Misplaced Pages

#941058

68-469: The phrase H ∞ control comes from the name of the mathematical space over which the optimization takes place: H ∞ is the Hardy space of matrix -valued functions that are analytic and bounded in the open right-half of the complex plane defined by Re( s ) > 0; the H ∞ norm is the supremum singular value of the matrix over that space. In the case of a scalar-valued function,

136-483: A n ) n ∈ Z with a n = 0 for every n < 0, then the element f of the Hardy space H associated to f ~ {\displaystyle {\tilde {f}}} is the holomorphic function In applications, those functions with vanishing negative Fourier coefficients are commonly interpreted as the causal solutions. Thus, the space H is seen to sit naturally inside L space, and

204-534: A transducer converting, for instance, signals from the control system in the electrical domain into motion in the mechanical domain. The control system also requires sensors that detect the motion and convert it back into the electrical domain through another transducer so that the motion can be properly controlled through a feedback loop. Other sensors in the system may be transducers converting yet other energy domains into electrical signals, such as optical, audio, thermal, fluid flow and chemical. Another application

272-587: A transfer function matrix , or just transfer matrix is a generalisation of the transfer functions of single-input single-output (SISO) systems to multiple-input and multiple-output (MIMO) systems. The matrix relates the outputs of the system to its inputs. It is a particularly useful construction for linear time-invariant (LTI) systems because it can be expressed in terms of the s-plane . In some systems, especially ones consisting entirely of passive components, it can be ambiguous which variables are inputs and which are outputs. In electrical engineering,

340-520: A common scheme is to gather all the voltage variables on one side and all the current variables on the other regardless of which are inputs or outputs. This results in all the elements of the transfer matrix being in units of impedance . The concept of impedance (and hence impedance matrices) has been borrowed into other energy domains by analogy, especially mechanics and acoustics. Many control systems span several different energy domains. This requires transfer matrices with elements in mixed units. This

408-557: A complex half-plane (usually the right half-plane or upper half-plane) are used. The Hardy space H ( H ) on the upper half-plane H is defined to be the space of holomorphic functions f on H with bounded norm, the norm being given by The corresponding H ( H ) is defined as functions of bounded norm, with the norm given by Although the unit disk D and the upper half-plane H can be mapped to one another by means of Möbius transformations , they are not interchangeable as domains for Hardy spaces. Contributing to this difference

476-582: A function in H . For example: every function in H is the product of two functions in H ; every function in H , p < 1, can be expressed as product of several functions in some H , q  > 1. Real-variable techniques, mainly associated to the study of real Hardy spaces defined on R (see below), are also used in the simpler framework of the circle. It is a common practice to allow for complex functions (or distributions) in these "real" spaces. The definition that follows does not distinguish between real or complex case. Let P r denote

544-462: A greater voltage at port 1 than was applied at port 2, an impossibility with a purely resistive circuit like this one. To correctly predict the behaviour of the circuit, the currents entering or leaving the ports must also be taken into account, which is what the transfer matrix does. The impedance matrix for the voltage divider circuit is, which fully describes its behaviour under all input and output conditions. At microwave frequencies, none of

612-399: A hybrid mix of units. Acoustic systems are a subset of fluid dynamics , and in both fields the primary input and output variables are pressure , P , and volumetric flow rate , Q , except in the case of sound travelling through solid components. In the latter case, the primary variables of mechanics, force and velocity, are more appropriate. An example of a two-port acoustic component

680-466: A kind of " complex convexity " remains, namely the fact that z → | z | is subharmonic for every q > 0. As a consequence, if is in H , it can be shown that c n = O( n ). It follows that the Fourier series converges in the sense of distributions to a distribution f on the unit circle, and F ( re ) =( f  ∗  P r )(θ). The function F ∈ H can be reconstructed from

748-412: A more accurate representation is achieved with a two-input, two-output MIMO transfer matrix. In the z-parameters, this takes the form, where F is the force applied to the actuator and v is the resulting velocity of the actuator. The impedance parameters here are a mixture of units; z 11 is an electrical impedance, z 22 is a mechanical impedance and the other two are transimpedances in

SECTION 10

#1732790450942

816-456: A perturbation: depending on the problem formulation, the impact will either be measured in terms of stabilization or performance. Simultaneously optimizing robust performance and robust stabilization is difficult. One method that comes close to achieving this is H ∞ loop-shaping , which allows the control designer to apply classical loop-shaping concepts to the multivariable frequency response to get good robust performance, and then optimizes

884-541: A subset of L ( T ). To every real trigonometric polynomial u on the unit circle, one associates the real conjugate polynomial v such that u + i v extends to a holomorphic function in the unit disk, This mapping u → v extends to a bounded linear operator H on L ( T ), when 1 < p  < ∞ (up to a scalar multiple, it is the Hilbert transform on the unit circle), and H also maps L ( T ) to weak- L ( T ) . When 1 ≤ p  < ∞,

952-595: A system or through it. This largely ends up the same as the mobility analogy except in the case of the fluid flow domain (including the acoustics domain). Here pressure is made analogous to voltage (as in the impedance analogy) instead of current (as in the mobility analogy). However, force in the mechanical domain is analogous to current because force acts through an object. There are some commonly used analogies that do not use power conjugate pairs. For sensors, correctly modelling energy flows may not be so important. Sensors often extract only tiny amounts of energy into

1020-420: A three-input, two-output system, one might write, where the u n are the inputs, the y m are the outputs, and the g mn are the transfer functions. This may be written more succinctly in matrix operator notation as, where Y is a column vector of the outputs, G is a matrix of the transfer functions, and U is a column vector of the inputs. In many cases, the system under consideration

1088-424: A transfer matrix is a useful thing to do because, like electrical circuits, the component can often be operated in reverse and its behaviour is dependent on the loads at the inputs and outputs. For instance, a gear train is often characterised simply by its gear ratio, a SISO transfer function. However, the gearbox output shaft can be driven round to turn the input shaft requiring a MIMO analysis. In this example

1156-403: Is a filter such as a muffler on an exhaust system . A transfer matrix representation of it may look like, Here, the T mn are the transmission parameters, also known as ABCD-parameters . The component can be just as easily described by the z-parameters, but transmission parameters have a mathematical advantage when dealing with a system of two-ports that are connected in a cascade of

1224-477: Is a linear time-invariant (LTI) system. In such cases, it is convenient to express the transfer matrix in terms of the Laplace transform (in the case of continuous time variables) or the z-transform (in the case of discrete time variables) of the variables. This may be indicated by writing, for instance, which indicates that the variables and matrix are in terms of s , the complex frequency variable of

1292-470: Is a norm when p ≥ 1, but not when 0 < p  < 1. The space H is defined as the vector space of bounded holomorphic functions on the disk, with the norm For 0 < p ≤ q ≤ ∞, the class H is a subset of H , and the H -norm is increasing with p (it is a consequence of Hölder's inequality that the L -norm is increasing for probability measures , i.e. measures with total mass 1). The Hardy spaces defined in

1360-417: Is an inner (interior) function if and only if | h | ≤ 1 on the unit disk and the limit exists for almost all θ and its modulus is equal to 1 a.e. In particular, h is in H . The inner function can be further factored into a form involving a Blaschke product . The function f , decomposed as f = Gh , is in H if and only if φ belongs to L ( T ), where φ is the positive function in

1428-411: Is defined (the subscript comes from lower ): Therefore, the objective of H ∞ {\displaystyle {\mathcal {H}}_{\infty }} control design is to find a controller K {\displaystyle \mathbf {K} } such that F ℓ ( P , K ) {\displaystyle F_{\ell }(\mathbf {P} ,\mathbf {K} )}

SECTION 20

#1732790450942

1496-417: Is defined as: where σ ¯ {\displaystyle {\bar {\sigma }}} is the maximum singular value of the matrix F ℓ ( P , K ) ( j ω ) {\displaystyle F_{\ell }(\mathbf {P} ,\mathbf {K} )(j\omega )} . The achievable H ∞ norm of the closed loop system is mainly given through

1564-424: Is desired to correctly model energy flows throughout the entire system then a pair of variables whose product is power (power conjugate variables) in one energy domain must map to power conjugate variables in other domains. Power conjugate variables are not unique so care needs to be taken to use the same mapping of variables throughout the system. A common mapping (used in some of the examples in this article) maps

1632-523: Is given further along somewhere below. For the same function F , let f r (e ) = F ( re ). The limit when r → 1 of Re( f r ), in the sense of distributions on the circle, is a non-zero multiple of the Dirac distribution at z = 1. The Dirac distribution at a point of the unit circle belongs to real- H ( T ) for every p  < 1 (see below). For 0 <  p  ≤ ∞, every non-zero function f in H can be written as

1700-400: Is in L ( T ). The function F defined on the unit disk by F ( re ) = ( f ∗ P r )(e ) is harmonic, and M f   is the radial maximal function of F . When M f   belongs to L ( T ) and p  ≥ 1, the distribution f   " is " a function in L ( T ), namely the boundary value of F . For p  ≥ 1, the real Hardy space H ( T ) is

1768-443: Is minimised according to the H ∞ {\displaystyle {\mathcal {H}}_{\infty }} norm. The same definition applies to H 2 {\displaystyle {\mathcal {H}}_{2}} control design. The infinity norm of the transfer function matrix F ℓ ( P , K ) {\displaystyle F_{\ell }(\mathbf {P} ,\mathbf {K} )}

1836-436: Is needed both to describe transducers that make connections between domains and to describe the system as a whole. If the matrix is to properly model energy flows in the system, compatible variables must be chosen to allow this. A MIMO system with m outputs and n inputs is represented by a m × n matrix. Each entry in the matrix is in the form of a transfer function relating an output to an input. For example, for

1904-451: Is often the case that the distinction between input and output variables is ambiguous. They can be either, depending on circumstance and point of view. In such cases the concept of port (a place where energy is transferred from one system to another) can be more useful than input and output. It is customary to define two variables for each port ( p ): the voltage across it ( V p ) and the current entering it ( I p ). For instance,

1972-426: Is represented by infinite sequences indexed by N ; whereas L consists of bi-infinite sequences indexed by Z . When 1 ≤ p < ∞, the real Hardy spaces H discussed further down in this article are easy to describe in the present context. A real function f on the unit circle belongs to the real Hardy space H ( T ) if it is the real part of a function in H ( T ), and a complex function f belongs to

2040-502: Is the fact that the unit circle has finite (one-dimensional) Lebesgue measure while the real line does not. However, for H , one has the following theorem: if m  : D → H denotes the Möbius transformation Then the linear operator M  : H ( H ) → H ( D ) defined by is an isometric isomorphism of Hilbert spaces. Transfer function matrix In control system theory, and various branches of engineering,

2108-471: Is the field of mechanical filters which require transducers between the electrical and mechanical domains in both directions. A simple example is an electromagnetic electromechanical actuator driven by an electronic controller. This requires a transducer with an input port in the electrical domain and an output port in the mechanical domain. This might be represented simplistically by a SISO transfer function, but for similar reasons to those already stated,

H-infinity methods in control theory - Misplaced Pages Continue

2176-501: The L spaces have some undesirable properties, and the Hardy spaces are much better behaved. There are also higher-dimensional generalizations, consisting of certain holomorphic functions on tube domains in the complex case, or certain spaces of distributions on R in the real case. Hardy spaces have a number of applications in mathematical analysis itself, as well as in control theory (such as H methods ) and in scattering theory . For spaces of holomorphic functions on

2244-432: The powers transmitted into, and reflected from a port which are readily measured in the transmission line technology used in distributed-element circuits in the microwave band. The most well known and widely used of these sorts of parameters is the scattering parameters , or s-parameters. The concept of impedance can be extended into the mechanical, and other domains through a mechanical-electrical analogy , hence

2312-457: The s-plane arising from Laplace transforms, rather than time. The examples in this article are all assumed to be in this form, although that is not explicitly indicated for brevity. For discrete time systems s is replaced by z from the z-transform, but this makes no difference to subsequent analysis. The matrix is particularly useful when it is a proper rational matrix , that is, all its elements are proper rational functions . In this case

2380-434: The state-space representation can be applied. In systems engineering, the overall system transfer matrix G ( s ) is decomposed into two parts: H ( s ) representing the system being controlled, and C ( s ) representing the control system. C ( s ) takes as its inputs the inputs of G ( s ) and the outputs of H ( s ) . The outputs of C ( s ) form the inputs for H ( s ) . In electrical systems it

2448-412: The z -parameters and their inverse, the admittance parameters or y -parameters. To understand the relationship between port voltages and currents and inputs and outputs, consider the simple voltage divider circuit. If we only wish to consider the output voltage ( V 2 ) resulting from applying the input voltage ( V 1 ) then the transfer function can be expressed as, which can be considered

2516-505: The Poisson kernel on the unit circle T . For a distribution f on the unit circle, set where the star indicates convolution between the distribution f and the function e → P r (θ) on the circle. Namely, ( f ∗ P r )(e ) is the result of the action of f on the C -function defined on the unit circle by For 0 < p  < ∞, the real Hardy space H ( T ) consists of distributions f such that M f  

2584-439: The circle belongs to real- H ( T ) iff it is the boundary value of the real part of some F ∈ H . A Dirac distribution δ x , at any point x of the unit circle, belongs to real- H ( T ) for every p < 1; derivatives δ′ x belong when p < 1/2, second derivatives δ′′ x when p < 1/3, and so on. It is possible to define Hardy spaces on other domains than the disc, and in many applications Hardy spaces on

2652-414: The effort and flow variables are torque T and angular velocity ω respectively. The transfer matrix in terms of z-parameters will look like, However, the z-parameters are not necessarily the most convenient for characterising gear trains. A gear train is the analogue of an electrical transformer and the h-parameters ( hybrid parameters) better describe transformers because they directly include

2720-435: The effort variables (ones that initiate an action) from each domain together and maps the flow variables (ones that are a property of an action) from each domain together. Each pair of effort and flow variables is power conjugate. This system is known as the impedance analogy because a ratio of the effort to the flow variable in each domain is analogous to electrical impedance. There are two other power conjugate systems on

2788-405: The elements of the Hardy space that extend continuously to the boundary and are continuous at infinity is the disk algebra . For a matrix-valued function, the norm can be interpreted as a maximum gain in any direction and at any frequency; for SISO systems, this is effectively the maximum magnitude of the frequency response. H ∞ techniques can be used to minimize the closed loop impact of

H-infinity methods in control theory - Misplaced Pages Continue

2856-486: The error signals z that we want to minimize, and the measured variables v , that we use to control the system. v is used in K to calculate the manipulated variables u . Notice that all these are generally vectors , whereas P and K are matrices . In formulae, the system is: It is therefore possible to express the dependency of z on w as: Called the lower linear fractional transformation , F ℓ {\displaystyle F_{\ell }}

2924-409: The following are equivalent for a real valued integrable function f on the unit circle: When 1 < p < ∞, H(f) belongs to L ( T ) when f ∈ L ( T ), hence the real Hardy space H ( T ) coincides with L ( T ) in this case. For p = 1, the real Hardy space H ( T ) is a proper subspace of L ( T ). The case of p = ∞ was excluded from the definition of real Hardy spaces, because

2992-448: The function Then F is in H for every 0 < p < 1, and the radial limit exists for a.e. θ and is in H ( T ), but Re( f ) is 0 almost everywhere, so it is no longer possible to recover F from Re( f ). As a consequence of this example, one sees that for 0 < p < 1, one cannot characterize the real- H ( T ) (defined below) in the simple way given above, but must use the actual definition using maximal functions, which

3060-417: The impedance parameters, and other forms of 2-port network parameters, can be extended to the mechanical domain also. To do this an effort variable and a flow variable are made analogues of voltage and current respectively. For mechanical systems under translation these variables are force and velocity respectively. Expressing the behaviour of a mechanical component as a two-port or multi-port with

3128-548: The matrix D 11 (when the system P is given in the form ( A , B 1 , B 2 , C 1 , C 2 , D 11 , D 12 , D 22 , D 21 )). There are several ways to come to an H ∞ controller: Hardy space In complex analysis , the Hardy spaces (or Hardy classes ) H are certain spaces of holomorphic functions on the unit disk or upper half plane . They were introduced by Frigyes Riesz ( Riesz 1923 ), who named them after G. H. Hardy , because of

3196-442: The maximal function M f   of an L function is always bounded, and because it is not desirable that real- H be equal to L . However, the two following properties are equivalent for a real valued function f When 0 < p < 1, a function F in H cannot be reconstructed from the real part of its boundary limit function on the circle, because of the lack of convexity of L in this case. Convexity fails but

3264-560: The open unit disk , the Hardy space H consists of the functions f whose mean square value on the circle of radius r remains bounded as r → 1 from below. More generally, the Hardy space H for 0 < p < ∞ is the class of holomorphic functions f on the open unit disk satisfying This class H is a vector space. The number on the left side of the above inequality is the Hardy space p -norm for f , denoted by ‖ f ‖ H p . {\displaystyle \|f\|_{H^{p}}.} It

3332-429: The output of one into the input port of another. In such cases the overall transmission parameters are found simply by the matrix multiplication of the transmission parameter matrices of the constituent components. When working with mixed variables from different energy domains consideration needs to be given on which variables to consider analogous. The choice depends on what the analysis is intended to achieve. If it

3400-416: The paper ( Hardy 1915 ). In real analysis Hardy spaces are certain spaces of distributions on the real line, which are (in the sense of distributions) boundary values of the holomorphic functions of the complex Hardy spaces, and are related to the L spaces of functional analysis . For 1 ≤  p  < ∞ these real Hardy spaces H are certain subsets of L , while for p < 1

3468-411: The preceding section can also be viewed as certain closed vector subspaces of the complex L spaces on the unit circle. This connection is provided by the following theorem ( Katznelson 1976 , Thm 3.8): Given f ∈ H , with p ≥ 1, the radial limit exists for almost every θ. The function f ~ {\displaystyle {\tilde {f}}} belongs to the L space for

SECTION 50

#1732790450942

3536-492: The product f = Gh where G is an outer function and h is an inner function , as defined below ( Rudin 1987 , Thm 17.17). This " Beurling factorization" allows the Hardy space to be completely characterized by the spaces of inner and outer functions. One says that G ( z ) is an outer (exterior) function if it takes the form for some complex number c with | c | = 1, and some positive measurable function φ {\displaystyle \varphi } on

3604-430: The real Hardy space iff Re( f ) and Im( f ) belong to the space (see the section on real Hardy spaces below). Thus for 1 ≤ p < ∞, the real Hardy space contains the Hardy space, but is much bigger, since no relationship is imposed between the real and imaginary part of the function. For 0 < p < 1, such tools as Fourier coefficients, Poisson integral, conjugate function, are no longer valid. For example, consider

3672-536: The real distribution Re( f ) on the circle, because the Taylor coefficients c n of F can be computed from the Fourier coefficients of Re( f ). Distributions on the circle are general enough for handling Hardy spaces when p  < 1. Distributions that are not functions do occur, as is seen with functions F ( z ) = (1− z ) (for | z | < 1), that belong to H when 0 < N   p  < 1 (and N an integer ≥ 1). A real distribution on

3740-399: The representation of the outer function G . Let G be an outer function represented as above from a function φ on the circle. Replacing φ by φ , α > 0, a family ( G α ) of outer functions is obtained, with the properties: It follows that whenever 0 < p , q , r < ∞ and 1/ r = 1/ p + 1/ q , every function f in H can be expressed as the product of a function in H and

3808-430: The response near the system bandwidth to achieve good robust stabilization. Commercial software is available to support H ∞ controller synthesis. First, the process has to be represented according to the following standard configuration: [REDACTED] The plant P has two inputs, the exogenous input w , that includes reference signal and disturbances, and the manipulated variables u . There are two outputs,

3876-522: The same sort occurs in the magnetic domain. This maps magnetic reluctance to electrical resistance, resulting in magnetic flux mapping to current instead of magnetic flux rate of change as required for compatible variables. The matrix representation of linear algebraic equations has been known for some time. Poincaré in 1907 was the first to describe a transducer as a pair of such equations relating electrical variables (voltage and current) to mechanical variables (force and velocity). Wegel, in 1921,

3944-512: The same variables that are in use. The mobility analogy maps mechanical force to electric current instead of voltage. This analogy is widely used by mechanical filter designers and frequently in audio electronics also. The mapping has the advantage of preserving network topologies across domains but does not maintain the mapping of impedances. The Trent analogy classes the power conjugate variables as either across variables, or through variables depending on whether they act across an element of

4012-514: The system. Choosing variables that are convenient to measure, particularly ones that the sensor is sensing, may be more useful. For instance, in the thermal resistance analogy, thermal resistance is considered analogous to electrical resistance, resulting in temperature difference and thermal power mapping to voltage and current respectively. The power conjugate of temperature difference is not thermal power, but rather entropy flow rate, something that cannot be directly measured. Another analogy of

4080-451: The transfer matrices based on port voltages and currents are convenient to use in practice. Voltage is difficult to measure directly, current next to impossible, and the open circuits and short circuits required by the measurement technique cannot be achieved with any accuracy. For waveguide implementations, circuit voltage and current are entirely meaningless. Transfer matrices using different sorts of variables are used instead. These are

4148-589: The transfer matrix of a two-port network can be defined as follows, where the z mn are called the impedance parameters , or z -parameters. They are so called because they are in units of impedance and relate port currents to a port voltage. The z-parameters are not the only way that transfer matrices are defined for two-port networks. There are six basic matrices that relate voltages and currents each with advantages for particular system network topologies. However, only two of these can be extended beyond two ports to an arbitrary number of ports. These two are

SECTION 60

#1732790450942

4216-418: The trivial case of a 1×1 transfer matrix. The expression correctly predicts the output voltage if there is no current leaving port 2, but is increasingly inaccurate as the load increases. If, however, we attempt to use the circuit in reverse, driving it with a voltage at port 2 and calculate the resulting voltage at port 1 the expression gives completely the wrong result even with no load on port 1. It predicts

4284-458: The turns ratios (the analogue of gear ratios). The gearbox transfer matrix in h-parameter format is, For an ideal gear train with no losses (friction, distortion etc), this simplifies to, where N is the gear ratio. In a system that consists of multiple energy domains, transfer matrices are required that can handle components with ports in different domains. In robotics and mechatronics , actuators are required. These usually consist of

4352-475: The unit circle such that log ⁡ ( φ ) {\displaystyle \log(\varphi )} is integrable on the circle. In particular, when φ {\displaystyle \varphi } is integrable on the circle, G is in H because the above takes the form of the Poisson kernel ( Rudin 1987 , Thm 17.16). This implies that for almost every θ. One says that h

4420-476: The unit circle, The space H ( T ) is a closed subspace of L ( T ). Since L ( T ) is a Banach space (for 1 ≤ p ≤ ∞), so is H ( T ). The above can be turned around. Given a function f ~ ∈ L p ( T ) {\displaystyle {\tilde {f}}\in L^{p}(\mathbf {T} )} , with p ≥ 1, one can regain a ( harmonic ) function f on

4488-459: The unit circle, and one has that Denoting the unit circle by T , and by H ( T ) the vector subspace of L ( T ) consisting of all limit functions f ~ {\displaystyle {\tilde {f}}} , when f varies in H , one then has that for p  ≥ 1,( Katznelson 1976 ) where the ĝ ( n ) are the Fourier coefficients of a function g integrable on

4556-429: The unit disk by means of the Poisson kernel P r : and f belongs to H exactly when f ~ {\displaystyle {\tilde {f}}} is in H ( T ). Supposing that f ~ {\displaystyle {\tilde {f}}} is in H ( T ), i.e. that f ~ {\displaystyle {\tilde {f}}} has Fourier coefficients (

4624-530: Was the first to express these equations in terms of mechanical impedance as well as electrical impedance. The first use of transfer matrices to represent a MIMO control system was by Boksenbom and Hood in 1950, but only for the particular case of the gas turbine engines they were studying for the National Advisory Committee for Aeronautics . Cruickshank provided a firmer basis in 1955 but without complete generality. Kavanagh in 1956 gave

#941058