Input Weighting Matrix Research Articles

Quantized Control Design for Linear Systems Using Reinforcement Learning

In this paper, a quadratic stabilizing controller minimizing a cost function is designed through model-free and online reinforcement learning for systems with logarithmic quantized input. By introducing a new gain dependent on quantization density, the input and related weighting matrix in the cost function are deviated from their original ones. Then, using these deviated parameters, the controller is trained through reinforcement learning such that the closed-loop system satisfies the quadratic stability condition with the cost function minimized. An inverted pendulum example is used to show the effectiveness and merits of the proposed method.

MPC closed-loop identification without excitation

This paper presents a method of closed-loop identification for multivariable systems without external excitation. The method is specially designed for model predictive control (MPC) systems. Without using external excitation (test signals), the method ensures the informativity of the closed-loop data and, at the same time, improve the control performance during the test period. The purpose of the study is to reduce the cost of identification test. The basic idea is to switch the input weighting matrix in the MPC controller which leads to the informativity of the data-set. A preliminary test is carried out in order to find a new input weighting matrix which improve the control performance; then a switching scheme is developed based on the two weighting matrices. Traditional simulation based model validation no longer works in closed-loop identification without excitation, and model error bounds on the frequency responses can be used instead. The effectiveness of the proposed method is demonstrated by a simulation study.

MPC closed-loop identification without excitation

This paper presents a method of closed-loop identification for multivariable systems without external excitation. The method is specially designed for model predictive control (MPC) systems. Without using external excitation (test signals), the method ensures the informativity of the closed-loop data and, at the same time, improve the control performance during the test period. The purpose of the study is to reduce the cost of identification test. The basic idea is to switch the input weighting matrix in the MPC controller which leads to the informativity of the data-set. A preliminary test is carried out in order to find a new input weighting matrix which improve the control performance; then a switching scheme is developed based on the two weighting matrixes. Traditional simulation based model validation no longer works in closed-loop identification without excitation, and model error bounds on the frequency responses can be used instead. The effectiveness of the proposed method is demonstrated by a simulation study.

Existence of a Mean-Square Stabilizing Solution to a Modified Algebraic Riccati Equation

In this paper, the existence of a mean-square stabilizing solution to a discrete-time modified algebraic Riccati equation (MARE), which arises in the study of some stochastic linear quadratic optimal control problems, is investigated. The theory of cone-invariant operators is employed as the mathematical tool to tackle this problem. We provide some criteria for the cone stability, cone observability, and cone detectability of a class of cone-invariant systems in terms of the associated distinguished eigenvalues. Then two scenarios concerning the problem of existence in terms of the positive definiteness of the input weighting matrix are considered. We first study the MARE under the assumption that the input weighting matrix in the cost function is positive definite. Then an explicit necessary and sufficient condition is obtained. Such a condition is derived for the very first time and it indicates that the common condition of observability or detectability of certain stochastic systems is unnecessary. Onl...

Suboptimal mixed vaccine and chemotherapy in finite duration cancer treatment: state-dependent Riccati equation control

The aim of this paper is to propose an optimal finite duration treatment method for avoiding tumor growth. For this purpose, a mathematical model in the form of ordinary differential equations is modified with combination vaccine therapy and chemotherapy treatments. Numerical simulations, using human parameters, show that there are two equilibrium points. The tumor-free equilibrium point is unstable while the high-tumor equilibrium point is stable. Hence, the dynamics of the cancer model must be changed to have finite duration treatment. Therefore, the vaccine therapy is used to change the parameters of the system and the chemotherapy is applied for pushing the system to the domain of attraction of the healthy state. It is shown that any treatment method without changing the dynamics of the system around the tumor-free equilibrium point is not an appropriate treatment method. For optimal chemotherapy, the State-Dependent Riccati Equation based optimal control is used to the nonlinear model. Different weighting matrices are used to show the flexibility of this method in design. Simulation results show that the performance of the treatment would be better if the matrix was state dependent. In this paper, the input weighting matrix depends on the tumor cell population. If the input matrix becomes less in the beginning of the treatment causes more tumor cells eradication. Also, the present study states that a proper treatment method should not only reduce the population of tumor cells, but also change the dynamics of the cancer.

Decoupling properties of singular LQ regulation problem

Some properties of a limiting solution derived from a standard LQ regulator problem are discussed in the case where the input weighting matrix tends to zero, for a discrete time system. The explicit limiting solution is given in some literature. It is shown that the limiting solution is closely connected with a disturbance decoupling problem. Moreover it is pointed out that the limiting feedback gain with the help of feedforward gain constructs the decoupled system with the maximum number of blocks.

Regularity and singularity in linear-quadratic control subject to implicit continuous-time systems

A linear-quadratic (LQ) control problem subject to a standard continuous-time system is called regular if the input weighting matrix is invertible, and singular if this is not the case. Consequently, optimal inputs for regular LQ problems are ordinary functions (state feedbacks), whereas optical controls for singular problems are in general distributions, e.g., impulses. We will show that regularity and singularity in LQ problems subject to ageneral (implicit) system depends not so much on the input weighting matrix, as on the property that the integrand of the cost criterion is a function only if inputs and state trajectories are, as is the case for LQ problems, subject to astandard system. In particular, we will provide a simple criterion for distinguishing between regularity and singularity in LQ problems subject to a general system. Our criterion is expressed in the system coefficients only and reduces to the classical one if the underlying system is standard.

Curvature properties of the algebraic Riccati equation

We prove that the solution to the algebraic Ricatti equation (ARE) is concave with respect to a nonnegative-definite symmetric state weighting matrix Q when the input weighting matrix R = R T > 0 . We also prove that the solution to the ARE is concave with respect to a positive-definite diagonal input weighting matrix R when Q = Q T ≥ 0 .

Explicit solution to the singular LQ regulation problem

An explicit solution to the multivariable discrete linear quadratic (LQ) regulation problem is obtained in the limiting singular case where the input weighting matrix tends to zero. Such a solution follows from a suitable spectral factorization of the input spectrum density matrix under the assumption that the system is stabilizable and detectable and that its transfer function matrix is of full rank. The suitable spectral factor is shown to be the product of the system's minimum-phase image and its unitary interactor matrix. The unitary interactor matrix defined is a special case of the nilpotent interactor matrix defined by M.W. Rogozinski et al. (1987).< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Some system considerations of neuron pools with feedback.

This paper describes an approach to the analysis of the inputoutput relationships present in a neuron pool that receives a number of inputs. These inputs consist of primary inputs to the neuron pool and inputs resulting from feedback of information from the neuron pool as well. Multiple input-output relationships are obtained in terms of the synaptic weightings of the inputs, the membrane response characteristics of the neurons and the conduction delays on the feedback pathways.y=(I+MHD) -1 ·MFx is the explicit representation of the cell pool behavior assuming quasi-linear conditions, wherey is the output vector of cell responses,I is the identity matrix,M is the response matrix of the cells,H is the feedback synaptic weighting matrix,D is the delay matrix,F is the input weighting matrix, andx is the input vector. It is shown that a solution to this formulation exists, is unique, is stable, and can be computed by specified algorithms. An insight gained from this formulation suggests that the output of each cell in the pool is related to virtually all of the inputs to the pool and the outputs of all cells in the pool.