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Mathematical Problems in Engineering
Volume 2011, Article ID 361640, 21 pages
http://dx.doi.org/10.1155/2011/361640
Research Article

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

1Federal University of São João del-Rei, Campus Alto Paraopeba, Rodovia MG 443 Km 7, 36420 Ouro Branco, MG, Brazil
2Department of Electronics Engineering, Federal University of Minas Gerais, Avenida Antônio Carlos 6627, 31270-010 Belo Horizonte, MG, Brazil

Received 23 August 2010; Accepted 24 January 2011

Academic Editor: Geraldo Silva

Copyright © 2011 Leonardo Amaral Mozelli and Reinaldo Martinez Palhares. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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 performance. Numerical tests and a practical experiment in Chua's circuit are presented to show the effectiveness.

1. 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 and 2 norms [813].

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 [1416] 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 [1719], 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 [2022], 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 [2431]. As to discrete-time fuzzy systems (DFS) see [12, 13, 32, 33].

The 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, 3437] 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 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 [3840] 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 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 and 𝒮, respectively; 𝑙2 is the discrete Lebesgue space; 2 is the 𝑙2 norm.

2. 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:𝑅𝑖IF𝑞1𝑘IS𝑖1ANDAND𝑞𝑠𝑘IS𝑖𝑠,THEN𝑥𝑘+1=𝐴𝑖𝑥𝑘+𝐵𝑖𝑢𝑘+𝐸𝑖𝑤𝑘,𝑧𝑘=𝐶𝑖𝑥𝑘+𝐷𝑖𝑢+𝑘+𝐹𝑖𝑤𝑘,(2.1) where 𝑅𝑖,𝑖, denotes the 𝑖th fuzzy inference rule. In rule 𝑅𝑖, the fuzzy sets are given by 𝑖𝑗,𝑗𝒮; 𝑞𝑗𝑘 are the premisse variables at instant 𝑘. 𝑞𝑘𝑠 is the premisse variables vector, stacking the premisse variables; 𝑥𝑘𝑛 is the state vector; 𝑢𝑘𝑚 is the control signal; 𝑤𝑘𝑝 is a disturbance input, belonging 𝑙2; 𝑧𝑘𝑞 is the regulated output. 𝐴𝑖, 𝐵𝑖, 𝐶𝑖, 𝐷𝑖, 𝐸𝑖, and 𝐹𝑖 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]𝑥𝑘+1=𝑟𝑖=1𝑖𝑞𝑘𝐴𝑖𝑥𝑘+𝐵𝑖𝑢𝑘+𝐸𝑖𝑤𝑘,𝑧𝑘=𝑟𝑖=1𝑖𝑞𝑘𝐶𝑖𝑥𝑘+𝐷𝑖𝑢𝑘+𝐹𝑖𝑤𝑘,(2.2) where 𝑖[𝑞𝑘] are weighting functions, denoting the normalized grade of membership of each rule, which satisfy𝑖𝑞𝑘0𝑖,𝑟𝑖=1𝑖𝑞𝑘=1.(2.3)

To avoid clutter, assume 𝑖[𝑞𝑘]𝑖 and 𝑖[𝑞𝑘+1]+𝑖.

2.1. 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 model𝑅𝑖IF𝑞1𝑘IS𝑖1ANDAND𝑞𝑠𝑘IS𝑖𝑠,THEN𝑢𝑘=𝐾𝑖𝑥𝑘,(2.4) where 𝐾𝑖 are the local gains.

Thus, the inferred nonlinear controller is given by𝑢𝑘=𝑟𝑖=1𝑖𝑞𝑘𝐾𝑖𝑥𝑘,(2.5) where 𝑖[𝑞𝑘] are the same membership functions of (2.2).

Taking (2.5) into account, the closed-loop description for (2.2) is given by𝑥𝑘+1=𝑟𝑟𝑖=1𝑗=1𝑖𝑗𝐺𝑖𝑗𝑥𝑘+𝐸𝑖𝑤𝑘,𝑧𝑘=𝑟𝑟𝑖=1𝑗=1𝑖𝑗𝐽𝑖𝑗𝑥𝑘+𝐹𝑖𝑤𝑘,(2.6) where 𝐺𝑖𝑗𝐴𝑖+𝐵𝑖𝐾𝑗 and 𝐽𝑖𝑗𝐶𝑖+𝐷𝑖𝐾𝑗.

2.2. Fuzzy Lyapunov’s Function

In order to obtain less conservative conditions, the following fuzzy Lyapunov function (FLF) [9] is adopted:𝑉𝑘=𝑥𝑘𝑟𝑖=1𝑖𝑃𝑖𝑥𝑘,(2.7) 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 𝑉𝑘=𝑥𝑘𝑃𝑖𝑥𝑘 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.

3. Less Conservative Conditions

In this section, new sufficient conditions for the stabilization and analysis of (2.6) are developed.

3.1. 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 𝑤𝑘 on 𝑧𝑘. In this paper, the -norm is adoptedsup𝑤𝑘𝑧2𝑤2𝛾,𝑧𝑘0.(3.1)

The -norm computes the 𝑙2-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 performance: one goal is to find a controller that guarantees 𝛾min, the minimum value of the -norm; on the other hand, it is possible to obtain a controller that stabilizes the system with a prescribed value 𝛾 or the -norm. In the following sections, both cases are addressed.

3.2. 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 𝛾min.

Theorem 3.1. The fuzzy controller (2.5) stabilizes the TS system in (2.6), with a guaranteed cost given by 𝛾min=𝛿, if there exist symmetric matrices 𝑋𝑖, 𝑅𝑖𝑗, 𝑇𝑖𝑗𝑡, and any matrices 𝐿, 𝑀𝑖, and 𝑆𝑖𝑗𝑡 satisfying the following optimization problem: min𝑋𝑖,𝑅𝑖𝑗,𝑇𝑖𝑗𝑡,𝐿,𝑀𝑖,𝑆𝑖𝑗𝑡,𝛿𝛿s.t.Ξ𝑡0,𝑋𝑖𝑇0,(𝑖,𝑡),𝑖𝑗𝑡0,(𝑖>𝑗,𝑖,𝑗,𝑡),(3.2) where Ξ𝑡𝑄11𝑡𝑍1𝑡𝑄21𝑡+𝑊21𝑡𝑄22𝑡𝑍2𝑡𝑄𝑟1𝑡+𝑊𝑟1𝑡𝑄𝑟2𝑡+𝑊𝑟2𝑡𝑄𝑟𝑟𝑡𝑍𝑟𝑡,(3.3)𝑄𝑖𝑗𝑡12Γ𝑖+Γ𝑗10𝛿𝐼2Ψ𝑖𝑗+Ψ𝑗𝑖12𝐸𝑖+𝐸𝑗𝑋𝑡12Φ𝑖𝑗+Φ𝑗𝑖12𝐹𝑖+𝐹𝑗,0𝐼(3.4)Γ𝑖𝑋𝑖𝐿𝐿,𝑊𝑖𝑗𝑡𝑉𝑖𝑗𝑡+𝑇𝑖𝑗𝑡+𝑆𝑖𝑗𝑡𝑆𝑖𝑗𝑡,(3.5)Ψ𝑖𝑗𝐴𝑖𝐿+𝐵𝑖𝑀𝑗,Φ𝑖𝑗𝐶𝑖𝐿+𝐷𝑖𝑀𝑗,(3.6)𝑉𝑖𝑗𝑡=12𝑅𝑖𝑗,if𝑖=𝑡or𝑗=𝑡,0,otherwise,𝑍𝑖𝑡=𝑅𝑖𝑡,if𝑅𝑖<𝑡,𝑡𝑖,if𝑖>𝑡,0,if𝑖=𝑡.(3.7)
Furthermore, the local gains are given by 𝐾𝑖𝑀𝑖𝐿1.

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 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 guaranteed cost given by 𝛾=𝛿, if there exist symmetric matrices 𝑋𝑖, 𝑅𝑖𝑗, 𝑇𝑖𝑗𝑡, and any matrices 𝐿, 𝑀𝑖, and 𝑆𝑖𝑗𝑡 satisfying the following LMIs: Ξ𝑡0,𝑋𝑖0,𝑇𝑖𝑗𝑡0,(𝑖>𝑗,𝑖,𝑗,𝑡),(3.8) where Ξ𝑡 is given as in (3.3). Furthermore, the local gains are given by 𝐾𝑖𝑀𝑖𝐿1.

Proof. See the appendix.

3.3. Analysis

Another problem is the analysis of fuzzy system when the control gains are already given. To search for 𝛾min provided by a given controller, the next theorem should be used.

Theorem 3.3. Let the gains 𝐾𝑖 be given. The fuzzy controller (2.5) stabilizes the TS system in (2.6), with a guaranteed cost given by 𝛾min=𝛿 if there exist symmetric matrices 𝑋𝑖, 𝑅𝑖𝑗, 𝑇𝑖𝑗𝑡, and any matrices 𝐿, and 𝑆𝑖𝑗𝑡 satisfying the following optimization problem: min𝑋𝑖,𝑅𝑖𝑗,𝑇𝑖𝑗𝑡,𝐿,𝑆𝑖𝑗𝑡,𝛿𝛿s.t.Ξ𝑡0,𝑋𝑖𝑇0,(𝑖,𝑡),𝑖𝑗𝑡0,(𝑖>𝑗,𝑖,𝑗,𝑡),(3.9) where Ξ𝑡 is given as in (3.3) and Ψ𝑖𝑗(𝐴𝑖+𝐵𝑖𝐾𝑗)𝐿 and Φ𝑖𝑗(𝐶𝑖+𝐷𝑖𝐾𝑗)𝐿.

Proof. See the appendix.

Finally, it is possible to check by the next theorem if a given controller guarantees a prescribed performance 𝛾.

Theorem 3.4. Let the scalar 𝛿>0 and the gains 𝐾𝑖 be given. The fuzzy controller (2.5) stabilizes the TS system in (2.6), with a prescribed guaranteed cost given by 𝛾=𝛿 if there exist symmetric matrices 𝑋𝑖, 𝑅𝑖𝑗, 𝑇𝑖𝑗𝑡, and any matrices 𝐿, and 𝑆𝑖𝑗𝑡 satisfying the following LMIs: Ξ𝑡0,𝑋𝑖0,𝑇𝑖𝑗𝑡0,(𝑖>𝑗,𝑖,j,𝑡),(3.10) where Ξ𝑡 is given as in (3.3) and Ψ𝑖𝑗(𝐴𝑖+𝐵𝑖𝐾𝑗)𝐿 and Φ𝑖𝑗(𝐶𝑖+𝐷𝑖𝐾𝑗)𝐿.

Proof. See the appendix.

4. 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 𝑥1𝑘. The local matrices are 𝐴1=1+𝑎0.510,𝐴2=,𝐵10.5101=11𝑏,𝐵2=12,𝐸1=0.20.3,𝐸2=,𝐶0.50.11=10.5,𝐶2=10.5,𝐷1=1,𝐷2𝐹=0.5,1=0.4,𝐹2=0.2.(4.1)
The membership functions for this system are 1𝑥1𝑘=𝑥1sin1𝑘2,2𝑥1𝑘=𝑥1+sin1𝑘2.(4.2)
Let a PDC controller (2.5) be given with the following gains: 𝐾1=0.650.30,𝐾2=.0.870.11(4.3) In this example, the objective is to determine the minimum guaranteed cost 𝛾min 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 𝛾min are found using Theorem 3.3.

tab1
Table 1: Comparison of 𝛾min by different strategies.

Example 4.2. Consider another DFS with 𝑥1𝑘 as premisse variable. The membership functions and local matrices are 1𝑥1𝑘=𝑥1𝑘+𝛽2𝛽,2𝑥1𝑘=11𝑥1𝑘,𝐴1=1𝛽10.5,𝐴2=,𝐵1𝛽10.51=5+𝛽2𝛽,𝐵2=,𝐸5𝛽2𝛽1=𝐸2=0.030.0100.01,𝐶1=𝐶2=0.10.05,𝐷1=𝐷2=0.5,𝐹1=𝐹2=.0.010.01(4.4)
In this example, the controllers are designed to provide the minimum value of the guaranteed cost, 𝛾min. Table 2 summarizes 𝛾min using different approaches for several fixed values of the parameter 𝛽 compared with Theorem 3.1. Note that for all strategies, 𝛾min 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 𝛽1.45, 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.

tab2
Table 2: 𝛾min computed for several values of 𝛽.

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: 𝑤𝑘=𝑒0.2540𝑘60,0.25100𝑘110,0.30110<𝑘120,1𝑘𝑒150𝑘175,2𝑘190𝑘220,(4.5) where 𝑒1𝑘, 𝑒2𝑘 are gaussian noises, with zero mean and standard deviations 0.05 and 0.10, respectively.
Assuming as initial conditions 𝑥0=[0.70.2] 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 𝑧𝑘2=0.6406 and that 𝑤𝑘2=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.

tab3
Table 3: Gains obtained by Theorem 3.1 considering 𝛾=0.5.
361640.fig.001
Figure 1: Closed-loop system, 𝑥1𝑘 trajectory.
361640.fig.002
Figure 2: Closed-loop system, 𝑥2𝑘 trajectory.
361640.fig.003
Figure 3: Disturbance signal (dashed) and system regulated output (continuous).
361640.fig.004
Figure 4: Control signal.

5. 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].

5.1. 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:̇𝑥11(𝑡)=𝐶1𝑥2(𝑡)𝑅𝑥1(𝑡)𝑅𝑥𝑔1,(𝑡)̇𝑥21(𝑡)=𝐶2𝑥1(𝑡)𝑅𝑥2(𝑡)𝑅+𝑥3,(𝑡)̇𝑥3𝑥(𝑡)=2(𝑡)𝐿𝑅0𝐿𝑥3(𝑡),(5.1) where 𝑔(𝑥1(𝑡)) is a nonlinear function given by𝑔2𝑥1𝑔(𝑡)+1𝑔2||𝑥1||||𝑥(𝑡)+𝐸1||(𝑡)𝐸2,(5.2) with 𝑔1 and 𝑔2 being conductance values.

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: 𝑥1𝑘[𝑑,𝑑]. The following fuzzy rules are able to exactly represent the dynamics of the discretized Chua's oscillator:𝑅1IF𝑥1𝑘IS11(near0),THEN𝑥𝑘+1=𝐴1𝑥𝑘+𝐵1𝑢𝑘+𝐸1𝑤𝑘,𝑦𝑘=𝐶1𝑥𝑘+𝐹1𝑤𝑘,𝑅2IF𝑥1𝑘IS21(near±𝑑),THEN𝑥𝑘+1=𝐴2𝑥𝑘+𝐵2𝑢𝑘+𝐸2𝑤𝑘,𝑦𝑘=𝐶2𝑥𝑘+𝐹2𝑤𝑘.(5.3)

The local matrices are given by𝐴𝑖=𝑇1𝑅𝐶1𝑇𝑔𝑖𝐶1𝑇𝑅𝐶10𝑇𝑅𝐶2𝑇1𝑅𝐶2𝑇𝐶2𝑇0𝐿1𝑇𝑅0𝐿,𝐵𝑖=001,𝐸𝑖=1×10300,𝐶𝑖=100,𝐷𝑖=0,𝐹𝑖=1×104,(5.4) where 𝑔2𝑈+(𝑔1𝑈)𝐸/𝑑.

The membership functions are1𝑥1𝑘=𝛼𝑥1𝑘+𝐸(1𝛼)𝑥1𝑘,𝑥1𝑘𝐸,𝛼𝑥1𝑘𝐸(1𝛼)𝑥1𝑘,𝑥1𝑘𝐸,1,otherwise,2𝑥1𝑘=11𝑥1𝑘,(5.5) where 𝛼𝐸/𝑑.

The parameters of the fuzzy model are given in Table 4.

tab4
Table 4: DFS parameters.
5.2. 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 𝑤𝑘=0 and that minimizes the guaranteed cost between 𝑤𝑘 and 𝑧𝑘. 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 𝐾𝑖=𝐶1𝐾𝑖/𝑇.

tab5
Table 5: Gains obtained by Theorem 3.1.

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 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 𝑥1(𝑡) is faster than the convergence of the remaining states. Possible reasons are the fact that the control action is applied only on 𝑥1(𝑡) and that the system output is given only by 𝑥1(𝑡). Nonetheless, the design controller accomplishes its goals. Notice in Figure 5 that the presence of disturbances in the output is negligible.

361640.fig.005
Figure 5: 𝑥1(𝑡) trajectory.
361640.fig.006
Figure 6: 𝑥2(𝑡) trajectory.
361640.fig.007
Figure 7: 𝑥3(𝑡) trajectory.

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.

361640.fig.008
Figure 8: Control signal.
361640.fig.009
Figure 9: Control signal, close view.

6. Conclusion

Different strategies to reduce numeric conservatism were combined with the fuzzy Lyapunov function to promote less conservative LMI conditions for 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.

Appendix

The following Lemmas are required in the development of the proof of Theorem 3.1.

Lemma A.1. If 𝑆0, then 𝐴𝑖𝑆𝐴𝑗+𝐴𝑗𝑆𝐴𝑖𝐴𝑖𝑆𝐴𝑖+𝐴𝑗𝑆𝐴𝑗.(A.1)

Proof. Notice that 𝐴𝑆0𝑖𝐴𝑗𝑆𝐴𝑖𝐴𝑗0,(A.2) leading to 𝐴𝑖𝑆𝐴𝑖𝐴𝑖𝑆𝐴𝑗𝐴𝑗𝑆𝐴𝑖+𝐴𝑗𝑆𝐴𝑗0,(A.3) which completes the proof.

Lemma A.2. Assume that 𝑖 (𝑖) satisfy (2.3) and let 𝑅𝑖𝑗 (𝑖<𝑗,𝑖,𝑗) be symmetric matrices of appropriate dimension. Define 𝐻𝐻11𝛼12𝑅12𝛼1𝑟𝑅1𝑟𝛼12𝑅12𝐻22𝛼2𝑟𝑅2𝑟𝛼1𝑟𝑅1𝑟𝛼2𝑟𝑅2𝑟𝐻𝑟𝑟,(A.4) where 𝐻𝑖𝑖𝑟𝑗=1𝑗>𝑖𝑗𝑅𝑖𝑗𝑟𝑗=1𝑗<𝑖𝑗𝑅𝑗𝑖,𝛼𝑘𝑙12𝑘+𝑙.(A.5)
Then, 12𝑟𝐻12𝑟=0.(A.6)

The proof of Lemma A.2 can be found in [15, Appendix A].

Lemma A.3. If 𝑊0, then 𝑆𝑊1𝑆(𝑆+𝑆𝑊).(A.7)

Proof. Since 𝑊0, there exists 𝑊1. Thus, (𝑊𝑆)𝑊1(𝑊𝑆)0,𝑊𝑊1𝑊𝑊𝑊1𝑆𝑆𝑊1𝑊+𝑆𝑊1𝑆0,𝑊𝑆𝑆+𝑆𝑊1𝑆0,(A.8) concluding the proof.

Proof of Theorem 3.1. Consider the induced 𝑙2-gain between the disturbance signal 𝑤𝑘 and the system output 𝑧𝑘 in (3.1) and the closed-loop TS system in (2.6). The stability and the performance can be achieved if the following inequality holds [11, 36, 42]: 𝒱=Δ𝑉𝑘+𝑧𝑘𝑧𝑘𝛾2𝑤𝑘𝑤𝑘<0,(A.9) where Δ𝑉𝑘𝑥𝑟𝑘+1𝑡=1+𝑡𝑃𝑡𝑥𝑘+1𝑥𝑘𝑟𝑖=1𝑖𝑃𝑖𝑥𝑘,(A.10) is the increment of the fuzzy Lyapunov function candidate 𝑉𝑘, shown in (2.7).
According to (2.6), it follows that 𝒱=𝜉𝑟𝑡=1+𝑡𝑟𝑖=1𝑟𝑟𝑗=1𝑟𝑙=1𝑣=1𝑖𝑗𝑙𝑣Λ𝑖𝑗𝑃𝑡Λ𝑙𝑣𝑃𝑖𝜉,(A.11) where 𝜉[𝑥𝑘𝑤𝑘] and Λ𝑎𝑏𝐺𝑎𝑏𝐸𝑎𝐽𝑎𝑏𝐹𝑎,𝑃𝑎𝑃𝑎0,𝑃0𝐼𝑎𝑃𝑎00𝛾2𝐼.(A.12) Equation (A.11) is equivalent to 𝒱=𝜉𝑟𝑡=1+𝑡𝑟𝑟𝑖=1𝑟𝑗=1𝑟𝑙=1𝑣=1𝑖𝑗𝑙𝑣18×Λ𝑖𝑗+Λ𝑗𝑖𝑃𝑡Λ𝑙𝑣+Λ𝑣𝑙+Λ𝑙𝑣+Λ𝑣𝑙𝑃𝑡Λ𝑖𝑗+Λ𝑗𝑖𝑃8𝑖𝜉.(A.13) By applying Lemma A.1, one obtains 𝒱𝜉𝑟𝑡=1+𝑡𝑟𝑟𝑖=1𝑟𝑗=1𝑟𝑙=1𝑣=1𝑖𝑗𝑙𝑣18Λ𝑖𝑗+Λ𝑗𝑖𝑃𝑡Λ𝑖𝑗+Λ𝑗𝑖+Λ𝑙𝑣+Λ𝑣𝑙𝑃𝑡Λ𝑙𝑣+Λ𝑣𝑙𝑃8𝑖𝜉=𝜉𝑟𝑡=1+𝑡𝑟𝑟𝑖=1𝑗=1𝑖𝑗14Λ𝑖𝑗+Λ𝑗𝑖𝑃𝑡Λ𝑖𝑗+Λ𝑗𝑖𝑃4𝑖𝜉=𝜉𝑟𝑡=1+𝑡𝑟𝑖=12𝑖Λ𝑖𝑖𝑃𝑡Λ𝑖𝑖𝑃𝑖+𝑟𝑖<𝑗𝑖𝑗Λ𝑖𝑗+Λ𝑗𝑖𝑃𝑡Λ𝑖𝑗+Λ𝑗𝑖𝑃𝑖𝜉.(A.14)
Now define Λ𝑎𝑏𝑐𝜆11𝑎𝑏𝑐𝜆21𝑎𝑏𝑐𝜆22𝑎𝑏𝑐,(A.15) where 𝜆11𝑎𝑏𝑐𝐺𝑎𝑏𝑃𝑐𝐺𝑎𝑏𝑃𝑎+𝐽𝑎𝑏𝐽𝑎𝑏,𝜆21𝑎𝑏𝑐𝐸𝑎𝑃𝑐𝐺𝑎𝑏+𝐹𝑎𝐽𝑎𝑏,𝜆22𝑎𝑏𝑐𝛾2𝐼+𝐸𝑎𝑃𝑐𝐸𝑎+𝐹𝑎𝐹𝑎,(A.16) and rewrite (A.14) as 𝒱𝒩=𝜉𝑟𝑡=1+𝑡𝑟𝑖=12𝑖Λ𝑖𝑖𝑡+2𝑟𝑖<𝑗𝑖𝑗Λ𝑖𝑗𝑡+Λ𝑗𝑖𝑡2𝜉.(A.17)
For stability of (2.6) with a guaranteed cost given by 𝛾, it is sufficient that Λ𝑖𝑖𝑡0 and Λ𝑖𝑗𝑡0(𝑖<𝑗,𝑖,𝑗,𝑡), since it follows that in (A.17), 𝒩<0, which implies that (A.9) holds.
At this point, the relaxation techniques from [11, 15, 3840] 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 Λ𝑎𝑏𝑐, obtaining Λ𝑎𝑏𝑐𝑃𝑎0𝛾2𝑃𝐼𝑐𝐺𝑎𝑏𝑃𝑐𝐸𝑎𝑃𝑐𝐽𝑎𝑏𝐹𝑎.0𝐼(A.18)
Define a slack matrix variable 𝐿𝑛×𝑛 (see [3840] for details). Take (A.17), replace Λ𝑎𝑏𝑐 with Λ𝑎𝑏𝑐, and then apply the following transformations: 𝒱𝑆2𝑆1(𝒩)𝑆1𝑆2,(A.19) where 𝑆1𝑃diag𝑎1,𝐼,𝑃𝑐1,𝐼,𝑆2𝐿diag𝑃𝑎,,𝐼,𝐼,𝐼𝒱𝑆2𝑆1𝒱𝑆1𝑆2(A.20) to obtain 𝒱<𝑁=𝜉𝑟𝑡=1+𝑡𝑟𝑖=12𝑖Λ𝑖𝑖𝑡+2𝑟𝑖<𝑗𝑖𝑗Λ𝑖𝑗𝑡+Λ𝑗𝑖𝑡2𝜉,(A.21) such that Λ𝑎𝑏𝑐𝐿𝑋𝑎1𝐿0𝛾2𝐺𝐼𝑎𝑏𝐿𝐸𝑎𝑋𝑐𝐽𝑎𝑏𝐿𝐹𝑎0𝐼,𝑋𝑎𝑃𝑎1.(A.22)
Applying Lemma A.3 results in 𝒩<𝑀=𝜉𝑟𝑡=1+𝑡𝑟𝑖=12𝑖̆Λ𝑖𝑖𝑡+2𝑟𝑖<𝑗𝑖j̆Λ𝑖𝑗𝑡+̆Λ𝑗𝑖𝑡2𝜉,(A.23) where ̆Λ𝑎𝑏𝑐𝑋𝑎𝐴𝐿𝐿0𝛿𝐼𝑎𝐿+𝐵𝑎𝑀𝑏𝐸𝑎𝑋𝑐𝐶𝑎𝐿+𝐷𝑎𝑀𝑏𝐹𝑎,0𝐼(A.24) and 𝑀𝑎𝐾𝑎𝐿,𝛿𝛾2 are linearizing change of variables.
Finally, the second relaxation technique is applied to add more slack matrix variables. Considering that 𝑇𝑖𝑗𝑡=𝑇𝑖𝑗𝑡0(𝑖,𝑗,𝑡) and using Lemma A.2, ones gets (A.25) as in the top of the next page =𝜉𝑟𝑡=1+𝑡𝑟𝑖=12𝑖̆Λ𝑖𝑖𝑡+2𝑟𝑖<𝑗𝑖𝑗̆Λ𝑖𝑗𝑡+̆Λ𝑗𝑖𝑡2𝜉𝜉𝑟𝑡=1+𝑡𝑟𝑖=12𝑖̆Λ𝑖𝑖𝑡+2𝑟𝑖<𝑗𝑖𝑗̆Λ𝑖𝑗𝑡+̆Λ𝑗𝑖𝑡2+𝑇𝑖𝑗𝑡+𝑆𝑖𝑗𝑡𝜉=𝜉𝑟𝑡=1+𝑡12𝑟̆Λ11𝑡V𝑡̆Λ22𝑡V𝑟1𝑡V𝑟2𝑡̆Λ𝑟𝑟𝑡12𝑟𝜉=𝜉𝑟𝑡=1+𝑡12𝑟Ξ𝑡12𝑟𝜉=𝜉𝑟𝑡=1+𝑡12𝑟Ξ𝑡+𝐻12𝑟𝜉=𝜉𝑟𝑡=1+𝑡12𝑟Ξ𝑡12𝑟𝜉.(A.25) where in (A.25), V𝑡̆Λ=(1/2)(12𝑡+̆Λ21𝑡)+𝑇21𝑡+𝑆21𝑡, Vr1t̆Λ=(1/2)(𝑟1𝑡+̆Λ1𝑟𝑡)+𝑇𝑟1𝑡+𝑆𝑟1𝑡, and V𝑟2𝑡̆Λ=(1/2)(𝑟2𝑡+̆Λ2𝑟𝑡)+𝑇𝑟2𝑡+𝑆𝑟2𝑡.
Matrices Ξ𝑡,𝑡 are given as in (3.3). 𝑆𝑖𝑗𝑡 are skew matrices which can be defined as the difference 𝑆𝑖𝑗𝑡𝑆𝑖𝑗𝑡, with 𝑆𝑖𝑗𝑡 being any matrices.
If the LMIs constraints given in (3.2) are satisfied, then <0. Because of (A.23), (A.17), and (A.9), it also implies that the TS system (2.6) is asymptotically stable with guaranteed cost 𝛿. Furthermore, since the optimization problem in (3.2) is convex, the minimization of 𝛿 produces the minimum disturbance attenuation level 𝛾min, 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 𝑀𝑖 must be dropped out.

Acknowledgments

This work was supported by the Brazilian agencies CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico) and FAPEMIG (Fundação de Amparo à Pesquisa do Estado de Minas Gerais).

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