Control of Vibrations

Transcription

1 Control of Vibrations M. Vidyasagar Executive Vice President (Advanced Technology) Tata Consultancy Services Hyderabad, India sagar

2 Outline Modelling Analysis Control Stabilization Optimization

3 Models for Vibrating Beams Euler-Bernoulli beam: 2 w(x, t) t 2 = 4 w(x, t) x 4. Closed-form solution in terms of eigenfunctions sin x, sinh x, cos x, cosh x. No closed-form expression for frequencies (unlike other old equations such as wave equation, diffusion equation etc.) With nonuniform beams (e.g., tapered beams) even the eigenfunctions are not known. Timoshenko beam: Useful when the beam length is < 10 times the other two dimensions.

4 Analysis of Vibrations Two competing approaches: Time domain (infinite-dimensional state spaces) Frequency domain Consider a differential equation f(t) = Af(t) + Bu(t), where f belongs to some infinite-dimensional normed space X, such as L 2 [0, l], the set of square-integrable (finite-energy) functions on a finite interval [0, l. A is an unbounded operator. For instance is unbounded. f(x, t) f(x, t) x

5 Time-Domain Approach to Vibration Modelling Seminal theorem of Yosida-Hille-Phillips (1957) gives necessary and sufficient conditions for a differential equation f = Af on an infinitedimensional Banach space X to have a (unique) solution (that depends continuously on the initial condition). Next advance (circa 1970 s): Operators with compact resolvent. If the inverse operator (λi A) 1 exists, then it is compact, i.e., maps bounded sequences into convergent sequences. The inverse of the differential operator f(x, t) f λf x is an integral operator and therefore compact.

6 Time-Domain Approach to Control and Optimization Practically all known state-space methods for linear quadratic optimal control extend to operators with compact resolvent. System model: f(t) = Af(t) + Bu(t). Objective function to be minimized: l I = f 2 (x, t)dxdt. 0 0 Familiar solution in terms of Riccati equation. Very pretty theory, but totally non-computable as it requires a knowledge of mode functions. References: Roberto Triggiani, Irena Lasiecka, Ruth Curtain, etc.

7 Modern Frequency-Domain Approach to Control and Optimization Big breakthrough: Youla parametrization of all controllers that can stabilize a given system. References: D. C. Youla, J. J. Bongiorno and H. A. Jabr, IEEE Trans. AC, (Very complex formulation; solution unclear) C. A. Desoer, R-W. Liu, J. J. Murray and R. Saeks, IEEE Trans. AC, (Clean solution but not very general.) MV, B. A. Francis and H. Schneider, IEEE Trans. AC, (Complete theory, what is used today).

8 Robustness The New Concept All models are imperfect, so: Does a controller designed using an inexact model work for the real system? Big breakthroughs: H -control theory (G. Zames, B. A. Francis, J. C. Doyle, K. Glover, et al., 1980 s) Graph topology : MV, 1980 s Made it possible to analyze the effect of unmodelled dynamics, e.g., using only the first few modes.

9 Newer Applications: Robotics, Large Space Structures Robotics until 1985 was completely rigid : Inelasticity of joints was ignored. Flexibility of robot arms was ignored. Key breakthrough: Feedback linearization theory.

10 Given a system described by the ODE x(t) = f(x(t)) + m i=1 u i (t)g i (x(t)), under suitable conditions it is possible to convert it to a linear system by using control feedback and/or redefining the state vector z(t) = η(x(t)). Robots with elastic joints are feedback linearizable. Robots with one single flexible arm are feedback linearizable. References: M. W. Spong, MV, D. W. L. Wong, During my days at Waterloo, we not only did the theory but also built teams and controlled them!

11 Large Space Structures Space antennas became very popular in 1980 s. Challenges: Two-dimensional oscillations (not one-dimensional). No inertial frame of reference Conservation of energy, angular momentum etc. Impossible to model, impossible to experiment with. Control using H theory has been fairly successful.

12 What Next? Earth-bound multi-dimensional vibration control problems, e.g., telescope at Hanle. A lens of a telescope is not glass these days! Similar problems in outer space. Key takeaway messages: No substitutes for knowing lots of mathematics, and for experimental verification of the theoretical predictions.