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Professor Walter W. OlsonDepartment of Mechanical, Industrial and Manufacturing EngineeringUniversity of ToledoLinear SystemsmkuxmkuxFixed Refe: yFixed Refe: ymkuxFixed Refe: ymmkuxFixed Refe: ymMode 1Mode 2Mixed Mode1Outline of Todays LectureReviewSystem ResponseHypothetical but common 2ne Order System ResponseStabilitystableunstablemarginal or neutral stabilityLinear SystemsFunctions of Square MatricesEigenvaluesStabilityModesHypothetical (but Common Form) 2nd Order System

Hypothetical (but Common Form) 2nd Order System 5 Phase Plots

Equilibrim PointsLimit Cycles2nd Order System Response

z

zz

System Response: Step InputThe time history of a systems outputs

Often called the system path, trajectory or time series

Transient period=settling time, tsSteady State

{OvershootMpRise time, tr

System Response: Frequency ResponseTime history with respect to a sinusoid:

Input Sin(t)Transient ResponsePhaseShift, DTAmplitudeAyPeriod,TAmplitudeAu

Types of Common Responses

General form of linear time invariant (LTI) system is expressed:The most general form of the response (the solution) is expressed:

Unstable Responses

Definition of StableA system described the solution (the response) is stable if that systems response stay arbitrarily near some value, a, for all of time greater than some value, tf.

Hypothetical (but Common Form) 2nd Order System 5 Phase Plots

Unstable

StableMarginally StableWhat is a Linear SystemsIn order to be Linear, a system f(x) must obey the rules

So why is this important for controls?

Linear Control SystemsIf we have two responses known from our system, say,

then we also can find the response to the sum of the imput:

Transformations of Linear SystemsDoes my system operate any differently is I observe it in a different coordinate system?x1x2z2z1

Transformations of Linear SystemsThe only changes are in the variables! For a linear transformation, we want it to be invertible such that we can move from one system to the other. These are known as similarity transformationsx1x2z2z1

Functions of Square MatricesWe can perform the following operations with the square matrix A:

Then we can form a polynomial in A:

We have create a function of our matrix which can be shown to to factor as

The Exponential Function of a Square Matrix AA Taylor series of the exponential function of x is

Thus we can define as the matrix exponential

In Matlab, the comand expm(A) computes And we can use this just as we would any other functionFor Example, the solution of is and

What Does this mean for a Linear System?The homogeneous solution is the solution where u(t)=0Therefore we can write a solution for the system with no input

In Matlab, this solution is displayed with the commandinitial(A,B,C,D,x0)

An Example

>> k = 250000; % Nose wheel spring stiffness N/mkt = 5000000; % Nose wheel tire stiffness in N/mb = 125000; % Nose wheel shock absorber in N/m/sm = 250000; % Mass allocated to the nose wheel kgmu = 50; % Mass of the landing gear kg>> A = [ 0 1 0 0; -k/m -b/m k/m b/m; 0 0 0 1; ...k/mu b/mu -(k+kt)/mu -b/mu];B = [0;0;0;kt/mu]; % Input MatrixC = [1,0,0,0]; % Output matrix for nose deflectionD = [ 0 ]; % Pass through matrix>> x0=[.05;0;0;0] %initial displaced 5 cmx0 = 0.0500 0 0 0>> initial(A,B,C,D,x0)

>> expm(A)ans = 0.6313 0.6874 -0.3236 0.0000 -0.6472 0.3077 -0.1556 -0.0000 0.0156 0.0401 -0.0191 0.0000 -0.0383 -0.0036 0.0013 -0.0000What is a measure of stability?If you have been paying attention you noted that if the system terms were such that

the system was stable!So, can we evaluate in out state space model? If the system is Linear, we can.

EigenvaluesAs we are about to find out, the eigenvalues are the key to determining stabilityFor a square matrix with n rows, the determinant will form an n degree polynomial of the form

Eigenvalues are the roots of this polynomial, that is

Eigenvectors x are the solution to the equation

Many methods exist to find the values of eigenvalues and eigenvectorsIn Matlab, the function eig(A) computes the eigenvaluesFor a 2nd order system

Determination of Stabilityfrom EigenvaluesContinuous Time

Discrete Time

Unstable

Stable

Asymptotic Stability

Hypothetical (but Common Form) 2nd Order System 5 Responses

>> A = [0 1;-1,-10];>> eig(A)ans = -0.1010 -9.8990>> A = [0 1;-1,-0.1];>> eig(A)ans = -0.0500 + 0.9987i -0.0500 - 0.9987i>> A = [0 1;-1,0];>> eig(A)ans = 0 + 1.0000i 0 - 1.0000i>> A = [0 1;-1,1];>> eig(A)ans = 0.5000 + 0.8660i 0.5000 - 0.8660i >> A = [0 1;-1,5];>> eig(A)ans = 0.2087 4.7913ModesMode: A pattern of motion, sometimes called a mode shapemkuxmkuxFixed Refe: yFixed Refe: ymkuxFixed Refe: ymmkuxFixed Refe: ymMode 1Mode 2Mixed ModeModesEach eigenvalue is associated with a mode of a systemEach eigenvalue is associated with an eigenvector, , such thatIf the eigenvalues are distinct, we can form the modal matrix, M, from the eigenvectors and use it to diagonalize the dynamics matrix A which will then separate each mode in the form of a differential equation:

When a set of eignevectors are repeated (equal to each other) a full set of n linear independent eignevectors may or may not exist. In that case we need to form the Jordan blocks for the repeated elements

ModesJordan blocks have the form

The number of rows and columns of the Jordan block is the number of times (the multiplicity) that the eigenvector is repeatedThen the modes are separated by using the modal matrix as before, but now producing the Jordan form

where each block now conforms to a mode.

ModesModes can be easily found using Matlab function Jordan:

The over decoupling of the system can be represented as>> AA = 1 -3 -2 -1 1 -1 2 4 5>> [V,D]=eig(A)V = -0.4082 + 0.0000i -0.4082 - 0.0000i -0.7071 -0.4082 - 0.0000i -0.4082 + 0.0000i 0.0000 0.8165 0.8165 0.7071 D = 2.0000 + 0.0000i 0 0 0 2.0000 - 0.0000i 0 0 0 3.0000>> J=jordan(A)J = 2 1 0 0 2 0 0 0 3

SummaryStabilitystableunstablemarginal or neutral stabilityLinear Systems

Functions of Square Matrices

EigenvaluesStabilityModes

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