New Forms of Riccati Equations and the Further Results of the Optimal Control for Linear Discrete-Time Systems

다국어 초록 (Multilingual Abstract)

For linear discrete-time systems, the traditional finite horizon optimal controller is proved to render the closed-loop systems asymptotically stable under some assumptions in literature. In this paper, a new form of finite horizon discrete-time Riccati equation is proposed. It is proved that the new form of fi-nite horizon discrete-time Riccati equation is equivalent to the other three old ones. Based on this new form of finite horizon discrete-time Riccati equation, the finite horizon optimal controller of linear discrete time systems is proved to render the closed-loop system exponentially stable without any assumptions. At the same time, a new form of infinite horizon discrete-time Riccati equation is proposed when the discrete system is controllable or stabilizable. It is proved that the new form of infi-nite horizon discrete-time Riccati equation is equivalent to the other three old ones too. Based on this new form of infinite horizon discrete-time Riccati equation, the infinite horizon optimal controller of linear discrete-time systems is proved to render the closed-loop system exponentially stable when the open-loop system is either controllable or stabilizable. Finally an unstable batch reactor and an unsta-ble inverted pendulum are used to verify the theory results of this paper.

참고문헌 (Reference)

1 Y. S. Shmaliy, "Time-variant linear optimal finite impulse response estimator for discrete state-space models" 26 (26): 95-104, 2012

2 M. Athans, "The uncertainty threshold principle : some fundamental limitations of optimal decision making under dynamic uncertainty" 22 (22): 491-495, 1977

3 A. A. Stoovogel, "The discrete-time Riccati equation related to the H ∞control problem" 39 (39): 686-691, 1994

4 D. L. Kleinman, "The discrete minimum principle with application to the linear regulator problem" Electron. Syst. Lab, Mass. Inst. Tech. 1966

5 B. Alan, "Swing-up control of inverted pendulum systems" 14 (14): 397-405, 1996

6 S. S. Hu, "Stochastic optimal control and analysis of stability of networked control systems with long delay" 39 (39): 1877-1884, 2003

7 D. L. Kleinman, "Stabilizing a discrete, constant, linear system with application to iterative methods for solving the Riccati equation" 19 (19): 252-254, 1974

8 Y. Engin, "Stabilization of deterministic and stochastic-parameter discrete systems" 42 (42): 33-41, 1985

9 R. E. Kalman, "Optimal synthesis of linear sampling control systems using generalized performance indexes" 80 : 1820-1826, 1958

10 D. Peter, "Optimal linear regulators : the discrete-time case" 16 (16): 613-620, 1971

11 T. Pappas, "On the numerical solution of the discrete-time algebraic Riccati equation" 25 (25): 631-641, 1980

12 K. Tohru, "On the matrix Riccati equation for linear systems with a random gain" 21 (21): 770-771, 1976

13 R. E. Kalman, "On the general theory of control" 481 (481): 481-492, 1960

14 A. H. Levis, "On the behavior of optimal linear sampled-data regulators" 13 (13): 343-361, 1971

15 A. H. Levis, "On the Optimal Sampled-data Control of Linear Processes" Inst. Tech., Cambridge 1968

16 R. Nils, "On Newton’s method for Riccati equation solution" 19 (19): 254-255, 1974

17 Xinghua Liu, "On Exponential Stability of Neutral Delay Markovian Jump Systems with Nonlinear Perturbations and Partially Unknown Transition Rates" 제어·로봇·시스템학회 12 (12): 1-11, 2014

18 J. J. Hench, "Numerical solution of the discrete-time periodic Riccati equation" 39 (39): 1197-1210, 1994

19 H. K. Wimmer, "Monotonicity of maximal solutions of algebraic Riccati equations" 5 (5): 317-319, 1985

20 Y. S. Shmaliy, "Linear optimal FIR estimation of discrete time-invariant state space models" 58 (58): 3086-3096, 2010

21 Y. Engin, "General suboptimal approach to the control and estimation of discrete-time system" 16 (16): 1095-1108, 1985

22 Q. X. Zhu, "Finite-horizon optimal control of discrete-time switched linear systems" 2012 : 1-12, 2012

23 B. J. Jackson, "Controllability, observability, realizability and stability of dynamic linear systems" 2009 (2009): 1-32, 2009

24 M. Aoki, "Control of linear discrete-time stochastic dynamic systems with multiplicative disturbance" 20 (20): 388-392, 1975

25 Y. Engin, "Bounds for the eigenvalue of the solution matrix of the algebraic Riccati equation" 16 (16): 815-820, 1985

26 Y. D. Ji, "Bounded sample path control of discrete time jump linear systems" 19 (19): 277-284, 1989

27 A. H. Gary, "An Iterative technique for the computation of the steady state gains for the discrete optimal regulator" 16 (16): 382-383, 1971

28 J. M. Davis, "Algebraic and dynamic Lyapunov equations on time scale" 329-334, 2010

29 P. Lancaster, "Algebraic Riccati Equations" Clarendon Press 1995