Control Design for a Perturbed System with an Ambiguous Nonlinearity

Abstract

The object of this study is an n-dimensional system of ordinary differential equations with an ambiguous relay nonlinearity under a continuous periodic perturbation. We consider continuous periodic solutions of the system with the state-space trajectory consisting of two parts connected at relay switching points. We develop an algorithm for selecting the nonlinearity parameters under which there is a unique asymptotically orbitally stable periodic solution of the system with given oscillation properties, including a given period and two switching points per period.

APPENDIX

Proof of Theorem 3. Starting its motion from a point on the hyperplane, the representative point of the canonical system solution moves in the prescribed sequence between the two switching hyperplanes along the coordinate axis x_s under condition (1) of Theorem 3. Moreover, the hyperplanes in the state space are orthogonal to the axis x_s.

Under condition (1) of Theorem 3, we write the canonical system (6) as the two systems

$$\dot {\bar {X}} = {{\bar {A}}_{0}}\bar {X} + {{\bar {B}}_{0}}{{m}_{\alpha }} + {{\bar {K}}_{0}}f(t),\quad \left\{ \begin{gathered} \sigma (t) = {{\gamma }_{s}}{{x}_{s}} \hfill \\ {{{\dot {x}}}_{s}} = {{\lambda }_{s}}{{x}_{s}} + {{m}_{\alpha }} + k_{s}^{0}f(t), \hfill \\ \end{gathered} \right.$$

where ${{\bar {A}}_{0}}$ is a diagonal matrix with the eigenvalues λ_j (j ≠ s) placed on the diagonal (the other elements are 0), $\bar {X}$ = (x₁, …, x_{s – 1}, x_{s + 1}, …, x_n)*, ${{\bar {B}}_{0}}$ = ($b_{1}^{0}$, …, $b_{{s - 1}}^{0}$, $b_{{s + 1}}^{0}$, …, $b_{n}^{0}$)*, ${{\bar {K}}_{0}}$ = ($k_{1}^{0}$, …, $k_{{s - 1}}^{0}$, $k_{{s + 1}}^{0}$, …, $k_{n}^{0}$)*, λ_j < 0, and $b_{j}^{0}$ = 1, where j = 1, …, s – 1, s + 1, …, n.

Provided that the real eigenvalues λ_j (j ≠ s) are negative, we can apply Lyapunov functions in the state space of system (6) to separate on the switching hyperplanes a bounded, closed, and convex set that is mapped into itself due to the canonical system solution. The trivial solution of the system $\dot {\bar {X}}$ = ${{\bar {A}}_{0}}\bar {X}$ is asymptotically stable, so there exists a positive definite quadratic form V($\bar {X}$) = $\bar {X}$*V$\bar {X}$. The equation V($\bar {X}$) = C_ν with constants C_ν (ν ∈ $\mathbb{N}$) describes cylindrical surfaces in the n-dimensional state space of the canonical system.

The attraction domain V($\bar {X}$) < $\mathop {\min }\limits_\nu$C_ν will intersect the switching hyperplanes (Γ, X) = ${{\ell }_{\eta }}$ (n = 1, 2) under the conditions

$$ - \,(\Gamma ,A_{0}^{{ - 1}}{{B}_{0}}{{m}_{2}}) < {{\ell }_{1}},\quad - (\Gamma ,A_{0}^{{ - 1}}{{B}_{0}}{{m}_{1}}) > {{\ell }_{2}}.$$

(A.1)

They have the following interpretation: in the case of no external perturbation, the virtual stability points X^(α) = –$A_{0}^{{ - 1}}$B₀m_α (α = 1, 2) of the canonical system are located outside the ambiguity domain of u(σ).

Selecting the elements of the vector Γ based on condition (1) of Theorem 3 simplifies inequalities (A.1) with λ_s ≠ 0 to

$$ - {{\gamma }_{s}}{{m}_{2}}{\text{/}}{{\lambda }_{s}} < {{\ell }_{1}},\quad - {{\gamma }_{s}}{{m}_{1}}{\text{/}}{{\lambda }_{s}} > {{\ell }_{2}}.$$

(A.2)

Obviously, the system of inequalities (A.2) holds if γ_sλ_s > 0 for m₁ < m₂ and ${{\ell }_{1}}$ < ${{\ell }_{2}}$. Note that we have γ_s > 0 for λ_s > 0 and γ_s < 0 for λ_s < 0. Thus, condition (2) of Theorem 3 must be imposed for the attraction domain to intersect the switching hyperplanes.

The intersection of the set described by the inequality V($\bar {X}$) $\leqslant$ $\mathop {\min }\limits_\nu$C_ν with the switching hyperplanes gives a compact and convex set Q, which is defined according to condition (4) of Theorem 3. If the initial points X₀ are taken from the domain bounded by the surface V($\bar {X}$) = $\mathop {\min }\limits_\nu$C_ν, the trajectory of the representative point of the solution due to the canonical system will remain in this domain of the state space. Therefore, starting its motion at X₀ ∈ Q on one of the hyperplanes σ(t) = ${{\ell }_{\eta }}$ (η = 1, 2), the representative point of the solution of system (6) will reach the other hyperplane in a finite time.

Next, let us establish conditions of no sliding mode occurrence. We write the condition under which the representative point of the solution of system (6) will reach the hyperplane without touching at the switching points X = X^β at the corresponding times t_β (β = 1, 2), where t₁ and t₂ = kT are the first and second switching times, respectively. The existence of a unique time t₁ for a given number k ∈ $\mathbb{N}$ is ensured by the conditions of Theorem 1 or Theorem 2, depending on the sign of the eigenvalue λ_s. Thus, we have the inequality (Γ, $\dot {X}$) ≠ 0. Considering condition (1) of Theorem 3, this inequality for λ_s ≠ 0 is rewritten as condition (3) of Theorem 3, which ensures the reachability of the switching hyperplanes without touching. Note that the function f(t) is T-periodic: its value at the second switching time kT coincides with that at zero. Therefore, we use t₂ = 0 in condition (3) of Theorem 3 for simplicity. The proof of Theorem 3 is complete.