Jump linear quadratic Gaussian control in continuous time

The optimal quadratic control of continuous-time linear systems that possess randomly jumping parameters which can be described by finite-state Markov processes is addressed. The systems are also subject to Gaussian input and measurement noise. The optimal solution for the jump linear-quadratic-Gaussian (JLQC) problem is given. This solution is based on a separation theorem. The optimal state estimator is sample-path dependent. If the plant parameters are constant in each value of the underlying jumping process, then the controller portion of the compensator converges to a time-invariant control law. However, the filter portion of the optimal infinite time horizon JLQC compensator is not time invariant. Thus, a suboptimal filter which does converge to a steady-state solution (under certain conditions) is derived, and a time-invariant compensator is obtained     

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.     

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.     

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.     

Invariants of optimal minimal-order observer-based compensators

The relations characterizing the optimal minimal-order observer-based compensators for a linear time-invariant multivariable system with a random initial state (or, equivalently, known initial state and white plant-driving noise) have been reported by Miller [1]. In this note we establish in general that the plant transfer function uniquely determines the optimal compensator transfer function, and that this characterizes precisely the degrees of freedom in the compensator design; computational implications of this result are indicated.     

Compensator design for multivariable linear systems

The problem of the specification of the order and structure of a linear dynamic compensator in order to obtain arbitrary pole placement in a closed-loop linear system comprised of the compensator in cascade with a linear plant is discussed. A significant application of the theory is to the design of optimal systems in those cases where not all the state variables of the plant can be measured. These results permit a completely algorithmized approach to the design of compensators for linear systems.     

Optimal state reconstruction algorithms for linear discrete-time systems

This paper deals with optimal time-invariant reconstruction of the state of a linear time-invariant discrete-time system from output measurements. The problem is analysed in two settings, depending on whether or not the present output measurement is available for the estimation of the present state. The results prove complete separation of observer and controller design for the optimal dynamic output feedback control with respect to a quadratic cost.     

Internal behaviour provided by nonlinear l1-optimal controllers

An l₁-optimal linear time-invariant (LTI) compensator may have an order significantly higher than that of the plant, even when the state is measurable. Recently there has been work exploring the use of nonlinear static feedback controllers which provide near optimal performance. Here we consider a class of nonlinear state feedback controllers and derive superposition-like bounds on both the plant state and the controlled output in the event that the plant initial condition, the disturbance, and the noise are all non-zero.     

Annihilator structure of a principal ideal : Relation to optimal compensators

Given a linear, time-invariant, discrete-time plant, we consider the optimal control problem of minimizing, by choice of a stabilizing compensator, the seminorm of a selected closed-loop map in a basic feedback system. The seminorm can be selected to reflect any chosen performance feature and must satisfy only a mild condition concerning finite impulse responses. We show that if the plant has no poles or zeros on the unit circle, then the calculation of the minimum achievable seminorm is equivalent to the maximization of a linear objective over a convex set in a low-dimensional Euclidean space. Hence, for a wide variety of optimal control problems, one can compute the answer to an infinite-dimensional optimization by a finite-dimensional procedure. This allows the use of effective numerical methods for computation.     

Pole placement using dynamic compensators

The problem of designing a compensator to obtain arbitrary pole placement in the system consisting of the plant and compensator in cascade is considered. The design uses only those state variables which can be measured. It is shown that for a controllable observable plant a compensator of orderbeta = min(nu_{c} - 1, nu_{o} - 1)is sufficient to achieve this result. Herenu_{c}(nu_{o})is the controllability (observability) index of the plant. This result is obtained by first showing that any multi-input multi-output linear time-invariant system may be made controllable (observable) from a single input (output) using only output feedback. The main result is then proved in a constructive manner which explicitly relates the compensator parameters to the coefficients of the desired characteristic polynomial.     

The generalized state space representation of the inverse of linear systems

In this paper a time-domain compensator written in generalized state space is shown to be the inverse of a square linear time-invariant plant with no transmission zeros at zero.     

On nonlinear state feedback controllers for persistent disturbance rejection

An l₁-optimal linear time-invariant (LTI) compensator may have an order significantly higher than that of the plant. In the state feedback case, there has been recent work exploring the use of nonlinear static feedback controllers which provide near optimal performance. In the case of D₁₂ square and invertible, we show that it is enough to have the nonlinearity on a particular substate, which can be used to significantly simplify the design problem.     

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.     

Disturbance-rejection problems with low-order dynamic compensatorfor linear ω-periodic discrete-time systems

In this note, necessary and sufficient conditions for the solvability of the disturbance-rejection problem with dynamic compensator (DRPDC), which has a time-varying state dimension, are given for linear ω-periodic discrete-time systems without assuming that the order of dynamic compensator is equal to that of system plant. Further, the minimal order of dynamic compensator, which is necessary for the solution of the problem, is also investigated under the assumption that the disturbance map is independent of time k     

Data-driven output-feedback fault-tolerant control for unknown dynamic systems with faults changing system dynamics

This paper studies the data-driven output-feedback fault-tolerant control (FTC) problem for unknown dynamic systems with faults changing system dynamics. In a framework of active FTC, two basic issues are addressed: the fault detection employing only the measured input–output information; the controller reconfiguration to achieve optimal output-feedback control in the presence of multiple faults. To detect faults and write the system state via the input–output data, an approach to data-driven design of a residual generator with a full-rank transformation matrix is presented. An output-feedback approximate dynamic programming method is developed to solve the optimal control problem under the condition that the unknown linear time-invariant discrete-time plant has multiple outputs. According to the above results and the proposed input–output data-based value function approximation structure of time-varying plants, a model-free output-feedback FTC scheme considering optimal performance is given. Finally, two numerical examples and a practical example of a DC motor control system are used to demonstrate the effectiveness of the proposed methods.     

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.     

Persistent disturbance rejection via static-state feedback

In contrast with ℋ_∞ and ℋ₂ control theories, the problem of persistent disturbance rejection (l¹ optimal control) leads to dynamic controllers, even when the states of the plant are available for feedback. Using viability theory, Shamma showed (1993), in a nonconstructive way, that in the state-feedback case the same performance achieved by any dynamic linear time-invariant controller can be achieved using memoryless nonlinear state feedback. In this paper we give an alternative, constructive proof of these results for discrete- and continuous-time systems. The main result of the paper shows that in both cases, the l¹ norm achieved by any stabilizing state-feedback linear dynamic controller can be also achieved using a memoryless variable structure controller     

基于动态补偿的线性系统最优干扰抑制

本文针对受外部干扰的线性时不变系统研究了基于动态补偿的最优干扰抑制问题,其中干扰信号为已知动态特性的扰动信号.首先,将原系统与扰动系统联立构成增广系统,进而转化为无扰动的标准线性二次最优问题.其次,给出了经具有适当动态阶的补偿器补偿后的闭环系统渐近稳定并且相关的Lyapunov方程正定对称解存在的条件,进一步给定的二次性能指标可写成一个与该解和闭环系统初值相关的表达式.为了得到系统的最优解,将该Lyapunov方程转化为一个双线性矩阵不等式形式,并给出了相应的路径跟踪算法以求得性能指标最小值以及补偿器参数.最后,通过数值算例说明应用本文方法可以不仅能够最小化线性二次指标,而且能够使得系统的干扰得到抑制.     

H∞ control of two-time-scale systems

H_∞ control of linear time-invariant singularly perturbed systems is considered. A sequential procedure is described to decompose the problem into slow and fast subproblems. The fast problem is solved first. Then the slow problem is solved under a constraint on the value of the compensator at infinity. A composite compensator is formed as the parallel connection of the fast compensator with the strictly proper part of the slow compensator. The asymptotic validity of the composite compensator is established.     

An adaptive controller which provides an arbitrarily good transientand steady-state response

A model reference adaptive control problem is posed. In the problem, the objective is not the usual one of forcing the error between the plant output and the reference model output asymptotically to zero, but instead, it is that of forcing this error to be less than a (arbitrarily small) prespecified constant after a (arbitrarily short) prespecified period of time, with a (arbitrarily small) prespecified upper bound on the amount of overshoot. It is shown that to achieve this goal for a stabilizable and detectable, single-input single-output linear time-invariant (LTI) plant, it is necessary and sufficient that the plant be minimum phase. Knowledge of an upper bound on the plant order, of the relative degree, and of the sign of the high-frequency gain is not required. The controller proposed consists of an LTI compensator together with a switching mechanism to adjust the compensator parameters. If an upper bound on the relative degree is available, the compensator has dynamics of order equal to this upper bound less one; otherwise, the order of the compensator is adjusted as well as its parameters