A quadratic programming approach for solving the l1multiblock problem

The authors present a new method to compute solutions to the general multiblock l₁ control problem. The method is based on solving a standard H₂ problem and a finite-dimensional semidefinite quadratic programming problem of appropriate dimension. The new method has most of the properties that separately characterize many existing approaches. In particular, as the dimension of the quadratic programming problem increases, this method provides converging upper and lower bounds on the optimal l₁ norm and, for well posed multiblock problems, ensures the convergence in norm of the suboptimal solutions to an optimal l₁ solution. The new method does not require the computation of the interpolation conditions, and it allows the direct computation of the suboptimal controller     

Optimal H2/l1 control via duality theory

In this paper we consider the problem of minimizing the H₂ -norm of the closed-loop map while maintaining its l₁-norm at a prescribed level. The problem is analyzed in the case of discrete-time, SISO closed-loop maps. Utilizing duality theory, it is shown that the optimal solution is unique, and, in the nontrivial case where the l₁ constraint is active, the optimal solution has a finite impulse response. A finite step procedure is given for the construction of the exact solution. This procedure consists of solving a finite number of quadratic programming problems which can be performed using standard methods. Finally, continuity properties of the optimal solution with respect to changes in the l₁-constraint are established     

A new approach to the solution of the l1 controlproblem: the scaled-Q method

We explore an approach for solving multiple input-multiple output (MIMO) l₁ optimal control problems. This approach, which we refer to as the scaled-Q approach, is introduced to alleviate many of the difficulties facing the numerical solution of optimal l₁ control problems. In particular, the computations of multivariable zeros and their directions are no longer required. The scaled-Q method also avoids the pole-zero cancellation difficulties that existing methods based on zero-interpolation face when attempting to recover the optimal controller from an optimal closed-loop map. Because the scaled-Q approach is based on solving a regularized auxiliary problem for which the solution is always guaranteed to exist, it can be used no matter where the system zeros are (including the stability boundary). Upper and lower bounds that converge to the optimal cost are provided, and all solutions are based on finite dimensional linear programming for which efficient software exists     

MIMO optimal control design: the interplay between the ℋ2 and the l1 norms

We consider controller design methods which can address directly the interplay between the ℋ2 and l₁ performance of the closed loop. The development is devoted to multi-input/multi-output (MIMO) systems. Two relevant multi-objective performance problems are considered each being of interest in its own right. In the first, termed as the combination problem, a weighted sum of the l₁ norms and the square of the ℋ₂ norms of a given set of input-output transfer functions constituting the closed loop is minimized. It is shown that, in the one-block case, the solution can be obtained via a finite-dimensional quadratic optimization problem which has an a priori known dimension. In the four-block case, a method of computing approximate solutions within any a priori given tolerance is provided. In the second, termed as the mixed problem, the ℋ₂ performance of the closed loop is minimized subject to an l₁ constraint. It is shown that approximating solutions within any a priori given tolerance can be obtained via the solution to a related combination problem     

An exact solution to general four-block discrete-time mixed ℋ2/ℋ∞ problems via convex optimization

The mixed ℋ₂ℋ_∞ control problem can be motivated as a nominal LQG optimal control problem subject to robust stability constraints, expressed in the form of an ℋ_∞ norm bound. This paper contains a solution to a general four-block mixed ℋ₂/ℋ_∞ problem, based upon constructing a family of approximating problems. Each one of these problems consists of a finite-dimensional convex optimization and an unconstrained standard ℋ_∞ problem. The set of solutions is such that in the limit the performance of the optimal controller is recovered, allowing one to establish the existence of an optimal solution. Although the optimal controller is not necessarily finite-dimensional, it is shown that a performance arbitrarily close to the optimal can be achieved with rational controllers. Moreover, the computation of a controller yielding a performance ϵ-away from optimal requires the solution of a single optimization problem, a task that can be accomplished in polynomial time     

Approximation methods of scalar mixed H2/l1problems for discrete-time systems

The scalar mixed H₂/l₁ problem for discrete-time systems is considered. The continuity property of the optimal value with respect to changes in the l₁ constraint is studied. An upper approximation method and a lower approximation method of the optimal value are given. Suboptimal values and superoptimal values of the problem can be obtained by solving a sequence of finite dimensional quadratic programming problems     

l2 state-feedback control with a prescribed rate ofexponential convergence

In this paper we consider the l₁-state feedback problem with an internal stability constraint. In particular, we establish the connection between controlled-invariant contractive sets and static control laws that achieve a level of l₁ performance as well as a desired unforced rate of convergence. We outline two algorithms for computing controlled-invariant contractive sets. The first is a modification of standard recursive techniques used in the literature, whereas the second is based on dynamic games and involves solving an appropriate discrete Isaacs recursion. The latter approach results in a min-max characterization of l₁-state feedback controllers. We point out that the Isaacs recursion provides a one-shot (as opposed to iterative) computation of the optimal l₁ performance     

Optimal solution and approximation to thel1/H∞ control problem

This paper addresses the l₁/H_∞ optimal control problem for a system described by linear time-invariant finite dimensional discrete-time equations. It is shown that a solution to this problem exists and can be approximated arbitrarily by real-rational transfer matrices. Perhaps more interesting from a computational point of view, a bound on the order of a δ-suboptimal solution is also given     

Mixed l1/H∞ control of MIMO systems viaconvex optimization

We present a methodology for designing mixed l₁/H_∞ controllers for MIMO systems. These controllers allow for minimizing the worst case peak output due to persistent disturbances, while at the same time satisfying an H_∞-norm constraint upon a given closed loop transfer function. Therefore, they are of particular interest for applications dealing with multiple performance specifications given in terms of the worst case peak values, both in the time and frequency domains. The main results of the paper show that: 1) contrary to the H₂/H_∞ case, the l₁/H_∞ problem admits a solution in l₁; and 2) rational suboptimal controllers can be obtained by solving a sequence of problems, each one consisting of a finite-dimensional convex optimization and a four-block H_∞ problem. Moreover, this sequence of controllers converges in the l₁ topology to an optimum     

Controller design with multiple objectives

In this paper we study multi-objective control problems that give rise to equivalent convex optimization problems. We develop a uniform treatment of such problems by showing their equivalence to linear programming problems with equality constraints and an appropriate positive cone. We present some specialized results on duality theory, and we apply them to the study of three multi-objective control problems: the optimal l₁ control with time-domain constraints on the response to some fixed input, the mixed H₂/l₁ -control problem, and the l₁ control with magnitude constraint on the frequency response. What makes these problems complicated is that they are often equivalent to infinite-dimensional optimization problems. The characterization of the duality relationship between the primal and dual problem allows us to derive several results. These results establish connections with special convex problems (linear programming or linear matrix inequality problems), uncover finite-dimensional structures in the optimal solution, when possible, and provide finite-dimensional approximations to any degree of accuracy when the problem does not appear to have a finite-dimensional structure. To illustrate the theory and highlight its potential, several numerical examples are presented     

混合l2/l1/2范数的局部组稀疏表示方法

当前基于稀疏表示的行人再识别都是通过松弛l₀正则项为l₁正则项以达到逼近l₀范数稀疏性的目的.在满足有限等距性质(RIP)条件下,l₁和l₀具有等价性,然而在具有杂乱背景、物体遮挡等众多干扰因素的行人再识别任务中,却很难满足RIP条件.因此,文中提出混合l₂/l_1/2范数的组稀疏表示方法,通过将gallery集中同一行人图像序列视为一组,利用l₂范数约束组内结构,l_1/2范数约束组间结构,对遮挡和杂乱背景等干扰因素具有更高的鲁棒性.为了进一步增强模型的判别性,引入人体结构约束,将行人图像划分为若干近邻块区域,针对每一区域分别构造适应性的混合l₂/l_1/2范数的组稀疏模型,最终融合全部稀疏模型得出再识别结果.在当前具有挑战性的2个多行人图像序列数据集PRID 2011和iLIDS-VID上的实验验证文中方法的有效性.     

On closed-loop adaptive noise cancellation

Given the mean limit ordinary differential equation for the stochastic approximation defining the adaptive algorithm for a closed-loop adaptive noise cancellation, we characterize the limit points. Under appropriate conditions, it is shown that as the dimension of the weight vector increases, the sequence of corresponding limit points converges in the sense of l₂ to the infinite-dimensional optimal weight vector. Also, the limit point of the algorithm is nearly optimal if the dimension of the weight vector is large enough. The gradient of the mean-square error with respect to the weight vector, evaluated at the limit, goes to zero in l₁ and l₂ as the dimension increases, as does the gradient with respect to the coefficients in the transfer function connecting the reference noise signal with the error output. Thus the algorithm is “nearly” a gradient descent algorithm and is error-reducing for large enough dimension. Under broad conditions, iterative averaging can be used to get a nearly optimal rate of convergence     

Optimal l∞ to l∞ estimation forperiodic systems

In this paper we consider the problem of finding a filter that minimizes the worst-case magnitude (l_∞) of the estimation error in the case of linear periodically time-varying systems subjected to unknown but magnitude-bounded (l_∞) inputs. These inputs consist of process and observation noises, and the optimization problem is considered over an infinite-time horizon. Lifting techniques are utilized to transform the problem to a time invariant l₁-model matching problem subject to additional constraints. Taking advantage of the particular structure of the estimation problem, it is shown how standard methods of l₁ optimization, in particular the delay augmentation technique, can be suitably modified to solve this nonstandard problem     

Preview tracking for discrete-time SISO systems

The preview tracking problem, that is, the problem of tracking a reference input when a window of its future values is known, is considered. This is done by showing how the set of all admissible regulated outputs of a system is modified when preview is incorporated. This allows the mechanism by which preview allows performance improvement to be seen more clearly . It also allows preview control systems to be designed using the powerful optimization techniques available for this framework. To exemplify this, some analytical results for the minimum l₁, l₂ and l_∞ norms of the tracking error in response to a step reference are derived for a simple system with preview. A worked example of some optimal compensator designs is also presented     

A robust adaptive controller with minimal modifications fordiscrete time-varying systems

The goal of this paper is to show that an indirect adaptive controller with parameter projection as the only modification on the basis of conventional adaptive control algorithms can globally stabilize systems having fast parasitics, bounded external disturbances, and time-varying parameters without any restriction on signals in the closed-loop system such as persistence of excitation. Further, the controller can still retain the properties of earlier unmodified conventional adaptive controllers when the controlled plant satisfies so-called “ideal assumptions” or the rates at which the plant parameters change belong to the l₁ (or l₂) space     

Distributed Sparse Signal Estimation in Sensor Networks Using H∞-Consensus Filtering

This paper is concerned with the sparse signal recovery problem in sensor networks, and the main purpose is to design a filter for each sensor node to estimate a sparse signal sequence using the measurements distributed over the whole network. A so-called l₁-regularized H_∞ filter is established at first by introducing a pseudo-measurement equation, and the necessary and sufficient condition for existence of this filter is derived by means of Krein space Kalman filtering. By embedding a high-pass consensus filter into l₁-regularized H_∞ filter in information form, a distributed filtering algorithm is developed, which ensures that all node filters can reach a consensus on the estimates of sparse signals asymptotically and satisfy the prescribed H_∞ performance constraint. Finally, a numerical example is provided to demonstrate effectiveness and applicability of the proposed method.      

Adaptive H∞ control with application to systemswith structural flexibility

We develop a robust adaptive control algorithm using a combination of H_∞ design and system identification. We derive frequency dependent bounds for the tolerated unmodeled dynamics and show that the approach gives a closed-loop system with bounded l_∞ , and l₂ gain when the model mismatch is small in the frequency range where the control gain is large. Our application focus is systems with structural flexibility. We present a parameter estimation algorithm that uses constrained least squares with prefiltering to overcome the problem of identifying lightly damped antiresonances (a common problem in identification of flexible systems). The estimation and control design are executed at a low frequency and only when parameter updating is needed. This allows us to apply computationally expensive control design and signal processing algorithms. It also eliminates many of the problems of earlier adaptive controllers (such as bursting, parameter drift, etc.) by turning off the estimator. We show results from the application of adaptive H_∞ control to a high-fidelity model of the Martin Marietta flexible beam testbed     

基于似零范数和混合优化的压缩感知信号快速重构算法

欠定系统(又称超完备系统)的稀疏信号恢复在压缩感知、源信号分离和信号采集等领域中被广泛研究. 目前这类问题主要采用l1范数约束结合线性规划优化或贪婪算法进行求解, 但这些方法存在收敛速度慢、 恢复精度不高等缺陷. 提出一种快速恢复稀疏信号的算法, 该算法采用一种新的近似l0范数代替l1范数构造代价函数, 并融合牛顿法和最陡梯度法推导出寻优迭代式,以获得似零范数代价函数的最优解. 仿真实验和真实数据实验结果表明, 与经典算法相比, 该算法在能提供相同精度、甚至更好精度的条件下, 收敛速度更快.     

H∞ multiple objective robust controllers forinfinite-horizon single measurement single control input problems

A graphical method is introduced that solves the robust infinite horizon H_∞ multiple-objective control problem for single measurement, single control input systems. The solution is obtained by describing boundaries on the Nichols chart. Each boundary defines the set of all admissible gain and phase values for the loop transmission at a given frequency. These boundaries are obtained by using the well-known parameterization of all the solutions for a single objective H_∞ control problem. The new method links between the theories of H_∞ and quantitative feedback theory (QFT). It can be used to design robust H_∞ controllers with almost no overdesign, and it provides a convenient solution of H_∞ multiple-objective problems that are difficult to solve by the standard four-block setting. It also extends the methods of SISO QFT to deal with a vector of disturbances. The latter may affect the controlled plant through any input coupling matrix and not necessarily through the controller input, or the measurement output     

Optimal and robust controllers for periodic and multirate systems

The problem of optimal rejection of bounded persistent disturbances is solved in the case of linear discrete-time periodic systems. The solution consists of solving an equivalent time-invariant standard l¹ optimization problem subject to an additional constraint. This constraint assures the causality of the resulting periodic controller. By the duality theory, the problem is shown to be equivalent to a linear programming problem, which is no harder than the standard l¹ problem. Also, it is shown that the method of solution presented applies exactly to the problem of disturbance rejection in the case of multirate sampled data systems. Finally, the results are applied to the problem of robust stabilization of periodic and multirate systems