1. Control Based on Instantaneous Linearization Eemeli Aro eemeli.aro@tkk.fi 16.11.2005

2. Structure Rationale Instantaneous linearization Controller implementation Discussion

3. Rationale Linear control better understood than nonlinear control Many well-established linear design techniques exist Linearization only valid in a limited operating range  Extract a linear model from the current sample

4. Instantaneous linearization Assume a deterministic neural network input-output model is available with a regression vector which is interpreted to define the state of the system

5. Instantaneous linearization Then linearize g around  (t=  ) to get an approximate model where

6. Instantaneous linearization Separating components of the current regression vector  (  ) into a bias term  (  ) : where

7. Instantaneous linearization The coefficients { a i } and { b i } are collected into the polynomials Thus the approximate model can be seen as a linear model affected by a constant disturbance  (  )

8. Instantaneous linearization For a multilayer perceptron network with one hidden layer on tanh units and a linear output,

9. Application to Control Extract linear model Control design ControllerSystem Linearized model parameters InputOutput Controller Parameters Reference

10. Application to Control At each sample, extract a linear model from a neural network model of the system and design a linear controller Can be seen as a gain scheduling controller with an infinite schedule

11. Application to Control Structurally equivalent to an indirect self- tuning regulator, only difference is in how the linear model is extracted Control design based on certainty equivalence principle – the controller is designed assuming that the linear model perfectly describes the system

12. Application to Control Can implement any linear control design Need to compensate for bias term  (  ) –e.g. by using integral action, which also compensates for other constant disturbances Need to keep in mind narrow operating range of linearized model

13. Application to Control Pole placement design –Assuming a linearized deterministic model –The objective is to select the three polynomials R, S, and T so that the closed loop system will behave as

14. Application to Control reference System inputoutput + –

15. Application to Control Minimum variance design –For regulation, not trajectory following –Design the controller to minimize a criterion J ( t ) –Generalized Minimum Variance controller where P, W and Q are rational transfer functions

16. Discussion Pro –Allows the use of linear design techniques –Reasonably simple implementation –Fast; linearization & design can be done between samples –Allows control of systems with unstable inverses (with approximate pole placement controller design without zero cancellation) –Can be used to understand the dynamics of the system (poles, zeros, damping, natural frequency)

17. Discussion Con –Linearized model often valid only in a narrow range –Can’t deal with hard nonlinearities –Requires understanding on linear control theory

18. References M. Nørgaard, O. Ravn, N. K. Poulsen, and L. K. Hansen, "Neural Networks for Modelling and Control of Dynamic Systems," Springer-Verlag, London, 2000 O. Ravn, "The NNCTRL Toolbox. Neural networks for control", Version 2, Technical University of Denmark, 2003, http://www.iau.dtu.dk/research/control /nnctrl.html http://www.iau.dtu.dk/research/control /nnctrl.html