Control of discrete-time systems with actuator nonlinearities: an LMI approach

Abstract

This paper considers the problem of stabilization of discrete-time systems with actuator nonlinearities. The proposed framework is based on a linear matrix inequality (LMI) approach and directly accounts for robust stability and robust performance over the class of actuator nonlinearities. Furthermore, it is directly applicable to actuator saturation control and provides state feedback controllers with guaranteed domains of attraction. IntroductionIn feedback control systems, actuator nonlinearities, such as saturation, arise frequently in practice and can severely degrade closed-loop system performance and in some cases drive the system to instability. For continuous-time systems, the problem of actuator saturation has been widely studied and an extensive literature is devoted to it (see, e.g., [4,9] and the numerous references therein). For discrete-time systems, Riccati equation-based global and semi-global stabilization techniques for actuator saturation have been developed in [7]. In addition, the ap lication of an anti-windup actuator saturation control {amework to discrete-time systems is given in [SI. Finally, in a recent paper [5], a Riccati equation-based global and local static, output feedback control design framework for discrete-time systems with general time-varying, sector-bounded, input nonlinearities was developed.In this aper, we develop an LMI formulation for fullstate feedtack control of systems with input nonlinearities. As demonstrated in this paper, static, full-state feedback control of discrete-time systems with timevarying, sector-bounded, actuator nonlinearities leads to a matrix inequality that is nonlinear in the decision variables; hence, circumventin a direct application of the LMI theory. By over-boun8ing several terms in the nonlinear matrix inequality (NMI), in this paper, we provide tractable sufficient conditions, in the form of LMIs, for feedback control of systems with time-varying, sectorbounded, actuator nonlinearities. Nomenclature

Optimal Control of Linear Systems With Input Saturation Using Iqcs

“…To demonstrate the potential of our approach, we consider two examples to compare the closed-loop performance, where the first one is from (Fu, 2000) (circle criterion) and the second one is from (Kapila et al, 1999) (Popov criterion).…”

“…In a similar way, we can also recover controllers based on Popov (e.g. (Kapila et al, 1999)) and Zames & Falb criteria if we choose respectively: . Then, we can writeF 12F…”

“…On the other hand, the regional stability problem of input constrained linear system has been addressed by some researchers via circle or Popov criteria and LMI constraints, e.g. (Hindi and Boyd, 1998;Kapila et al, 1999;Kiyama and Iwasaki, 2000). In this case, it should be noted that the LMI framework is probably more appropriate than the Riccatti approach since it enables the use of non-quadratic Lyapunov functions and multipliers via numerical procedure reducing the conservatism.…”

Robust State Feedback Control Through Actuators With Generalized Sector Nonlinearities And Saturation

“…Multiplying diag(P -1 , 1) from the left and right hand sides to (14) and letting Q = P -1 > 0, Y = KQ, we obtain…”

“…For certain applications, especially those that involve large amounts of power, if these nonlinearities are not properly accounted for, they will cause the overall performance to deteriorate, damage the system, or result in instability. Consequently, for many decades, control problems with nonlinear actuators have attracted considerable interest, and no less recently [8,11,12,14,15]. In the celebrated Lur'e problem [16], a general class of nonlinearities are described by means of the so-called sector condition, and many analytic or graphic stability criteria are derived.…”