Stable Compensator Research Articles

The periodic optimality of LQ controllers satisfying strong stabilization

An LQ strong stabilization problem is proposed. To determine when a controller with periodic gains is locally superior to a linear time invariant compensator for this problem, a Π test is presented. For systems with strictly proper transfer functions, it is proven that the frequency range where stable periodic controllers based on weak variations about the LTI case can give better performance than stable LTI compensators is finite. In the development, a means to evaluate the second partials of functions with respect to matrix-valued parameters is introduced. For those systems where periodic control is warranted, techniques for designing optimal periodic strongly stabilizing controllers are presented. Two examples detailing the application of the Π test are provided, as well as an optimal periodic controller design example.

Structural stability in discrete singular systems

In this paper, we study the structural stability of singular discrete systems with perturbations in coefficient matrices. Some sufficient and necessary conditions of structurally stable singular discrete systems are given. Two kinds of structurally stable normal compensators are discussed.

On the synthesis of controllers for continuous time LTI systems that achieve a non-negative impulse response

The problem of controlling the oscillatory response of a system has many practical applications. In this paper, I consider the problem of controlling the sign of the closed loop impulse response for a delay-free LTI system. If the impulse response of a stable system does not change sign, then its step response will neither undershoot nor overshoot. In this paper, I will show that such a synthesis is possible iff the open loop system does not have real, non-minimum phase zeros. In addition, I provide a stable compensator that achieves a stable, non-negative impulse response, if there exists one.

Comments on "An exponentially stable adaptive friction compensator"

This note points out a technique error in the above paper (Liao and Chien, IEEE Trans. Automat. Contr., Vol. 45, p. 977-80, 2000). It is shown that without persistence of excitation (PE) of the system signals, the estimated Coulomb friction parameter cannot be guaranteed to converge to its true value. An analysis is provided for the PE condition of parameter asymptotic convergence.

The Periodic Optimality of LQ Controllers Satisfying Strong Stabilization

A II test is presented for determining when a controller with periodic gains is superior to a LTI compensator for a class of LQ strong stabilization problems. For systems with strictly proper transfer functions, it is proven that stable high frequency periodic controllers based on weak variations from the LTI case cannot give better performance than stable LTI compensators. In the development, a means to evaluate the second partials of functions with respect to matrix valued parameters is introduced. Two examples detailing the application of the n test are provided.

An exponentially stable adaptive friction compensator

This note presents a novel adaptive compensation scheme for Coulomb friction in a servocontrol system. An adaptive observer for estimating the unknown Coulomb friction coefficient is also derived on the basis of the Lyapunov technique. In addition, a linearizing control law is developed to compensate for the friction force and obtain the tracking objective. The proposed adaptive compensation guarantees an exponential convergence for state errors and parameter error, and known adaptive schemes guarantee only an asymptotic (or stable) convergence. Simulation results demonstrate the effectiveness of the proposed method for a single-mass servocontrol system.

Adaptive eccentricity compensation

The paper is devoted to the problem of rejecting oscillatory position-dependent disturbances with unknown frequency and unknown amplitude. The considered disturbances are assumed to be produced by eccentricity in mechanical systems and drives. Most of the previous works on eccentricity cancellation assume a time-depending oscillation, we instead assume that the oscillatory disturbance is position-dependent. This leads us to formulate and to globally solve the adaptive cancellation problem using a velocity-dependent internal model of the eccentricity. The proposed control design results in an asymptotically globally stable adaptive eccentricity compensator (AEC). An apparatus with rolling eccentricity has been built to test the controller. The paper presents a series of experimental results showing the improvements of this controller. Also a comparative study with a simple proportional integral (PI) regulator is presented.

Stable ?2-optimal controller synthesis

This paper considers fixed-structure stable ℋ︁2-optimal controller synthesis using a multiobjective optimization technique which provides a trade-off between closed-loop performance and the degree of controller stability. The problem is presented in a decentralized static output feedback framework developed for fixed-structure dynamic controller synthesis. A quasi-Newton/continuation algorithm is used to compute solutions to the necessary conditions. To demonstrate the approach, two numerical examples are considered. The first example is a second-order spring–mass–damper system and the second example is a fourth-order two-mass system, both of which are considered in the stable stabilization literature. The results are then compared with other methods of stable compensator synthesis. Copyright © 2000 John Wiley & Sons, Ltd.

A Computational Method for Determining Strong Stabilizability of n-D Systems

This paper describes an algorithm for the following problem: given two multivariate complex or real polynomials f and g , decide whether there exist complex or real polynomials h and k such that both k and fh+gk have no zero in the unit polydisc. This problem, known as strong stabilizability, is fundamental in control theory, with important applications in designing stable feedback systems with a stable compensator. Our algorithm for solving the problem is formulated based on the cylindrical algebraic decomposition(cad) of an algebraic variety. While recent applications of cad to systems and control have been focused on those problems which have a quantifier elimination formulation, our method is novel in that it explicitly computes some topological properties of an algebraic variety based on the cad to solve the problem for which a quantifier elimination formulation is not readily available.

High-performance state feedback, robust, and output feedback stabilizing control-a systematic design algorithm

Improves a fundamentally new design approach which can generally realize the robustness properties of the state feedback control for the first time, by guaranteeing the resulting feedback system stability. To the best of the authors' knowledge, this improved design also provides the first simple and systematic design procedure and design solution to the strong stabilization problem-stabilize feedback system by a stable output feedback compensator. Finally, high-performance high robustness, and low-controller order can all be systematically achieved by this design.

On the Strong Stabilizability of MIMO n-Dimensional Linear Systems

A plant is strongly stabilizable if there exists a stable compensator to stabilize it. Based on some theorems in complex analysis of several variables proved in this paper, we present necessary conditions for the strong stabilizability of complex and real n-D multi-input multi-output (MIMO) shift-invariant linear plants. For the real case, the condition is a generalization of the parity interlacing property of Youla, Bongiorno, and Lu [ Automatica J. IFAC, 10 (1974), pp. 159--173] for the strong stabilizability of a real one-dimensional MIMO plant. These conditions are also sufficient for the cases of n-D plants with a single output (MISO) or with a single input (SIMO). For general n-D MIMO plants, we do not know if the conditions are sufficient or not. A useful sufficient, but not necessary, condition for the strong stabilizability of a class of n-D $(n\geq 2)$ MIMO plants is given.

Design of stable actuator saturation compensators in the frequency domain

An important practical control problem is the compensation for actuator saturation. Many saturation compensation techniques, widely known as antiwindup schemes, have been proposed, but few guidelines for the design of these techniques are available. Here, frequency domain methods for the design of a class of general saturation compensators are proposed. It is shown that the design of saturation compensators simply reduces to the design of a first order filter. Two methods, the phase shift method and the gain reduction method, are proposed to design the first order filter. The choice between these two methods depends on the shape of the frequency response of the system. An example is presented to illustrate the performance of the proposed methods. Although the proposed design methods are relatively simple, it is shown in the example that they may perform as good as, if not better than, the computational more complex ‘optimal’ controller that includes actuator saturation in its design, and that they are also more robust as actuator saturation becomes more severe.

On the existence of globally stable actuator saturation compensators

Although actuator saturation is a common nonlinear problem in practical control systems, the condition under which compensators exist that globally stabilize systems subject to actuator saturation, is still not well understood. In this paper, it is shown that actuator saturation compensators exist that globally stabilize systems containing up to one integrator. For systems with two integrators, it is established that the compensated system can only be Lagrange stable. The stability result presented in this paper can also be extended to analyse the stability of systems using existing compensation schemes, thus providing a much needed justification for the use of these schemes in practice. The results presented here are illustrated by examples.

Vibration Control of a Cantilever Beam Driven with a Piezoelectric Actuator Using Dynamic Compensator.

Unless the collocation requirement is satisfied, it is difficult to control the vibration of a mechanical system with direct velocity feedback control. In the control of a cantilever beam with a piezoelectric actuator and a displacement sensor, state-space methods are usually used because the actuator and sensor cannot be collocated. Though state-space methods are very useful, a large number of sensors at least equal to the number of eigenvalues concerned are needed to fulfill observability conditions. In this paper, a simple design for a stable dynamic compensator with which modal damping ratios of the mechanical system are greatly increased using a noncollocated actuator and sensor is proposed. Experimentally it is found that three natural frequencies of the cantilever beam are well damped with the designed compensator and calculated results of the mechanical system poles and forced response are consistent with experimental ones.

Coprime factorization approach to robust stabilization of control structures interaction evolutionary model

Stabilization in the presence of uncertainties is a fundamental requirement in the design of feedback compensators for flexible structures. The search for the largest neighborhood around a given design plant for which a single feedback controller produces closed-loop stability can be formulated as an H^ control problem. It has been shown that the use of normalized coprime factor plant descriptions, where the plant perturbations are defined as additive modifications to the coprime factors, leads to a closed-form expression for the maximal-neighborhood boundary allowing optimal and suboptimal HOC compensators to be computed directly without the usual 7iteration. This paper describes an application of normalized coprime factor stabilization theory to the computation of robustly stable compensators for the NASA Control Structures Interaction Evolutionary Model. Results indicate that the suboptimal version of the theory has the potential of providing low authority compensators that are robustly stable for significant regions of variations in design model parameters and additive unmodeled dynamics.

Feedback Synthesis via Interpolation Theory

We show in this paper how interpolation methods can be used in intro ductory feedback control courses for the analytic design of compensators. In partic ular we show how interpolation with units (BIBO stable functions whose inverses are also BIBO stable) can be used to design stable compensators, and how inter polation with BIBO stable functions (Q-parameterization) can be used to design unstable compensators when necessary. The mathematics required is very simple. The only major concept required is that of a BIBO stable transfer function.

Robust strong stabilization via modified Popov controller synthesis

In this paper the authors merge the parameter-dependent Lyapunov function framework used to construct robust Popov controllers with the a priori and a posteriori approaches for obtaining stable compensators. Specifically, the authors derive constructive sufficient conditions that yield robust Lyapunov and asymptotically stable stabilizing Popov dynamic compensators. >

Unified Output Feedback Design and Loop Transfer Recovery

Abastract This paper presents, for the first time, a systematic, general, and simple design procedure for a dynamic output feedback (DOF) compensator whose internal state z(t) converges to a linear transformation of open loop system states Tx(t). This procedure is valid for all observable systems {A,B,C} with more outputs than inputs, and is valid for arbitrary (but stable) compensator poles. The unique property z(t)→Tx(t) guarantees uniquely the following four significant advantages and developments. 1). this compensator implements a perfectly equivalent yet more powerful static output feedback (SOF), which is also equivalent of a constrained state feedback (SF). 2). this compensator guarantees exact and finite gain LTR for this constrained SF, and therefore significantly reduces the restrictions of the existing result of exact or finite gain LTR (restrictions of minimum-phase and rank(CB)=rank(B) are omitted). Therefore, our constrained SF is more reasonably deigned than the normally formulated SF which is designed independent of LTR observer implementation. 3). this compensator perfectly unifies SOF and arbitrary SF with exact LTR observer implementation, as its two extreme structures. 4). this compensator guarantees the critical separation principle of the corresponding closed loop system

Simultaneous stabilization of single-input single-output discrete time systems

The problem of finding one compensator which simultaneously stabilizes a family of single-input-single-output (SISO) discrete-time plants is considered. The family of plants is described by the transfer functions (Pz, q): q in Q), which are generated via a z-transformation. A number of assumptions describing the set of allowable plants are then given. These assumptions include some regularity conditions on the plants and a minimum-phase requirement. The satisfaction of these assumptions guarantees the existence of a strictly proper stable compensator C(z) for simultaneous stabilization. An iterative computation method is provided for control design.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Necessary and sufficient conditions for a nonminimum phase plant to have a recoverable target loop—A stable compensator design for LTR

Abstract In connection with loop transfer recovery of nonminimum phase systems, the purpose of this paper is two-fold: (1) to study the set of recoverable target loops and to establish necessary or/and sufficient conditions for a given plant to have at least one recoverable target loop, and (2) to show that the compensator structure developed earlier in Chen, Saberi and Sannuti [(1991) Automatica , 27, 257–280] for minimum phase systems, can also recover any recoverable target loop for nonminimum phase systems as well while retaining all its advantages over conventional observer based controllers.