Linear Periodic Control Systems Research Articles

Assignment of the upper Bohl exponent for linear periodic control systems in Hilbert spaces

A linear continuous-time control system with time-varying linear bounded operator coefficients in a Hilbert space is considered. The controller in the system has the form of linear state feedback with a time-varying linear bounded gain operator function. We study the problem of arbitrary assignment of the upper Bohl exponent by state feedback control. The property of exact controllability for the open-loop system is studied, some necessary and sufficient conditions are obtained for exact controllability. For periodic systems, it is proved that if the open-loop system is exactly controllable then one can shift the upper Bohl exponent of the closed-loop system by any pregiven number with respect to the upper Bohl exponent of the free system by means of linear state feedback with a periodic gain operator function. This implies arbitrary assignability of the upper Bohl exponent by linear state feedback. Finally, an illustrative example is presented.

Reflexive periodic solutions of general periodic matrix equations

Analysis and design of linear periodic control systems are closely related to the periodic matrix equations. The objective of this paper is to provide four new iterative methods based on the conjugate gradient normal equation error (CGNE), conjugate gradient normal equation residual (CGNR), and least‐squares QR factorization (LSQR) algorithms to find the reflexive periodic solutions (X1,Y1,X2,Y2,…,Xσ,Yσ) of the general periodic matrix equations urn:x-wiley:mma:media:mma5596:mma5596-math-0001 for i = 1,2,…,σ. The iterative methods are guaranteed to converge in a finite number of steps in the absence of round‐off errors. Finally, some numerical results are performed to illustrate the efficiency and feasibility of new methods.

Periodic conjugate direction algorithm for symmetric periodic solutions of general coupled periodic matrix equations

Analysis and design of linear periodic control systems are closely related to the periodic matrix equations. The conjugate direction (CD) method is a famous iterative algorithm to find the solution to nonsymmetric linear systems Ax=b. In this work, a new method based on the CD method is proposed for computing the symmetric periodic solutions (X1,X2,…,Xλ) and (Y1,Y2,…,Yλ) of general coupled periodic matrix equations ∑s=0λ−1(Ai,sXi+sBi,s+Ci,sYi+sDi,s)=Mi,∑s=0λ−1(Ei,sXi+sFi,s+Gi,sYi+sHi,s)=Ni,for i=1,2,…,λ. The key idea of the scheme is to extend the CD method by means of Kronecker product and vectorization operator. In order to assess the convergence properties of the method, some theoretical results are given. Finally two numerical examples are included to illustrate the efficiency and effectiveness of the method.

A Finite Iterative Method for Solving the General Coupled Discrete-Time Periodic Matrix Equations

Analysis and design of linear periodic control systems are closely related to the discrete-time periodic matrix equations. In this paper, we propose an iterative algorithm based on the conjugate gradient method on the normal equations (CGNE) for finding the solution group of the general coupled periodic matrix equations $$\begin{aligned} \left\{ \begin{array}{l} A_{1,i}X_iB_{1,i}+C_{1,i}X_{i+1}D_{1,i}=E_{1,i},\\ A_{2,i}X_iB_{2,i}+C_{2,i}X_{i+1}D_{2,i}=E_{2,i}, \end{array} \right. ~~~\mathrm {for}~~~i=1,2,3,\ldots . \end{aligned}$$ A 1 , i X i B 1 , i + C 1 , i X i + 1 D 1 , i = E 1 , i , A 2 , i X i B 2 , i + C 2 , i X i + 1 D 2 , i = E 2 , i , for i = 1 , 2 , 3 , ? . By proving some properties of the algorithm, we show that the solution group of the periodic matrix equations can be obtained within a finite number of iterations in the absence of roundoff errors. Numerical examples are given to illustrate the efficiency and accuracy of the proposed algorithm.

Developing CGNE algorithm for the periodic discrete-time generalized coupled Sylvester matrix equations

The discrete-time periodic matrix equations often arise in analysis and design of linear periodic control systems. This paper studies the problem of finding the solutions of the periodic continuous-time generalized coupled Sylvester matrix equations $$\begin{aligned} \left\{ \begin{array}{ll} A_kX_kB_k+C_kY_kD_k=M_k,&{} \hbox {} \\ E_kX_{k+1}F_k+G_kY_kH_k=N_k,&{} \hbox {} \end{array} \right. \quad {\mathrm {for}} \quad k=1,2,\ldots ,\phi . \end{aligned}$$ We propose an iterative method based on the idea of conjugate gradient method on the normal equations (CGNE) for solving this problem. By proving some properties of the iterative method, it is shown that the iterative method can solve the problem within a finite number of iterations in the absence of roundoff errors. Finally, numerical results are carried out to confirm the theoretical conclusions.

Asymptotic theory of attainable sets to linear periodic impulsive control systems

In this paper, asymptotic large-time dynamics of attainable sets to linear periodic control systems with bounded total impulse of control is studied. Asymptotic formulas for attainable sets and their shapes, i.e., the same sets considered up to an arbitrary nonsingular linear transformation, are found. It is shown that all limit shapes are obtained as the join of the limit shapes for neutral, stable-neutral, and unstable-neutral system components.

Asymptotic Behavior of the Attainable Sets of Linear Periodic Control Systems

We study in a more general setup a phenomenon concerning the asymptotic behavior of attainable sets for linear autonomous control systems. Oseevich's main result (1991) consists in discovering a simple behavior in a long run of shapes of attainable sets, unlike that of attainable sets itself. Here, shape stands for the entity of all images of a set under nonsingular linear transformations. More precisely, the shapes of attainable sets for linear autonomous control systems always possess a limit as t/spl rarr//spl infin/ in a natural metric of the infinite-dimensional space of forms. At present the range of this phenomenon is not clear-cut. In a search for its limits we consider here the asymptotic behavior of attainable sets for linear periodic control systems. Our main result establishes both a similarity and a distinction between the case under consideration and the autonomous case. We show that the curve t/spl rarr/D(t), t>0 of forms of attainable sets approaches, in general, not a point, but a limit cycle.

Optimal Solutions of Linear Periodic Control Systems with Convex Integrands

In this work we study the existence and asymptotic behavior of overtaking optimal trajectories for linear control systems with convex integrands. We extend the results obtained by Artstein and Leizarowitz for tracking periodic problems with quadratic integrands [2] and establish the existence and uniqueness of optimal trajectories on an infinite horizon. The asymptotic dynamics of finite time optimizers is examined.

Periodic Lyapunov equations: Some applications and new algorithms

The discrete-time periodic Lyapunov equation has several important applications in the analysis and design of linear periodic control systems. Specific applications considered are the solution of state- and output-feedback optimal periodic control problems, the stabilization by periodic state feedback and the square-root balancing of discrete-time periodic systems. Efficient and numerically reliable algorithms based on the periodic Schur decomposition are proposed for the solution of periodic Lyapunov equations. The proposed algorithms are extensions of the direct solution methods for standard discrete-time Lyapunov equations for the case of indefinite as well as non-negative definite solution.

Synthesis of Optimal Control for an Infinite Dimensional Periodic Problem

We prove an existence and uniqueness result on periodic solutions of an infinite dimensional Riccati equation.

Linear periodic control systems: Controllability with restrained controls

Consider the problem of null-controllability for a linear non-autonomous system of the form\(\dot x(t) = A(t)x(t) + B(t)u(t),x(t)\varepsilon X,u(t)\varepsilon \Omega \subset U\), whereX andU are Banach spaces,A(t) andB(t) are linear bounded operators for eacht ≥ 0, and Ω is a subset containing 0 (but not necessarily as an interior point). Some necessary and sufficient conditions for local and global null-controllability are proved whenA(·) andB(·) are periodic continuous functions. The proofs of the main results are based on discretization and on consideration of the corresponding linear discrete-time systems.

Periodicity, Detectability and the Matrix Riccati Equation

This paper discusses the periodic solution of matrix Riccati differential equations with periodic coefficients. Such equations arise in linear filtering and control and in many other applications. The principal result: the existence of a periodic solution is equivalent to detectability and stabilizability of certain coefficient pairs. This result generalizes the Kalman–Wonham–Kucera theorem for algebraic Riccati equations. Among the numerous preliminaries is a discussion, apparently new, of detectability for linear periodic control systems. Another important result, for a linear matrix differential equation, is the equivalence of a bounded solution, an exponentially stable solution and a periodic solution. Finally, the periodic solution is shown to be an equilibrium solution in the sense of Kalman.