DYNAMICAL SYSTEMS
EXAMPLES EXAMPLE - Temperature in building Energy balance: Rate of change = [Inflow of energy] [Outflow of energy] of stored energy where Rate of change of stored energy = cρv dt (c = specific heat capacity; ρ = density; V = volume) Inflow of energy = P (=power to heaters) Outflow of energy = k(t T y ) (k = heat conductance) 2
Hence: or Here: cρv cρv dt (t) dt (t) = P (t) k (T (t) T y (t)) + kt (t) = P (t) + kt y (t) T (t) = variable to be controlled (indoor temperature) P (t) = control variable (power to heaters) used to affect the system T y (t) = disturbance (ambient temperature), which cannot be affected 3
EXAMPLE: Speed controller Newton s law of motion: a(t)m = F (t) where a(t) = acceleration, a(t) = dy(t) where y(t) = speed, m = total mass of car, F (t) = total force acting in direction of road 4
Force: F (t) = F d (t) + F g (t) + F wind (t) + F f (t) where F d (t) = driving force of engine We can model F d (t) as F d (t) = ku(t) where k = a constant and u(t) = fuel flow to engine (control variable) F g (t) = mg sin(ϕ(t)) = gravitation force F wind (t) = b (y(t) v wind (t)) = drag from air F f (t) = friction forces Combining: or where m dy(t) is a disturbance. = F d (t) + F wind (t) + F g (t) + F f (t) m dy(t) + by(t) = ku(t) + d(t) d(t) = bv vind (t) + F g (t) + F f (t) 5
Here: y(t) = variable to be controlled (speed) u(t) = control variable (fuel flow to engine) used to affect the system d(t) = disturbance (gravitation, air drag, friction), which cannot be affected 6
GENERALIZATION In both cases the variable of interest (controlled variable) could be described by an equation of the form where: dy(t) + ay(t) = bu(t) + cd(t) (1) y(t): a measured output variable to be controlled u(t): a variable which can be used to affect the system behavior (control signal) d(t): a variable which affects the system behavior but which we cannot affect (disturbance) Equation (1) is an example of a differential equation (more precisely, a first-order differential equation) Many systems in technology and in the natural world are described by differential equations 7
Dynamical systems A system whose behavior is described by a differential equation such as (1) is an example of a dynamical system. A characteristic property of dynamical systems is that they have inertia, so that the dependent output variable (y(t)) depends on the past history of the input variables (u(t), d(t)), i.e., y(t) = F (u(τ)d(τ), τ t) Such systems may show complex responses to changes in input 1.2 1 0.8 u 0.6 0.4 0.2 0 0.2 1.5 y 1 0.5 0 2 1 0 1 2 3 4 5 6 7 8 tid 8
Compare with a static system, y(t) = f(u(t), d(t)) 1.2 1 0.8 u 0.6 0.4 0.2 0 0.2 0.6 0.5 0.4 y 0.3 0.2 0.1 0 0.1 2 0 2 4 6 8 10 tid 9
Signals and systems The generic nature of dynamical systems described by differential equations implies that if we, for example, know how to control the temperature in a building we also know how to control the speed of a car (although the practical implementation problems may, of course, be quite different). It follows that the study of the properties of dynamical system, feedback control etc, can be set in an abstract context. The basic structures are then signals and systems and their connections. Signal: a variable which depends on time, and often contains information of some kind (e.g., y(t), u(t), d(t)) Mathematically a signal is just a function of time. Physically a signal may consist of an electric signal (voltage) or a sequence of numbers (or bits) System: an element which is affected by some signals (input signals, e.g., u(t), d(t)), and generate other signals (output signals, e.g., y(t)). Mathematically a system is often described by a differential equation. Physically it may consist of a building, a car, etc. 10
Block diagrams The relations between signals and system can be depicted using block diagrams A general dynamical system can thus be depicted with a block diagram: u S d y 11
Examples: Series coupling: u S 1 y 1 S 2 y Output of system S 1 is the input to system S 2. Example: car model, where - u(t): amount of fuel injected to engine, - S 1 : engine dynamics, - y 1 (t): power generated by engine, - S 2 : car dynamics, - y(t): speed of car 12
Signal branching: u u u Two identical copies of signal u(t) are sent to different destination. Example: a radio broadcast is sent to all receivers, an audio signal may be sent to several loudspeakers, etc. 13
Addition and subtraction of signals: u 1 + + u 1 + u 2 u 1 u + 1 u 2 u 2 u 2 (a) (b) Example: noise (u 2 ) is added to a signal (u 1 ) 14
Qualitative analysis of the response of firstorder system Let s consider the first-order dynamical system dy(t) + ay(t) = bu(t) The following figure shows the output y(t) for stepwise changes of input u(t). 1.2 1 0.8 u 0.6 0.4 0.2 0 0.2 1.2 1 0.8 y 0.6 0.4 0.2 0 0.2 2 0 2 4 6 8 10 tid 15
The system response can be characterized in terms of two properties: the dynamic, or transient, response, in this case simply the speed of response, the static response, or value of output as t when u(t) is held constant These are determined by the two system parameters a and b. To see how a and b affect the system behavior, we write dy(t) + a y(t) b a u(t) = 0 Static response If u(t) is held constant, u(t) = u 0, we have that dy(t) = 0 if y(t) = b a u 0 Hence, if there exists a value y 0 such that y(t) y 0 when u(t) = u 0, then we should have y 0 = b a u 0. We define K = b a : static gain of the system 16
Stability For a constant u(t) = u 0 and y 0 = b a u 0, we can write the system equation as Hence: dy(t) = a (y(t) y 0 ) If a > 0, we have and = dy(t) dy(t) < 0 when y(t) > y 0 > 0 when y(t) < y 0 y(t) y 0 In this case the system is called stable. If a < 0, we have and = dy(t) dy(t) > 0 when y(t) > y 0 < 0 when y(t) < y 0 y(t) + or In this case the system is called unstable. 17
Example of a stable system: a hanging pendulum. Example of an unstable system: an inverted pendulum. The static gain of an unstable system is obviously infinite. Transient response Defining the deviation from the static value, y diff (t) = y(t) y 0 we have dy diff (t) = ay diff (t) Hence Large a = large derivative = fast response Small a = small derivative = slow response Often the system equation is written in a form where the transient and static properties are seen explicitly: T dy(t) where K = b/a: the static gain T = 1/a: the time constant + y(t) = Ku(t) Here T has dimension time, and it defines a characteristic time scale for the transient response: the smaller T, the faster the response is, and vice versa. 18