16.1.Robust Stabilization of Coprime Factors 303 8 -λ4 -λB det =0 -入B* (、-)D 台 det(N*N A)=0. 时器6大网o are Usngtmt fact we ave te followigoroly. CoroHary 16.fi Let K and Plbe any compatibly dimensioned compler matrices.Then ,28 IPK)-1I P I (+KP)1 I K h i; Proof.Define 28 128 M= +PIp,-0+m-K了， h -P h K T ents easy to ver fy trat M2=M and N =I-M.By Lemma 6.6 we ave kkkwk.Tre corollary follows by not ng t at 28 h 82 8 0-I (I+KP)-1 1 K 生01 Corollary 16.=Let P=M-IN=N M-1 be respectively the normalized left and right coprime factorizations. Then ,2 h1, (I+PK)-1M-1 =M-(I+KP)1 I K Proof. s follows from Corollary 6.7 and tre fact trat h i, ,28 i M-(+KP)1I K h P (I+KP)-1 I K ◇ corollary says trat any Hoo controller for tre normalzed left co r me factor 一T Hence
Robust Stabilization of Coprime Factors det A B B D detNN I This implies that all nonzero eigenvalues of MM and NN that are not equal to are equal ie iM iI M for all i such that iM Using this matrix fact we have the following corollary Corollary Let K and P be any compatibly dimensioned complex matrices Then I K I P K h I P i I P I KP h I K i Proof De ne M I K I P K h I P i N P I I KP h K I i Then it is easy to verify that M M and N I M By Lemma we have kMk kNk The corollary follows by noting that I P I KP h I K i I I N I I Corollary Let P M N NM be respectively the normalized left and right coprime factorizations Then K I I P KM M I KP h I K i Proof This follows from Corollary and the fact that M I KP h I K i I P I KP h I K i This corollary says that any H controller for the normalized left coprime factor ization is also an H controller for the normalized right coprime factorization Hence one can work with either factorization
304 H2 LOOP SHAPING For future reference<we shall dethe ≤2▣ 8, +PKS [I P if K stabilizes P bPK: 0 otherwise and then bPK=bK-P and oyV-A-Naf野 The number bpK can be related to the classical gain and phase margins T eorem-6>Let P be a SISO plant and K be a stabilizing controller.Then Gain Margin 6 1+bPK 1-bPK and Phase Margin 6 2 arcsin-bPKz Proof.Note that bPK≤ 1-PK +P不+☒元 3 So<at frequencies where P-j3K-j3>=k 1R+< 1- bPK≤ 1- 1+IP21+ PT≥ 1+P1+r which implies that 1-6PK k≤1+bPK or K6 1+6PK 1-bPK from which the gain margin result follows.Similarly <at frequencieswhere P-j3Kj3>= e.< 1-e3 2 sin bPK≤ 2in过 y++示V++ 2
H LOOP SHAPING For future reference we shall de ne bPK I K I P K h I P i if K stabilizes P otherwise and bopt sup K bPK then bPK bKP and bopt p maxY Q r h N M i H The number bPK can be related to the classical gain and phase margins Theorem Let P be a SISO plant and K be a stabilizing control ler Then Gain Margin bPK bPK and Phase Margin arcsinbPK Proof Note that bPK j P Kj p jP jp jKj So at frequencies where P jKj k R bPK j kj s jP j k jP j j kj s min P jP j k jP j k k which implies that k bPK bPK or K bPK bPK from which the gain margin result follows Similarly at frequencies where P jKj e j bPK j e j j s jP j jP j j sin j s min P jP j jP j j sin j
)y Loo Sra ing Desgn 305 which implies 06 sarcsin bp.K. 2 For example bp.K =1(og uarantees a gain margin of 3 and a phase marg in of 60. Illustrat ve MATLAB Com mandsA bp.kemarg n-P8K6%given P and K~conpute bp.K. >[Kopt8bp.=ncfsy n-P8 6<%find the optimal controller Kot. [K'sub8bp.ncfsy n-P8 6<%find a suboptimal controller Ku. 6Loos ayng Des.gn V This section considers the Hoo loop shaping design The obective of this approach is to incorporate the simple performance/robustness tradeoff obtained in the loop shaping with the guaranteed stability properties of Ho design methods Recall from Section 6.1 of Chapter 5 that good performance controller design requires that 万(H+PK61)8元(H+PK61P)8元(H+KP61)8元(KH+PK61)16.16 be made small~particularly in some low frequency range And good robustness requires that G(PK4+PK6)8 G(KP+KP6) -16.6 be made small-particularly in some high frequency range.These requirements in turn inply that good controller design boils down to achieving the desired loop -and con2 troller6gains in the appropriate frequency range aPK6>18¤KP6>18gK6>1 in some low frequency range and TPK6X18元KP6入18元K6≤M in some high frequency range where M is not too large. The H loop shaping design procedure is developed by McFarlane and Glover 1990-199 and is stated below. Loop Shaping Design Procedure -16 Loop Shaping:the singular values of the nominal plant are shaped~using a prec2 ompensator Wi and/or a postcompensator W2~to give a desired open2loop shape. The nominal plant P and the shaping functions Wi8W2 are combined to formthe shaped plant~P where P -W2PWi.We assume that Wi and W2 are such that P contains no hidden modes 1
Loop Shaping Design which implies arcsin bPK For example bPK guarantees a gain margin of and a phase margin of o Illustrative MATLAB Commands bpk emarginP K ! given P and K compute bPK Kopt bpk ncfsynP ! nd the optimal controller Kopt Ksub bpk ncfsynP ! nd a suboptimal controller Ksub Loop Shaping Design This section considers the H loop shaping design The ob jective of this approach is to incorporate the simple performance robustness tradeo obtained in the loop shaping with the guaranteed stability properties of H design methods Recall from Section of Chapter that good performance controller design requires that I P K I P KP I KP KI P K be made small particularly in some low frequency range And good robustness requires that P KI P K KP I KP be made small particularly in some high frequency range These requirements in turn imply that good controller design boils down to achieving the desired loop and con troller gains in the appropriate frequency range P K KP K in some low frequency range and P K KP K M in some high frequency range where M is not too large The H loop shaping design procedure is developed by McFarlane and Glover and is stated below Loop Shaping Design Procedure Loop Shaping the singular values of the nominal plant are shaped using a prec ompensator W and or a postcompensator W to give a desired openloop shape The nominal plant P and the shaping functions W W are combined to form the shaped plant Ps where Ps WPW We assume that W and W are such that Ps contains no hidden modes
306 H LOOP SHAPING Figure 16.3:Standard Feedback Configuration (2)Robust Stabilization:a)Calculate ema(i.e.,bopt(P)),where []a+)广 -N,a]<1 and Ms,Ns define the normalized coprime factors of P such that P=M.Ns and M+NN:=1. If emaz<1 return to (1)and adjust Wi and W2 b)Select e<emaz,then synthesize a stabilizing cont roller Koo,which satisfies (3)The final feedback controller K is then constructed by combining the Hoo con- troller Koo with the shaping functions Wi and W2 such that K WiKoo W2. A ty pical design works as follows:the designer inspects the open-loop singular values of the nominal plant,and shapes these by pre-and/or postcompensation until nominal performance (and possibly robust stability)specifications are met.(Recall that the open-loop shape is related to dlosed-loop objectives.)A feedback controller Koo with associated stability margin(for the shaped plant)e<emaz,is then sy nthesized.If emax is small,then the specified loop shape is incompatible with robust stability requirements, and should be adjusted accordingly,then Koo is reevaluated. In the above design procedure we have specified the desired loop shape by W2PWi. But,after Stage (2)of the design procedure,the actual loop shape achieved is in fact
H LOOP SHAPING e e e e n u y di d r K P Figure Standard Feedback Con guration Robust Stabilization a Calculate max ie boptPs where max inf K stabilizing I K I PsKM s r h N s M s i H and M s N s de ne the normalized coprime factors of Ps such that Ps M s N s and M sM s N sN s I If max return to and adjust W and W b Select max then synthesize a stabilizing controller K which satis es I K I PsKM s The nal feedback controller K is then constructed by combining the H con troller K with the shaping functions W and W such that K WKW A typical design works as follows the designer inspects the openloop singular values of the nominal plant and shapes these by pre and or postcompensation until nominal performance and possibly robust stability speci cations are met Recall that the openloop shape is related to closedloop ob jectives A feedback controller K with associated stability margin for the shaped plant max is then synthesized If max is small then the speci ed loop shape is incompatible with robust stability requirements and should be adjusted accordingly then K is reevaluated In the above design procedure we have speci ed the desired loop shape by WPW But after Stage of the design procedure the actual loop shape achieved is in fact
-6.2.Loop Shaping Design 307 P. Figuo 16.4:Th Loop Shaping Dosign Prooduo gin by Wik W2P at plant input and PWik.W2 at plant output.It is thofoo possibo that th inclusion of K in th opn-loop trans6r function will cauo o- rioration in th opon-loop shapo spcifod by P.In th xt ction,wo will show that th ogradation in th loop sharo cauod by th H controlor k is limiod at foqoncos wbo th spcifod loop shar is sufficontly larg or sufficontly small.In particular,show in thoxt oction that =can o inorpood as an indicator of th sucoss of th loop shaping in addition to providing a robust stability guaranto for th dood-loop sysoms.A small val of =az (a1)in Stag (2)always indicats incompatibility lotwon th spocifod loop shap,th nominal plant phao,and robust clood-loop stability. Remark -6.-No that,in contrast to th classical loop shaping approach,th loop shaping bo is do withoutoxplicit ogard for th nominal plant phao information. That is,clood-loop stability oquiononts ao disogard at this stago.Also,in con- trast with conntional H sign,tho robust stabilization is do without foqoncy wighting.Th sign prooduo scrilod bo is both simpb and sysomatic,and only assuns knowodgo ofoonontary loop shaping principos on th part of th osigor. a Remark -6.2 Th abov robust stabilization obocti can also bo inorpood as th moo standard H.probbm formulation of minimizing th H norm of th foqoncy wighd gain from disturbanos on th plant input and output to th controlor input
Loop Shaping Design e e W P W K W P W W K W P K Ps Figure The Loop Shaping Design Procedure given by WKWP at plant input and PWKW at plant output It is therefore possible that the inclusion of K in the openloop transfer function will cause dete rioration in the openloop shape speci ed by Ps In the next section we will show that the degradation in the loop shape caused by the H controller K is limited at frequencies where the speci ed loop shape is suciently large or suciently small In particular we show in the next section that can be interpreted as an indicator of the success of the loop shaping in addition to providing a robust stability guarantee for the closedloop systems A small value of max max in Stage always indicates incompatibility between the speci ed loop shape the nominal plant phase and robust closedloop stability Remark Note that in contrast to the classical loop shaping approach the loop shaping here is done without explicit regard for the nominal plant phase information That is closedloop stability requirements are disregarded at this stage Also in con trast with conventional H design the robust stabilization is done without frequency weighting The design procedure described here is both simple and systematic and only assumes knowledge of elementary loop shaping principles on the part of the designer Remark The above robust stabilization ob jective can also be interpreted as the more standard H problem formulation of minimizing the H norm of the frequency weighted gain from disturbances on the plant input and output to the controller input