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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
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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
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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
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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.
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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
Linearization method for solving BMIs in “low-authority” control
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Linearization method for solving BMIs in “low-authority” control
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
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Linearization method for solving BMIs in “low-authority” control
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:
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Linearization method for solving BMIs in “low-authority” control
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.
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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.
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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.
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Example: sparse linear constant output-feedback design
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:
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Example: sparse linear constant output-feedback design
• 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
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Example: sparse linear constant output-feedback design
With :
With open-loop we have:
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Example: sparse linear constant output-feedback design
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
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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:
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Example: simultaneous state-feedback stabilization with limits on the fedd back gains
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:
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Example: simultaneous state-feedback stabilization with limits on the fedd back gains
Example:
With and iterate 15 times we have: the three systems are simulaneously stabilizable
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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
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Example:H2/H∞ controller design
It equivalent to solve BMIs:
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Example:H2/H∞ controller design
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: •
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Example:H2/H∞ controller design
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.
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Example:H2/H∞ controller design
Example:
Result:
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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.
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