LOGO A Path –Following Method for solving BMI Problems in Control Author: Arash Hassibi Jonathan...

LOGO

A Path –Following Method for solving BMI Problems in

Control

Author: Arash Hassibi

Jonathan How

Stephen Boyd

Presented by: Vu Van PHong

American Control ConfedenceSan Diago, California –June 1999

Southern Taiwan University

www.themegallery.com

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Introduction1

Linearization method for solving BMIs in “Low-authority”

2

Path-Following method for solving BMIs in control

3

Example4

Inconclusion5


Introduction

Purpose to develop a new method is to formulate the analysis or synthesis problem in term of convex and bi-convex matrix optimization problems

We have some methods: Semi-definite Progamming problem(SDP), Linear matrix inequalities( LMIs).

Use “Bilinear matrix inequalities( BMIs)” to solve some control problems such as: synthesis with structured uncertainly, fixed-order controller design, output feed-back stabilization, …

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Introduction

This paper present a path-following method for solving BMI in control: BMI is linearized by using a first order

perturbation approximation Perturbation is computed to improve the

controller performance by using DSP. Repeat this process until the desired

performance is achieved


Linearization method for solving BMIs in “low-authority” control

It can predict the performance of the closed-loop system accurately.

BMIs can be solved as LMIs that can be solved very efficently.

To illustrate this method we consider the problems of linear output-feedback design with limits on the feedback gain.


Consider the linear time-invariant as below:

Open-loop system has a damping rate of at least .

Design feedback gain matrix in order to control law has an additional damping of

The constraints:

X: state variable, u: input, y output




According to Lyapunov theory, this problem is equivalent to the existence of that full-fill BMIs:

In order for linearization of BMIs we carry

following step:

Are variable



Step 1:• Consider open-loop system that has a decay rate at

least

• Compute Po >0 that satisfies:

Step 2:• Assign (2)

• Rewrite (1) we gain:



Step 3: • Assume that are small.• Ignore second order:• We obtain:

(4) is an LMI with variables which can solve efficiently for desired feedback matrix

Powerful method and can be applied in many other control problems.


Path-Following method for solving BMIs in control

Step 1: Carry out Linearization BMIs

Step 2: Starting from initial system( Open-loop system) Iterate many times until get result that satisfies

condition of BMIs.

The important thing to apply this method is choice initial value.


Example: sparse linear constant output-feedback design

We have to design sparse linear constant output-feedback u=Ky for system

Which results in a decay rate of at least Consider the BMIs optimization problem.



Step1:• Let K:=0

Step 2:• Calculate Lyapunov P0 by solving:

• With is the smallest negative real part of the eigenvalues of A,

Step 3: linearization (5) around P0 and K we have:



• Where• And such that the perturbation is small and

linear approximation is valid

Step 4: • .• Iteration will stop when exceeds the desired or if

cannot improved any further is feasible for any



With :

With open-loop we have:



The purpose is to design a sparse K so that decay rate at least is larger that 0.35.

Iteration 6 times with we get


Example: simultaneous state-feedback stabilization with limits on the fedd back gains

Consider system:

Compute K that satisfies so that The close-loop system below is stable:

It means that we have to solve BMIs as below:



Step 1: • compute the minimum condition number Lyapunov matrices

Pk, k=1,2,3

Step 2: • Linearization around K, and Pk

Step 3:• update K and Ak as:



Example:

With and iterate 15 times we have: the three systems are simulaneously stabilizable


Example:H2/H∞ controller design

Consider system:

Find a feedback gain matrix K such that for u=Kx the H2 norm from w to z2 is minimized while H∞

norm from w to z1 is less than some prescribed



It equivalent to solve BMIs:



Step 1:• Compute an initial K and suppose that P1 is Lyapunov matrix

obtained.

Step 2: • u=Kx, compute the H2 norm of close-loop system and P2 is

Lyapunov matrix.

Step 3:• Solve the linearized BMIs around and get

perturbation

Step 4: •



Step 5:• Solve the SDP:

• Get Lyapunov P which proves a level of in H∞ norm for closed-loop system. Let P1:=P and go to step 2.

Iterate until can not improved any further.



Example:

Result:


Conclusion

BMIs is a very powerful method to solve control problem in term of convex or bi-convex matrix optimization problems.

However its weakness is to select initial value.

Because if initial value is not good, it will not convergence to an acceptable solution.

LOGO A Path –Following Method for solving BMI Problems in Control Author: Arash Hassibi Jonathan...

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