Quadratic Cost Criterion Research Articles

Signaling Games for Log-Concave Distributions: Number of Bins and Properties of Equilibria

We investigate the equilibrium behavior for the decentralized cheap talk problem for real random variables and quadratic cost criteria in which an encoder and a decoder have misaligned objective functions. In prior work, it has been shown that the number of bins in any equilibrium has to be countable, generalizing a classical result due to Crawford and Sobel who considered sources with density supported on [0, 1]. In this paper, we first refine this result in the context of log-concave sources. For sources with two-sided unbounded support, we prove that, for any finite number of bins, there exists a unique equilibrium. In contrast, for sources with semi-unbounded support, there may be a finite upper bound on the number of bins in equilibrium depending on certain conditions stated explicitly. Moreover, we prove that for log-concave sources, the expected costs of the encoder and the decoder in equilibrium decrease as the number of bins increases. Furthermore, for strictly log-concave sources with two-sided unbounded support, we prove convergence to the unique equilibrium under best response dynamics which starts with a given number of bins, making a connection with the classical theory of optimal quantization and convergence results of Lloyd’s method. In addition, we consider more general sources which satisfy certain assumptions on the tail(s) of the distribution and we show that there exist equilibria with infinitely many bins for sources with two-sided unbounded support. Further explicit characterizations are provided for sources with exponential, Gaussian, and compactly-supported probability distributions.

Security tracking control for discrete-time stochastic systems subject to cyber attacks

This paper is concerned with the security tracking problem under the quadratic cost criterion for a class of discrete-time stochastic linear networked control systems (NCSs) exposed to cyber attacks, covering false data injection attacks as well as a class of DoS attacks. Taking into account factors such as network congestion and the defensive role of intrusion detection systems, successful attack events are modeled as a Bernoulli random sequence. To describe the transient trajectory of an NCS under the impact of a random attack, a probabilistic definition of secure trackability is taken. Therefore, an observer-based dynamic output feedback controller is designed in order to achieve the specified probabilistic secure trackability. Specifically, the probabilistic safety output tracking problem is transformed into an input-to-state stability problem in the probabilistic sense for closed-loop systems with some new sufficient conditions, provided that an augmented incremental model is utilized Then, the controller parameters and the upper bound of the quadratic cost function are determined by solving matrix inequalities, while easy-solution forms of the matrix inequalities to be solved are presented by the Schur complementary lemma. Both simulation studies and practical experiments demonstrate the effectiveness of the proposed control scheme.

Fault tolerant consensus control of multi-agent systems under dynamic event-triggered mechanisms

In this paper, the fault-tolerant consensus control (FTCC) problem is studied for multi-agent systems (MASs) with the dynamic event-triggered mechanism (DETM). To ease the communication burden, DETM governed by an additional internal dynamical variable is introduced. A novel fault-tolerant controller is presented to mitigate system performance degradation caused by failures, where a simple fault compensator is constructed through a protocol-based observer. Then, some sufficient conditions are established to examine the bounded consensus while optimizing the predetermined quadratic cost criterion. In addition, the explicit expression of the desired controller is also parameterized through orthogonal decomposition. Finally, two simulation results are made for the sake of verifying the effectiveness of the developed FTCC scheme, the fault compensation test as well as the quadratic cost one.

Observer-Based PID Security Control for Discrete Time-Delay Systems Under Cyber-Attacks

This article deals with the observer-based proportional-integral-derivative (PID) security control problem for a kind of linear discrete time-delay systems subject to cyber-attacks. The cyber-attacks, which include both denial-of-service and deception attacks, are allowed to be randomly occurring as regulated by two sequences of Bernoulli distributed random variables with certain probabilities. A novel observer-based PID controller is proposed such that the closed-loop system achieves the desired security level and the quadratic cost criterion (QCC) has an upper bound. Sufficient conditions are derived under which the exponentially mean-square input-to-state stability is guaranteed and the desired security level is then achieved. Subsequently, an upper bound of the QCC is obtained and the explicit expression of the desired PID controller is also parameterized. Finally, the validity of the developed design approach is verified via an illustrative example.

Hybrid [formula omitted] sub-optimal stochastic risk-sensitive control for polynomial systems of first degree

This paper presents the novelty RS (Risk-Sensitive) H2 optimal control problem with criterion J2, showing the energy of the control spent into the process, H∞ RS control approach with level of attenuation λ1 is obtained, with cost criterion J∞. Robustness is added when hybridized H2∕H∞ RS sub-optimal control problem is solved, as optimization approach, taking J2 as cost function criterion, with H∞ inequality constraint. The sub-optimal solution is obtained imposing that the H∞ constraint holds with the equality sign. Values of the criterion J2 versus exponential quadratic cost criterion J, (RS traditional criterion), illustrate advantages of exponential quadratic cost criterion J for different values of diffusion parameter ϵ into the state equation in the final time through an application to the model of the immune system. The experiments show the values of level of attenuation λ1 in H∞ RS control problem and hybrid H2∕H∞ RS sub-optimal control problem which holds with equality sign. The results of the different experiments that were carried out show that the proposed methodology is feasible and meets the conditions required in H2, and H∞ approaches.

A Note on the Separation of Optimal Quantization and Control Policies in Networked Control

For controlled $\mathbb{R}^n$-valued linear systems driven by Gaussian noise under quadratic cost criteria, we revisit the problem of the structure of optimal quantization and control policies. In ...

Security Control for Discrete-Time Stochastic Nonlinear Systems Subject to Deception Attacks

This paper is concerned with the security control problem with quadratic cost criterion for a class of discrete-time stochastic nonlinear systems subject to deception attacks. A definition of security in probability is adopted to account for the transient dynamics of controlled systems. The purpose of the problem under consideration is to design a dynamic output feedback controller such that the prescribed security in probability is guaranteed while obtaining an upper bound of the quadratic cost criterion. First of all, some sufficient conditions with the form of matrix inequalities are established in the framework of the input-to-state stability in probability. Then, an easy-solution version on above inequalities is proposed by carrying out the well-known matrix inverse lemma to obtain both the controller gain and the upper bound. Furthermore, the main results are shown to be extendable to the case of discrete-time stochastic linear systems. Finally, two simulation examples are utilized to illustrate the usefulness of the proposed controller design scheme.

Jointly Optimal LQG Quantization and Control Policies for Multi-Dimensional Systems

For controlled R <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> -valued linear systems driven by Gaussian noise under quadratic cost criteria, we investigate the existence and the structure of optimal quantization and control policies. For fully observed and partially observed systems, we establish the global optimality of a class of predictive encoders and show that an optimal quantization policy exists, provided that the quantizers allowed are ones which have convex codecells. Furthermore, optimal control policies are linear in the conditional estimate of the state, and a form of separation of estimation and control holds.

A stochastic controller for a scalar linear system with additive Cauchy noise

A stochastic control scheme is developed for scalar, discrete-time, and linear-dynamic systems driven by Cauchy distributed process and measurement noises. When addressing the optimal control problem for such systems, the standard quadratic cost criteria cannot be used. In this study we introduce a new objective function that is functionally similar to the Cauchy probability density function. The performance index, defined as the expectation of this objective function with respect to the Cauchy densities, exists. The dynamic programming solution to the fixed and finite horizon optimal control problem that uses this performance index appears to be intractable. Therefore, a moving horizon optimal model predictive control problem is implemented, for which the conditional expected value of the objective function and its gradients can be computed in closed form and without assumptions such as certainty equivalence. Numerical results are shown for this m-step model predictive optimal controller and compared to a similar, Linear-Exponential-Gaussian model predictive controller. An essential difference between the Cauchy and Gaussian controllers when applied to a system with Cauchy noises is that, while the Gaussian controller is linear and reacts strongly to all noise pulses, the Cauchy controller can differentiate between measurement and process noise pulses by ignoring the former while responding to the latter. This property of the Cauchy controller occurs when an impulsive measurement noise is more likely than an impulsive process noise. The Cauchy and Gaussian controllers react similarly when applied to a system with Gaussian noises, demonstrating the robustness of the proposed control scheme.

Stochastic optimal control for backward stochastic partial differential systems

This paper studies optimal controls for a class of backward stochastic partial differential systems in the abstract evolution form. Under the assumption of a convex control domain, necessary and sufficient conditions for an admissible control to be optimal are derived in the form of stochastic maximum principles by means of a convex variation method and a duality technique. As an application, the optimal control for a linear backward stochastic evolution equation (BSEE) with quadratic cost criteria (called BSEELQ problem) is discussed, and the corresponding optimal control is characterized via the stochastic Hamilton system which is a linear full-coupled forward–backward stochastic evolution equation (FBSEE) and consists of the state equation, the adjoint equation and the dual presentation of the optimal control.

Optimal variational principle for backward stochastic control systems associated with Lévy processes

The paper is concerned with optimal control of backward stochastic differential equation (BSDE) driven by Teugel’s martingales and an independent multi-dimensional Brownian motion, where Teugel’s martingales are a family of pairwise strongly orthonormal martingales associated with Lévy processes (see e.g., Nualart and Schoutens’ paper in 2000). We derive the necessary and sufficient conditions for the existence of the optimal control by means of convex variation methods and duality techniques. As an application, the optimal control problem of linear backward stochastic differential equation with a quadratic cost criteria (or backward linear-quadratic problem, or BLQ problem for short) is discussed and characterized by a stochastic Hamilton system.

Cooperative Distributed MPC using the Alternating Direction Multiplier Method

We consider cooperative distributed model predictive control of a linear, time-invariant, discrete-time plant, which consists of coupled subsystems. The cooperating controllers minimize a quadratic cost criterion subject to input and state constraints. We outline two approaches for such distributed controllers using the alternating direction multiplier method, which allows to obtain convergence guarantees. We outline suitable stability criterions and illustrate the applicability and performance of the proposed methods by a simple example.

A Fast Gradient method for embedded linear predictive control

AbstractThis work considers the fast solution of model predictive control problems for linear systems with input constraints and a quadratic cost criterion. If the resulting optimization problem arising from the model predictive control is solved online using the Fast Gradient method one needs to determine the gradient of the cost function. We propose a method, tailored for embedded control purposes, that efficiently calculates the gradient taking the underlying structure of the system into account. Moreover, we discuss how the stability of the plant influences the required number of iterations to obtain a solution within a prescribed accuracy.

Optimal Sliding Mode Control via Penalty Approach for Discrete-Time Linear Systems

AbstractThis paper introduces a penalty approach and quadratic cost criterion to deal with the optimal sliding mode control problem for linear discrete-time systems. The solution provided is a linear state feedback control law based on a recursive Riccati equation defined in a unified framework. A numerical example is shown to illustrate the effectiveness of this new approach.

Optimal and optimal-linear control over lossy, distributed networks

AbstractWe consider an optimal control problem for networked control systems, where the loop is closed via a lossy, distributed network with an acknowledgment mechanism. The network is distributed in the sense that there are different sets of sensors and actuators that each communicate individually with the controller. We assume that all packets, i.e., the measurement packets, the control packets and the acknowledgment packets are sent over the lossy network and thus are subject to loss. We derive suboptimal controllers with respect to a quadratic cost criterion for the general case and optimal controllers for the case that all states are perfectly measured over a single link. Additionally, we present stability criteria for both cases.

Scanning path optimization for ultrasound surgery

One of the problems in ultrasound surgery is the long treatment times when large tumour volumes are sonicated. Large tumours are usually treated by scanning the tumour volume using a sequence of individual focus points. During the scanning, it is possible that surrounding healthy tissue suffers from undesired temperature rise. The selection of the scanning path so that the tumour volume is treated as fast as possible while temperature rise in healthy tissue is minimized would increase the efficiency of ultrasound surgery. The main purpose of this paper is to develop a computationally efficient method which optimizes the scanning path. The optimization algorithm is based on the minimum time formulation of the optimal control theory. The developed algorithm uses quadratic cost criteria to obtain the desired thermal dose in the tumour region. The derived method is evaluated with numerical simulations in 3D which are applied to ultrasound surgery of the breast in simplified geometry. Results from the simulations show that the treatment time as well as the total applied energy can be decreased from 16% to 43% as compared to standard sonication. The robustness of the optimized scanning path is studied by varying the perfusion and absorption in the tumour region.

Feedforward and feedback control of ultrasound surgery

In this paper, a two stage method is used to control the thermal dose distribution in ultrasound surgery. In the first stage, a model-based nonlinear optimal control problem is solved, leading to an optimal feedforward thermal dose distribution. The control inputs are the real and the imaginary parts of the ultrasound transducer excitations and the feedforward control give the trajectories for control input and temperature that provide the optimal thermal dose distribution in tissues. A quadratic cost criteria is used to weight the thermal dose distribution and inequality constraints are used to limit the maximum input amplitude. In the second stage, the original nonlinear system equation is linearized with respect to the feedforward trajectories and an LQG feedback controller is derived from the linearized equations. The quadratic cost function is used to weight the error between the feedforward and measured temperature trajectories. With feedback, modeling errors in feedforward control can be corrected during the sonication and numerical simulations show that the derived method leads to a feasible solution, even when large modeling errors are present.

Quadratic guaranteed cost control of uncertain neutral delay‐differential systems

This paper is concerned with the robust stabilization problem for uncertain neutral delay‐differential systems. An integral quadratic cost criterion is employed to measure system performance. By checking the Hamiltonian matrix and solving an algebraic Riccati equation, a static state feedback control law is proposed. This simple state feedback controller not only stabilizes perturbed neutral systems but also guarantees an upper bound of system performance. Last, an example is included to illustrate the results developed in this paper.

Thermal dose optimization method for ultrasound surgery

In this paper, a model-based optimization method is derived to control the thermal dose in biological tissues for ultrasound surgery. The optimization method uses the bioheat equation as a system model and quadratic cost criteria for the desired thermal dose. Time-harmonic quasi-stationary ultrasound fields are used as the heat source. In this method the optimal phase and the amplitude trajectories are found directly by minimizing the associated cost function. The approach also allows for maximum input amplitude constraints. The method is based on the Hamiltonian form of the system and results in a large dimensional nonlinear optimization problem which is solved with a gradient-type iterative scheme. The performance of the optimization method is tested with 2D simulations and it is shown that the approach is able to yield a feasible nominal solution. This nominal evolution would then eventually be sought to be maintained with the help of a feedback controller during the actual sonication.

An optimal control approach for ultrasound induced heating

The absorbed energy of ultrasound and microwaves is changed into heat in a dissipative medium. This wave field induced heating has numerous applications in industry and medicine. In these applications the ability to control the resulting heat distribution is essential. In this paper we design a controller for quadratic cost criteria in the ultrasound induced heating problem in inhomogeneous media. In this approach the time-harmonic pressure fields are computed from individual wave sources by solving the Helmholtz equation with the ultra-weak variational formulation. The equations that define the controller are derived from the Hamiltonian form of the system, which results in a large dimension non-linear control problem. As an example we study a numerical simulation that resembles an ultrasound therapy problem in 2D. The simulation shows that feasible solutions can be obtained with this approach.