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Southern
Taiwan
University
LOGO
A Path –Following Method
for solving BMI Problems in
Control
American Control Confedence
San Diago, California –June 1999
Author: Arash Hassibi
Jonathan How
Stephen Boyd
Presented by: Vu Van PHong
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Contents
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1
Introduction
2
Linearization method for solving BMIs in
“Low-authority”
3
Path-Following method for solving BMIs
in control
4
5
Example
Inconclusion
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Introduction
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 Purpose to develop a new method is to formulate the
analysis or synthesis problem in term of convex and biconvex 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 feedback 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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Linearization method for solving
BMIs in “low-authority” control
 Consider the linear time-invariant as below:
X: state variable, u: input, y output
 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:
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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:


Are variable

 In order for linearization of BMIs we carry
following step:
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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
• Rewrite (1) we gain:
(2)
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Linearization method for solving
BMIs in “low-authority” control
 Step 3:
• Assume that
• Ignore second order:
• We obtain:
are small.
 (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
A,
is the smallest negative real part of the eigenvalues of
 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
cannot improved any further is feasible for any
or if
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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
perturbation

and get
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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LOGO