Improving Closed-Loop Signal Shaping of Flexible Systems with Smith Predictor and Quantitative Feedback

Transcription

1 Article proving Closed-Loop Signal Shaping of Flexible Systes with Sith Predictor and Quantitative Feedback Withit Chatlatanagulchai a, *, Puwadon Poedaeng b, and Nitirong Pongpanich c Control of Robot and Vibration Laboratory, Departent of Mechanical Engineering, Faculty of Engineering, Kasetsart University, 5 Nga Wong Wan Rd, Lat Yao, Chatuchak, Bangkok 9, Thailand E-ail: a fengwtc@ku.ac.th (Corresponding author), b oopuwadon@hotail.co, c aook-aik@hotail.co Abstract. nput shaping is a technique used to ove flexible systes fro point to point rapidly by suppressing the residual vibration at the destination. The vibration suppression is obtained fro the principle of destruction of ipulse responses. The input shaper, when placed before the flexible syste inside the control loop, proves to deliver several benefits. However, this so-called closed-loop signal shaping has one ajor disadvantage that it adds tie delays to the closed-loop syste. Being a transcendental function, the tie delays cause difficulty in analysis and design of the feedback controller. n ost cases, the tie delays also liit the axiu achievable bandwidth. n this paper, for the very first tie, Sith predictors were applied to the closed-loop signal shaping to reove the tie delay fro the loop. t was shown in siulation result that the detriental effect of the tie delays was copletely reoved in the case of perfect plant odel. The quantitative feedback control was used in the study to quantify the aount of achievable bandwidth and to suppress vibrations fro the plant-input disturbance. Keywords: Sith predictor, input shaping, closed-loop signal shaping, vibration suppression, quantitative feedback. ENGNEERNG JOURNAL Volue ssue 5 Received 7 Deceber 5 Accepted 8 March 6 Published 5 Noveber 6 Online at DO:.486/ej

2 DO:.486/ej ntroduction Flexible systes, when oved fro point to point rapidly, exhibit residual vibration at their destination. nput shaping technique suppresses this residual vibration by superposition and cancellation of ipulse responses. pulse sequence is fored as an input shaping filter and ipleented as an FR filter. The aplitudes and tie locations of the ipulses are coputed to obtain coplete cancellation of ipulse responses. The input shaping technique was originally proposed by Sith in [] under the nae Posicast control. t was later ade ore robust to uncertainty in the ode paraeters by Singer and Seering in [] and was given the nae nput shaping. Since its origin in the 5 s, the input shaping technique has found nuerous applications with flexible systes. Cranes are aong the ost popular applications of the input shaping technique due to the requireent to suppress payload swing. Notable literature includes those of gantry crane [3], bridge crane [4], tower crane [5], and boo crane [6]. Robot anipulators have flexibility in their joints and links that have to be taken into account during the design. Literature includes those of flexible-link robot [7] and flexible-joint robot [8]. nput shaping has also been used in suppressing the liquid sloshing in container [9], suppressing vibration of the flexible appendages of the flexible spacecraft [], suppressing probe vibration in coordinate easuring achine (CMM) [], designing ca profile that reduces vibration of the cafollower syste [], reducing vibrations during sall otion and unloading operations of a telescopic handler [3], suppressing vibration in a cherry picker due to its flexible joints [4], suppressing the oscillatory transients in a dual-solenoid positioning servo actuator [5], and eliinating bouncing in a icro-electro-echanical-syste (MEMS) contact switch [6]. The input shaping filter can be placed either outside the loop or inside the loop. Outside-the-loop input shaping filter is designed fro the ode paraeters (natural frequency and daping ratio) of the closed-loop syste. t can only suppress vibration fro the reference signal. However, there is no detriental effect fro the tie delay of the input shaper because the input shaper is placed outside the loop. When the input shaper is placed inside the loop before the flexible plant, so-called closed-loop signal shaping (CLSS) [7], the input shaper is designed fro the ode paraeters of the flexible plant. The CLSS offers several advantages over the outside-the-loop input shaping because, in CLSS, the input shaping filter becoes a part of the closed-loop syste. First, CLSS can suppress the vibrations induced by the reference signal, the plant-output disturbance, and noise. Second, CLSS can easily be designed for flexible plants that have hard nonlinearities. The hard nonlinearities, such as actuator saturation, deadzone, and backlash, can odify the input-shaped coand thus reduce its vibrationreducing perforance. n CLSS, the input shaper is placed directly before the hard nonlinearities; therefore, it is easy to design the input shaper such that its input-shaped coand will not be odified by the hard nonlinearities. Third, CLSS can be used to greatly iprove control perforance of the huan operator, perforing anual control of the flexible syste. n anual control, the huan operator is the feedback controller. t was shown by Potter and Singhose in [8] that huan can control the flexible syste better when there is an input shaper placed in front of the flexible syste to reduce its vibration. Nevertheless, because the input shaper contains tie-delay ters, the CLSS bring the tie delay into the closed-loop syste, which is the ajor disadvantage of the CLSS. The closed-loop characteristic equation of the syste with CLSS possesses exponential tie-delay function. Since the exponential function is transcendental, it is difficult to analyze the stability property or to design a controller for the syste. For exaple, Huey and Singhose [9] showed that, with CLSS, there are infinitely any root locus branches. Only in siple cases, a finite nuber of root locus branches closest to the real axis can be used in stability analysis. Fro the root locus stability analysis, Huey and Singhose [9] pointed out that CLSS requires quite an accurate plant odel to avoid unexpected instability. Huey and Singhose [] showed that CLSS cannot suppress the vibration induced by the plant-input disturbance or by non-zero initial conditions. Moreover, it is a well-known fact [] that tie delay in the loop can liit the axiu achievable bandwidth and phase argin and; therefore, liit the perforance of the control syste. Sith predictor was proposed by Sith in [] to copensate the effect of the tie delay. n the case of having perfect plant odel, that is, the plant odel is exactly the sae as the actual plant, the Sith predictor eliinates the tie delay fro the closed-loop characteristic equation. As a result, one can design the feedback controller as if the tie delay did not exist in the loop. However, the Sith predictor has several disadvantages of its own. First, when the plant odel isatches the actual plant, the tie delay will not be reoved copletely fro the loop; therefore, the Sith predictor can only iniize the effect of the tie delay, allowing tighter control to be used. Moreover, the reaining tie delay in the 56 ENGNEERNG JOURNAL Volue ssue 5, SSN 5-88 (

3 DO:.486/ej characteristic equation can affect the stability. Second, if the process has an integrator (as is norally the case of the flexible syste), a constant plant-input disturbance ay result in non-zero steady-state tracking error. Both disadvantages will be investigated in this paper. The Sith predictor was applied to the CLSS in our pioneering work [3]. This paper expands on the work in [3], yet it still reains one of the very first work that proposes this idea. Robust stability and robust perforance (the case when the plant odel differs fro the actual plant) of the proposed technique is explored in this paper. When the plant odel differs fro the actual plant, unexpected result such as instability ay occur fro using Sith predictor, depending on the type of plant and controller. n this paper, classical Sith predictor was applied to the CLSS. The quantitative feedback control [4] was used as the priary feedback controller. The paper offers the following advantages over the existing CLSS: The Sith predictor reoves the tie delay of the input shaper fro the characteristic equation. As a result, the feedback controller can be designed as if the tie delay was not present. To avoid unexpected instability caused by the CLSS when the plant odel is uncertain, a quantitative feedback control was used to ensure stability for all expected plant uncertainties. The quantitative feedback control was used to reject the vibration induced by the plant-input disturbance or by non-zero initial conditions. All the advantages of the CLSS are still preserved with the proposed technique. This paper is organized in the following way. Section introduces the input shaping technique, the classical CLSS, the CLSS with classical Sith predictor, and the flexible syste with one flexible ode that will be used in siulation. Section 3 contains the CLSS with Sith predictor and quantitative feedback control, together with siulation result. Section 4 presents conclusions and future work.. Closed-Loop Signal Shaping with Sith Predictor.. nput Shaping A floating oscillator plant with transfer function P s s n, nsn () where s n is the natural frequency and is the daping ratio, has two oscillatory poles at, n jd, where d n is the daped natural frequency. A zero-vibration (ZV) input shaper [] has transfer function where A K A K K ZV /,,, ZV /, t / d, S s A A e () ts, K exp /. t is easy to show that S s which eans that the input shaper uses its zeros to cancel the plant s two oscillatory poles to reove the vibratory ode fro the syste, resulting in zero residual vibration. n fact, the input shaper t () has infinitely any zeros, as can be shown by setting j A Ae, resulting in n and kd, k, 3, 5,... Note that the pair of input shaper s zeros that is the closest to the real axis is used to cancel the plant s two oscillatory poles, yielding the shortest input shaper duration t. t can be seen that the input shaper () requires the knowledge of the ode paraeters n and. For ore robustness to paraeter uncertainty, ultiple zeros of the input shaper can be placed over the plant s oscillatory poles, leading to the so-called zero-vibration-and-derivative (ZVD k ) input shaper with transfer functions ts k SZVDk s A A e, k,, 3,.... ENGNEERNG JOURNAL Volue ssue 5, SSN 5-88 ( 57

4 DO:.486/ej Classical CLSS Consider the classical closed-loop signal shaping [7] as shown in Fig.. P represents a flexible syste, whose transfer function is P s P s / P s, n where P n and P d are the nuerator and denoinator of P. S is the input shaper. C represents a priary controller. NL represents the hard nonlinearities. r is the reference signal; u is the control effort; d is the plant-input disturbance; d O is the plant-output disturbance; n is the noise, and y is the output of interest. d d d O r - C u S NL P y n Fig.. Classical closed-loop signal shaping. The closed-loop transfer functions fro each external signal to the output y of the CLSS in Fig. are P s P s S s C s y s r s d s d s P s S s C s P s S s C s P s S s C s O P s S s C s P s P s r s d s do s P s P s S s C s P s P s S s C s P s P s S s C s n n d d n d n d n. (3) By inspecting the transfer functions above, the first advantage of the CLSS that it can suppress the vibrations induced by r, d O, and n can be stated. Pd s contains oscillatory poles of the flexible plant P s ; therefore, the oscillatory poles are also the poles of the closed-loop transfer functions. For the closed-loop transfer function fro r to y, the zeros of 58 ENGNEERNG JOURNAL Volue ssue 5, SSN 5-88 ( S s are designed to cancel with those of P s, so the vibration induced by r is suppressed. For the closed-loop transfer function fro d O to y, the zeros of Pd s cancel with the oscillatory poles of the plant, so the vibration induced by d O is also suppressed. The transfer function fro n to y is erely the negative of that fro r to y ; hence, the vibration induced by the noise n can also be suppressed. The second advantage of the CLSS that it can easily be designed for flexible plants with hard nonlinearities can be explained by inspecting the CLSS diagra in Fig.. The hard nonlinearity, NL, can be actuator saturation, deadzone, or backlash. NL changes the shaped signal coing out of the input shaper, S, so the perforance of the input shaper is degraded. With the CLSS, an inverse of the nonlinearity, NL, can be placed before the input shaper, S, so the shaped output coing out of the input shaper will not be altered by the nonlinearity, NL ; therefore, the perforance degradation can be avoided. The third advantage of the CLSS that it can be used to iprove huan control of the flexible syste can also be shown by inspecting the CLSS diagra in Fig.. n anual control, huan is used as the priary controller, C, with eyes aiing at the output, y, of the flexible plant and hand gives the control effort, u. With CLSS, where an input shaper, S, is placed before the flexible plant, P, the output, y, will not oscillate; hence, the huan control perforance is iproved. This result was confired in [8]. The first disadvantage of the CLSS is that it cannot be used to suppress the vibration induced by the plant-input disturbance, d. This fact can be seen fro the closed-loop transfer function fro d to y. d

5 DO:.486/ej n The zeros of P s do not cancel with those of Pd s. Because non-zero initial conditions can be thought of as a plant-input disturbance [5], the CLSS cannot suppress the vibration induced by the non-zero initial conditions as well. The second disadvantage of the CLSS coes fro the tie-delay ter in the input shaper (). Fro Eq. (3), the transfer function of the input shaper, S, appears in the denoinator of the ts closed-loop transfer functions. Because the tie-delay ter, e, is a transcendental function of s, conventional control design and stability analysis do not apply. Soe rational-function approxiation (e.g. Pade approxiation) of the tie-delay ter ust be applied, introducing additional uncertainty into the syste. Moreover, the tie-delay ter in the CLSS results in non-iniu phase syste which, according to [], liits the perforance of the control syste by liiting the axiu achievable bandwidth. The third disadvantage of the CLSS was pointed out by Huey and Singhose [9] that the CLSS requires accurate plant odel in order to avoid unexpected instability..3. CLSS with Classical Sith Predictor Consider the CLSS with the classical Sith predictor as shown in Fig.. The classical Sith predictor, SM, has a transfer function SM s Ps Ps S s. (4) d r - C u S P y SM Fig.. Closed-loop signal shaping with classical Sith predictor. The closed-loop transfer functions fro r and d to y can be coputed as C s Ps C s P s C s S s P s C s S s P s y s r s Ps d s. (5) Coparing Eq. (5) to Eq. (3), the transfer function of the input shaper, S, is no longer present in the denoinator of the closed-loop transfer functions. As a result, conventional control design and stability analysis can be applied to the closed-loop syste. There are two disadvantages of the syste in Fig.. First, when the actual plant, P, and its odel, P ˆ, are different, the closed-loop transfer functions fro r and d to y becoe C s S s P s ˆ ˆ y s r s C s P s C s P s S s C s S s P s (6) d s. P s C s Pˆ s C s Pˆ s S s ˆ ˆ C s P s C s P s S s C s S s P s t can be seen that when the plant uncertainty exists, the tie delay fro the input shaper, S, is not reoved in its totality fro the denoinator of the closed-loop transfer function and therefore still affects the control syste. Second, if the disturbance, d, is present, the steady-state tracking error ay not be zero. ENGNEERNG JOURNAL Volue ssue 5, SSN 5-88 ( 59

6 DO:.486/ej Flexible Syste with One Flexible Mode As an exaple, consider a flexible syste with one flexible ode as shown in Fig. 3. n general, the syste represents two entities, connected via a flexible part, which encopasses a large ajority of actual rigidflexible systes. The driving one has an absolute position and ass of x and, and the driven one has x and. k, c, and c are spring stiffness and two daping constants. f is the control force. The objective is to ove both asses fro the origin to a displaceent X with zero residual vibrations and in a shortest tie possible T, that is, x X x,. x X x tt tt x x c c f k Fig. 3. Flexible syste with one flexible ode. The transfer function fro f s to x s is given by cs k Ps. 3 s s c c c s k k cc s ck (7) For siplicity, the ZV input shaper () and a P controller, C s K p, Ts i (8) where K p and T i are controller gains, are considered. For siulation purpose, let c 5 kg/s, c. kg/s, kg, kg, and k 8 kg/s. The plant (7) has one flexible ode with n rad/s and.5. The ZV input shaper () has the ipulse aplitudes A.54 and A.459 and the tie location t.57 s. The P controller (8) was designed as K p and Ti. The reference, r, is a unit-step signal, and there exists a step plant-input disturbance, d, of agnitude during 5 th to 3 th seconds. Figure 4 shows the tracking result where the dash lines are the reference signal and the solid lines are the output, y x, signal. Figure 4(a) shows the case without using the input shaper. The output is vibrating. Figure 4(b) shows the case when the CLSS with classical Sith predictor (4) was ipleented. The input shaper in the CLSS setting suppresses ost of the vibration fro the reference signal. However, it cannot suppress the vibration induced by the plant-input disturbance. Moreover, it is a well-known result that the classical Sith predictor yields non-zero steady-state tracking error when disturbance is present (Section 9.8 of [6]). 6 ENGNEERNG JOURNAL Volue ssue 5, SSN 5-88 (

7 DO:.486/ej x x.5 Vibration fro disturbance Non-zero error.5 Disturbance is present.5 Disturbance is present (a) Tie (s) (b) Tie (s) Fig. 4. Tracking result. (a) Without input shaper. (b) Use CLSS with classical Sith predictor as shown in Fig.. Dash lines are the reference signal; Solid lines are the output y x. To cope with the non-zero tracking error due to the disturbance, any researchers have already proposed the odifications to the classical Sith predictor. See, for exaple, two classical works by Watanabe and to [7] and Astro et al. [8], a two degrees-of-freedo ethod by Norey-Rico and Caacho [9], and a recent publication by [3]. This paper applies a schee, called feedforward Sith predictor [3], to eliinate non-zero tracking error due to the disturbance. n this schee, the disturbance, d, is easured and is fed to the Sith predictor. By doing so, the effect of the disturbance is counteracted before it can change the output. Our ethod is based on the quantitative feedback control [4]. The two disadvantages fro using the classical Sith predictor, naely the uncertainty in the plant odel causing unexpected perforance degradation and the non-zero steady-state tracking error under the presence of the disturbance, will be considered. 3. CLSS with Sith Predictor and Quantitative Feedback Figure 5 shows the proposed syste. The input shaper, S, is designed to suppress the vibration ode of the flexible plant, P. The classical Sith predictor, SM, reoves the tie delay of the input shaper fro the loop so that conventional feedback control design can be applied. The feedback controller, C, attenuates the effect of the disturbance, d, and ensures closed-loop stability. The controller, C, together with the pre-filter, F, enables good tracking perforance. The disturbance, d, is easured and fed to the Sith predictor to eliinate non-zero tracking error due to the presence of the disturbance. d r F - C u S P y SM Fig. 5. CLSS with classical Sith predictor and quantitative feedback control. Suppose the actual plant, P, and its odel, P ˆ, are different by a ultiplicative uncertainty, given by P Pˆ, (9) ENGNEERNG JOURNAL Volue ssue 5, SSN 5-88 ( 6

8 DO:.486/ej where is soe constant. Fro Eq. (6) and (9), the closed-loop transfer functions fro r and d to y are given by ˆ F sc s S s Pˆ s Pˆ s C s Pˆ s C s Pˆ s S s y s r s d s. C s Pˆ s S s C s P s S s () Note that the input shaper, S s, is siply a tie delay with unit steady-state aplitude. After the delay tie, the ter C s P ˆ s C s Pˆ s S s Eq. () is coparable to in () should go to zero. Using the fact that ˆ Pˆ s S s P s, ˆ ˆ ˆ F s C s S s P s Pˆ s S s y s r s d s. C s P s C s P s Hence, the proposed CLSS with Sith predictor and quantitative feedback control shown in Fig. 5 can be represented by a diagra in Fig. 6. The controller, C, and the pre-filter, F, will then be designed to cover the ultiplicative plant uncertainty in (9) and to eet the tracking, stability argin, and plant-input disturbance rejection specifications. d r F - C u P S y Fig. 6. Coparable CLSS with classical Sith predictor and quantitative feedback control. Consider again the plant (7). Suppose the plant paraeters, c, c, and k, have % uncertainties, 3 that is, c 45, 5, 55, c.9,.,., and k 7., 8, 8.8. There are 3 7 plant variations. t is easy to show, for exaple, in [] that these paraetric uncertainties can be changed to the ultiplicative uncertainties in the for (9). Three frequency-doain specifications iposed are: ) Stability argin C j P j C j P j 3 db,.,.,.5,, 3, 5 rad/s, () ) Tracking P jc j P j C j F j,.,.,.5, rad/s, () where the upper and lower bounds are given in Table. 3) Plant-input disturbance rejection 6 ENGNEERNG JOURNAL Volue ssue 5, SSN 5-88 (

9 DO:.486/ej P j C j db,.,.,.5, rad/s. P j (3) Note that the stability argin specification () is equivalent to a gain argin of 4.65 db and a phase argin of 4.46 degrees (see [4]). The tracking specification () was iposed to the axiu frequency of rad/s, which is the designed closed-loop bandwidth of the syste. This is because the natural frequency of the syste (7) is.9 rad/s, so the closed-loop bandwidth is liited by the natural frequency. The plant-input disturbance rejection specification (3) was also iposed at low frequencies, up to the bandwidth. The stability argin specification () was iposed for all frequencies, including the high frequencies beyond the bandwidth. Table. Upper and lower tracking bounds. rad/s...5 db db For each frequency,, the three frequency-doain specifications ()-(3) were converted into three bounds on the Nichols chart. The three bounds were intersected with one another to create one worst-case.,.,.5,, 3, 5 rad/s. bound. n Fig. 7, there are six worst-case bounds for six frequencies, The open-loop shape of L j P jc j, which is the plot between the open-loop phase, L j, versus the open-loop gain, L j, was shaped by altering the controller, Cs. Each frequency on the open-loop shape ust lie in the allowable region of the bounds. For our three specifications ()- (3), the allowable regions are either above or outside the bounds. Figure 7(a) shows the open-loop shape before the loop shaping, that is, the controller is Cs. t can be seen that an addition of 3-dB gain is required for the frequency.5 rad/s to satisfy its bound. However, lower gains are required for frequencies of rad/s to 3 rad/s, which are around the natural frequency, to avoid violating their bounds. To achieve this, an integrator, real poles and zeros, and coplex poles and zeros were appended to the controller, Cs. The final for of the controller is s s s s s s s 8s.644 s 4s C s. Figure 7(b) contains the open-loop shape after the loop shaping. The bounds are satisfied for all frequencies. The pre-filter, Fs, were designed to shift the closed-loop Bode agnitude plots to be within the tracking lower and upper bounds. The final for of the pre-filter is F s.7. s.7 ENGNEERNG JOURNAL Volue ssue 5, SSN 5-88 ( 63

10 Open-Loop Gain (db) Open-Loop Gain (db) DO:.486/ej db L j. rad/s.5 rad/s. rad/s 3 rad/s 5 rad/s, 3, 5 rad/s Open-Loop Phase (deg) (a). rad/s. rad/s.5 rad/s rad/s deg L j L j db rad/s.5 rad/s. rad/s. rad/s. rad/s, 3, 5 rad/s 5 rad/s 3 rad/s.5 rad/s rad/s Open-Loop Phase (deg) (b) deg L j Fig. 7. Loop shaping for Cs. (a) Before shaping. (b) After shaping. The frequency-doain siulation result is given in Fig. 8. The stability argin specification is shown in Fig. 8(a), the tracking specification in Fig. 8(b), and the plant-input disturbance rejection in Fig. 8(c). Solid lines are those of the 7 plant variations. The asterisks ark the upper and lower liits of each specification. t can be seen that all specifications are et by the quantitative feedback control for each frequency of interest. The tie-doain siulation result is given in Fig. 9. Figure 9(a) shows the second-ass position output, y x, when the proposed CLSS with Sith predictor and the quantitative feedback controller (shown in Fig. 5) was used. The reference, r, is the unit-step signal, and the disturbance is a step-signal with a agnitude of. The disturbance exists fro the 5 th second onwards. These reference and disturbance are the sae as those used to produce the result in Fig. 4. The plot is for the noinal plant only, that is, when the plant paraeters, c, c, and k take their iddle values. By coparing the result in Fig. 9(a) with the result in Fig. 4(b), when the CLSS was used with the Sith predictor and the P controller, the proposed syste reduces the settling tie by 4%, reduces the peak occurred fro the disturbance by 5%, and the non-zero steady-state tracking error is reoved. Note that the coparable block diagra in Fig. 6 ay be used during the design of the quantitative feedback controller; however, it should not be used to siulate the closed-loop syste in place of the original block diagra in Fig ENGNEERNG JOURNAL Volue ssue 5, SSN 5-88 (

11 Magnitude (db) Magnitude (db) Magnitude (db) DO:.486/ej C j db P j C j P j Bode Diagra P jc j P j C j F j Bode Diagra db (a) Frequency (rad/s) rad/s P j P j db C j Bode Diagra (b) Frequency (rad/s) rad/s (c) Frequency (rad/s) Fig. 8. Frequency-doain siulation result: (a) Stability argin; (b) Tracking; (c) Plant-input disturbance rejection. Figure 9(b) shows the siulation result of the proposed syste without the Sith predictor (Fig. 5 without the Sith predictor, SM ). Perforance of the control syste is degraded as seen fro higher level of vibration fro both the reference and disturbance signals. This is because, without the Sith predictor, the syste looses its capability to foresee its output after the tie delay. Hence, the controller can be isled by the current output signal causing perforance degradation. Figure 9(c) contains the siulation result of the case of Fig. 9(a) for all 7 plant variations. t can be seen that the robust stability and robust perforance when the plant is uncertain can be guaranteed using the proposed syste. Table suarizes the quantitative perforance of the four cases, which are without input shaper (Fig. 4(a)), traditional CLSS with Sith predictor (Fig. 4(b)), proposed syste without Sith predictor (Fig. 9(b)), and proposed syste with Sith predictor (Fig. 9(a)). The proposed syste with Sith predictor quantitatively outperfors the rest in ters of settling tie, peak overshoot, average residual vibration aplitude, and steady-state error. rad/s ENGNEERNG JOURNAL Volue ssue 5, SSN 5-88 ( 65

12 DO:.486/ej Table. Coparison of the quantitative perforance. Settling tie (s) Unit-step reference Peak overshoot () Average residual vibration aplitude () Settling tie (s) Unit-step disturbance Average Peak residual overshoot vibration () aplitude () Steadystate error () Without input shaper (Fig. 4(a)) Traditional CLSS with Sith predictor (Fig. 4(b)) Proposed syste without Sith predictor (Fig. 9(b)) Proposed syste with Sith predictor (Fig. 9(a)) Conclusions This paper presents a novel idea of integrating the Sith predictor into the closed-loop signal shaping (CLSS) syste. The Sith predictor reoves the tie delay, fro using the input shaper inside the loop, fro the closed-loop characteristic equation. As a result, standard feedback control design and analysis can be applied. t was shown in the paper that, when the actual plant and the plant odel are different by a ultiplicative uncertainty, the closed-loop syste can be casted as a standard quantitative feedback controller proble. Robust stability and good robust perforance (in tracking and disturbance rejection) can be obtained with the quantitative feedback control for all plant uncertainties. All the advantages of the CLSS are preserved whereas significant iproveents on the CLSS can be seen. Sith predictor is an old concept, but it is still not obsoleted. Most existing coercial controller products, especially those for process control, still use Sith predictor as their ain algorith for handling the tie delays. Using the Sith predictor in the Closed-Loop Signal Shaping is a novel concept, proposed by the authors of this paper. This paper is the very first paper that discusses this concept in details. The ost iportant benefits are the iproved control bandwidth and the eligibility of using various feedback controllers. Future work includes applying the proposed technique to experiental flexible systes, to anual control, and to flexible systes having ultiple flexible odes. n anual control, huan controls the flexible systes and is analogous to a feedback controller. t was shown in a recent work by [8] that anual control of flexible systes benefited fro the closed-loop signal shaping. t is believed that using the Sith predictor should increase the bandwidth of the huan control and should lead to iproved tracking perforance. n flexible systes having ultiple flexible odes, ultiple input shapers are required, each for one flexible ode. Each input shaper has different tie delay fro the others and ay require one Sith predictor to reove the tie delay fro the loop. 66 ENGNEERNG JOURNAL Volue ssue 5, SSN 5-88 (

13 DO:.486/ej x x Disturbance is present.5 Disturbance is present (a) x Tie (s) (b) Tie (s).5.5 Disturbance is present (c) Tie (s) Fig. 9. Tie-doain siulation result: (a) The proposed syste (CLSS with Sith predictor and quantitative feedback); (b) CLSS without Sith predictor; (c) The proposed syste for all 7 plant variations. References [] O. J. M. Sith, Posicast control of daped oscillatory systes, Proceedings of the RE, vol. 45, pp , Septeber 957. [] N. C. Singer and W. C. Seering, Preshaping coand inputs to reduce syste vibration, ASME J. of Dynaics Syste, Measureent and Control, vol., pp. 76-8, March 99. [3] K. L. Sorensen, W. Singhose, and S. Dickerson, A controller enabling precise positioning and sway reduction in bridge and gantry cranes, Control Engineering Practice, vol. 5, pp , July 7. [4] J. Stergiopoulos and A. Tzes, An adaptive input shaping technique for the suppression of payload swing in three-diensional overhead cranes with hoisting echanis, in Proc. EEE Conference on Eerging Technologies and Factory Autoation, Patras, Greece, 7, pp [5] D. Blackburn, J. Lawrence, J. Danielson, W. Singhose, T. Kaoi, and A. Taura, Radial-otion assisted coand shapers for nonlinear tower crane rotational slewing, Control Engineering Practice, vol. 8, pp ,. [6] E. Maleki and W. Singhose, Swing dynaics and input-shaping control of huan-operated doublependulu boo cranes, Journal of Coputational and Nonlinear Dynaics, vol. 7, pp. -,. [7] W. Chatlatanagulchai, S. Chotana, and C. Prutthapong, Extreu-seeking gain-scheduled adaptive input shaping applied to flexible-link robot, Kasetsart Journal: Natural Science, vol. 49, pp , March 5. [8] W. Chatlatanagulchai and K. Saeheng, Real-tie reference position shaping to reduce vibration in slewing of a very-flexible-joint robot, Journal of Research in Engineering and Technology, vol. 6, pp. 5-66, January 9. [9] Y. Baozeng and Z. Leei, Hybrid control of liquid-filled spacecraft aneuvers by dynaic inversion and input shaping, AAA Journal, vol. 5, pp , March 4. [] Y. Zhang and J.-R. Zhang, Cobined control of fast attitude aneuver and stabilization for large coplex spacecraft, Acta Mechanica Sinica, vol. 9, pp , Deceber 3. ENGNEERNG JOURNAL Volue ssue 5, SSN 5-88 ( 67

14 DO:.486/ej [] W. Singhose, W. Seering, and N. Singer, The effect of input shaping on coordinate easuring achine repeatability, in Proc. 995 FToMM World Congress on the Theory of Machines and Mechaniss, Milan, taly, 995, pp. -5. [] B. Pridgen and W. Singhose, Coparison of polynoial ca profiles and input shaping for driving flexible systes, Journal of Mechanical Design, vol. 34, pp. -7, Noveber. [3] J.-Y. Park and P.-H. Chang, Vibration control of a telescopic handler using tie delay control and coandless input shaping technique, Control Engineering Practice, vol., pp , June 4. [4] J. Hongxia, L. Wanli, and W. Singhose, Using two-ode input shaping to suppress the residual vibration of cherry pickers, in Proc. Third nternational Conference on Measuring Technology and Mechatronics Autoation, Shangshai, China,, pp [5] L. Yu and T. N. Chang, Zero vibration on-off position control of dual solenoid actuator, EEE Transactions on ndustrial Electronics, vol. 57, pp , July. [6] C. Do, M. Lishchynska, M. Cychowski, K. Delaney, and M. Hill, Energy-based approach to adaptive pulse shaping for control of RF-MEMS DC-contact switches, Journal of Microelectroechanical Systes, vol., pp , Dec.. [7] J. R. Huey and W. Singhose, Experiental verification of stability analysis of closed-loop signal shaping controllers, in Proc. EEE/ASME nternational Conference on Advanced ntelligent Mechatronics, Monterey, California, 5, pp [8] J. J. Potter and W. Singhose, proving anual tracking of systes with oscillatory dynaics, EEE Transactions on Huan-Machine Systes, vol. 43, pp. 46-5, 3. [9] J. R. Huey and W. Singhose, Trends in the stability properties of CLSS controllers: a root-locus analysis, EEE Transactions on Control Systes Technology, vol. 8, pp , Septeber. [] J. R. Huey, K. L. Sorensen, and W. E. Singhose, Useful applications of closed-loop signal shaping controllers, Control Engineering Practice, vol. 6, pp , 8. [] S. Skogestad and. Postlethwaite, Multivariable Feedback Control: Analysis and Design. Hoboken, NJ: Wiley, 5. [] O. J. M. Sith, Feedback Control Systes. New York, NY: McGraw-Hill, 958. [3] W. Chatlatanagulchai and P. Poedaeng, Closed-Loop input shaping with sith predictor and quantitative feedback control, in Proc. The 9th Conference of The Mechanical Engineering Network of Thailand, Nakhon Ratchasia, 5, pp [4] O. Yaniv, Quantitative Feedback Design of Linear and Nonlinear Control Systes, st ed. Norwell, Massachusetts, USA: Kluwer Acadeic Publishers, 999. [5] W. Chatlatanagulchai and T. Benjalersyarnon, Closed-loop input shaping with quantitative feedback controller applied to slewed two-staged pendulu, Walailak Journal of Science and Technology, to be published. [6] W. S. Levine, Control Syste Fundaentals. Boca Raton, FL: CRC Press,. [7] K. Watanabe and M. to, A process-odel control for linear systes with delay, EEE Trans. Auto. Control, vol. 6, pp. 6-69, 98. [8] K. J. Astro, C. C. Hang, and B. C. Li. A new Sith predictor for controlling a process with an integrator and long dead-tie, EEE Trans. Auto. Control, vol. 39, pp , 994. [9] J. E. Norey-Rico and E. F. Caacho, Robust tuning of dead-tie copensators for process with an integrator and long dead-tie, EEE Trans. Auto. Control, vol. 44, pp , 999. [3] M. Ajeri and A. Ali, Siple tuning rules for integrating processes with large tie delay, Asian Journal of Control, vol. 7, pp. 33-4, 5. [3] Z. J. Palor and D. V. Powers, proved dead-tie copensator controllers, AChE Journal, vol. 3, pp. 5-, ENGNEERNG JOURNAL Volue ssue 5, SSN 5-88 (