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Controller Design Under Fuzzy Pole-Placement Specifications: An Interval Arithmetic Approach Jorge Bondia, Antonio Sala, Member, IEEE, Jesús Picó, Member, IEEE, and Miguel A. Sainz
Abstract—This paper discusses fuzzy specifications for robust controller design, as a way to define different specification levels for different plants in a family and allow the control of performance degradation. Controller synthesis will be understood as mapping a fuzzy plant onto a desired fuzzy set of closed-loop specifications. In this context, a fuzzy plant is considered as a possibility distribution on a given plant space. In particular, pole placement in linear plants with fuzzy parametric uncertainty is discussed, although the basic idea is general and could be applied to other settings. In the case under consideration, the controller coefficients are the solution of a fuzzy linear system of equations with a particular semantics. Modal interval arithmetic is used to solve the system for each -cut. The intersection of the solutions, if not empty, constitutes the solution to the robust control problem. Index Terms—Fuzzy control, modal interval analysis, pole placement, robustness, uncertain systems.
I. INTRODUCTION
T
HE intelligent control approach has proposed different paradigms to achieve efficient control of complex processes. In particular, fuzzy control techniques have been successful in the application field mainly due to the parallelism with human reasoning. Most current fuzzy or neuro-fuzzy controllers act as universal function approximators (UFAs) [1], [2]. The UFA paradigm develops fuzzy models, such as Takagi–Sugeno ones [3], that are fully deterministic: uncertainty must be explicitly introduced to account for modeling errors [4]. However, fuzzy sets can be interpreted as possibility distributions [5]. In a control context, this interpretation allows to define fuzzy plants in a different way, associating a possibility value to every plant in a given family. Here uncertainty is, thus, implicit in the model [6]–[9]. On the other hand, robust control, based on classical foundations, has also reached maturity as a technique for linear systems capable of dealing with uncertainty in multivariable models under quite general assumptions. Available approaches and -synthesis tools [10], quantitative are, for instance, feedback theory [11], and parametric robust control [12] which considers interval parameters. Manuscript received January 18, 2005; revised September 7, 2005 and December 1, 2005. This work was supported in part by the Spanish government under Grants DPI-2004-07167-C02 and DPI-2004-07332-C02-02, and in part by the European Union through FEDER funds. J. Bondia, A. Sala, and J. Picó are with the Department of Systems Engineering and Control at the Technical University of Valencia, Valencia 46022, Spain (e-mail:
[email protected];
[email protected];
[email protected]). M. A. Sainz is with the Department of Computer Science and Applied Mathematics, the University of Girona, Girona 17071, Spain (e-mail:
[email protected]. es). Digital Object Identifier 10.1109/TFUZZ.2006.880002
Usually, robust control problems require that a particular specification is fulfilled for a full family of plants. Sometimes a solution cannot be found because the set of possible plants is too large. In that case, either specifications must be degraded or the family of systems must be reduced (i.e., leaving out infrequent cases, without performance guarantees for the discarded plants). In the latter case, a sort of “risk factor” or possibility of unacceptable performance may be computed [6]. In this paper, the issue of controlling this performance degradation is addressed. The problem is stated as achieving a different performance objective for different subsets of an uncertain family. For this purpose, both the specifications and the uncertain plants will be described by means of possibilistic fuzzy models. Fuzzy plant parameters will be assumed to be the result of identification (the core of the fuzzy plant set is expected to contain a good approximation to the true plant in the majority of cases and the support will be assumed to contain all the possible plants). Regarding the specifications, the core of the set will denote the requested most-cases behavior and the support of the set will denote the worst-case limit of acceptability (for example, very low stability margins). The control problem is cast as an inclusion of a fuzzy closedloop system (the image of a fuzzy open-loop plant model and a crisp controller) into a fuzzy specification set (in the sense of inclusion of the respective -cuts). A simpler case could be considering only the core and support of the sets, similar to a rough set approach [13]. In this paper, particularization to pole placement for possibilistic fuzzy plants will be discussed. However, some of the presented ideas may also be applied to different settings, for instance, in a frequency response framework [14]. When plant parameters are fuzzy numbers, as it will be the case in this work, -cuts correspond to intervals [15]. Then, for each -cut, an interval Diophantine equation must be solved. Interval arithmetic will be used for this purpose; however, classical interval arithmetic [16] is not suitable due to its implicit semantic interpretation (the evaluated interval encloses the actual solution). As specifications must be guaranteed for any chosen controller parameters, the resulting intervals must contain only feasible solutions (no overbouding is allowed). This is equivalent to obtaining inner estimations (i.e., subsets) of the so-called tolerable solution set [17], [18] of the interval linear system of equations associated to the Diophantine equation. In a general case, this system of equations can be solved by means of optimization methods [18]–[20] or constraint satisfaction solvers [21]. Compared to these methods, algebraic solutions [17], [22] offer a clear advantage from the computational point of view but they are not always applicable to a
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BONDIA et al.: CONTROLLER DESIGN UNDER FUZZY POLE-PLACEMENT SPECIFICATIONS: AN INTERVAL ARITHMETIC APPROACH
general case. Some common cases in which this is possible under the framework of modal interval analysis [23], [24] are investigated in this paper. Modal interval arithmetic associates a logical statement involving universal and existential quantifiers with an algebraic interval evaluation. This allows to correctly formulate the control problem under consideration. The structure of the paper is as follows. Section II introduces fuzzy parametric uncertainty as possibility distributions. Then, the concept of closed-loop fuzzy specifications (performance degradation) for fuzzy uncertain plants (possibility distribution on the parameters space) is defined: a general design problem is posed in terms of inclusion of -cuts. Section IV reviews basic concepts of pole-placement controller design. Section V generalizes the pole placement design when the characteristic polynomial has fuzzy coefficients. Sections VI and VII present an algebraic approach to the design problem under the framework of modal interval arithmetic. Some examples in Section VIII illustrate the methodology. A conclusion section summarizes the main ideas. The notation used will be the following. , ,
universe of plants; plant in ; fuzzy plant in (possibility distribution);
, , , ,
,
, , ,
,
,
, ,
,
,
,
universe of specifications; fuzzy specification in (possibility distribution); core/support of a fuzzy plant/specification, respectively; fuzzy number; weak -cut of ; real vector/matrix; vector/matrix of fuzzy numbers; vector/matrix of the weak -cut of its elements.
II. FUZZY PARAMETRIC UNCERTAINTY As discussed in the Introduction, the UFA paradigm provides a fully deterministic approximator and uncertainty must be explicitly introduced. An alternative approach, to be discussed in this paper, is considering fuzzy models as possibility distributions [5]. Given a universe set representing all possible plants in a given class, a fuzzy model will be understood as a possibility on . In this case, fuzzy models distribution actually represent uncertainty, in a totally different viewpoint from the UFA framework. These possibilistic fuzzy models can arise from identification experiments or from first-principle equations. For instance, in black-box identification, there are prior assumptions that constitute the tuning knobs, such as noise bounds, variances, etc. not precisely known which may be represented as fuzzy sets. In the second case, fuzzy sets may be used in order to represent unmodeled dynamics and parameter tolerances. In both cases, from an intuitive interpretation, the core of the resulting fuzzy model should contain the true plant in “most”
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Fig. 1. Block diagram of a one-degree-of-freedom control structure.
of the cases where the controller is to be applied. The support should, hopefully, contain the true plant in all conceivable cases. The usual families of plants in classical robust control [10] are a particular case, corresponding to crisp sets. 1) Fuzzy Parametric Uncertainty: A particular case of the above setup is fuzzy parametric uncertainty [8], [25]: A fuzzy model is obtained from possibility distributions defined on the parameters of a parameterized plant class. 2) Fuzzy Linear Plants: In this paper, the universe set of linear time-invariant plants of a particular order will be considered. This is the approach taken for instance in [9], where a machine tool structure is modeled as a second-order plant with fuzzy damping and natural frequency
(1) Other examples in the field of mechanical engineering, geotechnical engineering, and biomedical engineering can be found in [7]. In a practical setting, the fuzzy parameters in a parameterized model such as (1) will be fuzzy numbers, defined as follows. 3) Fuzzy Numbers: A fuzzy set in is defined to be a fuzzy number if its -cuts are a set of intervals. Fuzzy numbers admit the representation [15]
(2) where 1) 2)
; and are monotonic increasing and decreasing continuous functions, respectively.
III. CONTROLLER DESIGN FOR FUZZY UNCERTAIN PLANTS The previous section introduced families of plants (uncertainty models) expressed in the fuzzy set framework. The rest of this paper discusses controller design for such families of plants. Note that the designed controller must be a deterministic dynamical system, suitable to be implemented in a conventional algorithm. In the context of possibilistic models, the controller will not embody any fuzziness, contrary to the case of the UFA paradigm, where a fuzzy controller is usually considered (parallel distributed processing [4]). In controller design, the desired set of performances must be defined. This section states the problem of controller design for fuzzy plants as an inclusion of the closed-loop image of the fuzzy plant onto the defined fuzzy performance set. This image is obtained from the extension principle. Indeed, each of
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Fig. 2. Mapping a possibility distribution onto the specification space, assumed to be for simplicity. Cases (a)–(d) refer to crisp sets of plants (core and support). Light gray denotes the image of P and dark gray the image of P . Case (e) refers to a fuzzy set of plants.
the plants , when connected in closed loop to a (conventional, crisp) controller will yield a resulting closed loop dynamical system (see Fig. 1). Then, the closed-loop behavior must be evaluated in order to assess its acceptability. To do that, some numerical performance indexes may be computed, such as settling time, overshoot, system norms, etc. [26], conforming an -dimensional vector on a particular specification space . 1) Evaluation Map: The composition of the closed-loop system jointly with the computation of performance indexes may be expressed as a map from the plant space onto the specification space (for a fixed controller), i.e., the control block diagram jointly with the design specifications define, a . priori, the so-called evaluation map For instance, the evaluation map might map a process onto a real number, this number being any optimal-control related cost index. A (crisp) target specification would be a desired cost in . This is, indeed, a common setup [10]. In Secan interval tion V, the considered specification space will be the coefficients of the closed-loop characteristic polynomial. 2) Image Set: For a fuzzy plant , the evaluation map can be extended via the extension principle [15], as
For simplicity, the specification space is assumed to be . Consider two (crisp) sets of specifications, denoted as de, and relaxed (soft), , specifications. manding (hard), 1) The ideal case would be finding a fixed controller able to , under demanding perform satisfactorily in all plants, , depicted in Fig. 2(a). This can be exspecifications . pressed as the inclusion 2) If a solution cannot be found (maybe due to conservativeness of the chosen design techniques), one option is relaxing the requirements: the demanding bound is required [Fig. 2(b)]. Note that only for the most possible plants, in this case, the plants which do not belong to the core set are not considered in the design and thus nothing can be said about their performance: they might even become unstable with the resulting controller. 3) Alternatively, when a feasible solution in the first case does not exist, the specification bound can be relaxed to , trying to find a feasible solution for all the plants [Fig. 2(c)]. However, this may decrease the achieved performance on the core set. 4) The aforementioned loss of performance can be avoided by and a demanding one setting up a relaxed specification and trying to fulfill
(3) AND . This giving rise to the so-called fuzzy image set set is defined on the specification space and its core determines the variety of closed-loop performances achieved by the most possible plants. The support will denote the achieved worstcase performance. Intermediate cuts define how performance is degraded. The control objective is, thus, achieving a suitable which fulfills all the desired properties. Several possibilities may arise, as depicted in Fig. 2.
This situation is depicted in Fig. 2(d). 3) Fuzzy Target Specifications: This last approach can be generalized by defining a fuzzy target set of specifications, consisting on a fuzzy subset of , denoted as . The core of the set will represent the requested most-cases behavior and the support the worst-case limit of acceptability (for example very low stability margins). The membership function of will define how performance is allowed to degrade. An example of fuzzy
BONDIA et al.: CONTROLLER DESIGN UNDER FUZZY POLE-PLACEMENT SPECIFICATIONS: AN INTERVAL ARITHMETIC APPROACH
specification is a desired characteristic polynomial with fuzzy coefficients, as shown later. 4) Inclusion: In this context, the control design problem can be cast as an inclusion problem. A fuzzy set of plants must be mapped by the controller onto a subset of a fuzzy set of target specifications. That is, the resulting fuzzy image set must be a subset of the fuzzy specifications, indicating that the controller will yield good performance in the majority of plants (core) and a user-defined degradation of performance for the unlikely ones. The situation is depicted in Fig. 2(e). 5) Problem Statement: A general design problem can be formulated as follows: given a fuzzy plant and a fuzzy specifica. tion set , design a fixed controller such that The inclusion must be understood in terms of -cuts.1 Denoting and the corresponding -cuts of the fuzzy sets, the by designed controller must satisfy (4) IV. POLE PLACEMENT
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Consider a plant with numerator and denominator of order , a controller of order , and a closed-loop characteristic polynomial of order
(7)
(8) are the plant transfer where function coefficients, is a vector with the coefficients of the to-be-designed controller’s numerator and denominator. Note that the coefficients of in (5) are linear functions on the controller coefficients. Then, equality of polynomial coefficients in the Diophantine equation can be stated in matrix form as the linear system of equations [31] (9)
Henceforth, the introduced concepts will be applied to the particular case of pole-placement controller design. In this section, pole placement under no uncertainty is reviewed. In Section V, this design method will be extended to the fuzzy case. Consider the system depicted in Fig. 1, where is a known linear plant and is a linear controller. Important properties of the closed-loop behavior such as stability and settling time are determined by the poles of the closed-loop transfer function, i.e., the roots of the characteristic polynomial (5) To ensure stability, all the poles must be in the left open complex half-plane. Settling time in the reference or disturbance transient is inversely proportional to the real part of the poles. Control specifications can be translated to locations of the poles in the complex plane [27]. Under no plant uncertainty, this is equivalent to defining a desired closed-loop polynomial . Controller design is carried out by solving the equation , i.e.,
where
.. .
.. .
.. .
.. .
.. .
.. .
.. .
.. .
.. .
.. .
.. .
.. . (10)
is the so-called Sylvester matrix and is the vector of coefficients of the chosen characteristic polynomial. V. FUZZY POLE-PLACEMENT PROBLEM STATEMENT In a practical situation, uncertainty in the plant’s transfer function will appear. The problem stated in the previous section can be generalized as discussed below. Let an th-order linear time-invariant fuzzy parametric plant
(6) for the controller numerator and denominator polynomials and , respectively. This design methodology is widely known as pole placement [27], [28]. An equation of the form (6) is called a Diophantine equation (also Bézout equation) when defined over the elements of a ring, such as polynomials (which is the current case). These equations have been extensively studied in mathematics and control theory [29], [30]. 1Given
~ on a universe a fuzzy set A
:= fa 2 j (a ) > g) ,
~ , its -cut is defined as A
j (a) g), for 0 < 1, and A~
:=
cl(fa 2
for = 0, where cl denotes the topological closure.
(11)
be given. A linear controller with with (by assumption, nonfuzzy) tunable coefficients , with the same structure as that in (7) will be considered. Denote the closed-loop characteristic polynomial as
(12)
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where is a continuous function on its parameters, extended to fuzzy arithmetic via application of the extension principle [7], [15], yielding a polynomial whose coefficients are fuzzy sets. Following the pole placement procedure, a desired fuzzy characteristic polynomial
(13) must be defined. As each -cut of (13) is an interval polynomial, the Edge Theorem [12] can be used to evaluate the boundary of the region in the complex plane where the poles lie. The problem may then be stated as follows: Given a desired target fuzzy characteristic polynomial (13), find a controller pa, interpreting the inrameter vector so that clusion as the inclusion of the fuzzy polynomial coefficients
bers. If a discretization of the membership value is done, interval arithmetic can be directly applied to every -cut to obtain the resulting fuzzy set. Although there are other methods to operate with fuzzy sets, this is the most widely used due to its simplicity of implementation and known as standard fuzzy arithmetic [7]. A well-known drawback of standard fuzzy arithmetic, inherited from the defective properties of classical interval arithmetic [16], is overestimation. If multiple incidences of an argument occurs in , the -cuts of the output will be overestimated, resulting in a superset (hence, more uncertain) of that resulting from (16). Thus, classical interval arithmetic guarantees that the resulting interval encloses the real range of for all the possible values of the argument. The classical interval evaluation of a , , , where , are real function intervals is then interpreted as (17)
(14) As mentioned in Section III, the core of defines the required performance for the most likely plants, whereas its support determines the minimum performance for the whole family of plants. The membership function of determines how performance is allowed to degrade as the possibility of a candidate plant decreases. The inclusion (14) must be interpreted as the inclusion of the -cuts, as stated in (4), i.e., (15) On the following, it will be assumed that all fuzzy sets in consideration are fuzzy numbers, hence, (15) is an inclusion of intervals. As stated in Section IV, under no uncertainty, the controller design for pole placement involves the solution of the linear system of equations (9), where the th component of the vector corresponds to the th closed-loop coefficient . When fuzzy parametric uncertain plants and specifications are considered, a set of algebraic equations with fuzzy numbers will appear, derived from the fuzzy extension of (9) as discussed in Section VII. In Section VI, it is shown why standard fuzzy arithmetic [7], based on classical interval arithmetic, is not suitable for controller design in the problem under consideration. A more general arithmetic, modal interval arithmetic, will be introduced to solve it. VI. ALGEBRAIC EQUATIONS WITH FUZZY NUMBERS: SEMANTIC INTERPRETATION , The extension of an algebraic function , to fuzzy numbers is usually defined by means of the Zadeh’s extension principle [7]
A logical statement in the form (17) will be denoted as a semantics for the interval evaluation of . Different selections of existential and universal quantifiers will define other semantics, as discussed later. 1) Alternative Semantics: There exists many problems which do not correspond to the above semantics and thus cannot be solved by means of classical interval arithmetic. Consider, for instance, that in (17) is a tuning parameter of a controller to be designed, is an uncertain parameter of the plant, and is a parameter which defines the desired specifications. The objective here is to find a set of controller parameter values so that (18) i.e., that any controller parameter value chosen from the resulting interval should guarantee, for any plant in the uncer, a resulting behavior contained in the desired tain family . specifications set The above semantics (18), and any other, can be dealt with by means of modal interval arithmetic [23], [24]. A modal interval is defined as a couple formed by a classic interval (domain) and a quantifier, either universal or existential (modality). For instance, up to eight different semantics may be defined for . different modal interval arithmetic extensions of Expression (18) corresponds to the modal interval evaluation . This makes modal interval arithmetic a powerful tool to solve problems out of the scope of classical interval arithmetic, such as (18). A summary of the principal results of modal interval arithmetic is presented in Appendix. 2) Fuzzy Case: Some considerations must be taken into account when modal interval arithmetic is extended to the fuzzy case. When semantics (18) is considered for each -cut, the resulting set of intervals
(16) (19) The interval representation of the -cuts (2) allows to extend classical interval arithmetic [16], [32] to the case of fuzzy num-
may not define a fuzzy set, because they might not be nested. Such a case will occur in the examples in Section VIII (see, for
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(intervals are particular cases defines a fuzzy number of fuzzy sets). 3) Finite Precision Computations: To keep the desired semantics, rounding of the endpoints of the intervals resulting from each computation must be carefully carried out. Inner and outer roundings must be suitably used for this purpose (see Appendix). VII. FUZZY POLE-PLACEMENT CONTROLLER DESIGN The tools presented in the previous section will now be applied to solve the fuzzy pole-placement problem stated in Section V. A. General Case Consider the problem in Section V, where controller design was stated as the fuzzy set inclusion (14) and, hence, as the set of interval inclusions (15). For each -cut, a solution to (14) will be defined to be an interval vector
so that for
(22) that is, every element in the interval solution must lead to closedloop coefficients inside the specifications for all the plants in the cut considered. The aforementioned logical statement (22) precisely defines the semantics of the problem to be solved. As discussed in Section VI, it can be addressed by using modal interval arithmetic. From the closed-loop (6) and the extension of (9), the result is an interval linear system of equations2 [19]
(23)
Fig. 3. Solution of (40) for each -cut and intersection of them.
instance, in Fig. 3). Notwithstanding, this is not in disagreement with the problem posed. Indeed, given (19), with the associated semantics
(20) for all
, the interval
(21)
is shown in (24) at the bottom where the Sylvester matrix of the next page, and . must be found so that the semantics (22) A solution holds. Methods presented in [18], based on Jacobi iteration operators or mixed integer optimization, can be applied to each -cut to solve (23). Other approaches also based on optimization can be found in [19], [20]. In the cases described as follows, simple explicit algebraic expressions can be found for the controller parameters from (23). Then, a modal-algebraic approach solves the problem with very few arithmetic operations with numerically guaranteed results and no iterations. The optimization-based approach, although far less efficient, can be applied in a general setting. 2The notation 3 in (23) denotes the interval matrix product, i.e., the ordinary matrix product carried out with interval arithmetic in addition and multiplication. Note also that A(~ q ) is a matrix with intervals as its elements.
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Once (23) is solved for each , following (21), any controller with parameters
denotes the inner rounding of its argument (see where (Theorem.4 Appendix, (53)). In this case in Appendix) and semantics (22) applies for any interval fulfilling (28). As (26) is rational, can be easily evaluated as shown in the last section of Appendix. This arises, for instance, if the plant’s model can be expressed in what we will denote as fuzzy controllable canonical form
(25) will be a solution to the pole-placement problem for any cut of the plant and the specifications evaluated. If improper intervals are obtained in the evaluation of the controller parameters in the above cases for some , or the intersection is empty, the problem has no solution and specifications should be relaxed.
.. .
.. .
.. .
.. .
(29)
B. Canonical Form An algebraic solution can be obtained if, for a plant with real , explicit expressions on known arguments parameters , . This is can be obtained for each controller parameter the case if for each closed-loop polynomial coefficient only one controller parameter appears. Solving for it, a real rational function
where and are fuzzy parameters, and the state is measur, being able, then the state feedback a constant matrix, achieves a closed-loop whose characteristic polynomial is
(30) (26) For real , , solving for , yields which is in the form (26). The solution is then given by
will be obtained. uni-incident in . For a given -cut, the Consider now the fuzzy plant desired semantics (22) can be obtained from the modal interval -semantic extension of (26). It follows from Theorem 2 in the Appendix that the expression
(31) where and improper ones.
(27) is a proper inhas the desired semantic interpretation when , imterval (existential quantifier), proper ones (universal quantifier), and the interval proper can be ob(universal quantifier). The -semantic extension , which is uni-intained by means of the rational extension cident in its proper components
must be proper intervals and
.. .
and
C. Triangular Sylvester Matrix An analogous situation will apply if the interval linear system (23) is triangular (plant without zeros or discrete-time FIR model) and the controller parameters are solved sequentially. In this case, the first unknown corresponds to
(32)
(28)
.. .
,
.. .
.. .
.. .
.. . (24)
.. .
.. .
.. .
.. .
.. .
.. .
BONDIA et al.: CONTROLLER DESIGN UNDER FUZZY POLE-PLACEMENT SPECIFICATIONS: AN INTERVAL ARITHMETIC APPROACH
which is a real rational function, uni-incident in mantics (22) is equivalent to the interval inclusion
. The se-
yields the following linear system of equations:
(40)
(33) with , proper intervals and For the second unknown, given by
improper.
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The solution is, for fixed , given by
(34) Dual
the semantics (22) is equivalent to the interval inclusion Dual Dual
(41)
(35)
with , proper intervals, , improper is the interval obtained in (33). The rest of ones, and the controller parameters are obtained in the same way. Remark: Obviously, in practice, all approaches in this section must find a solution by evaluating a finite number of -cuts, . setting up a dense enough grid in the interval
with , , , , , proper intervals and , , improper ones. Fig. 3 shows the obtained results. Any controller in , and the set is feasible. For illustration, the Edge Theorem is applied to the central controller
VIII. EXAMPLES
(42)
1) Example 1: Diophantine equation. Let the plant
(36) be given with
, , and triangular fuzzy parameters3 and the
controller
to obtain the family of closed-loop poles shown in Fig. 4. Note that the plant’s core contains only one element. As expected, specifications are fulfilled. 2) Example 2: The methodology will be demonstrated on a position control experiment with a Feedback motor shown in Fig. 5. The output shaft is moved by the motor through a reduction belt. A magnetic brake and external passive electrical circuit is available to change the process dynamics and emulate a batch of different conditions. The model considered for the system is
(37) After inspection of the poles through the Edge Theorem, the following closed-loop characteristic polynomial is chosen:
(43)
(38)
The family of plants under consideration will be described by , the trapezoidal fuzzy parameters . The selected control structure is a PD controller
where
, and . The resulting Diophantine equation
(44) The desired closed-loop characteristic polynomial is defined (39) 3In the following, trap(a; b; c; d) will denote a trapezoidal fuzzy set with support [a; d] and core [b; c], and tri(a; b; c) = trap(a; b; b; c) a triangular fuzzy set.
as
(45)
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Fig. 4. Closed-loop poles. (a) Hard specifications—Plant’s core. (b) Soft specifications—Plant’s support. Black outlines denote the achieved closed-loop pole regions; gray ones denote the required specifications. Only the outer edges are relevant.
where and . Gray lines in Fig. 7 show the poles of the support (soft specifications) and core (hard specifications) of (45). Closed-loop coefficients are, for fixed ,
(46) Solving for the controller parameters for the real case
Fig. 5. Feedback motor 33-002.
(47)
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Fig. 6. Solution set for each -cut and their intersection.
As they are rational uni-incident functions, the desired semantics is, analogously to (31), equivalent to the interval inclusions
(48)
Remark: Note that in all of the examples considered, the solution of the problem for all -cuts would have been the same if only core and support problems were posed (cf. Figs. 6 and 3), i.e., even if the procedure may be applied to a fine grid of values for , sensible solutions in practice can be obtained by . just considering dual-specification settings
(49) with , , , and proper intervals, and and improper ones. Fig. 6 shows the result obtained for the defined plant and specifications. The thick line on the ordinate axis shows the intersecand tion of the solutions. Any controller with will fulfill the specifications. Fig. 7 shows the closed-loop poles for the support and core considering the controller (50) As it can be seen, specifications are fulfilled. Fig. 8 shows the simulated and experimental step responses for the following two sample conditions:
Case 1 Case 2 Case 1 belongs to the core and Case 2 to the support. As it can be seen, experimental response follows closely the expected theoretical one (initially, discrepancies occur due to backlash and unmodeled belt dynamics).
IX. CONCLUSION This paper discussed controller design for fuzzy uncertain plants, understood as possibility distributions. The controller maps each open-loop plant onto a closed-loop specification space. The problem is then posed as achieving the inclusion of the image of a fuzzy set of plants (obtained by the extension principle) into a user-defined fuzzy specification set or, equivalently, as an inclusion of the respective -cuts. In this way, a graceful degradation of specifications in a robust control problem can be enforced, describing a more general situation than usual robust control problems (with a single performance bound for all considered plants). Fuzzy pole placement has been addressed in this paper as a particular instance of the inclusion problem, considering the coefficients of the closed-loop characteristic equation. Under the premise that the plant is linear, described by fuzzy numbers as parameters of its transfer function, each -cut is an interval and, hence, interval arithmetic methods may be used in solving the resulting interval linear systems of equations. Some particular cases allow for an easy algebraic solution in terms of modal interval arithmetic, as the inclusion problem can be restated in terms of a particular semantic in this arithmetic. Evaluation of algebraic expressions via interval arithmetic has a much lower computational cost than a general iterative optimization-based approach.
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Fig. 7. Closed-loop poles. (a) Hard specifications—Plant’s core. (b) Soft specifications—Plant’s support. Black outlines denote the achieved closed-loop pole regions; gray ones denote the required specifications.
Some examples illustrate the possibilities of the presented techniques. The proposed framework of fuzzy plant uncertainty may be applied to other design techniques, such as frequency-response loop shaping, and sensible solutions in practice can be obtained by considering a reduced number of -cuts (even reducing to core and support dual-specification settings). APPENDIX Some basic definitions and results of modal interval arithmetic are presented in this Appendix. The reader is referred to [23] and [24] for a deeper coverage on this subject.
Modal Intervals: A modal interval is defined as a couple or where is its classic interval domain, , and the quantifiers and are a selection modality. The set of the modal intervals is represented by . Modal intervals of the type are called proper intervals or existential intervals, modal intervals of the are called improper intervals or universal intype tervals. A modal interval can be represented using its canonical coordinates in the form if if
.
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Fig. 8. Simulated and experimental unit step responses of Cases 1 and 2.
For example, the interval corresponds to and the interval corresponds to . The bounds and are named as the infimum, and the supremum, , of the interval. A point-wise interval , also represented as , can be considered as proper or improper and it is identifiable with the real number . The set of -dimensional modal interval vectors will be denoted by . , the operators Prop, Impr, and Dual For an interval are defined by if 1) ; if if ; 2) if 3) The process of construction of modal intervals is completed with the concept of modal quantifier defined by
if if which allows to define the set of real predicates accepted by a : modal interval
With the identification of a modal interval with the set of those arises the inclureal predicates which it accepts: sion of two intervals as the inclusion of the set of predicates that they accept, that is to say, if
Using their canonical coordinates and , this inclusion maintains the traditional modus operandi; that is to say
The lattice operations meet and join on for a bounded family of modal intervals ( is the index’s domain) are defined as the -maximum interval contained in all , for the meet, and the -minimum interval which contains all , for the join, i.e.,
Semantic Extensions: The extension of a given function to modal intervals will now be discussed. In modal interval analysis, the similar role to the “united extension” for the classical intervals [16], [32] is played by the semantic and -functions, denoted by and . Given a function and a vector of modal intervals , , split in proper components, , and improper ones, the semantic extensions of , are defined by
and
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where is the component-splitting corresponding to the proper and improper components of , . , (allowing for the abuse of language), Remark: If then
where, for a real number
(52) The quotient is defined by
If , then is said to be JM-commutable. An example of JM-commmutable functions are the arithmetic operators (see later). Two key results, named semantic theorems, give logical interpretation to these semantic extensions. and Theorem.1: ( -semantic theorem) Let be , then
Finite Precision: When working with finite precision, it is mandatory to control properly the rounding of the computations to preserve the validity of the associated semantics. Given a , the following functions will be demodal interval fined regarding the rounding of an interval with real endpoints:
(53)
Theorem.2: ( -semantic theorem) Let be and , then
Both semantic theorems are extremely important because they make equivalent any logical formula involving intervals, functional predicates and the universal quantifiers preceding the existential ones, to an interval inclusion. Arithmetic Operators: Important examples of JM-commutable functions are the one-variable continuous functions which is and every two-variable continuous function , like the arithmetic partially monotonic in a domain , , , and others, whose modal operators semantic extensions can be computed by means of arithmetic operations with the interval bounds. , , the arithmetic opGiven two modal intervals erations turn out to be as follows:
, , is the greatest value in a given digital scale where DI (e.g., the computer floating-point system) smaller than or is the smallest value in DI greater than equal to , and or equal to . Of course, and . Modal Rational Extensions: The semantic extensions and are out of reach for any direct computation except for simple real functions, since they correspond to multi-dimensional minimax optimization problems. When the continuous function is a rational function, there exist modal rational extensions which are obtained by using the computing program defined by the syntax tree of the expression of the function: if is a to rational function, its rational extension to the , represented by , modal intervals from to defined by the comis the function putational program indicated by the syntax of when the real operators (assumed JM-commutable) are transformed into their semantic extensions. Modal interval arithmetic provides a collection of results about the interpretability of modal rational extensions. Important results used in this paper are the following theorems. Theorem.3: ( -interpretability of modal rational functions) are uni-incident in , If the improper components of does exist, then and if
. A definition of the product given in [33] is the following:
(51)
and the -semantic theorem applies. Theorem.4: ( -interpretability of modal rational functions) If the proper components of are uni-incident in , and if does exist, then
and the
-semantic theorem applies.
BONDIA et al.: CONTROLLER DESIGN UNDER FUZZY POLE-PLACEMENT SPECIFICATIONS: AN INTERVAL ARITHMETIC APPROACH
Example: Consider the rational function
In this case, is said to be optimal. For instance, for the values of and in item b)
, , the modal rational extension is, a) For applying the aforementioned definitions for the arithmetic operators
As it is uni-incident in the proper parameters, the terpretability Theorem 4 applies and
which can be interpreted with the
which corresponds to the exact range of . As mentioned earlier, the reader is referred to the cited bibliographic references for in-depth coverage of the previously outlined concepts.
-inREFERENCES
-semantics
The same conclusion can be stated for any subset of . b) For , , the modal rational extension is
As there are no improper arguments, the -interpretability Theorem 3 applies and
which can be interpreted by the -semantics
The same applies for any superset of c) The function can be rewritten as
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which is uni-incident in and . In this case both the -interpretability and -interpretability theorem apply for any modal intervals and and
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[22] M. A. Sáinz, E. Gardenyes, and L. Jorba, “Interval estimations of solution sets to real-valued systems of linear or non-linear equations,” Reliable Comput., vol. 8, pp. 283–305, 2002. [23] E. Gardeñes, H. Mielgo, and A. Trepat, , K. Nickel, Ed., “Modal intervals: Reasons and ground semantics,” in Interval Mathematics. Berlin, Germany: Springer-Verlag, 1985, vol. 212, Lecture Notes in Computer Science, pp. 27–35. [24] M. A. Sáinz, “Modal intervals,” Reliable Comput., vol. 7, no. 2, pp. 77–111, 2001. [25] J. Bondia, Sistemas con incertidumbre paramétrica borrosa: Análisis y diseño de controladores Dept. Syst. Eng. Control, Valencia Tech. Univ. Valencia, Spain, 2002, Ph.D. dissertation. [26] P. Albertos and A. Sala, Multivariable Control Systems. New York: Springer-Verlag, 2004. [27] K. Ogata, Modern Control Engineering, 4th ed. Upper Saddle River, NJ: Prentice-Hall, 2002. [28] J. Antsaklis Panos and N. Michel Anthony, Linear Systems. New York: McGraw-Hill, 1997. [29] L. J. Mordel, Diophantine Equations. New York: Academic, 1969. [30] M. Green and D. J. N. Limebeer, Linear Robust Control. Englewood Cliffs, NJ: Prentice-Hall, 1995. [31] R. Isermann, Digital Control Systems (Volume II). New York: Springer-Verlag, 1989. [32] G. Alefeld and J. Herzberger, Introduction to Interval Computations. New York: Academic, 1983. [33] A. V. Lakeyev, “Linear algebraic equations in Kaucher arithmetic,” Reliable Comput., pp. 130–133, 1995.
Jorge Bondia was born in Valencia, Spain, in 1970. He received the M.Sc. degree in computer science and the Ph.D. degree in control engineering from the Valencia Technical University, Valencia, Spain, in 1994 and 2002, respectively. He is currently a Senior Lecturer at the Department of Systems Engineering and Control of the Technical University of Valencia. His main research interests are in robust control, parametric uncertainty, interval analysis, and applications in biomedicine.
Antonio Sala (M’03) was born in Valencia, Spain, in 1968. He received the B.Eng. degree (Hon. Deg.) from Coventry University, Coventry, U.K., in 1990, the 2nd Spanish National Graduation prize (M.Sc. degree in electrical engineering) in 1993, and the Ph.D. degree in control engineering from the Valencia Technical University, Valencia, Spain, in 1998. He has been teaching at the Valencia Technical University in the Systems Engineering and Control Department since 1993, in both undergraduate and Ph.D. courses. He has coauthored more than 60 papers in conference proceedings and journals. He is coauthor of the book Multivariable Control Systems (New York: Springer-Verlag, 2003), and coeditor of Iterative Identification and Control (New York: Springer-Verlag, 2002). Dr. Sala is a member of the IFAC Publications Committee and of the Technical Committee in Cognition and Control.
Jesús Picó (M’90) was born in Valencia, Spain, in 1964. He received the M.S.Eng. degree in industrial engineering and the Ph.D. degree in control engineering, both from the Valencia Technical University, Valencia, Spain, in 1989 and 1996, respectively. He is currently an Associate Professor in the Department of Systems Engineering and Control at the Technical University of Valencia. His main research interests are in nonlinear and intelligent control, modeling and control of biologic and biotechnological processes, and application of interval arithmetic to robust control.
Miguel A. Sainz was born in La Rioja, Spain, in 1942. He graduated in mathematics from the Universidad de Zaragoza, Zaragoza, Spain, in 1964, and received the Ph.D. degree from the Universidad de Madrid, Madrid, Spain, in 1970. He has been a Professor with the universities Politécnica de Cataluña, Autónoma de Cataluña, and Universidad de Gerona, Spain, since 1971. As member of the Institut d’Informàtica i Aplicacions, he develops his research in modal interval analysis and its applications to problems of simulation and control. Dr. Sainz has written several papers presented in international meetings (IFAC World Congress, SAFEPROCESS) and published in scientific journals (Journal of Process Control, Reliable Computing).
