On the relationship between the model order reduction problem and the simultaneous stabilization problem

This note points out the fact that, under certain conditions, the model order reduction problem is very closely tied to the simultaneous stabilization problem. These conditions occur when the reduced order model is to be used to design a suboptimal controller for a high-order plant. Under these conditions the controller must not only stabilize the low-order model, but must also stabilize the high-order plant. The result concerning the existence of a controller that will simultaneously stabilize two plants is used to conclude that the approximate inclusion of any unstable real modes of the high-order plant in the low-order model will guarantee the existence of such a controller.     

Control relevant estimation of plant and disturbance dynamics

Estimating models for both plant and disturbance dynamics is important in control design applications that focus on disturbance rejection. Several methods for low-order approximate model estimation on the basis of closed-loop data exist in the literature, but fail to address the simultaneous estimation of low-order approximate models of both plant and disturbance dynamics. In this paper a new extended two-stage methodology is proposed that allows for low-order approximate disturbance model estimation. In the proposed extended two-stage method the first stage is used to estimate high-order models for filtering purposes. In the second stage, filtered signals are used to provide the means for low-order approximate model estimation of both plant and disturbance dynamics.     

Tuning of Active Disturbance Rejection Control with application to power plant furnace regulation

Active Disturbance Rejection Control (ADRC) is emerging as a promising solution in dealing with the unmeasurable disturbances and unknown uncertainties, which are treated in a lumped manner and augmented as an extended state variable. Subsequently, an extended state observer (ESO) is designed to estimate and cancel the combined uncertain term in real time, modifying the uncertain plant to behave like a nominal model consisting of integrators. In the original ADRC formulation, the plant model is assumed to be of delay-free and its order is assumed to be equal to that of the real plant. However, a low-order ADRC is preferred and received a wide acceptance in practice because of its simplicity. Currently, the feasibility of such practice is not clearly revealed as well as its potential dangers. To this end, this paper analyzes the control mechanism from the perspective of the modified plant, which, in turn, would give guidance to parameter tuning. The effect of each parameter on the compensation efficiency and stability conditions of the modified plant is analyzed, based on which a complete tuning procedure for ADRC is developed where the initial settings is derived from the existing PI controller parameters. Finally, the proposed tuning method is experimentally used for a furnace pressure regulation of a 1000MW power plant, validating the feasibility of the low-order ADRC, even in the absence of both dynamic model and the information on the model order.     

Low-order stabilizing controllers

A method for designing low-order stabilizing controllers is described. It is shown that for some plants the order of stabilizing controllers may be less than or equal to l (respectively m), where l (respectively m) is the number of plant outputs (respectively inputs). An explicit characterization of a set of low-order stabilizing controllers is given and numerical examples are provided to illustrate the results of the note     

A simulation aid to gain estimates for robust tuning regulators

Gain estimates for Davison's robust tuning regulator are shown to be directly computable from plant open-loop step data using low-order simulation techniques. At no stage of the procedure is a detailed model of plant dynamics required.     

Integrated servo-mechanical design of high-performance mechatronics using generalized KYP Lemma

High-performance mechatronics have specifications which are difficult to achieve when the mechanical plant is non-minimum phase and a low-order controller is used. In this paper, an integrated servo-mechanical design algorithm is proposed for systematic finite frequency redesign of a mechanical plant using the generalized Kalman–Yakubovic–Popov Lemma. The synthesis of a minimum phase plant is carried out based on a predesigned low-order controller, positive realness constraints, and performance specifications of the overall control system. Our simulation results using the proposed algorithm achieve a high-bandwidth control system with disturbance attenuation capabilities at the phase-stabilized resonant modes of the plant with the low-order controller.     

Stability and robustness properties of a simple adaptive controller

A low-order adaptive tracking controller is proposed for linear time-invariant plants with a relative degree not exceeding two. The order of the plant is not required to be known a priori. The algorithm is robustly stable with respect to small linear time-invariant plant perturbations and bounded disturbances. The robustness is achieved by using a projection of the parameter estimates in the control law. An additional a priori information needed to design the controller is bounds on the plant parameters. In the absence of plant perturbations and disturbances, perfect tracking is obtained     

A Parametrized Controller Reduction Technique via a New Frequency Weighted Model Reduction Formulation

In this technical note, the solution to the controller reduction problem via a double-sided frequency weighted model reduction technique is considered. A new method for finding low-order controllers based on new frequency weights derived using closed-loop system approximation criterion is proposed. The formulas of frequency weights are obtained in terms of the plant, the original controller and a matrix of free parameters. By varying the free parameters in the resulting two-sided frequency weighted model reduction problem, frequency weighted error can be significantly reduced to yield more accurate low-order controllers.      

When is the discretization of a spatially distributed system good enough for control?

This paper describes a new and straightforward method for controlling spatially distributed plants based on low-order models obtained from spatial discretization techniques. A suitable level of discretization is determined by computing the sequence of ν-gaps between weighted models of successively finer spatial resolution, and bounding this by another sequence with an analytic series. It is proved that such a series forms an upper bound on the ν-gap between a weighted model in the initial sequence and the spatially distributed weighted plant. This enables the synthesis, on low-order models, of robust controllers that are guaranteed to stabilize the actual plant, a feature not shared by most model reduction methods where the gap between the high-order model and plant is often not known, and where the gap between high-order and reduced models may be too expensive to compute. Since the calculation of the current bound is based on weighted models of small state-dimension, the new method avoids the numerical problems inherent in large-scale model reduction based approaches. The ideas presented in this paper are demonstrated on a disturbance rejection problem for a 1D heat equation.     

Approximate pole placement in linear multivariable systems using dynamic compensators†

The problem of approximate pole placement in linear time-invariant systems is generalized to include the case in which the controllable, observable plant of order n is augmented by o, fixed order compensator. It is shown how a low-order compensator may be obtained such that the eigenvalues of the closed-loop system are close to a set of preassigned values.     

Controller reduction: concepts and approaches

The problem of passing from a linear time-invariant high-order controller designed for a linear time-invariant plant (of presumably high order) to a low-order approximation of the controller is discussed. The approximation problem is often best posed as a frequency-weighted L_∞ approximation problem. Many different controller representations are possible, giving different performances of the various reduction algorithms     

Boundary Feedback Control of the Burgers Equations by a Reduced-Order Approach Using Centroidal Voronoi Tessellations

In this paper, we study a reduced-order modelling for boundary feedback control problem of Burgers equations. Brief review of the CVT (centroidal Voronoi tessellation) approaches to reduced-order basis are provided. In CVT-reduced order modelling, we start with a snapshot set just as is done in a SVD (Singular Value Decomposition)-based setting. A weighted (nonuniform density) CVT is introduced and low-order approximate solution and compensator-based control design of Burgers equation is discussed. Through CVT-nonuniform method, the low-order basis is obtained. Then this low-order basis is applied to low-order approximate solution and functional gains to design a low-order controller, and by using the low-order basis order of control modelling was reduced. Numerical experiments show that a solution of reduced-order controlled Burgers equation performs well in comparison with a solution of full order controlled Burgers equation.     

Model reduction technique applied to a nuclear reactor turbine system

A method has been developed for determining a low-order model for a large multivariable system from its measured input-output data. It is based on the use of a matrix pseudo-inverse to estimate the parameters of the model which minimize the sum of the squares of the errors between the responses of the original system and the low-order model at the sampling instants. A recursive algorithm is proposed which makes it useful for on-line applications. The method is used to obtain low-order models for a nuclear reactor turbine system. The outputs of the low-order models and those of the large-scale model (consisting of 11 non-linear differential equations and 36 algebraic equations) are compared for the turbine system.     

On the synthesis of multivariable systems

The general question of compensating a linear multivariable system via linear state variable feedback (lsvf) in combination with input dynamics is considered. In particular, it is shown that, since lsvf and input dynamics can both be represented by proper feedforward compensation, any design based on feedforward compensation alone can usually be realized in a more efficient manner through the combined use of "low-order" feedforward compensation (input dynamics) and lsvf. A major result of this paper is the frequency-domain synthesis algorithm outlined in Section III for "separating" any proper feedforward transfer matrix in order to achieve the more efficient compensation scheme. Specific examples which illustrate the employment of the algorithm to the questions of decoupling and exact model matching are also included.     

Application of an iterative identification for control scheme to a neutralization process

In this paper an iterative scheme for identification and control is discussed. During the identification step a plant model which is suitable for the subsequent controller design step is obtained by estimation of the (dual) Youla-parameter from measurements of the input and output of the plant. Using the identified plant model, the frequency response of the ideal controller which perfectly realizes the desired closed-loop response for set-point changes is computed. This controller, in general, may not be realizable or is of high-order. A realizable, low-order controller is then calculated using frequency-weighted approximation. These steps are repeated until the performance of the closed-loop system is satisfactory or cannot be improved further. The proposed scheme is applied successfully to the identification and control of a continuous neutralization reactor.     

Identification with nonparametric uncertainty

An identification technique that is robust to nonparametric uncertainty (i.e. model mismatch) is presented. The identifier produces either a parameter set estimate or a frequency response set estimate. The estimates result from the inclusion of a model of the nonparametric uncertainty in the plant model. The frequency response set estimate is shown to always contain the frequency response of the plant as long as certain modeling conditions are met. This type of identifier would be useful in applications such as control where a property such as stability or performance level must be achieved in the face of low-order modeling and its associated nonparametric uncertainty     

The synthesis of linear multivariable systems by state-variable feedback

Design procedures are presented for the compensation of linear multivariable-feedback systems. The design objectives which are noninteraction and low-order compensators are obtained by state-variable feedback. Aqth order plant withminputs andmoutputs whereq>mis treated. All the state variables are assumed to be available and measurable. It is shown that it is often possible by state-variable feedback to obtain noninteraction without an increase in system order. When the determinant of the plant transfer-function matrix has right half plane zeros the state-variable feedback method cannot be used to obtain noninteraction since the resulting compensated system would be unstable. It is shown that it is not possible to change these right-half plane zeros by either the well-known transfer-function design methods or by the state-variable feedback method. A combination of both transfer-function and state-variable techniques is discussed and shown to lower the order of the compensators required for the compensation of certain systems.     

Variance analysis of L2 model reduction when undermodeling—the output error case

In this contribution, variance properties of L₂ model reduction are studied. That is, given an estimated model of high order we study the resulting variance of an L₂ reduced approximation. The main result of the paper is showing that estimating a low-order output error (OE) model via L₂ model reduction of a high-order model gives a smaller variance compared to estimating a low-order model directly from data in case of undermodeling. This has previously been shown to hold for Finite Impulse Response models, but is in this paper extended to general linear OE models.     

On the optimality of the Wilkie-Perkins low-order sensitivity model

Some comments on the low-order sensitivity model proposed by Wilkie and Perkins are presented. The structural controllability of this model and the analysis of its observability properties allow one to conclude that the model is optimum in the sense that no lower order sensitivity model exists that can generate all the state sensitivity functions for the class of systems for which the theory developed in Wilkie and Perkins holds.     

On selecting a low-order model using the dominant mode concept

A technique for selecting a low-order system to approximate a high-order model has been suggested by Davison [2]. A critical component in this technique is the criterion used to select the most appropriate order and modes for the low-order approximation. Criteria have been discussed and analyzed by Mahapatra [5], [6], Rao, Lamba, and Rao [7], and Elrazaz and Sinha [3]. In this note we overcome deficiencies in the criteria that have been proposed and we introduce a new criterion which is rigorously justified. The criterion we suggest is also applicable when eigenvalues of the system are nonreal.