The effects of time sampling and quantization in the discrete control of continuous systems

We investigate two types of discretization that can occur in control systems. The first type, time sampling, arises when a discrete time system (such as a computer) is used to control a continuous-time system. The second type, quantization, arises from the finite precision nature of a digital controller, or from the partial observation of the state of a continuous-variable system.;Time sampling destroys disturbance decouplability. As an alternative to completely decoupling the disturbance from the output we consider the sampled disturbance decoupling problem (SDDP) and the arbitrary disturbance attenuation problem (ADAP). In SDDP a controller is to be designed to eliminate the effect of an unknown disturbance at specified sample times. In ADAP a controller is to be designed to attenuate to any degree the effect of an unknown disturbance (including at intersample times.) For each of these problems a design method for a digital multirate controller is proposed and conditions for the solvability of the problem using this design method are given.;Quantization in the feedback loop destroys both stabilizability of the system and predictability of the state. We use the tools of ergodic theory to show that despite this, there are conditions under which we can drive the state of a first-order system into an arbitrarily small region using a static feedback controller, and keep it there. We draw some conclusions about the long term statistical behavior of the resulting system. We next examine the previously considered case where a continuous-variable static state controller is approximated by a quantized static state controller, and extend some available results.;We next extend a result in ergodic theory by proving that expansive higher-order piecewise-linear systems have constrictive Frobenius-Perron operators and we restate the spectral representation theorem for these systems. These results are used in the study of a continuous approximation to a simple buffer/server system. The state of the system, the amount of work in each buffer, behaves chaotically. The statistical properties of the system are analyzed using the new result from ergodic theory.