Control of a hydraulic system by means of a fuzzy approach

Non linear models can be represented conveniently by Takagi-Sugeno fuzzy models when nonlinearities are bounded. This approach uses a collection of linear models which are interpolated by non linear functions. Then the global control law is the interpolation by the same functions of each feedback associated to each linear model. A Lyapunov approach enables to compute these feedback gains. The number of linear models depends directly on the number of nonlinearities the system has. The more models there are, the more difficult it is to guarantee the stability of the closed loop. This paper proposes a method to reduce the number of linear models by assuming a number of nonlinearities considered as uncertainties and to guarantee the global exponential stability of the system. This method is applied on a hydraulic system.


Introduction
Takagi-Sugeno fuzzy models enable to represent precisely a wide class of nonlinear systems in a bounded domain of state variables.In brief, each nonlinearity is replaced by two linear models.Then these latters are interpolated by nonlinear functions to get back the initial nonlinear system [1].To guarantee the stability of the closed loop, a PDC (Parallel Distributed and Compensation) control law is often used: it assumes to link each linear model to a linear feedback.Using a Lyapunov approach, with a quadratic function, the whole control problem can be casted as Linear Matrix Inequalities problem (LMI) [1][2][3].We can remark that if the interpolating functions depend on non-measurable outputs, the problem is much more complex.In the opposite case, the observer and the controller can be designed separately [4].
Since the number of linear models is equal to 2 n with n the number of nonlinearities, it is clear that the more nonlinearities the system has, the more difficult it is to get the feedback gains because the number of LMI conditions will be so important that the solvers do not find a solution.For example in [5], there was a hydraulic system aimed at mixing two liquids into one tank.There were three nonlinearities, and feedbacks gains could be computed to guarantee the stability of the closed loop.However this approach was applied to a system based on three liquids to mix, but cannot be applied on a system based on four nonlinearities and the stability of the closed loop cannot be proved with the same LMI problem as previously.
This paper proposes to interpret some nonlinearities of the model as uncertainties and to guarantee the global exponential stability of the system.
However, the number of linear models is reduced to a half each time a non-linearity is suppressed.We hope to get a solution for that.
We will also show that this approach gives interesting results for our hydraulic system.
Section two recalls some basic principles of Takagi-Sugeno fuzzy models.Basic conditions to guarantee the stability of the closed loop will be evoked, as well as a few matrix properties used in the rest of the paper.
Section three presents the control problem when uncertainties in the model are taken into account.
Finally, section four exposes the results about our hydraulic system.

Fuzzy model
We begin by recalling what a Takagi-Sugeno (TS) model [2,6,7] is.Let be a nonlinear system: A Takagi-Sugeno (TS) model is obtained, if it's possible, to find r partial models of the type: such that (1) is equal to : Let us study the case of a single non linearity.To obtain a TS fuzzy model from a nonlinear model, we can use the following properties for every bounded function   Hence, we obtain 2 n r  partial models for a system having n nonlinearities.For example: let be the nonlinear model ( ) sin( ( )) x t x t  , then the nonlinear function considered is sin( ) () Then, the two rules TS model is:

Stabilization conditions
When dealing with TS fuzzy models the classical control law is PDC [8] : Determining the control law consists in finding r control gains i F .For continuous fuzzy models, a quadratic Lyapunov function : x t Px t  , with 0 P  , is mainly used.This lead to a LMI problem which enables to compute the gains i F [1][2][3][8][9][10]. Let us show this: Vx .This leads to: with : 1 XP   and 1 ii M F P   .This last inequation (12) is verified if we have: ,, with These conditions are very conservative.Current researches try to lower more and more the conservativity of the LMIs.The next section will present a series of matrix properties in order to reduce this conservativity.These properties can make conditions of stability into LMI if we obtain BMI (bilinear inequality of matrix) and/or maximize or minimize the number of free decisions variables to facilitate finding solution.
Please also note that in this paper only quadratic Lyapunov functions are used.They are currently researches trying to apply non quadratic Lyapunov functions for continuous fuzzy models.It is often required to use hypothesis about the derivative of the membership functions that must be checked a posteriori during experiments or simulations.Indeed terms   i hz appears in P .
But it is not possible to enforce such constraints in the control law through LMIs.This questions the whole validity of the approach because a valid simulation does not imply that the same validity would be obtained for other initial conditions.A thorough discussion on this issue can be found in [11], where the authors reject the whole approach due to this problem.Moreover non quadratic Lyapunov functions are often associated to bilinear matrix inequalities (BMIs).

Properties
In this section, some properties about matrices are given.
Lemma 1. (Schur's complement [6]) Matrices X , Y and R being of appropriate sizes, we have : ((*) represents all terms induced by symmetry in a symmetric matrix).Lemma 2. [12]: The two next problems are equivalent: (i) Find 0 P  , such that: Lemma 3. (Relaxation [5,13,14]) Let be the matrices ij  and the condition: Please remark that more advanced techniques have been proposed to obtain from (19) a finite sets of LMIs that are less restrictive than (20)-( 22), [10,[15][16][17].They can be applied in our approach, since the ij  terms are not modified.
However the associated LMIs may become too large so that the solvers are unable to find solutions.

Model
This approach consists of writing some nonlinearities as uncertainties [13,18]. With: Eb are constant matrixes and: .1 Let us consider the following augmented state vector , with c y consign on the outputs.
We suppose that the state is entirely accessible.In the case of the addition of an observer, there is no guarantee for convergence of the closed loop.We use an integral action in the forward path as indicated in figure 1.

Robust stabilization
The Lyapunov function is:   T V x x Px  A decay rate  is chosen, so we have the condition: The above inequality implies that the origin is globally exponentially stable. Indeed, Using the fact that: After pre and post multiplication by , we obtain : we have : By using this property we obtain with 0 By construction: Or equivalently: So the use of Schur's complement gives : The next matrix defined Theorem 2. Let be the uncertain TS model ( 28), the PDC law (29) and the ij  defined in (45).If there exist a matrix 0 X  , a scalars , ij  such that conditions (20), ( 21) and ( 22) are verified, then the closed loop is globally exponentially stable.
Proposition.Having Theorem 2 verified then, we can obtain an estimation as in (35), hence the system (28) is globally exponentially stable.
Proof.The application of these results will be done on a hydraulic system.

System
The system that we are studying is represented in The pumps supply the tanks respectively by variable flows.We suppose that these flows are proportional to the voltages applied to the Motopumps.The dynamic of these actioners are neglected.

Nonlinear model
The level of each tank depends on the difference of the liquid flowing into and off a tank.
Depending on the cross-cut of a valve i S that can be considered constant, the amount of liquid flowing off by an outlet valve   si qt according to Torricelli's law is :

 
1, , 1 ik  can be described by : ( ) ( ) q ek-1 (t) q s1 (t) From ( 46) and (47) we obtain the following differential equation: Based on these physical relations we get the k state equations to describe the nonlinear dynamic behaviour of the plant : The goal is to control the flow With matrices for the first model : .To avoid the problem of exponential augmentation of the number of rules according to the number of tanks we propose to reduce this number of rules by using a TS model with uncertainties.
We have said in the introduction that it's not possible to get a solution to LMI problem with four tanks.
We propose therefore to consider a single non linearity,

 
i nt, all others being considered as uncertainties.
We can write for the first tank: With the notation : Then, we consider the bounded functions


Finally we obtain : From the expression of uncertainties (54) we can write In the same way, from the expression (55), we can write Model TS with uncertainty is now explicit.Let us note that it is the "minimum" number of rules which prevailed with its obtaining, and which there exists obviously of other possible representatives of the nonlinear model (49).Now we will compare this method with a classic method of linearization.For example we use the Taylor linearization.

Taylor linearization:
We recall that with : .( , ) Taylor series can be writted as : ,, is is is n u y are the set points.
We also notice that for the change of set point for   1 yt there is an important error in the relative concentrations   2 yt .This corresponds to a transitional regime and the liquid produced in this lap of time will not be used.
To gain place, levels 1, 3 and 4 are suppressed.Level 2 are showed only with control variables, Figure 3.The comparison with Taylor linearization shows that the latter does not give satisfied result: on Figure 5,   1 yt is not better than in our approach.

Conclusion
The models of Takagi-Sugeno propose a rigorous approach allowing the treatment of nonlinear systems.In fact they allow their decomposition into linear systems, interpolated by nonlinear functions.It is possible with this approach to write a nonlinear system exactly as the convex sum of linear systems, according to weight determined in advance.The number of partial linear models determined from the number of nonlinearities is 2 k for a system having k nonlinearities.
Nevertheless, the study of stability of TS fuzzy models will be difficult in the case where the number of nonlinearities is important.This study proposed a methodology to reduce the number of rules and to guarantee the global exponential stability of the system.The new model included only two rules.All the other nonlinearities were considered as uncertainties.In this paper LMIs conditions were proposed to guarantee the exponential stability of the closed loop.These results were applied to a hydraulic system which was described by a nonlinear model.
that the Lemma 2 can be applied on the block (1,1) of ij  to further enhanced the results.

Figure 2 .
Figure 2. It's composed of k tanks interconnected with the th k and used in chemical, pharmaceutical and agroalimentary industries.The goal is to obtain a mixture of 1 k  liquids in the th k tank exit with a previously fixed concentration of every liquid in the mixture.The outputs of the system to control are the flow sk q and the relative concentrations 0 i   ,

Figure 2 .
Figure 2. Diagram of the of k tanks 4.2 Nonlinear model