LQ‐optimal control of positive linear systems

The LQ₊ problem, i.e. the finite‐horizon linear quadratic optimal control problem with nonnegative state constraints, is studied for positive linear systems in continuous time and in discrete time. Necessary and sufficient optimality conditions are obtained by using the maximum principle. These conditions lead to a computational method for the solution of the LQ₊ problem by means of a corresponding Hamiltonian system. In addition, the necessary and sufficient conditions are proved for the LQ₊‐optimal control to be given by the standard LQ‐optimal state feedback law. Then sufficient conditions are established for the positivity of the LQ‐optimal closed‐loop system. In particular, such conditions are obtained for the problem of minimal energy control with penalization of the final state. Moreover, a positivity criterion for the LQ‐optimal closed‐loop system is derived for positive discrete‐time systems with a positively invertible (dynamics) generator. The main results are illustrated by numerical examples. Copyright © 2010 John Wiley & Sons, Ltd.     

A µ‐dependent approach to H∞ control of uncertain switched linear systems with average dwell time

This paper concerns H_∞ control problem for a class of discrete‐time uncertain switched linear systems with average dwell time. The stability result for general discrete‐time switched systems is first explored, and a µ‐dependent approach is then introduced for the considered systems to the H_∞ controller solution. A mode‐dependent state‐feedback controller is designed such that the resulting closed‐loop system is robust exponentially stable and has a prescribed exponential H_∞ performance index. The µ‐dependent existence conditions of desired controller and admissible switching signals are derived and formulated in terms of linear matrix inequalities (LMIs). A numerical example is given to demonstrate the effectiveness of the developed theoretical results. Copyright © 2009 John Wiley & Sons, Ltd.     

Full delayed state feedback pole assignment of discrete‐time time‐delay systems

This paper studies the problem of pole assignment of discrete‐time time delay system with delayed state feedback. The problem is solved in this paper by requiring that the maximal delay in the feedback equals the maximal delay of the open‐loop system. A necessary and sufficient condition guaranteeing the existence of a solution is presented. By using the augmentation technique, the pole assignment problem is then transformed to the problem of solving a linear matrix equation such that certain conditions are satisfied. To solve the linear equation problem, when the desired closed‐loop eigenvalues are not prescribed, a parametric approach using real arithmetic is presented by using polynomial matrices associated with the system matrices. When the desired closed‐loop eigenvalues are prescribed, singular value decomposition can be adopted to solve the linear matrix equation. Both approaches can provide full degree of freedom, which can be further utilized to accomplish some other design objects. The robust pole assignment problem is considered to demonstrate the advantages of the method. Numerical examples are employed to illustrate the effectiveness of the proposed approaches. Copyright © 2009 John Wiley & Sons, Ltd.     

H∞ control of switched linear discrete‐time systems with polytopic uncertainties

In this paper, the problem of designing H_∞ state‐feedback controllers for switched linear discrete‐time systems with polytopic uncertainties is investigated. Two approaches on designing robust and parameter‐dependent H_∞ controllers are proposed and the existence conditions of the desired controllers are derived and formulated in terms of a set of linear matrix inequalities. By solving the corresponding convex optimization problem, the desired controllers are obtained, respectively, and different optimal H_∞ noise‐attenuation level bounds of corresponding closed‐loop systems are given as well. The designed controllers have their own advantages and disadvantages regarding the conservatism and realization complexity. An illustrative example emerging in networked control systems (NCS) and numerical simulations are presented to show the applicability and effectiveness of the obtained theoretic results. Copyright © 2006 John Wiley & Sons, Ltd.     

On the synthesis of time‐varying LQG weights and noises along optimal control and state trajectories

A general approach to control non‐linear uncertain systems is to apply a pre‐computed nominal optimal control, and use a pre‐computed LQG compensator to generate control corrections from the on‐line measured data. If the non‐linear model, on which the optimal control and LQG compensator design is based, is of sufficient quality, and when the LQG compensator is designed appropriately, the closed‐loop control system is approximately optimal. This paper contributes to the selection and computation of the time‐varying LQG weighting and noise matrices, which determine the LQG compensator design. It is argued that the noise matrices may be taken time‐invariant and diagonal. Three very important considerations concerning the selection of the time‐varying LQG weighting matrices are turned into a concrete computational scheme. Thereby, the selection of the time‐varying LQG weighting matrices is reduced to selecting three scalar design parameters, each one weighting one consideration. Although the three considerations seem straightforward they may oppose one another. Furthermore, they usually result in time‐varying weighting matrices that are indefinite, rather than positive (semi) definite as required for the LQG design. The computational scheme presented in this paper addresses and resolves both problems. By two numerical examples the benefits of our optimal closed‐loop control system design are demonstrated and evaluated using Monte Carlo simulation. Copyright © 2005 John Wiley & Sons, Ltd.     

H∞ control for sampled‐data linear systems with two Markov processes

This paper is concerned with the design problem of H_∞ digital switching control for linear continuous systems with Markovian jumping parameters. The controller is digital and monitored by the jumping parameters of the plant. The closed‐loop system is a hybrid one defined on a hybrid time space (composed of a continuous‐time and a discrete‐time) and a sample space. The sample space is specified by two separable continuous‐time discrete‐state Markov processes, one appearing in the open‐loop system, and the other appearing in control action, which is different with the traditional Markovian jumping process. Our attention is focused on designing digital output feedback controllers for the system with two Markovian jumping processes such that both stochastic stability and a prescribed H_∞ performance are achieved. The problem of robust H_∞ control for systems with parameters uncertainties is also studied. It is shown that the sampled‐data control problems for linear Markovian jumping systems with and without parameter uncertainties can be solved in terms of the solutions to a set of intercoupled matrix inequalities. Two numerical examples are given to show the design procedures. Copyright © 2005 John Wiley & Sons, Ltd.     

Singular linear quadratic optimal control for singular stochastic discrete‐time systems

The finite time horizon singular linear quadratic (LQ) optimal control problem is investigated for singular stochastic discrete‐time systems. The problem is transformed into positive LQ one for standard stochastic systems via two equivalent transformations. It is proved that the singular LQ optimal control problem is solvable under two reasonable rank conditions. Via dynamic programming principle, the desired optimal controller is presented in terms of matrix iterative form. One simulation is provided to show the effectiveness of the proposed approaches. Copyright © 2012 John Wiley & Sons, Ltd.     

Optimal guaranteed cost synchronization of coupled neural networks with Markovian jump and mode‐dependent mixed time‐delay

This paper is concerned with the optimal guaranteed cost synchronization problem for a class of coupled neural networks with Markovian jump parameters and mode‐dependent mixed time‐delay. The coupled neural networks contained N‐identical delayed neural nodes and M switch modes from one mode to another according to a Markovian chaining with known transition probability. All the coupled networks' parameters covering the coupled matrix and discrete and distributed time‐delay also depend on the Markovian jump mode. The associated optimal guaranteed cost function is a quadratic function; the activation function is supposed to satisfy sector‐bounded condition. By employing a new Lyapunov–Krasovskii functional and some analytic skills, the sufficiency conditions of guaranteed cost synchronization are derived to ensure that coupling neural network is asymptotically synchronized for related cost function in mean square. The exported sufficient condition is closely contacted with the distributed time‐delay, the mode transition probability, the discrete‐time delay, and the coupled structure of networks. The achieved conditions are given in the light of LMI that can be usefully solved by using the semi‐definite program scheme. Moreover, an LMI‐based approach to export the guaranteed cost synchronization controller is formulated to minimize the optimal guaranteed cost for closed‐loop dynamical networks. Numerical simulations are developed to further display the efficiency of the achieved theoretical results. Copyright © 2015 John Wiley & Sons, Ltd.     

Multiobjective fault‐tolerant fixed‐order/PID control of multivariable discrete‐time linear systems with unmeasured disturbances

In this paper, a multiobjective fault‐tolerant fixed‐order output feedback controller design technique is proposed for multivariable discrete‐time linear systems with unmeasured disturbances. Initially, a multiobjective fixed‐order controller is designed for the system by transforming the problem of tuning the parameters of the controller into a static output feedback problem and solving a mixed H₂/H_∞ optimization problem with bilinear matrix inequalities. Subsequently, the fixed‐order controller is used to construct the closed‐loop system and an active fault‐tolerant control scheme is applied using the input/output data collected from the controlled system. Motivated by its popularity in industry, the proposed method is also used to tune the parameters of proportional‐integral‐derivative controllers as a special case of structured controllers with the fixed order. Two numerical simulations are provided to demonstrate the design procedure and the flexibility of the proposed technique.     

Robust preview control for uncertain discrete‐time systems based on LMI

For a class of uncertain discrete‐time systems, a preview controller based on linear matrix inequality is proposed. A new method is derived to construct an augmented error system instead of taking the difference of the error signal and the system equation. The new approach avoids applying the difference operator to the time‐varying matrix and can simplify the augmented error system. For the augmented error system of the uncertain system, state feedback is introduced. The sufficient condition of asymptotic stability of the closed‐loop system is derived for the performance index by using the relevant theorems of robust control theory. The condition can be realised by solving a linear matrix inequality optimization problem. By incorporating the controller obtained into the original system, we obtain the preview controller. Moreover, introducing an integrator allows the closed‐loop system to robustly track the desired tracking signal without steady‐state error.     

Further results on H ∞ control for discrete‐time uncertain singular systems with interval time‐varying delays in state and input

This paper investigates the state feedback robust H _∞ control problem of a class of discrete‐time singular systems with norm‐bounded uncertainties and interval time‐varying delays in state and input. A new bounded real lemma for discrete‐time singular systems with a pair of time‐varying interval state delays is first investigated. Mathematical comparisons of the new bounded real lemma and two existing ones are presented. Then, on the basis of the bounded real lemma proposed here, a sufficient condition in the form of nonlinear matrix inequality, such that the considered state feedback robust H _∞ control problem is solvable, is given. In order to solve the nonlinear matrix inequality, a cone complementarity linearization algorithm is offered. Several numerical examples are presented to show the applicability of the proposed approach. Copyright © 2012 John Wiley & Sons, Ltd.     

Unified approach for open‐loop optimal control

We develop a unified approach for the necessary conditions for optimization of open‐loop control systems, starting from the basic principles of calculus of variations. The unified approach results are simultaneously applicable to both shift (q)‐operator‐based, discrete‐time systems and the derivative (d/dt)‐operator‐based, continuous‐time systems. It is shown that the optimal condition results that are now obtained separately for continuous‐time and discrete‐time systems can be easily obtained from the unified approach. An illustrative example is given. Copyright © 2006 John Wiley & Sons, Ltd.     

Event‐triggered and guaranteed cost finite‐time H∞ control for uncertain switched linear systems

This paper is concerned with the event‐triggered and guaranteed cost finite‐time H_∞ control problem for uncertain switched linear systems with exogenous disturbance. Instead of one common strategy, multiple event‐triggering strategies are first proposed, ie, each subsystem possesses its own corresponding event‐triggering substrategy. Then, by utilizing the multiple Lyapunov functions and average dwell‐time method, sufficient conditions for the finite‐time boundedness with an H_∞ performance level of the resulting event‐triggered switched closed‐loop system are derived. Meanwhile, a certain upper bound of the guaranteed cost function with respect to the system uncertainties is obtained. Subsequently, a set of sufficient conditions in terms of linear matrix inequalities is given for solving the event‐triggered and guaranteed cost finite‐time H_∞ state feedback controllers. Furthermore, the Zeno sampling behavior is excluded by presenting a positive lower bound estimation on the interexecution intervals. Finally, numerical simulations are provided to demonstrate the effectiveness of the proposed approach.     

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.     

Design of simultaneous static output feedback low‐gain H∞ controllers for a collection of time‐delay system

This paper considers the design of simultaneous static output feedback controllers for a finite collection of time‐delay linear systems. By solving a minimization problem, we try to find an output feedback low‐gain controller such that all resultant closed‐loop time‐delay systems are internally stable and satisfy a prespecified H_∞‐norm requirement. Based on the barrier method, necessary conditions for local optimum of the minimization problem are derived. An example is given for illustration.     

Controller design for optimal tracking response in discrete‐time systems

The problem of designing a controller, which results in a closed‐loop system response with optimal time‐domain characteristics, is considered. In the approach presented in this paper, the controller order is fixed (higher than pole‐placement order) and we seek a controller that results in closed‐loop poles at certain desired and pre‐specified locations; while at the same time the output tracks the reference input in an optimal way. The optimality is measured by requiring certain norms on the error sequence—between the reference and output signals—to be minimum. Several norms are used. First, l₂‐norm is used and the optimal solution is computed in one step of calculations. Second, l_∞‐norm (i.e. minimal overshot) is considered and the solution is obtained by solving a constrained affine minimax optimization problem. Third, the l₁‐norm (which corresponds to the integral absolute error‐(IAE)‐criterion) is used and linear programming techniques are utilized to solve the problem. The important case of finite settling time (i.e. deadbeat response) is studied as a special case. Examples that illustrate the different design algorithms and demonstrate their feasibility are presented. Copyright © 2007 John Wiley & Sons, Ltd.     

Robust finite‐time boundedness of H∞ filtering for switched systems with time‐varying delay

For switched system, switching behavior always affects the finite‐time stability property, which was neglected by most previous research. This paper investigates the problem of robust finite‐time boundedness of H_∞ filtering for switched systems with time‐varying delay. Sufficient conditions that can ensure finite‐time bounded and H_∞ filtering finite‐time stability are derived. Based on the average dwell‐time approach, the closed‐loop system trajectory stays within a prescribed bound. At last, numerical examples are given to illustrate the efficiency of the proposed method. Copyright © 2015 John Wiley & Sons, Ltd.     

Robust fixed‐order H2 controller for micro air vehicle—design and validation

Design and real‐time validation of a robust fixed‐order H₂ optimal controller for micro aerial vehicle, named Sarika‐2, is presented. Strengthened discrete optimal projection equations, which approximate the first‐order necessary optimality condition, are used for the controller design. Effect of low‐frequency gust disturbance and high‐frequency sensor noise is alleviated through the output sensitivity and control sensitivity minimization. The novelty of this paper is that a single robust H₂ controller, which is designed at the central operating point, ensures simultaneous stabilization of the radio‐controlled aircraft over the entire cruise speed range of 16–26 m/s. The controller is implemented on a digital‐signal‐processor‐based flight computer, and subsequently, it is validated through the real‐time hardware‐in‐loop‐simulation. The responses obtained from the hardware‐in‐loop‐simulation compares well with those obtained from the off‐line simulation. Copyright © 2006 John Wiley & Sons, Ltd.     

Analytical design of stable delta‐domain generalized predictive control

This paper addresses certain fundamental issues related to the discrete‐time design problem of the delta‐domain generalized predictive control (δ‐GPC) for both minimum phase and non‐minimum phase linear SISO plants including nominal stability and nominal performance of the closed‐loop system. The approach being presented is completely analytical, and the nominal performance of the control system is directly achieved by a prototype design of the closed‐loop system characteristics resulting in definite time‐domain specifications. Two design methods are offered in which a model‐based prediction paradigm is applied to achieve the future output and the future filtered output trajectory of the plant. Prediction of the first type is based on suitable emulations of the output δ‐derivatives and is used in the GPC controller design for minimum‐phase models of the plant. Prediction of the second type utilizes emulation of derivatives of the output filtered by the numerator polynomial of the transfer function of the controlled part of the plant. It can be employed both for minimum phase and non‐minimum phase plants. A numerical example is given that illustrates the δ‐GPC method for controller design. Copyright © 2002 John Wiley & Sons, Ltd.     

Near‐optimal control for multiparameter singularly perturbed stochastic systems

In this paper, the linear quadratic optimal stochastic control problem is investigated for multiparameter singularly perturbed stochastic systems in which N lower‐level fast subsystems are interconnected by a higher‐level slow subsystem. After establishing the asymptotic structure of the solution for the multiparameter stochastic algebraic Riccati equation (MSARE), a near‐optimal controller that is independent of small unknown parameters is obtained by neglecting these parameters. The stability of a closed‐loop stochastic system is investigated. Furthermore, it is shown that the resulting controller achieves an O(∥ν∥²) approximation to the optimal cost of the original optimal control problem. Finally, in order to demonstrate the efficiency of the proposed algorithm, a numerical example—a practical multi‐area power system—is solved. Copyright © 2010 John Wiley & Sons, Ltd.