On the simultaneous stabilization of three plants

Debowski and Kurylowicz (1986) made use of the method of Saeks et al. (1983) in an attempt to extend the necessary and sufficient conditions for two-plant simultaneous stabilization to the case of three plants. This paper demonstrates via counter-examples that certain aspects of these results are incorrect. An analysis is given which reveals flaws in the arguments used to develop these results. Finally, correct necessary and sufficient conditions are developed for the simultaneous stabilization of two and three plants.     

Strong, simultaneous, and reliable stabilization offinite-dimensional linear time-varying plants

Some issues related to the stabilization of linear, time-varying (LTV) plants are studied. The authors consider the class of finite-dimensional LTV plants that are internally stabilizable by output feedback, and show that every plant in the class can be stabilized by a stable LTV controller; moreover, they give a parameterization of all nonlinear time-varying controllers that stabilize a given LTV plant. Using these results, they show that every finite collection of LTV plants can be simultaneously stabilized by a stable LTV. Finally, the authors prove that every LTV plant can always be stabilized by a pair of simultaneously acting LTV controllers, each reliable against a failure of the other     

On minimal-order stabilization of minimum phase plants

In this note, the problem of minimal-order stabilization in the case where the plant is minimum phase is studied. A low bound on the order of stabilizers is derived and a set of minimal-order stabilizers are characterized. The low bound is related to the number and location of the plant's unstable and lightly damped poles and the number of zeros. How to construct a minimal-order or low-order stabilizer for a general case is also discussed and the algorithm is provided. Numerical examples are given to illustrate the proposed method.     

On minimal order stabilization of single loop plants

It is shown by example that a single loop plant of order n, relative degree 1, with one right half plane pole and one right half plane zero may not be stabilizable by a compensator of order n−2. A further example shows that a minimum phase plant of order n, relative degree r, with one right half plane pole may not be stabilizable by a compensator of order r−2.     

Restriction site mapping for three or more enzymes

Restriction site mapping requires a generator to put forward possible maps and a constraint checker to reject false maps. Ideally these combine to give an algorithm which calculates a sound and complete solution set. Three algorithms for generation are presented and compared. Two decompose a multi-enzyme problem (greater than or equal to 3) into subproblems. The constraint checker is based on separation theory. Some insights into the extent of constraint checking involved in and feasibility of more checking for three or more enzymes are discussed. The trade-off between computation time and the soundness of the solution set is examined.     

The transitivity in simultaneous stabilization

This paper deals with the transmission problem in simultaneous stabilization of time-varying linear systems. We establish two criteria for the transitivity in simultaneous stabilization. In particular, we give a necessary and sufficient condition for certain linear systems to satisfy a strengthened version of transitivity.     

On the relationship between the model order reduction problem and the simultaneous stabilization problem

This note points out the fact that, under certain conditions, the model order reduction problem is very closely tied to the simultaneous stabilization problem. These conditions occur when the reduced order model is to be used to design a suboptimal controller for a high-order plant. Under these conditions the controller must not only stabilize the low-order model, but must also stabilize the high-order plant. The result concerning the existence of a controller that will simultaneously stabilize two plants is used to conclude that the approximate inclusion of any unstable real modes of the high-order plant in the low-order model will guarantee the existence of such a controller.     

Characterization of controllers in simultaneous stabilization

This paper gives a convenient parametrization for the class of all stabilizing controllers for two or more plants. The result represent a generalization of the Youla parametrization of the class of all stabilizing controllers in terms of an arbitrary stable proper transfer function. Although, as expected, the additional constraints for simultaneous stabilization are not readily incorporated into H^∞, H² optimization procedures as in the standard case, there is immediate application of the theory for reduction of controllers which simultaneously stabilize two or more plants.     

A state-space interpretation of simultaneous stabilization

A state-space interpretation of some previously known results on simultaneous stabilization is given. The results reviewed were obtained using the Yuola lemma     

On multiple‐delay output feedback stabilization of LTI plants

We develop a control methodology for linear time‐invariant plants that uses multiple delayed observations in feedback. Using the special coordinate basis, we show that multiple‐delay controllers can always be designed to stabilize minimum‐phase plants, and identify a class of non‐minimum‐phase plants that can be stabilized using these controllers. Copyright © 2009 John Wiley & Sons, Ltd.     

A sufficient condition for simultaneous stabilization

The condition under which it is possible to find a single controller that stabilizes k single-input single-output linear time-invariant systems p_i(s) (i=1,. . .,k) is investigated. The concept of avoidance in the complex plane is introduced and used to derive a sufficient condition for k systems to be simultaneously stabilizable. A method for constructing a simultaneous stabilizing controller is also provided and is illustrated by an example     

Simultaneous stabilization of multiplicatively perturbed plants

Necessary and sufficient conditions are obtained for simultaneous stabilizability of a given linear, time-invariant, multi-input multi-output nominal plant and a multiplicatively perturbed plant, where the multiplicative perturbation is stable. These conditions are derived from the general parity-interlacing-property applicable to any two arbitrary plants and are expressed explicitly in terms of the real-axis poles and zeros of the nominal plant and the perturbation. The class of perturbations for which simultaneous stabilization is achievable are characterized using these conditions. A special class of unknown diagonal perturbations is also considered and simultaneously stabilizing controllers are designed for the nominal plant under any such unknown perturbation.     

On the simultaneous determination of zeros of analytic or sectionally analytic functions

It is shown how the total-step and single-step iterative methods, as well as their improvements, for the simultaneous determination of simple zeros of polynomials can be used (with one slight modification) for the determination of simple zeros of analytic functions (inside or outside a simple smooth closed contour in the complex plane) or sectionally analytic functions (outside their arcs of discontinuity). Numerical results, obtained by the single-step method, are also presented.     

Solution to the “Champagne problem” on the simultaneous stabilisation of three plants

In this paper, the problem proposed in Blondel et al. (SIAM J. Optim. 32 (2) 572–590) as an illustration of the difficulty of the simultaneous stabilization problem has been solved. The same problem was also mentioned in Blondel and Gevers (Math. Control, Signals, Systems 6 (1994) 135–145), where a bottle of good French champagne was offered for its solution. A new open problem is proposed based on the solution of the problem in this paper.     

Constrained stabilization problems for linear plants

Linear systems with magnitude and rate constraints on both the state and control variables are considered. For such systems, semi-global and global constrained stabilization problems are formulated when state feedback controllers are used. Necessary and sufficient conditions for the solvability of the formulated problems are developed. Moreover, design methodologies for such constrained stabilization problems are presented. An important aspect of our development here is a taxonomy of constraints to show clearly for what type of constraints what can or cannot be achieved.     

A geometric solution to the simultaneous bounded domain stabilization problem

This paper presents an algorithmic method for solving the two-plant simultaneous bounded domain stabilization problem for SISO LTI systems. This problem has no closed form solution. The solution provides robust performance in the presence of sensor or actuator failure, or other major parameter changes. Vidyasagar (1987) studied a similar problem involving partially bounded stability domains. However, stability with respect to partially bounded domains only partially bound performance characteristics, such as control energy and transient response. The current investigation gives necessary conditions for simultaneous bounded domain stability and demonstrates a geometry-based solution algorithm which can be automated. The possible solutions to the problem and the admissible solutions are represented as sets of points in Euclidean space. The solution to the problem is found by using computational geometric techniques to detect points in the intersection of these two sets, if there is one, and deducing the simultaneous stabilizing compensator design from the points found in the intersection.     

Fault tolerant control: a simultaneous stabilization result

This note discusses the problem of designing fault tolerant compensators that stabilize a given system both in the nominal situation, as well as in the situation where one of the sensors or one of the actuators has failed. It is shown that such compensators always exist, provided that the system is detectable from each output and that it is stabilizable. The proof of this result is constructive, and a worked example shows how to design a fault tolerant compensator for a simple, yet challenging system. A family of second order systems is described that requires fault tolerant compensators of arbitrarily high order.     

Rank-one LMI approach to simultaneous stabilization of linear systems

Following a polynomial approach to control design, the simultaneous stabilization by a controller of given fixed order of a family of SISO linear systems is interpreted as an NP-hard BMI feasibility problem. Upon formulating this BMI problem as an LMI problem with an additional non-convex rank constraint, two simultaneous stabilization methods are then proposed. The first method is a heuristic algorithm performing rank minimization by potential reduction. The second method hinges upon necessary conditions and sufficient conditions for simultaneous stabilization derived from geometric properties of the intersection of a set of ellipsoids. Both methods are then illustrated by numerical examples.     

Adaptive robust simultaneous stabilization of multiple n-degree-of-freedom robot systems

In this paper, the adaptive robust simultaneous stabilization problem of uncertain multiple n-degree-of-freedom (n-DOF) robot systems is studied using the Hamiltonian function method, and the corresponding adaptive L2 controller is designed. First, we investigate the adaptive simultaneous stabilization problem of uncertain multiple n-DOF robot systems without external disturbance. Namely, the single uncertain n-DOF robot system is transformed into an equivalent Hamiltonian form using the unified partial derivative operator (UP-DO) and potential energy shaping method, and then a high dimensional Hamiltonian system for multiple uncertain robot systems is obtained by applying augmented dimension technology, and a single output feedback controller is designed to ensure the simultaneous stabilization for the higher dimensional Hamiltonian system. On this basis, we further study the adaptive robust simultaneous stabilization control problem for the uncertain multiple n-DOF robot systems with external disturbances, and design an adaptive robust simultaneous stabilization controller. Finally, the simulation results show that the adaptive robust simultaneous stabilization controller designed in this paper is very effective in stabilizing multi-robot systems at the same time.