Robust Bounded Control Design and Experimental Verification for Permanent Magnet Linear Motor With Inequality Constraints

The robust bounded control problem for the permanent magnet linear motors with inequality constraints is studied. Firstly, the dynamical model of the system is built, through the state transformation is used to satisfy the inequality constraint of the position output. Thus, the controller after the state transformation can ensure that the control output of the linear motor stays the desired range. Selecting the appropriate boundary function, and then the upper and lower bounds of the control output can be set according to our control requirements. The control scheme can guarantee uniform boundedness and uniform ultimate boundedness. The results of experiments and simulation show that the proposed algorithm can assure the control output within the desired range regardless of the uncertainty.

The associate editor coordinating the review of this manuscript and approving it for publication was Shuai Liu . and other defects of the rotating motor. Theoretical travel is 23 not limited. Therefore, the permanent magnet linear motor 24 can satisfy the needs of high speed, high dynamic response, 25 and high precision when driving directly. In addition, the 26 control performance of PMLM is also influenced by all kinds 27 of uncertain factors, such as uncharted external disturbance, 28 friction [7], etc. In the study of PMLM, it is essential to 29 improve the control performance [8], [9]. 30 In the control technology of PMLM, the precise control of 31 linear motors is essential [10], but the influence of unknown 32 external disturbance and friction increases the difficulty of 33 linear motor position control. [11] designed a robust con-34 troller composed of adaptive compensator, PID feedback con-35 troller and feedforward compensator. [12] and [13] proposed 36 a typical adaptive robust control, which combines robust 37 control and traditional adaptive control to overcome various 38 under the conditions of uncertainties. And the displacement 95 of permanent magnet linear motor system can be controlled 96 within the defined range. The effect of control can meet the 97 requirements of some high applications. 99 A complete PMLM system includes cushion, linear motor, 100 displacement sensor, guide rail, and drag chain, etc [26]. 101 The cushion is used to prevent the motor body from being 102 damaged when the moving parts of the linear motor move 103 back and forth. And the displacement sensor is used to detect 104 the position of the motor in real time. 105 For PMLM, its dynamic model is usually approximated 106 by a second-order system [27], which can be expressed by 107 equation (1).

II. DYNAMIC MODEL OF PMLM
Here, y 1 represents the moving displacement of the linear 110 motor, y 1 ∈ (y m , y M ), y m represents the lower bound of 111 displacement and y M represents the upper bound of displace-112 ment. y 2 represents the linear velocity of PMLM, R represents 113 the impedance, m represents the mass of the motor, k f repre-114 sents the power constant, k e represents the back electromotive 115 force, and d(t) can be regarded as the disturbance including 116 ripple force and friction force. v(t) is the control signal.

117
It is assumed that the disturbance of the motor consists of 118 two terms, namely the friction force and the ripple force, and 119 friction varies with load. It is expressed as follows 120 Here, F ripple is the ripple force,F fric is the friction force.

122
F fric is expressed by the equation (3) 123 Here A 1 , A 2 , A 3 , ω are constants. 132 We choose y as the generalized coordinate, and y is the 133 displacement of linear motor. We can convert the linear motor 134 model of the approximate second-order system into the gen-135 eral form of the system dynamics model.

Furthermore,
⊂ R p is compact and unknown, which 145 represents the possible bounding of σ .

146
Therefore, the dynamical model of PMLM can be 147 expressed as follows: is not able to exceed the limitation (y m , y M ).

172
Assume that the motor position satisfies the bilateral 173 constraint as follows: 175 Let's set the state transformation equation as: Here y represents the actual position, y d represents the  From the equation (8), we can see that y → y M as 191 x → +∞ and y → y m as x → −∞. Thus, by selecting an 192 appropriate function x, the state transformation can convert 193 the state y with limitation to the state x without limitation. 194 From equation (8), we can get Take the derivative of the equation (10) to get the first 197 derivative Take the derivative of the equation (11) again to get the 200 second derivative Substituting the equation (11) and (12) into the dynamics 203 equation (6), we can get: Further simplified to the general form of the dynamical 207 model This transforms the bounded constraint problem on y into 211 an unbounded constraint problem on x. 212 Accordingly, we can get the inertia matrix H , Coriolis 213 force/centrifugal force matrix C and friction vector F of the 214 linear motor mechanical system after the state transformation. 215 is positive definite symmetric matrix, and is uni-219 formly bounded for all x. It can be expressed as where ς , ς are positive constant.

222
(ii) The matrixḢ (x) − 2C(x,ẋ) is skew symmetric for all 223 x,ẋ. That is, for any vector ζ , the output tracking error is: and hence 248 System equation (5) can be rewritten as where H , C and F are uncertain terms which depend 261 on σ , H (·), C(·) and F(·) are the nominal terms.

262
Assuming that the boundary of uncertainty is estimated 263 by ρ.
Note that for a given S >0 is constant. Apparently, if all 270 the uncertainties in the mechanical system disappear, then 271 ≡ 0. The problem of trajectory tracking is to design a controller 275 to guarantee the tracking error vector e(t) small enough.

276
The control torque τ (t) can be geted by where K p is the proportional control parameter and K v is 280 the differential control parameter. These control parameters 281 come from traditional PID control. The scalar γ > 0. K p , 282 K v and γ all are constant, which are the flexible parameter 283 variables.

284
For the mechanical system expressed in equation (5), the 285 control equation(25) adopted would make the tracking error 286 e(t) uniformly bounded and uniformly ultimately bounded. 287 The last item is a robust feedback item, mainly for uncer-288 tainty. In order to prove that the Lyapunov function V is a suitable 296 candidate, V must be proved to be positive definite and 297 decreasing.

298
With equation (16), H (x, σ, t) is bounded, so It can be easily proved that ϒ > 0. Therefore, V is positive Because the inertial matrix in equation (16) is bounded,

333
Consider the dynamic properties of the mechanical system 334 (17), hence With equation (22) 337 Substituting equation (37) into equation (33), it is easy to 342 get that The uniform boundedness performance follows. Upon 348 invoking the standard arguments as in [33], that is, given any 349 r > 0 with e(t 0 )) ≤ r, where t 0 is the initial time, there is 350 a d(r) given by such that e ≤ d(r) for all t ≥ t 0 .

354
Furthermore, uniform ultimate boundedness also follows. 355 That is e ≤ d, ∀t ≥ t 0 + T (d, r), with Please note that by adjusting the parameter variables K p , 359 K v and γ from equation (41), d can become arbitrarily small. 360

361
As shown in Figure(2), we summarize the flow chart of robust 362 bounded control with inequality constraints for PMLM. 363 In this paper, the dynamical system satisfying the output state 364 with inequality constraint for the linear motor is obtained. 365 By selecting an appropriate transformation function, the out-366 put state y with limitation is transformed into the state x 367 without limitation. Then we have robust control over the 368 unbounded state x. No matter how the uncertain external 369 disturbance is, the control output y can be kept within the 370 controllable and safe range. For the transformed state x, 371 we designed a robust controller based on PD. By selecting 372 proper K p , K v , γ , and S, the error of linear motor trajec-373 tory tracking is small enough. As K p increases, the system 374 response speeds up, and the steady-state error decreases. 375  We present the step signal as shown below: We present the sinusoidal signal, which enables the per-404 manent magnet linear motor system to achieve sinusoidal 405 trajectory tracking. We present the sine signal as shown 406 below: We analyze the results of the step signal response. The 409 comparison of response between the robust controller with 410 and without inequality constraints to control the PMLM sys-411 tem is shown in Figure 3. Next, we analyze the simulation results of the sinusoidal 420 signal response. The comparison of response between the 421 robust controller with and without inequality constraints to 422 control the PMLM is shown in Figure 4. The errors of 423 sinusoidal trajectory tracking are shown in Figure 5. The 424 comparison of the control inputs is shown in Figure 6.

425
As can be seen from Figure 4, the initial displacement 426 of the linear motor is 50mm, which satisfies the condition that 427 the initial value is incompatible. The linear motor without 428 the inequality constraint reaches stability at 2s when track-429 ing the sinusoidal curve. The maximum overshoot of the 430 sinusoidal tracking is 102mm, which exceeds the set upper 431 bound 101mm. The sine tracking of the linear motor with 432 inequality constraint reaches stability at 0.2 s. The boundary 433 condition is not exceeded and the error range is controlled 434 within [−0.02 mm, 0.02 mm]. From the simulation results, 435 it can be seen that the control effect with the inequality 436 constraint is significantly better than the effect without the 437 inequality constraint.    inequality constraints. Here, the resistance and inductance 454 will tend to become larger to some extent with increasing 455 temperature. But these changes are negligible and are not 456 considered in this paper at this time. We select the appropriate 457 controller parameters as follows, K p = 120; K v = 10; S = 1; 458 γ = 0.6954. 459 Figure 8 shows the step responses of PMLM with and 460 without inequality constraints. We can see that in the exper-461 iment of step response, The step time without the inequality 462 constraint is 1s to reach stability. The maximum value during 463 the step response is 102mm, which exceeds the set upper 464 bound 101mm. The step response with inequality constraint 465 reaches stability at 0.8 s, and the maximum value during 466 the response is close to 100mm. So the robust controller 467 with inequality constraints can limit the motor displacement 468 within the boundary.

469
The sinusoidal trajectory tracked by the linear motor is 470 x = 0.095sin( t 4 )m. The sinusoidal trace of the robust con-471 troller without inequality constraints is shown in Figure 9. 472 In the sinusoidal tracking experiment without inequality 473 constraint, the artificial external disturbances are applied at 474 7s,9s,11s,13s, respectively. We can clearly see that when 475 the motor is controlled by a robust controller without 476 inequality constraints, the displacement of the linear motor 477 reaches the minimum value −106mm and maximum value 478 105mm at 11s and 13s respectively, which exceed the set 479    Figure 14.

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In this paper, a robust bounded control algorithm for per-  Next, we will take the motor temperature compensation

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Her research interests include nonlinear system 683 control theory with applications to motor control, 684 analytical mechanics, multi-agent, and robotics.