STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Classifications of systems Lecture 2 Systems and Control Theory
STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Overview Based on the number of inputs and outputs ( SISO , SIMO, MISO, MIMO, autonomous ) Continuous vs. Discrete time Linear vs. Nonlinear Causal vs. Non-causal Time-invariant vs. Time-varying Lumped vs. Distributed bold : this course Systems and Control Theory 2
STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Based on the number of inputs and outputs SISO: Single Input Single Output SIMO: Single Input Multiple Outputs MISO: Multiple Inputs Single Output MIMO: Multiple Inputs Multiple Outputs Autonomous : No inputs ( one or more outputs) Systems and Control Theory 3
STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Continuous vs. Discrete time During this course we will discuss both simultaneously, in order to emphasize the similarities (and differences) A continuous system has continuous input and output signals We denote continuous time by 𝑢 ∈ ℝ We denote functions of continuous time with round brackets, e.g.: 𝑦 𝑢 A discrete system has discrete input and output signals We denote discrete time by 𝑙 ∈ ℤ We denote functions of continuous time with square brackets, e.g.: 𝑦 𝑙 Systems and Control Theory 4
STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Continuous vs. Discrete time Continuous Discrete For every moment 𝑢 ∈ ℝ , the For each timestep 𝑙 ∈ ℤ , the system has: system has: A vector of inputs 𝑣 𝑢 A vector of inputs 𝑣 𝑙 A vector of outputs 𝑧 𝑢 A vector of outputs 𝑧[𝑙] A vector of states 𝑦 𝑢 A vector of states 𝑦[𝑙] Systems and Control Theory 5
STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Linear vs. Nonlinear: A linear system Definition: A system is linear if 𝑣 1 𝑢 → 𝑧 1 𝑢 (input 𝑣 1 𝑢 results in output 𝑧 1 𝑢 ) and 𝑣 2 𝑢 → 𝑧 2 (𝑢) imply that 𝛽 𝑣 1 𝑢 + 𝛾 𝑣 2 𝑢 → 𝛽 𝑧 1 𝑢 + 𝛽 𝑧 2 (𝑢) Properties of a linear system (contained in the definition): Superposition Homogeneity Systems and Control Theory 6
STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Linear vs. Nonlinear: A linear system Properties of a linear system (contained in the definition): Superposition 𝑣 𝑏 𝑢 → 𝑧 𝑏 𝑢 , 𝑣 𝑐 𝑢 → 𝑧 𝑐 𝑢 ⇔ 𝑣 𝑏 𝑢 + 𝑣 𝑐 𝑢 → 𝑧 𝑏 𝑢 + 𝑧 𝑐 𝑢 This means the output of a system can be found by splitting up the input and solving it separately (analogous to the homogeneous part of an ordinary differential equation) Homogeneity 𝛽𝑣 𝑢 → 𝛽𝑧 𝑢 How to recognize a linear system: Linear in all of the variables No constant factors Systems and Control Theory 7
STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Linear vs. Nonlinear: A linear system Examples 𝑦 = 𝑣 𝑧 = 𝑦 + 2𝑣 Linearity of this system is easily verified, based on the linearity of the derivative: 𝛽 𝑦 𝑏 𝑢 + 𝛾 𝑦 𝑐 𝑢 = 𝛽𝑣 𝑏 𝑢 + 𝛾𝑣 𝑐 𝑢 𝛽 𝑧 𝑏 𝑢 + 𝛾 𝑧 𝑐 𝑢 = 𝛽𝑦 𝑏 𝑢 + 𝛾𝑦 𝑐 𝑢 + 2𝛽𝑣 𝑏 𝑢 + 2𝛾𝑣 𝑐 𝑢 𝑦 1 = 𝑣 3 𝑦 2 = 2 𝑦 1 + 𝑣 𝑧 = 𝑏𝑦 1 − 𝑦 2 + 2𝑣 Systems and Control Theory 8
STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Linear vs. Nonlinear: Autonomous linear systems Systems and Control Theory 9
STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Linear vs. Nonlinear: violating homogeneity Systems and Control Theory 10
STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Linear vs. Nonlinear: Nonlinear systems Systems and Control Theory 11
STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Linear vs. Nonlinear: Nonlinear systems Using a term like nonlinear systems is like referring to the bulk of zoology as the study of non-elephant animals. - S. Ulam Systems and Control Theory 12
STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Linear vs. Nonlinear Predominantly linear Inherently nonlinear Simple electrical systems Chemical systems Circuits with ideal Biological systems resistors, capacitors and Economical systems inductors More involved electrical or Simple mechanical systems mechanical systems Systems with ideal … springs Systems and Control Theory 13
STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Linear vs. Nonlinear Reality is nonlinear However, this course will only deal with linear systems (elephants) Why we prefer linear systems: The previously mentioned properties will allow for a thorough study of the system Why we are allowed to use linear systems, even in a nonlinear setting: You can linearize around an equilibrium point (we will do this in the next lecture) Systems and Control Theory 14
STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Causal vs. Non-causal A causal system only depends on the present and the past, not on the future An non-causal system (also) depends on the future (Almost) all physical systems are causal: A telephone: • It will not ring for future calls Any human: • Is a system that will only react on inputs it has already received • If we react because we expect something to happen in the future, then that expectation arose from past or present inputs Examples come from: http://www.deekshith.in/2013/03/causal-and-non-causal-systems-better-explained.html Systems and Control Theory 15
STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Causal vs. Non-causal How do non-causal systems arise? A possibility is by greatly reducing the complexity of a system, in which some causes of events are taken out of the equations: Take for instance an economics model in which we model the consumption (output) An incredible amount of factors determine this, but we only have the employment numbers (input) Current and past employment numbers determine consumption, but when someone gets fired, they will continue to work for several weeks in most instances, but their consumption will drop immediately a correct model for this relation would have to be non-causal The non-causal model for this input-output relation is not useful if you want to determine the level of consumption You could use the relation to see a drop in employment, before it is visible in the employment numbers Systems and Control Theory 16
STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Causal vs. Non causal Examples of non causal systems: expectations Modelling housing prices People are willing to offer more for houses if they expect rising prices It is hard to measure the expectations of housing prices Sometimes economists use their own predictions of housing prices to replace the expectations Systems and Control Theory 17
STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Causal vs. non causal Original image Removed details Highlight borders Systems and Control Theory 18
STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Causal vs. Non-causal The output only depends (directly and indirectly) on the input up to time 𝑙 Systems and Control Theory 19
STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Time-invariant vs. Time-varying Systems and Control Theory 20
STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Time-invariant vs. Time-varying Examples of time-varying systems: The properties of an electrical circuit slowly change over time The human body also has many changing properties Systems affected by night and day (Heating of buildings), when those aspects were ignored in the model Examples of (de facto) time-invariant systems: A system that describes a physical law, for instance a system with two masses as its input and their attractive force as an output In practice we approximate all systems whose properties change much slower than the variables as time-invariant Systems and Control Theory 21
STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Lumped vs. Distributed 𝑣 𝑢 𝑧 𝑢 Reactor Systems and Control Theory 22
Recommend
More recommend
