ENERGYSHAPING STABILIZATION OF DYNAMICAL SYSTEMS Romeo Ortega and - - PDF document

ENERGY–SHAPING STABILIZATION OF DYNAMICAL SYSTEMS

Romeo Ortega and Elo´ ısa Garc´ ıa

Laboratoire des Signaux et Syst` emes S U P E L E C Gif–sur–Yvette, FRANCE

• rtega,garcia@lss.supelec.fr
• ctober 2003

Contents

1 Motivation 5 1.1 Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Why? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Proposal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2 Introduction 9 2.1 Intelligent control paradigm revisited . . . . . . . . . . . . . . 9 2.2 Theoretical trend . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 Proposal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.4 Energy–shaping . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.5 Energy–shaping and passivity . . . . . . . . . . . . . . . . . . 11 2.6 Two approaches to PBC . . . . . . . . . . . . . . . . . . . . . 11 2.7 Basic references: Theory . . . . . . . . . . . . . . . . . . . . . 11 2.8 Application references . . . . . . . . . . . . . . . . . . . . . . 12 1

3 Mathematical preliminaries 14 3.1 Input–Output (I/O) theory . . . . . . . . . . . . . . . . . . . 14 3.2 Lq–spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.3 Lq–spaces are normed spaces . . . . . . . . . . . . . . . . . . . 14 3.4 Extended Lq–space . . . . . . . . . . . . . . . . . . . . . . . . 15 3.5 Operators: properties and examples . . . . . . . . . . . . . . . 16 3.6 Induced norms . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.7 Input–Output Stability . . . . . . . . . . . . . . . . . . . . . . 19 3.8 Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.9 Feedback systems or closed–loop systems . . . . . . . . . . . . 24 3.10 Small Gain Theorem . . . . . . . . . . . . . . . . . . . . . . . 26 3.11 Lure’s problem . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.12 Loop transformations . . . . . . . . . . . . . . . . . . . . . . . 29 3.13 The circle criterion . . . . . . . . . . . . . . . . . . . . . . . . 31 3.14 The passivity approach . . . . . . . . . . . . . . . . . . . . . . 32 3.15 Passivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.16 Dissipativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.17 Passivity and L2–gain . . . . . . . . . . . . . . . . . . . . . . 34 3.18 Passivity and feedback interconnections . . . . . . . . . . . . . 35 3.19 Kalman–Yacubovich–Popov’s lemma . . . . . . . . . . . . . . 36 3.20 Passivity and energy–shaping . . . . . . . . . . . . . . . . . . 37 3.21 Examples: Electrical circuits . . . . . . . . . . . . . . . . . . . 38 3.22 Examples: Mechanical systems . . . . . . . . . . . . . . . . . . 39 3.23 Examples: Electromechanical systems . . . . . . . . . . . . . . 40 3.24 Examples: Power converters . . . . . . . . . . . . . . . . . . . 41 4 Passivity–based control (PBC) 42 4.1 Feedback passivation . . . . . . . . . . . . . . . . . . . . . . . 42 4.2 Feedback passive systems . . . . . . . . . . . . . . . . . . . . . 42 4.3 Standard formulation of PBC . . . . . . . . . . . . . . . . . . 43 4.4 Connections with L2–gain assignment . . . . . . . . . . . . . . 44 5 Energy–balancing control (EBC) and dissipation 46 5.1 Stabilization via energy–balancing . . . . . . . . . . . . . . . . 46 5.2 Physical view: Mechanical systems . . . . . . . . . . . . . . . 46 5.3 Example: Pendulum . . . . . . . . . . . . . . . . . . . . . . . 48 5.4 Implications of EBE for (f, g, h) systems . . . . . . . . . . . . 49 5.5 EB controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 5.6 Caveat emptor . . . . . . . . . . . . . . . . . . . . . . . . . . 50 5.7 Dissipation obstacle for EBC . . . . . . . . . . . . . . . . . . . 50 5.8 Finite dissipation example . . . . . . . . . . . . . . . . . . . . 51 2

5.9 Infinite dissipation example . . . . . . . . . . . . . . . . . . . 52 5.10 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 6 Control by interconnection 54 6.1 Introduction to the control by interconnection . . . . . . . . . 54 6.2 Passive controllers . . . . . . . . . . . . . . . . . . . . . . . . 55 6.3 Invariant functions method . . . . . . . . . . . . . . . . . . . . 55 6.4 Series RLC circuit . . . . . . . . . . . . . . . . . . . . . . . . 56 6.5 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 6.6 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 6.7 Port–controlled Hamiltonian (PCH) systems . . . . . . . . . . 57 6.8 Examples: Series RLC Circuit . . . . . . . . . . . . . . . . . . 58 6.9 Mechanical systems . . . . . . . . . . . . . . . . . . . . . . . . 58 6.10 Electromechanical systems . . . . . . . . . . . . . . . . . . . . 58 6.11 Induction motor . . . . . . . . . . . . . . . . . . . . . . . . . . 59 6.12 Power converters . . . . . . . . . . . . . . . . . . . . . . . . . 59 6.13 Can dynamics overcome the dissipation obstacle? . . . . . . . 61 6.14 Admissible dissipation . . . . . . . . . . . . . . . . . . . . . . 62 7 IDA–PBC 63 7.1 Matching perspective . . . . . . . . . . . . . . . . . . . . . . . 63 7.2 When is IDA an EB–PBC? . . . . . . . . . . . . . . . . . . . . 64 7.3 IDA PBC for (f, g, h) systems . . . . . . . . . . . . . . . . . . 64 7.4 IDA PBC: Swapping the damping . . . . . . . . . . . . . . . . 65 7.5 New passivity property . . . . . . . . . . . . . . . . . . . . . . 65 7.6 Energy–balancing with new supplied power . . . . . . . . . . . 66 7.7 Interpretation in EM systems . . . . . . . . . . . . . . . . . . 66 7.8 Universal stabilizing property of IDA–PBC . . . . . . . . . . . 67 7.9 Integral action . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 7.10 Damping injection with ”dirty derivatives” . . . . . . . . . . . 69 7.11 IDA PBC as a state–modulated source . . . . . . . . . . . . . 69 7.12 Example: Parallel RLC circuit . . . . . . . . . . . . . . . . . . 70 7.13 Interconnection and damping assignment . . . . . . . . . . . . 71 7.14 Solving the PDE . . . . . . . . . . . . . . . . . . . . . . . . . 71 8 Examples 73 8.1 Some applications: . . . . . . . . . . . . . . . . . . . . . . . . 73 8.2 Magnetic levitation system . . . . . . . . . . . . . . . . . . . . 74 8.3 Mechanical systems . . . . . . . . . . . . . . . . . . . . . . . . 77 8.4 Strongly coupled VTOL aircraft . . . . . . . . . . . . . . . . . 78 8.5 Boost converter . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3

8.6 PM Synchronous Motor . . . . . . . . . . . . . . . . . . . . . 82 8.7 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 new practical 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, control design.

HOW?

Propose models that capture main physical ingredients: energy, dissipation, interconnection Attain classical control objectives (stability, performance) as by–products

• f:

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–processing viewpoint. Objectives: keep some error signals small and reduce the effect of certain disturbance 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...utmost impractical!: 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–transformation devices interconnected to achieve desired behaviour. Consider physical systems, i.e., that satisfy energy–conservation. Control problem is to assign a desired energy function.

c

Σ

• +
• +

Σ

uc yc y u

Σ 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 of passivity ⇒ 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 quadratic in the increments.) Problems: Not an energy function and stabilization mechanism akin to systems inversion. ii) Interconnection and damping assignment(IDA): Storage function –now a bona fide energy function– obtained as a result of our choice of desired subsystems interconnections and damping. Applications of IDA–PBC: mass–balance syst., electrical machines, power syst., underwater vehicules, magnetic levitation, underactuated mechanical syst., 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, IEEE

Control 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 mechanical

systems via IDA, IEEE Trans. Automat. Contr.(Regular Paper), Vol. AC–47, No. 8, August 2002, 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 of

Euler–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 modeling
• f switching power converters, Automatica, Vol. 1999.
• H. Rodriguez, R. Ortega, G. Escobar and N. Barabanov: A Robustly Stable Output

Feedback 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 assignment

approach 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 of

the 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 rigid

body 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–Chain

Systems via Energy Balancing, ACC 2000, Chicago, June 2000.

• H. Rodriguez, R. Ortega and I. Mareels: A Novel Passivity–Based Controller for

an Active Magnetic Bearing Benchmark Experiment, ACC 2000, Chicago, June 2000.

• H. Rodriguez, R. Ortega and G. Escobar: Energy–shaping control of switched

power 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 in

Power 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 input signal 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 ∞ |f(t)|q < ∞ (1) The set L∞[0, ∞) = L∞ consist of all functions f : R+ → R which are bounded, i.e. sup

t∈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 defined as f(t)q ∞ |f(t)|qdt 1/q , q ∈ [1, ∞) (3) respectively, the norm L∞ is given by f(t)∞ sup

t∈[0,∞]

|f(t)| (4)

14

Remarks: · q ∞ 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 function fT : R+ → R is defined by fT (t) = f(t), 0 ≤ t ≤ T 0, t > T (5) For each q = 1, 2, . . . , ∞, the set Lqe consists of all functions f : R+ → R such that 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) ∈ Lq, but sin(t) ∈ Lqe, ∀q ∈ [1, ∞).

Examples:

f f f f f f1

2 4 5 6

L L L

2 1

• 3

15

f1(t) = 1; f2(t) =

1 1+t

f3(t) =

1 1+t 1+t1/4 t1/4

f4(t) = e−t f5(t) =

1 1+t2 1+t1/4 t1/4

f6(t) =

1 1+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, and an output y ∈ Lqe with y = Gu.

Examples of operators:

Truncation: PT : Lqe → Lqe, and T > 0 (PT u)(t) u(t), t ≤ T 0, t > T (6) Delay DT : Lqe → Lqe, and T > 0 (DT u)(t) u(t − T) (7) Convolution G : Lqe → Lqe (Gu)(t) t g(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 PT u1 = PT u2 ⇒ PT (Gu1) = PT (Gu2) (9) that is, (PT u)(t) depends only on the input past values. Equivalently, PT Gu = PT (GPT u), ∀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(DT u) = DT (Gu), ∀T ≥ 0, u ∈ Lqe (12) Thus, the operator G commutate with the delay operator DT , i.e., DT G = GDT . Memoryless: A mapping G : Lqe → Lqe is said to be memoryless if the output (Gu)(t0) depends only on t0.

Algebra:

1 2 G 1 + + G 2 + G 1 G G 2 G 1G 2 G (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) = 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) = 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 Gq sup

u ∈ Lqe u = 0

Guq uq (13)

• r equivalently,

Gq = sup

uq ≤ 1

Guq (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 Gq < ∞ (15) if the initial conditions are different from zero, G is Lq stable if Guq ≤ γquq + 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 y2, 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 ˆ H(jω)2, where ˆ H(jω) is the frequency response. For finite dimensional linear time invariant systems, ˆ H(jω)2 can be evaluated resolving 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 ˆ H(s) = C(sI − A)−1B be the transfer function and L = LT > 0 the unique solution of AL + LAT = −BBT Hence, the norm L2 of the frequency response is given by ˆ 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 eAT t

= AeAtBBT eAT t + eAtBBT eAT tAT (18) integrating, we get eAtBBT eAT t

• ∞

0 = A

∞ eAtBBT eAT tdt + ∞ eAtBBT eAT tAT dt since A is stable, limt→∞ eAt = 0, we can write −BBT = AL + LAT where the controllability grammian is defined by L ∞ eAtBBT eAtdt (19)

20

Finally, applying Parseval’s theorem to the impulse response h(t) = CeAtB, we find ˆ H(jω)2

2

= 2πh(t)2

2

= 2π ∞ CeAtBBT eAT tCT dt = 2πCLCT

• H∞–control

The objective is to minimize the energy of the output signal, that is, min y2, considering that the energy of the input signal satisfies u2 ≤ 1. Using the definition of the induced norm, the objective of the H∞–control is to minimize H2 The name of H∞–control comes from the fact that the induced norm L2 of a linear time invariant operator in the time domain, corresponds to the norm L∞ in the frequency domain.

|Re |I ||H|| = sup |H(jw)|

2 w Norm L2 of the convolution operator

Theorem 2. Let H be a stable convolution operator to the single input–single

• utput (SISO), then

H2 = sup

ω | ˆ

H(jω)| (20)

21

Proof. Define R supω | ˆ H(jω)|. Let y = Hu with u ∈ L2, then y2

2 =

∞ y2dt (21) According to Parseval’s theorem we have y2

2

= 1 2π ∞

−∞

|ˆ y(jω)|2dω = 1 2π ∞

−∞

| ˆ H(jω)ˆ u(jω)|2dω ≤ 1 2π sup

ω | ˆ

H(jω)|2 ∞

−∞

|ˆ u(jω)|2dω ≤ R2u2

2

Hence, y2

u2 ≤ R for all u ∈ L2, where, according to the definition of the induced

norm, we have H2 ≤ R. Now, we choose ω0 such that, | ˆ H(jω0)| = R − ǫ 2, ǫ > 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 respectively with yss | ˆ H(jω0)| sin(ω0t + ∠ ˆ H(jω0)) (23) since H is a stable operator, we have ytr2 M < ∞ Now, choose an integer N such that

M

• Nπ

ω0

< ǫ

2 and consider the truncated input

uT = PT u ∈ L2 with T > 0 then, HuT 2 = HPT u2 ≥ PT HPT u2 since H is causal (PT Hu = PT HPT u, ∀ u ∈ Lqe, and ∀ T > 0) HuT 2 ≥ PT Hu2 ≥ PT yss2 − PT ytr2 ≥ | ˆ H(jω0)| T sin2(ω0t + ∠ ˆ H(jω0))dt 1/2 − M

22

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

ω0 then

uT 2 = T sin2(ω0t)dt 1/2 = T sin2(ω0t + ∠ ˆ H(jω0))dt 1/2 =

• Nπ

ω0 , thus, HuT 2 ≥ | ˆ H(jω0)|

• Nπ

ω0 − M HuT 2 uT 2 ≥ | ˆ H(jω0)| − M

• Nπ

ω0

≥ R − ǫ In this way, we have constructed an input uT ∈ L2 for which the operator H gives an energy amplification of R − ǫ. However, since ǫ > 0 could be arbitrarily small, H2 ≥ R. Finally, since we showed that H2 ≤ R, we conclude that H2 = R = sup

w | ˆ

H(jω)|

• L1–control

In this approach the objective is to minimize not the system energy but the peak

• f the response.

Hypothesis: bounded inputs, i.e. u ∈ L∞. In other words, we search for min 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 1 e e 2 G G2 1

Definition 7 (well–posedness condition). A feedback system is “well–posed” if the operator Φ(G1, G2) : u1 u2

• →

e1 e2

• (25)

allows the operators algebra, and if for all u1, u2 ∈ Lqe the equations e1 = 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 ∈ Lqe. The following lemma gives a sufficient condition in order to know if a feedback systems 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 PT u1 = PT u2 ⇒ PT+δ(G1u1) = PT+δ(G1u2),

• r

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

• →

e1 e2

• (26)

is said to be Lq–stable if Φ(G1, G2)q < ∞ Example

1 2 u y u y 1 2 G G2 1 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 = 0 the system becomes unstable. In this case, the

• perator 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 the

• perator

Φ(G1, G2) : u1 u2

• →

e1 e2

• (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 invariant linear operators. Then e1 = u1 − G2e2 = u1 − G2u2 − G2G1e1

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

25

e1 = (I + G2G1)−1u1 − (I + G2G1)−1G2u2 (28) similarly, e2 = u2 + G1e1 = u2 + G1u1 − G1G2e2

• r

e2 = (I + G1G2)−1u2 + (I + G1G2)−1G1u1 (29) Using (28) and (29), we can write e1 e2

• =

(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 that G1 and G2 are causal and Lq–stable. Then, if G1qG2q < 1 (30) the feedback system is Lq–stable. Proof. Take u1, u2 ∈ Lq. Since G1 is stable from lemma 2. We have to prove that Φ1(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 = PT u1 − PT G2e2 = PT u1 − PT G2PT e2 taking the norm in both sides PT e1q ≤ PT u1q + PT G2PT e2q ≤ PT u1q + G2qPT e2q (31) similarly, using the causality of G1 we have e2 = u2 + G1e1 PT e2 = PT u2 + PT G1e1 PT e2q ≤ PT u2q + G1qPT e1q (32)

26

Combining (31) and (32) yields PT e1q ≤ PT u1q + G2qPT u2q + G2qG1qPT e1q then (1 − G2qG1q)PT e1q ≤ PT u1q + G2qPT u2q taking the limit when T → ∞, and using the fact that u1, u2 ∈ Lq we get (1 − G2qG1q)e1q ≤ u1q + G2qu2q ≤

• u1

u2

• q

+ G2q

• u1

u2

• q

e1q

• u1

u2

• q

≤ 1 + G2q 1 − G2qG1q Elsewhere, by definition we know that Φ1(G1, G2)q = sup

u1, u2 ∈ Lqe u1, u2 = 0

e1q

• u1

u2

• q

≤ 1 + G2q 1 − G2qG1q ≤ ∞ that is, Φ1(G1, G2) is Lq–stable. Lemma 2 allows to conclude the stability of Φ(G1, G2)

• Remarks:

The quantity G1qG2q is called Lq–gain. The small gain theorem gives only sufficient conditions, that is, if the loop gain is greater than 1, nothing can be concluded. Corollary 1. Let G1 and G2 be two linear stationary operators, interconnected in a feedback system with G22 = γ < ∞. Then

• 1. If G12 < 1

γ then Φ(G1, G2)2 < ∞

• 2. ∃ G10(s) : G102 = supω | ˆ

G10(jω)| = 1

γ such that Φ(G1, G2)2 = ∞

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

27

3.11 Lure’s problem

y 1 a b

y u 1 y

− +

e 1 (s)

1

H

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) ˆ H1(s) making a 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 = 0, (33) The objective is to find the conditions for a, b, ˆ H1(s) so that the closed–loop system is stable. To this end, we can use the small–gain theorem to deduce that the gain of H1 has to be less than 1/b (this agrees with the definition of the norm of a nonlinearity). 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 defines a feedback system between H′

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

problem.

+ 1 +

• 1

y H K K H H’ y H’

1 1 2

(H2

2 1 1 +

e u

• 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)−1q(H2 − K)q < 1 An optimal selection is given by K = 1

2(a + b).

Define r 1

2(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–gain theorem, if H′

1q < 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

K m y1

’= (H

• K) y1

2

h2 r

• r

y1

1

y

2

= H h2 b a

Nonlinearity H′

2y1

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

12H′ 22 < 1, since H′ 22 = r, we get

H′

12

= H1(1 + KH1)2 = sup

ω

ˆ H1(jω) 1 + k ˆ H1(jω) < 1 r = 2 b − a From the Nyquist diagram (figure (a)), ˆ H′

1(jω) has to be inside Da, i.e. 1/ ˆ

H′

1(jω) =

1/ ˆ H1(jω) + K must be outside Db (figure (b)). Since K > 0, 1/ ˆ H1(jω) should not touch the circle Dc (figure (c)). ˆ H1(jω) should not touch, neither encircle Dd (figure (d))

30

H1(jw) H1(jw) 1+ k b-a 2 b-a 2 H1(jw) H1(jw) 1 H1(jw) Db b-a b-a 2

• 2

II IRe Da

(a) (b)

• IRe

II 1 IRe II Dc

(c)

• b-a

2 b-a 2 IRe II Dd

(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 to be 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 that the transfer function ˆ H1(s) has ρ > 0 poles with negative real part and with no poles 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 of the next conditions is satisfied: Case 1. b ≥ a > 0 : The Nyquist diagram of ˆ H1(jω) do not touch the disk D(a, b), encircling it ρ times in a counterclockwise sense. Case 2. b > a = 0 : ˆ H1(jω) is internally stable and inf

ω∈R Re{ ˆ

H1(jω)} > −1 b Case 3. a < 0 < b : ˆ H1(jω) is internally stable and its Nyquist diagram is strictly inside the disk D(a, b). Case 4. 0 > b > a : The Nyquist diagram of − ˆ H1(jω) do not touch the disk D(a, b), encircling it ρ times counterclockwise sense.

• Proof. See M. Vidyasagar, “Nonlinear system analysis”. New Jersey: Prentice
• Hall. 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 = αx, 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, yT = T x(t)y(t)dt (34) Notice that x2 x, x1/2. Also, x, y ≤ x2y2 (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 power delivered 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 passive if 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, and T ≥ 0 such that Gu, uT ≥ 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, uT ≥ δoGuT + 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, uT ≥ δiuT + 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 secured whenever 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 (an abstraction 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 rate w(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 w(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 cannot exceed the external supplied energy T

0 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 than the input power.

3.17 Passivity and L2–gain

Definition 14. System P is said to be passive if it is dissipative with supply rate w(u, y) = uT y. System P is said to have L2–gain less than or equal to γ > 0 if it is dissipative with supply rate w(u, y) = γ2uT u − yT y.

34

Lemma 3. Let G : L2e → L2e be an output strictly passive operator, then G2 < ∞

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

u, Gu ≥ δoGu2

2 + β

Using Schwartz’s inequality yields u2y2 ≥ u, Gu ≥ δoy2

2 + β

⇒ 0 ≥ δo 2

• y2

2 + y2 2

• − u2y2 + β

⇔ 0 ≥ δo 2 y2

2 +

• δo

2 y2 − 1 2

• 2

δo u2 2 − 1 2δo u2

2 + β

⇒ δo 2 y2

2 ≤

1 2δo u2

2 − β

⇒ y2

2 ≤ 1

δo u2

2 − 2β

δo

• 3.18

Passivity and feedback interconnections 1 2 + +

• +

u y u y 1 2 1 e e 2 G G2 1

Theorem 6. Consider the feedback system showed in the figure. Suppose that there exist real constants δ1, ǫ1, δ2, ǫ2 such that u, GiuT ≥ ǫiu2

T2 + δiGiu2 T2,

∀T ≥ 0, ∀u ∈ L2e, i = 1, 2, then, the system is L2–stable if δ1 + ǫ2 > 0, δ2 + ǫ1 > 0 (39)

35

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

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

passive, then the feedback interconnection with input (e1, e2) and output (y1, y2) is

• passive. 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 following conditions 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) is controllable and the pair (A, C) is observable. Under this conditions, the operator G : u → y is passive if and only if ∃P = P T > 0 ∈ Rn×n such that AT P + PA ≤ 0 BT P = C Remarks: Let ˆ G(jω) be the transfer function of a linear time invariant system, Re{ ˆ G(jω)} ≥ 0 ⇔ ˆ G(jω) is passive and positive real. If ∃ ǫ > 0 such that G(s − ǫ) 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 through power conjugated port variables u ∈ Rm and y ∈ Rm (product has units of power). They satisfy the energy–balance equation (EBE) H[x(t)] − H[x(0)]

• stored energy

= t u⊤(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 minimum energy. Control introduced to operate the system around some non–zero equilibrium point, say x∗. Rate of convergence increased if we extract energy u = −Kdiy, with Kdi = K⊤

di > 0.

− t

0 u⊤(s)y(s)ds ≤ H[x(0)] < ∞

⇔ amount of energy that can be extracted from a passive system is bounded. d(x(t), t) is non–decreasing; typically t

0(·)2ds.

37

3.21 Examples: Electrical circuits

1

1

C 1 L

1

C L

2

R q C ϕL

φL is flux, qc charge, u voltage Model, with x [qC, φL]⊤, Σ :    ˙ x1 =

1 Lx2

˙ x2 = − 1

C x1 − R Lx2 + u

y =

1 Lx2

Energy H(x) = 1 2C x2

1 + 1

2Lx2

2

Satisfies (EBE) with d(x(t), t) = R t [ 1 Lx2(s)]2ds

38

3.22 Examples: Mechanical systems

q2 q1 u

u is torque, q1 ball position, q2 beam angle Model m2¨ q1 + m3 sin(q2) − m1q1 ˙ q2

2

= (1 + m1q2

1)¨

q2 + 2m1q1 ˙ q1 ˙ q2 + m3q1 cos(q2) = u Energy H = 1

2 ˙

q⊤M(q1) ˙ q + m3q1 sin(q2), where M(q1) = m2 1 + m1q2

1

• 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 m¨ θ = F − mg F = 1 2 ∂L ∂θ (θ)i2 Total energy: H = 1 2 λ2 L(θ) + m 2 ˙ θ2 + mgθ Output y = i, same dissipation. Notice, however, that H is not bounded from below! 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 1 C x2 + E ˙ x2 = u 1 Lx1 − 1 RC x2 Total energy: H(x) =

1 2Lx2 1 + 1 2C x2 2.

Attention: “Input” and output: E → x1

L !

41

4 Passivity–based control (PBC)

4.1 Feedback passivation

Definition 15. Let a dynamical system be described in state space as ˙ x = f(x) + g(x)u, (40) y = h(x), where u and y are the input system and output system respectively. If it is possible to find a feedback transformation u = α(x) + β(x)v, (41) such that the system ˙ x = 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 the transformation (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 we have for all x ∈ Rn and for all u ∈ Rm ˙ E = ∂E ∂xf(x) + ∂E ∂xg(x)u ≤ hT (x)u (42)

• r

∂E ∂xf(x) + ((∂E ∂xg)T (x) − h(x))T u ≤ 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 and

thus 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 vT (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 origin

is said to be weakly minimum phase

2See: C. Byrnes , A. Isidori and J.C. Willems, “Passivity, feedback equivalence, and the

global 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 deleterious for performance. Automatically ensures some robust stability (e.g., frictions and parasitic resis- tances). Passivity can be used for stabilization independently of energy–shaping, finding a “detectable” z = h(x).

4.4 Connections with L2–gain assignment

Fact: If the dissipation is such that dd(t) ≥ δ t |z(s)|2ds for some δ > 0, then the map v → z has L2 gain smaller than 1

δ.

Achieved choosing damping injection v = Kdiz + w, Kdi = KT

di ≥ δI > 0

Proof.

From the new (EBE) and above t vT (s)z(s)ds ≥ δ t |z(s)|2ds − Hd[x(0)] which is equivalent to 1 2 t |z(s)|2ds ≤ ≤ −1 2 t |z(s)|2ds + 1 δ t vT (s)z(s)ds + Hd[x(0)] δ ≤ −1 2 t |z(s) − 1 δ v(s)|2ds + 1 2δ2 t v⊤(s)z(s)ds + Hd[x(0)] δ

44

Thus t |z(s)|2ds ≤ 1 δ2 t |v(s)|2ds + 1 δ Hd[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 β⊤(x(s))h(x(s))

• uT (s)y(s)

ds − d(x(t), t) Indeed, if − t β⊤(x(s))h(x(s))ds = Ha[x(t)] + κ (∗) for some Ha(x), then u = β(x) + v ensures v → 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 systems with total energy H(q, p) = 1 2p⊤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 ˙ q ˙ p

• =

In −In −R ∂H

∂q ∂H ∂p

• +
• G(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 is assumed of the form F( ˙ q) = 1 2 ˙ qT R ˙ q. Passivity: ˙ H = ∂H ∂p T G(q)u − ∂H ∂p T R∂H ∂p = pT M−1(q)G(q)u − pT M−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 any function of q with β(q) = −∂Ha ∂q (q) and passive output y = ˙ q. Energy–shaping: The potential energy must have a minimum at the desired equilibrium q = qd. Choose the desired storage function as: Hd(q, p) = 1 2pT M−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

• − pT M−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 = −Kdi ˙ q, Kdi = KT

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

5.3 Example: Pendulum

Energy H = 1

2 ˙

q2 − mgl cos(q) ¨ q + mgl sin(q) = u y = ˙ q We want − t β[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 → ˙ q is Hd(q, ˙ q) = 1 2 ˙ q2 + kp 2 (q − qd)2 which has a minimum in (qd, 0). Asymptotic stability with v = −kdi ˙ q, kdi > 0

48

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

Fact: If the system Σ : ˙ x = f(x) + g(x)u y = h(x) satisfies the (EBE) with energy function H(x). Then5 ∂H ∂x

T

(x)f(x) ≤ 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 restrictive for electrical or electromechanical syst.: power = v⊤i. For LTI systems (with |A| = 0) u⊤

∗ y∗ = 0 iff Σ(0) = 0

where Σ(s) = C(sI − A)−1B.

50

5.8 Finite dissipation example

1

1

C 1 L

1

C L

2

R q C ϕL

State x [qC, φL]⊤, energy H(x) =

1 2C x2 1 + 1 2Lx2 2.

Remarks: Equil: x∗ = [x1∗, 0]T ⇒ zero extracted power! Only need to “shape” x1 Dynamic equations Σ :    ˙ x1 =

1 Lx2

˙ x2 = − 1

C x1 − R Lx2 + u

y =

1 Lx2

PDE ⇔ 1 Lx2 ∂Ha ∂x1 − 1 C x1 + R L x2 − β(x) ∂Ha ∂x2 = − 1 Lx2β(x) Look solution Ha = Ha(x1) ⇒ β(x1) = −∂Ha

∂x1 (x1).

Propose Ha(x1) = 1 2Ca x2

1 −

1 C + 1 Ca

• x1∗x1 + κ ⇒

with Ca tuning parameter. Recalling H(x) =

1 2C x2 1 + 1 2Lx2 2, this yields

Hd(x) = 1 2 1 C + 1 Ca

• (x1 − x1∗)2 + 1

2Lx2

2 + κ

has a minimum at x∗ for all gains

1 Ca > − 1 C .

51

Control law u = − 1 Ca x1 + 1 C + 1 Ca

• x1∗
• = − 1

Ca (x1 − x1∗) + u∗

• is an EB–PBC.

5.9 Infinite dissipation example

1 L

1

C

2

R

1

L

1

C q C

ϕL

Remarks: Only the dissipation has changed. x∗ = [Cu∗, L

Ru∗]⊤ ⇒ nonzero power (∀u∗ = 0) ⇒

lim

t→∞ |

t u(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

c

Σ

• +
• +

Σ

uc yc y u

Σ 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 [yT (s), yT

c (s)]

u(s) uc(s)

• ds = 0
• Simplest example: Classical feedback interconnection

u uc

• =

−1 1 y yc

• 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, u uc

• = ΣI

y yc

• +

v vc

• with (v, vc) external inputs.

Then, [vT , vT

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

energy–function H(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

Say Ω {(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 ⇔ d

dtC ≡ 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 = − 1

C x1 − R Lx2 + u

y =

1 Lx2

Propose C(x1, ζ) F(x1) − ζ Now, d dtC = 1 Lx2 ∂F ∂x1 (x1) − 1

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

Hc(ζ) = 1 2Ca ζ2 − 1 C + 1 Ca

• 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 = n and other ΣI. Stabilization is ensured for all Ca > −C, but the system Σc is passive only for positive 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 the search 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)u

y = 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) = 1 2 x2

1

C + 1 2 x2

2

L PCH model ˙ x = 1 −1 −R

• J−R

x1

C x2 L

• ∂H

∂x

+ 1

• g

u

6.9 Mechanical systems

Assuming linear friction, F = R ˙ q, where, R = R⊤ ≥ 0 State x = q p

• , p D(q) ˙

q momenta. Total energy: H(q, p) = 1

2pT D−1(q)p + U(q)

PCH model, ˙ x = I −I −R ∂H ∂x (x) + I

• u

y = I ∂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, one mechanical d.o.f. θ ∈ R, u ∈ Rm voltages.

58

Total energy: H = 1

2λT L−1(θ)λ + m 2 ˙

θ2 + U(θ) State x = (λ, θ, m ˙ θ),

∂H ∂x (x) = (i, −τ, ˙

θ), τ force, τL ∈ R load torque, M ∈ Rn×m defines actuated coordinates. PCH model ˙ x =   −R 1 −1   ∂H ∂x (x) +   Mu −τL   y = M ∂H ∂x1 (x) (= Mi)

6.11 Induction motor

We have n = 4, m = 2, λ = λsI λrI

• ,

M = I

• ,

L(θ) =

• Ls

LsreJθ LsreJθ Ls

• ,

R = RsI 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) = 1

2xT 1 L−1x1 + 1 2xT 2 C−1x2, where L = diag{Li}, C =

diag{Ci}.

59

Cuk PCH model (x ∈ R4): ˙ x =     −(1 − u) 1 − u u −u −1 1 − 1

R

    ∂H ∂x (x) +     E    

+ L + + E u=0 u=1 C + R

Boost PCH model (x ∈ R2): ˙ x = −u u

−1 R

• J(u)−R

∂H ∂x (x) + E

• 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 = gT

c (ζ)∂Hc ∂ζ (ζ)

Jc(ζ) = −JT

c (ζ), Rc(ζ) = RT c (ζ) ≥ 0, and standard feedback interconnection,

i.e., u = −yc, uc = y. Closed–loop is still PCH: ˙ x ˙ ζ

• =

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 the Hamiltonian 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) = (DO) Rc(ζ) = ∂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 where there 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 Casimir functions 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 min Hd(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, then ˙ Ha = −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 :=

= 1 2zT ([J(x) − R(x)]−1 + [J(x) − R(x)]−T )z = 1 2zT [J(x) − R(x)]−1([J(x) − R(x)] + [J(x) − R(x)]T )[J(x) − R(x)]−T z = 1 2 ˜ zT (J(x) − R(x) + [J(x) − R(x)]T )˜ z = −˜ zT R(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 ˜ yT (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)]−1 ˙

x = ∇xH(x) + [J(x) − R(x)]−1g(x)u. Premultiplying by ˙ xT ˙ H(x) = ˙ xT ∇xH(x) = ˙ xT [J(x) − R(x)]−1 ˙ x − ˙ xT [J(x) − R(x)]−1g(x)u ≤ − ˙ xT [J(x) − R(x)]−1g(x)u = ˜ yT u,

• 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 uT (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 mechanical displacement 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, with

L(θ) = LT (θ) > 0 the inductance matrix. The natural power port variables u and y = B⊤L−1(θ)ψ. Now, ˜ yT u = uT BT R−1

e

˙ ψ where R−1

e Bu are the current sources obtained from the Norton equivalent

• f the Thevenin representation of the classical passivity property, with ˙

ψ the associated inductor voltages.

66

+ − + − R1 u1 uk Rk Electro- System

✻ ✻

R1 u1 R1 uk Rk Rk 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–PBC theorem. ⇔ IDA–PBC methodology generates all asymptotically stabilizing controllers for PCH 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) = −JT

d (x), Rd(x) = RT d (x) ≥ 0

such that f(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 ∂x fT (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 + 1 2vT K−1

I v

The closed–loop ˙ x ˙ v

• =

Jd − Rd gKI −KIgT ∂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 back the filtered signal, preserving stability. Of particular interest to obviate velocity measurements in mechanical systems where the passive output is ˙ q. Proposition 6. If u = ues + KdigT ∂Hd ∂x is asymptotically stable, then u = ues + udi, where ˙ udi = −1 τ udi − Kdi τ gT ∂Hd ∂x with τ > 0, also ensures convergence of x∗. Proof. With the energy W(x, udi) Hd + τ 2Kdi u2

di

we have ˙ x ˙ udi

• =
• Jd

Kdi τ g

−Kdi

τ gT

−Kdi

τ 2

∂W

∂x ∂W ∂udi

• yielding

˙ W = −uT

diKdi−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)

• =
• −β(x)

β(x) y(s) yc(s)

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

˙ x ˙ ζ

• =

J(x) − R(x) −g(x)β(x) βT (x)gT (x) ∂H

∂x (x) ∂Hc ∂ζ (ζ)

• 7.12

Example: Parallel RLC circuit

Total energy: H(x) = 1

2 x2

1

C + 1 2 x2

2

L

PCH model J = 1 −1

• , R =

1/R

• , g =

1

• (PDE) becomes

− 1 R ∂Ha ∂x1 (x) + ∂Ha ∂x2 (x) = −∂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 C

L R

• u∗

x∗

= arg min Hd(x) ⇐

• ∂Hd

∂x (x∗)

= 0 (EC)

∂2Hd ∂x2 (x∗)

> 0 (HC) (EC) equivalent ∂Ha ∂x (x∗) = R 1 ∂Φ ∂z (z∗) ≡ −∂H ∂x (x∗) = 1

1 R

• u∗

where z = Rx1 + x2. Thus, ∂Φ

∂z (z∗) = − 1 Ru∗.

Check (HC) ∂2Ha ∂x2 = ∂2Φ ∂z2 R2 R R 1

• 70

Let Φ(z) = Kp 2 (z − z∗)2 − 1 Ru∗z which yields Hd(x) = (x − x∗)T 1

C + R2Kp

RKp RKp

1 L + Kp

• (x − x∗) + κ

(HC) satisfied for Kp >

−1 (L+CR2)

Control u = −Kp[R(x1 − x1∗) + x2 − x2∗] + u∗. Procedure: From (PDE) results a family of “admissible” Ha(x), β(x). Choose

• ne that shapes the energy.

7.13 Interconnection and damping assignment

We aim at ˙ x = [Jd(x) − Rd(x)]∂Hd ∂x (x) for some new Jd(x) = −JT

d (x), Rd(x) = RT d (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′) where Ja(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) =

k 1−θ.

PCH model. State: x = [λ, θ, m ˙ θ]T Hamiltonian: H(x) =

1 2k(1 − x2)x2 1 + 1 2mx32 + mgx2. Matrices:

J =   1 −1   , R =   R   , g =   1   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)

=

∂Ha ∂x3 (x)

=

74

⇒ Ha = Ha(x1) can only depend on x1. The Hessian ∂2Hd ∂x2 (x) =  

(1−x2) k

+ ∂2Ha

∂x2 (x1)

−x1

k

−x1

k 1 m

  which is sign indefinite for all Ha(x1). Source of the problem: lack of effective coupling between the electrical and the mechanical subsystem.

IDA–PBC

Enforce a coupling between the flux x1 and the velocity x3 ⇒ Jd =   −α 1 α −1   (PDE’) ∂Ha ∂x3 (x) = −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 + 1

2kx2

1(x2 − 1) + Φ(x2 + 1

αx1), Suitable choice for Φ(x2 + 1 αx1) = mg[−(˜ x2 + 1 α ˜ x1) + b 2(˜ x2 + 1 α ˜ x1)2] Control law u = R k (1 − x2)x1

• Ri

− Kp( 1 α ˜ x1 + ˜ x2) − α mx3

• PD

− R α ( 1 2kx2

1 − mg)

• undesirable

75

To remove the high order term: shuffle the damping Ra =   −R Ra   This yields u = R

k (1 − x2)x1 − Kp( 1 α ˜

x1 + ˜ x2) − α

m + Kp

• x3

Comparison of various schemes

uFL =

• k

2F mvFL(θ, ˙ θ, F) + R(1 − θ)

• 2F

k uPB =

• k

2Fd mvFL(θ, ˙ θ, F) + R(1 − θ)

• 2Fd

k uIB = uPB − β √ 2mk [˜ θ, ˙ ˜ θ, t ˜ θ(s)ds]PB( √ F + √ F d) uIDA = −Kp( 1 α ˜ λ + ˜ θ) − α m + Kp

• ˙

θ + R(1 − θ)

• 2F

k where FL=feedback linearization, PB=standard PBC, IB=integrator backstepping, and F = 1 2λ2, Fd = m[¨ θ∗ − k2 bp p + a ˜ θ − k1˜ θ − k0 t ˜ θ(s)ds] vFL = θ(3)

∗

− k2[( 1 mF − g) − ¨ θ∗] − k1 ˙ ˜ θ − k0˜ θ

Experimental results

Linearized model based controller

−0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0.1 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 1.8 0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 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 1.8 0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 time [s]

IBC with integral term

−0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 time [s]

8.3 Mechanical systems

To stabilize some underactuated mechanical devices it is necessary to modify the total energy function. In open loop H(q, p) = 1 2p⊤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 ˙ q ˙ p

• =

In −In ∇qH ∇pH

• +
• G(q)
• u

Control u ∈ Rm, and assume rank(G) = m < n. Convenient to decompose u = ues(q, p) + udi(q, p) Target Dynamics: Desired (closed loop) energy function Hd(q, p) = 1 2p⊤M−1

d (q)p + Vd(q)

where Md = M⊤

d > 0 and Vd(q) is s.t. q∗ = arg min Vd(q). Thus,

˙ q ˙ p

• = [Jd(q, p) − Rd(q, p)]

∇qHd ∇pHd

• ,

where Jd = −JT

d =

• M−1Md

−MdM−1 J2(q, p)

• ,

Rd = RT

d =

GKvGT

• ≥ 0

77

8.4 Strongly coupled VTOL aircraft

Horizontal displacement: badly tuned ; well tuned ♥ Well tuned aggressive maneuver ♥ Model, ǫ = 0, possibly large ¨ x = − sin θv1 + ǫ cos θv2 ¨ y = cos θv1 + ǫ 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 ǫ sin θe3 where G(θ) =   1 1

1 ǫ cos θ 1 ǫ sin θ

  , e3   1  

Characterizing assignable energy functions

Potential energy Vd(q) = − g

ρ+

¸ Φ(η1(x, θ), η2(y, θ)) where Φ(η1, η2) = 1 2 η1 − η1(x∗, 0) η2 − η2(y∗, 0) T P η1 − η1(x∗, 0) η2 − η2(y∗, 0)

• ,

P = P T > 0 Kinetic energy Md(θ) =

78

 

k3 2 [tan2 θ − (1 + 2k1ǫ k3 ) log(1 + tan2 θ)] + k4

(k1ǫ + k3)θ − k3 tan θ k1 cos θ (k1ǫ + k3)θ − k3 tan θ

k3 2 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 > ˆ k4(k3, k1), k1

ǫ >

k2 > k1

2ǫ

Horizontal displacement: badly tuned ; well tuned ♥ Well tuned aggressive maneuver ♥

8.5 Boost converter

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

>0

˙ x = −u u

−1 R

• J(u)

∂H ∂x (x) + E

• Control objective: regulate 1

C x2 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 nonlinear

• utput feedback.

Proposition 7. For all R > 0 the IDA–PBC u = u∗ x2 x2∗ α , 0 < α < 1 yields (i) x∗ = ( L

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

Hd(x) = 1 2Lx2

1 + 1

2C x2

2 + κ1x2(1−α) 2

− (κ2 + κ3x1)x1−α

2

79

(ii) Domain of attraction: Ξα

△

= {x|x ∈ R2

>0

and Hd(x) ≤ Hd(0, x2∗)} is such 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 = diag{Ra, − 1

R} ⇒ Rd = diag{Ra, 0}.

Integrability Key PDE10 K ∂Ha ∂x (x) = 1 β(x2)

• − 1

RC x2

− 1

LRax1 − E + Ra RC x2 β(x2)

• PDE solvable ⇔ ∂K2

∂x1 (x) = ∂K1 ∂x2 (x) ⇔

dβ dx2 (x2) = α x2 β(x2) where α

△

= 1 − RaRC

L

. Thus, u = c1xα

2

Equilibrium assignment: c1 such that ∂Hd ∂x (x∗) = ∂H ∂x (x∗) + ∂Ha ∂x (x∗) = 0 This yields c1 = u∗

xα

2∗ .

Hessian condition ∂2Hd ∂x2 (x∗) =

• 1

L

− Ra

u∗L

− Ra

u∗L 1 C + (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 M ˙ z + J(u)z + Rz = g where z ∈ R2, M := diag{L, C}, g = [E, 0]T . An implicit definition of the controller is derived from a copy of the system with additional damping as M ˙ zd + J(u)zd + Rzd = g + Rdi˜ z where Rdi := diag{R1, 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 value V 2

d

RE . This leads to

C ˙ z2d = − 1 RL z2d + V 2

d

RLEz2d (E + R1˜ z1) u = 1 z2d (E + R1˜ z1) 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 enforcement

• f 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 Ld did dt = −Rsid + ωLqiq + vd Lq diq dt = −Rsiq − ωLdid − ωΦ + vq J dω dt = P · ((Ld − Lq)idiq + Φiq) − τl ω is angular velocity, vd, vq, id, iq are voltages and currents. P is the number of pole pairs, Ld and Lq are stator inductances, Rs is stator winding resistance, τl is a constant unknown load torque, and Φ and J are the dq back emf constant and the moment of inertia. Energy function H(x) = 1

2

• Ldi2

d + Lqi2 q + J P ω2

= 1

2x⊤D−1x

x = D   id iq ω   , D   Ld Lq

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 − τl

P ]

R = diag{Rs, Rs, 0} and J(x) =   x2 −(x1 + Φ) −x2 x1 + Φ   Desired equilibrium (“maximum torque per ampere” principle) x∗ = (x1∗, x2∗, x3∗) = (0, Lqτl

PΦ , J P ω∗)

Choice of interconnection

Isotropic (smooth rotor) synchronous motors are more efficient that indented rotor motors

82

Open loop J(x) =   x2 −(x1 + Φ) −x2 x1 + Φ   Φ is the dq back emf constant. Virtual behavior in closed–loop Jd(x) =   L0x3 −L0x3 −Φ Φ   L0 free parameter, represents stator 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 P τl Solution is Ha(x) = τl P arctan(x1 + Φ x2 ) + F(x2

2 + 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 + α2

2 ˜

x2

3, where ˜

(·) (·) − (·)∗, and α2 > 0. This choice yields K3 = −ω∗ + α2˜ x3. The equilibrium assignment and Lyapunov stability conditions reduce to f(¯ z) = − 1 2Lq ¯ z ¯ z + Φ2 , ∂f(z) ∂z

• z=¯

z

> 1 4Lq ¯ z − Φ2 (¯ z + Φ2)2

83

Propose f(z) = − 1

2Lq ¯ z z+Φ2 , which yields

∂Ha ∂x1 = τl/P x2

2 + (x1 + Φ)2

• x2 − Lqτl

PΦ2 (x1 + Φ)

• ∂Ha

∂x2 = − τl/P x2

2 + (x1 + Φ)2

• (x1 + Φ) + Lqτl

PΦ2 x2

• ∂Ha

∂x3 = −ω∗ + α2˜ x3 x∗ is asymptotically stable, but the initial conditions {(x1(0) + Φ)2 + x2(0)2 ≥ ǫ > 0}, and the load torque is different from zero. Load torque is unknown dˆ ω dt = P J

• γx1 + Φ

Lq

• x2 − l1(ˆ

ω − ω) − 1 J ˆ τl dˆ τl dt = l2(ˆ ω − ω) with γ

1 Lq − 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) =  

LP J x3

−LP

J x3

−Φ Φ   Propose Jd(x) =   L0x3 −L0x3 −Φ Φ   where L0 is a parameter to be defined.

84

Proposition 8. The control law vd vq

• =
• (L0

Lq − P J )x2x3 − Rsα1x1

−( L0

Ld − P J + L0α1)x1x3 + Φ( P J x3∗ − α2˜

x3)

• +

+ −Rs L0x3 −L0x3 −Rs

γ 2Φ(x2 2 − x2 2∗) γ Φx1x2 − 1 Lq x2∗

• where L0 is arbitrary α2 + P

J > 0, (α1 + 1 Ld ) 1 Lq > γ2 Φ2 x2 2∗, ensures x∗ is GAS with

energy–Lyapunov function Hd = 1 2x⊤D−1x + γ 2Φx1(x2

2 − x2 2∗) − 1

Lq x2∗x2 + α1 2 x2

1

−P J x3∗˜ x3 + α2 2 ˜ x2

3

• Connections with Current Practice

To recover a linear dynamics in the electrical subsystem it is common to cancel the 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∗ =

τl PΦ.

Asymptotically table if the load torque is known. In practice current reference generated 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 currents

are 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 of gravity of the body coincides with the center of buoyancy. Kirchhoff equations (Leonard, Automatica’97) ˙ Π1 =

• 1

J3 − 1 J2

• Π2Π3 +
• 1

M3 − 1 M2

• P2P3 + T1

˙ Π2 =

• 1

J1 − 1 J3

• Π3Π1 +
• 1

M1 − 1 M3

• P3P1 + T2

˙ Π3 =

• 1

J2 − 1 J1

• Π1Π2 +
• 1

M2 − 1 M1

• P1P2

˙ P1 =

P2Π3 J3

− P3Π2

J2

˙ P2 =

P3Π1 J1

− P1Π3

J3

˙ P3 =

P1Π2 J2

− P2Π1

J1

+ F3 Π1, Π2 and Π3 (P1, P2 and P3) angular (linear) momentum, J1 > 0, J2 > 0 and J3 > 0 principal moments of inertia and M1 > 0, M2 > 0 and M3 > 0 terms 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/reverse col(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 2

1 + 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 motion equilibrium 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 a damping 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, adapted for 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 physical

• considerations. 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, technique

for stabilization of general nonlinear systems Power shaping For a class of (nonlinear) RLC circuits, it is possible to formulate the 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. Some preliminary results on control by interconnection

88