Aaptive Gain-Scheule H Control of Linear Parameter-Varying Systems with ime-delaye Elements Yoshihiko Miyasato he Institute of Statistical Mathematics 4-6-7 Minami-Azabu, Minato-ku, okyo 6-8569, Japan el.: 8-3 - 54-875, Fax.: 8-3 - 54-8796 E-mail : miyasato@ism.ac.jp (MNS 4 REG-4) Abstract : his paper concerns with a new class of aaptive gain-scheule H control of linear parametervarying (LPV) systems with time-elaye elements. he plants in this manuscript are assume to be polytopic LPV systems which have time-elaye components, but the scheule parameters an time-elaye elements in those plants are not known a priori, an thus, the conventional gain-scheule control strategy cannot be applie. In the propose aaptive schemes, the estimates of the scheule parameters are obtaine successively, an the current estimates are fe to the gain-scheule controllers to stabilize the plants an to attain H control performance aaptively. Also, the control gain to compensate the effect of time-elaye elements, is tune recursively. Stability analysis of the aaptive control systems is carrie out by utilizing Lyapunov- Krasovskii functions base on linear matrix inequalities in the boune real lemma. Key Wors : aaptive control ; gain-scheule control; linear parameter-varying system; H control; linear matrix inequality; time-elay; Lyapunov-Krasovskii function Introuction Recently, there has been much progress in the fiel of gain-scheule control of linear parameter-varying (LPV) systems with guarantee control performances,, 3, 4, 5, 6. hose results are base on linear matrix inequalities (LMI) techniques in control engineering an computation tools solving LMI. Among several gainscheule control structures, the gain-scheule control schemes for polytopic LPV systems 3 has been one of the stanar techniques with useful computation tools 7. However, in those approaches, the time-varying process parameters are assume to be known a priori. hose parameters are fe to the gain-scheule controllers as scheule variables to attain stability an certain control performances. If those scheule parameters are unknown, or inaccurate, then even the stability of the resulting control systems is not assure. In orer to solve those problems, we propose aaptive gain-scheule H control of LPV systems in our previous research 8. he plants in that work are assume to be polytopic LPV systems, but the time-varying parameters (scheule parameters) in those plants are not available for measurement, an thus, the conventional gain-scheule control strategy cannot be applie. In the propose aaptive schemes, the estimates of the unknown scheule parameters are obtaine successively, an the current estimates are fe to the controllers as scheule variables to stabilize the plants an to attain H control performance aaptively. Stability analysis of the aaptive control systems is carrie out by utilizing Lyapunov function approaches base on LMI in the boune real lemma. By introucing projection-type aaptive laws, those control schemes can be applie to time-varying parameters an the bouneness of overall aaptive systems is assure. LPV systems are sometimes approximate moels of nonlinear systems, where scheule parameters reflect changing nonlinear characteristics of plants. Hence, the gain-scheule control schemes of LPV systems can be applie to the control of many nonlinear systems by introucing suitable LPV moels, an those approaches
give the clear prospect about the achievable control performance compare with the conventional approaches of nonlinear control. However, several systems which have time-elaye elements, are not inclue in the class of those LPV moels. ime elays are often seen in the control of chemical process, hyraulic process, an rolling mill processes, an even in social an economic phenomena, an the lack of consieration of such elaye elements, sometimes give rise to instability of overall feeback control systems. From that view point, we present a new class of aaptive gain-scheule H control of LPV systems which have also uncertain time-elaye elements. In our aaptive control schemes, tunings of scheule parameters are carrie out so as to satisfy L gain constraints erive from LMI formulas. Aitionally, the control gain to compensate the effect of time-elaye elements, is tune recursively. Stability analysis of the aaptive control systems is carrie out by utilizing Lyapunov-Krasovskii functions base on linear matrix inequalities in the boune real lemma. Aaptive Gain-Scheule H Control of LPV Systems In this section, our previous results 8 are summarize. he aaptive gain-scheule H controllers for polytopic LPV systems are constructe for the case where scheule parameters in those systems are not available for measurement.. Aaptive Gain-Scheule Control via State Feeback Consier the following LPV system t x = A p(α)x B p u B (α)w, () z = C (α)x D u D (α)w, () where w is an external isturbance. A p (α), B (α), C (α), D (α) epen affinely on the time-varying parameter α (scheule parameter) an satisfy Ap (α) B (α) Api B = α i C (α) D (α) i, (3) D i with time-invariant matrices A pi, B i,, D i. he parameter α ranges over a fixe polytope such that α α, α,, α r, α i =, α i. (4) For that LPV system, the system matrices A pi, B i,, D i ( r), B p, D are known, but the parameter α is not available for measurement. he control objective is to stabilize that polytopic LPV system an to make L gain from isturbances w to generalize outputs z less than (> ) for all possible α. Here we review the conventional gain-scheule control problem 3, where α is assume to be available for measurement. he following gain-scheule state feeback control is chosen for the control objective. u = F (α)x, F (α) = α i F i, (5) where F i are time-invariant matrices to be etermine later. hen the feeback system is escribe by where Acl (α) B (α) = C cl (α) D (α) t x = A cl(α)x B (α)w, (6) z = C cl (α)x D (α)w, (7) Acli B α i i = C cli D i Api B α p F i B i i. (8) D F i D i he controlle system is stabilize an the L gain from w to z is mae less than (> ), if there exists a positive efinite matrix P satisfying the following LMI for all possible α (4) (Boune Real Lemma). A cl(α) P P A cl (α) P B (α) C cl (α) B (α) P I D (α) = α i A cli P P A cli P B i Ccli Bi P I D i <. (9) C cl (α) D (α) I C cli D i I
he conition (9) is equivalent to the existence of the positive efinite P satisfying the next systems of LMIs (). A cli P P A cli P B i Ccli Bi P I D i <, ( i r). () C cli D i I he LMI-base solvability conitions an controller esigns were stuie in 3, an the computation algorithms to obtain P (Lyapunov matrices) an controller matrices F i simultaneously from (), were also evelope in the work 7. However, that approach is effective only for the case where α is available for measurement. When α is not available, ˆα (the current estimate of α) is fe to the gain-scheule state feeback controller as follows: u = F (ˆα)x = ˆα i F i x. () hen, the controlle system becomes Here we assume ẋ = z = α i A cli x α i C cli x Assumption α L an w L. an α i B i w B p (ˆα i α i }F i x, () α i D i w D (ˆα i α i )F i x. (3) Assumption LMI () is solvable for the LPV system (), (), an there exist a positive efinite matrix P an feeback matrices F i ( i r) satisfying (). Note that the unknown parameters α i are not inclue in the LMI (). On the basis of Assumption, the LMI (9) is ivie into the form A cli α i P P A cli P B i Ccli Bi P I D i C cli D i I P A cli P B i P A cli P B i = α i I C cli D i I I <, (4) C cli D i I an the following relation hols for any vector x, w, with proper imensions an δ, δ, δ 3 >. δ x δ w δ 3 α i x w P A cli P B i x I w C cli D i I ( ) ( = x P α i A cli x α i B i w α i C cli x By consiering (), (3), an by setting as the inequality (5) is rewritten into δ x δ w δ 3 z t (x P x) z w ) α i D i w w. (5) z, (6) (ˆα i α i ) x P B p F i x 3 ( ) z D F i x}. (7)
Here, efine the positive function V V = x P x (ˆα i α i ) /g i, (g i > ), (8) an take the time erivative of it along the trajectories of x, ˆα i an α i, by utilizing (7). V z ( ) z w (ˆα i α i ) x P B p F i x D F i x} (ˆα i α i ) ˆα i α i }/g i δ x δ w z δ 3. (9) From that, the aaptive laws of ˆα = ˆα,, ˆα r are etermine as follows (the projection-type aaptive laws 9). Gφ G ˆα(t) ξ(ˆα) ξ(ˆα) = ξ(ˆα) G ξ(ˆα) Gφ if ξ(ˆα) = & ξ(ˆα) Gφ <, () Gφ otherwise (ˆα() S), φ = φ φ φ r, ( ) z φ i = x P B p F i x D F i x, ( i r), () G = iag (g, g,, g r ), g i >, ( i r). () ξ(α) is a ifferentiable function of α satisfying ξ(α), α S ( R r ), (3) where S( R r ) is a boune region which contains the constraints of α (4). From the properties of projectiontype aaptive laws 9, it follows that ˆα S (α i L ) an that hen, V is evaluate by (ˆα α) G ˆα (ˆα α) φ. (4) V z w δ x δ w δ 3 z z w Since α, α L an ˆα L, the next relation is obtaine from (5), (α i ˆα i ) α i /g i (5) (α i ˆα i ) α i /g i. (6) V δv D, ( < δ, D < ), (7) an it is shown that V L, an that x, z L. Also, the next inequality is erive from (6). z(τ) τ ˆα i (t) α i } /g i x(t) P x(t) w(τ) τ ˆα i () α i } /g i x() P x() (α i ˆα i ) α i /g i τ. (8) hen, the L gain from w to z is prescribe by, where initial error of tuning parameters r ˆα i() α i } an time-varying elements of r (α i ˆα i ) α i /g i are also inclue (aaptive H control performance). heorem Consier LPV systems (), (). On Assumption an Assumption, the aaptive gain-scheule control schemes (), (), () stabilize the process; x, z, ˆα L, an attain aaptive H control performance (8). Especially, when α L an w L, then x, z (as t ). Remark Bouneess of the propose aaptive gain-scheule control schemes is erive from (7) (but not from (8)). he inequality (8) prescribes L gain property an hols for any finite t (> ). 4
. Aaptive Gain-Scheule Control via Dynamic Compensator Consier the following polytopic LPV system where A p (α), B (α), C (α), D (α) are efine by t x = A p(α)x B p u B (α)w, (9) y = C p x D w, (3) z = C (α)x D u D (α)w, (3) Ap (α) B (α) = C (α) D (α) Api B α i i, (3) D i with time-invariant matrices A pi, B i,, D i, an the time-varying parameter α (scheule parameter) satisfies the same conition as (4). For that LPV system, the system matrices A pi, B i,, D i ( r), B p, C p, D, D are known, but the parameter α is unknown. he control objective is to stabilize the process, an make L gain from isturbances w to generalize outputs z less than (> ), for all possible α. Here the conventional gain-scheule control problem 3 is reviewe, where α is assume to be available for measurement. he following gain-scheule ynamic compensator is introuce for the control objective. t x K = A k (α)x K B K (α)y, (33) u = C k (α)x K D K (α)y, (34) AK (α) B K (α) AKi B = α Ki C K (α) D K (α) i, (35) C Ki D Ki where A Ki, B Ki, C Ki, D Ki are time-invariant matrices. hen, the feeback system becomes t x cl = A cl (α)x cl B cl (α)w, (36) z = C cl (α)x cl D cl (α)w, (37) x x cl =, (38) x K where Acl (α) B cl (α) = C cl (α) D cl (α) A cli = Acli B α cli i, (39) C cli D cli Bp D Ki D B i, B Ki D Api B pi D Ki C p B p C Ki, B B Ki C p A cli = Ki C cli = D D Ki C p D C Ki, Dcli = D D Ki D D i. (4) he controlle process is stabilize, an L gain from w to z is mae less than (> ), if there exists a positive efinite P satisfying the following LMI for all possible α (4) (Boune Real Lemma). A cl(α) P P A cl (α) P B cl (α) C cl (α) B cl (α) P I D cl (α) = α i A cli P P A cli P B cli Ccli Bcli P I D cli <. (4) C cl (α) D cl (α) I C cli D cli I he conition (4) is equivalent to the existence of the positive efinite P satisfying the next systems of LMIs (4). A cli P P A cli P B cli Ccli Bcli P I D cli <, ( i r). (4) C cli D cli I he computation algorithms to obtain P (Lyapunov matrices) an controller matrices A Ki, B Ki, C Ki, D Ki simultaneously from (4), were also evelope in the previous work 7. However, that strategy can be applie only to the case where α is available for measurement. 5
When α is not available, ˆα (the current estimate of α) is fe to the gain-scheule ynamic compensators as follows: hen, the overall controlle process is written by Here we assume Assumption an t x K = A k (ˆα)x K B K (ˆα)y, (43) u = C k (ˆα)x K D K (ˆα)y, (44) AK (ˆα) B K (ˆα) AKi B = ˆα Ki C K (ˆα) D K (ˆα) i. (45) C Ki D Ki t x cl = A cl (α)x cl B cl (α)w z = C cl (α)x cl D cl (α)w Bp (C (ˆα i α i ) Ki x K D Ki y) A Ki x K B Ki y, (46) (ˆα i α i )D (C Ki x K D Ki y). (47) Assumption 3 LMI (4) is solvable for the LPV system (9), (3), (3), an there exist a positive efinite matrix P an controller matrices A Ki, B Ki, C Ki, D Ki ( i r) satisfying (4). Note that the unknown parameters α i are not inclue in the LMI (4). On the basis of Assumption 3, the LMI (4) is ivie into the form A cli α i P P A cli P B cli Ccli Bcli P I D cli C cli D cli I P A cli P B cli P A cli P B cli = α i I C cli D cli I I <, (48) C cli D cli I an the following relation hols for any vector x cl, w, with proper imensions an δ, δ, δ 3 >. δ x cl δ w δ 3 α i x cl w P A cli P B cli I x cl w C cli D cli I ( ) ( = x clp α i A cli x cl α i B cli w α i C cli x cl By consiering (46), (47), an by setting as the inequality (49) is reuce to ) α i D cli w w. (49) z, (5) δ x cl δ w δ 3 z t (x clp x cl ) z w (ˆα i α i ) x Bp (C Ki x K D Ki y) clp A Ki x K B Ki y ( ) z D (C Kix K D Kiy)}. (5) Here, efine V by V = x clp x cl (ˆα i α i ) /g i, (g i > ), (5) 6
an take the time erivative of it along the trajectories of x cl, ˆα i an α i, by utilizing (5). V z w (ˆα i α i ) x clp Bp (C Ki x K D Ki y) A Ki x K B Ki y (ˆα i α i ) ˆα i α i }/g i δ x cl δ w z δ 3 ( ) z D (C Kix K D Kiy)}. (53) For the tuning of ˆα = ˆα,, ˆα r, the same projection-type aaptive laws as () are chosen, but the efinition of φ = φ φ φ r is ifferent from the previous case. φ i = x clp Bp (C Ki x K D Ki y) A Ki x K B Ki y ( ) z D (C Kix K D Kiy), ( i r). (54) ξ(α) is the same as the previous one. hen, by consiering the property of projection-type aaptive laws, V is evaluate as follows: V z w δ x cl δ w δ 3 z z w (α i ˆα i ) α i /g i (55) (α i ˆα i ) α i /g i. (56) Since α, α L (Assumption ) an ˆα L (the property of projection-type aaptive laws), the next relation is obtaine from (55). V δv D, ( < δ, D < ). (57) Hence, it is shown that V L, an that x cl, z L. Also, the next inequality is erive from (56). z(τ) τ ˆα i (t) α i } /g i x cl (t) P x cl (t) w(τ) τ ˆα i () α i } /g i x cl () P x cl () (α i ˆα i ) α i /g i τ. (58) hen, the L gain from w to z is prescribe by, where initial error of tuning parameters r ˆα i() α i } an time-varying elements of r (α i ˆα i ) α i /g i are also inclue (aaptive H control performance). heorem Consier LPV systems (9), (3), (3). On Assumption an Assumption 3, the aaptive gainscheule control schemes (43), (44), (), (54) stabilize the process; x cl, z, ˆα L, an attain aaptive H control performance (58). Especially, when α L an w L, then x cl, z (as t ) 3 Aaptive Gain-Scheule H Control of LPV Systems with ime- Delaye Elements Many nonlinear systems can be seen as LPV systems with changing scheule parameters, an the gainscheule control schemes of LPV systems can be applie to the control of such nonlinear systems by introucing suitable LPV moels. However, several systems which have time-elaye elements, are not inclue in the class of those LPV moels, an the lack of consieration of time-elaye elements, sometimes give rise to instability of overall control systems. From that view point, we present a new class of aaptive gain-scheule H control of LPV systems which have also uncertain time-elaye elements. In our aaptive control schemes, tunings of scheule parameters are carrie out so as to satisfy L gain constraints erive from LMI formulas. Also, the control gain to compensate the effect of time-elaye elements, is tune recursively. Stability analysis of the aaptive control systems is carrie out by utilizing Lyapunov-Krasovskii functions base on linear matrix inequalities in the boune real lemma. 7
3. Aaptive Gain-Scheule Control via State Feeback Consier the following LPV system with time-elaye elements t x = A p(α)x B p u f i (x(t τ i ))} B (α)w, (59) z = C (α)x D u D (α)w, (6) where A p (α), B (α), C (α), D (α) an α are efine by (3) an (4). he parameter α is not available for measurement. f i (x(t τ i )) ( i m) are unknown time-elaye elements (f i ( ) an τ i (> ) are unknown), an are evaluate as follows: f i (x(t τ i )) ρ i x(t τ i ), ( i m), (6) where ρ i (> ) are unknown constant parameters. he initial conition is given by x(σ) = ψ(σ), (σ τ,, τ maxτ i }), (6) i where ψ(σ) is a continuous function. We assume Assumption an Assumption. he control input is compose of the gain-scheule state feeback term F (ˆα)x an an aitional signal v. u = F (ˆα)x v = u u, (63) u = F (ˆα)x, (64) u = v, (65) where v is a control signal which compensates the effects of time-elaye elements m f i(x(t τ i )). For the ivision of u, the controlle system is reefine as follows: t x = A p(α)x B p u f i (x(t τ i ))} B (α)w, (66) z = C (α)x D u D (α)w. (67) he substitution of the control schemes (63) (65) into the process (66), (67) yiels ẋ = A cl (α)x B p z = C cl (α)x D (ˆα i α i }F i x B (α)w B p v B p m f i (x(t τ i )), (68) (ˆα i α i )F i x D (α)w. (69) From Assumption, the same relation as (5) hols for any vector x, w, with proper imensions an δ, δ, δ 3 >. By consiering (68), (69), an by setting as = z, the inequality (5) is rewritten into δ x δ w δ 3 z x P ẋ B p (ˆα i α i )F i x B p v B p ( ) z z D ) (ˆα i α i )F i x z = t (x P x) z w x P B p v x P B p m m f i (x(t τ i )) w } (ˆα i α i ) x P B p F i x ( ) z D F i x} f i (x(t τ i )). (7) 8
Here, efine the positive function V by (8), an take the time erivative of it along the trajectories of x, ˆα i an α i, by utilizing (7). V z ( ) z w (ˆα i α i ) x P B p F i x D F i x} m x P B p v x P B p f i (x(t τ i )) (ˆα i α i )( ˆα i α i )/g i δ x δ w z δ 3. (7) he aaptive laws of ˆα i are the same as (), (). hen, from the properties of projection-type aaptive laws, we have V z w x P B p v x P B p δ x δ w δ 3 z he next inequality hols for arbitrary positive. f i (x(t τ i )) (α i ˆα i ) α i /g i. (7) x P B p f i (x(t τ i )) x P B p B p P x f i(x(t τ i )) x P B p B p P x ρ i x(t τ i ). (73) By utilizing, we efine the following positive function (Lyapunov-Krasovskii function). W = V t τ i ρ i x(σ) σ. (74) We take the time erivative of W along the trajectory of LPV systems with time-elaye elements. Ẇ = V ρ i x(t) z w x P B p v δ x δ w δ 3 z ρ i x(t τ i ) = z w x P B p v δ x δ w δ 3 z where (73) is consiere. Here we set by then, Ẇ is evaluate as follows: x P B p Bp P x C } i ρ i x(t τ i ) ρ i x(t) ρ i x(t τ i ) (α i ˆα i ) α i /g i x P B p Bp P x C } i C i ρ i x(t) (α i ˆα i ) α i /g i, (75) ρ i = δ m = δ ( i m), (76) m, Ẇ z w x P B p v ρ i δ x δ w δ 3 z 9 ρ i m δ x P B p B p P x (α i ˆα i ) α i /g i. (77)
An unknown positive constant k is efine by k ρ i m δ, (78) an the control signal v is etermine with the aaptive law of ˆk (the current estimate of k). v(t) = ˆk(t)B p P x(t), (79) g x(t) ˆk(t) = P B p Bp P x(t) when ˆk < k when ˆk = k, (g >, ˆk() < k ), (8) where k is an upper boun of k such that k k, an it is assume that k is known a priori. he positive function W is efine by W W ˆk(t) k} /g, (8) an take the time erivative of W. W (t) z w ˆk(t)x P B p Bp P x(t) kx P B p Bp P x δ x δ w z δ 3 (α i ˆα i ) α i /g i ˆk(t) k}x(t) P B p Bp P x(t) = z w δ x δ w δ 3 z (α i ˆα i ) α i /g i, (8) where the property of the projection-type aaptive laws are consiere. projection-type aaptive laws, it follows that ˆα i, ˆk L an that Since ˆα i an ˆk are tune by the (ˆα i α i ) /g i D α <, (83) ˆk(t) k} /g D k <, (84) where D α an D k are boune constants. Also the next evaluation is erive for a boune constant D w α. w δ w (α i ˆα i ) α i /g i D w α <. (85) hen we have W (t) δ x (ˆα i α i ) /g i ˆk(t) k} /g D α D k D w α, (86) an it follows that x an all other signals are uniformly boune. Aitionally, the next inequality is obtaine from (8). he main theorem is erive. z τ x P x (ˆα i α i ) /g i C t i ρ i x(σ) σ t τ i ˆk(t) k} /g w τ x() P x() (ˆα i () α i ) /g i τ i ρ i x(σ) σ ˆk(t) k} /g. (87)
heorem 3 Consier LPV systems with time-elaye elements (66), (67). hen, on Assumption an Assumption, the aaptive gain-scheule control schemes (63) (65), (), (), (79), (8) stabilize the process; x, z, ˆα, ˆk L. Furthermore, the inequality (87) hols. Especially, when α L an w L, then x, z (as t ). Remark Since k is unknown, k shoul be sufficiently large so as to satisfy k > k. hat k (fixe parameter) can be also utilize as the control parameter in the following form v(t) = k B p P x(t), (88) an there is no nee of the tuning of k for such case. However, that choice sometimes gives rise to the excessive control signal v because of the high gain k. Hence, the aaptation scheme of k is still necessary, where a moerate k can be obtaine compare with its upper boun k. 3. Aaptive Gain-Scheule Control via Dynamic Compensator Consier the following LPV system with time-elaye elements t x = A p(α)x B p u f i (x(t τ i ))} B (α)w, (89) y = C p x D w, (9) z = C (α)x D u D (α)w, (9) where A p (α), B (α), C (α), D (α) an α are efine by (3) an (4). he parameter α is not available for measurement. f i (x(t τ i )) ( i m) are unknown time-elaye elements (f i ( ) an τ i (> ) are unknown), an are evaluate by (6), where ρ i (> ) are unknown constant parameters. he initial conition is etermine by (6). We assume Assumption an Assumption 3. he control input is compose of the gain-scheule ynamic compensation term C K (ˆα)x K D K (ˆα)y an an aitional signal v. u = C k (ˆα)x K D K (ˆα)y v = u u, (9) t x K = A K (ˆα)x K B K (ˆα)y, (93) u = C k (ˆα)x K D K (ˆα)y, (94) u = v, (95) where v is an aitional signal which compensates the effects of time-elaye elements m f i(x(t τ i )). For the ivision of u, the controlle system is reefine as follows: t x = A p(α)x B p u hen the controlle system is escribe by f i (x(t τ i ))} B (α)w, (96) y = C p x D w, (97) z = C (α)x D u D (α)w. (98) t x cl = A cl (α)x cl B cl (α)w Bp v Bp (C (ˆα i α i ) Ki x K D Ki y) A Ki x K B Ki y f i (x(t τ i ))}, (99) z = C cl (α)x cl D cl (α)w D (ˆα i α i )(C Ki x K D Ki y). () From Assumption 3, the same relation as (49) hols for any vector x cl, w, with proper imensions an δ, δ, δ 3 >. By consiering (99), (), an by setting as = z, the inequality (49) is rewritten into δ x cl δ w δ 3
x clp ẋ cl (ˆα i α i ) ( ) z z Bp (C Ki x K D Ki y) A Ki x K B Ki y Bp (ˆα i α i )D (C Ki x K D Ki y) z = t (x clp x cl ) z w (ˆα i α i ) x Bp (C Ki x K D Ki y) clp A Ki x K B Ki y x Bp clp v } v w } f i (x(t τ i ))} ( ) z D (C Kix K D Kiy)} f i (x(t τ i ))}. () Here, efine the positive function V by (5) an take the time erivative of it along the trajectories of x cl, ˆα i an α i, by utilizing (). V z ( ) w (ˆα i α i ) x Bp (C Ki x K D Ki y) z clp D A Ki x K B Ki y (C Kix K D Kiy)} x Bp clp v f i (x(t τ i ))} (ˆα i α i )( ˆα i α i )/g i δ x δ w z δ 3. () he aaptive laws of ˆα i are the same as (), (54). hen, from the properties of projection-type aaptive laws, we have V z w x Bp clp v f i (x(t τ i ))} δ x cl δ w z δ 3 (ˆα i α i ) α i /g i. (3) he next inequality hols for arbitrary positive. x Bp clp f i (x(t τ i )) x C clp i x clp Bp Bp B p B p P x cl f i(x(t τ i )) P x cl ρ i x(t τ i ). (4) By utilizing, we efine the following positive function (Lyapunov-Krasovskii function). W = V t τ i ρ i x(σ) σ. (5) We take the time erivative of W along the trajectory of LPV systems with time-elaye elements. Ẇ = V ρ i x(t) ρ i x(t τ i ) z w x clp Bp δ x cl δ w z δ 3 = z w x Bp clp δ x cl δ w δ 3 z v v x Bp C clp i ρ i x(t) x clp B p ρ i x(t τ i ) B p Bp P x cl C } i ρ i x(t τ i ) (α i ˆα i ) α i /g i P x cl ρ i x(t) (α i ˆα i ) α i /g i, (6) }
where (4) is consiere. Here we set by then, Ẇ is evaluate as follows: Ẇ z w x Bp clp ρ i = δ m = δ ( i m), (7) m, δ x cl δ w δ 3 z where the following relation is also consiere. An unknown positive constant k is efine by v ρ i ρ i m x Bp δ clp B p P x cl (α i ˆα i ) α i /g i, (8) x x cl = x x K. (9) k ρ i m δ, () an the control signal v is etermine with the aaptive law of ˆk (the current estimate of k). v(t) = ˆk(t) Bp P x cl (t), () g x cl (t) ˆk(t) Bp P B p P x = cl (t) when ˆk < k when ˆk, (g >, ˆk() < k ), () = k where k is an upper boun of k such that k k, an it is assume that k is known a priori. he positive function W is efine by W W ˆk(t) k} /g, (3) an take the time erivative of W. W (t) z w ˆk(t)x cl (t) P kx cl (t) Bp P B p δ x cl δ w δ 3 z ˆk(t) k}x cl (t) P Bp Bp P x cl (t) B p B p (α i ˆα i ) α i /g i P x cl (t) = z w δ x cl δ w δ 3 z where the property of the projection-type aaptive laws are consiere. projection-type aaptive laws, it follows that ˆα i, ˆk L an that P x cl (t) (α i ˆα i ) α i /g i, (4) Since ˆα i an ˆk are tune by the (ˆα i α i ) /g i D α <, (5) ˆk(t) k} /g D k <, (6) where D α an D k are boune constants. Also the next evaluation is erive for a boune constant D w α. w δ w (α i ˆα i ) α i /g i D w α <. (7) 3
hen we have W (t) δ x cl (ˆα i α i ) /g i ˆk(t) k} /g D α D k D w α, (8) an it follows that x cl an all other signals are uniformly boune. Aitionally, the next inequality is obtaine from (4). z τ x clp x cl (ˆα i α i ) /g i t τ i ρ i x(σ) σ ˆk(t) k} /g w τ x cl() P x cl () he secon main theorem is erive. (ˆα i () α i ) /g i τ i ρ i x(σ) σ ˆk(t) k} /g (ˆα i α i ) α i /g i τ. (9) heorem 4 Consier LPV systems with time-elaye elements (96), (97), (98). hen, on Assumption an Assumption 3, the aaptive gain-scheule control schemes (9) (95), (), (54), (), () stabilize the process; x cl, z, ˆα, ˆk L. Furthermore, the inequality (9) hols. Especially, when α L an w L, then x cl, z (as t ). Remark (87) an (9) show the L gain property of the propose control scheme. he L gain from w to z is prescribe by. he initial errors of tuning parameters r ˆα i() α i } /g i, (ˆk() k) /g, time-varying elements of r (α i ˆα i ) α i /g i, an the effects of time-elaye elements m τ i ρ i x(σ) σ are also inclue in the inequalities (87), (9) (aaptive H control performance). 4 Numerical Example A numerical simulation stuy is performe to show the effectiveness of the propose control scheme. Consier the following LPV system with time-elaye element. ẋ (t) x (t) = u(t) f(x(t τ))} w(t), ( a ), ẋ (t) a x (t) y = x (t), z(t) = x (t), x (t), u(t), ( z(t) = x (t) x (t) u(t) ), where a is an unknown time-varying parameter. f( ) an τ are also unknown. he corresponing polytopic LPV system with time-elaye component is given by where Ap (α) B (α) = C (α) D (α) A p = t x = A p(α)x B p u f(x(t τ))} B (α)w, y = C p x D w,, A p = z = C (α)x D u C p =, D =, C = C = α i =, α i. D (α)w, Api B α i i, D i, B p = B = B =,, D = 4, D = D =,
he gain-scheule controller is compute via MALAB LMI oolbox (Math. Works Inc.) as follows: where u = C k (ˆα)x K D K (ˆα)y v = u u, t x K = A k (ˆα)x K B K (ˆα)y, u = C k (ˆα)x K D K (ˆα)y, u = v, AK (α) B K (α) AKi B = α Ki C K (α) D K (α) i, C Ki D Ki.39.7 A K = 4.366, B 3.484.65 K = 4, 5.967.396.74 A K = 4.37, B 3.49.6555 K = 4, 5.999 C K = C K =.59.884 4, D K = D K =. he achievable control performance an the Lyapunov matrix P are =.94, P = Other conitions are etermine in the following: 5.565.35 5.539.36.35.8.36.3 5.539.36 5.539.36.36.3.36.3 α (t) =.5.5 sin(3.4t), w = exp( t), f(x(t τ)) = x (t.3), g i =.. ˆα () =., ˆα () =.8, ˆk() =. 5. α (t) =.5.5 sin(3.4t), All other initial conitions are. Fig. shows the result where the propose aaptive control scheme is applie. 4 3.5 3.5.5.5 z w.5.5.5 3 Figure : Simulation Result (Propose Control Scheme). For comparison, Fig. shows the result of the conventional gain-scheule control scheme, where the exact α an α are utilize (non-aaptive), but the time-elaye element f(x(t τ)) is not consiere at all. 5
4.5 4 3.5 «3.5.5.5 属.5.5.5 3 Figure : Simulation Result (Conventional Gain-Scheule Control Scheme (Non-Aaptive) with No Consieration on ime-delaye Element). 5 Concluing Remarks A class of aaptive gain-scheule H control scheme of polytopic LPV systems having time-elaye elements is presente in this manuscript. Aaptive tuning of scheule parameters is carrie out so as to satisfy L constraints erive from LMI formulas. Also, the control gain is upate recursively so as to compensate the effect of time-elaye elements. Stability analysis of the aaptive control systems is carrie out by utilizing Lyapunov-Krasovskii functions base on LMI in the boune real lemma. It is shown that the overall control systems are uniformly boune, an that L gain properties incluing initial errors of tuning parameters, timevarying components, an the effects of time-elaye elements, are prescribe explicitly. Numerical examples also show the effectiveness of the propose metho. References A. Packar, Gain scheuling via linear fractional transformations, Sys. Contr. Lett.,, 436-44, 994. G. Becker, an A. Packar, Robust performance of linear parametrically-varying systems using parametrically-epenent linear feeback, Sys. Contr. Lett., 3, 436-44, 994. 3 P. Apkarian, P. Gahinet, an G. BeckersI, Self-scheule H control of linear parameter-varying systems : a esign example, Automatica, Vol.3, 5-6, 995. 4 P. Apkarian, an P. Gahinet, A convex characterization of gain-scheule H controllers, IEEE rans. Autom. Control, AC-4, 853-864, 995. 5 P. Gahinet, P. Apkarian an M. Chilali, Affine parameter-epenent Lyapunov functions an real parametric uncertainty, IEEE rans. Autom. Control, AC-4, 436-44, 996. 6 R. Watanabe, K. Uchia an M. Fujita, A new LMI approach to analysis of linear systems with scheuling parameter base on finite number of LMI conitions, Proceeings of the 35th IEEE Conference on Decision an Control, pp.3663-665, 996. 7 P. Gahinet, A. Nemirovski, A. J. Laub an M. Chilali., LMI Control oolbox User s Guie, Math. Work Inc. 995. 8 Y. Miyasato, Aaptive gain-scheule H control of linear parameter-varying systems, Preprints of the 5th IFAC Worl Congress, Barcelona,. 6
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