A multiple periodic disturbance rejection control for ...

Journal of Process Control 24 (2014) 1394–1401

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Journal of Process Control journal homepage: www.elsevier.com/locate/jprocont

A multiple periodic disturbance rejection control for process with long dead-time Ping Shen a , Han-Xiong Li b,∗ a b

The State Key Laboratory of High Performance Complex Manufacturing, Central South University, Changsha, Hunan, PR China Department of Systems Engineering and Engineering Management, City University of Hong Kong, Kowloon, Hong Kong, PR China

a r t i c l e

i n f o

Article history: Received 16 December 2013 Received in revised form 22 May 2014 Accepted 13 June 2014 Available online 15 July 2014 Keywords: Periodic disturbance rejection Dead-time Process control

a b s t r a c t Most of industrial processes have dead-time phenomenon that will affect the process performance. Though there are some disturbance rejection methods for the process with dead-time, however, most of them may not work well for multiple periodic disturbance. In this paper, a novel periodic disturbance rejection controller is designed based on the Smith predictor platform for process with long dead-time. By adding two feedback loops and the online spectrum analysis, multiple periodic disturbances can be suppressed effectively in existence of long dead-time. The rigorous analysis is conducted to prove that the robust stability can be maintained. Finally, the effectiveness of the proposed controller is demonstrated in the process simulations and industrial experiment. © 2014 Elsevier Ltd. All rights reserved.

1. Introduction Dead-time is a common phenomenon appearing in nearly all the industrial processes and deteriorates the performance greatly [1]. Smith predictor [2] is well known as an effective process control method to deal with the dead-time. However, there are several disadvantages in the traditional Smith predictor method. A major drawback is its poor disturbance rejection capability [3,4], including periodic disturbances. In industrial environment, the periodic disturbance widely exists in many processes [5], such as, rotational motion systems [6–8], casting process [9–11], disk driver process [12–15], motor systems [16], etc. In past decades, numerous extensions based on the Smith predictor have been proposed in order to improve the rejection ability of disturbances for the process with dead-time [17–19]. Another class of research is based on repetitive control [6–9,15,16], because of its good rejection of periodic disturbances, including multiple periodic disturbance [20]. However, it is still difficult to achieve satisfactory rejection control of periodic disturbances, especially multiple periodic disturbances, when the process has a long dead-time. Though there are some methods developed for the process with dead-time, however, most of them may be only for single frequency signal. A class of which is based on repetitive control. By adding an additional delay item, repetitive control is modified

∗ Corresponding author. Tel.: +852 34428435; fax: +852 34420172. E-mail address: [email protected] (H.-X. Li). http://dx.doi.org/10.1016/j.jprocont.2014.06.008 0959-1524/© 2014 Elsevier Ltd. All rights reserved.

for periodic disturbance suppressing in process with dead-time [9,21]. Other improved repetitive controllers are raised by modifying the internal model for dead-time process [22,23]. The other class of periodic disturbance rejection control for dead-time process is based on Smith predictor. Recently, several modified Smith predictor controllers are reported for periodic disturbance rejection for the situation with long dead-time [24–27], which are based on disturbance observer principle. There are several literatures about multiple periodic disturbance rejection control for process with dead-time. The combination of Åström’s modified Smith predictor and a gray predictor is one good example for rejecting both step and periodic disturbance [3]. Other examples include, designing a linear approximation method [26] and the artificial neural network [28] to estimate the inverse of dead-time for delay compensation. However, most of methods presented rely on accurate modeling of the inverse of dead-time, which will be difficult to achieve when the dead-time is long. Thus, these methods may be unsuitable for the process with long dead-time. To the best of our knowledge, until now, there are only a few effective control methods reported for multiple periodic disturbances for the process with long dead-time. In this paper, a novel control design is presented based on Smith predictor for multiple periodic disturbance rejection for the process with long dead-time, which is stable or with integral. After the system is stabilized, two more feedback loops will be added for the disturbance compensation. One is designed to reject multiple periodic disturbances after the online FFT analysis; another is to compensate the disturbance exerted in the forward

P. Shen, H.-X. Li / Journal of Process Control 24 (2014) 1394–1401

R (s)

D (s)

P (s)

C (s)

1395

Y (s)

M (s) Pm ( s )

e − Ls

Fig. 1. Modified Smith predictor by Åström.

control loop. After the design, a rigorous analysis will be conducted to evaluate the robustness of the proposed method. Finally, the numerical simulation, casting process simulation, and experimental control will be conducted to demonstrate the viability and effectiveness of the proposed control method. The proposed method has two major novel improvements: • By adding only one extra delay item and a novel tuning method to suppressing multiple periodic disturbance. • By adding a filter with comb feature to prevent inter-harmonic amplification of disturbance in the observer feedback.

2. Periodic disturbance rejection control

Y (s) C(s) · Pm (s) · e−Ls = R(s) C(s) · Pm (s) + 1

Gd (s) =

(s) · e−Ls

Y (s) Pm = D(s) 1 + M(s) · Pm (s) · e−Ls

2.1. Closed loop control design The closed-loop transfer function will be: G(s) =

Pm (s)C(s)P(s)[M(s)Pm (s)e−Ls + 1] L(s)

(3)

The disturbance transfer function will become: [1 − Q (s)FLP (s)e−(+L)s + C(s)Pm (s)(1 − e−(+L)s )]Pm (s)P(s) L(s) (4)

Gd (s) =

where:

Smith predictor is a very popular control method for process with dead-time. However, it may have poor disturbance rejection and may not work well for unstable processes [4]. The modified Smith predictor by Åström [29] is an effective method for unstable process control with dead-time under load disturbance. It is shown in Fig. 1, where C(s) is the controller, P(s) is the plant, Pm (s) is the nominal model of dead-time free part of plant, M(s) is disturbance rejection controller, R(s) represents the reference signal, and D(s) represents the external disturbance. The closed-loop transfer function and the disturbance transfer function are derived respectively as below. G(s) =

Fig. 2. Periodic disturbance rejection control.

(1)

L(s) = Pm (s)[Pm (s)C(s) + 1][P(s)M(s) + 1] + e−s [(Q (s)FLP (s) + Pm (s)C(s))(P(s) − Pm (s)e−Ls )] If the nominal model can be designed to have: P(s) = Pm (s)e−Ls

We will have the same closed-loop dynamics as that of the Smith predictor in (1), and the different disturbance transfer function. G(s) =

Pm (s)C(s)e−Ls Pm (s)C(s) + 1

Gd (s) =

(2)

(1) Feedback loop for periodic disturbance rejection – with the proper selection of Q(s) and e−s , the periodic disturbance can be compensated by the feedback signal U2 . (2) Feedback loop for performance enhancement – the feedback signal Y3 provides a fine adjustment of disturbance compensation to the forward controller C(s).

(6)

[1 − Q (s)FLP (s)e−(+L)s + C(s)Pm (s)(1 − e−(+L)s )] Pm (s)C(s) + 1 ×

The set-point response and the disturbance response are decoupled that can be adjusted separately. This structure provides more freedom for design of M(s) [29] for disturbance suppressing. However, it cannot suppress periodic disturbance effectively in the process with dead-time. Due to the phase deviation, the periodic disturbance cannot be compensated in the feedback loops. Integrated with the disturbance observer-based control [30], a novel disturbance rejection controller is proposed as shown in Fig. 2, where Q(s) is a modified comb filter, e−s is an additional dead-time item, Df (s) represents external periodic disturbance and FLP (s) is a low-pass filter. In difference to the Åström’s Smith predictor, two more feedback loops are added:

(5)

Pm (s)e−Ls Pm (s)M(s) + 1

(7)

2.1.1. Controller C(s) Suppose that the transfer function (6) can be approximated as a first-order plus dead-time (FOPDT) model in (8). G(s) =

Pm (s)C(s)e−Ls 1 = e−Ls Pm (s)C(s) + 1 s + 1

(8)

Then we will have the controller as: C(s) = Pm (s)−1

1 s

(9)

A larger means a better robustness, and a smaller means a faster response. Remark: In industry, many processes can be approximated as a FOPDT model as Eq. (10). P (s) ≈

K e−Ls Ts + 1

(10)

With Eqs. (9) and (10), the controller can be PI type and derived as: C(s) =

1 T + K Ks

(11)

1396


U2 U1

Y1

M (s)

D

U3

P( s )

Y

Fig. 3. The feedback loop of P(s) and M(s).

If the process is approximated as the integral plus dead-time model: P(s) ≈

K −Ls e Ts

(12)

The controller becomes a proportion type: C(s) =

T K

(13)

2.1.2. Stabilizer M(s) If the plant is unstable, the controller C(s) only may not be able to stabilize the system with disturbance. Then, M(s) can be designed to stabilize the unstable plant by approximating the output of P(s) with Pm (s)e−Ls , which can be clearly seen after the transformation of Fig. 2 into Fig. 3. Ideally, if the stabilizer M(s) is chosen as M(s) =

Pm (s)−1 1 + (s − e−Ls )

(14)

Actually, the multiple periodic disturbance Df (s) shows a group of peaks (harmonic frequencies) in its spectrum figure. Though it is multiplied by PD (s), it is enough for Q(s) design. Design of Q(s) requires frequencies of disturbance only (the amplitude of the disturbance is not needed). In general, the frequency information can be obtained through FFT. If the nominal model can match the plant perfectly, the disturbance can be figured out accurately in both frequency and amplitude through FFT for Q(s) design. If the nominal model does not match the plant well, amplitude at the harmonic frequency may vary due to modelplant mismatch. However, harmonic frequencies are generally not affected for most of industrial processes in practice. Thus, the proposed Q(s) design will be applicable to most of industrial processes. As a result, the spectrum analysis of e can help to find the frequencies of the disturbance. Based on the FFT conversion, the spectrum vector for the disturbance can be obtained: w k = (kw1 , kw2 . . .kwm )

(19)

The peak of the periodic disturbance at frequencies (wi ) can be easily determined with the criterion (20). If all these peaks can be suppressed effectively, the influence of the periodic disturbances will be minimized.

⎧ kw > Mthreshold ⎪ ⎨ i ⎪ ⎩

kwi > kwi−1

i ∈ N, Mthreshold is a preset value

(20)

kwi > kwi+1

The transfer function of Fig. 3 will be a FOPDT model as in (15).

Thus, the multiple periodic disturbances can be expressed as a group of single ones:

Y (s) M(s)P(s) 1 e−Ls = = s + 1 Y1 (s) 1 + M(s)P(s)

Df (s) =

(15)

−1 Though, when the process is integral, the Pm is unstable and difficult to be implemented. So, a low-pass filter is added to M(s) to make it realizable and internal stable. Then M(s) becomes:

M(s) =

Pm (s)−1 [1 + (s − e−Ls )](s + 1)

(16)

Here, makes a trade-off between response and robustness. 2.2. Periodic disturbance compensation

2.2.1. Disturbance spectrum analysis From Fig. 2, based on the disturbance observer, the error “e” can be estimated as: e=

(21)

where: Dwi (s) is the single frequency disturbance at the peak frequency wi . 2.2.2. Disturbance phase compensation The delay item can be figured out for phase compensation at the peak frequencies of the disturbance. From the disturbance transfer function (4), if the periodic disturbance can be rejected, one should have: [1 − Q (s)FLP (s)e−(+L)s + C(s)Pm (s)(1 − e−(+L)s )]Pm (s)P(s)Df (s) L(s)

≈0

(22)

The FLP (s) should be designed with higher cut-off frequency than the periodic disturbance Df (s). The Q(s) will be designed in the next part for the purpose that periodic disturbance Df (s) can pass through. So, one has: Q (s)FLP (s) · Df (s) ≈ Df (s)

(23)

Based on (22) and (23), for the purpose Yd (s) = 0, (24) must exist: Df (s) · e−Ls · e−s ≈ Df (s)

(24)

By taking Euler formula, (24) becomes:

P(s)D(s)(1 + C(s)Pm (s)) W (s)

(17)

where: W (s) = −(1 + C(s)Pm (s))(1 + P(s)M(s)) + C · (Pm (s)e

Df (jω) ≈ Df (jω) · (cos((L + )ω) − j · sin((L + )ω))

(25)

By the above derivation, if (25) is satisfied, the periodic disturbance can be suppressed. At ω = 2 wi , if we have:

−(L+)s

−P(s)e−s ) + Q (s)FLP (s)(e−(L+)s − P(s)Pm (s)−1 · e−s )

= (s)e−Ls ,

Suppose the nominal model and plant satisfy P(s)(1 +) = Pm the estimated disturbance can be easily derived with the design of M(s) in (14) from Fig. 2. e=−

Dwi (s)

i=1

Yd (s) =

In the disturbance compensation of the proposed method, the periodic of disturbance must be measured. To obtain this information, Fast Fourier Transformation (FFT) is applied in the disturbance observer loop. Through the FFT, the periodic of disturbance can be analyzed for design of e−s and Q(s) in the disturbance compensation.

n

ki − L, wi

(ki ∈ N, > 0, i = 1, 2, . . .n)

(26)

(25) can be satisfied. ki need to be as small as possible that make bigger than zero, because smaller shows faster disturbance suppression.

Df (s)Pm (s)e−Ls −(1 + ) − e−Ls /((1 + (s − e−Ls ))(s + 1)) + Df (s)Pm (s)e−Ls /(1 + ) + Q (s)FLP (s)e−(L+)s /(1 + )

Df (s)PD (s)

(18)


1397

2.2.3. Comb filter for disturbance reduction The additional delay in the disturbance rejection feedback loop will generate two kinds of frequency bands. One is harmonic frequencies at which the disturbance will be completely suppressed; another is the inter-harmonic frequencies where the disturbance will be amplified. From (22) to (25), it can be seen that the disturbance is suppressed at the harmonic frequency 2k /(L + ), and doubled at the inter-harmonic frequencies 2(k + 1) /(L + ). Thus, the filter Q(s) needs to be designed to suppress the inter-harmonic amplification as shown in (27).

⎧ 2k ⎪ , (k ∈ N) ⎨ |Q (s)Df (s)| ≈ |Df (s)|, ω → L+

(27)

⎪ ⎩ |Q (s)Df (s)| |Df (s)|, ω → 2(k + 1) , (k ∈ N) L+

That is:

Fig. 4. Multiplicative uncertainty based robust analysis.

⎧ 2k ⎪ ⎨ Q (s) ≈ 1, ω → L + , (k ∈ N)

where: (28)

⎪ ⎩ Q (s) 1, ω → 2(k + 1) , (k ∈ N) L+

Obviously, |Q(s)| = 1 will maintain the disturbance rejection at the peak frequencies wi ; while |Q(s)|1 will make the control system has no amplification of disturbance at non-peak frequencies. From (28), it is obvious Q(s) has a periodic peak and valley in its frequency response, which correspond to the feature of comb filter. Though, a comb filter can only deal with the even-order harmonic. A novel low order solution is to design Q(s) behaving as a comb-filter added an inverse comb filter as in (29), which can satisfy the characteristic of (28). Though, there may be other high order solutions with more complex expression. Q (s) =

(29)

Let: s = jω, the amplitude characteristic of Q(s) becomes:

2

2

(2a + 4ab cos2 ( ω) − 2ab) + 16a2 b2 cos2 ( ω) sin ( ω) (1 −

2b2

cos2 ( ω)

+

2 b2 )

+

4b2

2 cos2 ( ω) sin ( ω)

(30)

From d|Q(jω)|/dω = 0, and define 2a +b = 1 to make the peak amplification is 1, the optimum of |Q(jω)| can be easily obtained as follows.

Q (jω) →

1, 2a(1 − b) < 1, 1 + b2

ω → k , (k ∈ N)

ω → k + , (k ∈ N) 2

(31)

The a/b ratio affect the bandwidth of Q(s). As a/b ratio grows bigger, the pass-band of Q(s) becomes smaller and stop-band grows bigger. The smaller pass-band will lead to a poor disturbance rejection, and a bigger pass-band has a bigger inter-harmonic amplification in the disturbance rejection loop. Large amount of simulations conducted show that a = 0.25, b = 0.5 can have a good balance between disturbance rejection and inter-harmonic amplification suppressing. From (28) and (31), a solution for can be obtained in (32).

=

For a more conservative solution, considering the worst case when Q(s) FLP (s) = 1, then Eq. (33) becomes:

−s H(s) = M(s)Pm (s) + e M(s)Pm (s)e−Ls + 1

(34)

Here, the low-pass filter in M(s) is ignored to obtain a more practical and conservative condition. The M(s) is selected as: M(s) =

Pm (s)−1 1 + (s − e−Ls )

(35)

Then, (35) becomes:

a(1 + e− s ) a(1 − e− s ) + (1 + be− s ) (1 − be− s )

Q (jω) =

N(s) = Pm (s)C(s)e−s + M(s)Pm (s) + e−s FLP (s)Q (s) + Pm 2 (s)C(s)M(s)

L+ 2

(32)

3. Robust analysis The robust stability can be analyzed using the small gain theory [31]. Assuming that the multiplicative uncertainty (s)∈H∞ exists in Fig. 4 to have P(s)(1 + ) = Pm (s)e−Ls . In Fig. 4, the transfer function from v to u can be derived as:

e−Ls N(s) H(s) = (Pm (s)C(s) + 1)(M(s)Pm (s)e−Ls + 1)

(33)

−(L+)s − 1) −s H(s) = −e (s + 1) + (e ≤ e−s s + 1 e−(L+)s 1 1 + + = 1 + 2 s + 1 s + 1 s + 1

(36)

Using s = jω, we will have:

H(s) ≤ 1 +

2

(37)

1 + (ω)2

Thus, the robust stability can be obtained under the following condition.

H(s) ≤

1+

2 1 + (ω)2

|| < 1

(38)

From (32), “” will affect the robustness of the control system. Bigger “” makes better robustness. 4. Simulations and experiment In this section, simulations and experiment are conducted to verify the viability of the proposed control method. The performance index (ISE, IAE, ITAE) will be used for the experiment comparison. 4.1. Numerical simulation Many industrial processes can be modeled as an integral plus delay system as below. G(s) =

1 −Ls e s

(39)

1398

P. Shen, H.-X. Li / Journal of Process Control 24 (2014) 1394–1401 Single-Sided Amplitude Spectrum of disturbance

1.2

0.9

0.7

Amplitude

1

Output signal

0.8

0.6

0.8 0.6 Reference Modified Smith predictor by Astrom Disturbance compensation control Proposed method

0.4

0.5 0.2 0.4 0 0.3

0

5

10

15

20

25

35

40

45

50

5

0.2

Modified Smith predictor by Astrom Disturbance compensation control Proposed method

0

1

2

3

4

5

6

Frequency (Hz)

Control signal

4 0.1 0

30

Time

Fig. 5. The spectrum of the given disturbance.

3 2 1 0 -1 -2 -3

For the long dead-time situation, L = 8 s is taken for the example simulation. As Pm (s) = 1/s, from (13), the controller will be C(s) =

1

1 (s2 + (1 − e−Ls )s)(0.5s + 1)

Suppose unknown multiple frequency disturbances exist in the system, which is composed of a group of sine signals as follows.

d(t) = 0.8 sin(2 t) + 0.8 sin(8 t) − 0.2 sin

5

10

15

20

25

30

35

40

45

50

Time Fig. 6. Performance comparison under multiple periodic disturbance.

Through the trial and error process, = 0.5 can have a good balance between the process response and its robustness. The stabilizer M(s) can also be designed as follows according to (16), with = 1 for a better robustness to disturbance, and ˚ = 0.5 for better disturbance rejection effect. M(s) =

0

8 t 3

− 0.2 sin

16 t 3

for the casting efficiency and quality. The casting system can be modeled as the following transfer function [10]: P(s) ≈

The additional delay item e−s is designed as = 1. This disturbance is a typical example of the multiple frequency disturbance in industry with its spectrum shown in Fig. 5. The disturbance compensation control [26] and the Åström’ s Smith predictor [29] are compared. The simulations are compared in Fig. 6. The Åström’s Smith predictor is greatly affected by the periodic disturbance, for this control method has no periodic disturbance rejection ability. The disturbance compensation control [26] can give an incomplete disturbance rejection, for it can only suppressing one frequency disturbance and its harmonics. In this simulation, this control method is tuned to suppress the periodic disturbance: 0.5sin(2 t) + 0.5sin(8 t). The proposed control method can maintain a smooth response with little ripples after 26 s.

C(s) =

M(s) =

(0.5s + 10) 2.9184 s(0.5s + 10) (3.648 + 3.648(s − e−Ls ))(0.5s + 1)

The spectrum of the periodic disturbance in the system is shown in Fig. 8.

4.2. Casting process simulation In the steel industry, the continuous casting is a popular process to solidify the steel. As shown in Fig. 7, molten steel flows into the mold through the tundish, and then is solidified by the water. A solidified shell is thus formed and continuously withdrawn out of the mold. Through the cooling spray, the steel is fully solidified in the end. There are periodic oscillations in the process that are caused by an additional flow of liquid steel toward and beyond the strand [32]. An important work is to suppress the disturbance

(40)

Obviously, the casting process is a long dead-time process due to the large ratio of dead-time and time constant of the process. The controller C(s), stabilizer M(s) are designed with = 0.8, = 1 and ˚ = 0.5

Q(s) is designed as a = 0.25, b = 0.5 and ␴ = 4.5. 0.25(1 − e−4.5s ) 0.25(1 + e−4.5s ) Q (s) = + 1 + 0.5e−4.5s 1 − 0.5e−4.5s

3.648 e−0.5s s(0.5s + 10)

Fig. 7. Continuous casting system.


1399

Single-Sided Amplitude Spectrum of disturbance

0.9 0.8 0.7

Amplitude

0.6 0.5 0.4 0.3 0.2 0.1 0

0

1

2

3

4

5

6

Frequency (Hz) Fig. 10. The grip system of the heavy-duty forging manipulator.

Fig. 8. Spectrum of the periodic disturbance in the casting process.

Based on the spectrum in Fig. 8, Q(s) is designed as a = 0.25, b = 0.5 and = 2.5. Q (s) =

0.25(1 − e−2.5s ) 0.25(1 + e−2.5s ) + −2.5s 1 + 0.5e 1 − 0.5e−2.5s

The additional delay item e−s is designed as = 4.5. The proposed control method is compared with disturbance compensation control [26] and the Åström’ s Smith predictor [29] in Fig. 9. The Åström’ Smith predictor is effected by the disturbance greatly, the proposed method and disturbance compensation control can suppress the disturbance. Though, the proposed method can obtain a smoother response, which is shown in the local zoom graph in Fig. 9. 4.3. Experimental control of manipulator rotation Heavy-duty forging manipulator, shown in Fig. 10, is a important equipment in the forging process. The rotation movement is driven by two hydraulic motors. During forging, the manipulator grips the workpiece and rotates smoothly to a required position before the pressing. A smooth and accurate movement is required for quality operation. However, that is difficult to achieve and maintain because the forging manipulator is a slow-moving system with large inertia, and usually working in the complex

environment where lots of disturbance exist. Especially, eccentricity and periodic resistance in gear transmission generate strong periodic disturbance. The long transmission circle in the hardware control system generates more dead-time. The proposed control method will be applied to achieve a smooth and accurate movement. Large inertia system is insensitive to the high frequency signal. So, the high frequency signal measured can be considered as disturbance. A low-pass filter is used here to suppress the high frequency disturbance measured from the manipulator as below. FLP (s) =

1 0.2s + 1

Experimental configuration of the manipulator control is depicted in Fig. 11, where the control algorithm is implemented on PLC. Using the least squares method, the FOPDT model of the angular velocity response can be obtained: P(s) ≈

0.3103 e−0.8342s 0.8681s + 1

(41)

The forward controller C(s) can be designed as: C(s) = 5.595 + 6.445

1 s

0.8 0.6

Output signal

0.6 0.5

0.4 0.4

Reference

0.2

38

Modified Smith predictor by Astrom

40

42

Disturbance compensation control Proposed method

0 0

5

10

15

20

25

30

35

40

45

50

Time 1 Modified Smith predictor by Astrom

Control signal

Disturbance compensation control Proposed method

0.5

0

-0.5 0

5

10

15

20

25

30

35

40

Time

Fig. 9. Performance comparison in the casting simulation.

45

50

Fig. 11. PLC based rejection control of manipulator rotation. ((1) flood valve, (2) hydraulic pump, (3) electric motor, (4) servo valve, (5) check valve, (6) hydraulic motor, (7) encoder, (8) gear).

1400


5. Conclusion

Periodic Disturbance

angular velocity(rad/s)

1

0.8

0.6

0.4

0.2

0

0

Reference Smith predictor by Astrom Proposed method

5

10

15

20

25

Time(s) Fig. 12. Experimental comparison of the rotation control.

Then, = 1 and ˚ = 0.5 are selected, the stabilizer M(s) becomes: 3.223 + 2.798s (1 + (0.8s − e−0.8342s ))(0.5s + 1)

M(s) =

The additional delay item e−s is designed with = 3.5136. Initially, a = 0.25, b = 0.5 are selected for Q(s) in (24), and = 2.1739. Q (s) =

0.25(1 − e−2.1739s ) 0.25(1 + e−2.1739s ) + −2.1739s 1 + 0.5e 1 − 0.5e−2.1739s

The conventional PID control, Åström’s Smith predictor, and the proposed control method were working in the same condition. The experiment results are compared in Fig. 12 and Table 1. From Table 1, it is clear that the conventional PID control performs worst in the experiment. Since it is more close to the Smith predictor, PID performance is not plotted in Fig. 12 for a clearer comparison. Actually, the disturbance in this forging manipulator includes two parts: one is the periodic disturbance and the other is the random disturbance. Under the environment of dead-time, the random disturbance cannot be compensated with the above three methods; but the periodic disturbance can be suppressed by the proposed method as shown in Fig. 12. The proposed method can achieve a much smoother performance than the modified Smith predictor. The performance indexes ISE, IAE, ITAE are set up below for the comparison of the above three control methods.

(R(s) − Y (s))2 dt

ISE =

IAE =

|R(s) − Y (s)|dt

ITAE =

t|R(s) − Y (s)|dt

As shown in Table 1, the proposed method shows a better tracking performance than the Åström’s Smith predictor and the conventional PID control. Table 1 Performance comparison of the experiment. Item

ISE

IAE

ITAE

PID Åström’s Smith predictor Proposed control

32.5174 28.4816 25.9745

103.4403 80.3577 58.4525

957.7239 566.9601 275.2196

A multiple periodic disturbance rejection controller is designed for the process with long dead-time. The spectrum analysis of periodic disturbances is first conducted with FFT. Then, an additional delay item is designed and tuned for phase compensation for harmonic frequencies. A comb filter is further designed for suppressing inter-harmonic amplification. The proposed control method can deal with periodic disturbances in multiple frequencies for systems with long dead-time. A rigorous analysis is conducted to evaluate the robustness of the proposed method. The final simulations and experiment also demonstrate the viability and effectiveness of the proposed control method. Acknowledgement Authors would like to appreciate the valuable comments from editors and anonymous reviewers. The work presented in this paper is partially supported by a project from the National Basic Research Program 973 of China (2011CB013104) and a project from RGC of Hong Kong (CityU: 116212). References [1] A.S. Rao, M. Chidambaram, Enhanced Smith predictor for unstable processes with time delay, Ind. Eng. Chem. Res. 44 (2005) 8291–8299. [2] O.J. Smith, A controller to overcome dead time, ISA J. 6 (1959) 28–33. [3] Y.D. Chen, et al., Modified Smith predictor scheme for periodic disturbance reduction in linear delay systems, J. Process Control 17 (2007) 799–804. [4] J.E. Normey-Rico, E.F. Camacho, Dead-time compensators: a survey, Control Eng. Pract. 16 (2008) 407–428. [5] Q.-G. Wang, et al., Virtual feedforward control for asymptotic rejection of periodic disturbance, IEEE Trans. Ind. Electron. 49 (2002) 566–573. [6] C.-L. Chen, Y.-H. Yang, Spatially periodic disturbance rejection for uncertain rotational motion systems using spatial domain adaptive backstepping repetitive control, in: Industrial Electronics Society, 33rd Annual Conference of the IEEE, 2007, pp. 638–643. [7] C.L. Chen, G.T.C. Chiu, Spatially periodic disturbance rejection with spatially sampled robust repetitive control, J. Dyn. Syst. Meas. Control 130 (2008) 4117–4121. [8] Y.H. Yang, C.L. Chen, Spatially periodic disturbance rejection using spatialbased output feedback adaptive backstepping repetitive control, in: American Control Conference, 2008, pp. 4117–4122. [9] T.J. Manayathara, et al., Rejection of unknown periodic load disturbances in continuous steel casting process using learning repetitive control approach, IEEE Trans. Control Syst. Technol. 4 (1996) 259–265. [10] K. Jabri, et al., Suppression of periodic disturbances in the continuous casting process, in: IEEE International Conference on Control Applications, 2008, pp. 91–96. [11] J.Y. Kim, J. Bentsman, Disturbance rejection in a class of adaptive control laws for distributed parameter systems, Int. J. Adapt. Control Signal Process. 23 (2009) 166–192. [12] K. Ohno, et al., A comparative study of the use of the generalized hold function for HDDs, IEEE/ASME Trans. Mechatron. 10 (2005) 26–33. [13] M. Nagashima, et al., Rejection of unknown periodic disturbances in magnetic hard disk drives, IEEE Trans. Magn. 43 (2007) 3774–3778. [14] W. Kim, et al., Adaptive output regulation for the rejection of a periodic disturbance with an unknown frequency, IEEE Trans. Control Syst. Technol. 19 (2011) 1296–1304. [15] H. Fujimoto, RRO compensation of hard disk drives with multirate repetitive perfect tracking control, IEEE Trans. Ind. Electron. 56 (2009) 3825–3831. [16] R. Cao, K.-S. Low, A repetitive model predictive control approach for precision tracking of a linear motion system, IEEE Trans. Ind. Electron. 56 (2009) 1955–1962. [17] Q.-C. Zhong, J.E. Normey-Rico, Control of integral processes with dead-time. Part 1: Disturbance observer-based 2DOF control scheme, IEE Proc. Control Theory Appl. 149 (2002) 285–290. [18] Q.C. Zhong, H.X. Li, 2-degree-of-freedom proportional-integral-derivative-type controller incorporating the Smith principle for processes with dead time, Ind. Eng. Chem. Res. 41 (2002) 2448–2454. [19] R.C. Panda, et al., An integrated modified smith predictor with PID controller for integrator plus deadtime processes, Ind. Eng. Chem. Res. 45 (2006) 1397–1407. [20] H.-J. Chen (Ed.), Multiple Periodic Disturbance Rejection Techniques for Vibration Isolation, Columbia University, New York, 2001. [21] K.K. Tan, et al., A new repetitive control for LTI systems with input delay, J. Process Control 19 (2009) 711–716. [22] J. Na, et al., Discrete-time repetitive controller for time-delay systems with disturbance observer, Asian J. Control 14 (2012) 1340–1354.

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