A Survey on the Controllability of Bilinear Systems
-
摘要: 双线性系统是一类特殊的非线性系统,广泛存在于现实世界中,如工程、经济、生物、生态等领域,被认为是最接近于线性系统的非线性系统.对双线性系统的研究已历经了近半个世纪. 作为系统最基本的属性,双线性系统可控性的研究一直以来是热点和难点.本文分别对连续双线性系统可控性和离散双线性系统可控性进行讨论, 综述了双线性系统可控性的研究.特别地,报告了近来对离散双线性系统可控性研究的新成果.最后,例举了一些可控的双线性系统例子.Abstract: Bilinear systems are a special class of nonlinear systems, which are widely existing in real world, such as engineering, economics, biology, ecology, etc. Among nonlinear systems, bilinear systems are thought to be the most close to linear systems. The study on such systems has passed through nearly half a century. For the fundamental property, the controllability of bilinear systems has received considerable attention, while the difficulties and challenges still remain. The purpose of this paper is to give a survey on the controllability of bilinear systems through the discussion on the controllability of continuous-time bilinear systems and discrete-time bilinear systems, respectively. Particularly, new results on the controllability are reported for discrete-time bilinear systems. Finally, some examples of controllable bilinear systems are provided.
-
Key words:
- Bilinear systems /
- controllability /
- near-controllability /
- Lie groups /
- Lie algebras
-
[1] Mohler R R. Bilinear Control Processes. New York: Academic Press, 1973[2] Mohler R R, Ruberti A. Theory and Applications of Variable Structure Systems. New York: Academic Press, 1972[3] Bruni C, Pillo G D, Koch G. Bilinear systems: an appealing class of "nearly linear" systems in theory and applications. IEEE Transactions on Automatic Control, 1974, 19(4): 334-348[4] Mohler R R, Kolodziej W J. An overview of bilinear system theory and applications. IEEE Transactions on System, Man and Cybernetics, 1980, 10(10): 683-688[5] Mohler R R, Kolodziej W J. An overview of stochastic bilinear control processes. IEEE Transactions on Systems, Man and Cybernetics, 1980, 10(12): 913-918[6] Mohler R R. Nonlinear Systems (Vol.2): Applications to Bilinear Control. New Jersey: Prentice-Hall, 1991[7] Elliott D L. Bilinear systems. Wiley Encyclopedia of Electrical and Electronics Engineering. New York: Wiley, 1999. 308-323[8] Pardalos P M, Yatsenko V. Optimization and Control of Bilinear Systems: Theory, Algorithms, and Applications. New York: Springer, 2008[9] Elliott D L. Bilinear Control Systems: Matrices in Action. Dordrecht: Springer, 2009[10] Aoki M. Some examples of dynamic bilinear models in economics. Variable Structure Systems with Application to Economics and Biology. New York: Springer, 1975. 163-169[11] Elliott D L. Mathematical models of blood coagulation kinetics. Recent Developments in Variable Structure Systems, Economics and Biology. New York: Springer, 1978. 118-122[12] Sussmann H. Semigroup representations, bilinear approximations of input-output maps, and generalized inputs. Mathematical Systems Theory. Berlin: Springer-Verlag, 1976. 172-192[13] D'Alessandro D. Introduction to Quantum Control and Dynamics. Boca Raton: Taylor and Francis, 2007[14] Cong Shuang, Zheng Yi-Song, Ji Bei-Chen, Dai Yi. Survey of progress in quantum control system. Chinese Journal of Quantum Electronics, 2003, 20(1): 2-9(丛爽, 郑毅松, 姬北辰, 戴谊. 量子系统控制发展综述. 量子电子学报, 2003, 20(1): 2-9)[15] Huang G M, Tarn T J, Clark J W. On the controllability of quantum-mechanical systems. Journal of Mathematical Physics, 1983, 24(11): 2608-2618[16] D'Alessandro D. Small time controllability of systems on compact Lie groups and spin angular momentum. Journal of Mathematical Physics, 2001, 42(9): 4488-4496[17] Altafini C. Controllability of quantum mechanical systems by root space decomposition of su(N). Journal of Mathematical Physics, 2002, 43(5): 2051-2062[18] Albertini F, D'Alessandro D. Notions of controllability for bilinear multilevel quantum systems. IEEE Transactions on Automatic Control, 2003, 48(8): 1399-1403[19] Turinici G. Beyond bilinear controllability: applications to quantum control. International Series of Numerical Mathematics, 2007, 155: 293-309[20] Cong Shuang, Dong Ning. Comparative study on controllability relationship of quantum mechanical systems and bilinear systems. Chinese Journal of Quantum Electronics, 2006, 23(1): 83-92(丛爽, 东宁. 量子力学系统与双线性系统可控性关系的对比研究. 量子电子学报, 2006, 23(1): 83-92)[21] Kucera J. Solution in large of the control problem: x'=[A(1-u)+Bu]x. Czechoslovak Mathematical Journal, 1966, 16(4): 600-623[22] Kucera J. Solution in large of the control problem: x'=[Au+Bv]x. Czechoslovak Mathematical Journal, 1967, 17(1): 91-96[23] Rink R E, Mohler R R. Completely controllable bilinear systems. SIAM Journal on Control and Optimization, 1968, 6(3): 477-486[24] Kucera J. On accessibility of bilinear systems. Czechoslovak Mathematical Journal, 1970, 17(1): 160-168[25] Haynes G W, Hermes H. Nonlinear controllability via Lie theory. SIAM Journal on Control and Optimization, 1970, 8(4): 450-460[26] Elliott D L, Tarn T J. Controllability and observability for bilinear systems. In: Proceedings of the SIAM National Meeting. Seattle, USA: SIAM, 1971[27] Elliott D L. A consequence of controllability. Journal of Differential Equations, 1971, 10(2): 364-370[28] Mohler R R, Rink R E. Reachable zones for equicontinuous bilinear control processes. International Journal of Control, 1971, 14(2): 331-339[29] Brockett R W. System theory on group manifolds and coset spaces. SIAM Journal on Control and Optimization, 1972, 10(2): 265-284[30] Sussmann H J, Jurdjevic V. Controllability of nonlinear systems. Journal of Differential Equations, 1972, 12(1): 95-116[31] Jurdjevic V, Sussmann H J. Control systems on Lie groups. Journal of Differential Equations, 1972, 12(2): 313-329[32] Brockett R W. Lie theory and control systems defined on spheres. SIAM Journal on Applied Mathematics, 1973, 25(2): 213-225[33] Brockett R W. On the reachable set for bilinear systems. Variable Structure Systems with Application to Economics and Biology. New York: Springer, 1975. 54-63[34] Cheng G S J. Controllability of Discrete and Continuous-time Bilinear Systems [Master dissertation], Washington University, USA, 1974[35] Boothby W M. A transitivity problem from control theory. Journal of Differential Equations, 1975, 17(2): 296-307[36] Jurdjevic V, Quinn J. Controllability and stability. Journal of Differential Equations, 1978, 28(3): 381-389[37] Boothby W M, Wilson E N. Determination of the transitivity of bilinear systems. SIAM Journal on Control and Optimization, 1979, 17(2): 212-221[38] Jurdjevic V, Kupka I. Control systems subordinated to a group action: accessibility. Journal of Differential Equations, 1981, 39(2): 186-211[39] Boothby W M. Some comments on positive orthant controllability of bilinear systems. SIAM Journal on Control and Optimization, 1982, 20(5): 634-644[40] Bacciotti A. On the positive orthant controllability of two-dimensional bilinear systems. Systems and Control Letters, 1983, 3(1): 53-55[41] Jurdjevic V, Sallet G. Controllability properties of affine systems. SIAM Journal on Control and Optimization, 1984, 22(3): 501-508[42] Koditschek D E, Narendra K S. The controllability of planar bilinear systems. IEEE Transactions on Automatic Control, 1985, 30(1): 87-89[43] Piechottka U, Koditschek D E, Narendra K S. Comments, with reply, on "The controllability of planar bilinear systems". IEEE Transactions on Automatic Control, 1990, 35(6): 767-768[44] Piechottka U, Frank P M. Controllability of bilinear systems. Automatica, 1992, 28(5): 1043-1045[45] Khapalov A Y, Mohler R R. Reachable sets and controllability of bilinear time-invariant systems: a qualitative approach. IEEE Transactions on Automatic Control, 1996, 41(9): 1342-1346[45] Sachkov Y L. On positive orthant controllability of bilinear systems in small codimensions. SIAM Journal on Control Optimization, 1997, 35(1): 29-35[46] Cheng D Z. Controllability of switched bilinear systems. IEEE Transactions on Automatic Control, 2005, 50(4): 511-515[48] Tarn T J, Elliott D L, Goka T. Controllability of discrete bilinear systems with bounded control. IEEE Transactions on Automatic Control, 1973, 18(3): 298-301[49] Goka T, Tarn T J, Zaborszky J. On the controllability of a class of discrete bilinear systems. Automatica, 1973, 9(5): 615-622[50] Mullis C. On the controllability of discrete linear systems with output feedback. IEEE Transactions on Automatic Control, 1973, 18(6): 608-615[51] Cheng G S J, Tarn T J, Elliott D L. Controllability of bilinear systems. Variable Structure Systems with Application to Economics and Biology. New York: Springer, 1975. 83-100[52] Evans M E, Murthy D N P. Controllability of a class of discrete time bilinear systems. IEEE Transactions on Automatic Control, 1977, 22(1): 78-83[53] Evans M E, Murthy D N P. Controllability of discrete time inhomogeneous bilinear systems. Automatica, 1978, 14(2): 147-151[54] Funahashi Y. Comments on "Controllability of a class of discrete time bilinear systems". IEEE Transactions on Automatic Control, 1979, 24(4): 667-668[55] Hollis P, Murthy D N P. Controllability of two-input, discrete time bilinear systems. International Journal of Systems Science, 1981, 12(4): 485-494[56] Jakubczyk B, Sontag E D. Controllability of nonlinear discrete-time systems: a Lie-algebraic approach. SIAM Journal on Control and Optimization, 1990, 28(1): 1-33[57] Albertini F, Sontag E D. Discrete-time transitivity and accessibility: analytic systems. SIAM Journal on Control and Optimization, 1993, 31(6): 1599-1622[58] Albertini F, Sontag E D. Further results on controllability properties of discrete-time nonlinear systems. Dynamics and Control, 1994, 4(3): 235-253[59] Sontag E D. An eigenvalue condition for sampled weak controllability of bilinear systems. Systems and Control Letters, 1986, 7(4): 313-315[60] San Martin L A B. On global controllability of discrete-time control systems. Mathematics of Control, Signals, and Systems, 1995, 8(3): 279-297[61] Barros C J B, San Martin L A B. Controllability of discrete-time control system on the symplectic group. Systems and Control Letters, 2001, 42(2): 95-100[62] Sirotin A N. On controllability of homogeneous bilinear discrete systems with commutative matrices. Journal of Computer and Systems Sciences International, 2000, 39(2): 173-174[63] Sirotin A N. Zero-controllability sets of a bilinear discrete system with a bounded scalar control. Automation and Remote Control, 2000, 61b(10): 1681-1689[64] Sirotin A N. Characterization of the null-controllability sets of commutative bilinear discrete systems with bounded scalar control. Automation and Remote Control, 2002, 63(8): 1305-1321[65] Sirotin A N. On some problems of control of bilinear discrete systems with triangular matrices. Journal of Computer and System Sciences International, 2003, 42(1): 1-5[66] Djeridane B, Calvet J L. A necessary and sufficient local controllability condition for bilinear discrete-time systems. In: Proceedings of the American Control Conference. Boston, USA: IEEE, 2004. 1755-1757[67] Elliott D L. A controllability counterexample. IEEE Transactions on Automatic Control, 2005, 50(6): 840-841[68] Tie Lin, Cai Kai-Yuan, Lin Yan. Study on the controllability of a class of discrete-time bilinear systems. Journal of Control Theory and Applications, 2010, 27(5): 468-472(铁林, 蔡开元, 林岩. 一类离散双线性系统可控性研究. 控制理论与应用, 2010, 27(5): 468-472)[69] Tie L, Cai K Y, Lin Y, On controllability of discrete-time bilinear systems. Journal of the Franklin Institute, 2011, 348(5): 933-940[70] Tie L, Cai K Y, Lin Y. On uncontrollable discrete-time bilinear systems which are "nearly" controllable. IEEE Transactions on Automatic Control, 2010, 55(12): 2853-2858[71] Tie L, Cai K Y. New results on controllability of discrete-time bilinear systems. In: Proceedings of the 29th Chinese Control Conference. Beijing, China: IEEE, 2010. 520-523[72] Tie L, Cai K Y. On near-controllability of nonlinear control systems. In: Proceedings of the 30th Chinese Control Conference. Yantai, China: IEEE, 2011. 131-136[47] Sontag E D. Controllability is harder to decide than accessibility. SIAM Journal on Control and Optimization, 1988, 26(5): 1106-1118
点击查看大图
计量
- 文章访问数: 2906
- HTML全文浏览量: 67
- PDF下载量: 1382
- 被引次数: 0