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Control Systems

by: Lionel Hansen

Control Systems MECH 417

Lionel Hansen
GPA 3.87

David Alciatore

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David Alciatore
Class Notes
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This 5 page Class Notes was uploaded by Lionel Hansen on Tuesday September 22, 2015. The Class Notes belongs to MECH 417 at Colorado State University taught by David Alciatore in Fall. Since its upload, it has received 65 views. For similar materials see /class/210251/mech-417-colorado-state-university in Mechanical Engineering at Colorado State University.

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Date Created: 09/22/15
B Laplace Transform 39 4 his w ms 3 mm mew 0 9 TABLE 82 LAPLACE TRANSFORM THEOREMS Name Derivative nthMOrder derivative Integral Shifting Initial value Final value Frequency shift Convolution integral Theorem B sF3f0 43 513 squotFssquot foj dtquot fquot390 3 lime 555 0 of ff o Ijtol EMF mm 393 mm mm W 53 9 39ftl FSa rummun I frrgtfzrdx 0 I momtwm 0 who at NWMWM 5 P A M m mam vwaMwu 652 Laplace Transform and zTransform Tables Appendix C A 1 e cosbrgsmb7 B e39m39 e sin bT cos bT Time function Laplace transform zTransform et Es Ez I z quot2 E 2 1 Tz t z12 t2 1 T2zz 1 2 3 2z 13 1 k l lim 1 l 3 z tkm T a0 aak l Zane 11T z 1 l Tzea t quotat w nT 2 e sa2 he k ze a c m Away am 5ak 8a Z 8 41 a M lde ssa z l z ear he a zaT 1 e Tz l e T aTe Tl t 2 a7 a 32 30 az l z e a2 z 1 e T aTe Tz e 2 T aT 1e T I 1quot1atea ssa2 zl 239 e T2 aT 41 equot 39 equotb39 L39a e 18 3 2 30 sb 2 Ze b zsian sin bt m z2 2z cos bT 1 s zz cos bT cosbt m 2 2quot bT 1 324172 2 zcos 2bs ism bt W 32b22 2 2 tcos bt 45527 s2b2 m b 639 3111 b m 22 Zze aT cos bT eZaT 3 zzzemTcos bT e 39cos b m 22 ZZeMT cos bT 64 a a azIb2 Al3 1 9 COS btEsmbt W z z22zeu1 cos bTe2aT De nition The hyperbolic sine function denoted by ainh and the hyperbolic cosine function denoted by cosh are given by ex ewx x smhx 2 W and coshx m m 2 2 where x is any real number gtgt help residue RESIDUE Partialwfraction expansion residues RPK m RESIDUEBA finds the residues poles and direct term of a partial fraction expansion of the ratio of two polynomials BsAs If there are no multiple roots 35 R1 R2 Rn Ks As s P1 s P2 s Pn Vectors B and A specify the coefficients of the numerator and denominator polynomials in descending powers of s The residues are returned in the column vector R the pole locations in column vector P and the direct terms in row vector K The number of poles is n lengthAl e lengthR lengthP The direct term coefficient vector is empty if lengthB lt lengthA otherwise lengthK lengthBlengthAl If Pj Pjml is a pole of multplicity m then the expansion includes terms of the form Rj Rjl Rjml S Pj S PjA2 8 Pj m BA RESiDUERPK with 3 input arguments and 2 output arguments converts the partial fraction expansion back to the polynomials with coefficients in B and A Based on the preceding dct nnttons we may now stutc Mason s gain formula The 39qu mula gives the transfer function from a source input noth to a sink output node only and may he stated as i i z EiMiAtM2A1quot39MA 249 I 2 3 MA where I Is the gain transfer function from the input node to the output node I is the num ber ol lorward paths and t x 2 A z I sum ol all individual loop gains sum oi the products of the loop gains ot nll possible combinations of nontouch mg loops taken two at a time 31me the productsof loop gains of an possible Combinationsof nontouch ing loopstaken tines onetime sweet the ptodneteo gainsquot of alipoestbte of animation ing loops tekenfomatntime 2 L Mite path gain Menu tmara i l Agar vahmo A fem oi the 39not tou ching the kth inward path 41 TIME RESPONSE OF FIRSTORDER SYSTEMS In this section the time reSponse of rstvorder systems is investigated The transfer function of a general rst order system can be written as bo 39m 6 quot 123 Mao 4 wherems is the input function and 03 is the ohtpttt function this notation is common A more common notation for the firstorder transfer function is C S K 39 G no quot rs since physical meaning can be given to both K and t We call 42 the standard form of the rstorder system Of course in 41 and 42 42 bog f 4 3 t E 42 TIME Rasmussen mention sternum In this section we investigate the response of secotid order systems to certain inputs We assume that the system Wafer function is of the fOrm C at 1 bii 4 33 312 013 on However as in the firstorder case the coef cients are generally written in a manner such that they have physical The standardfonnof the secondorder transfer function is givenby quot t not a 4 17 A yogi G z y 4 18 839 ZCW 61 ROUTH HURWITZ STABILITY CRITERION The Routh Hurwitz criterion is an analytical procedure for determining if all roots of a polynomial have negative real parts and is used in the stability analysis of linear time invari ant systems The criterion gives the number of roots with positive real parts and applies to all LTl systemsfor which the characteristic equation is a polynomial set to zero This requirement excludes a system that contains an ideal time delay transport lag For this spe cial case which is covered later the Routh Hurwitz criterion cannot be employed The Routh Hurwitz criterion applies to a polynomial of the form Qs ansquot a squot 39 als an 64 where we can assume with no loss of generality that an t 0 Otherwise the polynomial can he expressed as a power of s multiplied by a polynomial in which an at 0 The power of s indicates roots at the origin the number of which is evident hence only the latter polyno mial need be investigated using the Routh Hurwitz criterion We assume in the following developments that a is not zero The first step in the applicationof the Routh Hurwitz criterion is to form the array below called the Routh array where the first two rows are the coef cients of the polyno mial in 64 S quotn M3 an 4 Mwh quot39 5 il It I Ind un5 1M7 s39quot 2 b I 73 I74 SH 3 39l 2 393 394 2 kl In S I I s In The column with the powers of s is included as a convenient accounting method The 1 row is calculated from the two rows directly above it the 39 row from the two rows directly ztbove it and so on The equations for the coefficients of the array are as follows bra quot2 172 1 aquot new a quot394 artl art 3 an arkI an5 65 i an 1 an 3 bl bl b2 1 anwl rt 5 cl no b1 b1 b3 023 and so on Note that the determinant in the expression for the ith coef cient in a row is formed from the first column and the i 1 column of the two preceding rows


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