Bounded inputs

Fig. 9.22 Transformation of a normalized bounded input V (s) into an actual input l/(s).

We also see another common definition—bounded input bounded output (BIBO) stability A system is BIBO stable if the output response is bounded for any bounded input. One illustration of this definition is to consider a hypothetical situation with a closed-loop pole at the origin. In such a case, we know that if we apply an impulse input or a rectangular pulse input, the response remains bounded. However, if we apply a step input, which is bounded, the response is a ramp, which has no upper bound. For this reason, we cannot accept any control system that has closed-loop poles lying on the imaginary axis. They must be in the LHP. 1... 

Fig. 40. Influent Si concentration (—) upper and lower bounds, (- -) input concentration...

S. Tarbouriech and G. Garcia. Stabilization of Linear Discrete-Time Systems with Saturating Controls and Norm-Bounded Time-Varying Uncertainty. Control of uncertain systems with bounded inputs Lecture Notes in Control and Information Science 221. Springer-Verlag Berlin, 1997. 

Definition 7—Bounded-input-bounded-output (BIBO) stability. A... 

Stability on the initial state of the system. For the case of a linear, unstable system with bounded inputs and without external disturbances, Zheng and Morari (1995) have developed an algorithm that can determine the domain of attraction for the initial state of the system. 

Choi, J., and Manousiouthakis V., Bounded input initial state bounded output stability over ball, AIChE Annual Meeting, paper 191a (1997). 

A dynamic system is considered to be stable if for every bounded input, the output is also bounded, irrespective of its initial state. The transfer function G(s) relates an output variable to an input variable or forcing function and is defined as ... 

Stability The stability of a system is determined by its response to inputs. A stable system remains stable unless it is excited by an external source, and it should return to its original state once the perturbation is removed and the system cannot supply power to the output irrespective of the input. The system is stable if its response to the impulse excitation approaches 0 at long times or when every bounded input produces a bounded output. Mathematically this means that the function does not have any singularities that caimot be avoided. The impedance Z(s) must satisfy the following conditions Z s) is real when s is real (that is, when 0) and Re[Z(5)] > 0 when v > 0 [ = v -i- ja>, see Section... 

How do we define a stable or unstable system There are different ways, depending on the mathematical rigorousness of the definition and its practical utility for realistic applications. In this text we employ the following definition, which is often known as bounded input, bounded output stability. 

A dynamic system is considered to be stable if for every bounded input it produces a bounded output, regardless of its initial state. 

We again employ the definition of stability outlined in Section 15.1, which is often called bounded input, bounded output stability. Thus ... 

Define what is known as bounded input, bounded output stability. 

Define the bounded input, bounded output stability of a discrete-time system. What is the rule for characterizing the stability of such a system ... 

Here we adopt the following definition valid for linear systems a system is stable if bounded input variations produce bounded output variations as t —>oo otherwise the system is unstable (Ogunnaike Ray, 1994). One of the main issues in designing feedback controllers is stability. Let consider the response of a closed-loop system under proportional control, as deviation in outputy vs. time (Fig. 12.5). If the controller gain is moderate then y goes to zero after some oscillations. By increasing gain... 

In this section we will look in more detail at system stability [587]. In control system theory, a stable system is one that produces a bounded response to a bounded input. In general, system stability depends on the proprieties of the transfer function, in this case of the impedance or the admittance [588]. Impedance and... 

Traditionally, a conservative approach to safety analysis has been employed. In this approach, pessimistic assumptions, bounding input data and even conservative physical models are used to obtain a pessimistic bounding analysis. This approach has been required to a greater or lesser extent by most regulators and for most reactor types. HWRs also followed this approach for licensing analysis, although the physical models used were realistic. 

A linear control system, such as the one expressed by Eq. (59), is said to be stable if and only if its output is bounded for every bounded input. 

An LBA A is a Turing machine where the R/W head never moves beyond the original input region. For the purpose of this paper, we require that the LBA works on bounded input i.e., input in the form of -6. As usual, we use Lang(A) to denote the set of input words that are accepted by A. 

Definition of Stability. An unconstrained linear system is said to be stable if the output response is bounded for all bounded inputs. Otherwise it is said to be unstable. 

By a bounded input, we mean an input variable that stays within upper and lower hmits for all values of time. For example, consider a variable u i) that varies with time. If u i) is a step or sinusoidal function, then it is bounded. However, the functions u t) = t and u i) = are not bounded. 

We conclude that the liquid storage system is open-loop unstable (or non-self-regulating) because a bounded input has produced an unbounded response. However, if the pump in Fig. 11.24 were replaced by a valve, then the storage system would be self-regulating (cf. Example 4.4). 

Stable A system whose response to a bounded input is also bounded. 

---