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In the current work, only continuous time systems will be discussed. However, the theory outlined in the following can be extended to the discrete time case.

4.2.1 Mathematical Preliminaries

A continuous time signal γ : R+→ Rm is inH (γ ∈ H ) if it has finite Lm2 -norm:

kγk2

2 =

Z ∞ 0

γ>(t)γ(t)dt <_∞. (4.1)

An extended signal space, Lm

2e, can be defined by introducing a truncation opera-

tor: γT(t) =    γ(t) t < T 0 t_{≥ T}. (4.2)

A continuous time signal γ : R+ → Rm is in He (γ ∈ He) if:

kγTk22 =

Z T 0

γ>(t)γ(t)dt <_{∞ ∀T ∈ R}+. (4.3)

The temporal inner product of two signal vectors y(t) and u(t) over the time interval [0, T] is defined:

hy(t), u(t)i[0,T] =

Z T 0

y>(t)u(t)dt. (4.4)

The spatiotemporal inner product of two signal vectors y(χ, t) and u(χ, t) over the time interval [0, T] is defined:

hy(χ, t), u(χ, t)i[0,T] =

Z T 0

Z

χ∈Ω

y>_{(χ, t)u(χ, t)dχdt. (4.5)}

Define a system Σ as a relation on He. For u ∈ He, Σu denotes an image of

u under Σ [113], and Σu(t) is the value of Σu at time t. The continuous time system Σ :He→ He is Lm2 stable if:

u∈ Lm

2 =⇒ Σu ∈ Lm2 . (4.6)

The continuous time system Σ : He → He is finite-gain Lm2 stable if there exist

 > 0 and κ such that:

4.2. DISSIPATIVITY, PASSIVITY AND POSITIVE REALNESS 65 The symbol ˇΣ shall be used if the continuous time system is LTI, otherwise the system shall be assumed general.

4.2.2 Dissipativity

The theory of dissipative dynamical systems, alternately referred to as dissipativ- ity, was first introduced by Willems [111, 112]. It can be viewed as a generalisation of passivity and is hence outlined first. Dissipativity analyses the dynamical re- lation between the internally stored energy of a system and the energy being supplied to that system.

For a general system Σ : He → He with input-output relation y(t) = Σu(t),

a locally integrable function called the supply rate can be defined as s(t) = s(u(t), y(t)). The system Σ is said to be dissipative with respect to its sup- ply rate s(t), if there exists a non-negative storage function V (x), such that for all t1 ≥ t0 ∈ R+:

V (x(t1))≤ V (x(t0)) +

Z t1 t0

s(u(t), y(t))dt, (4.8)

where x is a vector of system states. The inequality in (4.8) is called the dissi- pation inequality, and it states “if the energy stored within the system for time t1 is less than or equal to the amount of energy stored at earlier time t0 plus the

energy supplied to the system between these times, the system must either be lossless or dissipating energy”. This is intuitive. However, for many systems it is often difficult to determine the storage function uniquely [111]. This is not the case for LTI systems however, where both the storage function and supply rate can be easily defined.

For LTI system ˇ_{Σ ,{A, B, C, D}, defined such that:} ∂

∂tx(t) = Ax(t) + Bu(t), (4.9a)

y(t) = Cx(t) + Du(t), (4.9b)

it can be assumed without loss of generality that its storage function V (x) has quadratic form [112]:

V (x) = x>Qx, (4.10)

where matrix Q = Q> > 0. The system ˇΣ is (P, S, R)-dissipative [47] if it is dissipative with respect to the quadratic supply rate:

s(u, y) = y>Py + 2y>Su + u>Ru, (4.11) where P = P> and R = R>.

CHAPTER 4. PASSIVITY-BASED CONTROL 66 4.2.3 Passivity

Passivity is a simplification of dissipativity, in that it is purely an input-output property of a system and requires no explicitly defined storage function. A passive system can only store and dissipate energy and is unable to produce energy of its own. Examples of passive systems include passive components in a circuit such as a resistor, and as will be demonstrated - the nonlinearity in the Navier-Stokes equations for a channel flow.

For the general system y(t) = Σu(t), the following classifications of passivity can be defined [32, 72]:

1. Σ is passive if _{∃κ such that}

hy, ui[0,T] ≥ κ, (4.12)

2. Σ is strictly input passive (SIP) if_{∃ε > 0 and ∃κ such that}

hy, ui[0,T]≥ εkuTk22+ κ, (4.13)

3. Σ is strictly output passive (SOP) if_{∃δ > 0 and ∃κ such that} hy, ui[0,T] ≥ δkyTk

2

2+ κ, (4.14)

4. Σ is very strictly passive (VSP) if_{∃δ > 0, ε > 0 and ∃κ such that} hy, ui[0,T]≥ εkuTk22+ δkyTk

2

2+ κ. (4.15)

A connection can now be made between passivity and dissipativity for the LTI system case. Substituting P = 0, S = (1/2)I and R = _{−εI into the quadratic} supply rate in (4.11) yields:

s(u, y) = y>u− εu>

u. (4.16)

Then, substituting (4.16) into (4.8) results in the following dissipation inequality: V (x(t1))≤ V (x(t0)) +hy, ui[t0,t1]− εkuTk

2

2, (4.17)

which after rearranging becomes: hy, ui[t0,t1]≥ εkuTk 2 2+ (V (x(t1))− V (x(t0))) | {z } κ . (4.18)

4.2. DISSIPATIVITY, PASSIVITY AND POSITIVE REALNESS 67 The inequality in (4.18) is identical to the SIP inequality in (4.13). Similar results can be found for passive, SOP and VSP systems. Therefore, the passivity of a system identifies the boundedness of its supply rate.

4.2.4 Positive-Realness

Both passivity and dissipativity are time-domain concepts. Passivity has a frequency- domain equivalent named positive-realness. However, this only applies to LTI systems and is therefore less general.

Taking Laplace transforms of the state vector x(t), input vector u(t) and output vectory(t) such that:

X(s) = Z ∞ 0 x(t)e−stdt, (4.19a) U (s) = Z ∞ 0 u(t)e−st_{dt, (4.19b)} Y (s) = Z ∞ 0 y(t)e−stdt, (4.19c)

where s = σ + iω denotes complex frequency, the LTI system in (4.9) can be represented by the following transfer function matrix ˇΣ(s):

Y (s) = C(sI_{− A)}−1B + D

| {z }

ˇ Σ(s)

U (s). (4.20)

The transfer function matrix ˇΣ(s) is defined as positive real (PR) if the following conditions hold [72]:

1. All elements of ˇΣ(s) are analytic in Re(s) > 0.

2. ˇΣ(iω) + ˇΣ>(_{−iω) ≥ 0, ∀ω ∈ R, for which iω is not a pole for any element of} ˇ

Σ(s).

PR systems and passive systems are equivalent.

The transfer function matrix ˇΣ(s) is defined as strictly positive real (SPR) if the following conditions hold [72]:

1. All elements of ˇΣ(s) are analytic in Re(s)_{≥ 0.}

2. There exists ε > 0 such that ˇΣ(iω) + ˇΣ>(_{−iω) ≥ 2εI, ∀ω ∈ R.} SPR systems and SIP systems are equivalent, with identical bound ε.

The following connections can be made between passivity and positive realness [72]: 1. ˇΣ is passive iff ˇΣ(s) is positive real.

CHAPTER 4. PASSIVITY-BASED CONTROL 68

"

hy(t), u(t)i

ku(t)k

Figure 4.1: A schematic displaying the index of passivity ε of a system.

2. ˇΣ is strictly input passive iff ˇΣ(s) is strictly positive real.

The constant ε, which shall be referred to as a system’s index of passivity, can be determined directly from a system’s transfer function using the following for- mula [17]: ε := 1 2 n inf ω λmin ˇ Σ(iω) + ˇΣ>(_−iω)
o, (4.21) where λmin denotes smallest eigenvalue.

Thus far, the discussion has only been about passive systems, where ε _{≥ 0. In} these cases, ε acts as the lower bound on a system’s rate of energy dissipation. However, systems with index of passivity ε < 0 will not be passive, but their index gives us information regarding how non-passive the system is - ε acts as the upper bound on a system’s rate of energy production. A schematic of this is shown in Figure 4.1, where energy production can only occur in the dashed region of the graph. In later sections, feedback control will be used to make non-passive systems as close to passive as is possible by minimising_|ε|.

4.2.5 The Passivity Theorem

The passivity theorem determines the stability of the feedback interconnections of systems within the framework of passivity.

Theorem Consider the negative feedback interconnection of systems Σ1 and Σ2

in Figure 4.2, and set r2 ≡ 0. It is assumed that for any r1 ∈ H , there exists

solutionsu1,u2 ∈ He. Under these conditions, if Σ1 is passive and Σ2 is strictly

4.3. PASSIVITY OF CHANNEL FLOW 69