MSc in Computer Engineering, Cybersecurity and Artificial Intelligence, Fault Diagnosis and Estimation in Dynamical Systems (FDE), a.a. 2019/2020, Lecture 2 A brief review of analysis of continuous-time linear dynamical systems Prof. Mauro Franceschelli Dept. of Electrical and Electronic Engineering University of Cagliari, Italy Monday, 23rd March 2020 1 / 34
Outline Introductions to systems and models Analysis of IO dynamical systems Modes and their classification 2 / 34
What is a dynamical system? an example (water tank) • Input flow rate: q 1 ( t ) = a • Output flow rate: q 2 ( t ) = b • Water volume inside the tank: V ( t ) a c V h d b d dt V ( t ) = q 1 ( t ) − q 2 ( t ) . 3 / 34
Difference between systems and models • Systems are a collection of interacting parts, either natural or artificial, which evolve in time. • A model of a system is a mathematical abstraction intended to approximate and predict the behavior of the system under proper assumptions. • The term ”dynamical system” refers to a mathematical model. In this course we often call a ”dynamical system” simply a ”system” for brevity’s sake. • The properties of dynamical systems can be used for their classification (different models imply different methodologies for their analysis). Example : A dynamical system is linear or non-linear depending on whether its model has the property of linearity. On the other hand, a linear dynamical system can approximate the evolution in time of a nonlinear system under certain conditions. 4 / 34
Difference between systems and models • Systems are a collection of interacting parts, either natural or artificial, which evolve in time. • A model of a system is a mathematical abstraction intended to approximate and predict the behavior of the system under proper assumptions. • The term ”dynamical system” refers to a mathematical model. In this course we often call a ”dynamical system” simply a ”system” for brevity’s sake. • The properties of dynamical systems can be used for their classification (different models imply different methodologies for their analysis). Example : A dynamical system is linear or non-linear depending on whether its model has the property of linearity. On the other hand, a linear dynamical system can approximate the evolution in time of a nonlinear system under certain conditions. 4 / 34
Difference between systems and models • Systems are a collection of interacting parts, either natural or artificial, which evolve in time. • A model of a system is a mathematical abstraction intended to approximate and predict the behavior of the system under proper assumptions. • The term ”dynamical system” refers to a mathematical model. In this course we often call a ”dynamical system” simply a ”system” for brevity’s sake. • The properties of dynamical systems can be used for their classification (different models imply different methodologies for their analysis). Example : A dynamical system is linear or non-linear depending on whether its model has the property of linearity. On the other hand, a linear dynamical system can approximate the evolution in time of a nonlinear system under certain conditions. 4 / 34
Difference between systems and models • Systems are a collection of interacting parts, either natural or artificial, which evolve in time. • A model of a system is a mathematical abstraction intended to approximate and predict the behavior of the system under proper assumptions. • The term ”dynamical system” refers to a mathematical model. In this course we often call a ”dynamical system” simply a ”system” for brevity’s sake. • The properties of dynamical systems can be used for their classification (different models imply different methodologies for their analysis). Example : A dynamical system is linear or non-linear depending on whether its model has the property of linearity. On the other hand, a linear dynamical system can approximate the evolution in time of a nonlinear system under certain conditions. 4 / 34
Difference between systems and models • Systems are a collection of interacting parts, either natural or artificial, which evolve in time. • A model of a system is a mathematical abstraction intended to approximate and predict the behavior of the system under proper assumptions. • The term ”dynamical system” refers to a mathematical model. In this course we often call a ”dynamical system” simply a ”system” for brevity’s sake. • The properties of dynamical systems can be used for their classification (different models imply different methodologies for their analysis). Example : A dynamical system is linear or non-linear depending on whether its model has the property of linearity. On the other hand, a linear dynamical system can approximate the evolution in time of a nonlinear system under certain conditions. 4 / 34
Linear and non-linear systems Input/output (IO) models A model is linear if the principle of superposition principle holds: , � cause c 1 � effect e 1 cause ( ac 1 + bc 2 ) � = ⇒ cause c 2 � effect e 2 effect ( ae 1 + be 2 ) non-linear : if the superposition principle does not hold. Single Input-Single Output (SISO) Linear IO model : it is represented by a linear differential equation of order n with possibly time-varying coefficients: a 0 ( t ) y ( t ) + · · · + a n ( t ) y ( n ) ( t ) = b 0 ( t ) u ( t ) + · · · + b m ( t ) u ( m ) ( t ) . • y ( n ) ( t ) represents the n -th derivative of function y ( t ) 5 / 34
Linear and non-linear systems Input/output (IO) models A model is linear if the principle of superposition principle holds: , � cause c 1 � effect e 1 cause ( ac 1 + bc 2 ) � = ⇒ cause c 2 � effect e 2 effect ( ae 1 + be 2 ) non-linear : if the superposition principle does not hold. Single Input-Single Output (SISO) Linear IO model : it is represented by a linear differential equation of order n with possibly time-varying coefficients: a 0 ( t ) y ( t ) + · · · + a n ( t ) y ( n ) ( t ) = b 0 ( t ) u ( t ) + · · · + b m ( t ) u ( m ) ( t ) . • y ( n ) ( t ) represents the n -th derivative of function y ( t ) 5 / 34
Linear and non-linear systems State variable (VS) models Linear State Variable (SV) model : It is represented by a system of n first order differential equations and output which is linear combination of state variables and inputs: x 1 ( t ) ˙ = a 1 , 1 ( t ) x 1 ( t ) + · · · + a 1 , n ( t ) x n ( t ) + b 1 , 1 u 1 ( t ) + · · · + b 1 , r u r ( t ) . . . . . . x n ( t ) ˙ = a n , 1 ( t ) x 1 ( t ) + · · · + a n , n ( t ) x n ( t ) + b n , 1 u 1 ( t ) + · · · + b n , r u r ( t ) y 1 ( t ) = c 1 , 1 ( t ) x 1 ( t ) + · · · + c 1 , n ( t ) x n ( t ) + d 1 , 1 u 1 ( t ) + · · · + d 1 , r u r ( t ) . . . . . . y p ( t ) = c p , 1 ( t ) x 1 ( t ) + · · · + c p , n ( t ) x n ( t ) + d p , 1 u 1 ( t ) + · · · + d p , r u r ( t ) more compactly: � ① ( t ) ˙ = ❆ ( t ) ① ( t ) + ❇ ( t ) ✉ ( t ) ② ( t ) = ❈ ( t ) ① ( t ) + ❉ ( t ) ✉ ( t ) ❆ ( t ) = { a i , j ( t ) } matrix n × n ; ❇ ( t ) = { b i , j ( t ) } matrix n × r ; where ❈ ( t ) = { c i , j ( t ) } matrix p × n ; ❉ ( t ) = { d i , j ( t ) } matrix p × r 6 / 34
Linear and non-linear systems State variable (VS) models Linear State Variable (SV) model : It is represented by a system of n first order differential equations and output which is linear combination of state variables and inputs: x 1 ( t ) ˙ = a 1 , 1 ( t ) x 1 ( t ) + · · · + a 1 , n ( t ) x n ( t ) + b 1 , 1 u 1 ( t ) + · · · + b 1 , r u r ( t ) . . . . . . x n ( t ) ˙ = a n , 1 ( t ) x 1 ( t ) + · · · + a n , n ( t ) x n ( t ) + b n , 1 u 1 ( t ) + · · · + b n , r u r ( t ) y 1 ( t ) = c 1 , 1 ( t ) x 1 ( t ) + · · · + c 1 , n ( t ) x n ( t ) + d 1 , 1 u 1 ( t ) + · · · + d 1 , r u r ( t ) . . . . . . y p ( t ) = c p , 1 ( t ) x 1 ( t ) + · · · + c p , n ( t ) x n ( t ) + d p , 1 u 1 ( t ) + · · · + d p , r u r ( t ) more compactly: � ① ( t ) ˙ = ❆ ( t ) ① ( t ) + ❇ ( t ) ✉ ( t ) ② ( t ) = ❈ ( t ) ① ( t ) + ❉ ( t ) ✉ ( t ) ❆ ( t ) = { a i , j ( t ) } matrix n × n ; ❇ ( t ) = { b i , j ( t ) } matrix n × r ; where ❈ ( t ) = { c i , j ( t ) } matrix p × n ; ❉ ( t ) = { d i , j ( t ) } matrix p × r 6 / 34
Linear and non-linear systems Importance of linearity Why linearity is important? 1 Under certain conditions non-linear systems can be linearized around particular states of interest (equilibrium/operating points). 2 The qualitative behavior of linear systems approximates the behavior of many real systems. 3 Under certain conditions (small inputs) the superposition of effects holds for many non-linear systems (critical property in analogue electronics). 4 Powerful methodologies exist for the analysis, control and estimation of linear dynamical systems Example : a spring can be well approximated by a linear dynamical system for small deformations, i.e., the force F exerted by the string is proportional to its stretching x , F = − k · x . For large deformations the spring either fully compresses or stretches out. 7 / 34
Recommend
More recommend
