By Shiming Duan,  Last Update: 8/13/2010 5:03 PM

 

   V.         Eigenvalue Assignment via the Lambert W Function for Control of Time-Delay Systems

For linear ordinary differential equations (ODEs), if a system is completely controllable, its eigenvalues can be arbitrarily assigned via state feedback. For delay differential equations (DDEs), since the delay leads to an infinite spectrum of eigenvalues, the pole placement methods for ODEs cannot be directly applied to DDEs.

            As introduced in previous chapter, a closed-from solution for linear DDEs can be obtained using Lambert W function approaches. Using these results, one is able to determine the relationship between the eigenvalues and the parameters of the DDE through the Lambert W function. In this chapter, supplement examples of using the Lambert W function to realize stabilization and pole placement for DDEs are presented. Since assigning all the eigenvaules of a system of DDEs is practically infeasible due to its infinite spectrum, the focus is on placing the rightmost poles.

 

Eigenvalue Assignment

Systems with input delays

Consider a system of ODEs with input delays:

Then, the closed-loop system becomes a system of DDEs

The procedure for pole assignment for this closed-loop system can be summarized as follows

1        Select desired right most eigenvalues , where n is the order of system

2        Set initial condition

3        While  for  (external fsolve loop)

4                    Select

5                    let  solve for  such that  (internal fsolve loop)

6                    Calculate

7        end

 

Systems with state delays

Consider a system of DDEs with input delays:

with the generalized feedback structure

The closed-loop system becomes

 

The procedure for designing the gains and  can be described as

1        Select desired right most eigenvalues , where n is the order of system

2        Set initial conditions and

3        While  for  (external fsolve loop)

4                    Select  and

5                    let   solve for  such that  (internal fsolve loop)

6                    Calculate

7        end

 

Limitations on the rightmost eigenvalues

Due to the range limitation of the branches of the Lambert W function, the rightmost poles cannot be assigned to any arbitrary place in the s-plane.

 

* For scalar time-delay systems, this can be easily seen by examining the principal branch (see App.C of the book)

since . Thus the rightmost pole cannot be less than . This feasibility has to be consider in the design process for the method to succeed.

* For time-delay systems with higher order, the problem becomes complex. Depending on the structure and parameters of the system, there is a limiation on the rightmost eigenvalues. If improper values are assigned, the approach will fail to find a solution. An exact determination of the range can be difficult, but one can accomplish the design throught an iterative trial and error manner. For example, if an assignment of rightmost eigenvalues is not successful, one can try again with fewer desired eigenvalues, or pick different desired locations of them.

 

Example 5.1 Hydraulic System (Eigenvalue Assignment)

Consider the linear model for the hydraulic water tank system in Ex.1.1:

where A=10m2 is the cross-sectional area, =1000kgm-3  is the water density, g=10ms-2 is gravitational acceleration, h =1s is the delay, t is time, R=10m-1/2 kg-1/2  is the hydrodynamic resistance of the output pipe, uin is the input flow rate and x(t) is the water surface level. Here we want to design a proportional state feedback control uin(t)= - Kx(t) and assign the poles of the closed-loop system at a specific location.

 

First, recall that the closed-loop system can be described via a delay differential equation (DDE):

From the results in Chapter 3, we can obtain the analytical solution of the poles for the closed-loop systems        

For this scalar case, the right most pole is obtained using the pricipal branch (see Chapter 2).  Note that each branch of the Lambert W function has its own range. For the principal branch, 

which implies,

Therefore, we cannot select the desired rightmost pole to be less than −1.1. By picking different values of K, we can vary the location of the right more pole. A list of the K and the corrsponding right most poles are provided in the following table.

 

K

-3

-2

-1

0

1

2

3

Rightmost Poles

-0.710

-0.398

-0.225

-0.1

0

0.084

0.157

 

Furthermore, if a desired location is assigned, one can calculate the corresponding K using numerical methods (e.g., the fsolve function in matlab) to solve the equation:

For example, if we want to place the right-most pole at -0.5, we can use fsolve and obtain K = −2.426 using the following matlab code.

 

--------Matlab Code-----

function ex5_1

a = -0.1; % -g/A/rho

ad_res = 0.1; % 1/A/rho

lambda_des = -0.5; % place the right most pole at -0.5

k_ini = 0;

h = 1;

options=optimset('Display','iter'); % Option to display output

K_placed = fsolve(@assign_lambert,k_ini,options,a,ad_res,h,lambda_des)

display(['right most pole is placed at:' num2str(lambertw(0,ad_res*K_placed*h*exp(-a*h))/h+a)])

 

function F = assign_lambert(k,a,ad_res,h,lambda_des)

F = real(lambertw(0,ad_res*k*h*exp(-a*h))/h+a)-lambda_des;

----------End-------

 

 

Example 5.2 Pendulum System (Feedback Control Design)

Consider the linearized pendulum system in example 1.2 with parameters l=2m, m=1kg, g=10ms-2,  and h=0.2s

, where

In this example, assume that the measurements of both states are available and design a full state feedback control

Thus, the closed-loop system becomes

The objective is to find a proper gain  such that the right most poles of the closed-loop system are placed at  .

First, note this is a problem of system with input delays and apply the procedure to find

1        Select desired right most eigenvalues , where n is the order of system

2        Set initial condition

3        While  for  (external fsolve loop)

4                    Select a

5                    let  solve for  such that  (internal fsolve loop)

6                    Calculate

7        end

 

Using the following procedure, we are able to find the solution for the controller.

 

Remark

Note that here a controller with smaller delay is used for this example to have better flexibility (wider range of feasibility) for assigning rightmost eigenvalues. One can still possibly stabilize the system with a controller with larger delay. However, with a larger delay, the assignment of rightmost poles may become more restricted and one may not be able to place the rightmost eigenvalues at desired locations successfully.