系统的能控性和能观性英文版.doc

Unit 13 Controllability and Observability A system is said to be controllable at timeif it is possible by means of an unconstrained control vector to transfer the system from any initial stateto any other state in a finite interval of time. A system is said to be observable at timeif, with the system in state, it is possible to determine this state from the observation of the output over a finite time interval. The concepts of the controllability and observability were introduced by Kalman. They play an important role in the design of control systems in state space. In fact, the conditions of controllability and observability may govern the existence of a complete solution of the control system design problem. The solution to this problem may not exist of the system considered is not controllable. Although most physical systems are controllable and observable, corresponding mathematical models may not possess the property of controllability and observability.( Complete State Controllability of Continuous-Time Systems Consider the continuous-time system (13. 1) where X=state vector (n-vector) u=control signal (scalar) A= matrix B= matrix The system described by Equation (13. 1) is said to be state controllable atif it is possible to construct an unconstrained control signal that will transfer an initial state to any final state in a finite time interval. If every state is controllable, then the system is said to be completely state controllable. We shall now derive the condition for complete state of controllability. Without loss of generality, we can assume that the final state is the origin of the state space and that the initial time is zero,or. The solution of Equation (13. 1) is Applying the definition of complete state controllability just given, we have or (13. 2) Andcan be wr