|
Derivation Of The Routh Array
The Routh array is a tabular method permitting one to establish the stability of a system using only the coefficients of the characteristic polynomial. Central to the field of control systems design, the Routh–Hurwitz theorem and Routh array emerge by using the Euclidean algorithm and Sturm's theorem in evaluating 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(x)\mathfrak(x)big) \quad (6) from (2) note that : \theta(x) = \theta_(x)+\theta_(x)+\cdots+\theta_(x) \quad (7) where ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Routh–Hurwitz Stability Criterion
In control system theory, the Routh–Hurwitz stability criterion is a mathematical test that is a necessary and sufficient condition for the stability of a linear time-invariant (LTI) dynamical system or control system. A 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 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 Hurwitz matrix, and showed that the polynomial is stable if and only if the sequence of determinants of its principal submatrices are all positive. The two procedures are equivalent, with the Routh test providing a more efficient way to compute the Hurwitz determinants ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stable Polynomial
In the context of the characteristic polynomial of a differential equation or difference equation, a polynomial is said to be stable if either: * all its roots lie in the open left half-plane, or * all its roots lie in the open unit disk. The first condition provides stability for continuous-time linear systems, and the second case relates to stability of discrete-time linear systems. A polynomial with the first property is called at times a Hurwitz polynomial and with the second property a Schur polynomial. Stable polynomials arise in control theory and in mathematical theory of differential and difference equations. A linear, time-invariant system (see LTI system theory) is said to be BIBO stable if every bounded input produces bounded output. A linear system is BIBO stable if its characteristic polynomial is stable. The denominator is required to be Hurwitz stable if the system is in continuous-time and Schur stable if it is in discrete-time. In practice, stability is det ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example of a polynomial of a single indeterminate is . An example with three indeterminates is . Polynomials appear in many areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated scientific problems; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, which are central concepts in algebra and algebraic geometry. Etymology The word ''polynomial'' join ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Control Theory
Control theory is a field of mathematics that deals with the 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; often with the aim to achieve a degree of 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 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 which are also studied are controllability and observability. Control theory is used in control system eng ... [...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-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 test 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 ''f''(''z'') be a polynomial (with complex coefficients) of degree ''n'' with no roots on the imaginary axis (i.e. the line ''Z'' = ''ic'' where ''i'' is the imaginary unit and ''c'' is a real number). Let us define P_0(y) (a polynomial of degree ''n'') and P_1(y) (a nonzero polynomial o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean 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 (numbers), the largest number that divides them both without a remainder. It is named after the ancient Greek mathematician Euclid, who first described it in Euclid's Elements, his ''Elements'' (c. 300 BC). 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 Fraction (mathematics), fractions to their Irreducible fraction, 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sturm's Theorem
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 univariate ... [...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. 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''is direct (when neither ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tan(theta)
Tan or TAN may refer to: Businesses and organisations * Black and Tans, a nickname for British special constables during the Irish War of Independence. By extension "Tans" can now also colloquially refer to English or British people in general, especially disparagingly. * TAN Books, a Catholic publishing company * FC Rubin-TAN Kazan, a Russian professional ice hockey club in Kazan in 1991-94 * Transportes Aereos Nacionales, an airline based in Honduras known as TAN Airlines People * Tan (surname) (譚), a Chinese surname * Chen (surname) (陳), a Chinese surname, pronounced "Tan" in Min Nan languages * Laozi, posthumous name "Tan" or "Dān" (聃), philosopher of ancient China * Leborgne, nicknamed Tan, a patient of Paul Broca's, on whose autopsy he identified Broca's area * TAN (musician) (born 1990), Malaysian pop singer * Tan Sağtürk (born 1969), Turkish ballet Places China * Tan (state), an ancient viscountcy in eastern Shandong Province, China * Tai'an railway stat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cot(theta)
A cot is a camp bed or infant bed. Cot or COT may also refer to: In arts and entertainment * Chicago Opera Theater, an opera company In mathematics, science, and technology * Car of Tomorrow, a car design used in NASCAR racing * Cost of transport, an energy calculation * Cottage developed from the word cot, which can be seen in various forms in other languages meaning a tent / hut e.g. Goahti and Kohte * Cotangent, a trigonometric function, written as "cot" * Cyclooctatetraene, an unsaturated hydrocarbon * Finger cot, a hygienic cover for a single finger In government and military use * Colombian Time, the time zone used in Colombia (UTC−05:00) * Comando de Operações Táticas, a Brazilian counter-terrorism force * Commitments of Traders Report, US market report * Committee on Toxicity of Chemicals in Food, Consumer Products and the Environment, in the UK * RAF Cottesmore Flying Training Unit, United Kingdom (ICAO airline designator) People * Cot (surname) * Cot Deal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sturm Theorem
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 univariat ... [...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 (numbers), 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'' (c. 300 BC). 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
