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.     

The piecewise optimisation method for approximating uncertain optimal control problems under optimistic value criterion

In this paper, we introduce an approximate model and propose a piecewise optimisation method to simplify the expression of optimal control for an uncertain linear quadratic optimal control problem. First, we consider an optimal control problem of uncertain linear quadratic model under optimistic value criterion. Based on the equation of optimality, we deduce an analytic expression of optimal control. Then, we study an approximate model with control parameter and propose a piecewise optimisation method for solving the optimal parameter of such an approximate model. As an application, a four-wheel steering vehicle optimal control problem is given to show the utility of the proposed approximate model and the efficiency of the proposed piecewise optimisation method.     

Output feedback optimal guaranteed cost control of uncertain piecewise linear systems

This paper proposes the output feedback optimal guaranteed cost controller design method for uncertain piecewise linear systems based on the piecewise quadratic Lyapunov functions technique. By constructing piecewise quadratic Lyapunov functions for the closed‐loop augmented systems, the existence of the guaranteed cost controller for closed‐loop uncertain piecewise linear systems is cast as the feasibility of a set of bilinear matrix inequalities (BMIs). Some of the variables in BMIs are set to be searched by genetic algorithm (GA), then for a given chromosome corresponding to the variables in BMIs, the BMIs turn to be linear matrix inequalities (LMIs), and the corresponding non‐convex optimization problem, which minimizes the upper bound on cost function, reduces to a semidefinite programming (SDP) which is convex and can be solved numerically efficiently with the available software. Thus, the output feedback optimal guaranteed cost controller can be obtained by solving the non‐convex optimization problem using the mixed algorithm that combines GA and SDP. Numerical examples show the effectiveness of the proposed method. Copyright © 2008 John Wiley & Sons, Ltd.     

二次型最优控制问题中的权矩阵与最优控制律

二次型最优的权矩阵选择是一个包含大量经验与技巧的问题．对此，研究权矩阵与最优控制律的关系：一个最优控制律，其对应的性能指标（权矩阵）是否唯一的问题．不论是单输入还是多输入情形，在系统能控性指数大于2，对应权矩阵的对角线元素有2个不为零的元素，且满足一定条件时．该性能指标对应的最优控制律，必有另一性能指标与之对应．     

A team algorithm for robust stability analysis and control design of uncertain time‐varying linear systems using piecewise quadratic Lyapunov functions

A team algorithm based on piecewise quadratic simultaneous Lyapunov functions for robust stability analysis and control design of uncertain time‐varying linear systems is introduced. The objective is to use robust stability criteria that are less conservative than the usual quadratic stability criterion. The use of piecewise quadratic Lyapunov functions leads to a non‐convex optimization problem, which is decomposed into a convex subproblem in a selected subset of decision variables, and a lower‐dimensional non‐convex subproblem in the remaining decision variables. A team algorithm that combines genetic algorithms (GA) for the non‐convex subproblem and interior‐point methods for the solution of linear matrix inequalities (LMI), which form the convex subproblem, is proposed. Numerical examples are given, showing the advantages of the proposed method. Copyright © 2001 John Wiley & Sons, Ltd.     

Model predictive control for constrained uncertain piecewise linear systems

For constrained piecewise linear (PWL) systems, the possible existing model uncertainty will bring the difficulties to the design approaches of model predictive control (MPC) based on mixed integer programming (MIP). This paper combines the robust method and hybrid method to design the MPC for PWL systems with structured uncertainty. For the proposed approach, as the system model is known at current time, a free control move is optimized to be the current control input. Meanwhile, the MPC controller uses a sequence of feedback control laws as the future control actions, where each feedback control law in the sequence corresponds to each partitions and the arbitrary switching technique is adopted to tackle all the possible switching. Furthermore, to reduce the online computational burden of MPC, the segmented design procedure is suggested by utilizing the characteristics of the proposed approach. Then, an offline design algorithm is proposed, and the reserved degree of freedom can be online used to optimize the control input with lower computational burden.     

分段线性系统最优控制设计的一种混合算法

将分段线性系统的最优控制设计问题转化成以反馈增益为寻优参数，以最优控制性能上界为目标的一组双线性矩阵不等式(BMI)问题．将遗传算法与内点法相结合设计出一种混合算法，对BMI问题进行求解．算例仿真表明该算法是简便而有效的．     

On the robustness of jump linear quadratic control

This paper deals with the problem of how to render the jump linear quadratic (JLQ) control robust. Mainly, we present sufficient conditions for quadratic stabilization and guaranteed cost control of uncertain jump linear system using state feedback control. The proposed control law contains two components. The first one is a JLQ control law, while the second is a nonlinear bounded term to render the system robust and whose cost is not included in the performance index. © 1997 by John Wiley & Sons, Ltd.     

Discrete-time decentralized linear quadratic control for linear time-varying systems

This article addresses the problem of designing a decentralized control solution for a network of agents modeled by linear time-varying (LTV) dynamics, in a discrete-time framework. A general scheme is proposed, in which the problem is formulated as a classical linear quadratic regulator problem, for the global system, subject to a given sparsity constraint on the gain, which reflects the decentralized nature of the network. A method able to compute a sequence of well-performing stabilizing regulator gains is presented and validated resorting to simulations of two randomly generated LTV systems, one stable and the other unstable. Moreover, a tracking solution is developed, building on the solution to the regulator problem. Both methods rely on a closed-form solution, thus they can be computed very rapidly. Similarly to the centralized solution, both the presented methods require that a window of the future system dynamics is known. Both methods are validated resorting to simulations of: (i) a nonlinear network of four interconnected tanks; and (ii) a large-scale nonlinear network of interconnected tanks. When implemented to a nonlinear network, approximated by an LTV system, the proposed methods are able to compute well-performing gains that track the desired output. Finally, both algorithms are scalable, being adequate for implementation in large-scale networks.     

A generalised optimal linear quadratic tracker with universal applications. Part 2: discrete-time systems

Contrastive to Part 1, Part 2 presents a generalised optimal linear quadratic digital tracker (LQDT) with universal applications for the discrete-time (DT) systems. This includes (1) a generalised optimal LQDT design for the system with the pre-specified trajectories of the output and the control input and additionally with both the input-to-output direct-feedthrough term and known/estimated system disturbances or extra input/output signals; (2) a new optimal filter-shaped proportional plus integral state-feedback LQDT design for non-square non-minimum phase DT systems to achieve a minimum-phase-like tracking performance; (3) a new approach for computing the control zeros of the given non-square DT systems; and (4) a one-learning-epoch input-constrained iterative learning LQDT design for the repetitive DT systems.     

Progressive accommodation of parametric faults in linear quadratic control

In this paper, a strategy based on the linear quadratic design, which progressively accommodates the feedback control law, is proposed. It significantly reduces the loss of performance that results from the time delay needed by fault accommodation algorithms to provide a solution. An aircraft example is given to illustrate the efficiency of progressive accommodation.     

Designing parametric linear quadratic regulators for parametric LTI systems via LMIs

This paper addresses the problem of determining parametric linear quadratic regulators (LQRs) for continuous-time linear-time invariant systems affected by parameters through rational functions. Three situations are considered, where the sought controller has to minimise the best cost, average cost, and worst cost, respectively, over the set of admissible parameters. It is shown that candidates for such controllers can be obtained by solving convex optimisation problems with linear matrix inequality (LMI) constraints. These candidates are guaranteed to approximate arbitrarily well the sought controllers by sufficiently increasing the size of the LMIs. In particular, the candidate that minimises the average cost approximates arbitrarily well the true LQR over the set of admissible parameters. Moreover, conditions for establishing the optimality of the found candidates are provided. Some numerical examples illustrate the proposed methodology.     

Robust control for linear uncertain systems via linear quadratic state feedback

By the Lyapunov stability criterion and the algebraic Riccati equation, conditions of selecting the weighting matrices in the quadratic cost function are derived so that linear quadratic state feedback can exponentially stabilize a linear uncertain system, provided the uncertainties satisfy the so-called matching conditions and within a given bounding set. Furthermore, two simple but effective algorithms are proposed for systematically selecting the weighting matrices. The main features of this approach are that the uncertain system can be exponentially stabilized with prescribed exponential rate and no precompensator is needed. Two examples are given to illustrate the results.     

Modeling and quadratic stabilization of a class of linear uncertain time‐varying systems

This paper considers the quadratic stabilization of a class of uncertain linear time‐varying (LTV) continuous‐time plants. The state‐space representation of each plant is based on the physically meaningful assumption of a dynamical matrix containing uncertain elements whose time trajectories are sufficiently smooth to be well described by interval polynomial functions with arbitrarily time varying coefficients. At some isolated time instants, the parameters trajectories can exhibit some first‐kind discontinuities due for example to sharply varying operating conditions. Using a parameter independent Lyapunov function, a quadratically stabilizing dynamic output controller is directly obtained by the solution of some LMIs. A salient feature of the paper is that, unlike all the other existing methods, quadratic stabilization can be achieved over possibly arbitrarily large uncertain domains of parameters. Copyright © 2016 John Wiley & Sons, Ltd.     

A Riccati approach for constrained linear quadratic optimal control

An active-set method is proposed for solving linear quadratic optimal control problems subject to general linear inequality path constraints including mixed state-control and state-only constraints. A Riccati-based approach is developed for efficiently solving the equality constrained optimal control subproblems generated during the procedure. The solution of each subproblem requires computations that scale linearly with the horizon length. The algorithm is illustrated with numerical examples.     

An example in linear quadratic optimal control

We construct a simple example of a quadratic optimal control problem for an infinite-dimensional linear system based on a shift semigroup. This system has an unbounded control operator. The cost is quadratic in the input and the state, and the weighting operators are bounded. Despite its extreme simplicity, this example has all the unexpected features discovered recently by O. Staffans (and also by M. Weiss and G. Weiss). More precisely, in the formula linking the optimal feedback operator to the optimal cost operator, as well as in the Riccati equation, the weighting operator of the input has to be replaced by another operator, which can be derived from the spectral factorization of the Popov function.     

Solvability of indefinite stochastic Riccati equations and linear quadratic optimal control problems

A new approach to study the indefinite stochastic linear quadratic (LQ) optimal control problems, which we called the “equivalent cost functional method”, is introduced by Yu (2013) in the setup of Hamiltonian system. On the other hand, another important issue along this research direction, is the possible state feedback representation of optimal control and the solvability of associated indefinite stochastic Riccati equations. As the response, this paper continues to develop the equivalent cost functional method by extending it to the Riccati equation setup. Our analysis is featured by its introduction of some equivalent cost functionals which enable us to have the bridge between the indefinite and positive-definite stochastic LQ problems. With such bridge, some solvability relation between the indefinite and positive-definite Riccati equations is further characterized. It is remarkable the solvability of the former is rather complicated than the latter, hence our relation provides some alternative but useful viewpoint. Consequently, the corresponding indefinite linear quadratic problem is discussed for which the unique optimal control is derived in terms of state feedback via the solution of the Riccati equation. In addition, some example is studied using our theoretical results.     

A new PI-based optimal linear quadratic state-estimate tracker for discrete-time non-square non-minimum phase systems

A new proportional–integral (PI)-based optimal linear quadratic state-estimate tracker, derived using a proportional–integral–derivative (PID) filter-based frequency-domain shaping approach, is proposed in this paper for discrete-time non-square non-minimum phase multi-input-multi-output systems. Subsequently, a new integrated PID filter-shaped optimal PI state estimator is presented for the aforementioned systems, so that both the proposed state estimator and the state-estimate tracker are able to achieve satisfactory minimum phase-like tracking performance, for the case of arbitrary command inputs with significant variations at some isolated time instants.     

A new PI optimal linear quadratic state-estimate tracker for continuous-time non-square non-minimum phase systems

Based on the proportional-integral-derivative (PID) filter-shaping approach, this paper presents a new proportional-plus-integral (PI) optimal linear quadratic state estimator (LQSE) for the continuous-time non-square and non-minimum phase (NMP) multivariable systems. Together with the recently developed optimal linear quadratic tracker (LQT), the proposed LQSE-based tracker is able to optimally achieve good minimum phase-like tracking performances for a non-square NMP multivariable system with unmeasurable states and arbitrary command inputs.