function whee is an unknown constant epesents fo the un-modeled dynamics The pape investigates the position contol of electical moto dives that can be configued as stuctue of Fig 1 This poblem is fomulated as follows Conside the basic stuctue of a moto dive system assume that 1 The fequency esponse fom voltage u to oto position θ is available 2 is an unknown constant Unde these assumptions the contol objective is to calculate the family of PID contolles that stabilize the uncetain plants whee satisfies -nom consideation on complementay sensitivity function in the pesence of un-modeled dynamics u W(s)Δ Mechanical Pat T e Electical Pat θ whee ae polynomials with eal coefficients of degees m n, espectively Assume that have no zeo on axis Let detemine the numbe of open ight halp plane (RHP) (open left half plane (LHP)) zeos poles of Also let denotes the net change in phase of as uns fom to Then (1) Define the (Huwitz) signatue of as (2) since has no pole zeo on axis, it can be witten (3) The value of can be calculated fom the fequency data of can be calculated fom (3) Let whee denote the eal imaginay pats of, espectively Assume that the eal, distinct, finite zeos of denote as such that Fig 1 Moto dive system with un-modeled dynamics + _ e PID u P(s,q) Fig 2 Feedback stuctue with uncetain plant PID Fig 2 shows the feedback system with PID contolle in which is an uncetain plant can be substituted by In the next section, a suvey on the algoithm poposed fo nominal stability in [11] its genealization to uncetain plants is pesented it is illustated that the contol objectives of this pape on moto dive system can be satisfied in a same way III PID CONTROLLER DESIGN: MODEL FREE APPROCH θ conside the modified PID contolle as whee, ae the popotional, integal deivative coefficients of PID contolle T is a positive constant Lemma 1: Let Then the closed loop stability is equal to Let whee In this section, the poblem of achieving the family of stabilizing PID contolles fo nominal stable plant poposed in [11] is eviewed Then by veifying a theoem, this appoach is genealized to plants with an uncetain paamete Simila appoach is poposed fo unstable plants which ae omitted hee because of oom limitation The feedback system with PID contolle is shown in Fig 2 Fist, some mathematical peliminaies ae intoduced Conside a eal ational function http://apcaastedu Conside define whee (4) Theoem 1: Let denote the distinct fequencies of odd multiplicities which ae solutions of Detemine stings of integes whee such that: 45
fo n-m even: [ fo n-m odd: [ Also let (5) (6) Then fo, the values of coesponding to the closed loop stability ae given by (7) whee i t s ae taken fom stings satisfy (5) o (6) s ae taken fom the solutions of (4) Next theoem shows how to calculate the admissible ange of Theoem 2: The necessay condition of existence the stabilizing PID fo LTI plants is that thee exists such that has at least R distinct oots of odd multiplicities such that { (8) The pocedue fo calculation the family of stabilizing PID contolles is summaized in the following algoithm [11] Algoithm 1 1 Detemine the elative degee of plant fom high fequency slope of bode magnitude of 2 Detemine fom (2) 3 Detemine fom (3) 4 Detemine fo fom (4) 5 Apply Theoem 2 to detemine the ange of 6 Fo, solve (4) obtain 7 Let Detemine fom (5) o (6) 8 Fo, detemine the values fom (7) 9 Change go to step 6 to obtain the whole family of stabilizing PID contolles Now the main esult of this pape will be pesented Conside an uncetain eal ational plant whee is an uncetain paamete The contol goal is to calculate the family of obust stabilizing PID contolles fo the uncetain plant The feedback stuctue is shown in Fig 2 The only equied data is the set of fequency esponses including fo In the poposed appoach of Algoithm 1, oots of function in (8), can be obtained fom (4) the ange of is calculated based on the value of Also the ange of admissible values obtain fom (7) which have the slope equal to If by monotonic vaiation of the uncetain paamete q, vaies monotonically, some substantial esults could be concluded Fist of all, the ange of admissible fo stabilizing the uncetain plant is the common ange of admissible fo two plants Second, since the slope of linea inequalities in (7) vaies monotonically with monotonic vaying the uncetain paamete, the inequalities coesponding to have the maximum minimum slopes, not necessaily espectively Finally, since vaiation of the uncetain paamete, does not lead to monotonic vaiation in the the common space of inequality (7) fo is lage than the admissible values An appoach to find the exact ange of admissible paametes is pesented in the following emak Remak 1: A simple appoach to exclude these egions is to choose some test points in those egions analyzing thei stability So monotonic vaiation of the function makes it easy to calculate the family of stabilizing PID contolles fo plants with an uncetain paamete Theoem 3: Let be a eal ational tansfe function is an uncetain paamete If the Numbe of zeos poles the numbe of RHP zeos of ae fixed fo then the family of stabilizing PID contolles fo uncetain plant is a subspace of common space between two set of stabilizing PID contolles fo if one of the below constaints satisfy { (9) { (1) fo only one value of uncetain paamete q fo Poof: If monotonic vaying the uncetain paamete leads to monotonic vaying the function w, then the family of stabilizing PID contolles fo uncetain plant is a subspace of common space between two set of stabilizing PID contolles fo Then http://apcaastedu 46
Without loss of geneality, conside the case of monotonic inceasing in Inceasing the uncetain paamete leads to inceasing in if the following inequality holds: Assumption 1: Let (12) note that the symbol > could be invesed; but fo simplicity it is assumed as > Finally, conside the eal ational symbol of (12) So (11) can be tansfomed to (13) by change of vaiable as: the inequality (13) can be changed to: Assumption 2: Let (14) (15) (16) simila to Assumption 1, the eal ational symbol of (15) (16) will be consideed finally Fom (15) (16), the inequality (14) leads to: since, then that is one of the necessay conditions imply that inceasing the uncetain paamete leads to inceasing in Assumptions 1 2 imply anothe necessay condition as Remak 2: Bode diagams coesponding to could be ecognized fom the fequency esponses set of uncetain plant In fact if (9) o (1) holds, then fom monotonic vaiation of the function, it could be deduced that the uppemost lowemost plots of ae coesponding to, not necessaily espectively The following Coollay shows that the poblem can be solved easie when the uncetain paamete is in the feedfowad path of contol loop Coollay 1 Let P( s, q ) be a eal ational function http://apcaastedu q{ q, q } fo an uncetain LTI plant min max is an uncetain paamete that appeas in the feedfowad path Then the family of stabilizing PID contolles fo uncetain plant is a subset of common set between two set of stabilizing PID paametes fo P( s, qmin ) max P( s, q ) if one of the below sets satisfy ( P P) i : if g( ) is ascendant ( P P) i : if g( ) is decendant (17) fo fo only one value of uncetain paamete q poof: It can be witten P( j, q) qp ( j) qp( j) qp( j) qpi( j) g( ) 2 q ( P ( j) P( j)) Then i i dg 1 P 2 Pi 2 2 dq q P Pi So g( ) is monotonic if (17) satisfies The pocedue fo calculating the family of stabilizing PID contolles is summaized in the below algoithm Algoithm 2 1 Detemine fo fom plots using Remak 2 2 Using Algoithm 1, calculate the family of stabilizing PID contolles fo two plants Detemine the common space of PID paametes between two calculated family The family of stabilizing PID contolles fo uncetain plant is a subspace of this common space 3 Calculate the exact family of stabilizing PID contolles using the appoach poposed in Remak 1 Now it can be shown that the poblem of satisfying some pefomance specifications fo uncetain plant could be tansfomed to poblem of obust stabilizing the uncetain plant with additional vitual uncetain paamete Many pefomance attainment poblems fo uncetain plant can be cast as the simultaneously stabilization of the uncetain plant the family of eal complex plants [11] Fo example an -nom achievment on the complementay sensitivity function, that is, is equivalent to simultaneously stabilizing the uncetain plant as 47
(18) whee is a vitual uncetain paamete is a weight selected by designe Let the family of stabilizing PID paametes fo uncetain plant can be calculated fom Algoithm 1 Let in (18) be a eal ational function If coesponding to (18), vaies monotonically by monotonic vaying the vitual uncetain paamete, fo any any, then the family of stabilizing PID contolles fo uncetain plant that satisfy the -nom specification on the complementay sensitivity function, is a subspace of PID contolles obtained fom simultaneously stabilization of two plants family of obust PID contolles fo plants faced to loss of effectiveness can be calculated by the appoach poposed in the pevious section Fo this special case, the poblem of contolle synthesis could be hled easily using Coollay 1 Example 3: Robustness in plants with a dominant uncetain paamete Some plants that ae faced to paamete vaiations can be appoximated by plants with a dominant uncetain paamete Obviously the poposed appoach can be applied in this case to calculate the family of obust stabilizing PID contolle [ ] (19) [ ] (2) the exact family can be obtained using the appoach poposed in Remak 1 IV A REVIEW ON THE APPLICATIONS OF THE PROPOSED APPROACH In this section, some applications of the poposed method ae pesented In fact, it is illustated that some contol objectives can be cast as obust stabilizing of a plant with an uncetain paamete Fo example: Example 1: Pefomance achievement Many pefomance achievement poblems fo uncetain plant P( s, q) can be cast as the simultaneously stabilization of the uncetain plant the family of eal complex plants Some of these pefomance achievement poblems fo nominal plant ae listed in [11]; fo example, the poblem of H -nom achievement on the complementay sensitivity function is equivalent to simultaneously stabilizing the plant P(s) the family of eal plants C P ( s, q, ) : q{ q, q }, {,2 } min max the exact family can be obtained using some test points The othe specifications such as H - nom achievement on the sensitivity function phase magin can be satisfied by the same appoach Example 2: Robustness against loss of effectiveness in actuato Fig 3 Shows the feedback stuctue of a plant with PID faced to loss of effectiveness in actuato whee L is the paamete coesponding to loss of effectiveness belongs to (,1] Obviously this stuctue can be cast as a plant with an uncetain paamete, ie, P( j, L) with PID whee L q is the uncetain paamete Thee is simila case when loss of effectiveness happens in sensos Thus the Fig 3 The feedback stuctue of plant faced to loss of actuato effectiveness with PID Magnitude (db) Phase (deg) 15 5-5 54 45 36 27 18 9 1-2 1 1 2 Fequency (ad/sec) Fig 4 The fequency esponse of moto dive system V SIMULATION ON INDUCTION MOTOR The model studied in this section is an induction moto dive system intoduced in [9, 12] that has the simila stuctue to Fig 1 in [14] its nominal fequency esponse is shown in Fig 4 The fequency esponse of such systems can be obtained by vitual sine sweeping [13] The coesponding plot of is shown in Fig 5 Since (fom (8)), the admissible ange of is Calculating the values fo T=1 is esulted to the following inequalities { Bode Diagam http://apcaastedu 48
k p :=g() k p :=g() k p -2-4 -6 5 4 3 2 1 2 3 4 5 6 7 8 9 =715 g()=2 1 2 3 4 5 6 7 8 Fig 5 The plot of function g with the line kp=2 Fig 6 The whole family of stabilizing PID contolles fo nominal model of induction moto dive system Theefoe the family of obust PID contolles fo the induction moto dive with uncetainties can be calculated by stabilizing the following two plants { =181 g()=2 The ange of obust stabilizing PID paametes fo induction moto faced to un-modeled dynamics is illustated in Fig 7-a The admissible ange of fo all admissible is shown in Fig 7-b k p 35 3 25 2 15 6 4 2 6-5 5 k d 5 1 5 5 4 4 3 3 2 2 (a) -5 =812 g()=2-2 k d 5 2 k d 15 2 3 4 5 6 Fig 7 (a) Robust stabilizing PID paametes fo induction moto dive faced to un-modeled dynamics; (b) the admissible ange of (,k d) (b) Admissible ange of (,k d ) fo all admissible k p http://apcaastedu 14 12 8 Admissible ange 6 fo pefomance attainment 4 2 Test points: k p =5 Admissible ange fo nominal stability Admissible ange fo unmodeled dynamics -2-1 1 2 3 4 5 6 7 k d Fig 8 The ange of admissible (,k d)values that satisfy diffeent contol objectives fo k p=5 Now conside the poblem of satisfying -nom on the complementay sensitivity function The admissible ange of values that satisfy this pefomance citeion fo is shown in Fig 8 togethe with egions coesponding to nominal stability stability in the pesence of un-modeled dynamics The whole ange of values can be obtained similaly The step esponses of closed loop system with contolles coesponding to test points maked in Fig 8 ae shown in Fig 9 Applying the PID contolle that satisfies pefomance consideation is esulted to deceasing the oveshoot Also, the contol signals coesponding to selected contolles ae plotted in Fig 1 It can be seen that the contol signals dift to zeo afte the easonable times Fo bette tacking of the poposed appoach in this pape, the eades can efe to the many academic examples implemented in [15] VI CONCLUSION In this pape, a obust contol appoach pesented based on a geneal model fo diffeent types of electical moto dives The poblem of obustness against un-modeled dynamics in moto dives is tansfomed to the poblem of stabilizing a plant with an uncetain paamete It is shown that knowing the fequency esponses of moto dive system coesponding to maximum minimum values of uncetainty is sufficient to calculate the family of obust stabilizing PID contolles In fact, thee is no need to plant mathematical model Also it is illustated that the poblem of -nom achievement on the complementay sensitivity function can be solved by the same appoach Though the pape, it is assumed that the fequency esponse of plant is available Such an assumption is often valid in many pactical applications Also, this is an assumption that has aleady been used seveal times in othe papes dealing with contolle synthesis using fequency domain data [5,7,11,13,15-17] 49