Lecture IV-Nonparametric PDF

Modelling & Simulation of Control Systems SYSTEM IDENTIFICATION Dr. Hatice Doğan Nonparametric-Parametic Models  Nonparametric models are characterised as curves or functions, not a set of parameters. e.g. nonparametric model consist of a time record of the...

Modelling & Simulation of Control Systems SYSTEM IDENTIFICATION Dr. Hatice Doğan Nonparametric-Parametic Models  Nonparametric models are characterised as curves or functions, not a set of parameters. e.g. nonparametric model consist of a time record of the impulse or step response in the time domain, or a frequency record of the transfer function in the frequency domain. e.g. Bode diagram. Essentially, an infinite number of measurements would be needed to represent the system. Practically, a “sufficiently” large number is required to “acceptably” represent the system.  Parametric models concentrate all information in a model structure with a limited set of parameters. This makes the parametric model “economical” and powerful. Impulse Response  Impulse Response Model Representation In order to motivate the general applicability of the convolution model to LTI systems, first the unit impulse function has to be introduced.The unit impulse function or Dirac (δ) function at timezero is defined heuristically as and can be viewed as a rectangular, unit-area pulse with infinitesimally small width. Let the unit impulse function δ(t) be input to an LTI system and denote the impulse response by g(t).  Let the unit impulse function δ(t) be input to an LTI system and denote the impulse response by g(t). Then, due to the time-invariant behavior of the system, a time shifted impulse δ(t −τ) will result in an output signal g(t −τ).  Moreover, because of the linearity, the impulse δ(t −τ)u(τ) will result in the output g(t −τ)u(τ), and after integrating both the input and output impulses over the time interval [−∞,∞],that is,  Hence, in the derivation of the practically applicable convolution model only assumptions have been made with respect to the linearity and time invariance of the system. Thus the convolution model, fully characterized by the impulse response function g(t), is able to describe the input–output relationship of the large class of LTI systems. Consequently, if g(t) is known, then for a given input signal u(t), the corresponding output signal can be easily computed. This feature explains the interest in impulse response model representations, especially if there is limited prior knowledge about the system behavior. Transfer Function Model Representation Laplace transformation of the convolution model gives Y(s) =G(s)U(s) which defines an algebraic relationship between transformed output signalY(s) and transformed input signal U(s).The function G(s) is the Laplace transformed impulse response function, that is, G(s) ≡ L[g(t)], and is called the transfer function. Consequently, representationY(s) =G(s)U(s) is called the transfer function model representation. Direct Impulse Response Identification  In particular, for u(t) = 0, t < 0, and zero initial condition response, the convolution sum is given by  where g(0) is usually equal to zero, because no real system responds instantly to an input. Hence, if we are able to generate a unit pulse, the coefficients of g(t) can be directly found from the measured output. Let, for instance, the pulse input be specified as where α is chosen in accordance with the physical limitations on the input signal. The corresponding output will be where v(t) represents the measurement noise of the output signal. Consequently, an estimate of the impulse function, or better the unit-pulse response, is and the estimation errors are v(t)/α. Algorithm -I Identification of g(t) from a pulse input 1. Generate a pulse with maximum allowable magnitude, α 2. Apply this pulse to the system 3. To determine estimates of the components of the impulse response g(t), use: Impulse Response Identification Using Step Responses Applying the step input to an LTI system described by (2.5) gives with corresponding error equal to [v(t)−v(t −1)]/α. Algorithm -II Identification of g(t) from a step input 1. Generate a step with maximum allowable magnitude, α 2. Apply this step to the system 3. From the step response the dead time, dominant time constant, and static gain can be graphically determined 4. To determine estimates of the components of the impulse response g(t), use: Examples Example 1: Solution: Example 2: Solution: Example 3 Solution: Impulse Response Identification Using Input–Output Data Direct Step Response Identification of A First Order System: General Form of a first order system: 7 6 System Identification (For first and second order systems) Forward Shift Operator  Forward Shift Operator (q): q(t)=u(t+1)  Backward Shift Operator (q-1) q-1(t)=u(t-1) The convolution sum: y(t)=G(q)u(t) Discrete-Time Delta Operator References  Keesman, Karel J., System Identification:An Introduction, Springer, 2011  ELEC4410 Control System Design, Lecture 10: Elements of System Identification,The University of Newcastle