Stability

Stability - ME375 Handouts Stability • Stability Concept...

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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 left-half 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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