27-09-2012, 02:44 PM
Stability, controllability and observability
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Stability
Therefore, we can tell the stability of a system by calculating the eigenvalues of A or .
We can easily tell the stability of a system by inspecting its plant matrix A in continuous-time system or in discrete-time system if A or are in Diagonal or Jordan canonical form.
Liapunov stability analysis plays an important role in the stability analysis of control systems described by state space equations. From the classical theory of mechanics, we know that a vibratory system is stable if its total energy is continually decreasing until an equilibrium state is reached. Liapunov stability is based on a generalization of this fact.
Controllability
Controllability: A control system is said to be completely state controllable if it is possible to transfer the system from any arbitrary initial state to any desired state in a finite time period. That is, a control system is controllable if every state variable can be controlled in a finite time period by some unconstrained control signal. If any state variable is independent of the control signal, then it is impossible to control this state variable and therefore the system is uncontrollable.
Observability
Observability: A control system is said to be observable if every initial state X(0) can be determined from the observation of Y(k) over a finite number of sampling periods. The system, therefore, is completely observable if every transition of the state eventually affects every element of the output vector.
In the state feedback scheme, we require the feedback of all state variables. In practical, however, some of the state variables are not accessible for direct measurement. Then it becomes necessary to estimate the un-measurable variables in order to implement the state variable feedback scheme.
Effects of discretization on controllability and observability
When a continuous-time system with complex poles is discretized, the introduction of sampling may impair the controllability and observability of the resulting discretized system. That is, pole-zero cancellation may take place in passing from the continuous-time case to the discrete-time case. Thus, the discretized system may lose controllability and observability.