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A minimax design of robust I-PD controller for time-delay systems with parametric uncertainty
A MlNIMAX DESIGN OF ROBUST I-PD CONTROLLER FOR TlM...
14th World Congress oflFAC
C-2a-08-2
Copyright © 1999 IFAC 14th Triennial World Congress, Beijing, P.R. China
A MINIMA X DESIGN OF ROBUST I-PD
CONTROLLER FOR TIME-DELAY SYSTEMS WITH PARAMETRIC UNCERTAINTY K. Hirata* Y. Yanase u T. Katayama** T. Kawabe***
* Department of Electrical and Electronic Systems Osaka Prefecture University, Sakai, Osaka 599-8531, Japan [email protected]. osakafu.-u. ac.jp U Department of Applied Mathematics and Physics Graduate School of Informatics, Kyoto University Kyoto 606-8501, Japan, [email protected] *** Institute of Information Sciences and Electronics University of Tsukuba, Tsukuba, Ibaraki 305-8573, Japan kawabe@is. tsukuba. ac.jp
Abstract: This paper presents a design method of robust I-PD controller for SISO plants with a time-delay and parametric uncertainty by extending our earlier approach. It is assumed that the adjustable parameters of the I-PD controller are chosen so as to minimize the generalized integral of squared error (ISE) maximized by the plant parameters belonging to a given bounded set. Thus the design problem is formulated as a minimax optimization problem, which is solved by a genetic algorithm (GA). A novel feature of the present paper lies in the exact computation of the generalized ISE for time-delay systems without using Pade approximant. Numerical examples show the effectiveness of the present approach. Copyright © 1999 IFAC Keywords: Robust I-PD controller, Minimax optimization, Time-delay systems, Exact evaluation of lSE, Genetic algorithm
1.
~TRODUCTION
There has been a renewed interest in proportionalintegral-derivative (PID) controllers, since it has become easier to implement PID-based algorithms due to the recent development of digital process controllers. In fact, over 90% of industrial control problems are solved by PID controllers (or by their variants) in spite of the simple structures (Astrom and Hagglund 1995). Since industrial plants are too complex to obtain precise dynamics, the PID controllers must be robust for a wide range of operating conditions. There have been developed some works for robust PID tunings methods taking into account robust closed-loop stability and robust performance in
the presence of plant uncertainty. For example, the IMC based method of designing PID controller is considered in Morari and Zafiriou (1989), where a low-pass filter is introduced to achieve robust stability and robust performance under nonparametric uncertainty. In Chen et. al. (1995), the mixed H2/ Hoc, optimization is employed to design a rohust PID controller in the presence of nonparametric uncertainty, where the problem of minimizing the integral of squared error (ISE) subject to robust stability constraint is solved by using a genetic algorithm (GA). Motivated by a minimax approach of designing PID controller under parametric uncertainty due to Lu (1992a, 1992b), we have also developed a GA based method of designing robust I-PD
1272
Copyright 1999 IFAC
ISBN: 0 08 043248 4
A MlNlMAX DESIGN OF ROBUST I-PD CONTROLLER FOR TIM ...
controller (see Fig. 1) for time-delay systems with parametric uncertainty, where the time-delay is replaced by its Pade approximant (Kawabe and Katayama 1994, 1996, 1997). More specifically, the I-PD parameters are adjusted so as to minimize performance criterion maximized by plant parameters with bounded uncertainty. It is shown in Kawabe and Katayama (1994) that a minimax design based on the ISE leads to a very large overshoot in the optimal closed-loop response. A large overshoot has also been observed in earlier PID tunings based OD the ISE criterion (see e.g. Smith and Corripio (1985)). In order to circumvent this difficulty, we have applied the minimax design technique to a generalized ISE that includes a penalty on the derivative of the control variable (Kawabe and Katayama 1996, 1997). It is shown that the use of the generalized ISE cost function, which has been used in the LQ regulator based design of servomechanism, is very effective in shaping the closed-loop responses by adjusting a weighting parameter. The use of Pade approximants is, however, a drawback of (Kawabe and Katayama 1994, 1996, 1997), since the stability criterion with Pade approximants for a time-delay element may yield erroneous robust stability conditions and the evaluation of the (generalized) !SE is not exact (Marshall et. al. 1992). In this paper) we present a design method based on the exact evaluation of control performance for time-delay systems using the technique developed in Marshall et. al. (1992), in which the PID parameter settings are extensively studied based on the ISE. The performance criterion and the optimization technique of this paper are the same as those of Kawabe and Katayama (1996, 1997). The robust stability of the closed-loop system with a time delay is determined by checking the range of time-delay in the characteristic equation. Furthermore, we give a robust stability criterion based on zero exclusion principle (Kogan 1995) for a polytope of quasi-polynomials 1. This criterion gives a necessary and sufficient condition for stability of the polytope of quasi-polynomials , and is conveniently used in the present problem as well as in problems of adjusting parameters of controllers with a fixed configuration. 2. PROBLEM FORMULATION 2.1 Minimax Optimization Problem
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We assume that the plant model is described by P(s)e- h8 , where pes) is a strictly proper rational function with parametric uncertainties and h is a delay. I-PD Controller
r················································ r i
e
Fig. 1. Block diagram of control system Let (J be the parameter vector of pes) and belong to a bounded set e = {8IBI ~ 8 ~ Bu }. Let q := (Kc, Ti, Td)T be the I-PD controller parameter vector belonging to a bounded set Q = {qlql ~ q ~ qu} C R3 , which is the range of adjustable I-PD parameters. It is assumed that e and Q are given a priori. We define the generalized ISE (Kawabe and Katayama 1996, 1997) as J(q, (J):=
J={e
2
(t)
+ piJ?(t)} dt,
(1)
o
where p 2': 0 is an adjustable parameter. This performance index, employed in the LQ regulator based design of servomechanism, is a weighted sum of the ISE and the integral of squared derivative of the manipulated variable. Hence, this performance index not only evaluates the ISE but also gives a penalty to the velocity of manipulated variable. The robust I-PD controller design based on the generalized !SE is formulated as a minimax optimization problem: min max J (q, 9), under the qEQ iJE9
constraint that q stabilizes the closed-loop system for all B E 8. The designed controller guarantees the robust stability and the best performance under worst case scenario.
2.2 Description of the Closed-Loop System From the Parseval's theorem, (1) is written as
Consider the SISO control system shown in Fig. 1, where r is the step input, u the manipulated variable , y the measured variable, e the error, and K i , T i , Ta , I E R are parameters of the controller.
J(q,B)
~ 2!j L2=e(,)e(-')d' +p
1 The quasi-polynomial is a polynomial in one complex variable and exponential powers of the ·variable.
_2='0(') [-8«(-.)]
(2)
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Copyright 1999 IF AC
ISBN: 0 08 043248 4
A MTNIMAX DESIGN OF ROBUST I-PD CONTROLLER FOR TIM ...
where e(s) and u(s) are the Laplace transforms of e(t) and u(t), respectively. Let pes) = N(s)jD(s), and define A(s), Be(s), C(s), De(s) and B,..(s) as
C(s) = Kc {(1+-r)T,TdS2+(T;+'YTd)s+1}N(s),
De(s) = KcTi {(I Ru(s) = Ke(l
+ 'Y)TdS + I} N(s),
+ ")'TdS)D(s).
(3)
Then from Fig. 1, the error eCs) for the step input res) = l/s and the derivative of the manipulated variable are given by
Be(s) + De(s)C h8 e (s) - ........:::..-:---7---=7.::.,-:-----;-- A(s) + C(s)e- hs ' Bu(s) sues) - A(s) + C(s)e- hs '
(4) (5)
respectively. The characteristic equation of the closed-loop system is given by f(s)
;=
A(s)
+ C(s)
e- hs
and compute the corresponding h. Then analyze the behavior of the roots in the neighborhood of h. Step 4: Check the stability region of h by sorting destabilizing and stabilizing critical value of h. If a given time-delay h is in the stability region, the closed-loop system is stable. Otherwise, the system is unstable.
+ ,Tds)D(s), = Ti(l + ,Tds)D(s),
A(s) = TiS(l
Be(s)
14th World Congress ofTFAC
= O.
(6)
Since the plant is strictly proper, the degree of A(s) is always greater than that of C(s), so that this system is a retarded system.
3.2 Robust Stability of time-delay systems Next we consider the robust stability criterion against parametric uncertainties. Namely, we assume that N(s) and D(s) in (3) are the (real) interval polynomials. Let N denote the number of vertices of the hyperrectangle in the parameter space and
e
be the quasi-polynomials corresponding to each vertex. Then the characteristic quasi-polynomial is generated by the convex combination of these quasi-polynomials: r
f(s,k)
;=
L
(10)
k. fi(S),
;=1 N
3. ROBUST STABILITY CRITERION FOR TIME-DELAY SYSTEMS
s.t.
2::: k. = 1, O:S; k; :s; 1, i = 1,· .. ,N. i=l
3.1 Stability Criterion We first consider the stability criterion of the quasi-polynomial (6) when the coefficients of A(s) and C(s) are fixed. Let us regard h as a variable and write the characteristic equation (6) as f(s, h) := A(s)
+ C(s)e- hs
= O.
= 0.
Step 2: Consider infinitesimally small positive h. For retarded systems, all the new roots appear in the left half-plane. Step 3: Determine the positive zeros:' of the polynomial
W(w 2 ) := A(jw)A( -jw) - C(jw)C(-jw),(8) 2
K
(7)
Since a common factor of A(s) and C(s) is a root of(7) for any h?= 0, we assume that A(s) and C(s) have no common factors. A procedure of finding the stability regions for h consists of four steps (Marshall et. al. 1992). Step 1: Examine the stability of (7) at h
Let k := (k1 , · · · , kN)T and define the convex polyhedron K, the quasi-polynomial family S, and the value set Sw as :=
{k
I ~ k; = 1, 0 ~ ki ~ 1 } ,
I k E K}, {f(jw, k) I k E K}.
S := {f(s, k) Sw :=
(11)
The shape of Sw is a polygonal region and each quasi-polynomial segment corresponding to the boundary of Sw is called edge. From the edge theorem (FU et.al. 1989) and the zero exclusion principle (Kogan 1995), we see that the quasipolynomial family S is stable if and only if the boundary of Sw does not contain or pass through the origin for all w ;:::: O. Since the boundary of Sw is the value set of the segment quasi-polynomial corresponding to two generating points of S, we consider an edge connecting pes) and pes). Define the quasi-poLynomial segment
Each zero indicates the point where the root loci against
h touch or croS/l the imaginary axis. As is obvious from (8), such locations are independent of delay h. Another
with A E [0,1]. From (9) we can express (12) as
key fact is tha.t the number of zeros is finite, although (7) has infinitely ma.ny solutions.
f(8, A)
= A(8, >.) + C(8, A) e- hs ,
(13)
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Copyright 1999 IF AC
ISBN: 008 0432484
A MTNIMAX DESIGN OF ROBUST I-PD CONTROLLER FOR TIM ...
where A(s, A) and C(s, A) are defined as
+ AA2(8), C(S,A):= (1- A)C 1(S) + >.C2(8). A(s,A) := (1- A)Al(S)
A stability criterion of the quasi-polynomial segment (12) is given as follows: Theorem 3.1 Given>. E [0,1), let w).. be
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It is assumed that the system is stable, so that all the poles of (16) lie in the left half-plane. For the notational convenience, we suppress the Laplace variable s of each polynomial and denote its paraconjugate by (), for example, we write A := A(s) and A := A( -8). Then the integrand of (15) is expressed as
=
x(s)x( -s) w.\
= sup{w I A.(jw, >.)A( -jw, A)C(jw, A)C(-jw, A)
Also let
= a}.
sup{w.\
+ /3e- hs a + iJe hs + Ce-hs + A + Ce hs =: H(s) + H( -8),
I A E [0, I]}.
a x(s)x(-s) = A
Then, the quasi-polynomial segment (12) contains or passes through the origin for w ~ 0 if and only if there exist w E [0, w] satisfying the following condition 3
We can summarize the procedure examining stability of the quasi-polynomial family S as follows: Step 1: Examine the stability of one quasipolynomial in S. If the quasi-polynomial is stable, go to Step 2. If not, S is not stable. Step 2: Check whether 0 1. Swo for one Wo ~ o. IT rt Swo, go to Step 3. If not, S is not stable.
°
Step 3: For each edge, check the existence of w 2: 0 such that (14) holds. If there exist such an w on at least one edge, S is not stable. If not, S is stable.
4. EVALUATION OF 1SE FOR TIME-DELAY SYSTEMS
=
ACBE + DD) - 2BC15 2(AA - CC) ,
Q
=
r.I
= 2ADB - G(BE
+ Df5)
2(AA _ CC)
iJ
.
(19)
To compute (15), we consider the contour r shown in Fig. 2. Denote the segments on the imaginary axis and the semi-circle as r 1 and r 2 , respectively. If AA-CC = 0 has roots on the positive/negative imaginary axis, add infinitesimally small indentation to r 1 to the left/right around the root. Denote the integral of H(s) along r as I, Le., 1 =1-
21[j
J
0:
A
r
+ /3e- hs ds + Ce- hs -
(20)
Since the system is stable, A + Ce-hs = 0 ha.'! no root inside r. Then the singularities of the integrand of (20) inside r are limited to the roots of AA = 0, which are denoted by Si. It is crucial that {Si} is a finite set. It follows from the residue theorem that
cc
By using the technique of Marshall et. al. (1992), we develop a method of computing the ISE of the form
J
(18)
where
Re[fl(jw») Im[J2(jw»)Re[j2U w ») Im[fl(jw)] = 0, (14) Re[f1 (jw)] Re[f2(jW)] ::::; 0, Im[fl(jw)] Im[f'l(jw)] :::; 0,
=
(17)
It can be shown that (17) is additively decomposed as
w be w=
+ De- hs B + Dehs + Ce-hs A+Ce hs '
B A
I = -
Q + {3e- hs ] ~ Res [ A + Ce- hs ' Si
,
(21)
t
+i=
J
x'l(t) dt
=
2!j
o
J
Im
x(s)x( -s)ds, (15)
-joo > jro
where x(s) is expressed as
x(s)
=
B(8) A(s)
+ D(s)e- h8 + C{s)e-hs .
(16)
Re -jOl x
3 In contrast to the polynolllial family case, the first equation in (14) is not a polynomial but a nonlinear function of w. Therefore we must perfonn a line search
instead of solving polynomial roots. However, since the interval to be searched is finite, such a computation is tractable.
Fig. 2. Contour
r
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Copyright 1999 IF AC
ISBN: 008 0432484
14th World Congress oflFAC
A MINIMAX DESIGN OF ROBUST I-PD CONTROLLER FOR TIM...
where Res [f(s), Si] denotes the residlles of f(8) at Si. If Si is a singularity of H(s), -Si is also a singularity of H ( -s). \Vith respect to residlles, we have Res [H(s),
Si]
= -Res [H( -8),
-Si).
(22)
It follows from (22) that the integral of H(-s) along r is equal to I. Since X(8) has no singularity on the imaginary axis, we can replace the integral path of (18) by rI, i.e., J=
~f{H(s)+H(-s)}dS. 27rJ
r,
As the radius of P2 goes infinity, the integral along replace r 1 by
r 2 becomes zero and hence we can r. Thus we obtain J = 21.
As stated in Section 2, the design of robust I-PD controller is reduced to a minimax optimization problem: qEQ 9E8
The GA is one of the powerful tools for the noncovex optimization. To design this iterative random search algorithm, we must define some problem-dependent algorithm elements: Representation
In the present algorithm, we adopt the Gray-code string to represent candidate solutions. Each of the I-PD parameters Kc, T i , Td is represented by a binary string. Three such strings are concatenated into a binary string, which defines a point in the parameter space to be searched by the GA algorithm. Fitness function The linear scaling method is used as a fitness function, Le., the fitness is simply computed from the order of the cost function va lue of each individual among t he population.
5. OPTIMIZATION METHOD
min max J(q, 0),
5.2 Structure of Genetic Algorithm
(23)
where the closed-loop system is stable for all () E 8. It should be noted that the generalized ISE J(q,8) does not have saddle point in this case. Since there are many local optimal solutions for the controller parameters, we use a genetic searching algorithm to find a global solution.
Genetic operators
The genetic operators used in the present algorithm are uniform crossover, bit mutation and the roulette wheel selection. The uniform crossover generates two offspring by exchanging a predefined number of alternate subsections between two parent strings. The recombination operator, mutation, is implemented as altering bit values at randomly selected string positions.
6 . DESIGN EXAMPLES
We assume that q := (Kc, Ti, Td) E Q. For computational purpose, let Q d := {ql, ... , qN} be a discrete approximation of the set Q. Then the design algorithm of a robust I-PO parameters is summarized as follows.
In this section, we show a numerical example of robust I-PD controller design. The range of adjustable I-PD parameters and l' are given by Kc E [0.1,15.9]' Ti E [1.0, 100.0], Td E [1.0,100.0] and'Y 0.1. The parameters of the GA are given by population: 500, number of individual: 100, mutation rate: 0.2.
Check robust stability of qi for all
Example 6.1 We consider the second-order plant
5.1 Algorithm of Minimax Optimization' Problem
Step 1:
e E 8, where Qi is generated by the CA operations
=
G ( )
described b elow.
3 S
Step 2: If the closed-loop system with qi is robustly stable, compute
(24)
Step 3: H the GA algorithm stops, go to Step 4. Otherwise, go to Step 1. Step 4: Let the minimum of Mi be Afio ' Then the corresponding qio yields the minima x robust I-PD controller.
K
T2S2
+ T1s + le
-28
(25)
,
where the parameters belong to the set K E [0.8,1.2], Tl E \6.0, 20.0J, T2 E [225,625]. Table 1 shows the minimizing solution for nominal plant with p 0 and the minimax solutions for p 0.0, 1.0, 10.0. Fig. 3 shows the step responses for the worst case scenario with p = 10 where the solid
=
by using (21) and the GA. If the system is not robustly stable, set Mi = 00.
=
=
Table l' Robust I-PD controller design T; K Kc Td Tl T2 nominal 15.9 6 .90 8.30 12.0 1.0 400 0.0 15.9 17.4 5.44 0.8 6.0 625 1.0 9.30 9.03 625 19.9 0.8 6.0 10.0 5.45 23 .0 625 13.0 0.8 20.0 p
J
10.2 15.5 20.9 28.6
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Copyright 1999 IFAC
ISBN: 0 08 043248 4
A MTNIMAX DESIGN OF ROBUST I-PD CONTROLLER FOR TIM ...
line and the dashed line are the responses designed by the present method and by the earlier method (Kawabe and Katayama 1996, 1997), respectively. We see that the closed-loop response designed by
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stability criterion. The numerical example shows that • The robust I-PD controller guarantees the robust stability and the best performance under worst case scenario. • We can easily regulate the overshoot and oscillatory variation in the closed-loop responses by using the generalized ISE. • The control performance of present design is better than the earlier results based on the Pade approximant and the Kharitonov's theorem.
2
·:~--.·•i• • ;• • • 00
20
4Q
80
80
100
120
HO
160
8. REFERENCES
l·······l
180
200
11me
Fig. 3. Step responses of closed-loop system for the worse case scenario the present method is much better than the one designed by our earlier method based on Pade approximant. The improvement of the closed-loop performance is due to the exact treatment of the time-delay element. As shown in Fig. 4, the robust stability region of the I-PD parameters for Ti = 27 is much larger than that estimated by using the Pade approximant. Thus under the Pade approximation, the region of adjustable parameters becomes much smaller than the actual feasible o region.
. ,,.
SoIkI : f'l<>poood me1hod
,
D
,I ; '
15
10
Kc
a
10
Fig. 4. The stability region of I-PD parameters for T. = 27
7. CONCLUSION
We have developed a GA based design method of robust I-PD controller for time-delay systems with parametric uncertainty. A novelty of the present design method over the earlier ones is the exact evaluation of control performance and the robust
Astrom, K. J., and Hagglund, T.(1995). PID Control - Theory, Design and Tuning, 2nd ed. Instrument Society of America. Chen, B. S., Cheng, Y. M., and Lee, C. H.(1995). A genetic approach to mixed H 2 / Hoo optimal PID controL IEEE Control Systems Magazine, 15, 51-60. Fu, M., Olbrot, A. W., and Polis, M. P. (1989). Robust stability for time-delay systems: The edge theorem and graphical tests. IEEE Trans. Automat. Contr., 34,813-820. Kawabe, T., and Katayama, T. (1994). A minimax design of robust I-PD controller for system with time-delay. Proc. 1st Asian Control Conference, Tokyo, July, 3, pp. 495-498. Kawabe, T., and Katayama, T. (1996) A minimax design of robust I-PD controller based on a generalized integral-squared-error. Trans. SlCE, 32, 1226-1233 (in Japanese). Kawabe, T., and Katayama, T. (1997). A minimax approach to design of robust I-PD controller. Proc. 2nd Asian Control Conference, Seoul, July,!, pp. 661-664. Kogan, J. (1995). Robust Stability and Convexity. Springer-Verlag. Lu, J. P. (1992a). Robust process control on Honeywel1 TDC3000. Preprints, Honeywell Advanced Control Symposium. Lu, J. P. (1992b). A MIMO robust controller on the Honeywell TDC3000. Proc. 19th Annual Honeywell Industrial Automation and Control, User's Group Symposium. Marshall, J.E., Gorecki, H., Korytowski, A., and Walton, K. (1992). Time-Delay Systems: Stability and Performance Criteria with Applications. Ellis Horwood. Morari, M., and Zafiriou, E. (1989) Robust Process Control. Prentice-HalL Smith, C. A., and Corripio, A. B. {1985}. Principles and Practice of Automatic Process Control. Wiley.
1277
Copyright 1999 IF AC
ISBN: 008 0432484
14th World Congress oflFAC
C-2a-08-2
Copyright © 1999 IFAC 14th Triennial World Congress, Beijing, P.R. China
A MINIMA X DESIGN OF ROBUST I-PD
CONTROLLER FOR TIME-DELAY SYSTEMS WITH PARAMETRIC UNCERTAINTY K. Hirata* Y. Yanase u T. Katayama** T. Kawabe***
* Department of Electrical and Electronic Systems Osaka Prefecture University, Sakai, Osaka 599-8531, Japan [email protected]. osakafu.-u. ac.jp U Department of Applied Mathematics and Physics Graduate School of Informatics, Kyoto University Kyoto 606-8501, Japan, [email protected] *** Institute of Information Sciences and Electronics University of Tsukuba, Tsukuba, Ibaraki 305-8573, Japan kawabe@is. tsukuba. ac.jp
Abstract: This paper presents a design method of robust I-PD controller for SISO plants with a time-delay and parametric uncertainty by extending our earlier approach. It is assumed that the adjustable parameters of the I-PD controller are chosen so as to minimize the generalized integral of squared error (ISE) maximized by the plant parameters belonging to a given bounded set. Thus the design problem is formulated as a minimax optimization problem, which is solved by a genetic algorithm (GA). A novel feature of the present paper lies in the exact computation of the generalized ISE for time-delay systems without using Pade approximant. Numerical examples show the effectiveness of the present approach. Copyright © 1999 IFAC Keywords: Robust I-PD controller, Minimax optimization, Time-delay systems, Exact evaluation of lSE, Genetic algorithm
1.
~TRODUCTION
There has been a renewed interest in proportionalintegral-derivative (PID) controllers, since it has become easier to implement PID-based algorithms due to the recent development of digital process controllers. In fact, over 90% of industrial control problems are solved by PID controllers (or by their variants) in spite of the simple structures (Astrom and Hagglund 1995). Since industrial plants are too complex to obtain precise dynamics, the PID controllers must be robust for a wide range of operating conditions. There have been developed some works for robust PID tunings methods taking into account robust closed-loop stability and robust performance in
the presence of plant uncertainty. For example, the IMC based method of designing PID controller is considered in Morari and Zafiriou (1989), where a low-pass filter is introduced to achieve robust stability and robust performance under nonparametric uncertainty. In Chen et. al. (1995), the mixed H2/ Hoc, optimization is employed to design a rohust PID controller in the presence of nonparametric uncertainty, where the problem of minimizing the integral of squared error (ISE) subject to robust stability constraint is solved by using a genetic algorithm (GA). Motivated by a minimax approach of designing PID controller under parametric uncertainty due to Lu (1992a, 1992b), we have also developed a GA based method of designing robust I-PD
1272
Copyright 1999 IFAC
ISBN: 0 08 043248 4
A MlNlMAX DESIGN OF ROBUST I-PD CONTROLLER FOR TIM ...
controller (see Fig. 1) for time-delay systems with parametric uncertainty, where the time-delay is replaced by its Pade approximant (Kawabe and Katayama 1994, 1996, 1997). More specifically, the I-PD parameters are adjusted so as to minimize performance criterion maximized by plant parameters with bounded uncertainty. It is shown in Kawabe and Katayama (1994) that a minimax design based on the ISE leads to a very large overshoot in the optimal closed-loop response. A large overshoot has also been observed in earlier PID tunings based OD the ISE criterion (see e.g. Smith and Corripio (1985)). In order to circumvent this difficulty, we have applied the minimax design technique to a generalized ISE that includes a penalty on the derivative of the control variable (Kawabe and Katayama 1996, 1997). It is shown that the use of the generalized ISE cost function, which has been used in the LQ regulator based design of servomechanism, is very effective in shaping the closed-loop responses by adjusting a weighting parameter. The use of Pade approximants is, however, a drawback of (Kawabe and Katayama 1994, 1996, 1997), since the stability criterion with Pade approximants for a time-delay element may yield erroneous robust stability conditions and the evaluation of the (generalized) !SE is not exact (Marshall et. al. 1992). In this paper) we present a design method based on the exact evaluation of control performance for time-delay systems using the technique developed in Marshall et. al. (1992), in which the PID parameter settings are extensively studied based on the ISE. The performance criterion and the optimization technique of this paper are the same as those of Kawabe and Katayama (1996, 1997). The robust stability of the closed-loop system with a time delay is determined by checking the range of time-delay in the characteristic equation. Furthermore, we give a robust stability criterion based on zero exclusion principle (Kogan 1995) for a polytope of quasi-polynomials 1. This criterion gives a necessary and sufficient condition for stability of the polytope of quasi-polynomials , and is conveniently used in the present problem as well as in problems of adjusting parameters of controllers with a fixed configuration. 2. PROBLEM FORMULATION 2.1 Minimax Optimization Problem
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We assume that the plant model is described by P(s)e- h8 , where pes) is a strictly proper rational function with parametric uncertainties and h is a delay. I-PD Controller
r················································ r i
e
Fig. 1. Block diagram of control system Let (J be the parameter vector of pes) and belong to a bounded set e = {8IBI ~ 8 ~ Bu }. Let q := (Kc, Ti, Td)T be the I-PD controller parameter vector belonging to a bounded set Q = {qlql ~ q ~ qu} C R3 , which is the range of adjustable I-PD parameters. It is assumed that e and Q are given a priori. We define the generalized ISE (Kawabe and Katayama 1996, 1997) as J(q, (J):=
J={e
2
(t)
+ piJ?(t)} dt,
(1)
o
where p 2': 0 is an adjustable parameter. This performance index, employed in the LQ regulator based design of servomechanism, is a weighted sum of the ISE and the integral of squared derivative of the manipulated variable. Hence, this performance index not only evaluates the ISE but also gives a penalty to the velocity of manipulated variable. The robust I-PD controller design based on the generalized !SE is formulated as a minimax optimization problem: min max J (q, 9), under the qEQ iJE9
constraint that q stabilizes the closed-loop system for all B E 8. The designed controller guarantees the robust stability and the best performance under worst case scenario.
2.2 Description of the Closed-Loop System From the Parseval's theorem, (1) is written as
Consider the SISO control system shown in Fig. 1, where r is the step input, u the manipulated variable , y the measured variable, e the error, and K i , T i , Ta , I E R are parameters of the controller.
J(q,B)
~ 2!j L2=e(,)e(-')d' +p
1 The quasi-polynomial is a polynomial in one complex variable and exponential powers of the ·variable.
_2='0(') [-8«(-.)]
(2)
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where e(s) and u(s) are the Laplace transforms of e(t) and u(t), respectively. Let pes) = N(s)jD(s), and define A(s), Be(s), C(s), De(s) and B,..(s) as
C(s) = Kc {(1+-r)T,TdS2+(T;+'YTd)s+1}N(s),
De(s) = KcTi {(I Ru(s) = Ke(l
+ 'Y)TdS + I} N(s),
+ ")'TdS)D(s).
(3)
Then from Fig. 1, the error eCs) for the step input res) = l/s and the derivative of the manipulated variable are given by
Be(s) + De(s)C h8 e (s) - ........:::..-:---7---=7.::.,-:-----;-- A(s) + C(s)e- hs ' Bu(s) sues) - A(s) + C(s)e- hs '
(4) (5)
respectively. The characteristic equation of the closed-loop system is given by f(s)
;=
A(s)
+ C(s)
e- hs
and compute the corresponding h. Then analyze the behavior of the roots in the neighborhood of h. Step 4: Check the stability region of h by sorting destabilizing and stabilizing critical value of h. If a given time-delay h is in the stability region, the closed-loop system is stable. Otherwise, the system is unstable.
+ ,Tds)D(s), = Ti(l + ,Tds)D(s),
A(s) = TiS(l
Be(s)
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= O.
(6)
Since the plant is strictly proper, the degree of A(s) is always greater than that of C(s), so that this system is a retarded system.
3.2 Robust Stability of time-delay systems Next we consider the robust stability criterion against parametric uncertainties. Namely, we assume that N(s) and D(s) in (3) are the (real) interval polynomials. Let N denote the number of vertices of the hyperrectangle in the parameter space and
e
be the quasi-polynomials corresponding to each vertex. Then the characteristic quasi-polynomial is generated by the convex combination of these quasi-polynomials: r
f(s,k)
;=
L
(10)
k. fi(S),
;=1 N
3. ROBUST STABILITY CRITERION FOR TIME-DELAY SYSTEMS
s.t.
2::: k. = 1, O:S; k; :s; 1, i = 1,· .. ,N. i=l
3.1 Stability Criterion We first consider the stability criterion of the quasi-polynomial (6) when the coefficients of A(s) and C(s) are fixed. Let us regard h as a variable and write the characteristic equation (6) as f(s, h) := A(s)
+ C(s)e- hs
= O.
= 0.
Step 2: Consider infinitesimally small positive h. For retarded systems, all the new roots appear in the left half-plane. Step 3: Determine the positive zeros:' of the polynomial
W(w 2 ) := A(jw)A( -jw) - C(jw)C(-jw),(8) 2
K
(7)
Since a common factor of A(s) and C(s) is a root of(7) for any h?= 0, we assume that A(s) and C(s) have no common factors. A procedure of finding the stability regions for h consists of four steps (Marshall et. al. 1992). Step 1: Examine the stability of (7) at h
Let k := (k1 , · · · , kN)T and define the convex polyhedron K, the quasi-polynomial family S, and the value set Sw as :=
{k
I ~ k; = 1, 0 ~ ki ~ 1 } ,
I k E K}, {f(jw, k) I k E K}.
S := {f(s, k) Sw :=
(11)
The shape of Sw is a polygonal region and each quasi-polynomial segment corresponding to the boundary of Sw is called edge. From the edge theorem (FU et.al. 1989) and the zero exclusion principle (Kogan 1995), we see that the quasipolynomial family S is stable if and only if the boundary of Sw does not contain or pass through the origin for all w ;:::: O. Since the boundary of Sw is the value set of the segment quasi-polynomial corresponding to two generating points of S, we consider an edge connecting pes) and pes). Define the quasi-poLynomial segment
Each zero indicates the point where the root loci against
h touch or croS/l the imaginary axis. As is obvious from (8), such locations are independent of delay h. Another
with A E [0,1]. From (9) we can express (12) as
key fact is tha.t the number of zeros is finite, although (7) has infinitely ma.ny solutions.
f(8, A)
= A(8, >.) + C(8, A) e- hs ,
(13)
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where A(s, A) and C(s, A) are defined as
+ AA2(8), C(S,A):= (1- A)C 1(S) + >.C2(8). A(s,A) := (1- A)Al(S)
A stability criterion of the quasi-polynomial segment (12) is given as follows: Theorem 3.1 Given>. E [0,1), let w).. be
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It is assumed that the system is stable, so that all the poles of (16) lie in the left half-plane. For the notational convenience, we suppress the Laplace variable s of each polynomial and denote its paraconjugate by (), for example, we write A := A(s) and A := A( -8). Then the integrand of (15) is expressed as
=
x(s)x( -s) w.\
= sup{w I A.(jw, >.)A( -jw, A)C(jw, A)C(-jw, A)
Also let
= a}.
sup{w.\
+ /3e- hs a + iJe hs + Ce-hs + A + Ce hs =: H(s) + H( -8),
I A E [0, I]}.
a x(s)x(-s) = A
Then, the quasi-polynomial segment (12) contains or passes through the origin for w ~ 0 if and only if there exist w E [0, w] satisfying the following condition 3
We can summarize the procedure examining stability of the quasi-polynomial family S as follows: Step 1: Examine the stability of one quasipolynomial in S. If the quasi-polynomial is stable, go to Step 2. If not, S is not stable. Step 2: Check whether 0 1. Swo for one Wo ~ o. IT rt Swo, go to Step 3. If not, S is not stable.
°
Step 3: For each edge, check the existence of w 2: 0 such that (14) holds. If there exist such an w on at least one edge, S is not stable. If not, S is stable.
4. EVALUATION OF 1SE FOR TIME-DELAY SYSTEMS
=
ACBE + DD) - 2BC15 2(AA - CC) ,
Q
=
r.I
= 2ADB - G(BE
+ Df5)
2(AA _ CC)
iJ
.
(19)
To compute (15), we consider the contour r shown in Fig. 2. Denote the segments on the imaginary axis and the semi-circle as r 1 and r 2 , respectively. If AA-CC = 0 has roots on the positive/negative imaginary axis, add infinitesimally small indentation to r 1 to the left/right around the root. Denote the integral of H(s) along r as I, Le., 1 =1-
21[j
J
0:
A
r
+ /3e- hs ds + Ce- hs -
(20)
Since the system is stable, A + Ce-hs = 0 ha.'! no root inside r. Then the singularities of the integrand of (20) inside r are limited to the roots of AA = 0, which are denoted by Si. It is crucial that {Si} is a finite set. It follows from the residue theorem that
cc
By using the technique of Marshall et. al. (1992), we develop a method of computing the ISE of the form
J
(18)
where
Re[fl(jw») Im[J2(jw»)Re[j2U w ») Im[fl(jw)] = 0, (14) Re[f1 (jw)] Re[f2(jW)] ::::; 0, Im[fl(jw)] Im[f'l(jw)] :::; 0,
=
(17)
It can be shown that (17) is additively decomposed as
w be w=
+ De- hs B + Dehs + Ce-hs A+Ce hs '
B A
I = -
Q + {3e- hs ] ~ Res [ A + Ce- hs ' Si
,
(21)
t
+i=
J
x'l(t) dt
=
2!j
o
J
Im
x(s)x( -s)ds, (15)
-joo > jro
where x(s) is expressed as
x(s)
=
B(8) A(s)
+ D(s)e- h8 + C{s)e-hs .
(16)
Re -jOl x
3 In contrast to the polynolllial family case, the first equation in (14) is not a polynomial but a nonlinear function of w. Therefore we must perfonn a line search
instead of solving polynomial roots. However, since the interval to be searched is finite, such a computation is tractable.
Fig. 2. Contour
r
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A MINIMAX DESIGN OF ROBUST I-PD CONTROLLER FOR TIM...
where Res [f(s), Si] denotes the residlles of f(8) at Si. If Si is a singularity of H(s), -Si is also a singularity of H ( -s). \Vith respect to residlles, we have Res [H(s),
Si]
= -Res [H( -8),
-Si).
(22)
It follows from (22) that the integral of H(-s) along r is equal to I. Since X(8) has no singularity on the imaginary axis, we can replace the integral path of (18) by rI, i.e., J=
~f{H(s)+H(-s)}dS. 27rJ
r,
As the radius of P2 goes infinity, the integral along replace r 1 by
r 2 becomes zero and hence we can r. Thus we obtain J = 21.
As stated in Section 2, the design of robust I-PD controller is reduced to a minimax optimization problem: qEQ 9E8
The GA is one of the powerful tools for the noncovex optimization. To design this iterative random search algorithm, we must define some problem-dependent algorithm elements: Representation
In the present algorithm, we adopt the Gray-code string to represent candidate solutions. Each of the I-PD parameters Kc, T i , Td is represented by a binary string. Three such strings are concatenated into a binary string, which defines a point in the parameter space to be searched by the GA algorithm. Fitness function The linear scaling method is used as a fitness function, Le., the fitness is simply computed from the order of the cost function va lue of each individual among t he population.
5. OPTIMIZATION METHOD
min max J(q, 0),
5.2 Structure of Genetic Algorithm
(23)
where the closed-loop system is stable for all () E 8. It should be noted that the generalized ISE J(q,8) does not have saddle point in this case. Since there are many local optimal solutions for the controller parameters, we use a genetic searching algorithm to find a global solution.
Genetic operators
The genetic operators used in the present algorithm are uniform crossover, bit mutation and the roulette wheel selection. The uniform crossover generates two offspring by exchanging a predefined number of alternate subsections between two parent strings. The recombination operator, mutation, is implemented as altering bit values at randomly selected string positions.
6 . DESIGN EXAMPLES
We assume that q := (Kc, Ti, Td) E Q. For computational purpose, let Q d := {ql, ... , qN} be a discrete approximation of the set Q. Then the design algorithm of a robust I-PO parameters is summarized as follows.
In this section, we show a numerical example of robust I-PD controller design. The range of adjustable I-PD parameters and l' are given by Kc E [0.1,15.9]' Ti E [1.0, 100.0], Td E [1.0,100.0] and'Y 0.1. The parameters of the GA are given by population: 500, number of individual: 100, mutation rate: 0.2.
Check robust stability of qi for all
Example 6.1 We consider the second-order plant
5.1 Algorithm of Minimax Optimization' Problem
Step 1:
e E 8, where Qi is generated by the CA operations
=
G ( )
described b elow.
3 S
Step 2: If the closed-loop system with qi is robustly stable, compute
(24)
Step 3: H the GA algorithm stops, go to Step 4. Otherwise, go to Step 1. Step 4: Let the minimum of Mi be Afio ' Then the corresponding qio yields the minima x robust I-PD controller.
K
T2S2
+ T1s + le
-28
(25)
,
where the parameters belong to the set K E [0.8,1.2], Tl E \6.0, 20.0J, T2 E [225,625]. Table 1 shows the minimizing solution for nominal plant with p 0 and the minimax solutions for p 0.0, 1.0, 10.0. Fig. 3 shows the step responses for the worst case scenario with p = 10 where the solid
=
by using (21) and the GA. If the system is not robustly stable, set Mi = 00.
=
=
Table l' Robust I-PD controller design T; K Kc Td Tl T2 nominal 15.9 6 .90 8.30 12.0 1.0 400 0.0 15.9 17.4 5.44 0.8 6.0 625 1.0 9.30 9.03 625 19.9 0.8 6.0 10.0 5.45 23 .0 625 13.0 0.8 20.0 p
J
10.2 15.5 20.9 28.6
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line and the dashed line are the responses designed by the present method and by the earlier method (Kawabe and Katayama 1996, 1997), respectively. We see that the closed-loop response designed by
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stability criterion. The numerical example shows that • The robust I-PD controller guarantees the robust stability and the best performance under worst case scenario. • We can easily regulate the overshoot and oscillatory variation in the closed-loop responses by using the generalized ISE. • The control performance of present design is better than the earlier results based on the Pade approximant and the Kharitonov's theorem.
2
·:~--.·•i• • ;• • • 00
20
4Q
80
80
100
120
HO
160
8. REFERENCES
l·······l
180
200
11me
Fig. 3. Step responses of closed-loop system for the worse case scenario the present method is much better than the one designed by our earlier method based on Pade approximant. The improvement of the closed-loop performance is due to the exact treatment of the time-delay element. As shown in Fig. 4, the robust stability region of the I-PD parameters for Ti = 27 is much larger than that estimated by using the Pade approximant. Thus under the Pade approximation, the region of adjustable parameters becomes much smaller than the actual feasible o region.
. ,,.
SoIkI : f'l<>poood me1hod
,
D
,I ; '
15
10
Kc
a
10
Fig. 4. The stability region of I-PD parameters for T. = 27
7. CONCLUSION
We have developed a GA based design method of robust I-PD controller for time-delay systems with parametric uncertainty. A novelty of the present design method over the earlier ones is the exact evaluation of control performance and the robust
Astrom, K. J., and Hagglund, T.(1995). PID Control - Theory, Design and Tuning, 2nd ed. Instrument Society of America. Chen, B. S., Cheng, Y. M., and Lee, C. H.(1995). A genetic approach to mixed H 2 / Hoo optimal PID controL IEEE Control Systems Magazine, 15, 51-60. Fu, M., Olbrot, A. W., and Polis, M. P. (1989). Robust stability for time-delay systems: The edge theorem and graphical tests. IEEE Trans. Automat. Contr., 34,813-820. Kawabe, T., and Katayama, T. (1994). A minimax design of robust I-PD controller for system with time-delay. Proc. 1st Asian Control Conference, Tokyo, July, 3, pp. 495-498. Kawabe, T., and Katayama, T. (1996) A minimax design of robust I-PD controller based on a generalized integral-squared-error. Trans. SlCE, 32, 1226-1233 (in Japanese). Kawabe, T., and Katayama, T. (1997). A minimax approach to design of robust I-PD controller. Proc. 2nd Asian Control Conference, Seoul, July,!, pp. 661-664. Kogan, J. (1995). Robust Stability and Convexity. Springer-Verlag. Lu, J. P. (1992a). Robust process control on Honeywel1 TDC3000. Preprints, Honeywell Advanced Control Symposium. Lu, J. P. (1992b). A MIMO robust controller on the Honeywell TDC3000. Proc. 19th Annual Honeywell Industrial Automation and Control, User's Group Symposium. Marshall, J.E., Gorecki, H., Korytowski, A., and Walton, K. (1992). Time-Delay Systems: Stability and Performance Criteria with Applications. Ellis Horwood. Morari, M., and Zafiriou, E. (1989) Robust Process Control. Prentice-HalL Smith, C. A., and Corripio, A. B. {1985}. Principles and Practice of Automatic Process Control. Wiley.
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