Friction Compensation Methods in Position and

Friction Compensation Methods in Position and Speed Control Systems


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Joško Deur

University of Zagreb Faculty of Electrical Engineering and Computing Department of Control and Computer Engineering in Automation Unska 3, HR-10000 Zagreb, Croatia [email protected] [email protected]

University of Zagreb Faculty of Mechanical Engineering and Naval Architecture Department of Robotics and Automation of Manufacturing Systems -10000 Zagreb, Croatia [email protected]

Abstract - Friction appears in bearings and reduction gear of controlled electrical drives and affects the quality of the position, speed or force control. The servosystem performance can be improved by implementation of acceleration feedback. For the reason that acceleration measurement is often either not possible or practical, the implementation of acceleration estimation by observer or differentiation of speed is proposed. Another approach to friction influence compensation is based on the disturbance observer. Its advantage is that it can be applied generally for the estimation of various disturbances. This paper deals with these friction compensation methods which are not based on friction model, but on the compensation of friction as disturbance. The efficiency of these methods in a servodrive with Stribeck friction is compared by computer simulation and experimentally on a servosystem laboratory model.

II. POSITION AND SPEED CONTROLLER DESIGN

I. INTRODUCTION Disturbance torques, among which friction is the most frequent, are present in almost every motion control application. In position and speed control systems with high accuracy requirements friction torque in sliding bearings has a strong negative effect on the desirable control quality. The reasons are static friction (stiction) effect in stillstand, Coulomb friction, expressive nonlinearity and negative derivation of the static characteristic of friction during transition from stiction to Coulomb friction, called Stribeck curve [1]. The negative influence of friction can be decreased by introducing friction compensation into the control algorithm. Many different friction compensation techniques are described in [1], and some of non-model-based friction compensation methods are here described. Theirs advantages are that they don’t require friction model and they can be applied generally for the compensation of various disturbances. Efficient friction compensation can be achieved by closing a fast inner acceleration torque control loop [12]. Instead of acceleration torque measuring, it is possible to estimate acceleration in several ways. The simplest way is to differentiate speed signal [3]. Another solution is the use of a full-order acceleration observer described in [13]. There are friction compensation methods which are not based on acceleration torque feedback, but on estimation of the sum of all disturbance forces [10]. The method of feed forward friction compensation using disturbance estimator was clearly explained in [8]. The aim of this paper is to implement mentioned methods on a servodrive and to compare friction compensation efficiency by computer simulation and experimentally on a servosystem laboratory model.

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Three cascade control structures of servosystem are considered. The first one uses standard solution with PI speed controller and P position controller. The second control structure uses additional acceleration torque feedback which efficiently compensates friction influence. The third control structure is based on the disturbance observer. A. PI speed controller The standard cascade structure with P position controller and modified PI speed controller shown in Fig.1 (without acceleration torque feedback : Km = 0) is considered [7]. All system quantities are normalized [4]. The friction compensation methods were examined with Karnopp model of static + Coulomb friction [6]. With respect to the design of the slower speed control system, the inner current control loop of the DC or vector controlled AC drive can be approximated by the equivalent lag term Gei (s) [7]. It is assumed that the speed is reconstructed by time-differentiation of the measured position. The transfer function of the measuring term is [5]: ω m ( z ) TB z − 1 (1) , G ’mω ( z) = = α m (z) T z where: T - sampling time, TB - base quantity for time [5]. For the purpose of equivalent continuous-time control system design, the speed measuring element (1) and the series connection of the sampler and zero order hold (ZOH) may be approximated by equal lag terms Gmω(s)=GsE(s): T

Gmω ( s) =

ω m ( s) 1 − e −Ts −s 1 , = ≈e 2 ≈ 1 + 0.5Ts ω ( s) Ts

(2)

so that the dynamical part of the process with small time constants, including Gei(s), can be modeled as a single lag term [5]: G Σ ( s) = G sE ( s) ⋅ Gei ( s) ⋅ Gmω ( s) = =

1 1 1 1 ≈ , 1 + T 2 s 1 + Tei s 1 + T 2 s 1 + TΣ s

(3)

α

position controller

; α

8

ω

Kα

Approximated current control loop

speed controller

: ω

9

T TI

m1Rd

z z-1

GE (s)

1-e-sT s

m1R

Ms

friction

mf

Gei (s)

ma

1 1+Teis m 1

ZOH

Mc

1 TM s

m2

ω

α

1 TB s

T

Km Kω

torque feedback

T mam

mae Gma(z)

ω

=

α




Aα ( s) = TI T M TΣ TB s 4 +

(5)

+ TI T M TB (1 + K m ) s 3 + TI TB Kω s 2 + TB s + Kα .

The transfer function of the equivalent continuoustime closed-loop speed control system derived from Fig. 1 (with inserting Km=0) is: Gω ( s) =

ω ( s) 1 = ω R ( s) Aω ( s)

?

Aω ( s) = TI T M TΣ s 3 +

(6)

+ TI T M (1 + K m ) s + TI Kω s + 1. 2

The controller is designed according to the double ratios optimum (damping optimum) [9] with the aim of achieving a desirable critical damped position response or quasiaperiodic speed response. The optimization of the control system is done by equating the characteristic polynomial coefficients of the equivalent continuous-time speed or position control system with the corresponding coefficients of the polynomial of the damping optimum of the third or fourth order, respectively [5]: A( s) = 1 + Te s + + D3 D22 Te3 s 3 +

D2 Te2 s 2 + D4 D32 D23 Te4 s 4 ,

(7)

where: D2=D3=D4=0.5, except D2=0.38 for the position control system in order to achieve critical damped position response (without overshoot) [11].

The obtained equations for the position and speed controller parameters are given in Table I. The reduced-order state controller is obtained as a result of the decreased number of adjustable controller parameters (TI, Kω for speed control system, and TI, Kω, Kα for position control system). Consequently, as a result of optimization, the controller design yields the equation for equivalent speed or position control time constant Teω or Teα, respectively [11]. On the other hand, the controller design with acceleration torque feedback allows free choice of Teω or Teα (Table I). B. PI speed controller with acceleration torque feedback The control system behavior is improved by introducing acceleration torque ma feedback (Fig. 1). The method of controller design is analogous to the control system without acceleration torque feedback (section I.A). Because of the increased number of adjustable controller parameters, (Km is additional parameter), the control structure with acceleration feedback is in fact a full-order state controller, i.e. all states are controlled. Consequently, it is possible to choose the equivalent speed or position control time constant Teω or Teα (Table I) which are constrained by the noise influence [11]. The assumption in controller design is that the acceleration torque measurement is ideal, which is unrealizable on the observed experimental servosystem, so that it is replaced by acceleration estimation. The simplest way of acceleration estimation is to differentiate the speed as follows (Fig. 1): Gma ( z) =

mae ( z )

ω m ( z)

=

TM z − 1 , T z

(8)

where: mae - estimated acceleration torque used instead of measured mam , TM - mechanical time constant.

TABLE I. EQUATIONS FOR OPTIMAL PARAMETERS OF TWO CONTROLLER STRUCTURES.

Speed control

Position control

without acceleration torque feedback

with acceleration torque feedback

without acceleration torque feedback

with acceleration torque feedback

Te

TΣ D3 D2

-

TΣ D4 D3 D2

-

TI

Te2ω D2 TM

D3 D22 Te3ω T M TΣ

D3 D22 Te2α TM

D4 D32 D23 Te3α TM TΣ

Teω TI

Teω TI

D2 Teα TI

D2 Teα TI

Kω Km Kα

-

TΣ

-

−1

D3 D2 Teω

-

-

TM TM s z −1 , (9) ⇒ Gma ( z ) = TF 1 z − e −T TF 1 1 + TF 1 s

where the filter time constant TF1 must be included in the sum of small time constants (4). Because the speed signal is also reconstructed by differentiation of the measured position signal, it is possible to implement the filtered double derivative element derived from continuos-time element: Gma ′ ( s) =

mae ( s) T M TB s 2 = α m ( s) 1 + TF 2 s + 0,5TF22 s 2

.

(10)

A different solution for the problem of acceleration estimation is the use of Luenberger full-order observer with an integral term ensuring steady-state accuracy of the observer [13]. The observer parameters can be adjusted in accordance with the double ratios optimum procedure [11]. C. Disturbance observer based friction compensation Another approach to friction compensation is based on the disturbance estimator [8], shown in Fig. 2. It estimates not only friction mf but the entire disturbance torque m2* which includes the load torque m2 and the parasitic torques, the self-inertia variation, the gravitation [8], the cogging, the ripple and the magnetic hysteresis drag [3]. In the servodrive with negligible transmission elasticity the disturbance torque estimation is based on the motor torque and speed signal as follows (Fig. 2):

m2 *( s) = m1 − sTM ω .

(11)

In order to avoid the derivative element implementation problems, the estimated disturbance torque m2e is obtained by filtering m2* from (11):

−1

D4 D3 D2 Teα

TB Teα

The transfer function Gma(z) can also be derived by ZOH-discretization of the realizable derivative element: Gma ( s) =

TΣ

TB Teα

TM TM ω+ ω TF TF = 1 + TF s

m1R − sT M ω − m2 e ( s) =

(12)

TM ω TF TM ω . − TF 1 + TF s

m1R + =

Replacement of motor torque m1 in (11) by torque reference m1R in (12) is assumed because it is provided Tei /T