![]() We can say conclusively that the roots of the characteristic equation are the poles of the transfer function. We will look into this in more detail below.Ī system is defined to be BIBO Stable if every bounded input to the system results in a bounded output over the time interval [ t 0, ∞ ) It is important to note that even if all of the coefficients of the characteristic polynomial are positive the system may still be unstable. If any coefficient of the characteristic polynomial is zero or negative then the system is either unstable or at most marginally stable. Linear systems have an associated characteristic polynomial which tells us a great deal about the stability of the system. Readers without a strong mathematical background might want to review the necessary chapters in the Calculus and Ordinary Differential Equations books (or equivalent) before reading this material.įor most of this chapter we will be assuming that the system is linear and can be represented either by a set of transfer functions or in state space. ![]() The chapters in this section are heavily mathematical and many require a background in linear differential equations. Although we can design controllers that stabilize the system, it is first important to understand what stability is, how it is determined, and why it matters. Nonetheless, many systems are inherently unstable - a fighter jet, for instance, or a rocket at liftoff, are examples of naturally unstable systems. ![]() Also, systems that are unstable often incur a certain amount of physical damage, which can become costly. ![]() For instance, a robot arm controller that is unstable may cause the robot to move dangerously. This causes a number of practical problems. When a system is unstable, the output of the system may be infinite even though the input to the system was finite. ![]()
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