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Introduction to Coefficient Diagram Method
IFAC
Copyright © IF AC System Structure and Control. Prague . Czech Republic. 200 I
[: 0
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INTRODUCTION TO COEFFICIENT DIAGRAM METHOD Young Chol Kim* and Shunji Manabe**
* School of Electrical and Electronics Eng.
Chungbuk National University San 48, Gaesindong. Cheongju. Chungbuk, 361-763. Korea E-mail: [email protected] ** 1-8-12 Kataseyama, Fujisawa-shi, Kanagawa, 251-0033, Japan E-mail: [email protected]. jp
Abstract: We introduce a linear controller design method, called the Coefficient Diagram Method (CDM), which is a kind of model matching approach. In this paper, we have especially focused on the main role and characteristics of specific design parameters Yi and "l", which are defined by only coefficients of characteristic polynomial. The relationships between these parameters and the system response as well as the stability are investigated systematically. Based on the relations. we have proposed a prototype for target transfer functions of type I by which one is able to fit easily the desired transient response. It is ascertained that the CDM has some strong points in cases where fixed lower-order controllers wish to be designed. Copyright e 2001 IFAC Keywords: Model matching. Linear control design. Target transfer function 1. INTRODUCfION'
developed by Manabe (1998). He has suggested a new method by which we are able to easily build a target characteristic polynomial to meet the given time response specifications and has also found a graphical tool. referred to the Coefficient Diagram (CD). It is very easy for the CDM to solve the problems of designing a fixed order controller under the conditions of stability, performance and robustness. Moreover, it is possible to deal with them in CD simultaneously. The historical background and basic philosophy of CDM may refer to Manabe( 1998). In section 2, we explain some preliminary works for CDM. In section 3. we represent how to get the target transfer function of Type I and the design procedure of CDM.
The control system design problem with the model matching boils down to the problem determining either a proper target transfer function or a target characteristic polynomial that meets the specification. In many control systems. the controller is imposed some constraints such as low order, minimum-phase. bandwidth. maximum magnitude of control input, and so on. Thus it may be difficult to obtain a target characteristic polynomial to satisfy the specifications under these limitations. In this framework. the other question interested occurs at when we solve the Diophantine equation. The pole placement problem is usually converted to solving the Diophantine equation. There exists a solution but not unique if the order of controller is more than or equal to n-l (where n is the order of plant). However, there are many cases that we may achieve the control objective successfully by much lower order of controller. The ordinary Diophantine equation in this problem can not be solved. With this background, we intend in this paper to introduce a recent development of the Coefficient Diagram Method (CDM). which is based on the model matching mrthod. This paper will emphasize especially that the CDM can answer the above problems. Although the basic concept of the CDM has been known since 1960s (Kessler. 1960; Naslin. 1969), a systematic method has been
2. CDM PRELIMINARIES In CDM design. two-parameter configuration is used to implement an overall transfer function, as shown in Fig.I. Since the output feedback affects only the poles but not the zeros of plant, the zeros of plant remains to be the zeros of overall transfer function. Hence the CDM can be regarded basically as a pole placement method. In the first step. the transfer function of a feedback controller is computed by Diophantine equation. The second step is to find a feedforward controller so as to compensate the effect of plant zeros and to match the steady state gain of overall system. There are two distinctive remarks that reveal the feature of CDM. One is to express the characteristic polynomial in terms of the specific
I This work was supponed in pan by NSF under Grant HRD-9706268.
147
controller limitations when the target polynomial is chosen. We use a simple example to illustrate the basic feature. Consider a fourth-order plant with PlO controller as follows.
parameters, Y , and r . These parameters allow us to obtain a target polynomial so that meets the stability and performance. The other is that the target characteristic polynomial can be deterrruned and can be retuned easil y by shaping its coefficients In the graphical tool, so called the Coefficient Diagram (CD). The CD is a semi-log diagram with the coefficients of polynorrual.
4
Ap(s ) = 0.25s +s 3 + 2s 2 +0.5s, B p (s) = I , Ac(s)=lls,B c(s)=k 2 s 2 +k l s+k o ' where II = 1, k 2 =0.5 , kl =1, ko =0.2 . The Characteristic polynomial is expressed as P(s)=Q25s5 +i +2 s3+2i +s+Q2.
2.1 Definitions and Mathematical relations
Then from (2) - (4), we have
Consider two-parameter feedback configuration as shown in Fig .l.
a , = [as
a 2 al] = [0.25 I 2 2 I 0.2]
=[y~
Y2 h]=[2 2 2 2.5]. r=5
Y,
Y; =[Y: . .. Y2 y;]=[O.5 10.90.5] . 10
;---
V
..........
./
-..... i"'--.. 2
2
a,
I, ..........
- - T,
"
0.1
Fig. I. Two-parameter configuration
2
Some mathematical relations used in CDM will be introduced hereafter. The characteristic polynomial is given in the following form. P(s) = Ac (s)A p (s) + Bc (s)B p (s)
2
0 .0 1 0.5
0.00 1
-----
2
Y.
y:
0 .9
--
~:s: ~ 0.5
"'" "" 02
y, .
f
Y,' 0 .1
o
Fig. 2. Coefficient diagram (I)
The coefficient diagram is shown in Fig. 2, where coefficient a i is read by the left side scale, and
The stability index Yi' the equivalent time constant r , and stability limit
Y; are defined as follows.
Yi := a; l(ai+1ai-l)'
i
=1.2.· ·· . (n-I) .
r:=a1la o '
stability index Yi • equivalent time constant r, and
Y;
stability limit are read by the right side scale. The r is expressed by a line connecting I to r . In this example, the characteristic polynomial P(s) is decomposed into two component polynomials as follows. pes) = ~I (s) + Pt (s) (9)
(2) (3)
Y; :=lIYi+l +lIYi-l' i=l,2,···.n-l , Yn =Yo =00. (4) Also. define the pseudo-break point wi := a i I a i+1 • i = 0.1, .. ·• n -1. (5) We then have Y, =w;lw i _ 1 , i=I, .. ·.n-l. (6)
~1(s)=11(0.25ss +S4 +2S 3 +0.5s 2 )
Pk(s)=k2S2+kls+ko
(7)
(11) T(s) is
The complementary sensitivity function expressed as (12) T(s) = Pt (s)/ P(s) . Eq. (l0) is shown in Fig. 2 with small circles. Eq.(11) is depicted with small squares and dotted lines. Designer can visually assess the deformation of the coefficient diagram due to the parameter change of k2 ' kl and k o ' Then he can visualize the variation of stability and response. Also from (12), it is clear that robustness can be analyzed by comparison of coefficients a i and k i at the coefficient diagram. Taking logarithm values to both sides of (2). we have 10g(Yi ) = 210g(a i ) -log(a,+l) -log(a i _ 1 ) • (13) Therefore, it is obvious from the CD that the lager curvature of a i curve stands for the larger Yi 's for all
It is easy to show that characteristic polynomial is expressed by a o • rand Y as follows. 1
p(,)=a.[l~[ UIIY", },,) )+'>+1]
(10)
(8)
2.2 Coefficient Diagram(CD)
The CD is a semi-log diagram of the coefficients of polynomial in logarithmiC scale versus number of power of s in linear scale. As like the loop shape in the BodelNyquist plot gives information regarding stability and response. the coefficient shape in the CD has the similar relations. Because the coefficients of characteristic polynomial are related to the parameters of plant and controller in the form of linear combination, it is easy to consider the
i. The Yi can be graphically obtained on the CD.
148
This implies that the larger curvature of a, makes the system to be more stable in the sense that the larger damping pole is of more stable than the smaller damping poles. The reasons will be revealed in the section 2.3. If r is small, its response is fast. The equivalent time constant r specifies the response speed. The y, - y, * curves in the CD can be also used for parameter sensitivity analysis and robustness analysis. Note that the coefficient diagram indicates stability, response, and robustness, which are three essences in control design, in a single diagram, so that enables the designer to grasp the total picture of control system.
response is mainly used so far. The CDM is a polynomial method based on the model matching, which is very useful for the purpose of transient response design . In this approach, it is very important to find a target characteristic polynomial. Several standard forms have been proposed. Representative ones of them are the integral of time multiplied by absolute error (lTAE) prototype (Graham, 1953), Kessler standard form (Kessler, 1960). However, these prototypes are excessively rigid when they are applied to design problem. The ITAE and Kessler prototypes have unnecessary overshoot. In all of them, we see that its step response is delayed as the order of system increases.
2.3 Stability condition
3.1 How to get a target characteristic polynomial
The Routh-Hurwitz criterion provides the necessary and sufficient condition for pes) to be Hurwitz stable. However, this way can hardly work the problem designing controller because the Routh criterion is a highly nonlinear function with respect to controller's parameters. Lipatov (1978) proposed sufficient conditions for the Hurwitz stable and unstable. Since this condition is directly related to the CDM design parameters, y, and r, it allows us to consider both the response and the stability simultaneously. From the corollaries in Lipatov (1978), the sufficient conditions for Hurwitz stable and unstable, which are translated in terms of Y;, are as follows. Theorem I. pes) is Hurwitz stable if
As one see in many actual control systems, it is assumed here that the controller to be designed should meet the time-domain specifications such as rise time, settling time, and overshoot. Also, we consider the cases that the overall system can be expressed by a target transfer function of Type I. This means that it has zero steady state error to a step input and a constant steady state error due to a ramp reference input. A closed-loop transfer function of Type I here is
~Y; Y;+I >1.4656 for i=I,2,3,···,n-2 .
n n_~o ~ (19) ans +an_1s +· ··+a1s+ao pes) Thus, it is noted that this system has a DC gain of I, that is, T(O) = 1. Before introducing new prototypes, we represent some important relationships between T(s) and the
T(s) =
(14)
specific parameters Y; , r, and w;. It is obvious from (6) that for the coefficients of pes) in (19), if
Theorem 2. pes) is Hurwitz stable if y;~1.12374y;· for i=2,3, ··· ,n-2 (15)
y; >1, for all i = 1,2,· ··, n - I, then it has w;
or equivalently, if
all i = 0,1,2,·· ·, n - 1 . It is remarkable that
(16)
(20) logy; =logw; -Iogwi-l ' i=I,2,·· ·,n-l (21) With this context, the asymptotic line approximate plot for the Bode plot of a T(s) is shown in Fig.3. The straight-line approximation of Bode diagram used here is somewhat different from the ordinary way. The pseudo break points are defined by the ratio of the coefficients and not from the poles and zeros of the transfer function . However, both result in similar approximations. From low to high frequencies, the slope of the magnitude plot is 0 dB/dec and decreases by 20 dB/dec at each pseudo breakpoint. It is seen evidently that the bandwidth of target transfer function depends on absolutely the equivalent time constant r and its cut off rate near the cut off frequency, which is the region indicated by 10gYI
Theorem 3. pes) is unstable if Y;Y;+I:$1
for some i=I,2,·· · ,n-2
(17)
or equi valentl y, if (18) The above conditions can be graphically expressed in the CD (see Fig.2). In CDM, Theorem 2 is mainly used because of its closeness to the necessary condition. Theorem 1 implies that the overall system should be stable if all y;'s are greater than 1.4656. It turns out that the value can be used as a limit when we obtain a desired characteristic polynomial. a;a ;+I:$ a ;+2a;_1 .
3. CDM FOR LINEAR CONTROLLER DESIGN Even though the perfonnances of most control systems are judged by its time domain responses, there are few design methods available to this purpose because there are no simple relationships between time response characteristics and transfer function. Thus the loop shaping of frequency
and log Y2. Therefore, the settling time can be determined by r . If the cut off rate in the region of YI and Y2 decreases sharply, it makes overshoot in a step response. It is also certain that Y3 and higher
149
Y,' s give much less effect to the closed-loop response because the frequency response magnitudes corresponding to this area is relatively very small provided that it has large YI and Y2 • We now present important relationships between characteristic roots and the equivalent time constant.
.0r------------------------------------, - Ioo(r) '0
logO',) W
Wo
W,
Table I. Standard form of M-type prototype Proto .
type
logO',)
IOg0'2)
there are almost no overshoots and the same transient behaviors irrespective to the order of system. The designer can start from a simple controller designed by this form and then may move to more complicated one.
s' + 10 I r s' + 50 I r ' 5' + 1~5 I r ' s + 125 I r'
ai,
~
.§: f.
sS
~
' -6dB /oCl
·. 0
T( s) = _ _ _ _ _a...,:o'--_ _ __ sI'! + QIl _1Sn -1 + ... + Q1S + a o
.:
....
01
5'
. .
~~
1
M
+ 25 00 /r 4 5 +2500 /1"5
+ 40 /r 5' + 800/r's' + 8000/r'5' + 4xlO' Ir'5 ' +10' / r ' 5+10'/r'
+ 160 / T S1 + 12800 Ir 2/' + 5 . 12xlO ~ I r J s 5 + I .024xIO' 11'4s· + I.024xlO'/r' s' + 5.12xIO'/r's' + 1.28xlO' IT'S + 1.28xlO' IT'
SS
10
Fig.3. Frequency magnitude of a target model of Type I
Table 2. Relationships between " and t 5
Theorem 4 (Kim. 1999): Consider two stable polynomials of degree n, A(s) and 8(s) . A(s) =ans n + an_ls,,-1 + ... + als + a o 8(s)=b n s n +bn_ls n - 1 + .. ·+bls+bo
and
a Bi ' i = I, 2, .. ·, n respectively and let "A'" B be their equivalent time constants, respectively. If both polynomials have the same stability index Yi 's, then the following holds, aA, =aa B" i=I,2, .. ·,n,
+ 1000 /"["3 5 2
s' + 80 Ir 5' + 3200/r' s' + 6.4xlO' I r 's ' + 6.4xlO ' I r' s) +3 .2xlO'/r's' +8xlO'/r's+8xlO'/r'
logw Irad/s ec l
Let the roots of both polynomials be a Ai
+20/rs 4 +200/ 1' 2 5 3
(22)
Design specifications M
== t s /2.3095
1.2 r---~--~--~--
__- -
D.•
]
real, and negative. VVV In order to guarantee the stability by means of Lipatov's sufficient condition in Theorem I, all L's
TM
However, in some situations it may be difficult to choose a target transfer function so that meets the rise time and the settling time simultaneously by means of the Manabe form. Kim (1999) has proposed two other forms; F and S-types. The response of the F-type represents the fast rise time and slow settling time, whereas that of the S-type characterizes the slow rise time and fast settling time as shown in Fig.4.
where a=TA /T B • VVV Theorem 4 states that all the characteristic roots can be shifted at the same rate by only a single parameter ". It is noted that if only T is changed, then the damping ratios of all roots remain the same values. Theorem 5(Lipatov. 1978): For a polynomial, if all y, 's are greater than 4, then all its roots are distinct,
Settling Time ( t s )
~ «
06
3 Timel5 ec)
Fig. 4. Step responses of 3-types ( n = 4 - 8 )
should be greater than 1.4656. Thus Y,'s are usually chosen within interval, Y"
E
[1.5, 4). Based on the
above background, Manabe (1998) proposed a standard form for Type I system. It is Yl=2.5, Yi =2, for all i=2,3," ',n-1 (23) In Table I, for n =4,· .. ,8, the target transfer functions of M-type that are normalized by " are given. Also, the relationship between the settling time and T is given in Table 2. We call this value as the Manabe form or M-type. Although the Manabe form has been found by the heuristic approach, it is strongly recommended as a target characteristic polynomial due to its favorable properties. The step responses of the Manabe form for the order of n =4"" ,8 are shown in Fig. 4. It is remarkable that
150
~ . 50'--------.,. O~ 5 ----~----....J 15
Time!sec !
Fig. 5. Step response change rate (dy/dt) of 3 prototypes when n=5 and t s = 1 sec These curves have been specified by selection of both the times and the desired response values at 0, 10%, 50%, 90%, and 99% for the settling time, respectively. An integral of the squared error function
has been considered. Then solving numerically this optimization problem by a non linear interpolation method, they have obtained both F and S prototypes (Kim, 1999). They have almost the same response shapes irrespective to the order of systems . In Fig . 5, the step response change rates of 3 prototypes are compared. If the maximum value of the change rate occurs at late, this means the slow rise time. This change rate relates closely to the maximum magnitude of control input. So far, we have addressed three prototypes for Type I target system. In these forms, all Yi 's are fixed and only r is used so as to meet the desired settling time. There may be some cases that requires the other values of yi's and r . However, currently no constructive rules for this purpose have been published. We now explain about how to obtain the target characteristic polynomial by the coefficient shaping on the CD. As shown in Fig. 2, a CD includes five information such as numerator and denominator polynomials of plant, stability index Yi's, stability
Y;,
equivalent time constant 'l' , and the closedlimit loop polynomial. It is possible to consider the trade off among performance, stability and robustness. As seen in Fig.3, 'l' relates absolutely to the bandwidth of overall system. Since 'l' = a) / ao where a) and ao are coefficients of two lowest order of characteristic polynomial, it is represented by a slope on the coefficient diagram. Consequently, the line a) - ao on CD stands for the system response speed.
In particular, the stability index Yj is the most favorable parameter, which relates to the transient behavior as well as the stability of overall system. The distance between Yi and Y; indicates the degree of tolerable variance in the coefficients. A system will tend to remain the stability to larger perturbations of the system parameters if the distance is larger. In some cases, for example, a non-minimum phase system, the 'l' may be bounded by the lowest value. Thus although this way requires some or less empirical knowledge, it is not so much difficult to choose a target characteristic polynomial in a feasible range.
pes) = Ac(s)Ap(s)
=1::. =
Ba (s)B/s)
r
!\(s)-\(s) + Bc (s)Bp (s)
deg(Bc ) = deg ( Ap) -1
(25)
deg(Ac )~deg(Bc )
(26)
Example 1. (Franklin, 1994:P. 538 Example 7.30) Consider a third order plant transfer function G(s)= Bp(s) = Ap (s) s(s
10
+ 2)(s + 8)
Design a controller of first order by CDM that satisfies the settling time t s :5 2 sec and overshoot less than 2% . Solution: The controller can not be solved by the ordinary Diophantine equation because the condition (25) does not hold. Let the feedback controller transfer function be C(s) = B c (s) = k]s + ko ,(io =1) Ac(s) 11s+10
In CDM, there are two design approaches; the algebraic method under the matching condition and the coefficient shaping on the CD. We first state the algebraic method briefly. Consider a feedback system shown in Fig. 1. The closed-loop transfer function is
= Ba(s)Bp(s)
(24)
According to section 3.1. once a target transfer function res) is selected, the design problem boils down to the model matching one. Rewriting (l),
151
(27)
Then the characteristic polynomial becomes pes)
=a4s 4 + a3 s3 + a2s2 + a)s + ao
=~ s(s3 +loi + 1&)+(s3 +1(k2 + 16)+1
First, draw a coefficient diagram in the absence of control, that is, lo = I, l] = k] = ko = O. Then, the coefficients of Ap (s) would be depicted. The curve is shown in
pes)
(1)
For this case, since it is trivial to solve (I), the details refer to Chen (1993). In many control systems, it is true that a lower order controller may meet the given specifications. But, this problem can not be solved by any analytical approach. Furthermore, if a certain parameter of the controller will have to depend on the other one as a design constraint, then the number of equations becomes greater than the unknown variables. It turns out that the existence of solution can not be guaranteed. One of the main characteristics of CDM is that even such a problem can be dealt with in the manner which one adjusts Yi 's and 'l' so as to have a feasible region. Similar to the coefficient shaping for obtaining the target transfer function, it can be applied to the design process since the resulting characteristic polynomial is easily drawn by moving up and down as much as the sum of controller's parameter in the same order. Also, we can see directly by CD how much the overall system will be sensitive to the parameter perturbations. The basic references of coefficient shaping are to retain the original good nature of plant if any, and to make the ai's curve to be a smoothly concave shape. At the moment, the resulting controller can be immediately obtained from the CD. Now, we give an illustrative example.
3.2 Design procedure of CD M
res)
+ B c(s)Bp(s)
This Diophantine equation has a unique solution if the order of controller is given by
0---0---0,
denoted by lo in Fig. 6. From
CD, it is easily found that the selection of l] = 0.05 will not change P(s) much from Ap (s) (See l] curve). That is, it makes in part a sIOOOthly
concave shape around 10 , Writing the Diophantine equation of (28) in the Sylvester form, I
0
0
0
10
I
o o
0
16 10
0
0 16 10 0 0
0
o
10
a.
lH
,'jyJ Y;Y~
Cl:J ~
,J/ y2yI2 '"/Y
,
al
1
ao (29)
ao
Since the settling time t J :5 2 sec. using Table 2. computed below and thus we choose, = 0.8 .
,IS
t
(30) , :5 - -'- :5 0.866 2.3095 According to the Manabe form. we assume that y 2 = 2.0 . Now. the problem is to solve a non-square Sylvester matrix with the condition I1 = 0.05 . 10 = I .
Y2 =2.0.
,= 0.8. The unknown
variables are a o•
y I' k o • and k l · From (29). we have a J =ao,l jy2Y~ =1.5
, 2/
a 2 =a o Y1 =10.8. Solving these algebraic equations, we y 1 = 2.88 and finall y the feedback controller is Bc(s) 2.288s+4.86 C(s)=--= . Ac (s) 0.05s + 1
(31 ) (32) have
(33)
5. CONCLUDING REMARKS If we want to design a lower and fixed-order controller for given specifications. what methods are available? The model matching approach is a method possible for this issue. However. this method is difficult to get the target transfer function so that will meet the specifications subject to the constraints such as low order if possible. bandwidth. response rate. the maximum magnitude of control input and so on. The other problem is that it is difficult to solve the Diophantine equation when the order of controller is lower than full order required to ensure the existence of solution. The CDM has developed on these needs. In this paper. we have introduced a recent development of CDM. Especially. we have focused on the main role and characteristics of specific design parameters y, and, . The relationships between these parameters and the system response as well as the stability have been investigated. Based on the relations. we have proposed three prototypes for target transfer functions of type I by which one is able to fit easily the desired transient response. The CDM is at the stage of development and thus further effort is needed to make it completely. The most interesting problem is to find out rigorous relationships between ( Y j • , ) and stability. and also transient behaviors.
Ba (s) can be selected so that T(O) = I simply. Then Ba(s) = ko = 4.86. The step response is shown in Fig.
7. Note that in CDM. controller parameters li' k i and coefficients of a target polynomial are determined simultaneously under the consideration of plant nature of itself. Thus. it can be ascertained that the controller by CDM is properly fitted to the plant. ' oor-----,------r-----.,-----------~
a,
Fig.6. Coefficient diagram of Example 1
~08( .= 06
0.. E O'
""'" 02 o o
3
Time[secl
Fig. 7. Step response of Example I
152
REFERENCES Chen C. T. (1993). Analog and Digital Control System Design : Transfer-function. State space and Algebraic methods. Sauders College Publishing. Graham D. and R. C. Lathrop (1953). The synthesis of "optimum" transient response: criteria and standard forms. AlEE Transactions. 72. part 11. 273-288. Kessler C. (1960). Ein Beitrag zur Theorie mehrschleifiger Regelungen. Regelungstechnik•. 8.261-266. Kim S. K. and Y. C. Kim (1999). New Prototypes of Target Transfer Functions for Time Domain Specification. Journal of lCASE. Korea. 5, 889897. Lipatov A. V. and N. I. Sokolov (1979). Some Sufficient conditions for stability and instability of continuous linear stationary system. Automatic Remote Control. 39. 1285-1291. Manabe S. (1998). Coefficient Diagram Method. Proceeding of the 14th lFAC symposium on automatic control in aerospace. Seoul. Korea. 199-210.
Copyright © IF AC System Structure and Control. Prague . Czech Republic. 200 I
[: 0
C>
Publications www.elsevier.comllocatelifac
INTRODUCTION TO COEFFICIENT DIAGRAM METHOD Young Chol Kim* and Shunji Manabe**
* School of Electrical and Electronics Eng.
Chungbuk National University San 48, Gaesindong. Cheongju. Chungbuk, 361-763. Korea E-mail: [email protected] ** 1-8-12 Kataseyama, Fujisawa-shi, Kanagawa, 251-0033, Japan E-mail: [email protected]. jp
Abstract: We introduce a linear controller design method, called the Coefficient Diagram Method (CDM), which is a kind of model matching approach. In this paper, we have especially focused on the main role and characteristics of specific design parameters Yi and "l", which are defined by only coefficients of characteristic polynomial. The relationships between these parameters and the system response as well as the stability are investigated systematically. Based on the relations. we have proposed a prototype for target transfer functions of type I by which one is able to fit easily the desired transient response. It is ascertained that the CDM has some strong points in cases where fixed lower-order controllers wish to be designed. Copyright e 2001 IFAC Keywords: Model matching. Linear control design. Target transfer function 1. INTRODUCfION'
developed by Manabe (1998). He has suggested a new method by which we are able to easily build a target characteristic polynomial to meet the given time response specifications and has also found a graphical tool. referred to the Coefficient Diagram (CD). It is very easy for the CDM to solve the problems of designing a fixed order controller under the conditions of stability, performance and robustness. Moreover, it is possible to deal with them in CD simultaneously. The historical background and basic philosophy of CDM may refer to Manabe( 1998). In section 2, we explain some preliminary works for CDM. In section 3. we represent how to get the target transfer function of Type I and the design procedure of CDM.
The control system design problem with the model matching boils down to the problem determining either a proper target transfer function or a target characteristic polynomial that meets the specification. In many control systems. the controller is imposed some constraints such as low order, minimum-phase. bandwidth. maximum magnitude of control input, and so on. Thus it may be difficult to obtain a target characteristic polynomial to satisfy the specifications under these limitations. In this framework. the other question interested occurs at when we solve the Diophantine equation. The pole placement problem is usually converted to solving the Diophantine equation. There exists a solution but not unique if the order of controller is more than or equal to n-l (where n is the order of plant). However, there are many cases that we may achieve the control objective successfully by much lower order of controller. The ordinary Diophantine equation in this problem can not be solved. With this background, we intend in this paper to introduce a recent development of the Coefficient Diagram Method (CDM). which is based on the model matching mrthod. This paper will emphasize especially that the CDM can answer the above problems. Although the basic concept of the CDM has been known since 1960s (Kessler. 1960; Naslin. 1969), a systematic method has been
2. CDM PRELIMINARIES In CDM design. two-parameter configuration is used to implement an overall transfer function, as shown in Fig.I. Since the output feedback affects only the poles but not the zeros of plant, the zeros of plant remains to be the zeros of overall transfer function. Hence the CDM can be regarded basically as a pole placement method. In the first step. the transfer function of a feedback controller is computed by Diophantine equation. The second step is to find a feedforward controller so as to compensate the effect of plant zeros and to match the steady state gain of overall system. There are two distinctive remarks that reveal the feature of CDM. One is to express the characteristic polynomial in terms of the specific
I This work was supponed in pan by NSF under Grant HRD-9706268.
147
controller limitations when the target polynomial is chosen. We use a simple example to illustrate the basic feature. Consider a fourth-order plant with PlO controller as follows.
parameters, Y , and r . These parameters allow us to obtain a target polynomial so that meets the stability and performance. The other is that the target characteristic polynomial can be deterrruned and can be retuned easil y by shaping its coefficients In the graphical tool, so called the Coefficient Diagram (CD). The CD is a semi-log diagram with the coefficients of polynorrual.
4
Ap(s ) = 0.25s +s 3 + 2s 2 +0.5s, B p (s) = I , Ac(s)=lls,B c(s)=k 2 s 2 +k l s+k o ' where II = 1, k 2 =0.5 , kl =1, ko =0.2 . The Characteristic polynomial is expressed as P(s)=Q25s5 +i +2 s3+2i +s+Q2.
2.1 Definitions and Mathematical relations
Then from (2) - (4), we have
Consider two-parameter feedback configuration as shown in Fig .l.
a , = [as
a 2 al] = [0.25 I 2 2 I 0.2]
=[y~
Y2 h]=[2 2 2 2.5]. r=5
Y,
Y; =[Y: . .. Y2 y;]=[O.5 10.90.5] . 10
;---
V
..........
./
-..... i"'--.. 2
2
a,
I, ..........
- - T,
"
0.1
Fig. I. Two-parameter configuration
2
Some mathematical relations used in CDM will be introduced hereafter. The characteristic polynomial is given in the following form. P(s) = Ac (s)A p (s) + Bc (s)B p (s)
2
0 .0 1 0.5
0.00 1
-----
2
Y.
y:
0 .9
--
~:s: ~ 0.5
"'" "" 02
y, .
f
Y,' 0 .1
o
Fig. 2. Coefficient diagram (I)
The coefficient diagram is shown in Fig. 2, where coefficient a i is read by the left side scale, and
The stability index Yi' the equivalent time constant r , and stability limit
Y; are defined as follows.
Yi := a; l(ai+1ai-l)'
i
=1.2.· ·· . (n-I) .
r:=a1la o '
stability index Yi • equivalent time constant r, and
Y;
stability limit are read by the right side scale. The r is expressed by a line connecting I to r . In this example, the characteristic polynomial P(s) is decomposed into two component polynomials as follows. pes) = ~I (s) + Pt (s) (9)
(2) (3)
Y; :=lIYi+l +lIYi-l' i=l,2,···.n-l , Yn =Yo =00. (4) Also. define the pseudo-break point wi := a i I a i+1 • i = 0.1, .. ·• n -1. (5) We then have Y, =w;lw i _ 1 , i=I, .. ·.n-l. (6)
~1(s)=11(0.25ss +S4 +2S 3 +0.5s 2 )
Pk(s)=k2S2+kls+ko
(7)
(11) T(s) is
The complementary sensitivity function expressed as (12) T(s) = Pt (s)/ P(s) . Eq. (l0) is shown in Fig. 2 with small circles. Eq.(11) is depicted with small squares and dotted lines. Designer can visually assess the deformation of the coefficient diagram due to the parameter change of k2 ' kl and k o ' Then he can visualize the variation of stability and response. Also from (12), it is clear that robustness can be analyzed by comparison of coefficients a i and k i at the coefficient diagram. Taking logarithm values to both sides of (2). we have 10g(Yi ) = 210g(a i ) -log(a,+l) -log(a i _ 1 ) • (13) Therefore, it is obvious from the CD that the lager curvature of a i curve stands for the larger Yi 's for all
It is easy to show that characteristic polynomial is expressed by a o • rand Y as follows. 1
p(,)=a.[l~[ UIIY", },,) )+'>+1]
(10)
(8)
2.2 Coefficient Diagram(CD)
The CD is a semi-log diagram of the coefficients of polynomial in logarithmiC scale versus number of power of s in linear scale. As like the loop shape in the BodelNyquist plot gives information regarding stability and response. the coefficient shape in the CD has the similar relations. Because the coefficients of characteristic polynomial are related to the parameters of plant and controller in the form of linear combination, it is easy to consider the
i. The Yi can be graphically obtained on the CD.
148
This implies that the larger curvature of a, makes the system to be more stable in the sense that the larger damping pole is of more stable than the smaller damping poles. The reasons will be revealed in the section 2.3. If r is small, its response is fast. The equivalent time constant r specifies the response speed. The y, - y, * curves in the CD can be also used for parameter sensitivity analysis and robustness analysis. Note that the coefficient diagram indicates stability, response, and robustness, which are three essences in control design, in a single diagram, so that enables the designer to grasp the total picture of control system.
response is mainly used so far. The CDM is a polynomial method based on the model matching, which is very useful for the purpose of transient response design . In this approach, it is very important to find a target characteristic polynomial. Several standard forms have been proposed. Representative ones of them are the integral of time multiplied by absolute error (lTAE) prototype (Graham, 1953), Kessler standard form (Kessler, 1960). However, these prototypes are excessively rigid when they are applied to design problem. The ITAE and Kessler prototypes have unnecessary overshoot. In all of them, we see that its step response is delayed as the order of system increases.
2.3 Stability condition
3.1 How to get a target characteristic polynomial
The Routh-Hurwitz criterion provides the necessary and sufficient condition for pes) to be Hurwitz stable. However, this way can hardly work the problem designing controller because the Routh criterion is a highly nonlinear function with respect to controller's parameters. Lipatov (1978) proposed sufficient conditions for the Hurwitz stable and unstable. Since this condition is directly related to the CDM design parameters, y, and r, it allows us to consider both the response and the stability simultaneously. From the corollaries in Lipatov (1978), the sufficient conditions for Hurwitz stable and unstable, which are translated in terms of Y;, are as follows. Theorem I. pes) is Hurwitz stable if
As one see in many actual control systems, it is assumed here that the controller to be designed should meet the time-domain specifications such as rise time, settling time, and overshoot. Also, we consider the cases that the overall system can be expressed by a target transfer function of Type I. This means that it has zero steady state error to a step input and a constant steady state error due to a ramp reference input. A closed-loop transfer function of Type I here is
~Y; Y;+I >1.4656 for i=I,2,3,···,n-2 .
n n_~o ~ (19) ans +an_1s +· ··+a1s+ao pes) Thus, it is noted that this system has a DC gain of I, that is, T(O) = 1. Before introducing new prototypes, we represent some important relationships between T(s) and the
T(s) =
(14)
specific parameters Y; , r, and w;. It is obvious from (6) that for the coefficients of pes) in (19), if
Theorem 2. pes) is Hurwitz stable if y;~1.12374y;· for i=2,3, ··· ,n-2 (15)
y; >1, for all i = 1,2,· ··, n - I, then it has w;
or equivalently, if
all i = 0,1,2,·· ·, n - 1 . It is remarkable that
(16)
(20) logy; =logw; -Iogwi-l ' i=I,2,·· ·,n-l (21) With this context, the asymptotic line approximate plot for the Bode plot of a T(s) is shown in Fig.3. The straight-line approximation of Bode diagram used here is somewhat different from the ordinary way. The pseudo break points are defined by the ratio of the coefficients and not from the poles and zeros of the transfer function . However, both result in similar approximations. From low to high frequencies, the slope of the magnitude plot is 0 dB/dec and decreases by 20 dB/dec at each pseudo breakpoint. It is seen evidently that the bandwidth of target transfer function depends on absolutely the equivalent time constant r and its cut off rate near the cut off frequency, which is the region indicated by 10gYI
Theorem 3. pes) is unstable if Y;Y;+I:$1
for some i=I,2,·· · ,n-2
(17)
or equi valentl y, if (18) The above conditions can be graphically expressed in the CD (see Fig.2). In CDM, Theorem 2 is mainly used because of its closeness to the necessary condition. Theorem 1 implies that the overall system should be stable if all y;'s are greater than 1.4656. It turns out that the value can be used as a limit when we obtain a desired characteristic polynomial. a;a ;+I:$ a ;+2a;_1 .
3. CDM FOR LINEAR CONTROLLER DESIGN Even though the perfonnances of most control systems are judged by its time domain responses, there are few design methods available to this purpose because there are no simple relationships between time response characteristics and transfer function. Thus the loop shaping of frequency
and log Y2. Therefore, the settling time can be determined by r . If the cut off rate in the region of YI and Y2 decreases sharply, it makes overshoot in a step response. It is also certain that Y3 and higher
149
Y,' s give much less effect to the closed-loop response because the frequency response magnitudes corresponding to this area is relatively very small provided that it has large YI and Y2 • We now present important relationships between characteristic roots and the equivalent time constant.
.0r------------------------------------, - Ioo(r) '0
logO',) W
Wo
W,
Table I. Standard form of M-type prototype Proto .
type
logO',)
IOg0'2)
there are almost no overshoots and the same transient behaviors irrespective to the order of system. The designer can start from a simple controller designed by this form and then may move to more complicated one.
s' + 10 I r s' + 50 I r ' 5' + 1~5 I r ' s + 125 I r'
ai,
~
.§: f.
sS
~
' -6dB /oCl
·. 0
T( s) = _ _ _ _ _a...,:o'--_ _ __ sI'! + QIl _1Sn -1 + ... + Q1S + a o
.:
....
01
5'
. .
~~
1
M
+ 25 00 /r 4 5 +2500 /1"5
+ 40 /r 5' + 800/r's' + 8000/r'5' + 4xlO' Ir'5 ' +10' / r ' 5+10'/r'
+ 160 / T S1 + 12800 Ir 2/' + 5 . 12xlO ~ I r J s 5 + I .024xIO' 11'4s· + I.024xlO'/r' s' + 5.12xIO'/r's' + 1.28xlO' IT'S + 1.28xlO' IT'
SS
10
Fig.3. Frequency magnitude of a target model of Type I
Table 2. Relationships between " and t 5
Theorem 4 (Kim. 1999): Consider two stable polynomials of degree n, A(s) and 8(s) . A(s) =ans n + an_ls,,-1 + ... + als + a o 8(s)=b n s n +bn_ls n - 1 + .. ·+bls+bo
and
a Bi ' i = I, 2, .. ·, n respectively and let "A'" B be their equivalent time constants, respectively. If both polynomials have the same stability index Yi 's, then the following holds, aA, =aa B" i=I,2, .. ·,n,
+ 1000 /"["3 5 2
s' + 80 Ir 5' + 3200/r' s' + 6.4xlO' I r 's ' + 6.4xlO ' I r' s) +3 .2xlO'/r's' +8xlO'/r's+8xlO'/r'
logw Irad/s ec l
Let the roots of both polynomials be a Ai
+20/rs 4 +200/ 1' 2 5 3
(22)
Design specifications M
== t s /2.3095
1.2 r---~--~--~--
__- -
D.•
]
real, and negative. VVV In order to guarantee the stability by means of Lipatov's sufficient condition in Theorem I, all L's
TM
However, in some situations it may be difficult to choose a target transfer function so that meets the rise time and the settling time simultaneously by means of the Manabe form. Kim (1999) has proposed two other forms; F and S-types. The response of the F-type represents the fast rise time and slow settling time, whereas that of the S-type characterizes the slow rise time and fast settling time as shown in Fig.4.
where a=TA /T B • VVV Theorem 4 states that all the characteristic roots can be shifted at the same rate by only a single parameter ". It is noted that if only T is changed, then the damping ratios of all roots remain the same values. Theorem 5(Lipatov. 1978): For a polynomial, if all y, 's are greater than 4, then all its roots are distinct,
Settling Time ( t s )
~ «
06
3 Timel5 ec)
Fig. 4. Step responses of 3-types ( n = 4 - 8 )
should be greater than 1.4656. Thus Y,'s are usually chosen within interval, Y"
E
[1.5, 4). Based on the
above background, Manabe (1998) proposed a standard form for Type I system. It is Yl=2.5, Yi =2, for all i=2,3," ',n-1 (23) In Table I, for n =4,· .. ,8, the target transfer functions of M-type that are normalized by " are given. Also, the relationship between the settling time and T is given in Table 2. We call this value as the Manabe form or M-type. Although the Manabe form has been found by the heuristic approach, it is strongly recommended as a target characteristic polynomial due to its favorable properties. The step responses of the Manabe form for the order of n =4"" ,8 are shown in Fig. 4. It is remarkable that
150
~ . 50'--------.,. O~ 5 ----~----....J 15
Time!sec !
Fig. 5. Step response change rate (dy/dt) of 3 prototypes when n=5 and t s = 1 sec These curves have been specified by selection of both the times and the desired response values at 0, 10%, 50%, 90%, and 99% for the settling time, respectively. An integral of the squared error function
has been considered. Then solving numerically this optimization problem by a non linear interpolation method, they have obtained both F and S prototypes (Kim, 1999). They have almost the same response shapes irrespective to the order of systems . In Fig . 5, the step response change rates of 3 prototypes are compared. If the maximum value of the change rate occurs at late, this means the slow rise time. This change rate relates closely to the maximum magnitude of control input. So far, we have addressed three prototypes for Type I target system. In these forms, all Yi 's are fixed and only r is used so as to meet the desired settling time. There may be some cases that requires the other values of yi's and r . However, currently no constructive rules for this purpose have been published. We now explain about how to obtain the target characteristic polynomial by the coefficient shaping on the CD. As shown in Fig. 2, a CD includes five information such as numerator and denominator polynomials of plant, stability index Yi's, stability
Y;,
equivalent time constant 'l' , and the closedlimit loop polynomial. It is possible to consider the trade off among performance, stability and robustness. As seen in Fig.3, 'l' relates absolutely to the bandwidth of overall system. Since 'l' = a) / ao where a) and ao are coefficients of two lowest order of characteristic polynomial, it is represented by a slope on the coefficient diagram. Consequently, the line a) - ao on CD stands for the system response speed.
In particular, the stability index Yj is the most favorable parameter, which relates to the transient behavior as well as the stability of overall system. The distance between Yi and Y; indicates the degree of tolerable variance in the coefficients. A system will tend to remain the stability to larger perturbations of the system parameters if the distance is larger. In some cases, for example, a non-minimum phase system, the 'l' may be bounded by the lowest value. Thus although this way requires some or less empirical knowledge, it is not so much difficult to choose a target characteristic polynomial in a feasible range.
pes) = Ac(s)Ap(s)
=1::. =
Ba (s)B/s)
r
!\(s)-\(s) + Bc (s)Bp (s)
deg(Bc ) = deg ( Ap) -1
(25)
deg(Ac )~deg(Bc )
(26)
Example 1. (Franklin, 1994:P. 538 Example 7.30) Consider a third order plant transfer function G(s)= Bp(s) = Ap (s) s(s
10
+ 2)(s + 8)
Design a controller of first order by CDM that satisfies the settling time t s :5 2 sec and overshoot less than 2% . Solution: The controller can not be solved by the ordinary Diophantine equation because the condition (25) does not hold. Let the feedback controller transfer function be C(s) = B c (s) = k]s + ko ,(io =1) Ac(s) 11s+10
In CDM, there are two design approaches; the algebraic method under the matching condition and the coefficient shaping on the CD. We first state the algebraic method briefly. Consider a feedback system shown in Fig. 1. The closed-loop transfer function is
= Ba(s)Bp(s)
(24)
According to section 3.1. once a target transfer function res) is selected, the design problem boils down to the model matching one. Rewriting (l),
151
(27)
Then the characteristic polynomial becomes pes)
=a4s 4 + a3 s3 + a2s2 + a)s + ao
=~ s(s3 +loi + 1&)+(s3 +1(k2 + 16)+1
First, draw a coefficient diagram in the absence of control, that is, lo = I, l] = k] = ko = O. Then, the coefficients of Ap (s) would be depicted. The curve is shown in
pes)
(1)
For this case, since it is trivial to solve (I), the details refer to Chen (1993). In many control systems, it is true that a lower order controller may meet the given specifications. But, this problem can not be solved by any analytical approach. Furthermore, if a certain parameter of the controller will have to depend on the other one as a design constraint, then the number of equations becomes greater than the unknown variables. It turns out that the existence of solution can not be guaranteed. One of the main characteristics of CDM is that even such a problem can be dealt with in the manner which one adjusts Yi 's and 'l' so as to have a feasible region. Similar to the coefficient shaping for obtaining the target transfer function, it can be applied to the design process since the resulting characteristic polynomial is easily drawn by moving up and down as much as the sum of controller's parameter in the same order. Also, we can see directly by CD how much the overall system will be sensitive to the parameter perturbations. The basic references of coefficient shaping are to retain the original good nature of plant if any, and to make the ai's curve to be a smoothly concave shape. At the moment, the resulting controller can be immediately obtained from the CD. Now, we give an illustrative example.
3.2 Design procedure of CD M
res)
+ B c(s)Bp(s)
This Diophantine equation has a unique solution if the order of controller is given by
0---0---0,
denoted by lo in Fig. 6. From
CD, it is easily found that the selection of l] = 0.05 will not change P(s) much from Ap (s) (See l] curve). That is, it makes in part a sIOOOthly
concave shape around 10 , Writing the Diophantine equation of (28) in the Sylvester form, I
0
0
0
10
I
o o
0
16 10
0
0 16 10 0 0
0
o
10
a.
lH
,'jyJ Y;Y~
Cl:J ~
,J/ y2yI2 '"/Y
,
al
1
ao (29)
ao
Since the settling time t J :5 2 sec. using Table 2. computed below and thus we choose, = 0.8 .
,IS
t
(30) , :5 - -'- :5 0.866 2.3095 According to the Manabe form. we assume that y 2 = 2.0 . Now. the problem is to solve a non-square Sylvester matrix with the condition I1 = 0.05 . 10 = I .
Y2 =2.0.
,= 0.8. The unknown
variables are a o•
y I' k o • and k l · From (29). we have a J =ao,l jy2Y~ =1.5
, 2/
a 2 =a o Y1 =10.8. Solving these algebraic equations, we y 1 = 2.88 and finall y the feedback controller is Bc(s) 2.288s+4.86 C(s)=--= . Ac (s) 0.05s + 1
(31 ) (32) have
(33)
5. CONCLUDING REMARKS If we want to design a lower and fixed-order controller for given specifications. what methods are available? The model matching approach is a method possible for this issue. However. this method is difficult to get the target transfer function so that will meet the specifications subject to the constraints such as low order if possible. bandwidth. response rate. the maximum magnitude of control input and so on. The other problem is that it is difficult to solve the Diophantine equation when the order of controller is lower than full order required to ensure the existence of solution. The CDM has developed on these needs. In this paper. we have introduced a recent development of CDM. Especially. we have focused on the main role and characteristics of specific design parameters y, and, . The relationships between these parameters and the system response as well as the stability have been investigated. Based on the relations. we have proposed three prototypes for target transfer functions of type I by which one is able to fit easily the desired transient response. The CDM is at the stage of development and thus further effort is needed to make it completely. The most interesting problem is to find out rigorous relationships between ( Y j • , ) and stability. and also transient behaviors.
Ba (s) can be selected so that T(O) = I simply. Then Ba(s) = ko = 4.86. The step response is shown in Fig.
7. Note that in CDM. controller parameters li' k i and coefficients of a target polynomial are determined simultaneously under the consideration of plant nature of itself. Thus. it can be ascertained that the controller by CDM is properly fitted to the plant. ' oor-----,------r-----.,-----------~
a,
Fig.6. Coefficient diagram of Example 1
~08( .= 06
0.. E O'
""'" 02 o o
3
Time[secl
Fig. 7. Step response of Example I
152
REFERENCES Chen C. T. (1993). Analog and Digital Control System Design : Transfer-function. State space and Algebraic methods. Sauders College Publishing. Graham D. and R. C. Lathrop (1953). The synthesis of "optimum" transient response: criteria and standard forms. AlEE Transactions. 72. part 11. 273-288. Kessler C. (1960). Ein Beitrag zur Theorie mehrschleifiger Regelungen. Regelungstechnik•. 8.261-266. Kim S. K. and Y. C. Kim (1999). New Prototypes of Target Transfer Functions for Time Domain Specification. Journal of lCASE. Korea. 5, 889897. Lipatov A. V. and N. I. Sokolov (1979). Some Sufficient conditions for stability and instability of continuous linear stationary system. Automatic Remote Control. 39. 1285-1291. Manabe S. (1998). Coefficient Diagram Method. Proceeding of the 14th lFAC symposium on automatic control in aerospace. Seoul. Korea. 199-210.