S : Rnu× Rny → R, if there exists a storage function V : X × Rnx → R+, with V(¯x, 0) = 0, such that
V ¯x(t1), δx(t1)
− V ¯x(t0), δx(t0)
≤ Z t1
t0
S δu(t), δy(t)
dt, (2.7)
for all trajectories (¯x, ¯u)∈ πx,uB and for all t0, t1 ∈ R, with t0 ≤ t1.
Differential dissipativity can be interpreted as the energy dissipation in the trajectory vari-ations, which are not forced by the input. If the energy of these trajectory variations decreases over time, the trajectory variation will eventually only be determined by the input of the sys-tem. Hence, as the unforced variations vanish over time, the trajectory of the primal system will converge to an arbitrary forced equilibrium point or arbitrary reference trajectory, which may can be thought of as the particular solution of the nonlinear system. Therefore, diffential dissipativity is a global system property as well.
Remark 1. When the storage functions V(x, ˜x) and V(¯x, δx) are differentiable, it is possible to define the differentiated form of (2.5) and (2.7), respectively, similar to (2.4).
2.4 Dissipativity analysis results
This section gives an overview of the results obtained in [10] and discusses the extensions on [10], which are documented in [24]. The formal proofs and derivations of the analysis results are ommitted in this thesis, however for some results, the concept or intuition behind the proof is given. The aim of [24] was to have convex dissipativity analysis for nonlinear systems, therefore only quadratic storage and supply functions are considered.
Starting with differential dissipativity analysis, the differential storage function is chosen2 as V(¯x(t), δx(t)) = δx(t)>M (¯x(t))δx(t), (2.8) for which the following assumption holds
A1 The matrix function M (¯x(t))∈ C1 is real, symmetric, bounded and positive definite for all ¯x(t)∈ X .
The differential storage function represents the energy of the tangent variations of the state trajectory ¯x. The differential supply function is chosen as
S(δu(t), δy(t)) =
δu(t) δy(t)
>
Q S
S> R
δu(t) δy(t)
, (2.9)
with real, constant, bounded matrices R = R>, Q = Q> and S. The differential supply function represents how much energy is supplied to or extracted from the variations in a system trajectory. The following result is obtained from [24];
Theorem 1(Differential dissipativity [24]). Consider the system in primal form (2.1) and as-sume A1. This system is differentially dissipative with respect to the quadratic supply function
2One of the main extensions discussed in [24], compared to [10], is the use of a matrix function M (¯x), instead of a constant, bounded matrix M .
CHAPTER 2. INCREMENTAL DISSIPATIVITY ANALYSIS
(2.9), and the quadratic storage function (2.8) if and only if
I 0
Note that the time-dependence in (2.10) is omitted for brevity. Furthermore, one may also note that this (nonlinear) matrix inequality is very similar to the matrix inequalities for LPV systems with quadratic performance, see e.g. [38, Theorem 9.2].
Continuing with incremental dissipativity analysis, the incremental storage function is chosen as
V(x(t), ˜x(t)) = (x(t) − ˜x(t))>M (x(t), ˜x(t))(x(t)− ˜x(t)), (2.11) for which the following assumption holds
A2 The matrix function M (x(t), ˜x(t)) is real, differentiable, bounded, symmetric and pos-itive definite for all x(t), ˜x(t)∈ X .
Furthermore, the incremental supply function is considered in the quadratic form:
S u(t), ˜u(t), y(t), ˜y(t)
with Q = Q>, R = R> and S real, constant, bounded matrices. Furthermore, for the incremental dissipativity analysis, a non-unique mapping ζ :X × X → (0, 1) is required, such that for all x, ˜x∈ πxB
M (x, ˜x) := M ˜x + ζ(x, ˜x)(x− ˜x)
= M (¯x). (2.13)
The details for this part of the analysis are omitted, but can be found in [24]. The following assumption is made on ζ,
A3 ζ ∈ C1.
With the function ζ, the incremental dissipativity analysis is applied on the trajectory (¯x, ¯u, ¯y), which lies somewhere between3 the trajectories (x, u, y) ∈ B and (˜x, ˜u, ˜y) ∈ B. The con-sequence is that the analysis must be done in a convex setting, as shown in the next result from [24], which gives a condition for incremental dissipativity characterization with the con-sidered storage and supply function.
Theorem 2 (Incremental dissipativity [24]). Consider the system in primal form (2.1) and assume A2 and A3. The system is incrementally dissipative with respect to the quadratic
3Note that it is not guaranteed that (¯x, ¯u, ¯y) ∈B, as F might not be convex.
12 Incremental Dissipativity based Control of Nonlinear Systems
2.4. DISSIPATIVITY ANALYSIS RESULTS
supply function (2.12), with R = R> 4 0, and the quadratic storage function (2.11), if
I 0
A (¯x, ¯u) B (¯x, ¯u)
>
0 M (¯x) M (¯x) 0
I 0
A (¯x, ¯u) B (¯x, ¯u)
+ M (¯˙ x) 0
0 0
!
−
0 I
C (¯x, ¯u) D (¯x, ¯u)
>
Q S
S> R
0 I
C (¯x, ¯u) D (¯x, ¯u)
4 0, (2.14)
for all (¯x(t), ¯u(t)) ∈ πx,uC, with A (¯x, ¯u) = ∂ f∂x(¯x, ¯u), B (¯x, ¯u) = ∂ f∂u(¯x, ¯u), C (¯x, ¯u) = ∂ h∂x(¯x, ¯u) andD (¯x, ¯u) = ∂ h∂u(¯x, ¯u).
Note that the time-dependence in (2.14) is omitted for brevity.
Sketch of the proof of Theorem 2. The differentiated version of (2.5) is explicitly written out, which is an unattractive, non-quadratic form. Inspired by [34], the Mean-Value Theorem (MVT) is applied on the expression, which yields a quasi-quadratic form. Applying the MVT implies that there exists an equivalent expression in between the trajectories (x, u, y)∈ B and (˜x, ˜u, ˜y) ∈ B, i.e. (¯x, ¯u, ¯y), which is not necessarily an element of B, therefore, the analysis must done in C. The function ζ transforms the state-dependent matrix functions to a function of ¯x, such that the inequality can be written in terms of the (maybe non-existent) trajectory (¯x, ¯u, ¯y). By restricting R = R>4 0, it is possible to have a clever transformation, yielding the inequality in quadratic form, resulting in (2.14). The detailed proof is given in [24]. With the latter results, the connection between differential dissipativity and incremental dissipative is made.
Theorem 3 (Link differential and incremental dissipativity [24]). Consider a nonlinear sys-tem in its primal form (2.1), with its differential form (2.6) and assume A1–A3. If for all (¯x, ¯u, ¯y) ∈ CR, the differential form of the system is dissipative w.r.t. the storage function (2.8) and the supply function (2.9), with R4 0, then the primal form of the nonlinear system is incrementally dissipative w.r.t. the storage function (2.11) and the supply function (2.12), equally parametrized.
With this result, it is possible to conclude that the primal form of a nonlinear system is incrementally dissipative, whenever the differential form of the same system is dissipative for all points in C. Similarly, the implication holds between incremental dissipativity of a system and general dissipativity of a system. The following result from [24] connects incremental dissipativity and general dissipativity.
Theorem 4. Consider a nonlinear system in its primal form (2.1) and suppose (˘x, ˘u, ˘y)∈ B is a (forced) equilibrium point of the system, i.e. (˘x(t), ˘u(t), ˘y(t)) = (c1, c2, c3) for all t, with (c1, c2, c3)∈ (X × U × Y). Suppose the system is incrementally dissipative w.r.t. the storage function (2.11) and the supply function (2.12). Then for every (forced) equilibrium point, the system is dissipative w.r.t. the same, equally parametrized storage and supply function.
The intuition behind the proof for this last result comes from the fact that if a system is incrementally dissipative, then for a given input, its trajectories (with different initial state
CHAPTER 2. INCREMENTAL DISSIPATIVITY ANALYSIS
conditions) converge towards each other. If the given input is such that it yields a (forced) equilibrium point of the system, then all trajectories with the same input, but different initial state conditions, converge towards the forced equilibrium point.
The chain of implications that can be made using these theorems is depicted in Figure 2.1.
The black arrows point towards the implications and equalities that do hold only when the Differential dissipativity
Incremental dissipativity
General dissipativity
⇐⇒
⇒ =
(2.10)
0 and
R If
CR
πx,u
∈ ) u
¯ x,
∀(¯ =
⇒=⇒=
(2.14)
Figure 2.1: Chain of implications for the result on dissipativity analysis of nonlinear systems.
These implications hold for the considered storage and supply functions.
conditions in the box are fulfilled. It must be noted that the implications are always with respect to the considered quadratic storage and supply functions.
The next questions would be, how to apply these results? And is it possible to determine whether a nonlinear system is incrementally dissipative by simple computations? In [10, 24]
the results are applied to some well-known notions relating to dissipativity, e.g. incremental extensions of L2-gain and passivity, which are given (without proofs) in Appendix A.1 for completeness. The second question is answered in the next section.