A FEEDBACK STRUCTURE WITH HIGHER ORDER DERIVATIVES IN REGULATOR. Ryszard Gessing

A FEEDBACK STRUCTURE WITH HIGHER ORDER DERIVATIVES IN REGULATOR Ryszard Gessing Politechnika Śl aska Instytut Automatyki, ul. Akademicka 16, 44-101 Gliwice, Poland, fax: +4832 372127, email: gessing@ia.gliwice.edu.pl Abstract: The modified feedback structure is proposed, in which the regulator uses some approximations of higher order derivatives to assure stability under high gain. Additionally the high gain and dynamics of the regulator are separated by means of the summing junction with set point signal placed between them. It is shown that this structure implements the model reference control with the reference model transfer function equal to the inverse of that of the regulator dynamics. It becomes that the proposed structure is insensitive to large and fast plant parameter changes and works well both with linear, nonlinear and nonstationary plants. Keywords: Feedback structure, linear, nonlinear, nonstationary, control saturation. 1. INTRODUCTION System structure with feedback is commonly used in automatic control practice. Very well known properties of the system with feedback such as: disturbance influence compensation, dynamics shaping, robustness, linearization etc. are described in standard books e.g. (Franklin et al., 1994; Goodwin et al., 2001; Slotine and Lee, 1991). One from the most important problem is the control system design in which the quality of the result is dependent among others on such constrains as dynamics of the plant resulting from placement of its zeros and poles, as relative order of its transfer function (TF), as existence of saturation of the plant input etc (Goodwin et al., 2001). It is known that for the relative order of the open loop TF equal to one, it may be possible to obtain stable closed loop (CL) system for the proportional (P) regulator with high gain. But for the relative order higher than two, even PID regulator with derivative (D) action for high gain gives usually unstable CL system. In (Vostrikov, 1990; Yurkevich, 1995) an interesting approach is proposed, in which regulator uses higher order derivatives. The higher order derivatives applied in regulator are usually not accepted by control community, because they gain noises and cause impulsive and nervous control signal. I think that now, the latter view should be revised. There are actuators which work well under nervous control signals and the systems with higher order derivatives in regulator have very good properties. In contrary to (Vostrikov, 1990; Yurkevich,1995), where a more complicated case of nonlinear nonstationary plants is considered, in the present paper a simple approach based on linear theory is applied to justify appearance of higher order derivatives in regulator. Owing to this, in our approach the highest order of regulator derivative is smaller by one from that appearing in (Vostrikov, 1990; Yurkevich, 1995). In the present paper, first, we admit the application of approximations of higher order derivatives in regulator so that the CL system may be stable under high gain. Second, we separate the gain k of the regulator from its dynamics Q(s) and shift the dynamics before the summing junction with the set point signal. It becomes that modified in this manner

the structure of feedback loop realizes the model reference control which is insensitive to large parameter changes of the plant. The structure may be applied to linear, nonlinear and nonstationary plants. The results described in the present paper were obtained from many performed simulations. Programs like SIMULINK create new possibilities in researches. From many researched examples, with accounting nonlinearities and other difficult models, we may create some hypothesis which are usually valid, but which can not be generally proved. On the other hand from strict mathematical considerations we may obtain theorems which are always valid under some made assumptions. However these assumptions are usually very restrictive from the point of view of applications. Therefore it seems that the hypothesis obtained on the basis of simulations which are only usually valid are more interesting from engineering point of view. The contribution of the paper is in proposing the modified structure of the feedback loop in which the higher order derivatives are applied in regulator and the high gain and dynamics of the regulator are separated by the summing junction with the set point signal; the implemented in this manner model reference control is usually insensitive to large plant parameter changes and is applicable to linear, nonlinear and nonstationary plants. 2. MODIFIED FEEDBACK STRUCTURE Consider the system with feedback shown in Fig. 1 composed of the linear or nonlinear plant and linear regulator described by transfer function. R(s) = kq(s) (1) where k is gain and Q(s) is an appropriate transfer function with Q(0) = 1. Usually Q(s) is an adequate polynomial. In comparison to usual control system in the considered system the gain k and dynamics Q(s) are separated so that in the feedback loop k appears after summing junction while Q(s) before it. Assume that it is possible to find the regulator (1) with sufficiently high gain k so that the closed loop (CL) system is stable and has sufficiently fast decay of transients. The choice of Q(s) and k will be the subject of further considerations. From the assumptions it results that for limited values of w, under a stepwise or rapid change of w, for almost all times it is Accounting that e 0 (2) Fig. 1. Modified feedback structure. E(s) = W(s) Q(s)Y (s) (3) where E(s), W(s) and Y (s) are Laplace transforms of e(t), w(t) and y(t), respectively, we obtain from (2) and (3) Y (s) W(s) 1 (4) Q(s) From (4) it is seen that the transfer function of the CL system relating Y (s) with W(s) is determined approximately by the dynamics Q(s) of the regulator. It is important that under made assumptions the dependence (4) is valid both for linear and nonlinear plants which may be even non stationary. There is other more strict interpretation of the considered system with modified feedback. Denote by Y m (s) Laplace transform of the output y m (t) for which the error E(s) is equal to zero. Then we have Y m (s) W(s) = 1 (5) Q(s) Thus the output y m (t) may be interpreted as the reference output and the dependence (5) determines the reference model. The system with modified feedback implements the model reference control in which the reference model TF is equal to the inverse of Q(s). Since for the stable CL system, with fast decay of transients, the error y(t) y m (t) is almost always very small therefore the formula (4) determines also the approximate transfer function relating Y (s) with W(s). Now the problem which must be solved is the choice of the transfer function Q(s) for which the CL system for sufficiently high gain is stable and has fast decaying transients. That is the CL system for sufficiently high gain should have appropriate stability degree λ = min i Res i, where s i, i = 1, 2,..., n are the poles of the CL system. Additionally, Q(s) must be implementable. This problem will be discussed in the next sections. 3. THE CASE OF LINEAR PLANT Consider now the case of linear plant described by a strictly proper transfer function G(s) = L(s) (6) M(s) where L(s) and M(s) are polynomials of m-th and n-th order, respectively, m < n. Let z i, i = 1, 2,..., m and p j, j = 1, 2,..., n be zeros and poles

of G(s), respectively. Assume that the zeros are minimum phase. One can notice that the CL system may have appropriate stability degree, under sufficiently high gain if the TF kq(s)g(s) of the open loop system has relative order equal to one. For the stable open loop system this may be justified using Nyquist criterion of stability. Really for appropriately chosen numerator polynomial of TF kq(s)g(s), the frequency response kq(jw)g(jw) also for high gain lies in the first and second negative and/or positive quadrants in Nyquist plane not closing to the critical point ( 1, j0). This is the result of the fact that under comparable by order poles and zeros of the TF kq(s)g(s) the decrease of phase coming from the denominator factors related with the poles are compensated by the increase of phase of the numerator factors related with zeros. More exactly it is seen for the case when m = 0, the plant (6) is stable and say p 1 is real (of course p 1 < 0). Then choosing the polynomial Q(s) so that is has the roots p 2, p 3,..., p n we obtain (after cancelling the same factors in numerator and denominator) the following TF kq(s)g(s) = kk 0 /(s p 1 ), where k 0 = L(0)( p 1 )/M(0). The frequency response of kk 0 /(jw p 1 ) also for very high gain k lies in the first negative quadrant of Nyquist plane. It should be stressed that the choice of (n 1)-th order polynomial Q(s) is not critical. It becomes that any (n 1)-th order polynomial Q(s), with the roots of the same magnitude as that of the corresponding poles p j, j = 1, 2,..., n, gives usually a positive solution. Choosing the polynomial Q(s) with faster roots (i.e having smaller real parts) we obtain the faster reference model (5) and in accordance with (4) faster output response y(t). It may be shown that the plant may have even one nonstable pole, or one pair of complex conjugate nonstable poles. When the remaining poles are stable then it is possible to find Q(s)which under very high gain k stabilizes CL system (Gessing, 2005). In the case when m > 0, Q(s) may take the form of (n m 1)-th order polynomial. However if this kind of Q(s) not assures appropriate stability degree under high gain k, then we may try to apply Q(s) in the form of improper TF with e.g (n m 1 + i)-th order polynomial in numerator and i-th order polynomial in denominator, i = 1, 2,..., m. The latter polynomial may contain e.g. the zeros of the plant TF (6) to cancel the corresponding factors, or the roots of this polynomial are close to these zeros. Thus, in the presented approach, when m = 0 then the reference model (5) and approximate model (4) of the CL system has order smaller by one from that of the plant. When m > 0 then the reference model (5) and approximate model (4) may have the order smaller by (m + 1) from that of the plant. The last question which must be solved is how to implement Q(s) in the form of polynomial Q(s) = p 0 s q + p 1 s q 1 +... + p q 1 s + 1 (7) In the following simulations we will use the approximation s ds (8) s + d with high d (d = 100, 200,...). It becomes that for sufficiently high d the approximation (8) is very good and it influences the frequency response kq(jω)g(jω) in significantly. 3.1 Example 1 Let the linear plant is described by TF 13 G(s) = s 3 + 5s 2 (9) + 8.5s + 13 One may check that there is great freedom in choosing a second order polynomial Q(s) for the plant (9) so that the CL system has appropriate stability degree under relatively high gain. In Fig. 2 the results of simulations performed in SIMULINK for Q(s) = (0.5s + 1) 2, k = 20 (10) and w(t) = 1(t 1), (1(t) = 0 for t < 0 and 1(t) = 1 for t 0), are shown. The polynomial Q(s) was implemented using approximation (8) with d = 100. Not to high gain k has been chosen to show that even then the approximation (4) holds. By the way the considered CL system has acceptable stability degree even for k = 100 and of course then the approximation (4) is better. Since the gain k is limited then the output y st in steady state takes the value k/(k+1) (as Q(0) = 1 and G(0) = 1). Therefore, one can suppose that the step response y(t) of the CL system is more close to that of the following corrected reference model Y m (s) W(s) = k 1 (11) 1 + k Q(s) than to that of the reference model (5). The results of simulations confirms this supposition. In Fig. 2 the step response y(t) of the CL system (9), (10) is compared with that y m (t) of the corrected reference model (11) for k = 20. It is seen that both the responses are almost non distinguishable. Next, the simulations were repeated for the same regulator (10) and different parameters of the plant (for some of them the plant itself was even unstable). In all the cases the step response y(t)

Fig. 2. Step responses for y and y m in Example 1. was almost the same as in Fig. 2. Thus, the system is very robust. This property may be justified using Nyquist stability criterion. The plant model may be interpreted as in serious connection of two first order lag elements first of them has the gain 4(1+ u ) dependent on u and second the time constant (1 + x 2 ) dependent on x 2. In Fig. 3 the two time responses y(t) of the plant (14), for u(t) = u 0 1(t 1), u 0 = 1 and u 0 = 2, are shown. It is seen from them that both the gain and time constant are increasing when u 0 is increasing. 4. THE CASE OF NONLINEAR PLANT Now, consider the nonlinear plant described by the state and output equation ẋ = f(x, u), y = g(x) (12) where x is n dimensional state vector, u and y are scalar input and output of the plant andf(.,.), g(.) are appropriate given functions. Assume that ẏ = g x f(x, u) = g 1(x) ÿ = g 1 x f(x, u) = g 2(x) y (l 1) = g l 2 x f(x, u) = g l 1(x) y (l) = g l 1 x f(x, u) = g l(x, u) (13) that is g i (x) for i = 1, 2,..., l 1 are independent directly upon u, while g l (x, u) depends directly upon u. The nonlinear plant fulfilling assumption (13) corresponds to the linear plant (6) with m = n l. Therefore in this case we may first try to apply Q(s) as polynomial of (l 1)-th order. If the trial is non successful then similarly as in the linear plant with m > 0 we may try to apply Q(s) in the form of improper TF with e.g. (l 1 + i)- th order polynomial in numerator and i-th order polynomial in denominator, i = 1, 2,..., n l. This suggestion results from similarity to the linear case which may be partially supported by the method of describing function (Slotine and Lee, 1991). The latter may be treated as approximate use of the method based on frequency responses to nonlinear systems. Fig. 3. Time responses y of the plant (14) Checking the conditions (13) one can find that l = 2. Therefore we choose Q(s) = 0.25s + 1, k = 20 (15) though, one can check that also now, there is great freedom in choosing a first order polynomial Q(s). In simulations performed in SIMULINK the polynomial Q(s) (15) was approximated using formula (8) with d = 100. In Fig. 4 the responses of the CL system (14), (15) to the set point changes w(t) = 4 1(t 1) and w(t) = 12 1(t 1) are compared with the responses of the reference model (5), (15) to the same set point changes. It is seen that the responses of the CL system and of the reference model are very close each to other. The values 4 and 12 in set point changes were chosen to make possibility of comparison with time responses of the plant (14) shown in Fig. 3 which have the same values of y in steady state. From comparison of the scales on time axis of Fig. 3 and 4, it results that the CL system is significantly faster. 4.1 Example 2 Consider the nonlinear plant of second order described by the state and output equations ẋ 1 = 2x 1 + 4(1 + u )u, 1 ẋ 2 = (x 1 x 2 ) 1 + x 2, y = x 2 (14) Fig. 4. Time responses of the CL system (14), (15) in Example 2. 5. THE CASE OF NONSTATIONARY PLANT This case occurs, for example, when the plants parameters are varying. In this case applying an

adaptive control may be needed. It is known that different adaptive control systems have one common property: the loop for parameters tuning is significantly slower than the loop for plant output control. In contrary to that, it will be shown in the next examples that the proposed here feedback structure may work well also in the case when the velocity of the plant parameter changes is comparable with that of the plant output signal generated in the CL system. 5.1 Example 3 Consider the third order plant described by the following equations ẋ 1 = 2x 1 + k 0 u, ẋ 2 = x 3, ẋ 3 = ω 2 nx 2 2δω n x 3 + ω 2 nx 1, y = x 2 (16) where ω n and δ are undamped frequency and damping ratio, which together with gain k 0 are varying in accordance with the functions ω n = 4 4 sin(2π 2.5t) δ = 4 8 sin(2π 1 t) (17) k 0 = 4 2square(2π 0.4t) where the function square( ) is determined as follows 1 for 0 < t < π square(ωt) = ω 1 for π ω < t < 2π (18) ω square[ω(t + 2π ω )] = square(ωt) Then the parameters ω n, δ and k 0 are varied in large bounds from ω nmin = 0, δ min = 4, k 0min = 2 to ω nmx = 8, δ mx = 12, k 0mx = 6 with frequencies 2.5Hz, 1Hz and 0.4Hz, respectively. The CL system structure shown in Fig. 1 was researched with Q(s) = (0.2s + 1) 2, k=50 (19) where Q(s) was approximated using (8) with d = 300, while the set point w = square(πt/5). In Fig. 5a the time response y of the CL system is compared with that y m of the reference model 1/(0.04s 2 + 0.4s + 1) for the same w. It is seen that both the responses are very close one to other, which means that the large and relatively fast parameter variations determined by (17) are well compensated. In Fig. 5b the control signal u and its strong compensation action is shown. In the vicinity of t = 0 and t = 5 during fast changes of y the high values of signal u appears (about ±11000 and +1400, respectively) and high frequency oscillations appear. It becomes that for d = 500 these oscillations disappear and the high values of signal u are decreased to +600 and +1200, respectively, while the response of y remains almost unchanged. Of course, in practice, the implementation of so high u is impossible. Therefore in the next section the case of the control signal saturation will be considered. Fig. 5. Time responses y, y m and u for Example 3. 6. ACCOUNT OF CONTROL SATURATION One may suppose that in the proposed system structure shown in Fig. 1, when a stepwise change of the set point w occurs and a fast change of the output y is needed, then some high temporary values of control must be applied, which may be non implementable. In implementable control system a saturation of the control signal must be accounted as shown in Fig. 2. Fig. 6. Account of control saturation The difference in comparison to the system from Fig. 1 is that in the system shown in Fig.2 the element with saturation described by u mx for v u mx u = v for u min v u mx (20) u min for v u min has been introduced. It seems that the existence of appropriate saturation of the control signal should not influence the property of proposed CL system very much. Really, if the system has appropriate stability degree then the error which is not close to zero may occur shortly after a rapid change of the set point and only then the control u may attend saturation. After that e decreases and fulfills (2) giving (4). Our supposition will be confirmed in the following example. 6.1 Example 4 Now the structure shown in Fig. 2 will be researched with the same plant (16) as in Example

3. The varying parameters are now described by the functions ω n = 3 2 sin(2π 0.5t) δ = 2 4 sin(2π 0.3t) (21) k 0 = 4 2square(2π 0.4t) Thus, the parameters ω n and δ have somewhat smaller regions of variations from ω nmin = 1, δ min = 2 to ω nmx = 5, δ mx = 6 with somewhat lower frequencies 0.5Hz and 0.3Hz, respectively, while k 0 is the same as in Example 3. This is caused by the fact that the control signal u has now saturation u min = 30 and u mx = 30, which somewhat decreases its compensation ability. For the variations (17) the system with considered saturation is unstable. The function Q(s), gain k and set point w are the same as in Example 3. In Fig. 6a the time response y of the CL system is compared with that y m of the reference model. It is seen that both the responses are even closer one to other that those in Example 3 (since, now the parameters variations are slower and smaller). In Fig. 6b the compensation action of control u is shown. Now the high frequency oscillations disappear. Though the ability for compensation of varying parameters is somewhat smaller but also now the large and relatively fast parameter changes are compensated. Fig. 7. Time responses y, y m and u for Example 4. 7. CONCLUSIONS From many performed simulations (only some of them are described in the present paper) it results that the proposed structure is usually insensitive to relatively large and fast plant parameter changes. It works well with linear and nonlinear plants. The structure applied to nonlinear plant of n-th order gives the linearized approximate model described by 1/Q(s). Therefore it may be used for linearization. Since the structure is very robust it becomes that it may be also used to non stationary linear and nonlinear plants. Taking into account the large possible plant parameter changes, it is shown that the structure may replace some adaptive control systems. In implementations all the calculations related with approximation of the regulator dynamics Q(s) may be performed using appropriate microprocessor. Therefore the regulators implemented in appropriate microprocessors have created possibility of utilization of the proposed approach. Approximations of higher order derivatives in Q(s) gain noises and cause rapid and nervous change of control u. Therefore the proposed solution may be applied to actuators which accept these changes. It concerns e.g. an electric motor as actuator, in which the signal u is an electric voltage and during design the dynamics of the motor is included to that of the plant. During implementation of control the dynamics of the motor partially smoothes the nervous changes of the signal u. ACKNOWLEDGMENT The paper was realized in 2005 and was partially supported by the Science Research Committee (KBN), grant No. 3 T11A 029 28. 8. REFERENCES Franklin G. F., J. D. Powell and A. Emami Naeini. (1994). Feedback Control of Dynamic Systems. Addison Wesley, N.Y. Gessing R. (2004). Whether Feedback Itself May Replace Adaptation. Proceedings of International Symposium Large Scale Systems- Theory and Applications LSS 2004, July 25-28 2004, Osaka, Japan, pp. 560-565. Goodwin G. C., S. F. Graebe and M. E. Salgado. (2001). Control Systems Design. Prentice Hall, N. J. Phillips, C.L. and R.D. Harbor (1996). Feedback Control Systems, Prentice Hall, N. J. Slotine J.J.E. and W. Li. (1991). Applied Nonlinear Control. Prentice Hall Int., Inc. Vostrikov, A. S. (1990). Synthesis of nonlinear systems by means of localization method. Novosybirsk, Novosibirsk State University, (in Russian). Yurkevich, V. D. (1995). Decoupling of Uncertain Continuous Systems: Dynamic Contraction Method. Proceedings of the 34-th Conference on Decision and Control. New Orleans, LA- December 1995.