Control Designs for Linear Systems Using State-Derivative Feedback

From classical control theory, it is well-known that state-derivative feedback can be very useful, and even in some cases essential to achieve a desired performance. Moreover, there exist some practical problems where the state-derivative signals are easier to obtain than the state signals. For instance, in the following applications: suppression of vibration in mechanical systems, control of car wheel suspension systems, vibration control of bridge cables and vibration control of landing gear components. The main sensors used in these problems are accelerometers. In this case, from the signals of the accelerometers it is possible to reconstruct the velocities with a good precision but not the displacements. Defining the velocities and displacement as the state variables, then one has available for feedback the state-derivative signals. Recent researches about state-derivative feedback design for linear systems have been presented. The procedures consider, for instance, the pole placement problem (Abdelaziz & Valášek, 2004; Abdelaziz & Valášek, 2005), and the design of a Linear Quadratic Regulator (Duan et al., 2005). Unfortunately these results are not applied to the control of uncertain systems or systems subject to structural failures. Another kind of control design is the use of state-derivative and state feedback. It has been used by many researches for applications in descriptor systems (Nichols et al., 1992; A. Bunse-Gerstner & Nichols, 1999; Duan et al., 1999; Duan & Zhang, 2003). However, usually these designs are more complex than the design procedures with only state or state-derivative feedback. In this chapter two new control designs using state-derivative feedback for linear systems are presented. Firstly, considering linear descriptor plants, a simple method for designing a state-derivative feedback gain using methods for state feedback control design is proposed. It is assumed that the descriptor system is a linear, time-invariant, Single-Input (SI) or Multiple-Input (MI) system. The procedure allows that the designers use the well-known state feedback design methods to directly design state-derivative feedback control systems. This method extends the results described in (Cardim et al., 2007) and (Abdelaziz & Valášek, 2004) to a more general class of control systems, where the plant can be a descriptor system. As the first design can not be directly applied for uncertain systems, then a design considering LMI formulation is presented. This result can be used to solve systems with polytopic uncertainties in the plant parameters, or subject to structural failures. Furthermore, it can include as design specifications the decay rate and bounds on the output