Appunti Machine Learning

Stabilization. The stabilization problem is the following one. We have an equilibrium which is unstable (or not enough stable) without the use of

the control.

Stability

Discussion of transient response and steady-state error is moot if the system does not have stability. In order to explain stability, we start from

the fact that the total response of a system is the sum of the natural response and the forced response. When you studied linear differential

equations, you probably referred to these responses as the homogeneous and the particular solutions, respectively. Natural response describes

the way the system dissipates or acquires energy. The form or nature of this response is dependent only on the system, not the input. On the

other hand, the form or nature of the forced response is dependent on the input. Thus, for a linear system, we can write Total response ˆ

Natural response ‡ Forced response 1.1† 2 For a control system to be useful, the natural response

must (1) eventually approach zero, thus leaving only the forced response, or (2) oscillate. In some systems, however, the natural response

grows without bound rather than diminish to zero or oscillate. Eventually, the natural response is so much greater than the forced response that

the system is no longer controlled. This condition, called instability, could lead to self-destruction of the physical device if limit stops are not part

of the design. For example, the elevator would crash through the floor or exit through the ceiling; an aircraft would go into an uncontrollable roll;

or an antenna commanded to point to a target would rotate, line up with the target, but then begin to oscillate about the target with growing

oscillations and increasing velocity until the motor or amplifiers reached their output limits or until the antenna was damaged structurally. A

time plot of an unstable system would show a transient response that grows without bound and without any evidence of a steady-state

response. Control systems must be designed to be stable. That is, their natural response must decay to zero as time approaches infinity, or

oscillate. In many systems the transient response you see on a time response plot can be directly related to the natural response. Thus, if the

natural response decays to zero as time approaches infinity, the transient response will also die out, leaving only the forced response. If the

system is stable, the proper transient response and steady-state error characteristics can be designed. Stability is our third analysis and design

objective

Stability

In Chapter 1, we saw that three requirements enter into the design of a control system: transient response, stability, and steady-state errors.

Thus far we have covered transient response, which we will revisit in Chapter 8. We are now ready to discuss the next requirement, stability.

Stability is the most important system specification. If a system is unstable, transient response and steady-state errors are moot points. An

unstable system cannot be designed for a specific transient response or steady-state error requirement. What, then, is stability? There are

many definitions for stability, depending upon the kind of system or the point of view. In this section, we limit ourselves to linear, time-invariant

systems. In Section 1.5, we discussed that we can control the output of a system if the steadystate

response consists of only the forced response. But the total response of a system is the sum of the forced and natural responses, or c t† ˆ

cforced t† ‡ cnatural t† (6.1) Using these concepts, we present the following definitions of stability, instability, and marginal stability: A linear,

time-invariant system is stable if the natural response approaches zero as time approaches infinity. A linear, time-invariant system is unstable if

the natural response grows without bound as time approaches infinity. A linear, time-invariant system is marginally stable if the natural

response neither decays nor grows but remains constant or oscillates as time approaches infinity. Thus, the definition of stability implies that

only the forced response remains as the natural response approaches zero. These definitions rely on a description of the natural response.

When one is looking at the total response, it may be difficult to separate the natural response from the forced response. However, we realize

that if the input is bounded and the total response is not approaching infinity as time approaches infinity, then the natural response is obviously

not approaching infinity. If the input is unbounded, we see an unbounded total response, and we cannot arrive at any conclusion about the

stability of the system; we cannot tell whether the total response is unbounded because the forced response is unbounded or because the

natural response is unbounded. Thus, our alternate definition of stability, one that regards