This paper considers the problem of computing the input u.t/ of an internally asymptotically stable, possibly non-minimum phase, linear, continuous time system † yielding a very accurate tracking of a pre-specified desired output trajectory Q y.t/. The main purpose of the new approach proposed here is to alleviate some limitations that inherent the classical methods developed in the framework of the preview-based stable inversion, which represents an important reference context for this class of control problems. In particular, the new method allows one to deal with arbitrary and possibly uncertain initial conditions and does not require a pre-actuation. The desired output Q y s .t/ to be exactly tracked in steady state is here assumed to belong to the set of polynomials, exponential, and sinusoidal time functions. The desired transient response Q y t .t/ is specified to obtain a fast and smooth transition toward the steady-state trajectory Q y s .t/, without under and/or overshoot in the case of a set point reset. The transient control input u t .t/ is a priori assumed to be given by a piecewise polynomial function. Once Q y.t/ has been specified, this allows the computation of the unknown u t .t/ as the approximate least squares solution of the Fredholm's integral equation corresponding to the explicit formula of the output forced response. The steady-state input u s .t/ is analytically computed exploiting the steady-state output response expressions for inputs belonging to the same set of Q y s .t/. A MIXED NUMERICAL-ANALYTICAL STABLE PSEUDO-INVERSION METHOD 811 point reset. Also, the control input u.t / is partitioned in a transient .u t .t // and steady-state .u s .t // component defined as u.t / D´u t .t /, t 2 OE 0, t t 4 D Q y.t/ y t .t / has a minimum L 2 norm over the transient time interval T t 4 D OE0, t t , with a suitably chosen t t > 0.812 L. JETTO, V. ORSINI AND R. ROMAGNOLI Steady-state condition. The steady-state input u s .t / must yield a steady-state error e s .t / 4 Figure 4. Plots of the two control input components.
The purpose of this note is to consider the quadratic stabilization of LPV systems in the realistic case where only Gaussian noisy parameter measures are available. Though neglected in the actual literature on LPV systems, this question is particular important because in all situations of a practical interest the parameter measurements (or estimates) are never exact. The assumed noisy nature of physical parameter readings requires a specifically developed approach consisting of mixed robust and LPV control methods. In the present case, an approach based on a vertex result on interval time varying (ITV) matrices is proposed. This allows the solvability conditions to be stated in terms of a set of LMI's, whose number is independent of the number of time-varying parameters.
This paper considers the problem of achieving a very accurate tracking of a pre-specified desired output trajectorỹ y(k), k ∈ ∠ Z + , for linear, multiple input multiple output, non-minimum phase and/or non hyperbolic, sampled data, and closed loop control systems. The proposed approach is situated in the general framework of model stable inversion and introduces significant novelties with the purpose of reducing some theoretical and numerical limitations inherent in the methods usually proposed. In particular, the new method does not require either a preactuation or null initial conditions of the system. The desiredỹ(k) and the corresponding sought input are partitioned in a transient component (ỹ t (k) and u t (k), respectively) and steady-state (ỹ s (k) and u s (k), respectively). The desired transient componentỹ t (k) is freely assigned without requiring it to be null over an initial time interval. This drastically reduces the total settling time. The structure of u t (k) is a priori assumed to be given by a sampled smoothing spline function. The spline coefficients are determined as the least-squares solution of the over-determined system of linear equations obtained imposing that the sampled spline function assumed as reference input yield the desired output over a properly defined transient interval. The steady-state input u s (k) is directly analytically computed exploiting the steady-state output response expressions for inputs belonging to the same set ofỹ s (k). Key Words:Model stable inversion, sampled data non-minimum phase systems, optimal transient tracking.
SUMMARYThis paper presents a new sporadic control approach to the tracking problem for MIMO closed-loop systems. An LTI sampled data plant with unmeasurable state affected by external unknown disturbances is considered. The plant is interconnected to an event-based digital dynamic output-feedback controller via a network. Both the external reference and the unknown disturbance are assumed to be generated as the free output response of unstable LTI systems. The main feature of the new event-driven communication logic (CL) is that it works without the strict requirement of a state vector available for measurement. The purpose of the CL is to reduce as much as possible the number of triggered messages along the feedback and feedforward paths with respect to periodic sampling, still preserving internal stability and without appreciably degrading the control system tracking capability.The proposed event-driven CL is composed of a sensor CL (SCL) and of a controller CL (CCL). The SCL is based on the computation of a quadratic functional of the tracking error and of a corresponding suitably computed time-varying threshold: a network message from the sensor to the controller is triggered only if the functional equals or exceeds the current value of the threshold. The CCL is directly driven by the SCL: the dynamic output controller sends a feedforward message to the plant only if it has received a message from the sensor at the previous sampled instant. Formulation of the controller in discrete-time form facilitates its implementation and provides a minimum inter-event time given by the sampling period. An example taken from the related literature shows the effectiveness of the new approach. The focus of this paper is on the stability and performance loss problems relative to the sporadic nature of the control law. Other topics such as network delay or packets dropout are not considered.
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