Anders Hansson

Anders Hansson, Zhang Liu, Lieven Vandenberghe

We present a subspace system identification method based on weighted nuclear norm approximation. The weight matrices used in the nuclear norm minimization are the same weights as used in standard subspace identification methods. We show that the inclusion of the weights improves the performance in terms of fit on validation data. As a second benefit, the weights reduce the size of the optimization problems that need to be solved. Experimental results from randomly generated examples as well as from the Daisy benchmark collection are reported. The key to an efficient implementation is the use of the alternating direction method of multipliers to solve the optimization problem.

Mariette Annergren, Anders Hansson, Bo Wahlberg

We present an Alternating Direction Method of Multipliers (ADMM) algorithm for solving optimization problems with an l_1 regularized least-squares cost function subject to recursive equality constraints. The considered optimization problem has applications in control, for example in l_1 regularized MPC. The ADMM algorithm is easy to implement, converges fast to a solution of moderate accuracy, and enables separation of the optimization problem into sub-problems that may be solved in parallel. We show that the most costly step of the proposed ADMM algorithm is equivalent to solving an LQ regulator problem with an extra linear term in the cost function, a problem that can be solved efficiently using a Riccati recursion. We apply the ADMM algorithm to an example of l_1 regularized MPC. The numerical examples confirm fast convergence to moderate accuracy and a linear complexity in the MPC prediction horizon.

Jean Thomas, Anders Hansson

Direct torque control 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 paper 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 direct torque control.

Michel Verhaegen, Anders Hansson

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. This is illustrated using a data set from the DaSIy library. An efficient implementation in the framework of the Alternating Direction Method of Multipliers (ADMM) is presented that is used in the validation study.

Chengpu Yu, Michel Verhaegen, Anders Hansson

This paper considers the identification of large-scale 1D networks consisting of identical LTI dynamical systems. A new subspace identification method is developed that only uses local input-output information and does not rely on knowledge about the local state interaction. The identification of the local system matrices (up to a similarity transformation) is done via a low dimensional subspace retrieval step that enables the estimation of the Markov parameters of a locally lifted system. Using the estimated Markov parameters, the state-space realization of a single subsystem in the network is determined. The low dimensional subspace retrieval step exploits various key structural properties that are present in the data equation such as a low rank property and a {\em two-layer} Toeplitz structure in the data matrices constructed from products of the system matrices. For the estimation of the system matrices of a single subsystem, it is formulated as a structured low-rank matrix factorization problem. The effectiveness of the proposed identification method is demonstrated by a simulation example.

Seyyed Reza Jafari, Anders Hansson, Bo Wahlberg

Navigating a collision-free and optimal trajectory for a robot is a challenging task, particularly in environments with moving obstacles such as humans. We formulate this problem as a stochastic optimal control problem. Since solving the full problem is computationally demanding, we introduce a tractable approximation whose Bellman equation can be solved efficiently. The resulting value function is then incorporated as a terminal penalty in an online rollout framework. We construct a trade-off curve between safety and performance to identify an appropriate weighting between them, and compare the performance with other methods. Simulation results show that the proposed rollout approach can be tuned to reach the target in nearly the same expected time as receding horizon $A^\star$ while maintaining a larger expected minimum distance to the moving obstacle. The results also show that the proposed method outperforms the considered CBF-based methods when a larger obstacle clearance is desired, while achieving comparable performance otherwise.

Anders Hansson, Bo Wahlberg

This paper concerns rollout and certainty-equivalent rollout policies for stochastic shortest path problems with absorbing terminal states. The main result provides a direct non-asymptotic performance certificate for a fixed rollout policy: the loss relative to the optimal value is controlled by the uniform accuracy of the value approximation and by the expected time for which the rollout closed loop remains away from the terminal state. Thus, in the undiscounted transient setting, the expected hitting time plays the role of a discount or finite-horizon parameter in more standard approximate dynamic programming bounds. This paper also gives a performance-difference identity showing that suboptimality is exactly accumulated through the transient occupation measure, and a deterministic sharpness example showing that the hitting-time factor is unavoidable. Finally, consequences under uniform hitting-time and Foster-Lyapunov drift conditions are given, and extend the argument to certainty-equivalent rollout by adding a separate local model-mismatch term.

Yuwei Ying, Johan Löfberg, Anders Hansson

The computation of chance constraints in stochastic model predictive control is often numerically challenging due to the non-Gaussian nature of the disturbances. To overcome this problem, we propose an optimization computational framework applicable to non-Gaussian disturbances. This framework employs a numerical inversion method, utilizing the characteristic function of the disturbance distribution to compute the probability in the chance constraint as well as its gradient. To improve efficiency, it vectorizes integral points and reuses intermediate computations in Gauss-Kronrod quadrature. The framework is implemented within the YALMIP toolbox to perform chance constraint calculations for arbitrary non-Gaussian disturbances, applicable to both single-component distributions and mixture models. It allows the user to simply specify a distribution type and its parameters for the disturbance and directly compute the probability and its gradient to solve the optimization problem. The method is validated through a numerical example of a stochastic model predictive control application.

Lucas N. Egidio, Anders Hansson, Bo Wahlberg

We consider the problem of how 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 but currently avoided in large-scale problems for requiring step sizes to be provided at each iteration. Existing 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 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.

Arun Venkitaraman, Anders Hansson, Bo Wahlberg

This paper investigates the use of nonparametric kernel-regression to obtain a tasksimilarity aware meta-learning algorithm. Our hypothesis is that the use of tasksimilarity helps meta-learning when the available tasks are limited and may contain outlier/ dissimilar tasks. While existing meta-learning approaches implicitly assume the tasks as being similar, it is generally unclear how this task-similarity could be quantified and used in the learning. As a result, most popular metalearning approaches do not actively use the similarity/dissimilarity between the tasks, but rely on availability of huge number of tasks for their working. Our contribution is a novel framework for meta-learning that explicitly uses task-similarity in the form of kernels and an associated meta-learning algorithm. We model the task-specific parameters to belong to a reproducing kernel Hilbert space where the kernel function captures the similarity across tasks. The proposed algorithm iteratively learns a meta-parameter which is used to assign a task-specific descriptor for every task. The task descriptors are then used to quantify the task-similarity through the kernel function. We show how our approach conceptually generalizes the popular meta-learning approaches of model-agnostic meta-learning (MAML) and Meta-stochastic gradient descent (Meta-SGD) approaches. Numerical experiments with regression tasks show that our algorithm outperforms these approaches when the number of tasks is limited, even in the presence of outlier or dissimilar tasks. This supports our hypothesis that task-similarity helps improve the metalearning performance in task-limited and adverse settings.

Manon Kok, Sina Khoshfetrat Pakazad, Thomas B. Schön et al.

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.