ME375 Dynamic System Modeling and Control

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MESB374 System Modeling and AnalysisSystem
Stability and Steady State Response
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Stability
where the derivatives of all states are zeros

• Stability Concept
• Describes the ability of a system to stay at its
  equilibrium position in the absence of any inputs.

• A linear time invariant (LTI) system is stable if
  and only if (iff) its free response converges to
  zero for all ICs.

Ex Pendulum
Ball on curved surface
valley
plateau
hill
inverted pendulum
simple pendulum
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Examples (stable and unstable 1st order systems)

• Q free response of a 1st order system.

Q free response of a 1st order system.
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Stability of LTI Systems

• Stability Criterion for LTI Systems

Im
Re
Complex (s-plane)
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Stability of LTI Systems

• Comments on LTI Stability
• Stability of an LTI system does not depend on the
  input (why?)
• For 1st and 2nd order systems, stability is
  guaranteed if all the coefficients of the
  characteristic polynomial are positive (of same
  sign).
• Effect of Poles and Zeros on Stability
• Stability of a system depends on its poles only.
• Zeros do not affect system stability.
• Zeros affect the specific dynamic response of the
  system.

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System Stability (some empirical guidelines)

• Passive systems are usually stable
• Any initial energy in the system is usually
  dissipated in real-world systems (poles in LHP)
• If there is no dissipation mechanisms, then there
  will be poles on the imaginary axis
• If any coefficients of the denominator polynomial
  of the TF are zero, there will be poles with zero
  RP

• Active systems can be unstable
• Any initial energy in the system can be amplified
  by internal source of energy (feedback)
• If all the coefficients of the denominator
  polynomial are NOT the same sign, system is
  unstable
• Even if all the coefficients of the denominator
  polynomial are the same sign, instability can
  occur (Rouths stability criterion for
  continuous-time system)

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In Class Exercises
(1) Obtain TF of the following system (2) Plot
the poles and zeros of the system on the complex
plane. (3) Determine the systems stability.

• (1) Obtain TF of the following system
• (2) Plot the poles and zeros of the system on the
  complex plane.
• (3) Determine the systems stability.

• TF

• Poles

• Poles

• Zeros

• Zero

Img.
Img.

• Stable

Real
Real
Marginally Stable
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Example

• EOM

• Equilibrium position

• Inverted Pendulum
• (1) Derive a mathematical model for a pendulum.
• (2) Find the equilibrium positions.
• (3) Discuss the stability of the equilibrium
  positions.

• Assumption

• is very small

• Linearized EOM

• Characteristic
• equation

• Poles

Img.

• Unstable

Real
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Example (Simple Pendulum)

• EOM

• Equilibrium position

• How do the positions of poles change when K
  increases?
• (root locus)

• is very small

• Assumption

• Linearized EOM

• Characteristic
• equation

• Poles

Img.
Img.
Img.

• stable

• stable

• stable

Real
Real
Real
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Transient and Steady State Response

• Ex

Lets find the total response of a stable first
order system
to a ramp input
with I.C.
- total response
- PFE
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Transient and Steady State Response
In general, the total response of a STABLE LTI
system
to a input u(t) can be decomposed into two parts
where

• Transient Response
• contains the free response of the system plus a
  portion of forced response
• will decay to zero at a rate that is determined
  by the characteristic roots (poles) of the system
• Steady State Response
• will take the same (similar) form as the forcing
  input
• Specifically, for a sinusoidal input, the steady
  response will be a sinusoidal signal with the
  same frequency as the input but with different
  magnitude and phase.

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Transient and Steady State Response

• Ex

Lets find the total response of a stable second
order system
to a sinusoidal input
with I.C.
- total response
- PFE
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Steady State Response

• Final Value Theorem (FVT)

Given a signals LT F(s), if all of the poles of
sF(s) lie in the LHP, then f(t) converges to a
constant value as given in the following form
Ex.
A linear system is described by the following
equation
(1). If a constant input u5 is applied to the
sysetm at time t0, determine whether the output
y(t) will converge to a constant value? (2). If
the output converges, what will be its steady
state value?

• We did not consider the effects of IC since
• it is a stable system
• we are only interested in steady state response

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Steady State Response

• Given a general n-th order stable system

Transfer Function
Free Response
Steady State Value of Free Response (FVT)
In SS value of a stable LTI system, there is NO
contribution from ICs.