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79) guarantees that all the closed-loop signals are bounded and that the tracking error converges to zero exponentially fast, provided that the design constants ci satisfy ci > 2. In the case where the vector θ is not known, the certainty-equivalence principle can be employed in order to design an adaptive controller for the system. However, the problem of designing parameter estimators for the unknown parameters is not as easy as it was in the linear case. This can be seen from the fact that the “states” ζi , i > 1, are not available for measurement, since they depend on the unknown parameters.

Cl }. As we assume A and B are not linearly dependent, it is convenient to take C1 = A, C2 = B, C3 = [A, B]. If the entries of A and B are rational numbers, symbolic algebra computer programs can generate T (A, B) and produce B. It is known that generic pairs A, B will generate a tree of matrices whose span is Mn . In other words, if you ﬁll out the entries with randomly chosen real numbers, then for almost all sample pairs, {A, B}LA = gl(n, R). However, control systems have structural relationships among their components that may lead to smaller Lie algebras.

If the controls are not bounded, this condition is sufﬁcient for controllability, but if they are in some Uµ , the control may be small compared with Fx for large x and have an insufﬁcient effect. BILINEAR SYSTEMS: WHAT, WHY, WHERE? Precisely what are BLS and why should one use bilinear systems in control engineering? This section will give some answers to those questions, starting with a formal deﬁnition, and give some examples. Deﬁnition. A bilinear system is a ﬁrst-order vector differential equation x˙ = p(x, u) in which x is a state vector; u is a control signal (scalar or vector); and the components of p(x, u) are polynomials in (x, u) that are linear in x, u separately, but jointly quadratic, with constant real coefﬁcients.