Given the following circuit with unknown initial capacitor voltage v(0): X(s) Immediately, we know that the transfer function H(s) is

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1 EE 4G Note: Chapter 6 Intructor: Cheung More about ZSR and ZIR. Finding unknown initial condition: Given the following circuit with unknown initial capacitor voltage v0: F v0/ / Input xt 0Ω Output yt Simple meh analyi yield v v0 0 0 Immediately, we know that the tranfer function i 0 0 and the ZSR and ZIR are repectively 0 ZSR 0 0v0 ZIR 0 So, if you ue a tep function x t u t a input, you get an exponential decay 0 t output y t e u t Page 6-9 PDF Created with dekpdf PDF Writer - Trial ::

2 EE 4G Note: Chapter 6 Intructor: Cheung Page 6-0. Find out about an unknown ytem: ow do we find an unknown ytem and initial condition? Anwer: Input a tet ignal and meaure the output. The ue the I/O relationhip to figure out the ytem Example: The output of the ytem i 0 5 for an input So, i the tranfer function 0 5? NO, we ve forgotten the Initial Condition!! It hould be clear that ONE pair of I/O ignal will not be ufficient, we need two. Why? A we have learnt, the total output repone of a ytem i D C If we have two I/O pair, we have D C D C Subtracting from eliminate the initial condition and obtain the tranfer function: The initial condition can then be computed by ubtituting back to or. Try the above example again with another I/O pair: and and unknown IC PDF Created with dekpdf PDF Writer - Trial ::

3 EE 4G Note: Chapter 6 Intructor: Cheung Stability of ytem We firt need to introduce the concept of BOUNDED and TRANSIENT ignal. There are two type of well-behaved ignal: Bounded: x t M < for all t. Tranient: lim t x t 0 Are thee ignal bounded? Are they tranient? δ t co t u t 3 t in t u t 4 e t ut 5 e t ut Anwer: δ t Not bounded and tranient co t u t Bounded and not tranient 3 t in t u t Not bounded and not tranient 4 e t ut Bounded and tranient 5 e t ut Not bounded and not tranient Can you tell from their Laplace tranform? A ufficient condition for a ignal to be a tranient ignal : The pole on it Laplace Tranform mut be on the open left half plane. Thi condition i the ame a aying the ROC contain the imaginary axi. the ROC contain the origin 0 x t dt < lim x t 0 t Im Re A bounded ignal mut atify the following two condition:. It Laplace tranform mut be proper. N i.e. if, degree N <degree D D. The pole mut either be a/ on the open left half plane or b/ on the imaginary axi AND imple i.e. multiplicity. Page 6- PDF Created with dekpdf PDF Writer - Trial ::

4 EE 4G Note: Chapter 6 Intructor: Cheung Reaon for Condition : Conider δ t, L[ δ t ] i not proper. For general, if degree N degree D, applying long diviion: n n R n n an an... a0 L anδ t an δ t... a0δ t D [ ]... Thu, xt i not bounded due to the delta function. Reaon for Condition : If all the pole are on the open left half plane, we know the ignal decay to zero. If there are pole on the imaginary axi, there are two cae: Cae : Pole are imple, i.e. multiplicity uch a L[cot] /. In thi cae, the ignal i ocillating but till bounded. Cae : Pole are not imple, i.e. multiplicity > uch a L[tint]/. In thi cae, the ignal i not bounded. Now we come back to tudy the tability of a ytem. All ytem can be claified into table or untable. There are three different type of table ytem.. Bounded-Input Bounded-Output BIBO Sytem. Aymptotic Sytem 3. Marginally Sytem ere i a Venn diagram that decribe the claification. Marginally Aymptotically BIBO Untable Sytem Page 6- PDF Created with dekpdf PDF Writer - Trial ::

5 EE 4G Note: Chapter 6 Intructor: Cheung BIBO decribe ytem behavior ubjected to general input, auming zero initial condition. Aymptotic, marginally, and untable ytem refer to the long-term behavior of a ytem under any initial condition but no input. Marginally Aymptotically Untable N Recall: Given a ytem and input, the mot general form of D C N output i D D The firt term i the ZIR and the econd term i ZSR. BIBO Stability Definition: Aume zero initial tate, a ytem i BIBO table if it output a bounded output for any bounded input. A BIBO ytem mut atify the following two condition: N. If degree N degree D. D. The pole of mut be on the open left half plane. N A there i zero initial condition, the output i jut. To enure i D bounded, we need to check two thing: Page 6-3 PDF Created with dekpdf PDF Writer - Trial ::

6 EE 4G Note: Chapter 6 Intructor: Cheung. degree[numerator of ] < degree[denominator of ] degree[numerator of ] degree[n] degree[numerator of ] degree[denominator of ] degree[d] degree[denominator of ] Since i bounded, degree[numerator of ] < degree[denominator of ], thu all we need i degree[n] degree[d].. ha either open left half plane pole and/or imple pole on jω-axi. A i bounded, it pole mut either be on the open left half pole or imple on the imaginary axi. To enure atified the ame criteria, all the pole of mut be on the open left half plane. cannot have any pole on the imaginary axi, not even imple one becaue the input might alo have a imple pole at the ame location and the reulting will have DOUBLE imaginary pole, making it non-bounded. Aymptotic Definition: Aume zero input, a ytem i aymptotic table if it give a tranient output for AN initial tate. A ytem i aymptotically table if all the pole of the Laplace tranform are on the open left half plane. D C Thi i eay. A there i no input, the output i. In general, will D have the ame pole a unle there are cancellation of pole and zero. Thu D to enure yt i tranient, all we need i to enure that ha open left half D plane pole. Alo, it i eay to ee that BIBO tability Aymptotic tability. Page 6-4 PDF Created with dekpdf PDF Writer - Trial ::

7 EE 4G Note: Chapter 6 Intructor: Cheung Marginally Definition: Aume zero input, a ytem i marginally table if it give a bounded output for AN initial tate. A ytem i aymptotically table if all the pole of the Laplace tranform are either on the open left half plane or imple on the imaginary axi. D C To enure i bounded, we need to check: D. degree[c] < degree[d]. Thi i alway true. See page Pole of are either on the open left half plane or imple on the imaginary axi. Alo, it i eay to ee that BIBO tability Aymptotic tability Marginally tability. Untable ytem Definition: Aume zero input, a ytem i untable if it give a unbounded output for ome initial tate. An untable ytem i a ytem that i NOT marginally table. Page 6-5 PDF Created with dekpdf PDF Writer - Trial ::