Kyoto 625-0042 JAPAN (Tel: +81-773-62-8955; e-mail: [email protected]).
Abstract: A novel anti-windup PID controller design method under holonomic constraints is proposed for nonlinear Euler-Lagrange systems with actuator saturation. The controller design is based on passivity, quasi-natural potential and saturated-position feedback. According to four saturation cases, switching of four integrating functions in the control law is utilized and four Lyapunov functions, such as hybrid control, are derived. Global asymptotic stability is ensured by energy dissipation between the four Lyapunov functions. The control performance is verified by numerical simulations using a two-link robot arm. 1. INTRODUCTION Every plant system is subject to input limitations. When the controller requires an input that exceeds these limitations, the control performance is degraded and becomes unstable. Actuator saturation caused by exceeded input limitations gives rise to windup phenomena, which degrade the control performance. Over the past decade, several anti-windup controller design methods have been proposed for linear systems to suppress windup degradation (e.g., Kanamori et al., 2007). However, studies of specifically nonlinear robot systems are few. Several studies have investigated antiwindup controller design for robot systems, including those by Kanamori (2011, 2013a,b), Khan et al. (2010), LopezAraujo et al. (2012), Loria et al. (1997), Morabito et al. (2004), Suntibanez et al. (1996), Teo et al. (2009), and Zavara-Rio et al. (2009). Kanamori (2011) proposed an antiwindup PID position controller for nonlinear Euler-Lagrange systems with input saturation. In this work, global asymptotic stability was guaranteed by the Lyapunov theorem based on the passivity described by Arimoto (1996). This anti-windup method was extended to adaptive tracking control considering input saturation (Kanamori, 2013a), and the control performance was demonstrated by experimentation (Kanamori, 2013b). A series of such works based on passivity imply that passivity is extremely suitable to stabilization that considers input saturation for nonlinear Euler-Lagrange systems. Since, in general, holonomic constraints have given rise to almost all robot systems (Khan et al., 2010), an anti-windup control method based on passivity is expected.
ensured by energy dissipation between the four Lyapunov functions. The control performance of the proposed controller is verified by numerical simulations using a two-link robot arm under holonomic constraints. 2. PRELIMINARY Let us consider the case where the endpoint of the manipulator is constrained on a surface (Arimoto, 1996). The surface is described by a scalar function as (1)
x = [x1
(2)
where
x2
x3 ]
T
denotes the Cartesian coordinates fixed at the internal reference frame. The contact force arises in the direction of the normal vector to the surface at point x and the contact − x with friction arises in the direction of magnitude x ( x ) x , where x (⋅) is a positive scalar function. Then, the robot dynamics are described by 1 H (q ) + S (q, q )}q + g (q ) 2 = Jf (q )T f − x ( x ) J x (q )T x + σ (u) ,
H (q)q + {B0 +
(3)
where
In the present paper, the anti-windup adaptive law by Kanamori (2013a) is extended to anti-windup PID position control under holonomic constraints for nonlinear EulerLagrange systems with input saturation. According to four saturation cases, switching of four integrating functions in the control law is utilized and four Lyapunov functions, such as hybrid control, are derived. Global asymptotic stability is 978-3-902823-62-5/2014 © IFAC
f ( x1 , x2 , x3 ) = 0 ,
9321
∂x (q ) , i = 1,2,3., j = 1,..., n, J x (q ) = i ∂q j
(4)
T
∂f J x (q ) ∂x , J f (q ) = 2 ∂f ∂x
(5)
19th IFAC World Congress Cape Town, South Africa. August 24-29, 2014
and f is the contact force. In addition, q = [q1 .... qn ] T represents the angular vector of each joint, H (q ) ∈ R
n× n
is
where Vci and Wci are switched for i = 1,..., l . Adding (14) and (15), the following relation holds:
n× n
the positive definite inertia matrix, B0 ∈ R is the positive diagonal constant matrix derived from the Raleigh dissipation function, S (q, q ) ∈ R n× n is the skew-symmetric matrix, and
g (q ) is the gravity term derived from the relation ∂U (q ) = g (q ) , where U (q ) is the potential energy. ∂q Furthermore, u = [u1 .... un ] T is the control input torque vector, and σ (u) = [σ (u1 ) ..... σ (un )] T is the saturated input torque vector, defined as follows:
0≥ where
dVi + Wi , dt
(16)
Vi = V0 + Vci and Wi = W0 + Wci .
(17)
Assume that Vi and Wi for i = 1,..., l are positive definite and
V j is the minimum function, V j = min (V1 , ....., Vl ) . Let us
denote the difference between Vi and V j for i ≠ j by
U i = Vi − V j . For asymptotic stability, the following lemma may be introduced (Kanamori, 2013a).
if ui min ≤ ui ≤ ui max , σ (ui ) = ui , if 0 < ui max < ui , σ (ui ) = ui max ,
(6)
Lemma 1: If we have the following two states for i = 1,..., l :
if ui < ui min < 0 , σ (ui ) = ui min ,
Vi > V j > 0 and Vi < 0 for i ≠ j : state 1 V j > 0 and V j < 0 : state 2
where ui max = ui min for i = 1,..., n . The estimation value of the input saturation ψ (u) = [ψ (u1 ) ... ψ (u n )] T is introduced as
ψ ( u) = u − σ ( u) ,
(19)
the minimum function V j is a Lyapunov function of the whole trajectory of the closed-loop system and the equilibrium state is asymptotically stable.
(7)
where
Outline Proof of Lemma 1: For the proof, it is sufficient to show that V j is dissipative on state 1 shown in (18). Taking
if ui min ≤ ui ≤ ui max , ψ (ui ) = 0 , if 0 < ui max < ui , ψ (ui ) = ui − ui max > 0 ,
(18)
(8)
the time integral of (16), the following relation holds:
if ui < ui min < 0 , ψ (ui ) = ui − ui min < 0 .
t
∫
V j (t ) − V j (0) ≤ U i (0) − U i (t ) − Wi dt .
The closed-loop system is described as
(20)
0
L(q, q , q) + F ( g (q), σ (u), f ) = 0 ,
(9)
If U i increases, then V j decreases. If U i decreases, then V j might increase; however, the increase value is bounded by U i (0) . Since Wi is positive definite, the time integral of Wi , which is the dissipated energy, always increases with the integral of time so that the increase in V j turns into a
where 1 H (q ) + S (q, q )}q 2 + x ( x ) J x (q )T J x (q )q ,
L(q, q , q) = H (q )q + {B0 +
T
F ( g (q ), σ (u), f ) = g (q ) − σ (u) − J f (q ) f .
(10)
decrease soon. This implies that V j is a Lyapunov function
(11)
To achieve the state (q, q , f ) → (qd ,0, f d ) , where qd and f d are the target vectors of each joint angle and target contact force, respectively, the following assumption is introduced. Assumption 1: Assume that the following conditions are satisfied for i = 1,..., n for any target posture:
ui min < g i (q) < ui max , ui min < [ g (qd ) + J f (qd )T f d ]i < ui max .
(12)
and the equilibrium state of the closed-loop system is asymptotically stable. Let us choose the following output y for the evaluation in Arimoto (1996):
y = q + αQφ (q ) s (∆q ) ,
where α is a positive definite scalar, Qφ (q ) is defined as
(13)
Qφ (q ) = I −
Take the appropriate output y and assume that the inner product of y and the closed-loop system (9) has the following form: d (14) y T L(q, q , q) = V0 + W0 , dt d (15) y T F ( g (u), σ (u), f ) ≥ Vci + Wci , dt
(21)
J φ (q )T J φ (q ) J φ (q )
2
,
(22)
which is regarded as a projection that projects vectors in joint space onto the plane tangent to the surface f (q ) = 0 at the contact point q , and I denotes the identity matrix. The function s (∆q ) is the output saturation function described in Arimoto (1996) as follows:
9322
19th IFAC World Congress Cape Town, South Africa. August 24-29, 2014
s (∆q) = [s1 (∆q1 ) s2 (∆q2 ) ...... sn (∆qn )]T ,
holds so that the margins between input limits and the gravity term are utilized for the stability.
(23)
where ∆q is the control error defined by
Property 4: The following inequality holds:
∆q = q − qd .
(24)
If s (∆q )T Qf (q )ψ (u) < 0 , then
s (∆q )T Qf (q ) F ( g (q ), σ (u), f ) ,
Property 1: Concering the projection Qφ (q ) , the following relations hold.
≥ c0 min s (∆q )T Qf (q ) s (∆q ) ,
J φ (q )q = 0 ,
(25)
Qφ (q )q = q ,
(26)
Qφ (q ) J φ (q )T = 0 ,
(27)
Qφ (q )Qφ (q ) = Qφ (q ) .
(28)
where c0 min is the positive minimum value of c0i for i = 1,..., n as follows.
c0 min = min[c01 ...... c0 n ] .
Property 2: For the output saturation function in (23), the following properties are ensured. (a) There exists a positive convex function S i ( xi ) such that the following relation is satisfied: dSi ( xi ) = si ( xi ) . dxi
(29)
∑
(35)
Property 5: The following inequalities are satisfied for saturation cases (I) through (IV) so that the input saturation function is eliminated by the inequalities (Kanamori, 2013a): (I) If q T ψ (u) ≥ 0 and s (∆q )T Qf (q )ψ (u) ≥ 0 , then
y T F ( g (q), σ (u), f ) ≥ y T F ( g (q), u, f ) ,
(36)
T (II) If q T ψ (u) ≥ 0 and s ( ∆q ) Qf (q )ψ ( u) < 0 , then
y T F ( g (q), σ (u), f )
Then, the following relations hold:
Si (∆qi ) = si (∆qi )∆qi and
(34)
≥ q T F ( g (q ), u, f ) + αc0 min s (∆q )T Qf (q ) s (∆q ) , (37)
Si (∆qi ) = s (∆q ) T q . (30)
T
(III) If q T ψ ( u) < 0 and s ( ∆q ) Qf (q )ψ ( u) ≥ 0 , then
i
(b) There exists a positive real constant ci such that the following inequality is satisfied: 2
Si (∆qi ) ≥ ci si (∆qi ) .
y T F ( g (q ), σ (u), f ) ≥ as (∆q )T Qf (q ) F ( g (q ), u, f ) , T
(IV) If q ψ ( u) < 0 and s ( ∆q ) Qf (q )ψ ( u) < 0 , then
(31)
y T F ( g (q), σ (u), f ) ≥ αc0 min s(∆q)T Qf (q) s(∆q) .
(c) There exists a positive real constant c0 such that the following inequality is satisfied: − s(∆q)T Qφ (q) H (q)q 1 + s(∆q)T {− Qφ (q) H (q) − Q φ (q) H (q) + Qφ (q) S (q, q )}q 2 2
+ Qφ (q) S (q, q )}q ≥ −c0 q .
Property 6: V0 and W0 as shown in (14) become 1 T 2 q H (q )q + α (b0 min cmin − αγ M ) s (∆q ) , 4 1 2 W0 ≥ {b0 min − αc0 + x ( x ) J x (q )T J x (q )} q 2 1 2 1 T − α s (∆q ) Qf (q )[ ∆B0 2 2 T + x ( x ) J x (q) J x (q)]Qf (q) s(∆q) , V0 ≥
Property 3: Based on the characteristics of the output saturation function s (⋅) and condition (12), there exists a positive real constant c0i such that the following inequality is satisfied.
min{ ui max − g i (q ) , ui min − g i (q ) } i
(39)
The inner product, as shown in (14), is determined by the given robot system. V0 and W0 are independent of the controller design and are obtained as follows (Arimoto, 1996).
(32)
≥ c0i s (∆q) i max ≥ c0i Qφ (q) s (∆q) .
(38)
T
(40)
(41)
where cmin is the minimum value of ci as follows:
(33)
For the inner product, as shown on the left-hand side of (15), notice that the term y T J f (q )T f is zero with the relations (25) and (27) in Property 1 in Arimoto (1996). According to the polarity of the inner product y T ψ (u) , the input saturation function σ (⋅) is eliminated, as shown in Property 1 in
cmin = min{c1 ...... cn } ,
(42)
b0 min is the minimum value of the diagonal component of the positive diagonal matrix B0 in (3), γ M is the maximum eigenvalue of H (q ) , and ∆B0 is the difference between B0 and b0 min I as follows:
Kanamori (2013a). Replacing Qf (q ) s (∆q ) for y in Property 1 in Kanamori (2013a) and using (33), the following property
9323
∆B0 = B0 − b0 min I ≥ 0 .
(43)
19th IFAC World Congress Cape Town, South Africa. August 24-29, 2014
Assumption 2: There exists positive real numbers α and c0 min such that the following inequalities are satisfied:
b0 min cmin − αγ M > 0 ,
(44)
b0 min − αc0 > 0 ,
(45)
1 2c0 min I − α { ∆B0 + x ( x ) J x (q )T J x (q )} > 0 . 2
s (∆q )T Qφ (q ){g (q ) − g (qd )} +
a 2 ∆q > 0 , 2
(46)
[
φ = J φ (q ) H (q ) J φ (q ) where
] [− J
φ (q ) H (q )
−1
In CASE (IV), y (t ) = 0 .
(58)
y T F ( g (q ), σ (u), f ) ≥ y T F ( g (q ), u, f ) =
a 2 ∆q 2 c 2 2 + αbcmin s (∆q) + z , 2 T Wc1 ≥ α [ s (∆q) Qφ (q){g (q) − g (qd )}
Vc1 ≥ U (q) − U (qd ) − ∆qT g (qd ) +
]
2
y T Qφ (q )cz = cy T z = cy T z = cz T z =
Let us consider the following proportional, integral and derivative (PID) control law: t
u = −a∆q − bq − cQφ (q) ∫ y (t )dt 0
− J f (q ) f d + β J f (q )
T
∫ ∆f (t )dt ,
(51)
0
where a , b , c , and β are positive definite real numbers that are regarded as feedback gain, and ∆f is the error between contact force f and its target force f d represented as ∆f = f − f d . Using the notation
∫
(52)
τ
∫
∞
∫
(39) must be included in Wc 2 or Wc 4 , respectively. In CASE (IV), there is no inner product on the right-hand side in (39); however, it is regarded that there is the inner product of zero and F ( g (q ), u, f ) as follows:
0 T F ( g (q ), u, f ) = where Vc 4 =
∞
0
0
∫ y(τ )dτ = −c
−1
0
c 2 z : constant. 2
(63) (64)
(65)
It is confirmed that Vci for i = 1,...,4 is positive definite when
F ( g (q ), σ (u), f ) = g (q ) − Qf (q ) g (qd ) + a∆q
+ bq + Qf (q )cz − J f (q )T (∆f + b∆F ) + ψ (u) . (54) Let us take the following y (t ) according to the four saturation cases (Kanamori, 2013a):
y (t ) = q + αQφ (q) s (∆q) ,
dVc 4 = 0, dt
Wc 4 ≥ αc0 min s (∆q )T Qf (q ) s (∆q ) .
g (qd ) , (53)
F ( g (q), σ (u), f ) defined by (11) becomes
In CASE (I),
c d 2 z . (62) 2 dt
Then, Wc 4 becomes
0
z = y (τ )dτ − y (τ )dτ and
(61)
In the same manner, Vci and Wci for i = 2, 3 are obtained by using (56) and (57), respectively. Notice that the second term αc0 min s (∆q )T Qf (q) s (∆q) on the right-hand side of (37) or
t
∆F = ∆f (t )dt ,
(60)
The suffix i = 1 of Vc1 and Wc1 means saturation CASE (I). In the derivation, the following relation, (26) and (28) are utilized:
3. CONTROLLER DESIGN
T
2
+ a Qφ (q) s(∆q) ] + b q .
(50)
t
dVc1 + Wc1 , (59) dt
where Vc1 and Wc1 are represented as
X − Jφ (q )q , (49)
1 X = −{B0 + H (q ) + S (q, q )}q 2 − g (q) + σ (u) − x ( x ) J x (q)T x .
(57)
the inner product y T F ( g (q), σ (u), f ) in (36) is represented as follows:
(47)
2 a Qφ (q ) s (∆q ) > 0 . (48) 2
T −1
In CASE (III), y (t ) = αQφ (q ) s (∆q ) ,
and Vci and Wci in (15) are also easily derived. In CASE (I),
Property 8: The contact force f is given by the following form: −1
(56)
The inner products, as shown on the right-hand side of (36) through (39), can be taken easily because σ (⋅) is eliminated
Property 7: There exists a positive real number a such that the following inequalities are satisfied: U (q ) − U (qd ) − ∆q T g (qd ) +
In CASE (II), y (t ) = q ,
the conditions (47) are satisfied, and Vi in the first equation in (17) is also positive definite, since V0 is positive definite under (44). Wi for i = 1,...,4 in the second equation in (17) is also positive definite under (45), (46), (48) and the following condition:
(55) 9324
1 aI − a { ∆B0 + x ( x ) J x (q )T J x (q )} > 0 . 2
(66)
19th IFAC World Congress Cape Town, South Africa. August 24-29, 2014
The above results imply that we have the following two states according to the saturation cases:
Vi ≥ V4 > 0 , Vi < 0 , for i = 1, 2, 3 : state 1 : state 2 V > 0 , V < 0 . 4
4
(67) (68)
where Vi is switched according to saturation cases (I) through (IV). Based on Lemma 1, we have the following main theorem. Main Theorem (Anti-windup PID control law): Assume that conditions (44) through (46) are satisfied. Choose feedback gain a as the positive real number such that the conditions (47), (48) and (66) are satisfied. Adopt the switching function as (55) through (58) for the control law (51). Then, the equilibrium state of the closed-loop system is globally asymptotically stable based on Lemma 1. Proposition (Practical anti-windup PID control law): Assume that the conditions in the Main Theorem are satisfied. For the control law (51), choose the function y (t ) as follows:
y (t ) = y1 (t ) + αy 2 (t ) ,
saturation function si ( xi ) in (23) has also been shown in (52) in Kanamori (2013b). Input limits ±2 V are given for the shoulder joint input and ±1 V are given for the elbow joint input. The following values are used for the simulations.
f d = 10 N, x ( x ) = 0.3 Nm/s, γ M = 6.38 ,
(77)
α = 3.0 , a = 10.0 , b = 2.0 , c = 5.0 , β = 1.0 .
(78)
Fig. 2 shows the control performance using the normal PID controller. For the strict input limits, the windup phenomenon is observed at elbow joint input u2 , as shown in the upper plots in Fig. 2. As a result, considerable overshoots occur in shoulder joint angle q1 and elbow joint angle q2 , as shown in the lower plots in Fig. 2. As shown in Fig. 3, the contact force also degrades for the input saturation. In contrast, the windup phenomenon is restrained by using the proposed antiwindup PID controller, as shown in the upper plots in Fig. 4. As a result, considerable overshoots are restrained very well,
(69)
x2
q2
where if qiψ(ui ) ≥ 0 , then y1i = qi , if qiψ(ui ) < 0 , then y1i = 0 ,
q1
(70)
L0
0
θ
x1
Fig. 1 Two-link robot arm with holonomic constraint
y2i = {s (Δq )T Qf (q )}i , (72)
if {s (Δq ) Qf (q )}i ψ(ui ) < 0 , then y2i = 0 ,
L2
f
(71)
if {s (Δq)T Qf (q)}i ψ(ui ) ≥ 0 , then, T
L1
(73)
for i = 1,....., n , where {⋅}i denotes the i-th component of the vector, yki denotes the following vector component:
y k = [ yk1 ... yki ... ykn ]T and k = 1, 2 .
(74)
Then, the equilibrium state of the closed-loop system is globally asymptotically stable.
Solid: u1 Bloken: u2
Solid: σ(u1) Bloken: σ(u2)
4. CONTROL PERFORMANCE The control performance of the proposed anti-windup PID controller is examined by numerical simulations using a twolink robot arm. A schematic two-link robot arm with contact force is depicted in Fig. 1, where L1 and L2 denote the length of the arm. L0 is the coordinate value of x1 and the start point of the constraint surface. In this system, the constraint surface is represented as (75) and (1) becomes (76).
x2 = as x1 + bs , L0 ≤ x1 , f = x2 − as x1 − bs = 0 ,
Thick: q1 Thin: qd1
Thick: q2 Thin: qd2
Fig. 2 Control performance using normal PID controller
(75) (76)
where as and bs are real constants for the constraint surface. The system equation with actuators and the detail of the twolink robot system have been described in (48) through (51) and TABLE 1 in Kanamori (2013b), respectively. Output
9325
Fig. 3 Contact force using normal PID controller
19th IFAC World Congress Cape Town, South Africa. August 24-29, 2014
REFERENSES
solid: u1 Bloken: u2
Thick: q1 Thin: qd1
Solid: σ(u1) Bloken: σ(u2)
Thick: q2 Thin: qd2
Fig. 4 Control performance using the proposed controller
Fig. 5 Contact force using the proposed PID controller
Fig. 6 Confirmation plot of conditions (32), (47) and (48) as shown in the lower plots in Fig. 4. As shown in Fig. 5, the control performance of the contact force is improved in comparison with that in Fig. 3. Fig. 6 shows the validity of conditions (32), (47), and (48). The top plot shows the lefthand side of (32), where the right-hand side is transposed to the left-hand side. The middle and the lower plots correspond to the left-hand sides of (47) and (48), respectively. Since these values are positive or zero, it is confirmed that conditions (32), (47), and (48) are satisfied. These simulation results validate the analysis and design of the proposed antiwindup PID controller. 5. CONCLUSIONS An anti-windup PID controller design for Euler-Lagrange systems under holonomic endpoint constraints has been presented. The control performance was examined by numerical simulations using a two-link robot arm with contact force. The simulation results validate the analysis and design of the proposed anti-windup PID controller.
Arimoto, S. (1996), Control theory of non-linear mechanical systems: A passivity-based and circuit-theoretic approach, p. 93-113, Oxford University Press, Walton Street, Oxford OX2 6DP. Kanamori, M. and M. Tomizuka (2007), Dynamic antiintegrator-windup controller design for linear systems with actuator saturation, ASME Journal of Dynamic Systems, Measurement, and Control, vol. 129, no. 1, pp. 1–12. Kanamori, M. (2011), Global stabilization of anti-windup PID position control for Euler-Lagrange systems with actuator saturation, In: Proceedings of IEEE International Conference on Automation Science and Engineering, pp. 798–803, Trieste. Kanamori, M. (2013a), Anti-windup adaptive tracking for Euler-Lagrange systems with actuator saturation, In: Proceedings of 3rd International Conference on Instrumentation, Control and Automation, pp. 216-221, Bali. Kanamori, M. (2013b), Experimental study of anti-windup adaptive tracking control for Euler-Lagrange systems with actuator saturation, In: Proceedings of 6th IEEE International Conference on Robotics, Automation and Mechatronics , to be published. Khan, S. G., G. Herrmann, T. Pipe and C. Melhuish (2010), Adaptive multi-dimensional compliance control of a humanoid robotic arm with anti-windup compensation, In: Proceedings of IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 2218–2223, Taipei. Lopez-Araujo, D. J., A. Zavala-Rio, V. Santibanez and F. Reyes (2012), Global adaptive regulation of robot manipulators with bounded inputs, in Proc. 10th IFAC Sympo. Robot Control, Dubrovnik, pp. 806–813. Loria, A., R. Kelly, R. Ortega and V. Santibanez (1997), On global output feedback regulation of Euler-Lagrange systems with bounded inputs, IEEE Trans. Automatic Control, vol. 42, no. 8, pp. 1138–1143. Morabito, F., A. R. Teel and L. Zaccarian (2004), Nonlinear antiwindup applied to Euler-Lagrange systems, IEEE Trans. Robotics and Automation, vol. 20, no. 3, pp. 526– 537. Santibanez, V. and R. Kelly (1996), Global regulation for robot manipulators under SP-SD feedback, In: Proceedings of IEEE International Conference on Robotics and Automation, pp. 927–932, Minneapolis. Teo, J. and J. P. How (2009), Anti-windup compensation for nonlinear systems via gradient projection: Application to adaptive control, In: Proceedings of 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference, pp. 6910–6916, Shanghai. Zavala-Rio, A., E. Aguinaga-Ruiz and V. Santibanez, (2009), Global trajectory tracking through output feedback for robot manipulators with input saturations, In: Proceedings of IFAC Symposium on Robot Control, pp. 701–707, Gifu.
9326
Novel Anti-windup PID Controller Design under Holonomic Endpoint Constraints for Euler-Lagrange Systems with Actuator Saturation Mitsuru Kanamori* Keigo Iwagami* *Maizuru National College of Technology, Kyoto 625-0042 JAPAN (Tel: +81-773-62-8955; e-mail: [email protected]).
Abstract: A novel anti-windup PID controller design method under holonomic constraints is proposed for nonlinear Euler-Lagrange systems with actuator saturation. The controller design is based on passivity, quasi-natural potential and saturated-position feedback. According to four saturation cases, switching of four integrating functions in the control law is utilized and four Lyapunov functions, such as hybrid control, are derived. Global asymptotic stability is ensured by energy dissipation between the four Lyapunov functions. The control performance is verified by numerical simulations using a two-link robot arm. 1. INTRODUCTION Every plant system is subject to input limitations. When the controller requires an input that exceeds these limitations, the control performance is degraded and becomes unstable. Actuator saturation caused by exceeded input limitations gives rise to windup phenomena, which degrade the control performance. Over the past decade, several anti-windup controller design methods have been proposed for linear systems to suppress windup degradation (e.g., Kanamori et al., 2007). However, studies of specifically nonlinear robot systems are few. Several studies have investigated antiwindup controller design for robot systems, including those by Kanamori (2011, 2013a,b), Khan et al. (2010), LopezAraujo et al. (2012), Loria et al. (1997), Morabito et al. (2004), Suntibanez et al. (1996), Teo et al. (2009), and Zavara-Rio et al. (2009). Kanamori (2011) proposed an antiwindup PID position controller for nonlinear Euler-Lagrange systems with input saturation. In this work, global asymptotic stability was guaranteed by the Lyapunov theorem based on the passivity described by Arimoto (1996). This anti-windup method was extended to adaptive tracking control considering input saturation (Kanamori, 2013a), and the control performance was demonstrated by experimentation (Kanamori, 2013b). A series of such works based on passivity imply that passivity is extremely suitable to stabilization that considers input saturation for nonlinear Euler-Lagrange systems. Since, in general, holonomic constraints have given rise to almost all robot systems (Khan et al., 2010), an anti-windup control method based on passivity is expected.
ensured by energy dissipation between the four Lyapunov functions. The control performance of the proposed controller is verified by numerical simulations using a two-link robot arm under holonomic constraints. 2. PRELIMINARY Let us consider the case where the endpoint of the manipulator is constrained on a surface (Arimoto, 1996). The surface is described by a scalar function as (1)
x = [x1
(2)
where
x2
x3 ]
T
denotes the Cartesian coordinates fixed at the internal reference frame. The contact force arises in the direction of the normal vector to the surface at point x and the contact − x with friction arises in the direction of magnitude x ( x ) x , where x (⋅) is a positive scalar function. Then, the robot dynamics are described by 1 H (q ) + S (q, q )}q + g (q ) 2 = Jf (q )T f − x ( x ) J x (q )T x + σ (u) ,
H (q)q + {B0 +
(3)
where
In the present paper, the anti-windup adaptive law by Kanamori (2013a) is extended to anti-windup PID position control under holonomic constraints for nonlinear EulerLagrange systems with input saturation. According to four saturation cases, switching of four integrating functions in the control law is utilized and four Lyapunov functions, such as hybrid control, are derived. Global asymptotic stability is 978-3-902823-62-5/2014 © IFAC
f ( x1 , x2 , x3 ) = 0 ,
9321
∂x (q ) , i = 1,2,3., j = 1,..., n, J x (q ) = i ∂q j
(4)
T
∂f J x (q ) ∂x , J f (q ) = 2 ∂f ∂x
(5)
19th IFAC World Congress Cape Town, South Africa. August 24-29, 2014
and f is the contact force. In addition, q = [q1 .... qn ] T represents the angular vector of each joint, H (q ) ∈ R
n× n
is
where Vci and Wci are switched for i = 1,..., l . Adding (14) and (15), the following relation holds:
n× n
the positive definite inertia matrix, B0 ∈ R is the positive diagonal constant matrix derived from the Raleigh dissipation function, S (q, q ) ∈ R n× n is the skew-symmetric matrix, and
g (q ) is the gravity term derived from the relation ∂U (q ) = g (q ) , where U (q ) is the potential energy. ∂q Furthermore, u = [u1 .... un ] T is the control input torque vector, and σ (u) = [σ (u1 ) ..... σ (un )] T is the saturated input torque vector, defined as follows:
0≥ where
dVi + Wi , dt
(16)
Vi = V0 + Vci and Wi = W0 + Wci .
(17)
Assume that Vi and Wi for i = 1,..., l are positive definite and
V j is the minimum function, V j = min (V1 , ....., Vl ) . Let us
denote the difference between Vi and V j for i ≠ j by
U i = Vi − V j . For asymptotic stability, the following lemma may be introduced (Kanamori, 2013a).
if ui min ≤ ui ≤ ui max , σ (ui ) = ui , if 0 < ui max < ui , σ (ui ) = ui max ,
(6)
Lemma 1: If we have the following two states for i = 1,..., l :
if ui < ui min < 0 , σ (ui ) = ui min ,
Vi > V j > 0 and Vi < 0 for i ≠ j : state 1 V j > 0 and V j < 0 : state 2
where ui max = ui min for i = 1,..., n . The estimation value of the input saturation ψ (u) = [ψ (u1 ) ... ψ (u n )] T is introduced as
ψ ( u) = u − σ ( u) ,
(19)
the minimum function V j is a Lyapunov function of the whole trajectory of the closed-loop system and the equilibrium state is asymptotically stable.
(7)
where
Outline Proof of Lemma 1: For the proof, it is sufficient to show that V j is dissipative on state 1 shown in (18). Taking
if ui min ≤ ui ≤ ui max , ψ (ui ) = 0 , if 0 < ui max < ui , ψ (ui ) = ui − ui max > 0 ,
(18)
(8)
the time integral of (16), the following relation holds:
if ui < ui min < 0 , ψ (ui ) = ui − ui min < 0 .
t
∫
V j (t ) − V j (0) ≤ U i (0) − U i (t ) − Wi dt .
The closed-loop system is described as
(20)
0
L(q, q , q) + F ( g (q), σ (u), f ) = 0 ,
(9)
If U i increases, then V j decreases. If U i decreases, then V j might increase; however, the increase value is bounded by U i (0) . Since Wi is positive definite, the time integral of Wi , which is the dissipated energy, always increases with the integral of time so that the increase in V j turns into a
where 1 H (q ) + S (q, q )}q 2 + x ( x ) J x (q )T J x (q )q ,
L(q, q , q) = H (q )q + {B0 +
T
F ( g (q ), σ (u), f ) = g (q ) − σ (u) − J f (q ) f .
(10)
decrease soon. This implies that V j is a Lyapunov function
(11)
To achieve the state (q, q , f ) → (qd ,0, f d ) , where qd and f d are the target vectors of each joint angle and target contact force, respectively, the following assumption is introduced. Assumption 1: Assume that the following conditions are satisfied for i = 1,..., n for any target posture:
ui min < g i (q) < ui max , ui min < [ g (qd ) + J f (qd )T f d ]i < ui max .
(12)
and the equilibrium state of the closed-loop system is asymptotically stable. Let us choose the following output y for the evaluation in Arimoto (1996):
y = q + αQφ (q ) s (∆q ) ,
where α is a positive definite scalar, Qφ (q ) is defined as
(13)
Qφ (q ) = I −
Take the appropriate output y and assume that the inner product of y and the closed-loop system (9) has the following form: d (14) y T L(q, q , q) = V0 + W0 , dt d (15) y T F ( g (u), σ (u), f ) ≥ Vci + Wci , dt
(21)
J φ (q )T J φ (q ) J φ (q )
2
,
(22)
which is regarded as a projection that projects vectors in joint space onto the plane tangent to the surface f (q ) = 0 at the contact point q , and I denotes the identity matrix. The function s (∆q ) is the output saturation function described in Arimoto (1996) as follows:
9322
19th IFAC World Congress Cape Town, South Africa. August 24-29, 2014
s (∆q) = [s1 (∆q1 ) s2 (∆q2 ) ...... sn (∆qn )]T ,
holds so that the margins between input limits and the gravity term are utilized for the stability.
(23)
where ∆q is the control error defined by
Property 4: The following inequality holds:
∆q = q − qd .
(24)
If s (∆q )T Qf (q )ψ (u) < 0 , then
s (∆q )T Qf (q ) F ( g (q ), σ (u), f ) ,
Property 1: Concering the projection Qφ (q ) , the following relations hold.
≥ c0 min s (∆q )T Qf (q ) s (∆q ) ,
J φ (q )q = 0 ,
(25)
Qφ (q )q = q ,
(26)
Qφ (q ) J φ (q )T = 0 ,
(27)
Qφ (q )Qφ (q ) = Qφ (q ) .
(28)
where c0 min is the positive minimum value of c0i for i = 1,..., n as follows.
c0 min = min[c01 ...... c0 n ] .
Property 2: For the output saturation function in (23), the following properties are ensured. (a) There exists a positive convex function S i ( xi ) such that the following relation is satisfied: dSi ( xi ) = si ( xi ) . dxi
(29)
∑
(35)
Property 5: The following inequalities are satisfied for saturation cases (I) through (IV) so that the input saturation function is eliminated by the inequalities (Kanamori, 2013a): (I) If q T ψ (u) ≥ 0 and s (∆q )T Qf (q )ψ (u) ≥ 0 , then
y T F ( g (q), σ (u), f ) ≥ y T F ( g (q), u, f ) ,
(36)
T (II) If q T ψ (u) ≥ 0 and s ( ∆q ) Qf (q )ψ ( u) < 0 , then
y T F ( g (q), σ (u), f )
Then, the following relations hold:
Si (∆qi ) = si (∆qi )∆qi and
(34)
≥ q T F ( g (q ), u, f ) + αc0 min s (∆q )T Qf (q ) s (∆q ) , (37)
Si (∆qi ) = s (∆q ) T q . (30)
T
(III) If q T ψ ( u) < 0 and s ( ∆q ) Qf (q )ψ ( u) ≥ 0 , then
i
(b) There exists a positive real constant ci such that the following inequality is satisfied: 2
Si (∆qi ) ≥ ci si (∆qi ) .
y T F ( g (q ), σ (u), f ) ≥ as (∆q )T Qf (q ) F ( g (q ), u, f ) , T
(IV) If q ψ ( u) < 0 and s ( ∆q ) Qf (q )ψ ( u) < 0 , then
(31)
y T F ( g (q), σ (u), f ) ≥ αc0 min s(∆q)T Qf (q) s(∆q) .
(c) There exists a positive real constant c0 such that the following inequality is satisfied: − s(∆q)T Qφ (q) H (q)q 1 + s(∆q)T {− Qφ (q) H (q) − Q φ (q) H (q) + Qφ (q) S (q, q )}q 2 2
+ Qφ (q) S (q, q )}q ≥ −c0 q .
Property 6: V0 and W0 as shown in (14) become 1 T 2 q H (q )q + α (b0 min cmin − αγ M ) s (∆q ) , 4 1 2 W0 ≥ {b0 min − αc0 + x ( x ) J x (q )T J x (q )} q 2 1 2 1 T − α s (∆q ) Qf (q )[ ∆B0 2 2 T + x ( x ) J x (q) J x (q)]Qf (q) s(∆q) , V0 ≥
Property 3: Based on the characteristics of the output saturation function s (⋅) and condition (12), there exists a positive real constant c0i such that the following inequality is satisfied.
min{ ui max − g i (q ) , ui min − g i (q ) } i
(39)
The inner product, as shown in (14), is determined by the given robot system. V0 and W0 are independent of the controller design and are obtained as follows (Arimoto, 1996).
(32)
≥ c0i s (∆q) i max ≥ c0i Qφ (q) s (∆q) .
(38)
T
(40)
(41)
where cmin is the minimum value of ci as follows:
(33)
For the inner product, as shown on the left-hand side of (15), notice that the term y T J f (q )T f is zero with the relations (25) and (27) in Property 1 in Arimoto (1996). According to the polarity of the inner product y T ψ (u) , the input saturation function σ (⋅) is eliminated, as shown in Property 1 in
cmin = min{c1 ...... cn } ,
(42)
b0 min is the minimum value of the diagonal component of the positive diagonal matrix B0 in (3), γ M is the maximum eigenvalue of H (q ) , and ∆B0 is the difference between B0 and b0 min I as follows:
Kanamori (2013a). Replacing Qf (q ) s (∆q ) for y in Property 1 in Kanamori (2013a) and using (33), the following property
9323
∆B0 = B0 − b0 min I ≥ 0 .
(43)
19th IFAC World Congress Cape Town, South Africa. August 24-29, 2014
Assumption 2: There exists positive real numbers α and c0 min such that the following inequalities are satisfied:
b0 min cmin − αγ M > 0 ,
(44)
b0 min − αc0 > 0 ,
(45)
1 2c0 min I − α { ∆B0 + x ( x ) J x (q )T J x (q )} > 0 . 2
s (∆q )T Qφ (q ){g (q ) − g (qd )} +
a 2 ∆q > 0 , 2
(46)
[
φ = J φ (q ) H (q ) J φ (q ) where
] [− J
φ (q ) H (q )
−1
In CASE (IV), y (t ) = 0 .
(58)
y T F ( g (q ), σ (u), f ) ≥ y T F ( g (q ), u, f ) =
a 2 ∆q 2 c 2 2 + αbcmin s (∆q) + z , 2 T Wc1 ≥ α [ s (∆q) Qφ (q){g (q) − g (qd )}
Vc1 ≥ U (q) − U (qd ) − ∆qT g (qd ) +
]
2
y T Qφ (q )cz = cy T z = cy T z = cz T z =
Let us consider the following proportional, integral and derivative (PID) control law: t
u = −a∆q − bq − cQφ (q) ∫ y (t )dt 0
− J f (q ) f d + β J f (q )
T
∫ ∆f (t )dt ,
(51)
0
where a , b , c , and β are positive definite real numbers that are regarded as feedback gain, and ∆f is the error between contact force f and its target force f d represented as ∆f = f − f d . Using the notation
∫
(52)
τ
∫
∞
∫
(39) must be included in Wc 2 or Wc 4 , respectively. In CASE (IV), there is no inner product on the right-hand side in (39); however, it is regarded that there is the inner product of zero and F ( g (q ), u, f ) as follows:
0 T F ( g (q ), u, f ) = where Vc 4 =
∞
0
0
∫ y(τ )dτ = −c
−1
0
c 2 z : constant. 2
(63) (64)
(65)
It is confirmed that Vci for i = 1,...,4 is positive definite when
F ( g (q ), σ (u), f ) = g (q ) − Qf (q ) g (qd ) + a∆q
+ bq + Qf (q )cz − J f (q )T (∆f + b∆F ) + ψ (u) . (54) Let us take the following y (t ) according to the four saturation cases (Kanamori, 2013a):
y (t ) = q + αQφ (q) s (∆q) ,
dVc 4 = 0, dt
Wc 4 ≥ αc0 min s (∆q )T Qf (q ) s (∆q ) .
g (qd ) , (53)
F ( g (q), σ (u), f ) defined by (11) becomes
In CASE (I),
c d 2 z . (62) 2 dt
Then, Wc 4 becomes
0
z = y (τ )dτ − y (τ )dτ and
(61)
In the same manner, Vci and Wci for i = 2, 3 are obtained by using (56) and (57), respectively. Notice that the second term αc0 min s (∆q )T Qf (q) s (∆q) on the right-hand side of (37) or
t
∆F = ∆f (t )dt ,
(60)
The suffix i = 1 of Vc1 and Wc1 means saturation CASE (I). In the derivation, the following relation, (26) and (28) are utilized:
3. CONTROLLER DESIGN
T
2
+ a Qφ (q) s(∆q) ] + b q .
(50)
t
dVc1 + Wc1 , (59) dt
where Vc1 and Wc1 are represented as
X − Jφ (q )q , (49)
1 X = −{B0 + H (q ) + S (q, q )}q 2 − g (q) + σ (u) − x ( x ) J x (q)T x .
(57)
the inner product y T F ( g (q), σ (u), f ) in (36) is represented as follows:
(47)
2 a Qφ (q ) s (∆q ) > 0 . (48) 2
T −1
In CASE (III), y (t ) = αQφ (q ) s (∆q ) ,
and Vci and Wci in (15) are also easily derived. In CASE (I),
Property 8: The contact force f is given by the following form: −1
(56)
The inner products, as shown on the right-hand side of (36) through (39), can be taken easily because σ (⋅) is eliminated
Property 7: There exists a positive real number a such that the following inequalities are satisfied: U (q ) − U (qd ) − ∆q T g (qd ) +
In CASE (II), y (t ) = q ,
the conditions (47) are satisfied, and Vi in the first equation in (17) is also positive definite, since V0 is positive definite under (44). Wi for i = 1,...,4 in the second equation in (17) is also positive definite under (45), (46), (48) and the following condition:
(55) 9324
1 aI − a { ∆B0 + x ( x ) J x (q )T J x (q )} > 0 . 2
(66)
19th IFAC World Congress Cape Town, South Africa. August 24-29, 2014
The above results imply that we have the following two states according to the saturation cases:
Vi ≥ V4 > 0 , Vi < 0 , for i = 1, 2, 3 : state 1 : state 2 V > 0 , V < 0 . 4
4
(67) (68)
where Vi is switched according to saturation cases (I) through (IV). Based on Lemma 1, we have the following main theorem. Main Theorem (Anti-windup PID control law): Assume that conditions (44) through (46) are satisfied. Choose feedback gain a as the positive real number such that the conditions (47), (48) and (66) are satisfied. Adopt the switching function as (55) through (58) for the control law (51). Then, the equilibrium state of the closed-loop system is globally asymptotically stable based on Lemma 1. Proposition (Practical anti-windup PID control law): Assume that the conditions in the Main Theorem are satisfied. For the control law (51), choose the function y (t ) as follows:
y (t ) = y1 (t ) + αy 2 (t ) ,
saturation function si ( xi ) in (23) has also been shown in (52) in Kanamori (2013b). Input limits ±2 V are given for the shoulder joint input and ±1 V are given for the elbow joint input. The following values are used for the simulations.
f d = 10 N, x ( x ) = 0.3 Nm/s, γ M = 6.38 ,
(77)
α = 3.0 , a = 10.0 , b = 2.0 , c = 5.0 , β = 1.0 .
(78)
Fig. 2 shows the control performance using the normal PID controller. For the strict input limits, the windup phenomenon is observed at elbow joint input u2 , as shown in the upper plots in Fig. 2. As a result, considerable overshoots occur in shoulder joint angle q1 and elbow joint angle q2 , as shown in the lower plots in Fig. 2. As shown in Fig. 3, the contact force also degrades for the input saturation. In contrast, the windup phenomenon is restrained by using the proposed antiwindup PID controller, as shown in the upper plots in Fig. 4. As a result, considerable overshoots are restrained very well,
(69)
x2
q2
where if qiψ(ui ) ≥ 0 , then y1i = qi , if qiψ(ui ) < 0 , then y1i = 0 ,
q1
(70)
L0
0
θ
x1
Fig. 1 Two-link robot arm with holonomic constraint
y2i = {s (Δq )T Qf (q )}i , (72)
if {s (Δq ) Qf (q )}i ψ(ui ) < 0 , then y2i = 0 ,
L2
f
(71)
if {s (Δq)T Qf (q)}i ψ(ui ) ≥ 0 , then, T
L1
(73)
for i = 1,....., n , where {⋅}i denotes the i-th component of the vector, yki denotes the following vector component:
y k = [ yk1 ... yki ... ykn ]T and k = 1, 2 .
(74)
Then, the equilibrium state of the closed-loop system is globally asymptotically stable.
Solid: u1 Bloken: u2
Solid: σ(u1) Bloken: σ(u2)
4. CONTROL PERFORMANCE The control performance of the proposed anti-windup PID controller is examined by numerical simulations using a twolink robot arm. A schematic two-link robot arm with contact force is depicted in Fig. 1, where L1 and L2 denote the length of the arm. L0 is the coordinate value of x1 and the start point of the constraint surface. In this system, the constraint surface is represented as (75) and (1) becomes (76).
x2 = as x1 + bs , L0 ≤ x1 , f = x2 − as x1 − bs = 0 ,
Thick: q1 Thin: qd1
Thick: q2 Thin: qd2
Fig. 2 Control performance using normal PID controller
(75) (76)
where as and bs are real constants for the constraint surface. The system equation with actuators and the detail of the twolink robot system have been described in (48) through (51) and TABLE 1 in Kanamori (2013b), respectively. Output
9325
Fig. 3 Contact force using normal PID controller
19th IFAC World Congress Cape Town, South Africa. August 24-29, 2014
REFERENSES
solid: u1 Bloken: u2
Thick: q1 Thin: qd1
Solid: σ(u1) Bloken: σ(u2)
Thick: q2 Thin: qd2
Fig. 4 Control performance using the proposed controller
Fig. 5 Contact force using the proposed PID controller
Fig. 6 Confirmation plot of conditions (32), (47) and (48) as shown in the lower plots in Fig. 4. As shown in Fig. 5, the control performance of the contact force is improved in comparison with that in Fig. 3. Fig. 6 shows the validity of conditions (32), (47), and (48). The top plot shows the lefthand side of (32), where the right-hand side is transposed to the left-hand side. The middle and the lower plots correspond to the left-hand sides of (47) and (48), respectively. Since these values are positive or zero, it is confirmed that conditions (32), (47), and (48) are satisfied. These simulation results validate the analysis and design of the proposed antiwindup PID controller. 5. CONCLUSIONS An anti-windup PID controller design for Euler-Lagrange systems under holonomic endpoint constraints has been presented. The control performance was examined by numerical simulations using a two-link robot arm with contact force. The simulation results validate the analysis and design of the proposed anti-windup PID controller.
Arimoto, S. (1996), Control theory of non-linear mechanical systems: A passivity-based and circuit-theoretic approach, p. 93-113, Oxford University Press, Walton Street, Oxford OX2 6DP. Kanamori, M. and M. Tomizuka (2007), Dynamic antiintegrator-windup controller design for linear systems with actuator saturation, ASME Journal of Dynamic Systems, Measurement, and Control, vol. 129, no. 1, pp. 1–12. Kanamori, M. (2011), Global stabilization of anti-windup PID position control for Euler-Lagrange systems with actuator saturation, In: Proceedings of IEEE International Conference on Automation Science and Engineering, pp. 798–803, Trieste. Kanamori, M. (2013a), Anti-windup adaptive tracking for Euler-Lagrange systems with actuator saturation, In: Proceedings of 3rd International Conference on Instrumentation, Control and Automation, pp. 216-221, Bali. Kanamori, M. (2013b), Experimental study of anti-windup adaptive tracking control for Euler-Lagrange systems with actuator saturation, In: Proceedings of 6th IEEE International Conference on Robotics, Automation and Mechatronics , to be published. Khan, S. G., G. Herrmann, T. Pipe and C. Melhuish (2010), Adaptive multi-dimensional compliance control of a humanoid robotic arm with anti-windup compensation, In: Proceedings of IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 2218–2223, Taipei. Lopez-Araujo, D. J., A. Zavala-Rio, V. Santibanez and F. Reyes (2012), Global adaptive regulation of robot manipulators with bounded inputs, in Proc. 10th IFAC Sympo. Robot Control, Dubrovnik, pp. 806–813. Loria, A., R. Kelly, R. Ortega and V. Santibanez (1997), On global output feedback regulation of Euler-Lagrange systems with bounded inputs, IEEE Trans. Automatic Control, vol. 42, no. 8, pp. 1138–1143. Morabito, F., A. R. Teel and L. Zaccarian (2004), Nonlinear antiwindup applied to Euler-Lagrange systems, IEEE Trans. Robotics and Automation, vol. 20, no. 3, pp. 526– 537. Santibanez, V. and R. Kelly (1996), Global regulation for robot manipulators under SP-SD feedback, In: Proceedings of IEEE International Conference on Robotics and Automation, pp. 927–932, Minneapolis. Teo, J. and J. P. How (2009), Anti-windup compensation for nonlinear systems via gradient projection: Application to adaptive control, In: Proceedings of 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference, pp. 6910–6916, Shanghai. Zavala-Rio, A., E. Aguinaga-Ruiz and V. Santibanez, (2009), Global trajectory tracking through output feedback for robot manipulators with input saturations, In: Proceedings of IFAC Symposium on Robot Control, pp. 701–707, Gifu.
9326