Development and evaluation of methods for control and modelling of multiple-input multiple-output systems
Doctoral thesis, 2020

In control, a common type of system is the multiple-input multiple-output (MIMO) system, where the same input may affect multiple outputs, or conversely, the same output is affected by multiple inputs. In this thesis two methods for controlling MIMO systems are examined, namely linear quadratic Gaussian (LQG) control and decentralized control, and some of the difficulties associated with them.

One difficulty when implementing decentralized control is to decide which inputs should control which outputs, also called the input-output pairing problem. There are multiple ways to solve this problem, among them using gramian based measures, which include the Hankel interaction index array, the participation matrix and the Σ2 method.  These methods take into account system dynamics as opposed to many other methods which only consider the steady-state system. However, the gramian based methods have issues with input and output scaling. Generally, this is handled by scaling all inputs and outputs to have equal range. However, in this thesis it is demonstrated how this can cause an incorrect pairing. Furthermore, this thesis examines other methods of scaling the gramian based measures, using either row or column sums, or by utilizing the Sinkhorn-Knopp algorithm. It is shown that there are considerable benefits to be gained from the alternative scaling of the gramian based measures, especially when using the Sinkhorn-Knopp algorithm. The use of this method also has the advantage that the results are completely independent of the original scaling of the inputs and outputs.

An expansion to the decentralized control structure is the sparse control, in which a decentralized controller is expanded to include feed-forward or MIMO blocks. In this thesis we explore how to best use the gramian based measures to find sparse control structures, and propose a method which demonstrates considerable improvement compared to existing methods of sparse control structure design.

A prerequisite to implementing control configuration methods is an understanding of the processes in question. In this thesis we examine the pulp refining process and design both static and dynamic models for pulp and paper properties such as shives width, fiber length and tensile index, and various available inputs. We demonstrate that utilizing internal variables (primarily consistencies) estimated from temperature measurements yields improved results compared to using solely measured variables. The measurement data from the refiners is noisy, sometimes sparse and generally irregularly sampled. This thesis discusses the challenges posed by these constraints and how they can be resolved.  

An alternative way to control a MIMO system is to implement an LQG controller, which yields a single control structure for the entire system using a state based controller. It has been proposed that LQG control can be an effective control scheme to be used on networked control systems with wireless channels. These channels have a tendency to be unreliable with packet delays and packet losses. This thesis examines how to implement an LQG controller over such unreliable communication channels, and derives the optimal controller minimizing the cost function expressed in actuated controls.

When new methods of control system design and analysis are introduced in the control engineering field, it is important to compare the new results with existing methods. Often this requires application of the methods on examples, and for this purpose benchmark processes are introduced. However, in many areas of control engineering research the number of examples are relatively few, in particular when MIMO systems are considered. For a thorough assessment of a method, however, as large number of relevant models as possible should be used. As a remedy, a framework has been developed for generating linear MIMO models based on predefined system properties, such as model type, size, stability, time constants, delays etc. This MIMO generator, which is presented in this thesis, is demonstrated by using it to evaluate the previously described scaling methods for the gramian based pairing methods.

LQG control


Uncertain data sets

Unreliable communication links


Tensile Index


MIMO systems



Fiber length

Control configuration selection

Linear regression

Gramian based measures

Decentralized control


Input-output scaling


Opponent: Tore Hägglund, Department of Automatic Control, Lund University, Sweden. Password: 577727


Fredrik Bengtsson

Chalmers, Electrical Engineering, Systems and control, Automatic Control

LQG Control for Systems with Random Unbounded Communication Delay

Proceedings of the 55th IEEE Conference on Decision and Control (CDC 2016); Las Vegas; United States; 12-14 December 2016,; (2016)p. Art no 7798406, Pages 1048-1055

Paper in proceeding

Fredrik Bengtsson, Torsten Wik, “Stochastic optimal control over unreliable communication links”

Resolving issues of scaling for gramian-based input–output pairing methods

International Journal of Control,; Vol. In Press(2020)

Journal article

Fredrik Bengtsson, Torsten Wik, “Finding feedforward configurations using gramian based interaction measures”

Modeling of tensile index using uncertain data sets

Nordic Pulp and Paper Research Journal,; Vol. 35(2020)p. 231-242

Journal article

Fredrik Bengtsson, Anders Karlström, Torsten Wik, “On the modeling of pulp properties in CTMP processes”

In many control applications it is common that one simultaneously wishes to control numerous aspects (outputs) of the same process. This generally requires use of more than one actuator (input) and is thus called a multiple input multiple-output control problem. These types of problems are the focus of this thesis.

When attempting to control multiple outputs of the same process, a key challenge is that it is difficult to control one thing without affecting the others. One way to deal with this problem is to treat each output you wish to control individually and design a simple controller for each output. When implementing this, a central problem is to determine which of the available inputs should be used to control which output. In this thesis we propose and evaluate improvements to existing methods to solve this problem.

Another common issue when implementing the controllers over a network is unreliability in the communication. After a control input has been determined it may be that it arrives to the actuator late or not at all. Here, in this thesis, we examine how to best implement controllers in the case when we have statistics on the potential delays.

To implement and design controllers an understanding of the controlled process is highly important. Accurate models of the process are highly useful in this regard. Furthermore they give the possibility to complement unreliable manual measurements with model estimates which can then be used for control. In this thesis we examine the pulp refining process, and derive such models of how the process inputs affect the pulp quality.

Metoder och verktyg för utvärdering av dynamiska aspekter vid design av effektivare industriella energisystem

VINNOVA (2014-01945), 2014-09-01 -- 2017-08-31.

SiiS - Strategic industrial information Systems

Swedish Energy Agency (P42330-1), 2016-07-01 -- 2019-12-30.

Areas of Advance

Information and Communication Technology



Subject Categories

Paper, Pulp and Fiber Technology

Communication Systems

Control Engineering



Doktorsavhandlingar vid Chalmers tekniska högskola. Ny serie: 4848


Chalmers University of Technology


Opponent: Tore Hägglund, Department of Automatic Control, Lund University, Sweden. Password: 577727

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