Journal of Robotics, Networking and Artificial Life
Vol. 7(2); September (2020), pp. 129–132
DOI: https://doi.org/10.2991/jrnal.k.200528.012; ISSN 2405-9021; eISSN 2352-6386
https://www.atlantis-press.com/journals/jrnal

Research Article

A Design Method of Generalized Predictive Control
Systems in Consideration of Noise
Akira Yanou*
Faculty of Health Science and Technology, Kawasaki University of Medical Welfare, 288 Matsushima, Kurashiki, Okayama 7010193, Japan

ARTICLE INFOABSTRACT

Article History

Generalized Predictive Control (GPC) is one of the model-based control methods. The control law is derived by minimizing
performance index, which includes the squares of control input and the squares of the error between reference signal and output
prediction, in order to make the derived control system be stable. Although coprime factorization approach has been used toextend the conventional control law in the previous researches, there has been a possibility that the order of the derived control
law becomes high. Therefore, this paper extends GPC through newly-defined output prediction and proposes the method to redesign the control law or the disturbance response with keeping the closed-loop transfer function. Numerical example is shown
to check the characteristic of the proposed method.

Received 23 October 2019
Accepted 17 May 2020
Keywords
Generalized predictive control
closed-loop characteristics
noiseoutput prediction© 2020 The Authors. Published by Atlantis Press SARL.
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION
Generalized Predictive Control (GPC) [1] is one of the modelbased design methods and the closed-loop system is designed
through performance index. The performance index includes theerror between the reference signal and the output prediction, andthe control input. And the closed-loop system is derived by thedesign parameters for output prediction, future control input series
and weighting factor of inputs. For consideration of designing safe
systems, although coprime factorization approach [2] has beenused in order to extend the control law in the previous researches[3], the derived controller will become high order because a stable
polynomial is needed to derive the extended controller. Therefore,
this research directly extends generalized predictive control [4]by defining new output prediction and proposes the scheme tore-design the controller without changing the closed-loop characteristics. A numerical example is given to check the characteristicsof the proposed method.

2. EXTENSION OF GENERALIZEDPREDICTIVE CONTROL

Email: yanou-a@mw.kawasaki-m.ac.jp*

B[z -1] = b0 + b1z -1 +  + bmz -m (3)C[z -1] = 1 + c1z -1 +  + cl z -l (4)For (1), the following assumptions are hold.
(i) km is known.
(ii) The pairs of (A[z−1], B[z−1]) and (A[z−1], C[z−1]) are coprime.
(iii) C[z−1] is stable polynomial.
The following performance index J is minimized to derive thecontrol law.

åN2j = N1{ y(t + j) - w(t + j)}2 +Nu2ù

å l{Du(t + j - 1)} úû  (5)j =1

Ex[] means the expected value. The design parameters [N1, N2],
[1, Nu] and l are prediction horizon, control horizon and weighting
factor of control input. Their parameters should be chosen so thatthe closed-loop system become stable.

A single-input and -output system is considered for t = 0, 1, 2....

x (t )
DA[z -1] = 1 + a1z -1 +  + anz -n (2)éJ = Ex êë

2.1. Conventional GPC

A[z -1] y(t ) = z -km B[z -1]u(t ) + C[z -1]y(t) and u(t) are output and input respectively. km is time delay, x(t)is white Gaussian noise and Δ = 1 − z−1. A[z−1], B[z−1] and C[z−1] are
the following polynomials with known degrees n, m and l.

(1)

In order to derive the control law, the predicted output ŷ(t + j|t) forj = N1 … N2 is calculated by solving the following Diophantine equation.
C[z -1] = DA[z -1]E j[z -1] + z - j Fj[z -1] (6)

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Ej[z−1] and Fj[z−1] are expressed as follows.
E j[z -1] = 1 + e1z -1 +  + e j -1z -( j -1) (7)Fj[z -1] = f0j + f1j z -1 +  + fnj z -n (8)Multiplying zjΔEj[z−1] to (1) and substituting (6) into it, the following equation is obtained.
C[z -1] y(t + j) = E j[z -1]B[z -1]Du(t + j - km ) + Fj[z -1]y(t )+ E j[z -1]C[z -1]x (t + j)

 (9)2.2. Proposed Method
In conventional GPC, the prediction is defined without including
the future noise term Ej[z−1]C[z−1]x(t + j). On the other hand, thispaper proposes the use of the noise term to date, which can be calculated by (1). Concretely, the following output prediction is newly
defined by introducing constant parameter se.
y(t + j|t ) =1

C[z -1]{E j[z -1]B[z -1]Du(t + j - km )(10)-1-1+ Fj[z ]y(t ) + seC[z ]x [t ]}C[z -1]x (t ) = DA[z -1] y(t ) - z -km B[z -1]Du(t ) R j[z -1] = r0 + r1z -1 +  + rj -1z -( j -1) (13)S j[z -1] = s0j + s1j z -1 +  + snj z -n3 (14)3

where n3 = max(m, l ) - 1. Then the output prediction (10) can bere-expressed as follows.
1-1+(Fj[z ] + se DA[z ]) y(t )(15)+(S j[z -1] - se B[z -1])Du(t - km )}Next, the following equations are defined.
Fj¢[z -1] = Fj[z -1] + se DA[z -1] (16)S¢j[z -1] = S j[z -1] - se B[z -1] (17)The past and present signals with output and input are also definedby h¢j (t ) .
C[z -1]h¢j (t ) = Fj¢[z -1] y(t ) + S¢j[z -1]Du(t - km ) Y = [ y(t + N1|t ) y(t + N 2 |t )]T (20)U = [Du(t ) Du(t + N u - 1)]T (21)

H = [hN¢ 1 (t )hN¢ 2 (t )]T (22)

W = [w(t + N1 )w(t + N 2 )]T (23)é rN1 -km
êR=ê ê êêërN 2 -km

 r0ù
ú
ú
ú
ú
  rN 2 - N u -km -1 úû0r0

(24)

Y = RU + H (25)

By using the above equation, the performance index J can bedescribed as follows.

E j[z -1]B[z -1] = C[z -1]R j[z -1] + z - j S j[z -1] (12)-1(19)

The following vectors and coefficients at time t are defined for
j = N1,…, N2.

(11)

The following equations are considered.

{C[z -1]R j[z -1]Du(t + j - km )C[z -1]y(t + j|t ) = R [z -1]Du(t + j - k ) + h¢ (t ) jmj

Then the output prediction (15) can be expressed as followingvector form.

wherey(t + j|t ) =From these equations, the output prediction can be expressedas the following equation, which separates the input term
Rj[z−1]Δu(t + j − km) in the current and future time from the othersignal h¢j (t ).

(18)

J = (RU + H - W )T (RU + H - W ) + lU TU (26)Minimizing the performance index J for the input vector U, thecontrol input can be given.
Du(t ) = [10…0](RT R + l I )-1 RT (W - H ) (27)The following vector and polynomials are defined.
[ pN1  pN 2 ] = [100](RT R + l I )-1 RT (28)P[z -1] = pN 2 + pN 2 -1z -1 +  + pN1 z -( N 2 - N1 ) (29)Fp¢ [z -1] = pN1 FN¢ 1 [z -1] + ¼+ pN 2 FN¢ 2 [z -1] (30)¢¢S¢p[z -1] = pN1 SN[z -1] + ¼+ pN 2 SN[z -1] 12(31)Then the transfer function of control law is expressed as follows.

Du(t ) =C[z -1]P[z -1]

C[z -1] + z - km S¢p[z -1]-Fp¢ [z -1]

w(t + N 2 )y(t )C[z -1] + z -km S¢p[z -1]

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131

Moreover, the following polynomials are defined.
D p[z -1] = DA[z -1]S¢p[z -1] + B[z -1]Fp¢ [z -1] (33)T[z -1] = DA[z -1]C[z -1] + z -km D p[z -1] (34)Then the closed-loop system can be derived as follows.
y(t ) =z -km B[z -1]C[z -1]P[z -1]+T[z -1]

w(t + N 2 )C[z -1](C[z -1] + z -km S¢p[z -1])
T[z -1](35)x (t )
Figure 2 | Conventional input.

It means that the output response is the same as the conventional
GPC. On the other hand, it can find that the disturbance response
can be tuned through the design polynomial S¢p[z -1] including se.

3. NUMERICAL EXAMPLE
The following controlled plant is considered as Okazaki et al. [5].
A[z -1] = 1 - 0.998775z -1 , B[z -1] = 14.4
C[z -1] = 1, km = 1The number of simulation steps is 500, the initial values of input
and output are set to be 0, and the variance of x(t) is s 2 = 0.001(each data of noise x(t) is the same for the conventional and theproposed method). The reference signal w(t) is 0.1. The design
parameters of GPC are given as follows.
N1 = 1, N 2 = 6, N u = 6, l = 8500

Figure 3 | Proposed output.

In this example, the design parameter of the proposed method isgiven as se = −1.8.
Figures 1 and 2 show the conventional output and input. In Figure 1,
the dashed line and solid line mean the reference signal w(t) and theoutput signal y(t) respectively. In Figure 2, the upper figure shows
the control input u(t) and the lower one shows the input increment∆u(t). Figures 3 and 4 show the proposed output and input. Theirlines are the same meanings as conventional method. From these
Figure 4 | Proposed input.

figures, it can find that the proposed method can change the noise
influence on output.

4. CONCLUSION

Figure 1 | Conventional output.

This paper proposed a directly extended method of conventional
GPC through newly-defined output prediction. The derived
control law can re-design the characteristic from noise to output
with keeping the closed-loop transfer function. Numerical example
was given to check the characteristic of the proposed method.

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A. Yanou / Journal of Robotics, Networking and Artificial Life 7(2) 129–132

As future work, the selection method of design parameter se should
be considered.

CONFLICTS OF INTEREST
The author declares no conflicts of interest.

REFERENCES[1] S. Omatu. T. Yamamoto, Self-tuning control, Corona Publishing
Co. Ltd., Tokyo, Japan, 1996.

AUTHOR INTRODUCTION
Dr. Akira Yanou
He received his PhD in Engineering from
Okayama University, Japan in 2001. Heworked with School of Engineering, Kinki
University from 2002 to 2008, and Graduate
School of Natural Science and Technology,
Okayama University from 2009 to 2016.
He joined Department of RadiologicalTechnology, Kawasaki College of Allied Health Professions in2016, and Faculty of Health Science and Technology, Kawasaki
University of Medical Welfare in 2017, as an Associate Professor.
His research interests include adaptive control, strong stability
systems and estimation.

[2] M. Vidyasagar, Control system synthesis: a factorization
approach, MIT Press, Cambridge, MA, USA, 1985.
[3] A. Inoue, A. Yanou, Y. Hirashima, A design of a strongly stableself-tuning controller using coprime factorization approach, Preprintsof the 14th IFAC World Congress, vol. C, 1999, pp. 1225–1230.
[4] A. Yanou, A study on extension of generalized predictive control,
The Papers of Technical Meeting on “Control”, IEE Japan, 2019,CT–19–032.
[5] S. Okazaki, J. Nishizaki, A. Yanou, M. Minami, M. Deng, Strongly
stable generalized predictive control focused on closed-loop
characteristics, Trans. Soc. Instrum. Control Eng. 47 (2011),
317–325.