Self-adapting control element
O United States Patent [1 1 [1 11 3,876,871 Sinner Apr. 8, 1975 [54] SELF-ADAPTING CONTROL ELEMENT 3.463.908 8/1969 Connelly 235/l50.l  
 - v Inventor: Edouard n Grenoble, a c 3.795.799 3/l974 Courtrol ..35/l50.l  
 [73] Assignee: gtlicifte Genet-all: degonstructions Primary Emmmer Eugne G. Botz g xfii g gg ggz Attorney. Agent, or F irmTCraig &amp; Antonelli [22] Filed: Oct. 1, 1973 [2l] Appl. No.: 402,422 [57] ABSTRACT Self-adapting control element which effects the con- [30] Foreign Application priority Data trolling in a closed loop, of processes having parame- Se t 79 1977 France 72 4737 ters which vary in a way which cannot be foreseen or p measured, while maintaining unvarying performances, 7 characterized in that it comprises a determining of the coefficients of the regulator (R) by a Calculator (C) i receiving a data item (5) concerning the required [58] Field of Search 235/150 1 l50 2 formances and the parameters (4) of an adjustable model and a determining, by that adjustable model, of [56] SEZLTFE SZ the variables of state (7) entering the regulator (R).  
 3.431422 3/1969 Gerhardt et al. 235/l5().l X 19 Claims, 16 Drawing Figures PATENTEDAPR 876,871  
 SHEET ElflF 13 FIG/I PATENTEDAPR 8 I975 SHEET CEUF 13 FIGZ PATENTEUAPR&#39; 81975 SHEET CBUF 13 M t 9 mw M2 Ml m mm d d PA ENTEBAFR 8i975 3,876,871 SHEET mm 13 L(s) p l I l ,l I I. l 1 I l l. I. I I I PATENTEDAPR 8 1975 SHEET CSUF 13 lllllll-Illlll.  
 FIG.6  
 PATENTEUAPR 19m SHEET OEUF 13 PATENTEUAPR&#39; 8 ms SHEET O7UF 13 Q A a i w m a fl w m m w m F1.  
 PATEWED W 19 SHEET UBUF 13 k To  3,876,871 SHEET USUF 13 PATENTEBAPR 8 I975 PATENTEUAPR 8 I975 SHEET llUF l3 PATENTEUAPR 81875 SHEET LEUF 13 PATENTEDAPR s 1975 sum 1 3 (1F 13 4:53. NM mm k &#34;no N; a 1 J 5 Q A? 3 cm u E w I l c m .muwn. mm m? 8 \ll\ NJ mm &amp; NE NW rll 1 III |||1 ba l+llllllllllljl AH NE E SELF-ADAPTING CONTROL ELEMENT The invention concerns an adaptive regulator for effecting closed loop controlling of processes whose dynamic parameters vary in an unforeseen way and/or are not measurable during operation, or whose dynamic parameters are unknown or difficult to determine.  
  Processes having parameters which are variable in time are found in numerous branches.  
  In aeronautics, parameters such as gain, clamping, proper frequency...etc., come into play in the various transfer functions which describe the dynamic reaction of an aircraft, vary with the altitude and the speed of the latter. The thrust of a jet engine varies with the fuel output, the speed of the aircraft, the degree of humidity of the air, the mechanical condition of the jet engine, BIC.  
  In the navy the parameters coming into play in the transfer function which links the course of a ship to the deflection of the rudder vary with the speed, the mass and the moment of inertia of the ship and with the propulsion force of the propellers.  
  ln electrotechnics, the dynamic reaction of a direct current engine varies with the inertia and friction transferred to the engine shaft, which may vary greatly with the load driven. This is more particularly the case with engines used for electrical traction and drive engines for rolling mills. The dynamic reaction of an asynchronous engine with a squirrel cage rotor depends, moreover, on the variable amplitude and the frequency of the voltage applied to the engine. In drive systems for lifts, freight elevators and telepherics, the dynamic reaction of the process varies greatly with the load transported.  
  ln chemical and thermal processes, the various transfers which describe the dynamic reaction of these systems vary generally with the operation point. It is often very difficult to known beforehand by calculation the parameters which characterize these transfers (examples: distilling columns, heat exchangers, boilers. etc.).  
  It is known that, to be able to produce a regulating system or servo-system in a closed loop which has satisfactory performance, the structure and the parameters of the process to be regulated or servo-controlled must be known. Structure means the structure of the mathematical model which is used for describing the class of dynamic reactions to which the process belongs (for example, the dynamic reaction of the process may be described by a differential linear equation or a transfer function of the first order with two parameters, the gain and the time constant or else by an equation with partial derivatives; or else by a linear or nonlinear differential equation of the second order having differences, etc.). Parameter means the coefficients which come into play in the mathematical relations used to describe the dynamic reaction of the process. With that data, in one phase, the type of regulator which is best suited to the type of process to be controlled is chosen (for example, regulator P having a proportional action; or else, regulator PI having proportional and integral actions; or else, regulator PID having proportional, integral and derived actions) and in a second phase, the parameters of the regulator are adjusted (for example, the coefficients of proportional, integral, or derivative actions) so that the process-i regulator&#34; combination in a closed loop has satisfactory performance with respect to certain requirements, for example, stationary error zero, response and exceeding time less than an imposed limit, etc.  
  Now, if the parameters of the process vary in time as in the case of the previously listed processes and if a conventional regulator with fixed adjustments is used, the performance of the closed loop becomes degraded as the difference between the true parameters of the process and the parameters ofthe process for which the regulator had been adjusted increases. It would therefore be necessary to readjust the coefficients of the regulator as a function of the new values of the parameters of the process to avoid that deterioration of the performance of the regulating loopor servo-control loop.  
  In certain applications the way of carrying out the readjusting automatically is known if the relation between the variations of the parameters of the process and the factors which cause&#39;these variations are known beforehand. It is then sufficient to measure these factors, to deduce therefrom, by means of an equation or of a table, the values of the parameters of the process, to determine the value of the readjustments necessary for the regulator and to effect those readjustments automatically. Thus, it has been seen that the parametersgain, damping, proper frequency coming into play in the various transfer functions which describe the dynamic reaction of an aircraft vary with the altitude and the speed of the latter. If the variation law is known, it is sufficient to measure, by means of suitable sensors, the altitude and the speed, to deduce therefrom, by means of an automatic calculating device, the parameters-gain, damping and proper frequency of the aircraft and to readjust, from this calculated data, the coefficients of the automatic pilot.  
  A monitoring of the parameters of the regulator is thus effected starting from the measurement of certain magnitudes characterizing the operation of the process, but the implementing of that solution requires a great knowledge beforehand of the process to be controlled. It is defective as soon as the parameters of the process vary in an unforeseen way which cannot be measured, directly or indirectly. Such regulators are therefore not self-adpating, inasmuch as they are that capable of adjusting their own coefficients themselves during operation. They need outside data concerning the dynamic parameters of the process.  
  The present invention aims at effecting the controlling in a closed loop of processes whose dynamic parameters vary in an unforeseen way which cannot be measured during operation, while maintaining the dynamic performance of the closed loop which do not vary despite the variations of the dynamic parameters of the process.  
  For that purpose, it provides a self-adapting control element for processes whose parameters are unknown or variable, comprising a regulator for variables of states having adjustable coefficients, which receives an input value and variables representative of the state of the process and supplies an action signal for the process, characterized in that it comprises a subassembly for the identification of the parameters and for observing the variable of states of the process, constituted by a model having adjustable parameters, placed parallel to the process and by an adapting mechanism modifying the parameters of the model until the difference between the outputs of the process and the model be minimum and a subassembly for calculating the coefficients of the regulator, which receives the parameters of the model and data on the required performance and determines the coefficients of the regulator, whereas the variables which are representative of the state of the process, received by the regulator, are, besides the output of the process, variables of state sampled at the model.  
  That self-adapting control element does not require the previous knowledge of the laws of variation of the parameters of the process. It uses exclusively the input and output signals of the process to be controlled and, starting from that data, it calculates the values of the parameters of the regulator which are necessary for maintaining the performances of the closed loop invariable despite the great variations in the parameters of the process.  
  That self-adapting control element may also bring about a gain in performance and in time in all cases where it is difficult to obtain beforehand a sufficiently exact knowledge of the dynamic parameters of the process, in order to be able to effect an adjustment of the regulator which guarantees good performance of the closed loop when an installation is put into service. By way of example, complex chemical and thermal systems the theoretical and experimental examination of which is generally difficult, may be quoted. In these conditions, the adjusting of the regulators of all the loops may require a great deal of time, until they are all properly adjusted. In this case, the use of self-adapting control elements ensures an automatic adjusting of the coefficients of the regulators to the unknown parameters of each regulating loop. The result thereof is an improvement in dynamic performance, as well as a gain in time and labor.  
  Once the self-adapting control element has adjusted its coefficients as a function of the unknown parameters of the process which it controls, the value of these coefficients may be measured and the value of the parameters of the process may be deduced therefrom. That self-adapting control element therefore has also applications in the branch of identification in true time of closed loop processes.  
  With reference to the accompanying diagrammatic figures, the implementing of the invention will be described in a general way, then in particular cases where the adjustable model is respectively of the first order, of the first order with a pure delay and of the second order.  
  FIG. 1 shows the general diagram of a self-adapting control element.  
  FIG. 2 shows the regulator of that element as well as its connections with the other subassemblies.  
 FIG. 3 shows the identification subassembly.  
  FIG. 4 shows the use of low-pass derivating filters in that subassembly.  
  FIG. 5 shows a possible embodiment of such low pass derivation filters.  
  FIGS. 6, 7, 8, and 9 relate to the case of an adjustable model of the first order and show, respectively, the regulator, the identification subassembly, the calculation subassembly, and the general embodiment of a selfadapting control element.  
  FIGS. 10, 11, and 12 relate to the case of an adjustable model of the first order with a pure delay and show respectively, the regulator, the identification subassembly and the general embodiment of a self-adapting control element.  
 FIGS. 13, 14, 15, and 16 relate to the case of an adjustable model of the second order and show, respectively, the regulator, the identification subassembly, the calculation subassembly and the general embodiment of a self-adapting control element.  
  The general diagram of the self-adapting control element is shown in FIG. 2. That element consists essentially of three subassemblies:  
 a regulator subassembly referenced R;  
 an identification subassembly referenced I and connected up in parallel with a process P; a regulator coefficients calculation subassembly referenced C.  
 The process P is controlled by the regulator subassembly R. This latter calculates starting from an input 1 and from the output signal 2 of the process (also called measurement&#34;), an action signal 3 such that the performance of the closed loop be satisfactory. For that purpose the coefficients of the regulator subassembly R are adjusted by the regulator coefficients calculation subassembly C, as a function of data on the dynamic reaction of the process, supplied by 4 by the identification&#34; subassembly I and the required performances in a closed loop, brought in at 5.  
  The calculation subassembly C sends out, at 6, signals for the adjusting of the regulator coefficients R and the latter receives at 7, from the subassembly I, variables which are representative of the state of the process, estimated by that subassembly I.  
  All the operations shown above are effected in real time; if the dynamic parameters of the process vary during time, the identification subassembly detects that variation. The regulator coefficients calculation&#34; subassembly determines, consequently, the new values to be given to the coefficients of the regulator. With these new values and the variables of state of the process supplied by the identification&#34; subassembly, the regulator subassembly calculates an action signal such that the dynamic performance of the closed loop practically does not vary.  
  The subassembly R, which is shown in greater detail in FIG. 2, has the function of calculating the magnitude of action 3 which is applied to the input of the process P, starting from the following magnitudes. The input 1 (also called reference), the measurement 2 (process output) and other data on the dynamic reaction of the process which reach it from the subsystems I and C.  
 This data will be, on the one hand, estimated variable of state&#34; 2 to 3,. (coming from the identification subassembly I) and on the other hand, the coefficients of the regulator k and k to k, (coming from the regulator coefficients calculation subassembly C).  
  The regulator used is based on linear control by reaction of state. The control law most generally calculated by the regulator subassembly R has the form:  
  (equ. 1)  
  That is, the action is a linear combination of the input R and of n variables x x .x, called variables of state.  
  To ensure a zero stationary error in the closed loop in permanent state, the following must be so:  
 k =l so that the equation becomes: action=k (R x, k x k x knxn) (equ. 2) The coefficients k,,, k k are given by multipliers and the above subtraction is carried out in a subtractor 8.  
 action koj (R x1 k2 x2 kixiknxn )(equ. 3).  
  The coefficients of the regulatorf subassembly R are the coefficients k,,, k k,- k,,. They are adjusted automatically so that the performance of. the closed loop be satisfactory, due to the regulator coefficients calculation&#34; subassembly C.  
  The variables of state x x .r,, are variables which describe completely the process to be controlled. They form an assembly of n numbers such that the knowledge of these 11 variables of state of the process at a given instant t and of the input of that process for all instants t 2 t,, is sufficient for determining the development of the process at all instants t 2 t In the self-adapting control element, it is presumed that only the first variable of state of the process is directly measurable (physically, it may represent the output magnitude 0,, of the process, also called measurement) instead of the&#39;real variables of state of higher order: x x,,, the estimated variables of state i 5-,, 2,, supplied by the identification subassembly are used.  
  The identification subassembly, shown in greater detail in FIG. 3, has the function of determining, in real time, an estimation of the dynamic parameters and an estimation of the variables of state which characterize, at a given instant, the dynamic reaction of the process to be controlled.  
  That subassembly is based on the principle of identification by adjustable model and is composed of four parts: a model having adjustable parameters (which will be called, more simply adjustable model), which is generally a simplified representation of the transfer function of the process to be identified; a comparator 11, which compares the output 2 of the process received at 12 with the output 13 of the adjustable model and which supplies a signal 14 proportional to the difference 6 between these two outputs; an adapting mechanism which modifies the adjustable parameters of the adjustable model until the difference between the outputs of the process and of the model be minimum, that adapting mechanism itself being composed of two main parts: a linear-processing block 15 and a nonlinear calculating block 16; a compensation circuit 17 for the continuous component which is present at the output of the process and/or of the model.  
  The identification subassembly l provides two types of data to the other subassembly of the self-adapting control element: an estimate of the parameters of the process, the parameters a, and b, of the process (for example, gain, time constant, damping, proper frequency, etc.) are estimated by that subsystem; they are available at the output of integrators 1 ,1 1,, 1m and l,,, l I, l, and are sent towards the regulator parameter claculation subassembly C; an estimate of the variables of state of the process, the variables of state of the process are also estimated by that subassembly; the variables of state estimated, 3 i 1%,,  
  1%,, are available at various points of the model 10 and are sent towards the regulating subassembly R.  
  The parameters of the process may be unknown previously, or may vary under the effect of the outside environment (for example, a modification in altitude or in speed of an aircraft makes the gain, the damping and the proper frequency of the latter vary; or else, the change in the operation point of a distilling column or of a heat exchanger makes the gain, the time constants and the pure delays which come into play in the various transfers linking the input and output magnitudes of those units.) An attempt is made to follow these variations in real time by adjusting the parameters (3, and 13,- of the adjustable model in such a way that the output of the model be at all times as near as possible to the output of the process.  
  The adjustable model may be described by an integrodifferential relation of the following type between the input p of the model and the output 6 M of the model: (so 51 D sno +131 D in D&#34;) e M (equ. 4  
 Where D=d/dt is the differential operator in the distributive direction.  
  The adjustable parameters &amp;, and I of the model are monitored through integrators l,- and l,- by the adapting mechanism. the practical effecting of that monitoring depends on the structure of the adjustable model and on its technological embodiment.  
  Often a simplified representation of the transfer function of the process to be identified will be adopted as the adjustable model (for example, a well-known approximation is the representation of a process of a high order by a model of the first order with a pure delay). The approximation of a process by a model of lower order has the advantage of a simplification of the structure of the adjustable model and of a simplification of the adapting mechanism.  
  The linear process block 15 receives the signal e, obtained at 14, indicating the difference between the output 6 P of the process and the output 0 M of the model, calculated by the comparator 11; it provides at the output 18, a signal 11:  
  The transfer function M(s) of that linear block must be chosen in such a way that the following transfer function: 1  
  The nonlinear calculating block 16 contains multiplication circuits, such as 19 and 20, which produce the signals referenced dd ldt and db /dt at the inputs of the integrators l, and I,- monitoring the adjustable parameters 5i and bi of the model:  
 (equ. 9)  
 where A,- and ,-are positive or negative constants. Physically, these constants represent gains of the adapting loops of the model, which may be distributed among several distinct physical elements in a given embodiment. g  
  The magnitudes d&#39;PM/dt are the derivatives of order 1&#39; of the output of the, adjustable model, accessible, at certain points of the latter if the mathematical representation of the adjustable model is well chosen. They represent the variables of state of the adjustable model; they will be called estimated variables of state because they constitute an estimate of the real variables of state of the process which are inaccessible or cannot be measured. They are used by the regulating subassembly for calculating the magnitude of action 3 which controls the process in a closed loop. The magnitude d p/dt are the derivatives of order i of the action signal 3.  
  The laws for the adjustment of the parameters a,- and b given by the equations 9 and 10 may be given as follows:  
  The function Sgn which comes into play in equations 11 to 16 is defined by the following table:  
 x Sgn (x) where 8 is a small quantity which may even be zero.  
  Instead of using, in the block 15, exact derivation operations which are always very difficult to carry out in practice, approximate derivatives (filtered&#34; derivatives) may be used; M (s) then has the form:  
 M(s)= (equ. I7)  
  The technique is simple and ensures acceptable operation of the identification subassembly l as long as hs 2.  
  Another solution to bypass exact derivation operations is the inserting of the low-pass derivating filters 21 and 22 between the outputs 2 and 13 of the process and of the model and the comparator 11, as well as a lowpass derivating filter 23 between the action signal 3 and the block 16, as shown in FIG. 4. The transfer function of those filters has been reference by L(s).  
  FIG. 5 shows an embodiment of such low-pass derivating filters. The filter consists of a subtractor 24 receiving data 0 at its positive input and of a sequence of n integrators 25 in series, whose outputs are returned through connections 26 to the negative input of the substractor 24 with gains referenced g, to gf. The outputs of the various integrators 25 give the signals which may be used as approximate derivatives of the input signal 6.  
  The compensation circuit 17 0f the continuous component has the function of ridding the difference coming from the comparator ll of the non-zero average value which may result from the fact that the output signals of the process (0 p) and the of model (0 have different continuous components. That circuit constitutes a low-pass filter of the first order, having a great time constant and it is composed of an integrator 27 and of an adder 28. It calculates the average value of the difference signal, reference Vcc, which is obtained at the output of the integrator 27 and is added in the adder 28 at the output of the model so that the output of the comparator ll be at a zero average value.  
  The regulator coefficients calculation assembly C has the function of calculating the value of the adjustable coefficients k,,, k k k k,, provided at 6 at the regulator assembly R starting from: the parameters (i 47,, (i 11,513 13 ,11, B which are supplied to it at 4 by the identification subassembly l and data on the dynamic reaction which it is required to confer on the closed loop, this data being given at 5 in the form of a series of coefficient: A A A,, A and B B B,-, B  
  In the general case, the structure of that subassembly C is very complex and cannot be shown easily in the form ofa block diagram. The method to be used for determining that structure in the general case will be given herebelow and examples of embodiment thereof will be given subsequently in the particular case of adjustable models of the first order, of the first order with a pure delay, and the second order.  
 The general method is as follows:  
  The adjustable model is chosen in the identification subassembly l in such a way that it demonstrates quite faithfully the dynamic reaction of the process to be controlled (for example, model of the first order with adjustable gain and time constant; or else, model of the second order with three adjustable coefficients, with idle time, etc.).  
  An analog, digital, hybrid, stochastic, etc., calculating structure which produces the adjustable model chosen, is selected.  
  For the requirements of the synthesis of the regulator coefficients calculation subassembly, the real process and the adjustable model chosen are taken for one another; it is therefore presumed that the regulator subassembly controls only a model and that the real process and the model are identical, this implying that the parameters of the process and of the model. are identical: a =(i a i a,=ci,-, a,,,=ii,,,; [2 b =l3 b;=b,-. b,,=b,, and that the variables of state of the process and of the model are identical: .r =.i,,  
  If it is presumed, moreover, that all the parameters are constant (that is, that the model is no longer adjustable but has constant coefficients), the system thus obtained, called reduced system is a linear system having constant coefficients.  
  The transfer function of the reduced system, called W (s), is calculated. The coefficients of the various terms in s of W,,,,, (s) are a function of the parameters a,=i1,- of the model and the coefficient k k k,- k of the regulator.  
  The required reaction in a closed loop is demonstrated in the form of another transfer function W (5), having the same structure as W (s) namely:  
  a k 5,, 5,5 B252 5,5 B,,s&#34;  
 (equ. 18)  
 symbols of W,,,;(s), on the one hand, and the (6,, in, k,-)  
 symbols of W,,.,, (s) on the other hand are deduced therefrom.  
  By solving that system of algebraic equations obtained at the preceding phase, the algebraic relations which give the coefficients ki of the regulator as a function of the (A B,, (2,, 13, symbols, that is, as a function of the required reaction in a closed loop (demonstrated by A,, B,) and of the estimated dynamic parameters (13,-, 13,) are deduced therefrom.  
  The algebraic relations obtained at the preceding phase in the form of analog, digital, hybrid, stochastic, etc., calculation circuits are demonstrated,.effecting, if need be, simplifications.  
  The contents of the regulator parameters calculation subassembly is thus completely determined. Its complexity depends on the order of the model which has been chosen.  
  A self-adapting control element having an adjustable model of the first order will now be described, examining successively the regulator R, identificationl and calculation C subassemblies and giving the overall performance of the control element. The subsystem R, shown in FIG. 6, has the function of calculating the magnitude of action 3 which is applied to the input of the process and of the adjustable model, starting from the following magnitudes: the input 1, the measurement (output of the process) 2, an estimated variable of state .i&#39; supplied by the adjustable model of the first order and of the coefficients To and k supplied by the subassembly C.  
  The input 1 may be calculated by an auxiliary circuit of the self-adapting control element (for example, a potentiometer supplied by the inside reference voltage of the device), or else it may come from the outside.  
  In this latter case, it may enter directly on the input subtractor (cascade regulating) or else it may enter on the input subtractor across an alternating potentiometer (cascade regulating with a proportional factor).  
  The subassembly R comprises the subtractor 8, a multiplier 29, the integrator 9 having an integration constant which may be monitored. The subtractor carries out the operation:  
 e R 6p k 2, (equl 9) (equ. 19)  
 a formula in which: e is the signal provided by the subtractor 8, R is the input I, 0p is the output of the process (measurement) 2, 3&#39; is the second variable of state, estimated by the adjustable model and k is the coefficient of the regulator.  
  The multiplier 29 carries out the multiplication of the estimated variable of state i coming from the identification subassembly I with the coefficient k; coming from the regulator coefficients calculation subassembly C.  
  The integrator 9 having an integration constant which may be monitored To carries out the following operation:  
  The constant of integration To of that integrator is adjusted by a signal coming from the subassembly C; it represents the second adjustable coefficient of the subassembly R.  
  It is, more particularly, possible to provide the integrator 9 having an integration constant which may be monitored in the two following ways. In the first way, a conventional integrator having a fixed integration constant is precoded by a voltage dividing circuit whose voltage ratio is monitored by the signal To coming from the subassembly C. In the second way, a digital counting and reverse counting counter preceded by a ramp type voltage-to-frequency converter whose threshold voltage is monitoredby the signal T coming from the subassembly C is used. The digital counting and reverse counting counter is followed by a digitalto-analog converter.  
  The identification subassembly I, which is shown in FIG. 7, has the function ofdetermining, in real time, an estimate of the parameters which are, here: d (gain of the adjustable model) and 6) (cut f out pulse, equal to the reverse of the time constant b, of the adjustable model) and an estimate of the second variable of state of the process, referenced 3&#39; The input signals are the action 3, equal to the input of the process and the measurement, equal to the output 2 of the process.  
  The subassembly is based on the principle of identification by a model and consists of the four parts already seen, adjustable model 10, comparator l1, adaptation mechanism, which comprises only the non-linear calculating block 16 in the particular case where the adjustable model is of the first order and compensation circuit 17 for the continuous component.  
  The adjustable model is described by the following relation:  
 Output of the adjustable model M P1 with the notations. D d/dt differential operator in the distributive direction and p input signal of the p, p VM (equ. 22)  
  Where VM represents a constant approximately equal to the average value of the action signal p.  
  The subtraction operation defined by the equation 22 and effected in a subtractor 30 makes it possible to improve, in certain cases, the rapidity of covergence and the precision of estimation of the parameter (Ly, it is not indispensable and the subtractor 30 can be left out.  
  The adjustable model consists of a multiplier 31 in series with a negative reaction chain comprising an integrator 32 and a multiplier 33, looping being affected on a subtractor 34.  
  The multiplier 31 effects the multiplying of the input signal in the adjustable model p by the gain [1 parameter.  
  The loop consisting of the multiplier 33 receiving the signal (Ill of the integrator 32 calculates the adjustable dynamic response expression tbl/w,+D, so that the result obtained at the output of the adjustable model is p a,,-G:l/6) +D That structure leads to the most simple producing of the regulator coefficients calculation subassembly C, but these operations may also be carried out in the reverse order. Firstly, passing the input signal of the adjustable model p through the adjustable dynamic response loop; then multiplying that signal by the adjustable gain ?1 this leading to an embodiment which is a little more complex, of the regulator coefficients calculation subassembly (an extra multiplying operation), but may have advantages in certain applications inasmuch as concerns the rapidity of convergence of the identification subsystem.  
  The comparator 11 carries out the subtraction between the output 2 of the process, on the one hand the output 13 of the model to which is added the continuous component Vcc coming from the integrator 27, on the other hand.  
  The adaptation mechanism modifies the adjustable parameters 6 and 6, until the difference 65 between the output 0,, of the process 2 and the output 0,, of the adjustable model 13 be minimum.  
  in the particular case where the adjustable model is of the first order, the linear processing block of the adaptation mechanism such as defined previously is reduced to a simple gain, so that the adaptation mechanism no longer comprises anything other than the nonlinear calculation block 16.  
  That non-linear calculation block receives, as input signals, the difference 6 coming from 14, the comparator, the input signal p of the adjustable model and a signal referenced Z, proportional to the derivative dBM/dt of the output 0 of the adjustable model:  
  The proportionality factor X being equal to T /al where T is the integration constant of the integrator 32 and (I), is the estimated parameter which comes into play as a multiplying factor in the multiplier 33.  
  The non-linear caluation block supplies as an output signal the estimated parameters :3 and d), of the adjustable model, these signals being calculated as will now be explained.  
  The calculation chain for the parameter 12,, consists of a circuit 35, an integrator 36 and a limiter 37.  
  The circuit 35, whose exact function will be specified further on, receives the signals p (input of the adjustable model) and 6 (difference between the output of the process and the output of the adjustable model). it sends out a signal referenced di /(it; that signal represents the variation of the parameter (i and is integrated by the integrator 36 to give, at the output of the integrator, the parameter [1 By means of the limiter 37, it is possible to restrict to domain of variation of the parameter t) 50 that t) Iim min 0 n lim man with the restriction: 6 0. lndeed, that circuit must limit the excursion of [1 to a strictly positive domain, the value Z1 =0 not being admitted, for it gives rise to a stable state where the idenfication subassembly reamins blocked without being able to leave therefrom.  
  The chain for the calculation of the parameter (0 comprises, likewise, a circuit 38, an integrator 39 and a limiter 40.  
  The circuit 38 (whose exact function will be specified further on) receives the signals Z and 6 (difference between the output of the method and the output of the adjustable model); it sends out a signal referenced dbl/dt; that signal represents the variation of the parameter a, and is integrated by the integrator 39 to give, at the output of that integrator, the parameter (2) By means of the limiter 40, the domain of variation of the parameter 6), may be restricted in such a way that 1 Iim min l 1 lim lllllJ where Ilm min The circuits 35 and 38 may carry out various operations according to the possible variants of embodiment.  
  in a first variant, the circuit 35 carries out the operation: d21 /dt= AO&#39; p -e(equ. 23) and the circuit 38 carries out the operation tion: dfi ldt &#39;Sgn (pl)&#39;e (equ. 27) and the circuit 38 carries out the operation: dw /dt e(equ. 28)  
  In a fourth variant, the circuit 35 carries out the operation: dzi ldt ,-Sgn(p,)-Sgn (e) (equ. 29) and the circuit 38 carries out the operation: dul ldt ;1. &#39;Sgn (Z)&#39;Sgn (e) (equ. 30).  
  The function symbol Sgn which comes into play in the expressions (equation 23) to (equation 30) is that which has been defined previously with respect to equations 11 to 16; A and u, are positive or negative constants; physically, they represent the gains of the adapting loops of the model, which may be distributed among several distinct physical elements in a given embodiment.  
  The continuous compensation circuit is analogous to that which had already been seen with respect to FIG. 3.  
  The regulator coefficients calculation subassembly, which is shown in FIG. 8, has the function of calculating the coefficients To and of the regulator starting