Convergence analysis of a computational method for optimal control of non-linear differential-algebraic systems

The convergence analysis of a computational method for optimal control problems of non-linear differential-algebraic systems is considered. The class of admissible controls is taken to be the class of piecewise smooth functions. A control parametrizution technique is used to approximate the optimal control problem into a sequence of optimal parameter selection problems. The solution of each of these approximate problems gives rise to a suboptimal solution to the original optimal control problem in an obvious way. The gradients of the cost functional with respect to parameters are derived. Furthermore, the error bounds between the suboptimal costs and the true optimal cost are derived.     

Singular H∞ Suboptimal Control for a Class of Nonlinear Cascade Systems

In the present paper singular state feedback _∞ suboptimal control for a class of nonlinear cascade systems is addressed. Under the assumption that a regular state feedback _∞ suboptimal control problem is solvable for a particular subsystem of the cascade system, an auxiliary nonlinear system is defined. It is shown that a state feedback solution to the singular _∞ suboptimal control problem for the auxiliary system also applies to the original problem. The advantage of the auxiliary problem to the original problem is that the auxiliary penalty variable has lower dimension than the original penalty variable. It is shown how this fact can simplify the problem considerably for the case when the auxiliary system can be strongly input-output decoupled. The theory is applied to a problem of a rigid spacecraft with actuator dynamics. Application to the special case when a subsystem of the nonlinear cascade system is passive is also considered.     

Nonlinear model predictive control using deterministic global optimization

This paper presents a Nonlinear Model Predictive Control (NMPC) algorithm utilizing a deterministic global optimization method. Utilizing local techniques on nonlinear nonconvex problems leaves one susceptible to suboptimal solutions at each iteration. In complex problems, local solver reliability is difficult to predict and dependent upon the choice of initial guess. This paper demonstrates the application of a deterministic global solution technique to an example NMPC problem. A terminal state constraint is used in the example case study. In some cases the local solution method becomes infeasible, while the global solution correctly finds the feasible global solution. Increased computational burden is the most significant limitation for global optimization based online control techniques. This paper provides methods for improving the global optimization rates of convergence. This paper also shows that globally optimal NMPC methods can provide benefits over local techniques and can successfully be used for online control.     

Codesign of controller,routing and scheduling in WirelessHART networked control systems

In this paper, we provide a novel methodology to co‐design controller, scheduling and routing in a wireless control network compliant with the WirelessHART protocol. We both provide a modeling framework and derive a novel suboptimal solution to the linear‐quadratic regulator problem for a class of systems that extends Markov jump linear system considering both continuous and discrete inputs. To allow that, our results can be directly implemented in a real WirelessHART network, we setup a receding horizon optimization problem that takes into account the constraint for compliance with WirelessHART and validate our solution on a batch reactor control loop.     

On real-time robust model predictive control

High-speed applications impose a hard real-time constraint on the solution of a model predictive control (MPC) problem, which generally prevents the computation of the optimal control input. As a result, in most MPC implementations guarantees on feasibility and stability are sacrificed in order to achieve a real-time setting. In this paper we develop a real-time MPC approach for linear systems that provides these guarantees for arbitrary time constraints, allowing one to trade off computation time vs. performance. Stability is guaranteed by means of a constraint, enforcing that the resulting suboptimal MPC cost is a Lyapunov function. The key is then to guarantee feasibility in real-time, which is achieved by the proposed algorithm through a warm-starting technique in combination with robust MPC design. We address both regulation and tracking of piecewise constant references. As a main contribution of this paper, a new warm-start procedure together with a Lyapunov function for real-time tracking is presented. In addition to providing strong theoretical guarantees, the proposed method can be implemented at high sampling rates. Simulation examples demonstrate the effectiveness of the real-time scheme and show that computation times in the millisecond range can be achieved.     

Lossless convexification of a class of optimal control problems with non-convex control constraints

We consider a class of finite time horizon optimal control problems for continuous time linear systems with a convex cost, convex state constraints and non-convex control constraints. We propose a convex relaxation of the non-convex control constraints, and prove that the optimal solution of the relaxed problem is also an optimal solution for the original problem, which is referred to as the lossless convexification of the optimal control problem. The lossless convexification enables the use of interior point methods of convex optimization to obtain globally optimal solutions of the original non-convex optimal control problem. The solution approach is demonstrated on a number of planetary soft landing optimal control problems.     

A design technique for fast sampled-data nonlinear model predictive control with convergence and stability results

ABSTRACT

In this study, a sampled-data nonlinear model predictive control scheme is developed. The control algorithm uses a prediction horizon with variable length, a terminal constraint set, and a feedback controller defined on this set. Following a suboptimal solution strategy, a defined number of steps of an iterative optimisation routine improve the current input trajectory at each sampling point. The value of the objective function monotonically decreases and the state converges to a target set. A discrete-time formulation of the algorithm and a discrete-time design model ensure high computational efficiency and avoid an ad hoc quasi-continuous implementation. This design technique for a fast sampled-data nonlinear model predictive control algorithm is the main contribution of the paper. Based on a benchmark control problem, the performance of the developed control algorithm is assessed against state-of-the-art nonlinear model predictive control methods available in the literature. This assessment demonstrates that the developed control algorithm stabilises the system with very low computational effort. Hence, the algorithm is suitable for real-time control of fast dynamical systems.     

A uniqueness result for the isaacs equation corresponding to nonlinear H∞ control

The dynamic programming equation (DPE) corresponding to nonlinear H_∞ control is considered. When the cost grows quadratically in the state, it is well known that there may be an infinite number of viscosity solutions to the DPE. In fact, there may be more than one classical solution when a classical solution exists. For the case of fixed feedback control, it is shown that there exists a unique viscosity solution in the class of solutions meeting a certain growth condition, and a representation in terms of available storage is obtained. For the active control case, where the H_∞ problem is represented by a differential game, a similar representation result is obtained under the assumption of existence of a suboptimal feedback control. This research was partially supported by AFOSR under Grant F49620-95-1-0296 and by ONR under Grant N0014-96-1-0267.     

State waypoint approach to continuous‐time nonlinear optimal control problems

In this paper, we propose an optimal control technique for a class of continuous‐time nonlinear systems. The key idea of the proposed approach is to parametrize continuous state trajectories by sequences of a finite number of intermediate target states; namely, waypoint sequences. It is shown that the optimal control problem for transferring the state from one waypoint to the next is given an explicit‐form suboptimal solution, by means of linear approximation. Thus the original continuous‐time nonlinear control problem reduces to a finite‐dimensional optimization problem of waypoint sequences. Any efficient numerical optimization method, such as the interior‐reflection Newton method, can be applied to solve this optimization problem. Finally, we solve the optimal control problem for a simple nonlinear system example to illustrate the effectiveness of this approach. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Mathematical programs with complementarity constraints in the context of inverse optimal control for locomotion

In this paper an inverse optimal control problem in the form of a mathematical program with complementarity constraints (MPCC) is considered and numerical experiences are discussed. The inverse optimal control problem arises in the context of human navigation where the body is modelled as a dynamical system and it is assumed that the motions are optimally controlled with respect to an unknown cost function. The goal of the inversion is now to find a cost function within a given parametrized family of candidate cost functions such that the corresponding optimal motion minimizes the deviation from given data. MPCCs are known to be a challenging class of optimization problems typically violating all standard constraint qualifications (CQs). We show that under certain assumptions the resulting MPCC fulfills CQs for MPCCs being the basis for theory on MPCC optimality conditions and consequently for numerical solution techniques. Finally, numerical results are presented for the discretized inverse optimal control problem of locomotion using different solution techniques based on relaxation and lifting.     

Congestion control as a stochastic control problem with action delays

We consider the design of explicit rate-based congestion control for high-speed communication networks and show that this can be formulated as a stochastic control problem where the controls of different users enter the system dynamics with different delays. We discuss the existence, derivation and the structure of the optimal controller, as well as of suboptimal controllers of the certainty-equivalent type — a terminology that is precisely defined in the paper for the specific context of the congestion control problem considered. We consider, in particular, two certainty-equivalent controllers which are easy to implement, and show that they are stabilizing, i.e., they lead to bounded infinite-horizon average cost, and stable queue dynamics. Further, these controllers perform well in simulations.     

Optimal control problems with multiple characteristic time points in the objective and constraints

In this paper, we develop a computational method for a class of optimal control problems where the objective and constraint functionals depend on two or more discrete time points. These time points can be either fixed or variable. Using the control parametrization technique and a time scaling transformation, this type of optimal control problem is approximated by a sequence of approximate optimal parameter selection problems. Each of these approximate problems can be viewed as a finite dimensional optimization problem. New gradient formulae for the cost and constraint functions are derived. With these gradient formulae, standard gradient-based optimization methods can be applied to solve each approximate optimal parameter selection problem. For illustration, two numerical examples are solved.     

Affine and predictive control policies for a class of nonlinear systems

The input-state linear horizon (ISLH) for a nonlinear discrete-time system is defined as the smallest number of time steps it takes the system input to appear nonlinearly in the state variable. In this paper, we employ the latter concept and show that the class of constraint admissible N-step affine state-feedback policies is equivalent to the associated class of constraint admissible disturbance-feedback policies, provided that N is less than the system’s ISLH. The result generalizes a recent result in [Goulart, P. J., Kerrigan, E. C., Maciejowski, J. M. (2006). Optimization over state feedback policies for robust control with constraints. Automatica, 42(4), 523-533] and is significant because it enables one: (i) to determine a constraint admissible state-feedback policy by employing well-known convex optimization techniques; and (ii) to guarantee robust recursive feasibility of a class of model predictive control (MPC) policies by imposing a suitable terminal constraint. In particular, we propose an input-to-state stabilizing MPC policy for a class of nonlinear systems with bounded disturbance inputs and mixed polytopic constraints on the state and the control input. At each time step, the proposed MPC policy requires the solution of a single convex quadratic programme parameterized by the current system state.     

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.     

Regulator problems with a cost for applying or changing control

An examination is made of linear regulator problems with an additional cost for applying a nonzero control at each instant. The dual of this problem is the least-squares filtering problem with a cost for taking measurements. The regulator problem with a cost for performing a non-zero change in control is also examined by reducing it to the problem with a cost for applying control. Because a solution to the general problem involves solving a two-point boundary value problem in nonlinear matrix differential or difference equations, two easily computed suboptimal control schemes are proposed and their properties discussed. Bounds on the total length of time the optimum control is nonzero are derived, as well as upper bounds on the difference in performance between the optimum control and either of the proposed suboptimum schemes. For a class of continuous-time problems, a bound is derived on the number of times the control switches from zero to nonzero or vice versa. A class of problems is also delineated for which the optimum solution has some special properties which enable it to be computed simply and directly.     

非线性时滞系统次优控制的逐次逼近法

对状态变量含有时滞的非线性系统的次优控制问题进行了研究，提出了一种次优控制的逐次逼近设计方法．针对由最优控制理论导出的既含有时滞项又含有超前项的非线性两点边值问题，构造了其解序列一致收敛于原问题最优解的非齐次线性两点边值问题序列．从而将两点边值问题解序列的有限次迭代结果作为系统的次优控制律．仿真结果表明了所提出方法的有效性．     

Barrier function based model predictive control

A new formulation of nonlinear model predictive control (MPC) is developed by including a weighted barrier function in the control objective. While the barrier ensures that inequality constraints are strictly satisfied it also provides a smooth transition between points in the interior and those on the boundary of the constraint set. In addition, the resulting optimisation problem, to be solved at each control step, is effectively unconstrained and thus amenable to elegant optimisation techniques. The barrier must satisfy certain conditions in order that the state converges to the origin and we show how to construct such a barrier. Conventional MPC may be seen as a limiting case of the new class as the barrier weighting itself approaches zero. We pay particular attention to the novel approach of fixing the weighting parameter to some positive value—possibly large—and observe that this provides a degree of controller caution near constraint boundaries. We construct an ellipsoidal invariant set by exploiting the geometry of self-concordant functions and show nominal closed-loop stability for this class of controllers under full state feedback.     

Implicit improved vertex control for uncertain,time-varying linear discrete-time systems with state and control constraints

The problem of regulating an uncertain and/or time-varying linear discrete-time system with state and control constraints to the origin is addressed. It is shown that feasibility and a robustly asymptotically stable closed loop can be achieved using an interpolation technique. The design method can be seen as an alternative to optimization-based control schemes such as Robust Model Predictive Control. Especially for problems requiring complex calculations to find the optimal solution, the present method can provide a straightforward suboptimal solution. A simulation demonstrates the performance of this class of constrained controllers.     

非线性相似组合大系统最优控制的逐次逼近过程

研究一类仿射非线性相似组合大系统关于二次型性能指标的最优控制问题．首先通过模型简化,将非线性相似组合大系统化为若干个准解耦的子系统;然后利用非线性系统最优控制的逐次逼近设计方法,将求解高阶强耦合的非线性两点边值问题简化为求解一族解耦的线性两点边值问题序列．该线性两点边值问题序列的解一致收敛于非线性相似组合大系统的最优控制,得到的最优控制律由线性最优控制的解析项与非线性补偿序列的极限项组成．通过截取最优控制非线性补偿序列的有限次逼近值．得到了非线性组合大系统的次优控制律．