Copyright (j) IFAC Digital Control: Past, Present and Future of PlO Control, Terrassa, Spain, 2000
SELF-TUNING PID CONTROLLERS BASED ON THE STRONGLY STABLE GENERALIZED MINIMUM VARIANCE CONTROL LAW
Takao Sato' Akira Inoue' Toru Yamamoto" and Sirish L. Shah •••
• Department of Systems Engineering, Faculty of Engineering, Okayama University, Okayama, 700-8530, Japan. Phone and Fax: +81-86-251-8233, E-mail:
[email protected] •• Department of Information Science and Technology Education Faculty of Education, Hiroshima University, Kagamiyama, Higashi-Hiroshima, 739-8524 Japan. Phone and Fax: +81-824 -24 -7160, E-mail:
[email protected] ••• Department of Chemical and Materials Engineering, University of Alberta, Edmonton, Alberta, T6G 2G6, Canada. Phone: +1-780-492-5162, Fax: +1-780-492-2881, E-mail: sirish.
[email protected]
Abstract: This paper proposes a new design scheme for a self-tuning PID controller. Expressions to calculate PID parameters from the identified plant coefficients are given. These expressions are derived by approximating the PID controller to the generalized minimum variance control (GMVC) law. To obtain a better approximation, the PID controller has a time-varying proportional gain. The PID control law approximates the strongly stable GMVC resulting in a stable closedloop system and a stable control law. Using a numerical example, the control performance of the proposed method is compared with those of PID controllers designed by other schemes. Copyright@2000 IFAC Keywords: PID controller, self-tuning control, minimum variance control, stability, coprime factorization
1. INTRODUCTION
towards the automatic or adaptive selection of PID parameters. There are two main classes of methods to design PID parameters. The first is to design the parameters to satisfy given control specifications, such as phase and amplitude margins(Hiigglund and Astrom, 1995). Yu(1998) has given an auto-tuning method combining an identification scheme with relay feedback. Wang, et al(1999) have proposed a PID tuning method based on a frequency do-
The PID controller is the most widely used control law in industry. This is because the PID controller has a simple structure and its control parameters are intuitively understandable to practical engineers. However, it is rather difficult to find 'optimal' PID controller parameter values in implementing such controllers for practical industrial processes. Hence, many papers have been devoted 511
If the process is given by a first-order plus deadtime model, then there exists a GMVC law having a second-order numerator where the numerator is exact. In this case, since the obtained PID controller approximates both the denominator and the numerator of GMVC law exactly, it will provide the same control performance as a GMVC. Since most chemical processes are typically overdamped and can be sufficiently well approximated by a first-order process with a time-delay, it is expected that the proposed PID controller should yield good control results in most cases.
main approach. The second class of automatic PID tuning technique is based on the comparison and approximation of controllers designed by other methods by a PID controller transfer function. In this vein, Tjokro and Shah(1985) have given an adaptive PID controller based on a poleplacement scheme. Miller, et al(1995) have proposed an adaptive predictive PID control scheme based on a generalized predictive control scheme. Cameron and Seborg(1983) and Yamamoto et al(1998) have also considered a self-tuning PID controller by comparing it with a generalized minimum variance controllers(GMVC).
This paper is organized as follows. In section 2, the transfer function model to represent the controlled plant and the PID controller to be tuned are obtained. Section 3 discusses the strongly stable GMVC that has to be approximated. By comparing the PID controller with the strongly stable GMVC, a self-tuning method of PID controller is outlined in Section 4. Finally, a numerical example is given to evaluate the proposed method and compared to the results from a conventional PID control scheme.
The generalized minimum variance controller was based on the work of Clarke and Gawthrop in 1979 and can also be interpreted as a pole-placement controller. Since then, it also has been widely applied in industry. Yamamoto et al(1998) have applied the self-tuning PID controller to a temperature control system of an industrial-scale polybutene reactor and obtained successful control results. Yamamoto et al(1999) have extended the PID controller to a self-tuning pole-assignment controller with a PID structure. Generally the PID controller is expressed as a transfer function having a first-order denominator and a secondorder numerator, or in the digital velocity form ~u(t) = u(t) - u(t - 1), the transfer function of a PID law consists of a zeroth-order constant denominator and a second-order numerator. Hence, in order to approximate a GMVC law to a PID controller, the transfer function ofthe GMVC law in incremental form ~u(t) should approximate a transfer function with a constant denominator and a second-order numerator. To carry out this approximation, Yamamoto et al(1999) replaced the denominator of GMVC law by a fixed static gain.
Notations: ~
Z-l
denotes backward shift operator; ~ is the difference operator
y(t) = y(t-1) and = 1 - z-l.
Z-l
2. PLANT MODEL AND PID CONTROLLER Consider a plant described by the following discrete-time ARIMAX model:
where A(Z-l) and B(z-l) are polynomials of the form: A(Z-l) = 1 + a1z-1 + + anz- n
This paper improves this approximation by using a time-varying proportional gain to approximate the higher-order denominator of the GMVC law. Hence in the approximation for having the stable proportional gain, the denominator of GMVC must be stable. That is, the approximated GMVC must yield a stable closed-loop and a stable controller simultaneously. A control system in which both of the closed-loop system and the controller are stable is called "strongly stable" (Vidyasagar, 1985). In this study the approximated GMVC is required to be strongly stable. This paper uses the strongly stable GMVC given by Inoue et al(1999).
B(z-l) = bo + b1z- 1 +
+ bmz- m
(2)
and u(t) is the input, y(t) is the output and km is dead time. The following assumptions are required for the system described by eqn. (1):
[A.!] The dead time km and the degrees nand m of A(Z-l) and B(Z-l) are known.
[A.2] The coefficients a1,' .. ,an, bo, b1, ... ,bm of A(Z-l) and B(Z-l) are unknown, but the nominal values of these parameters are known. [A.3] The polynomials A(z-l) and B(Z-l) are coprime. [AA] The disturbance ((t) is a white Gaussian noise with zero mean.
The numerator of GMVC must also be approximated by the second-order numerator of a PID controller. Generally the numerator of GMVC is of higher-order than 2. This paper proposes to approximate the higher-order numerator of GMVC by a second-order numerator by choosing the first and second significant roots of the numerator.
The discrete-time PID control law considered in this paper has the following structure:
~u(t) 512
T TD = kc{~ + T; + T. ~2}(w(t) -
y(t)) (3)
where w(t) is the reference input to be followed by the output y(t). Parameters kc, T[ and TD are the proportional gain, the reset time and the derivative time respectively. The sampling time is denoted by T s .
G(Z-l) = E(Z-l )B(Z-l) R(Z-l) = F(Z-l)
kcL(z-l)(W(t) - y(t)),
(12) (13)
The poles of the controller are given by the roots of G(Z-l) = 0 and the poles of the closed-loop system with this controller are the roots of the following polynomial T(z-l):
In order to compare a PID controller (3) to the strongly stable GMVC later, the control law (3) is rewritten as: ~u(t) =
+ Q(z-l)
T(Z-l)
= p(Z-l )B(z-l) + Q(z-l )~A(Z-l) (14)
(4) The transfer function of the controller (9) is
where ~u(t)
L(Z-l) = (1
+ T s + T D ) _ (1 + 2TD )z-l + TD Z-2 (5) T[
Ts
Ts
-1
kcL(z
In this section, the design problem is discussed by using the nominal values of coefficients all .. " an, ba, b1 , " ' , bm . A GMVC law is derived to minimize the following variance of a generalized output:
~(t
+ km + 1) = p(Z-l)y(t + km + 1) - R(Z-1)W(t)(7)
(17)
and polynomials P(Z-l) and Q(Z-l) are
(18)
p(Z-l) = 1 + P1Z- 1 + Q(z-l) =
+ Pnl'z-nl' qo + qlZ-l + + qm_1Z-(m-l) (8)
where
These polynomials and the order n p of P(z-l) are designed to have a stable closed-loop system. The polynomial R(Z-l) is defined later.
where polynomials E(Z-l) and F(Z-l) are the solution of the Diophantine equation ~A(z-l )E(Z-l)
+ z-(km+l) F(Z-l )(10)
and have the forms:
E(Z-l)
F(Z-l) nf
= 1 + e1z- 1 + + ekmz-km = fa + !lz-l + + fnjz-n j = max{n,n p - km}
kc
is a constant.
To obtain the proportional gain kc(t), first the constant kc is derived from equation (17). Then kc is substituted in equation (18) and equation (18) is solved. For equation (18) to be solvable, the polynomial G(z-l) must be stable, that is, the controller (9) must be stable. Hence the GMVC must have a stable closed-loop characterized by T(Z-l) and a stable controller. That is, the GMVC should be strongly stable. In equations (17) and (18), polynomials F(Z-l) and G(z-l) are replaced by Fe(z-l) and Ge(z-l) of the strongly stable GMVC. Polynomials Fe(Z-l) and Ge(z-l) are obtained by the following equations(Inoue, Yanou and Hirashima, 1999):
The controller to minimize the variance (6) is given by
P(Z-l) =
(16)
In this paper, the proportional gain is made to be a time-varying gain kc(t) and the PID parameters are chosen to satisfy the following equations:
is the generalized output;
+Q(Z-l)~U(t)
F(Z-l) ):= G(Z-l)
The self-tuning controller given by Yamamoto et al(1998) is derived by replacing polynomial G(Z-l) by static gain G(I) in equation (16). They also restrict the plant to be of order 2 so that the numerator F(Z-l) of the right-hand side of equation (16) has the same order (2) as the order of the left-hand side of (16), L(z-l).
3. STRONGLY STABLE GMVC
+ 1)
(15)
To obtain an approximate PID controller (4) to a GMVC law (15), k c and L(Z-l) are selected to satisfy the following relation approximately,
Ts
The tuning problem is to design suitable values of parameters kc, T[ and TD for plant (1). This paper solves the design problem by approximating the controller (4) to a strongly stable GMVC.
where ~(t + km
F(z-l)
= G(Z-l) (w(t) - y(t))
Fe(Z-l)
(11)
= Ud(z-l )F(z-l) +Un(Z-l )~A(Z-l)
(19)
Ge(z-l) = Ud(z-l )G(z-l)
and G(Z-l) and R(z-l) are obtained by the following equations:
-Un(z-l )z-(km+1)B(z-1) 513
(20)
where polynomials Ud(z-l) and Un(z-l) are newly introduced design polynomials. And the controller of the strongly stable GMVC is
Fe(Z-l)y(t)
T[ =
TD
+ Ge(z-l)~u(t)
-Re(Z-l)W(t) Re(Z-l) = Ud(z-l )R(z-l)
=0
= -
11 + 2/2 T. 10 + h + 12 12
11 +212
(28)
T.
From equation (24), the proportional gain kc(t) is obtained from
(21) (22)
By using controller (21), the characteristic polynomial of the dosed-loop system is T(Z-l) as in equation (14), which is independent of Un(z-l) and Ud(z-l). The denominator of the controller is Ge(Z-l) (equation (20)) which will be designed by selecting Un(z-l) and Ud(z-l) without changing
where n g is the order of polynomial Ge(Z-l) and n g ~ km + m and go,' .. ,gn g are
T(Z-l ).
4. SELT-TUNING
pm CONTROLLER
If the coefficients a l , " ' , an, bo , bl , . . . ,bm of the plant are known, then the tuning parameters of the PID controller can be calculated through equations (26), (28) and(29). Since in this paper, the case with the unknown coefficients is considered, the recursive least squares identification law (Goodwin and Sin, 1984) is used to obtain the estimated values al(t), "', an(t), bo(t), bl(t), "', bm(t) of the unknown parameters aI, "', an, bo, bl , " ' , bm ·
In this section, we derive a self-tuning PID controller to approximate the strongly stable GMVC (21). Comparing the controller (4) with (21), the following relations are obtained:
The design scheme of the proposed self-tuning PID controller is summarized in the following steps: [Self-tuning PID controller]
The polynomial Fe(Z-l) is approximated as a second order polynomial because the order of L(z-l) is 2. This approximation of the polynomial Fe(z-l) is obtained as follows: first the roots of Fe(z-l) = 0 are calculated and the largest or the 'dominant' roots >'1 and A2 are selected such that they correspond to the first and the second largest absolute values IAll and IA21. Then the approximated polynomial Fe(z-l) is
Step 1: Using the known nominal values of the plant parameters, select the dosed-loop characteristic polynomial T(Z-l) and the controller's Ge(Z-l) so that the polynomial Fe(Z-l) of (19) has as small order as possible. Step 2: Obtain estimates Ul(t), "', un(t), bo(t), .. " bm(t) of the unknown plant parameters by using the recursive least square identification law. Step 3: Using the estimated values and then solving the Diophantine equation (14), obtain P(Z-l) and Q(Z-l). Step 4: Using P(z-l) and Q(Z-l) obtained in Step 3, and solving Diophantine equations (10) and (20) with fixed Ge(z-l) given in Step 1, obtain E(Z-l), F(z-l), Un(Z-l) and Ud(Z-l). Step 5: Calculate Fe(z-l) of (19) and obtain approximated Fe(Z-l) of (26). Step 6: Calculate the PID parameters using equations (28) and (29) Step 7: Obtain control input u(t) from the pm controller (3) Step 8: Repeat steps 2 to 7 at each sampling time.
- -1 (Al - Z-l )(A2 - Z-l) Fe(z ) = Fe(l) (Al _ 1)(A2 _ 1) (26) If A(z-l) is first order and one can find Ud(Z-l) to be first order and Un(Z-l) as zeroth order making Ge(z-l) of (20) stable, then Fe(Z-l) of (19) will have order 2 and the equation (23) holds without approximation of (26). In such case the pm controller is identical to the GMVC law.
From equations (5) and (23) and using notations for Fe(Z-l) of (26),
Fe(z-l) =
10 + IlZ- l + 12 Z- 2
(27)
the parameters of PID controller are given by
Remark: Step 1 is needed to have a good approximation of (26). 514
5. AN EXAMPLE
0.62 and all of them are stable. In the simulations, these polynomials Fe(z-l) and G e (Z-l) are unknown because the plant parameters are unknown. These polynomials are obtained by implementing step 2 to step 5 in the procedure of the self-tuning PID controller: first as in step 2, identified plant parameters 0. 1 (t),bo(t),·· .,b3 (t) are calculated; then in step 3 P(Z-l), Q(z-l) are obtained by solving the Diophantine equation (14) to have the fixed closed-loop poles, 0.4 and 0.7; in step 4, E(Z-l), F(Z-l) are calculated from equation (10) and Un(Z-l) and Ud(z-l) from (20) with the fixed Ge(z-l) of equation (37) and in step 5, Fe(Z-l) is calculated by equation (19) and if the order of Fe(z-l) is greater than 2, then the approximated Fe(z-l) is derived from (26). Finally, in step 6, PID gains are obtained.
In this section, the PID controller is designed for the following plant
=
(1 - 0.9z- 1 )y(t)
z-3(0.1 + 0.lz- + 2z- 2 )u(t) + ~(t)/tl (31) 1
The step response of the plant is given in Fig.1 and the response reaches steady state after 30 sample intervals and has a steady-state gain of 22. Also the plant parameter al changes from al = -0.9 to al = -0.7 at step 175 and to al = -0.8 at step 425. In this section, a controller will be designed so as to have a faster response, a unity gain (for asymptotic tracking) with the ability to adapt to parameter changes.
Simulations are conducted under the conditions that reference input w(t) is a rectangular wave with amplitude 1.0 over a period of 100 steps, the variance of random disturbance ~(t) is 0.03 and the recursive least square identification law having reset with the forgetting factor 0.99 is used. The initial value of the estimated covariance matrix is 101 and the initial values of identified coefficients is the nominal values which are true values of (31).
To design the controller, the GMVC law is first designed, that is, the controller to minimize the variance E[CI>(t + 3)2] of the generalized output:
CI>(t + 3)
= (0.40 -
0.32z- 1 )y(t + 3)
+(0.96 + 0.72z- 1 )tlu(t) - Re(z-l)w(t) (32)
Re(Z-l) is given by equation (22). This generalized output is designed such that the poles of the closed-loop system are 0.4 and 0.7 resulting in P(Z-l) and Q(z-l) polynomials from the Diophantine equation (14). Then the controller is
Simulation results of the proposed method are given in Fig.2. To compare with this result, the result obtained by self-tuning PID controller designed by Yamamoto et al(1998) is shown in Fig.3 and the result by PID controller designed by Chien, Rrones and Reswick (CRR) method (1952), which is a popular design method, is shown in Fig.4. The application of the proposed method is shown in Fig.2, which shows overshoot at only the initial stage of the identification and during the steps of the deviation of the plant parameters values. After the identification is completed, the response has no overshoot and stays within the range of the white noise disturbance.
tlu(t) = 0.52 - 0.43z- 1 1+ + 0.9z- 2 + 0.94z- 3 + 0.96z- 4 x(w(t) - y(t)) (33) O.8z- 1
But the poles of this controller are 0.35±0.96i and -0.75 ± 0.59i and include unstable poles (10.35 ± 0.96ij = 1.0441 > 1) . To redesign the controller to be stable, new design parameters Un(z-l) and Ud(z-l) are selected as
Un(Z-l) Ud(z-l)
= -0.3 = 1 - 0.9z- 1
=,
(34)
These parameters are chosen so that the polynomial Fe(Z-l) is second order and Ge(z-l) is a stable polynomial. That is, the controller is
Fe(Z-l)
~u(t) = Ge(Z-l) (w(t) - y(t))
,.
,
.
.,
(35)
and Fig. 1. Simulated output of step response 1
Fe(Z-l) = 0.22 - 0.33z- + O.12z-
2
(36) 3 Ge(z-l) = 1 - 0.lz- + 0.18z- + 0.16z1
+0.15z- 4
-
0.27z-:;
2
6. CONCLUSION
(37)
The poles of the controller, that is, the zeros of Ge(z-l) are 0.34 ± 0.78i,-0.60 ± 0.49i and
In this paper, a design scheme of a self-tuning PID controller is proposed. Expressions to calculate 515
tional popular PID design scheme based on the CHR scheme.
7. REFERENCES Camacho. E. F and C. Bordons (1995). Model Predictive Control in the Process Industry. Advances in Industrial Control, Springer. Cameron. F and D. E. Seborg (1983). A SelfTuning Controller with a PID Structure. 1nl. 1. Control. Vol.38, No.2, 401-417. Chien. K. L, I. A. Hrones and J. B. Reswick (1952). On the Automatic Control of Generalized Passive Systems; Trans on ASME, Vol.74,175-185. Clarke. D. Wand P. J. Gawthrop (1979). Self-Tuning Control. lEE Proc. Vo1.126D, No.6. 633-640. Goodwin. G. C and K. S. Sin (1984). Adaptive Filtering Prediction and Control, PrenticeHall. Hagglund. T and K. J. Astrom (1995). Automatic Tuning of PID Controllers. The Control Handbook. CRC Press, 817-826. Inoue. A, A. Yanou and Y. Hirashima(1999). A Design of a Strongly Stable Self-Tuning Controller Using Coprime Factorization Approach. 14th IFAC World Congress, Beijin, Vol.C, 211-216. Miller. R. M, K. E. Kwok, S. L. Shah and R. K. Wood (1995). Development of a Stochastic Predictive PID Controller. Proc. American Control Conference, Seattle. 4204-4208. Tjokro. Sand S. L. Shah (1985). Adaptive PlO Control. Proc. of American Control Conference, Boston. 1528-1534. Vidyasagar. M (1985). Control System Synthesis. A Factrization Approach, The MIT Press. Wang. Q. G, T. H. Lee, H. W. Fung, Q. Bi and Y. Zhang (1999). PID Tuning for Improved Performance. IEEE Trans on Control Systems Tech. Vol. 7, No. 4, 457-465. Yamamoto. T, A. Inoue and S. L. Shah(1999). Generalized Minimum Variance Self-Tuning Pole-Assignment Controller with a PID Structure. Pro. of 1999 IEEE Int Con/. on Control Applications, Hawaii. 125-130. Yamamoto. T, K. Fujii and M. Kaneda (1998). Design and Implementation of a Selt-Tuning PID Controller. Preprints of IFAC Workshop on Adaptive Control and Signal Processing. Glasgow. 58~63. Yu. C. C (1998). Auto-tuning of PID Controllers. Advances in Industrial Control. Springer.
t a.= -0.8 -'.:-------==c----=---""""'C-----7.;c--~-_____=. T ....
Fig. 2. Simulated output by the proposed method
.---a.= -0.7
-".!----7=--~~-~...~-~--~-_____=! T~.
Fig. 3. Simulated output by the self-tuning PlO controller with steady state gain
Fig. 4. Simulated output by PID controller designed by CHR method
PID parameters from the identified coefficients are given. These expressions are derived by approximating the PID controller to the GMVC law. To obtain a better approximation, the proposed PID controller has a time-varying proportional gain and a strongly stable GMVC algorithm is used as the approximated controller. This controller law simulataneouly results in a strongly stable closed-loop system and a stable control law. The proposed self-tuning PID controller includes a parameter identification algorithm. A numerical example is given to show the effectiveness of the proposed scheme and is compared with a self-tuning PID controller that approximates a GMVC with a static gain, and a conven516