System Identification Arun K. Tangirala Department of Chemical Engineering IIT Madras July 26, 2013 Module 9 Lecture 2 Arun K. Tangirala System Identification July 26, 2013 16
Contents of Lecture 2 In this lecture, we shall learn: Types of inputs for identification Pseudo-Random Binary Signal (PRBS) Procedure for input design Arun K. Tangirala System Identification July 26, 2013 17
Input design: Primary considerations Inputs for identification Input should be persistently exciting, i.e., should contain sufficiently many frequencies. A signal is persistently exciting of order n if its spectrum is non-zero at n distinct frequencies (or its covariance matrix consisting of ACFs up to lag n should be non-singular) The asymptotic properties of the estimate (bias and variance) depend only on the input spectrum - not on the actual waveform. The input must have limited amplitude: u min u(t) umax. Periodic inputs may have certain advantages. Remember: Covariance matrix is typically inversely proportional to the input power! Arun K. Tangirala System Identification July 26, 2013 18
Binary symmetric signals Crest Factor The desired property of the waveform is defined in terms of C 2 r = max t u 2 (t) 1 N u 2 (t) N lim N t=1 A good signal waveform is one that has a small crest factor. The theoretic lower bound of C r is 1, which is achieved for binary symmetric signals Arun K. Tangirala System Identification July 26, 2013 19
Types of inputs There are different kinds of inputs available for identification. To each its merits and demerits. 1 White-noise: Contains all frequencies uniformly. Theoretically a preferable input signal. Decouples the IR parameter estimation problem. Provides uniform fit at all frequencies However, possesses a high crest factor. 2 Random binary: Generated by starting with a Gaussian sequence and then passing it through a filter depending on the input spectrum requirements. The sign of the filtered signal is the RBS. No proper control over the spectrum. The sign operation distorts the spectrum of the input sequence. Has the lowest crest factor. Arun K. Tangirala System Identification July 26, 2013 20
Types of inputs... contd. 3 Pseudo-RBS: Generated using a Linear Feedback Shift Register (LFSR) of n bits. Maximum length PRBS are 2 n 1 sequences long. They possess white-noise like properties. Frequency content can be changed by altering the clock sampling rate. Has the lowest crest factor. Disadvantage: Only maximum length PRBS possess the desired properties. 4 Multisine: Multisines are a combination of sinusoids of different frequencies. Provides very good estimates of the t.f. at those frequencies. However, the spectrum is not continuous. Therefore, the estimates at other frequencies are not available. Arun K. Tangirala System Identification July 26, 2013 21
PRBS Binary signals have the lowest crest factor for a given variance. Remember: The input signal should also satisfy the condition of persistent excitation. Binary signals with a desired spectral shape can be generated in two ways 1 Random Binary signal: Generated by passing a random Gaussian signal through a sign function. Disadvantage: There is little control over the spectrum 2 Pseudo-Random Binary signal: These are deterministic binary signals that have white-noise like properties PRBS: u[k] = rem(a 1 u[k 1] + + a n u[k n], 2) (modulo 2) With n-coefficients, one can generate a 2n 1 full length sequence (zero is excluded) The choice of coefficients (which are zero / non-zero) determines if a full length or partial length sequence is generated Arun K. Tangirala System Identification July 26, 2013 22
Full-length PRBS For a n-coefficient PRBS, the maximum length sequence that can be generated without repetition is M = 2 n 1. The table lists the {a n}s that have to be non-zero. Order M = 2 n 1 Non-zero indices of {a n} 2 3 1,2 3 7 2, 3 4 15 1, 4 5 31 2, 5 6 63 1, 6 7 127 3, 7 8 255 1, 2, 7, 8 9 511 4, 9 10 1023 7, 10 11 2047 9, 11 Observe that the last coefficient has to be non-zero. Other choices of non-zero coefficients also exist. Only full-length PRBS have white-noise like properties! MATLAB: uk = idinput(2047, prbs,[0 1],[-1 1]); % full-length PRBS Arun K. Tangirala System Identification July 26, 2013 23
Band-limited PRBS To generate band-limited, for example, low-frequency content PRBS, the full-length sequence is subjected to a simple operation Re-sample P times faster than the frequency at which the PRBS is generated Idea is to elongate or stretch the constant portions of PRBS The resulting signal has the same properties as passing the PRBS through a simple moving average filter of order P ũ[k] = 1 (u[k] + u[k 1] + u[k P ]) P MATLAB: uk = idinput(1533, prbs,[0 0.3],[-1 1]); % full-length PRBS Q: Why not pass the full-length PRBS through a simple low-pass filter? Arun K. Tangirala System Identification July 26, 2013 24
Remarks on PRBS For a given amplitude range, PRBS packs the maximum variance or energy It is ideally suited only for linear systems Since it switches between two states, it cannot detect non-linearities Change in the initialization only produces a shift in PRBS This is due to the periodicity property of PRBS Therefore, PRBS is not directly suited for design of uncorrelated inputs for multivariable systems For time-varying and non-linear systems, some modifications exist such as Multi-valued PRBS and Amplitude-Modulated PRBS Arun K. Tangirala System Identification July 26, 2013 25
Preliminary tests Before exciting the process with the design input sequence, it is useful to perform preliminary tests: 1. Perform a step test (3% - 10% magnitude) on the system. The step response throws light on the gain, time constant, delay, inverse response, etc. 2. A step in one direction is insufficient. Perform a step at least test in two directions or of different magnitudes so as to check for the effects of non-linearities and the range of linearization. 3. From the step response, identify the effective time-constant τ of the c.t. process 4. Compute the effective bandwidth Ω BW = 1/τ. 5. Use sampling frequency F s = 1/T s anywhere between 10 20 times Ω BW and the discrete-time input frequency range as [0, 3.5Ω BW /F s]. 6. Design an input sequence of the appropriate type (white, rbs, multisine, prbs) accordingly. For systems with special frequency response characteristics, the frequency content of the inputs have to be determined carefully. Arun K. Tangirala System Identification July 26, 2013 26
Experimental design General guidelines Choose experimental conditions and inputs such that the predictor becomes sensitive to parameters of interest and importance. Choose excitation frequencies and use the input energy in those bands where a good model is intended and/or where the disturbance activity is significant. Open loop inputs: Binary, periodic signals with full control over the excitation energies. Remember Cov ĜN (eiω ) n N Φ vv(ω) Φ uu(ω) Sample 10-20 times the bandwidth frequency Arun K. Tangirala System Identification July 26, 2013 27
Summary Different types of inputs can be used in identification. While the actual choice depends on the application, some general guidelines are available: i. Input should have maximum power relative to noise (high SNR) ii. Low amplitude to prevent the process from getting into non-linear regimes iii. Maximum crest factor iv. Periodic inputs may be advantageous in certain applications In designing an input, the bandwidth of the system and ability to shape the spectral content are important Binary signals have maximum crest factor (of unity) for a given amplitude. PRBS is widely used in linear identification. For non-linear and time-varying systems, variants of PRBS are used. Some preliminary tests (unless prior process knowledge is available) are inevitable before designing an appropriate input Arun K. Tangirala System Identification July 26, 2013 28