1. Extension of a local linear controller
to a stabilizing semi-global piecewise
affine controller
Behzad Samadi and Luis Rodrigues
Hybrid Control Systems Laboratory
Department of Mechanical and Industrial Engineering
Concordia University
Montreal, Canada
Extension of a local linear controller to a stabilizing semi-global piecewise affine controller – p. 1/14

2. Outline
Introduction
Extension of a local linear controller to a stabilizing semi-global piecewise affine controller – p. 2/14

3. Outline
Introduction
Piecewise affine approximation
Extension of a local linear controller to a stabilizing semi-global piecewise affine controller – p. 2/14

4. Outline
Introduction
Piecewise affine approximation
Piecewise affine controller design
Extension of a local linear controller to a stabilizing semi-global piecewise affine controller – p. 2/14

5. Outline
Introduction
Piecewise affine approximation
Piecewise affine controller design
Example
Extension of a local linear controller to a stabilizing semi-global piecewise affine controller – p. 2/14

6. Outline
Introduction
Piecewise affine approximation
Piecewise affine controller design
Example
Conclusion
Extension of a local linear controller to a stabilizing semi-global piecewise affine controller – p. 2/14

7. Introduction
The class of nonlinear systems considered in this work is
described by
˙x = Ax + a + f(x) + Bu
where:
x ∈ Rn is the state vector
u ∈ Rm is the control input
f(x) ∈ Rn is a vector function
A ∈ Rn×n, a ∈ Rn, B ∈ Rn×m
Extension of a local linear controller to a stabilizing semi-global piecewise affine controller – p. 3/14

8. Introduction
Structure of the proposed method
Linear Design
PWA Design
Linearization
Nonlinear
Model
Linear
Model
PWA
Model
Linear
Control
Piecewise
Quadratic
Lyapunov
Function
PWA
Control
Anchored
PWA Approximation
Extension of a local linear controller to a stabilizing semi-global piecewise affine controller – p. 4/14

9. Uniform PWA Approximation
Uniform PWA approximation is made by connecting the
points on the nonlinear curve.
x1
Nonlinearfunction
Nonlinear function
Uniform PWA approximation
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Extension of a local linear controller to a stabilizing semi-global piecewise affine controller – p. 5/14

10. Anchored Uniform PWA Approximation
Anchored uniform PWA approximation is computed by
minimizing the approximation error subject to the condition
that the approximation coincides with the linear
approximation at the desired point.
x1
Nonlinearfunction
Nonlinear function
Anchored PWA approximation
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Extension of a local linear controller to a stabilizing semi-global piecewise affine controller – p. 6/14

11. PWA system
Nonlinear system
˙x = Ax + a + f(x) + Bu
PWA approximation of the nonlinear function
ˆfPWA(x) = Af
i x + af
i , if x ∈ Ri
To compute a PWA approximation of the nonlinear
system, f(x) is replaced by ˆfPWA(x)
˙x(t) = Aix(t) + ai + Biu(t), if x(t) ∈ Ri
where
Ai = A + Af
i , ai = a + af
i
Extension of a local linear controller to a stabilizing semi-global piecewise affine controller – p. 7/14

12. Controller Synthesis
Lyapunov based approach to stabilizing a nonlinear system
Lyapunov function (energy-like function)
V (x) = ¯xT ¯Pi¯x, x ∈ Ri
with ¯x and ¯Pi defined as
¯x =
x
1
, ¯Pi ≡
Pi −Pixcl
−xT
clPT
i ri
, Pi = PT
i > 0
Controller
u = ¯Ki¯x, x ∈ Ri
Extension of a local linear controller to a stabilizing semi-global piecewise affine controller – p. 8/14

13. Controller Synthesis Problem
Problem constraints
Continuous Lyapunov function (Equality)
Positive definite Lyapunov function (LMI)
Decreasing over time Lyapunov function (BMI)
Continuous controller (Equality)
Extension of a local linear controller to a stabilizing semi-global piecewise affine controller – p. 9/14

14. Example
Nonlinear system
˙x1
˙x2
=
x2 + x2
1
u
LQR controller parameters
Q =
2 0
0 1
, R = 0.1
LQR controller for the linear approximation at x = 0
KLQR = −4.4721 −4.3525
Extension of a local linear controller to a stabilizing semi-global piecewise affine controller – p. 10/14

15. Example
PWA controller
¯K1 = 5.9824 −4.3525 18.7150
¯K2 = −12.1841 −4.3525 −3.0848
¯K3 = −4.4721 −4.3525 0
¯K4 = −8.7094 −4.3525 1.6949
¯K5 = −22.3134 −4.3525 18.0198
Control gain for the center region
¯K3 = KLQR 0
Extension of a local linear controller to a stabilizing semi-global piecewise affine controller – p. 11/14

16. LQR Controller
Trajectories of the nonlinear system with LQR controller
x2
x1
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
-6
-4
-2
0
2
4
6
Extension of a local linear controller to a stabilizing semi-global piecewise affine controller – p. 12/14

17. PWA Controller
Trajectories of the nonlinear system with PWA controller
x2
x1
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
-6
-4
-2
0
2
4
6
Extension of a local linear controller to a stabilizing semi-global piecewise affine controller – p. 13/14

18. Conclusion
A Lyapunov-based method was proposed to extend a
linear controller to a PWA controller. This addresses the
problem of designing a nonlinear controller with good
local performance and a large region of attraction.
An optimization method was proposed to compute a
PWA approximation of a nonlinear system which is
identical to the linear approximation of the nonlinear
system at a desired point. This helps to make PWA
controller synthesis feasible in cases where the problem
was not feasible because of the uniform PWA
approximation error at the closed-loop equilibrium point.
Extension of a local linear controller to a stabilizing semi-global piecewise affine controller – p. 14/14