Control of discrete-time descriptor systems. An anisotropy-based approach (Q1753979)

The book presents recent results on the anisotropy based optimal control of discrete time descriptor systems. It will be of high value to all researchers interested in the field and its future developments. The main points covered in the book, as presented in each chapter, are as follows: Chapter 1 starts with a presentation of several applications of descriptor systems. Examples are taken from a variety of disciplines and include mechanical systems, like spring mass systems and constrained systems,, chemical models of isothermal reactions and evaporators, biological food chain models, economic models, large scale interconnected systems, robotic systems, electrical networks and discretized partial differential equations. Chapter 2 presents the basics of descriptor systems, including various equivalent forms, their solution formulas over a finite and infinite time horizon, and an algebraic analysis of controllability, observability, and stability. Chapter 3 introduces the anisotropy based analysis for descriptor systems. After the basic notions of anisotropy and \(\mathscr{H}_2, \mathscr{H}_\infty\) and \(a\)-anisotropic norms are defined, systems \((P)\) of the form \[ Ex(k+1)=Ax(k)+B w(k) \] \[ y(k)=Cx(k)+Dw(k) \] are considered, where \(x(k)\) is the system's state, \(w(k)\) is a random stationary sequence with bounded mean anisotropy, \(y(k)\) is the measurable output and \(E,A,B,C,D\) are matrices of appropriate dimensions, with \(E\) singular. Assuming that the system is admissible, that is, causal and stable, the problem of obtaining conditions for \(a\)-anisotropic norm bound \(||P||_a\leq\gamma\), \(\gamma>0\) is considered. In Chapter 4, optimal control for systems of the following form is considered \[ Ex(k+1)=Ax(k)+B_1 w(k)+B_2 u(k) \] \[ z(k)=C_1x(k)+D_{11} w(k)+D_{12} u(k) \] \[ y(k)=C_2x(k)+D_{21} w(k)+D_{22} u(k) \] where \(w(k)\) the input, \(u(k)\) the control, \(z(k)\) the controllable output, \(y(k)\) the measurable output and the matrices involved are of appropriate dimensions. The input signal is a colored Gaussian disturbance with known mean anisotropy level. The problem studied is the computation of a suitable feedback law that makes the closed loop system admissible and minimizes its \(a\)-anisotropic norm. Both state (\(u(k)=Kx(k)\)) and output (\(u(k)=Ky(k)\)) feedback is considered. The solution involves transforming the problem to a weighted \(\mathscr{H}_2\) optimization problem and solving generalized Riccati and Lyapunov equations. Chapter 5 studies the suboptimal control problem for system (2), which is finding a feedback law such that the closed loop system is admissible and its \(a\)-anisotropic norm is bounded. State feedback (\(u(k)=Kx(k)\)) as well as full feedback (\(u(k)=K_1w(k)+K_2u(k)\)) is considered and different approaches are used to establish necessary conditions for the existence of suitable feedback matrices. In Chapter 6, the anisotropy based analysis of descriptor systems (1) is reconsidered, for the case where the input signal \(w(k)\) has nonzero mean. For this case it is shown that the \(a\)-anisotropic norm is generally not monotonic and the monotonicity is satisfied only under some conditions on the mean. Chapter 7 considers systems with norm bounded uncertainties, that is \[ Ex(k+1)=A_\Delta x(k)+B_{\Delta1} w(k)+B_2u(k) \] \[ y(k)=C_\Delta x(k)+D_{\Delta2}w(k) \] with \(A_\Delta=A+M_A\Delta N_A\), \(B_{\Delta1} =B_1+M_B\Delta N_B\), \(C_\Delta=C+M_C\Delta N_C\), \(D_{\Delta1} =D_1+M_D\Delta N_D\) and \(||\Delta||_2\leq1\). The problem of computing a suitable state feedback law such that the closed loop system is admissible and has bounded \(a\)-anisotropic norm is considered and solved. In each chapter, the results are illustrated through numerous numerical and application examples. It should also be noted that in the control problems considered in each chapter, computationally efficient methods are applied for the computation of the feedback matrices.