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Combining Genetic Algorithms and Lyapunov-Based Adaptation for Online Design of Fuzzy Controllers Vincenzo Giordano, David Naso, and Biagio Turchiano, Senior Member, IEEE

Abstract—This paper proposes a hybrid approach for the design of adaptive fuzzy controllers (FCs) in which two learning algorithms with different characteristics are merged together to obtain an improved method. The approach combines a genetic algorithm (GA), devised to optimize all the configuration parameters of the FC, including the number of membership functions and rules, and a Lyapunov-based adaptation law performing a local tuning of the output singletons of the controller, and guaranteeing the stability of each new controller investigated by the GA. The effectiveness of the proposed method is confirmed using both numerical simulations on a known case study and experiments on a nonlinear hardware benchmark. Index Terms—Adaptive fuzzy control (AFC), genetic algorithms (GAs).

I. I NTRODUCTION

I

N THE LAST decade, a considerable amount of research focused on learning and tuning algorithms for both modelbased and model-free design of fuzzy controllers (FCs). The literature reports successful applications of methods based on, e.g., fuzzy clustering [21], neural networks (NNs) [3], reinforcement learning [26], and genetic algorithms (GAs) [13], just to mention the most widely adopted techniques. This paper focuses on two extensively investigated methods for the automated design of an FC, the genetic-based design (GBD) and the Lyapunov-based design (LBD). In GBD methods [9], [13], the configuration parameters of the controller are set up by means of an evolutionary algorithm. Such an approach can be applied to a variety of parameterized control laws (from PID [6] to linear transfer functions [10], [11], from variableorder discrete-time antiwindup controllers [5] to NNs [17]). The main motivation for GBD is the fact that GAs only require a minimal amount of assumptions on the characteristics of the objective function (parameterizations can simultaneously include binary, integer, and real-valued elements), and can take into account multiple design performance indexes, thus allowing a truly multiobjective design [10], [11]. On the other hand, as also acknowledged in the recent survey [9], the GBD still presents unaddressed problems, including the excessive slowness of the search (a GA generally needs at least several thousands of fitness evaluations to converge), and the difficulty to guarantee a priori the structural properties (closed-loop stability above all) of each new candidate solution (an FC) generated during the

Manuscript received July 15, 2005; revised November 4, 2005 and January 21, 2006. This paper was recommended by Associate Editor X. Wang. The authors are with the Dipartimento di Elettrotecnica ed Elettronica, Politecnico di Bari, 70125 Bari, Italy (e-mail: [email protected]). Digital Object Identifier 10.1109/TSMCB.2006.873187

search. These limitations make it difficult to employ GBD with hardware-in-the-loop (as described in [4] and [18]), restricting its main field of application to simulation models (e.g., [20]). The LBD of FCs is based on ideas drawn from classical adaptive control theory. In this research area, also known as adaptive fuzzy control (AFC) [22], [24], [28], [29] the FC is equipped with an adaptation algorithm derived from mathematical manipulations of the Lyapunov equation. The adaptation progressively drives the FC to approximate an unknown theoretically ideal control law. As in the case of GBD, the research on LBD is a very active field, and many variants and evolutions of the adaptation laws (see the recent survey in [2]) have been devised to: 1) improve the performances of basic schemes [15]; 2) reduce the effects of the inherent approximation error [25]; and 3) extend the applicability to broader classes of nonlinear processes [7]. With respect to GBD, Lyapunov adaptation laws are much faster in converging to the final solution, and make it also possible to define relatively simple supervisory mechanisms that guarantee closed-loop stability. On the other hand, the LBD is only able to optimize performance indexes related to the tracking error, and cannot take into account other control objectives (e.g., cost or oscillation of the control action). Furthermore, every LBD approach has a considerable number of configuration parameters that must be defined by hand, and tuned with trial-and-error procedures. In other words, the AFC design inherits, at least in part, the problem of appropriate configuration of the free parameters already discussed for basic FC design. Motivated by the complementary features of GBD and LBD strategies, this paper proposes a hybrid approach that offers a more systematic, reliable, and effective design method for FCs with guaranteed performances. Essentially, the idea is to devise an online design algorithm that simultaneously exploits the robust global-search capability of a GA, and the fast adaptation action supported by theoretical guarantees of the Lyapunov synthesis. Our approach separates the configuration parameters of the adaptive controller in two partially overlapping subsets, and assigns each set to one strategy. The first subset includes all the parameters defining the controller (e.g., number of membership functions (MFs) and rules, position of the input and output MFs, and learning rate), and is optimized by the GA, which works online to optimize a multiobjective aggregated fitness directly measured on the hardware process. During the evaluation of each controller, the second optimization strategy (the Lyapunov-based algorithm) is applied to the second subset of parameters (the output MFs) so as to: 1) improve the speed of convergence of the GA and 2) guarantee that each tested controller is stable. Even though the combination of GAs and

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Lyapunov theory is not completely new, the hybrid design strategy proposed here is significantly different from the other recent approaches proposed in literature. More specifically, Lam et al. [14] describes a fuzzy-model-based nonlinear control approach in which the conditions that guarantee closedloop stability are determined using Lyapunov and linear matrix inequality (LMI) approaches. In the second step, a GA is used to search for the maximum satisfaction of LMI stability conditions. However, this approach requires an accurate timeinvariant fuzzy Takagi–Sugeno (TS) model of the nonlinear plant. Leung et al. [16] extend the approach to the case of known parameter uncertainties. Note that this combination of GA and Lyapunov theory focuses on stabilization in the presence of uncertainties, rather than on hardware-in-the-loop optimization. Wang et al. [31] propose a direct control scheme in which the standard Lyapunov-based algorithm is replaced by a variant of a GA with a heuristic hill-climbing recombination. Since it is difficult to assess (both theoretically and empirically) whether the error-based objective function driving online adaptation is actually multimodal, and since the hill-climbing GA does not fully assure to avoid local minima, our approach uses the two tools in a different way. Essentially, the Lyapunovbased adaptation (introduced in Section II) is not replaced, but rather complemented by the GA, which searches for the optimal value of all the remaining free parameters of the controller (as illustrated in Section III). The advantages of the proposed hybrid scheme are assessed using both simulated examples derived from recent literature on AFC (Section IV), and a hardware benchmark consisting of the rotary speed control of a nonlinear set of servo drives (Section V). II. P ROBLEM S TATEMENT We consider the problem of controlling an unknown inputaffine single-input single-output (SISO) plant defined by the following nth-order model in normal form: x(n) = f (x) + bu (1) y=x where f (·) is an unknown function, b is the unknown positive input gain, u ∈ R and y ∈ R are plant input and output, respectively, and x = [x x˙ · · · x(n−1) ]T ∈ Rn is the state vector. As mentioned, many studies (e.g., [2], [8], and [29]) have extended AFC schemes to more general classes of models (e.g., state-dependent input gain, disturbances). Nevertheless, in this paper, we prefer to focus on this simple class of systems to keep the mathematical notation simple, underlining that the proposed approach can be used also with other more complex AFC schemes. A precise knowledge of f (·) and b would make it possible to implement the ideal control law guaranteeing a desired linear dynamic for the tracking error e(t) ≡ ym (t) − y(t), where ym (t) is the reference. The ideal law is defined as 1 (n) (n) = −f (x) + ym u∗ = u∗ x, e, ym + kT e b

(2)

where e = [e e˙ · · · e(n−1) ]T is the error vector, k = [kn kn−1 · · · k1 ]T ∈ Rn is the vector describing the de-

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(n)

sired closed-loop dynamic for the error, and ym is the nth derivative of the reference output. In some AFC literature (e.g., [8]), the reference signal ym (t) is obtained as the output of a reference model, excited with a given input signal w. When f (·) and b are unknown, a direct AFC scheme can be used to approximate the ideal law with a fuzzy interpolator (i.e., an FC) (n) (3) |φ uc = uc x, e, ym where φ is a vector of parameters that are continuously adjusted with an adaptation strategy that makes uc tend to u∗ with the minimum error. Various alternative schemes have been proposed to estimate and feed the information related to state, tracking error, and reference to the FC (e.g., the study in [8] only feeds x and w to the FC, and the studies in [23] and [27] use observers to estimate x in cases where it is not directly measurable). In this paper, we adopt the combination of inputs suggested in [8]. The paper shows that, if the reference plant has the same dynamic as the linearized error dynamic expressed by k, the adapted parameters converge to “almost” stable values (i.e., with negligible variations) also for reference signals changing in time. However, for the sake of generality, we will hereafter denote the vector of inputs for the FC with z = [z1 z2 . . . zN ]T , and the FC input space with Z. In this way, (3) can be rewritten as follows: uc = uc z|φ . (4) In addition to error minimization, the AFC scheme is requested to keep all the closed-loop system variables uniformly bounded, i.e., |x(t)| ≤ Mx < ∞, |φ(t)| ≤ Mφ < ∞, and |u(z|φ)| ≤ Mu < ∞, where Mx , Mφ , and Mu are userspecified design constants. Introducing the ideal parameter vector

∗ ∗ (5) φ ≡ arg min sup uc (z|φ) − u |φ|≤Mφ

z∈Z

the minimum approximation error can be defined as ω ≡ uc (z|φ∗ ) − u∗ . In the last decade, many different parameterizations have been proposed for (4) [1], [22]. The most common and straightforward solution is a zero-order TS system with singletons as output MFs. In such a case, defining φ ≡ θ, where θ = [θ1 θ2 · · · θM ]T is the vector of the output singletons and M is the number of rules (NoR), (4) can be written as M

uc (z|θ) =

θl l=1 M l=1

· αl (z)

= θT · ξ(z).

(6)

αl (z)

The notation αl (z) introduces the firing degree of rule l, defined as αl (z) =

N i=1

µli (zi )

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where µli (zi ) is the MF associated to the ith input in the lth rule. Finally, ξ(z) = [ξ1 (z) ξ2 (z) · · · ξM (z)]T is the vector of fuzzy basis functions αl (z)

ξl (z) = M

l=1

αl (z)

optimization strategy. Namely, we decompose the vector of FC parameters λ into two subvectors λ = [θT

.

(10)

(8)

This type of parameterization is commonly adopted because it makes the control action (6) a linear weighted combination of the adapted parameters θ, thus permitting very simple singleton-adaptation laws, such as the one proposed in [28] and many subsequent works θ˙ = γeT pn ξ(z)

ϕT ]T

(9)

where γ is the learning rate, and pn is the last column of a definite positive matrix P , satisfying the Lyapunov equation P · K + K T · P = −Q, where matrix K is the companion form of k and Q is a user-defined positive-definite matrix. The literature [28], [29] proves that the error asymptotically tends to zero provided that the minimum approximation error between uc and u∗ is squared integrable. The design of AFC schemes based on Lyapunov theory is very important in the context of fuzzy control, because this makes it possible to address closedloop stability issues. Namely, it can be easily shown [28] that a quadratic form Ve = eT P e/2 can be defined on the tracking error, and its derivative V˙ e can be made semidefinite negative by adding a stabilizing control action us to the output of the FC uc (for brevity, we omit the details about us , which can be found in [28]). A number of different strategies to guarantee stability have also been proposed (e.g., [22]), and could be used in our scheme in place of the stabilizing supervision us without particular difficulties. The AFC literature underlines that the inherent approximation error may significantly compromise the performance of the control loop. Despite the various efforts dedicated to this problem (e.g., [22] and [24]), it is generally not easy to determine how much these effects are influenced by the type of FC parameterization, and by the choice of the other configuration parameters that are not directly adjusted by the adaptation law. Theoretical results about fuzzy-data-driven identification [30] suggest that, in general, the approximation error can be reduced by increasing the number of MFs. While a certain amount of literature about AFC relies on this assumption as a means to disregard the actual effects of this error, our closed-loop experimental investigations show that increasing the number of MFs may not necessarily lead to performance improvements due to overparametrization. Experimental studies also show that the results of the AFC scheme are strongly influenced by many nonadapted configuration parameters of the FC, which must be set by hand.

III. O NLINE GA-B ASED D ESIGN OF A DAPTIVE FC The basic idea of this paper is to partition the set of configuration parameters in two subsets, each handled by a different

where θ is the vector of the output singletons, and ϕ is the vector of all the other configuration parameters of the FC, including: 1) the positions of the input MFs; 2) the input and output scaling factors; and 3) the learning rate of the adaptation law. The partition of λ separates the parameters that have a linear influence on uc [see (6)] from those that have a nonlinear influence of uc . If the value of the parameters in the vector ϕ is defined, the values of the corresponding θ minimizing the tracking error can be obtained with an adaptation law such as the integral of (9). Basically, while in AFC literature, ϕ is set a priori (generally as the result of preliminary hand tuning), here, we adopt a GA to optimize simultaneously ϕ and θ. In addition, the adaptation law (9) is applied to every new candidate solution to effectively adjust the values of θ. The choice of a GA allows us to also incorporate structural information about the FC in vector ϕ, such as the numbers of input MFs and output singletons. Our GA is based on the standard form well described in technical literature [19]. Therefore, for brevity, in the following, we only focus on the peculiar aspects of our implementation. A. Coding Strategy We adopt the zero-order TS fuzzy system specified in (6)–(9) with triangular input MFs and output singletons as general structure of the FC. The GA is allowed to modify both the structure (number of MFs and rules) and the parameters (position of MFs, scaling gains, and learning rate). In other words, the number of MFs that cover each input of the FC is not specified a priori, but optimized by the GA. To initialize the GA, the user must specify the maximum number of MFs admissible on each input of the FC. Indicating this number as mi (subscript i refers to the ith input, i = 1, . . . , N ), the maximum NoR forming a complete rule base for the FC is Mmax =

N

mi .

(11)

i=1

Fig. 1 illustrates the encoding scheme, showing that a chromosome contains both binary and real-valued genes. The coding strategy assumes that, for each input i, at least two MFs (the open-left and open-right triangular MFs centered in −1 and +1) are always present, while the remaining mi − 2 MFs can be freely added or removed by the GA. As a further constraint related to input coverage, the triangular MFs must always satisfy the condition of crossing the two adjacent MFs at 0.5 (this condition is referred to as having α-level = 0.5 in fuzzyset theory). This constraint is convenient both for the actual implementation of the adaptive FC, and for the solution encoding. In fact, the constraint on the α-level in conjunction with the use of triangular MFs (having limited support) guarantees

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Fig. 1. Schema of a generic chromosome used by the GA. The genes related to output singletons are depicted in gray to emphasize the fact that they are also processed by the LBD algorithm. For space limitations, λ, θ, and ϕ are represented as row vectors.

Fig. 2.

Examples of (portions of) chromosomes and the corresponding input MF partitions (mi = 5 in this example).

that no more than two MFs can be simultaneously fired on each input. Therefore, the maximum number of nonzero elements in the vector of fuzzy basis functions ξl (z) is lower or equal to 2N and, due to (9), no more than 2N singletons are actually adapted at each time step. This means that, independently of the actual number and placement of MFs, the computational effort for the singleton adaptation at each time step remains limited, which is a fundamental advantage for the experimental implementation of the scheme. Moreover, the α-level constraint also reduces the number of free parameters defining the MFs. In fact, the constraint makes the shape of each MF (except the first and the last on the universe of discourse) depend only on its center (i.e., the position of the upper vertex, as in the examples in Fig. 2). This strategy makes it possible to encode each MF in the chromosome by means of only two parameters: 1) the center of the MF on the corresponding universe of discourse and 2) the structural flag (SF) associated to the MF. SFs are control genes (inspired to the structural genes in the DNA of biological entities) often used in GAs to perform structural changes to the encoded solutions (see, e.g., [5] and [32]). If the SF of an MF is set to zero, the MF is removed from the input universe, and the rules in which it appears are removed from the rule base, thus reducing the complexity of the controller. Note that changing the SF of an MF also makes the two adjacent MFs change due to the mentioned constraints (see Fig. 2). Besides SFs and centers of input MFs, the chromosome specifies the position of the Mmax output singletons (only those corresponding to

nonremoved rules will actually take part in the inference), the scaling gains, and the learning rate. B. Fitness Evaluation The fitness of a control design problem is a scalar measure of the overall performance of the control loop, obtained by computing or performing a predefined and repeatable (simulated or experimental) evaluation test. In our approach, the fitness evaluation is also the phase in which the two learning techniques T T interact. Let us consider a generic chromosome λ0 = [θT 0 ϕ0 ] (e.g., obtained by random initialization, or by a crossover or mutation of existing solutions) univocally describing a controller FC0 . As Fig. 3 shows, the fitness evaluation of λ0 is composed of two consecutive phases: 1) the adaptation phase and 2) the performance evaluation phase. In the adaptation phase, a closed-loop experiment in which the plant is controlled by FC0 is carried out. During this phase, the position of the singletons of the controller θ0 is adjusted applying the Lyapunov adaptation algorithm. The role of the LBD is twofold: 1) it works as a local search algorithm that rapidly improves the tracking performance of the controller and 2) it guarantees the closed-loop stability of each experimented controller by means of the additive supervisory action us . As an effect of adaptation, the position of the singletons θ 0 at the end of the adaptation phase is, in general, different from the initial one, i.e., θ 0 = θ0 . In the subsequent evaluation phase, the adaptation is

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Fig. 3. Fitness evaluation. Adaptation and evaluation phases.

switched off, and the closed-loop experiment proceeds controlled by the nonadaptive FC having the singletons steadily constant in the positions θ 0 . Only during this second phase are the indexes of performance of the FC measured and used to compute the fitness at the end of the experiment. Once the T T fitness is computed, the updated chromosome λ0 = [θ T 0 ϕ0 ] replaces the original one λ0 , and the GA proceeds with the next operation. The interruption of the adaptation in the evaluation phase allows us to perform a fair comparison with purely GA-based design schemes in which the optimized parameters are not changed during the evaluation experiment. It is also important to observe that the GA allows one to define fitness functions that take into account not only the tracking error, but also any other measurable, observable, or calculable characteristic of the considered problem (e.g., see the fitness functions based on the oscillations of control action, and on the complexity of the FC used in the next sections). C. Crossover and Mutation All the binary and real-valued parameters (i.e., both θ and ϕ) undergo mutation and recombination. The crossover and mutation operators are variants of conventional operators suitable to deal with the proposed encoding. The mutation can be either binary or real depending on the parameter selected for mutation. Our preliminary experiments (and also previous research on the subject [5]) showed that mutation rates higher than the values usually suggested by literature are indispensable (especially for the binary mutation) for an effective exploration of alternative FC structures. Similar considerations can be extended to the crossover, which also consists of various operators applied with different frequencies. If the crossing point falls on the SFs, a multipoint crossover is applied (i.e., flags are randomly swapped between individuals). Otherwise, one of three different real-valued crossovers (simple, arithmetical, and heuristic crossover, see [19] for details) is applied. Readers may refer

to [5] for a discussion about the profitable interaction between these operators. IV. S IMULATED C ASE S TUDY To evaluate the effectiveness of the proposed scheme, we first focus on a simulated case study derived from technical literature [8]. The controlled plant is a second-order nonlinear dc motor described by the following model: x˙ 1 = x2 (12) 2) x˙ 2 = − f (x + C · Ju J where the state variable x1 is the angular position of the rotor in radians, x2 is the angular speed, u is the current fed in amperes, and the constants are defined as C = 10 N · m/A, and J = 0.1 kg · m2 . The nonlinear friction torque is defined as f (x2 ) = 5 arctan(5x2 ) N · m. The reference signal is a square wave of random amplitude filtered by a linear second-order system with two poles in s = −20 (s indicates the Laplace complex variable). The model is simulated in a Matlab/Simulink environment with a fixed-step Euler method (ode1) with sample time T = 0.001 s. As mentioned, in all the considered schemes, the FC is fed with the combination of inputs suggested in [8] uc = uc (x, w|λ).

(13)

We compare the three following design strategies. 1) LBD: This design strategy only optimizes the output singletons θ while all the other configuration parameters ϕ must be specified a priori. In our simulated or experimental investigations, all the input MFs are evenly distributed on the corresponding universe of discourse. The learning rate is γ = 1000, and input and output scaling gains are defined so as to map the ranges of the corresponding variables into the interval [−1,1].

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TABLE I SUMMARY OF THE CONFIGURATION PARAMETERS FOR THE GAs (UNLESS SPECIFIED, THE SAME PARAMETER IS USED FOR BOTH GAs)

2) GBD: This design strategy optimizes all the configuration T T parameters of the controller λ0 = [θT 0 ϕ0 ] using only the GA (no Lyapunov-based adaptation is performed). 3) Hybrid genetic- and Lyapunov-based design (HBD): This strategy combines the two design methods as described in the previous sections. Here, λ0 is optimized by the GA, and the Lyapunov algorithm adjusts the singletons θ0 . The bounds for each optimized parameter are the same used for GBD. Table I summarizes the final configurations of the two GAs used in GBD and HBD, respectively. For both GAs, we have chosen the setting that leads to the best observed performance. In particular, GBD needs a larger population to initially explore and determine satisfactory placements of output singletons (i.e., to find the coarse shape of the input–output law), while the HBD takes advantage of the singleton adaptation and works well with a small population. The first set of runs studies the sensitivity of the control performance with respect to the position of the input MFs when a fixed number of evenly distributed MFs and a complete rule base are used. The fitness is the integral absolute error (IAE) between the filtered reference ym and the motor output y = x1 . The stopping criterion for all the algorithms is the total amount of simulated time ST = 10 h. Simulation time is chosen very long in order to evaluate the actual results that would be obtained by each method if the design could be performed on the controlled plant without time limitations. The upper part of Table II shows the results of this set of simulated experiments. It can be immediately noted that the amount of allowed search time is still insufficient for the GBD to reach the performances of the other two strategies. Results also show that, as the number of input MFs decreases, the performance of the LBD strategy becomes unsatisfactory, whereas the HBD seems relatively less influenced by the reduction of the number of free parameters. The second set of simulations analyzes the tradeoff between complexity and performance. In this set, the two GA-based methods (GBD and HBD) are allowed to optimize also the number of input MFs (and rules). To reduce the complexity of the controller in terms of NoR, in this case, the fitness function is defined as a weighted combination of IAE and NoR, i.e., FF = a1 · IAE + a2 · NoR

(14)

TABLE II SIMULATION RESULTS (AVERAGE AND STANDARD DEVIATIONS ARE ROUNDED TO THE NEXT INTEGER)

where weights a1 (= 1.5) and a2 (= 1) are defined heuristically. The results (summarized in the last two rows of Table II) show that the HBD provides solutions having, in average terms: 1) less rules than the maximum allowed (7 × 7 × 7 = 343) and 2) a smaller IAE with respect to the case of fixed a priori structure (the average IAE obtained with a free structure is about 30% lower than the best IAE obtained with a fixed structure). To emphasize the twofolded advantage of this strategy, the last row of Table II reports two sample solutions, one with a relatively large NoR (294) and the smallest value for the IAE achieved in all the simulations, and a second one with fewer rules (180) and an IAE index comparable to the one obtained with the largest fixed structure (343 rules). It is also interesting to note that the tracking performance and the controller complexity are not always conflicting, and that the HBD allows one to automatically reduce the NoR while holding a satisfactory tracking performance. V. E XPERIMENTS The experimental benchmark consists of a couple of two nonlinear 370-W dc machines (AMIRA DR300) mounted on

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the same shaft. Both dc machines have a rated speed of 3000 rpm. The first one works as a motor and represents the controlled plant, whereas the second machine (the generator) is used to apply time-varying loads or increase the system nonlinearity. Both dc machines have built-in analog current controllers, whose reference is a voltage signal, which is provided by an external control board (dSPACE 1104) equipped with a 250-MHz Motorola microcontroller. The control board runs the FC, the adaptation law, and the stability supervisor, all automatically converted into discrete-time routines (Euler discretization, with sampling time T = 0.001 s). The board also processes the control signals applied to the load generator. Voltage references are fed to the dc drive using 16-bit D/A converters that are integrated in the dSPACE board. The GA (with the same configuration used in the simulations) and additional algorithms dealing with reference signals and process monitoring are written in Matlab and run in real time directly on the host PC. The electromechanical relation between the reference voltage u(t) and the angular speed ω(t) of the shaft can be written in a normal form as follows:

TV (x)R KKI KV KI R 1 dTV (x) x− + u(t) x− ˙ x ¨= − − L J dx LJ LJ LJ (15) where x = ω and x = [ω ω] ˙ T , R and L are armature resistance and inductance, respectively, J is the moment of inertia of the overall equipment, TV is an unknown nonlinear load torque (varied through the second dc machine), and K, KV , and KI are three constants. Frictions and other inherent nonlinearities make the servo motor nonlinear also when no additional load is applied by the generator. In this paper, the additional torque TV is chosen proportional to x3 , thus making the plant strongly nonlinear. In all the experiments, the desired output ym to be tracked is the output of a reference model having a double pole in s = −20. The reference signal w is a square wave obtained repeating a pattern consisting of six steps of 2 s (each pattern lasts 12 s and will be referred to as an adaptation cycle). The amplitude of each step in each adaptation cycle varies randomly between 0 and 2000 rpm, except for the cycles applied during fitness evaluation, which are all obtained by repeating the same identical pattern (called the evaluation cycle), so as to make fitness values comparable. The GA configuration is identical to the previous simulated benchmark (see Table I), except for the stopping criterion, which is set one order of magnitude shorter, i.e., ST = 1 h. A. Preliminary Sensitivity Analysis As in the previous simulations, we first compare several LBDs with different numbers of evenly placed input MFs on the input universes. The performance index chosen for evaluation is the IAE between the outputs of the reference model and the controlled plant. Each experiment (replicated ten times) uses 20 adaptation cycles with a learning rate γ = 5.0 e − 5, and a final evaluation cycle. The results in Table III substantially confirm the main conclusions drawn for the simulated case

TABLE III EXPERIMENTAL RESULTS WITH A FIXED STRUCTURE, USING IAE AS FITNESS FOR GAs (AVERAGE AND STANDARD DEVIATIONS ARE ROUNDED TO THE NEXT INTEGER)

study (merely using a higher number of MFs does not necessarily guarantee that the LBD obtains an improved tracking). Table III also shows the performance of GBD and HBD of FCs with a fixed predefined structure. It is evident that the position of the input MFs affects the overall tracking performances in a measure that also depends on the number of MFs. The largest structures are evidently overparametrized and cannot be properly optimized. B. Deﬁnition of Fitness Functions Another set of experiments compares various definitions of fitness functions combining indexes related to the tracking error, the smoothness of control action (i.e., two indexes describing the overall performance of the control loop), and the NoR used by the FC. The smoothness of the control action (SCA) is measured by filtering the control action with a linear second-order filter with two poles in s = −100, and comparing the actual control action with the filtered action by means of the IAE between the two signals. High SCA indexes always correspond to very nervous control actions that cause evident and undesirable stresses to the electromechanical equipment (e.g., compare the two actions shown in Fig. 4). The three indexes are combined using a linear weighted aggregation, so that the most general expression for the fitness function can be defined as FF = a1 · IAE + a2 · NoR + a3 · SCA.

(16)

Positive constants a1 , a2 , and a3 are chosen heuristically with a two-stage procedure. We preliminarily evaluate (for the IAE and SCA indexes) the ranges of values that correspond to satisfactory tracking and smoothness performances, and then use constants ai so as to: 1) normalize the contribution given by each weighted index in the sum and 2) emphasize the influence of tracking performance with respect to smoothness of control and FC complexity. Also, in these experiments, the stopping criterion is ST = 1 h for all the runs. Table IV reports the average values of the

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Fig. 4.

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Comparison of control actions obtained with different fitness functions.

TABLE IV EXPERIMENTS WITH AN OPTIMIZABLE STRUCTURE AND VARIOUS AGGREGATED FITNESS FUNCTIONS (AVERAGE AND STANDARD DEVIATIONS ARE ROUNDED TO THE NEXT INTEGER)

view at low and high reference speeds. Reducing the learning rates or increasing the NoR (two typical operations performed during AFC fine tuning to reduce the oscillations), do not help to overcome this flaw, as they produce smoother responses but significantly larger steady-state errors. The tracking of the HBD is indeed more satisfactory, both around low- and high-speed profiles, confirming the effectiveness of the obtained nonlinear control law. VI. C ONCLUSION

three indexes, and the standard deviation of the IAE for comparison with the results in Table III. It can be noted that the incorporation of multiple performance criteria in the fitness function can improve the results. Namely, even when the NoR and SCA are not used in the fitness, the HBD tends to use fewer rules than the maximum available, confirming that larger structures are unnecessary to optimize the tracking performance. When the NoR and SCA are introduced, the performances of the HBD are further improved (in this case, we obtain the best controller of the whole experimental campaign). Fig. 4 compares the control actions of the best FC obtained with IAE fitness, and with the three-term fitness, respectively. Note in Table IV that the smoother control action does not affect the tracking performance. Fig. 5 provides a comparison of the tracking performances of the best FCs obtained with HBD and LBD. In spite of the relatively low IAE value (IAE = 9102, see Table III), the best LBD exhibits a nervous tracking performance with oscillations that are clearly visible in the expanded

This paper has proposed a combination of existing methodologies to perform the design of an adaptive FC online. The suggested hybrid design approach achieves several advantages with respect to its single components. Namely, the adaptation law can be assimilated to a local search technique that contributes to make the genetic design process much faster. Additionally, the local adaptation process limits the inherent risks of experimenting badly performing controllers, simplifying the online implementation. On the other hand, the genetic approach greatly simplifies the design of the adaptive FC, overcoming the need of trial-and-error sessions and allowing the user to achieve the best approximation error without necessarily increasing the NoR. Finally, by defining a suitable fitness function for the overall tuning process, the genetic optimization can be used to accomplish other control objectives apart from those guaranteed by AFC. The integration of other efficient iterative discrete optimization algorithms (e.g., [12]) to accelerate the selection of the adequate FC structure, and the redesign of the prototype algorithms

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Fig. 5.

IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 36, NO. 5, OCTOBER 2006

Comparison of tracking performance. Detailed views of tracking at different reference speeds emphasize the differences between the LBD and the HBD.

for their integration on less expensive microcontroller-based hardware platforms (for self-tuning of industrial servo drives) are among the open research directions currently under investigation.

[15] [16]

ACKNOWLEDGMENT The authors wish to thank the Editor-in-Chief D. J. Cook and the reviewers for their valuable cooperation.

[17] [18]

R EFERENCES [1] M. Alata, C.-Y. Su, and K. Demirli, “Adaptive control of a class of nonlinear systems with a first-order parameterized Sugeno fuzzy approximator,” IEEE Trans. Syst., Man, Cybern. C, Appl. Rev., vol. 31, no. 3, pp. 410–419, Aug. 2001. [2] D. Bellomo, “Stable adaptive fuzzy control for input-affine nonlinear systems,” Ph.D. dissertation, Dipartimento di Elettrotecnica ed Elettronica, Politecnico di Bari, Bari, Italy, 2005. [3] J. J. Buckley and Y. Hayashi, “Fuzzy neural networks: A survey,” Fuzzy Sets Syst., vol. 66, no. 1, pp. 1–13, Aug. 1994. [4] F. Cupertino, V. Giordano, D. Naso, B. Turchiano, and L. Salvatore, “On-line genetic design of fuzzy controllers for DC drives with variable load,” Electron. Lett., vol. 39, no. 5, pp. 479–480, Mar. 2003. [5] F. Cupertino, E. Mininno, D. Naso, B. Turchiano, and L. Salvatore, “On-line genetic design of anti-windup unstructured controllers for electric drives with variable load,” IEEE Trans. Evol. Comput., vol. 8, no. 4, pp. 347–364, Aug. 2004. [6] W. G. da Silva, P. P. Acarnley, and J. W. Finch, “Application of genetic algorithms to the online tuning of electric drive speed controllers,” IEEE Trans. Ind. Electron., vol. 47, no. 1, pp. 217–219, Feb. 2000. [7] Y. Diao and K. M. Passino, “Adaptive neural/fuzzy control for interpolated nonlinear systems,” IEEE Trans. Fuzzy Syst., vol. 10, no. 5, pp. 583–595, Oct. 2002. [8] K. Fischle and D. Schroder, “An improved stable adaptive fuzzy control method,” IEEE Trans. Fuzzy Syst., vol. 7, no. 1, pp. 27–40, Feb. 1999. [9] P. J. Fleming and R. C. Purshouse, “Evolutionary algorithms in control system engineering: A survey,” Control Eng. Pract., vol. 10, no. 11, pp. 1223–1241, Nov. 2002. [10] C. M. Fonseca and P. J. Fleming, “Multiobjective optimization and multiple constraint handling with evolutionary algorithms—Part I: A unified formulation,” IEEE Trans. Syst., Man, Cybern. A, Syst., Humans, vol. 28, no. 1, pp. 26–37, Jan. 1998. [11] ——, “Multiobjective optimization and multiple constraint handling with evolutionary algorithms—Part II: Application example,” IEEE Trans. Syst., Man, Cybern. A, Syst., Humans, vol. 28, no. 1, pp. 38–47, Jan. 1998. [12] Y. C. Ho and J. T. Lee, “Granular optimization: An approach to function optimization,” in Proc. 39th IEEE Conf. Decision Control, Sydney, Australia, Dec. 2000, pp. 103–111. [13] F. Hoffmann, “Evolutionary algorithms for fuzzy control system design,” Proc. IEEE, vol. 89, no. 9, pp. 1318–1333, Sep. 2001. [14] H. K. Lam, F. H. Leung, and P. K. S. Tam, “Design and stability analysis of fuzzy model-based nonlinear controller for nonlinear systems using

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genetic algorithm,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 33, no. 2, pp. 250–257, Apr. 2003. Y.-G. Leu, W.-Y. Wang, and T.-T. Lee, “Robust adaptive fuzzy-neural controllers for uncertain nonlinear systems,” IEEE Trans. Robot. Autom., vol. 15, no. 5, pp. 805–817, Oct. 1999. F. H. F. Leung, H. K. Lam, S. H. Ling, and P. K. S. Tam, “Optimal and stable fuzzy controllers for nonlinear systems based on an improved genetic algorithm,” IEEE Trans. Ind. Electron., vol. 51, no. 1, pp. 172–182, Feb. 2004. C. T. Lin and C.-P. Jou, “Controlling chaos by GA-based reinforcement learning neural network,” IEEE Trans. Neural Netw., vol. 10, no. 4, pp. 846–859, Jul. 1999. D. A. Linkens and H. O. Nyongesa, “Genetic algorithms for fuzzy control, part 2: Online system development and application,” Proc. Inst. Elect. Eng.—Control Theory Appl., vol. 142, no. 3, pp. 177–185, May 1995. Z. Michalewicz, Genetic Algorithms + Data Structures = Evolution Programs. Berlin, Germany: Springer-Verlag, 1996. A. Rayapakse, K. Furuta, and S. Kondo, “Evolutionary learning of fuzzy logic controllers and their adaptation through perpetual evolution,” IEEE Trans. Fuzzy Syst., vol. 10, no. 3, pp. 309–321, Jun. 2002. M. Setnes and H. Roubos, “GA-fuzzy modeling and classification: complexity and performance,” IEEE Trans. Fuzzy Syst., vol. 8, no. 5, pp. 509–522, Oct. 2000. J. T. Spooner and K. M. Passino, “Stable adaptive control using fuzzy systems and neural networks,” IEEE Trans. Fuzzy Syst., vol. 4, no. 3, pp. 339–359, Aug. 1996. J. T. Spooner, M. Maggiore, R. Ordonez, and K. M. Passino, Stable Adaptive Control and Estimation for Nonlinear Systems. New York: Wiley, 2002. C.-Y. Su and Y. Stepanenko, “Adaptive control of a class of nonlinear systems with fuzzy logic,” IEEE Trans. Fuzzy Syst., vol. 2, no. 4, pp. 285–294, Nov. 1994. C.-S. Tseng, B.-S. Chen, and H.-J. Uang, “Fuzzy tracking control design for nonlinear dynamic systems via T-S fuzzy model,” IEEE Trans. Fuzzy Syst., vol. 9, no. 3, pp. 381–392, Jun. 2001. W. M. van Buijtenen, G. Schram, R. Babuska, and H. B. Verbruggen, “Adaptive fuzzy control of satellite attitude by reinforcement learning,” IEEE Trans. Fuzzy Syst., vol. 6, no. 2, pp. 185–194, May 1998. C. H. Wang, H. L. Liu, and T. C. Lin, “Direct adaptive fuzzy-neural control with state observer and supervisory controller,” IEEE Trans. Fuzzy Syst., vol. 10, no. 1, pp. 39–49, Feb. 2002. L. X. Wang, “Stable adaptive fuzzy control of nonlinear systems,” IEEE Trans. Fuzzy Syst., vol. 1, no. 2, pp. 146–155, May 1993. ——, “Stable adaptive fuzzy controllers with application to inverted pendulum tracking,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 26, no. 5, pp. 677–691, Oct. 1996. L.-X. Wang, “Design and analysis of fuzzy identifiers of nonlinear dynamic systems,” IEEE Trans. Autom. Control, vol. 40, no. 1, pp. 11–23, Jan. 1995. W.-Y. Wang, C.-Y. Cheng, and Y.-G. Leu, “An online GA-based outputfeedback direct adaptive fuzzy-neural controller for uncertain nonlinear systems,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 34, no. 1, pp. 334–345, Feb. 2004. C.-C. Wong and C.-C. Chen, “A GA-based method for constructing fuzzy systems directly from numerical data,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 30, no. 6, pp. 904–911, Dec. 2000.

GIORDANO et al.: GENETIC ALGORITHMS AND LYAPUNOV-BASED ADAPTATION

Vincenzo Giordano was born in Bari, Italy, in 1977. He received the M.S. degree (with honors) in electrical engineering and the Ph.D. degree in control engineering from the Polytechnic of Bari, Bari, in 2001 and 2005, respectively. In 2004, he was a Visiting Ph.D. student with the Automation and Robotics Research Institute, University of Texas, Arlington, under the supervision of Prof. F. Lewis. In 2005, he was a Visiting Researcher with the Singapore Institute of Manufacturing Technology, Singapore. As a Ph.D. student, he was coresponsible for the organization and startup of the Robotics Laboratory at the Polytechnic of Bari. He has published more than 20 international journal and conference papers. His research interests include intelligent control techniques applied to industrial automation, robotics, and discrete-event systems.

David Naso received the laurea degree (with honors) in electronic engineering and the Ph.D. degree in electrical engineering from the Polytechnic of Bari, Bari, Italy, in 1994 and 1998, respectively. He was a Guest Researcher with the Operation Research Institute, Technical University of Aachen, Aachen, Germany, in 1997. Since 1999, he has been an Assistant Professor of automatic control with the Department of Electric and Electronic Engineering, Polytechnic of Bari. He has published more than 60 journal and conference papers. His research interests include computational intelligence and its application to industrial automation and robotics, numerical and combinatorial optimization, discreteevent systems modeling and control, and distributed control of manufacturing systems.

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Biagio Turchiano (M’94–SM’01) was born in Bari, Italy. He received the degree (with honors) in electrical engineering from the University of Bari, Bari, in 1979. In 1984, he joined the Department of Electrical and Electronic Engineering, Polytechnic of Bari, Bari, as an Assistant Researcher and is currently a Full Professor of automatic control. His research interests include production automation, systems and control theory, and modeling and control of discreteevent systems.

IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 36, NO. 5, OCTOBER 2006

Combining Genetic Algorithms and Lyapunov-Based Adaptation for Online Design of Fuzzy Controllers Vincenzo Giordano, David Naso, and Biagio Turchiano, Senior Member, IEEE

Abstract—This paper proposes a hybrid approach for the design of adaptive fuzzy controllers (FCs) in which two learning algorithms with different characteristics are merged together to obtain an improved method. The approach combines a genetic algorithm (GA), devised to optimize all the configuration parameters of the FC, including the number of membership functions and rules, and a Lyapunov-based adaptation law performing a local tuning of the output singletons of the controller, and guaranteeing the stability of each new controller investigated by the GA. The effectiveness of the proposed method is confirmed using both numerical simulations on a known case study and experiments on a nonlinear hardware benchmark. Index Terms—Adaptive fuzzy control (AFC), genetic algorithms (GAs).

I. I NTRODUCTION

I

N THE LAST decade, a considerable amount of research focused on learning and tuning algorithms for both modelbased and model-free design of fuzzy controllers (FCs). The literature reports successful applications of methods based on, e.g., fuzzy clustering [21], neural networks (NNs) [3], reinforcement learning [26], and genetic algorithms (GAs) [13], just to mention the most widely adopted techniques. This paper focuses on two extensively investigated methods for the automated design of an FC, the genetic-based design (GBD) and the Lyapunov-based design (LBD). In GBD methods [9], [13], the configuration parameters of the controller are set up by means of an evolutionary algorithm. Such an approach can be applied to a variety of parameterized control laws (from PID [6] to linear transfer functions [10], [11], from variableorder discrete-time antiwindup controllers [5] to NNs [17]). The main motivation for GBD is the fact that GAs only require a minimal amount of assumptions on the characteristics of the objective function (parameterizations can simultaneously include binary, integer, and real-valued elements), and can take into account multiple design performance indexes, thus allowing a truly multiobjective design [10], [11]. On the other hand, as also acknowledged in the recent survey [9], the GBD still presents unaddressed problems, including the excessive slowness of the search (a GA generally needs at least several thousands of fitness evaluations to converge), and the difficulty to guarantee a priori the structural properties (closed-loop stability above all) of each new candidate solution (an FC) generated during the

Manuscript received July 15, 2005; revised November 4, 2005 and January 21, 2006. This paper was recommended by Associate Editor X. Wang. The authors are with the Dipartimento di Elettrotecnica ed Elettronica, Politecnico di Bari, 70125 Bari, Italy (e-mail: [email protected]). Digital Object Identifier 10.1109/TSMCB.2006.873187

search. These limitations make it difficult to employ GBD with hardware-in-the-loop (as described in [4] and [18]), restricting its main field of application to simulation models (e.g., [20]). The LBD of FCs is based on ideas drawn from classical adaptive control theory. In this research area, also known as adaptive fuzzy control (AFC) [22], [24], [28], [29] the FC is equipped with an adaptation algorithm derived from mathematical manipulations of the Lyapunov equation. The adaptation progressively drives the FC to approximate an unknown theoretically ideal control law. As in the case of GBD, the research on LBD is a very active field, and many variants and evolutions of the adaptation laws (see the recent survey in [2]) have been devised to: 1) improve the performances of basic schemes [15]; 2) reduce the effects of the inherent approximation error [25]; and 3) extend the applicability to broader classes of nonlinear processes [7]. With respect to GBD, Lyapunov adaptation laws are much faster in converging to the final solution, and make it also possible to define relatively simple supervisory mechanisms that guarantee closed-loop stability. On the other hand, the LBD is only able to optimize performance indexes related to the tracking error, and cannot take into account other control objectives (e.g., cost or oscillation of the control action). Furthermore, every LBD approach has a considerable number of configuration parameters that must be defined by hand, and tuned with trial-and-error procedures. In other words, the AFC design inherits, at least in part, the problem of appropriate configuration of the free parameters already discussed for basic FC design. Motivated by the complementary features of GBD and LBD strategies, this paper proposes a hybrid approach that offers a more systematic, reliable, and effective design method for FCs with guaranteed performances. Essentially, the idea is to devise an online design algorithm that simultaneously exploits the robust global-search capability of a GA, and the fast adaptation action supported by theoretical guarantees of the Lyapunov synthesis. Our approach separates the configuration parameters of the adaptive controller in two partially overlapping subsets, and assigns each set to one strategy. The first subset includes all the parameters defining the controller (e.g., number of membership functions (MFs) and rules, position of the input and output MFs, and learning rate), and is optimized by the GA, which works online to optimize a multiobjective aggregated fitness directly measured on the hardware process. During the evaluation of each controller, the second optimization strategy (the Lyapunov-based algorithm) is applied to the second subset of parameters (the output MFs) so as to: 1) improve the speed of convergence of the GA and 2) guarantee that each tested controller is stable. Even though the combination of GAs and

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GIORDANO et al.: GENETIC ALGORITHMS AND LYAPUNOV-BASED ADAPTATION

Lyapunov theory is not completely new, the hybrid design strategy proposed here is significantly different from the other recent approaches proposed in literature. More specifically, Lam et al. [14] describes a fuzzy-model-based nonlinear control approach in which the conditions that guarantee closedloop stability are determined using Lyapunov and linear matrix inequality (LMI) approaches. In the second step, a GA is used to search for the maximum satisfaction of LMI stability conditions. However, this approach requires an accurate timeinvariant fuzzy Takagi–Sugeno (TS) model of the nonlinear plant. Leung et al. [16] extend the approach to the case of known parameter uncertainties. Note that this combination of GA and Lyapunov theory focuses on stabilization in the presence of uncertainties, rather than on hardware-in-the-loop optimization. Wang et al. [31] propose a direct control scheme in which the standard Lyapunov-based algorithm is replaced by a variant of a GA with a heuristic hill-climbing recombination. Since it is difficult to assess (both theoretically and empirically) whether the error-based objective function driving online adaptation is actually multimodal, and since the hill-climbing GA does not fully assure to avoid local minima, our approach uses the two tools in a different way. Essentially, the Lyapunovbased adaptation (introduced in Section II) is not replaced, but rather complemented by the GA, which searches for the optimal value of all the remaining free parameters of the controller (as illustrated in Section III). The advantages of the proposed hybrid scheme are assessed using both simulated examples derived from recent literature on AFC (Section IV), and a hardware benchmark consisting of the rotary speed control of a nonlinear set of servo drives (Section V). II. P ROBLEM S TATEMENT We consider the problem of controlling an unknown inputaffine single-input single-output (SISO) plant defined by the following nth-order model in normal form: x(n) = f (x) + bu (1) y=x where f (·) is an unknown function, b is the unknown positive input gain, u ∈ R and y ∈ R are plant input and output, respectively, and x = [x x˙ · · · x(n−1) ]T ∈ Rn is the state vector. As mentioned, many studies (e.g., [2], [8], and [29]) have extended AFC schemes to more general classes of models (e.g., state-dependent input gain, disturbances). Nevertheless, in this paper, we prefer to focus on this simple class of systems to keep the mathematical notation simple, underlining that the proposed approach can be used also with other more complex AFC schemes. A precise knowledge of f (·) and b would make it possible to implement the ideal control law guaranteeing a desired linear dynamic for the tracking error e(t) ≡ ym (t) − y(t), where ym (t) is the reference. The ideal law is defined as 1 (n) (n) = −f (x) + ym u∗ = u∗ x, e, ym + kT e b

(2)

where e = [e e˙ · · · e(n−1) ]T is the error vector, k = [kn kn−1 · · · k1 ]T ∈ Rn is the vector describing the de-

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(n)

sired closed-loop dynamic for the error, and ym is the nth derivative of the reference output. In some AFC literature (e.g., [8]), the reference signal ym (t) is obtained as the output of a reference model, excited with a given input signal w. When f (·) and b are unknown, a direct AFC scheme can be used to approximate the ideal law with a fuzzy interpolator (i.e., an FC) (n) (3) |φ uc = uc x, e, ym where φ is a vector of parameters that are continuously adjusted with an adaptation strategy that makes uc tend to u∗ with the minimum error. Various alternative schemes have been proposed to estimate and feed the information related to state, tracking error, and reference to the FC (e.g., the study in [8] only feeds x and w to the FC, and the studies in [23] and [27] use observers to estimate x in cases where it is not directly measurable). In this paper, we adopt the combination of inputs suggested in [8]. The paper shows that, if the reference plant has the same dynamic as the linearized error dynamic expressed by k, the adapted parameters converge to “almost” stable values (i.e., with negligible variations) also for reference signals changing in time. However, for the sake of generality, we will hereafter denote the vector of inputs for the FC with z = [z1 z2 . . . zN ]T , and the FC input space with Z. In this way, (3) can be rewritten as follows: uc = uc z|φ . (4) In addition to error minimization, the AFC scheme is requested to keep all the closed-loop system variables uniformly bounded, i.e., |x(t)| ≤ Mx < ∞, |φ(t)| ≤ Mφ < ∞, and |u(z|φ)| ≤ Mu < ∞, where Mx , Mφ , and Mu are userspecified design constants. Introducing the ideal parameter vector

∗ ∗ (5) φ ≡ arg min sup uc (z|φ) − u |φ|≤Mφ

z∈Z

the minimum approximation error can be defined as ω ≡ uc (z|φ∗ ) − u∗ . In the last decade, many different parameterizations have been proposed for (4) [1], [22]. The most common and straightforward solution is a zero-order TS system with singletons as output MFs. In such a case, defining φ ≡ θ, where θ = [θ1 θ2 · · · θM ]T is the vector of the output singletons and M is the number of rules (NoR), (4) can be written as M

uc (z|θ) =

θl l=1 M l=1

· αl (z)

= θT · ξ(z).

(6)

αl (z)

The notation αl (z) introduces the firing degree of rule l, defined as αl (z) =

N i=1

µli (zi )

(7)

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where µli (zi ) is the MF associated to the ith input in the lth rule. Finally, ξ(z) = [ξ1 (z) ξ2 (z) · · · ξM (z)]T is the vector of fuzzy basis functions αl (z)

ξl (z) = M

l=1

αl (z)

optimization strategy. Namely, we decompose the vector of FC parameters λ into two subvectors λ = [θT

.

(10)

(8)

This type of parameterization is commonly adopted because it makes the control action (6) a linear weighted combination of the adapted parameters θ, thus permitting very simple singleton-adaptation laws, such as the one proposed in [28] and many subsequent works θ˙ = γeT pn ξ(z)

ϕT ]T

(9)

where γ is the learning rate, and pn is the last column of a definite positive matrix P , satisfying the Lyapunov equation P · K + K T · P = −Q, where matrix K is the companion form of k and Q is a user-defined positive-definite matrix. The literature [28], [29] proves that the error asymptotically tends to zero provided that the minimum approximation error between uc and u∗ is squared integrable. The design of AFC schemes based on Lyapunov theory is very important in the context of fuzzy control, because this makes it possible to address closedloop stability issues. Namely, it can be easily shown [28] that a quadratic form Ve = eT P e/2 can be defined on the tracking error, and its derivative V˙ e can be made semidefinite negative by adding a stabilizing control action us to the output of the FC uc (for brevity, we omit the details about us , which can be found in [28]). A number of different strategies to guarantee stability have also been proposed (e.g., [22]), and could be used in our scheme in place of the stabilizing supervision us without particular difficulties. The AFC literature underlines that the inherent approximation error may significantly compromise the performance of the control loop. Despite the various efforts dedicated to this problem (e.g., [22] and [24]), it is generally not easy to determine how much these effects are influenced by the type of FC parameterization, and by the choice of the other configuration parameters that are not directly adjusted by the adaptation law. Theoretical results about fuzzy-data-driven identification [30] suggest that, in general, the approximation error can be reduced by increasing the number of MFs. While a certain amount of literature about AFC relies on this assumption as a means to disregard the actual effects of this error, our closed-loop experimental investigations show that increasing the number of MFs may not necessarily lead to performance improvements due to overparametrization. Experimental studies also show that the results of the AFC scheme are strongly influenced by many nonadapted configuration parameters of the FC, which must be set by hand.

III. O NLINE GA-B ASED D ESIGN OF A DAPTIVE FC The basic idea of this paper is to partition the set of configuration parameters in two subsets, each handled by a different

where θ is the vector of the output singletons, and ϕ is the vector of all the other configuration parameters of the FC, including: 1) the positions of the input MFs; 2) the input and output scaling factors; and 3) the learning rate of the adaptation law. The partition of λ separates the parameters that have a linear influence on uc [see (6)] from those that have a nonlinear influence of uc . If the value of the parameters in the vector ϕ is defined, the values of the corresponding θ minimizing the tracking error can be obtained with an adaptation law such as the integral of (9). Basically, while in AFC literature, ϕ is set a priori (generally as the result of preliminary hand tuning), here, we adopt a GA to optimize simultaneously ϕ and θ. In addition, the adaptation law (9) is applied to every new candidate solution to effectively adjust the values of θ. The choice of a GA allows us to also incorporate structural information about the FC in vector ϕ, such as the numbers of input MFs and output singletons. Our GA is based on the standard form well described in technical literature [19]. Therefore, for brevity, in the following, we only focus on the peculiar aspects of our implementation. A. Coding Strategy We adopt the zero-order TS fuzzy system specified in (6)–(9) with triangular input MFs and output singletons as general structure of the FC. The GA is allowed to modify both the structure (number of MFs and rules) and the parameters (position of MFs, scaling gains, and learning rate). In other words, the number of MFs that cover each input of the FC is not specified a priori, but optimized by the GA. To initialize the GA, the user must specify the maximum number of MFs admissible on each input of the FC. Indicating this number as mi (subscript i refers to the ith input, i = 1, . . . , N ), the maximum NoR forming a complete rule base for the FC is Mmax =

N

mi .

(11)

i=1

Fig. 1 illustrates the encoding scheme, showing that a chromosome contains both binary and real-valued genes. The coding strategy assumes that, for each input i, at least two MFs (the open-left and open-right triangular MFs centered in −1 and +1) are always present, while the remaining mi − 2 MFs can be freely added or removed by the GA. As a further constraint related to input coverage, the triangular MFs must always satisfy the condition of crossing the two adjacent MFs at 0.5 (this condition is referred to as having α-level = 0.5 in fuzzyset theory). This constraint is convenient both for the actual implementation of the adaptive FC, and for the solution encoding. In fact, the constraint on the α-level in conjunction with the use of triangular MFs (having limited support) guarantees

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Fig. 1. Schema of a generic chromosome used by the GA. The genes related to output singletons are depicted in gray to emphasize the fact that they are also processed by the LBD algorithm. For space limitations, λ, θ, and ϕ are represented as row vectors.

Fig. 2.

Examples of (portions of) chromosomes and the corresponding input MF partitions (mi = 5 in this example).

that no more than two MFs can be simultaneously fired on each input. Therefore, the maximum number of nonzero elements in the vector of fuzzy basis functions ξl (z) is lower or equal to 2N and, due to (9), no more than 2N singletons are actually adapted at each time step. This means that, independently of the actual number and placement of MFs, the computational effort for the singleton adaptation at each time step remains limited, which is a fundamental advantage for the experimental implementation of the scheme. Moreover, the α-level constraint also reduces the number of free parameters defining the MFs. In fact, the constraint makes the shape of each MF (except the first and the last on the universe of discourse) depend only on its center (i.e., the position of the upper vertex, as in the examples in Fig. 2). This strategy makes it possible to encode each MF in the chromosome by means of only two parameters: 1) the center of the MF on the corresponding universe of discourse and 2) the structural flag (SF) associated to the MF. SFs are control genes (inspired to the structural genes in the DNA of biological entities) often used in GAs to perform structural changes to the encoded solutions (see, e.g., [5] and [32]). If the SF of an MF is set to zero, the MF is removed from the input universe, and the rules in which it appears are removed from the rule base, thus reducing the complexity of the controller. Note that changing the SF of an MF also makes the two adjacent MFs change due to the mentioned constraints (see Fig. 2). Besides SFs and centers of input MFs, the chromosome specifies the position of the Mmax output singletons (only those corresponding to

nonremoved rules will actually take part in the inference), the scaling gains, and the learning rate. B. Fitness Evaluation The fitness of a control design problem is a scalar measure of the overall performance of the control loop, obtained by computing or performing a predefined and repeatable (simulated or experimental) evaluation test. In our approach, the fitness evaluation is also the phase in which the two learning techniques T T interact. Let us consider a generic chromosome λ0 = [θT 0 ϕ0 ] (e.g., obtained by random initialization, or by a crossover or mutation of existing solutions) univocally describing a controller FC0 . As Fig. 3 shows, the fitness evaluation of λ0 is composed of two consecutive phases: 1) the adaptation phase and 2) the performance evaluation phase. In the adaptation phase, a closed-loop experiment in which the plant is controlled by FC0 is carried out. During this phase, the position of the singletons of the controller θ0 is adjusted applying the Lyapunov adaptation algorithm. The role of the LBD is twofold: 1) it works as a local search algorithm that rapidly improves the tracking performance of the controller and 2) it guarantees the closed-loop stability of each experimented controller by means of the additive supervisory action us . As an effect of adaptation, the position of the singletons θ 0 at the end of the adaptation phase is, in general, different from the initial one, i.e., θ 0 = θ0 . In the subsequent evaluation phase, the adaptation is

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Fig. 3. Fitness evaluation. Adaptation and evaluation phases.

switched off, and the closed-loop experiment proceeds controlled by the nonadaptive FC having the singletons steadily constant in the positions θ 0 . Only during this second phase are the indexes of performance of the FC measured and used to compute the fitness at the end of the experiment. Once the T T fitness is computed, the updated chromosome λ0 = [θ T 0 ϕ0 ] replaces the original one λ0 , and the GA proceeds with the next operation. The interruption of the adaptation in the evaluation phase allows us to perform a fair comparison with purely GA-based design schemes in which the optimized parameters are not changed during the evaluation experiment. It is also important to observe that the GA allows one to define fitness functions that take into account not only the tracking error, but also any other measurable, observable, or calculable characteristic of the considered problem (e.g., see the fitness functions based on the oscillations of control action, and on the complexity of the FC used in the next sections). C. Crossover and Mutation All the binary and real-valued parameters (i.e., both θ and ϕ) undergo mutation and recombination. The crossover and mutation operators are variants of conventional operators suitable to deal with the proposed encoding. The mutation can be either binary or real depending on the parameter selected for mutation. Our preliminary experiments (and also previous research on the subject [5]) showed that mutation rates higher than the values usually suggested by literature are indispensable (especially for the binary mutation) for an effective exploration of alternative FC structures. Similar considerations can be extended to the crossover, which also consists of various operators applied with different frequencies. If the crossing point falls on the SFs, a multipoint crossover is applied (i.e., flags are randomly swapped between individuals). Otherwise, one of three different real-valued crossovers (simple, arithmetical, and heuristic crossover, see [19] for details) is applied. Readers may refer

to [5] for a discussion about the profitable interaction between these operators. IV. S IMULATED C ASE S TUDY To evaluate the effectiveness of the proposed scheme, we first focus on a simulated case study derived from technical literature [8]. The controlled plant is a second-order nonlinear dc motor described by the following model: x˙ 1 = x2 (12) 2) x˙ 2 = − f (x + C · Ju J where the state variable x1 is the angular position of the rotor in radians, x2 is the angular speed, u is the current fed in amperes, and the constants are defined as C = 10 N · m/A, and J = 0.1 kg · m2 . The nonlinear friction torque is defined as f (x2 ) = 5 arctan(5x2 ) N · m. The reference signal is a square wave of random amplitude filtered by a linear second-order system with two poles in s = −20 (s indicates the Laplace complex variable). The model is simulated in a Matlab/Simulink environment with a fixed-step Euler method (ode1) with sample time T = 0.001 s. As mentioned, in all the considered schemes, the FC is fed with the combination of inputs suggested in [8] uc = uc (x, w|λ).

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We compare the three following design strategies. 1) LBD: This design strategy only optimizes the output singletons θ while all the other configuration parameters ϕ must be specified a priori. In our simulated or experimental investigations, all the input MFs are evenly distributed on the corresponding universe of discourse. The learning rate is γ = 1000, and input and output scaling gains are defined so as to map the ranges of the corresponding variables into the interval [−1,1].

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TABLE I SUMMARY OF THE CONFIGURATION PARAMETERS FOR THE GAs (UNLESS SPECIFIED, THE SAME PARAMETER IS USED FOR BOTH GAs)

2) GBD: This design strategy optimizes all the configuration T T parameters of the controller λ0 = [θT 0 ϕ0 ] using only the GA (no Lyapunov-based adaptation is performed). 3) Hybrid genetic- and Lyapunov-based design (HBD): This strategy combines the two design methods as described in the previous sections. Here, λ0 is optimized by the GA, and the Lyapunov algorithm adjusts the singletons θ0 . The bounds for each optimized parameter are the same used for GBD. Table I summarizes the final configurations of the two GAs used in GBD and HBD, respectively. For both GAs, we have chosen the setting that leads to the best observed performance. In particular, GBD needs a larger population to initially explore and determine satisfactory placements of output singletons (i.e., to find the coarse shape of the input–output law), while the HBD takes advantage of the singleton adaptation and works well with a small population. The first set of runs studies the sensitivity of the control performance with respect to the position of the input MFs when a fixed number of evenly distributed MFs and a complete rule base are used. The fitness is the integral absolute error (IAE) between the filtered reference ym and the motor output y = x1 . The stopping criterion for all the algorithms is the total amount of simulated time ST = 10 h. Simulation time is chosen very long in order to evaluate the actual results that would be obtained by each method if the design could be performed on the controlled plant without time limitations. The upper part of Table II shows the results of this set of simulated experiments. It can be immediately noted that the amount of allowed search time is still insufficient for the GBD to reach the performances of the other two strategies. Results also show that, as the number of input MFs decreases, the performance of the LBD strategy becomes unsatisfactory, whereas the HBD seems relatively less influenced by the reduction of the number of free parameters. The second set of simulations analyzes the tradeoff between complexity and performance. In this set, the two GA-based methods (GBD and HBD) are allowed to optimize also the number of input MFs (and rules). To reduce the complexity of the controller in terms of NoR, in this case, the fitness function is defined as a weighted combination of IAE and NoR, i.e., FF = a1 · IAE + a2 · NoR

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TABLE II SIMULATION RESULTS (AVERAGE AND STANDARD DEVIATIONS ARE ROUNDED TO THE NEXT INTEGER)

where weights a1 (= 1.5) and a2 (= 1) are defined heuristically. The results (summarized in the last two rows of Table II) show that the HBD provides solutions having, in average terms: 1) less rules than the maximum allowed (7 × 7 × 7 = 343) and 2) a smaller IAE with respect to the case of fixed a priori structure (the average IAE obtained with a free structure is about 30% lower than the best IAE obtained with a fixed structure). To emphasize the twofolded advantage of this strategy, the last row of Table II reports two sample solutions, one with a relatively large NoR (294) and the smallest value for the IAE achieved in all the simulations, and a second one with fewer rules (180) and an IAE index comparable to the one obtained with the largest fixed structure (343 rules). It is also interesting to note that the tracking performance and the controller complexity are not always conflicting, and that the HBD allows one to automatically reduce the NoR while holding a satisfactory tracking performance. V. E XPERIMENTS The experimental benchmark consists of a couple of two nonlinear 370-W dc machines (AMIRA DR300) mounted on

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the same shaft. Both dc machines have a rated speed of 3000 rpm. The first one works as a motor and represents the controlled plant, whereas the second machine (the generator) is used to apply time-varying loads or increase the system nonlinearity. Both dc machines have built-in analog current controllers, whose reference is a voltage signal, which is provided by an external control board (dSPACE 1104) equipped with a 250-MHz Motorola microcontroller. The control board runs the FC, the adaptation law, and the stability supervisor, all automatically converted into discrete-time routines (Euler discretization, with sampling time T = 0.001 s). The board also processes the control signals applied to the load generator. Voltage references are fed to the dc drive using 16-bit D/A converters that are integrated in the dSPACE board. The GA (with the same configuration used in the simulations) and additional algorithms dealing with reference signals and process monitoring are written in Matlab and run in real time directly on the host PC. The electromechanical relation between the reference voltage u(t) and the angular speed ω(t) of the shaft can be written in a normal form as follows:

TV (x)R KKI KV KI R 1 dTV (x) x− + u(t) x− ˙ x ¨= − − L J dx LJ LJ LJ (15) where x = ω and x = [ω ω] ˙ T , R and L are armature resistance and inductance, respectively, J is the moment of inertia of the overall equipment, TV is an unknown nonlinear load torque (varied through the second dc machine), and K, KV , and KI are three constants. Frictions and other inherent nonlinearities make the servo motor nonlinear also when no additional load is applied by the generator. In this paper, the additional torque TV is chosen proportional to x3 , thus making the plant strongly nonlinear. In all the experiments, the desired output ym to be tracked is the output of a reference model having a double pole in s = −20. The reference signal w is a square wave obtained repeating a pattern consisting of six steps of 2 s (each pattern lasts 12 s and will be referred to as an adaptation cycle). The amplitude of each step in each adaptation cycle varies randomly between 0 and 2000 rpm, except for the cycles applied during fitness evaluation, which are all obtained by repeating the same identical pattern (called the evaluation cycle), so as to make fitness values comparable. The GA configuration is identical to the previous simulated benchmark (see Table I), except for the stopping criterion, which is set one order of magnitude shorter, i.e., ST = 1 h. A. Preliminary Sensitivity Analysis As in the previous simulations, we first compare several LBDs with different numbers of evenly placed input MFs on the input universes. The performance index chosen for evaluation is the IAE between the outputs of the reference model and the controlled plant. Each experiment (replicated ten times) uses 20 adaptation cycles with a learning rate γ = 5.0 e − 5, and a final evaluation cycle. The results in Table III substantially confirm the main conclusions drawn for the simulated case

TABLE III EXPERIMENTAL RESULTS WITH A FIXED STRUCTURE, USING IAE AS FITNESS FOR GAs (AVERAGE AND STANDARD DEVIATIONS ARE ROUNDED TO THE NEXT INTEGER)

study (merely using a higher number of MFs does not necessarily guarantee that the LBD obtains an improved tracking). Table III also shows the performance of GBD and HBD of FCs with a fixed predefined structure. It is evident that the position of the input MFs affects the overall tracking performances in a measure that also depends on the number of MFs. The largest structures are evidently overparametrized and cannot be properly optimized. B. Deﬁnition of Fitness Functions Another set of experiments compares various definitions of fitness functions combining indexes related to the tracking error, the smoothness of control action (i.e., two indexes describing the overall performance of the control loop), and the NoR used by the FC. The smoothness of the control action (SCA) is measured by filtering the control action with a linear second-order filter with two poles in s = −100, and comparing the actual control action with the filtered action by means of the IAE between the two signals. High SCA indexes always correspond to very nervous control actions that cause evident and undesirable stresses to the electromechanical equipment (e.g., compare the two actions shown in Fig. 4). The three indexes are combined using a linear weighted aggregation, so that the most general expression for the fitness function can be defined as FF = a1 · IAE + a2 · NoR + a3 · SCA.

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Positive constants a1 , a2 , and a3 are chosen heuristically with a two-stage procedure. We preliminarily evaluate (for the IAE and SCA indexes) the ranges of values that correspond to satisfactory tracking and smoothness performances, and then use constants ai so as to: 1) normalize the contribution given by each weighted index in the sum and 2) emphasize the influence of tracking performance with respect to smoothness of control and FC complexity. Also, in these experiments, the stopping criterion is ST = 1 h for all the runs. Table IV reports the average values of the

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Fig. 4.

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Comparison of control actions obtained with different fitness functions.

TABLE IV EXPERIMENTS WITH AN OPTIMIZABLE STRUCTURE AND VARIOUS AGGREGATED FITNESS FUNCTIONS (AVERAGE AND STANDARD DEVIATIONS ARE ROUNDED TO THE NEXT INTEGER)

view at low and high reference speeds. Reducing the learning rates or increasing the NoR (two typical operations performed during AFC fine tuning to reduce the oscillations), do not help to overcome this flaw, as they produce smoother responses but significantly larger steady-state errors. The tracking of the HBD is indeed more satisfactory, both around low- and high-speed profiles, confirming the effectiveness of the obtained nonlinear control law. VI. C ONCLUSION

three indexes, and the standard deviation of the IAE for comparison with the results in Table III. It can be noted that the incorporation of multiple performance criteria in the fitness function can improve the results. Namely, even when the NoR and SCA are not used in the fitness, the HBD tends to use fewer rules than the maximum available, confirming that larger structures are unnecessary to optimize the tracking performance. When the NoR and SCA are introduced, the performances of the HBD are further improved (in this case, we obtain the best controller of the whole experimental campaign). Fig. 4 compares the control actions of the best FC obtained with IAE fitness, and with the three-term fitness, respectively. Note in Table IV that the smoother control action does not affect the tracking performance. Fig. 5 provides a comparison of the tracking performances of the best FCs obtained with HBD and LBD. In spite of the relatively low IAE value (IAE = 9102, see Table III), the best LBD exhibits a nervous tracking performance with oscillations that are clearly visible in the expanded

This paper has proposed a combination of existing methodologies to perform the design of an adaptive FC online. The suggested hybrid design approach achieves several advantages with respect to its single components. Namely, the adaptation law can be assimilated to a local search technique that contributes to make the genetic design process much faster. Additionally, the local adaptation process limits the inherent risks of experimenting badly performing controllers, simplifying the online implementation. On the other hand, the genetic approach greatly simplifies the design of the adaptive FC, overcoming the need of trial-and-error sessions and allowing the user to achieve the best approximation error without necessarily increasing the NoR. Finally, by defining a suitable fitness function for the overall tuning process, the genetic optimization can be used to accomplish other control objectives apart from those guaranteed by AFC. The integration of other efficient iterative discrete optimization algorithms (e.g., [12]) to accelerate the selection of the adequate FC structure, and the redesign of the prototype algorithms

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Fig. 5.

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Comparison of tracking performance. Detailed views of tracking at different reference speeds emphasize the differences between the LBD and the HBD.

for their integration on less expensive microcontroller-based hardware platforms (for self-tuning of industrial servo drives) are among the open research directions currently under investigation.

[15] [16]

ACKNOWLEDGMENT The authors wish to thank the Editor-in-Chief D. J. Cook and the reviewers for their valuable cooperation.

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GIORDANO et al.: GENETIC ALGORITHMS AND LYAPUNOV-BASED ADAPTATION

Vincenzo Giordano was born in Bari, Italy, in 1977. He received the M.S. degree (with honors) in electrical engineering and the Ph.D. degree in control engineering from the Polytechnic of Bari, Bari, in 2001 and 2005, respectively. In 2004, he was a Visiting Ph.D. student with the Automation and Robotics Research Institute, University of Texas, Arlington, under the supervision of Prof. F. Lewis. In 2005, he was a Visiting Researcher with the Singapore Institute of Manufacturing Technology, Singapore. As a Ph.D. student, he was coresponsible for the organization and startup of the Robotics Laboratory at the Polytechnic of Bari. He has published more than 20 international journal and conference papers. His research interests include intelligent control techniques applied to industrial automation, robotics, and discrete-event systems.

David Naso received the laurea degree (with honors) in electronic engineering and the Ph.D. degree in electrical engineering from the Polytechnic of Bari, Bari, Italy, in 1994 and 1998, respectively. He was a Guest Researcher with the Operation Research Institute, Technical University of Aachen, Aachen, Germany, in 1997. Since 1999, he has been an Assistant Professor of automatic control with the Department of Electric and Electronic Engineering, Polytechnic of Bari. He has published more than 60 journal and conference papers. His research interests include computational intelligence and its application to industrial automation and robotics, numerical and combinatorial optimization, discreteevent systems modeling and control, and distributed control of manufacturing systems.

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Biagio Turchiano (M’94–SM’01) was born in Bari, Italy. He received the degree (with honors) in electrical engineering from the University of Bari, Bari, in 1979. In 1984, he joined the Department of Electrical and Electronic Engineering, Polytechnic of Bari, Bari, as an Assistant Researcher and is currently a Full Professor of automatic control. His research interests include production automation, systems and control theory, and modeling and control of discreteevent systems.