Vladimír Havlena

Brief paper: Subspace like identification incorporating prior information

The subspace identification methods have proved to be a powerful tool, which can further benefit from the prior information incorporation algorithm proposed in this note. In the industrial environment, there is often some knowledge about the identified system (known static gains, dominant time constants, low frequency character, etc.), which can be used to improve model quality and its compliance with first principles. The proposed algorithm has two stages. The first one is similar to the subspace methods as it uses their interpretation as an optimization problem of finding parameters of an optimal multi-step linear predictor for the experimental data. This problem is reformulated in the Bayesian framework allowing prior information incorporation in the form of the mean value and the covariance of the impulse response, which is shown to be useful for the incorporation of several prior information types. The second stage with state space model realization from the posterior impulse response estimate is different from the standard subspace methods as it is based on the structured weighted lower rank approximation, which is necessary to preserve the prior information incorporated in the first stage.

Simultaneous parameter tracking and state estimation in a linear system

Abstract The task of simultaneous tracking of time-varying parameters and estimation of the state is treated for a linear system described by a time-varying input-output ARMAX or Delta model with known c (noise) parameters. First, a Bayesian approach-based conceptual solution is presented. Then it is shown that utilizing the properties of the observer canonical state model, algebraic recursion operating on the joint parameter and state mean and covariance matrix can be obtained with no approximation involved. Several illustrative examples are included.

Multiple-step active control with dual properties

Abstract This paper proposes multiple-step active control algorithms based on MPC approach that approximate persistent system excitation in terms of the increase of the lowest eigenvalue of the parameter estimate information matrix. It is shown how the persistent excitation condition is connected with a proposed concept of stability of a system with uncertain parameters. Unlike similar methods, the proposed algorithms predict the information matrix for more than one step of control. The problem is formulated as an MPC problem with an additional constraint on the information matrix. This constraint makes the problem non-convex, thus only locally optimal solutions are guaranteed. The proposed algorithm is derived for ARX system only, but it allows for future reformulation for a general ARMAX system with known moving average (MA) part.

An approach to out-of-sequence measurements in feedback control systems

Complex communication network architectures are becoming more frequent in control applications. In such networked control systems, it is often the case that information on process variables is received out-of-time-order. This paper presents a Bayesian approach to handling this out-of-sequence information problem. Such approach leads to a solution involving the joint probability density of current state and past measurements not yet received. Under linear Gaussian assumptions, the Bayesian solution reduces to an augmented state Kalman filter. Our approach augments the state dynamically based on the list of missing observations. As this solution can be time and memory consuming, two simplified implementations of the algorithm are presented.

Combined partial conjugate gradient and gradient projection solver for MPC

The objective of this paper is to present fast algorithm for large scale box constrained quadratic program which arises from linear model predictive control (MPC) with hard limits on the inputs. The presented algorithm uses the combination of the gradient projection method and the partial conjugate gradient method. The special structure of the MPC problem is exploited so that the conjugate gradient method converges in a few number of iterations and the algorithm well suitable for processes with thousands of inputs and small number of outputs is obtained.

Gradient projection based algorithm for large scale real time model predictive control

In model predictive control (MPC), the quadratic program (QP) is solved at each sampling time, thus a fast and effective on-line solver must be used for short sampling times. The multi-parametric quadratic programming (mp-QP) (explicit solution) is impossible to use for larger systems due to the memory limitation. The objective of this paper is to present an effective on-line solver for large-scale simple constrained quadratic programming which arises in the MPC framework. The presented algorithm uses the combination of gradient and Newton projection method to obtain super-linear convergent algorithm which is very close to optimum in very few iterations when many constraints are active in optimum and it does not involve the exact computation of the Newton step at each iteration.

Combined Gradient and Newton Projection Quadratic Programming Solver for MPC

Abstract The objective of this paper is to present an effective on-line solver for simple constrained quadratic programming (QP) which arises in linear model predictive control (MPC) framework. In MPC, the QP is solved at each sampling time, thus a fast solver must be used for short sampling times in real-time applications. The multi-parametric quadratic programming (mp-QP) approach (explicit solution) is impossible to use for larger systems due to the memory limitation. On the other hand, the presented approach is well suitable even for medium scale systems with short sampling time, since it is based on combination of gradient and Newton projection algorithm which is very close to optimum in a very few iterations and the computation of the Newton step is not involved at each iteration.

Noise covariances estimation for Kalman filter tuning

Abstract Kalman filter tuning is based on process and measurement noise covariances that are parameters of Riccati equation. Based on the Riccati equation solution, Kalman gain is calculated and further used for state estimator. Noise covariances are generally not known. Several methods have been published since 70s. The latest methods and their modifications were published in 2005 and later. In many parts of technical science the Bayesian approach can be used for various estimation problems. However many scientists and researchers a priori consider Bayesian principles to be unpractical because in most cases it is very difficult to work with probabilities or likelihood functions. The probability or likelihood functions cannot be solved analytically for most problems. In this paper we will discuss the performance of some published methods and compare them with the maximum likelihood approach using numerical methods. The key goal is to demonstrate quality of maximum likelihood approach that can effectivelly use information from given data. A very simple algorithm demonstrates using maximum likelihood approach for noise covariances estimation of a scalar system.

Tailored QP algorithm for Predictive Control with dynamics penalty

In order to reduce the computational complexity of solving Quadratic Programming (QP), related to linear Model Predictive Control (MPC), a new approximated formulation of the QP with simple bounds is introduced in this paper. This formulation is based on the idea not to consider model dynamics as a hard constraint but rather modify the objective function of MPC by penalty to capture the violation of model dynamics. The system dynamics is usually uncertain and then it does not make sense to design the control law based on the exact model. Furthermore, the specific sparse structure of the approximated simple bounded QP formulation of the MPC problem is exploited in the new type of combined gradient/Newton step projection algorithm with linear complexity of each iteration with respect to prediction horizon. It is shown by examples that the proposed method is faster on tested problem than other state-of-the-art solvers while retaining a high performance level.

Receding-horizon MIMO LQ controller design with guaranteed stability

To design an LQ-optimal receding horizon controller with guaranteed stability, the initialization of the underlying Riccati equation corresponding to an LQ problem with a fixed terminal state can be used. It is shown that the result can be generalized to MIMO systems and that the initialization is particularly simple if input multirate sampling is used.