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Input-Output (I/O) Stability I/O Stability ME689 Lecture Notes by B.Yao Input-Output (I/O) Stability -Stability of a System Outline: ¾ Introduction • White Boxes and Black Boxes • Input-Output Description ¾ Formalization of the Input-Output View • Signals and Signal Spaces • The Notions of Gain and Phase • The Notion of Passivity • The BIBO Stability Concept ¾ I/O Stability Theorems • The Small Gain Theorem • The Passivity Theorem • Positive Real Functions and Kalman-Yakubovich Lemma For LTI System 1 I/O Stability ME689 Lecture Notes by B.Yao 1. Introduction Systems can be modeled from two points of view: the internal (or state-space) approach and the external (or input-output) approach. The state-space approach takes the white box view of a system, and is based on a detailed description of the inner structure of the system. In the input-output approach, a system is considered to be a black box that transforms inputs to outputs. As a consequence, the system is modeled as an operator. The operator can be represented by either a verbal description of the input-output relationship, or a table look-up, or an abstract mathematical mapping that maps an input signal (or a function) in the input signal space (or a functional space) to an output signal (or a function) in the output signal space. 2. Formalization of I/O View 2.1 Signals and Signal Spaces 2.1.1 Single-Input Single-Output (SISO) Systems In SISO systems, the input and output signals can be described by real valued time functions, e.g., an input signal is denoted by a time-function u in R, i.e., u =Œ{}ut(), t R Note that to specify an input signal u, it is necessary to give its value at any time, not values at a point or an interval. The input and output signal spaces can be described by normed spaces; certain norms have to be defined to measure the size of a signal. For example, • The super norm or L• -norm u suput ( ) (I.1) • = t≥0 Application examples of using the super norm are the problem of finding maximum absolute tracking error and the problem of control saturation. • L2 -norm 1 • 2 u = Ê utdt2 () ˆ (I.2) 2 ˯Ú0 If we are interested in the energy of a signal, L2 -norm is a suitable norm to choose. • Lp -norm 1 • pp u = Ê ut() dtˆ , 1 £<•p (I.3) p ˯Ú0 Corresponding to different norms, different norm spaces are defined: • L• -space the set of all bounded functions, i.e., the set of all functions with a finite L• norm. LxxM=£: for some M >0 (I.4) • {}• 2 I/O Stability ME689 Lecture Notes by B.Yao • Lp -space the set of all functions with a finite Lp -norm., i.e., ϸ1 ÔÔ• p p LxxM=£: or Ê utdtM( )ˆ £ for some M >0 (I.5) p Ì˝p ËÚ0 ¯ Ó˛ÔÔ In general, these normed spaces may be too restrictive and may not include certain physically meaningful signals. For example, the L2 -space does not include the common sinusoid signals (why?). To be practically meaningful, the following extended space is defined to avoid this restrictiveness: Definition [ Extended Space] If X is a normed linear subspace of Y, then the extended space X e is the set XxYxXeT=Œ{}: Œ for any fixed T ≥0 , (I.6) where xT represents the truncation of x at T defined as Ïx(tt ) 0 ££T xtT ()= Ì (I.7) Ó0 tT> The extended L1 space is denoted as L1e , the extended L2 space as L2e , " . Example: u(t )=fiŒœ sinw t uL22e but uL y(tt )=fiŒœ yL••e but yL 2.1.2 Multi-Input Multi-Output (MIMO) Systems Multi inputs and multi outputs can be represented by vectors of time functions. For example, m inputs can be represented by a m-dimension of time functions u, i.e., u :0[ •Æ] Rm with È˘ut1() u()tRÍ˙# m (I.8) =ŒÍ˙ Î˚Í˙utm () Norms corresponding to (I.1)-(I.3) are defined as • The super norm u suput ( ) (I.9) • = t≥0 where u()t represents the norm of ut()in Rm space1. 1 ∞ m = The superior -norm of u(t) in R is u(t) ∞ max ui (t) . i 3 I/O Stability ME689 Lecture Notes by B.Yao • The L2 -norm ••2 u ==ut() dt uT () tutd () t (I.10) 22ÚÚ00 • The Lp -norm 1 • pp u = Ê ut() dtˆ , 1 £<•p (I.11) p ˯Ú0 where u()t represents the norm of ut()in Rm space2. m m m The corresponding norm spaces are represented by L• , L2 and Lp respectively. The extended m m m spaces are L•e , L2e and Lpe respectively. 2.1.3 Useful Facts The following lemmas state some useful facts about Lp -spaces. Lemma I.1 [Ref.1] (Holder’s Inequality) 11 If pq,1,Œ•, and +=1, then, fLŒ , g ŒL imply that fg ŒL , and [ ] pq p q 1 fg1 £ fpq g (I.12) i.e., 11 •••pqpq f() t g () t dt£ ÊÊ f () t dtˆˆ g () t dt (I.13) ÚÚÚ000Ë˯¯ Note: When pq==2 , the Holder’s inequality becomes the Cauchy-Schwartz inequality, i.e., 11 •••2222 f() t g () t dt£ ÊÊ f () t dtˆˆ g () t dt (I.14) ÚÚÚ000Ë˯¯ An extension of the Lemma I.1 to Lpe –space can be stated as Lemma I.2 11 If pq,1,Œ•, and +=1, then, fLŒ , g ŒL imply that fgLŒ , and [ ] pq pe qe 1e fg£ f g , "≥T 0 (I.15) ( )T 1 TTpq or 11 TTTpqpq ftgtdt() ()£ ÊÊ ft () dtˆˆ gt () dt "≥T 0 (I.16) ÚÚÚ000Ë˯¯ 1 m p p 2 The p-norm of u(t) in R m is u(t) = u (t) . p ∑ i i=1 4 I/O Stability ME689 Lecture Notes by B.Yao Lemma I.3 [Ref.1] (Minkowski Inequality) For p Œ•[1, ] , fg, Œ Lp imply that fgL+Œp , and fg+£ppp f + g (I.17) Note: (a) A Banach space is a normed space which is complete in the metric defined by its norm; this means that every Cauchy sequence is required to converge. It can be shown that Lp -space is a Banach space. (b) Power Signals: [Ref.5] The average power of a signal u over a time span T is 1 T u2 ()t dt (I.18) T Ú0 The signal u will be called a power signal if the limit of (I.18) exists as T Æ•, and then the square root of the average power will be denoted by pow(u ) : 1 2 ʈ1 T 2 pow(u )= Á˜ limutdt ( ) (I.19) ˯TÆ• T Ú0 (c) The sizes of common Lp spaces are graphically illustrated below [Ref.5] L pow 2 L∞ Note: L12«ÃLL• L1 Example: (verify by yourself) 1 ft()= fi f ()tLLŒ«2 , but f ()t œL1 , f ()tLŒ 1e 1+ t • Ï0 if t £ 0 Ô Ô 1 ft( )=<<Ì 0 t 1 fi f ()t ŒL1 , but f ()tLœ • , f ()tLœ 2 Ô t ÓÔ0 t > 1 5 I/O Stability ME689 Lecture Notes by B.Yao 2.2 Gain of A System 2.2.1 Definitions From an I/O point of view, a system is simply an operator S such that maps a input signal u into an output signal y. Thus, it can be represented by S : XXieÆ oe with y = Su (I.20) where the extended norm space X ie represents the input signal space (i.e., u Œ X ie ) and X oe is the output signal space. The gain of a linear system can thus be introduced through the induced norm of a linear operator defined below: Definition [Ref.2] [Gain of A Linear System] The gain g ()S of a linear system S is defined as Su g ()S = sup (I.21) uXŒ e u uπ0 where u is the input signal to the system. In other words, the gain g ()S is the smallest value such that SuS£ g ()u , "Œu X e (I.22) Nonlinear systems may have static offsets (i.e., Su π 0 when u = 0 ). To accommodate this particular phenomenon, a bias term is added in the above definition of gains: Definition [Gain of A Nonlinear System] The gain g ()S of a nonlinear system S is defined to be the smallest value such that Su£+g () S u b , "Œu X e (I.23) for some non-negative constant b . Definition [ Finite Gain Input-Output Stability] A nonlinear system S is called finite-gain Lp input-output stable if the gain g ()S defined in (I.23) is bounded (or finite), in which the Lp norm is used for input and output signals. Note: When p=∞, the above finite gain Lp stability, i.e., L∞ stability, results in bounded-input bounded-output (BIBO) stability. The converse is in general not true. For example, the static mapping y=u2 is BIBO stable but does not have a finite gain. 2.2.2 Linear Time Invariant Systems An LTI system with a TF G(s) can be described in input/output forms by Y ()sGsU= () ()s in s-domain t y()tgutgtud==- ( * )() (ttt ) ( ) in time-domain (I.24) Ú0 where g()tLGs= -1 {} () is the unit impulse response. 6 I/O Stability ME689 Lecture Notes by B.Yao Theorem I.1 [Ref.1] Let G(s) be a strictly proper rational function of s. (a) The system is stable (i.e., G(s) is analytic in Re[s ]≥ 0 or has all poles in LHP) if and only if gLŒ 1 .
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