Output Tracking and Stabilizing Control Problem Solution

Condition 11 The dimensions m of the vector q, N of the output vector y, and r of the control vector u of system (1), (2) obey: r ≥ m > N.

Let an auxiliary variablez_εand the induced matrixZ_εbe defined by (21):

z_ε= (z_ε1 z_ε2...z_εN)^T = ε⁽¹⁾+ Γε, Z_ε= diag {z_ε1 z_ε2...z_εN} . (21) Condition 12 A subsidiary function vs(.) : R^m→ R^p, p ∈ {1, 2, ..., m − N}, (22),

v_s(z) = [υ_s1(z) υ_s2(z) ... υ_sp(z)]^T, p ∈ {1, 2, ..., m − N}, (22) is a global differentiable positive definit vector function on the z-space R^m, i.e., it is a global differentiable positive definit vector function with respect to the set Sss, (3), if it is considered on the e¹-space R^2min view of (21) and (22).

The functionvs(.) will be used to solve the subproblems c) and d) of Problem 10.

Let its Jacobian be denoted byJs(.) : R^m → R^pxm, p ∈ {1, 2, ..., m − N}, Js(z) = [∂υsi(z)/∂zj] .

Condition 13 The Jacobians Jg(q) and Js(z) of the vector functions g(.), (2), and vs(.), (22), and Zε(.), (21), obey: rank [W (q, z, zε)] = N + p, ∀(q, z, zε) ∈ R^mx_R^m_xR^N _and Js(0) = 0, where W (q, z, zε) = [J_g^T(q)Z_ε^T(zε) J_s^T(z)]^T ∈ R^(N+p)xm.

LetL_εandT_sbe positive diagonal designpxp matrices, T_s= diag{τ_s1 τ_s2 ... τ_sp} ands(v_s) = [sign(υ_s1) sign(υ_s2) ... sign(υ_sp)]^T.

Theorem 14 If Conditions 7, 11 through 13 are satisfied if Tεis a positive diagonal Nx_N matrix such that

τ_RSSε(ε¹₀) = T_εε⁽¹⁾₀ + Λε₀ , ∀ε¹₀∈ R^2N, (23) and if the control vector function u(.) is in the form (24),

u(t, e¹, ε¹, q¹, q¹_d) = b^I

a) the sliding set Sssε, (16), is positive invariant with respect to ε¹(t), (17),

b) the sliding set Sssεis globally pairwise exponentially stable with the finit vector reachability time τrssε(ε¹₀) ∈ R^N₊, (18), (19), (23),

c) the sliding set Sss, (3), is positive invariant with respect to e¹(t), (4),

d) the sliding set Sssis globally asymptotically stable with the finit scalar reacha-bility time τrsss(e¹₀) = max{τsi(e¹₀) : i = 1, 2, ... , p}, (20), theorem statement hold. Let an extended vector function ve(.) : R^m+N → R^N+p be composed of the vector functionsv_ε(.) = (1/2)Z_ε(.)z_ε(.) and v_s(.):

It is global differentiable positive definite vector function on theze−space R^m+N. Its Ja-cobianJe(.) : R^m+N → R^(N+p)x(m+N)has the next form:

J_e(z_e) = blockdiag {Z_ε(z_ε) J_s(z)} . (27) Hence, along an output error response and a motion error of system (1), (2):

v⁽¹⁾_e (z_e¹) =

100 L.T. Gruyitch

This shows that v⁽¹⁾e (z_e¹) is majorized by a global negative definite vector function on R^mxR^N, which, together with (21),ve⁽¹⁾[z_e¹(e², ε²)] = 0 on the Cartesian product slid-ing set S_ss^xS_ssε (inR^2mxR^2N due to Z_ε(0) = 0 and J_s(0) = 0) and global positive definiteness ofv_e(.) with respect to the product sliding set S_ss^xS_ssε, proves global asymp-totic stability of the product sliding setS_ss^xS_ssεfor everyd(.) ∈ S_d. Hence, the product sliding setSssxSssεis also positive invariant relative to the error motions and error output responses of system (1), (2) for everyd(.) ∈ Sd. This proves the statements under a) and c). Integration of the preceding inequality, (21),vε(.) = (1/2)Zε(.)zε(.) and (26) yield:

This, (3), (16), and (23) complete the proof

6 CONCLUSION

The new vector development of the concepts of (semi-) definite scalar functions and decres-cent scalar functions, and the development of the concept of vector Lyapunov functions en-able an effective resolution of a basic complex problem of control synthesis in two different

spaces. Such a control ensures simultaneous global asymptotic stability of a system desired motion and a good quality of output tracking that is accepted to be exponential. Various kinds of vector Lyapunov functions, which are introduced herein, additionally permit an effective sliding control synthesis. Control guarantees that the motion error and the output error response of the controlled system are independent of the system internal dynamics over the sliding sets in the e¹-error space and in the outputε¹-error space. Asympotic (or exponential) stability of thee¹-sliding setSss and of theε¹-sliding setSssεwith pre-specified finite reachability times provide an engineering sense to the results and enable their effective applications. Synthesized controls are directly applicable to airplanes and to other nonlinear dynamical systems (autonomous vehicles, robots, ships, and spacecrafts).

All nonlinearities and parameters of their mathematical models can be unknown, except those of their inertia matrices and the ouput functiong(.). Besides, information about the form and real values of external disturbances is not requested. The synthesized controls, stability, and tracking properties are highly robust relative to such uncertainties. The paper opens new directions in studies of stability, tracking, and control of the nonlinear dynamical systems.

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Stabilization of Unstable Aircraft