Control Of Discrete-time Linear Systems Research Articles

The Design of Reduced Order Controllers for the Stabilization of Large Scale Linear Discrete-Time Control Systems

This paper investigates the design of reduced order controllers for the stabilization of large scale linear discrete-time control systems. Sufficient conditions are derived for the design of reduced order controllers by obtaining a reduced order model of the original large scale linear system using the dominant state of the system. The reduced order controllers are assumed to use only the state of the reduced order model of the original plant.

2-D analysis based iterative learning control for linear discrete-time systems with time delay

This paper investigates an iterative learning controller for linear discrete-time systems with state delay based on two-dimensional (2-D) system theory. It shall be shown that a 2-D linear discrete Roessor's model can be applied to describe the ILC process of linear discrete time-delay systems. Much less restrictive conditions for the convergence of the proposed learning rules are derived. A learning algorithm is presented which provides much more effective learning of control input, which enables us to obtain a control input to drive the system output to the desired trajectory quickly. Numerical examples are included to illustrate the performance of the proposed control procedures.

Design of Structured Static Output Feedback Controllers

AbstractWe address the design of structured controllers for linear discrete-time systems. Decentralized controllers are designed that use local measurements and a minimal number of additional measurement links between the subsystems and the controllers. The structure of the decentralized controller, i.e. the additional links between the subsystems and the controllers, is not specified in advance but included into the controller design. We consider static output feedback for multivariable subsystems and define a pattern matrix to deal with the block structure of the controller. We formulate this problem as a maximization of the degree of decentralization, subject to a given H∞-performance. For the resulting non-convex optimization problem, numerically tractable convex relaxations are provided and an example shows the effectiveness of this approach.

Robust Model Predictive Control through Dynamic State-feedback: An LMI Approach

AbstractWe propose a novel solution to the problem of robust model predictive control of linear discrete-time systems with bounded disturbances and polytopic constraints on the input and state. The proposed algorithm consists of an outer controller – which is responsible for steering the system state to a designed robust positively invariant set – and an inner controller which keeps the state within this set for all possible disturbances. The algorithm is novel in that the proposed outer controller has a dynamic state-feedback structure with feedback gains explicitly considered as decision variables in the on-line optimization. The non-linearities associated with the considered control structure are avoided by adopting a sequential approach in the formulation. The overall algorithm is linear in the problem data and consists of a series of LMI optimizations. Numerical examples demonstrate the effectiveness of the proposed scheme.

Robust energy-to-peak filtering for networked systems with time-varying delays and randomly missing data

This study is concerned with the robust energy-to-peak (l2–l∞) filtering problem for a class of uncertain linear discrete-time networked control systems with time-varying delays and randomly missing data. The system matrices are assumed to reside in a convex polytope, and the time-varying delays appearing in system states are assumed to lie in a given interval. Moreover, the random data missing is supposed to satisfy the Bernoulli random binary distribution. The authors' goal is to design a full-order filter such that the filtering error dynamic system is exponentially mean-square stable for all admissible time delays and randomly missing data while a prescribed l2–l∞ performance is achieved. Sufficient delay- and parameter-dependent conditions for the existence of the filter and for the solvability of the addressed problem are given in terms of a set of linear matrix inequalities. Finally, simulation examples and comparison studies are provided to demonstrate the effectiveness of the proposed approach.

H ∞ output feedback stabilisation of linear discrete-time systems with impulses

This article addresses the issue of designing an H ∞ output feedback controller for linear discrete-time systems with impulses. First, a new concept of H ∞ output feedback stabilisation for general linear discrete-time systems with impulses is introduced. Then sufficient linear matrix inequality conditions for the stabilisation and H ∞ performance of general discrete systems with impulses are proposed. In addition, the result is applied to resolve typical output feedback control problems for systems with impulses, such as the decentralised H ∞ output feedback control and the simultaneous H ∞ output feedback control. Finally, a numerical simulation is also presented to illustrate the effectiveness of the proposed results.

A BMI approach for ℋ︁∞ gain scheduling of discrete time‐varying systems

AbstractThe problem of gain‐scheduled state feedback control for discrete‐time linear systems with time‐varying parameters is considered in this paper. The time‐varying parameters are assumed to belong to the unit simplex and to have bounded rates of variation, which depend on the values of the parameters and can vary from slow to arbitrarily fast. An augmented state vector is defined to take into account possible time‐delayed inputs, allowing a simplified closed‐loop analysis by means of parameter‐dependent Lyapunov functions. A gain‐scheduled state feedback controller that minimizes an upper bound to the ℋ︁∞ performance of the closed‐loop system is proposed. No grids in the parametric space are used. The design conditions are expressed in terms of bilinear matrix inequalities (BMIs) due to the use of extra variables introduced by the Finsler's lemma. By fixing some of the extra variables, the BMIs reduce to a convex optimization problem, providing an alternate semi‐definite programming algorithm to solve the problem. Robust controllers for time‐invariant uncertain parameters, as well as gain‐scheduled controllers for arbitrarily time‐varying parameters, can be obtained as particular cases of the proposed conditions. As illustrated by numerical examples, the extra variables in the BMIs can provide better results in terms of the closed‐loop ℋ︁∞ performance. Copyright © 2009 John Wiley & Sons, Ltd.

Finite-time control of discrete-time linear systems: Analysis and design conditions

In this paper we deal with some finite-time control problems for discrete-time, time-varying linear systems. First we provide necessary and sufficient conditions for finite-time stability; these conditions require either the computation of the state transition matrix of the system or the solution of a certain difference Lyapunov inequality. Then we address the design problem. The proposed conditions allow us to find output feedback controllers which stabilize the closed loop system in the finite-time sense; all these conditions can be expressed in terms of LMIs and therefore are numerically tractable, as shown in the example included in the paper.

Gain-scheduled ℋ2 and ℋ∞ control of discrete-time polytopic time-varying systems

This study presents ℋ2 and ℋ∞ performance analysis and synthesis procedures for the design of both gain-scheduled and robust static output feedback controllers for discrete-time linear systems with time-varying parameters. The obtained controllers guarantee an upper bound on the ℋ2 or ℋ∞ performance of the closed-loop system. As an immediate extension, the mixed ℋ2/ℋ∞ guaranteed cost control problem is also addressed. The scheduling parameters vary inside a polytope and are assumed to be a priori unknown, but measured in real-time. If bounds on the rate of parameter variation are known, they can be taken into account, providing less conservative results. The geometric properties of the polytopic domain are exploited to derive finite sets of linear matrix inequalities (LMIs) based on the existence of a parameter-dependent Lyapunov function. An application of the methodology to a realistic vibroacoustic problem, with experimentally obtained data, illustrates the benefits of the proposed approach and shows that the techniques can be used for real engineering problems.

Optimal Controller Structure Reduction for Decentralized Control

This paper addresses the design of decentralized controllers for linear discrete-time systems. We consider state feedback control for networks of scalar interconnected systems. The structure of the decentralized controller is not specified in advance but included into the controller design. Decentralized controllers are designed based on local measurement with minimal number of additional measurement links between the subsystems and the controllers. We formulate this problem as one of maximizing the degree of decentralization subject to a given error performance in terms of the H∞-norm between the system controlled by a centralized controller and the system controlled by the decentralized controller. For the resulting non-convex optimization problem, numerically tractable convex relaxations are provided and an example shows the effectiveness of this approach.

Controller Structure Design for Decentralized Control of Coupled Higher Order Subsystems

We address the design of optimally decentralized controllers for linear discrete-time systems. Decentralized controllers are designed that use local measurements and a minimal number of additional measurement links between the subsystems and the controllers. The structure of the decentralized controller, i.e. the additional links between the subsystems and the controllers, is not specified in advance but included into the controller design. We consider state feedback for higher order subsystems and define a pattern matrix to deal with the block structure of the controller. We formulate this problem as a maximization of the degree of decentralization, subject to a given error performance in terms of the H∞-norm between the centralized and the decentralized closed loop. For the resulting non-convex optimization problem, numerically tractable convex relaxations are provided and an example shows the effectiveness of this approach.

Stability Analysis of Finite-Level Quantized Discrete-Time Linear Control Systems

This paper investigates the stability of discrete-time linear time-invariant systems subject to finite-level logarithmic quantized feedback. Both state feedback and output feedback are considered. A linear matrix inequality (LMI) approach is developed to estimate, for a given controller and a given finite-level quantizer, a set of admissible initial states and an associated attractor set in a neighborhood of the origin such that all state trajectories starting in the first set will converge to the attractor in a finite time and will never leave it. Furthermore, when two such sets are a priori specified, we develop sufficient conditions to design a finite-level logarithmic quantizer for a given stabilizing state or output feedback controller.

Robust multi-echelon multi-period inventory control

We consider the problem of minimizing the overall cost of a supply chain, over a possible long horizon, under demand uncertainly which is known only crudely. Under such circumstances, the method of choice is Robust Optimization, in particular the Affinely Adjustable Robust Counterpart (AARC) method which leads to tractable deterministic optimization problems. The latter is due to a recent re-parametrization technique for discrete time linear control systems. In this paper we model, analyze and test an extension of the AARC method known as the Globalized Robust Counterpart (GRC) in order to control inventories in serial supply chains. A simulation study demonstrates the merit of the methods employed here, in particular, it shows that a good control law that minimizes cost achieves simultaneously good control of the bullwhip effect.

Linear discrete-time global and regional anti-windup: an LMI approach

In this article we address linear anti-windup design for linear discrete-time control systems guaranteeing regional and global stability and performance. The techniques that we develop are the discrete-time counterpart of existing techniques for anti-windup augmentation which lead to convex constructions by way of linear matrix inequalities (LMIs) when adopting static and plant order anti-windup augmentation. Interesting system theoretic interpretations of the performance bounds for the non-linear closed loop can also be given. We show here that parallel results apply to the discrete-time case. We derive the corresponding conditions and prove their effectiveness by adapting the continuous-time approaches to the discrete-time case.

Finite-Level Quantized Feedback Control for Linear Systems

This technical note studies quantized output feedback control of discrete-time linear systems using a finite-level quantizer. The main objective is to find a quantization strategy which is easily implementable and achieves asymptotic stabilization. Based on a known logarithmic quantization scheme, we introduce a simple dynamic scaling method for the quantizer. A suboptimal approach for the optimization of the number of quantization levels and the design of a corresponding quantized dynamic output feedback controller is given. The robustness of the dynamic quantization scheme with respect to input disturbances is also examined.

Optimal control for linear discrete-time systems with Markov perturbations in Hilbert spaces

Optimal control for linear discrete-time systems with Markov perturbations in Hilbert spaces

Control of Systems With Uncertain Initial Conditions

This note deals with the control of linear discrete-time systems with uncertain initial conditions. Specifically, we consider the problem where the initial condition is known to reside in a norm ball of some radius, and the input disturbance is constrained to satisfy an independent norm condition. The paper focuses on eventually periodic systems; these include both finite horizon and periodic systems as special cases. The main theorem provides exact synthesis conditions for the existence of eventually periodic controllers which both stabilize and provide performance in closed-loop control systems. These conditions are given in terms of a finite-dimensional semidefinite programming problem. We also give a version of the main result for the special case of linear time-invariant systems with uncertain initial states, and conclude with an illustrative example.

The problem of absolute invariance of a linear discrete-time control system

The problem of absolute invariance of a linear discrete-time control system

Dynamic Output Feedback for Discrete-Time Systems Under Amplitude and Rate Actuator Constraints

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> This work proposes a technique for the design of stabilizing dynamic output feedback controllers for discrete-time linear systems with rate and amplitude saturating actuators. The nonlinear effects introduced by the saturations in the closed-loop system are taken into account by using a generalized sector condition, which leads to theoretical conditions for solving the problem directly in the form of linear matrix inequalities. </para>

Two controller reduction methods based on convex optimization

This article outlines two approaches for the computation of a suitable low order controller if a linear discrete-time control system with a high order controller is given. Actually this step is of significant importance in the case of the so-called Q-design procedure and especially in l 1-optimal control system design. Since the main focus during controller reduction is to maintain the performance of the closed loop system this task is quite different from the general order reduction problem and thus requires special methods. In this article two controller reduction methods are presented, which lead to linear programs. It is intended to augment the Q-design approach by a suitable controller reduction step and in this way it should become possible to utilize the full power of Q-design and to get controllers of reasonable order. The performance of the proposed methods is demonstrated by an example.