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Unformatted text preview: ME375 Handouts Stability
• Stability Concept
Describes the ability of a system to stay at its equilibrium position (for linear
systems: all state variables = 0 or y(t) = 0) in the absence of any inputs.
– A linear time invariant (LTI) system is stable if and only if (iff) its free
if
iff)
response converges to zero.
Ex: Pendulum
Pendulu Ball on curved surface School of Mechanical Engineering
Purdue University ME375 Transfer Functions  1 1 ME375 Handouts Stability and Pole Positions
The free response of a system can be represented by:
YFree ( s ) = C (s)
F ( s)
F (s)
=
=
n
n −1
D ( s ) an s + an −1s + + a1s + a0 an ( s − p1 )( s − p2 ) ( s − pn ) An
A1
A2
+
++
= Y1 ( s ) + Y2 ( s ) + ... + Yn ( s )
s − p1 s − p2
s − pn
≠ p2 ≠
≠ p n i .e. n d istinct real p oles o f
= Assume p1 D ( s ) = an s n + an −1s n −1 + ⇒ For a complex pole pair,
Y k ( s ) + Yk +1 ( s ) = + a1s + a0 = 0 y k ( t ) = L − 1 [Y k ( s ) ] = A k e p k ⋅t p k ,k +1 = α ± j β
Ak
Ak +1
Bs + C
+
=
s − α − jβ
s − α + jβ
(s − α )2 + β ⎡
Bs + C
y k ( t ) + y k +1 ( t ) = L −1 ⎢
2
⎣ (s − α ) + β 2 ⎤
αt
⎥=e
⎦ 2 ⎡
⎤
⎛ Bα + C ⎞
⎢ B cos β t + ⎜
⎟ s in β t ⎥
β
⎝
⎠
⎣
⎦ School of Mechanical Engineering
Purdue University ME375 Transfer Functions  2 2 ME375 Handouts Stability of LTI Systems
• Stability Criterion for LTI Systems
an y(n) + an−1 y(n−1) +
Stable + a1 y + a0 y = bm u(m) + bm−1 u(m−1) + ⇐⇒ all poles of D(s) = an sn + an−1sn−1 + + b1 u + b0 u + a1s + a0 lie in the lefthalf complex plane Characteristic Polynomial LHP Unstable: If any pole does not lie in the left half of the complex plane;
Marginally stable: If any pole lies on the imaginary axis with the rest of the poles being in the left
Marginally stable: If any pole lies on the imaginary axis with the rest of the poles being in the left
half
half of the complex plane. • Comments on LTI Stability
– Stability of an LTI system does not depend on the input. (why?)
(why?)
– Effect of Poles and Zeros on Stability
• Stability of a system depends only on its poles.
• Zeros do not affect system stability; zeros affect the transient response.
School of Mechanical Engineering
Purdue University ME375 Transfer Functions  3 3 ME375 Handouts In
In Class Exercises
(1) Find the transfer function of the
following
following I/O equation: − y − 2 y − 5y = 2u + u
(2) Determine the system’s stability.
(3) Plot the poles and zeros of the system on
the complex plane. (1) Find the transfer function of the
following I/O equation: y + y + 6 y = u − 3u + 4 u
(2) Determine the system’s stability.
(3) Plot the poles and zeros of the system on
the complex plane. School of Mechanical Engineering
Purdue University ME375 Transfer Functions  4 4 ME375 Handouts Example
(Inverted) Pendulum
Pendulum
(1) Derive a mathematical model for a
pendulum. (2) Find the equilibrium positions.
(3) Discuss the stability of the
equilibrium positions.
equilibrium positions. School of Mechanical Engineering
Purdue University ME375 Transfer Functions  5 5 ME375 Handouts Example (Pendulum)
(Pendulum) School of Mechanical Engineering
Purdue University ME375 Transfer Functions  6 6 ...
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 Fall '10
 Meckle

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