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We consider a class of nonlinear stochastic systems with square-root nonlinearities appearing in the diffusion terms. The optimal control problems with indefinite quadratic criteria in both finite ...

We study a kind of time-inconsistent linear-quadratic non-zero sum stochastic differential game problems with random jumps. The time-inconsistency arises from the presence of a quadratic term of th...

This paper studies the stability of the feedback interconnection of discrete-time negative imaginary (D-NI) systems through integral quadratic constraints (IQCs). Applying the latest IQC-based resu...

We revisit the issue of quadratic gain-scheduled and robust state-feedback control with a focus on linear matrix inequalities widely used in the literature. It has been established that for...

This paper provides a link between time-domain and frequency-domain stability results in the literature. Specifically, we focus on the comparison between stability results for a feedback interconnection of two nonlinear systems stated in terms of frequency-domain conditions. While the Integral Quadratic Constrain (IQC) theorem can cope with them via a homotopy argument for the Lurye problem, graph separation results require the transformation of the frequency-domain conditions into truncated time-domain conditions. To date, much of the literature focuses on "hard" factorizations of the multiplier, considering only one of the two frequency-domain conditions. Here it is shown that a symmetric, "doubly-hard" factorization is required to convert both frequency-domain conditions into truncated time-domain conditions. By using the appropriate factorization, a novel comparison between the results obtained by IQC and separation theories is then provided. As a result, we identify under what conditions the IQC theorem may provide some advantage.

This paper considers distributed systems arising when multiple agents cooperatively solve a quadratic optimisation problem. To maintain privacy of their states over time, agents implement a noise-a...

This technical note addresses the new nonlinear protocol class of doubly stochastic quadratic operators (DSQOs) for coordination of consensus problem in multi-agent systems (MAS). We derive the conditions for ensuring that every agent reaches consensus on a desired rate of the group's decision where the group decision value in its agent's initial statuses varies. Besides that, we investigate a nonlinear protocol sub-class of extreme DSQO (EDSQO) to reach a consensus for MAS to a common value with nonlinear low-complexity rules and fast time convergence if the interactions for each agent are not selfish. In addition, to extend the results to reach a consensus and to avoid the selfish case we specify a general class of DSQO for reaching a consensus under any given case of initial states. The case that MAS reach a consensus by DSQO is if each member of the agent group has positive interactions of DSQO (PDSQO) with the others. The convergence of both EDSQO and PDSQO classes is found to be directed tow...

Linear model predictive control (MPC) can be currently deployed at outstanding speeds, thanks to recent progress in algorithms for solving online the underlying structured quadratic programs. In contrast, nonlinear MPC (NMPC) requires the deployment of more elaborate algorithms, which require longer computation times than linear MPC. Nonetheless, computational speeds for NMPC comparable to those of MPC are now regularly reported, provided that the adequate algorithms are used. In this paper, we aim at clarifying the similarities and differences between linear MPC and NMPC. In particular, we focus our analysis on NMPC based on the real-time iteration (RTI) scheme, as this technique has been successfully tested and, in some applications, requires computational times that are only marginally larger than linear MPC. The goal of the paper is to promote the understanding of RTI-based NMPC within the linear MPC community.

Recent development of contraction theory-based analysis has opened the door for inspecting differential behaviour of singularly perturbed systems. In this paper, a contraction theory-based framework is proposed for stabilisation of singularly perturbed systems. The primary objective is to design a feedback controller to achieve bounded tracking error for both standard and non-standard singularly perturbed systems. This framework provides relaxation over traditional quadratic Lyapunov-based method as there is no need to satisfy interconnection conditions during controller design algorithm. Moreover, the stability bound does not depend on smallness of singularly perturbed parameter and robust to additive bounded uncertainties. Combined with high gain scaling, the proposed technique is shown to assure contraction of approximate feedback linearisable systems. These findings extend the class of nonlinear systems which can be made contracting.

This paper develops the problems of local stability and stabilisation for time-varying polytopic quadratic systems. The state-space data is assumed to be dependent on parameters that are measurable in real time and vary in a compact set with bounded variation rates. These problems are solved by utilising the parameter-dependent quadratic Lyapunov function and S-procedure approach. Sufficient conditions for local stability and stabilisation are first formulated as optimisation problems with ‘quasi’ linear matrix inequalities. Simulation examples are then provided to confirm the effectiveness of the given approach.

The output feedback robust model predictive control (MPC), for the linear parameter varying (LPV) system with norm-bounded disturbance, is addressed, where the model parametric matrices are only known to be bounded within a polytope. The previous techniques of norm-bounding technique, quadratic boundedness (QB), dynamic output feedback, and ellipsoid (true-state bound; TSB) refreshment formula for guaranteeing recursive feasibility, are fused into the newly proposed approaches. In the notion of QB, the full Lyapunov matrix is applied for the first time in this context. The single-step dynamic output feedback robust MPC, where the infinite-horizon control moves are parameterised as a dynamic output feedback law, is the main topic of this paper, while the multi-step method is also suggested. In order to strictly guarantee the physical constraints, the outer bound of the true state replaces the true state itself, so tightness of this bound has a major effect on the control performance. In order to ti...

The paper provides a direct solution to the determinantal assignment problem (DAP) which unifies all frequency assignment problems of the linear control theory. The current approach is based on the solvability of the exterior equation (Formula presented.) where (Formula presented.) is an n −dimensional vector space over (Formula presented.) which is an integral part of the solution of DAP. New criteria for existence of solution and their computation based on the properties of structured matrices are referred to as Grassmann matrices. The solvability of this exterior equation is referred to as decomposability of (Formula presented.), and it is in turn characterised by the set of quadratic Plücker relations (QPRs) describing the Grassmann variety of the corresponding projective space. Alternative new tests for decomposability of the multi-vector (Formula presented.) are given in terms of the rank properties of the Grassmann matrix, (Formula presented.) of the vector (Formula presented.), which is constructed by the coordinates of (Formula presented.). It is shown that the exterior equation is solvable ((Formula presented.) is decomposable), if and only if (Formula presented.) where (Formula presented.); the solution space for a decomposable (Formula presented.), is the space (Formula presented.). This provides an alternative linear algebra characterisation of the decomposability problem and of the Grassmann variety to that defined by the QPRs. Further properties of the Grassmann matrices are explored by defining the Hodge–Grassmann matrix as the dual of the Grassmann matrix. The connections of the Hodge–Grassmann matrix to the solution of exterior equations are examined, and an alternative new characterisation of decomposability is given in terms of the dimension of its image space. The framework based on the Grassmann matrices provides the means for the development of a new computational method for the solutions of the exact DAP (when such solutions exist), as well as computing approximate solutions, when exact solutions do not exist.

In this journal, Pantoja has described a deterministic optimal control problem in which his stagewise Newton procedure yields an exact optimal solution whereas differential dynamic programming (DDP) does not. This problem is also quoted by Coleman and Liao (in another journal) as a correct instance with some emphasis on the advantage of Pantoja's procedure over DDP. Pantoja argues that the problem involves nonlinear dynamics in his terminal-cost problem formulation, and therefore DDP and stagewise Newton methods are different. The purpose of this paper is to show that, while for a general nonlinear optimal control problem DDP and Pantoja's method differ, his problem has a special structure such that it is a false example of this claim; more specifically, the reason is twofold. First, he made an obvious algebraic error in his computation. Second, his example is equivalent to a problem of linear dynamics and quadratic criterion (LQ in short). It is true that when a general LQ that involves quadratic stage c...

International audience; This paper proposes a new predictive controller approach for nonlinear process based on a reduced complexity homogeneous, quadratic discretetime Volterra model called quadratic S-PARAFAC Volterra model. The proposed model is yielded by using the symmetry property of the Volterra kernels and their tensor decomposition using the PARAFAC technique which provide a parametric reduction compared to the conventional Volterra model. This property allows synthesizing a new nonlinear model based predictive control (NMBPC). We develop the general form of a new predictor and so, we propose an optimization algorithm formulated as a Quadratic Programming (QP) under linear and nonlinear constraints. The performances of the proposed quadratic S-PARAFAC Volterra model and the developed NMBPC algorithm are illustrated on a numerical simulation and validated on a benchmark as a continuous Stirred Tank Reactor (CSTR) system. Moreover the efficiency of the proposed quadratic SPARAFAC Volterra model and the NMBPC approach are validated on an experimental Communicating Two Tank system (CTTS).

The optimal feedforward torque control problem is tackled and solved analytically for synchronous machines while stator resistance and cross-coupling inductance are explicitly considered. Analytical solutions for the direct and quadrature optimal reference currents are found for all major operation strategies such as Maximum Torque per Current (MTPC) (or often called as Maximum Torque per Ampere (MTPA)), Maximum Current (MC), Field Weakening (FW), Maximum Torque per Voltage (MTPV) and Maximum Torque per Flux (MTPF). Numerical methods (approximating the optimal solutions only) or simplifying assumptions (neglecting stator resistance and/or cross-coupling inductance) are no longer necessary. The presented results are based on one simple idea: all optimization problems (e.g. MTPC, MTPV or MTPF) with their respective constraints (e.g. current or voltage limit) and the computation of the intersection point(s) of voltage ellipse, current circle, or torque, MTPC, MTPV and MTPF hyperbolas are reformulated implicitly as quadrics (quadratic surfaces) which allow to solve the feedforward torque control problem by invoking the Lagrangian formalism and by finding the roots of a fourth-order polynomial analytically. The proposed solutions are applicable to any anisotropic (or isotropic) synchronous machine independent of the underlying current control strategy.

The aim of this paper is to study regional quadratic control problem for a distributed bilinear system evolving in a spatial domain Ω. The question is to obtain a bounded feedback control that drives such a system from an initial state to a desired one in finite time, only on a subregion ω of Ω, and minimises a quadratic functional cost. Our purpose is to prove that an optimal control exists, and characterised as solution of an optimality system. The approach is successfully illustrated by simulations.

The goal of this paper is to study the switched stochastic control problem of discrete-time linear systems with multiplicative noises. We consider both the quadratic and the H∞ criteria for the performance evaluation. Initially we present a sufficient condition based on some Lyapunov–Metzler inequalities to guarantee the stochastic stability of the switching system. Moreover, we derive a sufficient condition for obtaining a Metzler matrix that will satisfy the Lyapunov–Metzler inequalities by directly solving a set of linear matrix inequalities, and not bilinear matrix inequalities as usual in the literature of switched systems. We believe that this result is an interesting contribution on its own. In the sequel we present sufficient conditions, again based on Lyapunov–Metzler inequalities, to obtain the state feedback gains and the switching rule so that the closed loop system is stochastically stable and the quadratic and H∞ performance costs are bounded above by a constant value. These results are illu...

In this article, we develop a predictive tracking controller for a nonlinear multi-compartment lung mechanics model. Specifically, for a given clinically plausible reference volume pattern, we use model predictive control to design a tracking controller that accounts for amplitude and rate input constraints. The predictive control law is derived by minimising a quadratic performance criterion involving a prediction of the system response over a prescribed time step. The proposed tracking control framework is applied to a two-compartment lung mechanics model with nonlinear lung compliance parameters.

Self-excited oscillation is becoming a major issue in low-emission, lean partially premixed combustion systems, and active control has been shown to be a feasible method to suppress such instabilities. A number of robust control methods are employed to obtain a feedback controller and it is observed that the robustness to system uncertainty is significantly better for a low complexity controller in spite of the norms being similar. Moreover, we demonstrate that closed-loop stability for such a complex system can be proved via use of the integral quadratic constraint method. Open- and closed-loop nonlinear simulations are provided.

This paper deals with the controller design problem of nonlinear quadratic Markov jumping systems with input saturation. Both mode-dependent and mode-independent state feedback controllers are designed. By using the concept of domain of attraction in mean square sense, sufficient conditions for stochastic stabilisation for nonlinear quadratic systems are derived in terms of linear matrix inequalities. Certain existing results are improved. Simulation examples are presented to illustrate the effectiveness of the proposed technique.

A key component in enabling the application of model predictive control (MPC) in fields such as automotive, aerospace, and factory automation is the availability of low-complexity fast optimisation algorithms to solve the MPC finite horizon optimal control problem in architectures with reduced computational capabilities. In this paper, we introduce a projection-free iterative optimisation algorithm and discuss its application to linear MPC. The algorithm, originally developed by Brand for non-negative quadratic programs, is based on a multiplicative update rule and it is shown to converge to a fixed point which is the optimum. An acceleration technique based on a projection-free line search is also introduced, to speed-up the convergence to the optimum. The algorithm is applied to MPC through the dual of the quadratic program (QP) formulated from the MPC finite time optimal control problem. We discuss how termination conditions with guaranteed degree of suboptimality can be enforced, and how the algorithm...

This paper considers model predictive control of a class of piecewise affine systems and introduces a method to reduce the complexity of the optimisation problem which is solved online. The optimisation problem is a mixed-integer quadratic program which is parameterised by the initial state of the system dynamics. The complexity of the problem solved online is reduced by moving part of the computational burden offline. In the offline computations, the structure of the mixed-integer problem is explored to determine a polytopic partition of the set of initial states. Each polytope of the partition is then considered and it is determined which integer combinations can/cannot be optimal, given that the initial state is contained in the current polytope. The result of the offline computations is that each polytope is associated with the set of potentially optimal integer combinations. The online solution of the mixed-integer problem consists of a two-step procedure: first, it is determined which polytope conta...

The paper describes a substantial extension of norm optimal iterative learning control (NOILC) that permits tracking of a class of finite dimensional reference signals whilst simultaneously converging to the solution of a constrained quadratic optimisation problem. The theory is presented in a general functional analytical framework using operators between chosen real Hilbert spaces. This is applied to solve problems in continuous time where tracking is only required at selected intermediate points of the time interval but, simultaneously, the solution is required to minimise a specified quadratic objective function of the input signals and chosen auxiliary (state) variables. Applications to the discrete time case, including the case of multi-rate sampling, are also summarised. The algorithms are motivated by practical need and provide a methodology for reducing undesirable effects such as payload spillage, vibration tendencies and actuator wear whilst maintaining the desired tracking accuracy necessary f...

This paper addresses the stability characterisation problem for a class of nonlinear discrete-time control systems subject to actuator saturation and energy bounded disturbance inputs. The considered class of systems covers all nonlinear discrete-time systems that can be modelled by rational difference equations. Based on quadratic and piecewise quadratic Lyapunov functions, conditions based on linear matrix inequalities are proposed to analyse the asymptotic stability (internal stability) and the l2 input-to-state stability (external stability) of the closed-loop system. The proposed conditions are then incorporated into convex optimisation problems to either maximise an estimate of the region of attraction or a bound on the admissible l2 disturbances, and also to obtain an estimate of the system l2-gain for an admissible set of disturbances.

In this article, we revisit the problem of inverse optimality for linear systems. By applying certain explicit formulae for coprime matrix fraction descriptions (CMFD) of linear systems, we propose a necessary and sufficient condition for a given stable state feedback law to be optimal for some quadratic performance index. Compared to existing results in the literature, the proposed condition is simpler to check and interpret. Moreover, it reduces the redundancy in the solutions of the associated algebraic Riccati equation (ARE). As a direct application of the proposed results, we consider the problem of inverse optimality of observer-based state feedback. To be specific, for the case where the state is not fully known, we consider the inverse optimality problem of an observer-based state feedback for the closed-loop system augmented by an observer. More precisely, it is shown that the observer-based state feedback is inverse optimal for the closed-loop system with some general forms of cost functions, on...

Minimum energy problems involving linear systems with quadratic performance criteria are classical in optimal control theory. The case where controls are constrained is discussed in Athans and Falb (1966) [Athans, M. and Falb, P.L. (1966), Optimal Control: An Introduction to the Theory and Its Applications, New York: McGraw-Hill Book Co.] who obtain a componentwise optimal control expression involving a saturation function expression. We show why the given expression is not generally optimal in the case where the dimension of the control is greater than one and provide a numerical counterexample.

This article is concerned with a class of continuous-time Markovian jump linear systems where the regime transition rates are determined through an initial policy. The purpose is to study the jump linear quadratic regulator for such class of systems. A two-level regulating method is adopted to design the state-feedback controller and the policy separately. The problem of tuning the existing policy with respect to a prescribed quadratic performance criterion is formulated as a gradient projection-based iterative optimisation. This method leads to a near-optimal policy whose performance is better than that of the initial one. Useful characteristics of such a policy are also developed. The efficiency of the main results is elucidated by two numerical examples.

This article studies the realisability property of discrete-time bilinear and quadratic input–output (i/o) equations in the classical state-space form. Constraints on the parameters of the i/o model are suggested that lead to realisable models. Using new formulae for computing basis vectors of certain vector spaces of differential one-forms, we present in this article the complete list of the third- and fourth-order realisable i/o bilinear models, and a new realisable subclass of an arbitrary order is suggested. Moreover, we provide the sufficient conditions of the second- and third-order realisable i/o quadratic models, respectively. All the developed theory and algorithms are illustrated by means of several examples.

In this article, the problem of the numerical computation of the stabilising solution of the game theoretic algebraic Riccati equation is investigated. The Riccati equation under consideration occurs in connection with the solution of the H ∞ control problem for a class of stochastic systems affected by state-dependent and control-dependent white noise and subjected to Markovian jumping. The stabilising solution of the considered game theoretic Riccati equation is obtained as a limit of a sequence of approximations constructed based on stabilising solutions of a sequence of algebraic Riccati equations of stochastic control with definite sign of the quadratic part. The proposed algorithm extends to this general framework the method proposed in Lanzon, Feng, Anderson, and Rotkowitz (Lanzon, A., Feng, Y., Anderson, B.D.O., and Rotkowitz, M. (2008), ‘Computing the Positive Stabilizing Solution to Algebraic Riccati Equations with an Indefinite Quadratic Term Viaa Recursive Method,’ IEEE Transactions on Automat...

In this article we investigate a Lyapunov approach to the stability of finite-dimensional 2D systems. We use the behavioural framework and consider a notion of stability following the ideas in Pillai and Shankar [H. Pillai and S. Shankar (1998). A Behavioral Approach to Control of Distributed Systems, SIAM Journal of Control and Optimization, 37, 388–408], Rocha [P. Rocha (2008). Stabilization of Multidimensional Behaviors, Multidimensional Systems and Signal Processing, 19, 273–286], Valcher [M. Valcher (2000). Characteristic Cones and Stability Properties of Two-dimensional Autonomous Behaviors, IEEE Transactions on Circuits and Systems, Part I, CAS-47, 290–302]. We characterise stability in terms of the existence of a (quadratic) Lyapunov function and provide a constructive algorithm for the computation of all such Lyapunov functions.

The purpose of this article is to provide a numerically efficient method for the quadratic stabilisation of a class of linear, discrete-time, uncertain, time-varying systems. The considered class of systems is characterised by an interval time-varying (ITV) matrix and constant sensor and actuator matrices. It is required to find a linear time-invariant (LTI) static output feedback controller yielding a quadratically stable closed-loop system independently of the parameter variation rate. The solvability conditions are stated in terms of linear matrix inequalities (LMIs). The set of LMIs includes the stability conditions for the feedback connection of a unique suitably defined extreme plant with an LTI output controller and the positivity of a closed-loop extremal matrix. A consequent noticeable feature of the article is that the total number of LMIs is independent of the number of uncertain parameters. This greatly enhances the numerical efficiency of the design procedure.

The objective of the work is twofold. First, we investigate the problem of observer design for a class of systems in cascade form. We suppose that an observer exists for each subsystem and then prove that this interconnection is an observer for the overall system. One of both nonlinear functions is assumed to be not Lipschitz (e.g. to satisfy a quadratic constraint on the unmeasured part of the state). This first main result may be viewed as an improvement of Besancon and Hammouri (Besancon, G., and Hammouri, H. (1998), ‘On Observers Design for Interconnected Systems’, Journal of Mathematical Systems, Estimation and Control, 8, 377–380) in which the authors discussed the problem of the synthesis of interconnected observers. The second part is a generalisation of Ammar, Mabrouk, and Vivalda (Ammar, S., Mabrouk, M., and Vivalda, J.-C. (2009), ‘Observer and Global Output Feedback Stabilisation for Some mechanical Systems’, International Journal of Control, 82, 1070–1081) in which the authors study the proble...

This article discusses feedback stabilisation of bilinear systems defined on a Hilbert state space. We show that stabilising such a system reduces stabilising only its projection on a suitable subspace. Then we give a new stabilising control that minimises a quadratic cost and allows the decay estimate of the optimal trajectories. An illustrating example is presented.

This article studies the realisability property of continuous-time quadratic input–output (i/o) equations in the classical state space form. Constraints on the parameters of the quadratic i/o model are suggested that lead to realisable models. The complete list of second- and third-order realisable i/o quadratic models is given and two subclasses of the n-th order realisable i/o quadratic systems are suggested. Our conditions rely basically upon the property that certain combinations of coefficients of the i/o equations are zero or not zero. We provide explicit state equations for realisable second-order quadratic i/o equations, and for one realisable subclass of quadratic i/o equations of arbitrary order.

The output feedback-based asymptotic stabilisation problem for a class of non-linear interconnected systems is considered, where the non-linear interconnections between subsystems satisfy quadratic constraints. A novel decentralised dynamic output feedback control strategy involving local projection operator-based reduced-order observers is proposed. This decentralised controller is characterised by the separation property. Two easily verifiable conditions that guarantee the asymptotic stability of the closed-loop system driven by the proposed decentralised controller are given. The effectiveness of the developed decentralised control strategy is illustrated with a numerical example and simulations.

This article revisits the problem of H ∞ filtering for discrete-time systems with a time-varying delay in the state and parameter uncertainties residing in a polytope. By utilising the polynomially parameter-dependent idea, a new filter design procedure is proposed, which formulates the existence of admissible robust H ∞ filters into a set of linear matrix inequalities. These conditions are developed based on homogeneous polynomially parameter-dependent matrices of an arbitrary degree. As the degree grows, test of increasing precision is obtained providing less conservative filter designs. It is established that the results in the quadratic framework (that entail fixed matrices for the entire uncertainty domain), and the linearly parameter-dependent framework (that use linear convex combinations of matrices) are special cases of the proposed conditions for the zeroth degree and the first degree, respectively. Moreover, in addition to parameter dependence, the obtained conditions are also dependent on both...

This paper proposes an algorithm for autonomous strategic mission planning of missions where multiple unhabitated underwater vehicles (UUVs) cooperate in order to solve one or more mission tasks. Missions of this type include multi-agent reconnaissance missions and multi-agent mine sweeping missions. The mission planning problem is posed as a receding horizon mixed–integer constrained quadratic optimal control problem. This problem is subsequently partitioned into smaller subproblems and solved in a parallel and decentralized manner using a distributed Nash-based game approach. The paper presents the development of the proposed algorithm and discusses its properties. An application example is used to further demonstrate the main characteristics of the proposed method.

This paper defines a class of system information—affine information—that includes both the dynamic residuals and some types of auxiliary information that can be used in system parameter estimation as special cases. The types of information that can be cast under the affine information format give rise to quadratic functions that measure the extent to which a model fits such information, and that can be aggregated in a single weighted quadratic cost functional. This allows the definition of a multiobjective methodology for parameter estimation in non-linear system identification, which allows taking into account any type of affine information. The results are presented in terms of a set of efficient solutions of the multiobjective estimation problem—such a solution set is more meaningful than a single model. Since any affine information leads to a convex (quadratic) functional, the whole set of efficient solutions is exactly accessible via the minimization of the quadratic functional with different weightings, via a least-squares minimization (a non-iterative, computationally inexpensive procedure). The decision stage, in which a single model is chosen from the Pareto-set, becomes well-defined with a single global solution. Residual variance, fixed point location, static function and static gain are shown to fit in the class of affine information. A buck DC-DC converter is used as example.

This paper presents properties of a control law which quantizes the unconstrained solution to a unitary horizon quadratic programme. This naive quantized control law underlies many popular algorithms, such as ΣΔ-converters and decision feedback equalities, and is easily shown to be globally optimal for horizon one. However, the question arises as to whether it is also globally optimal for horizons greater than one, i.e. whether it solves a finite horizon quadratic programme, where decision variables are restricted to belonging to a quantized set. By using dynamic programming, we develop sufficient conditions for this to hold. The present analysis is restricted to first order plants. However, this case already raises a number of highly non-trivial issues. The results can be applied to arbitrary horizons and quantized sets, which may contain a finite or an infinite (though countable) number of elements.

Quadratic Lyapunov functionals with a given time derivative for the case of neutral type time delay systems have been presented in Rodriguez et al. (“Robust stability of neutral systems: a Lyapunov–Krasovskii constructive approach”, International Journal of Robust and Nonlinear Control, 14, pp. 1345–1358, 2004). In this contribution a new form of the functionals is proposed. The functionals now do not include the time derivative of the system state. This modification provides new quadratic bounds for the functionals and makes them useful in computation of exponential estimates for solutions of the systems, as well as in calculation of the robustness bounds. Special attention has also been paid to the Lyapunov matrices which define the functionals. A new definition of the matrices is given, and their properties are analysed in detail.

In this paper, an algorithm is introduced for feasibility problems associated with frequency-dependent, linear matrix inequalities — these problems correspond to H 2 or H ∞ robust performance tests for a given controller, in the face of feedback perturbations. To this effect, an auxiliary H 2 cost-functional is introduced and a sequence of H 2 problems with a single linear constraint is considered in which the H 2 cost-functional is kept unchanged while the linear constraints are iteratively modified. It is shown that, in each step, the auxiliary optimal solution moves closer (in the sense of a weighted quadratic norm) to the target feasible set. Conditions are also established under which the sequence of solutions to the auxiliary problems yields a solution to the original feasibility problem. The method is illustrated by a numerical example corresponding to a H ∞ robust performance test for a multivariable system.

The existence of a canonical form for quadratic linearization of a linearly controllable system has been shown. However, if a linearly controllable system is amenable to quadratic linearization, it is not clear how to derive the transformation leading to quadratic linearization. Our approach to the derivation of necessary and sufficient conditions for quadratic linearization is constructive in nature in the sense that once the conditions are satisfied, the normalizing and input transformations are derived leading to quadratic linearization. We use state feedback to make the system eigenvalues non-resonant allowing the closed-form solution of the generalized homological equations which leads to the transformation solution. The proposed technique is illustrated with an example.

In this paper, a new method to estimate the initial condition set which guarantees the quadratic stability for a class of non-linear systems via dynamic surface control (DSC) in the presence of input saturation has been proposed. The estimated set is enlarged to allow some degree of input saturation, while achieving the simultaneous quadratic stability of the system in the domain. A convex optimization problem to obtain ellipsoidal initial condition sets via a linear matrix inequality (LMI) approach as well as an illustrative example will be presented.

In this paper parameter optimization through a quadratic performance index is introduced as a method to establish a new iterative learning control law. With this new algorithm, monotonic convergence of the error to zero is guaranteed if the original system is a discrete-time LTI system and it satisfies a positivity condition. If the original system is not positive, two methods are derived to make the system positive. The effect of the choice of weighting parameters in the performance index on convergence rate is analysed. As a result adaptive weights are introduced as a method to improve the convergence properties of the algorithm. A high-order version of the algorithm is also derived and its convergence analysed. The theoretical findings in this paper are highlighted with simulations.

This paper presents a Galerkin method to solve the mixed H2/H∞ optimal control problem. In this method the generator set is built step-by-step by an optimization procedure. Algorithm convergence is proved in the setting of weighted Hardy spaces. Some examples are discussed, showing the accelerated convergence obtained by this new algorithm and its numerical features. A particular quadratic optimal control problem on finite-dimensional linear systems under robustness constraints is developed as a motivation for the mixed H2/H∞ problem.

This paper considers the local stability and the quadratic performance of a feedback system with multiple saturation non-linearities. By combining the linear parameter-varying technique and the descriptor system representation, we develop a new local stability condition that is not only numerically tractable but also less conservative than the circle criterion. A sufficient condition for the quadratic performance based on this local stability condition is also derived. Numerical examples are included to show the effectiveness of the present stability condition.

We consider the problem of designing feedback control laws when a complete set of state variables is not available. For linear autonomous control systems with quadratic performance criterion, the design problem consists of choosing an appropriate static output feedback (SOF) gain matrix according to a certain objective function. The corresponding non-linear matrix optimization problem can be interpreted as an equality constrained minimization problem. For solving this problem, we propose a constrained trust region (CTR) algorithm, which presents a new and efficient numerical approach for this problem class. On the other hand, based on the formulation used in the past, the SOF problem can be also interpreted as an unconstrained programming problem. Thus, based on this interpretation, we also develop an unconstrained trust region (UTR) method. Finally, several numerical examples for optimal SOF problems demonstrate the applicability of the considered algorithm.

In this paper, we consider quadratic stabilizability via state feedback for both continuous-time and discrete-time switched linear systems that are composed of polytopic uncertain subsystems. By state feedback, we mean that the switchings among subsystems are dependent on system states. For continuous-time switched linear systems, we show that if there exists a common positive definite matrix for stability of all convex combinations of the extreme points which belong to different subsystem matrices, then the switched system is quadratically stabilizable via state feedback. For discrete-time switched linear systems, we derive a quadratic stabilizability condition expressed as matrix inequalities with respect to a family of non-negative scalars and a common positive definite matrix. For both continuous-time and discrete-time switched systems, we propose the switching rules by using the obtained common positive definite matrix.

In this paper we present sufficient conditions for the stability of a number of classes of time-varying systems. These results are of interest for two reasons. Firstly, the conditions presented take the form of matrix pencil eigenvalue criteria, and are therefore co-ordinate independent and easily verifiable. Secondly, we show a direct relationship between the form of our criteria, and the choice of Lyapunov function used to demonstrate stability (quadratic or unic). These results indicate the importance of the region of the complex plane known as the 45°-Region.

This paper discusses the matching conditions resulting from the controlled Lagrangians method and the interconnection and damping assignment passivity based control (IDA-PBC) method. Both methods have been presented recently in the literature as means to stabilize a desired equilibrium point of an Euler±Lagrange, respectively Hamiltonian, system. In the context of mechanical systems with symmetry, the original controlled Lagrangians method is reviewed, and an interpretation of the matching assumptions in terms of the matching of kinetic and potential energy is given. Secondly, both methods are applied to the general class of underactuated mechanical systems and it is shown that the controlled Lagrangians method is contained in the IDA-PBC method. The $\lambda$-method as described in recent papers for the controlled Lagrangians method, transforming the matching conditions (a set of non-linear PDEs) into a set of linear PDEs, is discussed. The method is used to transform the matching conditions obtained in the IDA-PBC method into a set of quadratic and linear PDEs. Finally, the extra freedom obtained in the IDA-PBC method (with respect to the controlled Lagrangians method) is used to discuss the integrability of the closed-loop system. Explicit conditions are derived under which the closed-loop Hamiltonian system is integrable, leading to the introduction of gyroscopic terms.