Modeling and control of complex systems: Some new ...

Modeling and control of complex systems: Some new challenges to deal with

Agenda • Behavioral Approach to systems and control. • Euler-Lagrange equations for mech.systems. • Bioengineering systems: Orthosis, EEG as dynamical systems/stochastic analysis.

BEHAVIORAL APPROACH TO SYSTEMS AND CONTROL (BA)

Behavioral Approach (Willems, NL) • Motivation: • Develop a suitable mathematical framework for discussing dynamical systems.

(And reflect on paradigms…)

BA: Paradigms in modeling and control • Typical paradigms in modeling and control of dynamical systems are: • Input/output approach (i/o). • Input/state/output app. (i/s/o).

• (Always…?)

BA: Cause/effect apps. Difficulties • (1) How to choose who is the input and who is the output…?

• Input:V, Output:I; Inputs: V,I; Outputs: V,I,…?

• (2) Dealing with interconections is awkward.

• Inputs: p1, p2; Outputs: f1, f2. • Interconnection: p1’=p2’’; f1’+f2’’=0.

?

• (3) I/O framework can not deal with initial conditions… • (Transfer/matrix functions assume IC=0).

• (Otherwise we would obtain different outputs for the same input)…

Impulse response matrix

• (4) I/O framework (i.e. block diagrams) imposes a rigid structure not always suitable.

?

• (5) Simulating a simple system implies a drastic change in its structure…

BA: I/S/O from I/O • (6) State construction: x’=Ax+Bu, x’=f(x,u); x0=x(0). • The state is a ‘construct‘… • But from what is the state constructed? • From the i/o behavior… • A ‘parametrized’ map… • Parametrized by the initial state ... • (The state should be constructed from the system model).

• I/O assignement should be deduced from a dynamical model. • Interconnection rather than input selection, is the basic mechanism by which a system interacts with its environment.

BA: Open physical systems • Therefore, we need a more general notion of ‘system', of `dynamical model'.

Interconecting systems: Plant+controller

BA: A more general framework • The behavior =All trajectories of the system variables… • Which, according to the mathematical model… • Are possible.

BA: Dynamical system • A dynamical system Σ is a triple Σ = (T,W,B) • With T a set, called the time axis… • W a set, called the signal space… • And B ⊆ WT is the behavior of the system.

BA: Linear differential systems • Lumped linear time-invariant dynamical systems. Σ = (R,Rw,B) • R is the time axis. • Rw is the signal space. • B ⊆ C∞(R,Rw) (the space of all infinitely often differentiable functions from R to Rw consisting of all solutions of a set of linear, constant coefficient differential equations)…

BA: Polynomial matrices(special case)

BA: Model+controller

Numerical issues of interconnecting systems

BA: Work recently done • • • • • • • •

Theoretical results+Numerical implemetation. Algorithms to: Design controllers. Determine stability. Determine controlability, observability. Study some numerical phenomena. However… BA still can not deal with nonlinear systems…

EULER-LAGRANGE EQUATIONS FOR MECH.SYSTEMS

Mech. Systems considered Serial robots

Parallel robots

Model deduction: EL equations L d  L  D      ui qi dt  qi  qi (Number of eqns=degrees of freedom)

2nd type eqn

1st type eqn.

j L d  L  D k      j  ui qi dt  qi  qi j 1 qi (Some times more generalized coordinates than degrees of freedom => Restriction fun set )

Mech. Systems considered Serial robots

Parallel robots j L d  L  D k      j  ui qi dt  qi  qi j 1 qi

L d  L  D      ui qi dt  qi  qi

Serial manipulators • A lot of things already done…

Parallel robots • Some difficulties: Closed kinematic chains: Coupled kinematic constraints.

• • • •

Plant: Serial robots Direct kinematics (Easy). Inverse kinematics (Difficult) Direct dynamics (Rel.easy). Inverse dynamics(Rel.easy)

• • • •

Plant: Parallel robots Direct kinematics (Difficult). Inverse kinematics (Easy) Direct dynamics (Difficult). Inverse dynamics(Rel.easy)

• (Analitically/numerically solved in our research)

Serial robots • Controller design: • Linear (=>Linearization). • Nonlinear(Lyapunov/Geom) • Sliding mode control (SMC) • Fuzzy,combined (FSMC),etc

Parallel robots • Controller design: • Linear (=>Linearization??). • Nonlin(Lyapunov/Geom??) • Sliding mode control (SMC?) • Fuzzy,combined (FSMC?),etc

• Not a lot of things done in general, but: • A 6-PUS (Parallel-Universal-Spherical joints) design has been proposed ). • Numerical algorithms to compute direct and inverse kinematics for this 6-PUS design prop. • Numerical algorithms to compute direct and inverse dynamics for this 6-PUS design prop. • PD, PID control implemented numerically.

• Things to do yet:

• Although powerful, EL equations can not deal with another kind of systems…

Bioengineering systems Orthosis EEG as dynamical systems/stochastic analysis.

An orthopedic appliance or apparatus used to Support, align, prevent, or correct deformities Or to improve function of movable parts of the body.

ORTHOSIS (ORTHOS=STRAIGHT)

Magnetorheological fluid • Fluid with the abitlity to switch back and forth from a liquid to a near – solid under the influence of a magnetic field.

Rheomagnetic orthosis

+

Gait identification:Artificial Intelligence

Generalized idiopathic epilepsy

EEG IN CHILDREN EPILLEPSY

EEG

Where does the seizure come from? • • • •

(Classsical) linear system identificaction. Neural networks. Nonlinear system identification. Stochastic (Correlation, cross correlation,etc).

PREDICTION???

• THANK YOU! • ¡GRACIAS!