Stable Inversion of MIMO Linear Discrete Time Non-Minimum Phase Systems (original) (raw)
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Automatica
In this paper, we propose a framework for output tracking control of both minimum phase (MP) and non-minimum phase (NMP) systems as well as systems with transmission zeros on the unit circle. Towards this end, we first address the problem of unknown state and input reconstruction of non-minimum phase systems. An unknown input observer (UIO) is designed that accurately reconstructs the minimum phase states of the system. The reconstructed minimum phase states serve as inputs to an FIR filter for a delayed non-minimum phase state reconstruction. It is shown that a quantified upper bound of the reconstruction error exponentially decreases as the estimation delay is increased. Therefore, an almost perfect reconstruction can be achieved by selecting the delay to be sufficiently large. Our proposed inversion scheme is then applied to solve the output-tracking control problem. We have also proposed a methodology to handle the output tracking prob! lem of systems that have transmission zeros on the unit circle in addition to MP and NMP zeros. Simulation case studies are also presented that demonstrate the merits and capabilities of our proposed methodologies.
A systematic and numerically efficient procedure for stable dynamic model inversion of LTI systems
1999
Output tracking via the novel Stable Dynamic model Inversion (SDI) technique, applicable to non-minimum phase systems, and which naturally takes into account the presence of noise in target time histories, is considered here. We are motivated by the typical need to replicate time signals in the automobile industry. The earlier approaches to stable inversion do not satisfactorily take into account the measurement and system noise, and the zeros of the system are restricted to certain regions in the complex plane.
Preview-Based Stable-Inversion for Output Tracking of Linear Systems
Journal of Dynamic Systems, Measurement, and Control, 1999
Stable Inversion techniques can be used to achieve high-accuracy output tracking. However, for nonminimum phase systems, the inverse is noncausal—hence the inverse has to be precomputed using a prespecified desired-output trajectory. This requirement for prespecification of the desired output restricts the use of inversion-based approaches to trajectory planning problems (for nonminimum phase systems). In the present article, it is shown that preview information of the desired output can be used to achieve online inversion-based output-tracking of linear systems. The amount of preview-time needed is quantified in terms of the tracking error and the internal dynamics of the system (zeros of the system). The methodology is applied to the online output tracking of a flexible structure and experimental results are presented.
Stable dynamic inversion of nonminimum-phase scalar linear systems
Proceedings of the 16th IFAC World Congress, 2005, 2005
The paper presents a transfer function approach to the stable dynamic inversion of nonminimum-phase scalar linear systems. The technique is based on the study of the structure of the causal unstable input obtained with the standard inversion technique. It is shown that the unbounded term of this input can be decomposed as the sum of a linear combination of unstable zero modes for which new formulae are provided. Then, a closed-form expression of the bounded noncausal solution of the input-output inversion problem is proposed. An automatic inversion scheme built by exploiting the new inversion formula is also presented.
Asian Journal of Control, 2014
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).
Data-Driven Inversion-Based Control: closed-loop stability analysis for MIMO systems
ArXiv, 2018
Data-Driven Inversion-Based Control (D$^{2}$-IBC) is a recently introduced control design method for uncertain nonlinear systems, relying on a two degree-of-freedom architecture, with a nonlinear controller and a linear controller running in parallel. In this paper, extending to the MIMO case a previous result holding for the SISO case, we derive a finite-gain stability sufficient condition for a closed-loop system formed by a nonlinear MIMO plant, connected in feedback with a D$^{2}$-IBC controller.
Stable input-output inversion for nondecoupable nonminimum-phase linear systems
2018 European Control Conference (ECC), 2018
Feedforward control can enhance the performances in the control and regulation of dynamic systems. With this aim, a new inversion formula to solve the stable input-output inversion problem is presented for multivariable nonminimum-phase linear systems. It is based on the computation of the system transfer function inverse and the splitting of the zero dynamics transfer function into stable and unstable parts. Differently from the known alternative statespace methods, the presented approach is applicable to systems that cannot be input-output decoupled by state feedback.
This paper presents a new control, namely additive-state-decomposition dynamic inversion stabilized control, that is used to stabilize a class of multi-input multi-output (MIMO) systems subject to nonparametric time-varying uncertainties with respect to both state and input. By additive state decomposition and a new definition of output, the considered uncertain system is transformed into a minimum-phase uncertainty-free system with relative degree one, in which all uncertainties are lumped into a new disturbance at the output. Subsequently, dynamic inversion control is applied to reject the lumped disturbance. Performance analysis of the resulting closed-loop dynamics shows that the stability can be ensured. Finally, to demonstrate its effectiveness, the proposed control is applied to two existing problems by numerical simulation. Furthermore, in order to show its practicability, the proposed control is also performed on a real quadrotor to stabilize its attitude when its inertia m...
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In this paper, we solve the exact output tracking problem for square nonlinear discrete-time systems through stable inversion. This stable inversion problem involves finding a bounded solution of a dynamical system driven by a bounded signal. Such a solution is given by a difference representation formula, whose existence, under the proper assumptions, can be proven using a Picard process. ).
International Journal of Robust and Nonlinear Control, 2013
This paper considers the problem of computing the input u(t) of an internally asymptotically stable, possibly non minimum phase, linear, continuous-time system Σ yielding a very accurate tracking of a pre-specified desired output trajectoryỹ(t). The main purpose of the new approach proposed here is to alleviate some limitations 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 outputỹs(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 responseỹ t (t) is specified to obtain a fast and smooth transition towards the steady-state trajectoryỹs(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. Onceỹ(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 us(t) is analytically computed exploiting the steady-state output response expressions for inputs belonging to the same set ofỹs(t).