Robustness bounds for LQ regulators

A robustification procedure for LQ state feedback design is presented. Such a procedure consists of choosing the state and input weighting matrices according to the kind of uncertainties on the system. Both structured and norm-bounded additive uncertainties are addressed, and upper bounds for the uncertainties that do not destabilize the closed-loop system are presented. Connections with the quadratic stabilizability problem are established     

Singular value properties of LQ regulators

The closed-loop singular values of an LQ regulator transfer function are proved to be no greater than the singular values of the open-loop transfer function and, in the case of output-weighted cost function, to be no greater than the output-weighting parameter     

Relation between pole assignment and LQ regulator

In this paper we demonstrate a connection between the pole assignment technique and the linear optimal regulator technique, namely there exist common Lyapunov functions between two closed-loop systems synthesized by each method. From this property, we derive a useful theorem for pole assignment.     

On improving the robustness of LQ regulators

Soroka and Shaked have shown that a linear quadratic state feedback regulator may suffer from poor robustness where small changes in the parameters of the system can lead to fast unstable closed-loop modes. It is shown there that the stability properties can be improved if possible nonminimum phase behavior is allowed for in modeling the system prior to the controller design.     

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].     

On the robustness of LQ regulators for discrete-time systems

A sufficient condition for quadratic stabilizability of uncertain linear discrete-time systems using LQ regulators is presented. The true system is represented by a nominal model plus additive terms representing the most common types of uncertainties in the state and input matrices. Sufficient robustness bounds as well as a robustification procedure for discrete-time LQ regulators are presented and the conservativeness of the proposed condition is discussed     

Stability of multiloop LQ regulators with nonlinearities--Part I: Regions of attraction

The closed-loop stability of linear, time-invariant systems controlled by linear quadratic (LQ) regulators is investigated when there are nonlinearities in the control channels which lie outside the (0.5, infty) stability sector in regions away from the origin (i.e., saturation-type nonlinearities). An estimate of the region of attraction is obtained which provides methods for selecting the performance function weights for more robust LQ designs.     

An iterative algorithm for pole placement by output feedback

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×t⩾n. Examples are provided to illustrate the applicability of the proposed method     

Efficient recursive adaptive pole placement algorithm

A recursive adaptive pole placement algorithm is presented. The stability and convergence of the algorithm are established respectively. Since a one-step iterative formulation in computing a controller's parameters is used, the on-line computation cost is greatly reduced with respected to the traditional algorithm. The algorithm with feed-forward can follow the arbitrarily bounded output. The algorithm is also extended to a multivariable case. Simulation examples show the efficiency and robustness of the algorithms.     

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.     

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     

Stability of multiloop LQ regulators with nonlinearities--Part II: Regions of ultimate boundedness

This note investigates the closed-loop stability of linear time-invariant systems controlled by linear quadratic (LQ) regulators when there are nonlinearities in the loops which escape the (0.5, infty) stability sector in a bounded region containing the origin. An estimate of the region of ultimate boundedness is obtained, which provides methods for selecting the performance function weights to get better regulator designs.     

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.     

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     

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.     

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.     

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     

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     

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.