Less Conservative H ∞ Fuzzy Control for Discrete-Time Takagi-Sugeno Systems

New analysis and control design conditions of discrete-time fuzzy systems are proposed. Using fuzzy Lyapunov's functions and introducing slack variables, less conservative conditions are obtained. The controller guarantees system stabilization and ℋ∞ performance. Numerical tests and a practical experiment in Chua's circuit are presented to show the effectiveness.


Introduction
Model-based fuzzy control is a widespread approach to deal with complex nonlinear dynamics 1 .Within this context, Takagi-Sugeno TS fuzzy model 2 is a landmark.It consists on fuzzy rules describing global semiglobal dynamics as linear models locally valid interpolated by membership functions.
From the modeling point of view, TS systems are known to be universal approximators 3, 4 and to possess a reduced number of rules, when compared with other fuzzy models 5 .Another interesting feature is the existence of a systematic procedure to obtain TS models from the nonlinear system equations, namely, the sector nonlinearity approach 3 .An advantage for control purposes is the possibility to use the Lyapunov stability theory and, simultaneously, to rely on tools from linear systems theory 3 .
Methodologies based on Lyapunov's functions provide a straightforward way to describe stability and control design issues of TS systems by means of linear matrix inequalities LMIs 6 , of which the solutions can be computed in polynomial-time by convex optimization techniques.There are several approaches to design fuzzy controllers that, besides stability, also guarantee some type of performance for the closed-loop nonlinear system as D-stability regions 7 , constraints over input/output signals 3 , and performance indexes, such as H ∞ and H 2 norms 8-13 .
In TS-based control, Lyapunov's function candidates are classified in three categories 1 : the common quadratic Lyapunov function CQLF , the piecewise Lyapunov function PLF , and the fuzzy Lyapunov function FLF .Most efforts deal with sufficient conditions for the existence of a CQLF 3 , a single quadratic function that guarantee stability for all fuzzy subsystems.However, as the number of rules increases, the CQLF turns out to be very conservative.
To keep obtaining solutions with the CQLF, many techniques were developed.Usually the underlaying strategy among them see 14-16 and references therein is the use of quadratic form relaxations right-hand side slack matrix variables into the LMI formulation.Extravariables improve the numeric behavior of LMI solvers, at the cost of higher computational time.Recently, sufficient and necessary conditions for the existence of CQLF were discussed in 17-19 , reaching the limits of quadratic stability.
Nonetheless, a CQLF might not exist even for a stable TS system, as demonstrated in 20 .An well-established alternative to overcome this problem is the PLF 20-22 , which consists of a finite combination of disjoint common quadratic Lyapunov's functions, each of them valid only into a compact domain.
The PLF is suitable whenever the TS model is not activating at each time the whole set of its linear models.Nevertheless, this assumption does not hold for many TS models.Another drawback is how to assess the behavior of the PLF at the boundaries of the partitions.Solutions range from considering boundary conditions 21 to introduce extra LMI constraints that guarantee continuity of the function across boundaries 20 and to use some methods ensuring that the function decreases when leaving one subspace for another closer to the equilibrium, relaxing the continuity assumption 22 .
More recently, approaches based on the FLF 9, 23 were developed.The FLF is a fuzzy blending of multiple quadratic functions, in the same way the TS model is constructed.Unlike the PLFs, continuity is an inherent feature of the FLF.Furthermore, the quadratic functions used to construct an FLF do not need to be Lyapunov's functions by themselves, just its fuzzy combination does.
Thus, the FLF has been attracting much attention.For continuous-time fuzzy systems CFS , recent results are given in 24-31 .As to discrete-time fuzzy systems DFS see 12, 13, 32, 33 .The H ∞ index is an adequate criterion to control and filter design in nonlinear systems with exogenous inputs with unknown spectral density and bounded energy see 11, 34-37 and references therein for examples of practical applications and computes the greatest ratio between the system output and the noisy input.Conservative conditions may not give a good tradeoff between disturbance attenuation level and a feasible controller, so the FLF is an interesting candidate 12, 13, 33 .This paper presents new sufficient conditions to H ∞ control for DFS in the TS form.To promote these improved conditions, three different strategies are employed.First, an FLF is adopted, since recent works reveal that this type of function is less conservative than the CQLF for TS systems.Then, a series of matrix transformations 38-40 are performed, allowing to obtain LMIs in which the controller gains are not directly dependent on the Lyapunov matrices, introducing extradegree of freedom for the optimization problem.Finally, 15 provides a successful approach to introduce slack matrix variables into the stabilization control, enhancing the numerical behavior of LMI solvers.This strategy was further extended to H ∞ control in 11 and is used in this paper.Numeric examples illustrate the improvement provided by merging these approaches.Smaller attenuation levels are achieved, and solutions are computed even for systems in which some other methods fail.A practical experiment using the Chuas's chaotic oscillator 41 is given to show the performance of the proposed methodology.Notation 1.The notation is standard.Transpose of vectors and matrices are indicated by the superscript ; the symbol • denotes transposed terms in symmetric matrices; the sets {1, 2, . . ., r} and {1, 2, . . ., s} are indicated by R and S, respectively; l 2 is the discrete Lebesgue space; • 2 is the l 2 norm.

Preliminaries on TS Systems
Consider nonlinear discrete-time systems that can be expressed as Takagi-Sugeno TS fuzzy models 2 , according to the following fuzzy rules: where R i , i ∈ R, denotes the ith fuzzy inference rule.In rule R i , the fuzzy sets are given by M i j , j ∈ S; q j k are the premisse variables at instant k. q k ∈ R s is the premisse variables vector, stacking the premisse variables; x k ∈ R n is the state vector; u k ∈ R m is the control signal; w k ∈ R p is a disturbance input, belonging l 2 ; z k ∈ R q is the regulated output.A i , B i , C i , D i , E i , and F i are the local matrices of proper dimensions.
By using a standard fuzzy inference method, that is, using a singleton fuzzifier, product fuzzy inference, and center-average defuzzifier, the global inferred TS model is given by 1, 3 where h i q k are weighting functions, denoting the normalized grade of membership of each rule, which satisfy To avoid clutter, assume h i q k h i and h i q k 1 h i .

Parallel Distributed Compensation
The control law adopted is given by the parallel distributed compensation PDC 3 , where the control signal consists in a fuzzy combination of linear-state feedbacks, likewise the TS model where K i are the local gains.Thus, the inferred nonlinear controller is given by where h i q k are the same membership functions of 2.2 .Taking 2.5 into account, the closed-loop description for 2.2 is given by where

Fuzzy Lyapunov's Function
In order to obtain less conservative conditions, the following fuzzy Lyapunov function FLF 9 is adopted: sharing the same membership functions of 2.6 .It is interesting to note see the discussion on Section II.B in 23 that the locally valid functions V k x k P i x k do not need to be Lyapunov's functions by themselves.Only its combination does, namely 2.7 , representing an advantage when compared to the CQLF and the PLF.

Less Conservative Conditions
In this section, new sufficient conditions for the stabilization and analysis of 2.6 are developed.

H ∞ Performance
In addition to stabilization, the designed controller must attenuate exogenous entries into the regulated output.There are several ways to quantify the effect of w k on z k .In this paper, the H ∞ -norm is adopted sup The H ∞ -norm computes the l 2 -gain between the noisy input and the system output.In other words, the greatest ratio between the output energy and the exogenous input energy, being suitable when no information about the spectral density of the disturbances is known a priori 42 .
There are two usual ways to apply the H ∞ performance: one goal is to find a controller that guarantees γ min , the minimum value of the H ∞ -norm; on the other hand, it is possible to obtain a controller that stabilizes the system with a prescribed value γ or the H ∞ -norm.In the following sections, both cases are addressed.

Control Design
The main result of this paper can be stated in the following theorem that provides a sufficient condition to obtain the gains of the fuzzy controller 2.5 that stabilizes the TS system 2.6 with the minimum guaranteed cost γ min .
Theorem 3.1.The fuzzy controller 2.5 stabilizes the TS system in 2.6 , with a H ∞ guaranteed cost given by γ min √ δ, if there exist symmetric matrices X i , R ij , T ijt , and any matrices L, M i , and S ijt satisfying the following optimization problem: where

3.7
Furthermore, the local gains are given by K i M i L −1 .
Proof.The proof of Theorem 3.1 is given in the appendix.
It is also desirable to design a stabilizing controller that provides a specific H ∞ guaranteed cost.In this case, the scalar γ should be provided beforehand and the next theorem should be applied.Theorem 3.2.Let δ > 0 be a scalar given.The fuzzy controller 2.5 stabilizes the TS system in 2.6 , with a prescribed H ∞ guaranteed cost given by γ √ δ, if there exist symmetric matrices X i , R ij , T ijt , and any matrices L, M i , and S ijt satisfying the following LMIs: where Ξ t is given as in 3.3 .Furthermore, the local gains are given by K i M i L −1 .
Proof.See the appendix.

Analysis
Another problem is the analysis of fuzzy system when the control gains are already given.To search for γ min provided by a given controller, the next theorem should be used.
Theorem 3.3.Let the gains K i be given.The fuzzy controller 2.5 stabilizes the TS system in 2.6 , with a H ∞ guaranteed cost given by γ min √ δ if there exist symmetric matrices X i , R ij , T ijt , and any matrices L, and S ijt satisfying the following optimization problem: where Ξ t is given as in 3.3 and Proof.See the appendix.
Finally, it is possible to check by the next theorem if a given controller guarantees a prescribed H ∞ performance γ.Theorem 3.4.Let the scalar δ > 0 and the gains K i be given.The fuzzy controller 2.5 stabilizes the TS system in 2.6 , with a prescribed H ∞ guaranteed cost given by γ √ δ if there exist symmetric matrices X i , R ij , T ijt , and any matrices L, and S ijt satisfying the following LMIs: where Ξ t is given as in 3.3 and Proof.See the appendix.

Numeric Results
In this section, numerical examples illustrate the performance of the proposed approach in comparison to some methods presented in the literature.The tests were performed using SeDuMi 43 together with Yalmip 44 in Matlab 7.4.0.
Example 4.1.The following example is borrowed from 10 .Consider DFS as in 2.2 which the premisse variable is x 1 k .The local matrices are

4.1
The membership functions for this system are Let a PDC controller 2.5 be given with the following gains: In this example, the objective is to determine the minimum H ∞ guaranteed cost γ min achieved when the PDC controller 4.3 is employed.Table 1 shows the results obtained considering b 0 and different fixed values for the parameter a.Note that the lower values of γ min are found using Theorem 3.3.

4.4
In this example, the controllers are designed to provide the minimum value of the H ∞ guaranteed cost, γ min .Table 2 summarizes γ min using different approaches for several fixed values of the parameter β compared with Theorem 3.1.Note that for all strategies, γ min increases as β increases.For |β| < 0.5 all approaches find very similar values for γ.When β is close to 1, the performance of 13, Theorem 4 deteriorates, whereas the remaining approaches calculate low values for γ.When β gets closer to 1.45, only the proposed approach and 13, Theorem 3 are feasible.For β ≥ 1.45, only the proposed approach remains feasible.Clearly the proposed approach always provides the best signal attenuation since smaller values of γ are found, as shown in boldface in Table 2.
Example 4.3.Consider the same DFS described in Example 4.1.Theorem 3.2 is applied in order to design a controller that guarantees an attenuation level γ 0.5 when b 0 and a 0.5.The gains obtained are given in Table 3.
Simulations were performed to show the response of the TS system under the designed PDC controller.A total of 250 iterations were considered and the disturbance signal applied is as follows: where e 1 k , e 2 k are gaussian noises, with zero mean and standard deviations 0.05 and 0.10, respectively.
Assuming as initial conditions x 0 0.7 − 0.2 the trajectory described by the controlled systems is depicted in Figures 1 and 2. Note that the designed controller is able to stabilize the TS system after the initial conditions and also after the presence of an exogenous entry both states converge asymptotically to zero .
Figure 3 reveals that the controller indeed reduces the effect of the noisy input into the regulated output.The computation revelas that z k 2 0.6406 and that w k 2 1.7919, resulting in an attenuation factor γ 0.3575.Notice that even for a rather complicated disturbance signal as the one in 4.5 , the attenuation provided by the controller was smaller than the upper bound prescribed.The control signal is shown in Figure 4.

Experimental Results
This section presents practical control experiments using the proposed methodology into the Chua's chaotic oscillator 41, 45 .This system is implemented on a laboratory setup called PCCHUA, with a complete set of tools to perform discrete-time control and data acquisition.
Constructive aspects and details can be found in 46 .

TS Model of Chua's System
The Chua's chaotic oscillator is continuous-time system with three unstable fixed points, and its trajectory in the space state is confined to a double-scroll attractor 45 .The following equations describe the system dynamics: where g x 1 t is a nonlinear function given by First, to use a discrete-time control strategy available in the PCCHUA framework, a discretized version of 5.1 is obtained by employing the methodology from 47 .Then, the sector nonlinearity approach 3 is applied to obtain a DFS.These modelling details can be found in 11 .
A single-premisse variable is chosen: The following fuzzy rules are able to exactly represent the dynamics of the discretized Chua's oscillator:

5.3
The local matrices are given by where

5.4
The membership functions are where α E/d.The parameters of the fuzzy model are given in Table 4.

Fuzzy Control of Chua's System
After obtaining a fuzzy model, it is possible to use Theorem 3.1 to design a PDC controller that guarantees asymptotically stability when w k 0 and that minimizes the H ∞ guaranteed cost between w k and z k .Since the PCCHUA contains a real Chua's circuit, the plant is subjected to noisy signals, quantization error, and random interferences, acting as disturbances inputs.Due to this reason, the proposed control scheme is suitable.
The gains shown in Table 5 are obtained after solving Theorem 3.1.These gains must be transformed to include the effect of the zero-order holder and to be compatible with the system actuators 36 .Therefore, the real gains are The elapsed time during this control experiment is 60 s.Only between 25 s and 35 s, the control action is performed.In the remaining time, the circuit runs freely.The system Mathematical Problems in Engineering    trajectory is shown in Figures 5, 6, and 7. Notice that the fuzzy controller is able to stabilize the system at the origin.The convergence of x 1 t is faster than the convergence of the remaining states.Possible reasons are the fact that the control action is applied only on x 1 t and that the system output is given only by x 1 t .Nonetheless, the design controller accomplishes its goals.Notice in Figure 5 that the presence of disturbances in the output is negligible.The control signal is depicted in Figures 8 and 9. Figure 9 is a close view pointing out the zero-order hold characteristic of the control signal.

Conclusion
Different strategies to reduce numeric conservatism were combined with the fuzzy Lyapunov function to promote less conservative LMI conditions for H ∞ fuzzy control design and analysis.Some numerical results showed that the proposed approach outperforms recent strategies.Also a practical experiment in the Chua's oscillator was performed to show the effectiveness of the proposed approach. where A.5 Then, The proof of Lemma A.2 can be found in 15, Appendix A .
Proof of Theorem 3.1.Consider the induced l 2 -gain between the disturbance signal w k and the system output z k in 3.1 and the closed-loop TS system in 2.6 .The stability and the H ∞ performance can be achieved if the following inequality holds 11, 36, 42 : where is the increment of the fuzzy Lyapunov function candidate V k , shown in 2.7 .

Mathematical Problems in Engineering
According to 2.6 , it follows that

A.13
By applying Lemma A.1, one obtains A. A.17 For stability of 2.6 with a H ∞ guaranteed cost given by γ, it is sufficient that Λ iit ≺ 0 and Λ ijt ≺ 0 i < j, i, j, t ∈ R , since it follows that in A.17 , N < 0, which implies that A.9 holds.
At this point, the relaxation techniques from 11, 15, 38-40 are combined to promote the less conservative LMIs proposed in Theorem 3.1.
First, some matrix transformations are employed to allow the decoupling of the Lyapunov matrices from the system matrices.Apply Schur's complement to Λ abc , obtaining Define a slack matrix variable L ∈ R n×n see 38-40 for details .Take A.17 , replace Λ abc with Λ abc , and then apply the following transformations: where and M a K a L, δ γ 2 are linearizing change of variables.Finally, the second relaxation technique is applied to add more slack matrix variables.Considering that T ijt T ijt 0 i, j, t ∈ R and using Lemma A.2, ones gets A. 25 A.25 where in A.25 , V t 1/2 Λ12t Λ21t T 21t S 21t , V r1t 1/2 Λr1t Λ1rt T r1t S r1t , and V r2t 1/2 Λr2t Λ2rt T r2t S r2t .Matrices Ξ t , t ∈ R are given as in 3.3 .S ijt are skew matrices which can be defined as the difference S ijt − S ijt , with S ijt being any matrices.
If the LMIs constraints given in 3.2 are satisfied, then M < 0. Because of A.23 , A.17 , and A.9 , it also implies that the TS system 2.6 is asymptotically stable with H ∞ guaranteed cost δ.Furthermore, since the optimization problem in 3.2 is convex, the minimization of δ produces the minimum H ∞ disturbance attenuation level γ min , which concludes the proof.
The proof of Theorem 3.2 follows the same steps as in Theorem 3.1.Because in Theorem 3.2 the value of the guaranteed cost is given beforehand, there is no need to impose the minimization constraint, since δ is not a variable anymore.Therefore, Theorem 3.2 is just a feasibility problem.
The proofs of Theorems 3.3 and 3.4 follow the same steps of the proof of Theorem 3.1 as well.However, since the gains are given, the linearizing variables M i must be dropped out.

Table 1 :
Comparison of γ min by different strategies.

Table 2 :
γ min computed for several values of β.
Example 4.2.Consider another DFS with x 1 k as premisse variable.The membership functions and local matrices are