Observer-based controller for robust pole clustering in a vertical strip and disturbance rejection in structured uncertain systems

Abstract

This paper presents an observer‐based multi‐objective robust feedback controller to achieve robust pole clustering within a vertical strip and disturbance rejection with an H∞‐norm constraint for the uncertain linear systems. The systems of interest include both matched and mismatched uncertain linear systems with structured uncertainties existing in both the system and input matrices. The controller is obtained by solving two Riccati equations (one for the controller and the other for the observer) and checking three conditions (two for robust pole clustering). A set of tuning parameters is incorporated to enhance flexibility in finding the controller. © 1998 John Wiley & Sons, Ltd.

<title>Robust control for structural systems with uncertainties</title>

“…Similar to the proof in Theorem 1, by using Lemma 3 and conditions r > 0.5, (28) and 5mbu <1 in (8) Thus, by Lemma 2 and some additional derivation, the controller (21) is robust and has a-degree relative stability (12) and a &-degree disturbance attenuation (1 1) for the uncertain structural system (5) with all admissible matched uncertainties normbounded as shown in (8). Similar to the proof in Theorem 1, by using Lemma 3 and conditions r > 0.5, (28) and 5mbu <1 in (8) Thus, by Lemma 2 and some additional derivation, the controller (21) is robust and has a-degree relative stability (12) and a &-degree disturbance attenuation (1 1) for the uncertain structural system (5) with all admissible matched uncertainties normbounded as shown in (8).…”

“…and AA = A+AA -BK -LBK, has a robust disturbance attenuation with a prescribed H,-norm constraint 5 (a specified disturbance attenuation index) that satisfies the following: (11) and a robust a-degree relative stability, i.e., Re{A(A)}<-c , (12) where (s) is a transfer function matrix from the disturbance vector w to the observation vector z of the structural system. and AA = A+AA -BK -LBK, has a robust disturbance attenuation with a prescribed H,-norm constraint 5 (a specified disturbance attenuation index) that satisfies the following: (11) and a robust a-degree relative stability, i.e., Re{A(A)}<-c , (12) where (s) is a transfer function matrix from the disturbance vector w to the observation vector z of the structural system.…”

“…In this section, we develop a state feedback controller in (9) to provide a robust a-degree relative stability in (12) and an H, disturbance attenuation with a prescribed index in (1 1) for uncertain structural system (5) and (10). The controller (9) is obtained by solving a Riccati equation as derived in this section.…”

“…For given scalars cx O and >O, if there exist a positive definite matrix P and an adjustable scalar >O such that (A +aI)TP+P(Ac +XI)4(1+cFw)2PFFTP+gcTc<o, (14) then the closed-loop system (10) is of the a-degree relative stable as (12) and ö-degree disturbance attenuation as (1 1). For given scalars cx O and >O, if there exist a positive definite matrix P and an adjustable scalar >O such that (A +aI)TP+P(Ac +XI)4(1+cFw)2PFFTP+gcTc<o, (14) then the closed-loop system (10) is of the a-degree relative stable as (12) and ö-degree disturbance attenuation as (1 1).…”

“…By extension of Lemma 2 in [10], it is known that the closed-loop system (10) is of the a-degree relatively stable as (12) and &degree disturbance attenuation as (11)if (Ac÷)TP+P(Ac+aI)4P(F÷F)(F÷iF)TP+._CTC<O. By extension of Lemma 2 in [10], it is known that the closed-loop system (10) is of the a-degree relatively stable as (12) and &degree disturbance attenuation as (11)if (Ac÷)TP+P(Ac+aI)4P(F÷F)(F÷iF)TP+._CTC<O.…”

<title>Robust control for structural systems with uncertainties</title>

“…Similar to the proof in Theorem 1, by using Lemma 3 and conditions r > 0.5, (28) and 5mbu <1 in (8) Thus, by Lemma 2 and some additional derivation, the controller (21) is robust and has a-degree relative stability (12) and a &-degree disturbance attenuation (1 1) for the uncertain structural system (5) with all admissible matched uncertainties normbounded as shown in (8). Similar to the proof in Theorem 1, by using Lemma 3 and conditions r > 0.5, (28) and 5mbu <1 in (8) Thus, by Lemma 2 and some additional derivation, the controller (21) is robust and has a-degree relative stability (12) and a &-degree disturbance attenuation (1 1) for the uncertain structural system (5) with all admissible matched uncertainties normbounded as shown in (8).…”

“…and AA = A+AA -BK -LBK, has a robust disturbance attenuation with a prescribed H,-norm constraint 5 (a specified disturbance attenuation index) that satisfies the following: (11) and a robust a-degree relative stability, i.e., Re{A(A)}<-c , (12) where (s) is a transfer function matrix from the disturbance vector w to the observation vector z of the structural system. and AA = A+AA -BK -LBK, has a robust disturbance attenuation with a prescribed H,-norm constraint 5 (a specified disturbance attenuation index) that satisfies the following: (11) and a robust a-degree relative stability, i.e., Re{A(A)}<-c , (12) where (s) is a transfer function matrix from the disturbance vector w to the observation vector z of the structural system.…”

“…In this section, we develop a state feedback controller in (9) to provide a robust a-degree relative stability in (12) and an H, disturbance attenuation with a prescribed index in (1 1) for uncertain structural system (5) and (10). The controller (9) is obtained by solving a Riccati equation as derived in this section.…”

“…For given scalars cx O and >O, if there exist a positive definite matrix P and an adjustable scalar >O such that (A +aI)TP+P(Ac +XI)4(1+cFw)2PFFTP+gcTc<o, (14) then the closed-loop system (10) is of the a-degree relative stable as (12) and ö-degree disturbance attenuation as (1 1). For given scalars cx O and >O, if there exist a positive definite matrix P and an adjustable scalar >O such that (A +aI)TP+P(Ac +XI)4(1+cFw)2PFFTP+gcTc<o, (14) then the closed-loop system (10) is of the a-degree relative stable as (12) and ö-degree disturbance attenuation as (1 1).…”

“…By extension of Lemma 2 in [10], it is known that the closed-loop system (10) is of the a-degree relatively stable as (12) and &degree disturbance attenuation as (11)if (Ac÷)TP+P(Ac+aI)4P(F÷F)(F÷iF)TP+._CTC<O. By extension of Lemma 2 in [10], it is known that the closed-loop system (10) is of the a-degree relatively stable as (12) and &degree disturbance attenuation as (11)if (Ac÷)TP+P(Ac+aI)4P(F÷F)(F÷iF)TP+._CTC<O.…”

<title>Robust control for structural systems with uncertainties</title>

“…Similar to the proof in Theorem 1, by using Lemma 3 and conditions r > 0.5, (28) and 5mbu <1 in (8) Thus, by Lemma 2 and some additional derivation, the controller (21) is robust and has a-degree relative stability (12) and a &-degree disturbance attenuation (1 1) for the uncertain structural system (5) with all admissible matched uncertainties normbounded as shown in (8). Similar to the proof in Theorem 1, by using Lemma 3 and conditions r > 0.5, (28) and 5mbu <1 in (8) Thus, by Lemma 2 and some additional derivation, the controller (21) is robust and has a-degree relative stability (12) and a &-degree disturbance attenuation (1 1) for the uncertain structural system (5) with all admissible matched uncertainties normbounded as shown in (8).…”

“…and AA = A+AA -BK -LBK, has a robust disturbance attenuation with a prescribed H,-norm constraint 5 (a specified disturbance attenuation index) that satisfies the following: (11) and a robust a-degree relative stability, i.e., Re{A(A)}<-c , (12) where (s) is a transfer function matrix from the disturbance vector w to the observation vector z of the structural system. and AA = A+AA -BK -LBK, has a robust disturbance attenuation with a prescribed H,-norm constraint 5 (a specified disturbance attenuation index) that satisfies the following: (11) and a robust a-degree relative stability, i.e., Re{A(A)}<-c , (12) where (s) is a transfer function matrix from the disturbance vector w to the observation vector z of the structural system.…”

“…In this section, we develop a state feedback controller in (9) to provide a robust a-degree relative stability in (12) and an H, disturbance attenuation with a prescribed index in (1 1) for uncertain structural system (5) and (10). The controller (9) is obtained by solving a Riccati equation as derived in this section.…”

“…For given scalars cx O and >O, if there exist a positive definite matrix P and an adjustable scalar >O such that (A +aI)TP+P(Ac +XI)4(1+cFw)2PFFTP+gcTc<o, (14) then the closed-loop system (10) is of the a-degree relative stable as (12) and ö-degree disturbance attenuation as (1 1). For given scalars cx O and >O, if there exist a positive definite matrix P and an adjustable scalar >O such that (A +aI)TP+P(Ac +XI)4(1+cFw)2PFFTP+gcTc<o, (14) then the closed-loop system (10) is of the a-degree relative stable as (12) and ö-degree disturbance attenuation as (1 1).…”

“…By extension of Lemma 2 in [10], it is known that the closed-loop system (10) is of the a-degree relatively stable as (12) and &degree disturbance attenuation as (11)if (Ac÷)TP+P(Ac+aI)4P(F÷F)(F÷iF)TP+._CTC<O. By extension of Lemma 2 in [10], it is known that the closed-loop system (10) is of the a-degree relatively stable as (12) and &degree disturbance attenuation as (11)if (Ac÷)TP+P(Ac+aI)4P(F÷F)(F÷iF)TP+._CTC<O.…”

<title>Robust control for structural systems with uncertainties</title>

“…Similar to the proof in Theorem 1, by using Lemma 3 and conditions r > 0.5, (28) and 5mbu <1 in (8) Thus, by Lemma 2 and some additional derivation, the controller (21) is robust and has a-degree relative stability (12) and a &-degree disturbance attenuation (1 1) for the uncertain structural system (5) with all admissible matched uncertainties normbounded as shown in (8). Similar to the proof in Theorem 1, by using Lemma 3 and conditions r > 0.5, (28) and 5mbu <1 in (8) Thus, by Lemma 2 and some additional derivation, the controller (21) is robust and has a-degree relative stability (12) and a &-degree disturbance attenuation (1 1) for the uncertain structural system (5) with all admissible matched uncertainties normbounded as shown in (8).…”

“…and AA = A+AA -BK -LBK, has a robust disturbance attenuation with a prescribed H,-norm constraint 5 (a specified disturbance attenuation index) that satisfies the following: (11) and a robust a-degree relative stability, i.e., Re{A(A)}<-c , (12) where (s) is a transfer function matrix from the disturbance vector w to the observation vector z of the structural system. and AA = A+AA -BK -LBK, has a robust disturbance attenuation with a prescribed H,-norm constraint 5 (a specified disturbance attenuation index) that satisfies the following: (11) and a robust a-degree relative stability, i.e., Re{A(A)}<-c , (12) where (s) is a transfer function matrix from the disturbance vector w to the observation vector z of the structural system.…”

“…In this section, we develop a state feedback controller in (9) to provide a robust a-degree relative stability in (12) and an H, disturbance attenuation with a prescribed index in (1 1) for uncertain structural system (5) and (10). The controller (9) is obtained by solving a Riccati equation as derived in this section.…”

“…For given scalars cx O and >O, if there exist a positive definite matrix P and an adjustable scalar >O such that (A +aI)TP+P(Ac +XI)4(1+cFw)2PFFTP+gcTc<o, (14) then the closed-loop system (10) is of the a-degree relative stable as (12) and ö-degree disturbance attenuation as (1 1). For given scalars cx O and >O, if there exist a positive definite matrix P and an adjustable scalar >O such that (A +aI)TP+P(Ac +XI)4(1+cFw)2PFFTP+gcTc<o, (14) then the closed-loop system (10) is of the a-degree relative stable as (12) and ö-degree disturbance attenuation as (1 1).…”

“…By extension of Lemma 2 in [10], it is known that the closed-loop system (10) is of the a-degree relatively stable as (12) and &degree disturbance attenuation as (11)if (Ac÷)TP+P(Ac+aI)4P(F÷F)(F÷iF)TP+._CTC<O. By extension of Lemma 2 in [10], it is known that the closed-loop system (10) is of the a-degree relatively stable as (12) and &degree disturbance attenuation as (11)if (Ac÷)TP+P(Ac+aI)4P(F÷F)(F÷iF)TP+._CTC<O.…”