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06,07 elec3114

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  • 1. 1 Stability• How to determine the stability of a system represented as a transfer function• How to determine the stability of a system represented in state space• How to determine system parameters to yield stability Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 2. 2 lntroductionDefinitions of stability for linear, time-invariant systems using the naturalresponse:1. A system is stable if the natural response approaches zero as timeapproaches infinity.2. A system is unstable if the natural response approaches infinity as timeapproaches infinity.3. A system is marginally stable if the natural response neither decays norgrows but remains constant or oscillates. Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 3. 3Bounded-input bounded-output (BIBO) definitions of stability for linear, time-invariant systems using the total response1. A system is stable if every bounded input yields a bounded output.2. A system is unstable if any bounded input yields an unbounded output. Stable systems have closed-loop transfer functions with poles only in the left half-plane Unstable systems have closed loop transfer functions with at least one pole in the right half-plane and/or poles of multiplicity greater than one on the imaginary axis Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 4. 4Marginally stable systems have closed loop transfer functions with onlyimaginary axis poles of multiplicity 1 and poles in the left half-plane Stable System Unstable System Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 5. 5 Routh-Hurwitz Criterion• yields stability information without the need to solve for the closed- loop system polesTwo steps:(1) Generate a data table called a Routh table(2) Interpret the Routh table to tell how many closed-loop system poles are inthe left half-plane and in the right half-plane. Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 6. 6Generating a Basic Routh TableNote: any row of the Routh table can be multiplied by a positiveconstant without changing the values of the rows below. Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 7. 7Interpreting the Basic Routh Table• the number of roots of the polynomial that are in the right half-plane is equal to the number of sign changes in the first column. - two sign changes, hence two poles in the right half-plane Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 8. 8Problem: Find range of gain K for which the system is stable, K>0Solution: System is stable for K<1386 Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 9. 9 Stability in State Space• we can determine the stability of a system represented in state space by finding the eigenvalues (poles) of the system matrix, A, and determining their locations on the s-plane • we can find the poles using equation: Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 10. 10 Steady-State Errors• How to find the steady-state error for a unity feedback system• How to specify a systems steady-state error performance• How to find the steady-state error for disturbance inputs• How to find the steady-state error for nonunity feedback systems• How to design system parameters to meet steady-state error performance specifications• How to find the steady-state error for systems represented in state space Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 11. 11 Steady-State ErrorsSteady-state error is the difference between the input and the output for aprescribed test input as t → ∞ Test waveforms for evaluating steady-state errors of control systems Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 12. 12Evaluating Steady-State Errors Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 13. 13Sources of Steady-State Errors - zero error in the steady state for a step input Error: Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 14. 14 Steady-State Error for Unity Feedback SystemsSteady-State Error in Terms of T(s) – closed loop transfer function … final value theorem Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 15. 15Steady-State Error in Terms of G(s) – open loop transfer function Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 16. 16Response to step, ramp and parabolic input Step input if (at least one pole must be at origin) Thus if n ≥ 1 then lim G ( s) = ∞ and e(∞) = 0 s →0 Ramp input Thus if n ≥ 2 then lim sG ( s) = ∞ and e(∞) = 0 s →0 Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 17. 17Parabolic input Thus if n ≥ 3 then lim s 2G ( s ) = ∞ and e(∞) = 0 s →0 Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 18. 18 Static Error Constants and System TypeStatic Error ConstantsThus, for a step input: for a ramp input: for a parabolic input: Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 19. 19System Type n=0 … Type 0 system n=1 … Type 1 system n=2 … Type 2 system Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 20. 20Problem Given the control system, find the value of K so that there is 10% errorin the steady state. Solution Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 21. 21Steady-State Error for Disturbances (1)E ( s) = R( s) − C ( s) Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 22. 22Steady-State Error for Disturbances (2)- let us assume a step disturbance - hence error can be reduced by increasing the DC gain of G1(s) or decreasing the dc gain of G2(s) Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 23. 23Steady-State Error for Nonunity Feedback Systems (1) • Control systems often do not have unity feedback because of the compensation used to improve performance or because of the physical model for the system. • When nonunity feedback is present, the plants actuating signal is not the actual error or difference between the input and the output. G1(s) … input transducer G2(s) … plant H1(s) … feedback G ( s ) = G1 ( s )G2 ( s ) H ( s ) = H1 ( s ) / G1 ( s ) Ea(s) … actuating signal (not the actual error) Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 24. 24 Steady-State Error for Nonunity Feedback Systems (2)Let us modify, the block diagram to show explicitly the actual error E(s)=R(s)-C(s) Ea(s) … actuating signal (not the actual error) E(s)=R(s)-C(s) actual error Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 25. 25Steady-State Error for Nonunity Feedback Systems (3) - both disturbance and nonunity feedbackIf we consider step input and step disturbance, then R(s)=D(s)=1/s and Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 26. 26For zero error:Above equation can be satisfied if: 1. System is stable 2. G1 is a Type 1system 3. G2 is a Type 0 system 4. H is a Type 0 system with DCgain=1 Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 27. 27Problem Determine the system type, error constant, and the steady state errorfor a unit step Solution E(s)=R(s)-C(s) actual error Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 28. 28 SensitivitySensitivity - the degree to which changes in system parameters affect systemtransfer functions, and hence performance.Sensitivity is the ratio of the fractional change in the function to the fractionalchange in the parameter as the fractional change of the parameter approaches zero. Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 29. 29Problem Find the sensitivity of the steady-state error to changes in parameter Kand parameter a for the system with a step input Solution Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 30. 30Steady-State Error for Systems in State Space(1) analysis via final value theorem(2) analysis via input substitutionAnalysis via Final Value Theorem closed-loop transfer function Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 31. 31Analysis via Input Substitution – avoids taking inverse of (sI - A)Step input (r = 1)If the input is a unit step r = 1 then a steady-state solution xSS for x is: where Vi is constant. yss = CV = −CA −1B Thus, for a unit step input: Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 32. 32Ramp inputs (r = t) We balance the equation: Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.