Rural/Urban Migration: The Dynamics of Eigenvectors
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1 * Analysis of the Dynamic Structure of a System * Rural/Urban Migration: The Dynamics of Eigenvectors EGR 326 April 11, Develop the system model and create the Matlab/Simulink model 2. Plot and interpret the system behavior and system evolution Find and interpret closed form solution x, y 3. Calculate the eigenvectors and eigenvalues 4. Interpret the eigenvectors and eigenvalues in terms of the state variables and the system behavior * Model Physical System * You need to know what the state variables are and their order in order to make sense out of the eigenanalysis Be familiar with the coefficients of the state variables and their physical significance In order to interpret the significance of the modes with respect to the physical system structure. (These are the elements in the A matrix) Migration Example R[k] and u[k]: Population of country divided into rural and urban, r[k] and u[k] a: Annual growth in both rural and urban areas, typically > 1 1
2 Migration Example R[k] and u[k]: Population of country divided into rural and urban, r[k] and u[k] a: Annual growth in both rural and urban areas, typically > 1 b: Migration factor from rural to urban areas Positive and < a g: The optimal rural base The percentage of population in rural areas to support the total population A measure of rural productivity Original Migration Model r[k +1] = αr[k] β{r[k] γ(r[k]+ u[k])} u[k +1] = αu[k]+ β{r[k] γ(r[k]+ u[k])} Expected Behavior? (handout) What do you anticipate for the system behavior under the following parameters? (α = growth rate, β = rural migration factor, γ = optimal rural base) α = 1, β = 0.5, γ = 0.7 α = 1, β = 0.7, γ = 0.7 α = 1, β = 0.5, γ = 0.95 α = 1, β = 0.5, γ = 0.7 α = 1, β = 0.7, γ = 0.7 α = 1, β = 0.5, γ = 0.95 α = 1.02, β = 0.5, γ = 0.7 α = 1.02, β = 0.5, γ = 0.95 α = 1.02, β = 2, γ = 0.7 2
3 Plotting the State Vector α = 1, β = 0.5, γ = 0.7 α = 1, β = 0.7, γ = 0.7 α = 1, β = 0.5, γ = 0.95 α = 1, β = 0.5, γ = 0.7 α = 1, β = 0.7, γ = 0.7 α = 1, β = 0.5, γ = 0.95 α = 1.02, β = 0.5, γ = 0.7 NEXT: α = 1.02, β = 0.5, γ = 0.7 3
4 Plotting the State Vector α = 1.02, β = 0.7, γ = 0.95 α = 1.02, β = 0.5, γ = 0.7 Deduce Eigenvalues -vectors Use deduction for this very stylized problem A characteristic of the system is that it grows at a rate of Try this as the first eigenvalue This is the growth rate of what element of the population? (This is the first left eigenvector f T =(1 1)x[k] = ) Verify with f T A = λf T ) Deduce 1 st Right Eigenvector The corresponding right eigenvector defines the state vector, the r[k] and u[k] values (i.e., the population distribution), required for these variables to grow at the rate λ 1 = For both r[k] and u[k] to grow at rate there must be no net migration Zero rural imbalance The ratio of r[k]:u[k] that will be maintained for all time Thus r[k] = γ(r[k] + u[k]), or the ratio of r[k]:u[k] must = γ : (1 γ) Therefore v 1 = r[k +1] = αr[k] β{r[k] γ(r[k]+ u[k])} u[k +1] = αu[k]+ β{r[k] γ(r[k]+ u[k])} 4
5 Original Migration Model Discussion à The second eigenvalue is (α β) ß 1. The total population grows at rate α each year 2. If initially more than a fraction γ of the population is rural, this rural imbalance changes by the factor (α β) à net migration to urban areas 3. Eventually the imbalance disappears A = ( α β(1 γ) ' β(1 γ) βγ ) + α βγ* r[k +1] = αr[k] β{r[k] γ(r[k]+ u[k])} u[k +1] = αu[k]+ β{r[k] γ(r[k]+ u[k])} Diagonalized Migration Model % 1 % 5
6 Interpret: Right Eigenvectors A mode of the system Represent a mode shape the relative activity of the state variables when a specific mode is excited Represent special directions in the state space once pointing in this direction, the state vector remains so Analyzed used as a vector Summary Eigenvalues are the characteristic values of the system Population growth factor, for example Right eigenvectors are interpreted as vectors, state vectors, in state space Left eigenvectors are interpreted as scalar valued functions of the state variables such as total_population = r[k] + u[k] Overview PART II: Introduction to Controllability EGR 326 April 11, 2019 Discussion: what is control? Open- and closed-loop control State- and output-feedback control Controllability (Chapter 3) Observability (Chapter 4) 6
7 Two Fundamental Control Problems 1) Controllability of a system Can we determine a sequence of inputs to control system behavior 2) Observability of a system Can an unknown initial state vector x(0) be determined (or observed) from monitoring the system output First Fundamental Control Problem 1) Can we determine a sequence of control inputs (u or u[k]) 2) That will move a known initial state vector x(0) to the origin of the state space 3) In a finite amount of time First step is to determine if we can do this. Second step is to actually do it. Controllable System Not Fully Controllable 7
8 Controllability Consider the discrete time system x[k + 1] = Ax[k] + Bu[k] y[k + 1] = Cx[k + 1] + Du[k + 1] Equation x[k] from previous slide: x[k] = A k x[0] + Bu[k-1] + ABu[k-2] + + A k-1 Bu[0] Define the following matrices U = P = Controllability Controllability Define the controllability matrix P = [ B AB A 2 B A k-1 B ] Controllability We know our initial state, and we define the final state to be the origin, so Set x[k] = 0 since the problem is defined as moving the state vector to the origin, Therefore, from x[k] = A k x[0] + PU, with x[k] = 0 Write this as PU = -A k x[0] 8
9 Controllability Note in the previous equation The matrix U is the unknown (what we want to find) A k x[0] is a constant We want: U = P -1 (-A k )x[0] Therefore P -1 must exist for U to exist P must span the state space There must be n independent columns in P Rank of P must equal n (with n = order of system) If all (any) of these conditions are true, then We can find the desired sequence of control inputs, U The system is controllable Controllability Summarized A property of the coupling between the input and the state, and so involves only matrices A and B A system is controllable if it is possible to find an input function, or input sequence, that will drive the state to the origin in a finite amount of time Controllability, Chapter 3 Controllability examples, 3.2 Ex 3.1: Using coordinate transformation for interpretation Ex 3.3: Controllable canonical form Ex 3.4: Sparse matrix example Trace through x coupling and how inputs affect the x Text Example 3.1 1) Construct P and calculate P 2) Investigate using transformation x=tz 3) Find z = A h z + B h u, and interpret x = 1 5 x+ " 8 4 % " " z 1 z 2 % = " 2 2 x 1 x 1 + x 2 % u % 9
10 A = 1 5 B = >> P = [B A*B] = [ ] Construct Matrix P >> P = [B A*B (A^2)*B] = [ ] >> P = [B A*B (A^2)*B (A^3)*B (A^4)*B] P = [ ] Text Example 3.1 Take 2 1) Construct P and calculate P à not controllable 2) Investigate using transformation x=tz 3) Find z = A h z + B h u, and interpret x = 1 5 x+ " 8 4 % " " z 1 z 2 % = " 2 2 u % x 1 x 1 + x 2 % = 1 0 x " 1 1 % Using Coordinate Transformation Given z = [1 0 * x = T -1 x 1 1] Therefore x = inv([1 0 ) * z = [1 0 * z ( 1 1]) -1 1] Using Coordinate Transformation Calculate A hat = T -1 AT = >> A = [1 5; 8 4] >> T = [1 0 ; -1 1] >> A hat = inv(t) * A * T = [ ] Calculate B hat = T -1 B = [1 0 * [-2 =? 1 1] 2] 10
11 Using Coordinate Transformation Example 3.1 Discussion Calculate A hat = T -1 AT = >> A = [1 5; 8 4] >> T = [1 0 ; -1 1] >> A hat = inv(t) * A * T = [-4 5 " z 1 z 2 % = 4 5 " 0 9 %" z 1 z 2 0 9] % + " 2 0 u % Calculation of P matrix and analysis Examine the role of A hat and B hat How are the modal state variables coupled? How are the input channels connected into the system dynamics? In this particular example, what does this transformation tell us about our ability to control (to affect the evolution of) z 2? " x = a 0 a 1 a 2 a 3... a n 1 y = " b 0 b 1... b n 1 Text Example 3.3 Recall: state space model from I/O equation y n + a n 1 y n a 2 y + a 1 y + a 0 y = u n + b n 1 u n b 0 u The state space model from an I/O equation will be in controllable canonical form % x = Cx % " ' ' ' ' x + ' ' ' % ' ' ' ' u = Ax + Bu ' ' ' Example 3.3 Results Read the example in Chapter 3 The determinant of P from this form of state-space model is P = -1 Which is independent of the A matrix coefficients, a i, (system matrix) so This form of state-space model is always controllable (Note also that the controllability matrix P is independent of C matrix coefficients) 11
12 " Example 3.4 Intuition Step through example on your own for details For now, examine the internal dynamic coupling and connections of inputs into the system à Do you expect this system to be controllable? Why? x 1 x 2 x 3 = x 4 x 5 % " x 1 x 2 x 3 + x 4 % " x 5 % " u 1 " u 2 % % Section 3.3: Modal Transformation Controllability using eigenanalysis Find: controllability is invariant with respect to coordinate transformations For the modal transformation (diagonal canonical form), the system is controllable iff: Every element of B m is non-zero (B DCF ) The eigenvalues are distinct Modal Transformation Explain why this condition must hold for a system to controllable Every element of B m is non-zero (The eigenvalues are distinct) " z 1 λ z 2 0 λ z 3 = 0 0 λ z λ 4 0 z 5 % " λ 5 % " z 1 z 2 z 3 + z 4 z 5 % " a b c d e u % 12
13 Controllability Questions Under what conditions can we find a control sequence, U, that will drive a given x[k] to the origin? Can we drive an arbitrary x[k] to zero? If we can find this solution, U, is it unique? U k = P kt [P k P kt ] -1 A k x[0] (minimum norm) To move x[0] to an arbitrary x[k] x[k] = A k x[0] + PU Then we need to solve x[k] A k x[0] = PU We will use Matlab to help us here. Required Control Effort The magnitude and number of the elements in U indicates the physical control effort needed (the energy required) to move the state vector to the origin As the number of time steps increases (k or t), the amount of time required to move the state vector increases, But each individual control input will require less energy (compared to moving the state vector in fewer steps) Controllability Summarized Two controllability criteria If A has distinct eigenvalues, then The system is controllable iff there are no zero rows in B m = M -1 B The system is controllable iff the controllability matrix P has rank n P = [ B AB A 2 B A n-1 B ] Recap of Controllability We developed this expression x[k] = A k x[0] + PU 1) Which value(s) are we solving for? 2) How do we know if there is solution? 3) How do we use the value(s) we find(assuming we can find them)? 4) How many input steps do we need? 13
14 (Roadmap for Chapter 3) Read discussion for continuous time pages 108 to 110 Read sections and examples indicated in these slides, at a minimum Skip pp Be comfortable using Matlab commands discussed in 3.5 forward Be familiar with the continuing examples Be comfortable using both discrete and continuous time notation and models 14
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