Transient and SteadyState Response Analysis

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Higher-Order Systems
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(Contd)
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(Contd)
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To remember

• The poles of the input R(s) yield the
  steady-state response terms in the solution.
• The poles of C(s)/R(s) yield the exponential
  transient-response terms and/or damped sinusoidal
  transient-response terms.
• The zeros of C(s)/R(s) affect the magnitudes and
  signs of the residues.

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Dominant Closed-Loop Poles

• The relative dominance of closed-loop poles is
  determined by the ratio of the real parts of the
  closed-loop poles, as well as by the relative
  magnitude of the residues.
• The magnitudes of the residues depend on both the
  closed-loop poles and zeros.
• If the ratios of the real parts exceed 5 and
  there are no zeros nearby.
• Quite often they occur in the form of a
  complex-conjugate pair.

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Stability Analysis in the s Plane

• The stability is determined from the location of
  the closed-loop poles in the s plane.
• For system with poles lying in the right-half s
  plane, it is unstable.
• Absolute stability can be achieved by choosing
  closed-loop poles lying to the left of the j?
  axis.

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Rouths Stability Criterion
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Rouths Stability Criterion (Contd)
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Rouths Stability Criterion (Contd)

• Statement
• The number of roots with positive real parts is
  equal to the number of changes in sign of the
  coefficients of the first column of the Rouths
  Table.

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Example 1
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Example 2
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Special Case Zero First-Column Term
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Special Case Zero Row
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Relative Stability Analysis

• Rouths stability criterion is useful for
  absolute stability analysis.
• To examine relative stability
• The number of roots that lies to the right of the
  vertical line s ?? can be obtained.

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Application
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Integral Control Action

• To eliminate the steady-state error, or offset.
• Possible oscillatory response is the problem.

Integral control.
Proportional control.
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Proportional Control of Systems
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Integral Control of Systems
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Response to Disturbances (P)
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Response to Disturbances (PI)
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Response to Disturbances (I)
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Derivative Control Action

• To provide high sensitivity when added to a
  proportional controller.
• Derivative control anticipates the actuating
  error, initiates an early corrective action, and
  tends to increase the stability of the system.
• It permits the use of a larger gain K, which
  results in improved steady-state error.
• It is normally used in PD or PID control action.

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Proportional Control of Systems
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PD Control of Systems
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PD Control of 2nd-Order Systems
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Classification of Control Systems

• Classified according to their ability to follow
  step inputs, ramp inputs, parabolic inputs, and
  so on.
• For a unity-feedback control system with the
  open-loop transfer function given by
• The system is called type N (based on the number
  of integrations).
• The accuracy is improved as N is increased.
• Increasing N aggravates the stability problem.

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Steady-State Errors ess
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Static Position Error Constant Kp
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Static Velocity Error Constant Kv
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A Summary on Kv
Type 1
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Static Acceleration Error Constant Ka
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A Summary on Ka
Type 2
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A Brief Summary