1 Stte spce systems nlysis (continued) Stbility A. Definitions A system is sid to be Asymptoticlly Stble (AS) when it stisfies ut () = 0, t > 0 lim xt () 0. t A system is AS if nd only if the impulse response of ll the system sttes goes to zero with time. The nturl modes tht pper in the impulse response of the system sttes depend on the loctions of the poles, defined s the roots of the chrcteristic eqution () s = si A = 0). A system is sid to be internlly stble or Asymptoticlly Stble (AS) if ll roots of the chrcteristic eqution () s = si A = 0 re in the open left side of the complex plne. A system is Mrginlly Stble (MS) when it stisfies ut () = 0, t > 0 xt () B with B constnt vector, (i.e. the sttes re bounded for ll time). A system is MS if nd only if the impulse response of ll tht sttes is bounded. Thus the system is MS if nd only if ll poles re in the left-hlf plne (i.e. they my be in the open left hlf plne or on the jω-xis), with non-repeted imginry poles. If the system hs simple poles on the imginry xis then the system is sid to be mrginlly stble. In this cse there exist some bounded inputs which will result in unbounded outputs. Tke for exmple the system G(s)=/s. If the input is unit step the output will become unbounded. If system hs multiple poles on the imginry xis or poles with rel prt positive then it is n unstble system. A system is sid to be input-output stble, or BIBO stble, if the poles of the trnsfer function (which is n input-output representtion of the system dynmics) re in the open left hlf of the complex plne. A system is BIBO stble if nd only if the impulse response goes to zero with time. If system is AS then it is lso BIBO stble (s the poles of the trnsfer function re subset of the poles of the system). However BIBO stbility does not generlly imply internl stbility. BIBO stbility implies internl stbility only when the system hs no trnsmission zeros (i.e. when the number of poles of the trnsfer function is equl to the number of poles of the stte-spce representtion of the system, or, in other words, when the stte vrible system is miniml representtion of the trnsfer function). Problem 0 0 Let x = Ax + Bu = x + u, y = Cx = [ ]x 0.
2 . The chrcteristic polynomil is s ( s ) = = s = ( s + )( s ). The poles re t s=-, s=, so the system is s t t not AS. It is unstble. The nturl modes re e, e. b. The trnsfer function is s H ( s) = C( si A) B = =, ( s )( s + ) s + which hs poles t s=-. Therefore, the system is BIBO stble. Note tht the unstble pole t s= hs cncelled with zero t s=. c. Does this system hve trnsmission zeros? d. Is this system controllble? e. Is this system observble? f. Does this system hve decoupling zeros? g. If YES, wht re their vlues nd wht is their nture (input, output or input/output decoupling zeros)? Problem Wht is the bounded input which will result in unbounded output if the system is s + described by the trnsfer function Gs () = 3 s + 4s + s+ 4? B. Routh-Hurwitz stbility test The Routh-Hurwitz test nswers the question Is given system stble? without ctully requiring clcultion of the roots of the chrcteristic eqution. Routh Test Given polynomil ps () the number of positive roots my be determined without finding the roots by using the Routh test. n n Given ps () = s 0 + s n s+ Build the Routh tble n The third row nd below re ech computed from the two rows immeditely preceding it by using reltions such s the next ones given, for exmple, for the third row b = 0 3 ; b = ; b =
3 The element is clled the pivot element for row two. To simplify computtions, by voiding frctions, one cn t ny point multiply ny row by positive constnt before proceeding to the next row. Routh Theorem. The number of roots of ps () in the right-hlf plne equls the number of sign chnges in column one. To exmine the input/output stbility (BIBO stbility) of system, one pplies the Routh test to the chrcteristic polynomil, the denomintor of the trnsfer function (fter pole/zero cnceltion). To exmine the AS of system, one pplies the Routh test to the chrcteristic polynomil () s = sin A. Problem A system hs the chrcteristic polynomil ps () = s + s + 6s + 0s+ 7. Is the system stble? If not how mny poles re in the right hlf plne? Problem 4 A system hs the chrcteristic polynomil how mny poles re in the right hlf plne? ps () = s 7s. Is the system stble? If not Problem 5 A system hs the chrcteristic polynomil ps () = s 7s. Wht is the condition tht the coefficients of the chrcteristic polynomil must stisfy such tht the system is stble? Problem 6 A system hs the chrcteristic polynomil is the system stble? 3 ps () = s + 3s + s+ k. For wht vlues of k While filling in the Routh tble one cn encounter two problems.. Routh test problem : Wht if one hs two successive rows which re proportionl? In this cse the next row will be ll zero. Solution: Tke out the polynomil corresponding to the lst nonzero row nd differentite it. Then plce it in the row which ws zero. And then continue filling in the Routh tble. Problem 7 A system hs the chrcteristic polynomil ps () = s + 4s + 5s + 5s + 6s + 9s + 8s+. Is the system stble? If not how mny poles re in the right hlf plne? 3
4 b. Routh test problem : Wht if n element in the first column is zero? In this cse the next row will be ll infinity. Solution: Plce letter indicting smll positive number (e.g. epsilon). Then continue filling in the Routh tble. You cn ssume tht the specific smll number (epsilon) is zero fter multiplying the entire next row with epsilon. Problem 8 A system hs the chrcteristic polynomil the system stble? ps () = s + 3s + s + 6s + 3s + 6s+ 3. Is Block digrms nd Mson s formul A liner time-invrint system cn be represented in mny wys, including: differentil eqution stte vrible form trnsfer function impulse response block digrm Ech description cn be converted to the others. In the following we will see how to determine the trnsfer function of system, which is described s block digrm, using Mson s formul. Simple system interconnection A. Series Interconnection The overll trnsfer function in Y (s) = H(s)U(s) is given by H() s = H() sh() s. B. Prllel interconnection The overll trnsfer function in Y (s) = H(s)U(s) is given by H() s = H() s + H() s 4
5 C. Feedbck interconnection H The overll trnsfer function in Y (s) = H(s)U(s) is given by () s H() s =. + H () sh () s Superposition For liner time-invrint systems superposition holds, so tht the effects of different inputs cn be dded together. To determine the effect of the input ut () on the output we consider tht dt () = 0. Then we hve HU () s = H() sh() s + H3() sh4() s. The trnsfer function between dt () nd the output yt () is HD () s = H() s. When the two inputs re both nonzero then the overll output is the sum of the effects of the two inputs. Mson's formul Mson's formul llows one to determine the trnsfer function of generl block digrms with multiple loops (creted by feedbck) nd multiple feed-forwrd pths. The formul uses some ides tht we now define. A block digrm consists of pths nd loops. A loop is ny pth where one cn go in circle nd return to the beginning point by following rrows in the direction in which they point. Two loops re sid to be disjoint if they hve no elements in common, i.e. if they do not touch. The determinnt of block digrm is defined s D(s)= - (sum of trnsmissions of ll loops) 5
6 + (sum of products of trnsmissions of ll pirs of disjoint loops)- - (sum of products of trnsmissions of ll triples of disjoint loops)+ + The cofctor of block digrm with respect to the i-th pth is defined s Di(s)= - (sum of trnsmissions of ll loops tht re disjoint from pth i) + (sum of products of trnsmissions of ll pirs of disjoint loops tht re disjoint from pth i ) - (sum of products of trnsmissions of ll triples of disjoint loops tht re disjoint from pth i) + We note tht the i-th cofctor is the sme s the determinnt, but does not include ny loops touching pth i. Also required is gi() s = trnsmission long pth i Using these notions one my write the trnsfer function of ny block digrm s H() s = gi() s i() s () n s i= where n is the number of pths in the block digrm. Problem 9 Use Mson s formul to find the trnsfer function for the feedbck interconnection Problem 0 Use Mson s formul to find the trnsfer function for the block digrm 6
7 Problem Use Mson s formul to find the trnsfer function for the block digrms A. B. Minimlity A stte vrible system (A,B,C,D) is miniml if it is system with the lest number of sttes giving its trnsfer function. If there is nother system (A,B,C,D) with fewer sttes (i.e. system of smller order) hving the sme trnsfer function, the given system is not miniml. For SISO systems, system is miniml if nd only if the trnsfer function hs the sme number of poles s the system. 7
8 Thus system (A,B,C,D) is miniml if nd only if it hs no input-decoupling zeros nd no output-decoupling zeros since, in fct, the decoupling zeros re exctly the zeros tht cuse pole-zero cncelltion in computing the trnsfer function. A block digrm is sid to be miniml if it relizes its trnsfer function with the minimum number of integrtors. From the result of Problem, one cn note tht the sme block digrm cn be miniml with respect to one input/output (I/O) pir but non-miniml with respect to nother. The poles re determined by the loops nd the zeros by the feed-forwrd pths. Note tht the zeros chnge s the input/output pir is chnged, but the poles depend on the bsic loop structure nd re independent of the selection of inputs nd outputs (i.e. the chrcteristic eqution of system in stte spce form does not chnge). Problem Consider the system from Problem in Lecture 3. Is the system miniml? Clculte the trnsfer function mtrix for the system described s x = 4x+ u = Ax+ Bu [ ]. y = x= Cx Note: This is miniml stte vrible representtion of the trnsfer function mtrix. The stte vrible representtion in Problem is not miniml representtion for the trnsfer function mtrix. A system which is miniml representtion of trnsfer function does not hve ny decoupling zeros. Problem 4 Consider the system from Exmple in Lecture 3 with C = [ ]. Determine the trnsmission zeros. Determine the decoupling zeros. Is the system miniml? Dominnt Mode Approximtion (quick note to remind you bout Lecture ) Though most systems of interest re of higher order, they often hve dominnt mode, which is complex pole pir of lower frequency thn the other poles. It is often useful to mke second-order pproximtion tht contins only the dominnt mode of the ctul system. This is chieved by testing the system experimentlly to obtin the step nd impulse response. This llows one to obtin the POV, dmping rtio, oscilltion frequency, settling time, nd stedy stte response, from which one cn find the dominnt mode pproximtion. The dominnt mode pproximtion cn be useful for quick, rough nlysis of the system, s well s for the design of simplified feedbck control systems. 8