Sum-of-Squares Applications in Nonlinear Controller Synthesis
Sum-of-Squares Applications in Nonlinear Controller Synthesis
Sum-of-Squares Applications in Nonlinear Controller Synthesis
- No tags were found...
You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
2.3 <strong>Sum</strong>-<strong>of</strong>-<strong>Squares</strong> Relaxation for CLFs Control Lyapunov Functions and <strong>Sum</strong>-<strong>of</strong>-<strong>Squares</strong>and by fix<strong>in</strong>g k 2 = 1 and s 2 = ŝ 2 l 2 , s 3 = ŝ 3 l 2 , p = ˆp l 2 the second equation is reduced to())( ))l 2 ŝ 2 + ŝ 3(L f V + l 2 + ˆp L g V = 0 ⇒ −(ŝ 3 L f V + l 2 + ˆp L g V = ŝ 2 .A sufficient condition for V to be a control Lyapunov function then is∃ s ∈ S, p, l 1 , l 2 ∈ P, l 1 , l 2 ≻0 such thatV − l 1 ∈ S( ( ))− s L f V + l 2 + p L g V ∈ S .(22)Us<strong>in</strong>g semidef<strong>in</strong>ite programm<strong>in</strong>g, it is possible to evaluate condition (22) and to verifywhether a given candidate function V is a control Lyapunov function. 2 This is stated asLemma 2: <strong>Sum</strong>-<strong>of</strong>-<strong>Squares</strong> Certificate for Control Lyapunov Functions.V is a control Lyapunov function for the polynomial system ẋ = f(x) + g(x) ifV − l( ( )) 1− s L f V + l 2 + p L g Vs∈ S∈ S∈ Sp, l 1 , l 2 ∈ Pl 1 , l 2 ≻ 0is a feasible semidef<strong>in</strong>ite programm<strong>in</strong>g problem.Note, that this can only be used to assess a candidate function V that has to be provided,e.g., by guess<strong>in</strong>g.A way to actually synthesize a control Lyapunov function is to allow the coeffcients <strong>of</strong>a parametric polynomial V , i.e., V = c z, c ∈ R 1×m , z ∈ M m (x), to be decision variables.The problem stated <strong>in</strong> lemma 2 can then be solved asProgram 1.F<strong>in</strong>d V such that:V − l( ( )) 1− s L f V + l 2 + p L g Vs∈ S∈ S∈ Sp, l 1 , l 2 ∈ Pl 1 , l 2 ≻ 0As both the polynomial multiplier’s and the CLF’s coefficients are now decision variables,2 For actual implementation, the positive def<strong>in</strong>ite polynomials l 1 and l 2 need to be specified. A commonapproach is to use l i = ɛ i x T x with a decision variable ɛ.10