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Inequalities
Published in John Bird, Basic Engineering Mathematics, 2017
If a quadratic expression does not factorise, then the technique of ‘completing the square’ is used. In general, the procedure for x2+bx+c $ x^{2}\,{+}{bx}\,{+}\,c $ is: x2+bx+c≡x+b22+c-b22 $$ x^{2} + bx + c \equiv \left(x + \frac{b}{2}\right)^{2} + c - \left(\frac{b}{2}\right)^{2} $$
Inequalities
Published in John Bird, Engineering Mathematics, 2017
If a quadratic expression does not factorise, then the technique of ‘completing the square’ is used. In general, the procedure for x2+bx+c $ x^{2}\,{+}{bx}\,{+}\,c $ is: x2+bx+c≡x+b22+c-b22 $$ x^{2} + bx + c \equiv \left(x + \frac{b}{2}\right)^{2} + c - \left(\frac{b}{2}\right)^{2} $$
Inequalities
Published in John Bird, Higher Engineering Mathematics, 2017
If a quadratic expression does not factorise, then the technique of ‘completing the square’ is used. In general, the procedure for x2 + bx + c is: x2+bx+c≡(x+b2)2+c−(b2)2
Data-driven optimal tracking control of discrete-time linear systems with multiple delays via the value iteration algorithm
Published in International Journal of Systems Science, 2022
Longyan Hao, Chaoli Wang, Guang Zhang, Chonglin Jing, Yibo Shi
Equation (31) has the following forms by substituting (29) and into (50) using (14) and (19) in (51) yields Equation (52) is the same as Equation (25); therefore, the policy improvement steps of Algorithms 1 and 3 are the same. Convergence of Algorithms 1 is shown in Valeria et al. (1972). The following shows that the control input of the design is optimal. Note that Using augmented ARE (21) in (53) and since , one has value function (16) in terms of and has the forms In fact, minimum value function (16) for system (14) is equivalent to minimum value function (55) for system (60). Multiplying the right-hand side of (54) by and adding its result to (55) yields Completing the square gives Since (55) achieves its minimisation when minimises value function (16), the control input is the optimal solution.
Finite-horizon H ∞ state estimation for time-varying complex networks based on the outputs of partial nodes
Published in Systems Science & Control Engineering, 2021
Wenhua Zhang, Li Sheng, Ming Gao
On the basis of Theorem 3.1, define , we obtain Adding the zero term to the right side of (41), one has Applying the completing-the-square technique, we have where Furthermore, it can be deduced that In the light of condition (36), one obtains If the matrix satisfies Equation (40), then the cost of can be minimized, which completes the proof.
Functional integral approach to the transfer function of a stochastic scattering channel
Published in Waves in Random and Complex Media, 2020
Octavio Cabrera, Damián H. Zanette
Replacing Equation (12) into Equation (10), the characteristic function turns out to be The argument of the exponential in the upper integral of this equation is a quadratic functional of , with linear terms coming from – see Equation (11) – and quadratic terms coming from the exponent in the Gaussian profile . The form of this quadratic functional can be simplified by ‘completing the square ’, just as in elementary algebra, with respect to the variable ρ. In fact, writing , with we get The change of variables is equivalent to a Popov–Faddeev canonical transformation [8], of standard use both in quantum and classical applications of functional integration [6,7]. It amounts to shifting the origin in ρ-space to , so that . Consequently, when the right-hand side of Equation (17) is replaced into Equation (15), the functional integral of its first term over ρ-space exactly cancels the denominator in the same equation. As a result of this cancelation we obtain [6,7]