International Journal of Electrical and Electronics Engineering Research (IJEEER) ISSN:2250-155X Vol.2, Issue 2 June 2012 27-38 TJPRC Pvt. Ltd., TWO- PHASE APPROACH TO DESIGN ROBUST CONTROLLER FOR UNCERTAIN INTERVAL SYSTEM USING GENETIC ALGORITHM 1 DEVENDER KUMAR SAINI & 2 RAJENDRA PRASAD Electrical Engineering Department, Indian Institute of Technology, Roorkee, India Electrical Engineering Department IIT Roorkee, Roorkee-247667, India ABSTRACT This Paper presents a atic two phase optimization approach to design a robust and stabilizing controller for high order linear interval. The approach has two phase design procedure; first phase involves a reduction algorithm which reduced the order of high order original and phase second design a robust and stabilizing controller for reduced interval which mimics the original high order interval. Reduction procedure based on minimization of integral square error between transient parts of original and reduced using genetic algorithm (GA), for to a unit step input. Designing of robust controller is done by GA after transforming the robust controller design problem into a nonlinear programming problem (NLP) using the Kharitonov and Hermite-Biehler stability theorems. The proposed approach guarantees the stability of reduced order interval and closed loop stability of controlled interval plant. A numerical example and simulation results successfully demonstrate the efficacy of proposed two phase approach for reduction and design of robust controller for uncertain interval s. KEYWORD : Robust Control, Uncertain Interval s, Reduced order modeling, Integral Square error (ISE), Genetic Algorithm (GA). INTRODUCTION In many situations it is desirable to represent a high- order by a lower order model. Model reduction seeks to replace a large-scale of differential or difference equations by a of substantially lower dimensions that has nearly the same response characteristics. In classical control theory there are some powerful methods and tools for reduction and design process, which were developed for fixed nominal s. From last two decades, much effort has been made in the field of model reduction of fixed coefficients linear dynamic s and several methods like: Aggregation method(aoki, 1968), Pade approximation (Shamash, 1974), Routh approximation (M.F. Hutton, 1999), Moment matching technique (N.K. Sinha, 1983), Routh stability technique (V. Krishnamurthy, 1978), and optimization technique (Glover, 1984), have been proposed. Among them Routh stability technique has been recognized as the simplest and powerful method because of its ability to yield stable reduced models for stable high-order s.
Devender Kumar Saini & Rajendra Prasad 28 However the parameters of the plant transfer function model will not be known exactly and will not be fixed, but may contain uncertainties due to simplification in the modeling process, uncertainties in the parameters of the and/or uncertainties due to working conditions. The uncertainty in a control causes degradation of performance and destabilization. A well known approach to solve this problem is based on expressing the characteristic polynomial by an interval polynomial whose coefficients vary independently in a prescribed interval (Barmish, 1994; L. Shieh, 1996). The determination of reduced order models and robust control of uncertain interval s is the major concern of control engineering. The control of s with uncertain models is known as Robust Control. Robust control has been received a great amount of attention after the pioneer work of Kharitonov (Kharitonov, 1978), during the last decade. As motivated by the seminal theorem of Kharitonov for determining the robust stability of polynomials with interval coefficients, there have been numerous interests in various problems of analysis, synthesis and design for interval plants. Barmish (Barmish B.R., 1992), explored some extremepoints results from the Kharitonov s theorem for robust stabilization of interval plants with first order compensators. Bernstein and Haddad (Bernstein D.S., 1992), applied the Kharitonov s theorem and a multiple-plant model formulation with a quadratic cost functional to synthesize a robust stabilizing feedback controller. Goh (Goh C.J., 1989), presented a numerical technique which takes the Kharitonov s condition into account to design a robust stabilizing controller for uncertain interval plants. Their technique is based on expanding the four Kharitonov polynomials in their factorized forms and formulating the design procedure as an optimization problem with the associated zeros being the decision variables while subject to stability constraints. Since the Kharitonov s polynomials are expressed in terms of their zeros, it is required to prescribe the configurations of complex- conjugate or real zeros. This results in increasing the dimensionality of the resulting NLP problem. In order to overcome the computational efforts and complexity in solving NLP problem, this paper proposed an evolutionary approach to design and solve NLP for robust controller. Approach has two phase but both phases are independent to each other. First phase deals with order reduction of high order using GA whereas second phase describe the design procedure. We can design a robust controller for high order with very less computational efforts by combining both phases. This also reduces the order of NLP problem arising in design procedure. PROCEDURE The procedure of presented paper is organized as follows. In section A, problem is formulated, section B describe the reduction algorithm for high order interval s. Section C reminds us the four Kharitonov polynomials and further solved condition for robust stability of interval polynomials. Section D illustrates the design procedure for robust controller for uncertain interval plant family.
29 Two-Phase Approach to Design Robust Controller for Uncertain Interval System using Genetic Algorithm A. Problem Formulation Given an original interval of order n that is described by the transfer function G(s) and its reduced interval model R(s) of order k be represented as: (1) (2) Where, and, are the interval coefficients of higher order numerator and denominator polynomials respectively. Where, and, are the interval coefficients of lower order numerator and denominator polynomials respectively The objective function is to find a order reduced model R(s) and design a robust controller C(s) for R(s), which mimics the important characteristic of G(s) for the same type of inputs. (3), and, are the unknown coefficients of robust controller. B. Determination of R(s) using GA The numerator and denominator polynomials of the reduced order model is determined by minimizing Integral square error between original and reduced using genetic algorithm pertaining to a unit step input. The deviation between the lower order response from the original is given by the error index ISE known as the Integral square error, which is given as follow: (4) Where and are the unit step response of the original and reduced order s, respectively. In the present paper, GA is employed to minimize the objective function ISE as given in Eq. (4), and the parameter to be determined are the coefficients of the numerator of the lower order. For the purpose of minimization of Eq. (4), routine from GA optimization toolbox are used. For different problems, it is possible that the same parameters for GA do not give the best solution and so these can be changed according to the situation. In Table 1, the typical parameters for GA optimization routines, used in the present study are given. The description of GA operators and their properties can be found in (Goldberg, 1989). The computational flow chart of the proposed algorithm is shown in Fig. 1.
Devender Kumar Saini & Rajendra Prasad 30 Table 1 : Typical parameters for GA optimization Name Value(type) Number of generations 100 Population size 80 Type of selection Type of crossover Type of mutation Termination method uniform Arithmetic uniform Maximum generation Create/Initialize Population Measure/Evalute Fitness Select Fitness Mutation Non Optimum Solution Crossover/Production Optimum solution Figure 1 : Flowchart of Genetic algorithm C. Condition for Robust stability of Interval Plant Family The strong Kharitonov theorem (Kharitonov, 1978), is an analysis tool which provides a necessary and sufficient condition result for determining the stability of polynomials with interval coefficient uncertainty. Consider the set of real polynomials of degree n of the form (5)
31 Two-Phase Approach to Design Robust Controller for Uncertain Interval System using Genetic Algorithm Where the coefficient lie within given ranges, (6) Kharitonov s Theorem states that every polynomial in the family following four extreme polynomials are Hurwitz: is Hurwitz if and only if the Further in (Deore, 2007), authors derived simplified conditions for robust stability from four Kharitonov vertex. Which are stated as follows The interval polynomial I(s) defined in (5) is Hurwitz for all where, if the following conditions are satisfied. Necessary conditions Sufficient conditions (8) D. Design Procedure of Robust Controller for Uncertain Interval Plant To state the design problem let the controller structure from Eq. (3) Controller C(s) is cascaded with reduced order interval model R(s) with unity negative feedback as shown in fig. 2. Figure 2 : Controlled closed- loop interval plant
Devender Kumar Saini & Rajendra Prasad 32 The transfer function of the closed- loop becomes: (9) Now controller C(s) will be robust controller if it stabilized the characteristic equation of closed loop. It is clear that closed loop is an interval, so its characteristic polynomial is given by (10) Now the problem can be formulated as Controller C(s) satisfy the polynomial, if it satisfy a set of inequality constraints given by eq. (7,8) and minimize the following objective function. (11) Subjected to Here and are the controller parameters obtained by minimizing the Eq. (13), given as below (12) (13) is the output of pertaining to a unit step input. GA is used to minimize Eq. (11, 12 & 13), which is a NLP problem getting after transforming the controller problem into robust controller problem for an interval plant family. The typical parameters for GA optimization routines, used to solve this NLP problem has been given in Table 1. NUMERICAL EXAMPLE A numerical example is chosen from the literature for the comparison of the lower order with the original high order. Consider a 7 th order taken from literature:
33 Two-Phase Approach to Design Robust Controller for Uncertain Interval System using Genetic Algorithm And a. Determination of 2 nd order reduced interval model is determined by minimizing eq. (4), along with typical parameter for GA optimization routines, used in present study as given in table 1. Thus becomes b. Design of Robust controller For the present example author design the PI type robust controller is designed shown in fig. 3 for. The transfer function for PI type robust controller is given by then closed loop for considered numerical example becomes c + 0 1 s c s Figure 3 : PI Controlled closed-loop reduced interval plant The characteristic equation of closed loop is given by After applying stability conditions on given closed loop characteristic polynomials from Eq. (11 &12), a NLP problem is formulated to determine unknown coefficients of controller. St.
Devender Kumar Saini & Rajendra Prasad 34 GA is applied to solve above NLP problem which gives The unit step responses and frequency responses are shown in fig 4 & 5 for the open loop and the closed loop responses are compared in fig 3 & 4 respectively. RESULTS a. Simulation Results 4 Step Response 3.5 3 2.5 Amplitude 2 1.5 1 original interval envelop reduced interval envelop 0.5 0 0 5 10 15 20 25 30 35 40 Time (sec) Figure 1 : Comparison of step response envelopes of original and reduced interval
35 Two-Phase Approach to Design Robust Controller for Uncertain Interval System using Genetic Algorithm 20 Bode Diagram Magnitude (db) 0-20 -40 0 reduced envelop Phase (deg) -45-90 -135 original envelop 10-2 10-1 10 0 10 1 10 2 Frequency (rad/sec) Figure 2 : Comparison of Bode envelopes of original and reduced interval 1.4 Step Response 1.2 1 Amplitude 0.8 0.6 0.4 Controlled reduced interval Controlled Original interval 0.2 0 0 0.5 1 1.5 2 2.5 Time (sec) Figure 3 : Comparison of step response envelopes of Robust Controlled original and reduced interval
Devender Kumar Saini & Rajendra Prasad 36 10 Bode Diagram Magnitude (db) 0-10 -20-30 -40 0 Controlled original interval Phase (deg) -45-90 Controlled reduced interval 10-1 10 0 10 1 10 2 10 3 Frequency (rad/sec) Figure 4 : Comparison of Bode envelopes of Robust Controlled original and reduced interval b. Test of Robust Hurwitz stability of Controlled original interval s Transfer function of controlled closed loop original interval plant is ] And ] To be a stable robust controlled original, necessary and sufficient conditions must be satisfied by characteristic polynomial ( ) of closed loop controlled. From Eq. (8), --- Satisfied Applying
37 Two-Phase Approach to Design Robust Controller for Uncertain Interval System using Genetic Algorithm 15918*24735 > 4099*21011 -- True 24735*19010 > 17593*8883.9 -- True 19010* 8037.8 > 27339*2178.8 -- True 8037.8* 1962.8 > 21.11*322.63 -- True 1962.8*288.36 > 8883.9*27.667 -- True 288.36*24.489 > 2178.8*1.05 -- True Same in this way condition is also satisfied by. So it s proven for present example that a robust controller which is designed for reduced interval also stabilized and controls the original interval plant. CONCLUSIONS This paper, proposed a design algorithm of Robust Controller for high order uncertain interval s. Controller design for a high order is a very tedious task. In this paper instead of fixed author consider uncertain interval, which makes the design procedure more difficult because the design problem gets transformed into a NLP problem after applying robust stability conditions of interval. Thus the proposed algorithm approaches the design procedure in two phases. First phase deals with reduced order modeling, in this phase an approximated reduced order interval model is obtained using GA. Second phase deals with designing a robust controller for reduced interval model which also stabilize and controls the original. The whole approach illustrated through a numerical example, simulation results and stability test for taken example shows that a PI type controller which was initially designed for 2 nd order reduced interval model also stabilized and controlled the 7 th order high order interval. The reliability of controller for high order depends upon that how much good approximated reduced order model is achieved to design a controller. REFERENCES 1. Aoki, M. (1968). Control of large- scale dynamic by aggregation. IEEE Trans. on Automatic Control, AC-13, 246-253. 2. Barmish B.R., H. C. (1992). Extreme point results for robust stabilization of interval plants with first compensators. IEEE Trans. Autom. Control, 37, 707-714. 3. Barmish, B. (1994). New Tools for Robustness of Linear Systems. New York: Macmillan Publishing Company. 4. Bernstein D.S., H. W. (1992). Robust Controller synthesis using Kharitonov's theorem. IEEE Trans. Autom. Control, 37, 129-132. 5. Dahleh M., T. A. An overview of extremal properties for robust control of interval plants. Automatica, 29, 707-721.
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