Performance analysis of switching systems

Performance analysis is an important aspect in the design of dynamic (control) systems. Without a proper analysis of the behavior of a system, it is impossible to guarantee that a certain design satisfies the system’s requirements. For linear time-invariant systems, accurate performance analyses are relatively easy to make and as a result also many linear (controller) design methods have appeared in the past. For nonlinear systems, on the other hand, such accurate performance analyses and controller design methods are in general not available. A main reason hereof is that nonlinear systems, as opposed to linear time-invariant systems, can have multiple steady-state solutions. Due to the coexistence of multiple steady-state solutions, it is much harder to define an accurate performance index. Some nonlinear systems, i.e. the so-called convergent nonlinear systems, however, are characterized by a unique steady-state solution. This steady-state solution may depend on the system’s input signals (e.g. reference signals), but is independent of the initial conditions of the system. In the past, the notion of convergent systems has already been proven to be very useful in the performance analysis of nonlinear systems with inputs. In this thesis, new results in the field of performance analysis of nonlinear systems with inputs are presented, based on the notion of convergent systems. One part of the thesis is concerned with the question "how to analyse the performance for a convergent system?" Since the behavior of a convergent system is independent of the initial conditions (after some transient time), simulation can be used to find the unique steady-state solution that corresponds to a certain input signal, but this can be very time-consuming. In this thesis, a computationally more efficient approach is presented to estimate the steady-state performance of harmonically forced Lur’e systems, in terms of nonlinear frequency response functions (nFRFs). This approach is based on the method of harmonic linearization. It provides both a linear approximation of the nFRF and an upper bound on the error between this linear approximation and the true nFRF. It is shown in several examples that the approximation of the nFRF is accurate, and that it provides more detailed information on the considered system than the often used ‘L2 gain’ performance index. An additional observation that is made, is that the method of harmonic linearization can sometimes be ‘misleading’ for Lur’e systems with a saturation-like nonlinearity: for the case that the harmonic balance equation has a unique solution, it is shown that the corresponding nonlinear system can have multiple distinct steady-state solutions. Another part of the thesis is concerned with the question "under what conditions is a system with inputs guaranteed to be convergent?" In particular two types of systems were investigated: switched linear systems and Lur’e systems with a saturation nonlinearity and marginally stable linear part. For the switched linear systems, it is assumed that the dynamics of all the separate linear modes are given. For this setting, it was investigated if it is possible to find a switching rule (which defines when to switch between the available modes) such that the closed-loop system is convergent. Both a state-based, an observer-based, and a time-based switching rule are presented that guarantee a convergent system, assuming some conditions on the linear dynamics are met. The second type of systems that are discussed, are Lur’e systems with a saturation nonlinearity and marginally stable linear part. For this type of systems, the goal was to find sufficient conditions under which the closed-loop system is convergent. Because of the marginally stable linear part, however, a quadratically convergent system cannot be obtained. Instead, sufficient conditions are discussed that guarantee uniform convergency of the system. The obtained theory is shown to be also applicable to a class of anti-windup systems with a marginally stable plant. For this class of systems, the results of the convergency-based performance analysis are compared with the analysis results of existing anti-windup methods. It is shown that the convergency-based performance analysis can in some cases provide more detailed information on the steady-state behavior of the system. The results of uniform convergency for anti-windup systems are shown to be also applicable in the field of production and inventory control of discrete-event manufacturing systems. Since a manufacturing machine has a certain production capacity and cannot produce at a negative rate, it can be seen as an integrator plant (input: production rate, output: amount of finished products) preceded by a saturation function. For this marginally stable plant, a controller was constructed in such a way that the closed-loop system is uniformly convergent. The controller was also implemented in the discrete-event domain and the results from discrete-event simulations were compared with those of continuous-time simulations. Similarly, the controller was also applied for the production and inventory control of a line of four manufacturing machines. For both the single machine and the line of four machines, the resulting controlled discrete-event systems are shown to have the desired tracking properties. Besides these theoretical and numerical results, also experimental results are presented in this thesis. By means of an electromechanical construction, several experimental results were obtained, and used to validate the theoretical results for both the switched linear systems and the anti-windup systems

Realization Theory for LPV State-Space Representations with Affine Dependence

In this paper we present a Kalman-style realization theory for linear parameter-varying state-space representations whose matrices depend on the scheduling variables in an affine way (abbreviated as LPV-SSA representations). We deal both with the discrete-time and the continuous-time cases. We show that such a LPV-SSA representation is a minimal (in the sense of having the least number of state-variables) representation of its input-output function, if and only if it is observable and span-reachable. We show that any two minimal LPV-SSA representations of the same input-output function are related by a linear isomorphism, and the isomorphism does not depend on the scheduling variable.We show that an input-output function can be represented by a LPV-SSA representation if and only if the Hankel-matrix of the input-output function has a finite rank. In fact, the rank of the Hankel-matrix gives the dimension of a minimal LPV-SSA representation. Moreover, we can formulate a counterpart of partial realization theory for LPV-SSA representation and prove correctness of the Kalman-Ho algorithm. These results thus represent the basis of systems theory for LPV-SSA representation.Comment: The main difference with respect to the previous version is as follows: typos have been fixe