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# DETERMINISTIC SYSTEMS EE 611

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This 62 page Class Notes was uploaded by Adaline Pollich on Friday October 23, 2015. The Class Notes belongs to EE 611 at University of Kentucky taught by Kevin Donohue in Fall. Since its upload, it has received 44 views. For similar materials see /class/228312/ee-611-university-of-kentucky in Electrical Engineering at University of Kentucky.

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Date Created: 10/23/15

EE61 l Deterministic Systems System Descriptions State Convolution Kevin D Donohue Electrical and Computer Engineering University of Kentucky InputOutput Systems 8180 systems with a single input and single output MIMO systems with multiple inputs and multiple outputs Continuoustime systems inputs and output are de ned over all time t 8180 input 211 output yt MIMO input ut output yt Discretetime systems inputs and output are de ned at discrete points in time t kT kE I With sampling interval T ukT uk ykT yk System Classes A system is memoryless instantaneous iff if and only if output y to depends only on input at ut0 A system is causal nonanticipatory iff output yt0 depends only on input 111 for 102 t State The state of a system at to Xt0 is the information required along with the input 111 for t 210 that uniquely determines the output yt for t Z to Example Find a state descriptions for the following system at to 0 Result 200uljgtl101yl120yl 101Llt0gt ylt0gt 100vc0 10iL0 MO System Classes Linear A system is linear iff for every 10 and inputoutput pair 139 12 Xi to t tZto uit tZto M then additivity holds Xlt0gtX2togt t t tZto ullttgtuZlttgt taro y1gt y2gt and homogeneity holds Xit0gt olt t tZto owlt taro y gt Where oce R ZeroState ZeroInput Response If input is zero the response that results is due to the system state known as the zeroinput response x to u t 0 tZt If state is zero the response that results is due to the system input known as the zerostate response Xt00 u t tZt 1Hyzitgtt2to 0 HyzstZt 0 0 In general for a linear system superposition holds between the contributions of the state and input to the response Therefore x00 ut rat 0 1Hyzitgtyzsgtt2to Response Classes Zeroinput response system output due only to system state or initial conditions N dquot O Z 0 orn 21 w system output due only to the input of the system H N d yzs uZnOoltn n X0 In general Total response Zeroinput response Zero state response yyziyzs Examples of Linearity Determination Determine Whether or not each system described below is linear Assume inputs and outputs are functions of time denoted by u and y and constants are denoted k 0 y kl u 5 7 y o yku10 InputOutput Description Convolution For a linear lumped or distributed system the inputoutput relationship for a zerostate response can be expressed in terms of the convolution integral and the system39s impulse response yltrfgltrrultrdr Where g t T is the system39s timevarying impulse response at time T If system is zerostate relaxed at to then integral can be written 00 yltrgtL gumMm If system also is causal impulse response must be zero for tgtts ylttgtfagtrurdr MIMO InputOutput Description For a p input and q output linear causal relaxed at to lumped or distributed system the inputoutput relationship for a zero state response can be expressed in terms of the convolution integral and the system39s impulse response matrix yrfacrTuTdT Where G t T is the system39s timevarying impulse response matrix describing the contribution of inputs at all p terminals to the q outputs g11ltZT g12ltZT g1plttT Carr g21fT g22ltFT g2plT gmm gq2l T gq StateSpace Description For a lumped system represented by an order N differential equation governing the state can be written as p inputs xtAtxtBtut Where A is an NxN matrix x is a le vector B is a pr matrix and u is a lxp vector The output q outputs is a linear combination of the states and inputs and can be written as yltrCltrxltrDlttulttgt Where C is an qu matrix x is an le vector D is a qxp matrix and u is a lxp vector Time Invariance If system is not changing overtime it is referred to as time invariant and results in signi cant simpli cations More formally stated A system is time invariant iff for every to and inputoutput pair x00 11er ut tat yltgt and any time shift T the following also holds Xt0T gt tTt2t0T ut T 1210T y gt Time Invariance Linear systems that are time invariant are referred to as linear time invariant LTI systems Their representations simplify t0 ytfaGtTuTdT gtytfaGt THTdT XtAtXtBtut gtXtAXtBut ylttCrxlttDlttultreylttCxrDultr Transfer Functions The transfer function TF of an LTI system can be derived from the Laplace Transform of its inputoutput description Show for a relaxed system the Laplace Transform of the impulse response is its transfer function LTlgltrgt1llslgltsgt Transfer Functions and State Space For a 8180 system derive the relationship between TF and the zerostate and zeroinput responses by taking the LT of a state space representation to obtain jscsI A1Xt0dcsI A1bits Find the formula to convert a statespace representation to a TF for the zerostate case Is it possible for a TF to represent the case when the state is not zero not a relaxed system What is the signi cance of the d parameter EE61 l Deterministic Systems Examples and Discrete Systems Descriptions Kevin D Donohue Electrical and Computer Engineering University of Kentucky StateSpace Description Example Find the statespace descriptions for the following circuit 0 2 IL VC ul iquotL 100 10 Vc 2 1 StateSpace Description A general approach Identify the independent voltages andor currents in the circuit These are values that you need initial conditions for in order to solve for the unique complete solution 39Find an equation for each independent value that relates its rst derivative to zeroorder derivatives of the other independent values and sourcesinputs Set up a system of rst order differential equations in a matrix form 39EXpress the particular output value in terms of a linear combination of the independent values Use matrix notation StateSpace Description Example Find the statespace descriptions for the following circuit 1 Vs T 5F v0 29 vol 718 16 v01 2 29 is v 39 14 14 v 0 0 v5 law 0251 StateSpace Description Example Find the statespace descriptions for the following circuit Example Find TF from SS Given SS description for a relaxed system nd the TF 1 100 10 1L 0W v6 2 1vc 2 y10 0M VC Show gltsgtj2s 200 Ms s2101s120 Example Find TF from SS Given SS description For a relaxed system nd the TF matrix v01 718 16 v01 2 29 1 vol 14 14 v02 0 0 v5 w om DiscreteTime Systems Dirac Delta impulse Function 6tlimhn0rect ytfy39r5t Td39r Kronecker Delta Function we 333 ylklZ wylml lk ml DiscreteTime Convolution Let gk m be the system impulse response The output yk can be computed from the input um with convolution summation ymZZ murmmm For a LTI system it reduces to ylkZ wumgk mz 00 m w uk mgm For a relaxed causal system it reduces to k ykZm0umgk mZOuk mgm ZTransform Analogous to the Laplace Transform for continuoustime systems is the Ztransform for discrete systems 00 lzl2k0ylklz k For a LTI system it can be shown that discrete convolution in time is multiplication in the zdomain ZTZ0utmigtk maltzgtgltzgt Delay Property ZTuk nl1ftzz quot Discrete Transfer Function Analogous to the differential equation for continuoustime systems is the difference equation for discrete systems ym2ff1anylk n2f0bmuk m For a relaxed LTI system the transfer function can be expressed as 9Z1blz 1sz M zN MzMble 1bM M2 1alz 1aNz N zNalzN 1aN Z Describe the relationship between causality and the orders of the numerator and denominator Discrete StateSpace Description Statespace description for a timevarying discrete linear system Xk1AkXkBkuk YkCkXkDkuk for a time invariant system Xk1AXkBuk ykCXkDuk Transfer Functions and State Space For a MIMO system the relation between TF and the zerostate and zeroinput responses of a statespace representation is given by jzCzI A1zx0DCzI A1B z For the zerostate case the TF is given by zDCzI A1B z DiscreteTime System Example For the 8180 system assume initial state XO l lT and input u 0 nd output for the k O l and 2 XMH 01 011 025 0675 PM yik1 leik Assume the system is relaxed at k 0 Find closedform expression for yk for k 2 0 given uk is the unit impulse function DiscreteTime System Example Zero input response 0 a A 25 0675 01 011 Define systems matrix c 1 O Define output matrix x11 139 Define initial state ktot 10 Define number of outputs to compute 0 2 Ampiitude for k1ktot We xk1 Axk W CXIk end kaxis 0Iengthy 1 Time index axis potkaxis y xabe39Time39 yabe39Ampitude39 ZeroState impulse response yepL ym 36 2 k1sin121k 1 for M 22 14z04 o for k0 Lecture Note Homework U21 Derive statespace and output equations for the analog computer circuit below voltage u is the input and y is the output le ZOkQ J Hint Use inputoutput integrator relationships to come up With a series of state variables also note that state equation should be third order Lecture Note Homework U22 Find the transfer function for the given the 8180 system 05 025Xm xkl O 01 ylklll lelk Assume the system is relaxed at k 0 Find closedform expression for yk for k 2 0 given uk is the unit step function Hint Use ztransform tables or Matlab s iztrans function from the symbolic toolbox EE61 1 Deterministic Systems Realizations StateTransition Matrices Kevin D Donohue Electrical and Computer Engineering University of Kentucky Realizations Every LTI system has an inputoutput description of the form A A A ysGsu1 If system is also lumped statespace descriptions also ISL XiAxiBui yiCXiDui which are referred to as realization of the transfeis matrix Cs is realizablAe if El a nitedimeps39onal state equation ABCD a Gs DCsI A Bl Transfer Matrix Realizations A transfer matrix 1 s is realizable iff C s is proper rational matrix Consider an element of G s that is not proper MgtO blsquot1b2squot2bn1sbn gpqsd0sMl d1sM1dM1sdMl n H H sl als a2s an1san How would the realization equations below have to change to accommodate the not proper G s XtAXtBut ytCXtDut Example Find the state equations for the circuit using 2 different state de nitions Show the transfer matrix is the same Assume 211 input and yt output ut L i L 1 C W T gs LC 1 SZ S L LC Realize a Strictly Proper TF For a proper TF long division can be applied to decompose it into a constant term for the d scalar and a strictly proper TF expressed below 313quot 113er 2Bn 1sBn SnO1Sn 1O2Sn 20n lsan The oparnp circuit for this TF can be realized as sps Realize a Strictly Proper TF The state equations for the strictly proper TF A 313quot 11325 2Bn 1sBn gspS n n 1 n 2 s ogs 095 0n lsan are written from x1x392 x2x393 x 1x39n x391u 01x1 02x2 0n 1x 1 ornxn yB1xlB2x239 Bn lxn 1ann olt1 x2 xn 1 xn 1 i l O O O X O u 3 0 O O l O 0 Transfer Matrix Realizations In general for a proper q x p transfer matrix 1 s its realization can be expressed as follows Find a common denominator ds for all element in G s and divide through by the denominator if necessary to separate each rational polynomial into a constant and strictly proper rational polynomial s w sps Then expand the strictly proper part into A 1 r r GspSmNlS 139l39leS 2Nr1SNr Where r is the order of ds Transfer Matrix Realizations A realization of f Scan therefore be written as 011p olt21p ar11p 091p Ip 113 9 9mm 0 0 I 0 0 Solution of Linear TimeVarying State Equation Given a lineartime varying statespace equation XtAtXtBtut yltrCltrxlttDltrultt the solution can be written as Xt tt0xt0f151TBuTdT ytCtd5t tOXtOCtj 45t TBuTdTDtut Where 15 t Tis called the statetransition matrix State Transition Matrix A fundamental matrix for the homogeneous equation XrAltrxltr is a matrix Xt 3 its columns are unique solutions of XrArX t and Xt is nonsingular for all t Then for any fundamental matrix of xtA t x t the state transition matrix is given by am r0gtXltrgtX 1ltrogt and is the unique solution of 0 t0Atd5t to for initial condition t03t01 Examples Find the state transition matrices for to the following 50 8Xt Special Case Solution For the case when At fAltTgtdT Then the solution for the state transition matrix becomes t fAltTgtdT to t fAltTgtdT to 00 1 t0texp 2 F k0 k Discrete Time Case The state transition matrix can be computed through recursion for the discrete time case Given k1k0Ak kk0 Substitute repeatedly into 45k k0Ak 1d5k 1k0 k k0Ak 1Ak 2lt15k 2k0 k k0Ak 1Ak 2 Ak0 l5k09k0 Then the solution for the state transition matrix becomes 15kk0Ak 1Ak 2Ak0 Useful Matrix Relationships dA 1t 1 dAt 1 dt A I ABAHB AB 113 1A1 ABTBTAT trABtrBA Homework U61 Find the op arnp circuit and statespace realization for the proper rational TF given below 2533s S 3 2 3 23 SS25 IQgt EE61 1 Deterministic Systems Controllability and Observability Discrete Systems Kevin D Donohue Electrical and Computer Engineering University of Kentucky Canonical Decompositions Given the Controllability matrix of an n dimensional system that is not controllable CB AB AZB A3B AHB where pCn1ltn De ne equivalence transformation Qq1 qz qnl qn PQ 1 Where the rst n1 columns of Q are the m li columns of C and the other columns are arbitrary vectors added such that Q is nonsingular Canonical Decompositions Then equivalence transformation 2sz partitions original statespace equations into 15c 2 Ac 12 Xe 13c i5 0 A6 is BE ylt c C5 Dultrgt XE Where Ac is m X m and AE is nnl X nnl and the m dimensional subequation is controllable and has the same transfer matrix as the original statespace equation Canonical Decompositions Given the Observability matrix of an n dimensional system that is not observable ohr AJC39AALC39 ALC39quot A r Where p0n2ltn De ne equivalence transformation V Pb p m p1h InT QP Where the rst 112 rows of P are the m li rows of O and the other rows are arbitrary vectors added such that P is nonsingular Canonical Decompositions Then equivalence transformation 2sz partitions original statespace equations into 0 A0 12 X0 1 30 i5 0 A6 i5 B6 y30 35 quot Dut X6 Where A0 is m X m and A6 is VI1 12 X VI1 12 and the m dimensional subequation is controllable and has the same transfer matrix as the original statespace equation QM NI E0ut Dut i0 0 0 0 yo MI Kalman Decomposition Theorem An equivalence transformation exists to transform any statespace equation into the following canonical form ico Aco 0 A13 0 gm Boo e 2 A21 Ac 23 A24 Kc c i o 0 0 A50 0 i o 0 i6 0 0 A43 Aer i6 0 1 K y3c0 0 360 0Dut X60 7 Where subscript co indicates the controllable and observable and the bar over the subscript indicates not Kalman Decomposition Example Perform a Kalman decomposition on 0 24 74 85 45 11 2 OOHOOO OHOOOO HOOOOO ooooo OOOHOO 39ooooHo39 y2 3 6 16 38 6OX Kalman Decomposition Theorem The controllable and observable subsystem is equivalent to the zerostate system given as i 3 meow CO CO X00 Controllability Discrete The state equation XklAXkBuk is controllable if for any pair of states Xk1 and Xk0 El an input uk that drives state Xk0 to Xk1 in a nite number of samples If the system is controllable then an input to transfer state Xko to Xk1 over the input samples in the interval k0 kll is given by uk B39A39k1 k 11wk1 k0 1Ak1 k xkO Xk1 Where n1 Wdcn lZ AmBB39A39m m0 Conditions for Controllability For an n state andp input system XA xBu This system is controllable if any one of the equivalent conditions are met 1 The n X n matrix Wdcn1 is nonsingular n l Wdcn 1 Z AmBB39A39m m0 2 The n X np controllability matrix Cd has full row rank n CdB AB AZB A3B A HBl Conditions for Controllability 3 The n X np matrix AAI B has full rovv rank for every eigenvalue 1 of A 4 All eigenvalues of A have magnitudes less than 1 and the unique solution Wdc is positive de nite WdC AWdCA39BB39 Where Wdcis the discrete controllability Gramian Wdci AmBB39A39m m0 Observability Discrete The discrete statespace equation xk1AxkBuk ykCxkDuk is observable if for any unknown initial state Xk0 there exists a nite integer k1 kg 3 knowledge of input uk and output input yk over k0 k1 is all that is required to uniquely determine Xk0 If the system is observable then an estimatorobserver to compute state Xk0 from the input and output over the time interval k0 k1 is given by k1 k 0 Xk0Wd0k1 k012A39quot C39yk0mk0 m0 n 1 Wan 1 Z A 39gt C39CA ykk0yk C Z Ak1mBum Duk Conditions for Observability For an n state and q output system xk1AxkBuk ykCxkDuk 1 This system is observable iff the nxn matrix W010 is nonsingular kl l Wd0k1 1Z AmCC39A39 m0 2 The nq X n observability matrix Od has full column rank n C CA 001 CA2 CAn l Conditions for Observability 3 The nq X n matrix All C3939 has full column rank for every eigenvalue 1 of A 4 All eigenvalues of A have magnitudes less than 1 and the unique solution Wdo is positive de nite Wdo A39Wd0AC39C Where Wdo is the observability Gramian Wdoz A39 C39CA m0 Controllability After Sampling Given a controllable continuoustime system a suf cient condition for controllability of its discretized system using sampling period T is that Im7u7tj 2 2am T for any ij pair of eigenvalues from continuous time system whenever Rem7g O For the singleinput system the above condition is necessary as well

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