A variational inequality for measurement feedback almost-dissipative control

In this paper, an optimal measurement feedback control problem that yields an almost- (or practically) dissipative closed loop system is considered. That is, the aim is to consider optimal control problems which, when solved, yield a closed loop system which almost satisfies a dissipation inequality. The main idea is that by weakening the required dissipation inequality, a broader class of open loop systems and controllers are admissible, leading to broader application. In obtaining the main results of this paper, dynamic programming is applied to the optimal control problem of interest to derive a variational inequality that generalizes the information state based partial differential equation associated with measurement feedback nonlinear dissipative control. This variational inequality can in principle be used to derive the optimal controller. In the special case of certainty equivalence, an explicit solution of the variational inequality exists and is a functional of the solution of the corresponding optimal state feedback almost-dissipative control problem.     

一类双线性随机系统M量测不定LQ控制

研究了带有乘性噪声和受扰动观测的离散时间随机系统不定线性二次(Linear quadratic, LQ) 最优输出反馈控制问题. 对此类问题而言,二次成本函数的加权矩阵不定号,并且最优控制具有对偶效果.为在最优性和计算复杂度间 进行折衷,本文采用了一种M量测反馈控制设计方法.基于动态规划方法,将未来的测量结合到当前控制 计算当中的M量测反馈控制可以通过倒向求解一类与原系统维数相同的广义差分Riccati方程(Generalized difference Riccati equation, GDRE)得到.仿真结果 表明本文提出的算法与目前普遍采用的确定等价性方法相比具有优越性.     

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.     

The drilling problem: A stochastic modeling and control example in manufacturing

A machining economics problem is considered where feed rate selection and tool replacement policies are to be determined. A new stochastic model for tool wear, called a diffusion-threshold model, is proposed. This tool wear model allows the machining economics problem to be formulated as a stochastic optimal control problem incorporating measurement feedback of tool wear. Two types of control policies are described. One is a traditional machining economics policy and the other utilizes tool wear feedback and allows on-line decision making. The optimal policy is described for both types. An example problem based on actual data is worked out that compares the two approaches and demonstrates the utility of information feedback and on-line control.     

Separation theorem for nonlinear measurements

General solutions to the optimal stochastic control problem, or the combined estimation and control problem, are extremely difficult to compute since dynamic programming is required. However, if the system is linear, if the measurements are linear, and if the cost is quadratic, then the optimal stochastic controller is separated into 1) a filter to generate the conditional mean of the state, and 2) the optimum (linear) controller that results when all uncertainties are neglected. By altering the system configuration a new separation theorem is derived for arbitrary nonlinear measurements, discrete-time linear systems, and a quadratic cost. If a feedback loop is placed around the nonlinear measurement device (e.g., an analog-to-digital converter), then the stochastic control can be found without dynamic programming and is computed by cascading a nonlinear filter and the optimum (linear) controller. The primary advantage is the significant saving in computation. The performance of this new system configuration relative to the system without feedback depends on the nonlinearity, and it is not necessarily superior. A numerical example is presented.     

Optimal distributed Kalman filtering fusion for a linear dynamic system with cross-correlated noises

In this article, we study the distributed Kalman filtering fusion problem for a linear dynamic system with multiple sensors and cross-correlated noises. For the assumed linear dynamic system, based on the newly constructed measurements whose measurement noises are uncorrelated, we derive a distributed Kalman filtering fusion algorithm without feedback, and prove that it is an optimal distributed Kalman filtering fusion algorithm. Then, for the same linear dynamic system, also based on the newly constructed measurements, a distributed Kalman filtering fusion algorithm with feedback is proposed. A rigorous performance analysis is dedicated to the distributed fusion algorithm with feedback, which shows that the distributed fusion algorithm with feedback is also an optimal distributed Kalman filtering fusion algorithm; the P matrices are still the estimate error covariance matrices for local filters; the feedback does reduce the estimate error covariance of each local filter. Simulation results are provided to demonstrate the validity of the newly proposed fusion algorithms and the performance analysis.     

Optimal structures for steady-state adaptive optimizing control of large-scale industrial processes

Novel structures are developed and discussed for the steady-state optimizing control of large-scale industrial processes. A new class of hierarchical structures is proposed involving iterative procedures which utilize available process mathematical models and feedback information. In contrast to the majority of existing structures, the new techniques are optimal in the sense that the control produced by each structure satisfies the Kuhn-Tucker necessary optimality conditions. The structures are adaptive where the process model, which is assumed to be point-parametric, contains parameters which are updated at each iteration. Depending on the real process measurement capabilities, some of the structures incorporate output information feedback only, while others utilize both input and output measurements. These measurements are used in different ways leading to structures with local and global feedback, together with alternative coordination strategies.     

On the determination of optimal costly measurement strategies for linear stochastic systems

This paper presents the formulation of a class of optimization problems dealing with selecting, at each instant of time, one measurement provided by one out of many sensors. Each measurement has an associated measurement cost. The basic problem is then to select an optimal measurement policy, during a specified observation time interval, so that a weighted combination of “prediction accuracy” and accumulated “observation cost” is optimized. The current analysis is limited to the class of linear stochastic dynamic systems and measurement subsystems. The problem of selecting the optimal measurement strategy can be transformed into a deterministic optimal control problem. An iterative digital computer algorithm is suggested for obtaining numerical results. It is shown that the optimal measurement policy and the associated “matched” Kalman-type filter can be precomputed, i.e. specified before the measurements actually occur. Numerical results for a third-order system with two possible measurements are presented.     

Quantum quasi-Markov processes in eventum mechanics dynamics, observation, filtering and control

Quantum mechanical systems exhibit an inherently probabilistic behavior upon measurement which excludes in principle the singular case of direct observability. The theory of quantum stochastic time continuous measurements and quantum filtering was earlier developed by the author on the basis of non-Markov conditionally-independent increment models for quantum noise and quantum nondemolition observability. Here this theory is generalized to the case of demolition indirect measurements of quantum unstable systems satisfying the microcausality principle. The exposition of the theory is given in the most general algebraic setting unifying quantum and classical theories as particular cases. The reduced quantum feedback-controlled dynamics is described equivalently by linear quasi-Markov and nonlinear conditionally-Markov stochastic master equations. Using this scheme for diffusive and counting measurements to describe the stochastic evolution of the open quantum system under the continuous indirect observation and working in parallel with classical indeterministic control theory, we derive the Bellman equations for optimal feedback control of the a posteriori stochastic quantum states conditioned upon these measurements. The resulting Bellman equation for the diffusive observation is then applied to the explicitly solvable quantum linear-quadratic-Gaussian problem which emphasizes many similarities with the corresponding classical control problem.     

Output feedback gains for a linear-discrete stochastic control problem

For the finite-horizon linear-discrete quadratic stochastic control problem, the control is restricted to be a memoryless linear transformation of the measurement. The two-point boundary value problem that specifies the feedback gain matrices is derived, and an algorithm for solving it is given. An example is solved comparing the cost of the suboptimal control to the optimal control.     

Optimal compensation of nonlinearities in the presence of noise

The problem of feedback linearization for control systems containing sign-type (jump) nonlinearities is considered. Sometimes it is theoretically possible to cancel such nonlinearities by feedback. But, even in that case, such cancellation involves the use of discontinuous feedback laws, which can behave poorly in the presence of noise. We formulate and solve a problem that allows us to obtain optimal feedback compensation of nonlinearities, assuming additive noise in the measurements. Some simple examples are included to illustrate the advantages of our approach.     

A new algorithm for suboptimal stochastic control

An apparently new stochastic control algorithm, calledM-measurement-optimal feedback control, is described for discrete-time systems. This scheme incorporatesMfuture measurements into the control computations: whenMis zero,it reduces to the well-known open-loop-optimal feedback control; whenMis the actual number of measurements remaining in the problem, it becomes the truly optimal stochastic control. This new algorithm may also be used to simplify computations when the plant is nonlinear, when the controls are constrained, or when the cost is nonquadratic. Simulation results are presented which show the superiority of the new algorithm over the open-loop-optimal feedback control.     

Coherent quantum LQG control

Based on a recently developed notion of physical realizability for quantum linear stochastic systems, we formulate a quantum LQG optimal control problem for quantum linear stochastic systems where the controller itself may also be a quantum system and the plant output signal can be fully quantum. Such a control scheme is often referred to in the quantum control literature as “coherent feedback control”. It distinguishes the present work from previous works on the quantum LQG problem where measurement is performed on the plant and the measurement signals are used as the input to a fully classical controller with no quantum degrees of freedom. The difference in our formulation is the presence of additional non-linear and linear constraints on the coefficients of the sought after controller, rendering the problem as a type of constrained controller design problem. Due to the presence of these constraints, our problem is inherently computationally hard and this also distinguishes it in an important way from the standard LQG problem. We propose a numerical procedure for solving this problem based on an alternating projections algorithm and, as an initial demonstration of the feasibility of this approach, we provide fully quantum controller design examples in which numerical solutions to the problem were successfully obtained. For comparison, we also consider the case of classical linear controllers that use direct or indirect measurements, and show that there exists a fully quantum linear controller which offers an improvement in performance over the classical ones.     

Robust end-point optimizing feedback for nonlinear dynamic processes

The paper suggests two novel approaches to the synthesis of robust end-point optimizing feedback for nonlinear dynamic processes. Classically, end-point optimization is performed only for the nominal process model using optimal control methods, and the question of performance robustness to disturbances and model-plant mismatch remains unaddressed. The present contribution addresses the end-point optimization problem for nonlinear affine systems with fixed final time through robust optimal feedback methods. In the first approach, a nonlinear state feedback is derived that robustly optimizes the final process state. This solution is obtained through series expansion of the Hamilton-Jacobi-Bellman PDE with an active opponent disturbance. As reliable measurements or estimates of all states may not always be available, the second approach also robustly optimizes the process end-point, but uses output rather than state information. This direct use of measurement information is preferred since the choice of a state estimator for robust state feedback is non-trivial even when the observability issue is addressed. A linear time-variant output corrector is obtained by feedback parametrization and numerical optimization of a nonlinear H _∞ cost functional. A number of possible variations and alternatives to both approaches are also discussed. As model-plant mismatch is particularly common with chemical batch processes, the suitability of the robust optimizing feedback is demonstrated on a semi-batch reactor simulation example, where robustness to several realistic mismatches is investigated and the results are compared against those for the optimal open-loop policy and the optimal feedback designed for the nominal model.     

Stabilizing a linear system with Saturation Through optimal control

We construct a continuous feedback for a saturated system x(t)=Ax(t)+B/spl sigma/(u(t)). The feedback renders the system asymptotically stable on the whole set of states that can be driven to 0 with an open-loop control. The trajectories of the resulting closed-loop system are optimal for an auxiliary optimal control problem with a convex cost and linear dynamics. The value function for the auxiliary problem, which we show to be differentiable, serves as a Lyapunov function for the saturated system. Relating the saturated system, which is nonlinear, to an optimal control problem with linear dynamics is possible thanks to the monotone structure of saturation.     

An optimal control approach to robust tracking of linear systems

In our early work, we show that one way to solve a robust control problem of an uncertain system is to translate the robust control problem into an optimal control problem. If the system is linear, then the optimal control problem becomes a linear quadratic regulator (LQR) problem, which can be solved by solving an algebraic Riccati equation. In this article, we extend the optimal control approach to robust tracking of linear systems. We assume that the control objective is not simply to drive the state to zero but rather to track a non-zero reference signal. We assume that the reference signal to be tracked is a polynomial function of time. We first investigated the tracking problem under the conditions that all state variables are available for feedback and show that the robust tracking problem can be solved by solving an algebraic Riccati equation. Because the state feedback is not always available in practice, we also investigated the output feedback. We show that if we place the poles of the observer sufficiently left of the imaginary axis, the robust tracking problem can be solved. As in the case of the state feedback, the observer and feedback can be obtained by solving two algebraic Riccati equations.     

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.     

Integral action in the optimal control of linear systems with some inaccessible state variables†

Optimal feedback control for linear systems with the usual quadratic performance index which penalizes the state and the magnitude of the control is often difficult to apply because of the requirement of complete state measurement and lack of integral mode. A technique is presented, which introduces integral mode into the optimal control law and does not require the measurement of all of the state variables during the dynamic period. The control produced is equivalent to the original optimal Kalman feedback when external disturbance is absent, and during the period of constant disturbance the method yields a trajectory which can be made arbitrarily close to the optimal trajectory without disturbance.     

Optimal linear control systems with incomplete state measurements

Optimal control laws usually require the complete measurement of the plant state. However, in practice one often has available only a small number of measurements. A procedure is developed that leads to a dynamic feedback control law which is a function of any given set of measurements. The resulting closed-loop system is optimal for all initial states of the system in the sense of minimizing a quadratic performance index. The order of the controller depends upon the observability properties of the plant. The development is extended to time-variable problems.     

Anti-windup via constrained regulation with observers

A method is presented for the output-feedback control of discrete-time linear systems with hard constraints on state and control variables. Prior work has shown that optimal controllers for constrained systems take the form of a nonlinear feedback law acting on a set-valued state estimate. In this paper, conventional state estimation schemes are used. A nonlinear control law is derived which views the state estimation error as a disturbance. The resulting control law is then used in conjunction with the conventional observer, rather than set-valued observer, to achieve the desired constrained regulation. The significantly reduced real-time computations come at the cost of restricting the controller structure and thereby introducing possible conservatism in the achievable performance. The results are specialized to the problem of anti-windup for systems with control saturations. A “measurement governor” scheme is introduced that alters plant measurements in such a way to improve performance in the presence of controller saturations.