Equilibrium

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1. 2. Equilibrium Equilibrium points (alias, working points or fixed points) are the points from which the the system will not evolve. Thus at STEADY CONDITION, in fact WP are considered in the response to constant inputs at the steady-state regime. Different WP have different evolutions. General case: Non-linear Systems Since the aim is to find the state from which the system will NOT EVOLVE, therefore the state that will constantly be kept, without variation of the input, we are looking for the point x in the State Space that corresponds to the null derivative of the function, with a fixed input u(t) The specific Non-Linear function that characterizes the system determines the number and the location of the fixed points: Mapping and characterizing the equilibrium points has many applications: analyze the system polarization, for example 1 equilibrium point non-linearly depending from the input expresses the system polarization (or bias) imposed by u. linearize the system around a given equilibrium to easily study the 2. 3. ● ● dynamics around it (small variations (in calculus), or small signals (in electronics), or “vibration dynamics” (in mechanics) Detect some important information about the overall non-linear behaviour. RECALL: Different equilibrium points correspond to: different linearization (the A,B,C state-space representation is different if the derivatives are applied in different equilibrium points). different results in stability (the some equilibrium points are stable and some not) Particular case: Linear systems STEADY-STATE WORKING POINT IN PHYSIOLOGICAL CLOSED LOOPS Physiological regulation is often the result of two mechanisms opposed in a feedforward vs. feedback fashion. It is also common that the static I/O laws are a) non-linear b) monotonic c)…