Name: system response and stability
Uploaded: 2024-11-21T14:54:10.000Z
Duration: 1 min 28 s
Channel: JoVE

The time response of a linear time-invariant system is divided into transient and steady-state responses.
The transient response diminishes to zero over time. The steady-state response persists after the transient response has faded.
The stability of such a system is directly linked to the roots of the system&#39;s characteristic equation or its poles.
A system maintaining bounded output for a bounded input, with roots of the characteristic equation in the left-half s-plane, is stable. However, if any root lies in the right-half s-plane, it becomes unstable.
Absolute stability confirms system stability, while relative stability measures its degree.
Consider a pendulum. When undisturbed, it rests in a state of static equilibrium, maintaining stability for minor motions around the equilibrium.
When external or frictional forces are added, the transient response of the system gradually decreases over time due to the influence of the damping. The pendulum achieves stable motion around the equilibrium position.
In contrast, an inverted pendulum is inherently unstable and requires active control to prevent it from toppling. When disturbed by an external force, it does not return to its original position, illustrating instability.

stabiliteit

De tijdrespons van een lineair tijdsinvariant (LTI) systeem kan worden onderverdeeld in transiënte en steady-state responsen. De transiënte respons is de initiële reactie van het systeem op een verandering in invoer en neemt in de loop van de tijd af tot nul. Daarentegen is de steady-state respons het gedrag dat blijft bestaan nadat de transiënte effecten zijn verdwenen.

De stabiliteit van een LTI-systeem wordt bepaald door de wortels van de karakteristieke vergelijking, de zogenaamde polen. Een systeem is stabiel als het een begrensde uitvoer produceert bij een begrensde invoer, wat het geval is als alle wortels van de karakteristieke vergelijking zich in de linkerhelft van het s-vlak bevinden. Als een wortel zich echter in de rechterhelft van het s-vlak bevindt, wordt het systeem instabiel. Absolute stabiliteit geeft aan of een systeem stabiel of instabiel is, terwijl relatieve stabiliteit de mate van stabiliteit meet.

Een illustratief voorbeeld van deze concepten is een slinger. Wanneer een slinger niet wordt verstoord, rust hij in een toestand van statisch evenwicht en vertoont hij stabiliteit bij kleine bewegingen rond deze evenwichtspositie. Deze stabiliteit blijft behouden wanneer de slinger wordt uitgelenkt en vrij kan slingeren. In dit geval zijn er externe of wrijvingskrachten aanwezig. Deze krachten zorgen voor demping, waardoor de transiënte respons van het systeem in de loop van de tijd geleidelijk afneemt. Uiteindelijk bereikt de slinger een stabiele beweging rond de evenwichtspositie, wat de kenmerken van een stabiel systeem toont.

Daarentegen, als de slinger wordt verplaatst vanuit een onstabiele evenwichtspositie, begint hij te slingeren met een tijdrespons die exponentieel groeit. Deze onbeperkte beweging wijst op instabiliteit, omdat het systeem niet terugkeert naar het evenwicht en de amplitude van de oscillaties blijft toenemen.

Het onderscheid tussen transiënte en steady-state responsen, samen met de stabiliteitsvoorwaarden, is essentieel voor het analyseren en ontwerpen van LTI-systemen. Door ervoor te zorgen dat alle polen van het systeem in de linkerhelft van het s-vlak liggen, kunnen ingenieurs stabiliteit garanderen en het systeemgedrag voorspellen bij verschillende inputs, wat leidt tot robuustere en betrouwbaardere systeemprestaties.

Stabiliteit

The time response of a linear time-invariant system is divided into transient and steady-state responses.
The transient response diminishes to zero over ...

Education

JoVE Core

Electrical Engineering

Elektrotechniek

Unit 1

Transient and Steady-state Response Analysis

Analyse van de overgangs- en stationaire respons

transient-and-steady-state-response-analysis

KEY TERMS AND CONCEPTS

BELANGRIJKSTE TERMEN EN CONCEPTEN

Stability

system response and stability

System response and stability describe how a linear time-invariant (LTI) system reacts to a change in input. The time response has two parts: transient response and steady-state response. The transient response is the first reaction and it fades to zero over time. The steady-state response remains after the transient effects have disappeared.

An LTI system is stable when it gives a bounded output for a bounded input. For these systems, stability depends on the roots of the characteristic equation, also called poles. The system is stable when all poles are in the left-half of the s-plane. If any pole is in the right-half of the s-plane, the system becomes unstable.

Engineers also use the terms absolute stability and relative stability. Absolute stability tells whether a system is stable or unstable. Relative stability describes how stable the system is. These ideas help show not only if a system will settle, but also how strongly it resists unwanted growth.

A pendulum gives a clear example of these ideas. When it is undisturbed, it rests in static equilibrium and is stable for small motions around that position. If it is displaced and allowed to swing, external or frictional forces add damping. The transient response then decreases over time, and the pendulum reaches stable motion around equilibrium.

A pendulum can also show instability. If it is displaced from an unstable equilibrium position, it begins to swing with exponential growth in its time response. The motion does not return to equilibrium. Instead, the amplitude of the oscillations keeps increasing, which shows an unstable system.

Knowing the difference between transient and steady-state responses is important for analyzing and designing LTI systems. Engineers use the location of the poles in the left-half s-plane to check stability and predict how a system will react to different inputs. This leads to more reliable system performance.

System Response and Stability

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stability

System response and stability describe how a linear time-invariant (LTI) system reacts to a change in input. The time response has two parts: transient response and steady-state response. The transient response is the first reaction and it fades to zero over time. The steady-state response remains after the transient effects have disappeared. An LTI system is stable when it gives a bounded output for a bounded input. For these systems, stability depends on the roots of the characteristic...

routh-hurwitz criterion i

Checking Stability with the Routh Table

Checking stability with the Routh table is a useful part of analyzing a closed-loop transfer function. It helps determine whether a system will remain stable when load conditions change or when a fault occurs. In a power grid, this kind of check is important because instability can lead to blackouts. The Routh-Hurwitz criterion uses a table built from the denominator polynomial of the transfer function. The rows of the Routh table are labeled by powers of the complex variable s, starting with...

routh-hurwitz criterion ii

Routh Table Special Cases

Routh table special cases can make stability analysis harder when using the Routh-Hurwitz criterion. Two situations need extra care. One is a zero in the first column. The other is an entire row of zeros. When a single zero appears in the first column, division by zero would occur. The usual fix is to replace that zero with a very small number, epsilon (ε). The stability test then continues by assuming a sign for ε. If ε is positive, any sign change in the first column means the system is...

key terms and concepts

Learn how LTI systems respond with transient and steady-state behavior, how poles affect stability, and how a pendulum illustrates stable and unstable motion.

System Response and Stability | JoVE K12

JoVE Core Electrical Engineering offers easy-to-understand animated videos that cover fundamental concepts of electric circuits along with circuit analysis techniques. It incorporates first-order and second-order circuits using operational amplifiers and circuit transformation concepts. The inclusion of practical design examples and real-world applications through scientist-in-action videos adds significant educational value, making complex concepts accessible.

electrical engineering

System Response and Stability

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