Conic Sector Research Articles

Spatial Asymptotic Expansions in the Navier–Stokes Equation

Abstract We prove that the Navier–Stokes equation for a viscous incompressible fluid in ${\mathbb {R}}^{d}$ is locally well-posed in spaces of functions allowing spatial asymptotic expansions with log terms as $|x|\to \infty $ of any a priori given order. The solution depends analytically on the initial data and time so that for any $0<\vartheta <\pi /2$ it can be holomorphically extended in time to a conic sector in ${\mathbb {C}}$ with angle $2\vartheta $ at zero. We discuss the approximation of solutions by their asymptotic parts.

Regional Eigenvalue Assignment in Cooperative Linear Output Regulation

The contribution of this paper is a regional eigenvalue assignment method for cooperative output regulation of heterogeneous linear multiagent systems with an internal model-based distributed dynamic state feedback control law. The proposed method offers an agent-wise local approach to synthesize distributed control gains while assigning, for example, the minimal decay rate and the minimal damping ratio of the overall closed-loop system as desired. Numerical examples demonstrate the efficacy of the proposed method by assigning the eigenvalues of a large-scale system to different regions, specifically, disks, shifted half-planes, and conic sectors.

The Problem of Generalized D-Stability in Unbounded LMI Regions and Its Computational Aspects

We generalize the concepts of D-stability and additive D-stability of matrices. For this, we consider a family of unbounded regions defined in terms of Linear Matrix Inequalities (so-called LMI regions). We study the problem when the localization of a matrix spectrum in an unbounded LMI region is preserved under specific multiplicative and additive perturbations of the initial matrix. The most well-known particular cases of unbounded LMI regions (namely, conic sectors and shifted halfplanes) are considered. A new D-stability criterion as well as sufficient conditions for generalized D-stability are analyzed. Several applications of the developed theory to dynamical systems are shown.

On approximating the nearest Ω‐stable matrix

SummaryIn this paper, we consider the problem of approximating a given matrix with a matrix whose eigenvalues lie in some specific region Ω of the complex plane. More precisely, we consider three types of regions and their intersections: conic sectors, vertical strips, and disks. We refer to this problem as the nearest Ω‐stable matrix problem. This includes as special cases the stable matrices for continuous and discrete time linear time‐invariant systems. In order to achieve this goal, we parameterize this problem using dissipative Hamiltonian matrices and linear matrix inequalities. This leads to a reformulation of the problem with a convex feasible set. By applying a block coordinate descent method on this reformulation, we are able to compute solutions to the approximation problem, which is illustrated on some examples.

Conic sector analysis using integral quadratic constraints

SummaryThis paper interprets the Conic Sector stability results of Zames, and the multivariable generalizations provided by Safonov and Athans, in an integral quadratic constraint context. The ideas are formulated for the case where one element of a feedback interconnection is linear and the other satisfies a conic sector condition. This scenario allows the main stability results to be formulated using integral quadratic constraints, with all finite‐sector cases being captured using this framework. The stability results are expressed using frequency domain inequalities or linear‐matrix inequalities. Some examples show how the main technical result can, in the multivariable case, be used to provide a reduction in conservatism over standard results.

Iterative ‐conic controller synthesis

SummaryThis paper proposes a method to synthesize controllers that minimize an upper bound on the closed‐loop ‐norm while imposing desired controller conic bounds. An initial conic controller is synthesized and iteratively improved. Conic sectors can be used to characterize a variety of input‐output properties, such as gain, phase, and minimum gain. If such plant properties hold robustly to uncertainty present, then closed‐loop stability can be ensured robustly via the Conic Sector Theorem by imposing desired controller conic bounds. Consequently, this paper provides a versatile optimal and robust controller synthesis method. Moreover, it relies only on the solution of convex optimization problems subject to linear matrix inequality constraints, making it readily implementable.

Input–output finite-time mean square stabilization of nonlinear semi-Markovian jump systems

This paper is concerned with the input–output finite-time mean square stabilization problem for a class of nonlinear semi-Markovian jump systems (SMJSs) with time-varying delay. Both time-varying factors of transition rates and states delay are considered to be bounded. Our main purpose is to derive some parameter-dependent sufficient criteria by designing a state feedback controller, to guarantee the input–output finite-time mean square stabilization for the underlying systems where the nonlinear term satisfies the conic sector constraint. Then, the stabilization conditions are obtained by employing a novel Lyapunov–Krasovskii functional, differential form Gronwall’s inequality, and an extra slack variable which is composed of the time-varying transition rates. Finally, a simulation example is provided to illustrate the effectiveness and merits of the proposed method.

Conic-Sector-Based Analysis and Control Synthesis for Linear Parameter Varying Systems

We present a conic sector theorem for linear parameter varying (LPV) systems in which the traditional definition of conicity is violated for certain values of the parameter. We show that such LPV systems can be defined to be conic in an average sense if the parameter trajectories are restricted so that the system operates with such values of the parameter sufficiently rarely. We then show that such an average definition of conicity is useful in analyzing the stability of the system when it is connected in feedback with a conic system with appropriate conic properties. This can be regarded as an extension of the classical conic sector theorem. Based on this modified conic sector theorem, we design conic controllers that allow the closed-loop system to operate in nonconic parameter regions for brief periods of time. Due to this extra degree of freedom, these controllers lead to less conservative performance than traditional designs, in which the controller parameters are chosen based on the largest cone that the plant dynamics are contained in. We demonstrate the effectiveness of the proposed design in stabilizing a power grid with very high penetration of renewable energy while minimizing power transmission losses.

A Comparative Study of Input–Output Stability Results

At present, many similar but disparate input–output (I–O) stability criteria exist. Without means for comparison, it is unclear which result is best used in any given application. This paper proposes a means for comparison between I–O stability results involving norms and inner products of inputs and outputs. The extended conic sector theorem provides a framework for determining which results are least conservative and most broadly applicable. In so-doing, numerous existing stability results are unified and revealed as more powerful than previously thought.

Conic Bounds for Systems Subject to Delays

Linear matrix inequality conditions implying conic bounds are developed for stable linear time-invariant systems with input, output, and/or state delays. Combined with the Conic Sector Theorem, this enables the design of output-feedback controllers ensuring closed-loop input-output stability that is robust to certain maximum delays. These contributions are used in a numerical example to design a nearly-optimal controller with guaranteed input-output-stability for a range of delays and improved performance relative to an $\mathcal{H}_{2}$ -optimal controller designed for the nominally undelayed system.

The Extended Conic Sector Theorem

Input-output stability theory is an indispensable tool in control engineering. However, classic results such as the Conic Sector Theorem, cannot be applied in some important cases. This technical note extends the original Conic Sector Theorem to apply to open-loop unstable systems and those with negative upper conic bounds and incorporates a careful treatment of degenerate conic systems and alterations accounting for nonzero initial conditions. Simulations highlight how the contributions facilitate robust control in the most challenging of circumstances.

A Graph Separation Stability Condition for Non-Planar Conic Systems

Abstract: Results related to an extension of the notion of nonlinear conic systems to the non-planar case are presented. Specifically, parametrization of the non-planar dynamic conic sectors in terms of their central subspace and radius is developed. Relationship between (Q,S,R)-dissipativity and non-planar conicity is studied; in particular, it is shown that any (Q,S,R)-dissipative system is non-planar conic, and a procedure for calculation of the central subspace and the radius is presented. A graph separation stability condition for interconnections of non-planar conic systems is formulated and proven. The results presented in this work will form a background for the future extensions of the scattering transformation-based stabilization techniques to the case of interconnections of non-planar conic systems.

The exterior conic sector lemma

The conic sector theorem is a powerful input–output stability analysis tool, providing a fine balance between generality and simplicity of system characterisations that is conducive to practical stability analysis and robust controller synthesis. Without means to systematically identify and impose exterior conic properties, existing analysis and synthesis methods are limited to employing a special case of the conic sector theorem which may prove overly conservative when considering very strictly passive plants. This paper removes the current limitation by establishing the exterior conic sector lemma, which provides matrix inequalities equivalent to exterior conic bounds for linear time-invariant systems.

Conic-Sector-Based Control in the Presence of Delay

Linear matrix inequality conditions implying interior conic bounds are developed for stable linear time-invariant systems with unknown input delay. Combined with the Conic Sector Theorem, these bounds enable the design of controllers ensuring closed-loop input-output stability that is robust with respect to input delay uncertainty. These contributions are used in a numerical example to design a nearly-optimal controller, which, for moderate delays, achieves improved performance relative to an H2–optimal controller designed for the nominally undelayed system. Moreover, the nearly-optimal controller has guaranteed input-output-stability for all delays, while in simulation the H2–optimal controller is observed to destabilize the closed-loop when adequately large delays are present.

A Generalization of the Scattering Transformation for Conic Systems

A generalized version of the scattering transformation is introduced which allows for rendering of the input-output characteristics of a nonlinear conic system into an arbitrary prescribed conic sector. The transformation is consequently applied to the problem of stabilization of conic systems interconnections. The stability is achieved by rendering the input-output characteristics of the subsystems into complementary cones such that the graph separation stability condition is satisfied. In the presence of communication delays, the scattering-based technique for stabilization of feedback interconnections is generalized to the case of arbitrary conic subsystems.

Conic-sector-based control to circumvent passivity violations

In order to ensure input–output stability for plants experiencing passivity violations, this paper explores the use of the conic sector theorem for both stability analysis and controller design. Given a previously designed controller and a plant that has experienced a (partially unknown) passivity violation, a novel conic bound selection procedure is presented. This procedure can be used to assess input–output stability of the violated plant and original controller via the conic sector theorem. Should input–output stability not be ensured, two original controller synthesis methods are suggested: one is designed to mimic the controller, and the other is inspired by strictly positive real controller synthesis. Both methods guarantee input–output stability by selecting controllers within appropriate conic sectors, and involve only the evaluation of readily solvable linear matrix inequalities and algebraic Riccati equations. Along with details on computational considerations, numerical simulations are provided, demonstrating the effectiveness of the proposed methods.

A Sequential Conic Programming Approach for the Coordinated and Robust Design of Power System Stabilizers

This paper shows that conic programming is an effective tool to solve robust power system stabilizer (PSS) design problems, namely coordinated gain tuning and coordinated phase and gain tuning. Design robustness is achieved by simultaneously considering several operating scenarios. The method is implemented through a sequence of conic programming runs that define a multivariable root locus along which the eigenvalues move. Specifically, the eigenvalues corresponding to the unstable and poorly damped modes are moved to a conic sector in the left half of the s-plane, whereas the eigenvalues corresponding to the well damped modes are constrained to stay within the boundaries of this conic sector. At each step of the solution, the PSS design parameters are restricted in a trust-region such that the computation of the eigenvalue shift based on the residue method holds valid. The proposed method is demonstrated on a 68-bus test system with nine different operating conditions. Comparisons are carried out between conic programming implementations for PSS coordinated gain tuning and for simultaneous tuning of gain and phase characteristics.

Adaptive quantized control for nonlinear uncertain systems

A direct adaptive control framework for nonlinear uncertain systems with input quantizers is developed. The proposed framework is Lyapunov-based and guarantees global ultimate boundedness (practical stability) of the closed-loop system. Specifically, the input quantizers are logarithmic and characterized by sector-bound conditions with the conic sector adjusted at each time instant by the adaptive controller in conjunction with the system response. Furthermore, for practical reasons we assume that the input logarithmic quantizers have deadzone around zero control input and we consider quantizer switching. Finally, a numerical example is provided to demonstrate the efficacy of the proposed approach.

Real-Time Control of Variable Air Volume System Based on a Robust Neural Network Assisted PI Controller

A neural network assisted proportional-plus-integral (PI) control strategy is proposed to improve the air pressure control performance of variable air volume (VAV) system. The neural network is trained online with a normalized training algorithm, which eliminates the requirement of a bounded regression signal to the system. To ensure the convergence of the training algorithm, an adaptive dead zone scheme is employed. Stability of the proposed control scheme is guaranteed based on the conic sector theory. To demonstrate the applicability of the proposed method, real-time tests were carried out on a pilot VAV air-conditioning system and good experimental results are obtained.

Adaptive quantized control for linear uncertain discrete-time systems

A direct adaptive control framework for linear uncertain systems for using communication channels is developed. Specifically, the control signals are to be quantized and sent over a communication channel to the actuator. The proposed framework is Lyapunov-based and guarantees partial asymptotic stability, that is, Lyapunov stability of the closed-loop system states and attraction with respect to the plant states. The quantizers are logarithmic and characterized by sector-bound conditions, with the conic sectors adjusted at each time instant by the adaptive controller, in conjunction with the system response. Furthermore, we extend the scheme to the case where the logarithmic quantizer has a deadzone around the origin so that only a finite number of quantization levels is required to achieve practical stability. Finally, a numerical example is provided to demonstrate the efficacy of the proposed approach.