CRONE control of continuous linear time periodic systems: Application to a testing bench

Sample and hold functions have been described for periodic linear system control applications where they are used to assign the eigenvalues of the system monodromy matrix. If the system is controllable in one period it was shown that the complete monodromy matrix can be assigned. In this paper we apply these methods to non periodic continuous time linear systems. When the system is controllable across each sampling interval, the state transition matrix associated with each interval can be completely assigned with full state feedback. Thus the resulting closed loop system resembles a closed loop periodic system whose monodromy matrix is completely specified.

This paper focuses on the parameter estimation of multi-input multi-output (MIMO), linear time-periodic (LTP) systems in the time domain. Discrete-time state-space models are selected for the description of LTP systems and the Covariance Based Realization Algorithm (CoBRA), one branch of subspace methods, is used for the identification of the generalized time-lifted state-space model for an LTP system. The use of the CoBRA method for estimation is motivated by the robustness against output noise with (high-order) noise dynamics and a focus on the estimation of low rank deterministic model for the LTP system. In addition to the use of CoBRA, a novel method is proposed to compute a topologically equivalent realization for the original LTP system. The method includes two different but theoretically equivalent approaches of calculating the state matrices, which can be also used as a measure for the estimation quality. Finally, a simulation is given to illustrate the efficiency of the proposed method.

In this paper, iterative learning control is applied to the continuous linear time invariant system with random noise term. The system with continuous time and discrete iteration number is thought as the hybrid system, which becomes more complex than exists systems based on iterative learning control. It is proved that the input error and tracking error along the iteration direction can converge to be bounded in the mean square sense using Gronwall inequality and stochastic differential equation theory. The results show that tracking error have closely relation with noise covariance. Finally, the effectiveness of the algorithm is verified by numerical simulation.

The epsilon-entropy was adopted as a measure of information in the analysis of linear Gaussian continuous time control systems under the consideration that the `virtual reproductions' of system input and output are subjected to a common high enough precision requirement. The function of system variety which describes the variation of time average information in system was defined as the difference between epsilon-entropy rates of system input and output. It formulates how does the information or uncertainty of input transmit through the dynamic system and impact output. Relations within variety, Bode sensitivity integral and H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">¿</sub> entropy were then derived.

Microgrids are gaining considerable attention due to the installation of renewable power generation resources in areas that are closer to the load centers. The multivariable droop idea, acknowledges active power sharing by concurrent drooping of both voltage and frequency, in resistance as well as weak grids. To develop a multivariable-droop synchronous current converter control strategy for current management using Lyapunov stability of microgrid applications. In this paper overview of Lyapunov stability of linear continuous systems is presented. The conditions for the existence of the solutions for both regular and irregular singular linear systems are introduced. To guarantee an asymptotical stability for linear continuous singular systems and new stability conditions are analyzed. A considerable attention is focused on the design of controllers for multivariable linear systems so that certain system properties could be preserved under various classes of perturbations occurring in the system. It presents a multivariable-droop synchronous current converter control strategy for current management of weak grid/microgrid applications, and offers a systematic and straightforward design approach using the loop shaping technique. The controller integrates current regulation, management, and limitation that are applicable to microgrids (MGs). The simulation is done by MATLAB/SIMULINK software.

This article presents the central finite-dimensional H ∞ controller for linear time-varying systems with unknown parameters, that is suboptimal for a given threshold γ with respect to a modified Bolza–Meyer quadratic criterion including the attenuation control term with the opposite sign. In contrast to the previously obtained results, this article reduces the original H ∞ controller problem to the corresponding H 2 controller problem, using the technique proposed in Doyle et al. [Doyle, J.C., Glover, K., Khargonekar, P.P., and Francis, B.A. (1989), ‘State-space Solutions to Standard H 2 and H Infinity Control Problems’, IEEE Transactions Automatic Control, 34, 831–847]. This article yields the central suboptimal H ∞ controller for linear systems with unknown parameters in a closed finite-dimensional form, based on the corresponding H 2 controller obtained in Basin and Calderon-Alvarez [Basin, M.V., and Calderon-Alvarez, D. (2008), ‘Optimal LQG Controller for Linear Systems with Unknown Parameters’, Journal of The Franklin Institute, 345, 293–302]. Numerical simulations are conducted to verify performance of the designed central suboptimal controller for uncertain linear systems with unknown parameters against the conventional central suboptimal H ∞ controller for linear systems with exactly known parameter values.

This paper proposes guaranteed cost design of robust output feedback controller for continuous linear parametric uncertain systems. New necessary and sufficient conditions for static output feedback stabilizability of linear continuous time systems underly the design procedure. Proposed algorithms are computationally simple and tightly connected with the Lyapunov stability theory and the LQR optimal state feedback design. The proposed approach allows for prescribing the structure of the output feedback gain matrix (including the decentralized one) by the designer. New design method proposed in this paper, exploit genetic algorithm to design robust controller with guaranteed cost for polytopic linear continuous time systems. Numerical example is given to illustrate the performance of the proposed robust controller.

This paper discusses the optimal preview control problem for a class of linear continuous stochastic control systems in the infinite horizon, based on the augmented error system method. Firstly, an assistant system is designed and the state equation is translated to the assistant system. Then, an integrator is introduced to construct a stochastic augmented error system. As a result, the tracking problem is converted to a regulation problem. Secondly, the optimal regulator is solved based on dynamic programming principle for the stochastic system, and the optimal preview controller of the original system is obtained. Compared with the finite horizon, we simplify the performance index. We also study the stability of the stochastic augmented error system and design the observer for the original stochastic system. Finally, the simulation example shows the effectiveness of the conclusion in this paper.

As per Floquet theory, a transformation matrix (Lyapunov Floquet transformation matrix) converts a linear time periodic system to a linear time-invariant one. Though a closed form expression for such a matrix was missing in the literature, this method has been widely used for studying the dynamical stability of a time periodic system. In this paper, the authors have derived a closed form expression for the Lyapunov Floquet (L-F) transformation matrix analytically using intuitive state augmentation, Modal Transformation and Normal Forms techniques. The results are tested and validated with the numerical methods on a Mathieu equation with and without damping. This approach could be applied to any linear time periodic systems.

A linear continuous time-varying system and a sample-data system with continuous time-varying linear part are considered. For both systems a two-dimensional control is formed which provides an invariant stabilization of a closed loop system. I. INTRODUCTION The invariant stabilization problem deals with such stabi- lization which provides an independence of output from outer action and boundedness of other variables. As demonstrated in (1-3) this problem remains one of the most actual in the theory of control systems. The main conclusion of these papers is: beginning from linear stationary systems the scalar control can provide either stability or invariance. A statement in (4) says that for a linear stationary system it is possible to combine the requirements of invariance and stability if the outer action is considered as one of the regulator's input. In the present report a linear continuous time-varying system and a sample-data system with continuous time- varying linear part are considered. For both systems a two- dimensional control is formed which provides an invariant stabilization of a closed loop system. If the vector of ob- servation is orthogonal to the vector of outer action the measurement of the action is not required. II. LINEAR NONSTATIONARY SYSTEM The following system

In this paper, new matrix inequality conditions on the terminal weighting matrices are proposed for linear continuous time-varying systems. Under these conditions, non-increasing and non-decreasing monotonicities of the saddle point value of a dynamic game are shown to be guaranteed. It is proved that the proposed terminal inequality conditions ensure the closed-loop stability of the receding horizon H/sub /spl infin// control (RHHC). The stabilizing RHHC guarantees an H/sub /spl infin// norm bound of the close-loop system. The proposed terminal inequality conditions for the monotonicity of the saddle point value and the closed-loop stability include most well-known existing terminal conditions as special cases. The results for time-invariant systems are obtained correspondingly from those in the time-varying case.

The optimal output tracking control(OOTC) problem is considered for continuous linear time-invariable systems, which include control delays and external sinusoidal disturbances. We first transform the original system into a new system without time-delay, then construct a new state vector, which containing a external system’s vector. The external system described the aim which we want to track. Following the law of internal model principle, we transform the original and external systems into a new augmented system. Finally we get an optimal control law according to the optimal performance index, and prove the method is effective by emulation.

A linear operator for LTP systems is presented, which is analogous to the LTI transfer function. Properties such as poles, transmission zeroes, and their associated directional properties are described in terms familiar to the multivariable control theory.

We present the experimental demonstration of the occurrence of exceptional points of degeneracy (EPDs) in a single resonator by introducing a linear time-periodic variation of one of its components. This is in contrast with the requirement of two coupled resonators with parity time-symmetric systems with precise values of gain and loss. In the proposed scheme, only the tuning of the modulation frequency is required, which is easily achieved in electronic systems. The EPD is a point in a system parameters’ space at which two or more eigenstates coalesce, and this leads to unique properties not occurring at other non-degenerate operating points. We show theoretically and experimentally the existence of a second-order EPD in a time-varying single resonator. Furthermore, we measure the sensitivity of the proposed system to a small structural perturbation and show that the two shifted system’s eigenfrequencies are well detected even for relative perturbations of 0.3%, with distinguished peaks well above the noise floor. We show that the regime of operation of the system at an EPD leads to a unique square-root-like sensitivity, which can devise new exceptionally sensitive sensors based on a single resonator by simply applying time modulation.

A linear control system with a periodic matrix of coefficients and program control is considered. The matrix under control is constant, rectangular (the number of columns does not exceed the number of rows) and its rank is not maximum, i. e. less than the number of columns. It is assumed that the control is periodic, and the module of its frequencies, i. e. the smallest additive group of real numbers, including all Fourier exponents of this control, is contained in the frequency module of the coefficient matrix. The following task is posed: to construct such a control from an admissible set that switches the system to asynchronous mode, i. e. the system must have periodic solutions such that the intersection of the frequency moduli of the solution and the coefficient matrix is trivial. The problem posed is called the problem of synthesis of asynchronous mode. The solution to the formulated problem significantly depends on the structure of the average value of the coefficient matrix. In particular, its solution is known for systems with zero average. In addition, solvability conditions were obtained in the case when the matrix under control has zero rows, the averaging of the coefficient matrix is reduced to the form with upper left diagonal block and with zero remaining blocks. In this paper we consider a more general case with a nontrivial left lower block. Assuming an incomplete column rank of the matrix function composed from the rows of oscillation path of the coefficient matrix, we construct the control explicitly. This control switches the system to asynchronous mode.