Anisochronic First Order Model and Its Application to Internal Model Control Vyhlídal, Tomáš Ing., Czech Technical University, Inst. of Instrumentation and Control Eng., Technická 4, Praha 6, 66 7, vyhlidal@student.fsid.cvut.cz Abstract: An original modelling approach for SISO process is presented based on first order system with delays in both input and state. By means of this anisochronic model it is possible truly to hit off properties of the plants which conventionally are used to be described by higher order models. With respect to its plain structure the anisochronic model is well suited to be applied in the framework of internal model control scheme (IMC). Despite its simplicity this controller has proved to be highly efficient and robust, more than PID. As regards implementation it can be easily implemented on PLC equipped with analog module. An implementation on the PLC Tecomat (product of TECO Kolín, CR) is presented. Keywords: time delay system, anisochronic model, internal model control, parameters identification, programmable controller Introduction Two approaches are generally used for the process description. The first is based on the good knowledge of inner structure of the process and physical laws that describe its behaviour. In this way created model mostly consists of set of non-linear differential equations or even partial differential equations. This model is necessary to simplify for practical use in classical control system theory due to desired linear system description. Finally, an identification of model parameters is performed. The other approach is based on universal model form and only its parameters are set or identified so that the model fits process properties. It is well known [] that most industrial processes can be adequately approximated by a first or higher - order rational function and a dead time as follows: K sτ p Gp( s) = e, T s + G p p K sτ p ( s) = e, T s + p where K is steady state gain, T p, T v are accumulative time constants and τ p, τ v are time delays. The advantage of the model () is its simplicity but sometimes it can describe the real process transient and frequency response only very roughly. The properties of the model (2) are much better in this aspect. But if a model-based method with high-order model is used the final controller is too complicated for practical use. () (2) 2 First order anisochronic model Instead of standard approximations (), (2) let us assume an alternative process model K sτ G( s) = e, Ts + e which corresponds to a differential equation with delayed argument dy( t) T + y( t φ ) = Ku( t τ ), dt (3) (4)
where u is system input and y is system output. The parameters of this model are closely related to basic properties of the process step response, namely K is process steady state gain T is time constant of the maximum slope φ serves to fit the actual inflex point position τ is pure delay (take-off delay) In the effect the feedback delay φ plays an analogous role as the n- th power in the denominator of (2). In combination with the input delay it allows to fit higher-order dynamics by means of the model (4). For the sake of model (3) generalization it is useful to introduce a reference model with the following relative parameters: K =, T =, φ = φ / T and τ = τ / T. sτ (5) G ( s) = e, s + e Significance of delay φ is demonstrated on frequency responses. For this purpose the model (5) is divided into two parts (6) Gi ( s) =, s + e sτ (7) G n = e, Frequency responses of the first part (6) are displayed in Fig. for several values of φ. It is obvious that increase of parameter φ causes an increase of G( jω) module and beginning from a certain value of φ the model (6) is even oscillatory. It is evident, mainly in Fig. 2 where are displayed frequency gain responses, that for φ =. 9 and φ =. 2 there exist an extremum point corresponding to the dominant natural frequency of G i ( jω). Specific significance of delay parameter φ in transfer functions (5) and (6) can be shown more distinctly by the solution of characteristic equation ( ) = + (8) M s s e =, The characteristic equation (8) is not algebraic, but transcendental, and in contrast to systems with polynomial characteristic function with n zeros it has infinitely many solutions (M(s) zeros). However only a few of M(s) zeros lying in the nearest position to the s-plane origin are of real significance in system behaviour [2], [3]. In any case, all of model poles (characteristic function zeros) unavoidably have to satisfy the stability condition, poles must not lie in the right-half complex domain. Pole placement of transfer functions (5) and (6) is shown in Fig. 3. Poles are marked with small black boxes. Trajectories of poles position (with respect to the continuous change of φ ) are marked with solid red curves, and some of poles with the same value of φ are joined with dashed blue lines. 2
φ Im -.5 -.3.6 -.5.9.5.5 Re Fig. Frequency responses of the model (6) φ.2 G(ω).9.6.3.. ω. Fig. 2 Frequency gain responses of the model (6) If φ = system has only one real pole. For increasing value of φ this first real pole moves to the left and in the opposite direction the second real pole is emerging near to the first one. Both these real poles are shown for φ =. 3 when both are dominant. Remaining poles are so far from the origin, that their influences to the system dynamic are negligible. For instance that φ =. 37 both real poles have the same position, system has double real pole. If φ >. 37 this pair is not furthermore real but complex conjugated (system becomes oscillatory) and with increasing value of φ it moves 3
back towards the imaginary axis. Also other system poles move in direction to the origin. The last important value of φ is φ =. 57, when the nearest complex conjugated pair to the origin lies in the imaginary axis and for higher values of φ system becomes unstable. 2.6.57 6 2 Im(s) 8.4 φ.9.2 4.3.37 - -8-6 -4 Re(s) Fig. 3 Pole positions of transfer functions (5) and (6) -2 Step responses of the investigated anisochronic model (5) are displayed in Fig. 4. These responses confirm how the increasing value of φ promotes the oscillatory system character. 4
y().8.6.4.2.8.6.2.9.6.3 φ =.4.2 2 4 6 8 t () Fig. 4 Step responses of the model (5) 3 Automatic tuning relay feedback A relay is used to provide on-off control of the process. For a large class of processes relay feedback gives an oscillating motion, the frequency of which is close to the process ultimate frequency ω u. The process ultimate gain is approximately given by Astrom a Hagglund as 4ua (9) ku =, π y a where u a a y a are the amplitudes of relay and process output, respectively [4], [5], [6]. From the limit cycle oscillation waveform, the process deadtime τ may be estimated from the time between an extremum in the variable to the preceding relay switch. When a biased relay feedback is used the steady state gain of the process can be computed from the following formula it u () y( t) dt K =, i =,2,3..., it u u( t) dt where T u is wave period [7]. As soon as ω u and k u are assessed from the relay feedback experiment the other parameters of the model (3) can be identified. Comparing the process and the model (3) at the ultimate frequencyω u = 2π / Tu, it follows that K () G( jω u ) = =, 2 2 T ω 2Tω sin( ω φ) ku u u u {sin( ωuτ )cos( ωuφ) + cos( ωuτ )[ Tω sin( ω φ)]} (2) u u arg[ G( jωu )] = arctan = π, {cos( ωuτ )cos( ωuφ) sin( ωuτ )[ Tωu sin( ωuφ)]} Simplifying () and (2), the remaining two parameters of model (3) are obtained as 5
(3) φ = { π arccos[ Kku cos( ωuτ )]}, ωu (4) T = { tan( ωuτ )cos( ωuφ) sin( ωuφ) }, ωu The above development can be summarised as the following identification procedure. Identification Procedure: The biased relay experiment is performed. The process input response u(t) and output y(t) are recorded, and the period and both amplitudes of y and u of the oscillation are measured. Step : Estimate τ from a time between an extreme of y(t) to the preceding relay switch. Step 2: Compute K from (). Step 3: Compute φ from (3). Step 4: Compute T from (4). 4 Application of the anisochronic model to internal model control Due to its simplicity and universality anisochronic first order model is applicable to internal model control (IMC) [8], [9]. A block diagram of the IMC idea is sketched in Fig. 5 w R () s P y G () s Fig. 5 Block diagram of internal model control With respect to feasibility of the expected results, the IMC method distinguishes between invertible and other dynamic properties of the process model factoring its transfer function as a product G( s) = Gi ( s) Gn ( s), (6) When the above introduced model (3) is used, noninvertible part is only delay τ, thus result of factorisation is following K (7) Gi ( s) =, Ts + e sτ Gn( s) = e (8) Controller R is according to the IMC design Ts + e (9) R ( s) = F( s) =, G ( s) K( T s + ) i f where F(s) is low-pass filter with time constant T f. The controller R and process model G can be represented by transfer function R ( s) Ts + e (2) R( s) = =, sτ R ( s) G( s) K( T s + e ) f 6
Control feedback arrangement with controller R then acquires a conventional structure. The obtained controller is very simple. In this point of view the controller is comparable with PID but with regards to its abilities, designed controller is better for its predictive properties. The controller has five parameters, but four of them are relay identification results. Only parameter T f is optional. Applying the controller (2) to the process (3) a feedback loop of quite simple and favourable dynamics is obtained. The closed loop transfer function is given by the product K Ts + e (2) sτ e sτ ( ) ( ) Ts + e K( T s + e ) sτ G s R s f e GL ( s) = = =, + G( s) R( s) K sτ Ts + e T f s + + e sτ Ts + e K( T s + e ) and its characteristic equation is free of any delay term. M ( s) = T f s + =, (22) f If the real process properties agree with the model (3) the control loop is not only always stable, but moreover its transients are without overshoot, given by the only one zero s = /T f. The IMC controller compensates most of undesirable effects of delays in the process response. In spite of encountering results (2) and (22) it should be noted that an exact agreement between the real process and its model (3) is impossible to achieve. For this case the parameter T f affects not only speed, but even robustness of the closed loop. Frequency gain responses of the IMC controller (6) are displayed in Fig. 6 for two values of T f. The remaining controller parameters are K =, T =, φ =. 37 and τ =. It is obvious that value of parameter T f affects gain responses mainly for higher frequencies. Whence it follows that closed loop transients are faster for higher values of T f, but less robust. Third frequency gain response in Fig. 6 belongs to the PID controller. Gain responses of IMC and PID are analogous for low frequencies, where the dominant controller action is integration. On the other hand for narrow frequency band behind the minimum of characteristics is dominant predictive part of the IMC (derivative part of the PID). The most important difference of IMC and PID is for high frequencies, when derivative behaviour is further dominant for PID while nearly proportional is dominant for IMC. Therefore the derivative part of PID have to be filtered for practical use while IMC need not additive filtering because it is predictive without derivation. R(jω) IMC, T f =. ideal PID IMC, T f = PID... ω Fig. 6 Frequency gain responses of controllers 7
5 Application examples Example. Suppose a process described by the fifth order model with time delay s (23) G( s) = e, 5 (5s + ) This process description may be substituted by a model of the form (3). The parameter values of such an approximation T =26 s, K =, φ =. 3s andτ = 2. 4 s are identified by the designed method, (see first part of Fig. 9). Step and frequency responses of process (23) and its model (3) with obtained parameters are displayed in Fig.7 and Fig. 8 respectively..2 y.8.6.4.2 2 4 6 8 t (s) Fig. 7 Step response of process example (23) and its model (3).4.2.5 Im.9.3.2 -.4.6 -.8.3 - -.8 -.4.4.8.2 Re Fig. 8 Frequency response of process example (23) and its model (3) 8
8 6 w 4 2 u y -2 2 3 4 5 6 7 8 9 t (s) Fig. 9 Process identification, set point response and disturbance rejection of the process model (23) Set point response and disturbance rejection of the process model (23) are displayed in second part of Fig.9 for T f = 2s. Both, system identification and IMC controller (uses identified parameters) are performed by one block in Simulink (S-function, Matlab). At first glance it is obvious that obtained results are very good. Model (3) fits very closely step and frequency responses and closed loop behaviour (see y in Fig. 9) is consistent with prescribed first order model (2). Different starts of the step responses are given by the first order model property and it is irremovable without change of model structure. Frequency responses correspond each other in first, second and third quadrants with only negligible deviations. The deflection of these characteristics for high frequencies is mostly unimportant from view of control synthesis. Example 2. The experimental set-up of the coupled-tanks system is shown in Fig.. The process consists of two tanks coupled to each other an orifice at the bottom of the tank wall. The inflow (control input) is supplied by a variable speed pump which pumps water from a reservoir into first tank through a long tube. The orifice between tanks allows the water to flow into second tank. In the experiments, it is desired to control the process with the voltage to drive the pump as input u and the water level in second tank as process output y. The result of process identification by biased relay ( ω u =. 92 s -, k u = 2.3, see Fig. ) is model (3) parameters K = 6, τ = 7 s, T = 6 s and φ = 8 s. For instance that optional parameter is chosen T f = 5 s the set point response and disturbance rejection of the process (performed by change of second tank outflow) are shown in Fig. 2. In this case the process approximation by model (3) is not such perfect like in the previous example. In spite of prescribed closed loop first order behaviour (2) the controlled variable overshoots its set point. Similar results, that model does not fit process perfectly, are likely to be expected for real processes, because most of them are non-linear and linear model is used. On the contrary the optional parameter T f can eliminate influence of these processmodel discrepancies on the process IMC control. 9
y d u Fig. Schematic of the coupled-tanks system 8 6 4 u y 2-2 -4-6 5 5 2 25 3 35 4 t (s) Fig. Biased relay oscillation of the second tank water level
4 2 y w 8 6 4 2-2 2 4 6 8 t (s) Fig. 2 Set point response and disturbance rejection of the process 6 Conclusion An implementation on the PLC Tecomat Anisochronic first order model with both input and state delay shows very good properties of anisochronic model formulation. In spite of its first order it is able to fit frequency properties of systems conventionally described by a high order model. The model is proper for description of non-oscillatory and oscillatory processes with arbitrary dead time. For model parameters identification may be used well known method based on closed loop with biased relay. With respect to its plain structure the anisochronic model is well suited to be applied in the framework of internal model control scheme (IMC). Despite its simplicity this controller has proved to be highly efficient and robust, more than PID. As regards implementation it can be easily implemented on PLC equipped with analog module. The author together with his supervisor Pavel Zítek and company TECO Kolín have designed a compact controller based on IMC scheme. It is implemented on both TECO systems - PLC Tecomat and Tecoreg, (see Fig. 3 and 4). The used algorithm offers a lot of function conventionally available in this class of controllers: optional sampling period saturation of both actuating signal u and its increment u antiwind-up optional control algorithm (IMC, PID, relay) bumpless switching of these algorithms bumpless switching from manual to automatic control and vice versa optional prescribed forms of set point change signal w (step, first order filter and ramp) optional dead band of actuating signal u and control error e fault detection of measurement failure biased relay identification of first order model automatic tuning of PID parameters
Fig. 3 PLC Tecomat TC6 Fig. 4 Controller Tecoreg TR3 7 References. LUYBEN, W.L LUYBEN, M.L., Essential of Process Control, McGraw-Hill, New York, 997. 2. ZÍTEK, P.:Time Delay Control Systems Using Functional State Models, CTU Reports. Czech Technical University in Prague, 998. 3. ZÍTEK, P. - VÍTEČEK, A.: Návrh řízení podsystémů se zpožděními a nelinearitami. ČVUT Praha, 999, 65 s. 4. ASTROM, K. J. HAGGLUND, T.: Automatic tuning of simple regulators with specifications on phase and amplitude margins, Automatica 2 (984), 645-65. 5. ASTROM, K. J. HAGGLUND, T.: New auto-tuning design. IFAC International Symposium on Adaptive Control of Chemical Processes, ADCHEM 88, Lyngby, Denmark, 988, 4. 6. MACHÁČEK, J.: Identifikace soustav pomocí nelinearity ve zpětné vazbě. 9 (998), 559-564 7. RAMIREZ, R. W.: The FFT Fundamentals and Concepts, Englewood Clifts, N.J., Prentice-Hall, 985. 8. MORARI, M. ZAFIRIOU, E.: Robust Process Control, Prentice-Hall, Englewood Clifts, N.J., 989 9. RIVERA, D. E. MORARI, M. SKOGESTAD, S.: Internal Model Control, 4. PID Controller Design, Industrial Eng. Process Design, Dev. 25, 252,986 2