H∞-optimal controller discretization

A redesign method for discretizing a continuous-time controller is proposed. The resulting hybrid control system, for example with continuous-time plant and discrete-time controller, is stable, and performance including the system's inter-sampling behaviour can be optimized by approximating some chosen reference transfer function of the continuous-time control system. In order to obtain a tractable problem, the continuous-time part of the hybrid system and the reference transfer function are approximated by a discrete-time system with arbitrary fast sampling. After lifting the resulting periodic system, the approximation problem can be formulated as a standard H_∞-problem which is solved using standard software for H_∞-controller design.     

Constrained controllability of discrete-time systems†

The constrained controllability of the discrete-time system x_k+1=A(k)x_k+B(k)u,_k is considered where the control uk is termed admissible if it satisfies specified magnitude constraints. Constrained controllability is concerned with the existence of an admissible control which steers the state x to a given target set from a specified initial state

Conditions for checking constrained controllability to a given target set from a specified initial state are presented. These conditions involve solving finite-dimensional optimization problems and can be checked via numerical computation. In addition, conditions for checking global constrained controllability to a given target set are presented. A system is globally constrained controllable if for every initial state, there exists an admissible control that steers the system to the target.

If a given discrete-time system is constrained controllable, it may be desirable to obtain an admissible control that steers the system to the target from a specified initial state. Such a control is called a steering control. Results for computing steering controls are also presented

This paper is concluded with a numerical example. In this example, it is shown that the constrained controllability of a continuous-time system which has been discretized is dependent on the discretization time. The set of states which can be steered to the target changes as the discretization time changes. Furthermore, the example shows that a discrete-time steering control cannot always be obtained by discretizing a continuous-time steering control; the steering control for the discrete-time system must be obtained directly from the discrete-time model.     

The generalized continuous algebraic Riccati equation and impulse-free continuous-time LQ optimal control

The purpose of this paper is to investigate the role that the so-called constrained generalized Riccati equation plays within the context of continuous-time singular linear–quadratic (LQ) optimal control. This equation has been defined following the analogy with the discrete-time setting. However, while in the discrete-time case the connections between this equation and the linear–quadratic optimal control problem has been thoroughly investigated, to date very little is known on these connections in the continuous-time setting. This note addresses this point. We show, in particular, that when the continuous-time constrained generalized Riccati equation admits a solution, the corresponding linear–quadratic problem admits an impulse-free optimal control. We also address the corresponding infinite-horizon LQ problem for which we establish a similar result under the additional constraint that there exists a control input for which the cost index is finite.     

Weak reachability and controllability of discrete-time nonlinear systems: generic approach and singular points

ABSTRACT

The paper finds the singular points from which (to which) the generically accessible system is not weakly reachable (controllable) in k steps. These points are found with the help of the space of vector fields, being the discrete-time analogue of the strong accessibility distribution. Unlike in the continuous-time case, a separate object is needed to find the singular points related to weak reachability.     

Singular perturbation approximation of balanced systems

This paper relates the singular perturbation approximation technique for model reduction to the direct truncation technique if the system model to be reduced is stable, minimal and internally balanced. It shows that these two methods constitute two fully compatible model-reduction techniques for a continuous-time system, and both methods yield a stable, minimal and internally balanced reduced-order system with the same L_∞-norm error bound on the reduction. Although the upper bound for both reductions is the same, the direct truncation method tends to have smaller errors at high frequencies and larger errors at low frequencies, while the singular perturbation approximation method will display the opposite character. It also shows that a certain bilinear mapping not only preserves the balanced structure between a continuous-time system and an associated discrete-time system, but also preserves the slow singular perturbation approximation structure. Hence the continuous-time results on the singular perturbation approximation of balanced systems are easily extended to the discrete-time case. Examples are used to show the compatibility and the differences in the two reduction techniques for a balanced system     

Optimal H∞ general distance problem with degree constraint

In this paper the optimal H_∞, general distance problem, for continuous-time systems, with a prescribed degree on the solution is studied. The approach is based on designing the Hankel singular values using an imbedding idea. The problem is first imbedded into another problem with desirable characteristics on the Hankel singular values, then the solution to the original problem is retracted via a compression. The result is applicable to both the one-block and the four-block problems. A special case is given for illustration.     

On H2 model reduction of bilinear systems

The H₂ model reduction problem for continuous-time bilinear systems is studied in this paper. By defining the H₂ norm of bilinear systems in terms of the state-space matrices, the H₂ model reduction error is computed via the reachability or observability gramian. Necessary conditions for the reduced order bilinear models to be H₂ optimal are given. The gradient flow approach is used to obtain the solution of the H₂ model reduction problem. The formulation allows certain properties of the original models to be preserved in the reduced order models. The model reduction procedure developed can also be applied to finite-dimensional linear time-invariant systems. A numerical example is employed to illustrate the effectiveness of the proposed method.     

Generalised tangential interpolation for model reduction of discrete-time MIMO bilinear systems

In this article, we discuss a model order reduction method for multiple-input and multiple-output discrete-time bilinear control systems. Similar to the continuous-time case, we will show that a system can be characterised by a series of generalised transfer functions. This will be achieved by a multivariate Z-transform of kernels corresponding to an explicit solution formula for discrete-time systems. We will further address the problem of generalised tangential interpolation which naturally comes along with this approach. We will introduce a reasonable generalisation of the linear ?₂-norm. Based on this concept, we discuss the choice of interpolation points. Furthermore, an efficient discretisation of continuous-time systems is provided. The performance of the proposed method is evaluated in some numerical examples and compared with the method of balanced truncation for bilinear systems.     

ρ-Stability and robustness: discrete-time case

We say that a discrete-time system is ρ-stable if, roughly speaking, ρ^k >X _k→0, where >X _k is the system state. General ρ-stability theorems are established in this paper. They concern systems governed by functional difference equations. Systems of this type are encountered in the robustness studies. These ρ-stability theorems are a generalization of the well-known Lyapunov criterion. These results are applied to the robustness quantification problem in the second part of the paper. The case of discrete-time LQ regulators is deeply investigated. Robustness properties of continuous-time LQ regulators are found as the limit when the sampling period >T tends to zero; robustness deteriorates as T increases. An upper bound is given for >T, under which the robustness remains satisfactory. The practical interest of these theoretical results is illustrated on the basis of an industrial example.     

On lp-stabilization of strictly unstable discrete-time linear systems with saturating actuators

For a continuous-time linear system with saturating actuators, it is known that, irrespective of the locations of the open-loop poles, both global and semi-global finite gain L_p-stabilization are achievable, by nonlinear and linear feedback, respectively, and the L_p gain can also be made arbitrarily small. In this paper we show that, these results do not hold for discrete-time systems. © 1998 John Wiley & Sons, Ltd.     

A computational method for simultaneous LQ optimal control designvia piecewise constant output feedback

The paper is concerned with simultaneous linear-quadratic (LQ) optimal control design for a set of LTI systems via piecewise constant output feedback. First, the discrete-time simultaneous LQ optimal control design problem is reduced to solving a set of coupled matrix inequalities and an iterative LMI algorithm is presented to compute the feedback gain. Then, simultaneous stabilization and simultaneous LQ optimal control design of a set of LTI continuous-time systems are considered via periodic piecewise constant feedback gain. It is shown that the design of a periodic piecewise constant feedback gain simultaneously minimizing a set of given continuous-time performance indexes can be reduced to that of a constant feedback gain minimizing a set of equivalent discrete-time performance indexes. Explicit formulas for computing the equivalent discrete-time systems and performance indexes are derived. Examples are used to demonstrate the effectiveness of the proposed method.