Jacobian Linearization and State-Space Equations

Problem 1

Consider the active Magnetic Suspension System in Fig. 1. To derive a mathematical model for this system the following assumptions were made:

(i) The suspended object moves only vertically.

(ii) The permeability of the magnetic core is infinity.

(iii) Magnetic hysteresis and eddy-currents are negligible.

Under these assumptions, the dynamic equations for the system are

mx¨ = mg - 1/2 i²d ∂L(x)/∂x + w          (1)

e = Ri + d/dt (i L(x))                              (2)

y = K_gx                                                 (3)

where x(t ) [m] is the gap between the magnet and suspended object, i (t ) [A] and e(t ) [V] are the magnet current and voltage, respectively, L(x) is the inductance of the magnetic circuit, and w (t ) [N] accounts for external disturbances. The measurement signal y (t ) [V] is the (voltage) output of the gap sensor. The inductance L(x) varies with the position of the suspended object and is given by the following equation

L(x) = Q/(x + X_∞)+ L_∞                            (4)

where the parameters Q, X_∞ and L_∞ are determined by the physical characteristics of the coil, the core and the suspended object (in practice these are determined experimentally).

The parameters of the systems are described in Table 1 (for this problem consider a gravitational constant g = 9.81m/s²).

Table 1: Physical Parameters of the Magnetic Suspension System

a) To simplify notation define xc = x + X_∞ and using as state vector (x_c, x?_c , i ) - in this order - rewrite explicitly the non-linear state-space equations (you need to evaluate explicitly the partial derivative with respect to x). In the rest of this assignment you will use these equations.

b) Find all the equilibrium points of the system. Use capital letters to denote equilibrium conditions, for instance E , I and X_c will be a constant voltage at equilibrium, etc. Note that this system does not have a unique equilibrium.

c) The linearized system of equations is given by

δ_x?_c = Aδx_c + Bδ_u
δ_y = Cδx_c + Dδ_u

Linearize the system around an equilibrium where the object is suspended at a distance of X_o (e.g. X_c = X_∞ + X_o ) and with zero disturbances. Give explicit symbolic expressions for the state matrices (in terms of the parameters of the system). Also give explicit expressions for the deviation variables x_c , u and y (note that the disturbance is also an input). (Hint: You can use mupad to verify or help you in the computation of the jacobians)

d) Write a matlab function, Magnet.m that takes a structure pars as input and returns the a state-space object corresponding to the linearized state-equations. The structure pars should have a fields the parameters of the system (similar to what you did in the DC motor problem).

e) Using the numerical state-space object obtained with your function find the poles ( pole). Then compare them with the eigenvalues of the state-matrix ? Are they the same ? Explain.

f) Using the numerical state-space object obtained with your function find the zeros ( zero) of the system. What do the zeros tell us ? Explain.

g) Convert the state-space object to transfer function in zero-pole-gain form (if G is your state-space object just do Gtf=zpk(G)) and then back to state-space form Gss=ss(Gtf). Now compare the state-space matrices of Gss with the state-space matrices of G (the original system). Are they the same ? Explain.

Problem 2

In this problem you will derive state-space formulas for linear interconnection of systems. Let Σ₁ = [ A₁, B₁,C₁, D₁] and Σ₂ = [ A₂, B₂,C₂, D₂] denote two system with corresponding state-space realizations. For ex- ample, the formulas to find the state-space matrices for the parallel interconnection of Σ₁ and Σ₂ can be found by direct algebraic manipulation to be

Σ = Σ₁ + Σ₂

So we conclude that a block diagonal structure of the state matrix is induced by the parallel interconnection of smaller subsystems (one extreme is when A is diagonal)

a) Derive explicit formulas for the state-space matrices of the Σ - [ A, B,C , D] that results when

   (a)Cascade or series interconnection of Σ₁ followed by Σ₂.

   (b) Parallel interconnection of Σ₁ and Σ₂.

   (c) Feedback interconnection with Σ₁ in the forward path and Σ₂ in the feedback path (consider negative feedback)

   (d) The inverse system Σ = (Σ₁)-1 when it exists.

b) Test your formulas in Matlab with the following numerical matrices. To this first write matlab functions myseries, myparallel, myfeedback and myinverse that take as input a state-space objects and return a state-space object.