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Robust proximate time-optimal servomechanism with speed constraint for rapid motion control
Robotics and Computer-Integrated Manufacturing 30 (2014) 379–388
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Robust proximate time-optimal servomechanism with speed constraint for rapid motion control Guoyang Cheng n, Jin-gao Hu College of Electrical Engineering and Automation, Fuzhou University, Fujian 350116, PR China
art ic l e i nf o
a b s t r a c t
Article history: Received 23 May 2013 Received in revised form 24 November 2013 Accepted 21 December 2013 Available online 24 January 2014
A control scheme is proposed to achieve fast and accurate set-point servo motion in typical double integrator systems subject to disturbances and speed constraint. The controller consists of a Proximate Time-optimal Servomechanism (PTOS) control law and a compensation term for the unknown disturbance. An extended state observer is adopted to estimate the un-measured velocity signal and the unknown disturbance. In the presence of speed constraint, a speed regulation stage is incorporated between the PTOS acceleration and deceleration stages. The closed-loop stability is analyzed theoretically. The control scheme is then applied to the position-velocity control loop in a permanent magnet synchronous motor servo system for set-point tracking. MATLAB simulation has been conducted, followed by experimental verification based on the TMS320F2812 DSP controller board. The results confirm that the proposed control scheme can track a wide range of target references fast and accurately, and has good performance robustness with respect to the disturbance and parameter variations. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Servo system Time-optimal control (TOC) State observer Disturbance rejection Speed regulation
1. Introduction One of the typical control applications is in servo systems, which essentially aim at fast and accurate tracking with a bounded control input, in the presence of plant uncertainty and disturbances (see e.g., [1–6]). For point-to-point tracking, one would first think of the time optimal control (TOC), which applies the maximum control authority to achieve the fastest acceleration followed by braking. However, the TOC bang-bang control law is not robust to plant uncertainty and will lead to control chattering. Hence TOC is rarely implemented on real systems. To overcome the shortcoming of TOC, the proximate time-optimal servomechanism (PTOS) was proposed in [7] and recently modified in [8], which implements a smooth transition to a linear control law when the tracking error is small, thus avoids the chattering problem, with some loss in transient performance to trade for improved robustness. Meanwhile, an alternative design, i.e., the near time optimal control, was proposed by Newman ([9]), which combines the traditional time-optimal control with methods of sliding-mode control. Both designs have found successful application in disk drives servo systems (see e.g., [10–12]). In [10], a third order velocity reference profile was proposed to replace the original second order profile in PTOS, and better settling performance could be achieved, as evidenced in the simulation study. In [11], the chattering problem in Newman0 s near time optimal control was analyzed, and a criterion for the selection of control
n
Corresponding author. Tel.: +86 591 22866596. E-mail addresses: [email protected] (G. Cheng), [email protected] (J.-g. Hu).
0736-5845/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.rcim.2013.12.002
parameters to prevent chattering in the discrete-time case was developed. A modified PTOS scheme with damping scheduling was proposed by Choi et al. [12], which resulted in faster settling in short span tracking, but the stability issue was not discussed. Late on, Salton et al. [13] proposed an improved design of PTOS with nonlinear feedback gains, and better performance has been reported. However, the design in [13] assumes that both state variables are available for feedback, and the compensation of disturbance was not included in its stability analysis. More recently, a discrete-time mode switching control scheme was developed in [14], which incorporates a composite nonlinear feedback control law into the PTOS framework to improve settling performance. This scheme uses the output position as the only measurable signal, and an extended state observer is designed to estimate the unmeasured velocity and unknown disturbance for feedback and compensation. Despite the performance improvement, none of the above work has considered the potential speed limitation in real application environment. For practical servo systems, a constraint will have to be imposed on the maximum speed of motion, so as to ensure the safety and smooth operation of the system. e.g., a motor is usually expected to run within its rated speed. However, with PTOS control, the servo system tracking a set-point target will undergo an acceleration process first and then a deceleration to rest. On the transition point, the speed reaches a peak value, which is increscent with the amplitude of target reference. For a large stroke motion, the peak speed can easily go far beyond the rated speed of servo mechanisms, leading to the breakdown of physical systems. Indeed, this problem motivated us to develop a modification to PTOS design for practical application in speed-constrained servo
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systems. Our basic idea is to introduce an interim stage of speed regulation as soon as the speed hits its constraint limit during PTOS acceleration, and then return to PTOS deceleration when it is appropriate. To form the PTOS control law, both the position and velocity signals are needed. Normally, the position signal is easily measurable. However, velocity sensors are relatively expensive and would be spared if possible in consideration of the cost, reliability and maintenance of the servo system. Although the velocity signal may be obtained by differentiating the position signal or integrating the acceleration signal, the former method might amplify the measurement noise while the latter is dependent on the precision of the installed accelerometer and will incur additional expenditure. In practical servo systems, there are always some unknown disturbances, such as friction and load torques, which will cause tracking error if they are not properly compensated. Usually, to remove the steady-state bias caused by unknown constant disturbances, one would consider using an additional integration action. The headache with integral control is that it can easily lead to the so-called integral-windup phenomenon, which in turn resorts to some anti-windup designs (see e.g., [15,16]). Moreover, it is found that integral control is not very robust to the amplitude of the disturbance and/or the target reference, e.g., a minor change in the amplitude of disturbance or reference may call for a retuning of the controller parameters in order to maintain a satisfactory performance (see e.g., [2]). This is undesirable for practical applications. A better solution is to estimate the unknown disturbance and then compensate it. The idea of using observer to estimate both state and disturbance dated back to the 1980s (see [17–20]), and the name of proportional-integral observer was adopted. Later, the observer-based disturbance rejection control scheme was developed (see e.g., [21–23]), in which the unknown disturbance is treated as an extended state variable to be augmented into the plant model for observer design, hence the name of extended state observer (ESO) came into being. The active disturbance rejection control (ADRC) scheme proposed in [24], which utilizes a nonlinear extended state observer to reconstruct the state variables and unknown input disturbance (see also [25]), had attracted a lot of attention. However, due to the difficulties in the implementation and stability analysis of the nonlinear ESO in ADRC, research interests are shifting towards the linear ESObased control schemes, see e.g., [2,26,27]. In this paper, a linear extended state observer is designed to estimate the un-measured velocity and unknown disturbance for feedback control and compensation within the PTOS framework. A speed regulation control law is also incorporated so as to satisfy the constraint of maximum allowable speed in practical systems. The control scheme, herein after referred to as robust PTOS control, not only maintains the fast transient performance of TOC as possible, but also has steady-state accuracy in the face of disturbance, with performance robustness with respect to the variation of target set-points and disturbance amplitude. The rest of this paper is organized as follows. The design of observer-based robust PTOS is presented in Section 2, followed by theoretical analysis of closed-loop stability in Section 3. Section 4 outlines the design of PTOS with speed constraint. In Section 5, we apply the robust PTOS control methodology to design a permanent magnet synchronous motor position servo system. MATLAB simulation and experimental results based on TMS320F2812 DSP are provided. Finally, we draw some concluding remarks in Section 6.
2. The observer-based robust PTOS control In this section, we present the design of robust PTOS control scheme for typical servo systems characterized by a double-integrator
as follows: ( y_ ¼ v; v_ ¼ b ðsatðuÞ þ dÞ;
ð1Þ
where y is the only measurable output (position), v represents the velocity signal, u is the control input of the plant, and d is the unknown constant or slowly varying disturbance. b is the acceleration constant and assumed to be positive for simplicity. sat : R-R represents the actuator saturation defined as sat½uðtÞ ¼ sign½uðtÞ minfumax ; juðtÞjg;
ð2Þ
with umax being the saturation level. The task of controller is to ensure that the system output y will track the target reference r fast and accurately, and the following PTOS control law is proposed in [8]: up ¼ satðk2 ½f p ðeÞ vÞ; where e ¼ r y is the tracking error. f p ðeÞ is given by 8 k > < 1 e; jejr yl ; f p ðeÞ ¼ k2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > : signðeÞð 2bαumax jej J Þ; jej4 y ; p l
ð3Þ
ð4Þ
where k1 and k2 are respectively the feedback gains for position and velocity, α A ð0; 1Þ is a constant referred to as the acceleration discount factor, and yl is the size of the linear region, Jp is an offset for velocity signal. The linear part of fp(e) must connect its nonlinear parts at the two joints, i.e., satisfy the constraints of continuity and smoothness: 8 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k1 > > > yl ¼ 2bαumax yl J p ; > < k2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð5Þ k1 bαumax > > > ¼ : > : k2 2yl Solving the above equation and applying pole placement for a conjugate pair of closed-loop poles with the damping ratio ζ , we can determine the values of k1, k2 and Jp as follows: rffiffiffiffiffi 2 2αumax ζ k1 k1 ; J p ¼ yl ; ; k2 ¼ 2ζ ð6Þ k1 ¼ yl b k2 where ζ and yl serve as two independent design parameters of the control law. The acceleration discount factor α can be chosen based on the model uncertainty and ζ so as to make a trade-off between the performance and robustness (see [8]):
α o 2β
1 2ζ
2
;
ð7Þ
where β A ð0; 1 is referred to as the uncertainty factor, which serves as some kind of measure for the accuracy of the nominal plant model. β ¼ 1 implies that the model is 100% accurate. Obviously, a larger value of α would lead to higher feedback gains k1 and k2, and a narrower unsaturated region for PTOS control, allowing more scope for fast acceleration with maximum control signal, thus a larger value of α is desirable for fast response. However, as the unsaturated region gets narrower, the robustness tends to get worse. Hence a trade-off should be made. In typical situations, we can consider choosing ζ ¼ 0:8 to obtain an output overshoot below 2%, meanwhile assuming a 90% accuracy for the plant model, i.e., β ¼ 0:9, consequently a sensible choice for α would be α ¼ 0:95. As for the length of the linear region, yl, it is related to the natural frequency of the conjugate closedloop poles: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2bαumax ωn ¼ ζ : ð8Þ yl The value of yl can then be determined such that ωn corresponds to the desired servo bandwidth.
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The PTOS control law in (3) needs to use the un-measured velocity v and does not compensate disturbances, hence is not easily applicable in practical servo systems. Since the unknown disturbance is assumed to be constant or slowly varying, it can be modelled by a differential equation d_ ¼ 0. Incorporating this equation into the plant model, we have the following augmented model: ( x_ ¼ A x þ B satðuÞ; ð9Þ y ¼ C x; where 0 1 y B C x ¼ @ v A; d
2
0
1
6 A¼40 0
0
3
0
7 b 5;
0
0
2 3 0 6 7 B ¼ 4 b 5; 0
C ¼ ½1 0 0:
Note that the output position y is measurable, we only need to estimate the velocity v and disturbance d. Hence a reduced-order observer can be designed as follows (see [2]): 8 x_ v ¼ ðA22 þ LA12 Þxv þ ðB2 þ LB1 Þ satðuÞ > > > > < þ ½A21 þ LA11 ðA22 þLA12 ÞL y; ! ð10Þ ^ v > > > ^ ¼ xv L y; > : d where v^ and d^ are the estimations of velocity v and disturbance d respectively. A11 ¼ 0; A12 ¼ ½1 0, 0 0 b b A21 ¼ ; A22 ¼ ; B1 ¼ 0; B2 ¼ : 0 0 0 0
where c 4 0 is the largest scalar parameter such that 8 z A ΩðP; cÞ ) jKzjr δumax ; for some positive scalar δ A ð0; 1Þ. The value of c can be estimated as (see [28]) c¼
ðδumax Þ2 KP 1 K T
:
ð14Þ
Theorem 1. For the system in (1), the observer-based robust PTOS control law of (11) can guarantee the closed-loop stability and the system output y will track the set-point reference r without steadystate error, subjected to the following conditions: 1. jdj o ð1 δÞumax ; 2. zð0Þ A ΩðP; cÞ. Proof. Similar to the situation shown in Fig. 1 (for double integrator plant without disturbance and both state variables are available), the state space for PTOS can be divided into the following four regions: n o ~ A R4 : umax r u rumax ; ~ dÞ U ¼ ðe; v; v; n o ~ A R4 : u o umax ; ~ dÞ S ¼ ðe; v; v; n o ~ A R4 : u 4 umax ; ~ dÞ S þ ¼ ðe; v; v; n o ~ A U : jej r y ~ dÞ L ¼ ðe; v; v; l ^ ^ d. Obviously, S þ [ U [ S ¼ R4 ; S þ \ with u ¼ k2 ½f p ðeÞ v S ¼ ϕ; L U. Here L is the linear control region for PTOS control. Considering the observer error dynamics (12), we define a Lyapunov function V 1 ðtÞ ¼ zT ðtÞPzðtÞ, and compute its time derivative:
The observer gain matrix L is designed such that the eigenvalues of A22 þ LA12 are stable. Based on the estimation of velocity, and using the estimated disturbance for compensation, we obtain the robust PTOS control law as follows:
V_ 1 ¼ z_ T Pz þ zT P z_ ¼ zT Qz r 0;
^ ^ d: u ¼ k2 ½f p ðeÞ v
then from the condition (2), we have
ð11Þ
381
zT ðtÞPzðtÞ rzT ð0ÞPzð0Þ oc Remark 2.1. The complete control law consists of Eqs. (4), (10) and (11). This control scheme belongs to the category of ESO-based disturbance rejection control. Although the observer in (10) is designed with a constant input disturbance in mind, the estimated disturbance actually is a lumped disturbance including not only the disturbance but also the system uncertainties that can be matched as an equivalent input disturbance. According to [27], the ESO is capable of tracking a time-varying disturbance, so long as the ESO bandwidth is chosen significantly larger than the frequency of the disturbance and ensuring that it is sufficiently smaller than unmodeled high frequency dynamics of the plant. Hence, by using the ESO-based control scheme, we can compensate the external disturbance and some model perturbations. 3. Analysis of closed-loop stability Define z ¼
v~ d~
¼
v^ v d^ d
) zðtÞ A ΩðP; cÞ; 8 t Z0 ) jKzj r δumax : Hence from the assumptions jd þ Kzj oumax :
ð15Þ
Note that the control law can be rewritten as u ¼ k2 ½f p ðeÞ v ðd þ KzÞ:
ð16Þ
It is now clear that the existence of disturbance and observer error does not change the direction of acceleration when the control is saturated. Hence for any initial state within the saturation region S þ or S , the saturated PTOS control law will be applied to drive the
, then it is easy to verify that
z_ ¼ ðA22 þ LA12 Þ z:
ð12Þ
Define K ¼ ½k2 1, and choose a symmetric positive definite matrix 2 Q such that Q 4 K T K=4k2 , and solve the following Lyapunov equation: ðA22 þ LA12 ÞT P þ PðA22 þ LA12 Þ ¼ Q :
ð13Þ
As ðA22 þ LA12 Þ is stable, the above Lyapunov equation has a unique solution of matrix P 4 0. Define a two-dimensional set ΩðP; cÞ≔ z A R2 : zT Pz o c ;
Fig. 1. Switching curve and operating regions of PTOS for double integrator system.
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system state into the unsaturation region U in finite time and stay therein. It remains to show that once the system state enters into the unsaturation region U, the closed-loop system will be stable, and the system output will track the target set-point asymptotically. For any trajectory in U, we have
weighted by the acceleration discount factor α to obtain an estimation of the maximum range of motion, i.e.,
satðuÞ þd ¼ k2 ½f p ðeÞ v Kz:
Furthermore, the minimum braking distance from speed vm to zero can be derived as r^ m =2. Now, it is clear that, to perform rapid motion with speed constraint, an additional stage of speed regulation may be needed in between the acceleration and the deceleration stages. To be specific, when tracking a large stroke target position using PTOS, the speed will reach the constraint value at some point during acceleration, then the control should be directed towards maintaining the current speed instead of continuing acceleration, and when the position error comes down to the minimum braking distance, the control is switched back to PTOS for deceleration and settling down. The work flow of this control scheme is demonstrated in Fig. 2. In this figure, em represents the distance reserved for braking and smooth settling from the given maximum speed vm, and is given by
(
Consider the system dynamics e_ ¼ v; v_ ¼ b ðk2 ½f p ðeÞ v KzÞ;
and the Lyapunov function: Z eðtÞ v2 ðtÞ þ f p ðsÞ ds þ zT ðtÞPzðtÞ: VðtÞ ¼ 2bk2 0
ð17Þ
ð18Þ
The time derivative of V(t) along the trajectory of closed-loop system comprising (12) and (17) can be calculated as follows: vv_ þ f p ðeÞe_ þ z_ T Pz þ zT P z_ V_ ¼ bk2 v½satðuÞ þ d þ f p ðeÞð vÞ zT Qz ¼ k2 k2 ½f p ðeÞ vv Kzv f p ðeÞv zT Qz ¼ k2 K ¼ v2 zv zT Qz k2 2 3 K
T 6 1 2k2 7 v 6 7 v : ¼ 6 T 7 5 z z 4K Q 2k2 By the choice of matrix Q, obviously V_ r 0 and the equality holds only at the region characterized by ðvzÞ ¼ 0. By the LaSalle theorem: v-0, z-0. From v_ ¼ b ½satðuÞ þ d ¼ b ½k2 f p ðeÞ k2 v Kz, we can obtain f p ðeÞ-0, which implies e-0, or y-r. Hence, the closedloop system has one single equilibrium at the origin, and is asymptotically stable. □ Remark 3.1. From the control law in (16), the control signal can be decomposed into three components for the disturbance, the observer error, and servo tracking respectively. Here the disturbance d is a lumped input disturbance, its amplitude should be smaller than umax , otherwise even the maximum control signal could barely reject the disturbance, and no control energy would be available for servo tracking. Theorem 1 assumes that the unknown disturbance is bounded by ð1 δÞumax , thus in the worst case we still have a remaining control energy of δ umax to overcome the observer error and perform servo tracking tasks.
r^ m ¼
em ¼
v2m : bαumax
v2m þ yl : 2bαumax
ð20Þ
For the PTOS acceleration and deceleration stages, the same control law of (11) with observer (10) is adopted. For the stage of speed regulation, the following simple control law can be designed: ^ ^ d; u ¼ kp ½signðeÞ vm v
ð21Þ
where kp 4 0 is the proportional gain for speed regulation. The term signðeÞ is used in the control law to allow for dual-direction motion. Define the speed tracking error v ¼ v signðeÞ vm , and note that the position error e does not change its sign and the control signal is not
Fig. 2. The work flow of PTOS control with speed constraint.
Fig. 3. Schematic diagram of PMSM position servo system.
4. Robust PTOS with speed constraint In this section, we consider the case when the servo system has a constraint on the motion speed. For the ideal double integrator system (with zero initial condition) under ffiTOC control, the speed pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi will reach the maximum value of bumax jrj at the transition point of acceleration and deceleration, which corresponds to the half distance of the set-point motion. Obviously, the maximum speed is increscent with the amplitude of target position. Suppose the maximum allowable speed is vm, then the range of motion should be no larger than r m ¼ v2m =bumax , otherwise the speed constraint will be breached. When the robust PTOS scheme is used instead of TOC, the situation is similar, although the strict quantity relations are hard to establish. Intuitively, the control limit umax should be
ð19Þ
Fig. 4. Experimental setup of PMSM servo system.
G. Cheng, J.-g. Hu / Robotics and Computer-Integrated Manufacturing 30 (2014) 379–388
saturated during speed regulation, it is easy to obtain
derivative as follows:
v_ ¼ bkp v bkp v~ bd~ ¼ bkp v Mz;
V_ 2 ¼ 2λv v_ þ z_ T Pz þ zT P z_ ¼ 2λbkp v 2 2λMzv zT Qz # T " 2λbk λM v
v p ¼ : z z λM T Q
where M ¼ ½bkp b. Now, define a Lyapunov function V 2 ðtÞ ¼ λv 2 ðtÞ þzT ðtÞPzðtÞ with 0 o λ o 2bkp =MQ 1 M T and calculate its time Table 1 PID parameters for current loops. Parameters
kp
ki
kd
iq loop id loop
19.053 43.301
0.1905 0.1443
18.1 0
383
Based on the choices of λ and Q, it is clear that V_ 2 is negative definite. Hence we can conclude that the speed regulation system is asymptotically stable, and the speed will track its target value accurately. Remark 4.1. Fig. 2 presents the typical work flow in target tracking with the speed-constrained PTOS control. In the case when the target distance is small, e.g., smaller than r^ m , the motion
Fig. 5. Simulation results for various target angles with speed constraint (vm ¼150 rad/s) and no disturbance (d ¼ 0A). (a) Position (normalized), (b) Speed, (c) Control current and (d) Disturbance.
Fig. 6. Simulation results for target angle 2π with different disturbance amplitudes. (a) Position (normalized), (b) Speed, (c) Control current and (d) Disturbance.
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speed may not hit the constraint limit vm, thus the stage of speed regulation will not be activated, and the servo system will undergo the PTOS acceleration followed by deceleration and then settle down on the target position.
5. Application in PMSM position control In this section we apply the robust PTOS control scheme to design a permanent magnet synchronous motor position servo system. PMSM has found extensive applications in high performance position/speed servo system (see e.g.,[26,29–33]), due to the advantages of compact structure, high power density and efficiency. PMSM servo systems typically adopt the vector control scheme, so that the
flux- and torque-producing components of the stator current are aligned along d (direct) and q (quadrature) axes respectively, thus enabling decoupled control of both the flux (d-axis) and torque (qaxis). In this work, we deal with a surface-mounted PMSM, with the dq model given by 8 dθ r > > ¼ ωr ; > > dt > > > > > d ωr kt iq kb ωr T L > > > < dt ¼ J J J ; ð22Þ diq U q Rs iq ωLd id ωψ > > ¼ ; > > > dt L L L L q q q q > > > > > did U d Rs id ωLq iq > > ¼ þ ; : dt Ld Ld Ld
Fig. 7. Simulation results for target angle 4π with different speed constraints (with disturbance d ¼ 0.2 A). (a) Position (normalized), (b) Speed, (c) Control current and (d) Disturbance.
Fig. 8. Simulation results for target angle π with respect to variations in system parameter b (with disturbance d ¼ 0.2 A). (a) Position (normalized), (b) Speed, (c) Control current and (d) Disturbance.
G. Cheng, J.-g. Hu / Robotics and Computer-Integrated Manufacturing 30 (2014) 379–388
Table 2 Comparison of settling time (s) in simulation. Target angle θ(rad) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi TOC: 2 θ=ðbumax Þ PTOS: vm ¼ 1 PTOS: vm ¼ 100 rad/s
2π
4π
8π
16π
0.122
0.173
0.245
0.346
0.131 0.131
0.172 0.181
0.235 0.296
0.325 0.535
385
where θr and ωr are the mechanical angle and angular speed, ω is the electrical angular speed, T L is the load torque, J is the moment of inertia of motor, kt is the torque constant, kb is the viscous friction coefficient. U d and U q are the voltage components in dq coordinate, id and iq are the electric currents, Ld and Lq are the inductances (Ld ¼Lq for surface-mounted PMSMs), Rs is the stator resistance, and ψ is the flux linkage established by permanent magnet.
Fig. 9. Experimental results for three target angles without load torque and speed constraint. (a) Position (normalized), (b) Speed, (c) Control current and (d) Disturbance.
Fig. 10. Experimental results for target angle 2π under various load conditions. (a) Position (normalized), (b) Speed, (c) Control current and (d) Disturbance.
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The SVPWM-based field oriented control scheme was adopted in this study, and the structure of the PMSM servo system is shown in Fig. 3. The inner electric current loops are regulated by PID control laws, but the position and speed loops have been unified and controlled by a robust PTOS controller. Taking the motor angular position θr (rad) as the system output y, the angular velocity ωr (rad/s)
as the state variable v, and iq as the control signal u (to serve as the target reference for the iq control loop), we obtain the double integrator model of (1), with the parameter b ¼ kt =J, and the lumped disturbance d ¼ ðkb ωr þ T L Þ=kt . The PMSM in our study is of model 60CB020C (as shown in Fig. 4), with 3000RPM as the rated speed of rotation, and a rated torque of 0.64 Nm, the torque constant is
Fig. 11. Experimental results for target angle 4π and load torque 0.1 Nm with different speed constraints. (a) Position (normalized), (b) Speed, (c) Control current and (d) Disturbance.
Fig. 12. Experimental results for target angle 4π and load torque 0.24 Nm with different speed constraints. (a) Position (normalized), (b) Speed, (c) Control current and (d) Disturbance.
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kt ¼0.712 Nm/A, the number of pole pairs is 4. It has an optical encoder with 2500 pulses per revolution for position measurement. The amplitude of the torque current iq is limited by 1.5 A, i.e., umax ¼ 1:5 A. The value of system parameter b has been identified as b¼1120. In this servo system, the load torque TL should be no larger than the rated toque, i.e., T L r 0:64 Nm, and note that the viscous friction coefficient kb usually has a small value (many motor manufacturers give an “approximately zero” for this parameter in their data
Fig. 13. Experimental results for target angle π and load torque 0.1 Nm with nominal parameter. (a) Position and speed, (b) Actual dq currents and (c) Actual dq voltages.
387
sheets) and is assumed to be 0 here for simplifying analysis, we have kb ωr þT L d ¼ r 0:64 Nm ¼ 0:9 A; 0:712 Nm=A kt then we can obtain δ ¼ 0:4 such that jdj o ð1 δÞumax , as required in Theorem 1. Thus even in the worst case we still have plenty of control energy (i.e., 0:4umax ) to cope with the observer error and servo tracking. For a servo system with an accurate model and reliable measurement, the observer error should be small, enabling more control energy to be devoted to improving the servo performance. With a sampling period of T¼0.002 s for digital implementation, the parameters of controller are chosen as ζ ¼ 0:8; yl ¼ 1:2; α ¼ 0:95 and kp ¼0.1. The observer for speed and disturbance is designed as 2 3 2 pffiffiffi 3 8 ! b ω20 > 2 ω0 b satðuÞ > > x_ ¼ 4 ffiffi p 4 5 5 > xv þ ; > v ω2 2ω3 > y > 0 b0 0 b 0 < 2 pffiffiffi 3 ð23Þ ! > 2ω0 > v^ > > 4 5 2 > y; ¼ xv þ ω0 > > : d^ b with ω0 ¼ 110. For the two control loops of id and iq, digital PID control laws are designed, with the sampling frequency of 20 kHz for control implementation and PWM generation. The parameters of PID controllers are summarized in Table 1. Note that we used a derivative gain kd for the iq loop so as to reduce the overshoot in its response, whereas no derivative was needed for the id loop. In typical PMSM servo systems, the PI control laws work well for the two inner current loops, and the derivative action can be spared to avoid noise amplification. To verify the design, simulations have been conducted in MATLAB/ Simulink, using the nominal double integrator (ignoring the electric current loops) as the plant model. Simulations were done for various target angles with and without disturbance. The results are shown in Figs. 5–8. Obviously the controller achieves fast as possible and smooth settling in all the tracking tasks, and has some robustness with respect to the disturbance and model perturbation. The performances in terms
Fig. 14. Experimental results for target angle π and load torque 0.1 Nm with parameter variations. (a) Position (normalized), (b) Speed, (c) Control current and (d) Disturbance.
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of settling time (for a 2% error bound) are summarized in Table 2. It is easy to see that the position response of PTOS with speed constraint is slower than the TOC optimal performance, especially when the target reference is large. However, in the absence of speed constraint, the performance of PTOS controller approaches and even transcends the theoretical TOC optimal performance as the target reference gets larger in amplitude, whereas for a small target reference, the performance of PTOS may lag behind the TOC, since the linear control plays a major role in such cases, resulting a slower transient process to trade for a better robustness. Next, control algorithms were implemented on a TM320F2812 DSP board from the TI corporation. Real-time experiments were subsequently carried out using the Code Composer Studio software system, and the collected data were then processed in MATLAB. The target reference r, or the angular position of motor, was provided in the form of square waves. Experiments were first carried out for three different target amplitudes (π , 2π and 4π ) without load torque (but there is some other disturbance in the system), the results are shown in Fig. 9. It is obvious that the system achieves superior performance in all the tracking tasks, with the overshoot kept within 2%, and the settling times are 0.106, 0.128 and 0.172 s respectively. Fig. 10 gives the results for the target angle 2π under various load torques, it is clear that the output response may get a bit more sluggish with a larger load torque, but the overall performance is still desirable and consistent, as the impact of disturbance is rejected effectively. Figs. 11 and 12 compare the results for the target angle 4π with different speed constraints and load torques. Obviously the output response will get faster with a larger speed upper bound. Fig. 13 presents the experimental results for target angle π with load torque 0.1 Nm. In this figure, the waveforms for the angular position, speed (converted into kRPM), the actual currents iq, id and voltages Uq, Ud are provided. Clearly, the fluxproducing current id is regulated towards zero, while the torqueproducing current iq is controlled to follow the reference command (i. e., the output of the position controller) and settles down to a nonezero value to counteract the load disturbance. To study the performance with respect to parameter variations, the value of b in the control law was replaced by 840 and 1400 respectively (the values of other tuning parameters remained unchanged) for experimental test, and the results are compared with the nominal case (b¼1120). As shown in Fig. 14, some minor degradation can be observed, but the overall performance is still acceptable for the range of variation up to 25%. Such kind of performance robustness against model uncertainty is beneficial for practical applications. 6. Concluding remarks A robust proximate time-optimal control scheme with speed constraint has been proposed by incorporating an extended state observer and a speed regulation law into the PTOS framework. The closed-loop stability has been analyzed theoretically. The method has been adopted to design a position servo controller for PMSMs. MATLAB simulation and experimental results based on TMS320F2812 DSP show that the proposed design is capable of fast and smooth target tracking in full range with a nearoptimal settling time, and has some robustness with respect to the amplitude variation of target reference and disturbance as well as the plant uncertainty. The proposed control method can be easily applied to other servo systems with a double integrator model. Acknowledgment This work was supported by the National Natural Science Foundation of PR China under Grant 61174051.
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Robotics and Computer-Integrated Manufacturing journal homepage: www.elsevier.com/locate/rcim
Robust proximate time-optimal servomechanism with speed constraint for rapid motion control Guoyang Cheng n, Jin-gao Hu College of Electrical Engineering and Automation, Fuzhou University, Fujian 350116, PR China
art ic l e i nf o
a b s t r a c t
Article history: Received 23 May 2013 Received in revised form 24 November 2013 Accepted 21 December 2013 Available online 24 January 2014
A control scheme is proposed to achieve fast and accurate set-point servo motion in typical double integrator systems subject to disturbances and speed constraint. The controller consists of a Proximate Time-optimal Servomechanism (PTOS) control law and a compensation term for the unknown disturbance. An extended state observer is adopted to estimate the un-measured velocity signal and the unknown disturbance. In the presence of speed constraint, a speed regulation stage is incorporated between the PTOS acceleration and deceleration stages. The closed-loop stability is analyzed theoretically. The control scheme is then applied to the position-velocity control loop in a permanent magnet synchronous motor servo system for set-point tracking. MATLAB simulation has been conducted, followed by experimental verification based on the TMS320F2812 DSP controller board. The results confirm that the proposed control scheme can track a wide range of target references fast and accurately, and has good performance robustness with respect to the disturbance and parameter variations. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Servo system Time-optimal control (TOC) State observer Disturbance rejection Speed regulation
1. Introduction One of the typical control applications is in servo systems, which essentially aim at fast and accurate tracking with a bounded control input, in the presence of plant uncertainty and disturbances (see e.g., [1–6]). For point-to-point tracking, one would first think of the time optimal control (TOC), which applies the maximum control authority to achieve the fastest acceleration followed by braking. However, the TOC bang-bang control law is not robust to plant uncertainty and will lead to control chattering. Hence TOC is rarely implemented on real systems. To overcome the shortcoming of TOC, the proximate time-optimal servomechanism (PTOS) was proposed in [7] and recently modified in [8], which implements a smooth transition to a linear control law when the tracking error is small, thus avoids the chattering problem, with some loss in transient performance to trade for improved robustness. Meanwhile, an alternative design, i.e., the near time optimal control, was proposed by Newman ([9]), which combines the traditional time-optimal control with methods of sliding-mode control. Both designs have found successful application in disk drives servo systems (see e.g., [10–12]). In [10], a third order velocity reference profile was proposed to replace the original second order profile in PTOS, and better settling performance could be achieved, as evidenced in the simulation study. In [11], the chattering problem in Newman0 s near time optimal control was analyzed, and a criterion for the selection of control
n
Corresponding author. Tel.: +86 591 22866596. E-mail addresses: [email protected] (G. Cheng), [email protected] (J.-g. Hu).
0736-5845/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.rcim.2013.12.002
parameters to prevent chattering in the discrete-time case was developed. A modified PTOS scheme with damping scheduling was proposed by Choi et al. [12], which resulted in faster settling in short span tracking, but the stability issue was not discussed. Late on, Salton et al. [13] proposed an improved design of PTOS with nonlinear feedback gains, and better performance has been reported. However, the design in [13] assumes that both state variables are available for feedback, and the compensation of disturbance was not included in its stability analysis. More recently, a discrete-time mode switching control scheme was developed in [14], which incorporates a composite nonlinear feedback control law into the PTOS framework to improve settling performance. This scheme uses the output position as the only measurable signal, and an extended state observer is designed to estimate the unmeasured velocity and unknown disturbance for feedback and compensation. Despite the performance improvement, none of the above work has considered the potential speed limitation in real application environment. For practical servo systems, a constraint will have to be imposed on the maximum speed of motion, so as to ensure the safety and smooth operation of the system. e.g., a motor is usually expected to run within its rated speed. However, with PTOS control, the servo system tracking a set-point target will undergo an acceleration process first and then a deceleration to rest. On the transition point, the speed reaches a peak value, which is increscent with the amplitude of target reference. For a large stroke motion, the peak speed can easily go far beyond the rated speed of servo mechanisms, leading to the breakdown of physical systems. Indeed, this problem motivated us to develop a modification to PTOS design for practical application in speed-constrained servo
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systems. Our basic idea is to introduce an interim stage of speed regulation as soon as the speed hits its constraint limit during PTOS acceleration, and then return to PTOS deceleration when it is appropriate. To form the PTOS control law, both the position and velocity signals are needed. Normally, the position signal is easily measurable. However, velocity sensors are relatively expensive and would be spared if possible in consideration of the cost, reliability and maintenance of the servo system. Although the velocity signal may be obtained by differentiating the position signal or integrating the acceleration signal, the former method might amplify the measurement noise while the latter is dependent on the precision of the installed accelerometer and will incur additional expenditure. In practical servo systems, there are always some unknown disturbances, such as friction and load torques, which will cause tracking error if they are not properly compensated. Usually, to remove the steady-state bias caused by unknown constant disturbances, one would consider using an additional integration action. The headache with integral control is that it can easily lead to the so-called integral-windup phenomenon, which in turn resorts to some anti-windup designs (see e.g., [15,16]). Moreover, it is found that integral control is not very robust to the amplitude of the disturbance and/or the target reference, e.g., a minor change in the amplitude of disturbance or reference may call for a retuning of the controller parameters in order to maintain a satisfactory performance (see e.g., [2]). This is undesirable for practical applications. A better solution is to estimate the unknown disturbance and then compensate it. The idea of using observer to estimate both state and disturbance dated back to the 1980s (see [17–20]), and the name of proportional-integral observer was adopted. Later, the observer-based disturbance rejection control scheme was developed (see e.g., [21–23]), in which the unknown disturbance is treated as an extended state variable to be augmented into the plant model for observer design, hence the name of extended state observer (ESO) came into being. The active disturbance rejection control (ADRC) scheme proposed in [24], which utilizes a nonlinear extended state observer to reconstruct the state variables and unknown input disturbance (see also [25]), had attracted a lot of attention. However, due to the difficulties in the implementation and stability analysis of the nonlinear ESO in ADRC, research interests are shifting towards the linear ESObased control schemes, see e.g., [2,26,27]. In this paper, a linear extended state observer is designed to estimate the un-measured velocity and unknown disturbance for feedback control and compensation within the PTOS framework. A speed regulation control law is also incorporated so as to satisfy the constraint of maximum allowable speed in practical systems. The control scheme, herein after referred to as robust PTOS control, not only maintains the fast transient performance of TOC as possible, but also has steady-state accuracy in the face of disturbance, with performance robustness with respect to the variation of target set-points and disturbance amplitude. The rest of this paper is organized as follows. The design of observer-based robust PTOS is presented in Section 2, followed by theoretical analysis of closed-loop stability in Section 3. Section 4 outlines the design of PTOS with speed constraint. In Section 5, we apply the robust PTOS control methodology to design a permanent magnet synchronous motor position servo system. MATLAB simulation and experimental results based on TMS320F2812 DSP are provided. Finally, we draw some concluding remarks in Section 6.
2. The observer-based robust PTOS control In this section, we present the design of robust PTOS control scheme for typical servo systems characterized by a double-integrator
as follows: ( y_ ¼ v; v_ ¼ b ðsatðuÞ þ dÞ;
ð1Þ
where y is the only measurable output (position), v represents the velocity signal, u is the control input of the plant, and d is the unknown constant or slowly varying disturbance. b is the acceleration constant and assumed to be positive for simplicity. sat : R-R represents the actuator saturation defined as sat½uðtÞ ¼ sign½uðtÞ minfumax ; juðtÞjg;
ð2Þ
with umax being the saturation level. The task of controller is to ensure that the system output y will track the target reference r fast and accurately, and the following PTOS control law is proposed in [8]: up ¼ satðk2 ½f p ðeÞ vÞ; where e ¼ r y is the tracking error. f p ðeÞ is given by 8 k > < 1 e; jejr yl ; f p ðeÞ ¼ k2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > : signðeÞð 2bαumax jej J Þ; jej4 y ; p l
ð3Þ
ð4Þ
where k1 and k2 are respectively the feedback gains for position and velocity, α A ð0; 1Þ is a constant referred to as the acceleration discount factor, and yl is the size of the linear region, Jp is an offset for velocity signal. The linear part of fp(e) must connect its nonlinear parts at the two joints, i.e., satisfy the constraints of continuity and smoothness: 8 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k1 > > > yl ¼ 2bαumax yl J p ; > < k2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð5Þ k1 bαumax > > > ¼ : > : k2 2yl Solving the above equation and applying pole placement for a conjugate pair of closed-loop poles with the damping ratio ζ , we can determine the values of k1, k2 and Jp as follows: rffiffiffiffiffi 2 2αumax ζ k1 k1 ; J p ¼ yl ; ; k2 ¼ 2ζ ð6Þ k1 ¼ yl b k2 where ζ and yl serve as two independent design parameters of the control law. The acceleration discount factor α can be chosen based on the model uncertainty and ζ so as to make a trade-off between the performance and robustness (see [8]):
α o 2β
1 2ζ
2
;
ð7Þ
where β A ð0; 1 is referred to as the uncertainty factor, which serves as some kind of measure for the accuracy of the nominal plant model. β ¼ 1 implies that the model is 100% accurate. Obviously, a larger value of α would lead to higher feedback gains k1 and k2, and a narrower unsaturated region for PTOS control, allowing more scope for fast acceleration with maximum control signal, thus a larger value of α is desirable for fast response. However, as the unsaturated region gets narrower, the robustness tends to get worse. Hence a trade-off should be made. In typical situations, we can consider choosing ζ ¼ 0:8 to obtain an output overshoot below 2%, meanwhile assuming a 90% accuracy for the plant model, i.e., β ¼ 0:9, consequently a sensible choice for α would be α ¼ 0:95. As for the length of the linear region, yl, it is related to the natural frequency of the conjugate closedloop poles: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2bαumax ωn ¼ ζ : ð8Þ yl The value of yl can then be determined such that ωn corresponds to the desired servo bandwidth.
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The PTOS control law in (3) needs to use the un-measured velocity v and does not compensate disturbances, hence is not easily applicable in practical servo systems. Since the unknown disturbance is assumed to be constant or slowly varying, it can be modelled by a differential equation d_ ¼ 0. Incorporating this equation into the plant model, we have the following augmented model: ( x_ ¼ A x þ B satðuÞ; ð9Þ y ¼ C x; where 0 1 y B C x ¼ @ v A; d
2
0
1
6 A¼40 0
0
3
0
7 b 5;
0
0
2 3 0 6 7 B ¼ 4 b 5; 0
C ¼ ½1 0 0:
Note that the output position y is measurable, we only need to estimate the velocity v and disturbance d. Hence a reduced-order observer can be designed as follows (see [2]): 8 x_ v ¼ ðA22 þ LA12 Þxv þ ðB2 þ LB1 Þ satðuÞ > > > > < þ ½A21 þ LA11 ðA22 þLA12 ÞL y; ! ð10Þ ^ v > > > ^ ¼ xv L y; > : d where v^ and d^ are the estimations of velocity v and disturbance d respectively. A11 ¼ 0; A12 ¼ ½1 0, 0 0 b b A21 ¼ ; A22 ¼ ; B1 ¼ 0; B2 ¼ : 0 0 0 0
where c 4 0 is the largest scalar parameter such that 8 z A ΩðP; cÞ ) jKzjr δumax ; for some positive scalar δ A ð0; 1Þ. The value of c can be estimated as (see [28]) c¼
ðδumax Þ2 KP 1 K T
:
ð14Þ
Theorem 1. For the system in (1), the observer-based robust PTOS control law of (11) can guarantee the closed-loop stability and the system output y will track the set-point reference r without steadystate error, subjected to the following conditions: 1. jdj o ð1 δÞumax ; 2. zð0Þ A ΩðP; cÞ. Proof. Similar to the situation shown in Fig. 1 (for double integrator plant without disturbance and both state variables are available), the state space for PTOS can be divided into the following four regions: n o ~ A R4 : umax r u rumax ; ~ dÞ U ¼ ðe; v; v; n o ~ A R4 : u o umax ; ~ dÞ S ¼ ðe; v; v; n o ~ A R4 : u 4 umax ; ~ dÞ S þ ¼ ðe; v; v; n o ~ A U : jej r y ~ dÞ L ¼ ðe; v; v; l ^ ^ d. Obviously, S þ [ U [ S ¼ R4 ; S þ \ with u ¼ k2 ½f p ðeÞ v S ¼ ϕ; L U. Here L is the linear control region for PTOS control. Considering the observer error dynamics (12), we define a Lyapunov function V 1 ðtÞ ¼ zT ðtÞPzðtÞ, and compute its time derivative:
The observer gain matrix L is designed such that the eigenvalues of A22 þ LA12 are stable. Based on the estimation of velocity, and using the estimated disturbance for compensation, we obtain the robust PTOS control law as follows:
V_ 1 ¼ z_ T Pz þ zT P z_ ¼ zT Qz r 0;
^ ^ d: u ¼ k2 ½f p ðeÞ v
then from the condition (2), we have
ð11Þ
381
zT ðtÞPzðtÞ rzT ð0ÞPzð0Þ oc Remark 2.1. The complete control law consists of Eqs. (4), (10) and (11). This control scheme belongs to the category of ESO-based disturbance rejection control. Although the observer in (10) is designed with a constant input disturbance in mind, the estimated disturbance actually is a lumped disturbance including not only the disturbance but also the system uncertainties that can be matched as an equivalent input disturbance. According to [27], the ESO is capable of tracking a time-varying disturbance, so long as the ESO bandwidth is chosen significantly larger than the frequency of the disturbance and ensuring that it is sufficiently smaller than unmodeled high frequency dynamics of the plant. Hence, by using the ESO-based control scheme, we can compensate the external disturbance and some model perturbations. 3. Analysis of closed-loop stability Define z ¼
v~ d~
¼
v^ v d^ d
) zðtÞ A ΩðP; cÞ; 8 t Z0 ) jKzj r δumax : Hence from the assumptions jd þ Kzj oumax :
ð15Þ
Note that the control law can be rewritten as u ¼ k2 ½f p ðeÞ v ðd þ KzÞ:
ð16Þ
It is now clear that the existence of disturbance and observer error does not change the direction of acceleration when the control is saturated. Hence for any initial state within the saturation region S þ or S , the saturated PTOS control law will be applied to drive the
, then it is easy to verify that
z_ ¼ ðA22 þ LA12 Þ z:
ð12Þ
Define K ¼ ½k2 1, and choose a symmetric positive definite matrix 2 Q such that Q 4 K T K=4k2 , and solve the following Lyapunov equation: ðA22 þ LA12 ÞT P þ PðA22 þ LA12 Þ ¼ Q :
ð13Þ
As ðA22 þ LA12 Þ is stable, the above Lyapunov equation has a unique solution of matrix P 4 0. Define a two-dimensional set ΩðP; cÞ≔ z A R2 : zT Pz o c ;
Fig. 1. Switching curve and operating regions of PTOS for double integrator system.
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system state into the unsaturation region U in finite time and stay therein. It remains to show that once the system state enters into the unsaturation region U, the closed-loop system will be stable, and the system output will track the target set-point asymptotically. For any trajectory in U, we have
weighted by the acceleration discount factor α to obtain an estimation of the maximum range of motion, i.e.,
satðuÞ þd ¼ k2 ½f p ðeÞ v Kz:
Furthermore, the minimum braking distance from speed vm to zero can be derived as r^ m =2. Now, it is clear that, to perform rapid motion with speed constraint, an additional stage of speed regulation may be needed in between the acceleration and the deceleration stages. To be specific, when tracking a large stroke target position using PTOS, the speed will reach the constraint value at some point during acceleration, then the control should be directed towards maintaining the current speed instead of continuing acceleration, and when the position error comes down to the minimum braking distance, the control is switched back to PTOS for deceleration and settling down. The work flow of this control scheme is demonstrated in Fig. 2. In this figure, em represents the distance reserved for braking and smooth settling from the given maximum speed vm, and is given by
(
Consider the system dynamics e_ ¼ v; v_ ¼ b ðk2 ½f p ðeÞ v KzÞ;
and the Lyapunov function: Z eðtÞ v2 ðtÞ þ f p ðsÞ ds þ zT ðtÞPzðtÞ: VðtÞ ¼ 2bk2 0
ð17Þ
ð18Þ
The time derivative of V(t) along the trajectory of closed-loop system comprising (12) and (17) can be calculated as follows: vv_ þ f p ðeÞe_ þ z_ T Pz þ zT P z_ V_ ¼ bk2 v½satðuÞ þ d þ f p ðeÞð vÞ zT Qz ¼ k2 k2 ½f p ðeÞ vv Kzv f p ðeÞv zT Qz ¼ k2 K ¼ v2 zv zT Qz k2 2 3 K
T 6 1 2k2 7 v 6 7 v : ¼ 6 T 7 5 z z 4K Q 2k2 By the choice of matrix Q, obviously V_ r 0 and the equality holds only at the region characterized by ðvzÞ ¼ 0. By the LaSalle theorem: v-0, z-0. From v_ ¼ b ½satðuÞ þ d ¼ b ½k2 f p ðeÞ k2 v Kz, we can obtain f p ðeÞ-0, which implies e-0, or y-r. Hence, the closedloop system has one single equilibrium at the origin, and is asymptotically stable. □ Remark 3.1. From the control law in (16), the control signal can be decomposed into three components for the disturbance, the observer error, and servo tracking respectively. Here the disturbance d is a lumped input disturbance, its amplitude should be smaller than umax , otherwise even the maximum control signal could barely reject the disturbance, and no control energy would be available for servo tracking. Theorem 1 assumes that the unknown disturbance is bounded by ð1 δÞumax , thus in the worst case we still have a remaining control energy of δ umax to overcome the observer error and perform servo tracking tasks.
r^ m ¼
em ¼
v2m : bαumax
v2m þ yl : 2bαumax
ð20Þ
For the PTOS acceleration and deceleration stages, the same control law of (11) with observer (10) is adopted. For the stage of speed regulation, the following simple control law can be designed: ^ ^ d; u ¼ kp ½signðeÞ vm v
ð21Þ
where kp 4 0 is the proportional gain for speed regulation. The term signðeÞ is used in the control law to allow for dual-direction motion. Define the speed tracking error v ¼ v signðeÞ vm , and note that the position error e does not change its sign and the control signal is not
Fig. 2. The work flow of PTOS control with speed constraint.
Fig. 3. Schematic diagram of PMSM position servo system.
4. Robust PTOS with speed constraint In this section, we consider the case when the servo system has a constraint on the motion speed. For the ideal double integrator system (with zero initial condition) under ffiTOC control, the speed pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi will reach the maximum value of bumax jrj at the transition point of acceleration and deceleration, which corresponds to the half distance of the set-point motion. Obviously, the maximum speed is increscent with the amplitude of target position. Suppose the maximum allowable speed is vm, then the range of motion should be no larger than r m ¼ v2m =bumax , otherwise the speed constraint will be breached. When the robust PTOS scheme is used instead of TOC, the situation is similar, although the strict quantity relations are hard to establish. Intuitively, the control limit umax should be
ð19Þ
Fig. 4. Experimental setup of PMSM servo system.
G. Cheng, J.-g. Hu / Robotics and Computer-Integrated Manufacturing 30 (2014) 379–388
saturated during speed regulation, it is easy to obtain
derivative as follows:
v_ ¼ bkp v bkp v~ bd~ ¼ bkp v Mz;
V_ 2 ¼ 2λv v_ þ z_ T Pz þ zT P z_ ¼ 2λbkp v 2 2λMzv zT Qz # T " 2λbk λM v
v p ¼ : z z λM T Q
where M ¼ ½bkp b. Now, define a Lyapunov function V 2 ðtÞ ¼ λv 2 ðtÞ þzT ðtÞPzðtÞ with 0 o λ o 2bkp =MQ 1 M T and calculate its time Table 1 PID parameters for current loops. Parameters
kp
ki
kd
iq loop id loop
19.053 43.301
0.1905 0.1443
18.1 0
383
Based on the choices of λ and Q, it is clear that V_ 2 is negative definite. Hence we can conclude that the speed regulation system is asymptotically stable, and the speed will track its target value accurately. Remark 4.1. Fig. 2 presents the typical work flow in target tracking with the speed-constrained PTOS control. In the case when the target distance is small, e.g., smaller than r^ m , the motion
Fig. 5. Simulation results for various target angles with speed constraint (vm ¼150 rad/s) and no disturbance (d ¼ 0A). (a) Position (normalized), (b) Speed, (c) Control current and (d) Disturbance.
Fig. 6. Simulation results for target angle 2π with different disturbance amplitudes. (a) Position (normalized), (b) Speed, (c) Control current and (d) Disturbance.
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speed may not hit the constraint limit vm, thus the stage of speed regulation will not be activated, and the servo system will undergo the PTOS acceleration followed by deceleration and then settle down on the target position.
5. Application in PMSM position control In this section we apply the robust PTOS control scheme to design a permanent magnet synchronous motor position servo system. PMSM has found extensive applications in high performance position/speed servo system (see e.g.,[26,29–33]), due to the advantages of compact structure, high power density and efficiency. PMSM servo systems typically adopt the vector control scheme, so that the
flux- and torque-producing components of the stator current are aligned along d (direct) and q (quadrature) axes respectively, thus enabling decoupled control of both the flux (d-axis) and torque (qaxis). In this work, we deal with a surface-mounted PMSM, with the dq model given by 8 dθ r > > ¼ ωr ; > > dt > > > > > d ωr kt iq kb ωr T L > > > < dt ¼ J J J ; ð22Þ diq U q Rs iq ωLd id ωψ > > ¼ ; > > > dt L L L L q q q q > > > > > did U d Rs id ωLq iq > > ¼ þ ; : dt Ld Ld Ld
Fig. 7. Simulation results for target angle 4π with different speed constraints (with disturbance d ¼ 0.2 A). (a) Position (normalized), (b) Speed, (c) Control current and (d) Disturbance.
Fig. 8. Simulation results for target angle π with respect to variations in system parameter b (with disturbance d ¼ 0.2 A). (a) Position (normalized), (b) Speed, (c) Control current and (d) Disturbance.
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Table 2 Comparison of settling time (s) in simulation. Target angle θ(rad) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi TOC: 2 θ=ðbumax Þ PTOS: vm ¼ 1 PTOS: vm ¼ 100 rad/s
2π
4π
8π
16π
0.122
0.173
0.245
0.346
0.131 0.131
0.172 0.181
0.235 0.296
0.325 0.535
385
where θr and ωr are the mechanical angle and angular speed, ω is the electrical angular speed, T L is the load torque, J is the moment of inertia of motor, kt is the torque constant, kb is the viscous friction coefficient. U d and U q are the voltage components in dq coordinate, id and iq are the electric currents, Ld and Lq are the inductances (Ld ¼Lq for surface-mounted PMSMs), Rs is the stator resistance, and ψ is the flux linkage established by permanent magnet.
Fig. 9. Experimental results for three target angles without load torque and speed constraint. (a) Position (normalized), (b) Speed, (c) Control current and (d) Disturbance.
Fig. 10. Experimental results for target angle 2π under various load conditions. (a) Position (normalized), (b) Speed, (c) Control current and (d) Disturbance.
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The SVPWM-based field oriented control scheme was adopted in this study, and the structure of the PMSM servo system is shown in Fig. 3. The inner electric current loops are regulated by PID control laws, but the position and speed loops have been unified and controlled by a robust PTOS controller. Taking the motor angular position θr (rad) as the system output y, the angular velocity ωr (rad/s)
as the state variable v, and iq as the control signal u (to serve as the target reference for the iq control loop), we obtain the double integrator model of (1), with the parameter b ¼ kt =J, and the lumped disturbance d ¼ ðkb ωr þ T L Þ=kt . The PMSM in our study is of model 60CB020C (as shown in Fig. 4), with 3000RPM as the rated speed of rotation, and a rated torque of 0.64 Nm, the torque constant is
Fig. 11. Experimental results for target angle 4π and load torque 0.1 Nm with different speed constraints. (a) Position (normalized), (b) Speed, (c) Control current and (d) Disturbance.
Fig. 12. Experimental results for target angle 4π and load torque 0.24 Nm with different speed constraints. (a) Position (normalized), (b) Speed, (c) Control current and (d) Disturbance.
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kt ¼0.712 Nm/A, the number of pole pairs is 4. It has an optical encoder with 2500 pulses per revolution for position measurement. The amplitude of the torque current iq is limited by 1.5 A, i.e., umax ¼ 1:5 A. The value of system parameter b has been identified as b¼1120. In this servo system, the load torque TL should be no larger than the rated toque, i.e., T L r 0:64 Nm, and note that the viscous friction coefficient kb usually has a small value (many motor manufacturers give an “approximately zero” for this parameter in their data
Fig. 13. Experimental results for target angle π and load torque 0.1 Nm with nominal parameter. (a) Position and speed, (b) Actual dq currents and (c) Actual dq voltages.
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sheets) and is assumed to be 0 here for simplifying analysis, we have kb ωr þT L d ¼ r 0:64 Nm ¼ 0:9 A; 0:712 Nm=A kt then we can obtain δ ¼ 0:4 such that jdj o ð1 δÞumax , as required in Theorem 1. Thus even in the worst case we still have plenty of control energy (i.e., 0:4umax ) to cope with the observer error and servo tracking. For a servo system with an accurate model and reliable measurement, the observer error should be small, enabling more control energy to be devoted to improving the servo performance. With a sampling period of T¼0.002 s for digital implementation, the parameters of controller are chosen as ζ ¼ 0:8; yl ¼ 1:2; α ¼ 0:95 and kp ¼0.1. The observer for speed and disturbance is designed as 2 3 2 pffiffiffi 3 8 ! b ω20 > 2 ω0 b satðuÞ > > x_ ¼ 4 ffiffi p 4 5 5 > xv þ ; > v ω2 2ω3 > y > 0 b0 0 b 0 < 2 pffiffiffi 3 ð23Þ ! > 2ω0 > v^ > > 4 5 2 > y; ¼ xv þ ω0 > > : d^ b with ω0 ¼ 110. For the two control loops of id and iq, digital PID control laws are designed, with the sampling frequency of 20 kHz for control implementation and PWM generation. The parameters of PID controllers are summarized in Table 1. Note that we used a derivative gain kd for the iq loop so as to reduce the overshoot in its response, whereas no derivative was needed for the id loop. In typical PMSM servo systems, the PI control laws work well for the two inner current loops, and the derivative action can be spared to avoid noise amplification. To verify the design, simulations have been conducted in MATLAB/ Simulink, using the nominal double integrator (ignoring the electric current loops) as the plant model. Simulations were done for various target angles with and without disturbance. The results are shown in Figs. 5–8. Obviously the controller achieves fast as possible and smooth settling in all the tracking tasks, and has some robustness with respect to the disturbance and model perturbation. The performances in terms
Fig. 14. Experimental results for target angle π and load torque 0.1 Nm with parameter variations. (a) Position (normalized), (b) Speed, (c) Control current and (d) Disturbance.
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of settling time (for a 2% error bound) are summarized in Table 2. It is easy to see that the position response of PTOS with speed constraint is slower than the TOC optimal performance, especially when the target reference is large. However, in the absence of speed constraint, the performance of PTOS controller approaches and even transcends the theoretical TOC optimal performance as the target reference gets larger in amplitude, whereas for a small target reference, the performance of PTOS may lag behind the TOC, since the linear control plays a major role in such cases, resulting a slower transient process to trade for a better robustness. Next, control algorithms were implemented on a TM320F2812 DSP board from the TI corporation. Real-time experiments were subsequently carried out using the Code Composer Studio software system, and the collected data were then processed in MATLAB. The target reference r, or the angular position of motor, was provided in the form of square waves. Experiments were first carried out for three different target amplitudes (π , 2π and 4π ) without load torque (but there is some other disturbance in the system), the results are shown in Fig. 9. It is obvious that the system achieves superior performance in all the tracking tasks, with the overshoot kept within 2%, and the settling times are 0.106, 0.128 and 0.172 s respectively. Fig. 10 gives the results for the target angle 2π under various load torques, it is clear that the output response may get a bit more sluggish with a larger load torque, but the overall performance is still desirable and consistent, as the impact of disturbance is rejected effectively. Figs. 11 and 12 compare the results for the target angle 4π with different speed constraints and load torques. Obviously the output response will get faster with a larger speed upper bound. Fig. 13 presents the experimental results for target angle π with load torque 0.1 Nm. In this figure, the waveforms for the angular position, speed (converted into kRPM), the actual currents iq, id and voltages Uq, Ud are provided. Clearly, the fluxproducing current id is regulated towards zero, while the torqueproducing current iq is controlled to follow the reference command (i. e., the output of the position controller) and settles down to a nonezero value to counteract the load disturbance. To study the performance with respect to parameter variations, the value of b in the control law was replaced by 840 and 1400 respectively (the values of other tuning parameters remained unchanged) for experimental test, and the results are compared with the nominal case (b¼1120). As shown in Fig. 14, some minor degradation can be observed, but the overall performance is still acceptable for the range of variation up to 25%. Such kind of performance robustness against model uncertainty is beneficial for practical applications. 6. Concluding remarks A robust proximate time-optimal control scheme with speed constraint has been proposed by incorporating an extended state observer and a speed regulation law into the PTOS framework. The closed-loop stability has been analyzed theoretically. The method has been adopted to design a position servo controller for PMSMs. MATLAB simulation and experimental results based on TMS320F2812 DSP show that the proposed design is capable of fast and smooth target tracking in full range with a nearoptimal settling time, and has some robustness with respect to the amplitude variation of target reference and disturbance as well as the plant uncertainty. The proposed control method can be easily applied to other servo systems with a double integrator model. Acknowledgment This work was supported by the National Natural Science Foundation of PR China under Grant 61174051.
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