Nonlinear Control Design for LDIs via Convex Hull Quadratic Lyapunov Functions

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Nonlinear Control Design for LDIs viaConvex Hull
Quadratic Lyapunov Functions

• Tingshu Hu
• University of Massachusetts, Lowell

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Outline

• Introduction
• Control design for LDIs, problems and background
• The convex hull quadratic Lyapunov function
• Definition, properties, applications
• Main results nonlinear control design for LDIs
• Robust Stabilization
• maximizing the convergence rate
• Robust disturbance rejection
• suppressing the L? gain
• suppressing the L2 gain, L2/L? gain
• Examples linear control vs nonlinear control
• Summary

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Problem statement
A polytopic linear differential inclusion (PLDI)
x state u control input w
disturbance y output.
Recall PLDIs can be used to describe
nonlinear uncertain systems in absolute
stability framework.
Objectives Design feedback law u f (x), to
achieve

• Stabilization with fast convergence rate
• Disturbance rejection for
• - magnitude bounded disturbance wT(t)w(t)
  1 for all t
• - energy bounded disturbance

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Background

• Linear feedback law u Fx
• Fully explored in Boyd et al, 1994
• Quadratic Lyapunov function was employed
• Design problems ? LMIs, e.g., to minimize the
  L2 gain, we obtain

• Observations and motivations
• The problem is convex and has a unique global
  optimal solution
• Why convex? The problem is obtained under two
  restrictions
• Linear feedback
• With quadratic storage/Lyapunov functions
• What if we consider nonlinear feedback?
  Nonquadratic functions?
• Nonlinear control may work better Blanchini
  Megretski, 1999
• Non-quadratic Lyapunov function will facilitate
  the construction of
• nonlinear feedback laws.

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The convex hull quadratic function
Given symmetric matrices
Denote
Definition The convex hull (quadratic) function
is

• The function is convex and differentiable

• The level set

Note The function was first defined in Hu
Lin, IEEE TAC, March, 2003 and used for
constrained control systems.
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Analysis with the convex hull function
Successfully applied to stability and
performance analysis of LDIs and saturated
systems. Significant improvement over quadratic
functions has been reported

• Hu, Teel, Zaccarian, Stability and performance
  for saturated systems
• via quadratic and non-quadratic Lyapunov
  functions," IEEE TAC, 2006.
• Goebel, Teel, Hu and Lin, Conjugate convex
  Lyapunov functions for dual
• linear differential equations," IEEE TAC
  51(4), pp.661-666, 2006
• Goebel, Hu and Teel, Dual matrix inequalities
  in stability and
• performance analysis of linear
  differential/difference inclusions," in
• Current Trends in Nonlinear Systems and
  Control, Birkhauser, 2005
• Hu, Goebel, Teel and Lin, Conjugate Lyapunov
  functions for saturated
• linear systems," Automatica, 41(11),
  pp.1949-1956, 2005.

When convex hull function is applied, the
analysis problem is transformed into BMIs. For
evaluation of the convergence rate of LDI, the
BMI is
When all Qks are the same, LMIs are obtained.
The bilinear terms injected extra degrees of
freedom.
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Control design linear vs nonlinear

• Design of linear controller problem easily
  follows from the
• analysis BMIs

When u Fx is applied, Ai Bi F ? Ai.
Stabilization problem choose F and Qks to
maximize b

• This work pursues the construction of a
  nonlinear controller.
• will be able to incorporate the structure of
  the Lyapunov function
• more degree of freedom for optimization
• simpler BMI problems. A typical BMI

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Main results

• Robust stabilization
• Maximizing the convergence rate
• Robust disturbance rejection
• For magnitude bounded disturbances,
• suppression the L? gain
• For energy bounded disturbances,
• suppression the L2 gain, L2/L? gain

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Robust stabilization
Theorem 1 If there exist Qk QkT gt0, Yk and
scalars lijk0, b gt 0 such that
Then a stabilizing control law can be
constructed via Qks such that every solution
x(t) of the closed-loop system satisfies
Optimization problem

• The path-following method Hassibi, How Boyd,
  1999 is effective on
• this problem and similar ones.
• Results at least as good as those from the LMI
  problem.

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Construction of the controller
The controller is constructed from the solution
to the optimization problem Qk, Yk, k 1, 2,
..., J. For x?Rn, define,
The key is to compute g, the optimal solution to
If J2, this is equivalent to computing the
eigenvalue of a symmetric matrix
Note
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Robust performance problems
The LDI

• Two types of disturbances

• Unit peak

• Unit energy

Objectives of disturbance rejection

• Keep the state or output close to the origin
• Minimize the reachable set
• Suppress the L? gain of the output

• Keep the total energy of the output small (for
  unit energy disturbance)
• Suppress the L2 gain or the mixed L2/L? gain

• Results for minimizing the L2 gain will be
  presented

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Suppression of the L2 gain
Theorem 4 If there exist Qk QkT gt0, Yk and
scalars lijk0, d gt 0 such that
Then a nonlinear control law can be constructed
via Qks such that y2/w2 d under zero
initial condition.

• The problem of minimizing the gain d translates
  into a BMI problem
• Again, when all Qks and Yks are the same, the
  BMIs reduce
• to LMIs
• Controller construction the same as that for
  stabilization

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Example Stabilization
A second-order LDI

• Cannot be stabilized via LMIs (Linear feedback
  quadratic function)

The maximal b is -0.6667

• Can be stabilized via BMIs (nonlinear feedback
  convex hull functions)

The maximal b is 0.4260

• Level set of the resulting convex hull
• function and a closed-loop trajectory

• The worst switching between (A1,B1)
• and (A2,B2) is produced so that dVc/dt is
• maximized.

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Example Suppression of L2 gain
A second-order LDI
Energy bounded
Objective minimize d such that y2 dw2

• For linear control design via LMI, minimal d is
  11.8886
• For nonlinear control design via BMI, minimal d
  is 1.8477

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Example Suppression of L? gain
Same second-order LDI as in last slide, with
w(t)1 for all t gt0.
Objective minimize d such that y(t) d

• With linear control design via LMI, minimal d
  is 12.8287
• With nonlinear control design via BMI, minimal
  d is 2.4573

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Summary

• Nonlinear control may work better than linear
  control
• Achieving faster convergence rate
• More effective suppression of external
  disturbances
• Nonlinear feedback law can be systematically
  constructed
• (optimized) via non-quadratic Lyapunov
  functions
• The convex hull quadratic function has been used
  for
• various design objectives
• Problems transformed into BMIs extensions to
  existing
• LMI results from Boyd et al, 1994
• Other nonquadratic Lyapunov functions
• Homogeneous polynomial Lyapunov function (HPLF,
  including
• sum of squares) obtained for the augmented
  system.
• More suitable for stability analysis.
• Piecewise quadratic Lyapunov function more
  applicable to
• piecewise linear systems