Fixed-order Output Feedback Controller Research Articles

External Positivity of Discrete-Time Linear Systems: Transfer Function Conditions and Output Feedback

This paper develops an optimization-based synthesis procedure for fixed-order output-feedback controllers, ensuring a stable closed-loop system with a monotonic step response and rejection of persistent disturbances. This result is based on a unified set of external positivity conditions for discrete-time linear systems. The conditions are derived using an alternative technique for the inverse <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${\mathcal {Z}}$</tex-math></inline-formula> -transform of rational transfer functions based on Bell polynomials and a convenient recurrence relation for the impulse response. Our approach yields a necessary and sufficient condition expressed as polynomial inequalities in the transfer function coefficients. Under additional assumptions, we derive a sequence of weaker but numerically more tractable conditions, leading to an efficient controller synthesis procedure.

Structured Feedback Synthesis for Stability and Performance of Switched Systems

This article addresses the synthesis of fixed-order output feedback controllers for stability and performance of continuous-time switched linear systems with dwell time constraints or arbitrary switching. Specifically, this article starts by considering the stabilization problem, which is addressed by searching for a family of homogeneous polynomial Lyapunov functions parameterized polynomially by the sought controller. In order to conduct this search, polynomials are introduced for approximating the matrix exponential and for quantifying the feasibility of the Lyapunov inequalities. It is shown that a stabilizing controller exists if and only if a condition built solving three convex optimization problems with linear matrix inequalities holds for polynomials of degree sufficiently large. Analogous conditions for the existence of a controller ensuring desired upper bounds on the H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> norm and on the rms gain of the closed-loop system are derived by searching for a family of homogeneous rational Lyapunov functions parameterized rationally by the sought controller.

Finite Frequency Memory Output Feedback Controller Design for T–S Fuzzy Dynamical Systems

In this paper, we will investigate the problem of finite frequency memory fixed-order output feedback controller design for Takagi–Sugeno (T–S) fuzzy affine systems. It is assumed that the disturbances reside in a finite frequency range, i.e., the low, middle, or high frequency range. The objective is to design a memory piecewise affine (PWA) controller by using past output measurements to guarantee the asymptotic stability of the resulting closed-loop system with a prescribed finite frequency $\mathscr H_{\infty }$ performance. Via the system state-input augmentation, a novel descriptor system approach is proposed to facilitate the controller design. All the design conditions are formulated in the form of linear matrix inequalities. It is also proven that the $\mathscr H_{\infty }$ performance can be improved with the memory control strategy. Finally, simulation studies are presented to show the effectiveness of the proposed design method.

LMI-Based Fixed Order Output Feedback Synthesis for Two-Dimensional Mixed Continuous-Discrete-Time Systems

This paper addresses the problem of designing stabilizing fixed order output feedback controllers for two-dimensional mixed continuous-discrete-time systems. The paper starts by addressing this problem in a basic formulation, where the plant is supposed strictly proper, and the output feedback controller is supposed static. It is shown that a necessary and sufficient condition for establishing the existence of a stabilizing feedback controller can be obtained by solving two convex optimization problems with linear matrix inequality (LMI) constraints. The condition is mainly obtained by exploiting a reformulation of the problem based on the stability of a matrix over the complex unit disc, and by introducing suitable tables for establishing stability of polynomials with complex coefficients. The condition is sufficient for any size of the LMIs, and is also necessary for a size sufficiently large. Then, the paper proceeds by addressing a more general formulation, showing that the proposed approach can be used also when the plant is allowed to be nonstrictly proper, and when the output feedback controller is allowed to be dynamic.

Stabilization and Entropy Reduction via SDP-Based Design of Fixed-Order Output Feedback Controllers and Tuning Parameters

This paper addresses the problem of designing fixed-order output feedback controllers and tuning parameters for reducing the instability of linear time-invariant (LTI) systems. Specifically, continuous-time (CT) and discrete-time (DT) LTI systems are considered, whose coefficients are rational functions of design parameters that are searched for in a given semi-algebraic set. Two instability measures are considered, the first defined as the spectral abscissa (CT case) or the spectral radius (DT case), and the second defined as the sum of the real parts of the unstable eigenvalues (CT case) or the product of the magnitudes of the unstable eigenvalues (DT case). Two sufficient conditions are given for establishing either the non-existence or the existence of design parameters that reduce the considered instability measure under a desired value. These conditions require to solve a semidefinite program (SDP), which is a convex optimization problem, and to find the roots of a multivariate polynomial, which is a difficult problem in general. To overcome this difficulty, a technique based on linear algebra operations is exploited, which easily provides the sought roots in common cases by taking into account the structure of the polynomial under consideration. Also, it is shown that these conditions are also necessary by increasing enough the size of the SDP under some mild assumptions. Lastly, it is explained how the proposed methodology can be used to search for design parameters that minimize a given cost function while reducing the instability.

Fixed-Order Output Feedback Control and Anti-Windup Compensation for Active Suspension Systems

Fixed-Order Output Feedback Control and Anti-Windup Compensation for Active Suspension Systems

A New Exterior-point Approach to a Fixed-order Output Feedback Controller Design

This paper deals with fixed-order controller design problems with output-feedback. It is well known that the control problems are difficult to be solved, because they become non-convex problems described by bilinear matrix inequality(BMI). In this paper, a new exterior-point approach to such BMI problems is proposed. This approach produces a controller sequence from a infeasible region to a feasible one. The advantage of this approach is that non-stabilizing controllers can be used as an initial controller. Numerical examples show the efficiency of our method.

Controller Design via Nonsmooth Multidirectional Search

We propose an algorithm which combines multidirectional search (MDS) with nonsmooth optimization techniques to solve difficult problems in automatic control. Applications include static and fixed-order output feedback controller design, simultaneous stabilization, $H_2/H_\infty$-synthesis, and much else. We show how to combine direct search techniques with nonsmooth descent steps in order to obtain convergence certificates in the presence of nonsmoothness. Our technique is efficient when small and medium size controllers for plants with large state dimension are sought. Our numerical testing includes several benchmark examples. For instance, our algorithm needs 0.41\,s to compute a static output feedback stabilizing controller for the Boeing 767 flutter benchmark problem [E. E. J. Davison, IFAC Technical Committee Reports, Pergamon Press, Oxford, 1990], a system with 55 states. The first static controller without performance specifications for this system was obtained in [J. Burke, A. Lewis, and M. Overton, SIAM J. Optim., 15 (2003), pp. 751-779].

Controller Design via Nonsmooth Multi-Directional Search

We propose an algorithm which combines multi-directional search (MDS) with nonsmooth techniques such as bundling to solve several difficult synthesis problems in automatic control. Applications include static and fixed-order output feedback controller design, simultaneous stabilization, H2/H∞ synthesis to cite just a few. We show in which way direct search techniques may be safeguarded by nonsmooth oracles in order to maintain convergence certificates in the presence of nonsmoothness. Our numerical testing includes numerous benchmark examples. For instance, our algorithm needs 0.41 seconds to compute a static output feedback stabilizing controller for the Boeing 767 flutter benchmark problem (E.J. Davison, 1990), a system with 55 states.

Computational complexity of a problem arising in fixed order output feedback design

This paper is concerned with a matrix inequality problem which arises in fixed order output feedback control design. This problem involves finding two symmetric and positive definitive matrices X and Y such that each satisfies a linear matrix inequality and that XY= I. It is well-known that many control problems such as fixed order output feedback stabilization, H ∞ control, guaranteed H 2 control, and mixed H 2/ H ∞ control can all be converted into the matrix inequality problem above, including static output feedback problems as a special case. We show, however, that this matrix inequality problem is NP-hard.

Nonlinear versus linear control in the robust stabilizability of linear uncertain systems via fixed-order output feedback

This paper considers the problem of robustly stabilizing an uncertain system via fixed-order output feedback control. This problem is considered by defining a notion of robust stabilizability with a quadratic storage function. This notion is closely related to the notion of quadratic stabilizability. The main result of the paper shows that if an uncertain system is robustly stabilizable with a quadratic storage function via a nonlinear time-varying controller, then it will also be robustly stabilizable with a quadratic storage function via a linear time-invariant controller of the same order.