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This work aims to introduce a tuning-free PID controller for stable FOPTD and FOLIPD systems. The objective was achieved using a partial model-matching strategy; matching the coefficients of corresponding powers of s of the closed loop response of the model to the desired closed loop transfer function. Closed-loop performance is evaluated by considering a unit step change in the process model and compared the results with other methods reported earlier in the literature. Input and output performances were evaluated using error analyses and total variation. The robustness of the proposed method was assessed via incorporation of uncertainties in the process model parameters. The proposed method was also experimentally validated for FOPTD model using a temperature controlled lab. It was concluded that the method was analytically derived and model based with a key advantage that the resultant PID required no tuning parameter.

Zrównoważony system rolny jest najlepszym sposobem na zaspokojenie potrzeb dzisiejszych i przyszłych pokoleń. W systemie tym wielkość plonu wzrasta wraz z zastosowaniem upraw międzyrzędowych, poprzez wyższy współczynnik wzrostu roślin, redukcję nasion chwastów, ograniczenie ilości szkodników i chorób oraz bardziej efektywne wykorzystanie zasobów. Uprawa międzyrzędowa jest jednym z najważniejszych sposobów zwiększenia różnorodności w ekosystemie rolniczym. Systemy międzyplonowe mogą być bardziej stabilnymi systemami praktyk rolniczych niż uprawy monokulturowe. Najważniejszymi zaletami uprawy międzyrzędowej są: zwiększenie produkcji rolnej i większe wykorzystanie zasobów środowiska. Zintegrowane stosowanie nawozów syntetycznych i organicznych może również prowadzić do rozwoju zrównoważonej produkcji roślinnej. Metoda ta poprawia wydajność działania nawozów chemicznych i jednocześnie ogranicza ich stosowanie. Zielony nawóz z roślin strączkowych, takich jak: koniczyna, lucerna i inne, które są bogatym źródłem azotu uwalnianego w glebie, jest w stanie zmniejszyć znacząco zapotrzebowanie na azot syntetyczny. Tego rodzaju biologiczne wiązanie azotu odgrywa bardzo ważną rolę w zrównoważonych systemach rolniczych. W zrównoważonym systemie rolnym nawozy, odchody zwierzęce i rośliny uprawne oraz chwasty są ważnymi parametrami dla zapewnienia stabilnej produkcji żywności.

In this paper, a finance system with delay is considered. By analyzing the corresponding characteristic equations, the local stability of equilibrium is established. The existence of Hopf bifurcations at the equilibrium is also discussed. Furthermore, formulas for determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are derived by applying the normal form method and center manifold theorem. Finally, numerical simulation results are presented to validate the theoretical analysis. Numerical simulation results show that delay can lead a stable system into a chaotic state.

The aim of this study was to determine the most suitable and practical definition of spatial and temporal proximities between grouped-housed horses by comparing GPS position data. In total, 43 horses held in a “HIT Active Stable®” were equipped with GPS loggers. Different definitions of spatial (≤ 1, 2, 3, 4, 5 m) and temporal proximity (≤ 10, 30, 60, 90, 120 s interruptions between recording points) were investigated for morning pasture periods of 30 days analyzing proximity lengths, number of proximities as well as the network parameters density and fragmentation. With increasing spatial proximity definition, the duration of proximities rose from 9 % (4 compared to 5 m) to 76 % (1 compared to 2 m). The number of proximities also increased accordingly. Ascending temporal proximity definitions provided a lower number of proximities by 13–21 % (10 s to 30 s) depending on the spatial definition and the proximities became longer. Further, with higher spatial proximity definition, the network density ascended and fragmentation decreased, which shows that more horses had proximities and group formation occurred. We identified a proximity definition of 3 m for at least 60 s as the most useful solution because of a decreasing change in proximity length and number due to biological and technical restrictions. Further, it is suspected that this approach helps to better characterize the group formation of horses kept in groups.

An algorithm of consistent implicit discretization for locally homogeneous finite-time stable system having discontinuity only at the origin is developed. It preserves finite-time stability property in the discrete-time models. The homogeneous domination approach is utilized for analysis of the discretized model. The scheme is demonstrated for a simplified model of a quadrotor control system.

We present a paradigm for developing thermodynamical consistent numerical algorithms for thermodynamically consistent partial differential equation (TCPDE) systems, called the supplementary variable method (SVM). We add a proper number of supplementary variables to the TCPDE system coupled with its energy equation and other deduced equations through perturbations to arrive at a consistent, well-determined, solvable and structurally stable system. The extended system not only reduces to the TCPDE system at specific values of the supplementary variables, but also allows one to retain consistency and solvability after a consistent numerical approximation . Among virtually infinite many possibilities to add the supplementary variables, we present two that maintain thermodynamical consistency in the extended system before and after the approximation. A pseudo-spectral method is used in space to arrive at fully discrete schemes. The new schemes are compared with the energy stable SAV scheme and the fully implicit Crank–Nicolson scheme. The numerical results favor the new schemes in the overall performance.

A common feature of pattern formation in both space and time is the destabilization of a stable equilibrium solution of an ordinary differential equation by adding diffusion or delay, or both. Here we study linear stability of general reactiondiffusion systems with off-diagonal time delays. We show that a delay-stable system cannot be destabilized by diffusion, and that a diffusion stable system is also stable with respect to delay, if the diffusion is sufficiently fast. A system with direct negative feedback which is strongly stable with respect to diffusion can be destabilized by off-diagonal delay.

A new approach was proposed to analyze the stability of the linear continuous stationary dynamic systems. It is based on the decomposition of a square H2 norm of the transfer function of the dynamic system into parts corresponding either to particular eigenvalues of the system matrix, or to pairwise combinations of these eigenvalues. The spectral decompositions of a square H2 norm of the transfer function with multiple poles were obtained using the residues of the transfer function and their derivatives. Exact analytical expressions for calculation of the quadratic forms of the corresponding expansions were derived for an arbitrary location of the eigenvalues in the left half-plane. The obtained decompositions allow one to characterize the contribution of individual eigen-components or their pairwise combinations into the asymptotic variation of the system energy. We propose the energy criterion for estimation of the system stability margins that uses an evaluation of energy accumulated in a group of weakly stable system modes. This approach is illustrated by calculating the energy of a band-pass filter.

In this paper, the challenging problem of determining the unknown parameters of an identifiable LTI state-space representation of a stable system is addressed by resorting to a specific H∞-norm-based optimization algorithm. More specifically, by assuming the availability of a reliable fully-parameterized representation of the system to identify, the algorithm developed herein consists in restructuring this initial black-box representation of the system dynamics via the optimization of a dedicated maximum eigenvalue-based criterion. This study shows that this H∞-norm-based approach can be seen as a good solution for the identification of gray-box LTI state-space representations and, by extension, as an interesting alternative or a reliable initialization step for the standard output-error techniques.

Consider a discrete-time linear time-invariant descriptor system $Ex(k+1)=Ax(k)$ for $k \in \mathbb Z_{+}$. In this paper, we tackle for the first time the problem of stabilizing such systems by computing a nearby regular index one stable system $\hat E x(k+1)= \hat A x(k)$ with $\text{rank}(\hat E)=r$. We reformulate this highly nonconvex problem into an equivalent optimization problem with a relatively simple feasible set onto which it is easy to project. This allows us to employ a block coordinate descent method to obtain a nearby regular index one stable system. We illustrate the effectiveness of the algorithm on several examples., Comment: 10 pages, 3 tables, 1 figure

Based on the results of Chap. 6, a square function is constructed as \(\mathscr {H}_2\) norm of the SD standard model depending on the choice of the system parameters. Using the Wiener–Hopf minimization method, the optimal value of a stable system function is determined and a \(\mathscr {H}_2\)-optimal causal control program is generated from it. The application of the described method can provide some important facts that are useful for the application: 1. The characteristic polynomial of the \(\mathscr {H}_2\)-optimal system is a divisor of a polynomial independent of delays. 2. A part of the poles of a regulated \(\mathscr {H}_2\) optimal system does not depend on the delays and the choice of the generalized digital-to-analog converter and is completely determined by the structure of the continuous part of the system and the properties of the elements it contains. These poles are called fixed. The existence of elements with unfavorable fixed poles can lead to the synthesis of a \(\mathscr {H}_2\)-optimal system with unsatisfactory quality. As an example, the construction of a set of fixed poles for a multidimensional single-loop SD system with a different selection of inputs and outputs is considered.

We propose a novel algorithm to decompose a 3D object into an atlas of disk-like charts. Decomposition into charts with controlled shape and topology is relevant in many engineering areas, such as spline fitting, compression and re-meshing. We produce our chartifications by jointly exploiting the Reeb graph of a guiding function and its gradient aligned flow paths. The key advancements of our method with respect to similar approaches are: (i) a novel strategy to provably remove all T-junctions; (ii) a stable system to trace flow paths starting far from critical points; (iii) the exploitation of the regularity of certain functions under isometries (e.g., harmonic ones) to produce structurally equivalent chartifications for families of objects posed differently. The charts produced by our system can be of two types: topological quads and topological octagons. Both of them can be easily gridded to produce full quadrilateral meshes, as we demonstrate in the second part of the article.

The idea of adiabatic invariance is presented in the context of simple classical mechanical models. The adiabatic invariant jumps across the separatrix — an attempt has been made to bring out the basic ideas underlying the Melnikov-Arnold integral. This becomes important as soon as a perturbation to a regular, stable system makes it dynamically unstable.

In this note, a novel robust control design for a class of pure unstable systems with time delay was addressed. In the algorithm, the time delay was not neglected as usual and was approximated as a stable component using improved Pade technique. The unstable system was mirror mapped into a stable system and subsequently the closed-loop gain shaping algorithm is used to design the robust control law. The proposed controller was easy to tune which had only one adjustable parameter with engineering sense. By virtue of the Nyquist stability criterion, the exact stabilization analysis has been obtained for the closed-loop system. Simulation results show that the control effects can tackle the input load disturbances and the delay perturbation more effectively. The method has the advantages of simple and efficient design.

The authors proposed a method for order reduction of linear dynamic systems using the advantages of the Interpolation criterion and Routh method. The denominator polynomial of the reduced order model is determined by using the Routh method while the numerator coefficients are computed by minimizing the integral square error between the original and the reduced system using Interpolation method. The proposed method guarantees stability of the reduced model, if the original high order system is stable system. A PID controller is designed for the high order original systems through its low order model proposed. Some numerical examples were considered to explain the effectiveness of the method.

We study the stability of a certain class of switched systems where discontinuous jumps (resets) on some of the state components are allowed, at the switching instants. It is known that, if all components of the state are available for reset, the system can be stabilisable by an adequate choice of resets. However, this question may have negative answer if there are forbidden state components for reset. We give a sufficient condition for the stabilisability of a switched system, under arbitrary switching, by partial state reset in terms of a block simultaneous triangularisability condition. Based on this sufficient condition, we show that the particular class of systems with partially commuting stable system matrices is stabilisable by partial state reset. We also provide an algorithm that allows testing whether a switched system belongs to this particular class of systems.

The problem of robust H∞ model reduction for uncertain stochastic systems with time delay is investigated in this paper. The attention of this paper is focused on the construction of a reduced-order model for a given stable system. Several sufficient conditions are obtained for ensuring the existence of the reduced-order model by means of linear matrix inequalities and a coupling non-convex rank constraint condition. Under the sufficient conditions, the desired reduced-order model can be constructed, and the error system between the original model and the reduced-order model is asymptotically stable and has a prescribed H∞ performance. Based on the proposed method, the construction approaches of reduced-order models with special structures, such as the zeroth-order model, the delay-free model, and the no-parameter-uncertainties model, are also developed. Finally, the effectiveness of the proposed model reduction method is illustrated by a numerical simulation example.

This article is concerned with Hankel-norm model approximation (HNMA) for linear parameter-varying systems with parameter-varying time delays. For a given stable system, our attention is focussed on the construction of reduced-order models, which approximate the original system well in a Hankel-norm sense. By applying the slack matrix approach, a delay-dependent sufficient condition is proposed for the robustly asymptotic stability with a Hankel-norm error performance for the error system. Then, the HNMA problem is solved by using the projection approach, which casts the model approximation into a sequential minimisation problem subject to linear matrix inequality constraints by employing the cone complementary linearisation algorithm. Finally, a numerical example is provided to illustrate the effectiveness of the proposed methods.

The problem of H∞ model simplification for singular systems with Markovian jumping parameters and time-delay is investigated. For a given stochastic stable system, our attention is focused on the construction of reduced-order models which guarantee the corresponding error system to be admissible and have a prescribed H∞ error performance. A robust admissibility condition and a robust bounded real lemma are developed based on which, sufficient conditions for the existence of desired reduced-order models are proposed. Desired reduced-order model can be constructed when the proposed conditions are satisfied. The effectiveness of the proposed model reduction method is illustrated via a numerical example. Copyright © 2010 John Wiley & Sons, Ltd.

The phenomenon of stochastic resonance (SR) in a biased mono-stable system driven by multiplicative and additive white noise and two periodic fields is investigated. Analytic expressions of the signal-to-noise ratio (SNR) for fundamental harmonics and higher harmonics are derived by using the two-state theory. It is shown that the SNR is a non-monotonic function of the intensities of the multiplicative and additive noises, as well as the bias of the mono-stable system and SR appears at both fundamental harmonics and higher harmonics. Moreover, the higher the order of mixed harmonics is, the smaller the SNR values are, that is, the suppression exists for higher harmonics.

This paper is concerned with the ℋ∞ model reduction for linear parameter-varying (LPV) systems with both discrete and distributed delays. For a given stable system, our attention is focused on the construction of reduced-order models, which approximate the original system well in an ℋ∞ norm sense. First, a sufficient condition is proposed for the asymptotic stability with an ℋ∞ performance of the error system by using the parameter-dependent Lyapunov functional method. Then, the decoupling technique is applied, such that there does not exist any product term between the Lyapunov matrices and the system matrices in the parametrised linear matrix inequality (PLMI) constraints; thus a new sufficient condition is obtained. Based on the new condition, two different approaches are developed to solve the model reduction problem. One is the convex linearisation approach and the other is the projection approach. Finally, a numerical example is provided to demonstrate the effectiveness of the proposed design method.

A new modification to the steepest-descent algorithm for discrete-time iterative learning control is developed for plant models with multiplicative uncertainty. A theoretical analysis of the algorithm shows that if a tuning parameter is selected to be sufficiently large, the algorithm will result in monotonic convergence provided the plant uncertainty satisfies a positivity condition. This is a major improvement when compared to the standard version of this algorithm, which lacks a mechanism for finding a balance between convergence speed and robustness. The proposed algorithm has been investigated experimentally on an industrial gantry robot and found to display a high degree of robustness to both plant modelling error and initial state error. The algorithm also exhibits both long-term performance and excellent tracking performance, as demonstrated by experimental tests of up to 4000 iterations. To further examine robustness, the plant has been approximated by simple models including one consisting of an integrator and a gain. A simple tuning rule for this reduced model is proposed, which generates a stable system with a good rate of convergence. The robustness properties of the steepest descent algorithm have then been experimentally verified.

Received 11 October 2014, Available online 26 January 2016 In deciding whether to join a coalition or not, an agent must consider both i) the expected power of the coalition and ii) her position in the vertical structure within the coalition. We establish the existence of a positive relationship between the degree of inequality in remuneration within coalitions and the number of coalitions to be formed endogenously in stable systems. We show that such coalitions can be mixed and balanced, rather than segregated, in terms of members' ability levels. In any stable system each coalition is of an efficient size and every agent is paid her marginal contribution. Article is version of EUI WP ECO; 2011/20

In this paper, some special decentralized control problems are addressed for discrete-time interconnected systems. First it is pointed out that some subsystems must be unstable to ensure stability of the overall system in some special cases. Then a special kind of decentralized control problem is studied. This kind of problem can be viewed as harmonic control among independent subsystems. Research results show that two unstable systems can generate a stable system through some effective cooperations. In addition, a decentralized controller design method based on linear matrix inequality is also given by using parameter-dependent Lyapunov function method developed for the study of robust stability. A special linear star coupled dynamical network is also considered. The central subsystem must be unstable to stabilize the whole network under a special coupling. Several examples are given to illustrate the results.

In this paper we present a right inversion scheme based on the essential concept of reachability. This scheme inverts a Hurwitz stable system one step ahead. We illustrate our procedure with the simulation of an ideal inverted pendulum.

It has been established that a class of bandpass sigma-delta modulators may exhibit state space dynamics which are represented by elliptical or fractal patterns confined within trapezoidal regions when the system matrices are marginally stable. In this brief, it is found that fractal or irregular chaotic patterns may also be exhibited in the phase plane when the system matrices are strictly stable

In this paper, some decentralized control problems composed of two subsystems are addressed from a special perspective. First, it is pointed out that some subsystems must be unstable to stabilize the overall interconnected system in some special cases. Then, a special kind of decentralized control problem is studied. This kind of problem can be viewed as harmonic control among independent subsystems. Research results show that two unstable systems can generate a stable system through some effective cooperations. And a linear matrix inequality method is provided for decentralized controller design by using the parameter-dependent Lyapunov function method. Two examples are given to illustrate the results.

In the chain scattering framework, it has been shown in Lee and Soh (2004) that the problem of simultaneous stabilization of a finite collection of l linear time-invariant plants such that each closed-loop transfer function has an norm less than a prescribed performance level is equivalent to a simultaneous strongly stabilizing problem, i.e. finding a single stable and norm bounded controller for associated l − 1 unimodular plants such that each resulting closed-loop system is stable and norm bounded. This paper attempts to provide a solution to the simultaneous strongly stabilization problem. A new sufficient condition in the form of the solution to only one algebraic Riccati equation is provided for the existence of a strongly stabilizing controller for a given stable system obtained from the parameterization of stabilizing controllers. With the parameterization of strongly stabilizing controllers provided in this paper, a solution is given for the simultaneous strongly stabilization problem and hence, ...

This paper investigates the problem of ℋ∞ model reduction for two-dimensional (2-D) discrete systems with parameter uncertainties residing in a polytope. For a given robustly stable system, our attention is focused on the construction of a reduced-order model, which also resides in a polytope and approximates the original system well in an ℋ∞ norm sense. Both Fornasini–Marchesini local state-space (FMLSS) and Roesser models are considered through parameter-dependent approaches, with sufficient conditions obtained for the existence of admissible reduced-order solutions. Since these obtained conditions are not expressed as strict linear matrix inequalities (LMIs), the cone complementary linearization method is exploited to cast them into sequential minimization problems subject to LMI constraints, which can be readily solved using standard numerical software. In addition, the development of zeroth order models is also presented. Two numerical examples are provided to show the effectiveness of the proposed theories. Copyright © 2005 John Wiley & Sons, Ltd.

The probabilities of various biological asymptotic dynamics are computed for a stable system that is invaded by another competing species. The asymptotic behaviors studied include extinction, weak extinction, permanence, and mutual exclusion. The model used is a discrete Lotka-Volterra system that models species that compete for the same resources. Among the results found are that the chance of permanence occurring in the invaded system is significantly higher than the probability of permanence in a purely random system, and that multiple extinctions that include the invading species and one of the original species are impossible.

For the global asymptotic stabilization of nonlinear—controllable in the first approximation—cascades consisting of a globally asymptotically—locally exponentially stable system driving a stable system, a Lyapunov-like design is proposed. This yields (bounded) control laws, where the control magnitude can be chosen arbitrarily large. This result provides an alternative to classical forwarding.

This paper proposes new fundamental limitations on the step response of the unity-feedback system based on its transfer function. Using the definition of Laplace transform, it is shown that time-domain integral equalities have to be satisfied by the step response of SISO (single-input, single-output) stable system, and shown that the maximum magnitude of its step response has some lower bounds with respect to the achievable settling time of the systems. It is also shown that the closed-loop system with open-loop RHP (right half plane) poles and/or zeros has some limitations in the unity-feedback control scheme. These results can be used to check in advance whether desired design specifications are achievable or not.

This paper investigates the problem of model reduction based on balanced truncation method for continuous-time linear switched systems with time-varying delay. For a given stable system, our attention is focused on the construction of reduced-order model, which not only approximates the original system well but also is translated into a linear lower-dimensional system. Specifically in the paper, we use the generalized dissipation inequalities that is in conjunction with a suitably defined storage function in [7] to obtain the reduced order models. Finally, an illustrative example is provided to state the feasibility of the obtained results.

Generally it is not easy task whether the stable systems governed by nonlinear ordinary differential equations are asymptotically stable or not. This problem often appears in studying a collision and avoidance control problem based on the stability theory. In this paper we devoted to finding conditions that the stable system obtained from the collision and avoidance control problem is asymptotically stable.

We consider the problem of approximating in H∞ norm a given discrete inverse-time stable system (possibly improper) T= T 11 T 12 T 21 T 22 by a discrete-time system S with no more than r poles outside the unit disk (possibly at infinity) such that T 11 +S T 12 T 21 T 22 ∞ Here r⩾0 is an integer and γ>0 is a prescribed tolerance. If r=0, this is the four-block Nehari problem while if we consider the “one block case” where T=T11 we obtain the well-known Hankel norm approximation problem. The theoretical developments are based on a frequency domain signature condition while the class of solutions is constructed in state-space in terms of the solutions to two Riccati equations.

A spatially explicit metapopulation model with positive density-dependent migration is analysed. We obtained conditions under which a previously stable system can be driven to instability caused by a density-dependent migration mechanism. The stability boundary depends on the rate of increase of the number of migrants on each site at local equilibrium, on the intrinsic rate of increase at local level, on the number of patches, and on topological aspects regarding the connectivity between patches. A concrete example is presented illustrating the dynamics on the dispersal-induced unstable regime.

In this paper we shall investigate the stability of a stochastic integro–differential equation which is regarded as a stochastically perturbed system of a given non-linear stable system . Several sufficient criteria are obtained under which the system can tolerate the stochastic perturbation without losing the property of stability

In this paper we analyze the behavior of system approximation processes for stable linear time-invariant (LTI) systems and signals in the Paley–Wiener space \(\mathcal{P}\mathcal{W}_{\pi }^{1}\). We study approximation processes, where the input signal is not directly used to generate the system output but instead a sequence of numbers is used that is generated from the input signal by measurement functionals. We consider classical sampling which corresponds to a pointwise evaluation of the signal, as well as several more general measurement functionals. We show that a stable system approximation is not possible for pointwise sampling, because there exist signals and systems such that the approximation process diverges. This remains true even with oversampling. However, if more general measurement functionals are considered, a stable approximation is possible if oversampling is used. Further, we show that without oversampling we have divergence for a large class of practically relevant measurement procedures.

This paper presents a case study for determining achievable performances in the basic form of extremum seeking control. Despite recent developments on more powerful versions of the algorithm, no detailed study that investigates this aspect thoroughly and provides clear guidelines for the tuning of this simpler, readily-implementable version is yet available. Interest remains for this basic version, for a variety of reasons. A review of extremum seeking fundamentals, with the support of a few simulations, allows to determine better means of selecting the elementary loop components, maximizing the gradient capture information. The choice of the excitation (dither) fundamental frequency is also better addressed under this optimal condition to effectively discuss achievable loop performance aspects. Most results apply to Wiener-Hammerstein classes of systems (strictly proper and stable system dynamics) and still allow other loop improvement modifications to be applied. Extensions to square wave dither signals and a few comparisons with other recently developed loop improvement techniques are also provided.

We give bounds on the self-sustained limit cycles in fixed-poin implementations of state-variable-form digital filters having general stable system matrices. These bounds, which are period independent and which concern limit cycles due to quantization errors in physical realizations of digital filters, are given in closed form for second-order sections. A numerical comparison shows that the bounds are smaller than corresponding ones reported in the literature. This is due to our use of real similarity transformations.

The paper deals with the problem of the robust D -stable system design. The proposed method solves the problem using the complex plane transformations. The aim of the paper is to point out some design limitations that appear when the combination of transformation techniques with the H ∞ method is used. Some characteristic effects are discussed and as the conclusions some suggestions concerning the choice of design parameters are formulated. The H ∞ synthesis technique used in calculations was the one proposed by Morari and Zafiriou (1989).

A generalized predictive controller (GPC) is derived based on a general state-space model. The equivalence of the predictive control problem to a perturbation problem is revealed. In the case of a small perturbation, the closed-loop poles are calculated with high accuracy. For the case of a general perturbation, an upper bound on the permissible perturbation norm is derived assuming an open-loop stable system. Both the plant-model match and plant-model mismatch cases are analysed. The controller is proven to be robust and an adaptive implementation is motivated. For open-loop stable systems, the convergence and stability of the control scheme are ensured by proper tuning of the control weight and prediction horizon. The results are applicable to a wide range of predictive controllers. The main contribution of this paper is not a new control algorithm, but new techniques to analyse the GPC as well as new stability and convergence results.

A large number of tuning methods have been developed for automatic tuning of Single-Input Single-Output (SISO) PID controllers. Many industrial processes are however multivariable. It is possible to design and use decoupling compensators to reduce the loop interactions and then use SISO methods to tune the PID controllers. This will however need to be repeated when the process operating condition changes. Furthermore, decoupling methods work well only for systems with weak loop interactions. Despite the wide popularity and extensive research for SISO tuning methods, the number of tuning methods for Multi-Input Multi-Output (MIMO) systems is limited. The MIMO control loops may also be tuned sequentially using scalar techniques. This is not often sufficient since tuning a loop may detune other loops in the system. Some sequential tuning methods are partially open loop and only applicable to open-loop stable system. Furthermore these methods are time consuming, expensive and not easily accepted by the plant operators. This chapter reviews and discusses some easy to use practical multivariable methods and demonstrates their application to an industrial example.

This paper is dealing with the problem of general conditions for the existence of the anticausal stabilizing solution to the discrete reverse-time Riccati equation. The connection between the γ-contracting property of the l2-input-output operator of a linear internally stable system and the existence of the anticausal stabilizing solution to the above-mentioned equation are pointed out.

Chapter 3 is devoted to answering the following question: do Lyapunov functionals always exist for a robustly globally asymptotically output stable system? The previous Chap. 2 showed that one of the most important ways of proving stability is the derivation of estimates which guarantee appropriate stability properties by means of Lyapunov functionals. The converse Lyapunov results obtained in this chapter show that such Lyapunov functionals always exist.

The instability of two superposed fluids in the presence of suspended particles is studied. It is found that the system is unstable for sufficiently long waves, however large the applied magnetic field may be, provided the initial configuration is top-heavy density wise. Also, the potentially stable system remains stable for the case of two superposed fluids in the presence of a variable magnetic field and suspended particles.

The spectral decomposition theorem, presented in Sect. 2.3 and first proved in (Smale, Bull. Am. Math. Soc. 73:747–817, 1967), provides a description of the non-wandering set of a structural stable system as a finite number of disjoint compact maximal invariant and transitive sets, each of these pieces being well understood from both the deterministic and from statistical viewpoints. Moreover such a decomposition persists under small C 1 perturbations. This naturally leads to the study of isolated transitive sets that remain transitive for all nearby systems (robustness).

In this paper, we develop a continuous age-dependent population model based on parity progression, and introduce the age-parity progression fertilities as the control variables of the system. The stationary situation is fully investigated and the distribution of age-parity progression fertilities for the stable system is studied.

A definition of constant steady-state output controllability of linear systems is presented based upon steady-state control. It shows that the constant steady-state output controllability and the output controllability are not equivalent, while the condition of the former is stricter. It is also proved that the necessary condition for the constant steady-state output controllability must be a stable system, on which the several important necessary and sufficient conditions for the constant steady-state output controllability are elaborated.