EP2558911A2

EP2558911A2 - Device and method for observing or controlling a nonlinear system

Info

Publication number: EP2558911A2
Application number: EP11719348A
Authority: EP
Inventors: Mathieu Grossard; Mehdi Boukallel
Original assignee: Commissariat a lEnergie Atomique CEA; Commissariat a lEnergie Atomique et aux Energies Alternatives CEA
Current assignee: Commissariat a lEnergie Atomique et aux Energies Alternatives CEA
Priority date: 2010-04-14
Filing date: 2011-04-11
Publication date: 2013-02-20
Also published as: FR2959030B1; WO2011128562A3; FR2959030A1; WO2011128562A2; US20130018612A1

Abstract

This device for observing a nonlinear system (20) comprises: - at least one sensor (14) for providing a measurement vector, each component of which is a measurable output parameter of the nonlinear system (20), - a state observer processor (12), based on a predetermined state representation of the nonlinear system (20), designed to provide an estimation of a state vector of the nonlinear system as a function of the measurement vector provided and of a control vector for the nonlinear system. Furthermore, the predetermined state representation comprising a model of nonlinearity of the system (20) in the form of a gain parameter, a component of the state vector is this gain parameter.

Description

Apparatus and method for observing or controlling a non-linear system

The present invention relates to a device for observing a non-linear system. It also relates to a control system including such an observation device, a corresponding method and its application to the observation of a non-linear Hammerstein type system.

By "nonlinear system" is meant a system that responds dynamically, but in a non-conforming manner to a linear model, to a command. This command takes the form of a set of parameters transmitted to the system which then constitute the components of a control vector. When this command can be provided by a processor transmitting electrical signals, the system is an automaton.

The nonlinear behavior of the system makes a state representation of this system more difficult to establish. Such a state representation must nevertheless make it possible to know the state of the system at any future time if one possesses the values of an initial state of the system and the command. The non-linearity of the system therefore makes its control more complex. In particular, the difficulty or impossibility of accurately modeling a nonlinear system by a state representation can cause a bias between the expected result of a command and the actual resultant state of the system.

One solution is to estimate this bias, to possibly correct it, using sensors to measure the state of the system at any time.

But the state of a system is not always directly measurable. On the basis of a state representation, it is then possible to define by extension a state observer which nevertheless makes it possible to estimate the state of the system from the state representation, of the control vector and a measurement vector of which each component is a measurable output parameter of the system. This is why a state observer is also called a "software sensor" of the system.

The invention thus applies more particularly to an observation device of a non-linear system, comprising:

at least one sensor for providing a measurement vector of which each component is a measurable output parameter of the nonlinear system, a state observer processor, based on a predetermined state representation of the nonlinear system, adapted to provide an estimate of a state vector of the nonlinear system according to the provided measurement vector and a control vector of the nonlinear system.

In particular, however, non-linear systems with hysteresis-type nonlinearity are particularly difficult to model by a state representation. More precisely, the Hammerstein-type systems, that is to say the systems that can be represented by a non-linear part, representing the static system, functionally in series with a linear part representing the dynamics of the system, are difficult to model, and even more, those whose static nonlinearity is of the hysteresis type.

Due to the presence of this hysteresis-type static nonlinearity, controlling such a system by known techniques is a difficult task in practice. Most existing solutions do not take into account the hysteresis phenomenon. This leads to degraded performance of the command. However, solutions for taking this phenomenon into account exist: some for control of the open-loop system (control without correction by a control law), others for closed-loop control (command controlled by a control law adjusted from measurements or estimates from sensors).

For example, numerical modeling techniques, such as the Preisach model, make it possible to describe hysteretic static nonlinearity, as described in the article by P. Ge et al, entitled "Generalized Preisach Model for Hysteresis nonlinearity of Piezoceramic Actuators", Precision Engineering, vol. 20, pp. 99-111, 1997. They can then be associated with an open-loop control method which consists in compensating, by inversion of the model, the phenomena of nonlinearity. But these models are generally complex and difficult to exploit for applications with real-time constraints.

More recently, it has been demonstrated in the article by UX Tan et al, entitled "Modeling Piezoelectric Actuator Hysteresis with Singularity Free Prandtl-lshlinskii Model," Proceedings of the IEEE International Conference on Robotics and Biomimetics, pages 251 -256, December 2006. , Kunming (China), that the Prandtl-lshlinskii operator, less complex than the Preisach model, has the advantage of having a more easily calculated computational inverse. This operator is better suited to a real-time implementation, for which the calculation time is always critical. But, as for most other models, some numerical cases that correspond to the singularities of the hysteresis curve prevent the mathematical existence of the inverse operator, or sometimes lead to poor numerical conditioning.

The hysteresis can also be taken into account by a generalized Maxwell model, by a set of differential equations describing the hysteresis curve (Bouc-Wen model for example), by polynomial or iterative approaches, using learning techniques by neural network, etc. But all these models or techniques still have a certain complexity and, in a more general context where the conditions of use are changing, the empirical estimation of the parametric variables of these models requires to multiply the identification protocols (ie ie shapes, amplitudes, signal frequencies), otherwise the model becomes unsuitable.

In general, a problem of precision of the model with respect to the reality of hysteresis always arises.

Finally, each of these models or techniques can also be used, for a closed-loop control, in the feedback loop to be taken into account in the control law by a control corrector. But despite the possible recourse to model reduction methods, these advanced closed-loop control techniques can sometimes lead to obtaining a high order corrector, which does not facilitate their implementation in a controller in the long run. with real-time constraints.

It may thus be desirable to provide an observation device that makes it possible to overcome at least some of the aforementioned problems and constraints.

The subject of the invention is therefore a device for observing a non-linear system, comprising:

at least one sensor for providing a measurement vector of which each component is a measurable output parameter of the nonlinear system,

a state observer processor, based on a predetermined state representation of the nonlinear system, adapted to provide an estimate of a state vector of the nonlinear system according to the provided measurement vector and a control vector of the nonlinear system, wherein, the predetermined state representation having a model of non-linearity of the system as a gain parameter, a component of the state vector is the gain parameter.

Thus, the state observer processor being designed to provide an estimate of the evolution of the state vector over time, thanks to the invention it also becomes able to provide an estimate of the temporal evolution of the parameter. gain that is integrated in the state representation as a parametric model of static nonlinearity. Indeed, the static nonlinearity can cleverly be simply seen at each moment as a static gain, the latter evolving moreover nonlinearly over time. Therefore, rather than a priori establishing a complex and approximate global model of this nonlinearity, the invention proposes to observe at each moment the corresponding gain parameter, that is to say the instantaneous effect of this non-linearity on the system. Since this model is simple, it allows the observation and therefore the monitoring of non-linearity in real time.

Optionally, the state observer processor is based on a state representation comprising a nonlinear portion modeled by the gain parameter representing the static of the nonlinear system, and a linear portion modeled by a predetermined transfer function. .

Also optionally, the state observer processor is an extended Kalman filter.

The invention also relates to a control system of a non-linear system comprising:

an observation device as defined above, and

a control vector corrector based on a control law comprising a gain adjusted over time as a function of the values taken by the gain parameter of the state vector.

Optionally, the corrector is of variable gain PID type.

Also optionally, the variable gain PID corrector has a proportional gain defined as inversely proportional to the gain parameter of the state vector.

The subject of the invention is also a method for observing a non-linear system comprising the following steps:

receiving, by a state observer processor based on a predetermined state representation of the nonlinear system, a vector for which each component is a measurable output parameter of the non-linear system,

estimating, by the state observer processor, a state vector of the non-linear system as a function of the measurement vector provided and a control vector of the non-linear system,

wherein, the predetermined state representation having a model of non-linearity of the system as a gain parameter, the estimation of the state vector includes an estimation of the gain parameter as a component of the vector of state.

The subject of the invention is also a method of controlling a non-linear system comprising the steps of an observation method as defined above and an updating step, by a control corrector based on a control law. of the non-linear system, a gain of this control corrector according to the values taken by the gain parameter of the state vector over time.

The subject of the invention is also the application of an observation or control method as defined above to the observation or control of a nonlinear Hammerstein static hysteresis type system, in particular a piezoelectric micro-actuator. or a robotic articulation with cable transmission manipulator arm.

Finally, the invention also relates to a computer program downloadable from a communication network and / or recorded on a computer readable medium and / or executable by a processor, including instructions for performing the steps of a observation or control method as defined above, when said program is executed on a computer.

The invention will be better understood with the aid of the description which follows, given solely by way of example and with reference to the appended drawings in which:

FIG. 1 schematically represents the general structure of a control system of a non-linear system, according to one embodiment of the invention,

FIG. 2 illustrates the successive steps of an observation and control method implemented by the system of FIG. 1,

FIG. 3 illustrates the use of the system of FIG. 1 for controlling a piezoelectric micro-actuator, FIG. 4 illustrates, by means of diagrams, an example of dependence of the static nonlinearity of the piezoelectric micro-actuator of FIG. 3 as a function of the frequency of excitatory signals,

FIG. 5 illustrates an exemplary Bode diagram of a transfer function capable of modeling the dynamic linearity of the piezoelectric micro-actuator of FIG. 3,

FIG. 6 illustrates by diagram a comparison of the performance of a control system according to the invention with respect to a conventional control system in the use of FIG. 3,

FIG. 7 diagrammatically illustrates the observed variations of a gain parameter modeling the static nonlinearity of the piezoelectric micro-actuator of FIG. 3 thanks to the implementation of an observation method according to the invention, and

FIG. 8 illustrates the use of the system of FIG. 1 for controlling a robotic articulation with a manipulator arm transmission,

The control system 10 diagrammatically shown in FIG. 1 comprises an observation device 12, 14, the latter comprising conventional software means (processor, dead and / or live memories, digital data transmission bus, etc.). implementation of a state observer processor 12 and at least one sensor 14 for providing at least one measured parameter Ym (t) to the state observer processor 12. It also includes a vector corrector 16 controller based on a control law and an input / output comparator 18 for supplying a servo signal e (t) to the corrector 16.

This control system 10 is connected to a non-linear system 20. In the remainder of the description, it is assumed that the non-linear system 20 is of the Hammerstein type, that is to say that its reaction to a command can be modeled by a nonlinear portion 22, representing the static system, operatively in series with a linear portion 24 representing the dynamics of the system.

More specifically, the control system 10 is designed so that its corrector 16 transmits a control vector U (t) at the input of the non-linear system 20. This transmission is for example electrical, the component or components of the control vector U (t). ) being composed of one or more exciter electrical signals of the system 20. In response to the excitation transmitted by the control vector U (t), the nonlinear system 20 reacts statically and dynamically and its state evolves. The sensor 14 of the control system 10 is then designed and placed to measure at least one output parameter of the nonlinear system. In a vector representation, the sensor 14 is placed at the measurable output Y (t) of the nonlinear system 20 to thereby provide a measurement vector Ym (t), each component of which is the measured value of a measurable output parameter of the nonlinear system 20.

Modeling of the non-linear system 20 and the evolution of its state can be performed on the basis of a predetermined state representation, in which the state of the non-linear system takes the form of a state vector X (t). As will be detailed below, according to the invention, the state vector X (t) defined to represent the state of the system 20 comprises, as a component, a gain parameter g modeling the static nonlinear part 22 This gain parameter g is not directly measurable at the output but can be estimated, via an estimation of the state vector X (t), thanks to the implementation of a software sensor constituted by the observer processor. state 12.

For this purpose, the state observer processor 12 receives as input the measurement vector Ym (t) coming from the sensor 14 and the control vector U (t) coming from the corrector 16 and provides, on the basis of the representation predetermined state of the non-linear system 20, an estimate X (t) of the state vector X (t). The operation of the state observer processor 12 will be detailed later on the basis of a non-limiting example of a generally used state observer of the extended Kalman filter type.

To allow closed-loop operation of the control system 10, the estimate X (t) provided by the state observer processor 12 is transmitted to the input / output comparator 18 which concretely compares the estimated value of least part of the components of the state vector X (t) to a reference signal E (t) to supply the servo signal e (t) to the corrector 16. In addition, according to the invention, the estimated values g in the course of time by the state observer processor 12 of the gain parameter g are provided to the controller 16 whose control law on which it is based has a gain adjusted over time according to these values.

We will now detail the modeling of the nonlinear system 20 and the establishment of its state representation. According to this modeling, and as previously indicated, a gain parameter g is integrated in the state representation as a parametric model of the static nonlinearity of the system 20. Indeed, from the moment when the nonlinear system 20 can be considered as having a separable static nonlinearity and dynamic linearity, which is particularly the case when it is of the Hammerstein type, including when its static nonlinearity is of the hysteresis type, the non-linearity can cleverly be seen at every moment as a static gain, the latter evolving otherwise non-linearly over time. The dynamics of the linear system 20 can then be represented independently of its statics by a transfer function F (s) in Laplace coordinates, that is to say the Laplace transform of the linear differential equation which represents the linear part between the input and the output of the system. For the sake of simplicity in the following description, the order of this transfer function F (s) is set arbitrarily to two. A higher order could be considered, but this would unnecessarily burden the calculations presented below. The following model is then obtained for the linear part 24 of the system 20:

F (s), where w _n and ξ represent the pulsation, respectively

natural and damping of the system 20.

By choosing to treat the static nonlinearity, in particular the hysteresis, as a simple static gain g likely to vary over time, the nonlinear system 20 finally boils down to a linearly modeled system resulting from the static gain serialization. variable g and transfer function F (s). The transfer function of the complete system 20 between the command u (s) and the measurable output y (s) is then written:

y (s) _ g

u (s) 1 ₂ 2 ^'

For the sake of simplicity again, it is assumed that the measurable output Y (t) of the system actually comprises only a position parameter x which is also a component of the state vector X (t), the latter further comprising the time derivative of this position x and the gain parameter g. More measurable parameters and / or components could be considered (eg an acceleration component) but this would unnecessarily burden the calculations presented below.

In state representation, that is to say in matrix form, the system 20 can then be defined by the following relation:

1

2, ie X = AX, where X is the state vector and A 0

state representation matrix of the system 20.

This relationship defines the temporal evolution of the system. It is not linear since it involves the command u in the matrix A.

The state observer processor 12, when it is of the extended Kalman filter type, can then be based on this state representation and defined by the following state observation model: x = [l 0 O ] or Y = CX, where Y is the measurement vector (here the position x) and C is the observation matrix. This relation defines the observation of the output of the system 20.

In order to be implemented in the state observer processor 12, this system model 20 needs to be discretized. For that, one uses for example the bilinear transformation of Tustin following:

2 z - 1

s =, where s is the Laplace variable and z is the Z transform

T _e z + l

of the sampled system. T _{e also} denotes the sampling period of the system.

After calculation (not detailed because classical), the discrete form of resentation and state observation is written:

^Y _{k + i} = ^H _{k +} x ^X k> WH

H _{¾ + 1} = [l 0 0]. These discrete recurrence equations are used to model the evolution of the state of the system at step (k + 1) knowing the state in step k and describe the fact that the output Y _{k + 1} is not otherwise that the position x in step k.

Once the discrete modeling is established, the extended Kalman observer is implemented in the processor 12 to estimate the state X (t) at each time step. Since the static nonlinearity, for example the static hysteresis, of the system 20 is present in the form of the gain g in the state, it is then possible to estimate the revolution of this static gain g over time by this observer.

The principle of extended Kalman filtering, which is also part of the general knowledge of those skilled in the art, is briefly recalled below.

On the basis of the previously defined model, noting w _k the state noise vector on the time interval [t _k , t _{k +} i], white, Gaussian, of zero mean and of covariance matrix Q _k = E [ w _k , w _k ^T ], noting also v _k the measurement noise vector at time t _k , white, Gaussian, of zero mean and covariance matrix

R _k = E [v _k , v _k ^T ], assuming known an initial state of the system, the extended Kalman filter implemented by the state observer processor 12 realizes the estimation of the state vector at each instant t _k by recurrence and more precisely by a calculation of prediction then a discounting calculation.

For this, the following notations are adopted:

the estimation of the state vector at time t _{k +} i is denoted by X _{k + k} after the prediction calculation but before updating by the knowledge of the measurement

^Ym k ₊ i>

the estimation of the state vector at time t _{k +} i is denoted X _{k + Uk + 1} after the discounting calculation,

the covariance matrix of the estimation error at time t _{k +} i is denoted P _{k + Uk} after the prediction calculation but before actualization by the knowledge of the measure Ym _{k + 1} ,

the covariance matrix of the estimation error at time t _{k +} i is denoted P _{k + Uk + 1} after the discounting calculation.

The prediction calculation is then done using the following equations:

X _{k + 1 / k} = f (X _{k / k} , U _k ), and

The discounting calculation is then done using the following equations: k + l ⁼ ^ k + l / k ^ k + l (H _{k + 1} P _{k + 1 / k} H _{k + 1} + R _{k + 1} ),

X k + l / k + l = X _{k +} uk + K _{k + l} (Ym _{k + l} - h (X _{k + k} , U _{k + l} )), and k + l / k + l - ( _x - ^K k + i ^H k ₊ i ^P k ₊ iik '° Ù H _{k + l} - H (X _{k + l / k} , U _{k + 1} )

It will be noted that, in accordance with the static gain model adopted to define the non-linearity of the system 20, F _k

are Jacobian matrices independent of the state vector X _k , which simplifies in practice the implementation in the state observer processor.

For a closed-loop operation of the control system 10, the corrector 16 must also be based on a control law incorporating the aforementioned model. Different types of control laws exist. PID-type control (for "proportional, integral, derivative") is quite suitable and is used widely in automatic mode. It is detailed below for purely illustrative purposes, knowing that other regulations can be applied within the scope of the invention.

According to the PID regulation and the state representation model retained in this embodiment, the corrector 16 respects the following canonical form in the Laplace domain:

K _PID (s) =, T _d , T _i and K represent the gains of the

regulation.

Thus, if it is desired to obtain a closed-loop transfer function, for the non-linear system corrected by this corrector 16, of the following form: where w ₀ and ξ _ΰ respectively represent the pulsation

natural and damping of the expected system, the gains of the regulation take the following values: 1 \ lw _n 1

and K =

N 2ξ ^ ₍ 2 ^ o

In particular, it is noted that the proportional gain K of the corrector 16 is a function of the static gain parameter g of the non-linear system 20. However, this gain parameter g varies because of the non-linearity of the system 20 and, as we have in view of the invention, the variations of this gain parameter can be estimated in real time by the state observer processor 12. We thus show that the PID regulation implemented by the corrector 16 can be made adaptive very simply by taking into account the gain parameter g in the calculation of its proportional gain K, without a complex global model of the static nonlinearity of the system 20 is necessary. This remains valid, of course, in the particular case where the static nonlinearity is of the hysteresis type.

The closed-loop operation of the control system 10 described above will now be detailed with reference to FIG. 2.

During a first servocontrol step 100, the comparator 18 of the control system 10 compares an instruction E (t) with the state of the known system 20 from a measurement of the output Y (t) of the system. . This state is for example derived from the estimate X (t) which is made by the state observer processor 12 as a function of the measurement Ym (t). The servo signal e (t) is supplied at the output of the comparator 18.

Then, during a step 102, the adaptive PID corrector 16 updates its proportional gain K as a function of the value g of the gain parameter g, this value g being supplied by the state observer processor 12 according to the measurement Ym (t) and the previous command, to provide a new control vector U (t).

In the next step 104, this control vector is applied to the non-linear system 20. In response, the non-linear system 20 evolves in a step 106.

Then, during a measurement step 108, the state observer processor 12 receives a new value of the measurement Ym (t) from the sensor 14. It derives a new estimate of the state vector. (Step 1 10), also according to the last value of the control vector, this estimate being supplied on the one hand to the comparator 18 so that it updates its knowledge of the state of the system 20 and on the other hand to the corrector 16 so that it adapts its regulation according to the new value of the gain parameter g.

Steps 100 to 110 are repeated in a loop. Note that they can be implemented as instructions of a computer program and be synchronized during their execution by the clock signal of a processor of the computer running the program.

As illustrated in FIG. 3, a possible application of the invention concerns the observation and possibly the closed-loop control of a piezoelectric micro-actuator 20A in the field of micro robotics.

Such an actuator is said to have a "unimorphous" structure, which is a structure commonly used in micro robotics. This means that when subjected to a voltage difference across its terminals, this type of actuator is able to produce a bending motion. By exploiting this movement, it is possible to grasp objects of very small sizes to carry out micro-manipulation tasks. A piezoelectric micro actuator behaves like a Hammerstein non-linear system with static hysteresis. According to the invention its control can be provided by the control system 10 described above, in which the static hysteresis is taken into account in the form of the gain parameter g integrated in the state vector estimated in real time by the state observer processor 12.

In the installation shown diagrammatically in FIG. 3, the piezoelectric micro-actuator 20A is fixed at one of its ends against a support 26. The sensor 14 of the control system 10, which is precisely in this application a large laser sensor resolution to be able to measure the micrometric displacements of the piezoelectric micro actuator 20A, is arranged so that the other end of the micro-actuator, free and mobile in bending, is in its laser emission beam.

The state observer processor 12, the corrector 16 and the comparator 18 are for example implemented in programmed form in a computer 28 which conventionally comprises at least one microprocessor, at least one RAM, ROM and / or another, and at least one data bus between the microprocessor and the memory. The computer supplies a control signal U (t), possibly amplified by an amplifier 30, to the piezoelectric micro-actuator 20A to which it is electrically connected. FIG. 4 illustrates the measured static hysteresis of the piezoelectric micro-actuator 20A and its dependence on the frequency of the control excitation signal: as a function of the value f of this frequency, 20 Hz, 300 Hz, 600 Hz or 900 Hz, it can it is indeed measured experimentally and reported on the diagrams of FIG. 4 that the bending D of the micro actuator as a function of the voltage U at its terminals describes four different hysteresis curves.

In accordance with the model defined previously, it is recalled that:

the static hysteresis of the piezoelectric micro-actuator 20A is modeled by the static gain parameter g that varies over time,

the dynamics of the piezoelectric micro-actuator 20A is modeled by a two-pole transfer function whose Bode diagram is illustrated in FIG.

In this FIG. 5, the curve A describes the values that can be found experimentally while the curve B results from a simulation. Note that these two curves are very close over a large frequency range so that the chosen dynamic linearity model can be considered as faithful to the actual dynamics of the piezoelectric micro actuator 20A.

Experimentally, for a sampling period T _e of 0.1 ms and adaptive PID corrector control gains 16 calculated for a zero static error, no setpoint overshoot and a rise time of 20 ms, a response of slave movement according to the curve B of the diagram of Figure 6, while the same PID corrector but not adaptive rendering by taking into account the gain parameter g would lead to the curve A of the same figure.

In particular, there are observed overshoots on the curve A while there is none on the curve B. This shows that the observation of the variations of the gain parameter g integrated in the state vector and their Consideration in regulation improves the control performance of Hammertein hysteresis systems.

FIG. 7 diagrammatically illustrates the corresponding variations of the gain parameter g as observed by the state observer processor 12.

Finally, as illustrated in FIG. 8, another conceivable application of the invention concerns the observation and closed-loop control of a robotic articulation 20B with a manipulator arm transmission.

The block diagram of FIG. 8 represents this hinge 20B in the form of a pulley 32 free to rotate about an axis 34. The rotation is controlled by a motor 36 which, via an arm 38, a worm 40, a cable 42 and a guide mechanism 44 and return 46 of the cable 42, causes the rotation 9 _a of the pulley 32. This chain of transmission of motion makes the cable transmission robotic articulation according to a Hammerstein type of system hysteresis.

In this application, the measured and observed displacement is rotation _9a of the pulley 32 about the axis 34. The extent of this rotation _9A is performed by an angle sensor 48 connected to a frame grabber 50. It is imperfect because of the existence of the hysteresis due mainly to the friction of the cable 42 on the pulley 32.

The control 9 _m carried out by the motor 36 is itself measured by an angular encoder 52 of the motor. This measurement is of good quality but the angular encoder 52 is remote from the articular transmission.

The aforementioned measurements are provided to the state observer processor 12 which, in this application also, provides in response an estimate of the variations of the gain parameter g defined to take into account the hysteresis phenomenon.

As in the previous application, the state observer processor 12, the corrector 16 and the comparator 18 are for example implemented in programmed form in a computer (not shown) which conventionally comprises at least one microprocessor, at least one memory of the RAM, ROM and / or other type, and at least one data transmission bus between the microprocessor and the memory.

Many other applications are possible as the field of automatic non-linear systems is vast. In particular, micromanipulation uses strongly non-linear actuators.

It is clear that a control system such as that described above makes it possible to identify, thanks to the state observer processor 12 associated with the sensor 14, and then possibly to take into account in the synthesis of a posterior adaptive control law. , thanks to the corrector 16, the phenomenon of static nonlinearity and in particular existing hysteresis for certain classes of nonlinear systems.

This identification is very simple since it is based on the observation of a state vector that includes a static gain parameter. It can therefore be performed in real time. But it is precisely this simple parameter of static gain observed at a course of time which accurately and comprehensively accounts for the non-linearity of the system, without the need to first establish a complete global model of this non-linearity.

By applying the result of this observation to a PID corrector, very commonly used in closed-loop control, to vary its proportional gain K, it is made adaptive and therefore efficient. The accuracy of the resulting command is improved.

For a hysteresis-type static nonlinearity system, the method presented above therefore has the major advantage of not requiring any particular prior modeling of the hysteresis. Indeed, most often, the shape of this hysteresis is a function of external and environmental parameters that are not controlled. It then becomes difficult to model all possible forms of hysteresis curves and to embark all of its possible forms in a control system. In addition, known strategies for modeling hysteresis are usually based on experimental testing of the system. However, as has been seen with reference to FIG. 4, the hysteresis is a phenomenon which depends in particular on the amplitudes and frequencies of the exciter signals at the input of the system. This therefore requires knowing in advance the range of amplitudes and frequencies of the signals that will be used a posteriori during the enslavement of the system. Conversely, the method described above does not require prior experimental protocols to characterize the hysteresis phenomenon. The modeling that is done is a simple gain observed by a state observer, for example an extended Kalman filter.

Note also that the invention is not limited to the embodiments and applications described above. It will be apparent to those skilled in the art that various modifications can be made to the embodiments and applications described above, in the light of the teaching just disclosed.

In particular, in the embodiments and applications described above, it has been made the choice of a PID type corrector, but some may prefer to adopt other methodologies for controlling the control. The principle of modeling and observation of the non-linearity of the controlled system presented previously remains compatible with other control laws. Finally, in the following claims, the terms used are not to be construed as limiting the claims to the embodiments set forth in this specification, but should be interpreted to include all the equivalents that the claims are intended to cover because of their formulation and whose prediction is within the reach of the person skilled in the art by applying his general knowledge to the implementation of the teaching which has just been disclosed to him.

Claims

1. Apparatus for observing a non-linear system (20; 20A; 20B), comprising:

at least one sensor (14) for providing a measurement vector of which each component is a measurable output parameter of the non-linear system (20; 20A; 20B),

a state observer processor (12), based on a predetermined state representation of the nonlinear system (20; 20A; 20B), adapted to provide an estimate of a nonlinear system state vector based on the provided measurement vector and a control vector of the nonlinear system,

characterized in that, the predetermined state representation comprising a model of non-linearity of the system (20; 20A; 20B) in the form of a gain parameter, a component of the state vector is the gain parameter.

An observation device according to claim 1, wherein the state observer processor (12) is based on a state representation comprising a non-linear portion (22) modeled by the gain parameter representing the static state of the state. non-linear system (20; 20A; 20B); and a linear portion (24) modeled by a predetermined transfer function.

An observation device according to claim 1 or 2, wherein the state observer processor (12) is an extended Kalman filter.

4. System (10) for controlling a non-linear system (20; 20A; 20B) comprising:

an observation device (12, 14) according to any one of claims 1 to 3, and

a control vector corrector (16) based on a control law having a gain adjusted over time according to the values taken by the gain parameter of the state vector.

5. Control system according to claim 4, wherein the corrector (16) is variable gain PID type.

The control system of claim 5, wherein the variable gain PID corrector has a proportional gain defined as inversely proportional to the gain parameter of the state vector.

7. A method of observing a nonlinear system (20; 20A; 20B) comprising the steps of:

receiving (108), by a state observer processor (12) based on a predetermined state representation of the non-linear system (20; 20A; 20B), a measurement vector of which each component is a parameter of measurable output of the nonlinear system, estimation (1 10), by the state observer processor (12), of a state vector of the nonlinear system according to the provided measurement vector and a vector of control of the non-linear system, characterized in that, the predetermined state representation comprising a model of non-linearity of the system (20; 20A; 20B) in the form of a gain parameter, the estimate (1 10) of the state vector includes an estimate of the gain parameter as a component of the state vector.

8. A control method of a non-linear system comprising the steps of an observation method according to claim 7 and an updating step (102), by a control corrector (16) based on a control law of the nonlinear system (20; 20A; 20B), a gain of this control corrector as a function of the values taken by the gain parameter of the state vector over time.

9. Application of an observation or control method according to claim 7 or 8 to the observation or the control of a nonlinear system of Hammerstein type with static hysteresis (20; 20A; 20B), in particular a microphone piezoelectric actuator (20A) or robotic articulation with manipulator arm transmission (20B).

Computer program downloadable from a communication network and / or recorded on a computer-readable and / or executable medium by a processor, characterized in that it comprises instructions for executing the steps of a method of observation or command according to claim 7 or 8, when said program is executed on a computer.