Nonlinear Control Design for LDIs via Convex Hull Quadratic Lyapunov Functions

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

2. 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 Outline

3. 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:

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 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.

5. 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.

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

7. 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 Qk’s 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

8. 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

9. Robust stabilization Theorem 1: If there exist Qk= QkT >0, Yk and scalars lijk≥0, b > 0 such that Then a stabilizing control law can be constructed via Qk’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. 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 J=2, this is equivalent to computing the eigenvalue of a symmetric matrix Note:

11. 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

12. Suppression of the L2 gain Theorem 4: If there exist Qk= QkT >0, Yk and scalars lijk≥0, d > 0 such that Then a nonlinear control law can be constructed via Qk’s such that ||y||2/||w||2≤ d under zero initial condition. • The problem of minimizing the gain dtranslates into a BMI problem • Again, when all Qk’s and Yk’s are the same, the BMIs reduce • to LMIs • Controller construction the same as that for stabilization

13. 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.

14. w Two output responses, both under the worst switching rule that maximizes dVc/dt Linear feedback, ||y||2>2.6858 Nonlinear feedback, ||y||2=0.7984 t Example: Suppression of L2 gain A second-order LDI: Energy bounded Objective: minimize d such that ||y||2≤d||w||2 • For linear control design via LMI, minimal d is 11.8886 • For nonlinear control design via BMI, minimal d is 1.8477

15. Two output responses, under the worst switch and w=±1 that maximizes dVc/dt Linear feedback, max{y(t)}>12 Nonlinear feedback, max{y(t)} < 2 Two reachable sets An ellipsoid Convex hull of two ellipsoids Example: Suppression of L gain Same second-order LDI as in last slide, with |w(t)|≤1 for all t >0. 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

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