Linear Time-varying Plants Research Articles

Simultaneous Coefficient Assignment of Discrete-Time Multi-Input Multi-Output Linear Time-Varying System: A New Approach for Compensator Design

In this paper, a linear time-varying input-output system is considered and its realization as a linear time-varying autoregressive moving average system (ARMA) is studied. A time-varying z-transform is also introduced and its properties are studied. Furthermore a time-varying version of the coefficient assignment problem well known in time invariant system theory as the pole placement problem is posed and analyzed. A r-tuple of discrete time, linear time-varying plants with m inputs and p outputs are considered together with a single p input m output linear time-varying compensator. The design objective is to construct a single compensator that “coefficient assigns,” and hence “bounded input bounded output stabilizes” under suitable additional technical assumptions, the set of r plants simultaneously in the closed loop. Such a problem is useful in robust design of linear time-varying control systems in the closed loop. Among the results, it is shown that a generic r-tuple of $p \times m$ plants (in a suitable topology) is simultaneously coefficient assignable, provided that $r < {m / p}$. The design procedure involves splitting the closed-loop system into an ARMA system in cascade with a moving average system. The coefficient assignment problem consists of assigning the coefficients of the autoregressive part of the ARMA subsystem. Thereby an algorithm is obtained that is nonrecursive and involves solving for each time instant a system of linear equations with time-varying coefficients. The associated time-varying matrix has the “Sylvester matrix structure.” Such a structure is well-known in pole placement of time-invariant systems by dynamic compensation. Additionally the problem of coefficient assignment of the autoregressive part of the ARMA system is considered in the closed loop, without splitting up into a cascade of two subsystems as before. A new recursive algorithm to analyze this problem has been introduced. The proposed algorithm has no counterpart in the time-invariant system design and thus represents a new design procedure. A special case of this algorithm for the single-input single-output system has been described in detail. An interesting feature of the proposed recursive algorithm is that one obtains a nonlinear recursion on the compensator parameters that would assign a prespecified sequence of coefficients for the closed-loop system. For a specific design problem it is shown that the dynamics of this nonlinear recursion is chaotic.

The graph topology for linear plants with applications to nonlinear robust stabilization

The graph topology is defined for the set of finite-dimensional linear time-varying plants that are internally stabilizable by output feedback. This follows some earlier results of M. Vidyasagar et al. (1982) on time-invariant systems. Feedback connections of linear time-varying plants and general nonlinear time-varying controllers are considered, and it is shown that the graph topology is the weakest topology in which feedback stabilization is a robust property. Using this result, necessary and sufficient conditions for robust stabilizability of families of linear time-varying plants with parametric uncertainty are derived.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A new approach to model reference control for a class of arbitrarily fast time-varying unknown plants

In this note, a new standard model reference control (NSMRC) to arbitrarily fast linear single-input single-output time-varying plants with relative degree one is presented. A modified version of standard model reference control (SMRC) structure is developed which incorporates the concept of variable structure design and null adaptation process for fast time-varying plants. It is shown that the output error will converge to zero in finite time without any restriction on the time-varying rate of the plant parameters while all signals inside the closed-loop system remain uniformly bounded so long as some design parameters are chosen properly (large). Superior transient responses as well as convergence properties can be observed from the simulation results provided.

On the robust adaptive control of a contour-following robotic system

To demonstrate the force sensing and control, a possible model of a contour-following system is represented by a fourth-order linear continuous-time time-invariant system, in which stiffness k t , robot natural frequency, ω n, linear accommodation gain k x , and angular accommodation gain k φ are all constants obtained by measurement and experiment. This model works well for following a short contour. To satisfactorily follow a longer contour, k t , ω n, k x and k φ can be treated as unknown constants or time-varying variables. When they are considered as unknown constants, a robust model reference adaptive controller can be used to achieve both stability and tracking without having to know or find the true values of those constants, given bounded input or output disturbances and stable unmodeled dynamics. If ω n is assumed to be a given constant but K t , k x , and k φ are assumed to be unknown variables, then one has a linear time-varying plant and other types of model reference adaptive controllers have to be used to achieve the same purpose. In this paper, the schemes from various robust model reference adaptive control design will be studied and comparison and suggestions will also be made based on the simulation results for the contour-following robotic system mentioned above.

Time-varying versus time-invariant compensation for rejection of persistent bounded disturbances and robust stabilization

It is shown that time-varying compensation does not improve the optimal rejection of persistent bounded disturbances. This result is obtained by exploiting a key observation that any time-varying compensator which yields a given degree of disturbance rejection must do so uniformly over time, thereby removing any advantage of time-variation. This key observation is exploited to show that time-varying compensation does not improve the optimal rejection of disturbances, regardless of the norm used to measure the disturbances. Thus, absolutely summable, finite-energy, or persistent bounded disturbances may be treated in the same manner. It is shown that time-varying compensation does not help in the bounded-input bounded-output robust stabilization of time-invariant plants with unstructured uncertainty. In doing so, it is also shown that the small-gain theorem is both necessary and sufficient for the bounded-input bounded-output stability of certain linear time-varying plants subject to unstructured linear time-varying perturbations.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The necessity of the small-gain theorem for time-varying and nonlinear systems

It is shown that an appropriately modified version of the small-gain theorem which uses the 'asymptotic gain' of a fading-memory system leads to a necessary and sufficient condition for the stable invertibility of certain feedback operators. This modified small-gain theorem is then used to obtain conditions for robust finite-energy input-output stability of: (1) nonlinear plants subject to nonlinear time-varying perturbations; and (2) linear time-varying plants subject to linear time-varying perturbations. These conditions are shown to be necessary as well as sufficient. >

Adaptive linear control of nonlinear systems

The theory of stable parameter adaptive control has advanced to allow linear time-varying plants. However, a more honest view of such systems is that they are often derived from inexact linearization about a trajectory of a nonlinear system. Standard adaptive control based on a linear model can then be interpreted as one way to realize a nonlinear controller for a nonlinear plant. The implications of this view are studied. Analytically, the stability problem is seen to be equivalent to showing robustness to time-varying parameters and a locally bounded model uncertainty. It is shown that if the trajectory is known to be within a bound, a parameter estimator with projection can ensure boundedness of departures from the trajectory. >

Adaptive control of linear time-varying plants: a new model reference controller structure

The problem of developing a control law which can force the output of a linear time-varying plant to track the output of a stable linear time-invariant reference model is discussed. It is shown that the standard model reference controller, used for linear time-invariant plants, cannot guarantee zero tracking error in general when the plant is time-varying. A new model reference controller is proposed which guarantees stability and zero tracking error for a general class of linear time-varying plants with known parameters. When the time-varying plant parameters are unknown but vary slowly with time, it is shown that the new controller can be combined with a suitable adaptive law so that all the signals in the closed loop remain bounded for any bounded initial conditions and the tracking error is small in the mean. The assumption of slow parameter variations in the adaptive case can be relaxed if some information about the frequency or the form of the fast varying parameters is available a priori. Such information can be incorporated in an appropriately designed adaptive law so that stability and improved tracking performance is guaranteed for a class of plants with fast varying parameters. >

Strong, simultaneous, and reliable stabilization of finite-dimensional linear time-varying plants

Some issues related to the stabilization of linear, time-varying (LTV) plants are studied. The authors consider the class of finite-dimensional LTV plants that are internally stabilizable by output feedback, and show that every plant in the class can be stabilized by a stable LTV controller; moreover, they give a parameterization of all nonlinear time-varying controllers that stabilize a given LTV plant. Using these results, they show that every finite collection of LTV plants can be simultaneously stabilized by a stable LTV. Finally, the authors prove that every LTV plant can always be stabilized by a pair of simultaneously acting LTV controllers, each reliable against a failure of the other. >

Adaptive Control of Linear Time-Varying Plants

In this paper we consider the model reference adaptive control (MRAC) problem of a class of linear time-varying (LTV) plants. The plant parameters are assumed to be smooth, bounded functions of time which satisfy the usual assumptions of MRAC for time-invariant plants, at each frozen time instant. We first show that if the plant parameters are sufficiently slowly-varying with time, a control parameter vector with smooth elements exists, such that the closed loop plant behaves almost like a linear time invariant reference model. We then use the robust adaptive law proposed in Ioannou and Tsakalis (1985) to adjust the controller parameters and establish boundedness for all signals in the adaptive loop for any bounded initial conditions. The bound for the residual tracking error depends on the speed of the plant parameter variations in such a way that as these parameters become constant the bound reduces to zero.

Adaptive control of linear time-varying plants

In this paper the model reference adaptive control (MRAC) problem of a class of linear time-varying (LTV) plants is considered. The plant parameters are assumed to be smooth, bounded functions of time which satisfy the usual assumptions of MRAC for time-invariant plants, at each frozen time instant. It is first shown that if the plant parameters are sufficiently slowly varying with time, a control parameter vector with smooth elements exists, such that the closed-loop plant behaves almost like a linear time-invariant reference model. The robust adaptive law proposed in Ioannou and Tsakalis (1986) IEEE Trans. Aut. Control, AC-31, 1033 is then used to adjust the controller parameters and establish boundedness of all signals in the adaptive loop for any bounded initial conditions. It is shown that the bound for the residual tracking error depends on the speed of the plant parameter variations in such a way that as these parameters become constant the bound reduces to zero.

Synthesis of linear time-varying non-minimum-phase systems with plant uncertainty†

The key element in a synthesis technique for uncertain linear time-varying systems, is the replacement of the linear time-varying plant set by an equivalent linear time-invariant plant set. Conditions under which the latter set must be so set up as to include non-minimum phase elements are presented. It is shown how to obtain the equivalent right half-plane zeros, which are sometimes accompanied by right half-plane poles. The synthesis technique previously usable only for uncertain ‘ minimum-phase ’ linear time-varying systems is extended to handle ‘ effectively non-minimum-phase ’ uncertain linear time-varying systems.

The Design of a Signal Synthesis Model Reference Adaptive Control System for Linear Unknown Plants

In this paper, a design of a signal synthesis adaptive controller for a class of linear time-varying uncertain plants using model reference adaptive control techniques is presented. Following a discussion of the general concept, the design of an adaptive controller for unconstrained and constrained control conditions is given. The small ultimate boundness of the state error is considered as the adaptation criterion, which is shown to be satisfied by a Liapunov type stability theorem. To handle the uncertainties that are associated with the plant dynamics and its environment, a min-max concept is employed in the design of the controller. The signal synthesis adaptive approach presented in this paper does not require as many structural assumptions as do most other adaptive approachers, and this is its principal advantage. The state vectors of the plant and model are assumed to be both accessible. Some simulation results are presented which illustrate the effectiveness of the design given here.