Robust eigenvalue assignment in generalized systems

The classic problem of robust pole placement for linear time invariant systems via state feedback has been studied for several decades, and involves obtaining a gain matrix that will assign a certain desired set of closed-loop poles, while also providing a robust eigenstructure that is insensitive to uncertainties in the system matrices. There are several ways of measuring the robustness of the eigenstructure, and numerous methodologies have appeared in the literature to address the problem. In this paper, results from extensive experiments comparing the performance of a number of methods—including variations on two methods previously proposed by the present authors—against a variety of robustness measures are reported. The size of the matrix gain, runtime and accuracy of the pole placement achieved by each method are also compared. The results show some notable differences between the methods surveyed.

Robust state feedback controllers for linear time-invariant systems with uncertain parameters are considered. Both matched and unmatched uncertainty structures are studied. A basic tool of analysis is a special auxiliary LQR problem without uncertainties. The solution to this problem guarantees unconditionally robust stabilization in the matched case. In the unmatched case, if some testable sufficient conditions are satisfied, the solution of the LQR problem is robustly stabilizing as well. Moreover, robust pole placement into an arbitrary left halfplane is constructively proved to be achievable in the matched case. In the unmatched case, our sufficient conditions for robust stabilization can be modified for the robust pole placement problem but they become void if the required set, into which the eigenvalues ought to be placed, is too far away from the imaginary axis.

We consider the classic problem of pole placement by state feedback. We adapt the Moore eigenstructure assignment algorithm to obtain a novel parametric form for the pole-placing gain matrix, and introduce an unconstrained nonlinear optimization algorithm to obtain a gain matrix that will deliver robust pole placement. Numerical experiments indicate the algorithm's performance compares favorably against several other notable robust pole placement methods from the literature.

Sign function and ANN based pole placement for computing interval controls

In this paper, we show that a linear system with distributed state delay can be approximated by a linear delay-free system that has the same state dimension as the original system if a so-called smallness condition holds. The smallness condition is an inequality which the maximum lag and norms of the system matrices have to satisfy. The eigenvalues of the approximate system correspond to the dominant eigenvalues of the original system with distributed delay. We provide a numerically stable, iterative algorithm to compute the state matrix of the approximate system. Furthermore, we show that, based on the proposed approximation, the stabilisation, pole placement and setpoint tracking control problems of the addressed class of distributed delay systems can be performed using methods developed for delay-free systems. Simulation results are provided to show the applicability of the proposed approximation and control design method.

In this paper, the pole placement problem by constant output feedback for linear time invariant systems is investigated. In particular, is proven that, for the class of linear time invariant systems with m inputs, p outputs, and McMillan degree n, the condition (m + p) > n is necessary for the solution of the arbitrary pole placement problem by constant output feedback.

Adaptive Pole Positioning in MIMO Linear Systems by Periodic Multirate-Input Controllers

In this chapter, a presentation of control theory and engineering as applied to ocean engineering is given. The chapter starts with the fundamentals of systems science and theory, i. e., descriptions of systems in state space and for linear, time-invariant ones in the frequency domain using the Laplace transform as well as with ordinary differential equations with respect to time. Then stability, controllability, and observability with an emphasis to linear, time-invariant systems are presented. Bode plots for sinusoidal steady-state analysis as well as the root locus technique for proportional gain feedback design are presented. For single-input, single-output systems, PID control is introduced as both a pole placement problem as well as in the framework of conventional Ziegler–Nichols methods. Pole placement design with the addition of Luenberger observers is presented for linear, time-invariant systems with any time of inputs and outputs. A brief presentation of digital controller implementations is given. Applications from ocean engineering include control of autonomous underwater vehicles and autopilots for surface vessels.

This paper investigates the stabilization of large scale discrete-time linear control systems by observer-based reduced order controllers. Sufficient conditions are derived for the design of observer-based reduced order controllers for the large scale discrete-time linear control systems by obtaining a reduced order model of the original linear plant using the dominant state of the system. A separation principle has been established in this paper which shows that the observer poles and controller poles can be separated and hence the pole placement problem and observer design problem are independent of each other.KeywordsModel reductionreduced-order controllersstabilizationdominant stateobserversdiscrete-time linear systems

The robust eigenvalue assignment of systems subject to parameter perturbations is addressed. Based on some essential properties of induced norms and matrix measures, the authors derive some sufficient conditions which ensure the assignment of the system's eigenvalues in the specified region irrespective of the system perturbations. The robustness bounds for eigenvalue assignment are obtained without the need to solve the Lyapunov equation. An example is given to illustrate the effectiveness and ease of the proposed analysis methods. >

The pole placement problem asks to find laws to feed the output of a plant governed by a linear system of differential equations back to the input of the plant so that the resulting closed-loop system has a desired set of eigenvalues. Converting this problem into a question of enumerative geometry, efficient numerical homotopy algorithms to solve this problem for general multiple-input-multiple-output systems have been proposed recently. Despite the wider application range of dynamic feedback laws, the realization of the output of the numerical homotopies as a machine to control the plant in the time domain has not been addressed before. In this note, we present symbolic-numeric algorithms to turn the solution to the question of enumerative geometry into a useful control feedback machine. We report on numerical experiments with our publicly available software PHCpack and illustrate its application on various control problems from the literature.

Dielectric electroactive polymer actuators are new important transducers in control system applications. The design of a high performance controller is a challenging task for these devices. In this work, a PI controller was studied for a dielectric electroactive polymer actuator. The pole placement problem for a closed-loop system with the PI controller was analyzed. The limitations of a PI controller in the pole placement problem are discussed. In this work, the analytic PI controller gain rules were obtained, and therefore extension to adaptive control is possible. To minimize the influence of unmodeled dynamics, the robust adaptive control law is applied. Furthermore, analysis of robust adaptive control was performed in a number of simulations and experiments.

A pole assignment algorithm was recently developed by considering the sensitivity of closed loop eigenvalues to each gain element. The theory is extended here to incorporate the Linear Quadratic Regulator formulation so that the closed loop system satisfies optimality as well as eigenvalue constraints. The approach is to compute the sensitivity of the Hamiltonian matrix to changes in the state weighting matrix, in contrast to the sensitivity of the closed loop system in the pole placement problem. Examples from a six degree-of-freedom lumped-mass model and an Euler-Bernoulli pinned beam system are investigated to demonstrate the procedure. It is shown that the resulting closed loop system behavior sufficiently approximates the desired response.

Since Diophantine equations play an important role in solving important real-world problems such as business investment problems, network flow problems, pole placement problems, and data privacy problems, researchers are increasingly interested in developing new techniques for analyzing the nature and solutions of the various Dio phantine equations. In this study we investigate the Diophantine problem 22x+40y = z2, where x, y, z are non-negative integers, and discover that it does not have a non-negative integer solution.

Pole placement by parametric output feedback