Some issues of structure and control for linear time-invariant systems

The objective of this paper is to emphasise the role played by particular structures in the solution of some control problems. The so-called “structural approach” relies on various indicators of dynamical systems such as, for instance, finite and infinite zeros, kernel indices, …. The fundamental invariance properties of these structures under the action of some transformations groups (e.g. feedback) are at the origin of their key role. Structural solutions to “classical” control problems, such as disturbance rejection, model matching and non-interaction are now rather well known: zeros at infinity play a role in the existence of “proper” solutions, while finite (invariant) zeros allow for the characterisation of “fixed poles”, whose location in the complex plane gives answer to pole placement limitations (including stability). Among the recent contributions to this structural approach, a particular attention is here devoted to:

- “Partial” versions of some of these control problems: The control objective only concerns a finite number of (and not necessarily all) the first Markov parameters of the transfer function matrix of the controlled system (e.g. to be zero for disturbance rejection or model matching, to be diagonal for non-interaction). Some interesting new issues in the dual context of failure detection are also sketched.

- Generalised solutions: Based on proportional and derivative feedback laws, with new issues in the context of systems with variable internal structures, and also for systems with delays.

Geometric concepts, such as invariant and almost invariant subspaces, and algebraic counterparts, such as factorisations on some special rings, are intermediary tools which support the characterisations of those particular structures and which allow for a structural treatment of the considered control and/or observation problems.

The results are here presented without proof: references are given to previous published results (in most cases in books and journals which are easily available), and some simple examples are used to illustrate non-standard notions (among which systems with variable internal structure, and time domain left invertibility).

Most of the results here presented rely on long and intensive collaborations between the author and various colleagues.     

Output feedback variable structure control for linear systems with uncertainties and disturbances

This paper proposes a dynamic output feedback variable structure controller for linear MIMO systems with mismatched and matched norm-bounded uncertainties and matched nonlinear disturbances. The proposed controller consists of nonlinear and linear parts, similar to the unit vector type of controller. The nonlinear part of the new controller takes care only of matched uncertainties and disturbances and, on the other hand, the linear part with full dynamics completely handles mismatched uncertainties. Designing such a linear part leads to use a Lyapunov function associated with the full states, which achieves the global stability against the mismatched uncertainties. The resulting criteria are, furthermore, converted into solvable ones, by using the so-called cone complementary linearization algorithm for bi-convex problems.     

The output regulation problem with stability for linear switching systems: A geometric approach

This work presents a solution for the output regulation problem with quadratic stability under arbitrary switching in linear switching systems. The extension to other stability requirements, like asymptotic stability in particular, is also considered and it is shown to affect only few specific features of the proposed solution. The main reason is that the geometric approach, which is at the basis of the developed methodology, establishes a neat separation between the structural aspects and the stability aspects of the problem. For the same reason, continuous-time systems and discrete-time systems are given a unified treatment as far as the structural issues are concerned, while different technicalities characterize the discussion of the stability issues.     

On almost invariant subspaces for implicit linear discrete-time systems

The concept of almost invariant subspace for an implicit linear discrete-time system is introduced and studied in detail. It is shown also that for regular homogeneous implicit systems the so-called deflating subspaces can be identified with almost invariant subspaces.     

A variable structure convex programming based control approach for a class of uncertain linear systems

A variable structure convex programming based control for a class of linear uncertain systems with accessible state is presented in this note. A convex programming problem is solved, on-line, by reformulating the problem in terms of a piecewise smooth penalty function, and relying on a suitable analog variable structure system implementing the gradient procedure. In this way, the controlled system reference movement, optimal with respect to a pre-specified scalar convex cost function and a set of suitable equality and inequality constraints, is generated. An inner control loop aimed at the finite time exact tracking of the reference movement is also designed. As a result, the controlled system trajectory starting in the feasible region there remains, and the optimal movement in the feasible region is proved to be an asymptotically stable equilibrium point of the controlled system.     

Feedback-reversibility and reachability of linear impulsive systems

For a linear impulsive system, the set of states that are reachable from the origin when the initial time, impulse times, and final time are fixed is contained in an invariant subspace determined by the system data. It is known that reversibility of the system is sufficient to yield, for a specified initial time, the existence of some impulse time set and final time for which the reachable set equals the invariant subspace. In this paper, we relax the reversibility requirement and present a condition that is necessary as well as sufficient under which this property holds. This new condition involves the property of achieving reversibility via feedback and admits an explicit geometric characterization. Moreover, this feedback-reversibility property only needs to hold for the subsystem defined as the full system restricted to the invariant subspace. We further show that feedback-reversibility of the restricted system ensures that the reachable set equals the invariant subspace for almost any impulse time set and final time for which the number of impulse times contained in the underlying time interval exceeds a lower bound.     

Fault Detection and Isolation of discrete-time Markovian jump linear systems with application to a network of multi-agent systems having imperfect communication channels

This paper deals with the problem of Fault Detection and Isolation (FDI) for discrete-time Markovian Jump Linear Systems (MJLSs). A geometric property related to the unobservable subspace of an MJLS is first presented and the concept of unobservability subspace is introduced. Sufficient conditions for designing an H_∞-based FDI algorithm for MJLSs subject to input disturbances and measurement noise are presented and developed. Our proposed approach is then applied to the problem of fault detection and isolation in a network of multi-agent systems when imperfect communication channels exist among the agents. A discrete-time communication link with a stochastic packet dropping effect is considered based on the Gilbert–Elliott model and the entire network is modeled as a discrete-time MJLS. Simulation results are presented for formation flight of satellites to demonstrate and verify the effectiveness and performance capabilities of our proposed FDI algorithm.     

On the stability of slowly time-varying linear systems

New conditions are given in both deterministic and stochastic settings for the stability of the system x=A(t)x when A(t) is slowly varying. Roughly speaking, the eigenvalues of A(t) are allowed to wander into the right half-plane as long as on average they are strictly in the left half-plane.This work was funded by the NSF under Grant ECS-8806063, and was completed while the author was with the Department of Electrical and Computer Engineering, Johns Hopkins University, Baltimore, MD 21218, U.S.A.     

On stability of linear all-pass systems

In this paper we analyze the class of all-pass, linear and stable systems. We give a condition on the zero distribution of the transfer function of such a system to guarantee stability and we show that stability is not preserved under cascade decomposition.     

On SISO and MIMO variable structure control of non linear distributed parameter systems: application to fixed bed reactors

This paper deals with the design of variable structure control of distributed parameter systems. The control problem is discussed in relation to a model consisting of a set of non-linear, time varying partial differential equations of hyperbolic type. A formulation of a Single Input–Single Output (SISO) variable structure controller based on the distributed parameter model (late lumping control) is given. An extension to the Multiple Input–Multiple Output (MIMO) case is derived when the control variables are coupled and located on boundary conditions. A theoretical proof of DPS convergence in sliding mode is given. A fixed bed bioreactor in which drinkable water is treated, was used as a simulated example to prove the effectiveness of the control design. The bioreactor must control the harmful component concentrations in such a way that the quality of water fulfils international standards.     

Reachability and observability of linear impulsive systems

Linear impulsive systems constitute a class of hybrid systems in which the state propagates according to linear continuous-time dynamics except for a countable set of times at which the state can change instantaneously. While in general these impulsive effects can be time-driven and/or event-driven, here we focus our attention on the time-driven case. For this class of systems, we address the fundamental concepts of reachability and observability. In particular, we present a geometric characterization of the reachable and unobservable sets in terms of invariant subspaces and provide algorithms for their construction.     

Adaptive control of linear time-varying systems

Single-input single-output uncertain linear time-varying systems are considered, which are affected by unknown bounded additive disturbances; the uncertain time-varying parameters are required to be smooth and bounded but are neither required to be sufficiently slow nor to have known bounds. The output, which is the only measured variable, is required to track a given smooth bounded reference trajectory. The undisturbed system is assumed to be minimum-phase and to have known and constant relative degree, known sign of the ‘high frequency gain’, known upper bound on the system order. An adaptive output feedback control algorithm is designed which assures: (i) boundedness of all closed-loop signals; (ii) arbitrarily improved transient performance of the tracking error; (iii) asymptotically vanishing tracking error when parameter time derivatives are L₁ signals and disturbances are L₂ signals.     

On the problem of general structural assignments of linear systems through sensor/actuator selection

A systematic method is developed for determining an output matrix C for a given matrix pair (A,B) such that the resulting linear system characterized by the matrix triple (A,B,C) has the pre-specified system structural properties, such as the finite and infinite zero structure and the invertibility structures. Since the matrix C describes the locations of the sensors, the procedure of choosing C is often referred to as sensor selection. The method developed in this paper for sensor selection can be applied to the dual problem of actuator selection, where, for a given matrix pair (A,C), a matrix B is to be determined such that the resulting matrix triple (A,B,C) has the pre-specified structural properties.     

Convolution profiles for right inversion of multivariable non-minimum phase discrete-time systems

The problem of the non-causal inversion of linear multivariable discrete-time systems is analyzed in the geometric approach framework and is solved through the computation of convolution profiles which guarantee perfect tracking under the assumption of infinite-length preaction and postaction time intervals. It is shown how the shape of the convolution profiles is related to both the relative degree and the invariant zeros of the plant. A computational setting for the convolution profiles is derived by means of the standard geometric approach tools. Feasibility constraints are also taken into account. A possible implementation scheme, based on a finite impulse response system acting on a stabilized control loop, is provided.     

On the stabilization of linear discrete time systems subject to input saturation

In this paper, an exponentially unstable linear discrete time system subject to input saturation is shown to be exponentially stabilizable on any compact subset of the constrained asymptotically stabilizable set by a linear periodic variable structure controller. We also point out that any marginally stable system² subject to input saturation can be globally asymptotically stabilized via linear feedback.     

On discrete-time linear systems over cones

The subject of this paper is the theory of linear discrete-time systems with input constrained to belong to a closed convex cone. After discussing the generality of this class of systems, a number of invariant cones of states are introduced. In the last section these latter are shown to allow the extension of some classical results of the theory of linear unconstrained systems.     

Robust control of linear discrete-time systems

Given a linear discrete-time system with additive disturbances, a general methodology for designing a stable control algorithm is shown. The approach is Lyapunov-based with certain liberty of tailoring the candidate function for a given problem. Comparison to the existing solutions based on a variable structure approach is given.     

Implicit linear discrete-time systems

This paper studies various properties of implicit linear discrete-time systems given by a linear difference equationEx _k+1 =Fx _k +Gu _k . The topics considered include a basic characterization of these subspaces which describe acceptance of all input sequences, the uniqueness property and regularity, and the notion of controllability. This work was performed under the auspices of Fund RP.I.02: Teoria sterowania i optymalizacji ciągłych układów dynamicznych i procesów dyskretnych.