Robust ℒ︁2‐gain feedforward control of uncertain systems using dynamic IQCs

We consider the problem of robust ??₂‐gain disturbance feedforward control for uncertain systems described in the standard LFT form. We use integral quadratic constraints (IQCs) for describing the uncertainty blocks in the system. For technical reasons related to the feedforward problem, throughout the paper, we work with the duals of the constraints involved in robustness analysis using IQCs. We obtain a convex solution to the problem using a state‐space characterization of nominal stability that we have developed recently. Specifically, our solution consists of LMI conditions for the existence of a feedforward controller that guarantees a given ??₂‐gain for the closed‐loop system. We demonstrate the effectiveness of using dynamic IQCs in robust feedforward design through a numerical example. Copyright © 2008 John Wiley & Sons, Ltd.     

A stabilizing solution to the algebraic Riccati equation the resolvent method

A new approach to the problem of analytic representation of the stabilizing solution to the algebraic Riccati equation is proposed. The quadratic matrix equation is reduced to a linear one using the resolvent (sI _2n -H)^?1 of the Hamilton matrix. The symmetric solution to the obtained linear equation defines a stabilizing solution to the Riccati equation. Matrix coefficients of the linear equation are defined by the integral of resolvent in the complex domain over the closed contour which contains all its right poles. This construction of the solution to the problem gives rise to the development of important parts of the analysis and of the corresponding computing procedures.     

C2 interpolation of spatial data subject to arc-length constraints using Pythagorean–hodograph quintic splines

In order to reconstruct spatial curves from discrete electronic sensor data, two alternative C² Pythagorean–hodograph (PH) quintic spline formulations are proposed, interpolating given spatial data subject to prescribed constraints on the arc length of each spline segment. The first approach is concerned with the interpolation of a sequence of points, while the second addresses the interpolation of derivatives only (without spatial localization). The special structure of PH curves allows the arc-length conditions to be expressed as algebraic constraints on the curve coefficients. The C² PH quintic splines are thus defined through minimization of a quadratic function subject to quadratic constraints, and a close starting approximation to the desired solution is identified in order to facilitate efficient construction by iterative methods. The C² PH spline constructions are illustrated by several computed examples.     

Algorithm/algorithmus 42 an algorithm for cubic spline fitting with convexity constraints

In this paper an algorithm is presented for fitting a cubic spline satisfying certain local concavity and convexity constraints, to a given set of data points. When using theL ₂ norm, this problem results in a quadratic programming problem which is solved by means of the Theil-Van de Panne procedure. The algorithm makes use of the well-conditioned B-splines to represent the cubic splines. The knots are located automatically, as a function of a given upper limit for the sum of squared residuals. A Fortran IV implementation is given.     

L 2-optimization and fixed poles for sampled-data systems with generalized hold

This paper considers the L ₂-optimization problem for a standard sampled-data system with generalized hold of an arbitrary order. The original problem is reduced to the minimization problem of a degenerate quadratic functional, where solution appears nonunique. And finally, the authors propose a polynomial procedure for constructing a set of optimal discrete causal controllers and establish some relevant properties of L ₂-optimal systems in the context of applications.     

Optimal LPV control with hard constraints

This paper considers the optimal control of polytopic, discrete-time linear parameter varying (LPV) systems with a guaranteed ?₂ to ?_∞ gain. Additionally, to guarantee robust stability of the closed-loop system under parameter variations, H _∞ performance criterion is also considered as well. Controllers with a guaranteed ?₂ to ?_∞ gain and a guaranteed H _∞ performance (?₂ to ?₂ gain) are a special family of mixed H ₂=H _∞ controllers. Normally, H₂ controllers are obtained by considering a quadratic cost function that balances the output performance with the control input needed to achieve that performance. However, to obtain an optimal controller with a guaranteed ?₂ to ?_∞ gain (closely related to the physical performance constraint), the cost function used in the H₂ control synthesis minimizes the control input subject to maximal singular-value performance constraints on the output. This problem can be efficiently solved by a convex optimization with linear matrix inequality (LMI) constraints. The main contribution of this paper is the characterization of the control synthesis LMIs used to obtain an LPV controller with a guaranteed ?₂ to ?_∞ gain and >H _∞ performance. A numerical example is presented to demonstrate the effectiveness of the convex optimization.     

Robust output feedback control against disturbance filter uncertainty described by dynamic integral quadratic constraints

Motivated by a robust disturbance rejection problem, in which disturbances are described by an uncertain filter at the plant input, a convex solution is presented for the robust output feedback controller synthesis problem for a particularly structured plant. The uncertainties are characterized by an integral quadratic constraint (IQC) with general frequency‐dependent multipliers. By exploiting the structure of the generalized plant, linear matrix inequality (LMI)‐synthesis conditions are derived in order to guarantee a specified ??₂‐gain or ??₂‐norm performance level, provided that the IQC multipliers are described by LMI constraints. Moreover, it is shown that part of the controller variables can be eliminated. Finally, the rejection of non‐stationary sinusoidal disturbance signals is considered. In a numerical example, it is shown that specifying a bound on the rate‐of‐variation of the time‐varying frequency can improve the performance if compared with the static IQC multipliers. Copyright © 2009 John Wiley & Sons, Ltd.     

A parameter set division and switching gain-scheduling controllers design method for time-varying plants

This paper presents a new technique to design switching gain-scheduling controllers for plants with measurable time-varying parameters. By dividing the parameter set into a sufficient number of subsets, and by designing a robust controller to each subset, the designed switching gain-scheduling controllers achieve a desired L₂-gain performance for each subset, while ensuring stability whenever a controller switching occurs due to the crossing of the time-varying parameters between any two adjacent subsets. Based on integral quadratic constraints theory and Lyapunov stability theory, a switching gain-scheduling controllers design problem amounts to solving optimization problems. Each optimization problem is to be solved by a combination of the bisection search and the numerical nonsmooth optimization method. The main advantage of the proposed technique is that the division of the parameter region is determined automatically, without any prespecified parameter set division which is required in most of previously developed switching gain-scheduling controllers design methods. A numerical example illustrates the validity of the proposed technique.     

Torsion problem for elastic cylinder with holes

An integral equation method is given to solve the classical torsion problem in elasticity theory for a multiply connected region. As is well known, the solution depends upon finding the solution of the two-dimensional Laplace's equation which takes the value $\text{1}\text{2}$(x²+y²)+c_i on the boundary, where x,y are the usual Cartesian coordinates, and c_i are unknown constants. The usual approaches are extremely cumbersome. In this paper a new approach is suggested to solve such problems. This method is simple and straightforward and requires the solution of simultaneous linear equations. An example is given which substantiates the theory. The method is very general and can take into account the discontinuity in the displacement component w. The result can therefore be applied to dislocation theory.     

Constrained smoothing of histograms by quadratic splines

For smoothing histograms under constraints like convexity or monotonicity, in this paper the functionalsK ₂ andK _∞ are proposed which can be considered as extensions of the Schoenberg functional known from data smoothing. When using quadratic splines we are led to structured finite dimensional programming problems. Occuring partially separable convex programs can be solved effectively via dualization.     

A memetic algorithm for cardinality-constrained portfolio optimization with transaction costs

A memetic approach that combines a genetic algorithm (GA) and quadratic programming is used to address the problem of optimal portfolio selection with cardinality constraints and piecewise linear transaction costs. The framework used is an extension of the standard Markowitz mean–variance model that incorporates realistic constraints, such as upper and lower bounds for investment in individual assets and/or groups of assets, and minimum trading restrictions. The inclusion of constraints that limit the number of assets in the final portfolio and piecewise linear transaction costs transforms the selection of optimal portfolios into a mixed-integer quadratic problem, which cannot be solved by standard optimization techniques. We propose to use a genetic algorithm in which the candidate portfolios are encoded using a set representation to handle the combinatorial aspect of the optimization problem. Besides specifying which assets are included in the portfolio, this representation includes attributes that encode the trading operation (sell/hold/buy) performed when the portfolio is rebalanced. The results of this hybrid method are benchmarked against a range of investment strategies (passive management, the equally weighted portfolio, the minimum variance portfolio, optimal portfolios without cardinality constraints, ignoring transaction costs or obtained with L₁ regularization) using publicly available data. The transaction costs and the cardinality constraints provide regularization mechanisms that generally improve the out-of-sample performance of the selected portfolios.     

Decentralized state feedback robust H ∞ control using a differential evolution algorithm

This paper presents a new method to synthesize a decentralized state feedback robust H ^∞ controller for a class of large‐scale linear uncertain systems satisfying integral quadratic constraints. The decentralized controller is constructed by taking only block‐diagonal elements of a nondecentralized state feedback controller and treating neglected off‐diagonal blocks as uncertainties. A solution to this controller synthesis problem is given in terms of a stabilizing solution to a parametrized algebraic Riccati equation where the parameters are obtained using a differential evolution algorithm.Copyright © 2012 John Wiley & Sons, Ltd.     

Delay-dependent stabilization of linear systems with time-varying state and input delays

The integral-inequality method is a new way of tackling the delay-dependent stabilization problem for a linear system with time-varying state and input delays: . In this paper, a new integral inequality for quadratic terms is first established. Then, it is used to obtain a new state- and input-delay-dependent criterion that ensures the stability of the closed-loop system with a memoryless state feedback controller. Finally, some numerical examples are presented to demonstrate that control systems designed based on the criterion are effective, even though neither (A,B₁) nor (A+A₁,B₁) is stabilizable.     

Decentralized state-feedback stabilization and robust control of uncertain large-scale systems with integrally constrained interconnections

This paper is concerned with a problem of stabilization and robust control design for interconnected uncertain systems. A new class of uncertain large-scale systems is considered in which interconnections between subsystems as well as uncertainties in each subsystem are described by integral quadratic constraints. The problem is to design a set of local (decentralized) controllers which stabilize the overall system and guarantee robust disturbance attenuation in the presence of the uncertainty in interconnections between subsystems as well as in each subsystem. The paper presents necessary and sufficient conditions for the existence of such a controller. The proposed design is based on recent absolute stabilization and minimax optimal control results and employs solutions of a set of game-type Riccati algebraic equations arising in H_∞ control.     

Robustness of trajectories with finite time extent

The problem of estimating perturbation bounds of finite trajectories is considered. The trajectory is assumed to be generated by a linear system with uncertainty characterized in terms of integral quadratic constraints. It is shown that such perturbation bounds can be obtained as the solution to a nonconvex quadratic optimization problem, which can be addressed using Lagrange relaxation. The result can be used in robustness analysis of hybrid systems and switched dynamical systems.     

An approach for minimizing a quadratically constrained fractional quadratic problem with application to the communications over wireless channels

Studies for the cognitive model are relatively new in the literature; however there is a growing interest in the communication field nowadays. This paper considers the cognitive model in the communication field as the problem of minimizing a fractional quadratic problem, subject to two or more quadratic constraints in complex field. Although both denominator and numerator in the fractional problem are convex, this problem is not so simple since the quotient of convex functions is not convex in most cases. We first change the fractional problem into a non-fractional one. Second, we consider the semi-definite programming (SDP) method. For the problem with m (m≤2) constraints, we use the SDP relaxation and obtain the exact optimal solution. However, for the problem with m (m>2) constraints, we choose the randomization method to gain an approximation solution in the complex case. At last, we apply this method to practical communications over wireless channels with good results.     

On delay-derivative-dependent stability of systems with fast-varying delays

Stability of linear systems with uncertain bounded time-varying delays is studied under the assumption that the nominal delay values are not equal to zero. An input-output approach to stability of such systems is known to be based on the bound of the L₂-norm of a certain integral operator. There exists a bound on this operator norm in two cases: in the case where the delay derivative is not greater than 1 and in the case without any constraints on the delay derivative. In the present note we fill the gap between the two cases by deriving a tight operator bound which is an increasing and continuous function of the delay derivative upper bound d?1. For d→∞ the new bound corresponds to the second case and improves the existing bound. As a result, for the first time, delay-derivative-dependent frequency domain and time domain stability criteria are derived for systems with the delay derivative greater than 1.     

An algorithm for robust performance tests based on frequency-dependent LMIs

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.     

Consistent tangent operators for rate-independent elastoplasticity

It is shown that for problems involving rate constitutive equations, such as rate-independent elastoplasticity, the notion of consistency between the tangent (stiffness) operator and the integration algorithm employed in the solution of the incremental problem, plays a crucial role in preserving the quadratic rate of asymptotic convergence of iterative solution schemes based upon Newton's method. Within the framework of closest-point-projection algorithms, a methodology is presented whereby tangent operators consistent with this class of algorithms may be systematically developed. To wit, associative J₂ flow rules with general nonlinear kinematic and isotropic hardening rules, as well as a class of non-associative flow rules are considered. The resulting iterative solution scheme preserves the asymptotic quadratic convergence characteristic of Newton's method, whereas use of the socalled elastoplastic tangent in conjunction with a radial return integration algorithm, a procedure often employed, results in Newton type of algorithms with suboptimal rate of convergence. Application is made to a set of numerical examples which include saturation hardening laws of exponential type.     

Numerical solution for the one-phase Stefan problem by Piecewise constant approximation of the interface

The classical one-phase one-dimensional Stefan problem is numerically solved on rectangles,R _j , of increasing size controlled by the Stefan condition. This approach is based on a scheme introduced by E. Di Benedetto and R. Spigler in 1983. The practical implementation rests on the representation viathermal potentials of the solutionu ^j (x, t) to the heat equation inR _j . The quantityu _x ^j (x _j ,jΔt) which determines the (j+1)-th rectangle is evaluatedanalytically by solving explicitly an integral equation. The solution inR _j+1 is then obtained bynumerically evaluating a further integral expression. The algorithm is tested by solving two problems whose solution is explicitly known. Convergence, stability and convergence rate as Δx→0, Δt→0 have been tested and plots are shown.