Model order-reduction for discrete-time switched linear systems

This article describes the problem of model order-reduction for a class of hybrid discrete-time switched linear systems composed of linear discrete-time invariant subsystems with a switching rule. This article investigates two novel approaches to model order-reduction. The first approach consists in evaluating the error approximation performance; the problems are solved using the robust stability results of the switched systems. The second approach presents the reachability and observability Gramians of the switched systems, which allows a balanced truncation model reduction procedure. A numerical example shows the effectiveness of the proposed approaches.     

Estimability and stochastic observability of quantised linear systems

The estimability and stochastic observability of quantised discrete-time linear dynamic systems are discussed from information theoretic viewpoint. Algebraic conditions of estimability and stochastic observability for quantised linear Gaussian systems, i.e., certain Gramians having full rank, are proposed based on the measure of mutual information. The obtained conditions of estimability and observability are consistent with the intuition and provide us with valuable hints on quantiser design. It is shown analytically that the Gramians of quantised systems converge to that of unquantised systems when the quantisation intervals turn to zero, and a well-designed quantiser can preserve the estimability and stochastic observability of the original system even if it is as coarse as one bit. Furthermore, the relation between estimability and stochastic observability is established for quantised stochastically autonomous systems. The analytical results are verified by illustrative simulations.     

Model reduction based on limited‐time interval impulse response Gramians

In this paper, three new Gramians are introduced namely ‐ limited‐time interval impulse response Gramians (LTIRG), generalized limited‐time Gramians (GLTG) and generalized limited‐time impulse response Gramians (GLTIRG). GLTG and GLTIRG are applicable to both unstable systems and also to systems which have eigenvalues of opposite polarities and equal magnitude. The concept of these Gramians is utilized to develop model reduction algorithms for linear time‐invariant continuous‐time single‐input single‐output (SISO) systems. In the cases of GLTIRG and GLTG based model reduction, the standard time‐limited Gramians are generalized to be applied to unstable systems by transforming the original system into a new system which requires the solution of two Riccati equations. Two numerical examples are included to illustrate the proposed methods. The results are also compared with standard techniques.     

A model validation approach to texture recognition and inpainting

A new approach to texture recognition and inpainting problems is proposed. The approach is based on the robust model validation and state estimation techniques. The proposed solutions require the modeling of textures by using uncertain dynamical systems. We propose a new modeling method which is efficient in terms of computational and memory requirements. The main aspects of the modeling method include system identification and order reduction of marginally stable uncertain discrete-time systems. To demonstrate the results, both static-image textures and video textures (also known as dynamic textures) are considered.     

State-space forms for higher-order discrete-time models

This paper presents a fundamental study of the connection between continuous- and discrete-time systems. Provided is a definition for discrete-time models, that is discrete-time systems with a continuous-time counterpart, whose order can be higher than that of the continuous-time system. This definition is based on a comparison in a certain sense on the time responses of continuous- and discrete-time systems. A theorem is presented for relating the higher-order discrete-time models to their continuous-time counterparts, which is an extension of a previous theorem for models with order equal to that of the continuous-time system. State-space forms are derived for models obtained through the use of a certain class of hold elements and through the use of mapping models, and these discrete-time systems are shown to be valid according to the definition. Special cases are models obtained using first-order and slewer hold devices, whose convergence to a continuous-time counterpart has not been shown mathematically before, and mapping models corresponding to two-step linear multi-step methods, which have not previously presented in the state-space form. The derived state-space forms provide a convenient way to implement these models for purposes of analysis, design, and implementation of discrete-time systems and finds applications in such areas as digital signal processing, digital simulation, and digital control.     

Structure-preserving model reduction for spatially interconnected systems with experimental validation on an actuated beam

A technique for model reduction of exponentially stable spatially interconnected systems is presented, where the order of the reduced model is determined by the number of truncated small generalised singular values of the structured solutions to a pair of Lyapunov inequalities. For parameter-invariant spatially interconnected systems, the technique is based on solving a pair of Lyapunov inequalities in continuous-time and -space domain with a rank constraint. Using log-det and cone complementarity methods, an improved error bound can be obtained. The approach is extended to spatially parameter-varying systems, and a balanced truncation approach using parameter-dependent Gramians is proposed to reduce the conservatism caused by the use of constant Gramians. This is done by considering two important operators, which can be used to represent multidimensional systems (temporal- and spatial-linear parameter varying interconnected systems). The results are illustrated with their application to an experimentally identified spatially interconnected model of an actuated beam; the experimentally obtained response to an excitation signal is compared with the response predicted by a reduced model.     

Frequency limited Gramians-based structure preserving model order reduction for discrete time second-order systems

A new technique for frequency limited model order reduction of discrete time second-order systems is presented. Discrete time frequency limited Gramians (DFLGs) and corresponding discrete algebraic Lyapunov equations are developed. An efficient technique for the computation of DFLGs and their Cholesky factors is presented. Computed DFLGs are partitioned to obtain position and velocity Gramians. These Gramians are balanced with different combinations to obtain various balanced transformations that yield Hankel singular values (HSVs) for order reduction. Frequency limited discrete time balanced truncation framework is proposed and truncation based on magnitudes of HSVs is applied to obtain the reduced order model. Moreover, stability conditions for reduced order models are stated. Results of the proposed technique are compared with infinite Gramians balancing scheme in order to certify the usefulness of the presented technique for frequency limited applications.     

Stability analysis and stabilization of networked linear systems with random packet losses

This paper is concerned with the stability analysis and stabilization of networked discrete-time and sampled-data linear systems with random packet losses. Asymptotic stability, mean-square stability, and stochastic stability are considered. For networked discrete-time linear systems, the packet loss period is assumed to be a finite-state Markov chain. We establish that the mean-square stability of a related discrete-time system which evolves in random time implies the mean-square stability of the system in deterministic time by using the equivalence of stability properties of Markovian jump linear systems in random time. We also establish the equivalence of asymptotic stability for the systems in deterministic discrete time and in random time. For networked sampled-data systems, a binary Markov chain is used to characterize the packet loss phenomenon of the network. In this case, the packet loss period between two transmission instants is driven by an identically independently distributed sequence assuming any positive values. Two approaches, namely the Markov jump linear system approach and randomly sampled system approach, are introduced. Based on the stability results derived, we present methods for stabilization of networked sampled-data systems in terms of matrix inequalities. Numerical examples are given to illustrate the design methods of stabilizing controllers.     

Model reduction of multidimensional and uncertain systems

Model reduction methods are presented for systems represented by a linear fractional transformation on a repeated scalar uncertainty structure. These methods involve a complete generalization of balanced realizations, balanced Gramians, and balanced truncation model reduction with guaranteed error bounds, based on solutions to a pair of linear matrix inequalities which generalize Lyapunov equations. The resulting reduction methods immediately apply to uncertainty simplification and state order reduction in the case of uncertain systems but also may be interpreted as state order reduction for multidimensional systems     

Solution of the Lyapunov matrix differential equations by the frequency method

Methods for solving the Lyapunov matrix differential and algebraic equations in the time and frequency domains are considered. The solutions of these equations are finite and infinite Gramians of various forms. A feature of the proposed new approach to the calculation of Gramians is the expansion of the Gramians in a sum of matrix bilinear or quadratic forms that are formed using Faddeev’s matrices, where each form is a solution of the linear differential or algebraic equation corresponding to an eigenvalue of the matrix or to a combination of such eigenvalues. An example illustrating the calculation of finite and infinite Gramians is discussed.     

Computation of contractive polyhedra for discrete-time linear systems with saturating controls

This paper deals with computational aspects of characterization and construction of polyhedral u -contractive sets with respect to discrete-time linear systems with saturating feedback control inputs. Using a piecewise-affine model of the saturating closed-loop system, new necessary and sufficient algebraic condition for convex closed polyhedra be u -contractive is derived. Based on linear programming formulation of this condition, an effective procedure is proposed for construction of as large as possible u -contractive convex polyhedra for estimation of the region of asymptotic stability of origin. The procedure starts with a u -contractive polyhedron, possibly contained in the region of linear control, and progressively expands it non-homothetically over the region of non-linear saturated control. The proposed approach is less conservative and computationally much more efficient than previously published ones.     

A new discrete impulse response Gramian and its application tomodel reduction

Some fundamental properties of an impulse response Gramian for linear, time-invariant, asymptotically stable, discrete single-input-single-output (SISO) systems are derived. This Gramian is system invariant and can be found by solving a Lyapunov equation. The connection with standard controllability, observability, and cross Gramians is proven. The significance of these results in model-order reduction is highlighted with an efficient procedure     

Observer design for nonlinear systems with discrete-timemeasurements

This paper focuses on the development of asymptotic observers for nonlinear discrete-time systems. It is argued that instead of trying to imitate the linear observer theory, the problem of constructing a nonlinear observer can be more fruitfully studied in the context of solving simultaneous nonlinear equations. In particular, it is shown that the discrete Newton method, properly interpreted, yields an asymptotic observer for a large class of discrete-time systems, while the continuous Newton method may be employed to obtain a global observer. Furthermore, it is analyzed how the use of Broyden's method in the observer structure affects the observer's performance and its computational complexity. An example illustrates some aspects of the proposed methods; moreover, it serves to show that these methods apply equally well to discrete-time systems and to continuous-time systems with sampled outputs     

On the balanced truncation and coprime factors reduction of Markovian jump linear systems

The paper deals with the balanced truncation and coprime factors reduction of Markovian jump linear (MJL) systems, which can have mode-varying state, input, and output dimensions. We develop machinery for balancing mean square stable MJL system realizations using generalized Gramians and strict Lyapunov inequalities, and provide an improved a priori upper bound on the error induced in the balanced truncation process. We also generalize the coprime factors reduction method and, in doing so, extend the applicability of the balanced truncation technique to the class of mean square stabilizable and detectable MJL systems. We provide tools to establish mean square stabilizability and detectability of the considered MJL systems. In addition, a notion of right-coprime factorization of MJL systems and methods to construct such factorizations are given. The error measure in the coprime factors reduction approach, while still norm-based, does not directly capture the mismatch between the nominal system and the reduced-order model, as is the case in the balanced truncation approach where mean square stable models are considered. Instead, the error measure is given in terms of the distance between the coprime factors realizations, and thus has an interpretation in terms of robust feedback stability. The paper concludes with an illustrative example which demonstrates how to apply the coprime factors model reduction approach.     

Quadratic Stabilization of Discrete-Time Bilinear Systems

We consider the problem of stabilization of discrete-time bilinear control systems. Using the linear matrix inequality technique and quadratic Lyapunov functions, we formulate a method for the construction of the so-called stabilizability ellipsoid having the property that the trajectories of the closed-loop system emanating from the points in the ellipsoid asymptotically tend to the origin. The proposed approach allows for an efficient construction of nonconvex domains of stabilizability of discrete-time bilinear control systems. The results are extended to the robust statement of the problem where the system matrix is subjected to structured uncertainties.     

Frequency Interval Gramians Based Structure Preserving Model Order Reduction for Second Order Systems

A new structure preserving model order reduction technique for second order systems in limited frequency interval is presented. Frequency limited Gramians (FLGs) and corresponding continuous time algebraic lyapunov equations (CALEs) are developed. For solution of CALEs and Cholesky factorization of FLGs, computationally efficient approximation scheme is proposed. Multiple transformations based on balancing of frequency limited position or velocity Gramians are defined in order to compute Hankel singular values (HSVs). Frequency limited second order balanced truncation based on magnitudes of HSVs is performed for order reduction. Moreover, stability conditions for reduced order models (ROMs) are stated and algorithms for achieving stability in ROMs are proposed. Results are compared with existing technique to certify the usefulness of the proposed technique.     

MPC of constrained discrete-time linear periodic systems — A framework for asynchronous control: Strong feasibility, stability and optimality via periodic invariance

State-feedback model predictive control (MPC) of discrete-time linear periodic systems with time-dependent state and input dimensions is considered. The states and inputs are subject to periodically time-dependent, hard, convex, polyhedral constraints. First, periodic controlled and positively invariant sets are characterized, and a method to determine the maximum periodic controlled and positively invariant sets is derived. The proposed periodic controlled invariant sets are then employed in the design of least-restrictive strongly feasible reference-tracking MPC problems. The proposed periodic positively invariant sets are employed in combination with well-known results on optimal unconstrained periodic linear-quadratic regulation (LQR) to yield constrained periodic LQR control laws that are stabilizing and optimal. One motivation for systems with time-dependent dimensions is efficient control law synthesis for discrete-time systems with asynchronous inputs, for which a novel modeling framework resulting in low dimensional models is proposed. The presented methods are applied to a multirate nano-positioning system.     

Exponential stabilization of discrete-time switched linear systems

This article studies the exponential stabilization problem for discrete-time switched linear systems based on a control-Lyapunov function approach. It is proved that a switched linear system is exponentially stabilizable if and only if there exists a piecewise quadratic control-Lyapunov function. Such a converse control-Lyapunov function theorem justifies many of the earlier synthesis methods that have adopted piecewise quadratic Lyapunov functions for convenience or heuristic reasons. In addition, it is also proved that if a switched linear system is exponentially stabilizable, then it must be stabilizable by a stationary suboptimal policy of a related switched linear-quadratic regulator (LQR) problem. Motivated by some recent results of the switched LQR problem, an efficient algorithm is proposed, which is guaranteed to yield a control-Lyapunov function and a stabilizing policy whenever the system is exponentially stabilizable.