LMI-Based Digital Redesign of Linear Time-Invariant Systems with State-Derivative Feedback Rodrigo Cardim, Marcelo C. M. Teixeira * , Member IEEE, Fl´ avio A. Faria and Edvaldo Assunc ¸˜ ao Abstract - A simple method for designing a digital state- derivative feedback gain and a feedforward gain such that the control law is equivalent to a known and adequate state feedback and feedforward control law of a digital redesigned system is presented. It is assumed that the plant is a linear controllable, time-invariant, Single-Input (SI) or Multiple-Input (MI) system. This procedure allows the use of well-known continuous-time state feedback design methods to directly design discrete-time state-derivative feedback control systems. The state-derivative feedback can be useful, for instance, in the vibration control of mechanical systems, where the main sensors are accelerometers. One example considering the digital redesign with state-derivative feedback of a helicopter illustrates the proposed method. Index Terms - Digital redesign, state-derivative feedback, control of mechanical systems, linear matrix inequalities. I. I NTRODUCTION In the last years, the proportional and state-derivative feedback have been very useful [1], for instance, to design controllers for the following problems: derivative feedback for multivariable linear systems using Linear Matrix Inequa- lities (LMIs) [2], robust state-derivative pole placement LMI- based designs for linear systems [3], [4], robust stabilization of descriptor linear systems [5], [6], feedback control of singular systems [7], nonlinear control with exact feedback linearization [8], and H ∞ -control of continuous-time sys- tems with state-delay [9]. There exist some practical problems where the state- derivative signals are easier to obtain than the state signals. For instance, for controlled vibration supression of mechani- cal systems, where the main sensors are accelerometers and it is possible to get the velocities with a good precision but not the displacements [10], [11]. Defining the velocities and displacements as the state variables, then one has available for feedback the state-derivative signals. In [11] a method to design a state-derivative feedback gain and a feedforward gain, such that the control law is equivalent to a known and suitable state feedback and feedforward control law was proposed. This method extends the results described in [10] to a more general class of control systems, such as the noninteracting control problem and also presents a theoretical analysis simpler and easier to understand. It was assumed that the plant is a linear controllable, time-invariant, single-input (SI) or multiple- input (MI) system. This procedure allows the designers to R. Cardim, M. C. M. Teixeira, F. A. Faria and E. Assunc ¸˜ ao are with the Department of Electrical Engineering, Faculdade de Engenharia de Ilha Solteira, UNESP-S˜ ao Paulo State University, Ilha Solteira, S˜ ao Paulo, Brazil. * Corresponding author: marcelo@dee.feis.unesp.br. use the well-known state feedback design methods to directly design state-derivative feedback control systems. The designs presented in [1]-[11] considered continuous- time state-derivative feedback. The authors did not find papers with results about the redesign of discrete-time state- derivative feedback. In this paper a new method to design a state-derivative feedback gain for digital control systems is proposed. This method is based on the digital redesign theory proposed in [12], and state-derivative feedback theory for continuous- time systems proposed in [11], described above. The so- called digital redesign problem ([12], [13]) is to design a suitable analogue controller first and then convert the obtained analogue controller to the equivalent digital con- troller maintaining the properties of the original analogously controlled system, by which the benefits of both continuous- time controllers and the advanced digital technology can be obtained [12]. In [12] a simple design methodology for the digital redesign of static state feedback controllers by using Linear Matrix Inequalities (LMI) was presented. The method provides close matching of the states between the original continuous-time system and those of the digitally redesigned system with a guaranteed stability. It is very useful for the solution of the proposed method in this paper. An example considering the pole-placement for the control problem of a helicopter illustrates the proposed design procedure. II. DIGITAL REDESIGN WITH STATE FEEDBACK This section describes the main results presented in [12]. These results will be used in the proof of the new method proposed in this paper. Consider a controllable linear time-invariant plant descri- bed by  ˙ x c (t)= Ax c (t)+ Bu c (t), x c (0) = x 0 , y c (t)= Cx c (t), (1) where x c (t) ∈ R n is the state vector, u c (t) ∈ R m is the control vector, y c (t) ∈ R p is the output vector, and A ∈ R n×n , B ∈ R n×m and C ∈ R p×n are time-invariant matrices. The control vector u c (t) is given by u c (t)= -K c x c (t)+ E c r, (2) where K c ∈ R m×n is the state feedback gain, E c ∈ R m×p is the feedforward gain, and r ∈ R m is the constant reference vector. Note that the gain K c can be specified using well- known methods available in the literature, for instance, such that the poles of the closed-loop of (1) and (2) are placed in the wanted positions [14], [15], [16]. 18th IEEE International Conference on Control Applications Part of 2009 IEEE Multi-conference on Systems and Control Saint Petersburg, Russia, July 8-10, 2009 978-1-4244-4602-5/09/$25.00 ©2009 IEEE 745