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1.
References
A MATLAB Toolbox for Piecewise-Aﬃne
Controller Synthesis
Mohsen Zamani Behzad Samadi Luis Rodrigues
HYCONS Lab, Concordia University
American Control Conference
Montreal, Canada
June 2012
2.
References
Outline
Motivation
PWA systems
PWA approximation
PWA controller synthesis
Stability analysis of the closed loop system
Example
Conclusion
3.
References
Motivation
Gain Scheduling Controller:
Design an autopilot to:
minimize steady state tracking error
maximize robustness to wind gust
subject to varying ﬂight conditions
For controller design, consider the following issues:
Theory of Linear Systems is very rich in terms of analysis and
synthesis methods and computational tools.
Real world systems, however, are usually nonlinear.
What can be done to extend the good properties of linear
systems theory to nonlinear systems?
4.
References
Motivation
Gain Scheduling Controller:
Gain scheduling is an attempt to address this issue.
Divide and conquer
Approximating nonlinear systems by a combination of local
linear systems
Designing local linear controllers and combining them
Started in 1960s, very popular in a variety of ﬁelds from
aerospace to process control
Problem: proof of stability!
Problem: By switching between two stable linear system, you
can create an unstable system.
5.
References
PWA Systems
Piecewise Smooth Systems
The dynamics of a piecewise smooth (PWS) is deﬁned as:
˙x = fi (x), x ∈ Ri
where x ∈ X is the state vector. A subset of the state space X is
partitioned into M regions, Ri , i = 1, . . . , M such that:
∪M
i=1
¯Ri = X, Ri ∩ Rj = ∅, i = j
where ¯Ri denotes the closure of Ri .
6.
References
PWA Systems
Piecewise Aﬃne Systems
The dynamics of a piecewise aﬃne (PWA) is deﬁned as:
˙x = Ai x + ai + Bi u, x ∈ Ri
where
Ri = {x|Ei x + ei ≥ 0}, i = 1, 2, . . . , M
PWA systems are:
nonlinear
locally linear
7.
References
PWA Systems
PWA Slab Systems
The regions of PWA slab system are intervals on a scalar variable:
Ri = {x|βi < cT
x < βi+1}
The regions of a PWA slab system can be written as degenerate
ellipsoids:
Ri = {x| |Ei x + ei | < 1}
8.
References
PWA Systems
PWA Diﬀerential Inclusions
In this toolbox, we have considered PWA diﬀerential inclusions
(PWADI) described by:
˙x ∈ conv{Ai1x + ai1 + Bi1u, Ai2x + ai2 + Bi2u}, x ∈ Ri
PWADIs are a form of uncertain PWA systems.
Nonlinear systems can be included by PWADIs.
9.
References
PWA Approximation
Given a nonlinear system:
˙x = Ax + a + f (x) + B(x)u
PWATOOLS can create a PWA approximation using the following
approaches:
Uniform: connecting the dots of a uniform grid on the
nonlinear surface
Optimal Uniform: minimizing the error for a uniform grid
Multi-resolution: recursively building a nonuniform grid based
on error minimization (divide and conquer)
10.
References
PWA Controller Synthesis
Global Lyapunov function
Find a PWA controller u = Ki x + ki such that:
V (x) = xTPx > 0
dV
dt < −αV for α > 0
the control signal is continuous (optional)
This problem is formulated as a set of Bilinear Matrix
Inequalities (BMI).
For PWA slab systems, the problem can formulated as a set of
Linear Matrix Inequalities (LMI).
11.
References
PWA Controller Synthesis
Piecewise quadratic Lyapunov function
Find a PWA controller u = Ki x + ki such that:
V (x) = xTPi x + 2qT
i x + ri > 0
dV
dt < −αV for α > 0
the Lyapunov function is continuous
the dynamics of the closed loop system is continuous
12.
References
Stability of the Closed Loop System
The dynamics of the closed loop system is described by:
˙x = (Ai + Bi Ki )x + ai + Bi ki
The same inequalities are used assuming that the controller
known. The BMIs become LMIs.
13.
References
Example
Flutter Phenomenon
Mechanism of Flutter
Inertial Forces
Aerodynamic Forces (∝ V 2
) (exciting the
motion)
Elastic Forces (damping the motion)
Flutter Facts
Flutter is self-excited
Two or more modes of motion (e.g. ﬂexural and torsional)
exist simultaneously
Critical Flutter Speed, largely depends on torsional and ﬂexural
stiﬀnesses of the structure
[1]
14.
References
Example
State Space Equations:
M
¨h
¨α
+ (C0 + Cµ)
˙h
˙α
+ (K0 + Kµ)
h
α
+
0
αKα(α)
= Bβo
Kα(α) = 2.82α − 62.322α2
+ 3709.71α3
− 24195.6α4
+ 48756.954α5
State variables: plunge deﬂection (h), pitch angle (α), and
their derivatives (˙h and ˙α)
Inputs: angular deﬂection of the ﬂaps (βo ∈ R2)
Constraints: on states and actuators
[2; 3]
15.
References
PWATOOLS
Create the system description (4 state variables, 2 inputs):
pwacreate(4,2,ActiveFlutter_non.m);
Edit the system description: ActiveFlutter non.m
model.A = [0 0 1 0; 0 0 0 1;
-M(Ko+Ku) -M(Co+Cu)];
model.Bx = [0 0; 0 0;muu*inv(M)*B];
model.aff = [0;0;0;0];
model.fx = @Flutter_Nonlinearity;
model.xcl = [0;0;0;0];
model.Domain ={[-1 1],[-1 1],[-1 1],[-5 5]}
model.NonlinearDomain = [0;1;0;0];
model.mtd = Uniform;
model.M =6; % Number of regions
16.
References
PWATOOLS
Create the nonlinear function:
function F = Flutter_Nonlinearity(x)
qx2 = 2.82*x(2)-62.322*x(2)^2+3709.71
*x(2)^3-24195.6*x(2)^4
+48756.954*x(2)^5;|
M = [12.387 0.418; 0.418 0.065];
F = [0;0;-inv(M)*[0;qx2]];
Run the system description: ActiveFlutter non.m. It created
two objects:
nlsys: the nonlinear system
pwainc: the PWADI
20.
References
Conclusions
PWATOOLS is an easy to use toolbox for PWA systems.
The plan is to extend PWATOOLS algorithms to piecewise
polynomial (PWP) systems.
Try PWATOOLS for your problem and let us know how it
works.
21.
References
References I
[1] M. R. Waszak. Modeling the benchmark active control technology
wind tunnel model for application to ﬂutter suppression. AIAA, 96 -
3437, http:
//www.mathworks.com/matlabcentral/fileexchange/3938.
[2] J. Ko and T. W. Strganacy,. Stability and control of a structurally
nonlinear aeroelastic system. Journal of Guidance, Control, and
Dynamics, 21, 718-725.
[3] S. Afkhami and H. Alighanbari. Nonlinear control design of an
airfoil with active ﬂutter suppression in the presence of disturbance.
Control Theory and Applications, 1(6):1638–1649, 2007.
[4] A. Hassibi and S. Boyd. Quadratic stabilization and control of
piecewise-linear systems. Proceedings of the American Control
Conference, 6:3659 – 3664, 1998.
[5] J. L¨ofberg. YALMIP : A Toolbox for Modeling and Optimization in
MATLAB. Proceedings of the CACSD Conference, Taipei, Taiwan,
2004.
22.
References
References II
[6] Manual. PWATOOLS.
http://hycons.encs.concordia.ca/Projects/PWAtoolbox, May2011.
[7] L. Rodrigues and S. Boyd. Piecewise-aﬃne state feedback for
piecewise-aﬃne slab systems using convex optimization. Systems
and Control Letters, 54:835–853, 2005.
[8] B. Samadi and L. Rodrigues. Extension of local linear controllers to
global piecewise aﬃne controllers for uncertain non-linear systems.
International Journal of Systems Science, 39(9):867–879, 2008.
[9] B. Samadi and L. Rodrigues. A duality-based convex optimization
approach to L2-gain control of piecewise aﬃne slab diﬀerential
inclusions. Automatica, vol. 45, no. 3, pp. 812 - 816, Mar. 2009.
[10] B. Samadi and L. Rodrigues. A uniﬁed dissipativity approach for
stability analysis of piecewise smooth systems. Automatica, vol. 47,
no. 12, pp. 2735 - 2742, Dec. 2011.
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