This paper studies the time-optimal path tracking problem for a team of cooperating robotic manipulators carrying an object. Considering the problem for rigidly grasped objects, we show that it can be cast as a convex optimization problem and solved efficiently with a guarantee of optimality. When formulating the problem, we avoid using a particular wrench distribution and exploit the full actuation available to the system. Then, we consider the problem for grasps using frictional forces and show that this problem also, under a force-closure grasp assumption, can be formulated as a convex optimization problem and solved efficiently and to optimality. To ensure a firm grasp, internal forces have been taken into account in this approach.

Mean square error optimal estimation requires the full correlation structure to be available. Unfortunately, it is not always possible to maintain full knowledge about the correlations. One example is decentralized data fusion where the cross-correlations between estimates are unknown, partly due to information sharing. To avoid underestimating the covariance of an estimate in such situations, conservative estimation is one option. In this paper the conservative linear unbiased estimator is formalized including optimality criteria. Fundamental bounds of the optimal conservative linear unbiased estimator are derived. A main contribution is a general approach for computing the proposed estimator based on robust optimization. Furthermore, it is shown that several existing estimation algorithms are special cases of the optimal conservative linear unbiased estimator. An evaluation verifies the theoretical considerations and shows that the optimization based approach performs better than existing conservative estimation methods in certain cases.

To optimally compensate for time-varying phase aberrations with adaptive optics, a model of the dynamics of the aberrations is required to predict the phase aberration at the next time step. We model the time-varying behavior of a phase aberration, expressed in Zernike modes, by assuming that the temporal dynamics of the Zernike coefficients can be described by a vector-valued autoregressive (VAR) model. We propose an iterative method based on a convex heuristic for a rank-constrained optimization problem, to jointly estimate the parameters of the VAR model and the Zernike coefficients from a time series of measurements of the point-spread function (PSF) of the optical system. By assuming the phase aberration is small, the relation between aberration and PSF measurements can be approximated by a quadratic function. As such, our method is a blind identification method for linear dynamics in a stochastic Wiener system with a quadratic nonlinearity at the output and a phase retrieval method that uses a time-evolution-model constraint and a single image at every time step. (c) 2019 Optical Society of America.

This note considers the identification of large-scale one-dimensional networks consisting of identical LTI dynamical systems. A subspace identification method is developed that only uses local input-output information and does not rely on knowledge about the local state interaction. The proposed identification method estimates the Markov parameters of a locally lifted system, following the state-space realization of a single subsystem. The Markov-parameter estimation is formulated as a rank minimization problem by exploiting the low-rank property and the two-layer Toeplitz structural property in the data equation, whereas the state-space realization of a single subsystem is formulated as a structured low-rank matrix-factorization problem. The effectiveness of the proposed identification method is demonstrated by simulation examples.

The identification of multivariable state space models in innovation form is solved in a subspace identification framework using convex nuclear norm optimization. The convex optimization approach allows to include constraints on the unknown matrices in the data-equation characterizing subspace identification methods, such as the lower triangular block-Toeplitz of weighting matrices constructed from the Markov parameters of the unknown observer. The classical use of instrumental variables to remove the influence of the innovation term on the data equation in subspace identification is avoided. The avoidance of the instrumental variable projection step has the potential to improve the accuracy of the estimated model predictions, especially for short data length sequences. (C) 2016 Elsevier Ltd. All rights reserved.

In this paper we describe an approach to maximum likelihood estimation of linear single input single output (SISO) models when both input and output data are missing. The criterion minimised in the algorithms is the Euclidean norm of the prediction error vector scaled by a particular function of the covariance matrix of the observed output data. We also provide insight into when simpler and in general sub-optimal schemes are indeed optimal. The algorithm has been prototyped in MATLAB, and we report numerical results that support the theory.

We propose a new method for generating semidefinite relaxations of optimal power flow problems. The method is based on chordal conversion techniques: by dropping some equality constraints in the conversion, we obtain semidefinite relaxations that are computationally cheaper, but potentially weaker, than the standard semidefinite relaxation. Our numerical results show that the new relaxations often produce the same results as the standard semidefinite relaxation, but at a lower computational cost.

Direct torque control (DTC) is considered as one of the most efficient techniques for speed and/or position tracking control of induction motor drives. However, this control scheme has several drawbacks: the switching frequency may exceed the maximum allowable switching frequency of the inverters, and the ripples in current and torque, especially at low speed tracking, may be too large. In this brief, we propose a new approach that overcomes these problems. The suggested controller is a model predictive controller, which directly controls the inverter switches. It is easy to implement in real time and it outperforms all previous approaches. Simulation results show that the new approach has as good tracking properties as any other scheme, and that it reduces the average inverter switching frequency about 95% as compared to classical DTC.

In many applications, design or analysis is performed over a finite-frequency range of interest. The importance of the H₂ norm highlights the necessity of computing this norm accordingly. This paper provides different methods for computing upper bounds of the robust finite-frequency H₂ norm for systems with structured uncertainties. An application of the robust finite-frequency H₂ norm for a comfort analysis problem of an aero-elastic model of an aircraft is also presented.

In this paper we derive the maximum likelihood problem for missing data from a Gaussian model. We present in total eight different equivalent formulations of the resulting optimization problem, four out of which are nonlinear least squares formulations. Among these formulations are also formulations based on the expectation-maximization algorithm. Expressions for the derivatives needed in order to solve the optimization problems are presented. We also present numerical comparisons for two of the formulations for an ARMAX model.

Many control-related problems can be cast as semidefinite programs. Even though there exist polynomial time algorithms and excellent publicly available solvers, the time it takes to solve these problems can be excessive. What many of these problems have in common, in particular in control, is that some of the variables enter as matrix-valued variables. This leads to a low-rank structure in the basis matrices which can be exploited when forming the Newton equations. In this article, we describe how this can be done, and show how our code, called STRUL, can be used in conjunction with the semidefinite programming solver SDPT3. The idea behind the structure exploitation is classical and is implemented in LMI Lab, but we show that when using a modern semidefinite programming framework such as SDPT3, the computational time can be significantly reduced. Finally, we describe how the modelling language YALMIP has been changed in such a way that our code, which can be freely downloaded, can be interfaced using standard YALMIP commands. This greatly simplifies modelling and usage.

The main objective in this work is to compare different convex relaxations for Model Predictive Control (MPC) problems with mixed real valued and binary valued control signals. In the problem description considered, the objective function is quadratic, the dynamics are linear, and the inequality constraints on states and control signals are all linear. The relaxations are related theoretically and the quality of the bounds and the computational complexities are compared in numerical experiments. The investigated relaxations include the Quadratic Programming (QP) relaxation, the standard Semidefinite Programming (SDP) relaxation, and an equality constrained SDP relaxation. The equality constrained SDP relaxation appears to be new in the context of hybrid MPC and the result presented in this work indicates that it can be useful as an alternative relaxation, which is less computationally demanding than the ordinary SDP relaxation and which often gives a better bound than the bound from the QP relaxation. Furthermore, it is discussed how the result from the SDP relaxations can be used to generate suboptimal solutions to the control problem. Moreover, it is also shown that the equality constrained SDP relaxation is equivalent to a QP in an important special case.

Semidefinite programs derived from the Kalman-Yakubovich-Popov (KYP) lemma are quite common in control and signal processing applications. The programs are often of high dimension which makes them hard or impossible to solve with general-purpose solvers. Here we present a customized preprocessor, KYPD, that utilizes the inherent structure of this particular optimization problem. The key to an efficient implementation is to transform the optimization problem into an equivalent semidefinite program. This equivalent problem has much fewer variables and the matrices in the linear matrix inequality constraints are of low rank. KYPD can use any primal-dual solver for semidefinite programs as an underlying solver.

The optimum multiuser detection problem can be formulated as a maximum likelihood problem, which yields a binary quadratic programming problem to be solved. Generally this problem is NP-hard and is therefore hard to solve in real time. In this paper, a preprocessing algorithm is presented which makes it possible to detect some or all users optimally for a low computational cost if signature sequences with low cross correlation, e.g., Gold sequences, are used. The algorithm can be interpreted as, e.g., an adaptive tradeoff between parallel interference cancellation and successive interference cancellation. Simulations show that the preprocessing algorithm is able to optimally compute more than 94,% of the bits in the problem when the users are time-synchronous, even though the system is heavily loaded and affected by noise. Any remaining bits, not computed by the preprocessing algorithm, can either be computed by a suboptimal detector or an optimal detector. Simulations of the time-synchronous case show that if a suboptimal detector is chosen, the bit error rate (BER) rate is significantly reduced compared with using the suboptimal detector alone.

In model predictive control an optimization problem has to be solved at each sampling instant. The objective in this article is to derive efficient methods to solve this optimization problem. The approach taken is to use interior point optimization methods. The model predictive control problem considered here has a quadratic objective and constraints which can be both linear and quadratic. The key to an efficient implementation is to rewrite the optimization problem as a second order cone program. To solve this optimization problem a feasible primal-dual interior point method is employed. By using a feasible method it is possible to determine when the problem is feasible or not by formalizing the search for strictly feasible initial points as yet another primal-dual interior point problem. There are several different ways to rewrite the optimization problem as a second order cone program. However, done carefully, it is possible to use very efficient scalings as well as Riccati recursions for computing the search directions. The use of Riccati recursions makes the computational complexity grow at most as script, O sign(N 3/2) with the time horizon, compared to script O sign(N3) for more standard implementations.

In this chapter parallel implementations of hybrid MPC will be discussed. Different methods for achieving parallelism at different levels of the algorithms will be surveyed. It will be seen that there are many possible ways of obtaining parallelism for hybrid MPC, and it is by no means clear which possibilities that should be utilized to achieve the best possible performance. To answer this question is a challenge for future research.

Results for stability analysis of the nonlinear rigid aircraft model and comfort and loads analysis of the integral aircraft model are presented in this chapter. The analysis is based on the theory for integral quadratic constraints and relies on linear fractional representations (LFRs) of the underlying closed-loop aircraft models. To alleviate the high computational demands associated with the usage of IQC based analysis to large order LFRs, two approaches have been employed aiming a trade-off between computational complexity and conservatism. First, the partitioning of the flight envelope in several smaller regions allows to use lower order LFRs in the analysis, and second, IQCs with lower computational demands have been used whenever possible. The obtained results illustrate the applicability of the IQCs based analysis techniques to solve highly complex analysis problems with an acceptable level of conservativeness.

This book summarizes the main achievements of the EC funded 6^th Framework Program project COFCLUO – Clearance of Flight Control Laws Using Optimization. This project successfully contributed to the achievement of a top-level objective to meet society’s needs for a more efficient, safer and environmentally friendly air transport by providing new techniques and tools for the clearance of flight control laws. This is an important part of the certification and qualification process of an aircraft – a costly and time-consuming process for the aeronautical industry.

We consider the problem to learn a step-size policy for the Limited-Memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) algorithm. This is a limited computational memory quasi-Newton method widely used for deterministic unconstrained optimization. However, L-BFGS is currently avoided in large-scale problems for requiring step sizes to be provided at each iteration. Current methodologies for the step size selection for L-BFGS use heuristic tuning of design parameters and massive re-evaluations of the objective function and gradient to find appropriate step-lengths. We propose a neural network architecture with local information of the current iterate as the input. The step-length policy is learned from data of similar optimization problems, avoids additional evaluations of the objective function, and guarantees that the output step remains inside a pre-defined interval. The corresponding training procedure is formulated as a stochastic optimization problem using the backpropagation through time algorithm. The performance of the proposed method is evaluated on the training of image classifiers for the MNIST database for handwritten digits and for CIFAR-10. The results show that the proposed algorithm outperforms heuristically tuned optimizers such as ADAM, RMSprop, L-BFGS with a backtracking line search, and L-BFGS with a constant step size. The numerical results also show that a learned policy can be used as a warm-start to train new policies for different problems after a few additional training steps, highlighting its potential use in multiple large-scale optimization problems.

In this paper we present a distributed second-order augmented Lagrangian method for distributed model predictive control. We distribute the computations for search direction, step size, and termination criteria over what is known as the clique tree of the problem and calculate each of them using message passing. The algorithm converges to its centralized counterpart and it requires fewer communications between sub-systems as compared to algorithms such as the alternating direction method of multipliers. Results from a simulation study confirm the efficiency of the framework. Copyright (C) 2021 The Authors.

In this paper we show that chordal structure can be used to devise efficient optimization methods for robust model predictive control problems. To this end, first the problem is converted to an equivalent robust quadratic programming formulation. We then illustrate how the chordal structure can be used to distribute the computations in a primal-dual interior-point method among computational agents, which in turn allows us to accelerate the algorithm by efficient parallel computations. We investigate performance of the framework in Julia using numerical examples.

In system identification, input selection is a challenging problem. Since less complex models are desireable, non-relevant inputs should be methodically and correctly discarded before or under the estimation process. In this paper we investigate an input selection extension in least-squares ARX estimation and show that better model estimates are achieved compared to the least-square ssolution, in particular, for short batches of estimation data. (C) 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.

In inertial motion capture, a multitude of body segments are equipped with inertial sensors, consisting of 3D accelerometers and 3D gyroscopes. Using an optimization-based approach to solve the motion capture problem allows for natural inclusion of biomechanical constraints and for modeling the connection of the body segments at the joint locations. The computational complexity of solving this problem grows both with the length of the data set and with the number of sensors and body segments considered. In this work, we present a scalable and distributed solution to this problem using tailored message passing, capable of exploiting the structure that is inherent in the problem. As a proof-of-concept we apply our algorithm to data from a lower body configuration. 

This paper presents identification of both network connected systems as well as distributed systems governed by PDEs in the framework of distributed optimization via the Alternating Direction Method of Multipliers. This approach opens first the possibility to identify distributed models in a global manner using all available data sequences and second the possibility for a distributed implementation. The latter will make the application to large scale complex systems possible. In addition to outlining a new large scale identification method, illustrations are shown for identifying both network connected systems and discretized PDEs.

Large-scale interconnected uncertain systems commonly have large state and uncertainty dimensions. Aside from the heavy computational cost of solving centralized robust stability analysis techniques, privacy requirements in the network can also introduce further issues. In this paper, we utilize IQC analysis for analyzing large-scale interconnected uncertain systems and we evade these issues by describing a decomposition scheme that is based on the interconnection structure of the system. This scheme is based on the so-called chordal decomposition and does not add any conservativeness to the analysis approach. The decomposed problem can be solved using distributed computational algorithms without the need for a centralized computational unit. We further discuss the merits of the proposed analysis approach using a numerical experiment.

We consider a class of convex feasibility problems where the constraints that describe the feasible set are loosely coupled. These problems arise in robust stability analysis of large, weakly interconnected uncertain systems. To facilitate distributed implementation of robust stability analysis of such systems, we describe two algorithms based on decomposition and simultaneous projections. The first algorithm is a nonlinear variant of Cimmino's mean projection algorithm, but by taking the structure of the constraints into account, we can obtain a faster rate of convergence. The second algorithm is devised by applying the alternating direction method of multipliers to a convex minimization reformulation of the convex feasibility problem. We use numerical results to show that both algorithms require far less iterations than the accelerated nonlinear Cimmino algorithm.

This paper considers robust stability analysis of a large network of interconnected uncertain systems. To avoid analyzing the entire network as a single large, lumped system, we model the network interconnections with integral quadratic constraints. This approach yields a sparse linear matrix inequality which can be decomposed into a set of smaller, coupled linear matrix inequalities. This allows us to solve the analysis problem efficiently and in a distributed manner. We also show that the decomposed problem is equivalent to the original robustness analysis problem, and hence our method does not introduce additional conservativeness.

The robust H₂ norm plays an important role in analysis and design in many fields. However, for many practical applications, design and analysis is based on finite frequency range. In this paper we review the concept of the robust finite frequency H₂ norm, and we provide an algorithmic method for calculating an upper bound for the mentioned quantity.

When designing robust controllers, H-infinity synthesis is a common tool to use. The controllers that result from these algorithms are typically of very high order, which complicates implementation. However, if a constraint on the maximum order of the controller is set, that is lower than the order of the (augmented) system, the problem becomes nonconvex and it is relatively hard to solve. These problems become very complex, even when the order of the system is low.

The approach used in this work is based on formulating the constraint on the maximum order of the controller as a polynomial (or rational) equation. By using the fact that the polynomial (or rational) is non-negative on the feasible set, the problem is reformulated as an optimization problem where the nonconvex function is to be minimized over a convex set defined by linear matrix inequalities.

The proposed method is evaluated together with a well-known method from the literature. The results indicate that the proposed method performs slightly better.

Many control related problems can be cast as semidefinite programs but, even though there exist polynomial time algorithms and good publicly available solvers, the time it takes to solve these problems can be long. Something many of these problems have in common, is that some of the variables enter as matrix valued variables. This leads to a low-rank structure in the basis matrices which can be exploited when forming the Newton equations. In this paper, we describe how this can be done, and show how our code can be used when using SDPT3. The idea behind this is old and is implemented in LMI Lab, but we show that when using a modern algorithm, the computational time can be reduced. Finally, we describe how the modeling language YALMIP is changed in such a way that our code can be interfaced using standard YALMIP commands, which greatly simplifies for the user.

Many control related problems can be cast as semidefinite programs but, even though there exist polynomial time algorithms and good publicly available solvers, the time it takes to solve these problems can be long. Something many of these problems have in common, is that some of the variables enter as matrix valued variables. This leads to a low-rank structure in the basis matrices which can be exploited when forming the Newton equations. In this paper, we describe how this can be done, and show how our code can be used when using SDPT3. The idea behind this is old and is implemented in LMI Lab, but we show that when using a modern algorithm, the computational time can be reduced. Finally, we describe how the modeling language YALMIP is changed in such a way that our code can be interfaced using standard YALMIP commands, which greatly simplifies for the user.

This paper presents the formulation of a flight clearance criterion as a convex optimization problem. The criterion is the stability margins criterion which is specified as an allowable phase and gain margin of a certain loop transfer function. The satisfaction of the criterion amounts to the Nichols plot of the loop transfer function being outside a specified trapezoidal region. It was shown previously that the exclusion condition from an ellipsoidal region is implied by using the generalized stability margin b_PC and its calculation was performed frequency-wise by solving a sequence of convex optimization problems. In this paper we formulate the calculation of a lower bound on b_PC as a convex optimization problem using Integral Quadratic Constraints (IQCs) and avoid the gridding procedure in the frequency domain. Furthermore, we formulate the problem of obtaining a lower bound on the perturbed stability margin, which is defined as the worst-case b_PC over variations in real uncertain parameters.

The objective of this work is to derive a QPalgorithm tailored for MPC. More specific, the primary targetapplication is MPC for discrete-time hybrid systems. A desiredproperty of the algorithm is that warm starts should be possibleto perform efficiently. This property is very important for online linear MPC, and it is crucial in branch and bound forhybrid MPC. In this paper, a dual active set-like QP methodwas chosen because of its warm start properties. A drawbackwith classical active set methods is that they often requiremany iterations in order to find the active set in optimum.Gradient projection methods are methods known to be ableto identify this active set very fast and such a method wastherefore chosen in this work. The gradient projection methodwas applied to the dual QP problem and it was tailored for theMPC application. Results from numerical experiments indicatethat the performance of the new algorithm is very good, bothfor linear MPC as well as for hybrid MPC. It is also noticed thatthe number of QP iterations is significantly reduced compared to classical active set methods.

Within the area of systems analysis there are several problem formulations that can be rewritten as semidefinite programs. Increasing demand on computational efficiency and ability to solve large scale problems make the available generic solvers inadequate. In this paper structure knowledge is utilized to derive tailored calculations and to incorporate adaptation to the different properties that appear in a proposed inexact interior-point method.

The objective of this work is to derive a QP algorithm tailored for MPC. More specifically, the primary target application is MPC for discrete-time hybrid systems. A desired property of the algorithm is that warm starts should be possible to perform efficiently. This property is very important for on-line linear MPC, and it is crucial in branch and bound for hybrid MPC. In this paper, a dual active set-like QP method was chosen because of its warm start properties. A drawback with classical active set methods is that they often require many iterations in order to find the active set in optimum. Gradient projection methods are methods known to be able to identify this active set very fast and such a method was therefore chosen in this work. The gradient projection method was applied to the dual QP problem and it was tailored for the MPC application. Results from numerical experiments indicate that the performance of the new algorithm is very good, both for linear MPC as well as for hybrid MPC. It is also noticed that the number of QP iterations is significantly reduced compared to classical active set methods. 

In this work, different relaxations applicable to an MPC problem with binary control signals are compared. The relaxations considered are the QP relaxation, the standard SDP relaxation and an alternative equality constrained SDP relaxation. The relaxations are related theoretically, and both the tightness of the bounds and the computational complexities are compared in numerical experiments. The result is that for long prediction horizons, the equality constrained SDP relaxation proposed in this paper provides a good trade-off between the quality of the relaxation and the computational time.

Semidefinite programs and especially those derived from the Kalman-Yakubovich-Popov lemma are quite common in control applications. KYPD is a dedicated solver for KYP-SDPs. It solves the optimization problem via the dual SDP. The solver is iterative. In each step a Hessian is formed and a linear system of equations is solved. The calculations can be performed much faster if we utilize sparsity and low rank structure. We show how to transform a dense optimization problem into a sparse one with low rank structure. A customized calculation of the Hessian is presented and investigated.

The objective of this work is to derive an MIQP solver tailored for MPC. The MIQP solver is built on the branch and bound method, where QP relaxations of the original problem are solved in the nodes of a binary search tree. The difference between the subproblems is often small and therefore it is interesting to be able to use a previous solution as a starting point in a new subproblem. This is referred to as a warm start of the solver. Because of its good warm start properties, a dual active set QP method was chosen. The method is tailored for MPC by solving a part of the KKT system using a Riccati recursion, which makes the computational complexity of the QP iterations grow linearly with the prediction horizon. Simulation results are presented both for the QP solver itself and when it is incorporated as a part of the MIQP solver. In both cases the computational complexity is significantly reduced compared to if a primal active set solver not utilizing structure is used.

In this paper a preprocessing algorithm for binary quadratic programming problems is presented. For some types of binary quadratic programming problems, the algorithm can compute the optimal value for some or all integer variables without approximations in polynomial time. When the optimal multiuser detection problem is formulated as a maximum likelihood problem, a binary quadratic programming problem has to be solved. Fortunately, the low correlation between different users in the multiuser detection problem enables the use of the preprocessing algorithm. Simulations show that the preprocessing algorithm is able to compute almost all variables in the problem, even though the system is heavily loaded and affected by noise.

Semidenite programs derived from the Kalman-Yakubovich-Popov lemma are quite common in control and signal processing applications. The programs are often of high dimension making them hard or impossible to solve with general-purpose solvers. KYPD is a customized solver for KYP-SDPs that utilizes the inherent structure of the optimization problem thus improving efficiency signicantly.

The problem of flight clearance of linear stability requirements in the complete flight evelope is studied. The approach is to use mu-analysis of linear fractional transformations description of the system dynamics augmented with an uncertainty model of the clearance criteria. Two methods to derive the linear fractional transformations are used and compared: the Trends and Bands method and rational approximations. The two approaches are applied to a UAV demostrator model. The resulting models are analyzed by mu-sensitivites in order to demonstrate how this can be used to reduce the order of the linear fractional transformations description. A number of algorithms to compute upper and lower bounds on mu are also compared. Results from applying the clearance criteria to the UAV demonstrator model are given.

In this paper a preprocessing algorithm for unconstrained mixed integer quadratic programming problems and binary quadratic programming problems is presented. The algorithm applies to problems with certain properties, which are further described in the paper. When the algorithm is applied to a problem with these properties, the optimal value for some or all integer variables can be computed without approximations in polynomial time. The algorithm is first derived for the binary quadratic programming problem and the result is then extended to the mixed integer quadratic programming problem by transforming the latter problem into the first problem. Both mentioned quadratic programming problems have several important applications. In this paper, the focus is on model predictive control problems with both real-valued and binary control signals. As an illustration of the method, the algorithm is applied to two different problems of this type.

As the semidefinite programs that result from integral quadratic contstraints are usually large it is important to implement efficient algorithms. The interior-point algorithms in this paper are primal-dual potential reduction methods and handle multiple constraints. Two approaches are made. For the first approach the computational cost is dominated by a least-squares problem that has to be solved in each iteration. The least squares problem is solved using an iterative method, namely the conjugate gradient method. The computational effort for the second approach is dominated by forming a linear system of equations. This systems of equations is used to compute the search direction in each iteration. If the number of variables are reduced by solving a smaller subproblem the resulting system has a very nice structure and can be solved efficiently. The first approach is more efficient for larger problems but is not as numerically stable.

We discuss fast implementations of primal-dual interior-point methods for semidefinite programs derived from the Kalman-Yakubovich-Popov lemma, a class of problems that are widely encountered in control and signal processing applications. By exploiting problem structure we achieve a reduction of the complexity by several orders of magnitude compared to general-purpose semidefinite programming solvers.

In this paper it is shown how to efficiently solve an optimal control problem with applications to model predictive control. The objective is quadratic and the constraints can be both linear and quadratic. The key to an efficient implementation is to rewrite the optimization problem as a second order cone program. This can be done in many different ways. However, done carefully, it is possible to use both very efficient scalings as well as Riccati recursions for computing the search directions.

Iterative feedback tuning (IFT) is a model free control tuning method using closed loop experiments. For single-input single-output (SISO) systems only 2 or 3, depending on the controller structure, closed loop experiments are required. However for multivariable systems the number of experiments increase to a maximum of 1 + m times p, where m times p is the dimension of the controller. In this contribution several methods are proposed to reduce the experimental time by approximating the gradient of the cost function. The local convergence for a method which uses the same technique as in IFT for SISO systems is analyzed. It is shown that even if there are commutation errors due to the approximation method, the numerical optimization may still converge to the true optimum.

We discuss how to implement an efficient interior-point algorithm for semi-definite programs that result from integral quadratic constraints. The algorithm is a primal-dual potential reduction method, and the computational effort is dominated by a least-squares system that has to be solved in each iteration. The key to an efficient implementation is to utilize iterative methods and the specific structure of integral quadratic constraints. The algorithm has been implemented in Matlab. To give a rough idea of the efficiencies obtained, it is possible to solve problems resulting in a linear matrix inequality of dimension 130 × 130 with approximately 5000 variables in about 5 minutes on a lap-top. Problems with approximately 20000 variable and a linear matrix inequality of dimension 230 × 230 are solved in about 45 minutes. It is not assumed that the system matrix has no eigenvalues on the imaginary axis, nor is it assumed that it is Hurwitz.

In this paper we derive the maximum likelihood problem for missing data from a Gaussian model. We present in total eight different equivalent formulations of the resulting optimization problem, four out of which are nonlinear least squares formulations. Among these formulations are also formulations based on the expectation-maximization algorithm. Expressions for the derivatives needed in order to solve the optimization problems are presented. We also present numerical comparisons for two of the formulations for an ARMAX model.

In this paper a specific class of convex feasibility problems are considered and tailored algorithms to solve this class of problems are introduced. First, the Nonlinear Cimmino Algorithm is reviewed. Then motivated by the special structure of the problems at hand, a modification to this method is proposed. Next, another method for solving the dual problem of the provided problem is presented. This leads to similar update rules for the variables as in the modified Nonlinear Cimmino Algorithm. Then an application for the proposed algorithms on the robust stability analysis of large scale weakly interconnected systems is presented and the performance of the proposed methods are compared.

We consider a class of convex feasibility problems where the constraints that describe the feasible set are loosely coupled. These problems arise in robust stability analysis of large, weakly interconnected systems. To facilitate distributed implementation of robust stability analysis of such systems, we propose two algorithms based on decomposition and simultaneous projections. The first algorithm is a nonlinear variant of Cimmino’s mean projection algorithm, but by taking the structure of the constraints into account, we can obtain a faster rate of convergence. The second algorithm is devised by applying the alternating direction method of multipliers to a convex minimization reformulation of the convex feasibility problem. We use numerical results to show that both algorithms require far less iterations than the accelerated nonlinear Cimmino algorithm.

When designing robust controllers, H-infinity synthesis is a common tool touse. The controllers that result from these algorithms are typically of very high order, which complicates implementation. However, if a constraint on the maximum order of the controller is set, that is lower than the order of the (augmented) system, the problem becomes nonconvex and it is relatively hard to solve. These problems become very complex, even when the order of the system is low.

The approach used in this work is based on formulating the constraint onthe maximum order of the controller as a polynomial (or rational) equation.This equality constraint is added to the optimization problem of minimizingan upper bound on the H-innity norm of the closed loop system subjectto linear matrix inequality (LMI) constraints. The problem is then solvedby reformulating it as a partially augmented Lagrangian problem where theequality constraint is put into the objective function, but where the LMIsare kept as constraints.

The proposed method is evaluated together with two well-known methodsfrom the literature. The results indicate that the proposed method hascomparable performance in most cases, especially if the synthesized con-troller has many parameters, which is the case if the system to be controlledhas many input and output signals.

The task of generating time optimal trajectories for a six degrees of freedom industrial robot is discussed and an existing convex optimization formulation of the problem is extended to include new types of constraints. The new constraints are speed dependent and can be motivated from physical modeling of the motors and the drive system. It is shown how the speed dependent constraints should be added in order to keep the convexity of the overall problem. A method to, conservatively, approximate the linear speed dependent constraints by a convex constraint is also proposed. A numerical example proves versatility of the extension proposed in this paper.

Here we present numerical results and timings obtained using our quasi-Newton interior point method on a set of 44 systems. We were not able to include these results in the article due to limited amount of space. Also results from our evaluation of HIFOO on the same systems are included.

This paper presents the formulation of a flight clearance criterion as a convex optimization problem. The criterion is the stability margins criterion which is specified as an allowable phase and gain margin of a certain loop transfer function. The satisfaction of the criterion amounts to the Nichols plot of the loop transfer function being outside a specified trapezoidal region. It was shown previously that the exclusion condition from an ellipsoidal region is implied by using the generalized stability margin b_PC and its calculation was performed frequency-wise by solving a sequence of convex optimization problems. In this paper we formulate the calculation of a lower bound on b_PC as a convex optimization problem using Integral Quadratic Constraints (IQCs) and avoid the gridding procedure in the frequency domain. Furthermore, we formulate the problem of obtaining a lower bound on the perturbed stability margin, which is defined as the worst-case b_PCover variations in real uncertain parameters.

The optimum multiuser detection problem can be formulated as a maximum likelihood problem, which yields a binary quadratic programming problem to be solved. Generally this problem is NP-hard and is therefore hard to solve in real time. In this paper, a preprocessing algorithm is presented which makes it possible to detect some or all users optimally for a low computational cost if signature sequences with low cross correlation, e.g., Gold sequences, are used. The algorithm can be interpreted as, e.g., an adaptive tradeoff between parallel interference cancellation and successive interference cancellation. Simulations show that the preprocessing algorithm is able to optimally compute more than 94,% of the bits in the problem when the users are time-synchronous, even though the system is heavily loaded and affected by noise. Any remaining bits, not computed by the preprocessing algorithm, can either be computed by a suboptimal detector or an optimal detector. Simulations of the time-synchronous case show that if a suboptimal detector is chosen, the bit error rate (BER) rate is significantly reduced compared with using the suboptimal detector alone.

The objective of this work is to derive a Mixed Integer Quadratic Programming algorithm tailored for Model Predictive Control for hybrid systems. The Mixed Integer Quadratic Programming algorithm is built on the branch and bound method, where Quadratic Programming relaxations of the original problem are solved in the nodes of a binary search tree. The difference between these subproblems is often small and therefore it is interesting to be able to use a previous solution as a starting point in a new subproblem. This is referred to as a warm start of the solver. Because of its warm start properties, an algorithm that works similar to an active set method is desired. A drawback with classical active set methods is that they often require many iterations in order to find the active set in optimum. So-called gradient projection methods are known to be able to identify this active set very fast. In the algorithm presented in this report, an algorithm built on gradient projection and projection of a Newton search direction onto the feasible set is used. It is a variant of a previously presented algorithm by the authors and makes it straightforward to utilize the previous result, where it is shown how the Newton search direction for the dual MPC problem can be computed very efficiently using Riccati recursions. As in the previous work, this operation can be performed with linear computational complexity in the prediction horizon. Moreover, the gradient computation used in the gradient projection part of the algorithm is also tailored for the problem in order to decrase the computational complexity. Furthermore, is is shown how a Riccati recursion still can be useful in the case when the system of equations for the ordinary search directino is inconsistent. In numerical experiments, the algorithm shows good performance, and it seems like the gradient projection strategy efficiently cuts down the number of Newton steps necessary to compute in order to reach the solution. When the algorithm is used as a part of an MIQP solver for hybrid MPC, the performance is still very good for small problems. However, for more difficult problems, there still seems to be some more work to do in order to get the performance of the commercial state-of-the-art solver CPLEX.

Within the area of system analysis there are multiple problem formulations that can be rewritten as semidefiniteprograms. Increasing demand on computational efficiency andability to solve large scale problems make the available genericsolvers inadequate. In this paper structure knowledge is utilizedto derive tailored calculations and to incorporate adaptationto the different properties that appear in a proposed inexactinterior-point method.

The objective of this work is to derive an MIQP solver tailored for MPC. The MIQP solver is built on the branch and bound method, where QP relaxations of the original problem are solved in the nodes of a binary search tree. The difference between the subproblems is often small and therefore it is interesting to be able to use a previous solution as a starting point in a new subproblem. This is referred to as a warm start of the solver. Because of its good warm start properties, a dual active set QP method was chosen. The method is tailored for MPC by solving a part of the KKT system using a Riccati recursion, which makes the computational complexity of the QP iterations grow linearly with the prediction horizon. Simulation results are presented both for the QP solver itself and when it is incorporated as a part of the MIQP solver. In both cases the computational complexity is significantly reduced compared to if a primal active set solver not utilizing structure is used.

In the CDIO-project course in Automatic Control, an Autonomous Unmanned Aerial vehicle (UAV) is constructed, utilizing an existing radio controlled model aircraft. By adding an inertial sensor measuring acceleration and rotation, together with a Global Positioning System (GPS) sensor, the aim is to construct an accurate positioning system. This is used by an on board computer to calculate rudder control signals to a set of DC-servos in order to follow a predefined way-point trajectory. The project involves 17 students, which is roughly three times as big as previous projects, and it comprises both positioning, control, and hardware design. Since the project is still ongoing some preliminary results and conclusions are presented.

An approach to model reduction of LTI systems using Linear Matrix Inequalities (LMIs) in an H-infinity framework is presented, where non-convex constraints are replaced with stricter convex constraints thus making it suboptimal. The presented algorithms are compared with the Optimal Hankel reduction algorithm, and are shown to achieve better results (i.elower H-infinity errors) in cases where some of the Hankel singular values are close, but not equal to each other.

Semidefinite programs and especially those derived from the Kalman-Yakubovich- Popov lemma are quite common in control applications. KYPD is a dedicated solver for KYP-SDPs. It solves the optimization problem via the dual SDP. The solver is iterative. In each step a Hessian is formed and a linear system of equations is solved. The calculations can be performed much faster if we utilize sparsity and low rank structure. We show how to transform a dense optimization problem into a sparse one with low rank structure. A customized calculation of the Hessian is presented and investigated

In this paper a preprocessing algorithm for unconstrained mixed integer quadratic programming problems and binary quadratic programming problems is presented. The algorithm applies to problems with certain properties, which are further described in the paper. When the algorithm is applied to a problem with these properties, the optimal value for some or all integer variables can be computed without approximations in polynomial time. The algorithm is first derived for the binary quadratic programming problem and the result is then extended to the mixed integer quadratic programming problem by transforming the latter problem into the first problem. Both mentioned quadratic programming problems have several important applications. In this paper, the focus is on model predictive control problems with both real-valued and binary control signals. As an illustration of the method, the algorithm is applied to two different problems of this type.

Semidenite programs derived from the Kalman-Yakubovich-Popov lemma are quite common in control and signal processing applications. The programs are often of high dimension making them hard or impossible to solve with general-purpose solvers. KYPD is a customized solver for KYP-SDPs that utilizes the inherent structure of the optimization problem thus improving efficiency signicantly.

As the semidefinite programs that result from integral quadratic contstraints are usually large it is important to implement efficient algorithms. The interior-point algorithms in this paper are primal-dual potential reduction methods and handle multiple constraints. Two approaches are made. For the first approach the computational cost is dominated by a least-squares problem that has to be solved in each iteration. The least squares problem is solved using an iterative method, namely the conjugate gradient method. The computational effort for the second approach is dominated by forming a linear system of equations. This systems of equations is used to compute the search direction in each iteration. If the number of variables are reduced by solving a smaller subproblem the resulting system has a very nice structure and can be solved efficiently. The first approach is more efficient for larger problems but is not as numerically stable.

This work discusses how to compute stability regions for nonlinear systems with slowly varying parameters using frozen stationary linearization. It is shown that larger stability regions can be obtained as compared to traditional approaches using recent stability results for linear parameter-varying systems.