Lecture A1 : Systems and system models

Transcription

1 Lecture A1 : Systems and system models Jan Swevers July 2006 Aim of this lecture : Understand the process of system modelling (different steps). Define the class of systems that will be considered in this course. How to derive a physical model of a system. 0-0

2 Lecture A1 : Systems and system models 1 Outline of this lecture System modelling physical and mathematical models input/output- vs. state space models Classification of systems Define the class of systems that will be considered in this course How to derive a physical model?

3 Lecture A1 : Systems and system models 2 System modelling Two basic steps can be distinguished : Derive a physical model : a hypothetical system model a good approximation of the main characteristics simpler and therefore more suited for a mathematical description. Derive a mathematical model : this is a set of equations solving them requires mathematical techniques it is important that you can interpret the solution (physical interpretation)

4 Lecture A1 : Systems and system models 3 Two types of mathematical models : Input/Output-models ( black-box ) Describes the response of the system (outputs) to external influences/excitations (inputs). State space models output is also a function of the internal state of the system state space models allow us to analyze internal variables of the systems System analysis based on system models aims at determining the system output based on a known system input (and the initial state of the system).

5 Lecture A1 : Systems and system models 4 Classification of systems linear and nonlinear systems continuous-time and discrete-time systems time-invariant and time-varying systems lumped-parameter and distributed-parameter systems static and dynamic systems

6 Lecture A1 : Systems and system models 5 Linear and nonlinear systems Assume that for the considered system, input u(t) yields output y(t). The considered system is linear if the following two requirements are fulfilled : homogeneity : input au(t) yields output ay(t). additivity : if u 1 (t) yields y 1 (t) and u 2 (t) yields y 2 (t), then u 1 (t) + u 2 (t) yields y 1 (t) + y 2 (t). Summarized: au 1 (t) + bu 2 (t) yields ay 1 (t) + by 2 (t). In this course: we consider mainly linear systems, because of: simplicity theory of linear systems is a closed theory existence of techniques to approximate nonlinear systems by linear models...

7 Lecture A1 : Systems and system models 6 Continuous-time and discrete-time systems continuous-time systems : input and output signals are function of the continuous time, and the system description is a (set of) differential equation(s). discrete-time systems : input and output signals change only at discrete time instances, that is, time is a discrete variable, and the system description is a (set of) difference equation(s). Example: continu systeem u(t) systeem y(t) u(t) T A/D u(kt) discreet systeem systeem y(kt) T D/A y(t)

8 Lecture A1 : Systems and system models 7 Example : Population growth If x i (k) represents the number of i-year-old people (counted at the beginning of year k); b i the yearly offspring of one i-year-old person; a i the fraction of the group of i-year-old people that survives the year; y(k) the total population at the beginning of year k. Discrete-time model : a set of difference equations: x 1 (k + 1) = b 1 x 1 (k) + b 2 x 2 (k) b N x N (k) x i+1 (k + 1) = a i x i (k), i = 1, 2...N 1 y(k) = x 1 (k) + x 2 (k) +... x N (k)

9 Lecture A1 : Systems and system models 8 Time-invariant and time-varying systems Time-varying systems : are systems of which the system operator changes with time. The system operator is defined as the set of rules for computing the system outputs given the inputs. These systems are described by differential of difference equations with time-varying coefficients. Time-invariant systems : are described by difference or differential equations with constant coefficients. If y(t) is the response of the time-invariant system to input u(t), then, y(t T) is the response to u(t T).

10 Lecture A1 : Systems and system models 9 Lumped-parameter and distributed-parameter systems Lumped-parameter systems : Are described by ordinary difference or differential equations. The effect of the input is felt simultaneously throughout the whole system The system operator is thereby independent of any spatial coordinates. Distributed-parameter systems : Are described by partial difference and differential equations. The effect of the input is not felt simultaneously throughout the system. A transmission line that must be modelled by a partial differential equation involving derivatives with respect to time and space, and chemical diffusion processes are two examples. A simple RLC circuit might have to be modelled as a distributed-parameter system if the frequency content of the excitation is high enough.

11 Lecture A1 : Systems and system models 10 Static and dynamic systems Static systems or memoryless systems: A static system is one for which the present values of the outputs depend only on the present values of the inputs. Example: an electrical resistor e = Ri. Dynamic systems or systems with memory : A dynamic system is one for which the present values of the outputs depend on the present and past values of the inputs. Example : an electrical capacitor C, the current i(t) is the input and the voltage e(t) across the capacitor is the considered output of this system: e(t) = 1 C t i(τ)dτ The output at any time t depends on the entire past history of the input.

12 Lecture A1 : Systems and system models 11 Define the class of systems that will be considered in this course This course considers almost exclusively the following class of systems: linear, time-invariant systems, with lumped-parameters, that are static or dynamic.

13 Lecture A1 : Systems and system models 12 How to derive physical models? A physical model of a dynamic system is a imaginary physical system, that is a sufficiently accurate simplification of the original system, allowing us to still perform valid and reliable system analysis (from and engineering point of view). Building a physical model requires simplification of the real system dynamics: neglecting side-effects, neglecting external influences, neglecting distributed-parameter effects, neglecting non-linear dynamics, neglecting time variations of parameters.

14 Lecture A1 : Systems and system models 13 Neglecting side-effects : Examples every electrical resistor has a certain inductance and capacitance that are however negligible if the frequency of the applied signals is not too high. every mechanical spring has a certain mass, that is negligible if the frequency of the applied forces is not too high....

15 Lecture A1 : Systems and system models 14 Neglecting external influences/loading : The sources that will be considered in this course are ideal, for example: the voltage of a battery does not drop as the supplied current increases....

16 Lecture A1 : Systems and system models 15 Neglecting distributed-parameter effects : Avoid partial differential equations, for example: neglect the mass of a leaf-spring approximate the dynamics of large flexible systems (antenna s, panels) with finite elements: a concatenation of small masses, springs and dampers.

17 Lecture A1 : Systems and system models 16 Linearize nonlinear elements : Linear differential equations with constant coefficients are much easier to solve. Therefore, for example linearize: non-linear spring characteristics hydraulic systems

18 Lecture A1 : Systems and system models 17 Linearize nonlinear elements (2) Pendulum: M L θ g θ F c g F From this figure it follows that: The moment due to gravity about the pivot point equals : +MgLsinθ The sum of moments about the pivot point: ML 2 d2 θ + MgLsinθ = 0, dt2 yielding the following equation of motion: d 2 θ dt 2 = g L sinθ

19 Lecture A1 : Systems and system models 18 Linearize nonlinear elements (3) Linearization about θ = 0, while approximating sinθ as: sinθ sin 0 + d dθ sinθ θ=0(θ 0) +... sin 0 + θ θ yields the following linear differential equation: d 2 θ dt 2 + g L θ = 0

20 Lecture A1 : Systems and system models 19 Time-invariant parameters : The time-variation of a parameter can be neglected if this time-variation is slow compared to the time-variations of the input/output signals, for example: the heating of an electrical motor changes is parameters changing friction properties as a function of time and place...

21 Lecture A1 : Systems and system models 20 Aim of this lecture : Understand the process of system modelling (different steps): derive a physical model based on that, derive a mathematical model We defined the class of systems that will be considered in this course. We explained how to derive a physical model of a system.