Real-μ bounds based on fixed shapes in the Nyquist plane: Parabolas, hyperbolas, cissoids, nephroids, and octomorphs

In this paper we introduce new bounds for the real structured singular value. The approach is based on absolute stability criteria with plant-dependent multipliers that exclude the Nyquist plot from fixed plane curve shapes containing the critical point − + _jO. Unlike half-plane and circle-based bounds the critical feature of the fixed curve bounds is their ability to differentiate between the real and imaginary components of the uncertainty. Since the plant-dependent multipliers have the same functional form at all frequencies, the resulting graphical interpretation of the absolute stability criteria are frequency independent in contrast to the frequency-dependent off-axis circles that arise in standard real-μ bounds.     

Robust discrete-time control design with time-varying parametric uncertainties

A novel multiplier approach for robust controller design in discrete-time systems with real, time-varying parametric uncertainty is presented. An important feature of our approach is that bounds on the rate of variation of the uncertain parameters are assumed and, unlike in most related approaches, dynamic multipliers are obtained that utilize this information. A convex minimization procedure formulated as an LMI problem is presented to obtain multipliers that satisfy the robustness conditions derived. Such conditions are transformed to an equivalent scaled H_∞ norm condition and a μ/k_m-synthesis approach is proposed to design robust controllers. Copyright © 1999 John Wiley & Sons, Ltd.     

A duality-based approach to the multiobjective H 2/H ∞ optimisation problem

In this article, a duality approach to multiobjective H ₂/H _∞ problems is pursued in which real-rational, para-Hermitian multipliers and real-valued ones are associated to H _∞ and (as usual) H ₂ constraints, respectively. It is shown that the maximisation of a dual functional over all such multipliers yields the optimal value of the original multiobjective H ₂/H _∞ problem. To compute lower bounds on the latter and the corresponding approximate solutions to the original problem, the maximisation of the dual functional over linearly-parameterised, finite-dimensional classes of real-rational multipliers is shown to be equivalent to semi-definite, linear programming problems – once the optimal multipliers in such a class are obtained, the corresponding approximate solutions can be computed from an unconstrained H ₂ problem. Iterative modification of such classes is discussed to obtain increasing sequences of lower bounds on the optimal value of the original problem. This is done on the basis of (locally) increasing directions for the dual functional which go beyond the finite-dimensional class of multipliers considered in a given step. Finally, a numerical example is presented to illustrate the way the presented results can lead to approximate solutions to the multiobjective H ₂/H _∞ problem together with tight estimates of the corresponding deviation from its optimal value.     

Robust H 2 control based on dynamic multipliers

In this article, upper bounds on the worst-case H ₂ performance index relative to structured, feedback perturbations are considered which are based on the minimisation of dual Lagrangean functionals over linearly-parametrised, finite-dimensional classes of dynamic multipliers. It is shown that the minimisation problems in question can be recast as optimisation problems with linear cost functional and Linear matrix inequality (LMI) constraints. An iterative scheme is suggested to generate linearly-parametrised classes of multipliers of increasing dynamic order so that progressively tighter upper bounds can be obtained, as illustrated by two simple numerical examples. Finally, with a view to synthesis procedures based on ‘D–K iterations’ relative to multipliers and controllers, it is shown that the minimisation of the upper-bounds corresponding to given multipliers with respect to linearly-parametrised classes of Youla parameters can also be cast as linear-cost/LMI-constraint problems.     

All stability multipliers for repeated MIMO nonlinearities

For block structured monotone or incrementally positive n-dimensional nonlinearities, the largest class of convolution operators (stability multipliers) that preserve positivity is derived. These multipliers can be used in conjunction with positivity and IQC stability criteria to evaluate stability and robustness of MIMO feedback systems.     

On stability of linear time-delay systems with multiple delays

In this article, delay-dependent stability criteria for linear retarded and neutral systems with multiple delays are proposed by employing the Lyapunov-Krasovskii functional approach and integral inequality. By the N-segmentation of delay length, we obtain more relaxed results on the delay bounds which guarantee the asymptotic stability of the linear retarded and neutral systems with multiple delays. Simulation results show that the stability boundary calculated by the proposed stability criteria are very near to the real stability boundary of the example systems. Moreover, it is shown that the proposed stability criteria are less conservative than several other existing criteria.     

On upper bounds on worst-case H 2 performance obtained with dynamic multipliers

In this note, an iterative procedure is presented for obtaining a decreasing sequence of upper bounds on the worst-case H ₂ performance of a given stabilizing controller in the presence of normalized coprime-factor perturbations. To obtain such bounds, a descent procedure is introduced for a dual Lagrangean functional which gives upper bounds on the worst-case H ₂ performance index and is defined on the set of real-rational and non-negative functions (dynamic multipliers). Specific ways are presented for selecting feasible and descent directions in this set and lower bounds are derived for the corresponding decreases on the dual functional. At any step, the dynamics of the current multiplier gives rise to a linear class over which the optimization of the dual functional is shown to be equivalent to linear optimization subject to linear matrix inequalities (LMI). This allows for a combination of function space and LMI techniques in the process of obtaining increasingly tighter upper bounds on worst-case H ₂ performance, as illustrated in a numerical example.     

Multi-periodic repetitive control system: a Lyapunov stability analysis for MIMO systems

A multi-input–multi-output (MIMO) repetitive control problem of tracking and disturbance rejection is considered when both reference and disturbance signals are finite linear combinations of periodic but not necessarily sinusoidal signals. Lyapunov stability analyses under a positive real condition (and a natural relaxation) and exponential stability under a strict positive real condition are provided together with bounds on the induced L ₂ and RMS gains of the closed loop system. It is shown that similar Lyapunov stability results apply when the plant is a positive real state-delay system. Extension of the analyses to a class of non-linear systems is discussed and indicates a good degree of robustness in the design.     

On control for linear systems with interval time-varying delay

This paper deals with the problem of delay-dependent robust H_∞ control for linear time-delay systems with norm-bounded, and possibly time-varying, uncertainty. The time-delay is assumed to be a time-varying continuous function belonging to a given interval, which means that the lower and upper bounds for the time-varying delay are available, and no restriction on the derivative of the time-varying delay is needed, which allows the time-delay to be a fast time-varying function. Based on an integral inequality, which is introduced in this paper, and Lyapunov–Krasovskii functional approach, a delay-dependent bounded real lemma (BRL) is first established without using model transformation and bounding techniques on the related cross product terms. Then employing the obtained BRL, a delay-dependent condition for the existence of a state feedback controller, which ensures asymptotic stability and a prescribed H_∞ performance level of the closed-loop systems for all admissible uncertainties, is proposed in terms of a linear matrix inequality (LMI). A numerical example is also given to illustrate the effectiveness of the proposed method.     

Liapunov functions and stability criteria for nonlinear systems with multiple critical eigenvalues

Efficient criteria are derived via explicit construction of Liapunov functions for local asymptotic stability inference of nonlinear systems, whose linearizations possessc 2 critical modes at an equilibrium point. The stability criteria are obtained in the context of two novel notions, relaxed definiteness and relaxed stability. A real symmetricc×c matrixQ isrelaxed negative definite ifw ^TQw<0 for any 0w ₊ ^c , ₊=[0, ); a matrixR isrelaxed stable if there is aP>0 such thatPR+R ^TP is relaxed negative definite. The construction leads to some characterizations of the nonlinear system's local structure,in the sense of Liapunov, and the so-called stability characteristic matrices and tensors. It is shown that a nonlinear system with multiple critical modes is locally asymptotically stable generically if the stability characteristic matrix is relaxed stable and less generically if the stability characteristic tensor is trivial or degenerate in certain way and the perturbed stability characteristic matrix is relaxed stable.     

On system gains for linear and nonlinear systems

System gains, and bounds for system gains, are determined for stable linear and nonlinear systems in different signal setups which include ℓ_p signal setups and certain persistent signal generalizations of the ℓ₂ and ℓ₁ signal setups. These results show that robust H_∞ and ℓ₁ control generalize to very versatile persistent signal settings. Relationships between different system gains are also derived. Finally, an application of nonlinear system gain bounds is given by establishing induced ℓ_∞ modelling error bounds for a class of (generalized) piecewise linear systems approximated with simpler linear time-invariant models.     

An implicit small gain condition and an upper bound for the real structured singular value

In this paper we develop an upper bound for the real structured singular value that has the form of an implicit small gain theorem. The implicit small gain condition involves a shifted plant whose dynamics depend upon the uncertainty set bound and, unlike prior bounds, does not require frequency-dependent scales or multipliers. Numerical results show that the implicit small gain bound compares favorably with real-μ bounds that involve frequency-dependent scales and multipliers.     

Robust Analysis of Discrete‐Time Lur'e Systems with Slope Restrictions Using Convex Optimization

This paper considers robust stability and robust performance analysis for discrete‐time linear systems subject to nonlinear uncertainty. The uncertainty set is described by memoryless, time‐invariant, sector bounded, and slope restricted nonlinearities. We first give an overview of the absolute stability criterion based on the Lur'e‐Postkinov Lyapunov function, along with a frequency domain condition. Subsequently, we derive sufficient conditions to compute the upper bounds of the worst case H₂ and worst case H∞ performance. For both robust stability testing and robust performance computation, we show that these sufficient conditions can be readily and efficiently determined by performing convex optimization over linear matrix inequalities.     

The dielectric properties prediction of the vegetation depending on the moisture content using the deep neural network model

In this paper, dielectric properties of citrus leaves are predicted with long short‐term memory (LSTM) which is one of the well‐known deep neural network (DNN) models and real‐time measurements for any moisture content (MC) values in the range of 4.90 to 7.05 GHz at a fixed temperature of 24°C for microwave applications, as a novelty. Firstly, S‐parameters of samples are measured with WR‐159 waveguide and Waveguide Transmission Line Method. In addition, the MCs of samples depending on their weights are calculated. Thus, the dataset depending on various MC and frequency is obtained with the measurement results to both training and testing the DNN model. Secondly, a total of 4000 datasets are obtained, 80% of which is used for training, and 20% for testing. The proposed DNN model consists of four inputs (f, MC, S₁₁, and S₂₁) and two outputs (ε′ and ε″). Finally, the dielectric parameters for the desired MC and f are displayed with the graphical user interface in real‐time. Success criteria for the prediction such as mean absolute error, root mean squared error, mean absolute percentage error, and R‐squared are calculated. The results indicated that there is good agreement between the measured and predicted ones. R‐squared are calculated as 0.962 and 0.968 for ε′ and ε″, respectively.     

Strong Minimax Lower Bounds for Learning

Minimax lower bounds for concept learning state, for example, that for each sample size n and learning rule g _n , there exists a distribution of the observation X and a concept C to be learnt such that the expected error of g _n is at least a constant times V/n, where V is the vc dimension of the concept class. However, these bounds do not tell anything about the rate of decrease of the error for a fixed distribution-concept pair.In this paper we investigate minimax lower bounds in such a (stronger) sense. We show that for several natural k-parameter concept classes, including the class of linear halfspaces, the class of balls, the class of polyhedra with a certain number of faces, and a class of neural networks, for any sequence of learning rules {g _n }, there exists a fixed distribution of X and a fixed conceptC such that the expected error is larger than a constant timesk/n for infinitely many n. We also obtain such strong minimax lower bounds for the tail distribution of the probability of error, which extend the corresponding minimax lower bounds.     

Lower Bounds for Dynamic Algebraic Problems

We consider dynamic evaluation of algebraic functions (matrix multiplication, determinant, convolution, Fourier transform, etc.) in the model of Reif and Tate; i.e., if f(x₁,…, x_n)=(y₁, …, y_m) is an algebraic problem, we consider serving online requests of the form “change input x_i to value v” or “what is the value of output y_i?” We present techniques for showing lower bounds on the worst case time complexity per operation for such problems. The first gives lower bounds in a wide range of rather powerful models (for instance, history dependent algebraic computation trees over any infinite subset of a field, the integer RAM, and the generalized real RAM model of Ben-Amram and Galil). Using this technique, we show optimal Ω(n) bounds for dynamic matrix–vector product, dynamic matrix multiplication, and dynamic discriminant and an Ω( ) lower bound for dynamic polynomial multiplication (convolution), providing a good match with Reif and Tate's O( ) upper bound. We also show linear lower bounds for dynamic determinant, matrix adjoint, and matrix inverse and an Ω( ) lower bound for the elementary symmetric functions. The second technique is the communication complexity technique of Miltersen, Nisan, Safra, and Wigderson which we apply to the setting of dynamic algebraic problems, obtaining similar lower bounds in the word RAM model. The third technique gives lower bounds in the weaker straight line program model. Using this technique, we show an Ω((log n)²/log log n) lower bound for dynamic discrete Fourier transform. Technical ingredients of our techniques are the incompressibility technique of Ben-Amram and Galil and the lower bound for depth-two superconcentrators of Radhakrishnan and Ta-Shma. The incompressibility technique is extended to arithmetic computation in arbitrary fields.     

Subcube Fault-Tolerance in Hypercubes

We consider the problem of determining the minimum number of faulty processors, K(n, m), and of faulty links, λ(n, m), in an n-dimensional hypercube computer so that every m-dimensional subcube is faulty. Best known lower bounds for K(n, m) and λ(n, m) are proved, several new recursive inequalities and new upper bounds are established, their asymptotic behavior for fixed m and for fixed n − m is analyzed, and their exact values are determined for small n and m. Most of the methods employed show how to construct sets of faults attaining the bounds. An extensive survey of related work is also included, showing connections to resource allocation, k-independent sets, and exhaustive testing.     

Some counterexamples in robust stability theory

This note deals with the problem of determining if a linear system whose characteristics polynomial depends multilinearly on n independent uncertain real parameters Δ_i, I = 1,…,n, is robustly stable. It is shown by example that a polynomial in n variables may have a unique real root, and that this observation disposes of several natural conjectures in robust stability theory. In particular, we show that, in a certain sense, there are no ‘edge’ or ‘m-dimensional face’ Kharitonov-like theorems for the general multilinear case. The result holds even when restricted to that subset of multilinear functions which can be written in the form f(Δ₁,…, Δ_n) = det(I + diag(Δ₁,…,Δ_n)M) for some complex matrix M.     

Worst‐case stability and performance with mixed parametric and dynamic uncertainties

This work deals with computing the worst‐case stability and the worst‐case H_∞ performance of linear time‐invariant systems subject to mixed real‐parametric and complex‐dynamic uncertainties in a compact parameter set. Our novel algorithmic approach is tailored to the properties of the nonsmooth worst‐case functions associated with stability and performance, and this leads to a fast and reliable optimization method, which finds good lower bounds of μ. We justify our approach theoretically by proving a local convergence certificate. Because computing μ is known to be NP‐hard, our technique should be used in tandem with a classical μ upper bound to assess global optimality. Extensive testing indicates that the technique is practically attractive. Copyright © 2016 John Wiley & Sons, Ltd.