Parametric optimal control for uncertain linear quadratic models

In recent few decades, linear quadratic optimal control problems have achieved great improvements in theoretical and practical perspectives. For a linear quadratic optimal control problem, it is well known that the optimal feedback control is characterized by the solution of a Riccati differential equation, which cannot be solved exactly in many cases, and sometimes the optimal feedback control will be a complex time-oriented function. In this paper, we introduce a parametric optimal control problem of uncertain linear quadratic model and propose an approximation method to solve it for simplifying the expression of optimal control. A theorem is given to ensure the solvability of optimal parameter. Besides, the analytical expressions of optimal control and optimal value are derived by using the proposed approximation method. Finally, an inventory-promotion problem is dealt with to illustrate the efficiency of the results and the practicability of the model.     

Optimal control of ε-coupled and singularly perturbed distributed-parameter systems

Many distributed-parameter systems consist of interconnected subsystems involving fast and slow physical phenomena or reducing to a number of independent subsystems when a scalar parameter ε is zero. The purpose of this paper is to treat the control of such systems by invoking the ε-coupling and singular perturbation approaches developed by Kokotovic and his co-workers for lumped-parameter large-scale systems. In the case of ε- coupled distributed-parameter systems it is shown that the optimal state feedback matrix can be approximated by a Volterra-MacLaurin series with coefficients determined by solving two lower-order decoupled Riccati and linear equations. By using an mth-order approximation of the optimal feedback matrix, one obtains a (2m+1)th order approximation of the optimal performance function. In the singular perturbation approach the result is that for an O(ε²) suboptimal control one must solve two decoupled Riccati equations, one for the fast and one for the slow subsystem, and then construct appropriately the composite control law. By using only the Riccati equation for the slow subsystem, one obtains an O(ε) suboptimal control. The singular perturbation technique is then used to treat interconnected distributed-parameter systems involving may strongly coupled slow subsystems and weakly coupled fast subsystems.     

Linear-quadratic tracking of coupled slow and fast targets

The paper examines a singular perturbation model of tracking two targets, one slow and a fast one which is assumed periodic. The plant is linear and the cost is quadratic. A limit problem is displayed. It corresponds to the limit behavior of the perturbed system when the small parameter tends to zero. This limit is not of reduced order, as the fast target is periodic. Rather, an infinite-horizon optimal averaging is performed on the fast scale, and serves as an input to slow-scale optimization. This limit design is used to approximate the optimal solution of the singularly perturbed system. Bounds for the rates of the approximation are given. The optimal limit design is computed in a feedback form in the general case, and for illustrative concrete examples.Zvi Artstein is an incumbent of the Hettie H. Heineman Professorial Chair in Mathematics. The work by Vladimir Gaitsgory was done while visiting the Department of Theoretical Mathematics, The Weizmann Institute of Science, Rehovot, Israel.     

Direct singular perturbation analysis of high-gain and cheap control problems

A new and direct approach to the analysis and design of high-gain feedback and cheap control problems is developed. Transformation of such singular problems to singular perturbation models of conventional type early at the onset of analysis or design allows a direct application of the readily available singular perturbation literature. The approach permits a simple characterization of (a) multivariable root loci under a high-gain feedback and (b) asymptotic behavior of optimal closed loop poles, state and control trajectories, performance index and optimal transfer function as the control cost coefficient in the performance index goes to zero.     

Constrained feedback control of imperfectly known singularly perturbed non-linear systems

Global attractors are investigated for a class of imperfectly known, singularly perturbed, dynamic control systems. The uncertain systems are modelled as non-linear perturbations to a known non-linear idealized system and are represented by two time-scale subsystems. The two subsystems, which depend on a scalar singular perturbation parameter, represent a singularly perturbed system which has the property that the system reduces to one of lower order when the singular perturbation parameter is set to zero. It is assumed that the full-order system is subject to constraints on the control inputs. A class of constrained feedback controllers is developed which assures global uniform attraction of a compact set, containing the state origin, for all values of the singular perturbation parameter less than some threshold value.     

First-order corrections in optimal feedback control of singularly perturbed nonlinear systems

In this paper first-order correction terms, developed by using the method of matched asymptotic expansions, are incorporated in the feedback solution of a class of singularly perturbed nonlinear optimal control problems frequently encountered in aerospace applications. This improvement is based on an explicit solution of the integrals arising from the first-order matching conditions and leads to correct the initial values of the slow costate variables in the boundary layer. Consequently, a uniformly valid feedback control law, corrected to the first-order, can be synthesized. The new method is applied to an example of a constant speed minimum-time interception problem. Comparison of the zeroth- and first-order feedback control laws to the exact optimal solution demonstrates that first-order corrections greatly extend the domain of validity of the approximation obtained by singular perturbation methods.     

REDUCED‐ORDER MODELS FOR FEEDBACK STABILIZATION OF LINEAR SYSTEMS WITH A SINGULAR PERTURBATION MODEL

The problem of output feedback stabilization of linear systems based on a reduced‐order model is addressed in this paper. New reduced‐order models are proposed for the output feedback design of linear systems with a singular perturbation model. An output feedback controller with a zero steady‐state gain matrix is proposed for stabilizing this kind of system. It is shown that with the proposed controller the reduced‐order model based feedback design can guarantee the actual closed‐loop stability for the sufficiently small perturbation parameter. This approach can overcome the difficulties in the existing design method using the so‐called zeroth‐order approximation model, whose validity is highly dependent on the value of the perturbation parameter.     

The Poisson disorder problem for large intensities†

The simplest Poisson disorder problem has as its solution a stopping rule which prescribes the best time to check for a change in the parameter of a Poisson process using only observations of the process. For additive costs of early and late checks, an optimal rule is the solution of an optimal stopping problem, and it has the form : check for a change when the conditional probability of one, given the observations, exceeds some constant level. When the parameter change is from one large value to another, our previous technique for finding the optimal level breaks down because of numerical difficulties. This paper describes an application of singular perturbation theory and a renewal argument to the avoidance of those difficulties and the derivation of a perturbation approximation to the optimal stopping level.     

On performance recovery of robust dynamic surface control

Robust dynamic surface control (RDSC) is effective in alleviating the implementation difficulty of backstepping‐based multiple‐surface sliding control (MSSC) for a class of strict‐feedback nonlinear systems with mismatched uncertainties. However, the synthesis and analysis of the classical RDSC are conservative, which not only may lead to an impractical control law but also cannot fully reveal the technical nature of RDSC. This article provides a comprehensive study of a preferred RDSC law with an approximation of the signum function based on the singular perturbation theory. By formulating the control problem into a singular perturbation form, it is proven that the preferred RDSC recovers the performance of MSSC as the decrease of a filter parameter. The attractive features of the preferred RDSC revealed during the proposed synthesis and analysis include: (a) the control gain can be significantly reduced resulting in a sharp decrease of control energy and (b) the closed‐loop stability can be guaranteed by only decreasing the filter parameter. Simulation results have been shown to be consistent with the theoretical findings.     

Optimal control of a particular class of singularly perturbednonlinear discrete-time systems

This note studies a class of discrete-time nonlinear systems which depend on a small parameter. Using the singular perturbation theory in a systematic way, we give a trajectory approximation result based on the decomposition of the model into reduced and boundary layer models. This decomposition is used to analyze optimal control via maximum principle of such systems. As a result, significant order reduction of optimal control problems is achieved     

不确定离散广义系统的D稳定鲁棒控制

研究了具有圆盘区域极点约束的一类不确定离散广义系统的鲁棒控制问题.首先,研 究了控制输入项不含扰动的不确定离散广义系统,提出了广义二次D镇定的概念,基于矩阵不 等式和广义Riccati方程,给出了一种广义二次D镇定器的设计方法,所得到的结论能够实现研 究目标;然后,讨论了控制输入项含有扰动的不确定离散广义系统,在一定的假设条件下,给出 了期望状态反馈增益阵的存在条件及其解析表达式.最后,用数值示例说明所给方法的有效性 及可行性.     

Singular perturbational approximations for discrete-time balanced systems

A new zero-order singular perturbational type approximation is developed for model reduction of discrete-time linear systems. This approximation is also suitable for model reduction of systems which have fast subsystems and which are represented in the internally balanced format. Subsystem elimination as suggested by Moore for continuous-time systems does not generalize for the discrete-time case, and the singular perturbational approximation gives a particular solution to the discrete-time internally balanced model reduction problem.     

Optimal singular control for nonlinear semistabilisation

The singular optimal control problem for asymptotic stabilisation has been extensively studied in the literature. In this paper, the optimal singular control problem is extended to address a weaker version of closed-loop stability, namely, semistability, which is of paramount importance for consensus control of network dynamical systems. Three approaches are presented to address the nonlinear semistable singular control problem. Namely, a singular perturbation method is presented to construct a state-feedback singular controller that guarantees closed-loop semistability for nonlinear systems. In this approach, we show that for a non-negative cost-to-go function the minimum cost of a nonlinear semistabilising singular controller is lower than the minimum cost of a singular controller that guarantees asymptotic stability of the closed-loop system. In the second approach, we solve the nonlinear semistable singular control problem by using the cost-to-go function to cancel the singularities in the corresponding Hamilton–Jacobi–Bellman equation. For this case, we show that the minimum value of the singular performance measure is zero. Finally, we provide a framework based on the concepts of state-feedback linearisation and feedback equivalence to solve the singular control problem for semistabilisation of nonlinear dynamical systems. For this approach, we also show that the minimum value of the singular performance measure is zero. Three numerical examples are presented to demonstrate the efficacy of the proposed singular semistabilisation frameworks.     

Robustness of dynamic output feedback control for singular perturbation systems

The problem of the robustness of dynamic output feedback control for singular perturbation systems is investigated. The solution to this problem is reduced to the simultaneous design of static output feedback controllers for the fast subsystem and the so-called auxiliary system of the slow subsystem. Some conditions are proposed to ensure the robustness of the actual closed-loop system. The exact upper bound of the parasitic parameter for the controlled system is also determined. Finally, an actual model which failed in dynamic output feedback control in [4] is reexamined successfully here.     

Singular Perturbation Method for Smart Building Temperature Control Using Occupant Feedback

In this work we propose a framework for incorporating occupant feedback towards temperature control of multi‐occupant spaces, and analyze it using singular perturbation theory. Such a system would typically have to accommodate occupants with different temperature preferences, and incorporate that with thermal correlation among multiple zones to obtain optimal control for minimization of occupant discomfort and energy cost. In current practice, an acceptable temperature set point for the occupancy level of the zone is estimated, and the control law is designed to maintain temperature at the corresponding set point irrespective of the changes in occupancy and the preferences of multiple occupants. Proposed algorithm incorporates active occupant feedback to minimize aggregate user discomfort and total energy cost. Occupant binary feedback in the form of hot/cold or thermal comfort preference input is used by the control algorithm. The control algorithm also takes the energy cost into account, trading it off optimally with the aggregate occupant discomfort. For convergence to the optimal, sufficient separation between the occupant feedback frequency and the temperature dynamics of system is necessary; in absence of which, the occupant feedback provided do not correctly reflect the effect of current control input value on occupant discomfort. Under sufficient time scale separation, using singular perturbation theory, we establish the stability condition of the system and show convergence of the proposed solution to the desired temperature that minimizes energy cost plus occupant discomfort. The occupants are only assumed to be rational in that they choose their own comfort range to minimize individual thermal discomfort. Optimization for a multi‐zone building also takes into account the thermal correlation among different zones. Simulation study with parameters based on our test facility, and experimental study conducted in the same building demonstrates performance of the algorithm under different occupancy and ambient conditions.     

Optimal bang-bang control of linear stochastic systems with a small noise parameter

This study considers the problem of determining optimal feedback control laws for linear stochastic systems with amplitude-constrained control inputs. Two basic performance indices are considered, average time and average integral quadratic form. The optimization interval is random and defined as the first time a trajectory reaches the terminal regionR. The plant is modeled as a stochastic differential equation with an additive Wiener noise disturbance. The variance parameter of the Wiener noise process is assumed to be suitably small. A singular perturbation technique is presented for the solution of the stochastic optimization equations (second-order partial differential equation). A method for generating switching curves for the resulting optimal bang-bang control system is then developed. The results are applied to various problems associated with a second-order purely inertial system with additive noise at the control input. This problem is typical of satellite attitude control problems.     

Stability constraints of active magnetic bearing control systems

Direct Lyapunov stability method is employed to analyse the closed-loop stability of an electromagnetically suspended rotor system in which non-linearity and singular perturbation nature are embedded. Five explicit sets of stability constraints are proposed so that the state feedback loop is not over-designed. Under impact of modeling inaccuracy, due to neglected higher-order terms, stability conditions for equilibrium point, slow-mode manifold and boundary layer are studied. In addition, the singular perturbation order-reduction technique is used to simplify the feedback loop and reduce the order of a state feedback controller. The stability margin can be numerically evaluated as long as the upper bound of the singular perturbation parameter is available. The proposed state feedback controller is verified by intensive computer simulations such that superior performance in terms of stiffness, rise time and overshoot is illustrated, in comparison to output feedback law and deadbeat control. The reported state feedback loop not only stabilizes the inherently unstable open-loop system, but also preserves the robustness with respect to the parasitic parameter variation and modeling error due to neglected higher-order terms of magnetic control force.     

High-gain feedback in non-linear control systems†

High-gain state and output feedback are investigated for non-linear control systems with a single additive input by using singular perturbation techniques.

Classical approximation results (Tihonov-like theorems) in singular perturbation theory are extended to non-linear control systems by defining a composite additive control strategy, a control-dependent fast equilibrium manifold and non-linear change of coordinates.

Those tools and an appropriate change of coordinates show that high-gain state feedback and variable structure control systems can be equivalently used for approximate non-linearity compensation in feedback-linearizable systems.

Next the effect of high-gain output feedback is shown to be related to the strong invertibility property and the relative order of invertibility. For strongly invertible systems the slow reduced subsystem coincides with the dynamics of the inverse system when zero input is applied and with the unobservable dynamics when a certain input-output feedback-linearizable transformation is applied.     

变采样周期网络化系统鲁棒容错控制

针对网络控制系统中存在的时变采样周期与时延,通过矩阵Jordan 变换与分解, 将采样周期和时延的不确定性转变为系统结构参数的不确定性,建立了离散时间凸多面体不 确定系统模型。在此基础上,首先设计观测器,保证系统状态和故障估计收敛于实际值。接 着,根据估计的故障,设计了主动动态输出反馈鲁棒容错控制器,给出了执行器发生故障时, 系统能保持渐近稳定的充分条件。将控制器设计问题转化为以线性矩阵不等式形式为约束条 件的凸优化问题,进而得出了最优H∞鲁棒容错控制器参数的具体表达式。数值仿真验证了提 出的设计方法的有效性。     

Synthesis of Optimal Closed Systems

Synthesis of optimal feedback, feedforward, and direct and combined coupling for nonlinear dynamic systems is considered in the paper. The method proposed is based on two procedures: a piecewise-linear approximation of the initial nonlinear problem and an asymptotic correction of the solution of a piecewise-linear problem. In its turn, the piecewise linear problem solution method is based on solution of a linear problem of optimal control and optimization for a finite parameter number. The results are illustrated by examples.