Rev. éc. Ing. Univ. Zulia. Vol. 39, Nº 8, 1-6, 16 doi:1.1311/1.39.8.3 Optimal Sliding Surface Design for a MIMO Distillation System Senthil Kumar B 1 *, K.Suresh Manic 1 Research Scholar, Faculty of Electrical Engineering, nna University,Chennai-65,India. *Corresponding uthor (Email: vlrsenthil@yahoo.co.in) Department of Electrical and Electronics Engineering, Sriram Engineering College, Chennai-64, India. (Email:ksureshmanic@gmail.com) bstract his Paper presents the Sliding Mode Control of a Distillation Colum System which are influenced by external disturbances or perturbations that are act along the input. he issue of robustness property to be studied only for Matched Perturbation or for Matched uncertainty. he switching surface design for Sliding Mode control to be interpreted as a static feedback selection problem.he switching hyper plane design to be carried out along with pole placement technique and Quadratic Minimization technique for regulatory problems or disturbance rejection problem.he results are obtained for the distillation column where the Eigen values are to be placed arbitrarily and optimally and the robustness property to be measured using the condition number or by obtaining norm of the state variables about where the Eigen values are placed in closed Loop Key words: Sliding Mode Control, Pole Placement Problem, Optimal Control, Matched Uncertainty, Distillation Colum 1. INRODUCION Sliding Mode Control is a robust control for both linear and nonlinear systems. In this paper perturbations or external bounded disturbances are to be studied only a vanishing disturbance. he vanishing disturbance is one where the disturbance vanishes about the equilibrium point. (nusha Rani, Kumar, and Manic 16)(Kumar and Manic 14). By properly applying high gain feedback in the closed loop the disturbance vanishes. State feedback controller can be used to place the closed loop Eigen values. For Multi Input the design of gain is not unique for a given set of Eigen values and for the same Eigen values different gain matrix will produce different Eigen vectors and different performance. Optimal values to be found for better performance. Switching hyper plane is designed by minimising cost functions in which quadratic terms of the states are used (KUSKY, NICHOLS, and DOOREN 1985). his method ensures weightings placed such that the control surface follows the modes(moore 1975). he Eigen vectors contain information about the interaction between the states which is arbitrarily fixed by pole placement design techniques. Optimal Eigen structure assignment design offers to minimise the effects of unmatched perturbations on the sliding mode dynamics optimally by designing the surface design(edwards andspurgeon 1988).. CONROL DESIGN.1. Regular Form regular form to be used for reduced order sliding mode dynamics Consider the system considered of the form x =x+bu (1) Where rank (B) =m and (, B) is a controllable pair and the switching function is represented as he system represented in regular form as [1] S ( =Sx ( () z1( 11z1( 1z( (3a) z( 1z1( z( Bu( nd the linear sliding manifold as S ( =S 1 z 1 ( +S z ( (4) nd the change of coordinates by an orthogonal matrix r Z( = r *x ( (5) (3b) 1
Rev. éc. Ing. Univ. Zulia. Vol. 39, Nº 8, 1-6, 16 1 I nm BB r = 1 (6) B and change of coordinate is Input matrix B in () may be partitioned (after reordering the state vector components) as B1 B (7) B Where B 1 ϵr n-m m, B ϵr m n with det B 11 1 r r = 1 (8) and nd the switching function after transformation r B= B S r = [S 1 S ] (1) he switching function S( to be identically equal to zero during sliding motion From Equation (4) S 1 z 1 ( +S Z ( =. Z ( =-S -1 S 1 z 1 ( (11) Z ( =-M Z 1 ( where M=S -1 S 1 which is the existence problem to give (n-m) negative poles to the closed loop system. he sliding system can become totally insensitive to matched uncertainty but it will be affected by unmatched uncertainty. he sliding mode is governed by the above equation (11). On substituting Z in the first equation it becomes a closed with feedback form Z1( ( 11 1M) Z1( the techniques for switching surface selection or M to found by using pole placement technique or by using quadratic minimisation technique. S r = [M I m ] which is same as Equation (1) he designed manifold can be linear or nonlinear, depends on our specified goal. he simplest manifold is a linear one..1.1 Existence Condition of Sliding Mode Objective here is to design a control input u. Such that, the sliding motion occurs infinite time. where K> (9) (1) v=-ksign(s) (13).. Pole Placement echnique Full state feedback (FSF), or pole placement, is a method employed in feedback control system theory to place the closed-loop poles of a plant in pre-determined locations in the s-plane. Placing poles is desirable because the location of the poles corresponds directly to the eigenvalues of the system, which control the characteristics of the response of the system. he system must be considered controllable in order to implement this method...1.principle If the closed-loop input-output transfer function can be represented by a state space equation hen the poles of the system are the roots of the characteristic equation given by x =x+bu (14 a) Y=Cx+Du (14 b)
Rev. éc. Ing. Univ. Zulia. Vol. 39, Nº 8, 1-6, 16 ǀSI-ǀ= (15) Full state feedback is utilized by commanding the input vector u. Consider an input proportional (in the matrix sense) to the state vector, System with state feedback (closed-loop) u=-kx (16) Substituting into the state space equations above, X ( BK) x (16a) Y=(C-DK) x he roots of the FSF system are given by the characteristic equation, det [SI-(-BK)].Comparing the terms of this equation with those of the desired characteristic equation yields the values of the feedback matrix K which force the closed-loop eigenvalues to the pole locations specified by the desired characteristic equation..3. Linear Quadratic Regulator Method he problem of minimising the quadratic performance index is given by positive definite Identity Matrix and t s is the time at which the sliding motion starts. Q11 Q1 r Qr 1 = Q Q Where Q 1 =Q 1, then the system is represented in the coordinate transformation as 1 J 1 11 1 1 1 Z Q Z Z Q Z Z Q Z dt ts Z 1 determines the system dynamics and the effective control input is determined by Z 1 [ Z1 ( Q11 Q1 Q Q1 ) z1 1 ( 1 1 1 1 ) ( J 1 1 ) Z Q Q Z Q Z Q Q z ts 1 Z Q Q Q Z ] 1 1 1 1 Q Q Q Q Q 1 11 1 1 (16b) 1 J ( ) ( ) x t Qx t dt where Q is a ts (17) (18) (19) () v Z Q Q Z 1 1 1 nd the equation written as (1) 1 J 1 1 Z QZ v Q vdt ts Q Q 1 11 1 1 () (3) he problem of Minimization of the standard linear quadratic optimal regulator problem ensures the positive definiteness of Q identity matrix which also ensures Q >, so that Q -1 exists and also Q >. he controllability of (, B) ensures ( 11, 1 ) Controllable and in turn ensures (, ) controllable Z MZ1 Q 1( 1 P1 Q1 ) where P 1 is a unique positive definite solution obtained from lgebraic Matrix Ricatti Equation. 3. NUMERICL EXMPLE Considering the distillation column (Kautsky, Nichols, and Dooren, 1985) Where the closed loop poles has been placed for n x n matrix. In this example the matrix considered was a 5 x 5 matrix.by reducing it into a 3 x 3 matrix and also the disturbance considered where the input appears. So the system has been formulated as a reduced order system. he condition numbers given in the example taken for full order matrix which is higher rather than for a reduced order system. 1 3
Rev. éc. Ing. Univ. Zulia. Vol. 39, Nº 8, 1-6, 16.194 1.36.638 B.838.14.63.68.13 1.595.355.3.1396.6.18.987 3.149.63 1.547 4.57.1636 Converting it into regular form where obtained from Equation (6) 1.855.165.1.13 1 B = 1 1 he system is obtained as in equation (3) 11 1 1.1.184.64.315.5184.175.369.659 1.377.73.66.467.88.194.68 1.36 18.3785 17.4915 6.416 36.4454.4 7.778 63.943 6.6156 15.618.315.6569.1584 1.377.497 1.41.88.4917.634.156 1.41.4917 he value obtained from equation (3).194.631.3 1.36 16.638 159.818 71.85 1.6 9.9633 and Kopt 115.643 3.9715 6.99 5.9995 34.17 hus the optimal value obtained is used for closed loop poles assignment. 4. RESUL ND DISCUSSION he Eigen values of matrix are [-5.3,-.914,-.8576,-.69,-.785].By pole placement technique the poles are placed as [-1,-,-3,-4,-5] and the condition number which is obtained as [13.8891, 65.334, 15.981, 37.954][5].By suing optimal quadratic minimization problem M is calculated. 4
Rev. éc. Ing. Univ. Zulia. Vol. 39, Nº 8, 1-6, 16.45.91.1734 M, which is a reduced order matrix and the gain k which is optimally.93.56.334 71.85 1.646 9.9633 placed for the Eigen values 11 =[,-6.7,6] and the gain computed isk opt =.In 115.6743 3.9715 6.999 the Input Matrix B=B it has no significance on the external bounded disturbance applied to the system. It becomes invariant to the input and when the system is in sliding phase. he condition number gives what will the maximum perturbation that an Eigen value can withstand. Figure 1 Control Input Figure Reduced order States Figure 3 Sliding Surface 5
Rev. éc. Ing. Univ. Zulia. Vol. 39, Nº 8, 1-6, 16 5. CONCLUSION For MIMO system considered the system has been transformed into a reduced order system and robustness properties has been obtained using sliding mode controller for matched disturbance acting along the input. he optimal gain calculated using linear quadratic method gives for the weighted values considered for anidentity matrix.he daptive Sliding Mode controller using Fuzzy and Heuristic lgorithms for nonlinear systems has been discussed in the preceding papers (Dinesh Kumar and Meenakshipriya 16)(Li et al. 16).he other papers deals with only parametric robustness and not on the unmodelled dynamics which has not been considered(vijayan and Panda, 1a)(Vijayan and Panda, 1b). REFERENCES nusha Rani, V., B. Senthil Kumar, and K. Suresh Manic(16) Sliding Mode Controlfor Robust Regulationof Chemical Processes,Indian Journal of Science and echnology, 9(1), 1-1. doi:1.17485/ijst/16/v9i1/89918. C.Edwards, and S.K.Spurgeon(1988) Sliding Mode Control: heory and pplications, Crc Pres. Dinesh Kumar, D, and B Meenakshipriya(16) Design of Heuristic lgorithm for Non-Linear System,Rev. éc. Ing. Univ. Zulia, 39(6),pp.1-15. Kautsky, J, N K Nichols, and P V N Dooren(1985) Robust Pole ssignment in Linear State Feedback,International Journal of Control, 41 (5), pp.119-1155. Kumar, B Senthil, and K.Suresh Manic(14) Sliding Mode Control for a Minimum Phase Unstable Second Order System. pplied Mechanics and Materials, pp.85-9. Li, Mengmeng, Dehui Qiu, Jinwen Zheng, Yuan Li, and Qinglin Wang(16) daptive Fuzzy Sliding Mode Control Based on a DEP Flexible ctuator,rev. éc. Ing. Univ. Zulia, 39(3),pp.55-68. Moore, B. (1975) On the Flexibility Offered by State Feedback in Multivariable Systems beyond Closed Loop Eigenvalue ssignment,ieee Conference on Decision and Control Including the 14th Symposium on daptive Processes, pp.7-14. Vijayan, V., and Rames C. Panda (1a) Design of a Simple Setpoint Filter for Minimizing Overshoot for Low Order Processes,IS ransactions, 51(), pp.71-76. V Vijayan, RC Panda(1b) Design of PID Controllers in Double Feedback Loops for SISO Systems with Set-Point Filters,IS ransactions, 51 (4), pp.514-51. 6