Robust stability of linear perturbed discrete systems with saturating actuators

This paper is concerned with the robust stability conditions for linear perturbed systems with saturating actuators in the discrete-time case. Both non-linear and linear perturbations are considered. Two types of controller, a state feedback and a dynamic controller, are introduced to guarantee asymptotic stability of the perturbed system.     

Robust stability bounds on time-varying perturbations for state-space models of linear discrete-time systems

The stability robustness of linear discrete-time systems in the time domain is addressed using the Lyapunov approach. Bounds on linear time-varying perturbations that maintain the stability of an asymptotically stable linear time-invariant discrete-time nominal system are obtained for both structured and unstructured independent perturbations. Bounds are also derived assuming that various elements of the system matrix are perturbed dependently. The result for the structured perturbation case is extended to the stability analysis of interval matrices.     

Robust stability of linear perturbed discrete large-scale systems with saturating actuators

Robust stability conditions are considered for linear perturbed large-scale systems with saturating actuators in the discrete-time case. Both non-linear and linear perturbations are considered. Two types of controllers, a state feedback and a dynamic controller, are introduced for each sub-system to guarantee asymptotic stability of the perturbed large-scale system.     

The explicit linear quadratic regulator for constrained systems

For discrete-time linear time invariant systems with constraints on inputs and states, we develop an algorithm to determine explicitly, the state feedback control law which minimizes a quadratic performance criterion. We show that the control law is piece-wise linear and continuous for both the finite horizon problem (model predictive control) and the usual infinite time measure (constrained linear quadratic regulation). Thus, the on-line control computation reduces to the simple evaluation of an explicitly defined piecewise linear function. By computing the inherent underlying controller structure, we also solve the equivalent of the Hamilton-Jacobi-Bellman equation for discrete-time linear constrained systems. Control based on on-line optimization has long been recognized as a superior alternative for constrained systems. The technique proposed in this paper is attractive for a wide range of practical problems where the computational complexity of on-line optimization is prohibitive. It also provides an insight into the structure underlying optimization-based controllers.     

Nonlinear regulation in constrained input discrete-time linear systems

The use of composite linear and non-linear feedback laws for the control of constrained input discrete-time linear systems is re-examined. By making use of the delta operator formulation of a discrete-time system, an apparent restriction on the magnitude of the non-linear control law is removed, and the similarities between the continuous and discrete-time solutions to the problem are elucidated. In order to develop the results, unconstrained systems are treated initially, but it is shown that, locally, the sufficient conditions for the stabilization of such systems are actually equivalent to those for the stabilization of the corresponding constrained systems.     

Transition probability bounds for the stochastic stability robustness of continuous- and discrete-time Markovian jump linear systems

This paper considers the robustness of stochastic stability of Markovian jump linear systems in continuous- and discrete-time with respect to their transition rates and probabilities, respectively. The continuous-time (discrete-time) system is described via a continuous-valued state vector and a discrete-valued mode which varies according to a Markov process (chain). By using stochastic Lyapunov function approach and Kronecker product transformation techniques, sufficient conditions are obtained for the robust stochastic stability of the underlying systems, which are in terms of upper bounds on the perturbed transition rates and probabilities. Analytical expressions are derived for scalar systems, which are straightforward to use. Numerical examples are presented to show the potential of the proposed techniques.     

Sliding mode state observers for discrete-time linear systems

In discrete-time systems, instead of having a hyperplane as in the continuous case, there is a countable set of points comprising a so-called lattice; and the surface on which these sliding points lie is named the latticewise hyperplane. In this paper an asymptotically stable observer for discrete-time systems is designed by using the properties of the sliding mode such that the stability of the error system in the sliding mode is maintained. Various techniques for finding the feedforward injection map are proposed. The stability of the reconstruction error system with a bounded perturbation signal is studied.     

Robust stability of discrete-time systems using delta operators

The robust stability of discrete-time systems formulated in terms of the delta (δ) operator is discussed. That is, given the nominal characteristic equation P(δ) of a discrete-time system, it is of interest to know how much the coefficients can be perturbed while preserving stability. A procedure to obtain the maximum intervals for a perturbed polynomial P(δ) to still be stable is presented     

Stability bounds of singularly perturbed discrete systems

The authors propose a systematic approach to determine the exact stability bound of singularly perturbed discrete time systems. By combining three conditions of the critical stability criterion with a bialternate product for discrete-time systems, the stability problems of two types of singularly perturbed discrete-time systems can be solved via the derived scheme. Several examples are given to illustrate the proposed results     

Asymptotically optimal controls of hybrid linear quadratic regulators in discrete time

This work develops asymptotically optimal controls for a class of discrete-time hybrid systems involving singularly perturbed Markov chains having weak and strong interactions. The state space of the underlying Markov chain is decomposed into a number of recurrent classes and a group of transient states. Using a hierarchical control approach, by aggregating the states in each recurrent class into a single state, a continuous-time quadratic limit control problem in which the resulting limit Markov chain has much smaller state space is derived. Using the optimal control for the limit problem, a control for the original problem is constructed, which is shown to be nearly optimal. Finally, a numerical example is given to demonstrate the effectiveness of the approximation scheme.     

Structural-and-parametric identification of linear stochastic plants using continuous fractions

A method of structural-and-parametric identification of linear dynamic plant is proposed on the basis of measured stochastic input and output signals. The method is based on the theory of continuous fractions and incorporates a stationarity test for input and output plant processes, stationarization (when necessary) and derivation of a discrete-time model of stochastic plant.     

A variational integrators approach to second order modeling and identification of linear mechanical systems

The theory of variational integration provides a systematic procedure to discretize the equations of motion of a mechanical system, preserving key properties of the continuous time flow. The discrete-time model obtained by variational integration theory inherits structural conditions which in general are not guaranteed under general discretization procedures. We discuss a simple class of variational integrators for linear second order mechanical systems and propose a constrained identification technique which employs simple linear transformation formulas to recover the continuous time parameters of the system from the discrete-time identified model. We test this approach on a simulated eight degrees of freedom system and show that the new procedure leads to an accurate identification of the continuous-time parameters of second-order mechanical systems starting from discrete measured data.     

State estimation for discrete-time markov jump linear systems based on orthogonal projective theorem

In this paper, state estimation problem for discrete-time Markov jump linear systems is considered. Based on orthogonal projective theorem, a novel suboptimal algorithm for state estimate of discrete-time Markov jump linear systems in the sense of minimum mean square error estimate is proposed. The proposed suboptimal algorithm is recursive and finite-dimensionally computable. Computer simulations are carried out to evaluate the performance of the proposed suboptimal algorithm.     

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.     

Suboptimal control for discrete linear constant stochastic systems

The regulation of discrete-time, constant, linear stochastic systems with quadratic performance criterion is considered. A fixed-dimensional, linear time-invariant controller is used. Algebraic matrix equations are derived whose solutions are the gains of the fixed-dimensional optimal controller.     

Comments on ‘Smallest destabilizing perturbations for linear systems’

A sufficient condition that guarantees the stability of a perturbed continuous time system is derived. It has the advantage that it does not require global minimiza- tion.     

H 2 optimal control for a wide class of discrete-time linear stochastic systems

In this article, the problem of H ₂-control of a discrete-time linear system subject to Markovian jumping and independent random perturbations is considered. Different H ₂ performance criteria (often called H ₂-norms) are introduced and characterised via solutions of some suitable linear equations on certain spaces of symmetric matrices. Some aspects specific to the discrete-time framework are revealed. The problem of optimisation of H ₂-norms is solved under the assumption that full state vector is available for measurements. One shows that among all stabilising controllers of higher dimension, the best performance is achieved by a zero-order controller. The corresponding feedback gain of the optimal controller is constructed based on the stabilising solution of a system of discrete-time generalised Riccati equations.     

Robust stabilisation of discrete-time time-varying linear systems with Markovian switching and nonlinear parametric uncertainties

In this paper, the problem of the robustness of the stability of a discrete-time linear stochastic system is addressed. The nominal plant is described by a discrete-time time-varying linear system subject to random jumping according with a non-homogeneous Markov chain with a finite number of states. The class of admissible uncertainties consists of multiplicative white noise type perturbations with unknown intensity. It is assumed that the intensity of white noise type perturbations is modelled by unknown nonlinear functions subject to linear growth conditions. The class of admissible controls consists of stabilising state feedback control laws. We show that the best robustness performance is achieved by the stability provided by a state feedback design based on the stabilising solution of a suitable discrete-time Riccati-type equation.     

Lexicographic perturbation for multiparametric linear programming with applications to control

Optimal control problems for constrained linear systems with a linear cost can be posed as multiparametric linear programs (mpLPs) with a parameter in the cost, or equivalently the right-hand side of the constraints, and solved explicitly offline. Degeneracy occurs when the control input, or optimiser, is non-unique, which can cause chattering of the control input and overlap of the polyhedral regions of the explicit solution. This paper introduces a new and efficient approach to the computation of the solution to a degenerate mpLP with the parameter in the cost. Rather than solve the degenerate problem directly, we solve a lexicographically (symbolically) perturbed version of it that is guaranteed to be non-degenerate. We show that every optimal solution of the perturbed problem is an optimal solution to the original and that the perturbed solution is continuous, unique and defined over a set of non-overlapping polyhedral regions. Furthermore, we introduce a new method for computing the optimal solution in an adjacent region, which is very efficient in all cases and reduces to a single simplex pivot for non-degenerate regions. The proposed algorithm is particularly suited for the calculation of the explicit solution to a class of constrained optimal control problems, since it ensures that the control input is everywhere continuous and unique, thereby removing the danger of chattering in problems with linear costs. The algorithm is compared through example to existing proposals and a significant complexity improvement is demonstrated.