|
Routh–Hurwitz Stability Criterion
In the control theory, control system theory, the Routh–Hurwitz stability criterion is a mathematical test that is a necessary and sufficient condition for the stable polynomial, stability of a linear time-invariant system, linear time-invariant (LTI) dynamical system or control system. A stability theory, stable system is one whose output signal is bounded; the position, velocity or energy do not increase to infinity as time goes on. The Routh test is an efficient recursive algorithm that English mathematician Edward John Routh proposed in 1876 to determine whether all the root of a function, roots of the characteristic polynomial of a linear system have negative real parts. German mathematician Adolf Hurwitz independently proposed in 1895 to arrange the coefficients of the polynomial into a square matrix, called the Routh–Hurwitz matrix, Hurwitz matrix, and showed that the polynomial is stable if and only if the sequence of determinants of its principal submatrices are all po ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Derivation Of The Routh Array
The Routh array is a Routh–Hurwitz_stability_criterion#Higher-order_example, tabular method permitting one to establish the stable polynomial, stability of a system using only the coefficients of the characteristic polynomial. Central to the field of control theory, control systems design, the Routh–Hurwitz theorem and Routh array emerge by using the Euclidean algorithm and Sturm's theorem in evaluating cauchy index, Cauchy indices. The Cauchy index Given the system: : \begin f(x) & = a_0x^n+a_1x^+\cdots+a_n & \quad (1) \\ & = (x-r_1)(x-r_2)\cdots(x-r_n) & \quad (2) \\ \end Assuming no roots of f(x) = 0 lie on the imaginary axis, and letting : N = The number of roots of f(x) = 0 with negative real parts, and : P = The number of roots of f(x) = 0 with positive real parts then we have : N+P=n \quad (3) Expressing f(x) in polar form, we have : f(x) = \rho(x)e^ \quad (4) where : \rho(x) = \sqrt \quad (5) and : \theta(x) = \tan^\big(\mathfrak[f(x)]/\mathfrak ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Control Theory
Control theory is a field of control engineering and applied mathematics that deals with the control system, control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to a desired state, while minimizing any ''delay'', ''overshoot'', or ''steady-state error'' and ensuring a level of control Stability theory, stability; often with the aim to achieve a degree of Optimal control, optimality. To do this, a controller with the requisite corrective behavior is required. This controller monitors the controlled process variable (PV), and compares it with the reference or Setpoint (control system), set point (SP). The difference between actual and desired value of the process variable, called the ''error'' signal, or SP-PV error, is applied as feedback to generate a control action to bring the controlled process variable to the same value as the set point. Other aspects ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jury Stability Criterion
In signal processing and control theory, the Jury stability criterion is a method of determining the stability of a discrete-time, linear system by analysis of the coefficients of its characteristic polynomial. It is the discrete time analogue of the Routh–Hurwitz stability criterion. The Jury stability criterion requires that the system poles are located inside the unit circle centered at the origin, while the Routh-Hurwitz stability criterion requires that the poles are in the left half of the complex plane. The Jury criterion is named after Eliahu Ibraham Jury. Method If the characteristic polynomial of the system is given by f(z) = a_n + a_z^1 + a_z^2 + \dots + a_1z^ + a_0z^n then the table is constructed as follows:Discrete-time control systems (2nd ed.), pg. 185. Prentice-Hall, Inc. Upper Saddle River, NJ, USA ©1995 \begin \underline\text & \ \underline \ & \ \underline \ & \ \underline \ & \ \cdots \ & \ \underline \ & \ \underline \ \\ pt 1 & a_0 & a_1 & a_2 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Liénard–Chipart Criterion
In control theory, the Liénard–Chipart criterion is a stability criterion modified from the Routh–Hurwitz stability criterion, proposed in 1914 by French physicists A. Liénard and M. H. Chipart. This criterion has a computational advantage over the Routh–Hurwitz criterion because it involves only about half the number of determinant computations. Algorithm The Routh–Hurwitz stability criterion says that a necessary and sufficient condition for all the roots of the polynomial with real coefficients f(z) = a_0 z^n + a_1 z^ + \cdots + a_n, \quad a_0 > 0 to have negative real parts (i.e. is Hurwitz stable A stable is a building in which working animals are kept, especially horses or oxen. The building is usually divided into stalls, and may include storage for equipment and feed. Styles There are many different types of stables in use tod ...) is that \Delta_1 > 0,\, \Delta_2 > 0, \ \ldots, \ \Delta_n > 0, where is the -th leading principal minor of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sturm Chain
In mathematics, the Sturm sequence of a univariate polynomial is a sequence of polynomials associated with and its derivative by a variant of Euclid's algorithm for polynomials. Sturm's theorem expresses the number of distinct real roots of located in an interval in terms of the number of changes of signs of the values of the Sturm sequence at the bounds of the interval. Applied to the interval of all the real numbers, it gives the total number of real roots of . Whereas the fundamental theorem of algebra readily yields the overall number of complex roots, counted with multiplicity, it does not provide a procedure for calculating them. Sturm's theorem counts the number of distinct real roots and locates them in intervals. By subdividing the intervals containing some roots, it can isolate the roots into arbitrarily small intervals, each containing exactly one root. This yields the oldest real-root isolation algorithm, and arbitrary-precision root-finding algorithm for univaria ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minor (linear Algebra)
In linear algebra, a minor of a matrix (mathematics), matrix is the determinant of some smaller square matrix generated from by removing one or more of its rows and columns. Minors obtained by removing just one row and one column from square matrices (first minors) are required for calculating matrix cofactors, which are useful for computing both the determinant and Inverse matrix, inverse of square matrices. The requirement that the square matrix be smaller than the original matrix is often omitted in the definition. Definition and illustration First minors If is a square matrix, then the ''minor'' of the entry in the -th row and -th column (also called the ''minor'', or a ''first minor'') is the determinant of the submatrix formed by deleting the -th row and -th column. This number is often denoted . The ''cofactor'' is obtained by multiplying the minor by . To illustrate these definitions, consider the following matrix, \begin 1 & 4 & 7 \\ 3 & 0 & 5 \\ -1 & 9 & 11 \\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sylvester Matrix
In mathematics, a Sylvester matrix is a matrix associated to two univariate polynomials with coefficients in a field or a commutative ring. The entries of the Sylvester matrix of two polynomials are coefficients of the polynomials. The determinant of the Sylvester matrix of two polynomials is their resultant, which is zero when the two polynomials have a common root (in case of coefficients in a field) or a non-constant common divisor (in case of coefficients in an integral domain). Sylvester matrices are named after James Joseph Sylvester. Definition Formally, let ''p'' and ''q'' be two nonzero polynomials, respectively of degree ''m'' and ''n''. Thus: :p(z)=p_0+p_1 z+p_2 z^2+\cdots+p_m z^m,\;q(z)=q_0+q_1 z+q_2 z^2+\cdots+q_n z^n. The Sylvester matrix associated to ''p'' and ''q'' is then the (n+m)\times(n+m) matrix constructed as follows: * if ''n'' > 0, the first row is: :\begin p_m & p_ & \cdots & p_1 & p_0 & 0 & \cdots & 0 \end. * the second row is the first row, shifted on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Routh–Hurwitz Stability Criterion
In the control theory, control system theory, the Routh–Hurwitz stability criterion is a mathematical test that is a necessary and sufficient condition for the stable polynomial, stability of a linear time-invariant system, linear time-invariant (LTI) dynamical system or control system. A stability theory, stable system is one whose output signal is bounded; the position, velocity or energy do not increase to infinity as time goes on. The Routh test is an efficient recursive algorithm that English mathematician Edward John Routh proposed in 1876 to determine whether all the root of a function, roots of the characteristic polynomial of a linear system have negative real parts. German mathematician Adolf Hurwitz independently proposed in 1895 to arrange the coefficients of the polynomial into a square matrix, called the Routh–Hurwitz matrix, Hurwitz matrix, and showed that the polynomial is stable if and only if the sequence of determinants of its principal submatrices are all po ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamental Theorem Of Algebra
The fundamental theorem of algebra, also called d'Alembert's theorem or the d'Alembert–Gauss theorem, states that every non-constant polynomial, constant single-variable polynomial with Complex number, complex coefficients has at least one complex Zero of a function, root. This includes polynomials with real coefficients, since every real number is a complex number with its imaginary part equal to zero. Equivalently (by definition), the theorem states that the field (mathematics), field of complex numbers is Algebraically closed field, algebraically closed. The theorem is also stated as follows: every non-zero, single-variable, Degree of a polynomial, degree ''n'' polynomial with complex coefficients has, counted with Multiplicity (mathematics)#Multiplicity of a root of a polynomial, multiplicity, exactly ''n'' complex roots. The equivalence of the two statements can be proven through the use of successive polynomial division. Despite its name, it is not fundamental for modern ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclid's Algorithm
In mathematics, the Euclidean algorithm,Some widely used textbooks, such as I. N. Herstein's ''Topics in Algebra'' and Serge Lang's ''Algebra'', use the term "Euclidean algorithm" to refer to Euclidean division or Euclid's algorithm, is an efficient method for computing the greatest common divisor (GCD) of two integers, the largest number that divides them both without a remainder. It is named after the ancient Greek mathematician Euclid, who first described it in his ''Elements'' (). It is an example of an ''algorithm'', a step-by-step procedure for performing a calculation according to well-defined rules, and is one of the oldest algorithms in common use. It can be used to reduce fractions to their simplest form, and is a part of many other number-theoretic and cryptographic calculations. The Euclidean algorithm is based on the principle that the greatest common divisor of two numbers does not change if the larger number is replaced by its difference with the smaller num ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Routh–Hurwitz Theorem
In mathematics, the Routh–Hurwitz theorem gives a test to determine whether all roots of a given polynomial lie in the left-half complex plane. Polynomials with this property are called Hurwitz stable polynomials. The Routh–Hurwitz theorem is important in dynamical systems and control theory, because the characteristic polynomial of the differential equations of a stable, linear system has roots limited to the left half plane (negative eigenvalues). Thus the theorem provides a mathematical test, the Routh–Hurwitz stability criterion, to determine whether a linear dynamical system is stable without solving the system. The Routh–Hurwitz theorem was proved in 1895, and it was named after Edward John Routh and Adolf Hurwitz. Notations Let be a polynomial (with complex coefficients) of degree with no roots on the imaginary axis (i.e. the line where is the imaginary unit and is a real number). Let us define real polynomials and by , respectively the real a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cauchy Index
In mathematical analysis, the Cauchy index is an integer associated to a real rational function over an Interval (mathematics), interval. By the Routh–Hurwitz theorem, we have the following interpretation: the Cauchy index of :''r''(''x'') = ''p''(''x'')/''q''(''x'') over the real line is the difference between the number of roots of ''f''(''z'') located in the right half-plane and those located in the left half-plane. The complex polynomial ''f''(''z'') is such that :''f''(''iy'') = ''q''(''y'') + ''ip''(''y''). We must also assume that ''p'' has degree less than the degree of ''q''. Definition * The Cauchy index was first defined for a pole ''s'' of the rational function ''r'' by Augustin-Louis Cauchy in 1837 using one-sided limits as: : I_sr = \begin +1, & \text \displaystyle\lim_r(x)=-\infty \;\land\; \lim_r(x)=+\infty, \\ -1, & \text \displaystyle\lim_r(x)=+\infty \;\land\; \lim_r(x)=-\infty, \\ 0, & \text \end * A generalization over the compact interval [''a'',''b' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
