PhD Thesis

Click HERE to receive the gzipped postscript file (622Kb).

Click HERE to receive the pdf file (1 Mb).

Summary

This thesis deals with an analysis of the contributions of dynamical elements to past and future energy of nonlinear state-space systems, and offers a tool for nonlinear model reduction.

A well-known tool to analyze linear state-space systems and to reduce the number of state-space components is balancing. Balancing has been introduced for stable linear systems, and is a method that measures at the same time the contributions of the state components of the linear system to the past input energy and to the future output energy (in an L2-sense). A state component that generates a relatively large amount of output energy, and needs a relatively small amount of input energy to be reached is considered to be of importance to the system, while a state component that generates a relatively small amount of output energy, and that needs a relatively large amount of input energy to be reached is considered to be of less importance to the system. Starting as an open-loop balancing method for stable linear systems the method has been developed further to open-loop and closed-loop balancing methods for (unstable) linear systems, and mechanical linear systems. Balancing for linear systems may be interpreted from several equivalent point of views, both in the time domain and in the frequency domain.
This thesis deals with balancing for nonlinear systems. Not all interpretations of balancing for linear systems are suitable for nonlinear systems, but the principal interpretation in terms of a past and future energy function may be carried over to balancing methods for nonlinear systems. In fact, from this point of view the linear balancing methods may be seen as a special case of the nonlinear balancing methods of this thesis.

For stable nonlinear systems an open-loop balancing method is developed. This method considers the past input and the future output energy function of the system. By a coordinate transformation these functions may be brought into a form that is called the balanced form. In balanced form both energy functions are at the same time a measure for the importance of the different state components, i.e., the past input energy function measures the amount of input energy that is needed to reach a state component, while the future output energy function measures the amount of output energy that is generated by this state component. The procedure that brings the system into balanced form is relying on Morse's Lemma, and a diagonalization procedure of state dependent matrices. It defines an ordered set of singular value functions that each correspond to a state component. A relatively large (or small, respectively) singular value function implies that the corresponding state component generates a relatively large (or small, respectively) amount of output energy, and needs a relatively small (or large, respectively) amount of input energy to be reached. The amount of input energy that is needed to reach a state component is a measure for the reachability of this state component, while the amount of output energy generated by a state component is a measure for the observability of this state component. Hence, a singular value function yields a characterization of the corresponding state component in terms of easy or difficult to reach and observe. A relatively large (or small, respectively) singular value function implies that the corresponding state component may be retained (or removed, respectively) to obtain the reduced order model.
In this thesis model reduction based on balancing is studied, as well as to what extent we may speak about similarity invariance of the singular value functions.

Balancing for stable nonlinear systems offers a procedure that may be used to define other balancing methods for possibly unstable nonlinear systems. Two open-loop balancing methods are derived from this procedure. Both methods are based on association of the nonlinear system with a stable nonlinear system that may be brought into balanced form. Furthermore, two other methods that consider a different pair of past and future energy functions (involving both inputs and outputs) are developed by following a similar procedure as for balancing stable nonlinear systems. These two methods may each be associated with a control problem (the nonlinear version of the LQG problem and the nonlinear normalized H-infinity control problem, respectively), and therefore may be seen as closed-loop balancing methods. For both closed-loop methods the characterization of a state component in terms of easy or difficult to reach and observe may be replaced by difficult or easy to control (to zero) and filter.
For possibly unstable nonlinear systems an open-loop balancing method is developed on the normalized right and left coprime factorizations of the nonlinear system. A normalized coprime representation is a stable nonlinear system that may be associated with the original unstable nonlinear system. Explicit expressions for the normalized coprime representations are derived. The normalized coprime representation of a system may be brought into balanced form, and this form offers a method for model reduction of the original unstable system.
A closed-loop balancing method for an unstable nonlinear system is considering a pair of past and future energy functions that are brought into a special form by a coordinate transformation. This special form is called the HJB (Hamilton-Jacobi-Bellman) balanced form. In the case of a linear system this HJB balanced form is the LQG balanced form, and has an interpretation in terms of the LQG control problem. In the general nonlinear case the method is related to the nonlinear version of the LQG problem, and we may give an interpretation in terms of this problem, but this is not as general as in the linear case. Furthermore, HJB balancing and balancing of the normalized coprime representation are being related via their singular value functions.
A second closed-loop balancing method is H-infinity balancing. Another pair of energy functions is defined, and again brought into a special form by a coordinate transformation. This form is called the H-infinity balanced form. Similarly to the relation between HJB balancing and the nonlinear version of the LQG problem, the H-infinity balanced form may be interpreted in terms of a normalized H-infinity control problem.
The last balancing method that is treated in this thesis is pseudo balancing. Pseudo balancing is the second open-loop balancing method that is developed for nonlinear systems, and is only defined for nonlinear Hamiltonian systems with positive internal energy. This balancing method takes the Hamiltonian structure of the system explicitly into account. A Hamiltonian system is not asymptotically stable, but it may be associated with a gradient system that is asymptotically stable, and that may be brought into balanced form. The Hamiltonian system is said to be in pseudo balanced form, if the associated gradient system is in balanced form. From a modelling and control point of view pseudo balancing has an advantage over other balancing methods for Hamiltonian systems, since model reduction based on pseudo balancing results again in a Hamiltonian system, and this is generally not true for other balancing methods.

Return to the home page of Jacquelien Scherpen