Analysis and Control of Multi-Robot Systems Elements of Passivity - - PowerPoint PPT Presentation

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Analysis and Control of Multi-Robot Systems Elements of Passivity - - PowerPoint PPT Presentation

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Elective in Robotics 2014/2015 Analysis and Control of Multi-Robot Systems Elements of Passivity Theory Dr. Paolo Robuffo Giordano CNRS, Irisa/Inria ! Rennes, France Interconnected Systems A way to look at interconnected systems:

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Elective in Robotics 2014/2015

Analysis and Control

  • f Multi-Robot Systems

Elements of Passivity Theory

  • Dr. Paolo Robuffo Giordano

CNRS, Irisa/Inria! Rennes, France

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  • A way to look at interconnected systems:
  • It is often very useful to consider “Input/Output”

characterizations of dynamical systems

  • e.g., passivity theory (not the only possibility!)
  • What if is made of a “network” of simpler systems?
  • Can we infer global features out of:
  • The network (graph) topology
  • The individual I/O properties of the single

subsystems?

  • Is this helpful for modeling and control
  • f multi-robot systems?

Interconnected Systems

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Σ

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Robuffo Giordano P ., Multi-Robot Systems: Elements of Passivity Theory

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  • A very useful tool: port-based network modeling
  • aka: Port-Hamiltonian Modeling, Generalized Bond-Graphs, etc.
  • General framework that captures
  • I/O - external - overall behavior
  • Internal interconnection (graph) of simpler subsystems
  • We will see how to embed within this machinery some of the graph-related topics

discussed so far

  • Mainly, graph theory, consensus/agreement protocols, distributed sensing

Interconnected Systems

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Robuffo Giordano P ., Multi-Robot Systems: Elements of Passivity Theory

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  • What is passivity?
  • Intuitively: something that does not produce internal

energy

  • Stems from circuit theory
  • Describes input/output (I/O) behaviors
  • Seamlessly applies to linear and nonlinear systems

Introduction to Passivity

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Robuffo Giordano P ., Multi-Robot Systems: Elements of Passivity Theory

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  • Passivity-like concepts are common to many scientific areas
  • Mathematics
  • Physics
  • Electronics
  • Control
  • Basic idea: most physical systems have common I/O characteristics dictated by
  • Energy conservation
  • Energy transportation
  • Energy dissipation
  • Energy plays a fundamental role
  • Common unifying language across all physical domains

Introduction to Passivity

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Robuffo Giordano P ., Multi-Robot Systems: Elements of Passivity Theory

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  • Consider this simple mechanical system with dynamics

and energy

  • How is the “energy flowing” within the system?
  • Integrating back, we get

Introduction to Passivity

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Input/Output Mechanical power Internal dissipated power Initial stored energy

Robuffo Giordano P ., Multi-Robot Systems: Elements of Passivity Theory

f

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  • if

no I/O “energy flow”, but still an internal dynamics

  • if

and since we have

  • The total extractable energy is limited by the initial stored energy

Introduction to Passivity

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Robuffo Giordano P ., Multi-Robot Systems: Elements of Passivity Theory

f

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  • Passivity: a property of a physical system (but also, more in general, of a linear/

nonlinear dynamical system)

  • Based on the concept of “energy”
  • Describes the energy flow (power) through the system
  • It is an I/O characterization
  • Usually, passivity is a robust property (e.g., w.r.t. parametric variations)
  • It is (of course) related to classical Lyapunov stability concepts
  • Proper compositions of passive systems are passive -> very useful property (later)

Introduction to Passivity

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Robuffo Giordano P ., Multi-Robot Systems: Elements of Passivity Theory

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  • A first definition of passivity can be given for memoriless (static) functions
  • The function is said to be passive if
  • “Power” flowing into the system is never negative
  • The system does not produce energy (can only absorb and dissipate)
  • Example: the familiar electrical resistance , the power is
  • For the scalar case, passivity imposes a constraint on the graph of
  • It must lie in the first and third quadrant
  • But we are interested in MIMO dynamical systems

Passivity: formal definitions

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y = Ru, R > 0 uT y = Ru2 ≥ 0 y = h(u) y = h(u)

Robuffo Giordano P ., Multi-Robot Systems: Elements of Passivity Theory

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  • Consider a generic nonlinear system (affine in the input)

with state/input/output

  • The system is dissipative if there exists a continuous (differentiable) lower bounded

function of the state (storage function) and a function of the input/output pair (supply rate) such that (~equivalently)

Passivity: formal definitions

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Robuffo Giordano P ., Multi-Robot Systems: Elements of Passivity Theory

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  • When the supply rate is

the system is said passive (w.r.t. the supply rate and with storage function )

  • In particular,
  • lossless if and
  • input strictly passive (ISP) if
  • output strictly passive (OSP)
  • very strictly passive (VSP)
  • If there exists a positive definite function such that

then the system is said strictly passive, and is called dissipation rate

Passivity: formal definitions

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Robuffo Giordano P ., Multi-Robot Systems: Elements of Passivity Theory

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  • Some (physical) interpretation:
  • The storage function represents the internal stored energy
  • The supply rate is the power (energy flow) exchanged with the external world
  • The basic passivity condition can be interpreted as

Current energy is at most equal to the initial energy + supplied energy from outside equivalent to “no internal generation of energy”

Passivity: interpretation

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Robuffo Giordano P ., Multi-Robot Systems: Elements of Passivity Theory

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  • Exctractable energy is bounded from below
  • One cannot extract an infinite amount of energy from a passive system
  • The maximum amount of extractable energy (net of the energy supplied from
  • utside) is the initial stored energy (recall the example before)
  • This yields an (additional) equivalent passivity condition: a system is passive if
  • This alternative definition is sometimes useful in proofs and general considerations
  • n the system at hand
  • No formal need of a storage function

Passivity: interpretation

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Robuffo Giordano P ., Multi-Robot Systems: Elements of Passivity Theory

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  • Consider again the initial example:
  • Take as the input and as the output
  • Take the total energy as

storage function

  • Is the system passive w.r.t. the input/output pair ?
  • By differentiating , we get
  • Therefore, the system is passive, in particular output strictly passive

Passivity: review of the example

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Robuffo Giordano P ., Multi-Robot Systems: Elements of Passivity Theory

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  • The integrator

is passive (lossless) w.r.t. the storage function since

  • Similarly, the integrator with nonlinear output (■)

with is passive (lossless) w.r.t. the storage function

  • This fact will be heavily exploited later on as (■) will constitute the fundamental

energy storage element with associated energy function

Passivity: another example

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Robuffo Giordano P ., Multi-Robot Systems: Elements of Passivity Theory

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  • Passivity, so far:
  • I/O characterization
  • Nice energetic interpretation
  • Used to describe how the “energy flows” within a system
  • Several equivalent definitions
  • But what is it good for? How can we use it?
  • Key features:
  • Strong link to Lyapunov stability
  • Proper (and useful) interconnections of passive systems are passive (modularity)
  • A system can be made passive
  • By a choice of the “right output”
  • By a feedback action
  • A passive system is “easily stabilizable” from the output
  • And... many real-world systems are passive

Passivity: what is it good for?

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Robuffo Giordano P ., Multi-Robot Systems: Elements of Passivity Theory

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  • Short summary about Lyapunov stability
  • Given a system (■)

the equilibrium is

  • Stable if
  • Unstable if it is not stable
  • Asymptotically stable if stable and
  • The Lyapunov Theorems allow to establish (asympt.) stability of (■) without

explicitly computing the solution of (■)

  • Pivotal is the concept of Lyapunov function, i.e., a positive definite function

Passivity vs. Lyapunov

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Robuffo Giordano P ., Multi-Robot Systems: Elements of Passivity Theory

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  • lf there exists a such that
  • then the system is stable
  • then the system is (locally) asympt. stable (LAS)
  • If is radially unbounded, i.e., and , and it still

holds ,, then the system is globally asympt. Stable (GAS)

  • Also in the case 1), let
  • LaSalle Th.:The system will converge towards , the largest invariant set in
  • If , i.e., only can stay identically in , then the system is

LAS (GAS)

Passivity vs. Lyapunov

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Robuffo Giordano P ., Multi-Robot Systems: Elements of Passivity Theory

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  • Let us go back to the passivity conditions
  • System dynamics and there exists a storage function

such that

  • Assume that , then is a Lyapunov candidate around and
  • If then , i.e., the system is stable
  • If then , i.e., the zero-dynamics of the system is stable
  • The system can be easily stabilized by a static output feedback

for instance

Passivity vs. Lyapunov

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Robuffo Giordano P ., Multi-Robot Systems: Elements of Passivity Theory

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  • By setting we obtain
  • Non increasing storage function bounded state

trajectories

  • Convergence to a manifold (the set of before)
  • Remember LaSalle: if the system is zero-state observable

then provides asympt. stability (LAS)

  • Global results (GAS) if the storage function is radially unbounded

Passivity: output feedback

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Robuffo Giordano P ., Multi-Robot Systems: Elements of Passivity Theory

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  • Passivity of a system is w.r.t. an input/output pair
  • One can also look for a good output w.r.t. which the system is passive
  • Consider the state evolution and assume we can find a

such that , i.e., a stable free evolution ( )

  • Then, the system is passive w.r.t. the output
  • The feedback makes the system LAS (GAS)

Passivity: output feedback

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Robuffo Giordano P ., Multi-Robot Systems: Elements of Passivity Theory

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  • Review of the example
  • Rewrite in canonical state-space form

with

  • Take the storage function (radially unbounded)
  • Passivity condition

Passivity: output feedback

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Robuffo Giordano P ., Multi-Robot Systems: Elements of Passivity Theory

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  • by setting we obtain , i.e.,
  • the state trajectories are bounded
  • the output (the velocity) will converge to 0:
  • Let us check the zero-state observability (i.e., LaSalle)
  • Zeroing the output means that
  • The dynamics restricted to this set become
  • Therefore, the only possible solution is
  • Or, in other words, zeroing the output implies zeroing the complete state
  • One can still feedback the output . This results in faster convergence

Passivity: output feedback

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Robuffo Giordano P ., Multi-Robot Systems: Elements of Passivity Theory

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  • Another example: consider the system without output
  • We want to look for an output that makes the system passive w.r.t. the

pair

  • Let us consider the (radially unbounded) Storage function
  • With it is (stable system). Remember slide 21...
  • We can take and stabilize the system by
  • Is the system zero-state observable? (exercise)

Passivity: output feedback

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Robuffo Giordano P ., Multi-Robot Systems: Elements of Passivity Theory

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  • Consider a manipulator arm with joint configuration
  • Its dynamical model (Euler-Lagrange form) takes the form
  • With these fundamental properties
  • Positive definite Inertia matrix
  • Skew-symmetry of
  • Apply a feedback pre-compensation (gravity compensation)
  • Prove passivity of the new system with storage function

w.r.t. the input/output pair

  • Prove zero-state observability

Robot Manipulators

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Inertia matrix Centrifugal/Coriolis terms Gravity vector Actuation torques (inputs)

˙ M(q) − 2C(q, ˙ q)

Robuffo Giordano P ., Multi-Robot Systems: Elements of Passivity Theory

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  • Fundamental property: proper interconnections of passive systems are again passive
  • This property opens the door to modularity (network modeling):
  • Identify subcomponents
  • Make them passive
  • Interconnect them in a “proper way”
  • The result will be a passive system (stable, etc.)
  • We will address two possible interconnections: parallel and feedback

Passivity: interconnection

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Robuffo Giordano P ., Multi-Robot Systems: Elements of Passivity Theory

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  • Given two passive systems with proper I/O dimensions and storage functions

and

  • For the parallel interconnection, set and
  • Let and be the storage function
  • Then
  • The new system is passive w.r.t. the pair

Passivity: interconnection

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Robuffo Giordano P ., Multi-Robot Systems: Elements of Passivity Theory

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  • Take
  • Prove that the interconnected system is passive with storage function

w.r.t. the (composed) input/output pair

Passivity: interconnection

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New (optional) inputs

Robuffo Giordano P ., Multi-Robot Systems: Elements of Passivity Theory

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  • Note the particular structure of the feedback interconnection
  • The coupling matrix is skew-symmetric
  • This is a fundamental property that allows to retain passivity of the composed system
  • We will see later that this is an example of a power-preserving interconnection

Passivity: interconnection

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Robuffo Giordano P ., Multi-Robot Systems: Elements of Passivity Theory

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  • Assume is a passive system with storage function w.r.t. the pair
  • Let be a (possibly state-dependent) matrix, and let and
  • Prove that passivity is preserved by a pre-multiplication of the input by and a

post-multiplication of the output by

Passivity: pre-post multiplication

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u ˜ u y ˜ y

M(x)

M T (x)

V (x) u = M(x)˜ u ˜ y = M T (x)y M(x)

M T (x)

M(x)

Robuffo Giordano P ., Multi-Robot Systems: Elements of Passivity Theory

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  • Passivity is a I/O property of a dynamical system, intuitively equivalent to
  • No internal production of “energy”
  • Bounded extractable “energy”
  • Seamlessly applies to linear and nonlinear systems
  • Linked to Lyapunov stability
  • Stability of the origin in free evolution (asympt. with some observability

properties)

  • Stable zero-dynamics
  • Easily stabilizable by static output feedback
  • Can be enforced by
  • Finding the “correct” output
  • A proper feedback (passifying) action
  • It is a modular property: proper interconnections of passive systems are passive

Summary

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  • Let us revisit the consensus protocol under the “passivity light”
  • Take a passive (lossless) system: single integrator
  • Consider a static function
  • This is a passive static function
  • Interconnect these two passive systems by means of a “feedback interconnection”
  • The resulting system will necessarily be passive….
  • And is nothing but the consensus closed-loop dynamics

Review of consensus protocol

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y2(u2) = Lu2 uT

2 y2 = uT 2 Lu2 ≥ 0

⇢ u2 = y1 u1 = −y2 ˙ x = −Lx Σ : ⇢ ˙ x = u1 y1 = x

Robuffo Giordano P ., Multi-Robot Systems: Elements of Passivity Theory

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  • As a block scheme
  • Another point of view: recall that . Then, the consensus protocol is just
  • Since the single integrator is passive and a pre-/post-multiplication preserves

passivity, we are just closing the loop of a passive with a negative unitary output feedback

Review of consensus protocol

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L = EET L

Z Z

ET E

Robuffo Giordano P ., Multi-Robot Systems: Elements of Passivity Theory