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# Module 5 - ECE 514 Nonlinear Adaptive Control Module 5-Part...

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1 ECE 514 Nonlinear & Adaptive Control Module 5-Part 1 INPUT-OUTPUT STABILITY

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2 Topics 1. Input-Output Stability 2. L-Stability of State-Space Models 3. Small-Gain Theorem This material covered in Chapter 5 of Khalil, and on my notes on Input-Output Stability, available under Additional Resources.
3 Motivation 1. So far, we have focused almost exclusively on state-space systems and Lyapunov stability which concerns the behavior of the state variables with respect to changes in the initial conditions. 2. There is however another way of modeling systems, namely by relating inputs to outputs without referring to the concept of a state.

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4 Input-Output Stability Lyapunov Stability with respect to initial condition. Input-Output Stability with respect to input. System u y Well-behaved (BI) Well-behaved? (BO) For linear system, asymptotic stability BIBO 0 1 0 Linear system is BIBO stable iff ( ) where { ( )} ( ) impulse response and ( ) iff all poles of ( ) are in the open left half plane asym. stable. gd LG s g t g d Gs ττ <∞ =→ < Remark :
5 Example 2 is GES (when 0). If we define the output , the resulting transfer function: () 1 is BIBO stable 1 However, consider , which is again GES when 0, but is not BIBO stable for xx u u yx Ys Us s u u =− + = = = + =− − = i i ,s i n ( ) . u t ==

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6 Nonlinear Systems I/O Stability How may input-output stability concepts be defined for nonlinear system ? q p n n p n R R R R h u x t h y R R R R f u x t f x × × = × × = + + : ) , , ( : ) , , ( ± Operator representation of systems: q p L y L u Hu y = , , normed linear spaces functions. measurable bounded y essentiall all of set the as defined is and ) , 1 [ for ) ( such that functions measurable all of set the as defined is where 0 < L p dt t f f L p p
7 Signal Norms H may be unstable, and y ( t ) might not have the same norm. To accomodate this situtaion, introduce extended space , : nm pe qe LL t ) ( t u τ ) ( t u > = t t t u t u 0 0 ) ( ) ( } 0 ) ( : { = n p n pe L t u u L Various norms on or can be used. pq () < = < = 2 1 0 2 2 0 ) ( ) ( sup d t u u t u u t

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8 Signal Norms Details In general Note that it does not matter which definition of || u(t)|| one uses .
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Module 5 - ECE 514 Nonlinear Adaptive Control Module 5-Part...

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