Note on 'Time-optimal control strategy for saturating linear discrete systems’

The control policy taking a linear system from a given state to the origin in a minimum number of time steps is not unique. Such flexibility also offers an opportunity to minimize the control effort in a straightforward manner.     

Robust stability of linear perturbed discrete systems with saturating actuators

This paper is concerned with the robust stability conditions for linear perturbed systems with saturating actuators in the discrete-time case. Both non-linear and linear perturbations are considered. Two types of controller, a state feedback and a dynamic controller, are introduced to guarantee asymptotic stability of the perturbed system.     

Robust stability of linear perturbed discrete large-scale systems with saturating actuators

Robust stability conditions are considered for linear perturbed large-scale systems with saturating actuators in the discrete-time case. Both non-linear and linear perturbations are considered. Two types of controllers, a state feedback and a dynamic controller, are introduced for each sub-system to guarantee asymptotic stability of the perturbed large-scale system.     

Time-optimal control algorithm for two-time-scale discrete system

A linear, shift-invariant discrete system exhibiting two-time-scale (slow and fast) character is addressed. An iterative algorithm based on linear programming for the computation of time-optimal control is presented by decoupling the original system into slow and fast subsystems. It is found that the computation time and storage capacity for the computation of time-optimal control of the slow and fast subsystems is less than that required for the original system. A numerical example is provided to illustrate the proposed algorithm.     

Design of linear systems with saturating linear control and bounded states

A theorem is developed, based on the circle criterion, which gives sufficient conditions for the stability of a saturating linear control for a linear system with bounded states. This theorem can be applied to determine a range of gains for the linear control which ensures stability. It is shown by example that this approach results in an increased speed of response over that obtained by previous methods which do not account for state bounds.     

Suboptimal control for discrete linear constant stochastic systems

The regulation of discrete-time, constant, linear stochastic systems with quadratic performance criterion is considered. A fixed-dimensional, linear time-invariant controller is used. Algebraic matrix equations are derived whose solutions are the gains of the fixed-dimensional optimal controller.     

In-the-large stability of relay and saturating control systems with linear controllers

It is shown that when the plant of a relay control system consists of a pure integrators, with n ≥ 3, and the control signal proceding the relay is a linear combination of state coordinates, the system is unstable in-the-large for all values of the controller coefficients. The same result holds if the relay is replaced by a. saturator.

The method used is to prove that (except for a degenerate case) state trajectories exist which go to infinity. This method enables more general systems to be treated than those discussed in a previous paper, which demonstrated the existence of a periodic solution.

Plants not consisting of pure integrators are also considered. The question of when it is worthwhile to incorporate non-linearities in the controller is discussed.     

Control of linear systems with saturating actuators

This paper deals with the design of linear systems with saturating actuators where the actuator limitations have to be incorporated a priori into control design. The authors take a semiglobal approach to solve some of the central control problems for such systems. These problems include stabilization, input-additive disturbance rejection, and robust stabilization in the presence of matched nonlinear uncertainties. The authors develop further the semiglobal design technology which was initiated in their earlier work (Lin and Saberi, 1995) and utilize it to deal with these control problems     

Time-optimal feedback control laws for certain third-order relay control systems

Time-optimal feedback control laws for six third-order plants having real, non-positive eigenvalues and a single saturable control input are calculated explicitly by direct application of Gulko's predictive control strategy. Errors in a previous treatment of three of the plants are discussed.     

Time-optimal control and high-gain linear state feedback

It is known that the fastest possible response of a linear system under input constraints is typically achieved using extreme control efforts at all times, or bang-bang control. However, closed-loop minimum-time control is very difficult to implement except for a few simple second-order systems, since a closed-form solution for the time-optimal switching hypersurface usually cannot be found for high-order systems. In this paper the connection between linear state feedback and time-optimal control is explored. It is shown that for a class of systems an equivalent time-optimal, linear, switching hypersurface exists corresponding to an initial condition. Based on these switching planes, a high-gain linear state feedback law can be used to achieve minimum-time control on systems of order higher than two, provided that the feedback coefficients are adjusted according to the initial conditions at the start of each operation. Three different cases are investigated in detail: a second-order DC servo drive, a third-order servo system including actuator dynamics, and a fourth-order flexible mechanism. Since the proposed algorithm resembles in form a sliding-mode controller, the similarities and differences between the two control algorithms are also discussed.     

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.     

Resilient guaranteed cost control for uncertain discrete linear jump systems

Robust stochastic stabilization and guaranteed cost control for a class of uncertain discrete-time linear system with Markovian jumping parameters are discussed. Two classes of controller gain perturbations, additive and multiplicative, are considered. The necessary and sufficient conditions for the above problems are derived, which are in terms of positive-definite solutions of a set of coupled linear matrix inequalities (LMIs). Furthermore, resilient guaranteed cost controllers are designed. Finally, numerical examples are presented to illustrate the solvability and effectiveness of the results.     

Iterative learning control for linear discrete time nonminimum phase systems

This paper investigates iterative learning control for linear discrete time nonminimum phase systems. First, iterative learning control with advanced output data is considered for maximum phase systems. Next, the results are extended to nonminimum phase systems. The stability of the inverse mapping from the desired output to the input is proven based on the results for maximum phase systems. The input should be updated with the output which is more advanced than the input by the sum of the relative degree of the system and the number of nonminimum phase zeros. An example is given to indicate the importance of proper advances of output in the input update law.     

滞后离散广义系统的鲁棒严格耗散控制

研究确定的及不确定的滞后离散广义系统的无记忆状态反馈严格耗散控制器设计问题.利用线性矩阵不等式(LMI)方法,首先给出滞后离散广义系统容许(即正则、稳定、因果)且严格耗散的条件,然后通过矩阵不等式(MIs)得到无记忆状态反馈严格耗散控制器的存在条件和设计方法;进而针对除E外其余系数矩阵均具有范数有界不确定性的滞后离散广义系统,利用矩阵不等式的解设计鲁棒严格耗散控制器,保证闭环系统广义二次稳定且严格耗散.     

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.     

具有饱和执行器线性系统的综合PLCöVSC 控制

基于LQ理论和变结构方法,导出一种带有饱和执行机构和不确定性线性系统的混合PLC／VSC控制律．引入一组嵌套的Lyapunov级集,并根据分段Lyapunov函数导出控制律,该控制律含有不连续控制项和趋近率项．利用Lyapunov函数方法分析了闭环系统的稳定性．结合简化单摆系统验证了设计方法的有效性,在抗干扰能力和减少超调量上反映出所提出的控制器具有比PLC和PLC／LHG更佳的性能．     

Piecewise-affine Lyapunov functions for discrete-time linear systems with saturating controls

This paper is concerned with piecewise-affine (PWA) functions as Lyapunov function candidates for stability analysis of time-invariant discrete-time linear systems with saturating closed-loop control inputs. Using a PWA model of saturating closed-loop system, new necessary and sufficient conditions for a PWA function be a Lyapunov function are presented. Based on linear programming formulation of these conditions, an effective algorithm is proposed for construction of such Lyapunov functions for estimation of the region of local asymptotic stability. Compared to piecewise-linear functions, like Minkowski functions, PWA functions are more adequate to capture the dynamical effects of saturation nonlinearities, giving strictly less conservative results. The complexity of the proposed approach is polynomial in state dimension and exponential in saturating control dimension, being hence appropriate for problems with large state dimension but with few saturating inputs.     

Suboptimal control of linear discrete stochastic systems with linear input constraints

Optimal control of linear discrete stochastic systems with linear input constraints is considered. A lower bound on the achievable minimum of the loss function is given. A suboptimal control strategy is derived by using a truncated Taylor series to approximate the expected future loss in the Bellman equation. The performance of the suboptimal controller is studied using Monte Carlo simulation and the obtained loss is compared with the lower bound.