Automatic control teaching can be fragmented, comprising

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1 LECURE NOES Frm Classical t State-Feedback-Based Cntrllers By Juan I Yuz and Mari E Salgad Autmatic cntrl teaching can be fragmented, cmprising different streams that seem t be discnnec t ed r even cmpeting he appeal f new theries and cncepts may encurage students t frget that the key issue in the discipline is the cntrl design prblem If tw r mre different appraches prvide a gd slutin t that prblem, then a strng cnnectin shuld exist between them If such cnnectins can be established, this shuld help students better understand the underlying cncepts in the cntrl design prblem In this artie, we cnsider cntinuus-time systems and explre the cnnectins between single-input, single-utput (SISO) linear assical cntrllers (ie, thse defined by transfer functins) and the cntrl design apprach based n feedback f an estimated system state he aim f this artie is t prvide insight fr autmatic cntrl teaching and cntrl system design he transitin frm estimated state feedback t assical cntrl is well knwn; hwever, t the authrs knwledge, the reverse transitin has nt been previusly articulated fr the general case As is well knwn, the cmbinatin f an bserver tgether with feedback f the bserved state can be reduced t an equivalent assical cntrl lp in which the cntrller is expressed in transfer functin frm [1] Fr instance, assume that the state-space plant mdel is given by the fur-tuple (A, B, C, 0), a full-rder state bserver is built with gain J, and the bserved state is fed back with gain K Hence, the equivalent assical cntrller has a transfer functin C( s)and an assciated state-space mdel given by the fur-tuple (A c, B c, C c, D c ), where 1 C( s) K( si A + B K+ JC ) J, (1) A A B K JC, B J, C K, D c 0 () c c c his equivalent assical cntrller in transfer functin frm has three distinctive features: Figure 1 Classical cntrl lp 1) he cntrller is strictly prper since a full-rder bserver is strictly prper (leading t D c 0) ) he cmplexity f the equivalent cntrller is tied t the cmplexity f the plant since the rder f the bserver is equal t the rder f the plant mdel 3) here is n guarantee that the equivalent cntrller has the necessary features t yield zer steady-state errr in the presence f certain types f disturbances (eg, cnstant, sinusid) he last prblem is the simplest t address; we will summarize the main results f hw that prblem is slved his step will be instrumental in tackling the first tw issues, which pse mre serius bstaes t understanding the cnnectin t linear assical cntrllers hse difficulties are highlighted by the fllwing bservatins: he mst cmmn cntrllers, such as thse belnging t the prprtinal-integral-derivative (PID) family, are biprper Sme cntrllers have a much simpler structure than the mdel f the plant under cntrl Sme cntrllers are mre cmplex than the mdel f the plant t be cntrlled We shw hw these issues can be reslved s that every linear assical cntrller can be assciated with a cntrller based n feedback f an bserved state In this endeavr, the key tl will be the use f reduced-rder bservers [] In additin, the technical link between bth asses f cntrllers will be established thrugh the sed-lp characteristic plynmial Althugh we will nly refer t linear, time-invariant, cntinuus-time, SISO systems, the reader will be able t appreciate that the extensin t the discrete-time case is straightfrward We illustrate this latter pint in the examples In summary, the main questin addressed in this artie is: Given a biprper r strictly prper cntrller, what is an equivalent bserver-state feedback scheme? Classical Cntrl Lp he basic assical cntrl lp f interest here is shwn in Figure 1 he plant is described by its nminal strictly prper transfer functin, G ( s) Further, we assume that the system has n time delay his is dne t keep the mdel f finite degree Of curse, plants with time delays can be easily dealt with in the discrete-time dmain he transfer functin can then be written as /03/$ IEEE 58 IEEE Cntrl Systems Magazine August 003

2 B ( s) G( s), A ( s) where B ( s )and A s ( )are plynmials in s, with A ( s)being a mnic plynmial f finite rder n Similarly, the cntrller is represented by the prper (nt necessarily strictly prper) transfer functin P( s) C( s), Ls ( ) where P( s)and Ls ( )are plynmials in s, with Ls ( )being a mnic plynmial f rder n c Under these cnditins, and regardless f the cntrl design methd being used, the sed-lp characteristic plynmial A ( s)is given by (3) (4) A ( s) A ( s) L( s) + B ( s) P( s) (5) If ple assignment is used t synthesize the cntrller, the first step is t specify A ( s ) fr G s ( ), fllwed by (5) slved fr P( s)and Ls ( ) his synthesis prcedure has the fllwing relevant features [1]: 1) arbitrarily specify A ( s), it must be f degree greater than r at least equal tn 1 When the degree takes the minimum value, a biprper cntrller f rder n 1 is btained ) If A ( s) is arbitrarily chsen with degree equal t n 1+ n d (n d + ), then the cntrller is strictly prper and has degree n 1 + n d 3) If the slutin Ls ( )must have a prespecified factr f degree n x, and A ( s)is arbitrarily chsen with degree equal tn 1+ n x, then the cntrller is biprper and f degree n 1 + n x 4) It might happen that fr a specific chice f A ( s), with degree n, the plynmials P( s)and Ls ( )have a cmmn factr W( s)(stable) f degree n w, which is als a factr f the chsen A ( s) In this case, the cntrller has degree n n nw If we use any ther design methd, such as rt lcus, lp shaping, r lead-lag cntrl, we always btain a cntrller transfer functin with structure resulting frm ne (r a cmbinatin) f the fur cases described abve Cnsider, fr instance, a plant with transfer functing ( s) having three ples (ie, n 3) Further assume that we want t design a PI cntrller, that is, a cntrller where Ls ( )has a prespecified first-rder factr [ie, n x 1 (case 3)] he reader can then verify that this crrespnds t a ple placement synthesis where A ( s) is a plynmial f degree n 1+ n x 6, leading t a biprper cntrller with Ls ( ) f degree 3, and where Ls ( )and P( s)(and A ( s)) share a cmmn stable factr W( s)f degree [ie, n w (case 4)] hus, any assical cntrller with any degree, either biprper r strictly prper, can be thught f as the utcme f a synthesis prcedure based n the specificatin f sed-lp ples his result will be used t frm a link with cntrller synthesis based n feedback f the estimated state he key link will be via the sed-lp characteristic plynmial Observers We will next review full and reduced-rder bservers We cnsider a cmpletely bservable system having state-space mdel x&( t) Ax( t) + B u( t), (6) y( t) Cx ( t), (7) where A m m, B m, and C m Figure shws a blck diagram representatin f this state-space mdel f the plant We intend t use the infrmatin prvided by the system state x( t ) t cntrl the utput y( t) Hwever, we knw the assumptin that all state signals are available is ften unrealistic due t physical r ecnmical limitatins [1], [3] Figure Blck diagram f state-space mdel, plant G ( 0 s ) Figure 3 Blck diagram fr a full-rder bserver August 003 IEEE Cntrl Systems Magazine 59

3 his prblem is typically addressed by estimating x( t ) using an bserver, which generates the state estimate x$( t )[] Full-Order Observers A full-rder bserver has the general frm crrectin term $ & x( t) Ax$( t) + Bu( t) + J( y( t) Cx$( t)), 13 where J m m is knwn as the bserver gain and x$( t ) is the estimated r bserved state We nte that the system defined in (8) is a cmbinatin f the system mdel (6) and a crrectin term, which is necessary due t the mismatch in the initial state, leading t an utput mismatch, since the system initial state is unknwn his system is knwn as a full-rder bserver because it estimates the mcmpnents f the state x( t ) Figure 3 shws the full-rder bserver in (8), with inputs u( t)and y( t)and utput x$( t ) he bserver gain J plays a fundamental rle in determining the speed at which the estimated state cnverges t the true system state Given the state-space mdel defined in (6)-(7) and the full-rder bserver (8), the dynamics f the estimatin errr x ~ ( t) x( t) x$( t) are described by $ y ( t ) (8) x ~& ( t) ( A JC) x ~ ( t) (9) Since the system is assumed t be cmpletely bservable, the eigenvalues f A JC can be placed anywhere in the cmplex plane by chsing a suitable J Equatin (8) can als be expressed as Reduced-Order Observers If we examine (7), we nte that there is infrmatin regarding the state x( t ) cntained in the (measured) utput, y( t) Furthermre, since the state-space mdel f a system is nt unique and, assuming cmplete bservability, there is always a state-space descriptin such that (6)-(7) can be written as [4] a A1 x&( ) ( ) ( ) A A x b t t + 11 u t 1 1 B, (1) y( t) [ 1 0 K 0] x ( t), (13) m where a11, b1 ; A1, A1, B, and A ( m ) ( m ) his descriptin can be btained using, fr example, either the bserver r the bservability cannical frms [5] he reader can nw appreciate that x1 ( t), the first cmpnent f the state x( t ),isy( t), the system utput herefre, nly the remaining m 1 state variables, gruped in m x r ( t ) 1, need t be estimated We nw fllw standard arguments utlined, fr example, in [6] By defining q( t) xr( t) Jy( t), it is shwn in [6] that the system 0 q&( t) [ J Im 1] A q( t) I m [ J Im ] A 1 y( t) J + J I [ m 1] Bu( t) (14) x(t) $ & ( A JC)$( x t ) + Bu( t ) + Jy ( t ), (10) and the bserver eigenvalues are the rts f the plynmial x$( t) 0 q( t) y( t) I + 1 m J 1 (15) E( s) det( sim A + JC) (11) is a reduced-rder bserver, where J m 1 is the bserver gain matrix his bserver is shwn as a blck diagram in Figure 4 Given the state-space mdel in (6)-(7) and the reduced-rder bserver (14)-(15), the dynamics f the estimatin errr x ~ ( t) x ( t) x$ ( t) are described by r r r ~& 0 x ( ) ~ [ J I ] A x ( ) ( A JA ) x ~ r t m 1 r t 1 r ( t ) I, m-1 (16) where matrix A is partitined as in (1) We can als see that the eigenvalues f the reduced-rder bserver (14)-(15) are the rts f the plynmial Figure 4 Blck diagram fr a reduced-rder bserver E( s) det( sim-1 A + JA1 ) (17) 60 IEEE Cntrl Systems Magazine August 003

4 It can be shwn that cmplete bservability f ( A, C) guarantees cmplete bservability fr ( A, A1 ) (see the Appendix) hus, the bserver eigenvalues can be placed anywhere n the cmplex plane by chsing the gain J apprpriately In particular, the reduced-rder bserver 1) estimates nly m 1 state variables ) has a direct link between its input y( t)and its utput x$( t )(ie, is nt strictly prper) Althugh the full-rder bserver is preferred fr applicatins (see, eg, [7]) and is less sensitive t nise in y( t), the secnd f the prperties abve is fundamental in understanding the link between biprper cntrllers and bserved-state feedback his aspect will be explained in the fllwing sectins Observed State Feedback We nw cnsider the cntrl f the plant utput, y( t), using the estimated state prvided by a state bserver he simplest way t d this (and the ptimal way under certain cnditins [1]) is t use a linear cntrl law, where the plant inputu( t)is a linear cmbinatin f estimated states in x$( t ), thrugh a gain matrix K Figure 5 shws the crrespnding feedback cntrl architecture, where r( t)is an external signal related t the reference r( t)in Figure 1 in a way t be arified sn he feedback law is then defined by Figure 5 Observed-state feedback F( s) det( sim A + BK) inudes the eigenvalues f the uncntrllable subspace Equivalence We next turn t an interpretatin f the bserver-state feedback strategy frm the assical cntrl standpint As already nted in the Intrductin, when a full-rder bserver is used, the equivalent cntrller is described by 1 C( s) K( si A + JC + BK) J (1) m Alternatively, the cntrller can be described in state-space frm: x& ( t) ( A JC BK) x ( t) + J( r( t) y( t)), () c c u( t) r( t) Kx $( t) (18) When the state estimate is btained using a full-rder bserver, the sed-lp ples, which determine the cntrl system behavir, are the m zers f the plynmial A( s) det( sim A + BK) det( sim A + JC) (19) F ( s) E ( s) We see that the separatin principle [1] hlds since the sed-lp ple set is independently determined by the feedback gain K and the bserver gain J Using (19), we can chse K and J t assign the sed-lp eigenvalues at desired lcatins in the cmplex plane (see [7] and [8]) On the ther hand, if we use the cntrl scheme shwn in Figure 5 with a reduced-rder bserver, it can be prved [4], [9] that the sed-lp ples are the m 1zers f A( s) det( sim A + BK) det( sim-1 A + JA ) (0) F ( s) E ( s) We again see that the separatin principle hlds fr the set f sed-lp ples since the reduced-rder bserver gain J and the feedback gain K independently determine the eigenvalues f the sed-lp system Nte that the system in Figure 5 may nt be cmpletely cntrllable In that case, the set f sed-lp ples due t u( t) Kx c ( t) (3) Nte that fr the full-rder bserver case: he cntrller has m eigenvalues, which are the m rts f det( si A + JC + BK) 0 (4) m Zer steady-state errr is nt guaranteed when the reference signal is cnstant his fllws, since the cntrller des nt generally have ples at s 0 he cntrller is strictly prper, since there is n direct link between the input and the utput he equivalence requires that r( t)in Figure 5 and r( t)in Figure 1 be related by Rs ( ) ( s + ) 1 K Im A JC J Rs ( ), (5) Fr ( s) where Fr ( s)is a stable reference prefilter On the ther hand, if a reduced-rder bserver is used, the equivalent cntrller takes the frm 1 C( s) C ( si A ) B + D, (6) c m 1 c c c August 003 IEEE Cntrl Systems Magazine 61

5 where 0 Ac [ J Im-1] ( A BK) I, m-1 (7) Bc [ J Im-1] ( A BK) 1 J, C c (8) K 0 I, m-1 (9) D c 1 K J (30) We nte that fr the reduced-rder bserver case: he cntrller has m 1 eigenvalues, given by the equatin 0 det sim-1 [ J Im-1] ( A BK) I m-1 det( sim-1 A + B K + JA1 b 1 JK ) 0 (31) where A and B are partitined as in (1) and the feedback gain matrix is K [ k 1 m K ], k 1, and K 1 he cntrller des nt necessarily have ples at the rigin, s zer steady-state errr is nt guaranteed when the reference signal (r any disturbance) is cnstant A similar situatin applies fr sinusidal references and disturbances with respect t (nnzer) cntrller ples n the imaginary axis In general, this cntrller is biprper because the state-space mdel (6) has a direct link between input and utput thrugh the matrix D c Hwever, this cntrller can als becme strictly prper when K and [ 1 J ] are rthgnal When the system under state estimate feedback cntrl, as in Figure 5, is the same as the plant in Figure 1, then m n and the equivalent cntrller, with reduced-rder bserver, will have n 1 ples he distinctin between nand mis necessary, as shwn in the next sectin he expressin fr the reference prefilter, analgus t (5), is derived in [4] he Internal Mdel Principle Disturbances are always present in a real cntrl system One way t reduce their deleterius effect n the cntrlled variable [the plant utput y( t)] is t use bservers his can be achieved by augmenting the state-space mdel t inude a mdel fr the disturbance Belw we present a brief sketch f hw that is usually dne We will cnsider the case shwn in Figure 6, where dt ( )is a disturbance present n the plant input We assume that the disturbances belng t knwn asses f signals Fr example, we can mdel a disturbance as a cnstant r as a sinusidal signal f specific frequency but unknwn amplitude and phase In these cases, we can build an uncntrllable state-space mdel withut input and whse utput is the disturbance x& ( t) A x ( t), (3) d d d dt ( ) C d x d ( t), (33) where A d n d n d n and C d d Next cnsider the nminal plant mdel x& ( t) A x ( t) + B u ( t), (34) p y( t) C x ( t), (35) where A n n n and B, C he state-space descriptin (34)-(35) can be then cmbined with (3)-(33) [10] t build a cmpsite mdel fr the system in Figure 6, where the disturbance crrespnds t a linear cmbinatin f uncntrllable states f the augmented system his yields x& ( t ) A B C x& d( t ) 0 Ad d x ( t ) x d( t ) + B 0 u( t), (36) y( t) C 0 x x [ ] d ( t ) ( t ) (37) Figure 6 Plant with input disturbance Nte that this cmpsite system is f rder m n+ n d A state bserver fr this system will prvide estimates fr the plant state and fr the state f the disturbance mdel hus, 6 IEEE Cntrl Systems Magazine August 003

6 we can cmpensate the disturbance effect n the plant input by selecting a suitable frm f the feedback gain matrix [ d] K K C (38) herefre, the cntrl signal is cnstructed as leading t the plant input u( t) r( t) Kx$ ( t) $ Cx d d( t) , (39) d $ ( t ) up( t) r( t) Kx$ ( t) ( d( t) $ + Cx d d ( t)) (40) S far, the abve results apply t full-rder as well as reduced-rder bservers Naturally, a different cntrller will arise in each case If a full-rder bserver is used, the equivalent cntrller ples are the m n+ n d slutins f the equatin det( si A + J C + B K ) det( si A ) 0 (41) n nd d On the ther hand, if a reduced-rder bserver is used, the cntrller ples are the m 1 n 1+ n d rts f det( sin-1 + J( A1 b1k ) A + BK) det( sin Ad) 0 d (4) his decmpsitin is cmmnly called the internal mdel principle [11] We see that t fully cmpensate any disturbance in a cntrl lp, its eigenvalues must be inuded as cntrller ples In summary, we see that if input disturbances are mdeled as nncntrllable states, and bserver-state feedback is used in an apprpriate way, then the disturbance mdel eigenvalues will appear as ples f the cntrller Fr example, if we assume a cnstant (r a step) input disturbance, the equivalent cntrller fr the bserver-state feedback scheme will be Figure 7 Unit step respnses fr Example 1 a cntrller having integral actin (ie, with a ple at s 0) he abve results are valuable per se We have summarized a useful strategy t cmpensate disturbances and t track references in a cntrl lp Hwever, it als sets up the necessary framewrk fr the main result f this artie his mechanism explains why the cntrller might have a number f ples that exceeds the rder f the plant mdel In ther wrds, this strategy will Figure 8 Unit step respnses fr Example allw us t interpret a assical cntrl lp in terms f bserver-state feedback, even when the cntrller is mre cmplex than the plant mdel Revealing the Equivalence Next we prceed t ur cre questin: Given a biprper r strictly prper cntrller, what is the equivalent bserverstate feedback scheme? Our subsequent develpment depends n the fllwing facts: F1) he plant mdel has a transfer functin (3) and a cmpletely cntrllable and bservable state-space descriptin f rder n, having the frm (34)-(35) F) he cntrller has a transfer functin as in (4) (pssibly biprper) f rder n c F3) As a cnsequence f F1 and F, all transfer functins in the assical sed lp have a ttal f n+ n c ples, given by the zers f the plynmial A ( s)defined in (5) We will cver all cases regarding the rder f the cntrller and its relative degree (prperness); hwever, as we have already seen, if a reduced-rder bserver is used, then the equivalent cntrller can be either biprper r strictly prper hus, we will be able t elucidate the equivalences nly when reduced-rder bservers are used herefre, the cases t deal with will be assified accrding t the rder f the cntrller with respect t the plant rder hree cases will be distinguished August 003 IEEE Cntrl Systems Magazine 63

7 he Prttype Case: n c n 1 In this situatin, the plynmials E( s) and F( s) can be directly cmputed by factring A ( s)defined in (5) Once we have E( s)and F( s), we can cmpute J and K frm (0) Hwever, we recgnize that there may be mre than ne state estimate feedback equivalent scheme, since there might be different ways t assign the sed-lp ples t E( s) and F( s) in (0) Whenever pssible, the bserver ples will be assigned in such a way that the estimatin errr will cnverge faster than the state feedback mdes Example 1 Cnsider a plant f rder n 3 having transfer functin G ( s s ) + 1 ( s + 1) 3, cntrlled by a strictly prper integrating cntrller (n c ) 09 ( s + 1) C( s) s( 011 s+ 1 ) In Figure 7 we shw the sed-lp respnse t a unit step reference and the plant pen-lp unit step respnse he sed-lp characteristic plynmial is cmputed using (5) his yields A( s) ( s+ 1)( s )( s )( s s+ 064 ) his is a prttype case since n c n 1 hus, the plynmials E( s) and F( s) can be evaluated immediately he E-plynmial is f rder and the F-plynmial is f rder 3 he fastest ples (at s 1and at s 9315 ) are assigned t E( s), and the remaining ples, t F( s) hus E( s) s s det( si A + JA1), F( s) ( s s+ 064 )( s ) det( si A + BK) cmpute J and K we use the state-space descriptin (bserver frm [5]) A A 3 0 1, B B, C C he abve state descriptin is used t cmpute J and K (use the MALAB cmmand place) [ ], [ ] J K 3 If ne uses (7)-(30) and the gains J and K btained, the riginal cntrller is btained, as expected he Lw-Cmplexity Cntrller Case: n < c n 1 he gap between this case and the prttype case can be bridged by assuming that n n c 1 stable cancellatins arise between ples and zers in the cntrller hus, t build the equivalence, we define the equivalent sed-lp characteristic plynmial, A ~ ( s), as ~ A ( s) A ( s) W( s), (43) where A ( s)is defined in (5) andw( s)is a stable plynmial f rder n n c 1 Since the chice f W( s)is arbitrary, there will be an infinite number f equivalent bserver-state feedback schemes Example Cnsider a plant f rder n 3 having the transfer functin G ( s s ) + 1 s + 5s + 7 s+ 4 cntrlled by PI cntrller (n c 1) 3, 14 s + C( s) s Figure 8 shws the sed-lp respnse t a unit step reference and the plant pen-lp unit step respnse he sed-lp plynmial can next be cmputed using (5) t btain 4 3 A( s) s + 5s + 86 s + 58 s+ Furthermre, since n c 1 and n 3, then n < c n 1 We thus have t add (n 1 n c ) stable cancellatins in the cntrller this end, the cntrller is assumed t have a ple (and a zer) at s 5; that is, a factr W( s) s+5 is appended t the equivalent sed-lp plynmial leading t ~ A ( s) W( s) A ( s) s 5 s 4 s s s + 10, having rts at { ± j0 4978, 0615 ± j 0 543, 5} Als, since the plant has three states, we require that F( s) shuld be f rder 3 and E( s)shuld be f rder We assign tw fast ples t the bserver plynmial E( s) (at s 0615 ± j0 543) and the remaining three t the feedback plynmial F( s) his yields E( s) s s det( si A + JA1), F( s) ( s s )( s+ 5) det( si A + BK) 3 64 IEEE Cntrl Systems Magazine August 003

8 cmpute J and K we use a state-space descriptin (bserver frm [5]) A A 7 0 1, B B 1, C C he abve state descriptin is used t cmpute J and K [ ], [ ] J K Again, if we use these gains J and K in (7)-(30), the riginal assical cntrller is btained he High-Cmplexity Cntrller Case: n > c n 1 When n > c n 1, the equivalence can be established if nd nc n+1cntrller ples are assumed t riginate frm the internal mdel principle his implies that these ples frm a fictitius disturbance mdel leading t an augmented plant f rder m n+ nd nc +1 Since the crrespnding eigenvalues crrespnd t uncntrllable but bservable states, they nly yield an increase in the degree f E( s)with- ut affecting the degree f F( s) Example 3 Cnsider a plant f rder n G ( s ) 10 s + 4s + 13 cntrlled by a PID cntrller (n c ) s + 5s+ 0 C( s) s( 0 003s+ 1 ) Figure 9 shws the sed-lp respnse t a unit step reference and the plant pen-lp unit step respnse he sed-lp characteristic plynmial is cmputed using (5) his yields A( s) ( s+ 33)( s )( s s+ 0 87) In this case, n ( c n 1 ) 1 We then use the internal mdel principle t augment the plant by n c n + 1 1uncntrllable states As already discussed, the ple assciated with this state must be a cntrller ple Hence, the degree f E( s)is set equal t tw, which crrespnds t m 1, where m is the number f states in the augmented plant On the ther hand, the degree f F( s) is tw, which crrespnds t the number f plant states (they are cntrllable states by assumptin) In this case, we will assciate the disturbance with a ple at the rigin; that is, we use the mdel (3)-(33) with A d 0 and C d 1 We assign the tw fastest ples (at s 33 and at s 9887 ) t the bserver plynmial E( s) and the remaining ples t the feedback plynmial F( s) E( s) ( s+ 33)( s ) det( si A + JA1), F( s) s s det( si A + B K ) cmpute J and K we use the state-space descriptin (bserver frm) fr the augmented system, as in (36)-(37) A , B 10, C he abve state-space descriptin can be used t cmpute J and K J K [ ] [ ], he uncntrllable ple at s 0 cannt be shifted by feedback and will be preserved using a feedback gain K f the fllwing frm: K [ ] If ne uses (7)-(30) and the gains J and K btained, the riginal assical cntrller is btained Figure 9 Unit step respnses fr Example 3 Discrete-ime Case In the previus sectins, we cvered nly cntinuus-time systems, but the August 003 IEEE Cntrl Systems Magazine 65

9 prir results apply directly t the discrete-time case We present an example t illustrate this Example 4 Cnsider the third-rder discrete-time plant G [] z + 05 z z( z 09 )( z 08 ) cntrlled by a discrete-time PI cntrller (n c 1): 0001 ( z 1) Cz [] z 1 he sed-lp plynmial can be cmputed using (5) t btain 4 3 A[ z] z 7z + 4 z 0 7z his is a lw-cmplexity cntrller since n c 1 and n 3 (ie, n < c n 1) herefre, we must add n 1 n c stable cancellatins in the cntrller this end, the cntrller is assumed t have a ple (and a zer) at z 05 ; that is, a factr Wz [] z 05 is appended t the equivalent sed-lp plynmial leading t ~ A [] z A [] z W [] z ( z )( z ) ( z 19 31z )( z 0 5) having zers at , ± j 0 079, , 0 5 Als, since the plant has three states, Fz []shuld be f rder 3 and Ez []f rder We assign tw fast ples t the bserver plynmial Ez [] (at z and at z 05 ) and the remaining three t the feedback plynmial Fz []: Ez [] z z det( zi A + JA1), 3 Fz [] z 7011z + 44 z det( si A + BK) cmpute J and K we use the state-space descriptin (bserver frm) A A , B B 1, C C he abve state descriptin can be used t cmpute J and K: [ ], [ ] J K Using these gains J and K in (7)-(30), the riginal assical cntrller is btained 3 Cnusins We have shwn that the use f reduced-rder bservers allws any SISO assical linear cntrller t be translated int a cntrl architecture based n feedback f an estimated system state his result allws shws that assical cntrllers add dynamics t the cntrl lp that riginate frm a plant state bserver plus pssible applicatin f the internal mdel principle It is als interesting t nte that an bserver-state feedback cntrl law leads t a unique equivalent assical cntrller (except fr a reference feedfrward blck) Hwever, the cnverse is nt true in general: given a assical cntrl lp and a fixed plant state-space descriptin, there might be many (indeed, an infinite number f) equivalent bserver-state feedback cntrl systems his applies particularly when we deal with lw-cmplexity cntrllers such as PID cntrllers In additin, the assical lp perfrmance regarding disturbance cmpensatin, nise attenuatin, and rbustness is cmpletely determined by the (assical) cntrller transfer functin herefre, there might exist an infinite number f estimated state-based cntrllers that yield the same sed-lp perfrmance his leads t an interesting related questin: Given that the equivalent bserver-based cntrller is nt unique, hw can sensible perfrmance specificatins be intrduced in the design f this cntrl law? Perhaps numerical r ther issues may be relevant in this regard his artie des nt advcate that assical cntrl nly be taught using the state-space synthesis methdlgy, but it shws that, at a mre advanced level, bth appraches can and shuld be brught tgether fr the benefit f a deeper understanding f the fundamental issues in cntrl system design he extensin f these ideas t the SISO discrete-time case is straightfrward In particular, sampled-data mdels allw ne t deal with systems with pure time delays Hwever, as investigated in [4], the extensin t multiple-input, multiple-utput (MIMO) systems (either cntinuus-time r discrete-time) is nt trivial and invlves mre cmplex issues Acknwledgments he authrs gratefully acknwledge Prf GC Gdwin f the Schl f Electrical Engineering and Cmputer Science, the University f Newcastle, fr his valuable cmments and suggestins References [1] GC Gdwin, S Graebe, and ME Salgad, Cntrl System Design Englewd Cliffs, NJ: Prentice-Hall, 001 [] D Luenberger, An intrductin t bservers, IEEE rans Autmat Cntr, vl 16, n 6, pp , June 1971 [3] K Ogata, State Space Analysis f Cntrl Systems Englewd Cliffs, NJ: Prentice-Hall, IEEE Cntrl Systems Magazine August 003

10 [4] JI Yuz, Equivalencia entre cntrladres ásics y cntrl del estad bservad, Master s thesis, Departament de Electrónica, UFSM, Valparais, Chile, Dec 001 [5] Kailath, Linear Systems Englewd Cliffs, NJ: Prentice-Hall, 1980 [6] H Kwakernaak and R Sivan, Linear Optimal Cntrl Systems New Yrk: Wiley-Interscience,197 [7] G Duan, S hmpsn, and G Liu, Separatin principle fr rbust ple assignment-an advantage f full-rder state bservers, in Prc 38th IEEE Cnf Decisin and Cntrl, 1999, vl 41, pp [8] J Kautsky, NK Nichls, and P van Dren, Rbust ple assignment in linear state feedback, Int J Cntr, vl 41, pp , 1985 [9] GF Franklin, JD Pwell, and A Emami-Naeini, Feedback Cntrl f Dynamic Systems Reading, MA: Addisn-Wesley, 1991 [10] K Zhu, J Dyle, and K Glver, Rbust and Optimal Cntrl Englewd Cliffs, NJ: Prentice-Hall, 1996 [11] BA Francis and W Wnham, he internal mdel principle f cntrl thery, Autmatica, vl 1, pp , 1976 Juan I Yuz received his BEng and MSc degrees in electrnics engineering frm the Universidad écnica Federic Santa María, Valparaís, Chile, in 001 He is currently a PhD student in electrical engineering at the University f Newcastle, Australia His research areas inude perfrmance limitatins, cntrl with cnstraints, and system identificatin Ax λ x, (44) Cx 0 (45) ii) he system is nt cmpletely cntrllable if and nly if there exists a nnzer vectr x n and a scalar λ such that x A λ x, (46) x B 0 (47) We will use the PBH test t prve that the cmplete bservability f ( AC, ) implies the cmplete bservability fr ( A, A ) 1 Lemma If the pair ( AC, ) is cmpletely bservable, then the pair ( A, A1 ) is cmpletely bservable t, where the matrices A and A 1 are defined in (1) Mari E Salgad received his prfessinal title f Ingenier Civil Electrónic frm the Universidad écnica Federic Santa María, Valparaís, Chile, in 1974 He received the MSc degree frm Imperial Cllege, Lndn, in 1979 and the PhD degree in electrical engineering frm the University f Newcastle, Australia, in 1989 He is currently an academic with the Department f Electrnic Engineering, Universidad écnica Federic Santa María His research areas inude cntrl system design and system identificatin He can be reached at the Department f Electrnic Engineering, Universidad écnica Federic Santa María (UFSM), Casilla 110-V, Valparaís, Chile, marisalgad@elutfsm Appendix Cmplete Observability One way t test cmplete cntrllability and bservability f a system is prvided by the fllwing lemma knwn as the Ppv-Belevitch-Hautus (PBH) test, which we recall withut prf (fr a cmplete prf see [5]) Lemma 1 Cnsider a state-space mdel given by the matrices ( ABC,, ) hen: i) he system is nt cmpletely bservable if and nly if there exists a nnzer vectr x n and a scalar λ such that Prf We use cntradictin We first assume that the pair ( AC, ) is cmpletely bservable and the pair ( A, A1 ) is nt cmpletely bservable hen the PBH test implies that there is a nnzer vectr x n 1 and a scalar λ such that Ax λ x (48) A x 1 0 (49) If we next cnsider the vectr x [, 0 x ], we have that A1 A1x Ax a A1 A x 11 0 Ax 0 x x λ (50) λ And if we write the matrix C as in (13), we have that Cx K x 0 [ ] (51) he results given in (50) and (51) shw that there is a vectr x n and a scalar λ satisfying (44), (45) s the pair ( AC, ) cannt be cmpletely bservable his cntradicts the initial assumptin hus, ( A, A1 ) must be cmpletely bservable August 003 IEEE Cntrl Systems Magazine 67