Asian Journal of Control, Vol. 8, No. 4, pp. 345-358, December 2006 345

Manuscript received January 21, 2005; revised July 12,2005; accepted October 28, 2005.

The authors are with the Department of Computer Scienceand Engineering, School of Engineering, Shiraz University,Shiraz, Iran (e-mails: katebi@shirazu.ac.ir; eftekhar@shirazu.ac.ir).

EVOLUTIONARY SEARCH FOR LIMIT CYCLE AND CONTROLLER DESIGN IN MULTIVARIABLE NONLINEAR

SYSTEMS

M. Eftekhari and S. D. Katebi

ABSTRACT

A feature of many practical control systems is a Multi-Input Multi-Output (MIMO) interactive structure with one or more gross nonlinearities. A primary controller design task in such circumstances is to predict and ensure the avoidance of limit cycling condi-tions followed by achieving other design objectives. This paper outlines how such a system may be investigated using the Sinusoidal Input Describing Function (SIDF) phi-losophy quantifying magnitude, frequency and phase of any possible limit cycle opera-tion. While Sinusoidal Input Describing function is a suitable linearization technique in the frequency domain for assessment of stability and limit cycle operation, it can not be employed in time domain. In order to be able to incorporate the time domain require-ments in an overall controller design technique, the appropriate linearization technique suggested here is the Exponential Input Describing Function (EIDF).

First, an evolutionary search based on a multi-objective formulation is employed for the direct solution of the harmonic balance system matrix equation. The search is based on Multi-Objective Genetic Algorithms (MOGA) and is capable of predicting specified modes of theoretically possible limit cycle operation.

Second, the design requirements in time as well as frequency domain are formu-lated by a set of constraint inequalities. A numerical synthesis procedure also based on Multi-Objective Genetic Algorithm is employed to adjust the initial compensator pa-rameters to meet the imposed constraints. Robust stability and robust performance are investigated with respect to linearization uncertainty within the context of multi- objective formulation. In order to make the Genetic Algorithm (GA) search more ame-nable to design trade-off between different and often contradictory specifications, a weighted sum of the functions is introduced. This criterion is subsequently optimized subject to the nonlinear system dynamics and a set of design requirements. Examples of use are given to illustrate the effectiveness of the proposed approach.

KeyWords: Multivariable, multi-objective, limit cycle, nonlinear, evolutionary.

I. INTRODUCTION

Nonlinear behaviors are very common in practice and are usually approximated by linearization around the oper-ating point. This procedure may not be acceptable for com-plex and highly reliable systems, particularly if the nonlin-earity is “hard”, such as a hard limit or saturation with a

dead zone. However, with the advent of fast and powerful digital computers, research for a more precise and accurate analysis of nonlinear systems has grown considerably [1-3]. One such method, which has traditionally been applied, is the replacement of nonlinear behavior with a quasi-linear gain called the Describing Function (DF)[4-6]. Describing function theory and techniques represent a powerful mathematical approach for analyzing and designing nonlinear systems.

The basic idea of the (DF) approach for modeling and studying nonlinear system behavior is to replace each nonlinear element with a quasi-linear descriptor or de-scribing function whose gain is a function of input ampli-

346 Asian Journal of Control, Vol. 8, No. 4, December 2006

tude. The functional form of such a descriptor is governed by several factors: the type of input signal, which is as-sumed in advance, and the approximation criterion, e.g., minimization of mean squared error. This technique is dealt with very thoroughly in a number of texts for the case of nonlinear systems with a single nonlinearity [4-6]. One category of DFs that have been particularly successful is the Sinusoidal-Input Describing Function (SIDF).

Extension of describing function techniques to multi-loop nonlinear systems is not new and followed the fre-quency domain multivariable linear theory developed by [7,8]. Mees and Ramani [9,10] proposed the extension of harmonic linearization based on Nyquist array in very early stages. Ramani and Woon [10,11] extended the characteris-tic loci to the describing function method. Gray [12] pro-posed a numerical technique with graphical interpretation for quantifying the limit cycle parameters in multivariable systems and later extended these techniques to account for higher harmonics [13]. Katebi [14] extended the graphical techniques to multi-loop systems with coupled multi- valued nonlinear elements. Patra, et al. [15] suggested an-other graphical method based on phasor diagram which particularly gave accurate limit cycle prediction for relay systems. Paoletti, et al. [16] presented a Computer Aided Design tool for limit cycle prediction in single loop nonlinear systems which are aimed at educational purposes. Pillai, et al. [17] presented a numerical technique based on deriving the least damped eigenvalue to the imaginary axis for single loop nonlinear feedback systems with multiple nonlinearities. Based on defining an appropriate error func-tion, Pillai used both eigenvalue and eigenvectors to for-mulate a generalized Newton-Raphson to solve for the state variable amplitude in a minimum norm sense.

Most of the above mentioned techniques are essen-tially dependent on the graphical displays in the frequency domain. While these are useful for getting insight into the subsequent compensator design, an efficient technique for accurately quantifying the limit cycle parameters and, more importantly, a technique which can be employed in a com-prehensive design procedure is highly desirable.

In this paper, firstly, a multi-objective formulation is presented to search numerically for limit cycles in a class of multi-loop nonlinear systems. The approach is computa-tionally efficient and is based on Multi-Objective Genetic Algorithms (MOGA) which results in accurate prediction of limit cycle parameters.

The second objective of this work is the subsequent controller design, aimed at eliminating the limit cycle be-havior and meeting additional design requirements. Pro-vided that a fast time response routine and an efficient multi-objective optimization method are available, any controller design can be formulated as a multi-objective problem [18-20]. A direct and an appropriate technique is to translate the design requirements into a set of con-strained functions representing the closed loop time domain specification, stability requirement, the avoidance of limit

cycling conditions or any other constraints of engineering or financial kind. Earlier, Zakian had shown that the time response requirements of linear control systems may be formulated as a set of inequalities, and had used the method of moving boundary process in conjunction with the Hill Climbing search to solve the set of inequalities [21]. Sev-eral other authors [19,22,23] have proposed various modi-fications to the method of inequalities. Fonseca, et al. [24] suggested the use of genetic algorithms for solving the set of inequalities. Tan, et al. reported on a Matlab-based Multi-objective Evolutionary Toolbox for Computer ⎯ Aided Multi-objective Optimization which can be em-ployed for design of multi-objective controller design problems [25]. The above mentioned works have primarily concentrated on linear systems.

However, in order to be able to extend the design and synthesis to multivariable nonlinear control systems with one or more gross nonlinearities, some form of lineariza-tion in frequency as well as time domain is required. The appropriate linearization technique suggested here for tran-sient evaluation of the class nonlinear multi-loop systems comprising single value sector bounded and separable nonlinear elements is the Exponential Input Describing Function (EIDF)[4]. In this work, the procedures for in-corporating the time domain requirements in an overall controller design is formulated as a multi-objective prob-lem and solved by MOGA techniques. The compensator design may additionally be subject to minimization of a weighted sum of the sensitivity, complementary, and the weighted time squared error functions, providing for robust performance and robust stability. In cases of limit cycle prediction and controller design, emphasis is placed on the multi-objective formulation and subsequent solution by the evolutionary MOGA method.

II. HARMONIC ANALYSIS

The harmonic balance equation for the autonomous feedback configuration of Fig. (1), in which the nonlinear-ity exists as a separable entity in an otherwise linear system is given as;

1 ( , ) ( ) 0N a G j+ ω ω = (1)

where N(a, ω) is the sinusoidal input describing function representing the nonlinear element and a, ω are the sinu-soidal input amplitude and input frequency respectively. G( jω) is the frequency transfer function of the linear part. In a simple harmonic analysis, some form of solution of the Eq. (1) is sought. However, the valid application of the SIDF requires that the input signal to the nonlinear element be essentially sinusoidal in form. This is a condition which imposes an overall low pass frequency characteristic on the linear system elements such that the super-harmonic sig-nals are attenuated around the feedback loop. Based on the assumption that the input to the nonlinear element is a si-nusoidal, the SIDF is derived as;

M. Eftekhari and S.D. Katebi: Evolutionary Search for Limit Cycle and Controller Design in Multivariable Nonlinear Systems 347

Fig. 1. Multivariable nonlinear autonomous feedback system.

1 11 1

( ) 1( , ) ( )j t

j tb ja eN a b ja

aae

ω

ω+ω = = + (2)

where a1 and b1 are the first harmonics in Fourier series of N, and a is the amplitude of the input sinusoidal.

III. LIMIT CYCLE PREDICTION IN NONLINEAR MULTIVARIABLE SYSTEMS

Provided that certain limitation is placed on the form of the linear system elements, the extension of the phi-losophy of harmonic linearization to multi-variable systems is conceptually straightforward. The equation governing limit cycle operation in the autonomous multi-variable nonlinear feedback system of Fig. 1 can be expressed as:

( )( , ) 0T a j I aω + = (3)

where T(a, jω) = N(a, jω) G( jω), and N(a, ω) is the matrix of sinusoidal input describing function corresponding to the nonlinear elements of N, and a is column vector of sinusoid at the inputs to these elements. The use of a single sinusoi-dal describing function analysis implies the following as-sumptions.

• The elements of G( jω) act as low pass system so that higher harmonic signal components are effectively sup-pressed.

• If a limit-cycle is present then all loops will oscillate at the same frequency. Experience indicates that this is a reasonable assumption, particularly if the nonlinear elements are similar and if the dominant linear ele-ments have approximately the same frequency charac-teristics.

Equation (3) will have a non-trivial solution, only if

det [ ( ) ( , ) ] 0G j N a Iω ⋅ ω + = (4)

Thus, for no limit-cycle to exist, no eigenvalue of G( jω) N(a, ω) can equal (−1, j0).

The conventional graphical frequency domain method for the solution of Eq. (4) for single input single output systems with a single nonlinear element may be extended to MIMO systems employing Nyquist or the Inverse Ny-quist Array. Following [9] and invoking the Inverse Ny-quist Array method, the possibility of limit cycle existence may be examined by studying the following inequalities.

ˆ ˆ| ( ) ( )| | ( ) ( )|kk k kk jk k jkj k

n a g j n a g j≠

+ ω > + ω∑ (5)

ˆ ˆ| ( ) ( )| | ( )| | ( )|kk k kk jk k jkj k j k

n a g j n a g j≠ ≠

+ ω > + ω∑ ∑ (6)

where njk(a) and ˆ ( )jkg jω are the jkth elements of the ma-

trices N(a) and G −1( jω) respectively. In general SIDF is a

function of both input amplitude and input frequency, however, for single valued, sector bounded nonlinearities which are the subject of this work, the describing function is real and only a function of input amplitude a, therefore for brevity N(a) is used to denote SIDF for single valued nonlinearities.

Inequality (5) implies that no limit cycle operation is theoretically possible if the Gershgorin band associated with each diagonal element of ( )1( ) ( )N a G j−+ ω does not encompass the origin in the complex frequency domain. Inequality (6) implies the same result provided that the bands traced out by the Gershgorin discs on the loci of ˆ ( )iig jω and nii(a) do not intercept for every i. The former

representation is computationally more demanding but gives less conservative results because of the weakening step between Inequalities (5) and (6). The Gershgorin discs set bounds to the eigenvalue locations and any method of limit cycle prediction based on band intersection must tend to be conservative.

An alternative approach lies in calculating the eigen-value of the harmonically linearised return ratio system equation and a simple graphical procedure for system con-taining only diagonal nonlinear elements has been given previously by Ramani [10]. For a more general system which contains both on and off diagonal nonlinear elements, the computational effort involved in determining the ei-genvalue becomes almost formidable as, at any frequency, the return ratio matrix is a function of the signal amplitude at the input to the nonlinear elements.

A numerically based technique called the sequential loop balance method had also been devised by Gray, et al. [13]. This method is based on Eq. (3) for which T(a, jω) has elements of the form tij (aj, jω) and a is a column vector at the input to the nonlinear elements such that aj = Aj exp( jφi), j = 1, 2, …, n, ∀j, i = 1, 2, …, m over an arbitrary ranges A, ω, and φ a possible infinite number of solutions may exist for Eq. (3). For a specified value of frequency and a specified range of discrete values of the reference signal a1 = A1e

j0, A1 > A2, A3, …, An a finite p sets of sinu-soids (a1k, a2k, …, ank), k = 1, 2, …, p which will satisfy the condition for harmonic signal balance in the nth system loop as given by the Eq. (7) are found;

1

1( 1) 0

n

nn n nj jj

t a t a−

=+ + =∑ (7)

Next the (n − 1)th loop is considered but using only those finite set of solution sinusoids derived from Eq. (7).

348 Asian Journal of Control, Vol. 8, No. 4, December 2006

If there are q solution sets of sinusoids (q ≤ p) which satisfy both equations, the (n − 2)th loop equation is next examined using only the q sets of solution sinusoids and the process is repeated in a sequential manner until loop 1 is reached. Clearly, this method is also computationally demanding due to undirected search over the specified ranges of pa-rameters. Further, it does not search for the phase differ-ence between the oscillating loops, and as discrete data are used in the computation there is the possibility of solution sets which exist over the data intervals.

One prime objective of this work is to derive an effi-cient and accurate method for limit cycle prediction which can also be utilised within the context of a comprehensive controller design procedure which avoids limit cycle in the frequency domain as well as achieving other design re-quirements.

IV. MULTI-OBJECTIVE GENETIC ALGORITHMS

Genetic Algorithms (GAs) are search procedures based on the evolutionary process in nature [26]. They dif-fer from other approaches in that they use probabilistic and non-deterministic criteria for search progression. The idea is that the GA operates on a population of individuals, each individual representing a potential solution to the problem, and applies the principle of survival of the fittest on the population, so that the individuals evolve towards better solutions to the problem. The individuals are usually given a genotypic representation. Three operations are performed on individuals in the population, selection, cross-over and mutation. These correspond to the selection of individuals in nature for breeding, where the fitter members of a popu-lation breed and, in doing so, pass on their genetic informa-tion. The cross-over corresponds to the combination of genes by mating, and mutation to genetic mutation in na-ture. The selection is biased so that the ‘fittest’ individuals are more likely to be selected for cross-over, the fitness being a function of the criterion which is being minimized. By means of these operations, the population will evolve towards a solution [27].

Most GAs have been used for single objective prob-lems, although several multi-objective schemes have been proposed [28]. Fonseca, et al. [29,30] have used an ap-proach called the multi-objective genetic algorithm (MOGA), which is an extension of an idea, by Goldberg [26]. This formulation maintains the genuine multi- objective nature of the problem, and is essentially the scheme used here. Further details of the MOGA can be found in [28,29] and only a brief description is given be-low.

An individual j with a set of objective functions φ j =

(φ1j, …, φm

j) is said to be non-dominated if, for a population of M individuals, there are no other individuals k = 1, 2, …, M; k ≠ j such that;

a) for all 1, 2, , and

b) for at least one .

k ji i

k ji i

i m

i

φ ≤ φ =

φ < φ

… (8)

The MOGA is set into a multi-objective context by means of the fitness function. The individuals are ranked on the basis of the number of other individuals they are dominated by for the unsatisfied inequalities. Each indi-vidual is then assigned fitness according to their rank. The mechanism is described in detail in [29]. To summarize, the MOGA problem could be stated as:

Find a set of M admissible points Pj; j = 1, 2, …, M such that

( 1, 2, , ), ( 1, 2, , )ji i j m i nφ ≤ ε = =… … (9)

and such that φ j( j = 1, 2, …, M) are non-dominated.

Genetic Algorithms are naturally parallel and hence lend themselves well to multi-objective settings. They also work well on non-smooth objective functions. Thus GA can be used to search for the existence of any possible limit cycle operation in nonlinear systems and subsequently for the controller parameters and even for the controller struc-ture, although the later is not attempted in this work. In this paper limit cycle prediction and the design requirements in the time as well as the frequency domain are formulated by a set of inequalities. A numerical synthesis procedure based on Multi-objective Genetic Algorithm is then employed to search for the possible limit cycle parameters and subse-quently for the compensator parameters which meet the imposed constraints.

However in order to be able to employ the time do-main synthesis technique, a computationally efficient and accurate transient response evaluation is developed based on the EIDF linearization in a generational and interactive MOGA environment. For the class of single-valued static nonlinearities the EIDF is used as equivalent linear but input dependent gain in the processes of time response evaluation.

V. TRANSIENT EVALUATION OF NONLINEAR SYSTEMS

There are several methods for prediction of transient in nonlinear systems [2,3,31]. Generally, for the class of separable nonlinear behavior, the methods attempt to evaluate an approximate transient gain when the nonlinear element is subject to some assumed input signal. The ex-ponential input describing function proposed by Gelb et al. [4] appears to be a simple and practical technique for evaluating the transient gain. For the nonlinear feedback configuration of Fig. 2, the input to the single valued nonlinear element is assumed to be an exponentially de-caying signal. The transient gain is derived in such a way that the integral squared error criteria are minimized, i.e.

M. Eftekhari and S.D. Katebi: Evolutionary Search for Limit Cycle and Controller Design in Multivariable Nonlinear Systems 349

(a)

(b)

Fig. 2. Nonlinear feedback system-EIDF linearization.

The representation error, e is;

( ) ( ) [ ( )]EIDFe t K x t y x t= − (10)

where KEIDF is a fixed linear gain. The corresponding squared error, integrated over all time is:

2 2 2

0 0

2

0 0

( ) ( ) ( )

2 ( ) [ ( )] [ ( )]

EIDF

EIDF

e t dt K x t dt

K x t y x t dt y x t dt

∞ ∞

∞ ∞

=

− ⋅ +

∫ ∫

∫ ∫

(11)

Minimizing this expression by differentiating with respect to KEIDF and setting the result to zero yields the EIDF gain.

0

2

0

( ) [ ( )]

( )

EIDFx t y x t dt

Kx t dt

∞

∞

⋅=∫

∫ For ( )

tx t E e

−τ= ⋅ (12)

E is the amplitude of the input to the nonlinear element. KEIDF is called the Exponential input Describing Function and for time independent and single valued nonlinearities, KEIDF is real and may only depend on the amplitude of the input.

VI. MULTI-OBJECTIVE FORMULATION FOR LIMIT CYCLE PREDICTION

The procedure for limit cycle prediction is formulated as a multi-objective problem. The objectives are the satis-faction of all harmonically linearised system loop equations simultaneously. An intelligent multi-objective genetic al-gorithm is used to search the space of possible limit cycle parameters; effectively solving the matrix Eq. (3) directly by the Genetic Algorithms. Inequality set (5) can be rear-

ranged in the form of the following inequalities which are subsequently considered as objectives for a square n di-mensional nonlinear system;

11 1 12 2 13 3 1

21 1 22 2 23 3 2

| (1 ) ( ) ( ) ( ) |

| ( ) (1 ) ( ) ( ) |

n n

n n

t a t a t a t a

t a t a t a t a

+ + + + + ≤ ε

+ + + + + ≤ ε

…

…

1 1 2 2 ( )( 1)| ( ) ( ) ... ( ) (1 ) |n n n n n nn nt a t a t a t a−+ + + + + ≤ ε (13)

where ε is a small positive number near zero and ai = Ai exp( jφi). The above n equations are the compact form of 2n equations representing the real and imaginary part of each equation or alternatively the magnitude and phase in each equation. The multi-objective GA is employed to search over the space ω, ai, and φi for Pareto minimal solution(s) of the Inequality set (13).

The idea behind the MOGA is to evolve a population of Pareto-optimal or near Pareto-optimal solutions. The aim is to find a set of solutions which is non-dominated and satisfy a set of Inequalities [29]. The MOGA differs from other optimization methods, in that a set of simultaneous solutions are sought, and the designer may then selects the best solution from the set. For the purpose of illustration consider a two dimensional nonlinear system shown in Fig. 3. The objectives are formulated as;

11 11 1 12 12 2Objective_1 = | (1 ) ( ) |jn g a n g a e φ+ + ≤ ε

21 21 1 22 22 2Objective_2 = | ( ) (1 ) |jn g a n g a e φ+ + ≤ ε (14)

Upper and lower bounds are specified for ω, a1, a2, and φ then the real generational MOGA with specified se-lection method, cross over, mutation rate and population size is called to search over the parameter space of ω, a1, a2, and φ. The required numerical accuracy may be achieved by specifying the number of genes (binary bits) for each individual in accordance with the value of ε. If the condi-tions for the Inequalities (14) exist, then the MOGA con-verges to the correct values of limit cycle parameters after a number of generations. If finer and more accurate parame-ter values are required, the bounds of the parameters ω, a1, a2, and φ may be tightened on a subsequent run of the MOGA program. In order to avoid the local minima and premature convergence the mutation probability rate may be taken higher at the beginning and decreased exponen-tially toward the end. One advantage of limit cycle predic-tion based on MOGA is that multiple solutions may be distinguished by using the Niching mechanism [28]. Hav-ing predicted the limit cycle parameters, the next step is the formulation of the controller design objectives.

350 Asian Journal of Control, Vol. 8, No. 4, December 2006

Fig. 3. A 2-input 2-output uncompensated nonlinear system.

VII. MULTI-OBJECTIVE FORMULATION OF THE CONTROLLER DESIGN

The next part of the design is to formulate the design requirements as a multi-objective optimization problem, that is, as a set of Inequality. The initial structure of the controller may be decided based on the domain knowledge obtained while searching for limit cycle. Additionally, for difficult behavior, reference can always be made to the Nyquist or Inverse Nyquist Array displays. Choosing a compensator structure without the use of domain knowl-edge and/or without an intuitive understanding of the sys-tem dynamics can lead to inefficient MOAG search. Fur-ther, the designer has to choose appropriate arrangement of the compensator, noting the theoretical effects on the steady state solution. Within the design program, diagonal or non diagonal, pre and/or post compensator of constant or dynamic forms may be chosen.

For a n × n square plant (shown in Fig. 4 for a 2 × 2), generally, the order of the compensator is related to the financial consideration and the value of its parameters are related to stability and physical limitations of the subse-quent measuring devices and instrumentations. The loca-tion of the compensator is dictated by accessibility to the desired part of the system. Analytically a pre or a post- compensator can be derived in a similar manner and will have the same effects on system stability. Physically they will differ due to different signal levels involved and gen-erally a pre-compensator is preferred.

Fig. 4. The two dimensional compensated nonlinear system.

In this work, the design is mainly aimed at utilizing a pre-compensator matrix, with elements of simple structure in the following form:

1 2

3 4( )

p s pP s

p s p⎛ ⎞+

= ⎜ ⎟⎜ ⎟+⎝ ⎠ (15)

where p1, p2, p3, p4 are the parameters to be found, and s is a Laplace operator. In the following, the design require-ments are formulated in both frequency and time domain and in practice either set or both sets of requirements may be utilized.

7.1 Frequency domain requirements

The first objective in the frequency domain is to de-sign a controller with parameters P which prevents the nonlinear system from limit cycling. This implies the fol-lowing inequalities;

1 1

2

( , )

( , ) , 1, 2, ,i i

P

a P i n

ϕ ω ≤ δ

ϕ ≤ δ = … (16)

where P is the vector of the controller parameters, ω is the frequency, and ai is the amplitude of loop 1 to loop n for any possible limit cycle and are positive numbers, δ1 and δi are positive numbers (1 + tnn) an near zero. These inequalities, in turn, imply that ε must be a negative num-ber in Equality set (13). Other frequency domain objectives such as gain margin, phase margin and bandwidth as well as bounds on the eigenvalue of linear systems elements can also be included, if desired.

7.2 Time domain requirements

The MOGA program may be linked to other programs, mainly, for evaluation of objectives. As in this design pro-cedure some objectives are defined in terms system’s step response, the time simulation setup in which each nonlinear element is replaced by its equivalent EIDF gain, provides these time response specifications.

Consider a n × n system, such that yii(i = 1, …, n) de-note the time outputs for loop1, loop2, …, loopn respec-tively, and yij (i = 1, …, n, j = 1, …, n, i ≠ j) denote the loop interaction effects, as time sequences that are available in each generation of MOGA for every member of population. Several routines are written to calculate the time response parameters, such as, rise time, settling time, overshoot, steady state error and loop interaction effects for each out-put. These time response parameters are calculated using the standard formula as shown below with the correspond-ing inequalities;

Rise time Rise time is the least value of t such that: y(t, P) = 0.9

× A

M. Eftekhari and S.D. Katebi: Evolutionary Search for Limit Cycle and Controller Design in Multivariable Nonlinear Systems 351

3 1( , )t Pϕ ≤ τ (17)

where y, ϕ3, and τ1 represent the step responses, the calcu-lated and the desired values of rise time for each loop re-spectively, and A is the amplitude of the step input.

Overshoot

( )Overshoot max ( , ) )y t P A= −

4 ( , )t Pϕ ≤ λ (18)

where ϕ4 and λ represent the calculated and the desired values of Overshoot for each loop respectively.

Settling time Settling time is the greatest value of t such that:

| ( , ) | 0.05A y t P A− > × 5 2( , )t Pϕ ≤ τ (19)

where ϕ5 and τ2 denote the calculated and the desired val-ues of the settling time for each loop respectively.

Steady State Error (SS)

| ( , ) |SS A y P= − ∞ 6 ( , )t Pϕ ≤ γ (20)

where y(∞, P) is the steady state value of outputs, and ϕ6 and γ denote the calculated and the desired values of the steady state error for each loop respectively.

Loop Interaction

( )max

( )ij

ijjj

y tI

y t∞

⎧ ⎫⎪ ⎪= ⎨ ⎬⎪ ⎪⎩ ⎭

or ~ ( )

( )ij

ijjj

y tI

y t∞

∞

⎧ ⎫⎪ ⎪= ⎨ ⎬⎪ ⎪⎩ ⎭

for all i ≠ j 7 ( , )t Pϕ ≤ σ (21)

where ϕ7 and σ are calculated and the desired loop interac-tion effects.

7.3 Robust stability and robust performance

There are several stability criteria given in the litera-ture [32,33] for asymptotic stability of the multiloop nonlinear systems. If the nonlinear elements can be ap-proximated by a set of linear gains, the method of Mees [34], which interprets stability behavior from the charac-teristic loci, can be used for systems with coupled nonlin-earities. In this method, the matrix of nonlinear elements is approximated as accurately as possible by a linear time invariant matrix, K, which gives rise to error function as follows:

( ) ( )E a N a Ka= − (22)

and it has been shown by Mees [34] that the choice of kij

that minimizes eij(a) is:

1 ( )2 ij ijK = β + α (23)

resulting in the constant gain matrix C having the elements 1 ( )2ij ij ijc = β − α , where α and β are the lower and upper

bounds of each sector bounded nonlinearity, respectively. The nonlinear system of configuration shown in Fig. 3 would be stable if the nonlinear matrix N were replaced by K such that the transfer matrix H defined as: H−1 = G−1 + K has characteristic function with its entire pole on the left half plane. G is the transfer matrix of the linear part. Then, in terms of the gain matrix C, the system is considered sta-ble in the sense that the output has the same boundedness, periodicity, and continuity as the input, provided H is a stable matrix and:

sup | ( ) | 1 for > 0H j Cω < ω (24)

The final task of any control design is to ensure that the system is robust with respect to stability, disturbance and noise rejection [35,36].

For system to be robustly stable, the ultimate constant nonlinear gain (upper bounds of each nonlinear element) can be used. However this is likely to result in an unneces-sarily conservative controller. Alternatively, more realistic values for constant nonlinear gains may be formulated as follows.

In each MOGA generation, for each string, the output of each nonlinear element is monitored and the corre-sponding maximum value of the EIDF gain is found. The uncertainty due to linearization is then taken as the differ-ence between the ultimate upper bound of the nonlinearity and its equivalent maximum EIDF gain. The matrix of the uncertainty Δ for each member of population is constructed with elements δij as follows.

( ) ( )max( )L Lij ij ijKδ = β − (25)

where βij is the upper bound for the elements of the nonlinear matrix, and ( )L

ijK is the EIDF gain for string L in a particular generation. The additive uncertainty may be formulated as:

( ) ( ) ( )( ) ( )L L LG s G s= + Δ where ( ) ( )( )L Lij ijG s g K=

(26)

where G(s) and ( )G s are the transfer function matrices of the nominal and the uncertain plant, respectively and Δ is the linearization uncertainty. Consider a system with closed loop transfer matrix CL(s) = y(s)/r(s). The sensitivity of the closed loop performance with respect to controller P is defined as:

352 Asian Journal of Control, Vol. 8, No. 4, December 2006

11 ( )

SG s+

(27)

G(s) has elements of the form ( )EIDFij ijK g s⋅ for the un-

compensated systems and of the form ( )( ) ( )L

ij ij ijP s k g s⋅ ⋅

ij+Δ for the compensated system for configuration of Fig. 4 and is different for each member of population at each generation. The complementary sensitivity function de-noted by T and is defined as:

( )1 ( )

G sT I SG s

− =+

(28)

Now the sensitivity function for each member of popula-tion at each generation is represented by:

( ) ( ) 1( )L LS I G −= + and ( ) ( )L LT I S= − (29)

The inequality constraint due to robust stability and robust performance can now be formulated as a weighted (w1, w2) sum as follows:

( ) ( )1

( ) ( )1

Robust Performance: || ||

Robust Stability: || ||

L Li i

L Li i

w S w ITSE

w T w ITSE

∞

∞

× + × ≤ ζ

× + × ≤ γ

(30)

where w1 + wi = 1 and are subject to genetic algorithms search and || ⋅ ||∞ denotes H-infinity norm, ITSEi is the inte-gral of time-weighted squared error for loopi (i = 1, 2, …, n) ξ and γ are a measure of robust performance and robust stability, respectively. Although the integral of square error (ISE) could be used, the use of ITSE results in an improved tracking behavior. In the evaluation of sensitivity and the complementary sensitivity function the uncertain model given by Eq. (29) is used. Since it is assumed that no mod-eling error is present in the linear part of the system and since the uncertainty due to linearization is not frequency dependent, the weights w1 and wi were not chosen to be of dynamic structure

7.4 Additional inequalities

Other inequalities can also be included as the experi-ence indicates. One such set of inequalities are related to a limit on the magnitude of the output signal from the con-troller, and subsequently bound to the controller parameter values (p1 to p4). This ensures that a physically realizable compensator is obtained. For example, for the compensator of the form [31]:

1 2

3 4( )

p s pP s

p s p⎛ ⎞+

= ⎜ ⎟⎜ ⎟+⎝ ⎠ (31)

The following additional inequalities are formulated:

1 1 2 2 3 3 4 4 , , , p c p c p c p c≤ ≤ ≤ ≤ (32)

Now the complete MOGA computational procedure may be outlined as follows: • Enter linear and nonlinear data. • Choose the compensator structure. • Specify bounds on the design parameters. • Initialize population (Controller Parameters). In the absence of any other information individuals are initialized randomly. • Specify the desired design criteria.

Start • Send individuals to the time simulation set up and other

programs. • Run Simulation Model and other programs. • Evaluate the Objectives from the time responses and

other programs. [Time domain performance (risetime1 risetime 2, settling 1 … ITSE, ISE as required), fre-quency domain objectives (avoidance of limit cycling condition, sensitivity, C-sensitivity, eigenvalue), oth-ers].

Run MOGA While objectives are not met Select and operate on new string Goto Start

VIII. APPLICATIONS

Example 1. Examination of the limit cycle behavior of the nonlin-

ear feedback system with configuration of Fig. 3 with lin-ear system matrix as given below and nonlinear data as shown in Fig. 5 is required.

2 2

3 2

1.2 2( 1.5 0.5) 2 1

3 1.63 2 ( 1.6 0.8)

s s s s sG

s s s s s

−⎡ ⎤⎢ ⎥+ + + +⎢ ⎥=⎢ ⎥⎢ ⎥

+ + + +⎣ ⎦

(33)

Fig. 5. Nonlinear matrix for example 1.

M. Eftekhari and S.D. Katebi: Evolutionary Search for Limit Cycle and Controller Design in Multivariable Nonlinear Systems 353

Linear and nonlinear data is first entered. In the limit cycle investigation phase, the nonlinear elements are re-placed by their corresponding SIDF gain and the harmonic balance matrix equations are constructed for this configura-tion. In order to make the MOGA search more realistic, upper and lower bounds are specified for all parameters of possible limit cycle operation, namely, the frequency and amplitude for each loop. Next, the parameters of the MOGA program such as population size, number of gen-erations, selection method, and crossover and mutation rate are specified, these are shown in Table 1. The program is run to search for those parameters (ω, a1, a2, φ) which sat-isfy the Inequality set (14). The search progress is moni-tored interactively and if, within the specified range of the parameters, feasible solutions exist, MOGA converges to solutions which are ranked according to their fitness values measured by the concept of non-dominance, that is, Pareto optimal. For this example, initially, the following ranges were specified;

1 20.1 2.0, 1.0 5.0 1.0 5.0 0.1 2a a≤ ω ≤ ≤ ≤ ≤ ≤ ≤ φ ≤ π

For a population size of 100, it was observed initially that a large number of solutions exist. However, as the search progressed it was observed that most members of population were concentrated in the following ranges (see Fig. 6):

1 20.5 1.0, 1.5 2.5 2 3.5 2.0 4.0a a≤ ω ≤ ≤ ≤ ≤ ≤ ≤ φ ≤

In order to reduce the number of solutions, this time, the weighted sum of the two objectives were formulated as one objective, i.e.

1 2

1 2 1

Objective Objective _1 Objective _ 2

Subject to : 0.1 1., and 1

w w

w w w

= × + ×

≤ ≤ = −

(34)

where Objective_1 and Objective_2 are as defined by the Inequality set (14) and w1 is subject to GA search. The refined ranges were specified and MOGA was called once more. It was observed that the number of solutions was reduced significantly. The algorithm converged after 15 generations and the first three solutions with the actual numerical simulation result are shown in Table 2 and Fig. 7 shows the actual limit cycle oscillation.

The next stage of the design is to devise a compensa-tor that would eliminate limit cycle operation and meet other requirements. First, a diagonal compensator with constant elements of the following form was tried.

11

22

00

pP

p⎡ ⎤

= ⎢ ⎥⎣ ⎦

(35)

The range of p11 and p22 were chosen to be between 0.1 and 1. A set of required time domain criteria, such as rise time, settling time, overshoot, steady state error and the interaction effects for each loop were specified as the ob-jectives and the MOGA with tuned parameters was called. It was observed that, although this compensator would eliminate limit cycle, it did not achieve an acceptable time response behavior. Next, the off-diagonal constant ele-ments were added and the desired time behavior was still not achieved. Finally, a diagonal dynamic compensator of the following form was chosen:

Table 1. MOGA parameters setting for example 1.

# of generation

Population Size

Selection Method

Representation Method

Multi Objective method

Mutation Probability

Cross Over Probability

50 100 Tournament Binary 8 bit for each parameter Pareto 0.02 0.7

Fig. 6. Individuals (genes) for frequency and amplitude of

loop 1 of possible limit cycle.

Table 2. Results of limit cycle prediction with MOGA for example 1.

Best Ranked

ω Radians/sec. a1 a2

ϕ Radians Obj 1

1 0.8366 2.1910 2.9009 2.3504 5.3851E-7

2 0.8366 2.1910 2.9009 2.3504 5.3851E-7

3 0.8366 2.1910 2.9009 2.3504 5.3851E-7

Simulation Results 0.8437 2.123 2.96 2.38

354 Asian Journal of Control, Vol. 8, No. 4, December 2006

Fig. 7. Limit cycle oscillation for example 1.

1 2

3 4

5 6

7 8

0

0

p s pp s p

Pp s pp s p

+⎡ ⎤⎢ ⎥+⎢ ⎥= ⎢ ⎥+⎢ ⎥⎢ ⎥+⎣ ⎦

(36)

After the bounds for all parameters were set to be be-tween 0.1 and 1.0 and a set of unit step response require-ments shown in Table 3 were specified, the MOGA was run to search for those values of p1 to p8 that would meet the required criteria. With these settings, after 27 genera-tions, MOGA converged and the first three set of compen-sator parameters with some of the resulting unit step re-sponse specifications are shown in Table 4. It is seen that every compensators given in Table 4 and many other in the population of 50, meet all the requirements except the loop interaction effects. The step response of the linearised

compensated system for the first ranked compensator pa-rameters is shown in Fig. 8.

One advantage of the Pareto optimal method is that the designer is provided with a set of possible solutions from which an appropriate one may be selected. For exam-ple, the compensator on line 3 of Table 4 resulted in a bet-ter speed of response at the expense of reduced accuracy (increased overshoot). In order to reduce the loop interac-tion and achieve robust stability and robust performance, a new design based on the following weighted sum was for-mulated:

1 2 3 4Objective || || || || 1 2w S w T w ITSE w ITSE∞ ∞= × + × + × + ×

1 2 3 4 1w w w w+ + + = (37)

where ITSE 1 and ITSE 2 are the integral of time weighted squared error for loop 1 and loop 2, respectively ⎯ al-though the upper bounds of the nonlinear elements could be used instead of the EIDF as the constant gain. However, in order to obtain a less conservative controller, the sensi-tivity and complementary sensitivity functions were evalu-ated based on the uncertain plant given by Eq. (26). It was observed that the dynamic diagonal compensator could not achieve the desired performance. Clearly, a more compre-hensive controller is now required to ensure the robust sta-bility. A controller matrix of the following form was cho-sen:

1 25

3 4

7 86

9 10

p s pp

p s pP

p s pp

p s p

+⎡ ⎤⎢ ⎥+⎢ ⎥= ⎢ ⎥+⎢ ⎥⎢ ⎥+⎣ ⎦

(38)

Table 3. Step response indices for example 1 with compensator (36).

Objectives Rise Time 1

Rise Time 2

Over Shoot 1

Over Shoot 2

Settling Time 1

Settling Time 2

Steady State

Error 1

Steady State

Error 2

Interaction 1 2

Interaction 2 1

Specified 5 5 10% 10% 15 15 5% 5% 10% 10%

Obtained 3.25 1.62 0.0% 6% 12.7 5.75 0.0% 0.0% 40.5% 20.4%

Table 4. First five Pareto optimal solutions for diagonal dynamic compensator of example 1. rt1, rt2 denote rise time for loop 1 and loop 2 and similarly os 1, os 2, st 1, st 2 denote overshoot and settling time for loop 1 and loop 2 re-spectively.

Compensator parameters Time domain performance indices

[p1, p2, p3, p4, p5, p6, p7, p8] rt1 rt2 os1 os2 st1 st2

[0.952, 0.198, 0.211, 0.784, 0.971, 0.604, 0.205, 0.914] 3.257 1.621 0.0 0.060 12.708 5.751

[0.995, 0.198, 0.210, 0.805, 0.981, 0.556, 0.376, 0.958] 3.282 1.736 0.0 0.051 13.255 6.205

[0.952, 0.191, 0.282, 0.641, 0.979, 0.556, 0.204, 0.945] 3.701 1.675 0.1 0.018 12.659 5.841

M. Eftekhari and S.D. Katebi: Evolutionary Search for Limit Cycle and Controller Design in Multivariable Nonlinear Systems 355

Fig. 8. Step response for compensated system of example 1

with actual nonlinear elements (no linearization). Solid curves are responses of loop 1 and loop 2 (the interac-tion of loop 1 on loop 2) respectively, when only loop 1 is subjected to a unit step input and the doted curves are those for loop 2.

After 28 generations the following parameters were obtained and the step response of the system is shown in Fig. 9 for different amplitudes of the step input with actual nonlinear elements.

0.89 0.4 0.310.12 0.91( )

0.75 0.390.290.10 0.71

ssP s

ss

+⎡ ⎤⎢ ⎥+⎢ ⎥=

+⎢ ⎥⎢ ⎥+⎣ ⎦

(39)

The value of the weighted sum was reduced to 1.14 and further generation did not reduce this value. Clearly the robust stability is ensured when the uncertainty due to lin-earization is accounted for, on the expense of a more so-phisticated controller.

Example 2. In this example nonlinear system with the configura-

tion shown in Fig. 10 is considered. The nonlinear matrix is diagonal with identical saturation elements, shown in Fig. 11 with the coupled linear matrix given as:

0.1

2 2

2 2

(3 6) 0.5( 1.4 0.4) 1

1 4 21 ( 1.4 0.4)

se ss s s s s

Gs

s s s s s

−⎡ ⎤+ −⎢ ⎥+ + + +⎢ ⎥= ⎢ ⎥+⎢ ⎥

+ + + +⎢ ⎥⎣ ⎦

(40)

In this example, the element g11 of G is associated with a 0.1 second transport delay for which a first order Pade approximation is used in the simulation setup but used directly in the limit cycle prediction. The appropriate harmonic balance equation was constructed for this con-figuration and similar experiments as in example 1 were performed.

Fig. 9. Time responses with compensator (39) for different

amplitude of step input for loop 1 and loop 2.

Fig. 10. System configuration for example 2.

Fig. 11. Saturation characteristic for elements n11 and n22 of

example 2.

The results for the limit cycle prediction are shown in Table 5. A set of time domain requirements was specified subject to simultaneous minimization of the H∞ norm of the weighted sum of the sensitivity and its complement. In the calculation of the H∞ norms, the uncertainty was taken to be Δ as given by Eq. (23). MOGA was run and after 15 generations the design requirements were met as shown in Table 6 and the evolved compensator parameters are as follows. The resulting step responses are shown in Fig. 12 with the actual nonlinear elements.

0.87 0.13 00.15 0.77( )

0.59 0.3600.14 0.17

ssP s

ss

+⎡ ⎤⎢ ⎥+⎢ ⎥=

+⎢ ⎥⎢ ⎥+⎣ ⎦

(41)

356 Asian Journal of Control, Vol. 8, No. 4, December 2006

Table 5. Results of limit cycle prediction with MOGA for example 2.

Solutions ω a1 a2 ϕ Obj 1 Obj 2 1 0.9873 5.5694 0.5412 3.7555 0.0237 0.0246 2 0.9874 5.5700 0.5420 3.7774 0.0108 0.0873 3 0.9873 5.5698 0.5338 3.7550 0.0206 0.0408

Simulation Results 1.006 5.5331 0.5886 3.6400

Table 6. Step response indices for example 2.

objectives Rise Time 1

Rise Time 2

Over Shoot 1

Over Shoot 2

Settling Time 1

Set-tling

Time 2

Steady State

Error 1

Steady State

Error 2

Interac-tion 1 2

Interac-tion 2 1

Specified 4 4 10% 10% 10 10 5% 5% 30% 30% Obtained 1.12 0.89 5% 3% 7 2.4 0.0% 0.0% 20% 8%

Fig. 12. Step response for compensated system of example 2

with actual nonlinear elements (no linearization). Solid curves are responses of loop 1 and loop 2 (the interaction of loop1 on loop 2) respectively, when only loop1 is subjected to a unit step input and the doted curves are those for loop 2.

This example illustrates that the limit cycle prediction and the controller design approach presented in this paper is applicable for different MIMO nonlinear configurations. Further, it gives highly accurate results for limit cycle pa-rameters even in the presence of time delay.

IX. CONCLUSIONS

Multi-objective Genetic Algorithms was employed to predict limit cycle and subsequently design a near optimal controller to meet a set of specified design criteria in a class of nonlinear MIMO systems. The extension of the Single Sinusoidal Input Describing Function design phi-losophy to multi loop nonlinear systems was described. For the class of separable nonlinear element the harmonic bal-

ance equations were derived. The search for limit cycle was formulated as a multi-objective problem. The MOGA was employed to solve the multi-objective formulation and obtain quantitative values for parameters of any possible limit cycle operation. The MOGA search space is the space of the possible limit cycle parameters, such as amplitudes, frequency and phase difference between the interacting loops.

Next, it is shown, how a complex nonlinear multi-variable controller design may be formulated in a multi- objective optimization framework. Both frequency and time domain requirements were formulated as a set of ine-qualities. The SIDF linearization was used to facilitate the limit cycle examination and the EIDF linearization proce-dures was employed to facilitate the time response evalua-tion. Robust stability and robust performance were exam-ined based on minimizing the sensitivity and the comple-mentary sensitivity function in the presence of linearization uncertainty for the elements of the nonlinear matrix in a multi-objective context.

The proposed approach for limit cycle prediction and compensator design for the class of nonlinear systems con-sidered is shown to be effective and promising. Further, it is applicable to different MIMO nonlinear system configu-ration and could be used where the linear system elements comprise time delay. Experimental results obtained show that the multi-objective GA optimization technique devel-oped in this paper is capable of producing accurate quanti-tative values for limit cycle parameters and result in Pareto optimal and practical controllers that meet the design re-quirements in an efficient computational manner.

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358 Asian Journal of Control, Vol. 8, No. 4, December 2006

Mehdi Eftekhari received his B.Sc. in computer engineering from the Department of Computer Science and Engineering, Shiraz University, Iran in September 2000. He then was admitted to the M.Sc. course in AI in the same department and defended his M.Sc. thesis with distinction in September

2003. He is now a Ph.D. student in the same department.

S. D. Katebi graduated with an honor degree in Computer Systems Engineering from the Coventry University, England in 1972. He ob-tained his M.Sc. and Ph.D. from the Control Systems Center, University of Manchester Institute of Science and technology (UMIST) in 1973 and

1976 respectively. He has been a faculty member of the department of Computer Science & Engineering, Shiraz University since 1976, teaching undergraduate and graduate courses and conducting research in various aspects of nonlinear control and AI. He is the author of several papers in cited journals and has been a full pro-fessor since 1993.