Linear Time-invariant Control Systems Research Articles

Experimental demonstration of reset control design

Using the describing function method, engineers in the 1950s and 1960s conceived of novel nonlinear compensators in an attempt to overcome performance limitations inherent in linear time-invariant (LTI) control systems. This paper is concerned with a subset of such devices called “reset controllers”, which are LTI systems equipped with mechanisms and laws to reset their states to zero. This paper reports on a design procedure and a laboratory experiment, the first to be reported in the literature, in which the resulting reset controller provides better design tradeoffs than LTI compensation. Specifically, it is shown that reset control increases the level of sensor-noise suppression without sacrificing either disturbance-rejection performance or gain/phase margins.

Characterization and selection of global optimal output feedback gains for linear time-invariant systems

The problem of global optimal static output feedback control for linear time-invariant systems with linear quadratic index is investigated. The contributions of this paper are two-fold. One is to investigate the dependence of the global optimal output feedback gain on the system initial conditions. The other is to construct a globally optimal feedback under a certain output measurement structure. Copyright © 2000 John Wiley & Sons, Ltd.

Finite-Dimensional Compensators for the $H^\infty$-Optimal Control of Infinite-Dimensional Systems via a Galerkin-type Approximation

We study the existence of general finite-dimensional compensators in connection with the $H^{\infty}$-optimal control of linear time-invariant systems on a Hilbert space with noisy output feedback. The approach adopted uses a Galerkin-type approximation, where there is no requirement for the system operator to have a complete set of eigenvectors. We show that if there exists an infinite-dimensional compensator delivering a specific level of attenuation, then a finite-dimensional compensator exists and achieves the same level of disturbance attenuation. In this connection, we provide a complete analysis of the approximation of infinite-dimensional generalized Riccati equations by a sequence of finite-dimensional Riccati equations. As an illustration of the theory developed here, we provide a general procedure for constructing finite-dimensional compensators for robust control of flexible structures.

Comparison of two feedforward design methods aiming at accurate trajectory tracking of the end point of a flexible robot arm

This paper discusses the design and properties of two trajectory tracking controllers for linear time-invariant systems, and compares their implementation and experimental results on a flexible one-link robot equipped with a velocity-controlled actuator. High positioning accuracy and low tracking errors within a specified bandwidth are their performance specifications. Both controllers use the same state feedback controller, but have a different feedforward design approach. Both feedforward methods design stable prefilters which approximate the unstable inverse system model. The first method designs a stable prefilter using the extended bandwidth zero phase error tracking control (EBZPETC) method. The second feedforward method adds delay to the inverse model and then uses common filter design techniques to approximate this delayed frequency response.

Multirate Adaptive Control of a Servo Motor System

This paper deals with adaptive control of linear time-invariant systems with unknown parameters and with two sampling rates: a slower one for the output and a faster one for the input. Recently, adaptive control design methods with multirate sampling have been developed by some researchers. However, stable plants only are treated in the previous work. The algorithm proposed in this paper is applicable to a class of unstable plants. Experimental results for a servo motor system are shown.

A Behavioral Approach By Using Pencil Based System Models

Abstract In this article we consider the problem of analysis and design of linear time-invariant control systems by using the system models which are based on matrix pencils. Behaviors, poles, zeros, stability, controllability, observability, transfer function, and other traditional concepts are discussed mainly besed on the Willems’ behavioral approach. Moreover the extensions to LQ optimal control problem and computation of H norm are investigated.

A New Approach for Designing Robust Controllers for the Servomechanism Problem

In this paper, we provide an alternative method for solving the more general servomechanism problem. The new approach based on designing a simple structure of a dynamic output feedback controller to achieve asymptotic tracking, disturbance rejection and pole assignment in linear time-invariant multivariable control systems. In addition, the resulting dynamic compensator is robust in the sense that asymptotic regulation take place for some or all disturbances and reference signals independent of any nondestabilizing perturbations in the system parameters. The main results of this paper are illustrated by an example.

Controllability is not necessary for adaptive pole placement control

The key issue for adaptive pole-placement control of linear time-invariant systems is the possible singularity of the Sylvester matrix corresponding to the coefficient estimate. However, to overcome the difficulty, the estimate is modified by several methods which are either nonrecursive and with high computational load or recursive but with random search involved. All of the previous works are done under the assumption that the system is controllable. This paper gives the necessary and sufficient condition, which is weaker than controllability, for the system to be adaptively stabilizable. First, a nonrecursive algorithm is proposed to modify the estimates, and the algorithm is proved to terminate in finitely many steps. Then, with the help of stochastic approximation, a recursive algorithm is proposed for obtaining the modification parameters; it is proved that these modification parameters turn out to be a constant vector in a finite number of steps. This leads to the convergence of the modified coefficient estimates. For both algorithms the Sylvester matrices corresponding to the modified coefficient estimates are asymptotically uniformly nonsingular; thus, the adaptive pole-placement control problem can be solved, i.e., the system can be adaptively stabilized.

Least squares pole assignment by memoryless output feedback

In this paper, a pole assignment problem for a linear time-invariant control system by memoryless output feedback is posed as a least squares optimisation problem and analysed. The cost functions are appropriately modified so as to obtain convergence of the corresponding gradient flow and existence of the global minimum. This approach is also extended to accommodate the output pole assignment problem with insensitivity against disturbances in system parameters. The relation between the modified cost functions and the original pole assignment problem is revealed. The proposed approach is compared with other existing approaches and illustrated by numerical results.

A design scheme of variable structure adaptive control for uncertain dynamic systems

A design scheme of variable structure adaptive control for linear time-invariant system with uncertain dynamics is proposed. Both additive and multiplicative unmodeled dynamics are taken into consideration. The transfer function of the modeled part of the plant may have unstable zeros and unstable poles. A sign-following system and logic switchings are introduced into the control system. The global stability of the overall system is proved. Simulation results show the effectiveness of the proposed method.

Using decomposition approach to estimate the stable regions of variable structure control for linear time-invariant system with bounded control variables

The stable region of dynamic systems is discussed in this paper. A new approach to estimate the stable region is developed based on the decomposition of the system. The stable region of a variable structure system (VSS) is studied in detail, since the structure of a variable structure system is changed according to the states of the system. Furthermore, a useful method is proposed to estimate the stable region when the control signals are bounded owing to a physical constraint. A numerical example is given to illustrate the result obtained in the paper.

Robust performance of decentralized control systems by expanding sequential designs

This paper considers designs of decentralized controllers for continuous-time linear time-invariant multivariable control systems. The purpose is to show how to attain robust stability and robust performance of decentralized control systems by sequential design procedure. Compared with independent designs in which each controller block is designed independently of others, sequential designs can reduce conservatism associated with the controller design since they can exploit information about the other loops which have been already designed. A novel method of sequential designs for the robust performance are proposed. Moreover, an approach to a kind of failure tolerance is considered. The proposed sequential design procedure is based on expanding construction of decentralized control systems, so that the proposed methods can deal with extension of subsystems.

A canonical parameterization of the Kronecker form of a matrix pencil

The Kronecker form is the classical canonical form for matrix pencils under strict equivalence transformation. Consider a matrix pencil whose entries are smooth functions of a parameter vector. The Kronecker form of the parameterized pencil will, in general, be a discontinuous function of the parameters. For a linear time-invariant control system these discontinuities correspond to a change in the finite and infinite invariant zero structure. Since many control strategies require knowledge of, or place restrictions on, the zero structure, these points of discontinuity are of considerable interest. In this paper a general approach to the study of such points is developed in the framework of singularity theory. We derive a ‘miniversal’ parameterization of a given pencil. That is, a parameterized family of pencils that: (i) includes the given pencil, (ii) is locally equivalent to any other family up to a change of parameters, and (iii) uses the fewest number of parameters to achieve this property. All ‘nearby’ zero structures can be obtained by varying parameter values in the miniversal parameterization. From all miniversal parameterizations, one having a particularly uncluttered representation is selected as canonical. However, some restrictions on the finite elementary divisors are then required.

Positive real control problem for uncertain linear time-invariant systems

This paper focuses on positive real control of linear time-invariant systems which are subjected to norm-bounded uncertainties in the state equation. We address the problem of designing a linear dynamic output feedback controller that robustly stabilizes the uncertain system and achieves the extended strict positive realness property for a given closed-loop transfer function. It is shown that a solution to the above problem can be obtained by solving a scaled strict positive real control problem for which no parameter uncertainty occurs.

Placing zeroes and the Kronecker canonical form

Given a linear time-invariant control system, it is well known that the transmission zeroes are the generalized eigenvalues of a matrix pencil. Adding outputs to place additional zeroes is equivalent to appending rows to this pencil to place new generalized eigenvalues. Adding inputs is likewise equivalent to appending columns. Since both problems are dual to each other, in this paper we only show how to choose the new rows to place the new zeroes in any desired locations. The process involves the extraction of the individual right Kronecker blocks of the pencil, accomplished entirely with unitary transformations. In particular, when adding one new output, i.e., appending a single row, the maximum number of new zeroes that can be placed is exactly the largest right Kronecker index.

Numerical solution of positive control problem via linear programming

The reachability problem for linear time-invariant discrete-time control systems with sign-restricted input is considered. The time-optimal control is constructed by an iterative procedure. Each step of the iteration is defined as a linear programming problem. This problem is solved by the simplex algorithm. The initial feasible solution for the simplex algorithm is provided by the preceding step of the iteration. The inversion of the basis matrix is reduced to a bordering procedure. The structural stability of the solution is investigated.

077 Periodic control of linear time-invariant systems—A critical point of view

077 Periodic control of linear time-invariant systems—A critical point of view

On the maximum feedback delay in a linear/nonlinear control system with input disturbances caused by controller-computer failures

Electromagnetic interferences or other environmental disturbances may cause transient failures to the controller computer of a real-time control system. Such a faulty controller either fails to update the control input for one or more sampling periods, or generates erroneous control inputs until the failure is handled properly or disappears. The goal of this paper is to derive the maximum duration of controller's faulty behavior, called the hard deadline, a real-time control system can tolerate without losing stability or leaving its allowed state space. For linear time-invariant control systems, one can derive hard deadlines by testing the stability of their state difference equations which account for the effects of stationary occurrences of disturbances to, as well as the random delays in, the control input. Similarly, one can derive deadlines for nonlinear time-invariant control systems by linearizing their nonlinear state equations and using the Lyapunov's first method. In addition to this stationary model, a one-shot event model is considered for linear/nonlinear time-invariant control systems by using their state trajectories and allowed state spaces. The hard deadline information that represents the knowledge of the controlled process's inertia and timing constraints is applied to the design and evaluation of controller computers.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A convex parameterization of robustly stabilizing controllers

This paper treats synthesis of robust controllers for linear time-invariant systems. Uncertain real parameters are assumed to appear linearly in the closed loop characteristic polynomial. The main contribution is to give a convex parameterization of all controllers that simultaneously stabilize the system for all possible parameter combinations. With the new parameterization, certain robust performance problems can be stated in terms of quasi-convex optimization.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Unfolding of Singular System Pencils with Application to Regulation

The system pencil is a convenient way of representing a linear time invariant control system. It is particularly appropriate when system zeros are important — in regulator design, for example. This paper presents an unfolding of certain system pencils, possibly, but not necessarily, singular. That is, it demonstrates a minimal parameterization of the nominal pencil such that any nearby pencil, up to strict equivalence, corresponds to some value of the parameter vector. This unfolding has potential practical and theoretical applications to systems with parameter dependence.