Optimal preview-based stable-inversion for output tracking of nonminimum-phase linear systems

In this article, a new approach to output tracking of nonminimum-phase systems is proposed. The proposed technique extends the preview-based stable-inversion method to optimally utilize finite-preview (in time) of the future desired output trajectory to find the feedforward input (called the inverse input) for achieving precision output tracking for nonminimum-phase systems. It has been shown that having a large enough preview time is critical to ensure the precision in the preview-based output tracking. The available preview time, however, can be limited due to the physical constraints, and more generally, the associated cost and/or hardware limits. Therefore, we propose obtaining the optimal preview-based inverse input by minimizing, within the preview time window, the predicted tracking error (under the preview-based inverse input) relative to the input energy. A simulation study on a piezoelectric actuator model is used to illustrate the proposed technique.     

Optimal semistable control for continuous-time linear systems

In this paper, we develop a new H₂ semistability theory for linear dynamical systems. Specifically, necessary and sufficient conditions based on the new notion of weak semiobservability for the existence of solutions to the semistable Lyapunov equation are derived. Unlike the standard H₂ optimal control problem, a complicating feature of the H₂ optimal semistable control problem is that the semistable Lyapunov equation can admit multiple solutions. We characterize all the solutions using matrix analysis tools. With this theory, we present a new framework to design H₂ optimal semistable controllers for linear coupled systems by converting the original optimal control problem into a convex optimization problem.     

H Optimal control for linear stochastic systems

The aim of the present paper is to provide an optimal solution to the H² state-feedback and output-feedback control problems for stochastic linear systems subjected both to Markov jumps and to multiplicative white noise. It is proved that in the state-feedback case the optimal solution is a static gain which is also optimal in the class of all higher-order controllers. In the output-feedback case the optimal H² controller has the same order as the given stochastic system. The realization of the optimal controllers depend on the stabilizing solutions of some appropriate systems of Riccati-type coupled equations. An effective iterative convergent algorithm to compute these stabilizing solutions is also presented. The paper gives some illustrative numerical example allowing to compare the results obtained by the proposed design approach with the ones presented in the recent control literature.     

Optimal control of continuous-time linear systems with a time-varying, random delay

We study a finite horizon optimal control problem for a class of linear systems with a randomly varying time-delay. The systems of this type may arise in embedded control applications and in certain applications in economics. The delay value is treated as an unknown variable but with known statistical properties, modelled by a Markov process with a finite number of states. The “probabilistic delay averaging” approach is employed to determine the optimal control in the form which is independent of the delay value.     

Optimal linear estimation for systems with multiple packet dropouts

This paper is concerned with the optimal linear estimation problem for linear discrete-time stochastic systems with multiple packet dropouts. Based on a packet dropout model, the optimal linear estimators including filter, predictor and smoother are developed via an innovation analysis approach. The estimators are computed recursively in terms of the solution of a Riccati difference equation of dimension equal to the order of the system state plus that of the measurement output. The steady-state estimators are also investigated. A sufficient condition for the convergence of the optimal linear estimators is given. Simulation results show the effectiveness of the proposed optimal linear estimators.     

Optimal linear estimation for networked systems with communication constraints

This paper is concerned with the optimal linear estimation problem for discrete time-varying networked systems with communication constraints. The communication constraint considered is that only one network node is allowed to gain access to a shared communication channel, then the various network nodes of the networked systems are scheduled to transmit data according to a specified media access control protocol, and a remote estimator performs the estimation task with only partially available measurements. The channel accessing processes of those network nodes are modeled by Bernoulli processes, and optimal linear filters are designed by using the orthogonal projection principle and the innovation analysis approach. It is shown that the optimal estimation performances critically depend on the channel accessing probabilities of the network nodes and the packet loss probability, and the optimal filters can be obtained by solving recursive Lyapunov and Riccati equations. An illustrative example is finally given to show the effectiveness of the proposed filters.     

Optimal design of linear control systems by an interactive optimization method

When designing linear control systems, one of the most difficult problems is that the designer almost has no theoretical basis for the determination of proper parameters in order to obtain a system with desired specifications. Poles and directions of eigenvectors in the pole assignment method or weighting matrices of the quadratic criterion function in the optimal regulator method are such parameters. The designer has to determine them by trial-and-error using computer simulation. The purpose of this paper is to propose an approach to helping determine proper parameters in linear control system design by the state space methods. In the case where the desired specifications are not given explicitly, the approach applies an interactive optimization method called the Interactive Simplex method to search the most suitable parameters directly in the parameter space. But, if the specifications are given explicitly, the design problem can be formulated as a multiobjective optimization problem. In this case, weights which indicate relative importance of different specifications are introduced and the Interactive Simplex method is applied in the weight space to indirectly find the most appropriate parameters. The approach is implemented as part of a CAD system. The designer has only to make pairwise comparisons of response curves which are shown on a graphics display terminal in order to obtain the most preferred control system. Two illustrative examples are demonstrated to indicate the efficiency of the approach.     

Optimal distributed linear averaging

A new optimal distributed linear averaging (ODLA) problem is presented in this paper. This problem is motivated by the distributed averaging problem which arises in the context of distributed algorithms in computer science and coordination of groups of autonomous agents in engineering. The aim of the ODLA problem is to compute the average of the initial values at nodes of a graph through an optimal distributed algorithm in which the nodes in the graph can only communicate with their neighbors. Optimality is given by a minimization problem of a quadratic cost functional under infinite horizon. We show that this problem has a very close relationship with the notion of semistability. By developing new necessary and sufficient conditions for semistability of linear discrete-time systems, we convert the proposed ODLA problem into an equivalent, constrained optimization problem and then derive a solvable, fixed-structure convex optimization problem.     

一种新的混沌系统线性反馈最优控制

为了实现一种新的混沌系统的镇定控制且满足给定的性能指标达到极小值。描述了一种新的混沌系统的动力学行为;设计一个二次目标函数,同时根据Pontryagin极小值原理为混沌系统设计一个线性状态反馈最优控制器,所得到的控制律是最优控制,控制器能将混沌系统的轨道从任意初始点出发稳定到零点,控制器结构简单且易于实现。基于Lyapunov稳定性理论证明了系统的稳定性,同时通过对施加闭环控制后的系统状态变化的数值模拟证明了所建议的方法的正确性。     

Optimal control of distributed bilinear systems

The optimal control problem for a bilinear distributed parameter system subject to a quadratic cost functional is solved. It is shown that the optimal control is given by a convergent power series in the state with tensor coefficients.     

轿车动力总成主动隔振最优控制研究

介绍压电作动器与被动悬置串联组成的轿车动力总成主动悬置系统,分析了隔振系统的数学模型,采用最优控制原理提出了实施方法。     

Optimal estimation of linear discrete-time systems with stochastic parameters

This paper considers optimal linear state estimation in the general case of linear discrete-time systems with stochastic parameters which are statistically independent with respect to time. The estimator is derived by transforming the system to one with deterministic parameters and state dependent additive system and observation noise. It is shown that mean square stability of the system is a sufficient and almost necessary condition for the existence, uniqueness and stability of the time invariant estimator.     

Output feedback controller design for uncertain piecewise linear systems

This paper proposes output feedback controller design methods for uncertain piecewise linear systems based on piecewise quadratic Lyapunov function. The α-stability of closed-loop systems is also considered. It is shown that the output feedback controller design procedure of uncertain piecewise linear systems with α-stability constraint can be cast as solving a set of bilinear matrix inequalities （BMIs）. The BMIs problem in this paper can be solved iteratively as a set of two convex optimization problems involving linear matrix inequalities （LMIs） which can be solved numerically efficiently. A numerical example shows the effectiveness of the proposed methods.     

Optimal control for linear systems with state equality constraints

This paper deals with the optimal control problem for linear systems with linear state equality constraints. For deterministic linear systems, first we find various existence conditions for constraining state feedback control and determine all constraining feedback gains, from which the optimal feedback gain is derived by reducing the dimension of the control input space. For systems with stochastic noises, it is shown that the same gain used for constraining the deterministic system also optimally constrains the expectation of states inside the constraint subspace and minimizes the expectation of the squared constraint error. We compare and discuss performance differences between unconstrained (using penalty method), projected, and constrained controllers for both deterministic and stochastic systems. Finally, numerical examples are used to demonstrate the performance difference of the three controllers.     

Optimal controlled variables for polynomial systems

We present a method for finding optimal controlled variables, which are polynomial combinations of measurements. Controlling these variables gives optimal steady state operation. Our work extends the concept of self-optimizing control; starting from the first-order necessary optimality conditions, any unknown variables are eliminated using elimination theory for polynomial systems to obtain invariant variable combinations, which contain only known variables (measurements). If a disturbance causes the active constraints to change, the invariants may be used to identify, and switch to the right region. This makes the method applicable over a wide disturbance range with changing active sets. The procedure is applied to two case studies of continuous stirred tank reactors.     

Optimal control of switching systems

This paper considers an optimal control problem for a switching system. For solving this problem we do not make any assumptions about the number of switches nor about the mode sequence, they are determined by the solution of the problem. The switching system is embedded into a larger family of systems and the optimization problem is formulated for the latter. It is shown that the set of trajectories of the switching system is dense in the set of trajectories of the embedded system. The relationship between the two sets of trajectories (1) motivates the shift of focus from the original problem to the more general one and (2) underlies the engineering relevance of the study of the second problem. Sufficient and necessary conditions for optimality are formulated for the second optimization problem. If they exist, bang-bang-type solutions of the embedded optimal control problem are solutions of the original problem. Otherwise, suboptimal solutions are obtained via the Chattering Lemma.     

Adaptive dynamic programming for linear impulse systems

We investigate the optimization of linear impulse systems with the reinforcement learning based adaptive dynamic programming （ADP） method. For linear impulse systems, the optimal objective function is shown to be a quadric form of the pre-impulse states. The ADP method provides solutions that iteratively converge to the optimal objective function. If an initial guess of the pre-impulse objective function is selected as a quadratic form of the pre-impulse states, the objective function iteratively converges to the optimal one through ADP. Though direct use of the quadratic objective function of the states within the ADP method is theoretically possible, the numerical singularity problem may occur due to the matrix inversion therein when the system dimensionality increases. A neural network based ADP method can circumvent this problem. A neural network with polynomial activation functions is selected to approximate the pr~impulse objective function and trained iteratively using the ADP method to achieve optimal control. After a successful training, optimal impulse control can be derived. Simulations are presented for illustrative purposes.     

Optimal control of stable weakly regular linear systems

The paper extends quadratic optimal control theory to weakly regular linear systems, a rather broad class of infinite-dimensional systems with unbounded control and observation operators. We assume that the system is stable (in a sense to be defined) and that the associated Popov function is bounded from below. We study the properties of the optimally controlled system, of the optimal cost operatorX, and the various Riccati equations which are satisfied byX. We introduce the concept of an optimal state feedback operator, which is an observation operator for the open-loop system, and which produces the optimal feedback system when its output is connected to the input of the system. We show that if the spectral factors of the Popov function are regular, then a (unique) optimal state feedback operator exists, and we give its formula in terms ofX. Most of the formulas are quite reminiscent of the classical formulas from the finite-dimensional theory. However, an unexpected factor appears both in the formula of the optimal state feedback operator as well as in the main Riccati equation. We apply our theory to an extensive example.Part of the results reported here were obtained while the second author was visiting FUNDP Namur, under the Belgian Program on Inter-University Poles of Attraction initiated by the Belgian state, Prime Minister's Office, Science Policy Programming. The scientific responsibility is assumed by the authors.     

Optimal complementary control for positive stable LTI systems

In this paper, positive stable MIMO LTI systems under output feedback control are studied. In particular, the results of the paper focus on the solution and implementation of output feedback optimal complementary control strategies for the case of constant tracking signals and rejection of constant unmeasurable disturbance signals under positive stable MIMO LTI systems.     

Adaptive learning control of linear systems by output error feedback

This paper addresses the problem of designing an output error feedback tracking control for single-input, single-output uncertain linear systems when the reference output signal is smooth and periodic with known period T. The considered systems are required to be observable, minimum phase, with known relative degree and known high frequency gain sign. By developing in Fourier series expansion a suitable unknown periodic input reference signal, an output error feedback adaptive learning control is designed which ‘learns’ the input reference signal by identifying its Fourier coefficients: bounded closed-loop signals and global exponential tracking of both the input and the output reference signals are obtained when the Fourier series expansion is finite, while global exponential convergence of the input and output tracking errors into arbitrarily small residual sets is achieved otherwise. The structure of the proposed controller depends only on the relative degree, the reference signal period, the high frequency gain sign and the number of estimated Fourier coefficients.