Applying Experienced Self-Tuning PID Controllers to Position Control of Slider Crank Mechanisms

Transcription

1 (00/7) 7 80 Applying Experienced Self-Tuning Controllers to osition Control of Slider Crank Mechaniss Chih-Cheng ao cckao@cc.kyit.edu.tw epartent of Electrical Engineering ao Yuan nstitute of Technology 181 Chung-Shan Road, Lu-Chu Hsiang, aohsiung, Taiwan, ROC 7 Abstract This paper proposes an experienced self-tuning control ethod to the position control of slider-crank echanis. The atheatical odel of a slider crank echanis coupled with M actuator is described, firstly. By the Hailton principle and Lagrange ultiplier ethod, the atheatical forula is derived. Secondly, according to the experience of engineers, the initial paraeters under noral operating condition can be found out. By the sae way, the best paraeters of controller under full-load condition can be found, too. The proposed self-tuning controller will autoatically tune its paraeters under these ranges according to the position error and error derivation. Moreover, the C-based controller is ipleented to control the position of the otor echanis coupling syste. The siulation and experiental results will show the potential of the proposed controller. 1. ntroduction The slider crank echanis is a basic structure in echanical application. t is also widely used in practical application. For exaples, fretsaws, petrol and diesel engines are the typical application of velocity control. ue to its echanical coupling, the physical sense is not enough to derive its dynaic equations. Jasinski et al. [1], Zhu and Chen [] and Badlani et al. [3] have solved the steady state solutions of a slider crank few years ago. According to reference [4], the response of slider crank is dependent on length, ass, daping, external piston force and frequency. Based on the viewpoints of the ratios, length and speeds of the crank to the connecting rod, the transient responses have been investigated []. Mostly, the slider crank echaniss are actuated by the field oriented control M synchronous [-8]. The slider crank echanis driven by M synchronous is to transfer otion to translation otion. Hence, the coputed torque based robust and adaptive controllers are designed to control the otor echanis [8]. With Hailiton principle and Lagrange ultiplier ethod, odel equations of the slider crank echanis coupling with M synchronous are forulated [8-11]. Observing these dynaic equations, highly coupling is existed in these nonlinear equations. Hence, the traditional control schee usually is not suitable to this echanis.

2 8 Applying Experienced Self-Tuning Controllers to osition Control of Slider Crank Mechaniss The ethod is the ost popular controller up to now. espite the progression of any control theories, the controller is still the ajority of industrial processes [1-14]. ue to the easily understanding of the physical sense for paraeters of controller, engineers used to apply it to practical objects. However, the controller is not robust to wide paraeter varying and large external disturbance. Especially for the highly coupling nonlinear syste of slider crank echaniss, the controller is lack of adaptive capability. Usually, the paraeters of controller are anually tuned under ideal condition, that is the operating point without load. However, these paraeters are ostly not suitable for the condition with full load. To achieve practical requireent, engineers have to adjust the paraeters under different operating conditions. However, the robustness is liited with a sall range. A rule to overcoe this disadvantage is called self-tuning rule. Many researches and reports of self-tuning have been published. The paraeter tuning at any tie instance is usually based on a structurally fixed atheatical odel produced by on-line identification procedure [1-17]. Unfortunately, recent plants are ostly difficult to obtain their fixed atheatical odels. However, experts easily obtain the ost proper paraeters for no-load and full load. This paper proposed an intelligent self-tuning controller. The tuning ethod is based on the skilled engineers experience. Engineers will easily accept the straightforward design procedure. At the sae tie, the robustness will be extended. This paper is arranged as follows. Firstly, the atheatical odel of a slider crank echanis coupled with a M synchronous is derived. Second, the tuning ethod of the proposed controller will be discussed. Section 4 will show siulation results. n section, the C-based controller is set up for experiental. The proposed controller is applied to position control of a slider crank. The experiental results are also presented in this section.. Slider Crank Actuated by a M Synchronous.1 M Synchronous A odel of a M synchronous otor can be siplified to the following block diagra. Fig 1 Block iagra of a M synchronous otor Usually, the M synchronous otor is coupled with a gear speed reducer with a gear ratio of n. Hence, the applied torque can be described as τ = n i nj θ nb θ (1) ( ) t q where τ is the torque in the direction of r r ω r, t is the torque constant, J and B

3 9 are the inertia and viscous daping ration, respectively.. Slider Crank Mechanis n this section, Hailton's principle and Lagrange ultiplier are used to derive the differential equation for the slider-crank echanis. The slider crank echanis syste is shown in Fig.. Y r O A 3 1 θ φ R l B τ l F E F B, X b X Fig A Slider Crank Mechanis Syste The slider-crank echanis consists of three parts: crank, rod and slider. The holoonic constraint equation [1] is Φ ( Ψ) = r sinθ l sin φ = 0 () Ψ = θ φ. where [ ] T The kineatic velocity is obtained by the first derivation of eq. (), as Ψ = r θ cos θ l φ cosφ = 0 (3) Φ Ψ The kineatic acceleration is obtained by the second derivatives of eq. (), as Ψ = r θ sinθ l φ sinφ = λ. (4) Φ Ψ where λ is the Lagrange ultiplier as λ = r θ sinθ l φ sinφ () The Lagrangian L, that is the total kinetic energy inus the potential energy, is L = 1R θ + l φ + r θ sin θ + rl θ sinθ sinφ r θ sin θ + 3rl θφ sinθ sinφ + 3l φ sin φ gl sinφ The virtual work δw A includes the applied torque τ with the virtual angle δθ, the friction force F B and the external force F E with the virtual displaceent δx B,that is A δw τ δθ + ( FB FE ) δx B = τ δθ + FBE ( r sinθδθ l sinφδφ ) where F = µ g sgn( ) and V B is the velocity of the slider B. = (7) B 3 V B ()

4 70 Applying Experienced Self-Tuning Controllers to osition Control of Slider Crank Mechaniss Rewrite eq. () in ters of the generalized coordinate Ψ, then δ where A T A W = δψ Q (8) Q A Q is the generalized force and given as A F = BE r sinθ n ( i J θ B θ ) F BE t q l sinφ r The generalized constraint force can be fored in ter of Lagrange ultiplier as r C T Q = Φ Ψ λ (9) where = [ r cosθ l cosφ ] Φ Ψ. Thus, the virtual work by all constraint reaction forces is δw C T C = δψ Q (10) The general for of Hailton's principle is 0 = = t t1 t t1 A C [ δl + δw + δw ] T L δψ Ψ dt d L Q dt Ψ A + Q The eq. (11) ust hold for all C dt, (11) δ Ψ and δ Ψ( t ) = δψ( t ) 0. Thus, according to 1 = the Euler-Lagrange equations of otion, the dynaic equation can be obtained as T M( Ψ) Ψ + N( Ψ, Ψ ) BU ( Ψ) + Ψλ = 0 (1) where M N ( Ψ) = ( Ψ, Ψ ) 1 1R 1 = 1 n = t 0 ( + ) ( + ) B ; = [ ] r l sinθ sinφ 3 3 r sin θ n J 1 3r l sinθ sinφ 1 φ l 3l sin r θ sinθ cosθ + 3rl φ sinθ cosφ n B θ 1 + θ cosθ sinφ φ sinφ cosφ cosφ 3 rl 3l gl BE U ; ( Ψ) = i q F F BE r sinθ l sin φ where 1, and 3 are the ass of crank, rod and slider, respectively, and r and l are length of crank and rod, and θ and φ are angle of crank and rod. The translation position X b can be obtained by transforing θ, that is X b = r cosθ + l cosφ + l' = r cosθ + ( sin ) 1 l r θ + l' (13) 3. Self-Tuning controller n ost researches, the paraeter tuning is based on known atheatical odels.

5 Even the odels unknown, the on-line identification or paraeter estiation is proposed to establish an estiated odel. However, the odeling error usually causes the unexpected conditions. Fortunately, for any coplex syste, the experienced engineers can easily obtain the ost proper paraeters under no-load and full-load conditions. Take the proportional controller for exaple. Under operating conditions between no-load and full load, the proportional gain should be in the range of the gains of full-load and no-load conditions. Let the proportional gains are respectively 71 in and ax for no-load and full-load conditions. The proportional gain should be decreased fro to along with the error and error derivation. According the coon sense [19], if in the proportional () controller s gain increases, then the rising tie and steady state error will be reduced. Too large gain will ake the great overshoot and extree oscillation. Too sall gain will ake steady state error existed. While the absolutions of error and error derivation are large, the proportional gain should be working on largest nuber, that is ax ax. While the error and error derivation are sall enough, the proportional gain should be in to reduce the steady state error. The relationship can be shown as figure 3. Hence, let the tuning rule is defined as where ( t) = e E ( t ) + E ( t ) γ 0 = in, = ax in (14) and γ is the adjusting rate. E(t) is error function and E(t) is variety of the error function. Fig 3 Tuning Rule of roportional Gain Fig 4 Tuning Rule of ntegral Gain By the sae conception, the integral controller is helpful for steady state and hurtful for transient state. Hence, the integral () controller s gain should be increased

6 7 Applying Experienced Self-Tuning Controllers to osition Control of Slider Crank Mechaniss along with the error and error derivation decreasing. The curve is shown as figure 4. Let the tuning rule is defined as Let ( t) ( ) ( ) E t + E t 0 γ = 1 e (1) and ax and in are the gains under no-load and 0 = in, = ax in full-load conditions respectively. E(t) is error function and E(t) is variety of the error function. Large differential controller can increase the speed of response. At the sae tie, large differential controller will cause large steady state error. Therefore, the differential () controller s gain should be decreased along with the error and error derivation decreasing. The tuning rule can be shown as following figure and equation. Let ( t) ( ) ( ) E t + E t 0 γ = + 1 e (1) and ax and in are the gains under full-load 0 = in, = ax in and no-load conditions respectively. E(t) is error function and E(t) is variety of the error function. Fig Tuning Rule of ifferential Gain ue to the proposed self-tuning ethod based on the engineer s experience, the experienced engineers will easily accept it. 4. Siulation Results n this section, nuerical siulation results are used to deonstrate the potential of the proposed control rule. To deonstrate the perforance, the self-tuning ethod is copared with a fixed control. The actual slider crank echanis diensions are 1 =3.4, =1.18, 3 =1.8, r =0.1, R=0.1, l =0.30, l ' =0.0, t =0.73, J =0.000 and B = The initial

7 angle is ( 0) π rad! " # 1 θ =. The objective is to control the desired translation position to Figures -7 show the response for syste with the fixed controller. The paraeters of controller is anually setup for optial perforance under no-load no condition, where =3.9, specifications are settling tie overshoot M <% and steady state error no =.97 and t =0.sec, rising tie s no 73 = The desired t r =0. sec, axiu e <1%. Figure shows the responses of translation position and angular under no load condition. These curves show this controller can achieve the desired specification. ss Tie(sec) (a) Response of Translation osition Fig Siulation Response of Fixed Controller (No-Load Condition) Figure 7 shows the responses under different operating conditions. The paraeter variation is appeared by the ass of slider crank 3 changed fro 1.8g to 7.g. The external force is changed fro 0Nt to Nt. Obviously, the fixed obtained under no-load condition cannot achieve the robustness with paraeter varying and load existed Noral 3=1.8 Fe=0 3=7. Fe=0 3=1.8 Fe= 3=7. Fe= Noral 3=1.8 Fe=0 3=7. Fe=0 3=1.8 Fe= 3=7. Fe= (a) Response of Translation osition Fig 7 Siulation Response of Fixed Controller (No-Load Condition)

8 74 Applying Experienced Self-Tuning Controllers to osition Control of Slider Crank Mechaniss Let the paraeters of controller is anually setup for optiization with full-load existed. The optial paraeters for controller with full-load are full =4.194, full =1.997 and full = The response of the syste with full-load is shown in figure 8. As see to this curve, the fixed controller can satisfy the request of paraeter and external load. However, once the paraeter varying and external load are reoved, the responses will cause the large steady-state error and unexpected transient state shown in figure (a) Response of Translation osition Fig 8 Siulation Response of Fixed Controller (Full-Load Condition) Noral 3=1.8 Fe=0 3=7. Fe=0 3=1.8 Fe= 3=7. Fe=. Noral 3=1.8 Fe=0 3=7. Fe=0 3=1.8 Fe= 3=7. Fe= (a) Response of Translation osition Fig 9 Siulation Response of Fixed Controller (Full-Load Condition) The proposed self-tuning controller is based on these two optial paraeters. Let the noral paraeters are based on no-load condition, that is = 0 no full =.97, and = =0.47, no = 0 no = = 0 no =3.9, = The tuning ranges are selected as full no =1.70 and = =. full no

9 ! " # Applying eqs (14)~(1), the responses of the proposed control rule are shown in figure 10. Obviously, the proposed self-tuning controller displays the robust characteristic. Under different operating conditions, it can still aintain the desired perforance Noral 3=1.8 Fe=0 3=7. Fe=0 3=1.8 Fe= 3=7. Fe=. Noral 3=1.8 Fe=0 3=7. Fe=0 3=1.8 Fe= 3=7. Fe= (a) Response of Translation osition Fig 10 Siulation Response of Self-Tuning Controller. Experiental Results n order to deonstrate the proposed control rule, a C-based experiental equipent is setup in this paper. The experiental instruent of slider crank is divided into three parts: actuator, slider crank and controller. The photographic is shown in figure 11. The first part consists of a M Synchronous otor, driver. The driver is worked on 3-phase, 0 V and 0 Hz. The slider crank is coupled with the M otor. The translation position is easured by a potentioeter. The output of potentioeter is 0~V, which apped to real translation position is 0~0.. The controller is based on a C with entiu-8 CU. The data acquisition interface card (Advantech CO., CL-1800) is installed in the SA bus to handle the A/ and /A process. The graphical software of Siulink is used to ipleent the proposed control rule. At the sae tie, the linear converts between the physical scale and voltage fro sensors are also worked in this software. To carry out the paraeter varying and external load is to add an external ass (7.4g) on slider.

10 7 Applying Experienced Self-Tuning Controllers to osition Control of Slider Crank Mechaniss Fig 11 Experiental nstruent of Slider Crank Based on the sae requireent of siulation, the experiental results are shown in figures 1~17. Firstly, the fixed setup under no-load is experiented. Under no-load condition, the experiental result is shown in figure 1. However, the response shown in fig 13 has large overshoot when the load added. t shows the bad robustness of fixed controller (a) Response of Translation osition Fig 1 Experiental Response of Fixed Controller (No-Load Condition) with No-Load

11 ! " # (a)response of Translation osition Fig 13 Experiental Response of Fixed Controller (No-Load Condition) With 7. g External Load The second experient is fixed controller whose paraeters are obtained under full-load condition. Figure 14 shows the experiental result. t shows the anually chosen paraeters can achieve desired requireent under full-load condition. However, figure 1 shows the sae controller applied when load reoved. Obviously, the fixed controller cannot overcoe the large change in operating situation (a) Response of Translation osition (b)response of Angular Fig 14 Experiental Response of Fixed Controller (Full-Load Condition) With 7. g External Load (a) Response of Translation osition Fig 1 Experiental Response of Fixed Controller (Full-Load Condition) With No-Load

12 78 Applying Experienced Self-Tuning Controllers to osition Control of Slider Crank Mechaniss Finally, applying the proposed self-tuning controller to the sae experiental instruent. The responses of translation position and angular with no-load existed are shown in figure 1. t can achieve the sae perforance with the fixed controller under no-load condition. The response with load existed is shown in figure 17. The dynaic response is alost the sae as the result of no-load. t also approves that the experienced self-tuning controller has the robustness to paraeter variation and external load changed (a) Response of Translation osition Fig 1 Experiental Response of Self-Tuning Controller With No-Load (a) Response of Translation osition Fig 17 Experiental Response of Self-Tuning Controller With 7. g External Load.Conclusion This paper proposed a self-tuning approach for controllers. Based on the experienced engineer, the noinal values and tuning ranges of paraeters can be assigned. According to the coon sense, an intelligent self-tuning rule is proposed in this paper. A C-based controller is ipleented and applied to the

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