Further Results on the Synthesis of PID Controllers

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Index Terms—Dead time, parameter space, PID controllers, second- order plants, stabilizing regions. I. INTRODUCTION. In recent years, there has been a trend ...
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 52, NO. 6, JUNE 2007

Further Results on the Synthesis of PID Controllers De-Jin Wang

Abstract—This note focuses on a graphical approach to determine the stability region, in the parameter space, of PID-controlled second-order plants with dead time. This work is the extension of and further results on the same topic of the previous paper in the literature on first-order plants with dead time. The main characterization of the new method is the utilization of a graphical stability criterion in the parameter space, which releases a necessary and sufficient stability condition. Once the range of proportional gain is known, the complete stabilizing set in the plain of integral-derivative gains can be drawn and identified immediately, not to be computed mathematically. A practical algorithm of drawing the stabilizing boundaries in the integral-derivative space is proposed. Index Terms—Dead time, parameter space, PID controllers, secondorder plants, stabilizing regions.

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The staring point of our research in this note stems from a parameter space approach developed in [13] which presents a simple, direct characterization in determining the stability regions of quasi-polynomials. We first compute the range of admissible proportional gain in parallel to the arguments for the first-order plants with dead time in [1]. Then, pick up a value of proportional gain in this range, we draw and identify the stabilizing boundaries of PID controllers in integral-derivative space, not to calculate analytically. We relate our consideration with the second-order plants with dead time, because many processes in control fields can be approximated by the second-order model with dead time, especially the higher-order systems possessing a pair of dominant second-order poles. II. GRAPHICAL CRITERION

()

In recent years, there has been a trend towards the computation of stabilizing regions in the parameter space of PID controllers [1]–[7]. It is well-known that the PID type of controller is most widely used in industrial applications, because of its simplicity in structure and capability in controlling many practical processes. The theory, design, and tuning of PID controllers were reviewed and summarized in [8]. In practice, the tuning of proportional, integral and derivative gains is usually based on the empirical techniques such as the popular Ziegler–Nichols rules [9]. In most cases, the empirical methods can give a satisfactory response behavior of the controlled systems. However, the complete stability region remains unknown, which is important to the performance design of the processes [10]. On the other hand, in some cases, the test of a process is not allowed from the safety consideration, such as the unstable processes. The theoretical determination of stabilizing PID controllers reveals a complete stability region in the parameter space, and the research in this direction can close the gap between the theory and practice of automatic control [1]. In the earlier attempt to determine the entire stabilizing region of PID parameters, a generalization of the Hermite–Biehler Theorem was derived for a given linear, time-invariant plant with rational transfer function [3], [4]. Based on this characterization, H1 optimal PID controller design was completed [10]. However, this result cannot be applied to the plants with dead time, in which case the characteristic equation of the closed-loop system is a quasi-polynomial instead of a rational one. In [1], the first-order plants with dead time were investigated using a version of the Hermite–Biehler Theorem derived by Pontrayagin [11], [12] applicable to quasi-polynomials. An alternative simple way to determine the stabilizing region of PID controllers in [2] gave the same results as those in [1] by means of the classical Nyquist stability criterion. Manuscript received March 10, 2006; revised September 1, 2006 and January 7, 2007. Recommended by Associate Editor M. Fujita. The author is with the School of Electronic Information and Automation, Tianjin University of Science and Technology, Tianjin 300222, China (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAC.2007.899045

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second-order form with dead time k 0Ls ( ) = (1 + T s)(1 + T s) e

G s

I. INTRODUCTION

()

Consider SISO unity feedback system with G s being the plant, C s the controller. Throughout this note, we suppose that G s is of

0 ()

1

(1)

2

0

where k > is the steady-state gain of the plant, L > the dead time, and T1 ; T2 the time constants of the plant. We also suppose that the controller C s is of PID type

( ) = kp + ksi + kd s:

C s

(2)

(

)

The synthesis objective is to determine the parameter set kp ; ki ; kd of the PID controller such that the unity feedback closed-loop system is stable. The characteristic quasi-polynomial multiplied by eLs of the closedloop system is given by

1(s) = (1 + T s)(1 + T s)seLs + k(ki + kp s + kd s ): 1

2

2

(3)

Now, define the change of variables s

= z 0 ;

z

= x + jy;

>

0

(4)

0

i.e., we consider the problem of stability degree  > in the s-plane. Then, substitute (4) into (3), and express the characteristic quasi-polynomial in terms of z , we have

1(z) = [1 + T (z 0 )][1 + T (z 0 )](z 0 )eL z0 +k[ki + kp (z 0 ) + kd (z 0 ) ]: (5) Letting z = jy in (5), i.e., along the imaginary axis in the z -plane, and partitioning the corresponding 1(jy ) into real and imaginary com1

(

2

)

2

ponents yields

1(jy) = r (y) + ji (y) where

( ) = 0A(; y)( cos(Ly) + y sin(Ly)) 0 B(; y)(y cos(Ly) 0  sin(Ly)) + kkd ( 0 y ) 0 kkp  + kki i (y ) = 0B (; y )( cos(Ly ) + y sin(Ly )) + A(; y)(y cos(Ly) 0  sin(Ly)) 0 2kkd y + kkp y

r y

0018-9286/$25.00 © 2007 IEEE

2

2

(6a)

(6b)

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Fig. 1. (a) Unstable region swept by the straight line (11). (b) Initial pieces of the curve given by (10).

As the coefficients of the characteristic polynomial (5) are real, if

where

0L (1 0 T1 )(1 0 T2 ) 0 T1 T2 y2 A(; y ) = e 0L ((1 0 T1 )T2 y + (1 0 T2 )T1 y): B (; y ) = e

z is a root of (5), then also the complex conjugate of it. Hence, it is sufficient to consider y [0; ). (0; ), In the following, we discuss two cases, y = 0 and y

2

Note that both r and i are dependent on (ki ; kd ; kp ; ; y ). Then, the stability property of the characteristic quasi-polynomial (5) can be investigated in the parameter space (ki ; kd ), as described below, taking the parameters kp and  to be fixed. Suppose that we have found, in one way or another, a point (ki0 ; kd0 ; kp ; ; y ) on the imaginary axis, x = 0, such that 0

0

0

0

r = r ki ; kd ; kp ; ; y = 0 i = i ki ; kd ; kp ; ; y = 0

(7)

i.e., there is a root on the imaginary axis. According to the Implicit Function Theorem, if the Jacobi matrix @ @k @ @k

J =

@ @k @ @k

(8) (k ;k ;k ;;y )

is nonsingular, the (7) has a local unique solution curve (ki (y ); kd (y )). Moreover, we have the following proposition [13]. Proposition 1: The critical roots are in the right-half plane if the point in the parameter space, relative to the selected values of ki and kd , lies at the left side of the curve (ki (y ); kd (y )) when we follow this curve in the direction of increasing y , whenever det J < 0 and at the right side when det J > 0. Here, J is the Jacobi matrix defined in (8). From (6a) and (6b), it follows that det J =

02k

2

y < 0;

8 y > 0:

(9)

Solving for ki and kd in terms of (kp ; ; y ) yields kd =

kp

2

+

1

2ky

[(yA(; y )

0 B(; y)) cos(Ly)

0 (A(; y) + yB(; y)) sin(Ly)] 0 )

k i = k p  + kd (y

2

1

[(A(; y ) + yB (; y )) cos(Ly ) k + (yA(; y ) B (; y )) sin(Ly )]: +

(10a)

2

0

(10b)

1

2

1

respectively, and recognize the stable and unstable regions in the (ki ; kd )-plane accordingly. First, from (5), we observe that when z is real and positive, i.e., x  0; y = 0, (5) determines a linear relation between ki and kd kd =

0 (x 01 )

2

ki

+ T (x 0  ))e 0 kkp + (1 + T (x 0k())(1 x 0 ) 1

2

0

L(x )

(11)

which always has a minus slope in the (ki ; kd )-plane. When these lines are plotted for fixed values of  and kp , the lines sweep over the unstable region in the (ki ; kd )-plane, as shown in Fig. 1(a), where the highlighted line corresponds to the principal line, x = 0. Next, we sketch the curve (ki (y ); kd (y )) for y > 0 using (10). This is shown in Fig. 1(b). Fig. 2(a) is the zoomed-in version of Fig. 1(b) around the origin combined with Fig. 1(a). From Fig. 2(a), we observe that the starting point, y = 0, of the curve (ki (y ); kd (y )) is on the principal line of (11), with the increase of frequency y to big enough, the curve forms a closed region to the right of the principal line. From the sign of the determinant of the Jacobi matrix (9) and the Proposition 1, this region can be identified as the stabilizing region of parameters (ki ; kd ) with stability degree  = 0:5. As  decreases, the stabilizing region is enlarged as shown in Fig. 2(b). When  is small enough, the exact stabilizing region is recovered as shown in Fig. 2(c) by the triangle. In fact, when  becomes zero, the curve is transformed in two straight lines, whose one [see (10a)] is parallel to the axis kd and at infinity and the other is expressed by (10b). Therefore, the stabilizing zone, according to the Proposition 1, is above this line if the derivative dki =dy is negative or under if positive. From (10a) and (10b), it can be shown that the derivative of ki , independent of kd , becomes infinite for  = 0, but since only its sign is needed, it is convenient to multiply

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Fig. 2. Stabilizing regions. (a)

= 0 5. (b)

= 0 15. (c)

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= 0 0001

Fig. 3. (Solid line) Plot of the curve (10), (dashed line) the principal line of (11), and (dotted line) the tangent line of (10).

ki by  and then to assume  = 0 in the right-hand side of the relative equation. Along this line, we obtain

+6T1 T2 (1

0 L) + L2 (3 0 L)(1 0 T1 )(1 0 T2 )

n = (1 0 T1 )(1 0 T2 )(1 + L 0 L  ) 0 2T1 T2  + (1 0 2L )((1 0 T1  )T2 + (1 0 T2  )T1 ): 2 2

y2 dki j=0 = 0 ((2T1 T2 + LT1 + LT2 )y cos(Ly) dy 2k 2 +(T1 + T2 + L 0 LT1 T2 y ) sin(Ly )): Lemma 1: A necessary and sufficient condition for the existence of a stabilizing PID controller with stability degree  > 0 for the plant (1) is

m

6(n + kkp eL ) +  2 m


is

Case 2:

T1 T2 < 0. In this case, if

L ( T 1 + T 2 ) + T 1 T 2 < 0 0 :5 L 2

0

0 6(n + kkpm > 0 12 eL ) + 2 m 

and the equation

tan = 0 L2 02TT11TT22+ 2L+(TL1 (+T1T+2 )T2 )

which is equivalent to (12). This completes the proof. Remark 1: As a matter of fact, as kp decreases, the stabilizing region shrinks also. Hence, the condition (12) provides a relation between the maximum allowed stability degree  > and the minimum allowed proportional gain kp . Remark 2: Note that the slope of the principal line of (11) is identically less than zero. From Lemma 1 and Fig. 3, we claim that there exists an " > such that if

0

0

"
0 kki > 0 kkd + T1 + T2 > 0 i 1 + kkp > kkdT+1 TT2kk >0 (15a) 1 + T2 or T1 T2 < 0 kki < 0 kkd + T1 + T2 < 0 i 1 + kkp < kkdT+1 TT2 kk 1 + T2 < 0: (15b) 3

III. STABILIZING REGIONS From the expressions of ki and kd in (10), we see that both ki and kd are dependent on the proportional gain kp . In order to determine the stability boundaries in the ki ; kd -plane, the admissible range of stabilizing kp must be given a priori. The following theorem provides such results for second-order plants with dead time. Theorem 1: The range of kp values for which a second-order plant, with transfer function G(s) as in (1), can be stabilized by a PID controller is summarized as follows. Case 1: T1 T2 > . In this case, if

)

0 L (T 1 + T 2 ) + T 1 T 2 > 0 0: 5L 2

has a solution 2 in the interval

kl < kp < 0 1

(21)

(0; ), then

(22)

k

where

(14)

then the stabilizing region is nonempty. Remark 3: When L , i.e., there is no delay,

(

(20)

kl =

1

T1 + T2 sin 0 2 2 L

k

1 0 TL1T2 2 22 cos 2

:

For kp values outside this range, there are no stabilizing PID controllers. Proof: As mentioned in the previous section, when  ! , the complete stabilizing region is recovered. In this case, the sufficient condition for the existence of a stabilizing PID controller in (14) reduces to the following:

0

T 1 T 2 + 0 :5 L 0 < " < L(T1 + T21) ++ kk p

2

< +1

which is equivalent to

or

L ( T 1 + T 2 ) + T 1 T 2 + 0 :5 L 2 > 0 1 + kkp >0

(24)

L ( T 1 + T 2 ) + T 1 T 2 + 0 :5 L 2 < 0 < 0: 1 + kkp

(25)

Case 1: Consider (15a) in Remark 3 and the inequalities (24), the condition (16) and the lower bound of kp in (18) are yielded immediately. The condition (17) and the upper bound of kp are obtained as follows. can With the change of variables z Ly , (6b) with  be written as

=

i (z ) = z kkp + L

1 0 TL1T2 2 z2 cos z 0(T1 + T2 ) Lz sin z

To compute the roots of i equations:

(16)

=0

has a solution 1 in the interval where

ku =

0 1 < kp < ku

1

k

z=0

(17)

(0; ), then

or (18)

T1 + T2 sin 0 1 0 T1 T2 2 cos : (19) 1 1 1 L L2 1 For kp values outside this range, there are no stabilizing

1 0 TL1T22 z2 cos z 0 (T1 + T2 ) Lz sin z = 0

which yield

k

PID controllers.

kkp +

z=0 or

:

(z), we arrive at the following

and the equation

tan = 0 L2 02TT11TT22+ 2L+(TL1 (+T1T+2 ) T2)

(23)

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Fig. 4. Stabilizing regions of unstable second-order plants.

kkp +

1

0 TLT sin z

z2

cos z

=

T 1 + T2 z: L

(26)

From these equations, we can deduce, along the similar lines as those in [1] (Lemma 4.1 and Theorem 2.1 therein), and by observing the plots of the functions on both sides of (26), that if the conditions (16) and (17) are satisfied simultaneously then the upper bound of kp value is given by (18) and (19). Case 2: Combine (15b) in Remark 3 with the inequalities (25), the condition (20), and the upper bound of kp can be obtained, and the proof of the condition (21) and the lower bound of kp is similar to Case 1. The proof is completed. Remark 4: When T1 = 0 or T2 = 0, the results in Theorem 1 are simplified to those in [1] for first-order plants with dead time. In light of Theorem 1, we propose the following algorithm to draw and identify the stabilizing regions of PID controllers in the (ki ; kd )-plane. Algorithm 1: For determining the stabilizing regions of PID controllers. Step 1: For a given plant (1), verify whether it can be stabilized by means of (16)–(17) for T1 T2 > 0 or (20)–(21) for T1 T2 < 0. Step 2: Calculate the bound of kp according to Theorem 1. Step 3: Pick up a kp value in the range given by (18) and (19) or (22) and (23). Step 4: Compute the principle line given by (11), and calculate the couples of straight lines, given by (10a) and (10b) for a small enough  , for a big enough y until the stability region becomes well defined. IV. CASE STUDIES Given the second-order plant (1) with k = 1; L = 0:2 and  = 0:0001, we study the stabilizing regions of PID controllers in the (ki ; kd )-plane for the following cases.

Case 1: T1 = 0:2; T2 = 0:5. From (18), we have01 < kp < 5:56. Choosing kp = 0 and applying Algorithm 1 gives the stabilizing region shown in Fig. 2(c) by the triangle. Case 2: T1 = 01; T2 = 00:5, i.e., the open-loop plant has two unstable poles. In this case, the conditions (16) and (17) are satisfied, and01 < kp < 0:58. Choose kp = 00:5, we plot the stabilizing zone using Algorithm 1. The result is shown in Fig. 4(a) by the triangle. Case 3: T1 = 1; T2 = 00:5, i.e., the open-loop plant has one unstable pole. For this plant, the conditions (20) and (21) are satisfied, and the admissible range of kp value is 05:21 < kp < 01. Fixing kp = 01:5 and employing Algorithm 1 yields the stabilizing region shown in Fig. 4(b) by the triangle. Case 4: T1 = 01; T2 = 0:5, i.e., the open-loop plant has one unstable pole. In this case, the conditions (20) and (21) are true, and we have 09:79 < kp < 01. Select kp = 01:5, the stabilizing region is shown in Fig. 4(c) by the triangle. Remark 5: It can be shown that, for both stable and unstable firstorder plants with dead time, using our method, the stabilizing regions have exactly the same shapes as those in [1].

V. CONCLUTION We have discussed a technique for determining the stabilizing parameter set of PID controllers via a graphical stability criterion. This criterion exhibits a necessary and sufficient stability condition, and, thus, the result provided in this note does not have any conservatism, i.e., the complete stabilizing area in the parameter space is obtained. From the design procedure given in the note, we conclude that the method can be applied not only to the first and second-order plants with dead time, but also to other arbitrary plants with dead time, provided that the range of proportional gain is known. Also, the new method can be applied to synthesize other forms of controllers besides PID type.

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REFERENCES

I. INTRODUCTION

[1] G. J. Silva, A. Datta, and S. P. Bhattacharyya, “New results on the synthesis of PID controllers,” IEEE Trans. Autom. Control, vol. 47, no. 2, pp. 241–252, Feb. 2002. [2] G. Martelli, “Comments on “New results on the synthesis of PID controller”,” IEEE Trans. Autom. Control, vol. 50, no. 9, pp. 1468–1469, Sep. 2005, Comments on. [3] M.-T. Ho, A. Datta, and S. P. Bhattacharyya, “A linear programming characterization of all stabilizing PID controllers,” in Proc. Amer. Control Conf., Albuquerque, NM, Jun. 1997, pp. 3922–3928. [4] A. Datta, M.-T. Ho, and S. P. Bhattacharyya, Structure and Synthesis of PID Controllers. London, U.K.: Springer, 2000. [5] M. T. Soylemez, N. Munro, and H. Baki, “Fast calculation of stabilizing PID controllers,” Automatica, vol. 39, no. 1, pp. 121–126, Jan. 2003. [6] J. Ackermann and D. Kaesbauer, “Stable polyhedra in parameter space,” Automatica, vol. 39, no. 5, pp. 937–943, May 2003. [7] H. Xu, A. Datta, and S. P. Bhattacharyya, “PID stabilization of LTI plants with time- delay,” in Proc. 42nd IEEE Conf. Decision Control, Maui, HI, Dec. 2003, pp. 4038–4053. [8] K. Astrom and T. Hagglund, PID Controllers: Theory, Design, and Tuning. Research Triangle Park, NC: Instrument Society of America, 1995. [9] J. G. Ziegler and N. B. Nichols, “Optimum setting for automatic controller,” Trans. ASME, vol. 64, pp. 759–768, 1942. PID controllers: A parametric approach,” [10] M.-T. Ho, “Synthesis of Automatica, vol. 39, no. 6, pp. 1069–1075, Jun. 2003. [11] L. S. Pontryagin, “On the zeros of some elementary transcendental function,” Amer. Math. Soc. Transl., vol. 2, pp. 95–110, 1955. [12] J. K. Hale and S. M. verduyn Lunel, “Introduction to functional differential equations,” in Applied Mathematical Sciences. New York: Springer-Verlag, 1993. [13] O. Diekmann, S. A. van Gils, S. M. verduyn Lunel, and H.-O. Walther, “Delay equations: Functional-, complex- and nonlinear analysis,” in Applied Mathematical Sciences. New York: Springer-Verlag, 1995.

Robust Filtering for Linear Systems With Convex-Bounded Uncertain Time-Varying Parameters Carlos E. de Souza, Fellow, IEEE, Karina A. Barbosa, and Alexandre Trofino

Abstract—This note addresses the design of robust filters for linear systems with a state-space model subject to time-varying uncertain parameters with limited variation. The uncertain parameters and their rate of variation are assumed to belong to a given convex-bounded polyhedral domain. A method based on a parameter-dependent Lyapunov function is proposed for designing a linear stationary asymptotically stable filter with a guaranteed average error variance, irrespective of the uncertain parameters. The proposed design is formulated in terms of linear matrix inequalities. Index Terms—Convex-bounded uncertainties, parameter-dependent filtering, time-varying parameters, uncerLyapunov function, robust tain systems.

In recent years, the design of robust H2 (minimum variance) filters for linear dynamic systems with parametric uncertainty has received special attention. The problem consists of designing a linear filter that ensures a bound on the H2 norm of the operator from the noise signals to the filtering error in spite of the uncertainties. In the case of linear systems with norm-bounded parameter uncertainty in the state-state model, Riccati equation approaches [3], [8], [9], [12], [13], [15] have been proposed, whereas systems with polytopic-type parameter uncertainty have been recently treated in [5] and [7] using LMI methodologies. The aforementioned methods are based on the notion of quadratic stability and have the advantage that they are computationally simple. However, they have the drawback that stability and the guaranteed bound on the H2 performance are based on a parameter-independent Lyapunov function, and, thus, the uncertain parameters are allowed to vary arbitrarily fast, which can be quite conservative. Very recently, LMI approaches of robust H2 filtering for linear systems with convex-bounded uncertain time-invariant parameters have been proposed in [2], [14], and [16] using parameter-dependent Lyapunov functions. This note proposes an LMI based method for the design of robust H2 filters for linear systems with uncertain time-varying parameters which appear affinely in the matrices of the system state-space model. The parameters value and rate of variation are assumed to belong to a given polytope. The problem addressed is the design of a linear stationary asymptotically stable filter which achieves a guaranteed average error variance. The proposed robust H2 filter design method is based on a parameter-dependent Lyapunov function with quadratic dependence on the uncertain parameters and incorporates information on known bounds on the rate of variation of the uncertain parameters. In particular, the new filtering method includes the quadratic stability approach as a special case. Notation n and n2m are the set of n-dimensional real vectors and n 2 m real matrices, respectively, In is the n 2 n identity matrix, 0n and 0m2n are the n 2 n and m 2 n matrices of zeros, respectively, diagf1 1 1g stands for a block-diagonal matrix and Trf1g denotes matrix trace. For a real matrix S; S T denotes its transpose, HerfS g stands for S + S T and S > 0 (S < 0) means that S is symmetric and positive-definite (negative-definite). For a symmetric block matrix, the symbol ? denotes the transpose of the blocks outside the main diagonal block. E[ 1 ] stands for mathematical expectation and V (D) denotes the set of all vertices of the polytope D . II. PROBLEM STATEMENT Consider the following uncertain linear system: x_ (t) = A((t))x(t) + B ((t))w(t); y (t) = C ((t))x(t) + D((t))w(t) z (t) = Cz x(t)

with  := [1 ; . . . ; p ]T

Manuscript received September 4, 2006; revised January 30, 2007 and February 11, 2007. Reommended by Associate Editor L. Xie. This work was supported in part by the “Conselho Nacional de Desenvolvimento Científico e Tecnológico—CNPq,” Brazil, under Grants PRONEX 0331.00/00, 30.2317/02-3/PQ, and 30.5665/03-0/PQ. K. A. Barbosa was supported by CAPES/PRODOC. C. E. de Souza and K. A. Barbosa are with the Department of Systems and Control, Laboratório Nacional de Computação Científica—LNCC/MCT, Petrópolis, RJ 25651-075, Brazil (e-mail: [email protected]; [email protected]). A. Trofino is with the Department of Automation and Systems, Universidade Federal de Santa Catarina, Florianópolis, SC 88040-900, Brazil (e-mail: [email protected]). Digital Object Identifier 10.1109/TAC.2007.899043 0018-9286/$25.00 © 2007 IEEE

2

x(0) = x0

(1)

p and p

A((t)) = A0 + B ((t)) = B0 + C ((t)) = C0 + D((t)) = D0 +

i=1 p i=1 p i=1 p i=1

 i ( t) A i  i ( t) B i

(2)

 i ( t) C i  i ( t) D i

(3)