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 ortega,garcia@lss.supelec.fr october 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 L q –spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.3 L q –spaces are normed spaces . . . . . . . . . . . . . . . . . . . 14 3.4 Extended L q –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 L 2 –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 L 2 –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 of: 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
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