On the regular implementability of nD systems

In this short paper we study the problem of finding necessary and sufficient conditions for regular implementability by partial interconnection for nD system behaviors. In [M.N. Belur, H.L. Trentelman, Stabilization, pole placement and regular implementability, IEEE Trans. Automat. Control 47(5) (2002) 735–744.] such conditions were obtained in the context of 1D systems. In the present paper we show that the conditions obtained in [M.N. Belur, H.L. Trentelman, Stabilization, pole placement and regular implementability, IEEE Trans. Automat. Control 47(5) (2002) 735–744.] are no longer valid in general in the nD context. We also show that under additional assumptions, the conditions still remain relevant. We also reinvestigate the conditions for regular implementability by partial interconnection in terms of the canonical controller that were obtained in [P. Rocha, Canonical controllers and regular implementation of nD behaviors, Proceedings of the 16th IFAC World Congress, Prague, Czech Republic, 2005.]. Using the geometry of the underlying modules we generalize a result on regular implementability from the 1D to the nD case. Finally, we study how, in the 1D context, the conditions from [M.N. Belur, H.L. Trentelman, Stabilization, pole placement and regular implementability, IEEE Trans. Automat. Control 47(5) (2002) 735–744; P. Rocha, Canonical controllers and regular implementation of nD behaviors, Proceedings of the 16th IFAC World Congress, Prague, Czech Republic, 2005.] are connected.     

Almost disturbance decoupling and pole placement

We revisit here the Almost Disturbance Decoupling Problem (ADDP) (Willems, 1981) by state feedback with the objective to solve ADDP and simultaneously place the maximal number of poles in the closed-loop solution. Indeed, when ADDP is solvable, we show that, whatever be the choice of a particular feedback solution, the obtained closed-loop system always has a set of fixed poles. We characterize these Fixed Poles of ADDP. The other (non-fixed) poles can be placed freely, and we characterize the “optimal” solutions (in terms of ad hoc subspaces and feedbacks) which allow us to solve ADDP with maximal pole placement. From our contribution, which treats the most general case for studying ADDP with maximal, usually partial, pole placement, directly follow the solutions of ADDP with complete pole placement (when there are no ADDP Fixed Poles) and ADDP with internal stability (when all the Fixed Poles of ADDP are stable), without requiring the use of stabilizability subspaces, as in Willems (1981). We extend the concept of Self-Bounded Controlled-Invariant Subspaces (Basile & Marro, 1992) to almost ones. An example is proposed that illustrates our contributions.     

Robustness in partial pole placement

The robustness of state feedback solutions to the problem of partial pole placement obtained by a new projection procedure is examined. The projection procedure gives a reduced-order pole assignment problem. It is shown that the sensitivities of the assigned poles in the complete closed-loop system are bounded in terms of the sensitivities of the assigned reduced-order poles, and the sensitivities of the unaltered poles are bounded in terms of the sensitivities of the corresponding open-loop poles. If the assigned poles are well-separated from the unaltered poles, these bounds are expected to be tight. The projection procedure is described in [3], and techniques for finding robust (or insensitive) solutions to the reduced-order problem are given in [1], [2].     

Controllability, pole placement, and stabilizability for homogenous polynomial systems

In linear system theory the concepts of controllability, pole assignment, and stabilizability are very familiar. These topics are now addressed for the class of homogeneous polynomial systems. First, general results on necessary conditions for the state controllability are derived using results from linear algebra and algebraic geometry. Then pole placement and stabilizability results are developed for the two-dimensional case. Finally, practical examples illustrate the results.     

A note on robust pole placement

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     

The pole placement map, its properties, and relationships to systeminvariants

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

Continuous pole placement for delay equations

In this paper, we describe a stabilization method for linear time-delay systems which extends the classical pole placement method for ordinary differential equations. Unlike methods based on finite spectrum assignment, our method does not render the closed loop system, finite dimensional but consists of controlling the rightmost eigenvalues. Because these are moved to the left half plane in a (quasi-)continuous way, we refer to our method as continuous pole placement. We explain the method by means of the stabilization of a linear finite dimensional system in the presence of an input delay and illustrate its applicability to more general stabilization problems.     

Robust pole placement direct adaptive control

A pole-placement direct adaptive control for not necessarily minimum-phase plants is presented, which is robust with respect to unmodeled dynamics and bounded disturbances. The robustness is achieved by using both a normalized least-squares algorithm with dead zone and appropriate nonlinear feedback. The resulting closed-loop system is shown to be globally stable subject to standard assumptions     

On decentralized dynamic pole placement and feedback stabilization

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     

Output feedback pole placement with dynamic compensators

Derives 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 the authors establish several new sufficiency conditions which ensure the arbitrary pole assignability of a generic system. The authors' proofs also come with a concrete numerical procedure to construct a particular compensator which assigns a given set of closed-loop poles     

Guaranteed dominant pole placement with PID controllers

Pole placement is a well-established design method for linear control systems. Note however that with an output feedback controller of low-order such as the PID controller one cannot achieve arbitrary pole placement for a high-order or delay system, and then partially or hopefully, dominant pole placement becomes the only choice. To the best of the authors’ knowledge, no method is available in the literature to guarantee dominance of the assigned poles in the above case. This paper proposes two simple and easy methods which can guarantee the dominance of the two assigned poles for PID control systems. They are based on root locus and Nyquist plot respectively. If a solution exists, the parametrization of all the solutions is explicitly given. Examples are provided for illustration.     

Adaptive pole placement control of MIMO systems

An adaptive pole-assignment controller design for an MIMO (multi-input-multi-output) system is with unknown observability indexes or with an overparameterized system model is presented. The controller design algorithm and stability issues are addressed     

Feedback norm minimisation with regional pole placement

This article proposes a convex algorithm for minimising an upper bound of the state feedback gain matrix norm with regional pole placement for linear time-invariant multi-input systems. The inherent non-convexity in this optimisation is resolved by a combination of two separate approaches: (1) an inner convex approximation of the polynomial matrix stability region due to Henrion and (2) a novel convex parameterisation of column reduced matrix fraction system representations. Using a sequence of approximations enabled by the above two methods, it is shown that the constraints on closed-loop poles (both pre-specified exact locations and regional placement) define linear matrix inequalities. Finally, the effectiveness of the proposed algorithm is compared with similar pole placement algorithms through numerical examples.     

Global stability of adaptive pole placement algorithms

This paper presents direct and indirect adaptive control schemes for assigning the closed-loop poles of a single-input, single-output system in both the continuous- and discrete-time cases. The resulting closed-loop system is shown to be globally stable when driven by an external reference signal consisting of a sum of sinusoids. In particular, persistent excitation of the potentially unbounded closed-loop input-output data, and hence convergence of a sequential least-squares identification algorithm is proved. The results are applicable to standard sequential least squares, and least squares with covariance reset.     

Adaptive pole placement without excitation probing signals

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     

Dominant pole placement for multi-loop control systems

This paper proposes a method for multi-loop PI controller design which can achieve dominant pole placement for two input two output processes. It is an extension of the original dominant pole design (PID Controllers: Theory, Design, and Tuning, Instrument Society of America, Research Triangle park, NC, 1995.) for SISO systems. Unlike its SISO counterpart, where the controller parameters can be obtained analytically, the multi-loop version amounts to solving some coupled nonlinear equation with complex coefficients, for which closed-form formulae are not possible. A novel approach is developed to solve the equation using a “root trajectory” method, in which the solution to our pole placement problem is found from intersection points between the “root trajectories” and the positive real axis. The design procedure is given and simulation examples are provided to show the effectiveness of the proposed method and comparisons are made with the BLT method.     

Singularity-free adaptive pole placement using periodic controllers

A globally convergent adaptive control scheme for nonminimum phase systems is presented. The approach is based on a particular periodic pole-placement controller. An ad-hoc parameter estimate 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     

Limitations of robust adaptive pole placement control

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     

Stabilization of linear system behaviour by pole shifting

The problem of determining the output feedback control matrix such that the system is as stable as possible is considered from the point of view of minimizing the real part of the least stable eigenvalue of the system. To ensure reasonable control and system behaviour, minimum and maximum constraints are placed on each element of the feedback matrix. An effective algorithm is given to carry out the pole shifting. The feasibility of the approach is illustrated with an eight-state three-feedback linearized model of a divided winding rotor synchronous machine and a linear six-plate gas absorber.     

Multivariable Nyquist criteria, root loci, and pole placement: A geometric viewpoint

Classical control theory is concerned with the topics in our title in the context of single-input/single-output systems. There is now a large and growing literature on the extension of these ideas to the multiinput/multioutput case. This development has posed certain difficulties, some due to the intrinsic nature of the problem and some, we would argue, due to an inadequate reflection on what the multivariable problem calls for. In this paper we describe what seems to us to be the natural multivariable analogs of these concepts from classical control theory. A rather satisfactory generalization of the Nyquist criterion will be described, and a clear analog of the asymptotic properties of the root locus will be obtained in the "multiparameter" case. However, an example is given which illustrates the quite surprising fact that the root locus map is not always continuous at infinite gains. This calls for a new ingredient, a compactification of the space of gains, and perhaps the most interesting new feature in this circle of ideas comes in the area of pole placement. This problem is difficult in the multivariable case, but by establishing a correspondence with a classical set of problems in geometry, we are able to understand its main aspects and to derive results on pole placement by output feedback over the real field.