- Email: [email protected]
Discussion on: “Adaptative Variable Structure Maneuvering Control and Vibration Reduction of Three-axis Stabilized Flexible Spacecraft”
671
Discussion on: ‘‘ADVS Maneuvering Control and Vibration Reduction of TASFS’’
20. Singh SN. Robust nonlinear attitude control of flexible spacecraft. IEEE Trans Aerosp Electron Syst 1987; AES-23(3): 380–387 21. Singh SN. Nonlinear adaptive attitude control of spacecraft. IEEE Trans Aerosp Electron Syst 1987; AES-23(3): 371–379 22. Singh SN. Rotational maneuver of nonlinear uncertain elastic spacecraft. IEEE Trans Aerosp Electron Syst 1988; 24(2): 114–123
23. Wen JT, Kreutz-Delgado K. The attitude control problem. IEEE Trans Autom Control 1991; AC-36(10): 1148–1162 24. Wie B, Barba PM. Quaternion feedback for spacecraft large angle maneuvers, AIAA J Guid 1985; 8(3): 360–365
Discussion on: ‘‘Adaptative Variable Structure Maneuvering Control and Vibration Reduction of Three-axis Stabilized Flexible Spacecraft’’ Xing-Gang Yan Control & Instrumentation Research Group, Department of Engineering, University of Leicester, University Road, Leicester, LE1 7RH, UK
Sliding mode control is a typical variable structural control, which has received much attention in recent years due to its high robustness to uncertainties. In paper [1], Hu and Ma consider the following class of systems x_ ¼ ðA þ AÞxðtÞ þ ðB þ BÞðuÞ þ fðtÞ
ð1Þ
y ¼ Cx
ð2Þ
where x 2 Rn, u 2 Rm and y 2 Rp are system states, inputs and outputs respectively. An output feedback sliding mode control is proposed to stabilize the system asymptotically and an exponential stabilization result is also presented. Under the assumption that the bounds on the uncertainties are constants, traditional adaptive techniques are employed to estimate/identify the bounds and based on this a robust output feedback sliding mode control is given as well. The developed results are interesting and are applied to a flexible spacecraft. Some nice comparisons are also shown. From my own view, paper [1] belongs to theoretical work. The approach employed is mainly based on linear system theory although the considered control inputs involve nonlinear term but it can not bring any challenging under the standard Assumption 3 which is a limitation on (u). In paper [1], Assumption 2 is imposed on the considered system (1)–(2), which implies that all the uncertainties experienced by the system are matched. E-mail: [email protected]
It is well known that sliding mode control is insensitive to matched uncertainty and thus the associated results about the stability of sliding motion are available without difficulty. It should be noted that mismatched uncertainty can really destroy stability of sliding mode dynamics unless some limitation is imposed on the uncertainty. Recently, some authors have exploited sliding mode control to deal with mismatched uncertainties and it has shown that sliding mode control can deal with mismatched uncertainties under certain conditions. By using sliding mode techniques, a static output feedback control scheme is given in [2] and a dynamical output feedback control strategy is proposed in [3], which show how to deal with mismatched uncertainties when sliding mode techniques are employed. Following the approaches given in [2,3], the results obtained in [1] can be directly extended to the case when system (1)–(2) experiences mismatched uncertainties. It should be noted that a real system may experience an uncertainty bounded by a nonlinear function (see, e.g. [4,5]). Therefore, it is meaningful to study an uncertainty which has nonlinear bound. This implies that it will be more useful if Assumption 2 is extended to the following Assumption 20 Assumption 20 There exist function matrices H(x, t), E(x, t) and d(t) such that A ¼ BHðx, tÞ B ¼ BEðx, tÞ fðtÞ ¼ BdðtÞ
672
Discussion on: ‘‘ADVS Maneuvering Control and Vibration Reduction of TASFS’’
where kHðx, tÞk 1 ðx, tÞ kEðx, tÞk 2 ðx, tÞ < 1 kdðtÞk 3 ðtÞ where 1( ) and 2( ) are known nonnegative functions in Rn R þ and 3( ) is a known nonnegative function in R þ with R þ :¼ {t jt 0}. Obviously, Assumption 20 includes Assumption 2 as a special case. It should be point out that if Assumption 20 is used to replace Assumption 2 in paper [1], the adaptive output feedback control result given in [1] may not be applicable but the results in section III in [1] can be extended to this case only by a little corresponding change using approach proposed in [2,3]. Notably, to study the nonlinear bounds is very useful in reality because normally the designed control is closely connected with the bounds on uncertainties (except the adaptive control where it may require that the uncertain bound is constant and thus it can be identified/estimated based on an adaptive updated law). It is supposed that a system is affected by an uncertainty which has a nonlinear bound (x, t) ¼ sin2(0.1t). A specific control law, for example, u ¼ 5ðx, tÞ has been designed for the system and the corresponding closed-loop system has already have the desired performance. This implies that the control law u ¼ 5 sin2 ð0:1tÞ will satisfy the design requirement. Of course, we may use u ¼ 5 as the control input because the uncertainty is also bounded by a constant ¼ 1 ((x, t) ¼ sin2(0.1t) 1). However, this will result
in unnecessary control consumption if u ¼ 5 instead of u ¼ 5 sin2(0.1t) is employed as the control input. The last point I am going to talk is something relevant to practical engineering issues. As the paper title suggested, this paper seems focused on a real system-flexible spacecraft. However, I think this paper is theoretical since there is no experiment results and how to implement the designed control in flexible spacecraft has not been touched. I understand that some nice simulation results have been presented. However, simulation results some times can not represent what the real situation is. Also, real system always exhibits some nonlinear behavior. Actually system (53) exploited in this paper is an approximation of the real mode. Therefore, to study the nonlinear control for the more accuracy model – nonlinear model for flexible spacecraft seems more valuable and full of challenging. Further study on control implementation in a specific flexible spacecraft is also more interesting.
References 1. Hu Q, Ma G. Maneuvering and vibration reduction of three-axis stabilized flexible spacecrafts using variable structure strategies with improved adaption law 2. Yan XG, Edwards C, Spurgeon SK. Decentralised robust sliding mode control for a class of nonlinear interconnected systems by static output feedback. Automatica 2004; 40: 613–620 3. Yan XG, Spurgeon SK, Edwards C. Dynamic sliding mode control for a class of systems with mismatched uncertainty. Eur J Control 2005; 11: 1–10 4. Shu X, Rigopoulos K, Cinar A. Vibrational control of an exothermic CSTR: Productivity improvement by multiple input oscillations. IEEE Trans Autom Control 1989; 34: 193–196 5. Guo Y, Hill DJ, Wang Y. Nonlinear decentralized control of large-scale power systems. Automatica 2000; 36: 1275–1289
Final Comments by the Authors Q. Hu, G. Ma We thank the discussers for their thoughtful and incisive comments. We agree with the discussers that it would be desirable to study the nonlinear control for more accuracy nonlinear flexible spacecraft model during large slewing maneuver, especially, when the unknown unmatched uncertainties/ disturbances and input saturation are explicitly considered in the near future. Further study on active
vibration control, implementation in a specific flexible spacecraft, spillover problem and comparison with some classical control methods (such as PID control) is more interesting and important as well. While we believe our initial work would initiate further research in application of adaptive variable structure control to flexible spacecraft attitude control.
Discussion on: ‘‘ADVS Maneuvering Control and Vibration Reduction of TASFS’’
20. Singh SN. Robust nonlinear attitude control of flexible spacecraft. IEEE Trans Aerosp Electron Syst 1987; AES-23(3): 380–387 21. Singh SN. Nonlinear adaptive attitude control of spacecraft. IEEE Trans Aerosp Electron Syst 1987; AES-23(3): 371–379 22. Singh SN. Rotational maneuver of nonlinear uncertain elastic spacecraft. IEEE Trans Aerosp Electron Syst 1988; 24(2): 114–123
23. Wen JT, Kreutz-Delgado K. The attitude control problem. IEEE Trans Autom Control 1991; AC-36(10): 1148–1162 24. Wie B, Barba PM. Quaternion feedback for spacecraft large angle maneuvers, AIAA J Guid 1985; 8(3): 360–365
Discussion on: ‘‘Adaptative Variable Structure Maneuvering Control and Vibration Reduction of Three-axis Stabilized Flexible Spacecraft’’ Xing-Gang Yan Control & Instrumentation Research Group, Department of Engineering, University of Leicester, University Road, Leicester, LE1 7RH, UK
Sliding mode control is a typical variable structural control, which has received much attention in recent years due to its high robustness to uncertainties. In paper [1], Hu and Ma consider the following class of systems x_ ¼ ðA þ AÞxðtÞ þ ðB þ BÞðuÞ þ fðtÞ
ð1Þ
y ¼ Cx
ð2Þ
where x 2 Rn, u 2 Rm and y 2 Rp are system states, inputs and outputs respectively. An output feedback sliding mode control is proposed to stabilize the system asymptotically and an exponential stabilization result is also presented. Under the assumption that the bounds on the uncertainties are constants, traditional adaptive techniques are employed to estimate/identify the bounds and based on this a robust output feedback sliding mode control is given as well. The developed results are interesting and are applied to a flexible spacecraft. Some nice comparisons are also shown. From my own view, paper [1] belongs to theoretical work. The approach employed is mainly based on linear system theory although the considered control inputs involve nonlinear term but it can not bring any challenging under the standard Assumption 3 which is a limitation on (u). In paper [1], Assumption 2 is imposed on the considered system (1)–(2), which implies that all the uncertainties experienced by the system are matched. E-mail: [email protected]
It is well known that sliding mode control is insensitive to matched uncertainty and thus the associated results about the stability of sliding motion are available without difficulty. It should be noted that mismatched uncertainty can really destroy stability of sliding mode dynamics unless some limitation is imposed on the uncertainty. Recently, some authors have exploited sliding mode control to deal with mismatched uncertainties and it has shown that sliding mode control can deal with mismatched uncertainties under certain conditions. By using sliding mode techniques, a static output feedback control scheme is given in [2] and a dynamical output feedback control strategy is proposed in [3], which show how to deal with mismatched uncertainties when sliding mode techniques are employed. Following the approaches given in [2,3], the results obtained in [1] can be directly extended to the case when system (1)–(2) experiences mismatched uncertainties. It should be noted that a real system may experience an uncertainty bounded by a nonlinear function (see, e.g. [4,5]). Therefore, it is meaningful to study an uncertainty which has nonlinear bound. This implies that it will be more useful if Assumption 2 is extended to the following Assumption 20 Assumption 20 There exist function matrices H(x, t), E(x, t) and d(t) such that A ¼ BHðx, tÞ B ¼ BEðx, tÞ fðtÞ ¼ BdðtÞ
672
Discussion on: ‘‘ADVS Maneuvering Control and Vibration Reduction of TASFS’’
where kHðx, tÞk 1 ðx, tÞ kEðx, tÞk 2 ðx, tÞ < 1 kdðtÞk 3 ðtÞ where 1( ) and 2( ) are known nonnegative functions in Rn R þ and 3( ) is a known nonnegative function in R þ with R þ :¼ {t jt 0}. Obviously, Assumption 20 includes Assumption 2 as a special case. It should be point out that if Assumption 20 is used to replace Assumption 2 in paper [1], the adaptive output feedback control result given in [1] may not be applicable but the results in section III in [1] can be extended to this case only by a little corresponding change using approach proposed in [2,3]. Notably, to study the nonlinear bounds is very useful in reality because normally the designed control is closely connected with the bounds on uncertainties (except the adaptive control where it may require that the uncertain bound is constant and thus it can be identified/estimated based on an adaptive updated law). It is supposed that a system is affected by an uncertainty which has a nonlinear bound (x, t) ¼ sin2(0.1t). A specific control law, for example, u ¼ 5ðx, tÞ has been designed for the system and the corresponding closed-loop system has already have the desired performance. This implies that the control law u ¼ 5 sin2 ð0:1tÞ will satisfy the design requirement. Of course, we may use u ¼ 5 as the control input because the uncertainty is also bounded by a constant ¼ 1 ((x, t) ¼ sin2(0.1t) 1). However, this will result
in unnecessary control consumption if u ¼ 5 instead of u ¼ 5 sin2(0.1t) is employed as the control input. The last point I am going to talk is something relevant to practical engineering issues. As the paper title suggested, this paper seems focused on a real system-flexible spacecraft. However, I think this paper is theoretical since there is no experiment results and how to implement the designed control in flexible spacecraft has not been touched. I understand that some nice simulation results have been presented. However, simulation results some times can not represent what the real situation is. Also, real system always exhibits some nonlinear behavior. Actually system (53) exploited in this paper is an approximation of the real mode. Therefore, to study the nonlinear control for the more accuracy model – nonlinear model for flexible spacecraft seems more valuable and full of challenging. Further study on control implementation in a specific flexible spacecraft is also more interesting.
References 1. Hu Q, Ma G. Maneuvering and vibration reduction of three-axis stabilized flexible spacecrafts using variable structure strategies with improved adaption law 2. Yan XG, Edwards C, Spurgeon SK. Decentralised robust sliding mode control for a class of nonlinear interconnected systems by static output feedback. Automatica 2004; 40: 613–620 3. Yan XG, Spurgeon SK, Edwards C. Dynamic sliding mode control for a class of systems with mismatched uncertainty. Eur J Control 2005; 11: 1–10 4. Shu X, Rigopoulos K, Cinar A. Vibrational control of an exothermic CSTR: Productivity improvement by multiple input oscillations. IEEE Trans Autom Control 1989; 34: 193–196 5. Guo Y, Hill DJ, Wang Y. Nonlinear decentralized control of large-scale power systems. Automatica 2000; 36: 1275–1289
Final Comments by the Authors Q. Hu, G. Ma We thank the discussers for their thoughtful and incisive comments. We agree with the discussers that it would be desirable to study the nonlinear control for more accuracy nonlinear flexible spacecraft model during large slewing maneuver, especially, when the unknown unmatched uncertainties/ disturbances and input saturation are explicitly considered in the near future. Further study on active
vibration control, implementation in a specific flexible spacecraft, spillover problem and comparison with some classical control methods (such as PID control) is more interesting and important as well. While we believe our initial work would initiate further research in application of adaptive variable structure control to flexible spacecraft attitude control.