Act and Wait Concept in Force Controlled Systems With ... - CiteSeerX

Proceedings of IDETC/CIE 2005 Proceedings of IDETC 2005 ASME 2005 International Design Engineering Technical Conferences 2005 ASME International Design Engineering Technical Conferences & Computers and Information Engineering Conference Long Beach, California, in USA, September 24-28, 2005 September 24-28, 2005, Long Beach, California USA

DETC2005-85036

DETC2005-85036

ACT AND WAIT CONCEPT IN FORCE CONTROLLED SYSTEMS WITH DISCRETE DELAYED FEEDBACK

´ Insperger£ Tamas Department of Applied Mechanics Budapest University of Technology and Economics H-1521, Budapest, Hungary Email: [email protected]

´ ´ an ´ Gabor Step Department of Applied Mechanics Budapest University of Technology and Economics H-1521, Budapest, Hungary Email: [email protected]

ABSTRACT A 1 DOF model of force control is considered with discrete delayed feedback. The act and wait control concept is introduced as a special case of periodic control: the feedback gain is constant for a sampling period (act), then it is zero for a certain number of periods (wait), then it is constant again, etc. It is shown that by applying the act and wait concept, the stability properties of the system improve and the force error caused by the Coulomb friction decreases. If the period of gain variation is larger than the feedback delay, then the system performance changes radically: the stability properties improve significantly, and the optimal control parameters can provide dead beat control. This way, by using the act and wait concept, the delay effects are eliminated from the system.

systems controlled through the internet [3] or in robotic applications with time-consuming control force computation [4]. Caused by the delay of the control feedback, the governing equation is a delay-differential equation (DDE). DDE’s usually have infinite dimensional phase spaces [5, 6], therefore the linear stability conditions for the system parameters are complicated and often do not have an analytical form. However, there exists several methods to analyze control systems with delayed feedback [7-10]. An effective way for analyzing DDEs is the semidiscretization method introduced and developed by Insperger and Stépán [7, 11] and also improved by Elbeyly and Sun [12]. The point of the method is that the continuous system is approximated by a semi-discrete system, where only the delayed terms are discretized, the actual time domain terms are left in the original form. This way, the DDE is approximated by a series of ordinary differential equations (ODEs). Digital control of time-continuous systems provides also a semi-discrete nature due to the sampling effect and the corresponding zero order hold. The time-continuous system is forced by the time-discrete feedback of the controller and the connection between states at discrete sampling instants can be described by ODEs [13, 14]. This serves a finite dimensional discrete map model similarly to the semi-discretization method. Gain parameters and sampling time do not always provide stable motion and fast settling time, since these parameters are often hedged by other technical conditions. For these cases, application of periodic controllers (e.g., time varying feedback

1

INTRODUCTION Force control is a frequent mechanical controlling problem in robotics. The aim is to provide a desired force between the actuator and the environment. Time delay often arise in feedback control systems due to acquisition of response and excitation data, information transmission, on-line data processing, computation and application of control forces. In spite of the efforts to minimize time delays, they can not be eliminated totally even with today’s advanced technology due to physical limits. The information delay is often negligible, but for some cases, it still may be crucial, for example, in space applications [1, 2], in

Address

all correspondence to this author.

1

Copyright c 2005 by ASME

is governed by the differential equation mx¨t kx˙t sxt Q

(1)

If the Coulomb friction C is not neglected, then the system’s behavior can be analyzed in two parts. If x˙t 0 and sxt Q C (the body is moving), then the system is governed by the differential equation

Figure 1.

Mechanical model

mx¨t kx˙t sxt Q

If x˙t 0 and sxt friction), then

gains) may stabilize or speed up the control. The idea of stabilizing by parametric excitation comes from the classical example of the pendulum: the upper position of a pendulum can be stabilized by vertically vibrating its pivot point [15]. For memoryless feedback control systems, several papers have been published on the stabilization effect of periodic feedback for both time-discrete [16, 17] and time-continuous systems [18-20]. If the feedback is delayed and periodic at the same time, then the system is governed by a periodic DDE and the Floquet theory of DDEs should be used to derive stability properties. Usually, stability conditions can not be determined in closed form even for simple time periodic delayed systems, like the delayed Mathieu equation [21]. However, for computer controlled systems, the feedback is discrete (even if the sampling period is very short), the system can be transformed into a periodic discrete map, and the system performance can be analyzed this way. In this paper, a 1 DOF model of force control is considered with discrete delayed feedback. In order to improve stability performance, the act and wait control concept is introduced as a special case of periodic control: the feedback gain is constant for the first sampling period (act), then it is zero for a certain number of samplings (wait), then it is constant again, etc. The effect of time delay and periodic gains are investigated via stability charts and optimal control parameters are given.

xt

x

Csgnx˙t

(2)

Q C (the block is stuck due to the C Q C Q x s s

s

s

(3)

According to Stépán [23], the control force can be written as

Q Psxt

F sxt d

(4)

where P is the proportional gain and Fd is the desired force. Now, the equation of motion reads

mx¨t kx˙t Psxt

F Csgnx˙t d

(5)

Clearly, if C 0, then the trivial solution is given as xt x 0 Fd s. Introduce the perturbation ξt so that xt x 0 ξt and substitute into Eq. (5) to get

mξ¨ t kξ˙ t Psξt Csgnξ˙ t

(6)

C satisfy, then the block sticks and

If ξ˙ t 0 and sPξt its position can be given as

2 CONTINUOUS MODEL Figure 1 presents an ideal 1 DOF mechanical model of the force control, where m stands for the mass modeling the inertia of the robot, k and s denotes the damping and the stiffness which models the elastic force sensor and the elastic environment. Similar models are often used in basic textbooks to analyze force control [22, 23]. The single coordinate x is chosen in a way that the spring is relaxed at x 0. The control force Q is provided by the ideal actuator, and calculated by the digital processor from the contact force error Fe Fm Fd , where Fm and Fd are the measured and the desired contact forces, and the contact force is measured via the deformation x of the spring: Fm sx. If the Coulomb friction is neglected (C 0), then the system

ξt

ξ

s

ξs

sPC

C sP

(7)

Consequently, the maximum possible force error can be characterized by ∆Fe CP. Thus, the larger the gain is, the smaller the force error is. Theoretically, there is no upper limit for the gain P, since the zero solution of Eq. (6) is always asymptotically stable when C 0. Experiments show, however, that the real system is not stable for large gain P [13]. The instability is caused by the digital effects and delays of the control as explained below.

2


3 DISCRETE MODEL Assume that the digital control samples the force signal at time instant t j j∆t, j 0 1 , where ∆t is the sampling period. Assume, furthermore, that the feedback delay is N∆t where integer N will be called as delay parameter. The control force can be given as

Q Psxt j N

F sxt d

j N

t

t

j t j 1

3.1 Solution map Due to the discrete feedback, the state at time instant T can be given for any known values of x j and y j n:

x j 1 Px j Ry j

The equation of motion

F sxt t t

j N

d

j t j 1

(9)

mξ¨ t kξ˙ t sξt s1 Pξt j N

t

t

j t j 1

ω2n ∆t 21 Pξ j N

T

j j 1

N Cx j N

T

j j 1

(12)

A

0 ωn∆t 2

1 2ζωn ∆t

0 B ω ∆t 1 P y ξ

2

y j

P 0 0 R x j C 0 0 0 y j 1 0 I 0 0 y j 2 .. . .. ..

.

. ..

. 0 0 I 0 N 1 y j N

0

D

zj

(15)

P12 P22 0 0 0 .. . 0

0 0 0 1 0 .. .

0 0 0 0 1 .. .

0 0

0 0 0 0 0 .. .

R1 R2 0 0 0 .. .

(16)

1 0

where Pi j and Ri , i j 1 2 are the corresponding elements of matrices P and R. Introduce the decay index ρ µ 1 , where µ1 is the critical characteristic multiplier (i.e., the largest in modulus) of D. Decay

C 1 0

z j 1

P11 P21 1 D 0 0 ..

.

n

ξ ξ¼

(14)

3.2 Stability charts Equation (11) is described by four dimensionless parameters, the ratio of the system and the sampling frequencies f n ∆t ωn ∆t 2π, the proportional gain P, the damping ratio ζ and the delay parameter N. If all the parameters are fixed, then matrices P and R can be computed, and D can be written as

where x

A 1 B

(11)

sm is the angular natural frequency. Equation (11) can be written in the state space form

N

I

where the coefficient matrix D is actually the Floquet transition matrix over the period ∆T 1. If all of the eigenvalues of D are in modulus less than one, then Eq. (15) and, consequently, Eq. (11) are asymptotically stable. The identity matrices below the diagonal in D represents the delay effect. The larger the delay parameter N is, the larger the system’s dimension is (since D is a N 2 N 2 matrix).

where ζ is the damping ratio defined by 2ζω n km and ωn

x¼ T AxT By j

x j 1 y j y j 1 .. .

(10)

Introduce the dimensionless time T t ∆t and denote the derivatives with respect to T by prime. The equation of motion gets the form ξ¼¼ T 2ζωn∆tξ¼ T ω2n ∆t 2 ξT

R eA

and I denotes identity matrix. Equation (13) implies the discrete map in N 2

is a delayed differential equation. From now on, the Coulomb friction will be neglected, and we will concentrate on the digital effects. The use of the perturbation ξt xt x 0 in Eq. (9) results in

y j

P eA

mx¨t kx˙t sxt Psxt j N

(13)

where

(8)

N

j

3


2

2

N=1, ζ=0

N=1, ζ=0.02 ρ=1

ρ=0.9

1

ρ=0.8

0.5

P *

1

0

0 (f ∆ t) * n

A3

0.5

1

1

ρ=0.8

P *

A4 1.5

1

0

0 (f ∆ t) *

2

n

f ∆t

A2

A

1

1

1

1.5

2

1.5

2

1.5

2

f ∆t n

2

N=2, ζ=0

N=2, ζ=0.02

1.5

1.5

1

1

P

P

A3 A4

0.5

n

2

0.5

0.5

0

0 0

0.5

1

1.5

2

0

0.5

f ∆t

1

f ∆t

n

n

2

2

N=3, ζ=0

N=3, ζ=0.02

1.5

1.5

1

1

P

P

ρ=0.9

1 0.5

A2

A

1

ρ=1

1.5

P

P

1.5

0.5

0.5

0

0 0

0.5

1

1.5

2

0

0.5

f ∆t

Figure 2.

1

f ∆t

n

n

Stability charts for Eq. (11) for different delay parameters N and relative damping ζ

The optimal parameters f n ∆t i£ , Pi£ and the corresponding decay indices ρ£i (i 1 2 3 4), are presented in Table 1. For the undamped cases (ζ 0), the optimal parameters are dis£ £ tributed periodically along the axis f n ∆t: fn ∆t 2i 1 fn ∆t 1 £ £ i, fn ∆t 2i fn ∆t 2 i. It can be seen that the addition of a slight damping ζ 002 changes the stability charts (destroys its symmetry), but it does not have significant effect on the optimal parameters. For both the damped and the undamped cases, the larger the time delay parameter is, the larger the optimal decay parameter is, the slower the convergence of the system is.

index ρ is a kind of measure of the average error decay over a single sampling period. The control is optimal, if the decay index is minimal. In Figure 2, stability charts are presented in the plane of the frequency ratio f n ∆t and the gain P for different delay parameters N and relative damping ζ. The charts were determined via point-by-point evaluation of the eigenvalues of the transition matrix (16) over a 400 100 sized grid of frequency ratio f n ∆t and gain P. Contour plot was used to obtain the transition curves associated to different decay indices ρ 1, 0.9, 0.8, etc. Obviously, the stability boundaries are the transition curves where ρ 1. Stable domains are denoted by grey color. The undamped (ζ 0) cases in Figure 2 shows a periodic structure in f n ∆t of period 1. A special symmetry can also be observed. Damping, however, destroys this periodicity and symmetry. The closed form stability limits and the optimal parameters for the case N 1 ζ 0 can be found in [23]. The optimal gain parameters, where the decay index is minimal, are denoted by black dots marked by A 1 , A2 , A3 and A4 .

4 ACT AND WAIT CONTROL CONCEPT According to the act and wait control concept, the control force is given as

Q gt Psxt j N 4

F sxt d

t t (17) c 2005 by ASME Copyright j N

t

j j 1

N and damping ζ.

Table 1. Optimal control parameters and the corresponding decay indices for Eq. (11) with different delay parameters

N

ζ

fn ∆t £1

P1£

ρ£1

fn ∆t 2£

P2£

ρ£2

fn ∆t 3£

P3£

ρ£3

fn ∆t £4

1

0

0.103

0.254

0.532

0.897

0.254

0.532

1.103

0.254

0.532

1

0.02

0.103

0.269

0.525

0.881

0.364

0.438

1.119

0.397

2

0

0.063

0.260

0.677

0.937

0.260

0.677

0.063

2

0.02

0.063

0.275

0.671

0.919

0.457

0.527

3

0

0.045

0.262

0.754

0.955

0.262

3

0.02

0.350

0.903

0.733

0.936

0.542

P4£

ρ£4

1.897

0.254

0.532

0.426

1.867

0.478

0.353

0.260

0.677

1.937

0.260

0.677

1.081

0.488

0.518

1.902

0.636

0.328

0.754

1.045

0.262

0.754

1.955

0.262

0.754

0.549

1.064

0.571

0.544

1.918

0.767

0.478

where gt is a K∆t-periodic switching function: gt

1 if t 0 if t

t hK∆t t hK∆t h t hK∆t t hK∆t h 0

1

1

K

Using the dimensionless time T tion gets the form

g j ω2n ∆t 21 Pξ j N

Here, integer K will be called period parameter. If K 1, then gt 1. This corresponds to the traditional control with constant gains. If K 2, then gt alternates between zero and one: in the first sampling interval, it is one, in the following K 1 number of intervals, it is zero, in the K 1 st interval, it is one again, etc. This control concept is a special case of periodic feedback controllers called act and wait control. In the first interval, gt 1, the control is active (act), in the following K 1 number of intervals, gt 0, the control term is switched off (wait), then, in the K 1st interval, the control is active again, etc. The equation of motion for the act and wait control system is

gj

y j

0

1sx t t

0 j t j 1

(21)

1 0

if

j hK h otherwise

(22)

N Cx j N

N

T

j j 1

(23)

4.1 Solution map Similarly to Eq. (15), the following discrete map can be constructed

(19)

x , the resulted variational

mξ¨ t kξ˙ t sξt gt s1 Pξt j N gt

where the vectors x, y and the matrices A, B and C are defined at Eq. (12).

Using the perturbation ξt xt equation reads

x¼ T AxT g j By j

mx¨t kx˙t sxt

j j 1

The state space representation of Eq. (21) is

T

where

gt Psxt j N Fd sxt j N t t j t j1

∆t, the equation of mo-

ξ¼¼ T 2ζωn ∆tξ¼ T ω2n ∆t 2 ξT

(18)

t

(20)

The term gt 1sx 0 is a periodic excitation, it does not affect the stability of the system, therefore it will be eliminated in the next step.

x j 1 y j y j 1 .. .

z j1

P 0 C 0 0 I .. ..

. . N 1 0 0

y j

0 g jR 0 0 0 0 .. . I 0

Dj

x j y j 1 y j 2 ..

.

y j

(24)

N

zj

where matrices P and R are defined in Eq. (14). As in Eq. (15), the identity matrices below the diagonal represents the delay effect. Since the coefficient matrix is K-periodic (D jK D j ), the 5


2

2

N=1, K=1

1.5

ρ=0.9

1

P

P

N=2, K=1

ρ=1

1.5

ρ=0.8

0.5

P * 1

0.5

B

1

0 0 (f ∆ t) * n

1

B1

0 0.5

1

1.5

2

0

0.5

1

fn ∆ t

1.5

2

1.5

2

1.5

2

fn ∆ t

N=1, K=2

2

1

N=2, K=2

2

P

P

B3 1

B1

0

0

B2

−1 0

1

B1

−1

0.5

1

1.5

2

0

0.5

1

fn ∆ t

fn ∆ t 4

N=1, K=3

2

B3 P

1

P

N=2, K=3 2

0

B

2

−1 −2 0

3

0

B2

−2

B1

B4

B

B

5

B

1

−4 0.5

1

1.5

2

0

0.5

1

f ∆t

f ∆t

n

Figure 3. Stability charts for Eq.

n

(21) with ζ 0 for different delay parameters N and period parameters K

Floquet transition matrix can be given by coupling the solutions over K number of samplings: Φ DK DK 1 D1

Similarly to Fig. 2, contour plot was used to obtain the transition curves associated to different decay indices ρ 1, 0.9, 0.8, etc, and he optimal gain parameters are denoted by black dots marked by B 1 , B2 , etc. The optimal parameters f n ∆t i£ , Pi£ and the corresponding decay indices ρ £i (i 1 2 ) are presented in Table 2. For the sake of simplicity, only the optimal parameters in the interval f n ∆t 0 05 are presented in table. It can be seen that if the period parameter K is larger than the delay parameter N, then numerous optimal parameters arise with zero decay index resulting dead beat control as it will be shown in the next section. For the cases N 1 with K 2 and N 1 with K 3, there are three optimal parameter pairs in f n ∆t 0 05 denoted by B 1 , B2 and B3 . For the case N 2, K 3, there are five optimal points in f n ∆t 0 05 denoted by B 1 , B2 , B3 , B4 and B5 . Note, that the optimal gains P corresponding to points B 3 for N 1 with K 2 3 or B3 and B4 for N 2 with K 3 are larger then those of the cases K 1. This implies that, for the optimal cases, the maximum possible force error ∆Fe CP caused by

(25)

Now, define the decay index as ρ µ 1 1K , where µ1 is the critical characteristic multiplier (i.e., the largest in modulus) of Φ . By this definition, decay index ρ characterize the average error decay over a single sampling period as it was in the case of traditional control with constant gain. This way, decay index can be used to compare systems with different periods K of the act and wait control for a given delay parameter N of the system.

4.2

Stability charts In Figure 3, stability charts are presented for Eq. (21) with ζ 0 for different delay parameters N and period parameters K. Due to the zero damping, these stability charts are periodic in fn ∆t of period 1. 6


Table 2.

Optimal control parameters and the corresponding decay indices for Eq.

parameters

K in the interval fn ∆t

0 0 5.

(21) with ζ 0 and with different delay parameters N and period

N

K

f n ∆t 1

P1

ρ1

f n ∆t 2

P2

ρ2

f n ∆t 3

P3

ρ3

f n ∆t 4

P4

ρ4

f n ∆t 5

P5

ρ5

1

1

0.103

0.254

0.532

-

-

-

-

-

-

-

-

-

-

-

-

1

2

1/7

-0.182

0

2/7

-0.474

0

3/7

1.656

0

-

-

-

-

-

-

1

3

1/9

-0.688

0

2/9

0.102

0

4/9

1.586

0

-

-

-

-

-

-

2

1

0.063

0.260

0.677

-

-

-

-

-

-

-

-

-

-

-

-

2

2

0.061

-0.497

0.692

-

-

-

-

-

-

-

-

-

-

-

-

2

3

1/11

-0.793

0

2/11

-2.283

0

3/11

1.727

0

4/11

2.017

0

5/11

0.332

0

the delay effect and the system is N 2 dimensional. If the act and wait concept is used with K N, then these identity matrices all disappear, and the system is described by the left upper 2 2 submatrix M PK RCPK N 1 . This shows that by using the act and wait control concept with K N, the order of the system is reduced radically, this way, the delay effect is eliminated from the system in a certain meaning. Clearly, stability properties of matrix (27) is determined by the 2 2 submatrix M. Consider the undamped case (ζ 0). Substitution of the matrices P, R and C gives

the Coulomb friction can be decreased by using the act and wait control concept.

Analytical stability analysis for K N and ζ 0 Due to the act and wait concept, the coefficient matrices in Eq. (24) read

4.3

D1

P 0 C 0 DK 1 0. I ..

.. . 0 0

00 0 0 0 0 .. .

(26)

I0

M

cK 1 c1 cK ωn ∆t sK s1 cK

and

P 0

0R

P P

1 ωn ∆t sK 1 c1 sK N 1 1 cK s1 sK N 1 1 P

P

(29)

C 0 0 0 DK 0 I 0 0 .. . . . . ..

.

0 0

N 1 1 N 1 1

where (27)

c j cos jωn ∆t s j sin jωn ∆t

I 0

j j

If K N, then Eq. (25) results in the transition matrix

Φ

PK RCPK N 1 CPK 1 CPK 2 .. . CPK N

0 0 0 0 0 0 .. .. . . 0 0

It can be shown by algebraic calculation that the control is dead beat (i.e., matrix M has two zero eigenvalues), if

0 0 0 .. .

fn ∆t £ 2N h2K 1

(28)

P£ 1

0

As it can be seen, all the columns except for the first one consist of zero matrices. In system (15) with constant control gain (or in system (24) with K 1), the identity matrices below the diagonal represent

cos

(30)

2 cos 2NhK2π 2K 1

hK N 2π 2N 2K 1

cos

hK N 12π 2N 2K 1

(31)

h 1 2

These optimal values can also been seen in Table 2. 7


5 CONCLUSIONS A 1 DOF model of force control was considered with discrete delayed feedback. The act and wait control concept was introduced in order to improve control performance. The feedback gains are constant (the delayed feedback is switched on – act) for the first sampling period, then they are zero for a certain number of sampling periods (the delayed feedback is switched off – wait). It was shown that if the period of gain modulation is larger than the time delay itself, then the delay effect is eliminated from the system in a certain meaning. The act and wait concept provides an alternative for control systems with significant feedback delays. The traditional way is the continuous use of the control gain P according to the K 1 case. The other, alternative way is the act and wait control concept, when a constant control gain is used for a short time (for a sampling period) and zero gain for long time (for a period equal to the time delay itself). By using the act and wait concept, the stability properties improves significantly (the convergence of the system gets faster), and, for the optimal parameters, the maximum possible force error caused by the Coulomb friction decreases. It was shown that for a certain pairs of gain and sampling frequency, dead beat control can be achieved.

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ACKNOWLEDGMENT This research was supported in part by the Magyary Zoltán Postdoctoral Fellowship of Foundation for Hungarian Higher Education and Research (TI), and by the Hungarian National Science Foundation under grants no. OTKA F047318 (TI) and OTKA T043368 (GS, TI).

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