A New Closed-loop Identification Method of a Hammerstein-type System with a Pure Time Delay

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1 A New Closed-loop Identification Method of a Hammerstein-tpe Sstem with a Pure Time Dela Edin Drljević a, Branislava Peruničić b, Željko Jurić c, Member of IEEE Abstract A procedure for the closed-loop identification of a class of Hammerstein tpe nonlinear plants with a pure time dela in the linear part, based on a generalization of the well-known Ziegler-Nichols' (ZN) experiment, is proposed. It provides a simultaneous estimation of the linear plant dnamic, input-output approximate model of the non-linear part and value of the time dela. As in the ZN method, onl a suitable controller is needed for experiments. Tuning of PID controller based on the obtained plant model is compared with PID tuning recommended b Ziegler-Nichols method and clear advantage of knowledge of the plant model is demonstrated. Kewords Closed-loop, Hammerstein, identification, nonlinear, pure time dela, oscillations, PID. T I. INTRODUCTION HE experimental methods for the identification of the transfer function of an unknown linear plant ma be classified into two categories: open-loop methods, which require the isolation of the plant from the control loop, and closed-loop methods, which work in a closed loop with a controller or some other device. The oldest closed-loop method is well-known Ziegler-Nichols' (ZN) method [1] that is mainl a PID controller tuning method, although it allows a rough estimation of the basic dnamic parameters of a plant. Åström and Hägglund proposed a similar method replacing the P controller with a rela [2]. It has an advantage that the amplitude of the oscillations can be controlled. However, it does not provide an more information than ordinar ZN method. For a better tuning of a PID controller, or for design and tuning of more advanced controllers, more information about the plant is usuall required. In order to solve this problem, new closed-loop methods are proposed in the past providing more detailed models of the plant. Among such methods are the Rela and Hsteresis method [2], the Two Channel Rela (TCR) method [4], the Auto Tune Variation (ATV) method [5], a Facult of Electrical Engineering, Universit of Sarajevo, Bosnia and Herzegovina (phone: , e.drljevic@lol.ba). b Facult of Electrical Engineering, Universit of Sarajevo, Bosnia and Herzegovina (phone: , brana_p@hotmail.com). c Facult of Science, Universit of Sarajevo, Bosnia and Herzegovina (phone: , zjuric@utic.net.ba). and its improvement, known as ATV+ method [6]. All these methods are based on relas, which can cause considerable errors due to unavoidable presence of the nonlinearit in the loop. Also, non-standard additional equipment is needed to implement the method. A new method for the closed-loop identification based on enforced oscillations that requires onl standard equipment in the control loop is described in [7] [10]. It does not use nonlinear elements at all, so there are no errors introduced b their presence. One important limitation of practicall all closed-loop identification methods is the assumption that the plant itself is linear. If the plant is not linear, these methods ma show completel incorrect results, especiall if the method is based on the frequenc response of the plant. This paper describes a frequenc-based closed-loop identification method that ma be applied for some plants having linear dnamic without finite zeros but with pure time dela, and a non-inertial nonlinearit at the plant input. The paper gives an elaborate approach to the frequenc-based closed-loop identification of such plants, and it is an extension of ideas presented in [11]. II. THE BASIC APPROACH FOR THE LINEAR CASE Although the aim of this paper is the identification of the nonlinear Hammerstein-tpe models with pure time dela, it is necessar to briefl recall the procedure for purel linear case first, as described in detail in [7]. Suppose that the plant controller has a transfer function G R (s, Λ) where Λ is a vector of tunable parameters. It will be assumed here that the controller is able to stabilize the plant. The plant itself ma be unstable. If G(s) is the transfer function of the plant, the transfer function of the closed-loop sstem with unit feedback is given b: W(s, Λ) = G(s) G R (s, Λ) / [1 + G(s) G R (s, Λ)] (1) To appl the method, the boundar of the closed-loop stabilit region must be reachable b changing the controller parameters. Here it will be also assumed that undamped oscillations are permitted in the sstem. The procedure when this assumption does not appl is described in detail in [7]. If sustained undamped oscillations with a frequenc ω = ω arise for the controller setting Λ = Λ, then W(s, Λ ) has a pole s = jω. Therefore, we can use the following equation:

2 G(jω ) = 1 / G R (jω, Λ ). (2) Since the value of the controller transfer function ma be calculated, the outcome of the experiment identifies uniquel a point on the Nquist curve G(jω ). It is eas to show that if Eq. (2) ma be satisfied for one setting Λ = Λ, then there exists an infinite set of pairs (ω k, Λ k) that also satisf Eq. (2). Hence, it is possible to identif as man points on the Nquist curve as the controller settings allow. For example, with a PI controller it is possible to identif points in the third quadrant, and another tpe of controllers ma be used to obtain points in other quadrants. Sometimes, the knowledge of few characteristic points on the Nquist curve is all that is needed for design for simple tpes of plants that do not demand a high qualit tuning. But, these points ma be also used for the parametric identification of the transfer function as well, assuming that a model of the transfer function is chosen in advance. Such procedures for some general models of transfer functions with various degrees of complexit are described in detail in [7] [10]. Here, it will be recalled the procedure for a simple but still a quite general case, when an all-poles transfer function of the plant ma be expressed as G(s) = 1 / P N (s, Π) assuming that P N (s, Π) is a polnomial given b P N (s, Π) = p 0 + p 1 s + p 2 s p N s N. (3) Here, Π is a vector of unknown model coefficients p i, i = 0..N. Under such assumptions, Eq. (2) becomes P N (jω, Π) = G R (jω, Λ ). (4) Eq. (4) is linear in all unknown coefficients that are represented b Π. In fact, after separating real and imaginar parts of Eq. (4), two independent equations are obtained. So, it is enough to collect N/2 + 1 experimental pairs (ω k, Λ k) to determine uniquel all unknown coefficients. In the next section, this approach will be extended to some nonlinear plants having the nonlinear element onl at its input. One such example is a linear plant with a nonlinear actuator. III. THE EXTENSION FOR NON-LINEAR CASES WITH PURE TIME DELAY It is known that all frequenc-based methods ma produce completel wrong results even in a presence of relativel weak nonlinearit. The situation is even worse if a pure time dela is introduced in the control loop. This is also true for the approach described in the previous paragraph. But, it ma be extended to some non-linear plants where pure time dela is present. It will be assumed that the plant ma be represented as a non-linear model of Hammerstein tpe, with a pure time dela i.e. b a non-inertial nonlinearit described b a input-output relation = f(x), followed b a pure time dela block e -τs, and a linear all-poles block with transfer function G(s). The nonlinearit ma be described b its describing function 2π 1 N(a) = π f ( asinu)sinudu (5) a 0 It is known that the describing function concept ma be applied if the plant input is zero (i.e. when the controller set point is zero, and the plant is not astatic), else the definition of N(a) should be slightl modified, as shown in [12]. Structure of the control loop mentioned above is shown on Fig Controller G R (s, Λ) a Plant Nonlinear N(a) Linear G(s) Fig. 1: Principal structure of the control loop If a controller setting Λ = Λ producing sustained periodic and nearl sinusoidal oscillations at the plant input with a frequenc of ω = ω and amplitude a = a, the principle of harmonic balance gives the following equation: 1 + N(a ) G(jω ) e -τjω G R (jω, Λ ) = 0. (6) Note, however, that Eq. (6) holds onl approximatel, because the principle of harmonic balance is just an approximate principle, and it works quite well onl under the assumption that the plant dnamic is strongl lowpass dnamic. It is assumed that the plant satisfies this condition also. From Eq. (6) it follows G(jω ) = 1 / [N(a ) G R (jω, Λ ) e -τjω ]. (7) If the plant nonlinearit f(x) is known in advance, e.g. when f(x) is a known input-output relation of an actuator, or when f(x) is obtained b some stead-state open-loop experiment, Eq. (7) ma be used to identif a point on a Nquist curve of the linear part of the plant. Namel, N(a ) ma be calculated from known f(x) using numerical integration. Afterwards, the obtained points ma be used for the parametric identification of the linear part of the plant and pure time dela. So, it is possible to perform the identification of the linear part and pure time dela of the plant without inserting additional elements in the loop to compensate the nonlinearit. Basicall, this is the same approach as used in rela-based identification methods (TCR, ATV, etc.), except that here the nonlinearit is not caused b inserting an extra equipment (e.g. a rela), but it is a part of the plant itself. However, Eq. (6) ma be used also for the parametric identification even when f(x) is not known in advance. In this case it is needed to choose in advance some model of the describing function N(a) = N(a, Θ) that depends of the unknown vector of parameters Θ. Under such assumptions, Eq. (6) can be written as: P(jω, Π) + N(a, Θ) G R (jω, Λ ) e -τjω = 0. (8) Using the obtained experimental data, it is possible to collect enough equations to calculate all unknown coefficients of unknown vectors Π and Θ, as well as value of pure time dela τ. Of course, these equations ma be hard to solve for arbitrar models of N(a). Also, e -τs A

3 the solving of these equations ma be quite complicated due to presence of unknown time dela τ. So, suppose for a moment that the value of τ is known in advance. Under such assumptions, the obtained equations ma be solved easil for some appropriate models of N(a). For example, it is ver reasonabl to choose the model N(a, Θ) = θ 1 N 1 (a) + θ 2 N 2 (a) θ K N K (a) (9) where N k (a), k = 1..K are some suitable pre-selected functions. Assuming that τ is known in advance, such model leads to the linear set of equations that ma be solved ver easil. Now, the first problem is how to find an adequate set of basis functions N k (a), k = 1..K. If the function f(x) is continuous, it ma be approximated reasonabl well with a polnomial in a range of interest, especiall if f(x) is a smooth function. Straightforward calculation shows that the corresponding describing function N(a) will be also a polnomial in a. Additionall, if the f(x) is an odd function, N(a) will be a polnomial of even order. So, it ma be assumed that in most cases, for the purpose of parametric identification, the basis functions N k (a) ma be taken as N k (a) = a 2(k 1), k = 1..K. The polnomial approximation of N(a) in a range of interest ma be suitable even if f(x) is not continuous, since N(a) is alwas a continuous function. Assuming the chosen polnomial model of N(a, Θ), Eq. (8) breaks into a real and an imaginar part providing in this wa two independent equations. Further, it is possible to set p 0 = 1 without an loss of the generalit, because the overall plant gain will be incorporated in N(a). In order to identif all unknown parameters it is needed to perform several experiments so that a determined of equations can be formed. The second problem is how to find the right value of τ, as it is not known in advance. To solve this problem, the obtained set of equations has to be solved for various preselected values of τ from a range of expected values of τ in steps of a suitable size. Such range ma be determined from some a priori knowledge of a plant. For example, a ZN experiment ma be used to estimate an upper bound of τ. Now, the question is how to determine when a right value of τ is found. If there is just a minimal amount of experimental data that is necessar to find all unknowns uniquel, there is no chance to find a right value of τ, because for each assumed value of τ, calculated unknowns will satisf Eq. (8) perfectl. Thus, it is necessar to collect more experimental data, and solve the obtained overdetermined sstem of equations in MLS sense. Then, the total mean square error (MSE) will be small onl if experimental data satisf (23) well in MLS sense, which ma be true onl when assumed value of τ is close to the true value of plant pure time dela. In ideal case, it would have minimum just for the exact value of τ. So, it is reasonable to choose the value of τ that gives the smallest value of MSE as the true value of τ. The method described above is illustrated using two MATLAB SIMULINK simulation examples. IV. EXAMPLES The presented theor will be demonstrated b finding the sstem models that include all-poles linear transfer function, the value of the pure time dela and the polnomial approximation of the nonlinearit. To demonstrate the usefulness of a knowledge of a valid plant model, a PID controller will be preliminar tuned at first using the Ziegler-Nichols recommendation. Afterwards, more advanced tuning based on Haalman method will be used, based on identified linear part of the plant, ignoring the nonlinearit. Finall, the additional fine tuning is performed based on the simulation of the identified plant, including the non-linear part. In all cases, the obtained responses will be compared to show the obvious advantage of acquired knowledge of the plant model. Note also that when the plant model is known, it is possible to use man other known techniques for PID loop optimization, so that the result of advanced controller tuning mentioned above represents onl one of man possible cases. In the first example, the dnamic of the plant is described b a second-order all-poles zero-tpe linear block with pure time dela G(s) = e 0.5 s / (8 s s + 1) (10) preceded b a non-inertial nonlinear block having the following input-output relation: =1.2 x 0.01 x 3. (11) This input-output relation, illustrated on Fig. 2, is intended to model a saturation caused b an actuator. Fig. 2: The quasi-saturation nonlinearit Strictl speaking, this is not a true saturation-tpe characteristic, because the gain d/dx becomes negative for x > and, in this region, the output falls when the input rises. Such behavior is not realistic in practice, so it will be assumed here that the value of the plant input will never allow such values of x. Under such an assumption, the used input-output relation ma model the saturation behavior reasonabl well. In this example, it is assumed that no information about plant behavior is available in advance, i.e. linear part of plant as well as nonlinearit and plant pure time dela are all the aims of the identification. x

4 The experiments are performed using an ideal PI controller with the transfer function G R (s, Λ) = λ 1 + λ 2 /s, Λ = {λ 1, λ 2 }. All four experiments are performed using zero set point of the controller, and the disturbance in the loop is introduced b making a short pulse change in the controller set point. The obtained data are given in Table 1. TABLE 1. Experiment λ 1 λ 2 T A I II III IV The parameter T is determined b inspecting a plant output and then ω is calculated as ω = 2π/T, while a is determined b inspection from the graph of the plant input. Assuming that the plant order is N = 2 and that the describing function of unknown nonlinearit is assumed in form N(a) = θ 1 + θ 2 a 2 + θ 3 a 4 (12) the following equations are obtained from Eq. (8) after separating its real and imaginar parts: (ω ) 3 p 1 = C R N(a ) (13) (ω ) 4 p 2 (ω ) 2 p 0 = C I N(a ) (14) where C R and C I are given in the following form: C R = λ 2 cos ω τ + λ 1 ω sin ω τ (15) C I = λ 1 ω cos ω τ λ 2 sin ω τ (16) Using the obtained results from four experiments, it is possible to form an over-determined set of equations. Solving this sstem of equations b MLS method gives the following results: p 2 = , p 1 = and p 0 = 1 for the linear part of plant; θ 1 = , θ 2 = and θ 3 = 0 for the coefficients of the describing function of the nonlinearit modeled b Eq. (12) and τ = 0.5 as the value of the pure time dela. The actual nonlinearit that corresponds to the obtained model of N(a) ma be calculated as = x x 3 using a simple numerical calculation. The power of the proposed identification method is apparent from the results presented above. Namel, both linear and nonlinear parts of plant are identified with high precision, while the pure time dela is identified exactl. The sstem response to the step function using three different tuning are compared on Fig. 3. First, an ideal PID controller is tuned using Ziegler-Nichols rules based on induced sustained oscillations with the P control. Next, the initial tuning is performed using the Haalman method, ignoring the nonlinear part of the sstem. The initial tuning is followed b a fine tuning using the obtained plant model. It is possible to do this due to fact that simulation model can be created using obtained results so the real plant is not involved in the process of the fine tuning. Finall, an additional nonlinear element (compensator) is added after the controller to compensate the nonlinearit, as shown on Fig. 4. The compensator is based on a piece-wise linear approximation of the inverse function x = f 1 ( ) of the identified model of the nonlinear part of the plant. This makes the overall sstem linear as much as the identification on the nonlinearit in the sstem is precise. The step response of the sstem with the compensator is presented b third curve on Fig. 3. The compensator additionall improves the sstem response, and thus advantage of the plant identification is clearl shown again. Fig. 3: Sstem response on step function (Ex. 1) It is evident from Fig. 3 that the response obtained b usage of ZN rules has an overshot of 60%. Note that in this case even the claim of ZN experiment, i.e. 25% overshot condition is not fulfilled. 0 Controller G R (s, Λ) Compensator Plant Fig. 4: Structure of control loop with compensator Even more noticeable results of the comparison are gained if we compare the response of the same sstem to the short disturbance in the controller set point. This is shown on Fig. 5. Fig. 5: Sstem response to the disturbance in set point (Ex. 1)

5 In the first example, the advantage of the compensator is not ver significant, since the nonlinearit is of the saturation tpe. In the second example, another nonlinearit is studied, having an increasing marginal gain. The plant dnamic is again described b a secondorder zero-tpe all-poles linear block with a pure time dela G(s) = e 0.35 s / (1.2 s s + 1) (17) preceded b an unknown non-inertial nonlinear block with the following input-output relation: =x x 3 (18) This input-output relation is shown on Fig. 6. is obviousl inferior in comparison with other two responses. Fig.6: The nonlinearit with an increasing marginal gain The experiments are again performed with an ideal PI controller, with the zero set point. The disturbance in the loop is introduced b making a short pulse change in the controller set point. The obtained experimental data are summarized in Table 2. TABLE 2. Experiment λ 1 λ 2 T a I II III IV It will be assumed again that the plant order is N = 2 and that describing function of unknown nonlinearit is modeled b Eq. (12). Using a MLS method to solve equations based on the data from these four experiments gives p 2 = , p 1 = and p 0 = 1 for the linear part of plant; θ 1 = , θ 2 = and θ 3 = 0 for the coefficients of the model of the describing function and τ = as the value of the pure time dela. The actual nonlinearit that corresponds to the obtained model of N(a) ma be expressed as = x x 3. Again, if we compare the sstem response to the step function with the PID controller tuned using ZN rules and then using Haalman method for PID tuning after linear part and pure time dela of plant is identified, with and without the compensation of the nonlinear part, the superiorit of the proposed method is quite clear. Such comparison is shown on Figure 7. In this case compensator does not affect significantl the overshoot, but the settling time is improved. The response of the closed loop when the controller is tuned b ZN rules in this case is not bad as much as in previous example but it x Fig. 7: Sstem response on step function (Ex. 2) To summarize the comparison of the sstem responses on different PID tuning techniques, the sstem response to the short disturbance at the controller set point is shown on Fig. 8. It is evident that PID controller tuned b ZN rules continues to oscillate long after the disturbance is gone, while both compensated and non-compensated case of PID tuning using the proposed method is able to stabilize sstem as soon as the disturbance is eliminated. Also, it is noticeable that the response of the sstem when the nonlinearit compensation is introduced gives smaller error in amplitude of the controller set point in comparison with the non-compensated case. Similarl, the settling time in the compensated case is better too. Fig. 8: Sstem response on disturbance in controller set point (Ex. 2) V. CONCLUSION A new method for the closed-loop identification of both the linear and the nonlinear part of a Hammersteintpe nonlinear model with a pure time dela is proposed. As presented before, the method assumes the following conditions: The linear part of the plant contains onl poles and a dela, and it behaves as a low-pass filter. The nonlinearit has a describing function in a form of a linear combination of some pre-selected basis functions.

6 The proposed method has some advantages in comparison to other closed-loop methods. It takes into the consideration unknown non-inertial nonlinearities in the plant, which usuall cause severe errors in the most of the other frequenc based methods. Its application usuall requires onl PI or PID controller, without an additional equipment. Also, the method itself uses onl linear blocks, so there are no unavoidable errors introduced b presence of non-linear elements, which exist in all closed-loop rela methods. However, the method requires more experimental runs than other methods to obtain reliable results. Due to the lack of space, this paper shows onl how to identif a plant whose dnamic ma be described using a simple model without finite zeros, and where the noninertial nonlinearit, if not known in advance, ma be described well using the polnomial describing function (in a range of interest). The described procedure ma possibl be generalized for more complex models with finite zeros. Such generalizations will be the topic of the follow-up papers. REFERENCES [1] J. G. Ziegler, N. B. Nichols, Optimal Settings for Automatic Controllers, Trans ASME, vol. 64, pp , [2] J. K. Åström, T. Hägglund, Automatic Tuning of Simple Regulators with Specifications on Phase and Amplitude Margins, Automatica 20, pp , [3] K. J. Åström, T. Hägglund, PID controllers, 2 nd ed., Instrument Societ of America, [4] M. Friman, K. V. Waller, A Two-Channel Rela for Autotuning, Ind. Eng. Chem. Res. 36(7), pp , [5] W. Li, E. Eskinat, W. Luben, An Improved Autotune Identification Method, Ind. Eng. Chem. Res. 30(7), pp , [6] C. Scali, G. Marchetti, D. Semino, Rela with Additional Dela for Identification and Autotuning of Completel Unknown Processes, Ind. Eng. Chem. Res. 38(5), pp , [7] Ž. Jurić, B. and Peruničić, A new method for the closed-loop identification based on the enforced oscillations, IASTED MIC proceedings, pp , Grindelwald, Switzerland, [8] Ž. Jurić, B. Peruničić, An Extension of the Ziegler-Nichols' method for Parametric Identification of Stanard Plants. IEEE MELECON 2004 proceedings, Dubrovnik, Croatia, [9] Ž. Jurić, B. Peruničić, A method for Parametric Closed-loop Identification of Plants with Finite Zeros,.IEEE MED'04 proceedings, Kuşadasi, Turke, [10] Ž. Jurić, B. Peruničić, A method for Closed-loop Identification of Plants with Unknown Dela, IFAC TDS'04 proceedings, Leuven, Belgium, [11] Ž. Jurić, B. Peruničić, B. Lačević, Closed-loop Identification of Some Linear Plants with a Static Nonlinearit, REDISCOVER 2004 proceedings, pp , Cavtat, Croatia, [12] Ž. Jurić, B. Peruničić, Simultaneous Closed-loop Identification of Nonlinear and Linear part of a Hammerstein-tpe Nonlinear Model, EUROCON 2005 proceedings, Belgrade, Serbia & Montenegro, 2005.

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