Real-valued Control Research Articles

Modeling and Path-Following Control for Dubins Dynamics in Complex State Space

A path-following control design in the complex state space for the Dubins vehicle is proposed. The vehicle dynamics are modeled as a bilinear system of two complex ordinary differential equations with a scalar complex control input. The developed control solution represents a feedback linearizing proportional–integral–derivative servomechanism, with real valued control gains applied to complex valued tracking error components. The design is performed within the model reference control framework. It is applied to solving path-following and waypoint routing problems for curves with bounded curvature and vehicles with bounded speeds. The path-following controller regulates the system turn rate and the reference model speed. The vehicle speed dynamics are independently maintained as desired. In that regard, the developed control policy forces the system position to track a virtual target point defined by the reference model. Verifiable sufficient conditions are formulated to enable steering the Dubins vehicle along the designated routes in the presence of unknown bounded environmental disturbances, such as wind and gust. Application examples of the developed path-following control solution to waypoint routing guidance for unmanned aerial platforms are discussed.

On the Fractal interpolation functions associated with Matkowski contractions

<abstract><p>In this paper we investigate an iterated function system that defines a fractal interpolation function, where ordinate scaling, that is Lipschitz constant in Banach contraction principle is substituted by real-valued control function. In such a manner, fractal interpolation functions associated with Matkowski contractions are obtained and provide a new framework of approximating experimental data. Furthermore, given a data generating function $ f $, we study a new class of fractal interpolation functions which converge to $ f $.</p></abstract>

Non-stationary $$\phi$$-contractions and associated fractals

In this study we provide several significant generalisations of Banach contraction principle where the Lipschitz constant is substituted by real-valued control function that is a comparison function. We study non-stationary variants of fixed-point. In particular, this article looks into “trajectories of maps defined by function systems” which are regarded as generalizations of traditional iterated function system (IFS). The importance of forward and backward trajectories of general sequences of mappings is analyzed. The convergence characteristics of these trajectories determined a non-stationary variant of the traditional fixed-point theory. Unlike the normal fractals which have self-similarity at various scales, the attractors of these trajectories of maps which defined by function systems that may have various structures at various scales. In this literature we also study the sequence of countable IFS having some generalized contractions on a complete metric space.

Ensemble qubit controllability with a single control via adiabatic and rotating wave approximations

In the physics literature it is common to use “in cascade” the rotating wave approximation and the adiabatic approximation for chirped pulses of two-level quantum systems driven by one external field, in particular when the resonance frequency is not known precisely. Both approximations need relatively long time and are based on averaging theory of dynamical systems. Unfortunately, the two approximations cannot be done independently since, in a sense, the two time scales interact. We study how the cascade of the two approximations can be justified, while preserving the robustness of the adiabatic strategy. Our first result, based on high-order averaging techniques, gives a precise quantification of the uncertainty interval of the resonance frequency for which the population inversion works. As a by-product, we prove controllability of an ensemble of spin systems by a single real-valued control, providing an extension of a celebrated result with two controls by Khaneja and Li.

HEVC Rate-Distortion Optimization with Source Modeling

The Rate-Distortion adaptive mechanisms of MPEG-HEVC (High Efficiency Video Coding) and its derivatives are an incremental improvement in the software reference encoder, providing a selective Lagrangian parameter choice which varies by encoding mode (intra or inter) and picture reference level. Since this weighting factor (and the balanced cost functions it impacts) are crucial to the RD optimization process, affecting several encoder decisions and both coding efficiency and quality of the encoded stream, we investigate an improvement by modern reinforcement learning methods. We develop a neural-based agent that learns a real-valued control policy to maximize rate savings by input signal pattern, mapping pixel intensity values from the picture at the coding tree unit level, to the appropriate weighting-parameter. Our testing on reference software yields improvements for coding efficiency performance across different video sequences, in multiple classes of video.

Technical Note: Parameterised-Response Zero-Intelligence (PRZI) Traders

This brief technical note introduces PRZI (Parameterised-Response Zero Intelligence), a new form of zero-intelligence trader intended for use in simulation studies of auction markets. Like Gode & Sunder's classic Zero-Intelligence Constrained (ZIC) trader, PRZI generates quote-prices from a random distribution over some specified domain of discretely-valued allowable quote-prices. Unlike ZIC, which uses a uniform distribution to generate prices, the probability distribution in a PRZI trader is parameterised in such a way that its probability mass function (PMF) is determined by a real-valued control variable s in the range [-1.0, +1.0] that determines the strategy for that trader. When s is zero, a PRZI trader behaves identically to the ZIC strategy, with a flat/rectangular PMF; but when s is close to plus or minus one the PRZI trader's PMF becomes asymptotically maximally skewed to one extreme or the other of the price-range, thereby enabling the PRZI trader to act in the same way as the Shaver strategy (SHVR) or the Giveaway strategy (GVWY), both of which have recently been demonstrated to be surprisingly dominant over more sophisticated, and supposedly more profitable, trader-strategies that incorporate adaptive mechanisms and machine learning. Depending on the value of s, a PRZI trader will behave either as a ZIC, or as a SHVR, or as a GVWY, or as some hybrid strategy part-way between two of these three previously-reported strategies. The novel smoothly-varying strategy in PRZI has value in giving trader-agents plausibly useful market impact responses to imbalances in an auction-market's limit-order-book, and also allows for the study of co-adaptive dynamics in continuous strategy-spaces rather than the discrete spaces that have traditionally been studied in the literature.

Synchronization in finite/fixed time of fully complex-valued dynamical networks via nonseparation approach

In this paper, a nonseparation approach is developed to explore the problem of finite-time and fixed-time synchronization for fully complex-valued dynamical networks. Firstly, to avoid the traditional separation method, the signum function of complex-valued vectors is proposed as an extension of the real-valued signum function and several essential properties of it are derived. Based on the introduced signum function, several complex-valued controllers are directly designed on the networks, which are totally different from the existing real-valued control designs on the decomposed subsystems. Furthermore, by directly constructing Lyapunov functions in the complex field and developing some innovative inequalities, several criteria described by matrix inequalities are derived to achieve finite-time or fixed-time synchronization, which are much simpler and more convenient compared with the previous algebraic conditions. Finally, some numerical simulations are provided to support the validity of the derived results.

Control Law Realification for the Feedback Stabilization of a Class of Diagonal Infinite-Dimensional Systems With Delay Boundary Control

Recently, a predictor feedback control strategy has been reported for the feedback stabilization of a class of infinite-dimensional Riesz-spectral boundary control systems exhibiting a finite number of unstable modes by means of a delay boundary control. Nevertheless, for real abstract boundary control systems exhibiting eigenstructures defined over the complex field, the direct application of such a control strategy requires the embedding of the control problem into a complexified state-space which yields a complex-valued control law. This letter discusses the realification of the control law, i.e., the modification of the design procedure for obtaining a real-valued control law for the original real abstract boundary control system. The obtained results are applied to the feedback stabilization of an unstable Euler–Bernoulli beam by means of a delay boundary control.

LMI-based robust mixed-integer model predictive control for hybrid systems

ABSTRACT This paper proposes a linear matrix inequality (LMI) approach to mixed-integer model predictive control (MPC) of uncertain hybrid systems with binary and real valued control inputs. The stability condition of the hybrid system is obtained by using the Lyapunov function and then a sufficient state feedback control law is achieved so that guarantees the closed-loop stability and also minimises an infinite horizon performance index. The primal optimisation problem is convex, therefore, a convex relaxation is investigated by introducing the Lagrange dual function. The real and binary control inputs are obtained by solving the dual of the dual problem in the framework of LMIs. The performance and effectiveness of the proposed MPC approach and the duality gap of the convex relaxation are studied through simulation results of a hybrid system with mixed real and binary inputs.

Approximate model predictive building control via machine learning

Many studies have proven that the building sector can significantly benefit from replacing the current practice rule-based controllers (RBC) by more advanced control strategies like model predictive control (MPC). However, the optimization-based control algorithms, like MPC, impose increasing hardware and software requirements, together with more complicated error handling capabilities required from the commissioning staff. In recent years, several studies introduced promising remedy for these problems by using machine learning algorithms. The idea is based on devising simplified control laws learned from MPC. The main advantage of the proposed methods stems from their easy implementation even on low-level hardware. However, most of the reported studies were dealing only with problems with a limited complexity of the parametric space, and devising laws only for a single control variable, which inevitably limits their applicability to more complex building control problems. In this paper, we introduce a versatile framework for synthesis of simple, yet well-performing control strategies that mimic the behavior of optimization-based controllers, also for large scale multiple-input-multiple-output (MIMO) control problems which are common in the building sector. The approach employs multivariate regression and dimensionality reduction algorithms. Particularly, deep time delay neural networks (TDNN) and regression trees (RT) are used to derive the dependency of multiple real-valued control inputs on parameters. The complexity of the problem, as well as implementation cost, are further reduced by selecting the most significant features from the set of parameters. This reduction is based on straightforward manual selection, principal component analysis (PCA) and dynamic analysis of the building model. The approach is demonstrated on a case study employing temperature control in a six-zone building, described by a linear model with 286 states and 42 disturbances, resulting in an MPC problem with more than thousand of parameters. The results show that simplified control laws retain most of the performance of the complex MPC, while significantly decreasing the complexity and implementation cost.

Decomposition and Mean-Field Approach to Mixed Integer Optimal Compensation Problems

Mixed integer optimal compensation deals with optimization problems with integer- and real-valued control variables to compensate disturbances in dynamic systems. The mixed integer nature of controls could lead to intractability in problems of large dimensions. To address this challenge, we introduce a decomposition method which turns the original n-dimensional optimization problem into n independent scalar problems of lot sizing form. Each of these problems can be viewed as a two-player zero-sum game, which introduces some element of conservatism. Each scalar problem is then reformulated as a shortest path one and solved through linear programming over a receding horizon, a step that mirrors a standard procedure in mixed integer programming. We apply the decomposition method to a mean-field coupled multi-agent system problem, where each agent seeks to compensate a combination of an exogenous signal and the local state average. We discuss a large population mean-field type of approximation and extend our study to opinion dynamics in social networks as a special case of interest.

Towards a Generic Constructive Nonlinear Control Design Tool Using Relaxed Control

In this technical note, we revisit a control design approach for general (non-affine) nonlinear systems using relaxed control. Using the notion of relaxed input, where the ordinary real-valued control input is replaced by a measure-valued control input, we are able to manipulate the original system such that the resulting relaxed system is amenable to control. We show the applicability of using relaxed input for transforming a non-affine nonlinear system into an affine one such that the relaxed system has desirable control properties. Finally, we present a practical implementation of relaxed input via ordinary control input which is based on the sampled-data of the relaxed input.

Internal null controllability of a linear Schrödinger–KdV system on a bounded interval

The control of a linear dispersive system coupling a Schrödinger and a linear Korteweg–de Vries equation is studied in this paper. The system can be viewed as three coupled real-valued equations by taking real and imaginary parts in the Schrödinger equation. The internal null controllability is proven by using either one complex-valued control on the Schrödinger equation or two real-valued controls, one on each equation. Notice that the single Schrödinger equation is not known to be controllable with a real-valued control. The standard duality method is used to reduce the controllability property to an observability inequality, which is obtained by means of a Carleman estimates approach.

A new fixed point theorem in the fractal space

In this paper we present some important generalizations of the Banach contraction principle in which the Lipschitz constant k is replaced by some real-valued control function.For the applications to the fractal, we obtain the fixed point theorem of the some generalized contraction in the fractal space.

Building Temperature Control by Simple MPC-like Feedback Laws Learned from Closed-Loop Data

We show how to synthesize simple, yet well-performing feedback strategies that mimic the behavior of optimization-based controllers, such as those based on model predictive control (MPC). The approach is based on employing regression trees to derive dependence of real-valued control inputs on measurements. Quality of classical regression policies is improved by finding, simultaneously, optimal affine splits and optimal local affine regressors. We furthermore illustrate how to refine the local regressors such that the overal feedback strategy guarantees satisfaction of input constraints. The main advantage of the proposed regression-based control strategy stems from its fast implementation even on very simple hardware. The approach is demonstrated on a case study that assumes control of temperature in a one-zone building. Here, the data used in the learning process are generated by MPC. We show that the simple feedback law attains almost the same level of performance as the complex MPC controller.

Actor-Critic Based Ink Drop Spread as an Intelligent Controller

This paper introduces an innovative adaptive controller based on the actor-critic method. The proposed approach employs the ink drop spread (IDS) method as its main engine. The IDS method is a new trend in soft-computing approaches that is a universal fuzzy modeling technique and has been also used as a supervised controller. Its process is very similar to the processing system of the human brain. The proposed actor-critic method uses an IDS structure as an actor and a 2-dimensional plane, representing control variable states, as a critic that estimates the lifetime goodness of each state. This method is fast, simple, and away from mathematical complexity. The proposed method uses the temporal differences (TD) method to update both the actor and the critic. Our system: 1) learns to produce real-valued control actions in a continuous space without regarding the Markov decision process, 2) can adaptively improve performance during the lifetime, and 3) can scale well to high-dimensional problems. To show the effectiveness of the method, we conduct experiments on 3 systems: an inverted pendulum, a ball and beam, and a 2-wheel balancing robot. In each of these systems, the method converges to a pertinent fuzzy system with a significant improvement in terms of the rise time and overshoot compared to other fuzzy controllers.

Convex relaxations for mixed integer predictive control

The main objective in this work is to compare different convex relaxations for Model Predictive Control (MPC) problems with mixed real valued and binary valued control signals. In the problem description considered, the objective function is quadratic, the dynamics are linear, and the inequality constraints on states and control signals are all linear. The relaxations are related theoretically and the quality of the bounds and the computational complexities are compared in numerical experiments. The investigated relaxations include the Quadratic Programming (QP) relaxation, the standard Semidefinite Programming (SDP) relaxation, and an equality constrained SDP relaxation. The equality constrained SDP relaxation appears to be new in the context of hybrid MPC and the result presented in this work indicates that it can be useful as an alternative relaxation, which is less computationally demanding than the ordinary SDP relaxation and which often gives a better bound than the bound from the QP relaxation. Furthermore, it is discussed how the result from the SDP relaxations can be used to generate suboptimal solutions to the control problem. Moreover, it is also shown that the equality constrained SDP relaxation is equivalent to a QP in an important special case.

Real-valued error control coding by using DCT

The discrete cosine theorem (DCT)-based real-valued linear code is derived for the first time in the literature. A BCH-like subclass of DCT linear codes is also developed, for which fast decoding algorithms exist. It is shown that the DCT-based error control codes proposed by the authors can be viewed as a bridge to link the fields of source coding and channel coding.

Learning and tuning fuzzy logic controllers through reinforcements

A method for learning and tuning a fuzzy logic controller based on reinforcements from a dynamic system is presented. It is shown that: the generalized approximate-reasoning-based intelligent control (GARIC) architecture learns and tunes a fuzzy logic controller even when only weak reinforcement, such as a binary failure signal, is available; introduces a new conjunction operator in computing the rule strengths of fuzzy control rules; introduces a new localized mean of maximum (LMOM) method in combining the conclusions of several firing control rules; and learns to produce real-valued control actions. Learning is achieved by integrating fuzzy inference into a feedforward network, which can then adaptively improve performance by using gradient descent methods. The GARIC architecture is applied to a cart-pole balancing system and demonstrates significant improvements in terms of the speed of learning and robustness to changes in the dynamic system's parameters over previous schemes for cart-pole balancing.

Global controllability of nonlinear systems in two dimensions

LetM be a connected real-analytic 2-dimensional manifold. Consider the system $$\dot x(t) = f(x(t)) + u(t)g(x(t)),x(0) = x_0 \in M,$$ (t) = f(x(t)) + u(t)g(x(t)),x(0) =x 0 ∈ M, wheref andg are real-analytic vector fields onM which are linearly independent at some point ofM, andu is a real-valued control. Sufficient conditions on the vector fieldsf andg are given so that the system is controllable fromx 0. Suppose that every nontrivial integral curve ofg has a pointp wheref andg are linearly dependent,g(p) is nonzero, and that the Lie bracket [f,g] andg are linearly independent atp. Then the system is controllable (with the possible exception of a closed, nowhere dense set which is not reachable) from any pointx 0 such that the vector space dimension of the Lie algebraL A generated byf,g and successive Lie brackets is 2 atx 0.