Geometric characterization on the solvability of regulator equations

The solvability of the regulator equation for a general nonlinear system is discussed in this paper by using geometric method. The ‘feedback’ part of the regulator equation, that is, the feasible controllers for the regulator equation, is studied thoroughly. The concepts of minimal output zeroing control invariant submanifold and left invertibility are introduced to find all the possible controllers for the regulator equation under the condition of left invertibility. Useful results, such as a necessary condition for the output regulation problem and some properties of friend sets of controlled invariant manifolds, are also obtained.     

Application of discrete-time quantized data controllers to systems with disturbances

In a recent paper, a new method for the digital control of a continuous-time system based on a table look-up technique was presented. This digital controller is called a discrete-time quantized data (DTQD) controller. In order to minimize the number of table entries, a grid embedding process is proposed. This control technique proved successful when applied to the regulator problem. However, when a DTQD controller using the grid embedding process is applied to a continuous-time system with additive disturbances, the performance of the DTQD controller degraded, causing oscillations and possible limit cycle phenomena. In this paper, the grid embedding process is replaced with a continuous scaling process. The continuous scaling process retains the advantages of the grid embedding process in reducing the number of table entries. In addition, it reduces the amount of oscillation in the system response when disturbances are present.     

A feedback solution for the large infinite stage discrete regulator and servomechanism problem

In this paper a simple way of solving the discrete infinite stage regulator and servomechanism problem is suggested for large dimensional systems. The basis of the approach is that a solution for the finite stage case can be obtained by duality and decomposition within a two level optimisation structure except that the resulting control is open loop. However, given a precalculated open loop solution for one initial condition, it is possible to calculate the gains for the regulator and servomechanism for the infinite stage case by inverting one n × n matrix for a composite system of order n. The approach is demonstrated on a river pollution problem using some data on the river Cam near Cambridge.     

Stochastic optimal LQR control with integral quadratic constraintsand indefinite control weights

A standard assumption in traditional (deterministic and stochastic) optimal (minimizing) linear quadratic regulator (LQR) theory is that the control weighting matrix in the cost functional is strictly positive definite. In the deterministic case, this assumption is in fact necessary for the problem to be well-posed because positive definiteness is required to make it a convex optimization problem. However, it has recently been shown that in the stochastic case, when the diffusion term is dependent on the control, the control weighting matrix may have negative eigenvalues but the problem remains well-posed. In this paper, the completely observed stochastic LQR problem with integral quadratic constraints is studied. Sufficient conditions for the well-posedness of this problem are given. Indeed, we show that in certain cases, these conditions may be satisfied, even when the control weighting matrices in the cost and all of the constraint functionals have negative eigenvalues. It is revealed that the seemingly nonconvex problem (with indefinite control weights) can actually be a convex one by virtue of the uncertainty in the system. Finally, when these conditions are satisfied, the optimal control is explicitly derived using results from duality theory     

Variance constrained self-tuning control

A self-tuning regulator for a variance constrained optimal control problem is given. The criterion for control is to minimize the stationary variance of the output. In the cases when the regulator which gives minimum variance requires too large control signals an inequality constraint on the input variance is introduced. In practice it is easier to select a constraint on the variance of the input than to choose the relative weights in a quadratic loss function. The self-tuning regulator applies the Robbins-Monro scheme to adjust the Lagrange multiplier of the variance constrained control problem. The behaviour of the algorithm is illustrated by a simulated example. The asymptotic behaviour of the regulator is studied using a set of associated ordinary differential equations.     

State controllability and optimal regulator control of time-delayed systems

An important recent advance in the solution of the optimal regulator control problem for time-delayed systems is extended here to multivariable systems and to systems which exhibit multiple time delays. The state equations are partitioned into discrete and continuous portions through a state transformation such that the solution of the optimal regulator problem reduces to finding a steady-state controller gain based on both a discrete and continuous Riccati matrix. The discrete Ricatti matrix is found independently of the continuous solution due to the partitioning of the state equations, and it is not necessary to solve the system of partial differential Riccati equations which arise in the traditional solution of the linear quadratic regulator (LQR) problem for time-delayed systems. In addition, through this state transformation it becomes possible to extend the standard state controllability tests to time-delayed systems. It is shown that the controllability of the transformed state space is necessary for a feasible solution to the optimal regulator problem for time-delayed systems. This is an important test to determine the practicality of various time-delayed system realizations. Numerical examples illustrate the application of the technique to systems exhibiting multiple time delays, multivariable systems and time-series models. It is shown that the classic Wood-Berry distillation model realization does not possess state controllability properties which explains why this system has been historically difficult to control using feedback techniques.     

A state-space description for GPC controllers

Generalized predictive control (GPC)-type control algorithms traditionally derived in the polynomial domain are derived in this paper in the state-space domain, but following the polynomial approach due to Clarke et al. (1987). Relations between the polynomial and state-space parameters are presented. Some possible state-space representations which were used earlier in different publications are discussed. The problem of deriving the GPC algorithm in the state-space domain is solved for the unrestricted case as well as for the case of restricted control and output horizons. Some properties of the state estimate for this problem are presented; in particular, two methods of Kalman filtering—optimal and asymptotic—are proposed. The solution is valid for any possible (minimal or non-minimal) state-space representation. Another approach to this problem is by the ‘dynamic programming method’ and solving the Riccati equation (Bitmead et al. 1990). This approach is also presented in this paper but the method differs from this earlier work and does not require extending the state dimension. Ultimately, certain features of the state-space approach are discussed, such as (a) the opportunity for straightforward analysis of the transient states produced by switching on the regulator, by changing the set-point or by changing the regulator parameters; (b) easy extension to the multidimensional case; and (c) the possibility of introducing nonlinearities into the model     

Regulator problem for infinite-dimensional systems

In this paper a regulator problem is investigated for an infinite-dimensional system with constant disturbances. The regular problem is to determine a feedback control law which stabilizes and regulates the system. A design procedure of a regulator is proposed which can be realized in finite-dimensional theories and techniques for an infinite-dimensional system.     

基于风扰动预测的阵风减缓最优控制研究

目前大多数阵风减缓控制方法都是等到飞机到达风场之后才起作用,由此带来了时滞与舵面速率饱和等问题。为了解决这一问题,提出了一种基于风扰动预测的阵风减缓控制系统方案。首先,对风扰动预测技术进行了研究,利用二阶互补滤波器实现了一种基于激光测风雷达获取的阵风信息与其它渠道获取的阵风信息的数据融合算法。其次,以某型民用飞机模型为对象,采用LQR方法设计最优状态调节器使得性能指标最小。接着,引入基于风扰动预测的前馈补偿,使得在未来阵风到达时飞机状态要尽可能保持不变。仿真结果表明,基于风扰动预测的阵风减缓最优控制系统能大幅度地减少阵风干扰对飞机法向过载和俯仰角速度的影响,证明了所设计的控制系统方案的正确性和有效性。     

Regulator design for continuous-time distributed parameter systems by discrete-time controls

In this paper, a regulator problem is investigated regarding a continuous-time parabolic distributed parameter system with constant disturbances. The regulator problem is to determine a feedback control law which stabilizes and regulates the system. A new feedback control law is presented which stabilizes the continuous-time system and regulates the output, using only discrete-time controls. The design procedure of a regulator can be realized in finite-dimensional theories and techniques.     

A sub-optimal algorithm to synthesize control laws for a network of dynamic agents

We study the synthesis problem of a linear quadratic regulator (LQR) controller when the matrix describing the control law is constrained to lie in a particular vector space. Our motivation is the use of such control laws to stabilize networks of autonomous agents in decentralized fashion, with the information flow being dictated by the constraints of a pre-specified topology. In this paper, we consider the finite-horizon version of the problem and provide both a computationally intensive optimal solution and a sub-optimal solution that is computationally more tractable. Then we apply the technique to the decentralized vehicle formation control problem. It is numerically illustrated that while the loss in performance due to the use of the sub-optimal solution is not huge, the topology can have a large effect on performance.     

Comments on ‘ Dynamic feedback methodology for interconnected control systems ’ and ‘ Dynamic multilevel optimization for a class of non-linear systems ’

The paper considers the Kalman-Bucy filter for a linear system when the measurement noise covariance matrix is singular. It is shown that the problem of infimizing the square of a linear functional of the state estimation error is the dual of the optimal singular linear regulator problem. Furthermore there is an optimal reduced-order Kalman-Bucy filter for minimization of the trace of the state error covariance matrix, when all extremal controls for a dual regulator have finite order of singularity, and no Luenberger observer is needed. The proof is constructive. Necessary and sufficient conditions for the existence of a reduced-order optimal estimator are derived.     

A recursive deconvolution approach to disturbance reduction

Active noise control (ANC) uses an estimate of the noise affecting a system in order to remove its effect from the output. In some applications, it is possible to directly measure the noise and a feedforward compensator can be used. Other applications require instead that the noise be estimated from its effect on the system. This results in an adaptive feedback ANC, see a previous paper by Gan and Kuo. The identification of the disturbance from the output of the linear system is a deconvolution problem. In this paper, we study a deconvolution technique for the active reduction of the influence of the disturbance on the output of a linear control system. The regulator that we present does not affect the behavior of the system in the absence of disturbances.     

Reliable regulation in decentralised control systems

This article addresses the design of decentralised regulators which supply the control systems with signal tracking and disturbance rejection. This property has to be attained, to the maximum possible extent, even when instrumentation faults occur, thus causing the opening of some feedback loops. The problem is tackled for LTI asymptotically stable plants, subject to perturbations, under the assumption that the Laplace transforms of the exogenous signals have multiple poles on the imaginary axis. The proposed regulator is composed of an LTI nominal controller supervised by a reconfiguration block. Once the actions of the reconfiguration block have been settled, the synthesis of the nominal controller is reformulated as a suitable regulation problem. A constructive sufficient condition for its solvability is established. This condition turns out to be also necessary if the exogenous signals are polynomial in time.     

Optimal damping of forced oscillations in discrete-time systems

In this paper we consider a linear discrete-time control system affected by an additive sinusoidal disturbance with known frequencies but unknown amplitudes and phases. The problem is to damp this forced oscillation in an optimal fashion. We show that the natural solution from the point of view of optimal control is neither robust with respect to errors in the frequencies, and thus not optimal in practice, nor independent of the unknown amplitudes and phases. The main result of this paper concerns the existence and design of a realizable, robust optimal regulator, which is universal in the sense that it does not depend on the unknown amplitudes and phases and is optimal for all choices of such parameters. The regulator allows for a considerable degree of design freedom to satisfy other design specifications. Finally, it is shown that this regulator is optimal also for a wide class of stochastic control problems     

Multi-model predictive control of an industrial C3/C4 splitter

In this paper, the application of a linear predictive controller to an industrial distillation column that presents a nonlinear behavior is described. The system is represented by a set of linear approximating models, where each model corresponds to a possible operating point of the system. The control sequence computed by the control algorithm is based on a min–max optimization problem where the controller cost is minimized for the worst process model. The control algorithm makes use of a particular form of the state-space model, which preserves the structure of conventional model predictive control controllers that are based on the step response model. The performance of the proposed controller applied to an industrial system is illustrated with results of the real system at typical plant conditions with the controller performing as a regulator and as an output reference tracker.     

不确定噪声下确保控制性能的鲁棒LQG调节器

本文讨论了一类具有不确定噪声的离散时间随机线性系统的鲁棒ＬＱＧ问题，文章给出了确保控制性能的不确定噪声协方差矩阵的扰动上界，以及极小极大鲁棒ＬＱＧ调节器的设计方法，采用这种调节器不仅能极小化不确定下的最坏性能，而且也能确保控制性能指标达到给定的自由度内。     

A general one-dimensional search algorithm for computing approximate solutions of some non-linear optimal control problems

Barr and Gilbert (1966, 1969 b) have presented computing algorithms for converting a brood class of optimal control problems (including minimum time, and fixed-time minimum fuel, energy and effort problems) to a sequence of optimal regulator problems, using a one dimensional search of the cost variable. These Barr and Gilbert algorithms, which use quadratic programming algorithms by the same authors (1969 a) to solve the resulting optimal regulator problems, are restricted to dynamic equations linear in state by virtue of using the convexity and compactness (Neustadt 1963) and contact function (Gilbert 1966) of the reachable set

This paper extends the above approach to a class of terminal cost optimal control problems similar to those considered by Barr and Gilbert (including quite general control constraints, but only allowing initial and final state constraints), having differential equations non-linear instate and control (where the convexity-compactness results do not hold), by converting each such problem to a sequence of optimal regulator problems, with non-linear differential equations. These, in turn, are solved by one of the author's earlier algorithms (Katz 1974) that makes use of the above convexity, compactness, and contact function results by repeatedly linearizing the regulator problems. The approach of this paper differs from that of Halkin (1964 b), in that Halkin directly linearizes the original problem (e.g. converting a non-linear minimum fuel problem to a linear minimum fuel problem) and then solves the linearized version by a doubly iterative procedure

The computing algorithm presented here is based on the definition of an appropriate approximate solution of the terminal cost problem. A local-minimum convergence proof is given, which is weak in the sense that it assumes convergence of the substep algorithm (Katz 1974) for non-linear optimal regulator problems, whose convergence has not been proved. A subsequent paper (Katz and Wachtor, to appear) shows good convergence of the (overall) terminal cost problem algorithm in examples having singular arcs, with no prior knowledge of the solution or its singular nature, other than an initial upper bound on the cost.     

The digital optimal regulator and tracker for stochastic time-varying systems

The general approach to solve digital control problems is to approximate them by discrete-time control problems, which consider the system behaviour at the sampling instants only, completely disregarding the inter-sample behaviour. In this paper we consider digital control problems without making any approximations, i.e. we solve problems involving continuous-time criteria taking explicitly into account the inter-sample behaviour, which relaxes the demand for a 'small' sampling time. We solve what we call the digital optimal regulator and tracking problem where the continuous-time system is linear time-varying, and disturbed by additive white noise, and the state information at the sampling instants incomplete, and corrupted by additive white noise. The control is piecewise constant, and the continuous-time criteria are quadratic. Both the regulator and tracking problem turn out to be certainty equivalent. The solutions to both the regulator and tracking problem therefore consist of the well-known discrete-time Kalman one step ahead predictor, and a feedback generated by a Riccati type recursion that runs backward in time. In the case of the tracking problem the feedforward is also generated by a recursion that runs backward in time. Both recursions can be computed off-line. Expressions for the minimum cost of both problems, explicit in the system, criterion and covariance matrices, are derived. In a companion paper we treat the numerical computation which is not straightforward.     

Minimal time control of a discrete system with a nonlinear plant

This paper treats the problem of minimal time control of a class of pulse-width-modulated (PWM) regulator systems with plants described by a nonlinear second-order differential equation. For the type of nonlinearity considered, it is found possible to carry out the necessary phase plane constructions by making use of the geometric properties of the solutions of the plant differential equation. The resulting optimal strategy is quite simple and is similar to the one for PWM systems with linear second-order plants whose transfer functions have nonpositive poles and no finite zeros [3].