Regional stabilisation of polynomial non-linear systems using rational Lyapunov functions

In this article, we propose a new non-linear stabilisation approach based on the popular linear parameter-varying control techniques. The regional state-feedback control problem of polynomial non-linear systems will be studied using rational Lyapunov functions of states. By bounding the variation rates of each state, the domain of attraction will be embedded in the region specified by the non-linear vector field. As a result, the state-feedback stabilisation conditions will be formulated as a set of polynomial matrix inequalities and can be solved efficiently by sum-of-squares programming. The resulting Lyapunov matrix and state-feedback gains are typically state-dependent rational matrix functions. This approach is also extended to a class of output-dependent non-linear systems where the stabilising output-feedback controller can be synthesised using rational Lyapunov functions of outputs. Finally, several examples will be used to demonstrate the proposed stabilisation approach and clarify the effect of various choices of Lyapunov function forms and state constraints.     

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.     

基于动态规划的约束优化问题多参数规划求解方法及应用

结合动态规划和单步多参数二次规划, 提出一种新的约束优化控制问题多参数规划求解方法. 一方面能得到约束线性二次优化控制问题最优控制序列与状态之间的显式函数关系, 减少多参数规划问题求解的工作量; 另一方面能够同时求解得到状态反馈最优控制律. 应用本文提出的多参数二次规划求解方法, 建立无限时间约束优化问题状态反馈显式最优控制律. 针对电梯机械系统振动控制模型做了数值仿真计算.     

Infinite horizon optimal control of affine nonlinear discrete switched systems using two-stage approximate dynamic programming

In this article, a novel iteration algorithm named two-stage approximate dynamic programming (TSADP) is proposed to seek the solution of nonlinear switched optimal control problem. At each iteration of TSADP, a multivariate optimal control problem is transformed to be a certain number of univariate optimal control problems. It is shown that the value function at each iteration can be characterised pointwisely by a set of smooth functions recursively obtained from TSADP, and the associated control policy, continuous control and switching control law included, is explicitly provided in a state-feedback form. Moreover, the convergence and optimality of TSADP is strictly proven. To implement this algorithm efficiently, neural networks, critic and action networks, are utilised to approximate the value function and continuous control law, respectively. Thus, the value function is expressed by the weights of critic networks pointwise. Besides, redundant weights are ruled out at each iteration to simplify the exponentially increasing computation burden. Finally, a simulation example is provided to demonstrate its effectiveness.     

Optimal control of discrete-time linear fractional-order systems with multiplicative noise

A finite horizon linear quadratic (LQ) optimal control problem is studied for a class of discrete-time linear fractional systems (LFSs) affected by multiplicative, independent random perturbations. Based on the dynamic programming technique, two methods are proposed for solving this problem. The first one seems to be new and uses a linear, expanded-state model of the LFS. The LQ optimal control problem reduces to a similar one for stochastic linear systems and the solution is obtained by solving Riccati equations. The second method appeals to the principle of optimality and provides an algorithm for the computation of the optimal control and cost by using directly the fractional system. As expected, in both cases, the optimal control is a linear function in the state and can be computed by a computer program. A numerical example and comparative simulations of the optimal trajectory prove the effectiveness of the two methods. Some other simulations are obtained for different values of the fractional order.     

一类离散线性切换系统的最优控制

研究了一类离散线性切换系统在切换时间、切换次数固定的情况下的二次最优控制问题.利用离散动态规划的方法,将多级决策过程分解成一系列易于求解的单级决策过程,求出最优控制序列和最优切换序列,并给出算法.最后通过一个数值例子来说明所提出的方法的有效性.     

Infinite horizon indefinite stochastic linear quadratic control for discrete-time systems

This paper discusses discrete-time stochastic linear quadratic (LQ) problem in the infinite horizon with state and control dependent noise, where the weighting matrices in the cost function are assumed to be indefinite. The problem gives rise to a generalized algebraic Riccati equation (GARE) that involves equality and inequality constraints. The well-posedness of the indefinite LQ problem is shown to be equivalent to the feasibility of a linear matrix inequality (LMI). Moreover, the existence of a stabilizing solution to the GARE is equivalent to the attainability of the LQ problem. All the optimal controls are obtained in terms of the solution to the GARE. Finally, we give an LMI -based approach to solve the GARE via a semidefinite programming.     

Optimal LQG control across packet-dropping links

We examine two special cases of the problem of optimal linear quadratic Gaussian control of a system whose state is being measured by sensors that communicate with the controller over packet-dropping links. We pose the problem as an information transmission problem. Using a separation principle, we decompose the problem into a standard LQR state-feedback controller design, along with an optimal encoder–decoder design for propagating and using the information across the unreliable links. Our design is optimal among all causal algorithms for any arbitrary packet-drop pattern. Further, the solution is appealing from a practical point of view because it can be implemented as a small modification of an existing LQG control design.     

A new approach on designing l_1 optimal regulator with minimum order for SISO linear systems

For a SISO linear discrete-time system with a specified input signal, a novel method to realize optimal l1 regulation control is presented. Utilizing the technique of converting a polynomial equation to its corresponding matrix equation, a linear programming problem to get an optimal l1 norm of the system output error map is developed which includes the first term and the last term of the map sequence in the objective function and the right vector of its constraint matrix equation, respectively. The adjustability for the width of the constraint matrix makes the trade-off between the order of the optimal regulator and the value of the minimum objective norm become possible, especially for achieving the optimal regulator with minimum order. By norm scaling rules for the constraint matrix equation, the optimal solution can be scaled directly or be obtained by solving a linear programming problem with l\ norm objective.     

A new approach on designing l1 optimal regulator with minimum order for SISO linear systems

For a SISO linear discrete-time system with a specified input signal, a novel method to realize optimal l1 regulation control is presented. Utilizing the technique of converting a polynomial equation to its corresponding matrix equation, a linear programming problem to get an optimal l1 norm of the system output error map is developed which includes the first term and the last term of the map sequence in the objective function and the right vector of its constraint matrix equation, respectively. The adjustability for the width of the constraint matrix makes the trade-off between the order of the optimal regulator and the value of the minimum objective norm become possible, especially for achieving the optimal regulator with minimum order. By norm scaling rules for the constraint matrix equation, the optimal solution can be scaled directly or be obtained by solving a linear programming problem with l\ norm objective.     

Inf–sup control of discontinuous piecewise affine systems

This paper considers the worst‐case optimal control of discontinuous piecewise affine (PWA) systems, which are subjected to constraints and disturbances. We seek to pre‐compute, via dynamic programming, an explicit control law for these systems when a PWA cost function is utilized. One difficulty with this problem class is that, even for initial states for which the value function of the optimal control problem is finite, there might not exist a control law that attains the infimum. Hence, we propose a method that is guaranteed to obtain a sub‐optimal solution, and where the degree of sub‐optimality can be specified a priori. This is achieved by approximating the underlying sub‐problems with a parametric piecewise linear program. Copyright © 2008 John Wiley & Sons, Ltd.     

Constrained control for uncertain discrete-time linear systems

In this paper the problem of stabilizing uncertain linear discrete-time systems under state and control linear constraints is studied. Many formulations of this problem have been given in the literature. Here we consider the case of finding a linear state feedback control law making a given polytope in the state space positively invariant while the control remains bounded within prefixed values under the effect of all the uncertainty sequences belonging to a given polytope in the perturbations space. A necessary and sufficient condition for the existence of a solution of this problem is first given. This condition leads to a set of linear constraints which can be solved using linear programming tecniques by defining an appropriate objective function. A worked example shows the effectiveness of the proposed algorithm. © 1998 John Wiley & Sons, Ltd.     

带ε误差限的近似最优控制

近似动态规划方法求解非线性系统最优控制, 需要迭代无限步才能得到最优控制律. 本文提出了一种ε–近似最优控制算法, 选择ε误差限, 通过自适应迭代不断逼近哈密顿– 雅可比– 贝尔曼(HJB)方程的解, 应用神经网络实现在有限步迭代后得到带ε误差限的近似最优控制律. 计算机仿真结果表明了该算法的有效性.     

结合分支定界法和线性规划的摄像机位姿估计

介绍了一种新的利用对应点估计摄像机位姿的算法。通常情况下,摄像机位姿估计可以转化为一个最优化问题,现有算法将问题转换成一个序列二阶锥规划问题,通过对旋转矩阵所在空间进行分支定界搜索来求取全局最优解。对现有算法进行改进,通过将二阶锥约束松弛为线性约束,提出了一种结合分支定界法和线性规划方法的全局优化算法。该算法不仅能够求得全局最优解,而且算法速度较现有算法提高了一倍以上。最后通过模拟数据和真实数据对该算法进行了验证,结果表明了该算法的准确性和高效性。     

线性参变过驱动系统鲁棒控制分配策略

针对一类具有不确定时变参量的线性参变(linear parameter-varying,LPV)过驱动系统的控制分配问题,考虑系统的不确定参量扰动和执行器物理约束,利用伪控指令分配误差和控制量误差的1--范数,建立了含有时变不确定因子的控制分配优化模型.根据鲁棒优化思想,采用矢量变换技术处理时变不确定因子,得到了一种基于有约束锥二次凸优化模型的鲁棒控制分配算法,实现对LPV过驱动系统伪控指令的在线优化分配.最后,对某4轮电动汽车时变二自由度转向过驱动控制系统的对比仿真实验表明,相比常规4WS和伪逆控制分配方法,本文的鲁棒控制分配算法有效地降低了系统参变量不确定扰动的影响,得到更合理的控制分配解,有效改善了车辆的操纵稳定性.     

Adaptive differential dynamic programming for multiobjective optimal control

An efficient numerical solution scheme entitled adaptive differential dynamic programming is developed in this paper for multiobjective optimal control problems with a general separable structure. For a multiobjective control problem with a general separable structure, the “optimal” weighting coefficients for various performance indices are time-varying as the system evolves along any noninferior trajectory. Recognizing this prominent feature in multiobjective control, the proposed adaptive differential dynamic programming methodology combines a search process to identify an optimal time-varying weighting sequence with the solution concept in the conventional differential dynamic programming. Convergence of the proposed adaptive differential dynamic programming methodology is addressed.     

An efficient algorithm for optimal control of PWA systems with polyhedral performance indices

We present an algorithm for the computation of explicit optimal control laws for piecewise affine (PWA) systems with polyhedral performance indices. The algorithm is based on dynamic programming (DP) and represents an extension of ideas initially proposed in Kerrigan and Mayne [(2003). Optimal control of constrained, piecewise affine systems with bounded disturbances. In Proceedings of the 41st IEEE conference on decision and control, Las Vegas, Nevada, USA, December], and Baoti? et al. [(2003). A new algorithm for constrained finite time optimal control of hybrid systems with a linear performance index. In Proceedings of European control conference, Cambridge, UK, September]. Specifically, we show how to exploit the underlying geometric structure of the optimization problem in order to significantly improve the efficiency of the off-line computations. An extensive case study is provided, which clearly indicates that the algorithm proposed in this paper may be preferable to other schemes published in the literature.     

Optimal control of investments for quality of supply improvement in electrical energy distribution networks

This paper considers the problem of deciding multi-period investments for maintenance and upgrade of electrical energy distribution networks. After describing the network as a constrained hybrid dynamical system, optimal control theory is applied to optimize profit under a complex incentive/penalty mechanism imposed by public authorities. The dynamics of the system and the cost function are translated into a mixed integer optimization model, whose solution gives the optimal investment policy over the multi-period horizon. While for a reduced-size test problem the pure mixed-integer approach provides the best optimal control policy, for real-life large-scale scenarios a heuristic solution is also introduced. Finally, the uncertainty associated with the dynamical model of the network is taken care of by adopting ideas from stochastic programming.     

Control applications of nonlinear convex programming

Since 1984 there has been a concentrated effort to develop efficient interior-point methods for linear programming (LP). In the last few years researchers have begun to appreciate a very important property of these interior-point methods (beyond their efficiency for LP): they extend gracefully to nonlinear convex optimization problems. New interior-point algorithms for problem classes such as semidefinite programming (SDP) or second-order cone programming (SOCP) are now approaching the extreme efficiency of modern linear programming codes. In this paper we discuss three examples of areas of control where our ability to efficiently solve nonlinear convex optimization problems opens up new applications. In the first example we show how SOCP can be used to solve robust open-loop optimal control problems. In the second example, we show how SOCP can be used to simultaneously design the set-point and feedback gains for a controller, and compare this method with the more standard approach. Our final application concerns analysis and synthesis via linear matrix inequalities and SDP.     

参数不确定线性系统混合H2/H∞状态反馈控制

对一类含有范数有界参数不确定线性系统的混合H2/H∞状态反馈控制问题进行了研究.给出了系统一个输出在满足给定的H∞干扰衰减约束条件下,其另一个输出的H2性能指标所满足的上界,并利用Lagrange乘子法导出了使得该界达到最小的"最优的"状态反馈控制器.结果仅需求解一个含有两个尺度参数的修正代数Riccati方程.而且数值结果表明该方法是非常有效的.