The design of optimal compensators for linear constant systems with inaccessible states

This paper considers the problem of designing an optimal linear time-invariant dynamic compensator for the regulation of annth-order linear time-invariant plant. The usual quadratic cost on the state and control is averaged over initial plant state values. The globally optimal compensator gains and dynamic order are determined by showing that this problem is mathematically identical to a steady-state stochastic control problem whose optimal solution is known.     

Estimation and control of systems with unknown covariance andmultiplicative noise

The problem of estimation and control for systems with multiplicative noise and unknown second-order statistics is considered. Conditions are found for the existence of a solution based on game theoretic ideas. The conditions for the existence of a saddle point for the time-invariant filtering problem are necessary and sufficient, whereas for all other cases only necessary. The central idea of the solution is to convert the stochastic problem to a deterministic optimal control problem whose minimax point is sought with respect to the control, filter, and unknown statistics parameters. The results that are derived show that the problem of estimation for systems with unknown covariances depends on the costate matrix, which in turn is a function of the performance measure. Thus, the filter loses one of its best known properties, that of independence of the performance functional. This property holds not only for the classical Kalman filter but also for multiplicative systems     

On the design of optimal constrained dynamic compensators for linear constant systems

The design of linear time-invariant dynamic compensators of fixed dimensionalitys, which are to be used for the regulation of annth-order linear time-invariant plant, is dealt with. A modified quadratic cost criterion is employed in which a quadratic penalty on the system state as well as all compensator gains is used; the effects of the initial state are averaged out. The optimal compensator gains are specified by a set of simultaneous nonlinear matrix algebraic equations. The numerical solution of these equations would specify the gain matrices of the dynamic compensator. The proposed method may prove useful in the design of low-orderscompensators for high-ordernplants that have fewroutputs, so that the dimension of the compensator is less than that obtained through the use of the associated Kalman-Bucy filternor the Luenberger observern - r.     

Robust output regulation under uncertainties of physical parameters

In this paper the robust output regulation problem is solved for linear time-invariant systems whose matrices are assumed to depend on some parameters, each of which possibly affects all the elements of the matrices describing the system, thus playing the role of a ‘physical’ parameter. The robustness here obtained is the preservation of the output regulation property under perturbations of such parameters. Both the conditions for the existence of a solution and a design procedure of the compensator are given.     

Equivalent optimal compensation problem in the delta domain for systems with white stochastic parameters

The equivalent representation in the delta domain of the digital optimal compensation problem is provided and computed in this article. This problem concerns finding digital optimal full and reduced-order output feedback controllers for linear time-varying and time-invariant systems with white stochastic parameters. It can subsequently be solved in the delta domain using the strengthened optimal projection equations that we recently formulated in this domain as well. If the sampling rate becomes high, stating and solving the problem in the delta domain becomes necessary because the conventional discrete-time problem formulation and solution become ill-conditioned. In this article, by means of several numerical examples and compensator implementations, we demonstrate this phenomenon. To compute and quantify the improved performance when the sampling rate becomes high, a new delta-domain algorithm is developed. This algorithm computes the performance of arbitrary digital compensators for linear systems with white stochastic parameters. The principle application concerns nonconservative robust digital perturbation feedback control of nonlinear systems with high sampling rates.     

A new approach to adaptive control: no nonlinearities

In adaptive control the goal is to design a controller to control an uncertain system whose parameters may be changing with time. 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 good asymptotic behaviour is achieved, such as model matching (for minimum phase systems) or stability. However, a typical adaptive controller does not track time-varying parameters very well, and it is often highly nonlinear, which can result in undesirable behaviour, such as large transients or a large control signal. Furthermore, most adaptive controllers provide only asymptotic tracking, with no ability to design for a pre-specified settling time.Here we propose an alternative approach, which yields a linear periodic controller. Rather than estimating the plant or compensator parameters, instead we estimate what the control signal would be if the plant parameters were known. In this paper we argue the utility of this approach and then examine the first order case in detail, including a simulation. We also explore the benefits and limitations of the approach.     

On the optimal unbiased functional filtering

Presents a simple solution to the optimal unbiased reduced-order (or functional) filtering problem for linear time-varying systems. Necessary and sufficient conditions for the existence of the obtained filter are given. Stability and convergence conditions are developed for the time-invariant systems. Both continuous- and discrete-time cases are considered     

Some results on minimum magnitude regulated response

Analytical results are obtained for the minimum peak tracking error magnitude achievable by some finite settling time control systems in response to a step reference input. The limits of these results as the settling time approaches infinity are also obtained. These represent performance bounds which apply for any finite order linear time-invariant (LTI) controller of a given configuration (i.e., one-parameter or two-parameter). The systems considered are one-parameter discrete-time single-input, single-output (SISO) where the plant has one unstable pole and one non-minimum-phase zero. The result for a two-parameter compensator for plants with one non-minimum-phase zero is presented     

OPTIMAL AND ROBUST SLIDING MODE CONTROL FOR LINEAR SYSTEMS WITH MULTIPLE TIME DELAYS IN CONTROL INPUT

This paper presents the optimal regulator for a linear system with multiple time delays in control input and a quadratic criterion. The optimal regulator equations are obtained using the duality principle, which is applied to the optimal filter for linear systems with multiple time delays in observations. Performance of the obtained optimal regulator is verified in the illustrative example against the best linear regulator available for linear systems without delays. Simulation graphs and comparison tables demonstrating better performance of the obtained optimal regulator are included. The paper then presents a robustification algorithm for the obtained optimal regulator based on integral sliding mode compensation of disturbances. The general principles of the integral sliding mode compensator design are modified to yield the basic control algorithm oriented to time‐delay systems, which is then applied to robustify the optimal regulator. As a result, the sliding mode compensating control leading to suppression of the disturbances from the initial time moment is designed. The obtained robust control algorithm is verified by simulations in the illustrative example.     

A parametric extension of mixed time/frequency robustidentification

A parametric extension to the time/frequency robust identification framework is presented. The results can be applied to stable linear time-invariant systems on which time and/or frequency experiments have been performed. The parametric portion of the model should be affine in the unknown parameters, which includes practical applications such as flexible structures. The consistency problem is cast as a constrained finite-dimensional convex optimization problem that can be formulated as a linear matrix inequality. The proposed procedure provides an interpolating identification algorithm, convergent and optimal up to a factor of two (with respect to central algorithms)     

Controllability, stabilizability, and continuous-time Markovianjump linear quadratic control

Consideration is given to the control of continuous-time linear systems that possess randomly jumping parameters which can be described by finite-state Markov processes. The relationship between appropriately defined controllability, stabilizability properties, and the solution of the infinite time jump linear quadratic (JLQ) optimal control problems is also examined. Although the solution of the continuous-time Markov JLQ problem with finite or infinite time horizons is known, only sufficient conditions for the existence of finite cost, constant, stabilizing controls for the infinite time problem appear in the literature. In this paper necessary and sufficient conditions are established. These conditions are based on new definitions of controllability, observability, stabilizability, and detectability that are appropriate for continuous-time Markovian jump linear systems. These definitions play the same role for the JLQ problem as the deterministic properties do for the linear quadratic regulator (LQR) problem     

一类时变系统中补偿器的功能和设计

PID控制器由于其结构简单和参数整定容易,具有良好的控制功能,被广泛应用在工程上的线性确定性系统中.但是,在非线性系统和不确定性系统中, PID控制器由于其固定的控制参数和结构限制,很难克服系统中内外干扰的影响,控制器鲁棒性能较差.本文按照被控对象中的标称模型设计常规PID控制器优化参数,对非线性系统中多个线性化模型和不确定性系统中的不同工作点模型,设计了一类补偿器,使得整个闭环系统具有与标称模型下相同的控制品质.工程应用例子论证了本文提出的补偿器能提高不确定与非线性系统的控制品质和鲁棒性能.     

Robust multi-parametric model predictive control for LPV systems with application to anaesthesia

We present a multi-parametric model predictive controller (mpMPC) for discrete-time linear parameter-varying (LPV) systems based on the solution of the mpMPC problem for discrete-time linear time-invariant (LTI) systems. The control method yields a controller that adapts to parameter changes of the LPV system. This is accomplished by an add-on unit to the implementation of the mpMPC for LTI systems. No modification of the optimal mpMPC solution for LTI systems is needed. The mpMPC for LPV systems is entirely based on simple computational steps performed on-line. This control design method could improve the performance and robustness of a mpMPC for LPV systems with slowly varying parameters. We apply this method to process systems which suffer from slow variation of system parameters due, for example, to aging or degradation. As an illustrative example the reference tracking control problem of the hypnotic depth during intravenous anaesthesia is presented: the time varying system matrix mimics an external disturbance on the hypnotic depth. In this example the presented mpMPC for LPV systems shows a reduction of approximately 60% of the reference tracking error compared to the mpMPC for LTI systems.     

Relatively optimal control and its linear implementation

Motivated by the fact that determining a feedback solution for the optimal control problem under constraints is a hard task we introduce the concept of relative optimality, roughly optimality for a specific (nominal) plant initial condition. We consider a generic discrete-time finite-horizon constrained optimal control problem for linear systems, and we seek for a state feedback (possibly dynamic) controller. As a fundamental requirement, we do not admit preactions or controller-state initialization based on the plant initial state and we assume our controller to be time-invariant. In particular, we do not consider controllers simply achieved by the feedforward and tracking of the optimal trajectory. A relatively optimal control is a stabilizing controller such that, if initialized at its zero state, produces the optimal (constrained) trajectory for the nominal initial condition of the plant. We show that one of such controllers is linear, dead-beat, and its order is equal to the length of the horizon minus the plant order, thus, of complexity which is known a priori. Some additional features such as the assignment of the compensator poles to achieve strong stabilization are proposed. We show that, by means of the proposed approach, we can face several problems such as optimal point-to-point operations, optimal impulse response and optimal tracking.     

Riccati differential equation in optimal filtering of periodic non-stabilizable systems

This paper discusses the periodic solutions of the matrix Riccati differential equation in the optimal filtering of periodic systems. Special emphasis is given to non-stabilizable systems and the question addressed is the existence and uniqueness of a steady-state periodic non-negative definite solution of the periodic Riccati differential equation which gives rise to an asymptotically stable steady-state filter. The results presented show that the stabilizability is not a necessary condition for the existence of such a periodic solution. The convergence of the general solution of the periodic Riccati differential equation to a periodic equilibrium solution is also investigated. The results are extensions of existing time-invariant systems results to the case of periodic systems     

Two-stage estimation of time-invariant and time-varying parameters in linear systems†

A two-stage method for estimating time-invariant and time-varying parameters in linear systems is developed. The linear system is decomposed into two subsystems which have time-invariant and time-varying parameters, respectively. The unknown time-varying parameters are considered as control inputs and a linear state regulator quadratic cost function dynamic optimization problem is formulated. The solution of the associated two-point boundary-value problem for the optimum control results in an estimate for the time-varying parameters. The time-invariant parameters are estimated by a minimum mean-square error solution of a set of linear equations obtained by discretization of augmented state equations. The method is computationally simple and its effectiveness is illustrated by numerous examples.     

An LMI approach to design robust fault detection filter for uncertain LTI systems

In this paper, the robust fault detection filter design problem for uncertain linear time-invariant (LTI) systems with both unknown inputs and modelling errors is studied. The basic idea of our study is to use an optimal residual generator (assuming no modelling errors) as the reference residual model of the robust fault detection filter design for uncertain LTI systems with modelling errors and, based on it, to formulate the robust fault detection filter design as an H_∞ model-matching problem. By using some recent results of H_∞ optimization, a solution of the optimization problem is then presented via a linear matrix inequality (LMI) formulation. The main results include the development of an optimal reference residual model, the formulation of robust fault detection filter design problem, the derivation of a sufficient condition for the existence of a robust fault detection filter and a construction of it based on the LMI solution parameters, the determination of adaptive threshold for fault detection. An illustrative design example is employed to demonstrate the effectiveness of the proposed approach.     

Stability robustness bounds for discrete-time linear regulators

Stability robustness analysis and design for linear multivariable discrete-time systems with bounded uncertainties are discussed. Robust stability of the full-state feedback linear quadratic (LQ) regulator in the presence of perturbations (modelling errors) of the system matrices is investigated. These results are based on a recently developed bound on elemental (structured) time-varying perturbations of an asymptotically stable linear time-invariant discrete-time system. Lyapunov theory and singular value decomposition techniques are employed in deriving these bounds. Extensions of these results to linear stochastic systems with the Kalman filter as the stale estimator (LQG regulators) and to reduced-order dynamic compensator feedback are described. A state feedback control design method is presented for LQ regulators, using a quantitative measure called the Stability Robustness Index. Simple examples illustrate these new results.     

The optimal projection equations for fixed-order dynamic compensation

First-order necessary conditions for quadratically optimal, steady-state,fixed-order dynamic compensation of a linear, time-invariant plant in the presence of disturbance and observation noise are derived in a new and highly simplified form. In contrast to the pair of matrix Riccati equations for the full-order LQG case, the optimal steady-state fixed-order dynamic compensator is characterized by four matrix equations (two modified Riccati equations and two modified Lyapunov equations) coupled by a projection whose rank is precisely equal to the order of the compensator and which determines the optimal compensator gains. The coupling represents a graphic portrayal of the demise of the classical separation principle for the reduced-order controller case.