Linear Time-invariant Compensator Research Articles

Characterization and Optimization of <inline-formula> <tex-math notation="LaTeX">$l_{\infty}$</tex-math> </inline-formula> Gains of Linear Switched Systems

In this paper, we consider the $l_{\infty}$ gain characterizations of linear switched systems (LSS) and present various relevant results on their exact computation and optimization. Depending on the role of the switching sequence, we study two broad cases: first, when the switching sequence attempts to maximize, and second, when it attempts to minimize the $l_{\infty}$ gain. The first, named as worst-case throughout the paper, can be related to robustness of the system to uncontrolled switching; the second relates to situations when the switching can be part to the overall decision making. Although, in general, the exact computation of $l_{\infty}$ gains is difficult, we provide specific classes, the input-output switching systems, for which it is shown that linear programming can be used to obtain the worst-case $l_{\infty}$ gain. This is a sufficiently rich class of systems as any stable LSS can be approximated by one. Certain applications to robust control design are provided where we show that a switched compensation independently of the plant has no advantage over a linear time invariant (LTI) compensation, and further, if the plant is strictly causal, even a switched compensation which has a matched switching with the plant does not provide a better performance over an LTI compensation. Also, we present a new necessary and sufficient condition to check the stability of LSS in form of a model matching problem. On the other hand, if one is interested in minimizing the $l_{\infty}$ gain over the switching sequences, we show that, for finite impulse response (FIR) switching systems the minimizing switching sequence can be chosen to be periodic. For input-only or output-only switching an exact, readily computable, characterization of the minimal $l_{\infty}$ gain is provided, and it is shown that the minimizing switching sequence is constant, which, as also shown, is not true for input-output switching.

Consensus of Multi-Agent Systems Under Periodic Time-Varying Network

In this paper, we address the consensus problem for a class of feedback linearizable nonlinear multi-agent systems under a dynamically changing and periodic network communication. The information delivered throughout the communication network is the output of each identical MIMO autonomous agent which can be of any order. First of all, with a local-level control strategy, we show that the consensus is reached by only using a linear time-invariant compensator if the network changes sufficiently fast in time and the compensator is designed so that the multi-agent system achieves the consensus for the average of the communication network. In this case, it is also shown that the state trajectories are close to those of average system if one of the agents is. Finally, in the case of integrator chains, we relax the fast switching condition of the network and give an example that verifies the result.

The periodic optimality of LQ controllers satisfying strong stabilization

An LQ strong stabilization problem is proposed. To determine when a controller with periodic gains is locally superior to a linear time invariant compensator for this problem, a Π test is presented. For systems with strictly proper transfer functions, it is proven that the frequency range where stable periodic controllers based on weak variations about the LTI case can give better performance than stable LTI compensators is finite. In the development, a means to evaluate the second partials of functions with respect to matrix-valued parameters is introduced. For those systems where periodic control is warranted, techniques for designing optimal periodic strongly stabilizing controllers are presented. Two examples detailing the application of the Π test are provided, as well as an optimal periodic controller design example.

A new approach to model reference adaptive control

In classical model reference adaptive control, the goal is to design a controller to make the closed-loop system act like a prespecified reference model in the face of significant plant uncertainty. Typically, the controller consists of an identifier (or tuner) which is used to adjust the parameters of a linear time-invariant (LTI) compensator, and under suitable assumptions on the plant model uncertainty it is proven that asymptotic matching is achieved. However, the controller is typically highly nonlinear, and the closed loop system can exhibit undesirable behavior, such as large transients or a large control signal, especially if the initial parameter estimates are poor. Furthermore, its ability to tolerate time-varying parameters is typically limited. Here, we propose an alternative approach. Rather than estimating the plant or parameters, instead we estimate what the control signal would be if the plant parameters and plant state were known and the ideal LTI compensator were applied. We end up with a linear periodic controller. Our assumptions are reasonably natural extensions of the classical ones to the time-varying setting; we allow rapid parameter variations, although we add a compactness requirement. We prove that we can obtain arbitrarily good tracking, explore the benefits and limitations of the approach, and provide a simulation study.

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.

Self‐oscillating adaptive design of systems with dry friction and significant parameter uncertainty

AbstractMotion control systems with dry friction and high‐order uncertain plants suffer from one or more limit cycles which can be predicted with describing function theory. In this paper quantitative design theory is presented to satisfy given frequency domain tolerances in these systems. The case of linear time‐invariant (LTI) compensation is studied first. Then it is shown how, in those systems with a limit cycle, self‐oscillating design with an adaptive loop (SOAL) can be used to reduce the LTI compensator bandwidth for a large class of problems.

Application of a subspace model identification technique to identify LTI systems operating in closed-loop

In this paper we reformulate the identification of linear time-invariant (LTI) systems operating in a closed-loop with an LTI compensator to an open-loop multi-input-multi-output (MIMO) (state space model) identification problem, followed by a model reduction step. The open-loop identification problem consists of the identification of a MIMO state space model, describing the deterministic transfer from freely excitable inputs to measurable (constrained) signals in the loop. This problem is solved by the recently proposed MOESP ( MIMO output- error state s pac]e model) identification technique, described in Verhaegen and Dewilde ( Int. J. Control, 56, 1187–1210, 1992). The described state space approach does not constrain the delay structure of the controller-plant combination; it delivers an estimate of the controller and it allows an analytical reduction of the obtained state space model of the system (or controller) to half its original system order. The open-loop identification problem only requires the selection of the combined model and controller order; the individual model and controller order selection is performed in the subsequent model reduction step. Furthermore, specializing the presented approach to the identification of a SISO system provided useful information regarding where to inject the signal in the loop in order to enhance numerical robustness. The reliability of the outlined approach and the usefulness of the obtained insights are highlighted in two illustrative numerical examples.

Simultaneous stabilization using an LTI compensator with a sampler and hold

This paper considers problems of simultaneous stabilization using a linear time-invariant compensator incorporated with a sampler and zeroth-order hold function. It is shown that this controller strongly stabilizes a number of plants which may not be stabilizable by a conventional linear time-invariant controller. A controller design procedure is given which is illustrated by an example.

Rejection of persistent bounded disturbances: Nonlinear controllers

This paper considers nonlinear time-varying (NLTV) compensation for linear time-invariant (LTI) plants subject to persistent bounded disturbances. It is shown using two different approaches that using NLTV compensation instead of LTI compensation does not improve the optimal rejection of persistent bounded disturbances. The first approach is to derive a bound on the achievable performance over all stabilizing NLTV controllers. Using results from l 1-optimal control; it follows that in some special cases this bound can be achieved by LTI compensation. This approach involves the introduction of an operator analogous to the Hankel operator in H ∞-optimal control and is of independent interest. The second approach is to assume the NLTV controller is sufficiently smooth to admit a time-varying linearization. This time-varying linearization is then used to construct an LTI controller which achieves the same performance as the original NLTV controller. These results extend previous work by the authors regarding linear time-varying compensation.

An adaptive controller which provides an arbitrarily good transient and steady-state response

A model reference adaptive control problem is posed. In the problem, the objective is not the usual one of forcing the error between the plant output and the reference model output asymptotically to zero, but instead, it is that of forcing this error to be less than a (arbitrarily small) prespecified constant after a (arbitrarily short) prespecified period of time, with a (arbitrarily small) prespecified upper bound on the amount of overshoot. It is shown that to achieve this goal for a stabilizable and detectable, single-input single-output linear time-invariant (LTI) plant, it is necessary and sufficient that the plant be minimum phase. Knowledge of an upper bound on the plant order, of the relative degree, and of the sign of the high-frequency gain is not required. The controller proposed consists of an LTI compensator together with a switching mechanism to adjust the compensator parameters. If an upper bound on the relative degree is available, the compensator has dynamics of order equal to this upper bound less one; otherwise, the order of the compensator is adjusted as well as its parameters. >

Quantitative feedback theory for multiple-input-multiple output feedback systems with control input failures†

In quantitative feedback theory (QFT), plant parameter and disturbance uncertainties are the reasons for using feedback. The system design is tuned to quantitative statements of these parameters and of the performance tolerances. Available design freedom is used to minimize the cost of feedback which is in the bandwidths of the loop transfer functions. This paper extends QFT to 2 × 2 linear time invariant (LTI) multiple-input-multiple-output plants, in which total failure of some control inputs is possible. Maximum possible achievement of the performance specifications is determined, with single fixed LTI compensation networks. A detailed design example is included.

Feedback systems with non-linear uncertain plants†

Two non-linear feedback problems are considered. In one, a non-linear plant of net order e, with highly uncertain parameters, is subjected to external disturbances of bounded magnitude, and of bounded variation over any finite interval [0, T]. It is shown how causal LTI (linear time invariant) compensation can guarantee the resulting output y(t), and its derivatives y(u)(t) for u≤e-1 can be made arbitrarily small in magnitude. The applicable non-linear class is greatly increased by use of non-linear compensation. Owing to the finite [0, T] interval, there is no theoretical difference between minimum and non-minimum-phase plants, both for LTI and nonlinear systems. In the second problem there is an infinite set of command inputs 𝒸= {c(trpar; }. It is required that the closed-loop system, with its uncertain non-linear plant, has an output ylpar;trpar;, which can be written as y=φ*c, a linear convolution, for any cϵ𝒸, with φϵΨa set of acceptable response functions. The tolerances on Φcan be arbitrarily narro...