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

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 ℋ_∞ 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 ℋ_∞ and ℋ₂ 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; 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:𝑅_𝑖∶⎧⎪⎨⎪⎩IF𝑞^1𝑘ISℳ^𝑖1AND⋯AND𝑞^𝑠𝑘ISℳ^𝑖𝑠,THEN𝑥𝑧_𝑘+1𝑘=𝐴_𝑖𝑥_𝑘+𝐵_𝑖𝑢_𝑘+𝐸_𝑖𝑤_𝑘,=𝐶_𝑖𝑥_𝑘+𝐷_𝑖𝑢+𝑘+𝐹_𝑖𝑤_𝑘,(2.1) where 𝑅_𝑖,𝑖∈ℛ, denotes the 𝑖th fuzzy inference rule. In rule 𝑅_𝑖, the fuzzy sets are given by ℳ^𝑖𝑗,𝑗∈𝒮; 𝑞^𝑗𝑘 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 𝑙₂; 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]𝑥_𝑘+1=^𝑟_𝑖=1𝑧_𝑘ℎ_𝑖𝑞_𝑘𝐴_𝑖𝑥_𝑘+𝐵_𝑖𝑢_𝑘+𝐸_𝑖𝑤_𝑘,=^𝑟_𝑖=1ℎ_𝑖𝑞_𝑘𝐶_𝑖𝑥_𝑘+𝐷_𝑖𝑢_𝑘+𝐹_𝑖𝑤_𝑘,(2.2) where ℎ_𝑖[𝑞_𝑘] are weighting functions, denoting the normalized grade of membership of each rule, which satisfyℎ_𝑖𝑞_𝑘≥0𝑖∈ℛ,^𝑟_𝑖=1ℎ_𝑖𝑞_𝑘=1.(2.3)

To avoid clutter, assume ℎ_𝑖[𝑞_𝑘]≜ℎ_𝑖 and ℎ_𝑖[𝑞_𝑘+1]≜ℎ^+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 model𝑅_𝑖∶IF𝑞^1𝑘ISℳ^𝑖1AND⋯AND𝑞^𝑠𝑘ISℳ^𝑖𝑠,THEN𝑢_𝑘=𝐾_𝑖𝑥_𝑘,(2.4) where 𝐾_𝑖 are the local gains.

Thus, the inferred nonlinear controller is given by𝑢_𝑘=^𝑟_𝑖=1ℎ_𝑖𝑞_𝑘𝐾_𝑖𝑥_𝑘,(2.5) 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 by𝑥_𝑘+1=^𝑟_𝑟𝑖=1_𝑗=1𝑧_𝑘ℎ_𝑖ℎ_𝑗𝐺_𝑖𝑗𝑥_𝑘+𝐸_𝑖𝑤_𝑘,=^𝑟_𝑟𝑖=1_𝑗=1ℎ_𝑖ℎ_𝑗𝐽_𝑖𝑗𝑥_𝑘+𝐹_𝑖𝑤_𝑘,(2.6) where 𝐺_𝑖𝑗≜𝐴_𝑖+𝐵_𝑖𝐾_𝑗 and 𝐽_𝑖𝑗≜𝐶_𝑖+𝐷_𝑖𝐾_𝑗.

2.2. Fuzzy Lyapunov’s Function

In order to obtain less conservative conditions, the following fuzzy Lyapunov function (FLF) [9] is adopted:𝑉_𝑘=𝑥′_𝑘^𝑟_𝑖=1ℎ_𝑖𝑃_𝑖𝑥^𝑘,(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 𝑉_𝑘=𝑥^𝑘𝑃_𝑖𝑥_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.

3. Less Conservative Conditions

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

3.1. 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 adoptedsup_𝑤𝑘‖𝑧‖₂‖𝑤‖₂≤𝛾,∀𝑧_𝑘≠0.(3.1)

The ℋ_∞-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 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.

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 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: min_{𝑋𝑖,𝑅𝑖𝑗,𝑇𝑖𝑗𝑡,𝐿,𝑀𝑖,𝑆𝑖𝑗𝑡,𝛿}𝛿s.t.𝑇Ξ_𝑡≺0,𝑋_𝑖≻0,(𝑖,𝑡∈ℛ),_𝑖𝑗𝑡⪰0,(𝑖>𝑗,𝑖,𝑗,𝑡∈ℛ),(3.2) where Ξ_𝑡≜⎡⎢⎢⎢⎣𝑄^11𝑡𝑄−𝑍_1𝑡•…•_21𝑡𝑄+⋮𝑊_21𝑡𝑄_22𝑡⋮−𝑍_2𝑡⋱…⋮_𝑟1𝑡•+𝑊_𝑟1𝑡𝑄_𝑟2𝑡+𝑊_𝑟2𝑡…𝑄_𝑟𝑟𝑡−𝑍_𝑟𝑡⎤⎥⎥⎥⎦,(3.3)𝑄_𝑖𝑗𝑡≜⎡⎢⎢⎢⎢⎢⎢⎣12Γ^𝑖10+Γ_𝑗−•𝛿𝐼••••2Ψ^𝑖𝑗+Ψ_𝑗𝑖12𝐸^𝑖1+𝐸_𝑗−𝑋_𝑡•2Φ^𝑖𝑗+Φ_𝑗𝑖12𝐹^𝑖+𝐹_𝑗0−𝐼⎤⎥⎥⎥⎥⎥⎥⎦,(3.4)Γ_𝑖≜𝑋_𝑖−𝐿−𝐿′,𝑊_𝑖𝑗𝑡≜𝑉_𝑖𝑗𝑡+𝑇_𝑖𝑗𝑡+𝑆_𝑖𝑗𝑡−𝑆_𝑖𝑗𝑡′,(3.5)Ψ_𝑖𝑗≜𝐴_𝑖𝐿+𝐵_𝑖𝑀_𝑗,Φ_𝑖𝑗≜𝐶_𝑖𝐿+𝐷_𝑖𝑀_𝑗,(3.6)𝑉_𝑖𝑗𝑡=21𝑅^𝑖𝑗0,,if𝑖=𝑡or𝑗=𝑡,otherwise,𝑍_𝑖𝑡=⎧⎪⎨⎪⎩𝑅^𝑖𝑡𝑅,if𝑖<𝑡,_𝑡𝑖0,,if𝑖>𝑡,if𝑖=𝑡.(3.7)
Furthermore, the local gains are given by 𝐾_𝑖≜𝑀_𝑖𝐿⁻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: Ξ_𝑡≺0,𝑋_𝑖≻0,𝑇_𝑖𝑗𝑡⪰0,(𝑖>𝑗,𝑖,𝑗,𝑡∈ℛ),(3.8) where Ξ_𝑡 is given as in (3.3). Furthermore, the local gains are given by 𝐾_𝑖≜𝑀_𝑖𝐿⁻¹.

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 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: min_{𝑋𝑖,𝑅𝑖𝑗,𝑇𝑖𝑗𝑡,𝐿,𝑆𝑖𝑗𝑡,𝛿}𝛿s.t.Ξ𝑇_𝑡≺0,𝑋_𝑖≻0,(𝑖,𝑡∈ℛ),_𝑖𝑗𝑡⪰0,(𝑖>𝑗,𝑖,𝑗,𝑡∈ℛ),(3.9) where Ξ_𝑡 is given as in (3.3) and Ψ_𝑖𝑗≜(𝐴_𝑖+𝐵_𝑖𝐾_𝑗)𝐿 and Φ_𝑖𝑗≜(𝐶_𝑖+𝐷_𝑖𝐾_𝑗)𝐿.

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: Ξ_𝑡≺0,𝑋_𝑖≻0,𝑇_𝑖𝑗𝑡⪰0,(𝑖>𝑗,𝑖,j,𝑡∈ℛ),(3.10) where Ξ_𝑡 is given as in (3.3) and Ψ_𝑖𝑗≜(𝐴_𝑖+𝐵_𝑖𝐾_𝑗)𝐿 and Φ_𝑖𝑗≜(𝐶_𝑖+𝐷_𝑖𝐾_𝑗)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 𝑥^1𝑘. The local matrices are 𝐵𝐴₁=1+1𝑎−00.5,𝐴₂=−11−00₁.5,=𝐶1−1𝑏,𝐵₂=−12,𝐸₁=00..23,𝐸₂=−00.5.1₁,=01.5^,𝐶₂=01.5^,𝐷¹𝐹=1,𝐷₂=0.5,₁=0.4,𝐹₂=0.2.(4.1)
The membership functions for this system are ℎ₁𝑥^1𝑘=1−sin𝑥^1𝑘2,ℎ²𝑥^1𝑘=1+sin𝑥^1𝑘2.(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 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.

Table 1 

Comparison of 𝛾_min by different strategies.

Example 4.2. Consider another DFS with 𝑥^1𝑘 as premisse variable. The membership functions and local matrices are ℎ₁𝑥^1𝑘=𝑥^1𝑘+𝛽𝐴₁2𝛽,ℎ²𝑥^1𝑘=1−ℎ₁𝑥^1𝑘,=𝐵−11−−0𝛽.5,𝐴₂=−11−0𝛽.5₁,𝐸=52+𝛽𝛽,𝐵₂=5−−2𝛽₁𝛽,=𝐷𝐸₂₁=−00.0300..0011,𝐶₁=𝐶₂=−−00.0.15^,=𝐷₂=0.5,𝐹₁=𝐹₂=0.010.01.(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.

Table 2 

𝛾_min 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: 𝑤_𝑘=⎧⎪⎪⎪⎨⎪⎪⎪⎩𝑒−0−.002..5235041100100≤≤<𝑘𝑘𝑘≤≤≤6011,1200,,_1𝑘𝑒150≤𝑘≤175,_2𝑘190≤𝑘≤220,(4.5) where 𝑒^1𝑘, 𝑒^2𝑘 are gaussian noises, with zero mean and standard deviations 0.05 and 0.10, respectively.
Assuming as initial conditions 𝑥₀=[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.

Table 3 

Gains obtained by Theorem 3.1 considering 𝛾=0.5.

Figure 1 

Closed-loop system, 𝑥1𝑘 trajectory.

Figure 2 

Closed-loop system, 𝑥2𝑘 trajectory.

Figure 3 

Disturbance signal (dashed) and system regulated output (continuous).

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:𝑥̇₁(𝑡)=1𝐶₁𝑥²(𝑡)𝑅−𝑥¹(𝑡)𝑅−𝑔𝑥¹𝑥̇₂(𝑡),(𝑡)=1𝐶₂𝑥¹(𝑡)𝑅−𝑥²(𝑡)𝑅+𝑥³𝑥̇₃(𝑡),(𝑡)=−𝑥²(𝑡)𝐿−𝑅𝐿⁰𝑥³(𝑡),(5.1) where 𝑔(𝑥₁(𝑡)) is a nonlinear function given by𝑔₂𝑥₁(𝑡)+𝑔¹−𝑔₂||𝑥₁(𝑡)+𝐸||−||𝑥₁(𝑡)−𝐸||2,(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: 𝑥^1𝑘∈[−𝑑,𝑑]. The following fuzzy rules are able to exactly represent the dynamics of the discretized Chua's oscillator:𝑅₁∶⎧⎪⎨⎪⎩IF𝑥^1𝑘ISℳ¹¹(near0),THEN𝑥𝑦_𝑘+1𝑘=𝐴₁𝑥_𝑘+𝐵₁𝑢_𝑘+𝐸₁𝑤_𝑘,𝑅₂=𝐶₁𝑥_𝑘+𝐹₁𝑤_𝑘,∶⎧⎪⎨⎪⎩IF𝑥^1𝑘ISℳ²¹(near±𝑑),THEN𝑥𝑦_𝑘+1𝑘=𝐴₂𝑥_𝑘+𝐵₂𝑢_𝑘+𝐸₂𝑤_𝑘,=𝐶₂𝑥_𝑘+𝐹₂𝑤_𝑘.(5.3)

The local matrices are given by𝐴_𝑖=⎡⎢⎢⎢⎢⎢⎣1−𝑇𝑅𝐶₁−𝑇𝑔^𝑖𝐶₁𝑇𝑅𝐶₁𝑇0𝑅𝐶₂1−𝑇𝑅𝐶₂𝑇𝐶₂0−𝑇𝐵_𝑖𝐿1−𝑇𝐿𝑅⁰⎤⎥⎥⎥⎥⎥⎦,𝐶_𝑖=⎡⎢⎢⎣−001⎤⎥⎥⎦,𝐸_𝑖=⎡⎢⎢⎣1×0010⁻³⎤⎥⎥⎦,=⎡⎢⎢⎣100⎤⎥⎥⎦^,𝐷_𝑖=0,𝐹_𝑖=1×10⁻⁴,(5.4) where 𝑔₂≜𝑈+(𝑔₁−𝑈)𝐸/𝑑.

The membership functions areℎ₁𝑥^1𝑘=⎧⎪⎪⎨⎪⎪⎩−𝛼𝑥^1𝑘+𝐸(1−𝛼)𝑥^1𝑘−𝛼𝑥_1𝑘,𝑥^1𝑘≥𝐸,−𝐸(1−𝛼)𝑥^1𝑘1,,𝑥^1𝑘≤𝐸,ℎ₂otherwise,𝑥^1𝑘=1−ℎ₁𝑥^1𝑘,(5.5) where 𝛼≜𝐸/𝑑.

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

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 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 𝐾_𝑖=𝐶₁𝐾_𝑖/𝑇.

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.

Figure 5 

𝑥1(𝑡) trajectory.

Figure 6 

𝑥2(𝑡) trajectory.

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.

Figure 8 

Control signal.

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 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.

Appendix

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

Lemma A.1. If 𝑆≻0, then 𝐴^𝑖𝑆𝐴_𝑗+𝐴^𝑗𝑆𝐴_𝑖⪯𝐴^𝑖𝑆𝐴_𝑖+𝐴^𝑗𝑆𝐴_𝑗.(A.1)

Proof. Notice that 𝑆≻0⟹𝐴_𝑖−𝐴_𝑗^𝑆𝐴_𝑖−𝐴_𝑗⪰0,(A.2) leading to 𝐴^𝑖𝑆𝐴_𝑖−𝐴^𝑖𝑆𝐴_𝑗−𝐴^𝑗𝑆𝐴_𝑖+𝐴^𝑗𝑆𝐴_𝑗⪰0,(A.3) which completes the proof.

Lemma A.2. Assume that ℎ_𝑖 (𝑖∈ℛ) satisfy (2.3) and let 𝑅_𝑖𝑗 (𝑖<𝑗,𝑖,𝑗∈ℛ) be symmetric matrices of appropriate dimension. Define 𝐻≜⎡⎢⎢⎢⎣𝐻¹¹𝛼₁₂𝑅₁₂…𝛼_1𝑟𝑅_1𝑟𝛼₁₂𝑅₁₂𝐻₂₂…𝛼_2𝑟𝑅_2𝑟𝛼⋮⋮⋱⋮_1𝑟𝑅_1𝑟𝛼_2𝑟𝑅_2𝑟…𝐻_𝑟𝑟⎤⎥⎥⎥⎦,(A.4) where 𝐻_𝑖𝑖≜−^𝑟_{𝑗=1𝑗>𝑖}ℎ_𝑗𝑅_𝑖𝑗−^𝑟_{𝑗=1𝑗<𝑖}𝛼_𝑘𝑙ℎ_𝑗𝑅_𝑗𝑖,≜12ℎ^𝑘+ℎ_𝑙.(A.5)
Then, ℎ₁ℎ₂⋯ℎ_𝑟𝐻ℎ₁ℎ₂⋯ℎ_𝑟^=0.(A.6)

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

Lemma A.3. If 𝑊≻0, then −𝑆𝑊⁻¹𝑆′⪯−(𝑆′+𝑆−𝑊).(A.7)

Proof. Since 𝑊≻0, there exists 𝑊⁻¹. Thus, 𝑊𝑊₋₁(𝑊−𝑆)𝑊⁻¹(𝑊−𝑆)^⪰0,𝑊−𝑊𝑊−𝑊𝑆′⁻¹−𝑆𝑆′−+𝑆𝑆𝑊𝑊⁻¹₋₁𝑊+𝑆𝑊⁻¹𝑆′⪰0,𝑆′⪰0,(A.8) concluding the proof.

Proof of Theorem 3.1. Consider the induced 𝑙₂-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]: 𝒱=Δ𝑉_𝑘+𝑧^𝑘𝑧_𝑘−𝛾²𝑤^𝑘𝑤_𝑘<0,(A.9) where Δ𝑉_𝑘≜𝑥^{𝑟𝑘+1}_𝑡=1ℎ^+𝑡𝑃_𝑡𝑥_𝑘+1−𝑥^𝑘𝑟_𝑖=1ℎ_𝑖𝑃_𝑖𝑥_𝑘,(A.10) is the increment of the fuzzy Lyapunov function candidate 𝑉_𝑘, shown in (2.7).
According to (2.6), it follows that 𝒱=𝜉^{∑𝑟𝑡=1ℎ+𝑡∑𝑟𝑖=1𝑟}_𝑟𝑗=1_𝑟𝑙=1_𝑣=1ℎ_𝑖ℎ_𝑗ℎ_𝑙ℎ_𝑣Λ^𝑖𝑗𝑃_𝑡Λ_𝑙𝑣−𝑃_𝑖𝜉,(A.11) where 𝜉≜[𝑥^𝑘𝑤^𝑘] and Λ_𝑎𝑏≜𝐺^𝑎𝑏𝐸_𝑎𝐽_𝑎𝑏𝐹_𝑎,𝑃_𝑎≜𝑃^𝑎0𝐼0,𝑃_𝑎≜𝑃^𝑎0𝛾0₂𝐼.(A.12) Equation (A.11) is equivalent to 𝒱=𝜉′^𝑟_𝑡=1ℎ^+𝑡𝑟_𝑟𝑖=1_𝑟𝑗=1_𝑟𝑙=1_𝑣=1ℎ_𝑖ℎ_𝑗ℎ_𝑙ℎ_𝑣1×Λ_𝑖𝑗8+Λ_𝑗𝑖^𝑃_𝑡Λ_𝑙𝑣+Λ_𝑣𝑙+Λ_𝑙𝑣+Λ_𝑣𝑙^𝑃_𝑡Λ_𝑖𝑗+Λ_𝑗𝑖−8𝑃_𝑖𝜉.(A.13) By applying Lemma A.1, one obtains 𝒱≤𝜉′^𝑟_𝑡=1ℎ^+𝑡𝑟_𝑟𝑖=1_𝑟𝑗=1_𝑟𝑙=1_𝑣=1ℎ_𝑖ℎ_𝑗ℎ_𝑙ℎ_𝑣18Λ^𝑖𝑗=𝜉′_𝑟+Λ_𝑗𝑖^𝑃_𝑡Λ_𝑖𝑗+Λ_𝑗𝑖+Λ_𝑙𝑣+Λ_𝑣𝑙^𝑃_𝑡Λ_𝑙𝑣+Λ_𝑣𝑙−8𝑃_𝑖𝜉_𝑡=1ℎ^+𝑡𝑟_𝑟𝑖=1_𝑗=1ℎ_𝑖ℎ_𝑗14Λ^𝑖𝑗=𝜉′_𝑟+Λ_𝑗𝑖^𝑃_𝑡Λ_𝑖𝑗+Λ_𝑗𝑖−4𝑃_𝑖𝜉_𝑡=1ℎ^+𝑡^𝑟_𝑖=1ℎ^2𝑖Λ^𝑖𝑖𝑃_𝑡Λ_𝑖𝑖−𝑃_𝑖+^𝑟_𝑖<𝑗ℎ_𝑖ℎ_𝑗Λ^𝑖𝑗+Λ_𝑗𝑖^𝑃_𝑡Λ_𝑖𝑗+Λ_𝑗𝑖−𝑃_𝑖𝜉.(A.14)
Now define Λ_𝑎𝑏𝑐≜𝜆^{11𝑎𝑏𝑐}𝜆_{21𝑎𝑏𝑐}•𝜆^{22𝑎𝑏𝑐},(A.15) where 𝜆𝜆^{11𝑎𝑏𝑐}_{21𝑎𝑏𝑐}≜𝐺^𝑎𝑏𝑃_𝑐𝐺_𝑎𝑏−𝑃_𝑎+𝐽^𝑎𝑏𝐽_𝑎𝑏,𝜆_{22𝑎𝑏𝑐}≜𝐸^𝑎𝑃_𝑐𝐺_𝑎𝑏+𝐹^𝑎𝐽_𝑎𝑏,≜−𝛾²𝐼+𝐸^𝑎𝑃_𝑐𝐸_𝑎+𝐹^𝑎𝐹_𝑎,(A.16) and rewrite (A.14) as 𝒱≤𝒩=𝜉′^𝑟_𝑡=1ℎ^+𝑡⎡⎢⎢⎣^𝑟_𝑖=1ℎ^2𝑖Λ_𝑖𝑖𝑡+2^𝑟_𝑖<𝑗ℎ_𝑖ℎ_𝑗Λ_𝑖𝑗𝑡+Λ_𝑗𝑖𝑡2⎤⎥⎥⎦𝜉.(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 Λ_𝑎𝑏𝑐≜⎡⎢⎢⎢⎢⎣−𝑃^𝑎0−𝛾•₂••𝑃𝐼••_𝑐𝐽𝐺_{𝑎𝑏𝑎𝑏}𝑃_𝑐𝐸_𝑎−𝑃_𝑐•𝐹_𝑎0−𝐼⎤⎥⎥⎥⎥⎦.(A.18)
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: 𝒱≤𝑆₂𝑆₁(𝒩)𝑆^1𝑆^2,(A.19) where 𝑆₁≜diag𝑃^𝑎−1,𝐼,𝑃^𝑐−1,𝐼,𝑆₂≜diag𝐿^𝑃_𝑎,𝐼,𝐼,𝐼,𝒱≜𝑆₂𝑆₁𝒱𝑆^1𝑆^2(A.20) to obtain 𝒱<𝑁=𝜉′^𝑟_𝑡=1ℎ^+𝑡⎡⎢⎢⎣^𝑟_𝑖=1ℎ^2𝑖Λ_𝑖𝑖𝑡+2^𝑟_𝑖<𝑗ℎ_𝑖ℎ_𝑗Λ^𝑖𝑗𝑡+Λ_𝑗𝑖𝑡2⎤⎥⎥⎦𝜉,(A.21) such that Λ_𝑎𝑏𝑐≜⎡⎢⎢⎢⎢⎣−𝐿′𝑋^𝑎−10𝐿−𝛾•₂••𝐺𝐼••_𝑎𝑏𝐽_𝑎𝑏𝐿𝐸_𝑎−𝑋_𝑐•𝐿𝐹_𝑎0−𝐼⎤⎥⎥⎥⎥⎦,𝑋^𝑎≜𝑃^𝑎−1.(A.22)
Applying Lemma A.3 results in 𝒩<𝑀=𝜉′^𝑟_𝑡=1ℎ^+𝑡^𝑟_𝑖=1ℎ^2𝑖Λ̆_𝑖𝑖𝑡+2^𝑟_𝑖<𝑗ℎ_𝑖ℎ_jΛ̆_𝑖𝑗𝑡+Λ̆_𝑗𝑖𝑡2𝜉,(A.23) where Λ̆_𝑎𝑏𝑐≜⎡⎢⎢⎢⎣𝑋^𝑎𝐴−𝐿0′−𝐿−•𝛿𝐼•••_𝑎•𝐶_𝑎𝐿+𝐵_𝑎𝑀_𝑏𝐸_𝑎−𝑋_𝑐•𝐿+𝐷_𝑎𝑀_𝑏𝐹_𝑎0−𝐼⎤⎥⎥⎥⎦,(A.24) and 𝑀_𝑎≜𝐾_𝑎𝐿,𝛿≜𝛾² 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) as in the top of the next page ℳ=𝜉′^𝑟_𝑡=1ℎ^+𝑡^𝑟_𝑖=1ℎ^2𝑖Λ̆_𝑖𝑖𝑡+2^𝑟_𝑖<𝑗ℎ_𝑖ℎ_𝑗Λ̆^𝑖𝑗𝑡+Λ̆_𝑗𝑖𝑡≤𝜉′_𝑟2𝜉_𝑡=1ℎ^+𝑡^𝑟_𝑖=1ℎ^2𝑖Λ̆_𝑖𝑖𝑡+2^𝑟_𝑖<𝑗ℎ_𝑖ℎ_𝑗Λ̆^𝑖𝑗𝑡+Λ̆_𝑗𝑖𝑡2+𝑇^𝑖𝑗𝑡+𝑆_𝑖𝑗𝑡=𝜉′_𝑟𝜉_𝑡=1ℎ^+𝑡⎡⎢⎢⎢⎣ℎ¹ℎ₂ℎ⋮_𝑟⎤⎥⎥⎥⎦^⎡⎢⎢⎢⎢⎢⎢⎣Λ̆^11𝑡•…•V⋮_𝑡Λ⋮̆_22𝑡⋱…⋮•V_𝑟1𝑡V_𝑟2𝑡⋯Λ̆_𝑟𝑟𝑡⎤⎥⎥⎥⎥⎥⎥⎦⎡⎢⎢⎢⎣ℎ₁ℎ₂ℎ⋮_𝑟=𝜉′_𝑟⎤⎥⎥⎥⎦𝜉_𝑡=1=𝜉′_𝑟ℎ^+𝑡ℎ₁ℎ₂⋯ℎ_𝑟Ξ_𝑡ℎ₁ℎ₂⋯ℎ_𝑟^𝜉_𝑡=1=𝜉′_𝑟ℎ^+𝑡ℎ₁ℎ₂⋯ℎ_𝑟Ξ_𝑡+𝐻ℎ₁ℎ₂⋯ℎ_𝑟^𝜉_𝑡=1ℎ^+𝑡ℎ₁ℎ₂⋯ℎ_𝑟Ξ_𝑡ℎ₁ℎ₂⋯ℎ_𝑟𝜉.(A.25) where in (A.25), V_𝑡=(1/2)(Λ̆_12𝑡+Λ̆_21𝑡)+𝑇_21𝑡+𝑆_21𝑡, V_r1t=(1/2)(Λ̆_𝑟1𝑡+Λ̆_1𝑟𝑡)+𝑇_𝑟1𝑡+𝑆_𝑟1𝑡, and V_𝑟2𝑡=(1/2)(Λ̆_𝑟2𝑡+Λ̆_2𝑟𝑡)+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.

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).