points maxima and minima of linear functions

Ngày tải lên: 13/05/2014, 12:24

... on maxima and minima can be stated as problems for finding the maxima or the minima of certain functions depending on several variables For example, the problem of inscribing a triangle of maximal ... points C1 and A2 , the second line intersects the sides BC and C A at points A1 and B2, and the third line intersects the sides C A and AB at points B1 and C2 Find the least possible value of the ... cube 1.2.17 Let α be a given plane in space and A and B two points on different sides of α Describe the sphere through A and B that cuts off a disk of minimal area from α 26 Chapter Methods for...

Ngày tải lên: 05/06/2014, 19:01

... g(y) ≥ g(x + y − xy), and the proof of the first part of Theorem 3.6 is complete The proof of the second part is similar Note that the condition c ∈ (1/2, ∞), d ≥ c/(2c − 1) of Lemma 2.11 is equivalent ... equivalent to the condition 1/c + 1/d ≤ of Theorem 3.6 4.8 Proof of Theorem 3.7 (1) The left-hand side of this inequality is a direct consequence of part (1) of Theorem 3.6 Next, from Lemma 2.11, ... since parts (1) and (4) of Lemma 2.11 give R − B + < and B − < Because the right-hand inequality is a consequence of Theorem 3.6, part (2), the proof is complete 4.9 Proof of Theorem 3.9 Putting...

Ngày tải lên: 18/06/2014, 15:20

... g(y) ≥ g(x + y − xy), and the proof of the first part of Theorem 3.6 is complete The proof of the second part is similar Note that the condition c ∈ (1/2, ∞), d ≥ c/(2c − 1) of Lemma 2.11 is equivalent ... equivalent to the condition 1/c + 1/d ≤ of Theorem 3.6 4.8 Proof of Theorem 3.7 (1) The left-hand side of this inequality is a direct consequence of part (1) of Theorem 3.6 Next, from Lemma 2.11, ... since parts (1) and (4) of Lemma 2.11 give R − B + < and B − < Because the right-hand inequality is a consequence of Theorem 3.6, part (2), the proof is complete 4.9 Proof of Theorem 3.9 Putting...

Ngày tải lên: 20/06/2014, 23:20

... restricted power of motors, precision of captors and variability of the behavior of the processes, more than from their possible non-linearity It is necessary to know the basics of linear automated ... Cataloging-in-Publication Data [Analyse des systèmes linéaires/Commande des systèmes linéaires eng] Analysis and control of linear systems analysis and control of linear systems/edited by Philippe de Larminat p ... 2.2.4 Calculation method of the transition matrix e A(t -t0 ) 2.2.5 Application to the modeling of linear discrete systems 2.3 Scalar representation of linear and invariant systems...

Ngày tải lên: 09/08/2014, 06:23

... series of joint impulses of width τ and amplitude x(kτ ) x(t) τ Figure 1.6 Step approximation Transfer Functions and Spectral Models By applying the linearity and invariance hypotheses of the ... at 5%) and ω t m according to the damping ξ Figure 1.21 ω0 tr and ω0 tm according to the damping ξ 30 Analysis and Control of Linear Systems The alternation of slow and fast variations of product ... jf0 20 Analysis and Control of Linear Systems Table 1.1 sums up the features of a system’s transfer function, the existence conditions of its frequency response and the possibility of performing...

Ngày tải lên: 09/08/2014, 06:23

... Analysis and Control of Linear Systems 3.2.2 Delay and lead operators The concept of an operator is interesting because it enables a compact formulation of the description of signals and systems ... development of X (z) by polynomial division according to 86 Analysis and Control of Linear Systems the decreasing powers of z −1 or apply the method of deviations, starting from the definition of the ... ) and y f (k ) designate respectively the free response and the forced response of the system Unlike the continuous case, the solution involves a sum, and not an integration, of powers of A and...

Ngày tải lên: 09/08/2014, 06:23

... and W t a basis of the canceller at the left of W (i.e a maximal solution of equation WtW = {0}), a basis of LV is obtained by directly preserving only the independent columns of LV A basis of ... illustrates the invariance properties of the various structures of these canonical forms (indices of controllability, of observability, finite and infinite zeros) and of the associated transformation ... conclusions and lists the main references 4.1.1 Vector spaces, linear applications Let X and Y be real vector spaces of finite dimension and V ⊂ X and W ⊂ Y, two sub-spaces Let L: X → Y be a linear...

Ngày tải lên: 09/08/2014, 06:23

... proximity of ν0 152 Analysis and Control of Linear Systems Figure 5.1 Typical power spectrum of an MA (left) or AR (right) model In the case of a single denominator (nc = 0), we talk of an AR ... 144 Analysis and Control of Linear Systems On the other hand, the Fourier transform preserves the energy (Parseval theorem) Indeed, the energy of a continuous-time signal y(t) or of a discrete-time ... are not periodic and have infinite energy The mathematical characterization of such signals is particularly well formalized in the case of stationary and ergodic random signals: – random: this means...

Ngày tải lên: 09/08/2014, 06:23

... Analysis and Control of Linear Systems u (t ) = − K x(t ) + e(t ) [6.2] where K is an m × n matrix (Figure 6.1) and signal e(t ) represents the input of the looped system The equations of the looped ... eigenvalues of the looped system are the totality of eigenvalues of sub-matrices A11 − B1 K1 and A22 The eigenvalues of the non-controllable part are thus, by all means, eigenvalues of the looped ... control, whereas the increase of ξ leads to better dynamics 164 Analysis and Control of Linear Systems Figure 6.2 Stabilization by pole placement 6.3 Reconstruction of state and observers 6.3.1 General...

Ngày tải lên: 09/08/2014, 06:23

... one hand, the problem of identification (and also the problem of simulation and control) is made much easier by the computing tool and is already well known in data analysis (linear or non -linear ... 196 Analysis and Control of Linear Systems in performing the roles of the procedure, in connection with a structure and behavior of components They are used for the design of procedure monitoring, ... objective of modeling is the research of a (simple) linear model in view of the control, we can use direct methods based on the use of experimental recordings Two methods are available: the use of the...

Ngày tải lên: 09/08/2014, 06:23

... requirements of speed and accuracy accessible in simulation 8.2 Standard linear equations 8.2.1 Definition of the problem We will adopt the notations usually used to describe the state forms and linear ... the integration of [8.1] is done from [8.5] The calculation of Φ and Γ is ˆ ˆ obtained via the estimation N of N Finally, the calculation of N goes through that of the upper bound of sampling interval ... B and C are constant and verify A ∈ Rn×n , B ∈ Rn×m As for X and U , their size is given by X ∈ Rn×m and U ∈ Rm×m To establish the solution of these equations, we examine the free state, and...

Ngày tải lên: 09/08/2014, 06:23

... equation of n order The roots of this equation are of strictly negative real part if and only if the terms of this first column of the table have the same sign and are not zero Statement of Routh’s ... number of unstable poles of equation [9.18] and hence of µ ( p ) β ( p) Figure 9.7 Contour and image of the Nyquist curve Contour C is chosen in such as way as to surround the poles of possible ... → ∞ and the phase is of −π / When ω → +∞ , the gain → and the phase is of −π The image of the ray ] − ∞, `− [ is the symmetric curve, with respect to the axis of real numbers The image of the...

Ngày tải lên: 09/08/2014, 06:23

... transfer functions In these points we have: µ1 µ µ β = µ1 µ β C i.e.: µ2 = C or µ C = Points A and B are thus the separations of cases and Synthesis of Closed Loop Control Systems Between A and ... variations of these two transfer functions in the Bode plane The curves are cut in points A and B as indicated in Figure 10.35 (one of these points can be infinitely rejected) Figure 10.35 Variation of ... response time and limited precision) On the other hand, irrespective of the linearity of the system, it is not always interesting to have a too extended bandwidth, due to the amplification of noises...

Ngày tải lên: 09/08/2014, 06:23

... degree of polynomial A We will assume that the concepts of polynomials division are known, as well as those of PGCD and PPCM of polynomials If G and L are respectively the PGCD and the PPCM of A and ... and R and thus5: number of unknown factors = ∂S + ∂R + A polynomial of degree n has n + coefficients [11.36a] 338 Analysis and Control of Linear Systems The number of equations is the number of ... in the model 366 Analysis and Control of Linear Systems Figure 11.17 Evolution of sA Ao (⎯ο⎯) .of sAS ' Ao Am ⎛ ⎞ ⎟ Ao = ⎜ s + ⎜ To ⎟ ⎝ ⎠ and of B Am (⎯⎯) in the case of From these answers we notice...

Ngày tải lên: 09/08/2014, 06:23

... j , J j are single solutions of Diophantus equations, which are obtained by equality of the inputs and output of transfer functions of equations [12.1] and [12.3] and they are solved recursively: ... big ideas of the method, starting with the form of the model, the quadratic criterion and up to the examination of adjustment parameters The formalism and the 390 Analysis and Control of Linear ... definition of coefficient α of the reference trajectory; t – nb and u b,k : the number of basic functions and their nature These parameters are set as soon as the nature of the setting signal and the...

Ngày tải lên: 09/08/2014, 06:23

... formulated in the standard form of Figure 13.1 408 Analysis and Control of Linear Systems Figure 13.1 Standard feedback diagram The quadripole G , also called a standard model, and feedback K are ... observable part by y is solution of Let us give the idea of the equivalence proof of properties and 2., the equivalence of properties and resulting by duality Methodology of the State Approach Control ... functions X (s) in Re(s) ≥ and of integrable square correspond to Ln Parseval’s theorem makes it possible to connect the standard of a temporal signal of Ln to the standard n of its Laplace transform...

Ngày tải lên: 09/08/2014, 06:23

... 14.5 Area of the complex plane corresponding to the desired performances and to the constraints on the bandwidth 460 Analysis and Control of Linear Systems 14.4.2 Choice of eigenvectors of the closed ... Shift of a set of poles with minimum dispersion 464 Analysis and Control of Linear Systems EXAMPLE 14.4 To illustrate these points, let us take a set of models pertaining to the lateral side of ... is of size m, it is possible to impose m − decoupling constraints (number of rows of E0 and F0 ) 14.4.2.2 Considering the insensitivity of eigenvalues The concept of insensitivity consists of...

Ngày tải lên: 09/08/2014, 06:23

... Analysis and Control of Linear Systems where N R and N S form a base of cores of ( Bu T Deu T ) and (C y Dyw ) respectively Inequalities [15.70], calculated on inequalities [15.20.a, b, c] of the ... the looping of an invariant system of transfer matrix K (s ) and of a response of matrix Θ(t ) (Figure 15.15) Robust H∞/LMI Control 513 Figure 15.15 Structures of the system and of the corrector ... of τ in dotted line): we see that stability is ensured for any value of τ less than 0.2 Figure 15.6 Determination of a bound value of the neglected time constant 488 Analysis and Control of Linear...

Ngày tải lên: 09/08/2014, 06:23

... λ + t and H (λ ) = λ−1 so ∑ y(t ) = u(t ) 1 530 Analysis and Control of linear Systems 16.4.2 Parallel systems Let ∑1 and ∑ be two systems of transfer functions H1 (λ) = Q1 (λ )−1 P (λ) and H ... t Linear Time-Variant Systems 531 16.5 Applications In this section, two types of usage of this algebra are presented in the field of modeling and control 16.5.1 Modeling One of the methods of ... solution of the Diophantine equation; ~ ~ – search of RLCM M (λ) of Q(λ), P(λ) ; ~ ~ – P(λ ) and Q(λ ) are quotient polynomials of M (λ ) by Q(λ) and P(λ) respectively The resolution of the Diophantine...

Ngày tải lên: 09/08/2014, 06:23