Introduction Eigenmodes Convolution and Response ODEs and Linear - PowerPoint PPT Presentation

BENG 221 Lecture 1 Introduction BENG 221 Mathematical Methods in Bioengineering Overview Ordinary Differential Equations Lecture 1 Linear Time-Invariant Systems Introduction Eigenmodes Convolution and Response ODEs and Linear Systems Functions Further Reading Gert Cauwenberghs Department of Bioengineering UC San Diego 1.1

BENG 221 Course Objectives Lecture 1 Introduction Overview 1. Acquire methods for quantitative analysis and prediction of Ordinary biophysical processes involving spatial and temporal Differential Equations dynamics: Linear ◮ Derive partial differential equations from physical principles; Time-Invariant Systems ◮ Formulate boundary conditions from physical and operational Eigenmodes constraints; Convolution and ◮ Use engineering mathematical tools of linear systems Response Functions analysis to find a solution or a class of solutions; Further Reading 2. Learn to apply these methods to solve engineering problems in medicine and biology: ◮ Formulate a bioengineering problem in quantitative terms; ◮ Simplify (linearize) the problem where warranted; ◮ Solve the problem, interpret the results, and draw conclusions to guide further design. 3. Enjoy! 1.2

BENG 221 Today’s Coverage: Lecture 1 Introduction Overview Ordinary Differential Equations Ordinary Differential Equations Linear Time-Invariant Systems Eigenmodes Linear Time-Invariant Systems Convolution and Response Functions Further Reading Eigenmodes Convolution and Response Functions 1.3

BENG 221 ODE Problem Formulation Lecture 1 Introduction Solve for the dynamics of n variables x 1 ( t ) , x 2 ( t ) , . . . x n ( t ) in time Overview (or other ordinate) t described by m differential equations: Ordinary Differential Equations ODE Linear Time-Invariant Systems Eigenmodes dt , . . . d k x 1 � x 1 , dx 1 F i dt k , . . . Convolution and Response Functions dt , . . . d k x 2 x 2 , dx 2 Further Reading dt k , . . . (1) dt , . . . d k x n x n , dx n � = 0 dt k for i = 1 , . . . m , where m ≤ n and k ≤ n . Solutions are generally not unique. A unique solution, or a reduced set of solutions, is determined by specifying initial or boundary conditions on the variables. 1.4

BENG 221 ODE Examples Lecture 1 Introduction Kinetics of mass m with potential V ( x ) : Overview Ordinary Differential � 2 1 � dx Equations 2 m + V ( x ) = 0 (2) Linear dt Time-Invariant Systems Eigenmodes Two masses with coupled potential V ( x ) : Convolution and Response Functions � 2 � 2 � dx 1 � dx 2 Further Reading 1 + 1 2 m 1 2 m 2 + V ( x 1 , x 2 ) = 0 (3) dt dt Second order nonlinear ODE: � 2 x d 2 x dt 2 = 1 � dx (4) 2 dt 1.5

BENG 221 ODE in Canonical Form Lecture 1 Introduction In canonical form , a set of n ODEs specify the first order derivatives of each of n single variables in the other variables, Overview without coupling between derivatives or to higher order Ordinary Differential derivatives: Equations Linear Canonical ODE Time-Invariant Systems Eigenmodes Convolution and dx 1 Response = f 1 ( x 1 , x 2 , . . . x n ) Functions dt Further Reading dx 2 = f 2 ( x 1 , x 2 , . . . x n ) (5) dt . . . dx n = f n ( x 1 , x 2 , . . . x n ) . dt Not every system of ODEs can be formulated in canonical form. An important class of ODEs that can be formulated in canonical form are linear ODEs . 1.6

BENG 221 Canonical ODE Examples Lecture 1 Introduction Overview Ordinary Differential Amplitude stabilized quadrature oscillator: Equations Linear Time-Invariant − y − ( x 2 + y 2 − 1 ) x dx � = Systems dt (6) x − ( x 2 + y 2 − 1 ) y dy Eigenmodes = dt Convolution and Response Functions Any first-order canonical ODE without explicit time dependence Further Reading can be solved by separation of variables, e.g. , dx dt = ( 1 + x 2 ) / x (7) 1.7

BENG 221 Initial and Boundary Conditions Lecture 1 Introduction Initial conditions are values for the variables, and some of their Overview derivatives of various order, specified at one initial point in time Ordinary t 0 , e.g. , t = 0: Differential Equations Linear IC Time-Invariant Systems Eigenmodes d i x j Convolution and dt i ( 0 ) = c ij , i = 0 , . . . m , j = 1 , . . . n . (8) Response Functions Further Reading Boundary conditions are more general conditions linking the variables, and/or their first and higher derivatives, at one or several points in time t k : BC g l ( . . . , d i x j dt i ( t k ) , . . . ) = 0 . (9) 1.8

BENG 221 ICs in Canonical Form Lecture 1 Introduction Overview For ODEs in canonical form, initial conditions for each of the Ordinary Differential variables are specified at initial time t 0 , e.g. , t = 0: Equations Linear Time-Invariant Canonical IC Systems Eigenmodes Convolution and Response x 1 ( 0 ) = c 1 Functions Further Reading x 2 ( 0 ) = c 2 (10) . . . x n ( 0 ) = c n ICs for first or higher order derivatives are not required for canonical ODEs. 1.9

BENG 221 Linear Canonical ODEs Lecture 1 Introduction Linear time-invariant (LTI) systems can be described by linear canonical ODEs with constant coefficients: Overview LTI ODE Ordinary Differential Equations d x Linear dt = A x + b Time-Invariant (11) Systems Eigenmodes with x = ( x 1 , . . . x n ) T , and with linear initial conditions: Convolution and Response Functions LTI IC Further Reading x ( 0 ) = e (12) or linear boundary conditions at two, or more generally several, time points: LTI BC C x ( 0 ) + D x ( T ) = e (13) 1.10

BENG 221 LTI Systems ODE Examples Lecture 1 Introduction Overview Examples abound in biomechanical and electromechanical Ordinary Differential systems (including cardiovascular system, and MEMS Equations biosensors), and more recently bioinformatics and systems Linear Time-Invariant biology. Systems Eigenmodes A classic example is the harmonic oscillator ( k = 0), and more Convolution and generally the damped oscillator or resonator : Response Functions Further Reading � du = v dt (14) m dv = − k u − γ v + f ext dt where u represents some physical form of deflection, and v its velocity. Typical parameters include mass/inertia m , stiffness k , and friction γ . The inhomogeneous term f ext represents an external force acting on the resonator. 1.11

BENG 221 LTI Homogeneous ODEs Lecture 1 Introduction In general, LTI ODEs are inhomogeneous . Homogeneous LTI ODEs are those for which x ≡ 0 is a valid solution. This is the Overview case for LTI ODEs with zero driving force b = 0 and zero IC/BC: Ordinary Differential LTI Homogeneous ODE Equations Linear Time-Invariant Systems d x dt = A x (15) Eigenmodes Convolution and Response Functions LTI Homogeneous IC Further Reading C x ( 0 ) = 0 (16) LTI Homogeneous BC C x ( 0 ) + D x ( T ) = 0 . (17) Eigenmodes , arbitrarily scaled non-trivial solutions x � = 0, exist for under-determined IC/BC (rank-deficient C and D ). 1.12

BENG 221 Eigenmode Analysis Lecture 1 Introduction Overview Eigenvalue-eigenvector decomposition of the matrix A yields the Ordinary Differential eigenmodes of LTI homogeneous ODEs. Let: Equations Linear Time-Invariant A x i = λ i x i (18) Systems Eigenmodes with eigenvectors x i and corresponding eigenvalues λ i . Then Convolution and Response Functions Eigenmodes Further Reading x ( t ) = c i x i e λ i t (19) are eigenmode solutions to the LTI homogeneous ODEs (15) for any scalars c i . There are n such eigenmodes, where n is the rank of A (typically, the number of LTI homogeneous ODEs). 1.13

BENG 221 Orthonormality and Inhomogeneous IC/BCs Lecture 1 Introduction The general solution is expressed as a linear combination of Overview eigenmodes: Ordinary n Differential Equations � c i x i e λ i t x ( t ) = (20) Linear Time-Invariant i = 1 Systems For symmetric matrix A ( A ij = A ji ) the set of eigenvectors x i is Eigenmodes orthonormal : Convolution and Response x T i x j = δ ij (21) Functions Further Reading so that the solution to the homogeneous ODEs (15) with inhomogeneous ICs (12) reduces to c i = x T i x ( 0 ) , or: LTI inhomogenous IC solution (symmetric A ) n � x T i x ( 0 ) x i e λ i t x ( t ) = (22) i = 1 1.14

BENG 221 Superposition and Time-Invariance Lecture 1 Introduction Overview Linear time-invariant (LTI) homogeneous ODE systems satisfy Ordinary the following useful properties: Differential Equations Linear LTI ODE Time-Invariant Systems 1. Superposition: If x ( t ) and y ( t ) are solutions, then Eigenmodes A x ( t ) + B y ( t ) must also be solutions for any constant A Convolution and Response and B . Functions Further Reading 2. Time Invariance: If x ( t ) is a solution, then so is x ( t + ∆ t ) for any time displacement ∆ t . An important consequence is that solutions to LTI inhomogeneous ODEs are readily obtained from solutions to the homogeneous problem through convolution . This observation is the basis for extensive use of the Laplace and Fourier transforms to study and solve LTI problems in engineering. 1.15