Nonlinear Systems with Output Measurement
The present chapter extends the results of Chapter 3 to the nonlinear case. As remarked earlier, the case where the whole state vector is not directly measured and the measurements provide values only for an output of the system is very important for man
- PDF / 989,874 Bytes
- 58 Pages / 439.37 x 666.142 pts Page_size
- 101 Downloads / 204 Views
Nonlinear Systems with Output Measurement
1 Introduction The present chapter extends the results of Chapter 3 to the nonlinear case. As remarked earlier, the case where the whole state vector is not directly measured and the measurements provide values only for an output of the system is very important for many reasons. We consider only the case where the input is applied with ZOH; the reader can easily modify the suggested methodologies and results to the case of continuously applied input. We also consider only the case where the measurement is sampled for the same reasons. The second section of the present chapter shows that when the predictor mapping is available then we are in a position to construct a discrete-time system, which can help us to reconstruct the state of the system after an initial transient period. When, the predictor mapping is not available, then conventional observers have to be used. This is a complication. However, as remarked in Chapter 3, although conventional observers do not provide the exact value of the state vector after some finite time but instead the error of the provided estimate converges to zero at an exponential rate (in the absence of disturbances), conventional observers offer an important feature: they can handle sampled measurements even in the case of uncertain sampling schedules. Therefore, we are in a position to consider the ISP-O-P-DFC control scheme, already introduced in Chapter 3, which consists of: 1. An Inter-Sample Predictor (ISP), which uses the sampled and delayed measurements and gives a continuous signal that approximates the “nominal” continuous (but delayed) output signal, 2. An Observer (O), which uses the continuous signal from the ISP and provides a continuous estimate of the delayed state vector xðt r Þ, 3. An Approximate Predictor (P), which exploits the state estimate given by the observer in order to predict the future value of the state vector xðt þ τÞ, and
© Springer International Publishing AG, CH 2017 I. Karafyllis, M. Krstic, Predictor Feedback for Delay Systems: Implementations and Approximations, Systems & Control: Foundations & Applications, DOI 10.1007/978-3-319-42378-4_5
155
156
5
Nonlinear Systems with Output Measurement
4. A Delay-Free Controller (DFC) (used with ZOH), i.e., a “nominal” feedback law, which is stabilizing for the delay-free version of the system and uses the prediction of the future value of the state vector. Sections 3 and 4 of the present chapter are devoted to two important classes of nonlinear where the ISP-O-P-DFC control scheme can be applied with success: the class of globally Lipschitz systems and the class of nonlinear systems with a compact absorbing set.
2 Solution Map Explicitly Known As in the previous chapter, we consider nonlinear systems of the form: x_ ðtÞ ¼ f ðxðtÞ, uðt τÞÞ xðtÞ 2 ℜn , uðtÞ 2 ℜm
ð2:1Þ
where τ 0 is the input delay and f : ℜn ℜm ! ℜn is a locally Lipschitz mapping with f ð0; 0Þ ¼ 0. The assumption that the state vector is measured is seldom realistic. Instead, measureme
Data Loading...
