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I. INTRODUCTION

CONTROL OF MULTIVARIABLE SYSTEM BY MULTIRATE FAST OUTPUT SAMPLING TECHNIQUE B. Bandyopadhyay and Jignesh Solanki Indian Institute of Technology Mumbai, India. I. INTRODUCTION. II. REVIEW ON FAST OUTPUT SAMPLING TECHNIQUE.

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I. INTRODUCTION

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  1. CONTROL OF MULTIVARIABLE SYSTEM BY MULTIRATE FAST OUTPUT SAMPLING TECHNIQUEB. Bandyopadhyay and Jignesh Solanki Indian Institute of Technology Mumbai, India

  2. I. INTRODUCTION II. REVIEW ON FAST OUTPUT SAMPLING TECHNIQUE III. PROPOSED MULTIRATE FAST OUTPUT SAMPLING FEEDBACK TECHNIQUE IV. NUMERICAL EXAMPLE V. CONCLUSION

  3. INTRODUCTION • Multirate sampled-data control system has been an area of research interest since 1957 • The poles of a linear time invariant controllable system can be assigned by linear state feedback • But for cases where only the outputs but not the states are available, it is desirable to go for an output feedback design • Static output feedback problem, an open question in control theory

  4. It represents the simplest closed loop control than can be realized in practical situations. • Here a multirate fast output sampling feedback control is proposed for multivariable system where each output is sampled at different time instants where as, each control input is applied at a fixed instant. • Also shown here is that the fast output sampling feedback gains can be obtained for the multivariable system.

  5. REVIEW ON FAST OUTPUT SAMPLING TECHNIQUE • Consider a plant dx/dt = Ax + Bu y = Cx (1) where x n, u m, and y p. • Assumed that (A, B) pair is controllable and (C, A) pair is observable. • Plant is to be controlled, with sampling time  and zero order hold • A sampled data state feedback is carried out to find state feedback gain F such that the closed loop system x (k + ) = ( + F) x (k) has desired properties

  6. Here  = eA and =  e As ds B and output measurements, at time instants t = l , are taken whereas constant control signal is applied over a period  which is generated according to u(t) = L yk • The matrix L represents output feedback gains, 1/ is rate at which loop is closed whereas output samples are taken at N times faster rate 1/. • If discrete time system is considered with input uk, states xk and output yk at time t = k. xk+1 = xk + uk yk+1 = Co xk + Do uk ,

  7. For the state feedback, where state feedback gain F has been designed such that ( + F) has no eigen values at the origin the fictitious measurement matrix is • C(F, N) = (C0 + D0F) ( + F)1, satisfying • yk = C xk • And the feedback law u = Lyk can be interpreted as static output feedback for the system with the measurement matrix C. For L to realize the effect of F, it must satisfy • xk+1 = ( + F) xk = ( + LC) xk • i.e. LC = F

  8. PROPOSED MULTIRATE FAST OUTPUT SAMPLING FEEDBACK TECHNIQUE Multirate Output Sampling Mechanism

  9. Let 1 = /N1, 2 = /N2…… p = /Np where p is the number of outputs and assume that multirate fast output feedback control is as or u(t) = L yk

  10. 1/ is the rate at which the loop is closed, whereas ith output samples are taken at Ni-times faster rate 1/i. • From the output equation of the plant y = Cx , we get y1 = c1x ; y2 = c2x ………… yp =cpx • From the first output it can be shown, y1,k+1 = C1xk + D1uk yk = C x(k) where C = [C1; C2; …….; Cp](N1+N2+..Np  n)

  11. u = L C x(k) x(k+1) =  x(k) + u(k) x(k+1) = ( + LC) x(k) L C = F

  12. INTEGRAL ACTION • Integral action is included in to the control law to reject step disturbances. • For this a new state vector k+1 = k + rk - yk, representing the integrated error is introduced • A state space representation for the augmented system is Multirate fast output sampled data system

  13. Let • xk = [xTkTk] and to yield control law with the closed loop matrix is searched for the augmented system F and FI • FI is then implemented directly where as F is realized by multirate output feedback

  14. NUMERICAL EXAMPLE [F FI] FI

  15. Fast output sampling gain L is calculated with sampling intervals 1 = /N1 and 2 = /N2 with N1 = 8 and N2 = 10. • Mutirate fast output feedback control law has been designed showing that the system is stable Results after applying control law u = L yk

  16. Results after applying control law with integral action

  17. CONCLUSION • A new control law for a multirate fast output feedback control was presented which, allows to realize the effect of a state feedback gain • In particular, it has been shown that integral action can be included without disturbing the original design. • Results show that the design of control law gives satisfactory result

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