Discrete-time design of state-derivative feedback control laws

State-derivative feedback control laws can be very useful in the control of systems using accelerometers as sensors. Moreover, in cases where both state and state derivative measurements are available, a state-derivative feedback controller can be employed as a backup alternative in the case of sensor failure. The present work is concerned with the design of such a controller in a discrete-time framework, assuming that the plant input is kept constant between sampling times, which is typically the case in digital control implementations. More specifically, this paper proposes a method to design a state-derivative feedback gain matrix in order to obtain equivalence to a given discrete-time state feedback control law. It is assumed that the plant is linear and time-invariant, and that the sampling of the state-derivative occurs just before the update of the control value. The proposed method consists of a direct digital design in the sense that it does not require the preliminary design of a continuous-time controller. For illustration, a simulated example involving the suppression of vibrations in a mechanical system is presented. The results show that the state-derivative feedback controller provides suitable damping of the vibrations in the case of failure of a displacement sensor employed by the conventional state feedback controller, even in the presence of measurement noise and parameter variations.