On observers in multi-variable control systems

Abstract

Many feedback control system designs require the availability of the state of the controlled plant. Since the plant state variables are not generally available for direct measurement in practice a means for estimating these variables is required. Attention is restricted to plants which are linear, time-invariant, finite-dimensional, dynamic systems in this paper. In this case, necessary and sufficient conditions are derived for which a dynamical observer can be employed to estimate the plant state variables. An observer which satisfies these conditions is said to be compatible. When a compatibly observer is employed, and when the plant input vector is made equal to a linear, time-invariant, matrix multiplication on the estimated plant state vector, it is shown hero that the eigenvalues of the overall system can be made equal to any desired values (complex eigenvalues occurring in conjugate pairs) provided the plant is completely controllable. In particular, the plant can always be stabilized. The use of compatible observers in regulator control systems which for error-free observers are optimal with respect to a quadratic cost index is also investigated. When the observer order is equal to the difference between the dimension of the plant state vector and the dimension of the plant output vector, it is shown that an increase in the cost index generally results. Moreover, the increase in cost cannot be made arbitrarily small by choosing observer eigenvalues with highly negative real parts (this choice causes the exponentially decaying errors in the estimate of the plant state vector to disappear more rapidly). Indeed, there exists a class of initial states for which this procedure leads to arbitarily large increases in cost. When the order of the observer is made equal to the dimension of the plant state vector, it is possible to make the increase in cost arbitrarily small. In order to achieve this result, however, high gains are required in the feedback circuitry. The possibility that a feedback law different from the optimal one is better when a compatible observer is employed in regulators of the kind described above is also studied. It is established that there exists no other feedback law which gives a smaller increase in cost for all initial states when the overall system is required to be stable.