Infinite horizon linear quadratic tracking problem: A discounted cost function approach

The discounted cost function approach is one of the main approaches to the infinite time horizon problems in economics and market. This paper introduces a causal and optimal solution based on the discounted cost function approach for infinite horizon linear quadratic tracking problem with disturbance rejection and the problem of disturbance tracking by presenting the theoretical foundations. It is shown that the proposed method has the ability to solve the problems where the reference inputs and disturbance signals are not asymptotically stable. Two numerical examples, a grid connection of voltage source power electronic converter as a SISO system and a load frequency control of a 2‐area nonreheat thermal power system as a MIMO example, are presented to illustrate the effectiveness of the proposed method.     

LQ control of unknown discrete‐time linear systems—A novel approach and a comparison study

In this paper, we propose a novel approach to the linear quadratic (LQ) optimal control of unknown discrete‐time linear systems. We first describe an iterative procedure for minimizing a partially unknown static function. The procedure is based on simultaneous updates in the estimation of unknown parameters and in the optimization of controllable inputs. We then use the procedure for control optimization in unknown discrete‐time dynamic systems—we consider applications to the finite‐horizon and the infinite‐horizon LQ control of linear systems in detail. To illustrate the approach, an example of the pitch attitude control of an aircraft is considered. We also compare our proposed approach to several other approaches to finite/infinite‐horizon LQ control problems with unknown dynamics from the literature, including extremum seeking and adaptive dynamic programming/reinforcement learning. Our proposed approach is competitive with these approaches in speed of convergence and in implementation and computational complexity.     

Optimal control of two‐dimensional parabolic partial differential equations with application to steel billets cooling in continuous casting secondary cooling zone

Our work is devoted to an optimal control problem for two‐dimensional parabolic partial differential equations(PDEs) and its application in engineering sciences. An adjoint problem approach is used for analysis of the Fréchet gradient of the cost functional, and we prove the gradient is Lipschitz continuous. An improved conjugate gradient method is proposed to solve this problem. Based on Lipschitz continuity of the gradient, the convergence analysis of the conjugate gradient algorithm we proposed is studied. Results of some computational experiments obtained by the conjugate gradient algorithm are illustrated. The results show that the improved conjugate gradient algorithm is effective. Copyright © 2016 John Wiley & Sons, Ltd.     

Structure‐preserving local optimal control of mechanical systems

While system dynamics are usually derived in continuous time, respective model‐based optimal control problems can only be solved numerically, ie, as discrete‐time approximations. Thus, the performance of control methods depends on the choice of numerical integration scheme. In this paper, we present a first‐order discretization of linear quadratic optimal control problems for mechanical systems that is structure preserving and hence preferable to standard methods. Our approach is based on symplectic integration schemes and thereby inherits structure from the original continuous‐time problem. Starting from a symplectic discretization of the system dynamics, modified discrete‐time Riccati equations are derived, which preserve the Hamiltonian structure of optimal control problems in addition to the mechanical structure of the control system. The method is extended to optimal tracking problems for nonlinear mechanical systems and evaluated in several numerical examples. Compared to standard discretization, it improves the approximation quality by orders of magnitude. This enables low‐bandwidth control and sensing in real‐time autonomous control applications.     

An improved smoothing technique‐based control vector parameterization method for optimal control problems with inequality path constraints

An improved control vector parameterization (CVP) method is proposed to solve optimal control problems with inequality path constraints by introducing the l₁ exact penalty function and a novel smoothing technique. Both the state and control variables are allowed to appear explicitly in the inequality path constraints simultaneously. By applying the penalty function and smoothing technique, all the inequality path constraints are firstly reformulated as non‐differentiable penalty terms and incorporated into the objective function. Then, the penalty terms are smoothed by using a novel smooth function, leading to a smooth optimal control problem with no inequality path constraints. With discretizing the control space, a corresponding nonlinear programming (NLP) problem is derived, and error between the NLP problem and the original problem is discussed. Results reveal that if the smoothing parameter is sufficiently small, the solution of the NLP problem is approximately equal to the original problem, which shows the convergence of the proposed method. After clarifying some theories of the proposed approach, a concomitant numerical algorithm is put forward with furnishing the updating rules of both the penalty parameter and smoothing parameter. Simulation examples verify the advantages of the proposed method for tackling nonlinear optimal control problems with different inequality path constraints. Copyright © 2016 John Wiley & Sons, Ltd.     

A constrained H∞ smooth optimization technique

In H^∞ optimal control the cost function is the maximum singular value of a transfer function matrix over a frequency range. The optimization is over all stabilizing controllers. In constrained H^∞ control the controllers typically have a fixed structure, perhaps conveniently parametrized in terms of a parameter vector. Also, there may be functional constraints involving singular values representing, for example, robustness requirements. Such problems are usually cast as non-smooth optimization problems. In this paper we consider a general class of constrained H^∞ optimization problems and show that these problems can be approximated by a sequence of smooth optimization problems. Thus each of the approximate problems is readily solvable by standard optimization software packages such as those available in the NAG or IMSL library. The proposed approach via smooth optimization is simple in terms of mathematical content, easy to implement and computationally efficient.     

A robust control system scheme based on model predictive controller (MPC) for continuous‐time systems

In this paper, an LMI framework based on model predictive strategy is addressed to design a robust dynamical control law in a typical control system. In the proposed method, instead of traditional static controller, a dynamic control law is used. With a suitable matrix transformation, the controller parameters selection are translated into an optimization problem with some LMI constraints. The plant input and output constraints are also handled with another LMIs. The controller is represented in state space form, and its parameters are computed in real‐time operation. For achieving this goal, by solving an optimization problem, a dynamic controller is designed, which meets the required plant performances. These results are used in 2 numerical examples to demonstrate the effectiveness of the proposed approach.     

Feedback predictive control strategies for investment in the financial market with serially correlated returns subject to constraints and trading costs

In this paper, we consider the optimal portfolio selection problem subject to hard constraints on trading amounts, trading costs, and different rates for borrowing and lending when the risky asset returns are serially correlated. We consider both explicit and implicit trading costs. No assumptions about the correlation structure between different time points or about the distribution of asset returns are needed. The problem is stated as a dynamic tracking problem of a reference portfolio with a desired return. We leverage the methodology of model predictive control (also known as receding horizon control) to design feedback portfolio optimization strategies and to provide a numerically tractable algorithm for practical applications. All expressions are presented in terms of first‐ and second‐order conditional moments. Our approach is tested on sets of real data from the Russian Stock Exchange Moscow Interbank Currency Exchange and New York Stock Exchange.     

A numerical method for computing optimal controls in feedback and digital forms and its application to the blowing‐venting control system of manned submarines

On the basis of the classical variational reformulation of optimal control problems, we introduce a numerical scheme for solving those problems where the goal is the computation of optimal controls in feedback and digital forms defined on a discrete time mesh. The algorithm reduces the computation of such controls to solving a suitable nonlinear mathematical programming problem where the unknowns are the controls and slope of the state variable of the original problem. The motivation for this study comes from the real‐world engineering problem which consists of maneuvering a manned submarine by using the blowing‐venting control system of the ballast tanks of the vehicle. After checking the proposed algorithm in an academic example, we apply it to the maneuvering problem of submarines whose mathematical model includes a state law which is composed of a system of twenty‐four nonlinear ordinary differential equations. Numerical results illustrate the performance of the numerical scheme. Copyright © 2012 John Wiley & Sons, Ltd.     

Exact penalty functions for optimal control problems I: Main theorem and free‐endpoint problems

In this two‐part study, we develop a general approach to the design and analysis of exact penalty functions for various optimal control problems, including problems with terminal and state constraints, problems involving differential inclusions, and optimal control problems for linear evolution equations. This approach allows one to simplify an optimal control problem by removing some (or all) constraints of this problem with the use of an exact penalty function, thus allowing one to reduce optimal control problems to equivalent variational problems and apply numerical methods for solving, eg, problems without state constraints, to problems including such constraints, etc. In the first part of our study, we strengthen some existing results on exact penalty functions for optimisation problems in infinite dimensional spaces and utilise them to study exact penalty functions for free‐endpoint optimal control problems, which reduce these problems to equivalent variational ones. We also prove several auxiliary results on integral functionals and Nemytskii operators that are helpful for verifying the assumptions under which the proposed penalty functions are exact.     

Globalized and bounded Nelder‐Mead algorithm with deterministic restarts for tuning controller parameters: Method and application

This paper develops and examines an optimization algorithm for simulation‐based tuning of controller parameters. The proposed algorithm globalizes the Guin augmented variant of Nelder–Mead's nonlinear downhill simplex by deterministic restarts, linearly growing memory vector, and moving initial simplex. First, the effectiveness of the algorithm is tested using 10 complex and multimodal optimization benchmarks. The algorithm achieves global minima of all benchmarks and compares favorably against the evolutionary, swarm, and other globalized local‐search multimodal optimization algorithms in probability of finding global minimum and numerical cost. Next, the proposed algorithm is applied for tuning sliding mode controller parameters for a servo pneumatic position control application. The experimental results reveal that the system with sliding mode controller parameters tuned using the proposed algorithm targeting smooth position control with maximum possible accuracy, performs as desired and eliminates the need of manual online tuning for desired performance. The results are also compared with the performance of the same servo pneumatic system with parameters tuned using manual online tuning in an earlier published work. The system with controller parameters tuned using the proposed algorithm shows improvement in accuracy by 28.9% in sinusoidal and 42.2% in multiple step polynomials tracking.     

Constrained linear parameter‐varying control using approximate multiparametric programming

We develop an approximate multiparametric convex programming approach with its application to control constrained linear parameter‐varying systems. Recently, the application of the real‐time model predictive control (MPC) for various engineering systems has been significantly increased by using the multiparametric convex programming tool, known as explicit MPC approach. The main idea of explicit MPC is to move the major parts of the computations to offline phase and to provide an explicit piecewise affine solution of the constrained MPC problem, which is defined over a set of convex polyhedral partitions. In the proposed method, the idea of convex programming and partitioning is applied for linear parameter‐varying control systems. The feasible space of the time‐varying parameters is divided into simplices in which approximate solutions are calculated such that the approximation error is kept limited by solving sequences of linear programs. The approximate optimal solution within each simplex is obtained by linear interpolation of the optimal solutions in the simplex vertices, and then multiparametric programming tool is utilized to compute an explicit state feedback solution of linear quadratic optimal control problem for simplex vertices subject to state and input constraints. The proposed method is illustrated by a numerical example and the simulation results show the advantages of this approach.     

Optimal fixed‐levels control for nonlinear systems with quadratic cost‐functionals

This paper is devoted to general optimal control problems (OCPs) associated with a family of nonlinear continuous‐time switched systems in the presence of some specific control constraints. The stepwise (fixed‐level type) control restrictions we consider constitute a common class of admissible controls in many real‐world engineering systems. Moreover, these control restrictions can also be interpreted as a result of a quantization procedure appglied to the inputs of a conventional dynamic system. We study control systems with a priori given time‐driven switching mechanism in the presence of a quadratic cost functional. Our aim is to develop a practically implementable control algorithm that makes it possible to calculate approximating solutions for the class of OCPs under consideration. The paper presents a newly elaborated linear quadratic‐type optimal control scheme and also contains illustrative numerical examples. Copyright © 2015 John Wiley & Sons, Ltd.     

Model‐free online tuning of controller parameters using a globalized local search algorithm

In an earlier work, the authors proposed a globalized bounded Nelder‐Mead algorithm with deterministic restarts and a linearly growing memory vector. It was shown that the algorithm was a favorable option for solving multimodal optimization problems like controller tuning because of the greater probability of finding the global minimum and lesser numerical cost. Therefore, the algorithm was successfully used for model‐based offline tuning of sliding mode controller parameters for a servo‐pneumatic position control application. However, such offline tuning requires a sufficiently adequate system model, which, in some applications, is difficult to attain. Moreover, it is not generally appreciated as an essential requirement for controller tuning by the end user like the industry. An improvement in performance of optimization algorithm for tuning is expected if it relies on measurements coming directly from an actual physical system and not just its mathematical model. Therefore, in this paper, we apply the aforementioned algorithm for model‐free online optimization of controller parameters. The application involves the programmatic control of a real‐time interface of a physical system by the algorithm for data flow and logical decisions for optimization. For comparison with the results of the model‐based offline tuning suggested in earlier work, the sliding mode controller parameters are tuned online for the same position control application. The experimental results reveal that the system performance with controller parameters tuned online using the algorithm compares favorably to the one with model‐based offline tuning especially at higher priority level for accuracy. The improvement in system performance amounts to 21%.     

Real‐time scheduling of multiple uncertain receding horizon control systems

In this paper, a new scheduling approach is proposed that considers the effect of modeling uncertainty for multiple continuous time receding horizon control (RHC) systems. This is accomplished by combining a scheduling approach with results from the continuous time nonlinear systems theory. It is shown that using a rate monotonic priority assignment method combined with analytical bounds on the prediction error, the problem of scheduling multiple uncertain plants can be cast into an appropriate constrained optimization problem. The constraints guarantee that the processes will be schedulable. The optimization provides optimized performance and balanced resource allocation in the presence of uncertainty. The proposed method was applied to a real‐time simulation of RHC trajectory tracking for two hovercraft vehicles demonstrating the validity of the approach. Copyright © 2008 John Wiley & Sons, Ltd.     

Robust deadbeat pole assignment with gain constraints: an LMI optimization approach

This paper considers a kind of robust pole assignment problem for discrete‐time systems. A recently developed a posteriori measure of closed‐loop eigenvalue sensitivity suitable for MIMO deadbeat regulator design is minimized during the search of the state feedback gain. The problem is cast into an LMI‐based convex optimization task in which constraints on the size of feedback gain can be catered naturally. The proposed pole assignment algorithm can also be used to find the minimum feedback gain with respect to different norms. A numerical example is then employed to illustrate the effectiveness of the method. Copyright © 2000 John Wiley & Sons, Ltd.     

Hamiltonian two‐degrees‐of‐freedom control of chemical reactors

A novel unified approach to two‐degrees‐of‐freedom control is devised and applied to a classical chemical reactor model. The scheme is constructed from the optimal control point of view and along the lines of the Hamiltonian formalism for nonlinear processes. The proposed scheme optimizes both the feedforward and the feedback components of the control variable with respect to the same cost objective. The original Hamiltonian function governs the feedforward dynamics, and its derivatives are part of the gain for the feedback component. The optimal state trajectory is generated online, and is tracked by a combination of deterministic and stochastic optimal tools. The relevant numerical data to manipulate all stages come from a unique off‐line calculation, which provides design information for a whole family of related control problems. This is possible because a new set of PDEs (the variational equations) allow to recover the initial value of the costate variable, and the Hamilton equations can then be solved as an initial‐value problem. Perturbations from the optimal trajectory are abated through an optimal state estimator and a deterministic regulator with a generalized Riccati gain. Both gains are updated online, starting with initial values extracted from the solution to the variational equations. The control strategy is particularly useful in driving nonlinear processes from an equilibrium point to an arbitrary target in a finite‐horizon optimization context. Copyright © 2010 John Wiley & Sons, Ltd.     

Solution of some model predictive control problems using slopes between unconstrained and constrained optimums

At times, the number of controlled variables equals the number of manipulated variables and the objective of the control system is to minimize the difference in the desired and predicted output trajectories subject only to constraints on the manipulated variables. If a simplified model predictive control algorithm is used for such applications, then solution to the optimization problem can be obtained by using the slopes between the unconstrained and constrained optimums. The solution procedure is described for a two‐input–two‐output case. A comparison with a linear programming (LP) formulation showed that the computational time for the proposed solution was about 35 times less than the time for the LP solution. Copyright © 1999 John Wiley & Sons, Ltd.     

A primal‐dual active‐set method for distributed model predictive control

We present a novel distributed primal‐dual active‐set method for model predictive control. The primal‐dual active‐set method is used for solving model predictive control problems for large‐scale systems with quadratic cost, linear dynamics, additive disturbance, and box constraints. The proposed algorithm is compared with dual decomposition and an alternating direction method of multipliers. Theoretical and experimental results show the effectiveness of the proposed approach for large‐scale systems with communication delays. The application to building control systems is thoroughly investigated. Copyright © 2016 John Wiley & Sons, Ltd.     

Optimal control of one‐dimensional cellular uptake in tissue engineering

A control problem motivated by tissue engineering is formulated and solved, in which control of the uptake of growth factors (signaling molecules) is necessary to spatially and temporally regulate cellular processes for the desired growth or regeneration of a tissue. Four approaches are compared for determining one‐dimensional optimal boundary control trajectories for a distributed parameter model with reaction, diffusion, and convection: (i) basis function expansion, (ii) method of moments, (iii) internal model control, and (iv) model predictive control (MPC). The proposed method of moments approach is computationally efficient while enforcing a nonnegativity constraint on the control input. Although more computationally expensive than methods (i)–(iii), the MPC formulation significantly reduced the computational cost compared with simultaneous optimization of the entire control trajectory. A comparison of the pros and cons of each of the four approaches suggests that an algorithm that combines multiple approaches is most promising for solving the optimal control problem for multiple spatial dimensions. Copyright © 2012 John Wiley & Sons, Ltd.