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This technical note deals with the robust exact pole placement problem: pole placement algorithms that guarantee a small variation of the assigned poles against possible perturbations. The solution to this problem is related to the solvability of a Sylvester-like equation. Thus, the main issue is to compute a well-conditioned solution to this equation. Also, the best candidate solution must possess the minimal condition number, to reduce sensitivity to perturbation. This problem is cast as a global optimization under linear matrix inequality constraints, for which a numerical convergent algorithm is provided and compared with the most attractive methods in the literature. Index Terms--Condition number, global optimization, linear matrix inequality (LMI), pole placement, Sylvester equation.

This technical note presents two closely related algorithms for the problem of pole placement via static output feedback. The algorithms are based on two different trust region methods and utilize the derivatives of the closed loop poles. Extensive numerical experiments show the effectiveness of the algorithms in practice though convergence to a solution is not guaranteed for either algorithm. While desired poles must be distinct, strategies for dealing with repeated poles are also presented. Index Terms--Eigenvalue derivatives, Levenberg-Marquardt method, Newton's method, pole placement, static output feedback, trust region method.

This note comes back to the hard problem of pole placement by static output feedback: let a triplet of matrices {A; B; C} be given with n state variables, m inputs and p ouputs, find a matrix K such that the spectrum of A + B K C equals a specified set. When mp > m + p, a simple noniterative technique based upon the notion of eigenstructure that, in most cases, assigns m + p roots is proposed. It, therefore, enables the designer to assign the whole of the desired spectrum when m + p = n mp. Index Terms--Eigenstructure, Kimura's condition, pole placement.

This note focuses on the control of continuous-time linear systems subject to time-domain constraints (input amplitude limitation, output overshoot) on closed-loop signals. Using recent results on positive polynomials, it is shown that finding a Youla-Kucera polynomial parameter of fixed degree (hence, a controller of fixed order) such that time-domain constraints are satisfied amounts to solving a convex linear matrix inequality (LMI) optimization problem as soon as distinct strictly negative closed-loop poles are assigned by pole placement. Proceeding this way, time-domain constraints are handled by an appropriate choice of the closed-loop zeros. Index Terms--Linear matrix inequality (LMI), linear systems, pole placement, polynomials, time-domain constraints.

In this paper, we study control by interconnection of linear differential systems. We give necessary and sufficient conditions for regular implementability of a given linear differential system. We formulate the problems of stabilization and pole placement as problems of finding a suitable, regularly implementable sub-behavior of the manifest plant behavior. The problem formulations and their resolutions are completely representation free, and specified only in terms of the system dynamics. Control is viewed as regular interconnection. A controller is a system that constrains the plant behavior through a distinguished set of variables, namely, the control variables. The issue of implementation of a controller in the feedback configuration and its relation to regularity of interconnection is addressed. Freedom of disturbances in a plant and regular interconnection with a controller also turn out to be inter-related. Index Terms--Behaviors, controller implementation, interconnection, pole placement, regular implementability, stabilization.

The key issue for adaptive pole-placement control of linear time-invariant systems is the possible singularity of the Sylvester matrix corresponding to the coefficient estimate. However, to overcome the difficulty, the estimate is modified by several methods which are either nonrecursive and with high computational load or recursive but with random search involved. All of the previous works are done under the assumption that the system is controllable. This paper gives the necessary and sufficient condition, which is weaker than controllability, for the system to be adaptively stabilizable. First, a nonrecursive algorithm is proposed to modify the estimates, and the algorithm is proved to terminate in finitely many steps. Then, with the help of stochastic approximation, a recursive algorithm is proposed for obtaining the modification parameters; it is proved that these modification parameters turn out to be a constant vector in a finite number of steps. This leads to the convergence of the modified coefficient estimates. For both algorithms the Sylvester matrices corresponding to the modified coefficient estimates are asymptotically uniformly nonsingular; thus, the adaptive pole-placement control problem can be solved, i.e., the system can be adaptively stabilized. Index Terms - Adaptive pole-placement control, controllability, parameter modification, stochastic approximation.

A generic procedure for designing a M-periodic controller (sought in the controller canonical form) for the simultaneous placement of the closed-loop poles of M(= 2, 3, 4, ...) discrete, time-invariant plants is presented. The procedure is a two-stage one: first, a set of M simultaneous, linear, polynomial equations, arising out of the M given plants and the corresponding desired closed-loop pole locations, are solved via a generalized Sylvester matrix approach to obtain a set of (M + 1) intermediate polynomials; and next, the controller parameters are obtained solving another set of simultaneous, linear polynomial equations that involve the above intermediate polynomials. Thus, both the computational steps are linear algebraic in nature. A list of the isolated plant configurations for which solutions do not exist is given. An example illustrates the procedure. Index Terms--Periodic control, simultaneous pole-placement/stabilization.

In this paper, we consider indirect adaptive pole-placement control (APPC) of linear multivariable stochastic systems. Instead of the canonical representation often used in the literature, we propose using a nonminimal but otherwise uniquely identifiable pseudo-canonical parameterization that is more suitable for multivariable ARMAX model identification. To identify the plant, we use the weighted extended least-squares (WELS) algorithm, a least-squares method with slowly decreasing weights which was introduced in [1]. The pole-placement controller parameters are then calculated by using a certain perturbation of the parameter estimates such that the linear models corresponding to the perturbed estimates are uniformly controllable and observable. We prove that with a reasonable amount of prior information, the resulting APPC scheme is globally stabilizing and asymptotically self-tuning regardless of the degree of persistency of external excitation. These results represent the most complete study of stochastic multivariable APPC systems to this date. Index Terms - Certainty-equivalent adaptive control, persistency of excitation, self-tuning, stochastic gradient estimation, weighted least squares.

Vibrational control modifies the system dynamics by introducing fast zero-average parametric oscillations. However, this property is constrained by the trace invariance condition which limits the pole placement capabilities. This paper provides the necessary and sufficient conditions under which pole assignment by vibrational control in certain parts of the complex plane is possible. Index Terms - Complex pole assignment, real pole assignment, vibrational control.

This paper addresses the design of state- or output-feedback [H.sub.[infinity]] controllers that satisfy additional constraints on the closed-loop pole location. Sufficient conditions for feasibility are derived for a general class of convex regions of the complex plane. These conditions are expressed in terms of linear matrix inequalities (LMI's), and our formulation is therefore numerically tractable via LMI optimization. In the state-feedback case, mixed [H.sub.2]/[H.sub.[infinity]] synthesis with regional pole placement is also discussed. Finally, the validity and applicability of this approach are illustrated by a benchmark example.

In this paper we study the feedback control problem using an r-channel decentralized dynamic feedback control scheme. We will develop the theory in the behavioral framework. Using this framework we introduce an algebraic parameterization of the space of all possible feedback compensators having a bounded McMillan degree, and we show that this parameterization has the structure of an algebraic variety. We define the pole-placement map for this problem, and we give exact conditions when this map is onto, and almost onto. Finally we provide new necessary and sufficient conditions which guarantee that the set of stabilizable plants is a generic set.

Pole placement algorithm in single input-single output (SISO) systems is discussed with respect to the corresponding, real stability radius of the resulting closed loop polynomial.

In this paper we investigate the limitations of adaptive pole placement control of AR systems when an additive disturbance is present. The magnitude of the disturbance is dependent on past input and output values. It is shown that the control objective of robust stabilization restricts the set of systems one can deal with. We subsequently propose and analyze an adaptive algorithm which can be applied to convex, not necessarily bounded, subsets of the aforesaid set. In particular, it is shown that all signals in the control loop remain bounded.

This paper presents an indirect adaptive control scheme for linear systems which may possibly be a nonminimum phase. The control scheme achieves asymptotical pole placement without either introducing persistent excitation probing signals into the systems or assuming any a priori knowledge on the plant parameters. The system order is the only a priori knowledge required on the plant. The adaptive control law is free from singularities in the sense that the estimated plant model is always controllable. The singularities are overcome by a suitable parameter estimates modification which is based upon standard least squares covariance matrix properties. The analysis of the stability and the global convergence of a closed-loop system is given in detail for both discrete-time and continuous-time systems.

A number of new properties of the complex and real pole placement map (PPM) are derived which relate to the dimension of their images and relate them to known system invariants. It is shown that those two dimensions are equal and that their computation is equivalent to a rank determination of the corresponding differential. A new expression of the differential of the PPM, allows the derivation of relationships between the Markov parameters and the Plucker matrix invariant of the system. Finally, conditions for pole assignability are derived, based on the relationships between the rank of the Plucker matrix and the rank of the differential of the PPM.

To avoid the windup problem in adaptive pole-placement controllers in the presence of a saturating input, an antiwindup compensator based on a generalization to the conditioning technique (Hanus [1], [2]) is introduced to the controller. This modification to the controller also provides a unified approach to investigate the asymptotic properties of the adaptive controllers with a class of antiwindup compensators, at least, when applied to stable processes.

This note presents a globally convergent adaptive control scheme for nonminimum phase systems. The approach is based on a particular periodic pole-placement controller. An ad-hock parameter estimates correction procedure is used to prevent any possible singularities in the control law computation. Furthermore, a lower bound on the estimated model controllability is ensured. The a priori knowledge on the system is reduced to the process order.

In this paper, we incorporate the placement of one real closed-loop pole into a compensator design framework based upon the Youla parameterization, duality theory, and linear programming. This framework has been used to design discrete-time compensators to solve the [l.sub.1] controller design problem as well as other related time-domain optimization problems. Previous work on these problems has focused on deadbeat systems. It is known that these can require high-order controllers. Part of the motivation for this work is to improve the tradeoff between controller order and performance over that using deadbeat control.

In this paper we derive a new rank condition which guarantees the arbitrary pole assignability of a given system by dynamic compensators of degree at most q. By using this rank condition we establish several new sufficiency conditions which ensure the arbitrary pole assignability of a generic system. Our proofs also come with a concrete numerical procedure to construct a particular compensator which assigns a given set of closed-loop poles.

This paper studies the robustness properties of a basic adaptive control algorithm with respect to plant parameter variation as well as modeling errors and bounded disturbances. The algorithm consists of a projected gradient estimator and a pole assignment controller. A unified method of proof is presented for robust stability of both discrete and continuous-time adaptively controlled time-varying systems. The robust performance of an adaptive pole assignment controller is also discussed.

This note proves that the problem of pole placement via static output feedback for linear time-invariant systems is NP-hard. Index Terms--Computational complexity, pole placement, static output feedback.

In this note, under assumptions that the nominal linear implicit system is regular, impulse-free, with all its generalized eigenvalues lying inside some specified regions, sufficient conditions are proposed to preserve the assumed properties when structured perturbations are added into the nominal system. Explicit bounds on perturbations with different structures are given in terms of the spectral radii of some nonnegative matrices.

In this note, an iterative algorithm is proposed for solving the pole assignment by output feedback problem for an nth-order l-input m-output linear system. The algorithm uses a full rank feedback gain matrix and requires only linear equations of low dimension to be solved at each iteration. Application of the proposed algorithm allows all n closed-loop poles to be almost exactly assigned provided that the iterations converge so that deviations of the resulting closed-loop poles from the respective desired values become less than some user-specified tolerance, and m X l is greater than or equal to n. Examples are provided in the note to illustrate the applicability of the proposed method.

In this note, an adaptive pole-assignment controller design for MIMO system with unknown observability indexes or with an over-parameterized system model will be presented. The controller design algorithm and stability issues will be addressed.

This paper discusses analysis and synthesis techniques for robust pole placement in linear matrix inequality (LMI) regions, a class of convex regions of the complex plane that embraces most practically useful stability regions. The focus is on linear systems with static uncertainty on the state matrix. For this class of uncertain systems, the notion of quadratic stability and the related robustness analysis tests are generalized to arbitrary LMI regions. The resulting tests for robust pole clustering are all numerically tractable because they involve solving linear matrix inequalities LMI's and cover both unstructured and parameter uncertainty. These analysis results are then applied to the synthesis of dynamic output-feedback controllers that robustly assign the closed-loop poles in a prescribed LMI region. With some conservatism, this problem is again tractable via LMI optimization. In addition, robust pole placement can be combined with other control objectives, such as [H.sub.2] or [H.sub.[infinity]] performance, to capture realistic sets of design specifications. Physically motivated examples demonstrate the effectiveness of this robust pole clustering technique.