1. 1 Nonlinear Control Design for LDIs via Convex Hull Quadratic Lyapunov Functions Tingshu Hu University of Massachusetts, Lowell

2. 2 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 L 2 gain, L 2 /L  gain  Examples: linear control vs nonlinear control  Summary

3. 3 A polytopic linear differential inclusion (PLDI) x – state; u – control input; w – disturbance; y – output.  Stabilization with fast convergence rate;  Disturbance rejection for - magnitude bounded disturbance: w T (t)w(t) ≤ 1 for all t ; - energy bounded disturbance: Objectives: Design feedback law u = f (x), to achieve Recall: PLDIs can be used to describe nonlinear uncertain systems in absolute stability framework. Problem statement

4. 4 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 L 2 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.

5. 5 The convex hull quadratic function Given symmetric matrices : Denote Definition: The convex hull (quadratic) function is  The level set: Note: The function was first defined in [Hu & Lin, IEEE TAC, March, 2003] and used for constrained control systems.  The function is convex and differentiable

6. 6 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 Q k ’s are the same, LMIs are obtained. The bilinear terms injected extra degrees of freedom.

7. 7 Control design: linear vs nonlinear  Design of linear controller: problem easily follows from the analysis BMIs When u = Fx is applied, A i +B i F  A i. Stabilization problem: choose F and Q k ’s to maximize   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

8. 8  Robust stabilization  Maximizing the convergence rate  Robust disturbance rejection  For magnitude bounded disturbances,  suppression the L  gain  For energy bounded disturbances,  suppression the L 2 gain, L 2 /L  gain Main results:

9. 9 Robust stabilization Theorem 1: If there exist Q k = Q k T >0, Y k and scalars ijk ≥0,  such that Then a stabilizing control law can be constructed via Q k ’ s 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.

10. 10 Construction of the controller The controller is constructed from the solution to the optimization problem: Q k, Y k, k =  J. For x  R n, define, Note: The key is to compute    the optimal solution to If J=2, this is equivalent to computing the eigenvalue of a symmetric matrix

11. 11 Robust performance problems  Two types of disturbances: The LDI:  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 L 2 gain or the mixed L 2 /L  gain  Results for minimizing the L 2 gain will be presented

12. 12 Suppression of the L 2 gain Theorem 4: If there exist Q k = Q k T >0, Y k and scalars ijk ≥0,  such that Then a nonlinear control law can be constructed via Q k ’s such that ||y|| 2 /||w|| 2 ≤  under zero initial condition.  The problem of minimizing the gain  translates into a BMI problem  Again, when all Q k ’s and Y k ’s are the same, the BMIs reduce to LMIs  Controller construction the same as that for stabilization

13. 13 Example: Stabilization A second-order LDI:  Cannot be stabilized via LMIs (Linear feedback + quadratic function) The maximal  is   Can be stabilized via BMIs (nonlinear feedback + convex hull functions) The maximal  is   Level set of the resulting convex hull function and a closed-loop trajectory  The “worst” switching between ( A 1,B 1 ) and ( A 2,B 2 ) is produced so that dV c /dt is maximized.

14. 14 Example: Suppression of L 2 gain A second-order LDI: Objective: minimize  such that  y|| 2 ≤  ||w|| 2  For linear control design via LMI, minimal  is 11.8886  For nonlinear control design via BMI, minimal  is 1.8477 Two output responses, both under the worst switching rule that maximizes dV c /dt Linear feedback, ||y|| 2 >2.6858 Nonlinear feedback, ||y|| 2 =0.7984 t w Energy bounded

15. 15 Example: Suppression of L  gain Same second-order LDI as in last slide, with |w(t)|≤1 for all t >0. Objective: minimize  such that  y ( t)| ≤   With linear control design via LMI, minimal  is 12.8287  With nonlinear control design via BMI, minimal  is 2.4573 Two output responses, under the worst switch and w=±1 that maximizes dV c /dt Linear feedback, max{y(t)}>12 Nonlinear feedback, max{y(t)} < 2 Two reachable sets An ellipsoid Convex hull of two ellipsoids

16. 16 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