Comments and References

Piecewise linear systems is an interesting system class, and many impor-tant remarks can be made to the developments described so far. Rather

2.5 Comments and References

PwL PwL PwL

1 s 1

s ⁺

+

−

u x y

−1 0 0 1

−1 0 1

x y

Figure 2.7 Series connection of two piecewise linear systems is piecewise linear (^left). The interconnected system has the partition shown to the right.

than obstructing the general presentation with long discussions, we have chosen to collect such remarks in a special section. Some of the material presented here are pure remarks, while others discuss related work, give alternative perspectives on the material or present issues that are not otherwise covered in the thesis.

Piecewise Linear or Piecewise Affine?

The term piecewise linear may at first appear inappropriate for the system (^2.3), since the dynamics is in fact affine in the state. However, since the name is generally accepted, we have chosen not to make a stronger point out of this. One may motivate the name piecewise linear by the fact that around any trajectory inside a cell, the dynamics will behave linearly.

Memory Efficient Representations

Modeling and simulation of piecewise linear systems has attracted a large interest in the circuit theory community during the last decades[^{30, 84}]^. Driven by the need to simulate large-scale circuits with piecewise linear components, a large research effort has been focused on deriving mem-ory efficient representations for piecewise linear systems. More compact descriptions than the matrix representation(^2.7) can be obtained when the vector field of the system is continuous across cell boundaries. To see this, consider the situation in Figure 2.8.

To obtain continuity on the hyperplane

H

A_j
^Ai+^cijh^T_ij aj
^ai+^cijgij

for some c_ij ∈ ^Rn. Since the boundary equations of the cells reappear in the description of feasible changes in the mapping, there is a certain

Chapter 2. Piecewise Linear Modeling

X_i X_j

˙x
^Aix+^ai ˙x
^Ajx+^aj

h^T_ijx+^gij
⁰

Figure 2.8 Continuity of vector fields allows parameter savings.

redundancy in the data given by the simple model (^2.7). The argument above indicates that it should be sufficient to store one linear system description, the boundary equations, and the update vectors c_ij.

The first compact parameterization that appeared was the canonical piecewise linear function description introduced in[³⁰]. For piecewise lin-ear dynamic systems, it takes the form

˙x
^Ax+^a+ Xp

i
1

c_it^gTi x+^hit. (^2.12)

This representation stores only a single affine system description, the boundary hyperplane equations, and the update vectors. It has also elim-inated the need for explicit storage of cell descriptions, and no cell identi-fication is necessary to evaluate the mapping. The representation(^2.12) is very efficient compared to the simple matrix parameterization (^2.7)^. However, it can only represent a subset of the continuous piecewise lin-ear mappings (^cf. [^{71, 77}]). To overcome this problems, various higher order basis function expressions have been suggested. These are much more complicated than the simple model (^2.12), but can be put in the general form

˙x
^Ax+^a+ Xq

i
1

c_iϕi(^x)

Here, the ϕi(^x) are piecewise linear functions constructed from nested absolute or maximum functions, see[^{71, 43, 77}]

An alternative formulation, which is closely related to the matrix rep-resentation(^2.7), is the implicit piecewise linear function description

re-2.5 Comments and References ported in[¹³⁹]









˙x
^Ax+^a+^Bu i
^Gx+^g+^Cu 0
^{uTi u}⁰,ⁱ⁰

(^2.13)

This representation was derived from a static linear network where some ports have been terminated by negative ideal diodes [⁷⁷]. The variables i and u correspond to currents and voltages respectively, and the last equation of(^2.13)describes the characteristic of an ideal negative diode.

Vectors u and i that satisfy this equation are called complementary. To see the close connection to our matrix parameterization, consider the region where u
0. Then, the above model reduces to

˙x
^Ax+^a for x such that Gx+^g⁰.

Given a vector x, an evaluation of the mapping needs the associated diode voltages u. This requires that we solve the linear complementary problem of finding i and u that satisfy the last two equations of(^2.13)^{. In} principle, a solution to this problem can be obtained through a sequence of pivoting operations around the C-matrix. Each such pivoting step forces one entry of the vectors u or i to zero, while the corresponding entry in the other vector is allowed to be non-zero. The non-zero entries of u and i define affine inequalities in x (since the corresponding entries of the other vector are zero), and hence closed halfspaces in the state space.

The zero entries in the i-vector forces the corresponding entries of u to be affine functions of x. These affine expressions are used to describe the relative changes in the local dynamics via the first equation of(^2.13)^{. In} this way, the matrix C encodes the changes in the cell descriptions, while the matrix B encodes the changes in the affine dynamics. The best way to understand the implicit model(^2.13)in further detail is to work through some of the examples given in[¹³⁹] (^{see also}[^{28, 110}])^.

One drawback with the implicit model(^2.13)is that a solution to the linear complementary problem may require a number of pivoting opera-tions that is exponential in the number of entries of the vectors u and i.

As we have seen above, solving the linear complementary problem is re-lated to performing a cell identification in our framework. The exponential complexity is related to the fact that we may have to check membership to all cells when evaluating the piecewise linear mapping. However, once a feasible set of complementary vectors u and i has been found, only one pivoting operation is needed in order to determine the new set-up when one constraint has been violated. This has allowed the development of fast

Chapter 2. Piecewise Linear Modeling

and memory efficient simulation programs based on this model[²⁸]^{. We} also note that complementary conditions occur naturally in the modeling of impulse and contact forces in mechanics, see[^{87, 141}]^.

3

Structural Analysis

The main aim of this thesis is to provide quantitative methods for anal-ysis of piecewise linear dynamical systems. This chapter will present the first building blocks of such an analysis. The methods described here are mainly of a static nature, and can all be obtained through vector field considerations. We will treat equilibrium computations and static gain analysis, as well as detection of sliding modes and verification of affine state constraints. These results are useful for ruling out degeneracies in piecewise linear models, give important engineering insight and are valuable complements to the Lyapunov-based methods developed in the subsequent chapters.

3.1 Equilibrium Points and Static Gain Analysis

An initial problem in the study of a nonlinear system is to determine its equilibrium points. In this context, we will understand the term equilib-rium point as a constant trajectory(in the sense of Definition 2.1)^. Con-trary to a linear system which always has an equilibrium at the origin, a general nonlinear system

˙x
^f(^x,^u) y
^g(^x,^u)

may have any number of equilibrium points. We will make a distinction between equilibrium point computations and static gain analysis. In equi-librium point computations, we let u
0 and consider the problem of finding the solutions x^∗ to the equation f(^x∗,⁰)
0. By static gain analy-sis, we refer to the problem of computing the outputs y^∗ that correspond to the equilibrium points obtained for a constant input signal u
^{u∗. This} problem can be solved by first computing the equilibrium point x
^{x∗ for}