Optimal control for linear systems with time delay in control input

This paper presents the optimal regulator for a linear system with time delay in control input and a quadratic cost function. The optimal regulator equations are obtained using the duality principle, which is applied to the optimal filter for linear systems with time delay in observations, and then proved using the maximum principle. Performance of the obtained optimal regulator is verified in the illustrative example against the best linear regulator available for linear systems without delays. Simulation graphs and comparison tables demonstrating better performance of the obtained optimal regulator are included.     

Adaptive robust approximate constraint-following control for mechanical systems

We formulate control problems of mechanical systems as constraint following. The system contains uncertainty which is (possibly fast) time-varying. It is unknown but bounded. The bound is, however, unknown. The objective is to design control which renders approximate constraint following. Adaptive laws are constructed. The controls are then based on the adaptive parameters. We are able to demonstrate guaranteed system performance, regardless of the uncertainty.     

Optimal LQG controller for linear stochastic systems with unknown parameters

This paper presents the optimal LQG controller for linear systems with unknown parameters. The optimal controller equations are obtained using the separation principle, whose applicability to the considered problem is substantiated. Performance of the obtained optimal controller is verified in the illustrative example against the conventional LQG controller that is optimal for linear systems with known parameters. Simulation graphs verifying overall performance and computational accuracy of the designed optimal controller are included.     

Optimal control of linear delay systems via hybrid of block-pulse and Legendre polynomials

A method for finding the optimal control of a linear time varying delay system with quadratic performance index is discussed. The properties of the hybrid functions which consists of block-pulse functions plus Legendre polynomials are presented. The operational matrices of integration, delay and product are utilized to reduce the solution of optimal control to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.     

Numerical solutions of optimal control for linear time-varying systems with delays via hybrid functions

By applying hybrid functions of general block-pulse functions and Legendre polynomials, the linear-quadratic problem of linear time-varying systems with delays are transformed into the optimization problem of multivariate functions. The approximate solutions of the optimal control and state as well as the optimal value of the objective functional are derived. The numerical examples illustrate that the algorithms are valid.     

Optimal controller for uncertain stochastic polynomial systems

This paper presents the optimal quadratic-Gaussian controller for uncertain stochastic polynomial systems with linear control input and a quadratic criterion over linear observations. The optimal closed-form controller equations are obtained using the separation principle, whose applicability to the considered problem is substantiated. As intermediate results, the paper gives closed-form solutions of the optimal regulator and controller problems for stochastic polynomial systems with linear control input and a quadratic criterion. Performance of the obtained optimal controller is verified in the illustrative example against the conventional quadratic-Gaussian controller that is optimal for stochastic polynomial systems with known parameters. Simulation graphs demonstrating overall performance and computational accuracy of the designed optimal controller are included.     

Observer-based sliding mode control for a class of discrete systems via delta operator approach

In this paper, an observer-based sliding mode control (SMC) problem is investigated for a class of uncertain delta operator systems with nonlinear exogenous disturbance. A novel robust stability condition is obtained for a sliding mode dynamics by using Lyapunov theory in delta domain. Based on a designed sliding mode observer, a sliding mode controller is synthesized by employing SMC theory combined with reaching law technique. The robust asymptotical stability problem is also discussed for the closed-loop system composed of the observer dynamics and the state estimation error dynamics. Furthermore, the reachability of sliding surfaces is also investigated in state-estimate space and estimation error space, respectively. Finally, a numerical example is given to illustrate the feasibility and effectiveness of the developed method.     

Sliding mode regulator as solution to optimal control problem for non-linear polynomial systems

This paper presents the optimal control problem for a non-linear polynomial system with respect to a Bolza-Meyer criterion with a non-quadratic non-integral term. The optimal solution is obtained as a sliding mode control, whereas the conventional polynomial-quadratic regulator does not lead to a causal solution and, therefore, fails. Performance of the obtained optimal controller is verified in the illustrative example against the conventional polynomial-quadratic regulator that is optimal for the quadratic Bolza-Meyer criterion. The simulation results confirm an advantage in favor of the designed sliding mode control.     

Bond graph formulation of an optimal control problem for linear time invariant systems

A recent communication has proposed a conjectural procedure for representing a category of optimal control problems in bond graph language [W. Marquis-Favre, B. Chereji, D. Thomasset, S. Scavarda, Bond graph representation of an optimal control problem: the dc motor example, in: ICBGM’05 International Conference of Bond Graph Modelling and Simulation, New Orleans, USA, January 23-27, 2005, pp. 239-244]. This paper aims at providing a fundamental theory for proving the effectiveness of this procedure. The class of problem that the procedure can deal with has been extended. Its application was formerly restricted to linear time invariant siso system. The systems considered now are linear time invariant mimo systems. The optimization objective is the minimization of dissipation and input. The developments concerning the optimal control problem are based on the Pontryagin maximum principle and the proof of the effectiveness of the procedure makes a broad use of the port-Hamiltonian concept. As a result, the bond graph representation of the given optimization problem enables the analytical system, which provides the optimal solution, to be derived. The work presented in this paper is the first step in research with perspectives towards formulating dynamic optimization problems in bond graph and, towards coupling this formulation with a sizing methodology using bond graph language and a state-space inverse model approach. This sizing methodology, however, is not the topic of this paper and thus is not presented here.     

Optimal control in unobservable integral Volterra systems

This paper presents solution of the optimal linear-quadratic controller problem for unobservable integral Volterra systems with continuous/discontinuous states under deterministic uncertainties, over continuous/discontinuous observations. Due to the separation principle for integral systems, the initial continuous problem is split into the optimal minmax filtering problem for integral Volterra systems with deterministic uncertainties over continuous/discontinuous observations and the optimal linear-quadratic control (regulator) problem for observable deterministic integral Volterra systems with continuous/discontinuous states. As a result, the system of the optimal controller equations are obtained, including the linear equation for the optimally controlled minmax estimate and two Riccati equations for its ellipsoid matrix (optimal gain matrix of the filter) and the optimal regulator gain matrix. Then, in the discontinuous problems, the equation for the optimal controller and the equations for the optimal filter and regulator gain matrices are obtained using the filtering procedure for deriving the filtering equations over discontinuous observations proceeding from the known filtering equations over continuous ones and the dual results in the optimal control problem for integral systems. The technical example illustrating application of the obtained results is finally given.     

An intelligent self-repairing control for nonlinear MIMO systems via adaptive sliding mode control technology

In this paper, an intelligent self-repairing control scheme is proposed for a class of nonlinear MIMO system. A direct self-repairing controller of a nonlinear SISO system is firstly designed, and then the control scheme is promoted to a nonlinear MIMO system. The error signals are replaced by the state variables to deal with the high derivate problems of the desired signals and a nonlinear regulating function is brought in to improve the performances of the sliding mode. The self-repairing controller is made up of four parts: the nonlinear regulator, the equal controller, the compensator I and the compensator II. The control method is applied to a helicopter flight control system with loss-in-effectiveness faults. Some simulation results illustrate the effectiveness and feasibility of the proposed control scheme in the paper.     

Unknown input observer for linear non-minimum phase systems

An unknown input observer is to estimate the system state of a dynamic system subject to unknown input excitations. In this note, by assuming that at each time instant, the unknown input can be approximated by a polynomial over a local time interval, a finite-time observer is proposed to achieve approximate joint state and input estimation. Both the obtained state and input estimates are moving averages of the present and past output signals. The advantage of the proposed design is that it can be applied to non-minimum phase systems or systems with non-unity relative degree. Notice that most previous unknown input observer designs require the system to be minimum-phase and relative degree one.     

Robust adaptive sliding mode control for uncertain discrete-time systems with time delay

This paper focuses on robust adaptive sliding mode control for discrete-time state-delay systems with mismatched uncertainties and external disturbances. The uncertainties and disturbances are assumed to be norm-bounded but the bound is not necessarily known. Sufficient conditions for the existence of linear sliding surfaces are derived within the linear matrix inequalities (LMIs) framework by employing the free weighting matrices proposed in He et al. (2008) [3], by which the corresponding adaptive controller is also designed to guarantee the state variables to converge into a residual set of the origin by estimating the unknown upper bound of the uncertainties and disturbances. Also, simulation results are presented to illustrate the effectiveness of the control strategy.     

Robust stable pole-placement-based adaptive control of continuous linear systems with two parametrical estimation schemes

This paper deals with the pole-placement-type robust adaptive control of continuous linear systems in the presence of bounded noise and a common class of unmodeled dynamics provided that two estimation schemes are used in parallel. Both estimation schemes are introduced in order to minimize the plant identification error by selecting, as plant parameter estimates, a convex combination of both parameter estimates which leads to the selection of one of the estimation schemes, via a switching rule, on time intervals of at least a minimum prefixed residence duration. The weights of the individual parameter vector estimates are provided at each time by an optimization or suboptimization scheme for a quadratic loss function of the possibly filtered tracking error and/or control input. The robust stability of the overall adaptive scheme is ensured by an adaptation relative dead zone which takes into account the contribution of the unmodeled dynamics and bounded noise. The basic results are derived for two different estimation strategies which have either a shared regressor with the plant or individual regressors for the input contribution and its contributed derivatives. In this second case, the plant input is obtained from a similar convex combination rule as the one used for the estimators in the first approach. An extension of the basic strategies is also pointed out including a combined use of the (sub) optimization scheme with a supervisor of past measures for the on-line calculation of the estimator weights in the convex combination. Finally, the extension of the scheme for the use of any number of parametrical estimators is focused on.     

Mixed order robust adaptive control for general linear time invariant systems

We provide a solution to the adaptive control problem of an unknown linear system of a given derivation order, using a reference model or desired poles defined in a possibly different derivation order and employing continuous adjustment of parameters ruled by possibly another different derivation order. To this purpose, we present an extension for the fractional settings of the Bezout’s lemma and gradient steepest descent adjustment. We analyze both the direct and indirect approaches to adaptive control. We discuss some robustness advantages/disadvantages of the fractional adjustment of parameters in comparison with the integer one and, through simulations, the possibility to define optimal derivation order controllers.     

Model reference control of LPV systems

This paper deals with the problem of model reference control for linear parameter varying (LPV) systems. The LPV systems under consideration depend on a set of parameters that are bounded and available online. The main contribution of this paper is to design an LPV model reference control scheme for LPV systems whose state-space matrices depend affinely on a set of time-varying parameters that are bounded and available online. The design problem is divided into two subproblems: the design of the coefficient matrices of the controller and the design of the gain of the state feedback controller for LPV systems. The singular value decomposition is used to obtain the coefficient matrices, while the linear matrix inequality methodology is used to obtain the parameter-dependent state feedback gain of the control scheme. A simple numerical example is used to illustrate the proposed design and a coupled-tank process example is used to demonstrate the usefulness and practicality of the proposed design. Simulation and experimental results indicate that the proposed scheme works well.     

Optimal boundary control of heat conduction problems on an infinite time domain by control parameterization

The present paper deals with an optimal boundary control problem in which the process of systems under consideration is governed by a linear parabolic partial differential equation over an infinite time interval. The objective of the paper is to determine the optimal boundary control that minimize a given energy-based performance measure. The performance measure is specified as a quadratic functional of displacement and a suitable penalty term involving the boundary controls. In order to determine the optimal boundary controls, the problem with boundary controls are converted into a problem with distributed controls. The modal space technique is then used to reduce the system into the optimal control of time invariant lumped parameter system. The associated system of uncoupled first order initial value problems is solved in terms of controllers. Next step deals with the computation of the control and trajectory of the linear time-invariant lumped parameter. For this we approximate the controllers by a finite number of orthogonal exponential zero-interpolants over the interval [0,∞). The resultant performance index after using the optimality condition leads to a system of linear algebraic equations. The suggested technique is easy to implement on digital computer. We provide a numerical example to demonstrate the applicability and efficiency of the proposed approach.     

Robust control for discrete-time singular large-scale systems with parameter uncertainty

This paper addresses the problem of robust stabilization for uncertain discrete-time singular large-scale systems with parameter uncertainties. The system under consideration is not necessarily regular. The parameter uncertainties are assumed to be time invariant, but norm-bounded. The purpose of the robust stabilization problem is to design state feedback controllers such that, for all admissible uncertainties, the closed-loop system is regular, causal and stable. In terms of strict LMIs, sufficient conditions for the solvability of the problem is presented, and the parameterization of desired state feedback controllers is also given. A numerical example is given to demonstrate the applicability of the proposed design approach.     

Stochastic fault tolerant control of networked control systems

A problem of stabilization about uncertain networked control systems (NCSs) with random but bounded delays is discussed in this paper. By using augmented state-space method, this class of problems can be modeled as discrete-time jump linear systems governed by finite-state Markov chains. A new switched model based on probability is proposed to research problems of reliable control when actuators become ageing or partially disabled. Using improved V-K iteration algorithm, a class of reliable controllers are designed to make systems asymptotically mean square stable under several stochastic disturbances such as random time-delay and stochastic actuator failure and the maximal redundancy degree is given through this method.