ENERGY–SHAPINGSTABILIZATION OF DYNAMICAL

SYSTEMS

Romeo Ortega and Eloısa GarcıaLaboratoire des Signaux et Systemes

S U P E L E CGif–sur–Yvette, FRANCE

ortega,garcia@lss.supelec.fr

october 2003

Contents

1 Motivation 51.1 Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Why? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Proposal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Introduction 92.1 Intelligent control paradigm revisited . . . . . . . . . . . . . . 92.2 Theoretical trend . . . . . . . . . . . . . . . . . . . . . . . . . 92.3 Proposal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.4 Energy–shaping . . . . . . . . . . . . . . . . . . . . . . . . . . 102.5 Energy–shaping and passivity . . . . . . . . . . . . . . . . . . 112.6 Two approaches to PBC . . . . . . . . . . . . . . . . . . . . . 112.7 Basic references: Theory . . . . . . . . . . . . . . . . . . . . . 112.8 Application references . . . . . . . . . . . . . . . . . . . . . . 12

1

3 Mathematical preliminaries 143.1 Input–Output (I/O) theory . . . . . . . . . . . . . . . . . . . 143.2 Lq–spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.3 Lq–spaces are normed spaces . . . . . . . . . . . . . . . . . . . 143.4 Extended Lq–space . . . . . . . . . . . . . . . . . . . . . . . . 153.5 Operators: properties and examples . . . . . . . . . . . . . . . 163.6 Induced norms . . . . . . . . . . . . . . . . . . . . . . . . . . 183.7 Input–Output Stability . . . . . . . . . . . . . . . . . . . . . . 193.8 Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 193.9 Feedback systems or closed–loop systems . . . . . . . . . . . . 243.10 Small Gain Theorem . . . . . . . . . . . . . . . . . . . . . . . 263.11 Lure’s problem . . . . . . . . . . . . . . . . . . . . . . . . . . 283.12 Loop transformations . . . . . . . . . . . . . . . . . . . . . . . 293.13 The circle criterion . . . . . . . . . . . . . . . . . . . . . . . . 313.14 The passivity approach . . . . . . . . . . . . . . . . . . . . . . 323.15 Passivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.16 Dissipativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.17 Passivity and L2–gain . . . . . . . . . . . . . . . . . . . . . . 343.18 Passivity and feedback interconnections . . . . . . . . . . . . . 353.19 Kalman–Yacubovich–Popov’s lemma . . . . . . . . . . . . . . 363.20 Passivity and energy–shaping . . . . . . . . . . . . . . . . . . 373.21 Examples: Electrical circuits . . . . . . . . . . . . . . . . . . . 383.22 Examples: Mechanical systems . . . . . . . . . . . . . . . . . . 393.23 Examples: Electromechanical systems . . . . . . . . . . . . . . 403.24 Examples: Power converters . . . . . . . . . . . . . . . . . . . 41

4 Passivity–based control (PBC) 424.1 Feedback passivation . . . . . . . . . . . . . . . . . . . . . . . 424.2 Feedback passive systems . . . . . . . . . . . . . . . . . . . . . 424.3 Standard formulation of PBC . . . . . . . . . . . . . . . . . . 434.4 Connections with L2–gain assignment . . . . . . . . . . . . . . 44

5 Energy–balancing control (EBC) and dissipation 465.1 Stabilization via energy–balancing . . . . . . . . . . . . . . . . 465.2 Physical view: Mechanical systems . . . . . . . . . . . . . . . 465.3 Example: Pendulum . . . . . . . . . . . . . . . . . . . . . . . 485.4 Implications of EBE for (f, g, h) systems . . . . . . . . . . . . 495.5 EB controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.6 Caveat emptor . . . . . . . . . . . . . . . . . . . . . . . . . . 505.7 Dissipation obstacle for EBC . . . . . . . . . . . . . . . . . . . 505.8 Finite dissipation example . . . . . . . . . . . . . . . . . . . . 51

2

5.9 Infinite dissipation example . . . . . . . . . . . . . . . . . . . 525.10 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

6 Control by interconnection 546.1 Introduction to the control by interconnection . . . . . . . . . 546.2 Passive controllers . . . . . . . . . . . . . . . . . . . . . . . . 556.3 Invariant functions method . . . . . . . . . . . . . . . . . . . . 556.4 Series RLC circuit . . . . . . . . . . . . . . . . . . . . . . . . 566.5 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576.6 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576.7 Port–controlled Hamiltonian (PCH) systems . . . . . . . . . . 576.8 Examples: Series RLC Circuit . . . . . . . . . . . . . . . . . . 586.9 Mechanical systems . . . . . . . . . . . . . . . . . . . . . . . . 586.10 Electromechanical systems . . . . . . . . . . . . . . . . . . . . 586.11 Induction motor . . . . . . . . . . . . . . . . . . . . . . . . . . 596.12 Power converters . . . . . . . . . . . . . . . . . . . . . . . . . 596.13 Can dynamics overcome the dissipation obstacle? . . . . . . . 616.14 Admissible dissipation . . . . . . . . . . . . . . . . . . . . . . 62

7 IDA–PBC 637.1 Matching perspective . . . . . . . . . . . . . . . . . . . . . . . 637.2 When is IDA an EB–PBC? . . . . . . . . . . . . . . . . . . . . 647.3 IDA PBC for (f, g, h) systems . . . . . . . . . . . . . . . . . . 647.4 IDA PBC: Swapping the damping . . . . . . . . . . . . . . . . 657.5 New passivity property . . . . . . . . . . . . . . . . . . . . . . 657.6 Energy–balancing with new supplied power . . . . . . . . . . . 667.7 Interpretation in EM systems . . . . . . . . . . . . . . . . . . 667.8 Universal stabilizing property of IDA–PBC . . . . . . . . . . . 677.9 Integral action . . . . . . . . . . . . . . . . . . . . . . . . . . . 687.10 Damping injection with ”dirty derivatives” . . . . . . . . . . . 697.11 IDA PBC as a state–modulated source . . . . . . . . . . . . . 697.12 Example: Parallel RLC circuit . . . . . . . . . . . . . . . . . . 707.13 Interconnection and damping assignment . . . . . . . . . . . . 717.14 Solving the PDE . . . . . . . . . . . . . . . . . . . . . . . . . 71

8 Examples 738.1 Some applications: . . . . . . . . . . . . . . . . . . . . . . . . 738.2 Magnetic levitation system . . . . . . . . . . . . . . . . . . . . 748.3 Mechanical systems . . . . . . . . . . . . . . . . . . . . . . . . 778.4 Strongly coupled VTOL aircraft . . . . . . . . . . . . . . . . . 788.5 Boost converter . . . . . . . . . . . . . . . . . . . . . . . . . . 79

3

8.6 PM Synchronous Motor . . . . . . . . . . . . . . . . . . . . . 828.7 Underactuated Kirchhoff’s equations . . . . . . . . . . . . . . 86

9 Concluding remarks and future research 88

4

1 Motivation

1.1 Facts

Modern (model–based) control theory is not providing solutions to newpractical control problems

Prevailing trend in applications: data–based “solutions”

Neural networks, fuzzy controllers, etc

They might work but we will not understand why/when

New applications are truly multidomain

There is some structure hidden in “complex systems”

Revealed through physical laws

Pattern of interconnection is more important than detail

1.2 Why?

Signal processing viewpoint is not adequate:

= Input-Output-Reference-Disturbance.

Classical assumptions not valid:

linear + “small” nonlinearities

interconnections with large impedances

time–scale separations

lumped effects

Methods focus on stability (of a set of given ODEs)

5

no consideration of the physical nature of the model.

1.3 Proposal

Reconcile modelling with, and incorporate energy information into, controldesign.

HOW?

Propose models that capture main physical ingredients:

energy, dissipation, interconnection

Attain classical control objectives (stability, performance) as by–productsof:

energy–shaping, interconnection and damping assignment.

Confront, via experimentation, the proposal with current practice.

Some examples:

Ball and Beam

6

Ball and Beam

Vertical take-off and landing aircraft

(Passive) walking

7

Contents

1. Introduction.

2. Tools (mathematical preliminaries).

3. Passivity based control (PBC).

4. Energy balancing (EBC) and dissipation .

5. Control by interconnection

6. Interconnection and damping assignment control (IDA–PBC).

7. Examples.

8

2 Introduction

2.1 Intelligent control paradigm revisited

Control design problems traditionally approached adopting a signal–processingviewpoint.

Objectives: keep some error signals small and reduce the effect of certaindisturbance inputs in spite of unmodeled dynamics.

Discriminated via filtering.

Very successful for linear time–invariant (LTI) systems

Impossible in nonlinear case:

far from obvious computations,

nonlinear systems “mix” the frequencies.

2.2 Theoretical trend

“Crank–up” the gain to quench the (large set of) undesirable signals...utmostimpractical!:

Intrinsically conservative

amplifies noise

energy consumption...

How to incorporate prior structural information?

Our inability is inherent to the signal–processing viewpoint.

Attempts for a monolithic theory doomed to failure.

9

2.3 Proposal

To incorporate energy principles in control

adopt a control–as–interconnection framework

view dynamical systems (plant and controller) as energy–transformationdevices interconnected to achieve desired behaviour.

Consider physical systems, i.e., that satisfy energy–conservation.

Control problem is to assign a desired energy function.

cΣ-

+

-

+ Σu c

yc y

uΣ I

2.4 Energy–shaping

Advantages of adopting an energy–shaping perspective

Aim at, not just stabilization, but also performance objectives.

Energy is a fundamental concept that can serve as a lingua franca

to communicate with practitioners,

incorporate prior knowledge and

provide physical interpretations to the control action.

There’s a clear geometrical characterization of passifiable systems

10

2.5 Energy–shaping and passivity

The idea has its roots in robot control (Takegaki/Arimoto,’81). Also (Jon-ckheree,’81).

Principle formalized in (Ortega/Spong,’89), via definition of passivity–based control (PBC):

Control as interconnections of passive systems

⇒ energy–balancing interpretation of stabilization;

Approach hinges upon the fundamental (and universal) property ofpassivity

⇒ can be extended to many applications.

Passivation is “easier” than stabilization

2.6 Two approaches to PBC

i) Standard: Fix a priori the desired storage function (typically quadraticin the increments.) Problems:

Not an energy function and

stabilization mechanism akin to systems inversion.

ii) Interconnection and damping assignment(IDA): Storage function –nowa bona fide energy function– obtained as a result of our choice of desiredsubsystems interconnections and damping.

Applications of IDA–PBC: mass–balance syst., electrical machines, powersyst., underwater vehicules, magnetic levitation, underactuated mechanicalsyst., and power converters.

2.7 Basic references: Theory

(Material of the course)

11

R. Ortega, A. van der Schaft, I. Mareels and B. Maschke: Putting energy back in control, IEEEControl Syst. Magazine, Vol. 21, No. 2, April 2001, pp. 18–33.

(Basic theory of IDA–PBC)

R. Ortega, A. van der Schaft, B. Maschke and G. Escobar: IDA-PBC of port–controlled hamil-tonian systems, Automatica (Regular Paper), , Vol. 38, No. 4, April 2002.

R. Ortega, M. Spong, F. Gomez and G. Blankenstein: Stabilization of underactuated mechanicalsystems via IDA, IEEE Trans. Automat. Contr.(Regular Paper), Vol. AC–47, No. 8, August2002, pp. 1218–1233.

(General I/O theory)

A. J. van der Schaft, L2–Gain and Passivity Techniques in Nonlinear Control,Springer–Verlag, Berlin, 1999.

(”Standard” PBC approach)

R. Ortega, A. Loria, P. J. Nicklasson and H. Sira–Ramirez, Passivity–based control ofEuler–Lagrange systems, Springer-Verlag, Berlin, Sept. 1998.

2.8 Application references

G. Escobar, A. van der Schaft and R. Ortega: A Hamiltonian viewpoint in the modelingof switching power converters, Automatica, Vol. 1999.

H. Rodriguez, R. Ortega, G. Escobar and N. Barabanov: A Robustly Stable OutputFeedback Saturated Controller for the Boost DC–to–DC Converter, Systems and Con-trol Letters, Vol. 40, No. 1, pp. 1 -8, May 2000.

V. Petrovic, R. Ortega and A. Stankovic: Interconnection and damping assignmentapproach to control of PM synchronous motor, IEEE Trans. Control Syst. Techn, Vol.9, No. 6, pp. 811–820, Nov. 2001.

R. Ortega, V. Petrovic and A. Stankovic: Extending passivity–based control beyond me-chanics: a synchronous motor example, Automatisierungstechnik, Oldenbourg Verlag,Vol. 48, No. 3, March 2000, pp. 106–115.

A. Astolfi, D. Chhabra and R. Ortega: Asymptotic stabilization of selected equilibria ofthe underactuated Kirchhoff’s equations, Systems and Control Letters, Vol 45, No. 3,pp. 193–206, April 2002.

12

A. Astolfi and R. Ortega: Energy based stabilization of the angular velocity of a rigidbody operating in failure configuration, J of Guidance Control and Dynamics, Vol 25,No. 1, pp. 184–187, Jan–Feb 2002.

R. Ortega, A. Astolfi, G. Bastin and H. Rodriguez: Stabilization of Food–ChainSystems via Energy Balancing, ACC 2000, Chicago, June 2000.

H. Rodriguez, R. Ortega and I. Mareels: A Novel Passivity–Based Controller foran Active Magnetic Bearing Benchmark Experiment, ACC 2000, Chicago, June2000.

H. Rodriguez, R. Ortega and G. Escobar: Energy–shaping control of switchedpower converters, IEEE Conf. ISIE’01, Pusan, Korea, June 13–15, 2001.

M. Galaz, R. Ortega, A. Bazanella and A. Stankovic: An Energy–Shaping Ap-proach to Excitation Control of Synchronous Generators, ACC 2001, Arlington,VA, USA, June 25–27, 2001.

A. Stankovic, G. Escobar, R. Ortega and S. Sanders, Energy–based Control inPower Electronics, chapter in book: Nonlinear Control in Electrical Systems, ed.G. Verghese, IEEE Press, Piscataway, NJ, 2001.

13

3 Mathematical preliminaries

3.1 Input–Output (I/O) theory

The I/O approach describes the system like an operator that relates the inputsignal with the output signal without regarding the internal system structure.

Advantages:

Each system is an operator

Gives a general representation

It is based on the physical properties of the system (like passivity)

3.2 Lq–spaces

Definition 1. For each q ∈ 1, 2, . . . the set Lq[0,∞) = Lq consists of all func-tions f : R+ → R (R+ = [0,∞)) which satisfy∫ ∞

0|f(t)|q <∞ (1)

The set L∞[0,∞) = L∞ consist of all functions f : R+ → R which are bounded,i.e.

supt∈R+

|f(t)| <∞ (2)

Remarks:

Lq is a linear space

3.3 Lq–spaces are normed spaces

Definition 2. The function ‖·‖q : Lq → R+ is called the norm Lq and it is definedas

‖f(t)‖q ,

[∫ ∞

0|f(t)|qdt

]1/q

, q ∈ [1,∞) (3)

respectively, the norm L∞ is given by

‖f(t)‖∞ , supt∈[0,∞]

|f(t)| (4)

14

Remarks:

‖ · ‖q 9 ∞

limt→∞ ‖ · ‖q exists and it is finite, ex. f(t) = exp−t

3.4 Extended Lq–space

Definition 3. Let f : R+ → R. Then for each T ∈ R+, the truncated functionfT : R+ → R is defined by

fT (t) =f(t), 0 ≤ t ≤ T0, t > T

(5)

For each q = 1, 2, . . . ,∞, the set Lqe consists of all functions f : R+ → R suchthat fT ∈ Lq for all T with 0 ≤ T <∞. Lqe is called the extended Lq–space.

Remark:

It is possible to have limT→∞ ‖fT (t)‖q = ∞

Lq ⊂ Lqe,ex. f(t) = sin(t) 6∈ Lq, but sin(t) ∈ Lqe, ∀q ∈ [1,∞).

Examples:

f ff

ff

f1

2

45

6

L

L

L2

1

oo

3

15

f1(t) = 1; f2(t) = 11+t f3(t) = 1

1+t1+t1/4

t1/4

f4(t) = e−t f5(t) = 11+t2

1+t1/4

t1/4 f6(t) = 11+t2

1+t1/2

t1/2

3.5 Operators: properties and examples

Definition 4. An operator is a map G : Lqe → Lqe relating an input u ∈ Lqe, andan output y ∈ Lqe with y = Gu.

Examples of operators:

Truncation: PT : Lqe → Lqe, and T > 0

(PTu)(t) ,

u(t), t ≤ T0, t > T

(6)

Delay DT : Lqe → Lqe, and T > 0

(DTu)(t) , u(t− T ) (7)

Convolution G : Lqe → Lqe

(Gu)(t) ,∫ t

0g(t− T )u(T )dT , t ≥ 0 (8)

Properties:

16

Causality: A mapping G : Lqe → Lqe is said to be causal if ∀u1, u2 ∈ Lqe and∀T > 0

PTu1 = PTu2 ⇒ PT (Gu1) = PT (Gu2) (9)

that is, (PTu)(t) depends only on the input past values. Equivalently, PTGu =PT (GPTu),∀u ∈ Lqe, and ∀T > 0.

Linearity: A mapping G : Lqe → Lqe is said to be linear if

G(au) = aG(u), ∀a ∈ R (10)G(u1 + u2) = G(u1) +G(u2). (11)

Time Invariance: A mapping G : Lqe → Lqe is said to be time invariant if

G(DTu) = DT (Gu), ∀T ≥ 0, u ∈ Lqe (12)

Thus, the operator G commutate with the delay operator DT , i.e., DTG =GDT .

Memoryless: A mapping G : Lqe → Lqe is said to be memoryless if the output(Gu)(t0) depends only on t0.

Algebra:

1

2

G1+

+G2

+

G1

G

G

2 G1G2G

(b)

(a)

Addition:

(G1 +G2)(u) = G1u+G2, ∀u ∈ Lqe

(G1 +G2) = (G2 +G1), (commutativity)

(G1 +G2) +G3 = G1 + (G2 +G3), (associativity)

17

G+ (−G) = 0, (negative)

Composition:

(G1G2)(u) = G1(G2u), ∀u ∈ Lqe

In general G1(G2u) 6= G2(G1u), (non commutative)

G1(G2G3u) = (G1G2)(G3u), (associativity)

Iu = u, ∀u ∈ Lqe (identity)

0u = u, ∀u ∈ Lqe (zero operator)

G−1G = GG−1 = I (inverse)

(G1 +G2)G3 = G1G3 +G2G3, (distributive by the right)

G1(G2 +G3) 6= G1G2 +G1G3, (non distributive by the left)

3.6 Induced norms

Definition 5. Let G : Lqe → Lqe. Then, the induced norm of G is defined by

‖G‖q , supu ∈ Lqe

u 6= 0

‖Gu‖q

‖u‖q(13)

or equivalently,

‖G‖q = sup‖u‖q ≤ 1

‖Gu‖q (14)

Remark:

The induced norm quantifies the maximum amplification of the input signal norm.

Example: Bounded non linearity

18

u

y

b a

(u)

Norm L2 of a nonlinearity y = Gu

3.7 Input–Output Stability

Definition 6 (Lq Stability). A mapping G : Lqe → Lqe with zero initial condi-tions is said to be Lq stable if and only if

γq , ‖G‖q <∞ (15)

if the initial conditions are different from zero, G is Lq stable if

‖Gu‖q ≤ γq‖u‖q + βq (16)

where γq > 0 and βq ∈ R.

Remarks:

A Lq–stable operator G is a mapping Lq → Lq. The inverse is false.

Example: An operator G such that y = Gu = u2 is a mapping G : L∞ → L∞but ‖G‖∞ = ∞, thus, G is not Lq–stable.

3.8 Linear Systems

LQ–control

The objective is to minimize the energy of the output signal, that is, min ‖y‖2,taking into account that the input signal is a white noise.

19

From a deterministic point of view, the white noise is an impulse. Therefore,according to the Parseval’s theorem, the LQ–control, search min ‖H(jω)‖2,where H(jω) is the frequency response.

For finite dimensional linear time invariant systems, ‖H(jω)‖2 can be evaluatedresolving a Lyapunov equation.

Theorem 1. Consider the system

x = Ax+Bu

y = Cx

where A ∈ Rn×n, B, C ∈ Rn. Suppose that A is stable, the pair (A,B) is control-lable and (A,C) is observable. Let H(s) = C(sI −A)−1B be the transfer functionand L = LT > 0 the unique solution of

AL+ LAT = −BBT

Hence, the norm L2 of the frequency response is given by

‖H(jω)‖2 =√

2π(CLCT )1/2 (17)

Proof.

The proof is based on Parseval’s theorem. Knowing that

eAt = I +At+A2t2

2+ · · ·

we can write

d

dt

eAtBBT eA

T t

= AeAtBBT eAT t + eAtBBT eA

T tAT (18)

integrating, we get

eAtBBT eAT t∣∣∣∞0

= A

∫ ∞

0eAtBBT eA

T tdt+∫ ∞

0eAtBBT eA

T tATdt

since A is stable, limt→∞ eAt = 0, we can write

−BBT = AL+ LAT

where the controllability grammian is defined by

L ,∫ ∞

0eAtBBT eAtdt (19)

20

Finally, applying Parseval’s theorem to the impulse response h(t) = CeAtB, wefind

‖H(jω)‖22 = 2π‖h(t)‖2

2

= 2π∫ ∞

0CeAtBBT eA

T tCTdt = 2πCLCT

H∞–control

The objective is to minimize the energy of the output signal, that is, min ‖y‖2,considering that the energy of the input signal satisfies ‖u‖2 ≤ 1.

Using the definition of the induced norm, the objective of the H∞–control is tominimize ‖H‖2

The name of H∞–control comes from the fact that the induced norm L2 of alinear time invariant operator in the time domain, corresponds to the norm L∞in the frequency domain.

|Re

|I

||H|| = sup |H(jw)|2w

Norm L2 of the convolution operator

Theorem 2. Let H be a stable convolution operator to the single input–singleoutput (SISO), then

‖H‖2 = supω|H(jω)| (20)

21

Proof.

Define R , supω |H(jω)|. Let y = Hu with u ∈ L2, then

‖y‖22 =

∫ ∞

0y2dt (21)

According to Parseval’s theorem we have

‖y‖22 =

12π

∫ ∞

−∞|y(jω)|2dω

=12π

∫ ∞

−∞|H(jω)u(jω)|2dω

≤ 12π

supω|H(jω)|2

∫ ∞

−∞|u(jω)|2dω

≤ R2‖u‖22

Hence, ‖y‖2‖u‖2 ≤ R for all u ∈ L2, where, according to the definition of the induced

norm, we have ‖H‖2 ≤ R.Now, we choose ω0 such that,

|H(jω0)| = R− ε

2, ε > 0

let u(t) = sin(ω0t) ∈ L2e, writing the ouptut y(t) = (Hu)(t) as

y(t) , yss(t) + ytr(t) (22)

where yss and ytr are the steady state and the transient responses respectivelywith

yss , |H(jω0)| sin(ω0t+ ∠H(jω0)) (23)

since H is a stable operator, we have ‖ytr‖2 , M <∞Now, choose an integer N such that M√

Nπω0

< ε2 and consider the truncated input

uT = PTu ∈ L2 with T > 0 then,

‖HuT ‖2 = ‖HPTu‖2

≥ ‖PTHPTu‖2

since H is causal (PTHu = PTHPTu, ∀ u ∈ Lqe, and ∀ T > 0)

‖HuT ‖2 ≥ ‖PTHu‖2

≥ ‖PT yss‖2 − ‖PT ytr‖2

≥ |H(jω0)|[∫ T

0sin2(ω0t+ ∠H(jω0))dt

]1/2

−M

22

It is important to notice that if we choose T = Nπω0

then

‖uT ‖2 =[∫ T

0sin2(ω0t)dt

]1/2

=[∫ T

0sin2(ω0t+ ∠H(jω0))dt

]1/2

=√Nπ

ω0,

thus,

‖HuT ‖2 ≥ |H(jω0)|√Nπ

ω0−M

‖HuT ‖2

‖uT ‖2≥ |H(jω0)| −

M√Nπω0

≥ R− ε

In this way, we have constructed an input uT ∈ L2 for which the operator H givesan energy amplification of R − ε. However, since ε > 0 could be arbitrarily small,‖H‖2 ≥ R. Finally, since we showed that ‖H‖2 ≤ R, we conclude that

‖H‖2 = R = supw|H(jω)|

L1–control

In this approach the objective is to minimize not the system energy but the peakof the response.

Hypothesis: bounded inputs, i.e. u ∈ L∞. In other words, we search formin ‖H‖∞.

Theorem 3. Let H be the convolution operator, SISO, stable with impulse re-sponse h(t), then

‖H‖∞ = ‖h(t)‖1 (24)

23

3.9 Feedback systems or closed–loop systems

1

2

+

+

-

+u y

u

y

1

2

1e

e 2

G

G2

1

Definition 7 (well–posedness condition). A feedback system is “well–posed”if the operator

Φ(G1, G2) :[u1

u2

]→[e1e2

](25)

allows the operators algebra, and if for all u1, u2 ∈ Lqe the equationse1 = u1 −G2e2

e2 = u2 +G1e1

have a unique solution [e1, e2]T ∈ Lqe and Φ(G1, G2) is causal.

The operator interconnections are not always well–posed

Example: A feedback system can diverge towards the infinity in a finite time,i.e. u1, u2 ∈ Lqe but e1 or e2 6∈ Lqe.

The following lemma gives a sufficient condition in order to know if a feedbacksystems is well–posed or not

Lemma 1. If either one of the operators G1 or G2 has a delay different from zero,i.e. if ∃ δ > 0 such that

PTu1 = PTu2 ⇒ PT+δ(G1u1) = PT+δ(G1u2),

or PT+δ(G2u1) = PT+δ(G2u2)

then the feedback system is well–posed.

24

Definition 8 (Lq–stability of feedback systems). Consider a well–posed feed-back system. Then the operator

Φ(G1, G2) :[u1

u2

]→[e1e2

](26)

is said to be Lq–stable if

‖Φ(G1, G2)‖q <∞

Example

1

2

uy

u

y

12

G G21

s - 2

s - 2

s + 3

s + 1

For u2 = 0, the unstable pole of G1 is cancelled and the system G1G2 : u1 → y1

is stable. Contrariwise, if u2 6= 0 the system becomes unstable. In this case, theoperator Ψ1 : [u1 u2]T → y1 is Lq–stable, but the operator Ψ2 : [u1 u2]T → [y1 y2]T

is unstable.

Lemma 2. Consider a well–posed feedback system with G1 Lq–stable, then theoperator

Φ(G1, G2) :[u1

u2

]→[e1e2

](27)

is also Lq–stable if and only if Φ1(G1, G2) : [u1 u2] → e1 is Lq–stable.

Particular case: linear operators

Consider a well–posed linear system and suppose that G1 and G2 are time invariantlinear operators. Then

e1 = u1 −G2e2 = u1 −G2u2 −G2G1e1

or (I + G2G1)e1 = u1 − G2u2. Since the system is well–posed, (I + G2G1) isinvertible. Thus

25

e1 = (I +G2G1)−1u1 − (I +G2G1)−1G2u2 (28)

similarly,e2 = u2 +G1e1 = u2 +G1u1 −G1G2e2

ore2 = (I +G1G2)−1u2 + (I +G1G2)−1G1u1 (29)

Using (28) and (29), we can write[e1e2

]=[

(I +G2G1)−1 −(I +G2G1)−1G2

(I +G1G2)−1G1 (I +G1G2)−1

] [u1

u2

]Thus, the feedback system is stable if the four operators (I + G2G1)−1, −(I +

G2G1)−1G2, (I +G1G2)−1G1 and (I +G1G2)−1 are stable.

3.10 Small Gain Theorem

Theorem 4 (Small gain). Consider a well–posed feedback system. Suppose thatG1 and G2 are causal and Lq–stable. Then, if

‖G1‖q‖G2‖q < 1 (30)

the feedback system is Lq–stable.

Proof.

Take u1, u2 ∈ Lq. Since G1 is stable from lemma 2. We have to prove thatΦ1(G1, G2) : [u1 u2]T → e1 is also stable. Using the causality of G2 we have

Using the causality of G2 we have

e1 = u1 −G2e2

PT e1 = PTu1 − PTG2e2

= PTu1 − PTG2PT e2

taking the norm in both sides

‖PT e1‖q ≤ ‖PTu1‖q + ‖PTG2PT e2‖q

≤ ‖PTu1‖q + ‖G2‖q‖PT e2‖q (31)

similarly, using the causality of G1 we have

e2 = u2 +G1e1

PT e2 = PTu2 + PTG1e1

‖PT e2‖q ≤ ‖PTu2‖q + ‖G1‖q‖PT e1‖q (32)

26

Combining (31) and (32) yields

‖PT e1‖q ≤ ‖PTu1‖q + ‖G2‖q‖PTu2‖q + ‖G2‖q‖G1‖q‖PT e1‖q

then(1− ‖G2‖q‖G1‖q)‖PT e1‖q ≤ ‖PTu1‖q + ‖G2‖q‖PTu2‖q

taking the limit when T →∞, and using the fact that u1, u2 ∈ Lq we get

(1− ‖G2‖q‖G1‖q)‖e1‖q ≤ ‖u1‖q + ‖G2‖q‖u2‖q

≤∥∥∥∥ u1

u2

∥∥∥∥q

+ ‖G2‖q

∥∥∥∥ u1

u2

∥∥∥∥q

‖e1‖q∥∥∥∥ u1

u2

∥∥∥∥q

≤ 1 + ‖G2‖q

1− ‖G2‖q‖G1‖q

Elsewhere, by definition we know that

‖Φ1(G1, G2)‖q = supu1, u2 ∈ Lqe

u1, u2 6= 0

‖e1‖q∥∥∥∥ u1

u2

∥∥∥∥q

≤ 1 + ‖G2‖q

1− ‖G2‖q‖G1‖q≤ ∞

that is, Φ1(G1, G2) is Lq–stable. Lemma 2 allows to conclude the stability of Φ(G1, G2)

Remarks:

The quantity ‖G1‖q‖G2‖q is called Lq–gain.

The small gain theorem gives only sufficient conditions, that is, if the loop gainis greater than 1, nothing can be concluded.

Corollary 1. Let G1 and G2 be two linear stationary operators, interconnected ina feedback system with ‖G2‖2 = γ <∞. Then

1. If ‖G1‖2 <1γ then ‖Φ(G1, G2)‖2 <∞

2. ∃ G10(s) : ‖G10‖2 = supω |G10(jω)| = 1γ such that ‖Φ(G1, G2)‖2 = ∞

Remarks:

In other words, there always exist a stable operator G10 with norm L2 such that theL2–gain is equal to 1 and for which the closed–loop system becomes unstable.

27

3.11 Lure’s problem

y 1

a

b

yu 1

y

−

+ e 1(s)1H

2

1

H 2

y 2

This problem is known as the absolute stability problem

According to the figure, consider an operator (stable or unstable) H1(s) makinga feedback system with a static nonlinear operator H2 whose output y2 = H2y1

belongs to the sector [a, b], i.e. y2(0) = 0 and

a ≤ y2(σ)σ

≤ b, ∀σ 6= 0, (33)

The objective is to find the conditions for a, b, H1(s) so that the closed–loopsystem is stable.

To this end, we can use the small–gain theorem to deduce that the gain ofH1 has to be less than 1/b (this agrees with the definition of the norm of anonlinearity). However, this is quite restrictive.

We can get less restrictive conditions using loop transformations.

28

3.12 Loop transformations

This transformation consists in the introduction of a gain K. The new system definesa feedback system between H ′

1 and H ′2, which is equivalent to the system of Lure’s

problem.

+

1

+

-

1

yH

K

K

HH’ y

H’

1

1

2(H2

2

1

1

+ eu

- K)

-

-

Applying the small–gain theorem, we get that the closed–loop system (H ′1,H

′2)

(equivalently, the system (H1,H2)) is Lq–stable if

(i) H1(1 +KH1)−1 is causal and Lq–stable

(ii) ‖H1(1 +KH1)−1‖q‖(H2 −K)‖q < 1

An optimal selection is given by K = 12(a+ b).

Define r , 12(b − a). With this choice, the nonlinearity is brought to the real

axis (see next figure)

The norm of H ′2 = H2 − K becomes r = b − K, which is the smallest gain

that we can get using this transformation. Then, according to the small–gaintheorem, if ‖H ′

1‖q < 1/r the closed–loop system is Lq–stable.

29

However, we have the stability conditions for H ′1 = H1(I + KH1)−1 and not

for H1

Km

y1

’= (H - K) y12h2

r

-r

y1

1y2= Hh2

b

a

Nonlinearity H ′2y1

To find the stability conditions for H1, we have that, according to condition (ii)it is enough that ‖H ′

1‖2‖H ′2‖2 < 1, since ‖H ′

2‖2 = r, we get

‖H ′1‖2 = ‖H1(1 +KH1)‖2

= supω

H1(jω)1 + kH1(jω)

<1r

=2

b− a

From the Nyquist diagram (figure (a)), H ′1(jω) has to be insideDa, i.e. 1/H ′

1(jω) =1/H1(jω) +K must be outside Db (figure (b)).

Since K > 0, 1/H1(jω) should not touch the circle Dc (figure (c)).

H1(jω) should not touch, neither encircle Dd (figure (d))

30

H1(jw)

H1(jw)1+ k

b-a2

b-a2

H1(jw)

H1(jw)

1H1(jw)

Db

b-a b-a2-2

II

IRe

Da

(a) (b)

-

IRe

II

1

IRe

IIDc

(c)

- b-a2

b-a2

IRe

IIDd

(d)

1-

1-a

1-b

+K

K

+K-K

Small gain and loop transformation

If a = b = 1, in order to guarantee condition (ii), (i.e. H1(1 +KH1)−1 has tobe stable according to Nyquist theorem), H1 should not encircle the point −1(circle Dd in figure (d)).

Similar conditions exist for the case where H1 is internally unstable.

3.13 The circle criterion

Theorem 5 (circle criterion). Consider the system defined by the Lure’s prob-lem, with a memoryless nonlinearity H2 that belongs to sector [a, b]. Suppose thatthe transfer function H1(s) has ρ > 0 poles with negative real part and with nopoles in the imaginary axis jω; consider the disk D(a, b) with center in the real

31

axis and passing by the points −1/a, −1/b, then the system is L2–stable if one ofthe next conditions is satisfied:

Case 1. b ≥ a > 0 : The Nyquist diagram of H1(jω) do not touch the diskD(a, b), encircling it ρ times in a counterclockwise sense.

Case 2. b > a = 0 : H1(jω) is internally stable and

infω∈R

ReH1(jω) > −1b

Case 3. a < 0 < b : H1(jω) is internally stable and its Nyquist diagram isstrictly inside the disk D(a, b).

Case 4. 0 > b > a : The Nyquist diagram of −H1(jω) do not touch the diskD(a, b), encircling it ρ times counterclockwise sense.

Proof. See M. Vidyasagar, “Nonlinear system analysis”. New Jersey: PrenticeHall. 1993.

3.14 The passivity approach

Inner product

Definition 9. The inner product is a function 〈·, ·〉 : L2 × L2 → R, satisfying

(a) 〈x+ y, z〉 = 〈x, z〉+ 〈y, z〉(b) 〈αx, z〉 = α〈x, z〉 ∀ α ∈ R(d) 〈x, x〉 ≥ 0, and 〈x, x〉 = 0 ⇔ x ≡ 0

Similarly, in the L2e–space, the inner product is defined by

〈x, y〉T =∫ T

0x(t)y(t)dt (34)

Notice that ‖x‖2 , 〈x, x〉1/2. Also,

〈x, y〉 ≤ ‖x‖2‖y‖2 (35)

Motivation of the passivity theory

32

i(t)

+

-

v(t) H

Motivation comes from electric circuit theory

Consider the circuit shown in the figure. The powerdelivered to H at the instant t is v(t)i(t) where v(t)are i(t) the voltage and current respectively. Let E(t0)be the energy in H at the instant t0, then H is passiveif and only if

E(t0) +∫ t

t0

v(T )i(T )dT ≥ 0, ∀t ≥ t0

3.15 Passivity

Definition 10. An mapping G : L2e → L2e is passive if and only if ∃ β ∈ R, andT ≥ 0 such that

〈Gu, u〉T ≥ β, ∀ u ∈ L2e

Definition 11. A mapping G : L2e → L2e is output strictly passive if and only if∃ β ∈ R, δo > 0 and T ≥ 0 such that

〈Gu, u〉T ≥ δo‖Gu‖T + β, ∀ u ∈ L2e

Definition 12. A mapping G : L2e → L2e is input strictly passive if and only if∃ β ∈ R, δi > 0 and T ≥ 0 such that

〈Gu, u〉T ≥ δi‖u‖T + β, ∀ u ∈ L2e

3.16 Dissipativity

It provides a state–space interpretation of passivity and of small gain (Willems,1978).

We consider state–space systems described by equations of the form

(P )x = f(x, u), x ∈ Rn,y = h(x, u), u, y ∈ Rm (36)

33

Assume that the system (P ) with input u = 0 possesses an equilibrium at x = 0,that is f(0, 0) = 0, and that h(0, 0) = 0.

Note that the issue of well-posedness of feedback interconnections is securedwhenever the output of one of the two systems contains no throughput, that is,is of the form y = h(x).

The definition of dissipativity involves a supply rate w : Rm × Rm → R (anabstraction of the physical notion of power) and a storage function E : Rn×R+

(an abstraction of the physical notion of energy stored in the system).

Definition 13 (dissipativity). The system P is dissipative with the supply ratew(u, y) if there exists a storage function E(x), E(0) = 0, such that for all x ∈ Rn

E(x(T ))− E(x(0)) ≤∫ T

0w(u(t), y(t))dt, E(x) ≥ 0 (37)

for all u ∈ U and all T ≥ 0 such that x(t) ∈ Rn for all t ∈ [0, T ].

The dissipation inequality expresses that the increase of stored energy cannotexceed the external supplied energy

∫ T0 w(u(t), y(t))dt at any instant of time.

If the storage function E(x) is differentiable, we can write

E(x(t)) ≤ w(u(t), y(t)) (38)

Again, the interpretation is that the rate of increase of energy is not bigger thanthe input power.

3.17 Passivity and L2–gain

Definition 14. System P is said to be passive if it is dissipative with supply ratew(u, y) = uT y. System P is said to have L2–gain less than or equal to γ > 0 if itis dissipative with supply rate w(u, y) = γ2uTu− yT y.

34

Lemma 3. Let G : L2e → L2e be an output strictly passive operator, then ‖G‖2 <∞

Proof. From definition 11 we have that for all u ∈ L2

〈u,Gu〉 ≥ δo‖Gu‖22 + β

Using Schwartz’s inequality yields

‖u‖2‖y‖2 ≥ 〈u,Gu〉 ≥ δo‖y‖22 + β

⇒ 0 ≥ δo2(‖y‖2

2 + ‖y‖22

)− ‖u‖2‖y‖2 + β

⇔ 0 ≥ δo2‖y‖2

2 +

(√δo2‖y‖2 −

12

√2δo‖u‖2

)2

− 12δo

‖u‖22 + β

⇒ δo2‖y‖2

2 ≤1

2δo‖u‖2

2 − β

⇒ ‖y‖22 ≤

1δo‖u‖2

2 −2βδo

3.18 Passivity and feedback interconnections

1

2

+

+

-

+u y

u

y

1

2

1e

e 2

G

G2

1

Theorem 6. Consider the feedback system showed in the figure. Suppose thatthere exist real constants δ1, ε1, δ2, ε2 such that

〈u,Giu〉T ≥ εi‖u‖2T2 + δi‖Giu‖2

T2,

∀T ≥ 0, ∀u ∈ L2e, i = 1, 2, then, the system is L2–stable if

δ1 + ε2 > 0, δ2 + ε1 > 0 (39)

35

Theorem 7. Consider a feedback interconnection. Assume that for any e1, e2 inLm

2e there are solutions u1, u2 in Lm2e (well-posedness condition). If G1 and G2 are

passive, then the feedback interconnection with input (e1, e2) and output (y1, y2) ispassive. It is output strictly passive if both G1 and G2 are output strictly passive.

Corollary 2. A feedback system is L2 with finite gain if one of the followingconditions is satisfied:

1. G1 is input strictly passive with finite gain and G2 is passive.

2. G2 is input strictly passive with finite gain and G1 is passive.

3. Both G1, and G2 are output strictly passive.

3.19 Kalman–Yacubovich–Popov’s lemma

Theorem 8. Let a linear system be described in state–space as

x = Ax+Bu

y = Cx

where A ∈ Rn×n, B, C ∈ Rn. Suppose A is strictly stable, the pair (A,B) iscontrollable and the pair (A,C) is observable. Under this conditions, the operatorG : u→ y is passive if and only if ∃P = P T > 0 ∈ Rn×n such that

ATP + PA ≤ 0BTP = C

Remarks:

Let G(jω) be the transfer function of a linear time invariant system,

ReG(jω) ≥ 0 ⇔ G(jω) is passive and positive real.

If ∃ ε > 0 such that G(s− ε) is positive real, then G(s) is strictly positive real.

Strictly positive realness implies output strictly passivity.

36

3.20 Passivity and energy–shaping

Lumped parameter systems interconnected to the external environment throughpower conjugated port variables u ∈ Rm and y ∈ Rm (product has units ofpower).

They satisfy the energy–balance equation (EBE)

H[x(t)]−H[x(0)]︸ ︷︷ ︸stored energy

=∫ t

0u>(s)y(s)ds︸ ︷︷ ︸supplied

− d(x(t), t)︸ ︷︷ ︸dissipated ≥0

Systems that satisfy EBE with

H(x) ≥ c

are passive, and y is called the passive output.

Key observations

With u ≡ 0, we have H[x(t)] ≤ H[x(0)]. Will stop in a point of minimumenergy.

Control introduced to operate the system around some non–zero equilibriumpoint, say x∗.

Rate of convergence increased if we extract energy u = −Kdiy, with Kdi =K>

di > 0.

−∫ t0 u

>(s)y(s)ds ≤ H[x(0)] <∞

⇔ amount of energy that can be extracted from a passive system isbounded.

d(x(t), t) is non–decreasing; typically∫ t0 (·)2ds.

37

3.21 Examples: Electrical circuits

1

1C

1L

1C

L

2R

q C

ϕL

φL is flux, qc charge, u voltage

Model, with x , [qC , φL]>,

Σ :

x1 = 1

Lx2

x2 = − 1Cx1 − R

Lx2 + uy = 1

Lx2

Energy

H(x) =1

2Cx2

1 +1

2Lx2

2

Satisfies (EBE) with

d(x(t), t) = R

∫ t

0[1Lx2(s)]2ds

38

3.22 Examples: Mechanical systems

q2

q1

u

u is torque, q1 ball position, q2 beam angle

Model

m2q1 +m3 sin(q2)−m1q1q22 = 0

(1 +m1q21)q2 + 2m1q1q1q2 +m3q1 cos(q2) = u

Energy H = 12 q>M(q1)q +m3q1 sin(q2), where

M(q1) =[m2 00 1 +m1q

21

]

Satisfies (EBE) with d(x(t), t) = 0 and y = q2.

39

3.23 Examples: Electromechanical systems

u

i

y

g

m

λ

λ is flux, y position, u voltage

Model (Assuming linear magnetics, i.e., λ = L(θ)i)

λ+Ri = u

mθ = F −mg

F =12∂L

∂θ(θ)i2

Total energy:

H =12λ2

L(θ)+m

2θ2 +mgθ

Output y = i, same dissipation. Notice, however, that H is not bounded frombelow! Thus it is not a passive system.

40

3.24 Examples: Power converters

+

L+

+E

u=0

u=1

C+

R

u switch position: 0, 1

Model, with x , [φL, qC ]>,

x1 = −u 1Cx2 + E

x2 = u1Lx1 −

1RC

x2

Total energy: H(x) = 12Lx

21 + 1

2Cx22.

Attention: “Input” and output: E 7→ x1L !

41

4 Passivity–based control (PBC)

4.1 Feedback passivation

Definition 15. Let a dynamical system be described in state space asx = f(x) + g(x)u, (40)y = h(x),

where u and y are the input system and output system respectively. If it is possibleto find a feedback transformation

u = α(x) + β(x)v, (41)

such that the systemx = f(x) + g(x)α(x) + g(x)β(x)v,y = h(x)

is passive, then the original system (40) is said to be feedback passive and thetransformation (41) is called feedback passivation.

4.2 Feedback passive systems

Suppose the (f, g, h) system (40) is passive with a C1 storage function E. Then wehave for all x ∈ Rn and for all u ∈ Rm

E =∂E∂xf(x) +

∂E∂xg(x)u ≤ hT (x)u (42)

or∂E∂xf(x) + ((

∂E∂xg)T (x)− h(x))Tu ≤ 0, ∀u ∈ Rm,∀x ∈ Rn (43)

This yields the following two passivity conditions

∂E∂xf(x) ≤ 0 (44)

(∂E∂xg)T (x) = h(x) (45)

Remarks:

Conditions (44) and (45) are the nonlinear version of the fundamental KYP lemma

Differentiating (45) along the vector field g(x) and evaluating at x = 0 yields

gT (0)∂2E∂x2

(0)g(0) = Lgh(0)

42

If E > 0 and ∂2E∂x2 (0) > 0, then gT (0)∂2E

∂x2 (0)g(0) is a positive definite matrix andthus Lgh(0) must be nonsingular.

That means that the relative degree of the (f, g, h) system at x = 0 is 1.

The relative degree of a system cannot be changed by regular feedback.

If y = h(x) does not satisfy ∂h∂xg(0) > 0, then it does not qualify for feedback

passivation designs.

Theorem 9 (Feedback passivity). Assume that rank ∂h∂x(0) = m. Then the

(f, g, h) system is feedback passive with a C2 positive definite storage function E(x)if and only if it has relative degree one at x = 0 and is weakly minimum phase1.

Proof.2

4.3 Standard formulation of PBC

Select a control action3 u = β(x) + v so that

Hd[x(t)]−Hd[x(0)] =∫ t

0vT (s)z(s)ds− dd(x, t)

where

Hd(x), the desired total energy function, has a minimum at x∗,

dd(x, t) ≥ 0 desired damping, and

z (which may be equal to y) is the new passive output

⇔

Energy–shaping plus damping injection.

1A nonlinear system whose zero dynamics have a Lyapunov stable equilibrium at the originis said to be weakly minimum phase

2See: C. Byrnes , A. Isidori and J.C. Willems, “Passivity, feedback equivalence, and theglobal stabilization of minimum phase nonlinear systems”, IEEE Trans Aut. Cont., Vol.36,no.11, pp. 1228–1240, 1991.

3State feedback, for ease of presentation

43

Discussion

Labeling inputs u and outputs y is restrictive, and the “control–as–interconnection”perspective is needed to cover a wider range of applications.

u may contain some external variables like disturbances or sources.

Control may not enter at all in u! (e.g., converter example)

The choice of the desired damping is far from obvious and maybe deleteriousfor performance.

Automatically ensures some robust stability (e.g., frictions and parasitic resis-tances).

Passivity can be used for stabilization independently of energy–shaping, findinga “detectable” z = h(x).

4.4 Connections with L2–gain assignment

Fact:

If the dissipation is such that

dd(t) ≥ δ

∫ t

0|z(s)|2ds

for some δ > 0, then the map v 7→ z has L2 gain smaller than 1δ .

Achieved choosing damping injection

v = Kdiz + w, Kdi = KTdi ≥ δI > 0

Proof.

From the new (EBE) and above∫ t

0vT (s)z(s)ds ≥ δ

∫ t

0|z(s)|2ds−Hd[x(0)]

which is equivalent to

12

∫ t

0|z(s)|2ds ≤

≤ −12

∫ t

0|z(s)|2ds+

1δ

∫ t

0vT (s)z(s)ds+

Hd[x(0)]δ

≤ −12

∫ t

0|z(s)− 1

δv(s)|2ds+

12δ2

∫ t

0v>(s)z(s)ds+

Hd[x(0)]δ

44

Thus ∫ t

0|z(s)|2ds ≤ 1

δ2

∫ t

0|v(s)|2ds+

1δHd[x(0)]

45

5 Energy–balancing control (EBC) and dissi-

pation

5.1 Stabilization via energy–balancing

For a class of systems, including mechanical, the solution is very simple:4

Find β(x) s. t. the energy supplied by the controller is a function of the syst.state.

H[x(t)]−H[x(0)] =∫ t

0β>(x(s))h(x(s))︸ ︷︷ ︸

uT (s)y(s)

ds− d(x(t), t)

Indeed, if

−∫ t

0β>(x(s))h(x(s))ds = Ha[x(t)] + κ (∗)

for some Ha(x), thenu = β(x) + v

ensures v 7→ y is passive with new energy function

Hd(x) , H(x)︸ ︷︷ ︸stored

+ Ha(x)︸ ︷︷ ︸−supplied

⇔ EB − PBC

5.2 Physical view: Mechanical systems

EBC has been first employed in the robotics literature (Takegaki & Arimoto,1981) for potential energy–shaping on fully actuated simple mechanical systemswith total energy

H(q, p) =12p>M−1(q)p+ V (q)

with

q ∈ Rn/2, the generalized coordinates;

p = M(q)q ∈ Rn/2, the generalized momenta;

M(q) = M>(q) > 0 the generalized mass matrix, and

4Assume y = h(x).

46

V (q) the systems potential energy.

Hamiltonian model: From

q = M−1(q)p =∂H

∂p

we get the model [qp

]=[

0 In−In −R

] [ ∂H∂q∂H∂p

]+[

0G(q)

]u

where R = RT ≥ 0 accounts for the presence of linear friction terms, u ∈ Rm

are the external forces, and G(q) is an interconnection matrix. Dissipation isassumed of the form

F(q) =12qTRq.

Passivity:

H =(∂H

∂p

)T

G(q)u−(∂H

∂p

)T

R∂H

∂p

= pTM−1(q)G(q)u− pTM−1(q)RM−1(q)p

Hence, passive outputs y = G>(q)q

EB controllers: If m = n,G(q) = I (fully actuated case) we can assign anyfunction of q with

β(q) = −∂Ha

∂q(q)

and passive output y = q.

Energy–shaping: The potential energy must have a minimum at the desiredequilibrium q = qd.

Choose the desired storage function as:

Hd(q, p) =12pTM−1(q)p+ Vd(q)

where Vd(q) has been shaped to have a minimum at q = qd.

New energy–balance equation (in differential form)

Hd = yT

(u+

∂(Vd − V )∂q

)− pTM−1(q)RM−1(q)p

47

Passivation:

passive output: y

new storage function: Hd

feedback transformation:

u = −∂(Vd − V )∂q

+ v

Asymptotic stability: we might need to inject damping with v = −Kdiq, Kdi =KT

di > 0, which results in the well–known PD+gravity compensation control.

5.3 Example: Pendulum

Energy H = 12 q

2 −mgl cos(q)

q +mgl sin(q) = u

y = q

We want

−∫ t

0β[q(s), q(s)]q(s)ds = Ha[q(t), q(t)] + κ

⇔ Ha = −β(q, q)q.

If Ha is only function of q, we can assign it with

β(q) = −∂Ha

∂q(q)

Let Ha(q) = mgl cos(q) + kp

2 (q − qd)2, yielding

β(q) = mgl sin(q)− kp(q − qd)

New total energy for v 7→ q is

Hd(q, q) =12q2 +

kp

2(q − qd)2

which has a minimum in (qd, 0).

Asymptotic stability with

v = −kdiq, kdi > 0

48

5.4 Implications of EBE for (f, g, h) systems

Fact:If the system

Σ :x = f(x) + g(x)uy = h(x)

satisfies the (EBE) with energy function H(x). Then5

∂H

∂x

T

(x)f(x) ≤ 0

gT (x)∂H

∂x(x) = h(x)

Proof.

Differential form of (EBE)H ≤ uT y

equivalent to∂H

∂x

T

(x)f(x)︸ ︷︷ ︸≤0

+(∂H

∂x

T

(x)g(x)− h(x))

︸ ︷︷ ︸=0

u ≤ 0

5.5 EB controllers

Corollary

(i) Σ satisfies the (EBE).(ii) The partial differential equation(

∂Ha

∂x(x))T

[f(x) + g(x)β(x)] = −(∂H

∂x(x))T

g(x)β(x)

can be solved for Ha(x).(iii) The total energy Hd(x) = H(x) +Ha(x) has a minimum at x∗.Then,

u = β(x) + v is an EB–PBC

Proof.PDE ⇔ Ha = u>y.

5Actually, iff as shown in (Hill/Moylan’76)

49

5.6 Caveat emptor

Limited interest because:

(f, g, h) are cryptic models

PDE parameterized in terms of β(x)!

difficult to incorporate prior information to solve the PDE.

Besides mechanical systems, the applicability of EB–PBC is severely stymied by

the systems natural dissipation.

5.7 Dissipation obstacle for EBC

Fact: A necessary condition for the solvability of the PDE is

f(x) + g(x)β(x) = 0 ⇒ h>(x)β(x) = 0.

Extracted power (= h>β) should be zero at equilibrium.⇒ EB–PBC applicable only for systems

without pervasive damping.

OK in regulation of mechanical syst. where power = F>q, but very restrictivefor electrical or electromechanical syst.: power = v>i.

For LTI systems (with |A| 6= 0)

u>∗ y∗ = 0 iff Σ(0) = 0

where Σ(s) = C(sI −A)−1B.

50

5.8 Finite dissipation example

1

1C

1L

1C

L

2R

qC

ϕL

State x , [qC , φL]>, energy H(x) = 12Cx

21 + 1

2Lx22.

Remarks:

Equil: x∗ = [x1∗, 0]T ⇒ zero extracted power!

Only need to “shape” x1

Dynamic equations

Σ :

x1 = 1

Lx2

x2 = − 1Cx1 − R

Lx2 + uy = 1

Lx2

PDE ⇔ (1Lx2

)∂Ha

∂x1−[

1Cx1 +

R

Lx2 − β(x)

]∂Ha

∂x2= − 1

Lx2β(x)

Look solution Ha = Ha(x1) ⇒ β(x1) = −∂Ha∂x1

(x1).

Propose

Ha(x1) =1

2Cax2

1 −(

1C

+1Ca

)x1∗x1 + κ⇒

with Ca tuning parameter. Recalling H(x) = 12Cx

21 + 1

2Lx22, this yields

Hd(x) =12

(1C

+1Ca

)(x1 − x1∗)2 +

12Lx2

2 + κ

has a minimum at x∗ for all gains 1Ca

> − 1C .

51

Control law

u = − 1Cax1 +

(1C

+1Ca

)x1∗

(= − 1

Ca(x1 − x1∗) + u∗

)is an EB–PBC.

5.9 Infinite dissipation example

1L

1C 2R 1

L

1CqC

ϕL

Remarks:

Only the dissipation has changed.

x∗ = [Cu∗, LRu∗]

> ⇒ nonzero power (∀u∗ 6= 0) ⇒

limt→∞

|∫ t

0u(s)y(s)ds| = ∞

for any stabilizing controller (run down the battery!)

5.10 Remarks

The dissipation obstacle is “coordinate–free”.

In the LTI case we can design an EB–PBC on incremental states. Not feasible–and actually unnatural– for the general nonlinear case.

We will propose a method (IDA–PBC) that

handles pervasive dissipation,

52

does not rely on incremental dynamics, and

energy functions will be (in general) non–quadratic.

53

6 Control by interconnection

6.1 Introduction to the control by interconnection

To characterize admissible dissipations, and overcome the obstacle:

(i) Adopt a “control–as–interconnection” viewpoint,

(ii) Give more structure to system

cΣ-

+

-

+ Σu c

yc y

uΣ I

Subsystems:

Σc control

ΣI interconnection and

Σ plant.

Principle: Select ΣI such that we can “add” the energies of Σ and Σc.

Definition: The interconnection is power preserving if∫ t

0[yT (s), yT

c (s)][u(s)uc(s)

]ds = 0

Simplest example: Classical feedback interconnection[uuc

]=[

0 −11 0

] [yyc

]

54

6.2 Passive controllers

Proposition 1. ΣI power preserving, Σ, Σc passive6 with states x ∈ Rn, ζ ∈ Rnc,and energy–functions H(x), Hc(ζ), resp. Let,[

uuc

]= ΣI

[yyc

]+[vvc

]with (v, vc) external inputs. Then, [vT , vT

c ]T 7→ [yT , yTc ]T is passive with new

energy–functionH(x) +Hc(ζ).

Problem: Although Hc(ζ) is free, not clear how to affect x?

6.3 Invariant functions method

Principle:7 restrict the motion to a subspace of (x, ζ),

1

x (t)

x (o) x 2

ξ

6They satisfy the EBE.7(Marsden/Ratiu,’94; Dalsmo/van der Schaft,’99)

55

SayΩ , (x, ζ)|ζ = F (x) + κ

κ determined by the controllers ICs, w.l.o.g. κ = 0.

Then,Hd(x) , H(x) +Hc[F (x)]

It can be shaped selecting Hc(ζ).

Find F (·) that renders Ω invariant ⇔ ddtC ≡ 0, where

C(x, ζ) , F (x)− ζ

is invariant function candidate.

6.4 Series RLC circuit

Controller: an integrator

Σc :

ζ = uc

yc = ∂Hc∂ζ (ζ)

with negative feedback interconnection u = −yc, uc = y.

Recalling

Σ :

x1 = 1

Lx2

x2 = − 1Cx1 − R

Lx2 + uy = 1

Lx2

ProposeC(x1, ζ) , F (x1)− ζ

Now,d

dtC =

1Lx2

(∂F

∂x1(x1)− 1

)Thus, we take F (x1) = x1. With the controller energy

Hc(ζ) =1

2Caζ2 −

(1C

+1Ca

)x1∗ζ

we recover the previous Hd(x).

56

6.5 Remarks

EB–PBC is constant voltage source in series with a capacitor Ca.

The control action can be implemented without the addition of dynamics.

We have assumed that nc = n. We also have considered nc 6= n and other ΣI .

Stabilization is ensured for all Ca > −C, but the system Σc is passive only forpositive values of Ca.

6.6 Questions

Will the method work for pervasive damping?

How to select the controller structure?

Finding F (·) that renders Ω invariant involves the solution of a PDE. Can thesearch for a solution of the PDE be made systematic?♥ YES: choice of a suitable system/controller representation.

6.7 Port–controlled Hamiltonian (PCH) systems

Model of a PCH system

Σ :x = [J(x)−R(x)]∂H

∂x (x) + g(x)uy = gT (x)∂H

∂x (x)

J(x) = −J>(x) is the interconnection matrix,

R(x) = R>(x) ≥ 0 damping matrix, and

g(x) is input matrix.

PCH systems clearly satisfy the EBE

d

dtH[x(t)] = −∂

>H

∂x[x(t)]R[x(t)]

∂H

∂x[x(t)] + uT (t)y(t)

57

Nice geometric structure formalized with notion of Dirac structures.

6.8 Examples: Series RLC Circuit

State: x = [qC , φL]T

total energy:

H(x) =12x2

1

C+

12x2

2

L

PCH model

x =[

0 1−1 −R

]︸ ︷︷ ︸

J−R

[x1Cx2L

]︸ ︷︷ ︸

∂H∂x

+[

01

]︸︷︷︸

g

u

6.9 Mechanical systems

Assuming linear friction, F = Rq, where, R = R> ≥ 0

State x =[qp

], p , D(q)q momenta.

Total energy: H(q, p) = 12p

TD−1(q)p+ U(q)

PCH model,

x =[

0 I−I −R

]∂H

∂x(x) +

[0I

]u

y =[

0I

]∂H

∂x(x)

(= D−1(q)p

)6.10 Electromechanical systems

Assuming linear magnetics, i.e., λ = L(θ)i ∈ Rn, L(θ) = LT (θ) ≥ 0, onemechanical d.o.f. θ ∈ R, u ∈ Rm voltages.

58

Total energy: H = 12λ

TL−1(θ)λ+ m2 θ

2 + U(θ)

State x = (λ, θ,mθ), ∂H∂x (x) = (i,−τ, θ), τ force, τL ∈ R load torque, M ∈

Rn×m defines actuated coordinates.

PCH model

x =

−R 0 00 0 10 −1 0

∂H∂x

(x) +

Mu0−τL

y = M

∂H

∂x1(x) (= Mi)

6.11 Induction motor

We have n = 4, m = 2,

λ =[λsIλrI

], M =

[I0

],

L(θ) =[

Ls LsreJθ

LsreJθ Ls

], R =

[RsI 00 RrI

]

6.12 Power converters

More general class of PCH models:

x = [J(x, u)−R(x)]∂H

∂x+ g(x, u)

The control u modifies the interconnection

Assuming: fast switching, linear Ri, Li, Ci.

State x ,

[φL

qC

]

Total energy: H(x) = 12x

T1 L

−1x1 + 12x

T2 C

−1x2, where L = diagLi, C =diagCi.

59

Cuk PCH model (x ∈ R4):

x =

0 −(1− u) 0 0

1− u 0 u 00 −u 0 −10 0 1 − 1

R

∂H∂x (x) +

E000

+

L+

+E

u=0

u=1

C+

R

Boost PCH model (x ∈ R2):

x =[

0 −uu −1

R

]︸ ︷︷ ︸

J(u)−R

∂H

∂x(x) +

[E0

]

60

6.13 Can dynamics overcome the dissipation obstacle?

Consider PCH controllers

Σc :

ζ = [Jc(ζ)−Rc(ζ)]∂Hc

∂ζ (ζ) + gc(ζ)uc

yc = gTc (ζ)∂Hc

∂ζ (ζ)

Jc(ζ) = −JTc (ζ), Rc(ζ) = RT

c (ζ) ≥ 0, and standard feedback interconnection,i.e., u = −yc, uc = y.

Closed–loop is still PCH:[xζ

]=[J(x)−R(x) −g(x)gT

c (ζ)gc(ζ)gT (x) Jc(ζ)−Rc(ζ)

] [ ∂H∂x (x)

∂Hc∂ζ (ζ)

]with total energy H(x) +Hc(ζ).

Casimir functions: conserved quantities of the system for any choice of theHamiltonian

completely determined by the geometry (i.e., the interconnection struc-ture) of the system.

Look for C(x, ζ) = F (x)− ζ, such that

d

dtC = 0

for all H(x), hence[∂F T

∂x −Im] [ J(x)−R(x) −g(x)gT

c (ζ)gc(ζ)g>(x) Jc(ζ)−Rc(ζ)

]= 0.

Proposition 2. C(x, ζ) is a Casimir function if and only if F (x) satisfies(∂F

∂x(x))T

J(x)∂F

∂x(x) = Jc(ζ)

R(x)∂F

∂x(x) = 0 (DO)

Rc(ζ) = 0(∂F

∂x(x))T

J(x) = gc(ζ)gT (x)

Dynamics reduced to Ω :

Σd : x = [J(x)−R(x)]∂Hd

∂x(x)

with Hd(x) = H(x) +Hc[F (x) + κ].

61

6.14 Admissible dissipation

(DO) implies that, for any Hc(·),

R(x)∂Hc(F )∂x

(x) = 0

Assume R(x) diagonal,8 then Hc should not depend on coordinates wherethere is damping. Consequently:

Dissipation only in “non–shaped” coordinates.

How to overcome dissipation obstacle?

Explicitly incorporating information on the state, obviates the need of the Casimirfunctions and still shape the energy function.

8Must often encountered in applications.

62

7 IDA–PBC

7.1 Matching perspective

Starting from a PCH model, define desired dynamics

x = [J(x)−R(x)]∂Hd

∂x

New total energy Hd(x) = H(x) +Ha(x)

Find β(x) such that the PDE is solved (for Ha(x))

(PDE) [J(x)−R(x)]∂Ha

∂x(x) = g(x)β(x)

Setting u = β(x), yields

x = [J(x)−R(x)]∂H

∂x(x) + g(x)β(x)

= [J(x)−R(x)](∂H

∂x(x) +

∂Ha

∂x(x))

︸ ︷︷ ︸∂Hd∂x

If, further, x∗ = arg minHd(x) then x∗ is stable.

Key properties:

Remove the dependence on β(x). Indeed, (PDE) ⇔

g⊥(x)[J(x)−R(x)]∂Ha

∂x(x) = 0

where g⊥(x)g(x) = 0. Control:

β(x) = [gT g]−1gT

[J −R]

∂Ha

∂x

Show later how to change J(x),R(x) invoking physical considerations.

63

7.2 When is IDA an EB–PBC?

Compute

Hd = uT y −[∂H

∂x(x)]T

R(x)∂H

∂x(x)︸ ︷︷ ︸

H

+Ha

= −[∂Hd

∂x(x)]T

Rd(x)∂Hd

∂x(x)

and Rd(x) = Ra(x) +R(x), we have that

Ha = −uT y −[2∂H

∂x+∂Ha

∂x

]T

R∂Ha

∂x−[∂Hd

∂x

]T

Ra∂Hd

∂x.

Consequently, if Ra(x) = 0 and R(x) satisfies

R(x)∂Ha

∂x(x) = 0,

thenHa = −uT y ⇔ IDA–PBC is energy balancing.

7.3 IDA PBC for (f, g, h) systems

Fix the matrices Jd(x),Rd(x).

Solve

g⊥(x)f(x) = g⊥(x)[Jd(x)−Rd(x)]∂Hd

∂x(x)

The closed–loop system with

u = β(x)

= [gT (x)g(x)]−1gT (x)

[Jd(x)−Rd(x)]∂Hd

∂x(x)−f(x)

,

will be a PCH system with dissipation of the form

x = [Jd(x)−Rd(x)]∂Hd

∂x(x)

with x∗ a (locally) stable equilibrium.

64

7.4 IDA PBC: Swapping the damping

Lemma 4. Assume rank[J(x)−R(x)] = n9, then

zT [J(x)−R(x)]−1z ≤ 0,

for all z ∈ Rn.

Proof. zT [J(x)−R(x)]−1z :=

=12zT ([J(x)−R(x)]−1 + [J(x)−R(x)]−T )z

=12zT [J(x)−R(x)]−1([J(x)−R(x)] +

[J(x)−R(x)]T )[J(x)−R(x)]−T z

=12zT (J(x)−R(x) + [J(x)−R(x)]T )z

= −zTR(x)z ≤ 0.

where z = [J(x)−R(x)]−T z.

7.5 New passivity property

Proposition 3. PCH systems satisfy the new EB inequality

H[x(t)]−H[x(0)] ≤∫ t

0yT (s)u(s)ds,

y = h(x, u)= −gT (x)[J(x)−R(x)]−T [J(x)−R(x)]∇xH(x) + g(x)u .

Proof. [J(x)−R(x)]−1x = ∇xH(x) + [J(x)−R(x)]−1g(x)u. Premultiplying byxT

H(x) = xT∇xH(x)= xT [J(x)−R(x)]−1x− xT [J(x)−R(x)]−1g(x)u≤ −xT [J(x)−R(x)]−1g(x)u = yTu,

9If not ⇒ equilibria at points which are not extrema of the energy function.

65

7.6 Energy–balancing with new supplied power

Corollary 3. IDA–PBC transforms the PCH system into

x = [J(x)−R(x)]∇xHd(x).

with

Hd[x(t)] = H[x(t)]−∫ t

0uT (s)y(s)ds.

Proof. Matching equation

[J(x)−R(x)]−1g(x)β(x) = ∇xHa(x)

7.7 Interpretation in EM systems

The new passivity property is a corollary of Thevenin and Norton equivalence.

x = col(ψ, θ, p) ∈ Rne+2, ψ ∈ Rne magnetic fluxes, θ, p ∈ R mechanicaldisplacement and momenta, u external voltages.

Electrical equations of this system are of the form ψ = −Rei + Bu, Re =RT

e > 0 ∈ Rne×ne resistors, i ∈ Rne currents on the inductors, ψ = L(θ)i, withL(θ) = LT (θ) > 0 the inductance matrix.

The natural power port variables u and y = B>L−1(θ)ψ. Now,

yTu = uTBTR−1e ψ

where R−1e Bu are the current sources obtained from the Norton equivalent

of the Thevenin representation of the classical passivity property, with ψ theassociated inductor voltages.

66

+

−

+

−

R1

u1

uk

Rk

Electro-

System

R1u1

R1

uk

RkRk

Electro-

System

ψ1

L1

ψk

Lk

⇐⇒

ψ1

ψk

mechanical mechanical

7.8 Universal stabilizing property of IDA–PBC

Proposition 4. If ∃β(x) ∈ C1 that asymptotically stabilizes the PCH system, then∃Ja(x),Ra(x) ∈ C0 and Ha(x) ∈ C1 which satisfy the conditions of the IDA–PBCtheorem.

⇔ IDA–PBC methodology generates all asymptotically stabilizing controllers forPCH systems.

Lemma 5. If x∗ is asymptotically stable for x = f(x), f(x) ∈ C1 then ∃Hd(x) ∈

67

C1, positive definite, and C0 functions

Jd(x) = −JTd (x),Rd(x) = RT

d (x) ≥ 0

such thatf(x) = [Jd(x)−Rd(x)]

∂Hd

∂x

Proof. Converse Lyapunov theorem ⇒ ∃Hd(x) s.t.[∂Hd

∂x(x)]T

f(x) ≤ 0.

Define

Rd(x) := − 1|∂Hd

∂x |4∂Hd

∂x

[∂Hd

∂x

]T

fT (x)∂Hd

∂x

Jd(x) :=1

|∂Hd∂x |2

f(x)

[∂Hd

∂x

]T

− ∂Hd

∂xfT (x)

7.9 Integral action

Proposition 5. IDA–PBC with integral action u = ues + udi + v where

v = −KIgT ∂Hd

∂x

with KI = K>I > 0, preserves stability.

Proof. Let

W (x, udi) , Hd +12vTK−1

I v

The closed–loop [xv

]=[Jd −Rd gKI

−KIgT 0

] [ ∂W∂x∂W∂v

]is clearly PCH.

68

7.10 Damping injection with ”dirty derivatives”

To inject additional damping, instead of the passive output, we can feed backthe filtered signal, preserving stability.

Of particular interest to obviate velocity measurements in mechanical systemswhere the passive output is q.

Proposition 6. If

u = ues +KdigT ∂Hd

∂x

is asymptotically stable, then u = ues + udi, where

udi = −1τudi −

Kdi

τgT ∂Hd

∂x

with τ > 0, also ensures convergence of x∗.

Proof.

With the energy

W (x, udi) , Hd +τ

2Kdiu2

di

we have [xudi

]=[

JdKdiτ g

−Kdiτ gT −Kdi

τ2

] [ ∂W∂x∂W∂udi

]yielding

W = −uTdiKdi−1udi

7.11 IDA PBC as a state–modulated source

For infinite dissipation systems need non–passive controllers!

1. Controller as an (infinite energy) source

Σc :

ζ = uc

yc = ∂Hc∂ζ (ζ)

with energy function Hc(ζ) = −ζ.

69

2. State–modulated (power–preserving) interconnection[u(s)uc(s)

]=[

0 −β(x)β(x) 0

] [y(s)yc(s)

]Overall interconnected PCH system (with total energy H(x) +Hc(ζ))[

xζ

]=[J(x)−R(x) −g(x)β(x)βT (x)gT (x) 0

] [ ∂H∂x (x)

∂Hc∂ζ (ζ)

]

7.12 Example: Parallel RLC circuit

Total energy: H(x) = 12

x21

C + 12

x22

L

PCH model

J =[

0 1−1 0

], R =

[1/R 00 0

], g =

[01

]

(PDE) becomes

− 1R

∂Ha

∂x1(x) +

∂Ha

∂x2(x) = 0

−∂Ha

∂x1(x) = β(x)

First equation is a PDE with solution Ha(x) = Φ(Rx1 + x2), where Φ(·) arbi-trary.

Choose Φ(·) so that[CLR

]u∗︸ ︷︷ ︸

x∗

= arg minHd(x) ⇐

∂Hd∂x (x∗) = 0 (EC)

∂2Hd∂x2 (x∗) > 0 (HC)

(EC) equivalent

∂Ha

∂x(x∗) =

[R1

]∂Φ∂z

(z∗) ≡ −∂H∂x

(x∗) =[

11R

]u∗

where z = Rx1 + x2. Thus, ∂Φ∂z (z∗) = − 1

Ru∗.

Check (HC)∂2Ha

∂x2=∂2Φ∂z2

[R2 RR 1

]

70

Let

Φ(z) =Kp

2(z − z∗)2 −

1Ru∗z

which yields

Hd(x) = (x− x∗)T

[1C +R2Kp RKp

RKp1L +Kp

](x− x∗) + κ

(HC) satisfied for Kp >−1

(L+CR2)

Controlu = −Kp[R(x1 − x1∗) + x2 − x2∗] + u∗.

Procedure: From (PDE) results a family of “admissible” Ha(x), β(x). Chooseone that shapes the energy.

7.13 Interconnection and damping assignment

We aim at

x = [Jd(x)−Rd(x)]∂Hd

∂x(x)

for some newJd(x) = −JT

d (x), Rd(x) = RTd (x) ≥ 0

(PDE) becomes

[J(x)+Ja(x)−R(x)−Ra(x)]∂Ha

∂x= −[Ja(x)−Ra(x)]

∂H

∂x+g(x)β(x) (PDE′)

whereJa(x) , Jd(x)− J(x), Ra(x) , Rd(x)−R(x)

are new degrees of freedom.

7.14 Solving the PDE

Pre-multiply by g⊥(x) to eliminate β(x)

Defining a PDE directly for β

71

Lemma 6. Given K(x) : Rn → Rn, ∃Ha(x) : Rn → R s.t.

K(x) =∂Ha

∂x⇔ ∂K

∂x=[∂K

∂x

]T

(IC)

Consequently, we can equivalently check that

K(x) , −[Jd −Rd]−1

([Ja −Ra]

∂H

∂x− gβ

)satisfies (IC).

72

8 Examples

8.1 Some applications:

Mass–balance systems (ACC, 2000),

electrical motors (IEEE CST, 2001),

power systems (ACC, 2001; Automatica 2002),

magnetic levitation systems (MTNS, 2000; ACC 2001),

underactuated mechanical systems (IEEE TAC, 2002),

power converters (SCL, 99),

rigid body dynamics (AIAA, 2000),

electromechanical systems (IJRNLC, 2003),

underwater vehicles (SCL, 2001).

73

8.2 Magnetic levitation system

u

i

y

g

m

λ

Approximate the inductance L(θ) = k1−θ .

PCH model. State: x = [λ, θ,mθ]T

Hamiltonian: H(x) = 12k (1− x2)x2

1 + 12mx3

2 +mgx2. Matrices:

J =

0 0 00 0 10 −1 0

, R =

R 0 00 0 00 0 0

, g =

100

Equilibrium x∗ = [

√2kmg, x2∗, 0]T .

Structural limitation

(PDE) without changing J or R:

(J −R)∂Ha

∂x(x) = gβ(x) ⇔

−R∂Ha

∂x1(x) = β(x)

∂Ha∂x2

(x) = 0∂Ha∂x3

(x) = 0

74

⇒ Ha = Ha(x1) can only depend on x1. The Hessian

∂2Hd

∂x2(x) =

(1−x2)k + ∂2Ha

∂x2 (x1) −x1k 0

−x1k 0 0

0 0 1m

which is sign indefinite for all Ha(x1).

Source of the problem: lack of effective coupling between the electrical and themechanical subsystem.

IDA–PBC

Enforce a coupling between the flux x1 and the velocity x3 ⇒

Jd =

0 0 −α0 0 1α −1 0

(PDE’)

∂Ha

∂x3(x) = 0

−R∂Ha

∂x1(x)(x) =

α

mx3 + β(x)

α∂Ha

∂x1(x)− ∂Ha

∂x2(x) = −α

k(1− x2)x1

Solving the latter (e.g. Maple)

Ha(x) =1

6kαx3

1 +12kx2

1(x2 − 1) + Φ(x2 +1αx1),

Suitable choice for

Φ(x2 +1αx1) = mg[−(x2 +

1αx1) +

b

2(x2 +

1αx1)2]

Control law

u =R

k(1− x2)x1︸ ︷︷ ︸

Ri

−Kp(1αx1 + x2)−

α

mx3︸ ︷︷ ︸

PD

− R

α(

12kx2

1 −mg)︸ ︷︷ ︸undesirable

75

To remove the high order term: shuffle the damping

Ra =

−R 0 00 Ra 00 0 0

This yields u = R

k (1− x2)x1 −Kp( 1α x1 + x2)−

(αm +Kp

)x3

Comparison of various schemes

uFL =

√k

2FmvFL(θ, θ, F ) +R(1− θ)

√2Fk

uPB =

√k

2FdmvFL(θ, θ, F ) +R(1− θ)

√2Fd

k

uIB = uPB −β√2mk

[θ, ˙θ,

∫ t

0θ(s)ds]PB(

√F +

√F d)

uIDA = −Kp(1αλ+ θ)−

( αm

+Kp

)θ +R(1− θ)

√2Fk

where FL=feedback linearization, PB=standard PBC, IB=integrator backstepping,and

F =12λ2, Fd = m[θ∗ − k2

bp

p+ aθ − k1θ − k0

∫ t

0θ(s)ds]

vFL = θ(3)∗ − k2[(

1mF − g)− θ∗]− k1

˙θ − k0θ

Experimental results

Linearized model based controller

−0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Feedback linearization control

−0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80.10

0.10.20.30.40.50.60.70.80.9

time [s]

76

Passivity based control with inte-gral term

−0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80.10

0.10.20.30.40.50.60.70.80.9

time [s]

IBC with integral term

−0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80.10

0.10.20.30.40.50.60.70.80.9

time [s]

8.3 Mechanical systems

To stabilize some underactuated mechanical devices it is necessary to modifythe total energy function. In open loop

H(q, p) =12p>M−1(q)p+ V (q)

where q ∈ Rn, p ∈ Rn are the generalized position and momenta, respectively,M(q) = M>(q) > 0 is the inertia matrix, and V (q) is the potential energy.

Model [qp

]=[

0 In−In 0

] [∇qH∇pH

]+[

0G(q)

]u

Control u ∈ Rm, and assume rank(G) = m < n. Convenient to decomposeu = ues(q, p) + udi(q, p)

Target Dynamics: Desired (closed loop) energy function

Hd(q, p) =12p>M−1

d (q)p+ Vd(q)

where Md = M>d > 0 and Vd(q) is s.t. q∗ = arg minVd(q). Thus,[

qp

]= [Jd(q, p)−Rd(q, p)]

[∇qHd

∇pHd

],

where

Jd = −JTd =

[0 M−1Md

−MdM−1 J2(q, p)

],

Rd = RTd =

[0 00 GKvG

T

]≥ 0

77

8.4 Strongly coupled VTOL aircraft

Horizontal displacement: badly tuned⊗

; well tuned ♥

Well tuned aggressive maneuver ♥

Model, ε 6= 0, possibly large

x = − sin θv1 + ε cos θv2y = cos θv1 + ε sin θv2 − g

θ = v2

Objective: Characterize assignable energy functions with

(x∗, y∗, 0, 0, 0, 0) asymptotically stable

With q = [x, y, θ]T , p = [x, y, θ]T and an input change of coordinates

q = p

p = G(θ)u+g

εsin θe3

where

G(θ) =

1 00 1

1ε cos θ 1

ε sin θ

, e3 ,

001

Characterizing assignable energy functions

Potential energy Vd(q) = −gρ+Φ(η1(x, θ), η2(y, θ)) where

Φ(η1, η2) =12

[η1 − η1(x∗, 0)η2 − η2(y∗, 0)

]T

P

[η1 − η1(x∗, 0)η2 − η2(y∗, 0)

],

P = P T > 0

Kinetic energyMd(θ) =

78

k32 [tan2 θ − (1 + 2k1ε

k3) log(1 + tan2 θ)] + k4 (k1ε+ k3)θ − k3 tan θ k1 cos θ

(k1ε+ k3)θ − k3 tan θ k32 log(1 + tan2 θ) + k4 k1 sin θ

k1 cos θ k1 sin θ k2

positive and bounded ∀θ ∈ (−π/2, π/2), k1 > 0, and k4 > k4(k3, k1), k1

ε >

k2 >k12ε

Horizontal displacement: badly tuned⊗

; well tuned ♥

Well tuned aggressive maneuver ♥

8.5 Boost converter

Model (under fast switching), x(0) ∈ R2>0

x =[

0 −uu −1

R

]︸ ︷︷ ︸

J(u)

∂H

∂x(x) +

[E0

]

Control objective: regulate 1Cx2 to a desired constant value V∗ > E, verifying

C.1 Only x2 measurable.

C.2 u ∈ (0, 1).

C.3 x ∈ R2>0.

C.4 R is unknown.

Main contribution: Stabilization via IDA–PBC with a simple static nonlinearoutput feedback.

Proposition 7. For all R > 0 the IDA–PBC

u = u∗

(x2

x2∗

)α

, 0 < α < 1

yields

(i) x∗ = ( LREV

2∗ , CV∗) is asymptotically stable with Lyapunov function

Hd(x) =1

2Lx2

1 +1

2Cx2

2 + κ1x2(1−α)2 − (κ2 + κ3x1)x1−α

2

79

(ii) Domain of attraction: Ξα4= x|x ∈ R2

>0 and Hd(x) ≤ Hd(0, x2∗) issuch that x(0) ∈ Ξα ⇒ x(t) ∈ Ξα and limt→∞ x(t) = x∗

(iii) Saturation: ∀x∗, ∃α ∈ (0, 1) s.t.

x(0) ∈ Ξα ⇒ 0 < u(t) < 1

Proof.

IDA: Select Ra = diagRa,− 1R ⇒ Rd = diagRa, 0.

Integrability Key PDE10

K ,∂Ha

∂x(x) =

1β(x2)

[− 1

RCx2

− 1LRax1 − E + Ra

RCx2

β(x2)

]

PDE solvable ⇔ ∂K2∂x1

(x) = ∂K1∂x2

(x) ⇔

dβ

dx2(x2) =

α

x2β(x2)

where α4= 1− RaRC

L . Thus,u = c1x

α2

Equilibrium assignment: c1 such that

∂Hd

∂x(x∗) =

∂H

∂x(x∗) +

∂Ha

∂x(x∗) = 0

This yields c1 = u∗xα2∗

.

Hessian condition

∂2Hd

∂x2(x∗) =

[1L − Ra

u∗L

− Rau∗L

1C + (Rax1∗+EL)α

u∗Lx2∗+ (1−2α)Ra

u∗2RC

]

Positive definite ⇔ −1 < α < 1.

10Assuming J(β(x))−Rd is invertible.

80

Comparison with the Standard PBC

The model of the system is written as

Mz + J(u)z +Rz = g

where z ∈ R2, M := diagL, C, g = [E, 0]T .

An implicit definition of the controller is derived from a copy of the system withadditional damping as

Mzd + J(u)zd +Rzd = g +Rdiz

where Rdi := diagR1, 0, R1 > 0, zd ∈ R2 is an auxiliary vector, z := z−zd,and zd will be defined later.

The idea is that, for all u, the error equation

M ˙z + [J(u) +Rd]z = 0

with Rd := R+Rdi, is exponentially convergent, that is, z → 0 (exp.).

Find a control u such that z → 0 ⇒ z2 → Vd.

Fix z1d to its desired valueV 2

dRE . This leads to

Cz2d = − 1RL

z2d +V 2

d

RLEz2d(E +R1z1)

u =1z2d

(E +R1z1)

z2d is the state of our dynamic controller.

To complete the stability analysis we must show that z2d remains bounded,which follows from the minimum phase properties of the system with output z1.

Comparing the solutions

(i) Obvious complexity reduction. (ii) IDA is more “natural”: no enforcementof quadratic storage functions. (iii) No need for stable invertibility.11

11Brings along robustness problems inherent to linearization.

81

8.6 PM Synchronous Motor

Model

dq model

Lddiddt

= −Rsid + ωLqiq + vd

Lqdiqdt

= −Rsiq − ωLdid − ωΦ + vq

Jdω

dt= P · ((Ld − Lq)idiq + Φiq)− τl

ω is angular velocity, vd, vq, id, iq are voltages and currents. P is the number ofpole pairs, Ld and Lq are stator inductances, Rs is stator winding resistance, τlis a constant unknown load torque, and Φ and J are the dq back emf constantand the moment of inertia.

Energy function H(x) = 12

(Ldi

2d + Lqi

2q + J

P ω2)

= 12x>D−1x

x = D

idiqω

, D ,

Ld 0 00 Lq 00 0 J

P

PCH model x = [J(x)−R(x)]∂H

∂x (x) + g(x)u+ ζ with

g = [ 1 00 10 0 ] , u = [ vdvq ] , ζ = [ 00− τlP ]

R = diagRs, Rs, 0 and

J(x) =

0 0 x2

0 0 −(x1 + Φ)−x2 x1 + Φ 0

Desired equilibrium (“maximum torque per ampere” principle) x∗ = (x1∗, x2∗, x3∗) =(0, Lqτl

PΦ ,JP ω∗)

Choice of interconnectionIsotropic (smooth rotor) synchronous motors are more efficient that indented rotor

motors

82

Open loop

J(x) =

0 0 x2

0 0 −(x1 + Φ)−x2 x1 + Φ 0

Φ is the dq back emf constant.

Virtual behavior in closed–loop

Jd(x) =

0 L0x3 0−L0x3 0 −Φ

0 Φ 0

L0 free parameter, representsstator inductance (Ld = Lq).

Natural Interconnection Controller

Without modification, that is, Ja(x) = Ra(x) = 0.

vd = −Rs∂Ha

∂x1+ x2

∂Ha

∂x3, vq = −Rs

∂Ha

∂x2− (x1 + Φ)

∂Ha

∂x3

−x2∂Ha

∂x1+ (x1 + Φ)

∂Ha

∂x2= − 1

Pτl

Solution is

Ha(x) =τlP

arctan(x1 + Φx2

) + F (x22 + x2

1 + 2x1Φ) + h(x3)

F and h are differentiable functions to be chosen.

Since ∂Ha∂x3

only depends on x3,h(x3) = −ω∗x3 + α22 x

23, where (·) , (·) − (·)∗,

and α2 > 0. This choice yields K3 = −ω∗ + α2x3.

The equilibrium assignment and Lyapunov stability conditions reduce to

f(z) = − 12Lq

z

z + Φ2,∂f(z)∂z

∣∣∣∣z=z

>1

4Lq

z − Φ2

(z + Φ2)2

83

Propose f(z) = − 12Lq

zz+Φ2 , which yields

∂Ha

∂x1=

τl/P

x22 + (x1 + Φ)2

[x2 −

LqτlPΦ2

(x1 + Φ)]

∂Ha

∂x2= − τl/P

x22 + (x1 + Φ)2

[(x1 + Φ) +

LqτlPΦ2

x2

]∂Ha

∂x3= −ω∗ + α2x3

x∗ is asymptotically stable, but the initial conditions

(x1(0) + Φ)2 + x2(0)2 ≥ ε > 0,

and the load torque is different from zero.

Load torque is unknown

dω

dt=

P

J

(γx1 +

ΦLq

)x2 − l1(ω − ω)− 1

Jτl

dτldt

= l2(ω − ω)

with γ , 1Lq− 1

Ld, and l1, l2 some positive design parameters.

Isotropic Interconnection Controller

Modify the interconnection matrix to “emulate” an isotropic machine (Ld =Lq = L) which is easier to control.

PCH model with g(x), ζ, R and u as before, and

J(x) =

0 LPJ x3 0

−LPJ x3 0 −Φ0 Φ 0

Propose

Jd(x) =

0 L0x3 0−L0x3 0 −Φ

0 Φ 0

where L0 is a parameter to be defined.

84

Proposition 8. The control law[vd

vq

]=[

(L0Lq− P

J )x2x3 −Rsα1x1

−(L0Ld− P

J + L0α1)x1x3 + Φ(PJ x3∗ − α2x3)

]+

+[−Rs L0x3

−L0x3 −Rs

] [ γ2Φ(x2

2 − x22∗)

γΦx1x2 − 1

Lqx2∗

]where L0 is arbitrary α2 + P

J > 0, (α1 + 1Ld

) 1Lq

> γ2

Φ2x22∗, ensures x∗ is GAS with

energy–Lyapunov function

Hd =12x>D−1x+

γ

2Φx1(x2

2 − x22∗)−

1Lqx2∗x2 +

α1

2x2

1

−PJx3∗x3 +

α2

2x2

3

Connections with Current Practice

To recover a linear dynamics in the electrical subsystem it is common to cancelthe nonlinear terms.

vd = −ωLiq + vd1, vq = ωLid + ωΦ + vq1

Drawback is lack of robustness. An alternative

vd = Rsid∗ − ωLiq∗, vq = Rsiq∗ + ωLid∗ + ω∗Φ

where iq∗ = τlPΦ .

Asymptotically table if the load torque is known. In practice current referencegenerated by PI controller in the outer loop. Stability?

In the isotropic rotor case γ = 0, and IDA has the same form by setting α1 =α2 = 0 and L0 = PL

J . Only difference that the desired values for the currentsare generated by the nonlinear observer.

85

8.7 Underactuated Kirchhoff’s equations

T1

T2 F3

Model

Ellipsoidal rigid body submerged in an ideal fluid and assume that the center ofgravity of the body coincides with the center of buoyancy.

Kirchhoff equations (Leonard, Automatica’97)

Π1 =(

1J3− 1

J2

)Π2Π3 +

(1

M3− 1

M2

)P2P3 + T1

Π2 =(

1J1− 1

J3

)Π3Π1 +

(1

M1− 1

M3

)P3P1 + T2

Π3 =(

1J2− 1

J1

)Π1Π2 +

(1

M2− 1

M1

)P1P2

P1 = P2Π3J3

− P3Π2J2

P2 = P3Π1J1

− P1Π3J3

P3 = P1Π2J2

− P2Π1J1

+ F3

Π1, Π2 and Π3 (P1, P2 and P3) angular (linear) momentum, J1 > 0, J2 > 0and J3 > 0 principal moments of inertia and M1 > 0, M2 > 0 and M3 > 0terms of the inertia matrix,

T1, T2 and F3 manipulated variables ⇒ the system is locally strongly accessible.

86

Stabilization of equilibria

xe =

col(0, 0, 0, 0, P2, 0), forward/reversecol(0, 0, 0, P1, 0, P3), diving/rising with f/r

Port-controlled Hamiltonian (PCH) description

x = (J(x)−R(x))(∂H

∂x

)T

+Gu.

x = col(Π1,Π2,Π3, P1, P2, P3) ∈ R6, u = col(T1, T2, F3) ∈ R3.

Selective damping

Proposition 9. Solutions of PDE

G⊥J(x)(∂Ha

∂x

)T

= 0,

ensure

u(x)=[G>G]−1G>[(J(x)−Ra(x))

(∂Ha

∂x

)>−Ra(x)

(∂H

∂x

)>]

stabilizes xe, for all Ra(x) = R>a (x) ≥ 0 and

Im(Ra(x)) ⊆ Im(G).

All solutions of the PDE are of the form∏i∈I

φi(ψj),

with I a finite set of indexes, j = 1, 2, 3, differentiable φi(·) and

ψ1 = P3

ψ2 = P 21 + P 2

2

ψ3 = Π1P1 + Π2P2 + Π3P3

The equilibrium is almost globally asymptotically stable.

Suppose M1 > M2. Then, the steady rising/diving with forward/reverse motionequilibrium is locally asymptotically stable.

87

9 Concluding remarks and future research

Tracking The basic IDA–PBC is restricted to stabilization of fixed points

Tracking general signals remains an essentially open (relevant?) issue

Tracking exosystem–generated references may be cleanly recasted as adamping injection problem, but unfortunately with “unmatched” signals

Stabilization of periodic orbits with a “Mexican sombrero” Hd(x)

Solving the PDE For mechanical systems:

the λ–method developed for the Controlled Lagrangian method, adaptedfor IDA–PBC, yielding a bilinear PDE

PDE reduced to ODE if level of underactuation is one

Better understanding of the role of J2(q, p) needed

Robustness and adaptation Current framework based on contrived (but math-ematically convenient) uncertainty structures difficult to justify from physicalconsiderations. Need to develop a theory that would accommodate intercon-nection of (partially uncertain) parameterized PCH systems to reverse this trend

Asymptotic matching Attaining the model matching of IDA–PBC only asymp-totically. Led to the development of a new, immersion and invariance, techniquefor stabilization of general nonlinear systems

Power shaping For a class of (nonlinear) RLC circuits, it is possible to formulatethe stabilization problem in terms of power (as opposed to energy) shaping.Advantages:

Adding “derivative” actions in the control that naturally yield faster re-sponses

Nice geometric formalization (in terms of Dirac structures)

Clear connection with power–balancing (always!)

Infinite dimensional systems The PCH modelling framework available. Somepreliminary results on control by interconnection

88