A LINEAR PROGRAMMING APPROACH FOR REGIONAL POLE PLACEMENT UNDER POINTWISE CONSTRAINTS

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1 A LINEAR PROGRAMMING APPROACH FOR REGIONAL POLE PLACEMENT UNDER POINTWISE CONSTRAINTS Max M.D. SANTOS, Eugêno B. CASTELAN, Jean-Claude HENNET Abstract Ths paper consders the problem of control of dscrete-tme lnear systems under several complcatng features frequently encountered n practce : uncertan parameters n the model, addtve nose, lnear symmetrcal state and nput constrants. The man control desgn objectve, beyond local stablzaton, s to locate the closed-loop poles n a good regon of the unt dsc of the complex plane. The orgnalty of the proposed approach s to use Lnear Programmng as the basc desgn tool. It s shown that the matrx condtons derved from postve nvarance relatons and from desgn constrants can be formulated as lnear matrx equaltes and scalar nequaltes. The control desgn problem can then be solved by Lnear Programmng, wth an objectve functon descrbng the search for a trade-off between several desgn objectves. Keywords State Feedback, Dsturbance Rejecton, Lnear systems, Invarance, Uncertan systems. 1 Introducton Many practcal control problems are characterzed by several complcatng features n the system model, such as persstent dsturbances, parameters uncertanty, pontwse constrants on nput and output varables. Moreover, the control objectves often go beyond stablzaton or dsturbance attenuaton, seekng for a satsfactory trade-off between performance and robustness. When the system model admts a lnear descrpton n a lmted doman of the state space, most of these features and requrements can be translated nto lnear matrx condtons. Usng the approach by postve nvarance of polyhedral domans, only lnear matrx equaltes and scalar nequaltes are nvolved. The use of Lnear Programmng algorthm s then possble, wth the advantage of a great computatonal effcency. Ths paper descrbes ths approach n the case of a dscrete-tme lnear model wth addtve dsturbances and convex parameters uncertantes, subject to symmetrcal lnear constrants on ts nput vector. The addressed control problem s of the l 1 -type wth addtonal constrants and pole locaton requrements for the closed-loop state matrx. Classcally 13, the effects of the addtve dsturbance on the system output are measured n terms of the l -nduced norm. Here, they wll be represented Laboratóro de Controle e Mcronformátca (LCMI / EEL / UFSC), Floranópols, S.C., Brazl, E-mal eugeno@lcm.ufsc.br Contact Author, Laboratore d Analyse et d Archtecture des Systèmes du C.N.R.S, 7, Avenue du Colonel Roche, Toulouse Cédex, FRANCE, E-mal hennet@laas.fr, Fax (+33)

2 by pontwse symmetrcal constrants on the output vector. Addtonnally, a major concern for the consdered control problem s to locate the closed-loop poles n a partcular regon of the unt dsc of the complex plane. Lnear formulatons of ths problem are provded n the cases of pole regons made of the ntersecton of the unt dsc wth a vertcal strp and of centered rngs n the unt dsc. Notatons. The followng notatons are used throughout the paper. Matrx I n s the dentty t matrx n R n n ; vector 1 n s defned as 1 n = R n. For any matrx M R n n, M denotes { the matrx of the absolute values of ts components: M j = M j. ˆM s defned ˆM = M by :. The nfnty norm of M s classcally defned by: M = ˆM j = M j for j. n m max M j. The nfnty matrx measure of M s defned by : µ (M) = max M + M j. =1,..,n j=1 2 Problem statement 2.1 Lnear control under output and nput constrants Consder the lnear dscrete-tme system represented by the followng tme-nvarant equatons: x k+1 = Ax k + Bu k + Dd k, k N (1) y k = Gx k (2) where A R n n, B R n m, wth rank(b) = m, D R n l, G R g n, rank(g) = n. Vectors x k R n, u k R m d k R l and y k R g are, respectvely, the current state, control, dsturbance and output vectors, at tme k. As n most practcal cases, all the varables of ths model are subject to constrants. In the proposed model, all the constrants are assumed to be lnear and symmetrcal : - the dsturbance vector, d k, s assumed to belong to a closed convex set : j=1 d k S(P, ξ) R l ; k N S(P, ξ) = {d R l ; ξ P d ξ} (3) wth: P R p l, ξ R p, ξ > 0 ; - the control nputs are assumed to be symmetrcally bounded : u k U R m ; k N U = {u R m ; ρ u k ρ}. (4) wth: ρ R m and ρ > 0 ; - the dsturbance rejecton requrement mposes that the states of the system should reman n a compact symmetrcal polyhedron, denoted S(G, γ) : x k S(G, γ) R n ; k N S(G, γ) = {x R n ; γ Gx k γ} (5) wth γ R g, γ > 0. 2

3 A class of canddate solutons of ths problem s constructed wth a lnear state feedback control law u k = F x k (6) such that the dsturbed lnear closed-loop model x k+1 = (A + BF )x k + De k (7) admts a set Ω that s S(P, ξ)-nvarant n the state space and satsfes 14) : { Ω S(G, γ) Ω S(F, ρ) = {x R n ; ρ F x k ρ}. Defnton 1 3 A set Ω R n s S(P, ξ)-nvarant, f for any ntal condton belongng to Ω the correspondng trajectory remans n Ω for any sequence of dsturbances d S(P, ξ). The partcular choce of S(G, γ) as the canddate set Ω corresponds to the followng condtons : H R g g, K R g p, M R m g such that : HG = GA + GBMG (8) KP = GD (9) M γ ρ (10) H γ + K ξ γ (11) If these condtons are satsfed, the closed-loop system trajectory satsfes constrants (4), (5) from any ntal state x 0 S(G, γ) and for any sequence of dsturbances d k S(P, ξ), under the control law (6) wth F = MG. 2.2 Constrants on the closed-loop pole locaton Clearly, S(P, ξ)-nvarance of a set Ω wth respect to system (7) s a suffcent condton for postve nvarance of Ω wth respect to the assocated determnstc system (12) defned by : x k+1 = (A + BF )x k. (12) Then, classcally 1, 7, postve nvarance of a compact doman n the state space havng the zero state n ts nteror s a suffcent condton for local stablty of the constraned determnstc system around the zero state. Ths property mples, n partcular that, under condtons (8), (9), (10), (11), the closed-loop state matrx (A+BF ) has ts poles located n the unt crcle of the complex plane. Ths elementary stablty condton may not be suffcent for the desgner. In many practcal applcatons, t s desrable to locate the closed-loop poles n a partcular regon, Σ, of the unt crcle, and thus to mpose the addtonal spectral constrants : σ(a + BF ) Σ (13) Two types of spectral regons Σ wll be consdered n the sequel. The assocated regonal pole locaton condtons wll be characterzed and ncorporated n the control desgn problem, ether as hard constrants or as desgn objectves. 3

4 2.3 Model uncertantes If some of the model parameters n (1) are uncertan, the system dynamcs can be represented by a famly of models (1): A A ; B B (14) A general representaton of uncertantes (see e.g. 6, 10) s obtaned when assumng that the sets A and B are convex and compact polyhedral domans defned by ther vertces (A R n n, B j R n m ), through the followng relatons : N A A = {A : A = ζ A, ζ 0, ζ = 1} (15) =1 N B B = {B : B = κ j B j, κ j 0, j=1 N A =1 N B j=1 κ j = 1} (16) If na and nb denote the numbers of uncertan entres n A and B, the number of vertces defnng the convex polyhedral doman of uncertantes for the par (A, B) has the order of magntude 2 na+nb. Therefore, t s manly for small values of na + nb that such a representaton of uncertantes may be consdered. It has been shown by B.E.A.Mlan, A.N.Carvalho 10 (see also 11) that due to the convexty and compactness of the doman of uncertantes, a necessary and suffcent condton for relatons (8), (9), (11) to be satsfed by any possble par of matrces (A, B) n the doman of uncertantes s that for any par of vertces (A, B j ) : H j R g g, K R g p such that : H j G = GA + GB j F (17) KP = GD (18) H j γ + K ξ γ (19) It s not dffcult to show a smlar result for the pole placement relaton (13) : σ(a + BF ) Σ for any possble par of matrces (A, B) f and only f for any par of vertces (A, B j ), the followng relaton s verfed : σ(a + B j F ) Σ. (20) 3 Some lnear representatons of regonal pole placement The basc technque whch wll be used for obtanng suffcent pole placement condtons s derved from the two followng stablty theorems due to A.P. Molchanov, V.S. Pyatnsky 12. In the sequel, λ (H) denotes the th egenvalue of a matrx H and λ (H) the correspondng modulus Theorem 1 12 : The lnear tme-nvarant dscrete-tme system x k+1 = Hx k s asymptotcally stable, that s, λ (H) < 1 for = 1,..., n, f and only f there exst a matrx Q R q n wth rank(q) = n and a matrx Γ R q q, such that : ΓQ = QH (21) Γ < 1 (22) 4

5 Theorem 2 12 : The lnear tme-nvarant contnuous-tme system ẋ(t) = Hx(t) s asymptotcally stable, that s, R(λ (H)) < 0 for = 1,..., n, f and only f there exst a matrx Q R q n wth rank(q) = n and a matrx Γ R q q, such that : ΓQ = QH (23) µ (Γ) < 0 (24) The followng remarks can be made about the above theorems : Equatons (21) and (23) can be seen as generalzed smlarty relatons. In partcular, snce rank(q) = n, we have: σ(h) σ(γ)) (25) Furthermore, from basc propertes of matrx norms and matrx measures, nequaltes (22) and (24) guarantee the asymptotc stablty of the egenvalues of Γ (respectvely n the dscrete-tme and the contnuous-tme cases) and, n consequence of (25)), the asymptotc stablty of the egenvalues of H. For computatonal purposes, equatons (22) and (24) can be respectvely replaced by : Canddate Lyapunov functons are descrbed by: Γ 1 q 1 q and ˆΓ 1 q 0 q. V (x) = max (Qx) The correspondng domans of stablty and nvarance are convex and polyhedral. 3.1 Pole locaton n a vertcal strp Let F(α, β) represent a vertcal strp n the complex plane, where α and β are real numbers satsfyng α > β : λ (H) F(α, β) β < Rλ (H) < α (26) The objectve s to obtan algebrac condtons whch quarantee that the egenvalues of a matrx H belong to F(α, β). The followng proposton gves necessary and suffcent condtons for (26) to be verfed. Proposton 1 The egenvalues of a matrx H R n n verfy λ (H) F(α, β), f and only f there exst matrces Q 1 R m 1 n and Q 2 R m 2 n wth rank(q 1 ) = rank(q 2 ) = n and matrces Γ 1 R m 1 m 1 e Γ 2 R m 2 m 2 such that: Γ 1 Q 1 = Q 1 H (27) µ (Γ 1 ) < α (28) Γ 2 Q 2 = Q 2 H (29) µ ( Γ 2 ) < β (30) Proof : It wll frst be shown that the exstence of matrces Q 1 and Γ 1 satsfyng (27) and (28) s a necessary and suffcent condton for verfyng Rλ (H) < α. To ths end, consder the matrx H α = H αi n, whose egenvalues verfy λ (H α ) = λ (H) α. Then, Rλ (H) < α f and only f 5

6 Rλ (H α ) < 0. From Theorem 2, ths s verfed f and only f there exst Q 1 R m1 n of rank n, and Γ 1 R m 1 m 1 such that Γ 1 Q 1 = Q 1 H α (31) µ Γ 1 < 1 (32) Thus, equatons (27) and (28) are obtaned from (31) and (32) by defnng Γ 1 = Γ 1 +αi n and by usng the fact that µ (Γ 1 ) = µ (Γ 1 ) + α. To complete the proof, t must be shown that the verfcaton of (29) and (30) s a necessary and suffcent condton for Rλ (H) > β to be satsfed. Ths can be done by remarkng that Rλ (H) > β s equvalent to Rλ ( H) < β, whch s verfed f and only f there exst m 2 > n, Q 2 R m2 n of full rank, and Γ 2 R m 2 m 2 such that Γ 2 Q 2 = Q 2 ( H) (33) µ ( Γ 2 ) < β (34) Thus, (33) and (34) can be equvalently replaced by (29) and (30) when settng Γ 2 = Γ 2. Notce that even for a gven H, relatons (27) and (29) are non-lnear and cannot be used as constrants n lnear programmng. The result that wll be used n the sequel for characterzng the locaton of the egenvalues of H n F(α, β) s the followng one : Property 1 If µ (H) < α and µ ( H) < β then λ (H) F(α, β). 3.2 Pole locaton n a centered rng Let A(δ, τ) defne a centered rng n the complex plane, where δ and τ are real numbers satsfyng δ > τ > 0 : { ρ(h) < δ λ (H) A(δ, τ) (35) ϱ(h) > τ where: ρ(h) = max λ (H) e ϱ(h) = mn λ (H). Smlarly to the prevous case, the objectve s to derve algebrac condtons whch guarantee that the egenvalues of a matrx H belong to A(δ, τ). Snce τ > 0 guarantees the exstence of H 1, we have: λ (H 1 ) = 1 1 λ (H), whch mples ϱ(h) =. Thus, (35) can be replaced by : ρ(h 1 ) λ (H) A(δ, τ) { ρ(h) < δ ρ(h 1 ) < 1 τ (36) The followng result gves a necessary and suffcent condton for (36) to be verfed : Proposton 2 The egenvalues of a matrx H R n n verfy λ (H) A(δ, τ), f and only f there exst matrces Q 1 R m 1 n and Q 2 R m 2 n, wth rank(q 1 ) = rank(q 2 ) = n and matrces Γ 1 R m 1 m 1 and Γ 2 R m 2 m 2 such that : Γ 1 Q 1 = Q 1 H (37) Γ 1 < δ (38) Γ 2 Q 2 = Q 2 H 1 (39) Γ 2 < 1 τ (40) 6

7 Proof : Consder the matrx H δ = 1 δ H, wth δ > 0, whose egenvalues satsfy λ (H δ ) = 1 δ λ (H). Then, ρ(h δ ) < 1 f and only f ρ(h) < δ, whch s verfed f and only f there exst Q 1 R m1 n of rank n, and Γ 1 R m 1 m 1 such that: Γ 1 Q 1 = Q 1 H δ (41) Γ 1 < 1 (42) Relatons (37) and (38) are obtaned from (41) and (42) by defnng Γ 1 = 1 δ Γ 1 and by usng the fact that Γ 1 = 1 δ Γ 1, because δ s postve. To complete the proof, t suffces to apply (41) and (42) to obtan ρ(h 1 ) < 1 τ. As n the case of vertcal strps, t s possble to derve suffcent condtons for characterzng the locaton of egenvalues n A(δ, τ). In addton, the followng result (see 8, page 167) can be used : Lemma 1 : If Z R w w satsfy: w 0 < z Zz j = α j 1 = 1,..., w (that s, Z s strctly dagonally domnant by rows), then ρ(z 1 ) Z 1 1 mn α Thus, the followng characterzaton s obtaned for the locaton of the egenvalues of H n A(δ, τ) Property 2 : If H < δ and mn H H j > τ then λ (H) A(δ, τ). j 4 A Lnear Programmng Formulaton of the Control Problem 4.1 An LP formulaton to satsfy state and nput constrants The basc constraned control problem whch has been stated n secton 2.1 may be solved by fndng matrces (H, K, M) whch satsfy relatons (8), (9), (10), (11). The two frst relatons are lnear matrx equaltes. The thrd one and the fourth one nvolve the absolute values of matrces H, K and M. A classcal way to lnearly reformulate relaton (11) s through the decomposton of these matrces nto ther postve and negatve elements (see e.g. 7). Any matrx L and, n consequence, the matrx L, can be decomposed nto: L = L + L and L = L + + L (43) wth:l + j = max(l j, 0) and L j = max( L j, 0). Usng matrces H +, H, K +, K,M +, M, relatons (10), (11) are then formulated as lnear relatons, and the satsfacton of the second relaton n (43) s forced by the choce of the objectve functon. 7

8 The LP formulaton of the constraned dsturbance rejecton problem takes the followng form, wth the components of matrces H +, H, K +, K, M +, M as unknown varables : Mnmze under p 1 ɛ + p 2 ζ (H + H )G GB(M + M )G = GA (K + K )P = GD (M + + M )γ ζρ 0 (H + + H )γ + (K + K )ξ ɛγ 0 0 ɛ 1 ; 0 ζ 1 H +, H, K +, K, M +, M 0 (44) If ths problem has no soluton, then the choce Ω = S(G, γ) s not admssble. Ths does not always mean that the performance bound vector γ s mpossble to acheve. To be sure that ths bound s not achevable, one has to show that the maxmal controlled nvarant set ncluded n S(G, γ) s empty (see 13). The study of the maxmal controlled nvarant set ncluded n a gven compact polyhedron has been made by several authors 9, 2, and numercal technques have been proposed for computng ths doman 4, 5. If the model s uncertan, as descrbed by (14), (15) and (16), unknown matrces H j +, H j, correspondng to each par of vertces (A, B j ), must be computed to verfy equatons (17) to (19). 4.2 An LP formulaton ntegratng spectral specfcatons In order to ntegrate spectral specfcatons (pole locaton n vertcal strps, F(α, β), or n centered rngs, A(δ, τ)) n the desgn procedure, we propose to add the condtons stablshd n propertes 2 and 1 to the LP desgn procedure. It s worth notcng that the desgn methodology s based on the search of a constant state feedback control law whch should leave a gven polyhedral set nvarant wth respect to the dsturbed system (7). Furthermore, the condtons that wll be used for guaranteeng the regonal pole locaton are only suffcent condtons. Thus, n order to provde some addtonal degrees of freedom n the desgn technque, we shall consder that : 1) the lmts that descrbe the regons of pole locaton (α and β for vertcal strps, F(α, β), and δ and τ for centered rngs, A(δ, τ) ) can be used as varables n the LP formulatons 2) the lnear objectve functons constructed from these varables descrbe dfferent possbltes of locatng the poles wthn regons wth the desred shape. From the above dscusson, we propose the followng two LP formulatons for regonal pole locaton 8

9 - Locaton n vertcal strps F(α, β) : Mnmze subject to J = p 1 α + p 2 β (H + H )G GB(M + M )G = GA (K + K )P = GD (M + + M )γ ρ (H + + H )γ + (K + K )ξ γ n H + H + (H j + + Hj ) < α j n H + + H + (H j + + H j ) < β j α > β H +, H, K +, K, M +, M 0 (45) - Locaton n centered rngs A(δ, τ) : Mnmze subject to J = p 1 δ + p 2 τ (H + H )G GB(M + M )G = GA (K + K )P = GD (M + + M )γ ρ (H + + H )γ + (K + K )ξ γ (H + + H )γ δγ n (H + + H ) + (H j + + H j )1 n < τ1 n j δ > τ H +, H, K +, K, M +, M 0 (46) In both LP formulatons gven above, the weghts p 1 and p 2 should be used by the desgner n order to fnd a soluton that approprately locates the closed-loop poles n the complex plane A numercal Example Consder the followng matrces descrbng the model of a DC motor 3 11: A l = B = A u = The elements of matrces A l and A u descrbe, respectvely, the lower and upper bounds of each element of the system matrx A. Matrx B s supposed perfectly known. For the sake of smplcty, t s assumed that the control nput varables are not constraned. Only the dsturbance rejecton specfcaton s mposed, under the form of a doman of state constrants S(G, γ) gven by: G = γ =

10 The spectral control objectve s to locate the poles of the uncertan closed-loop n a regon of type F(α, β), and a constant state feedback soluton s nvestgated. Usng frst the lnear program (45) wth the weghts p 1 = 1 and p 2 = 1, both α and β tend to be mnmzed. The followng values of F, α, β have been computed : F = ; α = ; β = The egenvalues correspondng to each corner A o = A + BF, for = 1,..., 8 are the followng ones : λ (A o1 ) = (A o2 ) = λ (A o3 ) = (A o4 ) = λ (A o5 ) = (A o6 ) = λ (A o7 ) = ± j λ (A o8 ) = ± j Now, usng the weghts p 1 = 1 and p 2 = 1, α tends to be maxmzed whle β tends to be mnmzed. The followng values of F, α, β have been obtaned for ths case : F = ; α = ; β = The egenvalues correspondng to each corner A o = A + BF, for = 1,..., 8 are the followng ones : λ (A o1 ) = (A o2 ) = Concluson λ (A o3 ) = (A o4 ) = λ (A o5 ) = (A o6 ) = λ (A o7 ) = ± j (A o8 ) = ± j The approach by controlled nvarance of polyhedral sets seems to be partcularly well ftted to the resoluton of some dsturbance rejecton problems havng complcatng features such as control nput constrants, and addtonal pole placement requrements. The man advantage of ths approach s ts computatonal effcency whch derves from the use of standard Lnear Programmng as the basc computatonal tool. Some extensons of the work presented n ths study are currently under study, wth the objectve of decreasng the performance gap between smple solutons nvolvng constant lnear state feedback laws and dynamc or nonlnear l 1 -optmal solutons. 10

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