worse, limit cycles whereby the system oscillates, often in an irregular fashion,
with a ..... For an electrical DC-motor, Tm = kii where ki (Nm=A) is the known or ...
Automatica 38 (2002) 1633 – 1649
www.elsevier.com/locate/automatica
Survey Paper
Controlling mechanical systems with backlash—a survey Mattias Nordina; ∗ , Per-Olof Gutmanb b Faculty
a Rolling Mills Department, ABB Process Industries, 721 67 V Sweden asteras, of Agricultural Engineering, Technion—Israel Institute of Technology, Haifa 32000, Israel
Received 23 February 2000; received in revised form 29 January 2002; accepted 4 March 2002
Abstract Backlash is one of the most important non-linearities that limit the performance of speed and position control in industrial, robotics, automotive, automation and other applications. The control of systems with backlash has been the subject of study since the 1940s. This survey reveals that surprisingly few control innovations have been presented since the early path breaking papers that introduced the describing function analysis of systems with backlash. Promising developments are however taking place using adaptive and non-linear control strategies. ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Backlash compensation; Mechanical systems; Motor control; Describing functions; Non-linear control
0. Introduction The de9nition of backlash relevant for this survey paper is “the play between adjacent movable parts (as in a series of gears)” and is found in a Webster dictionary. A simple schetch of a backlash is found in Fig. 5. Backlash is present in every mechanical system where a driving member (motor) is not directly connected with the driven member (load). This is the case in many, if not most driven mechanical systems, notably those with gears, e.g. the drive train in cars, rolling mills, printing presses, and industrial robots. It is immediately clear that control of a load behind a backlash is complicated in particular if high precision is desired. There are instances when the backlash gap opens, and the motor loses contact with the load. This may happen when a disturbance acts on the load, or when the motor has to take corrective action in the opposite direction to where the load is moving or is positioned at the moment. When the backlash gap is open, the movement of the load is autonomous, and in addition, the force or moment generated by the motor drives only the motor itself (and the parts of the transmission before the backlash) and not the load. One might claim that in those instances the load is “uncontrollable” This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Editor Manfred Morari. ∗ Corresponding author. Tel.: +46-21-340527; fax: +46-21-181724. E-mail addresses:
[email protected] (M. Nordin),
[email protected] (P.-O. Gutman).
and that the controlled dynamics is diDerent. Controlled systems with backlash often exhibit steady-state errors or, even worse, limit cycles whereby the system oscillates, often in an irregular fashion, with a peak–peak amplitude that may exceed the total size of the backlash gap. The control of systems with backlash has been the subject of study since 1940s. Linear controllers have been investigated, including P-, PI-, and PID-controllers, high-order linear controllers, state feedback controllers, and observer based controllers. The main analytical tool to describe the backlash has been the describing function technique. The preload, i.e. an approximate inverse of the backlash, has often been suggested as a backlash remedy, both in a non-adaptive and an adaptive setting. Other non-linear controllers have also been proposed, including fuzzy controllers and switched linear controllers. This survey is the result of extensive computer searches and contains 96 references. Of course there are more out there but we hope that we have not missed any references that present essentially diDerent ideas. The paper is organized as follows: In Section 1 the control of elastic systems without backlash is brieGy described i.e. systems where the motor and load are connected with a shaft or axle without backlash. Although not the main focus of the survey, some of the well known control algorithms for such systems are reviewed critically since they form the basis around which backlash compensation is built. Section 2 contains a review of backlash models with a discussion of their usefulness for control design purposes.
0005-1098/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 5 - 1 0 9 8 ( 0 2 ) 0 0 0 4 7 - X
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M. Nordin, P.-O. Gutman / Automatica 38 (2002) 1633–1649
Internal backlash is covered in Sections 3 and 4 where speed control and postion control are treated, respectively, while systems with backlash at the input and output are found in Section 5. Where appropriate, Sections 3 and 4 are divided into subsections according to where from the feedback signal is sensed: from the motor side, from the load side, or from both, and according to what type of controllers are used: linear or non-linear. A few simulated design examples are included in Section 3.2.1 for the industrially interesting case of linear speed feedback from the motor side. For other con9gurations we refer to simulations and experiments in the surveyed papers. A short conclusion in Section 6 summarizes the survey.
1. Control of elastic systems Many motion control systems of practical interest may be modeled as a multimass system with the masses connected with Gexible shafts or springs, see e.g. Hori, Iseki, and Sigiura (1994). In many cases the modeling is further simpli9ed by considering a two-mass system where the 9rst mass represents the motor, the second mass represents the load, and the shaft is considered mass or inertia free. Such a K om & Witsystem serves as a model e.g. for a robot arm (AstrL tenmark, 1989; Brandenburg, Hertle, & Zeiselmair, 1987; Ahmad, 1985), or a rolling mill drive (Pollman, Tosetto, & Brea, 1991; Dhaouadi, Kubo, & Tobise, 1993). The realism of the two-mass model can be increased by assuming that some parameters are uncertain. We therefore select the following linear model of an uncertain two-mass system as the standard plant of this paper. For simplicity we assume that the gear ratio is one. In Section 2 below, the linear model will be complemented with backlash, while non-linear friction will in general be neglected as not belonging to the main topic of this paper. The surveyed control laws will be discussed in relation to this standard plant. Variations of the model can be found in e.g. Oldak, Baril, and Gutman (1994) or Nordin and Gutman (1995). Jm !˙ m = −cm !m − Ts + Tm ; Jl !˙ l = −cl !l + Ts − Td ;
(1)
!d = !m − !l with ˙m = !m ;
˙l = !l ;
2
k s cs ωm
Tm
where Jm (kg m ) is the motor moment of inertia, cm (Nm=(rad=s)) is the viscous motor friction, Ts (Nm) is the transmitted shaft torque, Tm (Nm) is the motor torque,
Td
Fig. 1. A schematic diagram of the linear two-mass system (1), (2).
Tm
+
+ 1 Jm s + c m m
−
−
cs s +ks s
Ts
+ −
l Jl s + c l
l
Td
Fig. 2. A block diagram for the linear two-mass system (1), (2).
Jl (kg m2 ) is the load moment of inertia, cl (Nm=(rad=s)) is the viscous load friction, Td (Nm) is the load torque disturbance, ks (Nm=rad) is the shaft elasticity, and cs (Nm=(rad=s)) is the inner damping coeScient of the shaft. Note that the shaft damping cannot be incorporated with the viscous motor or load friction, since in general !d = !l = !m . The angles m , l , d are the motor angle, load angle, and diDerence angle (rad), respectively, while !m , !l , !d are their respective time derivatives; the motor angular velocity the load angular velocity, and the diDerence angular velocity (rad=s). Notice that for the linear system (1), (2) the diDerence angle equals the shaft angle. In some examples below we will use the numerical values Jm = 0:4 kg m2 ; cm ∈ [0; 0:1] Nm=(rad=s); Jl ∈ [5:5; 6:0] kg m2 ; cl ∈ [0; 1] Nm=(rad=s);
(3)
ks ∈ [3000; 4000] Nm=rad; cs ∈ [1; 20] Nm=(rad=s): The nominal case is de9ned by the parameter combination cm = 0:1, Jl = 5:6, cl = 1, ks = 3300, and cs = 1. A schematic diagram of the system (1), (2) is found in Fig. 1. Its block diagram is found in Fig. 2. For the plant (1), (2), let Pum (s) denote the open loop, uncompensated transfer function from Tm to !m , Pul (s) the transfer function from Tm to !l , Pdm (s) the transfer function from Td to !m , and Pdl (s) the transfer function from Td to !l . Eq. (1) gives Jl s2 + (cl + cs )s + ks ; d(s) cs s + ks Pul (s) = ; d(s)
(2)
ωl
Ts
Pum (s) =
˙d = !d
and Ts = k s d + c s !d ;
Jl
Jm
Pdm (s) = − Pdl (s) = −
cs s + ks ; d(s)
Jm s2 + (cm + cs )s + ks d(s)
M. Nordin, P.-O. Gutman / Automatica 38 (2002) 1633–1649
while the anti-resonance of Pum ( j!) at the lower frequency ks !2 = (rad=s) (5) Jl
[dB]
0
−50
−100
1
2
10
10 [rad/sec]
Phase [degree]
1635
0
−200 −400 1
2
10
10 freq [rad/sec]
Fig. 3. Bode diagrams for the frequency functions Pum ( j!) for 1024 cases of the two-mass system (1), (2), (3) with actuator dynamics modelled as a 10 ms time delay.
has no correspondance for Pul ( j!). This fact implies that the control design for a high-bandwidth closed-loop system will depend on whether the position or velocity sensor is located on the motor side or on the load side, see Brandenburg and Kaiser (1995), Pettersson (1997). Another important issue is the actuator dynamics, i.e. how the motor torque is produced. Without loss of generality we assume that the torque actuator has limited bandwidth, 1= e , where e (s) is its time constant. It is further assumed that, due to delays, saturations, etc., it is impossible to completely cancel the actuator dynamics by inverse regulator dynamics, so that arbitrary open-loop cross-over frequency is not achievable. In our designs and simulations below, we will let the actuator dynamics be e−s e =2 =(1 + s e ) which is consistent with the assumptions. In such a way unrealistic control solutions are avoided. See also the discussion in Section 1.1. 1.1. PI-control
Fig. 4. Bode diagrams for the frequency functions Pul ( j!) for 1024 cases of the two-mass system (1), (2), (3) with actuator dynamics modelled as a 10 ms time delay.
with d(s) = Jm Jl s3 + (Jl (cm + cs ) + Jm (cm + cs ))s2 + ((Jl + Jm )ks + cm cl + cm cs + cl cs )s + (cm + cl )ks By gridding each uncertain parameter interval in (1) – (3) equidistantly into three sub-intervals, and selecting all parameter combinations, 1024 plant cases are de9ned. The frequency functions Pum ( j!), where ! (rad=s) is the frequency variable, for these 1024 plant cases are displayed in the Bode diagram in Fig. 3. The Bode diagrams of Pul (s) for the same cases are found in Fig. 4. The Bode diagrams reveal that Pum ( j!) and Pul ( j!) have a common resonance peak at ks (Jm + Jl ) !1 = (rad=s); (4) Jm J l
The most common controller for elastic systems, and in particular two-mass systems, is the continuous time or sampled PI-regulator, or some variation thereof, see e.g. Koyama and Yano (1991), Brandenburg and SchLafer (1989), Ji and Sul (1995), Joos and Sicard (1992), de Santis (1994), Pollman et al. (1991). PI-control suDers from severe performance limitations. In Brandenburg and SchLafer (1990), SchLafer and Brandenburg (1991), ShLafer (1993), it is stated that for so called “stiD” systems where the time constant, e (s), of the torque actuator, or in this case the current control loop, is long, e ¿ 1:5=!1 , and hence the actuator is too slow to place the poles of the resonance (4) at a position of signi9cantly better damping, incomplete state feedback, or a PI-regulator (Brandenburg, 1986), will achieve satisfactory damping. For “soft” systems, where the actuator is fast enough to damp the resonance ( e ¡ 1:5=!1 ), complete state feedback control is said to be preferable. Considering the case of feedback from !m , Koyama and Yano (1991) 9nds that for a high ratio between the load moment of inertia and the motor moment of inertia, Jl =Jm , it is diScult to get a satisfactory closed-loop bandwidth with a PI-regulator. On the opposite end, Hori et al. (1994) claims that when Jl =Jm is very low, the control problem is very diScult. Hence the control problem is easiest when Jm ≈ Jl . The latter result is corroborated in Baril and Galic (1994): with Jl =Jm 1 it is very diScult, even impossible, to get a well damped response with a linear speed controller when only !m is measured. Note that the measured motor speed amplitude of the oscillations may be very small although the load oscillates with a large amplitude. Instead a cascaded controller, with an additional feedback signal from the load speed, is shown to be able to damp the oscillations.
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M. Nordin, P.-O. Gutman / Automatica 38 (2002) 1633–1649
The report (Baril & Galic, 1994) also extends the above stated results for motor speed control by investigating the joint inGuence of the inertia ratio Jl =Jm and the quantity
d !1 , where d (s) stands for the delay introduced by the current control, speed sensor, and sampling, and hence corresponds to e in SchLafer and Brandenburg (1991). When
d !1 1 a PI-regulator will give satisfactory performance, and the achievable improvement with a higher-order controller is insigni9cant. For an increasing d !1 (but still with
d !1 ¡ 1), the stability margins begin to be unacceptable and it is necessary to use higher order controllers. The transition between PI and a high-order controller depends on the inertia ratio Jl =Jm , such that for high inertia ratios a high-order controller is mandatory, while for lower inertia ratios, a PI might still be acceptable. The results in Baril and Galic (1994), SchLafer and Brandenburg (1991), and Brandenburg (1986) are thus consistent with each other. In Brandenburg et al. (1987) the symmetric optimum is referred to as a suitable method to tune a PI-regulator for an elastic two-mass system. An analysis of PI-control for motor speed control of uncertain elastic two-mass systems with time delay is carried out in Galardini, Nordin, and Gutman (1997), with the following results: (i) If undamped oscillations are to be avoided, the achievable cross over frequency is approximately !2 =2; and (ii) the gains of the PI-regulator kP + kI =s should be tuned such that kI 6 ks (Jm + Jl )=(4Jl ) and Jm !1 ¡ kP 6 2 (kI (Jm + Jl )), i.e. the PI-regulator is given as a function of the plant parameters. It can also be shown that for the case when the load speed is measured, the cross over frequency is limited to ≈ !1 =2 with PI-control. Similarly, in Brandenburg and Kaiser (1995), the inGuence of Jl =Jm is investigated with feedback of !m and !l , respectively, using PI speed control. Additionally, a notch 9lter to compensate for the resonance peak is considered. It was found that feedback of !m facilitates a faster control than feedback of !l also when Jl Jm , though in that case the anti-resonance is shifted towards very low frequencies and thus con9nes the reachable crossover frequency of the open speed control loop. In the case of Jl Jm , weakly damped oscillations of both motor and load mass occur when applying a PI controller. It was found that a state feedback controller can govern this case. A notch 9lter—which is often applied in industrial plants—is suitable to improve the command step response of the motor speed but not of the load speed. The notch 9lter deteriorates considerable the disturbance step response: due to the pole cancellation the natural frequency of the system becomes uncontrollable. 1.2. Higher order linear control Many authors have suggested various solutions to overcome the limitations of PI-control with motor speed feedback. In Brandenburg (1986), and in Brandenburg’s subsequent works (Brandenburg, Unger, & Wagenpfeil, 1986; Brandenburg et al., 1987; Brandenburg & SchLafer,
1987; Brandenburg & SchLafer, 1989; SchLafer & Brandenburg, 1990; SchLafer & Brandenburg, 1991), a disturbance or load torque observer is proposed to estimate the torque transmitted through the shaft, Ts in (2). Let cˆm be an a priori estimate of cm in (1), and Tm the known or estimated motor torque. For an electrical DC-motor, Tm = ki i where ki (Nm=A) is the known or estimated motor torque constant and i (A) is the motor current. Then a Luenberger observer, giving the estimate Tˆ s (Nm) has the form Jˆm !ˆ˙ m = −cˆm !ˆ m − Tˆ s + Tm ; Tˆ s = ko (!m − !ˆ m );
(6)
where ko is the observer gain to be chosen such that a suitable observer bandwidth is achieved. With Tc being the control torque generated by the “ordinary” feedback regulator (e.g. the PI), the total torque applied to the motor, as a torque reference to the torque actuator, is chosen to be Tm = Tc + k1 Tˆ s . If the constant k1 equals to one, the shaft torque is completely compensated, and the motor would be entirely decoupled from the load and no resonance peak would appear. The load oscillations would however not be inGuenced by the motor speed control. For this reason, only a partial compensation of Ts is successful. In e.g. Brandenburg et al. (1986) k1 = 0:3 is proposed leading to a reference step response settling time reduction by 20% compared with a PI-regulator without shaft torque compensation. A “diDerential” shaft torque estimator, identical to (6) is also suggested by Brandenburg in Brandenburg et al. (1986). Among other researchers proposing shaft torque estimation, in many cases similarly or identically to Brandenburg, we have the authors of Baril and Galic (1994), Cohen and Cohen (1994), Dhaouadi, Kubo, and Tobise (1994), Ji and Sul (1995), Koyama and Yano (1991), Ohishi, Ohnishi, and Miyachi (1983), most of whom note that the load is unobservable from the motor in case of complete shaft torque compensation. Shaft torque compensation is useful also when backlash and other non-linearities are present. Clearly, the combination of PI-regulation and shaft torque estimation and compensation can be extended to include feedback from more measured or estimated states. In Brandenburg and SchLafer (1989), Koyama and Yano (1991), the PI-control with shaft torque compensation is extended with a 9xed parameter, linear model reference controller. State feedback from the states of a two-mass or three-mass model observer is proposed in SchLafer and Brandenburg (1991), Dhaouadi et al. (1993), Hori et al. (1994). The shaft torque compensation is complemented with state feedback computed by linear quadratic optimal control in Ohishi, Myooi, Ohnishi, and Miyaji (1986), Ohishi et al. (1983), and with pole placement computed with the Diophantine equation in Cohen and Cohen (1994). Multi-model pole placement is proposed in Grundmann (1995). LQG=LTR control is proposed in Pettersson (1997) and Gurian (1998). H∞ -designs are proposed in Nordin (1992).
M. Nordin, P.-O. Gutman / Automatica 38 (2002) 1633–1649
Frequency domain passband loop shaping is proposed in Chang, Ji, Shimanovich, and Caudill (1996). Full two degree-of-freedom loop shaping by quantitative feedback theory, QFT (Horowitz, 1992), yielding a robust 9xed-parameter linear controller is used in Boneh and Yaniv (1999), Yaniv and Horowitz (1990), Nordin and Gutman (1996), Kidron and Yaniv (1996). In e.g. Yaniv and Horowitz (1990), Nordin and Gutman (1996) it is demonstrated that a non-minimum phase controller gives the required robustness. The Third European Control Conference in Rome 1995 (ECC 95) sported a session organized by I.D. Landau, entitled “Design of Robust Digital Controllers—A Flexible Transmission Benchmark” where the load position of an uncertain three-mass system, connected with springs was to be controlled by the competing designs. The results are reported in the European Journal of Control, no. 2, 1996, see Landau, Rey, Karimi, Voda, and Franco (1996). Both the winning entry (Nordin & Gutman, 1996), and the runner up (Kidron & Yaniv, 1996) used QFT. Loop shaping was also used by Oustaloup, Mathieu, and Lanusse (1996), while Landau, Karimi, Voda, and Franco (1996) combined pole placement and sensitivity function shaping. H∞ -control was proposed by Jones and Limebeer (1996). The other entries with linear designs include Hjalmarsson, Gunnarsson, and Gevers (1996), Decker, Ehrlinger, Boucher, and Dumur (1996).
1637
2α Jm
Tm
ωm
Jl k s cs
Ts
ωl
Td
Fig. 5. Backlash or gear play in a two-mass system. s (rad) is the twist of the shaft (not explicitly shown in the 9gure), d = m − l (rad) is the displacement angle between the motor angle m and the load angle l (1), and b = d − s (rad) is the backlash angle, (10), with the restriction that − 6 b 6 . ks , cs are the elasticity and damping coeScients, respectively, of the shaft, giving the shaft torque Ts (Nm), see (9).
during adaptation, while after adaptation, the gain, and hence the bandwidth, of the adaptive system is lower than the gain of the robust linear system. The choice between adaptive and robust linear control will then depend on the amount of uncertainty and on the weight that is placed on transient behavior versus noise sensitivity during steady-state operation. The scale shifts in favor of the inclusion of adaptation when backlash is present, which can be understood by the fact that from a linear point of view, backlash introduces more uncertainty into the linearized model of the plant. For equal performance, more uncertainty demands either a higher feedback gain which may be unachievable due to stability constraints, or adaptive or some other non-linear control, see Horowitz (1992), Horowitz (1963).
1.3. Towards backlash compensation: non-linear and adaptive control
2. Backlash models
Several authors have proposed non-linear or adaptive control of elastic plants similar to (1), (2), also in the absence of backlash. In Pan and Furuta (1996) a variable structure control law is suggested for the ECC 95 benchmark problem mentioned above. Variable Structure control is also used by Harashima, Ueshiba, and Hashimoto (1987) to control a multi-joint manipulator, while adaptive variable structure control is proposed by Dumitrache, Dumitriu, and Monteano (1987) for the speed control of a DC-motor. In Yeh, Fu, and Yang (1996), an adaptive computed torque control is proposed for a gun-turret system. By far the most common non-linear alternative is adaptive model reference control, proposed in Hahn and Unbehauen (1987), SchLonfeld (1987). Also in Recker et al. (1991), Tao and Kokotovic (1993), Tao and Kokotovic (1996), model reference adaptive control in the backlash free case forms the basis of the control algorithms when backlash is present. K om and WittenAdaptive pole placement is suggested in AstrL mark (1989), Han and de la Sen (1996), Ji and Sul (1995). See also Ma and Tao (1999), Tao (2000). K om and Wittenmark (1989, p. 278), there is a In the AstrL comparison between adaptive pole placement control for a system similar to (1), (2) and linear robust control designed with QFT by the second author of this paper. One may conclude that adaptive control seems to yield a bad transient
Backlash or gear play is a common non-linearity in mechanical systems. Depending on the mechanical surrounding of the backlash, and the operating conditions, diDerent mathematical models must be utilized to model the behavior. In this section, we concentrate on the mechanical model depicted in Fig. 5. This is the two-mass model described in Section 1, where the shaft is replaced by a shaft including backlash, see Fig. 5. The maximal backlash angle is denoted (rad). Eq. (1) remains the same, but Eq. (2) modeling the shaft torque, as a function of d and !d needs to be modi9ed. In Section 2.1, we present the commonly used dead zone model, valid only when the shaft damping cs equals zero. In Section 2.2 models valid for non-zero shaft damping which require dynamic equations for the shaft, are presented. In Section 2.3 describing functions for the above models are computed. A more thorough expos\e of these models is found in Nordin, Galic, and Gutman (1997), Nordin (2000). The backlash model in Taylor and Lu (1995) is equivalent to the model of backlash with an inertia-free stiD shaft in Nordin et al. (1997). In Section 2.4, the hysteresis model is presented for completeness. Here the shaft is assumed stiD, while the input of the backlash model is the motor position m , and the output is the load position l .
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M. Nordin, P.-O. Gutman / Automatica 38 (2002) 1633–1649
2.1. The dead zone models An often proposed model, found in almost any basic control course, for analysis and simulations of an elastic shaft with backlash is the classical dead zone model, see e.g. Tustin (1947), Liversidge (1952), Gelb and Vander Velde (1968), CosgriD (1958), Chestnut and Mayer (1995), where the shaft torque, being proportional to the shaft twist s (rad), is given by Ts = ks s = ks D (d ):
(7)
The dead zone function x − x ¿ ; |x| ¡ ; D (x) = 0 x+ x¡−
(8)
gives s directly as a static function of the displacement d . Note that without backlash, s =d . For this dead zone model to be valid it is essential that the shaft damping cs is zero, i.e. the shaft is modeled as a pure spring. The dead zone model is a static, scalar non-linear function, which means that it is relatively easy to analyze. Such an analysis is given in Goodman (1963) with some simpli9cations for simulation purposes. 2.2. Exact models When there is damping in the shaft, i.e. cs = 0, the shaft torque is given by Ts = ks s + cs !s
(9)
With b and ˙b known from (11), and d (t); ˙d (t) given, the torque Ts is found by (9,10). We now have a non-linear dynamical system and not a static function that gives the torque Ts with given d and ˙d . It is possible to approximate this dynamic model with a static function of d and !d , which simpli9es analysis and simulation signi9cantly (Nordin, 1995; Nordin et al., 1997). In the same papers, models for coupling with the gap partly 9lled with rubber are introduced. 2.3. Describing functions A common method used for analysis and synthesis of non-linear systems is the describing function method, see e.g. Chestnut (1954), Gelb and Vander Velde (1968), Atherton (1975), Atherton (1981). The describing function model of the dead zone model for backlash is found already in Thomas (1954). As pointed out by Brandenburg in e.g. Brandenburg et al. (1986), Brandenburg and SchLafer (1987), Brandenburg and SchLafer (1989), a constant output torque, here denoted T0 , best describes the operating point for a shaft with backlash. Using this idea, 9rst calculate the dual-input describing function DIDF, where the input is a sine wave with a constant oDset B given as d (t) = B + A sin(!t)
(12)
and the output is approximated by a constant oDset NB B and the 9rst harmonic (see Gelb & Vander Velde, 1968 for a theoretical background): Ts (t) = Ts (d ; ˙d ) ≈ NB B + ANp sin(!t) + ANq cos(!t)
(13)
with !s = ˙s , see Nordin (1995), Nordin et al. (1997). In Nordin (1995), Nordin et al. (1997) it is shown that the widely used dead zone model (7), (8) is erroneous for the case cs = 0. De9ning the backlash angle
from which the DIDFs NA (A; B; !) = Np (A; B; !) + iNq (A; B; !) and NB (A; B; !) are calculated. Then consider the equation
b = d − s
T0 = BNB (A; B; !)
(10)
it is possible to obtain the dynamic equation (Nordin, 1995; Nordin et al., 1997) b = ks ˙ max 0; d + (d − b ) cs ks ˙d + (d − b ) cs k min 0; ˙d + s (d − b ) cs
if b = − (Ts 6 0) if |b | ¡ (Ts = 0) if b =
(Ts ¿ 0): (11)
This state equation can be interpreted as a limited integrator with the time derivative ˙d + k=c(d − b ) and limit .
(14)
with B as the independent variable. Since there exists a unique solution B∗ (A; T0 ; !) for (14), the system around a working point T0 can now be described by B∗ (A; T0 ; !) and NA (A; B∗ (A; T0 ; !); !) in a DIDF setting. Note that with T0 = 0 this reduces to the standard sinusoidal input DF (SIDF). Applying this procedure to the dead zone model and to the exact model we obtain the curves in Fig. 6. The curves display the describing functions of the shaft with backlash, divided by the transfer function of the shaft without backlash, in order to show the diDerence with respect to the linear case. Note here especially the extra phase lag caused by the non-linearity, and the fact that this is not properly modeled by the dead zone model. Hence the dead zone model should be used for control systems design only when the shaft damping is negligible.
M. Nordin, P.-O. Gutman / Automatica 38 (2002) 1633–1649
Bode plot of N (A,T ,ω)/(k +jc ω) A
s
s
(3)
0 magnitude [dB]
0
−10
(2)
−20 −30 −40 (1)
−50 −60
0
1
2 A/α
3
4
phase [dB]
0 (3)
−10 −20
(1) 0
The inertia driven and friction driven hysteresis models are described in e.g. CosgriD (1958), Gelb and Vander Velde (1968). Their describing functions are found in Gelb and Vander Velde (1968). Using (15), the transient response of a closed-loop system with a backlash element in the feedback path is analyzed in Kreer (1956) by replacing the backlash with a modi9ed forcing function. The hysteresis models do not account for damping in the shaft, like the dead zone model, but in addition, the shaft is assumed to be totally stiD, i.e. ks = ∞. In the friction driven hysteresis model, no knowledge of the friction itself is required. Since the hysteresis models are not natural extensions of the backlash free case (1), they have found less use for control design except when the backlash is found at the plant input or output, see Section 5, or when the design is based on the describing function methods, see Gelb and Vander Velde (1968). 3. Speed control of elastic systems with internal backlash
(2)
−30 −40
1639
1
2 A/α
3
4
Fig. 6. Bode plot of NA (A; T0 ; !)=(ks + j!cs ) and for ks = 3000 Nm=rad, cs = 20 Nm=(rad=s). 0 Hz is represented by bold, 5 Hz by solid and 15 Hz by dotted curve. (1): T0 = 0:001k, (2): T0 = 0:001k, (3): T0 = k. Note that the 0 Hz case is equivalent to the dead zone model case.
2.4. The hysteresis model As described in e.g. Tao and Kokotovic (1996), the hysteresis model relates the output angle of the backlash, l , to the input angle, m under the assumption of a stiD shaft. Under the assumption that the load torque disturbance equals zero, the hysteresis model is given by ˙m (t) if ˙m (t) ¿ 0 and l (t) = m − ; ˙l (t) = ˙m (t) if ˙m (t) ¡ 0 and l (t) = m + ; (15) 0 otherwise; where it has been assumed, as everywhere in this paper, that the gear ratio equals 1. Eq. (15) represents the so-called friction driven hysteresis model for backlash, i.e. the driven member retains its position when the backlash gap is not closed as if kept in place by strong friction. The alternative is the inertia driven hysteresis model, where the friction in the backlash gap is assumed zero, and the driven member continues to move with constant velocity: ˙l (t) = ˙m (t) if ˙m (t) ¿ 0 and l (t) = m − ; ˙l (t) = ˙m (t)
if ˙m (t) ¡ 0
Ll (t) = 0
otherwise:
and
l (t) = m + ; (16)
Almost all the papers we found on speed control assume feedback from the motor side only. This is the most relevant case from an industrial point of view since often it is impossible or very expensive to measure the velocity of the environmentally harsh load side. Examples of such systems where high precision speed control is mandatory include rolling mills, and paper machines. It is however somewhat surprising that feedback from the load side for speed controlled systems seems to be very scantily treated in the literature (Brandenburg, Geissenberger, Kink, Schall, & Schramm, 1999) being an exception, since in many important systems there are sensors on the load side: the wheel speed in trucks and cars is measured, or the end eDector in robots may contain several sensors. The lack of interest might be due to the fact that high precision speed control is not of relevance for many of these systems, but also, following (Brandenburg & Kaiser, 1995; ShLafer, 1993), due to the fundamental diSculties to speed control elastic systems with backlash from the load side. Many of the designs reported in Section 1, in particular the linear ones, e.g. those by Brandenburg and co-authors, are also suggested to be used when the plant includes backlash. Sometimes it is suggested that the design be retuned, or modi9ed, to accommodate the non-linearity. 3.1. Control based on describing functions The most common way to integrate a non-linear element into a linear control design problem is to describe the non-linearity by its describing function, N , see Section 2.3, and then designing the linear part of the open loop such that its frequency function does not intersect −1=N in a Nyquist or Nichols chart, see e.g. Gelb and Vander Velde (1968), Slotine and Li (1991). The describing function method gives at most approximate conditions for limit cycle avoidance.
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2α Jl
Jm C
k s cs
Tm ωm
G
− +
Ts
ωl
Td
ωref
Fig. 7. Linear feedback from motor speed. C denotes the actuator dynamics, and G the linear feedback control.
Often the non-linearity is described by a set of describing functions, e.g. when the describing function depends on the operating point, or, as in Brandenburg (1986), on an unknown load torque disturbance, Td in (1), or if the non-linearity itself is uncertain. Another case of multiple describing functions is when the non-linear element includes dynamics, such as the exact backlash model in Section 2.2. Then the describing function becomes frequency dependent, see e.g. Chestnut (1954), Slotine and Li (1991). The design method remains the same: the linear part of the open loop must not intersect the negative inverse of any one of the describing functions. When working with uncertain plants described by frequency dependent value sets like in Oldak et al. (1994), or with Taylor’s method (Taylor & Lu, 1995), the describing function (or set of describing functions) may be integrated into the value sets, giving rise to enlarged open-loop value sets, see e.g. Yang (1992), Oldak et al. (1994). The necessary design criterion for stability is then that the sequence of value sets encircle −1 according to the Nyquist criterion. If some circle or Popov criterion is used instead of the describing function, a suScient condition for asymptotic stability is achieved (Slotine & Li, 1991). These alternatives often give conservative designs (Brandenburg & SchLafer, 1987). 3.2. Linear feedback from the motor side Linear feedback from the measured motor speed, see Fig. 7, is the most common control con9guration. C denotes the actuator dynamics, and G the linear feedback control. In Brandenburg (1986), Brandenburg et al. (1986) the describing function of the dead zone backlash model is used to ascertain that PI-control of (1) with backlash may cause limit cycles whose amplitudes depend on the size of the load torque disturbance Td . A supplementary shaft torque observer (6) with partial torque compensation may make the closed loop system asymptotically stable with a transient response similar to that of a linear system. In Dhaouadi et al. (1994) a three mass system is considered: motor, gear, and load, with a backlash between the motor and the gear. The backlash is modeled as a dead zone. The controller contains an integral term, partial feedback from a gear torque observer, (6), and state feedback from a
two-mass-model reduced order observer that estimates the second shaft torque, the load torque disturbance, and the load angular velocity. The reduced order observer and state feedback gains are tuned by pole placement. The backlash is replaced by an uncertain gain, 0 6 N 6 1, which is equivalent to using the describing function of a shaft modeled as a pure spring, integrated into the frequency function of the linear part of the open loop, see Nordin et al. (1997). Closed-loop stability is investigated by drawing the root locus of the closed-loop poles with respect to N , and by drawing Bode plots of the open compensated loop for several N -values. Experiments show that feedback from the gear torque observer suppresses the high frequency limit cycle that appears otherwise, at the price of a higher overshoot and longer settling time of a step response on the load side. In Nordin (2000) it is demonstrated how to use QFT for uncertain systems with backlash, described by a static function of d and !d mentioned in Section 2.2. In Hori et al. (1994) a simulation study shows that PI-control of a two-mass, or three-mass system with each backlash between the masses modeled as a dead zone is accompanied by limit cycles on the load side. If the PI is complemented with feedback from observed states, the limit cycle is eliminated. In the simulation study (Cohen & Cohen, 1994) a worm gear is considered in place of the backlash element, modeled such that Ts in the 9rst equation of (1), is multiplied by a gear ratio that depends on the direction of motion. The controller includes a shaft torque compensator, (6), together with a pole placement design using the Diophantine equation K om & Wittenmark, 1984). (AstrL 3.2.1. Design examples Here we compare three linear designs for the two-mass system described in Sections 1 and 2, see Fig. 7, with a ◦ backlash gap of 1 . The actuator dynamics include a time delay of 4 ms and a lowpass 9lter with time constant 6 ms (and are hence non-minimum phase), in order to have realistic bandwidth limitations. All of the designs are based on describing function stability, and are tuned to have the same gain margin of at least 6 dB, for all plant cases. The speed overshoots for a load torque disturbance step are similar with a slightly lower value for the QFT-design, the latter enabled by QFT’s greater loop shaping design freedom. The three designs are: 1. PI-controller. In Fig. 8 a simulation of a PI-design is shown. Using G = Kp (1 + 1=sTi ) with Kp = 38 and Ti = 0:35 s, the gain margin is 6 dB. The integral part was tuned to have an overshoot of 1 rpm. 2. PI-controller with load observer. In Fig. 9 a simulation of a Brandenburg observer design is shown, see Eq. (6), in combination with a PI-controller. The designed parameter values are kp = 28, Ti = 0:18 s, k0 = 40, k1 = 0:25.
M. Nordin, P.-O. Gutman / Automatica 38 (2002) 1633–1649
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Load torque step response with PI−control
Motor speed (dashed), Load speed (solid) [ rpm]
120 115 110 105 100 95 90 85 80 0
500
1000
1500
2000
2500
3000
3500
4000
0
500
1000
1500
2000
2500
3000
3500
4000
Shaft torque (dashed), Motor torque (solid) [ kNm]
0.15
0.1
0.05
0
−0.05 time [ ms] Fig. 8. Simulation of load torque step response with PI-controller.
3. QFT-design. In Fig. 10 a simulation of a QFT-design is shown with G = Kp (1 + 1=sTi )(1=!02 + 2 0 =!0 + 1)=(1=!p2 + 2 p =!p + 1) using Kp = 84, Ti = 0:228 s, !0 = 211:7, 0 = 0:44, !p = 28:06, p = 1.
the nominal plant case, but the variation is small over all plant cases, indicating the robustness of all three design methods.
As seen in the 9gures, it seems as if the load observer in our example only gives a minor diDerence in performance, compared with the standard PI-controller, although it is one of the most commonly proposed solutions, see Brandenburg et al. (1999) for a successful application. In our example the QFT-design has superior performance, in terms of recovery time and maximum speed deviation. The simulations show
3.3. Adaptive and non-linear control In Brandenburg and SchLafer (1989), Brandenburg complements his previous works by considering systems with backlash and friction, claiming that model reference adaptive control with adaptation of the disturbance model is of great bene9t both for speed and position control: a PI
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Motor speed (dashed), Load speed (solid) [ rpm ]
120 115 110 105 100 95 90 85 80 0
500
1000
1500
0
500
1000
1500
2000
2500
3000
3500
4000
2500
3000
3500
4000
Shaft torque (dashed), Motor torque (solid) [ kNm ]
0.15
0.1
0.05
0
−0.05
2000 time [ ms ]
Fig. 9. Simulation of load torque step response with PI-controller and load observer.
disturbance controller, active only when a disturbance occurs, compensates for steady-state speed and position errors on the motor side, in such a way that it suSces to use a linear speed controller of P rather than of PI-type, thus reducing the order of the closed-loop system and achieving a faster response. Similar ideas to those of Johansson and Rantzer (1997), Johansson and Rantzer (1998), Johansson (1999), Indri and Tornamb]e (1997) are used in Nordin (2000) to prove set convergence when viscous friction, shaft damping, and actuator dynamics are present, with a deadzone backlash model, and a switched high-order controller. By using smart switching between two linear controllers, one tuned optimally for the system without backlash, and one tuned for robust performance for the system with backlash, almost the same performance as
for the system without backlash can be achieved (Nordin, 2000; Nordin & Gutman, 2000). A similar solution is suggested in ShLafer (1993) where two linear observers are used for the system with open and closed backlash, respectively, and switched according to the state of the backlash element. Two non-linear schemes are also proposed in ShLafer (1993): if only a motor sensor is available, a compensation of the steady-state error due to the backlash is recommended using a simple hysteresis element in the controller. If there is a second sensor at the load side, a solution is suggested in which, at the instant of the backlash opening, a trajectory is predicted which ensures that the backlash closes smoothly, and a limit cycle free operation results. Dynamic errors due to backlash are reduced. This method requires utmost fast calculations and a high sam-
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Load torque step response with QFT controller
Motor speed (dashed), Load speed (solid) [ rpm ]
120 115 110 105 100 95 90 85 80 0
500
1000
1500
0
500
1000
1500
2000
2500
3000
3500
4000
2000 2500 time [ ms ]
3000
3500
4000
Shaft torque (dashed), Motor torque (solid) [ kNm ]
0.15
0.1
0.05
0
−0.05
Fig. 10. Simulation of load torque step response with QFT-controller.
pling frequency if the backlash angle is small (SchLafer & Brandenburg, 1991). 4. Position control of elastic systems with internal backlash A mechanical solution to the backlash problem in high precision position control systems is the so called anti-backlash gear that contains two cog wheels on the motor side connected with a stiD spring such that the backlash gap is always closed. The price for this solution is the appearance of an additional resonance that limits the achievable closed-loop bandwidth.
A very useful idea is presented in Choi, Nakamura and Kobayashi (1996) to replace the mechanical anti-backlash gear: let the gear be driven by two motors with preloads of opposite signs such that the backlash gap is closed. The price for this solution is increased energy usage in steady state. Experiments show that the limit cycle disappears and that the steady state position error is zero. 4.1. Linear control 4.1.1. Feedback from the motor side One of the earliest papers using the describing function technique is (Chestnut, 1954). A generalized describing function is used for the deadzone model of backlash together
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with the equations of a two-mass system with position feedback either from the motor or from the load. Conditions for limit cycles are clearly analyzed. Similarly to the speed controlled systems treated in Brandenburg (1986), Brandenburg et al. (1986), Brandenburg and SchLafer (1987) shows that a load observer (6) together with a cascaded motor position PI- or P-control with inner motor speed P-control, respectively, of (1) avoids high-frequency limit cycles. Describing function analysis, using the dead zone model (7) reveals that in the no-load disturbance case Td = 0 there might still appear low-frequency low-amplitude limit cycles. These were however never discovered neither in simulations nor in experiments, and, in any case will be “destroyed” by any non-zero load disturbance that is always present in real systems, e.g. due to friction. It is also shown in Brandenburg and SchLafer (1987) that the circle criterion demands a much larger load disturbance than the describing function analysis as a suScient condition for limit cycle avoidance. The paper (Oldak et al., 1994) develops QFT for uncertain systems with backlash. The describing function of the backlash, in this case modeled as a dead zone, augments the plant value set. Moreover, the deviation of the backlash from a linear function is seen as a disturbance that gives rise to a disturbance rejection speci9cation. A simulated example demonstrates a closed loop system without limit cycles. In Ahmad (1985) the reference signal to the motor position PID controller is complemented with an oDset, called preload, to compensate the backlash gap such that the load position will presumably be correct in steady state. No dynamic eDects are considered. The backlash compensation is thus done in open loop, since there is no feedback from the load position. Under the assumption that the motor loop is fast, the backlash compensation is similar to inverse compensation of a backlash at the plant input, as proposed in Tao and Kokotovic (1996). For the model (1), (7) with cm = cl = cs = Td = 0, the reference angle ref = 0 and ideal PD-control Tm = −kP m − kV !m , i.e. with neglected actuator dynamics, Indri and Tor2 + Jl !l2 + namb]e (1997) uses the energy function V = Jm !m 2 2 ks D (d ) +kP m as a Lyapunov function candidate to prove that m converges to 0 and that |l | 6 asymptotically. It can be shown that for stability analysis PD position control is equivalent to PI speed control, see Nordin (2000) where stability for the speed controlled case is treated. 4.1.2. Feedback from the load side In Azenha and Tenreiro Machado (1996) 9rst- and second-order variable structure controllers (VSS) are tested with position feedback from the load side. In this case the variable structure controllers turn out to be saturated 9rst and second-order controllers. In the paper they are presented as non-proper, and it is not clear how they were
simulated. By describing function analysis of the VSS for these controllers, and for the inertia driven hysteresis model for the backlash, the existence of limit cycles is indicated, and the 9nding is corroborated by simulations. This comes as no surprise, since it is known that linear feedback from the load side results either in limit cycles or a steady-state error if the controller contains an integrator (Thomas, 1954; Oldak et al., 1994). 4.1.3. Feedback from the motor and load side, or state feedback In Freeman (1959), Freeman analyzes, using the generalized describing function method (Slotine & Li, 1991), a two-mass system with linear friction, modeled as a second-order system, with a stiD axis and backlash (modeled as a dead zone) and feedback from the load position and=or motor position or velocity. The analysis is compared with simulations on an analog computer. With the chosen model, limit cycles were found for all cases where the load position was measured. In Freeman (1957) Freeman analyzes the transients of his system, considering also non-linear friction. In Freeman (1960) he continues to propose measuring both the motor position, and the load position and apply a preload in form of an extra step signal at each backlash separation in order to close the backlash gap. An analog computer study shows that limit cycles are reduced if the preload is smaller than the backlash gap. In Boneh and Yaniv (1999) a cascaded controller is proposed with an inner high-bandwidth motor position loop, and an outer load position loop. Since both motor and load positions are measured, it is possible to know whether the backlash is active or not, if the backlash gap is known. Inverse compensation for the backlash is proposed, similar to Freeman (1960) and Tao and Kokotovic (1996). In simulations it is shown that the limit cycle amplitude decreases also when the backlash gap is underestimated. 4.2. Adaptive and non-linear control No works were found with adaptive or non-linear position control with feedback from the motor side only. 4.2.1. Feedback from the load side Fuzzy control is proposed in Lin, Yu and Xu (1996) with feedback from the load position error and its derivative. The backlash is modeled as a dead-zone and the linear part of the plant as a second-order system but the model is not explicitly used when setting up the fuzzy control rule. The result is a slightly non-linear discrete-time PD-controller with saturation. The controller was implemented on a laboratory test bed with a sampling period of 5 ms. From the presented 9gures it seems as if the output position converges to the reference with almost no error, and without limit
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cycling. The latter seems to be in conGict with the above stated observation in Thomas (1954), ShLafer (1993), Brandenburg and SchLafer (1987), Oldak et al. (1994) that linear feedback from the load side always causes limit cycles if the controller contains an integrator. However it seems as if the included non-linear friction in the experiment of Lin et al. (1996) kills the limit cycles. The authors discuss the tradeoD between friction and backlash in a control system. A general result of Brandenburg’s work, e.g. Brandenburg and SchLafer (1988a), Brandenburg and SchLafer (1988b), is that Coulomb friction with non-decreasing friction torque-to-speed characteristics stabilizes a system with backlash while backlash destabilizes a system with (or without) friction. In Taylor and Lu (1995), the authors apply Taylor’s SIDF synthesis method to a system with backlash. The drive subsystem of a simulation model of a gun-turret test bed includes an exact backlash model, see Section 2.2. The input of the non-linear drive subsystem is the motor torque, Tm and the output, l is the position of the driven gear after the backlash, as in (1), (9), (10), (11) with cm = cl = 0 and Td , representing non-linear friction. In addition there is non-linear friction acting on the load side. A set of equivalent linear frequency functions, or generalized sinusoidal input describing functions (SIDFs) (Slotine & Li, 1991), Pi ( j!) are identi9ed empirically by inserting sinusoidal inputs of various amplitudes and computing the frequency function values with the Fourier integral method. The feedback controller is chosen to be a non-linear PIDcontroller t Tm (t) = fP (lref (t)) + fI (lref (t)) dt 0
+
dfD (lref (t)) dt
(17)
where lref (t) is the load reference position signal, and fP , fI , and fD are non-linear functions, chosen piece-wise linear such that they interpolate the constant PID gains found for each plant case Pi , when making the frequency functions of the compensated open loops approximately equal to a desired open loop frequency function. The procedure is called SIDF inversion. The resulting PID-controller is almost linear with saturating P- and I-terms. The simulations results show well damped reference step responses without limit cycles which, we believe, are killed by the friction, as above in Lin et al. (1996). 4.2.2. Feedback from the motor and load side, or state feedback Friedland (Friedland, 1997) proposes to control the two mass system with backlash, modeled as a dead-zone with a very soft spring in the backlash gap and a stiD axis with SDARE, the state dependent algebraic Riccati equation. This leads to a two gain state feedback control law with
switching:
−gsoft x; d 6 ; u= −ghard x; d ¿ ;
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(18)
where x = [l ˙l d ˙d ]T . When only l and ˙l are available for measurement, Friedland proposes a reduced order observer to estimate d and ˙d , under the assumption that the parameters of the plant such as the backlash gap and the moments of inertia are well known. Simulations show that limit cycles disappear in steady state. It is important to note that the gains gsoft and ghard are such that the feedback from d and ˙d is considerably smaller when the backlash is active. This is in contrast with the inverse compensation of Tao and Kokotovic (1996) or the high bandwidth backlash closing inner loop proposed in Boneh and Yaniv (1999) where the control action increases when the backlash is active. 5. Control of systems with backlash at the plant input or output The central idea in Recker et al. (1991), Tao and Kokotovic (1993), in the book Tao and Kokotovic (1996), and in other papers by the same authors is the inverse compensation of non-linearities at the plant input or output, i.e. the non-linearity is compensated for by a control element in series that realizes the inverse of the non-linearity such that the net eDect is ideally a pure gain. The idea has been around for a long time under the name of preload, see e.g. Gelb and Vander Velde (1968). The novelty in Recker et al. (1991), Tao and Kokotovic (1993), Tao and Kokotovic (1996) is that when the type of non-linearity is known but its parameters unknown (e.g. the size of the backlash gap and the slopes in the hysteresis or dead-zone model) and=or some or all of the parameters of the linear part of the plant are unknown, adaptive inverse control is developed based on model reference adaptive control for linear plants. Thus, these works extend the preload idea to the case of dynamic backlash and hysteresis. A more extensive review of Tao and Kokotovic (1996) is found in Gutman (1998). For backlash at the plant input, inverse compensation implies that the backlash gap has to be traversed instantaneously— something that is impossible when the backlash is sandwiched between the motor and the load, as in our example in Section 3.2.1. Although not strictly applicable for a sandwiched con9guration, but still of interest for many real systems, a backlash inverter of the type proposed in Tao and Kokotovic (1996) was experimentally evaluated in Dean, Surgenor, and Iordanou (1995). It was found that a linear controller alone performed better than a controller including the selected backlash inverter with a correctly estimated or overestimated backlash gap, the reason being that measurement noise induced chattering in the inverter.
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It was noted that the linear controller alone (which can be said to represent a backlash inverter with an underestimated backlash gap) also traverses the backlash gap rapidly since only the motor moment of inertia (and not the load) is driven inside the backlash gap. The results in Recker et al. (1991), Tao and Kokotovic (1993), Tao and Kokotovic (1996) have been extended to the sandwiched case in Kokotovi\c, Ezal, and Tao (1997), Ling and Tao (1999), Tao and Ma (1999), and more research is in progress how to make the inverse compensation approach advantageous. In Han and de la Sen (1996) a MIMO plant with backlash at the plant input, (modeled with the discrete time hysteresis model, see Tao & Kokotovic, 1996) is proposed to be controlled with an adaptive controller that includes the adaptation of a preload whose initial value is over-estimating the backlash gap. Under some assumptions it is shown that the system is globally stable. A simulation of a SISO system is presented from which it is diScult to judge to what extent limit cycles are avoided. Dither is a high frequency signal, added to the output of the feedback controller that eDectively linearizes certain non-linear elements. Using the hysteresis model for backlash at the plant input (Desoer & Sharuz, 1986) gives conditions for dither frequency and amplitude to stabilize the position controlled closed loop system. 6. Conclusions The survey reveals that surprisingly few control innovations have been presented since the early path breaking papers. The most common design methods seem to be based on describing functions (Chestnut, 1954; Thomas, 1954; Gelb & Vander Velde, 1968; Brandenburg et al., 1987; Oldak et al., 1994; Slotine & Li, 1991; Nordin & Gutman, 1996). The survey also reveals that there are essentially two main non-linear backlash compensation practices. The 9rst is “strong action in the backlash gap” (Azenha & Tenreiro Machado, 1996; Boneh & Yaniv, 1994; Boneh & Yaniv, 1999), the idea of which is that the backlash gap be should traversed quickly, so that regular, engaged control action can be resumed. This method includes inverse compensation (Recker et al., 1991; Tao & Kokotovic, 1993; Dean et al., 1995; Tao & Kokotovic, 1996; Gutman, 1998) and preload (Freeman, 1960; Gelb & Vander Velde, 1968; Ahmad, 1985; Han & de la Sen, 1996), The other practice is “weak action in the backlash gap” implying that the bandwidth is lowered when the gap is open. Works in this category include switched control (ShLafer, 1993; Nordin, 2000; Nordin & Gutman, 2000), Taylor’s SIDF method (Taylor & Lu, 1995), the results in Lin et al. (1996), Friedland (1997), and in fact, dithered control (Desoer & Sharuz, 1986). Since (Dean et al., 1995) in experiments found that the “strong control action in the gap” inverse compensation does not work well, the authors of this survey tend to believe that the concept of “weak action in the gap” might be advantageous.
The survey seems to indicate that much research on backlash compensation remains to be done, both in synthesis and analysis. References Ahmad, S. (1985). Second order nonlinear kinematic eDects and their compensation. In Proceedings of CDC’85 (pp. 307–314). K om, K. J., & Wittenmark, B. (1984). Computer controlled systems: AstrL theory and design. Englewood CliDs, NJ: Prentice-Hall. K om, K. J., & Wittenmark, B. (1989). Adaptive control. Reading, MA: AstrL Addison-Wesley. Atherton, D. P. (1975). Nonlinear control engineering. New York: Van Nostrand Reinhold Co. Ltd. Atherton, D. P. (1981). Stability of nonlinear systems. New York: Wiley. Azenha, A., & Tenreiro Machado, J. R. (1996). Variable structure control of systems with nonlinear friction and dynamic backlash. In Proceedings of IFAC World Congress, San Francisco. Baril, C., & Galic, J. (1994). Speed control of an elastic two-mass system. Technical Report TRITA=MA-94-29T, Optimization and Systems Theory, Royal Institute of Technology, 10044 Stockholm, Sweden. Boneh, R., & Yaniv, O. (1994). Control of an elastic two-mass system with large backlash. Master’s thesis, Department of Electronic Systems, Tel Aviv University, Tel Aviv, Israel. Boneh, R., & Yaniv, O. (1999). Control of an elastic two-mass system with large backlash. Journal of Dynamic Systems, Measurement and Control, 121(2), 278–284. Brandenburg, G. (1986). Stability of a speed controlled elastic two-mass system with backlash and Coulomb friction and optimization by a load observer. In Symposium: Modelling and simulation for control of lumped and distributed parameter systems, Lille (pp. 107–113). New Brunswich, NJ: IMACS-IFACS. Brandenburg, G., Geissenberger, S., Kink, C., Schall, N. -H., & Schramm, M. (1999). Multi-motor electronic line shafts for rotary printing presses: A revolution in printing machine techniques IEEE=ASME Transactions on Mechatronics, 4(1), 25–31. Brandenburg, G., Hertle, H., & Zeiselmair, K. (1987). Dynamic inGuence and partial compensation of coulomb friction in a position- and speed-controlled elastic two-mass systems. In IFAC World Congress, Munich (pp. 91–99). Brandenburg, G., & Kaiser, W. (1995). On pi motion control of elastic systems with notch 9lters in comparison with advanced strategies. In Proceedings of IFAC conference motion control, Munich, Germany (pp. 863–872). IFAC. Brandenburg, G., & SchLafer, U. (1987). InGuence and partial compensation of backlash for a position controlled elastic two-mass system. In Proceedings of the European conference on power electronics and applications, Grenoble, France (pp. 1041–1047). EPE. Brandenburg, G., & SchLafer, U. (1988a). InGuence and partial compensation of simultaneously acting backlash and coulomb friction in a speed- and position-controlled elastic two-mass system. In Proceedings of the second international conference on electrical drives, Poiana Brasov, Rumania: ICED. Brandenburg, G., & SchLafer, U. (1988b). Stability analysis and optimization of a position-controlled elastic two-mass system with backlash and coulomb friction. In Proceedings of the 12th IMACS world congress on scientiCc computation, Paris (pp. 220 –223). New Brunswick, NJ, USA: IMACS. Brandenburg, G., & SchLafer, U. (1989). InGuence and adaptive compensation simultaneously acting backlash and Coulomb friction in elastic two-mass systems of robots and machine tools. In Proceedings of ICCON ’89, Jerusalem (pp. WA-4-5). New York: IEEE Press. Brandenburg, G., & SchLafer, U. (1990). Design and performance of diDerent types of observers for industrial speed- and position controlled electromechanical systems. In Proceedings of the international
M. Nordin, P.-O. Gutman / Automatica 38 (2002) 1633–1649 conference on electrical drives and power electronics, Strbske Pleso, Slovakia, (pp. 1–10). EPE. Brandenburg, G., Unger, H., & Wagenpfeil, A. (1986). Stability problems of a speed controlled drive in an elastic system with backlash and corrective measures by a load observer. In Proceedings of the international conference on electrical machines (pp. 523–527). Technische UniversitLat MLunchen, MLunchen. Chang, T. N., Ji, Z., Shimanovich, M., & Caudill, R. (1996). Vibration control of contactless drive system using passband shaping. In Proceedings of IFAC world congress, San Francisco (pp. 357–362). Chestnut, H. (1954). Approximate frequency-response methods for representing saturation and dead band. Transactions of the ASME, pp. 1345 –1363. Chestnut, H., & Mayer, R. W. (1995). Servomechanisms and regulating system design (pp. 251–257, 301–321). New York: Wiley. Choi, G.-H., Nakamura, H., & Kobayashi, H. (1996). Calibration of servo systems with redundant actuators. In Proceedings of IFAC world congress, San Francisco (pp. 169 –174). Cohen, M., & Cohen, I., (1994). The use of worm gear transmissions in electromechanical systems: Analysis, digital control and application. In The 25th Israel conference on mechanical engineering (pp. 175 –182). CosgriD, R. L. (1958). Nonlinear control systems (pp. 121–123, 193–194). Reading, MA: McGraw-Hill. de Santis, R. M. (1994). A novel PID con9guration for speed and position control. Transactions of the ASME, 116, 542–549. Dean, S. R. H., Surgenor, B. W., & Iordanou, H. N. (1995). Experimental evaluation of a backlash inverter as applied to a servomotor with gear train. In Proceedings of the fourth IEEE conference control applications (pp. 580 –585). Decker, C., Ehrlinger, A. U., Boucher, P., & Dumur, D. (1996). Application of constrained receding horizon control predictive control to a benchmark problem. European Journal of Control 1(2), 157–165. Desoer, C. A., & Sharuz, S. M. (1986). Stability of dithered non-linear systems with backlash or hysterisis. International Journal of Control, 43(4), 1045–1060. Dhaouadi, R., Kubo, K., & Tobise, M. (1993). Two-degree-of-freedom robust speed controller for high performance rolling mill drives. IEEE Transactions on Industry Applications, 29(5), 919–927. Dhaouadi, R., Kubo, K., & Tobise, M. (1994). Analysis and compensation of speed drive systems with torsional load. IEEE Transactions on Industry Applications, 30(3), 760–766. Dumitrache, I., Dumitriu, S., & Monteano, Fl. (1987). Adaptive structure algorithm for microprocessor control of DC motors. In Proceedings of the European conference on power electronics and applications, Grenoble (pp. 1041–1047). EPE. Freeman, E. A. (1957). An approximate transient analysis of a second order position-control system when backlash is present. Institution Monograph 254, The Institution of Electrical Engineers. Freeman, E. A. (1959). The eDect of speed dependent friction and backlash on the stability of automatic control systems. Transaction of the American IEE, pp. 680 – 691. Freeman, E. A. (1960). Stabilization of control systems with backlash using a high-frequency on-oD loop. Institution Monograph 356, The Institution of Electrical Engineers. Friedland, B. (1997). Feedback control of systems with parasitic eDects. In Proceedings of ACC-97, Albuquerque, USA, American Automatic Control Council. Galardini, D., Nordin, M., & Gutman, P.O. (1997). Robust PI tuning for an elastic two-mass system. In Proceedings of the Cfth European control conference, Brussels. Gelb, A., & Vander Velde, W. E. (1968). Multiple-input describing functions and nonlinear system design. New York: McGraw-Hill. Goodman, T. P. (1963). How to calculate dynamic eDects of backlash. Machine Design, pp. 150 –157.
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Grundmann, S. (1995). Robust vibration control by multi model pole assessment. In Proceedings of IFAC motion control, Munich (pp. 876 –881). Gurian, S. (1998). Resonance compensation in large drive systems. In Proceedings of 1998 AISE annual convention, www.aise.org, Pittsburg, USA: AISE. Gutman, P. O. (1998). Book review of adaptive control of systems with actuators and sensor nonlinearities. In G. Tao, & P. V. Kokotovi\c (Eds.), International Journal of Robust and Nonlinear Control, 12(6), 537–538. Hahn, V., & Unbehauen, H. (1987). Cascade adaptive speed control of a thyristor driven dc-motor. In Proceedings of the European conference on power electronics and applications, Grenoble (pp. 1041–1047). EPE. Han, C. W., & de la Sen, M. (1996). Multivariable adaptive control of backlash nonlinear systems with arbitrary interactor matrix. In Proceedings of IFAC world congress, San Francisco (pp. 79 –84). Harashima, F., Ueshiba, T., & Hashimoto, H. (1987). Sliding mode control for robotic manipulator. In Proceedings of European conference on power electronics and applications, Grenoble (pp. 1041–1047). Grenoble, France: EPE. Hjalmarsson, H., Gunnarsson, S., & Gevers, M. (1996). Model free tuning of a robust regulator for a Gexible transmission system. European Journal of Control, 1(2), 97–103. Hori, Y., Iseki, H., & Sigiura, K. (1994). Basic consideration of vibration supression and disturbance rejection control of multi-inertia system using SFLAC (state feedback and load acceleration control). IEEE Transactions on Industry Applications, 30(4), 889–896. Horowitz, I. M. (1963). Synthesis of feedback systems. New York: Academic Press. Horowitz, I. M. (1992). Quantitative feedback design theory (QFT). Boulder CO, QFT Publications. Indri, M., & Tornamb]e, A. (1997). Application of a PD controller on two mating gears with elasticity and backlash. In Proceedings of the 36th conference on decision and control, San Diego, USA (pp. 4363– 4368). New York: IEEE Press. Ji, J. -K., & Sul, S. -K. (1995). DSP-based self-tuning IP speed controller with load torque compensation for rolling mill DC drive. IEEE Transactions on Industrial Electronics, 42(4), 382–386. Johansson, M. (1999). Piecewise linear control systems. Ph.D. thesis, Department of Automatic Control, Lund University, Sweden. Johansson, M., & Rantzer, A. (1997). Computation of piecewise quadratic Lyapunov functions for hybrid systems. In Proceedings of the 36th conference on decision and control, San Diego, USA, Vol. 4 (pp. 3515 –3520). New York: IEEE Press. Johansson, M., & Rantzer, A. (1998). Computation of piecewise quadratic Lyapunov functions for hybrid systems. IEEE Transactions on Automatic Control, 45(4), 555–559. Jones, N. W., & Limebeer, D. J. N. (1996). A digital h∞ controller for a Gexible transmission system. European Journal of Control, 1(2), 93–96. Joos, G., & Sicard, P. (1992). A comparision of microcomputer-based implementations of cascades and parallel speed and current loops in DC motor drives. IEEE Transactions on Industry Applications, 28(1), 136–143. Kidron, O., & Yaniv, O. (1996). Robust control of uncertain resonant systems. European Journal of Control, 1(2), 66–72. Kokotovi\c, P. V., Ezal, K., & Tao, G. (1997). Optimal control of tracking systems with backlash and Gexibility. In Proceedings of the 36th IEEE conference on decision and control, San Diego, CA (pp. 1749 –1754). New York: IEEE Press. Koyama, M., & Yano, M. (1991). Two degrees of freedom controller using reference system model for motor drives. In Proceedings of the European conference on power electronics and applications, Firenze (pp. 596 – 602). Firenze, Italy: EPE. Kreer, J. B. (1956). Transient response of pseudo-linear feedback control systems. Ph.D. thesis, University of Illinois, Urbana.
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M. Nordin, P.-O. Gutman / Automatica 38 (2002) 1633–1649
Landau, I. D., Karimi, A., Voda, A., & Franco, A. (1996). Robust digital control of Gexible transmissions using the combined pole placement=sensitivity function shaping method. European Journal of Control, 1(2), 19–25. Landau, I. D., Rey, D., Karimi, A., Voda, A., & Franco, A. (1996). A Gexible transmission system as a benchmark for robust digital control. European Journal of Control, 1(2), 12–18. Lin, C.-F., Yu, T.-J., & Feng, X. (1996). Fuzzy control of a nonlinear pointing testbed with backlash and friction. In Proceedings of the 35th conference on decision and control, Kobe, Japan (pp. 4363– 4368). New York: IEEE Press. Ling, Y., & Tao, G. (1999). Numerical design and analysis of backlash compensation for a multivariable nonlinear tracking system. In Proceedings of the 1999 American control conference (pp. 3539 – 3543). New York: IEEE Press. Liversidge, J. H. (1952). Backlash and resilience within closed loop of automatic control systems. New York: Academic Press. Ma, X. L., & Tao, G. (1999). An algorithm for addition and multiplication of value sets of uncertain transfer functions. In Proceedings of the 38th IEEE conference on decision and control, Phoenix, Arizona (pp. 1246 –1247). New York: IEEE Press. Nordin, M. (1992). Robust control of a rolling mill with minimal impact drop. Master’s thesis, Optimization & Systems Theory, Department of Mathematics, The Royal Institute of Technology, Stockholm, Sweden, TRITA=MAT-92-0037. Nordin, M. (1995). Uncertain systems with backlash: Analysis, identiCcation and synthesis. Licenciate thesis, trita=mat-95-os3, Royal Institute of Technology. Nordin, M. (2000). Nonlinear backlash compensation for speed controlled elastic systems. Ph.D. thesis, Division of Optimization and Systems Theory, Royal Institute of Technology, 10044 Stockholm, Sweden, ISSN 1401–2294. Nordin, M., Galic, J., & Gutman, P. O. (1997). New models for backlash and gear play. International Journal of Adaptive Control and Signal Processing, 1, 9–63. Nordin, M., & Gutman, P.O. (1995). A robust linear design of an uncertain two-mass system with backlash. In Proceedings of the Crst IFAC workshop in automotive control, Ascona, Schweiz (pp. 183–188). IFAC. Nordin, M., & Gutman, P. O. (1996). Digital QFT-design for the benchmark problem. European Journal of Control, 1(2), 97–103. Nordin, M., & Gutman, P.O. (2000). Nonlinear speed control of elastic systems with backlash. In Proceedings of the 39th IEEE conference on decision and control, Sydney (pp. ThP13-4). New York: IEEE Press. Ohishi, K., Myooi, T., Ohnishi, K., & Miyaji, K. (1986). An approach to speed control of DC-motor with two inertias. Electrical Engineering in Japan, 106(2), 115–120. Ohishi, K., Ohnishi, K., & Miyachi, K. (1983). Torque-speed regulation of DC-motor based on load torque estimation method. In Proceedings of IPEC-Tokyo (pp. 1209 –1218). Oldak, S., Baril, C., & Gutman, P. O. (1994). Quantative design of a class of nonlinear systems with parameter uncertainty. International Journal of Robust and Nonlinear Control, 4, 101–117. Oustaloup, A., Mathieu, B., & Lanusse, P. (1996). The CRONE control of resonant plants: Application to a Gexible transmission European Journal of Control, 1(2), 113–121. Pan, Y., & Furuta, K. (1996). VSS approach to the design of robust digital controller. European Journal of Control, 1(2), 166–173. Pettersson, M. (1997). Driveline modeling and control. Ph.D. thesis, Department of Electrical Engineering, LinkLoping University, Sweden. Pollman, A., Tosetto, A., & Brea, P. (1991). Micro processor based control systems for rolling mills. In Proceedings of the European conference on power electronics and applications, Firenze (pp. 64 – 67). Firenze, Italy: EPE. Recker, D. A., et al. (1991). Adaptive nonlinear control of systems containing a dead zone. In Proceedings of the 30th CDC, Brighton, England (pp. 2111–2115). New York: IEEE Press.
SchLafer, U., & Brandenburg, G. (1990). State position control for elastic pointing and tracking systems with gear play and Coulomb friction—a summary of results. In Proceedings of the sixth Conference on power electronics and motion control (PEMC), Budapest (pp. 797–801). EPE. SchLafer, U., & Brandenburg, G. (1991). State position control for elastic pointing and tracking systems with gear play and Coulomb friction—a summary of results. In Proceedings of the European conference on power electronics and applications, Firenze (pp. 596 – 602). Firenze, Italy: EPE. SchLonfeld, R. (1987). Adaptive digital speed and position control of DC servo drives. In Proceedings of the European conference on power electronics and applications, Grenoble (pp. 1041–1047). Grenoble, France: EPE. ShLafer, U. (1993). Entwicklung von nichlinearen Drehzahl- und lageregelungen zur Kompensation von Coulomb-Reibung und Lose bei einem elektrisch angetriebenen, elastischen Zweimassensystem. Dr. ing, dessertation, Technische Hochschule MLunchen. Slotine, J. J., & Li, W. (1991). Applied nonlinear control. Englewood CliDs, NJ: Prentice-Hall International. Tao, G. (2000). A two-step adaptive controller for a two-body system. IEEE Transactions on Automatic Control, 45(5), 1011–1016. Tao, G., & Kokotovic, P. (1993). Adaptive control of systems with backlash. Automatica, 29(2), 323–335. Tao, G., & Kokotovic, P. V. (1996). Adaptive control of systems with actuator and sensor nonlinearities. New York: Wiley. Tao, G., & Ma, X. L. (1999). Backlash compensation for multivariable nonlinar systems with actuator dynamics. In Proceedings of the 38th IEEE conference on decision and control, Phoenix, Arizona (pp. 596 – 602). New York: IEEE Press. Taylor, J. H., & Lu, J. (1995). Robust nonlinear control system synthesis method for electro-mechanical pointing systems with Gexible modes. Journal of Systems Engineering, 5, 192–204. Thomas, C. H. (1954). Stability characteristics of closed loop systems with dead band. Transactions of ASME, pp. 1365 –1382. Tustin, A. (1947). EDects of backlash and of speed dependent friction on the stability of closed-cycle control systems. Journal IEE, 94(IIA), 143–151. Yang, J. S. (1992). Design of sampled-data control systems with one memoryless, time-invariant nonlinearity using QFT technique. In Proceedings of the 1992 American control conference, Piscataway, N.J., USA. Vol. 1 (pp. 915 –916). New York: IEEE Press. Yaniv, O., & Horowitz, I. (1990). Quantitative feedback theory for active vibration control systems. International Journal of Control, 51(6), 1251–1258. Yeh, C.-S., Fu, L.-C., & Yang, L.-J. (1996). Nonlinear adaptive control of a two-axis gun-turret system with backlash. In Proceedings of IFAC world congress, San Fransisico, June 1996, number 3b-03 4 in 1 (pp. 91–96).
Mattias Nordin was born in Lund, Sweden, in 1967. He received a civ. ing, degree in engineering physics in 1992 and the Ph.D. degree in Optimization and Systems Theory in 2001 from the Royal Institute of Technology, Stockholm, Sweden. His civ. ing. Diploma work earned him the Volvo Award to the Volvo Royal Institute Technologist of the Year 1993. Since 1995 he is employed by ABB Process Industries, VLasteraK s, Sweden, where he is presently the head of the development department for rolling mills. His current research interest is the development of robust control for rolling mill drives, speci9cally minimizing mechanical wear while maximizing the dynamic performance.
M. Nordin, P.-O. Gutman / Automatica 38 (2002) 1633–1649 Per-Olof Gutman was born in HLoganLas, Sweden on May 21, 1949. He received the civ. ing. degree in Engineering Physics in 1973, the Ph.D. degree in automatic control, and the title of docent in automatic control in 1988, all from the Lund Institute of Technology, Lund, Sweden. As a Fulbright grant recipient, he received the M.S.E. degree in 1977 from the University of California, Los Angeles. He taught mathematics in Tanzania, 1973–1975. From 1983–1984 he held a post-doctoral position with the Faculty of
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Electrical Engineering, Technion—Israel Institute of Technology, Haifa, Israel. From 1984 –1990 he was a scientist with the Control Systems Section, El-Op Electro-Optics Industries, Rehovot, Israel, where he designed high precision electro-optical and electro-mechanical control systems. Since 1990 he is with the Faculty of Agricultural Engineering, Technion, Haifa, where he now holds the position of Associate Professor. He has spent several periods as a guest researcher at the Division of Optimization and Systems Theory, Royal Institute of Technology, Stockholm, Sweden, and a sabbatical year (1995 –1996) at LAG, INPG, Grenoble, France. His research interest include robust and adaptive control, control of non-linear systems, computer aided design and control and modeling of agricultural systems.
