# SYSTEMS AND CONTROL ZAK PDF

- Contents:
- A variable structure approach to robust control of VTOL aircraft
- Proceedings and Books 2013
- Integrated fault estimation and fault‐tolerant control for uncertain Lipschitz nonlinear systems
- Introduction
- Detection of Zak phases and topological invariants in a chiral quantum walk of twisted photons

Contents

˙Zak, Stanislaw H. Systems and Control / Stanislaw H. ˙Zak. p. cm. Includes bibliographical references and index. ISBN 1. Linear control systems. Systems and Control (The Oxford Series in Electrical and Computer Engineering) [Stanislaw H. Zak] on norinkgibipen.gq *FREE* shipping on qualifying offers. Zak, Stanislaw H. Systems and Control / Stanislaw H. Zak. p. cm. Includes bibliographical references and index. ISBN 1. Linear control systems.

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INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL. Int. J. Robust Nonlinear other elegant parts concerning systems and control. Chapter 1 introduces the . norinkgibipen.gq$zak/systems/norinkgibipen.gq'. REFERENCES. 1. Jan 11, (Required) Stanislaw H. Zak, Systems and Control, Oxford Uni- versity Press mal control methods; linear quadratic regulator, dynamic pro-. Sign in. Loading Main menu.

Historically, topology was first proven to have a key role in explaining algebraically decaying order, transport and coherence of two-dimensional Bose liquids, XY models and crystals 1. Shortly after, the quantization of Hall conductance 2 was shown to be rooted in current-carrying edge states, protected by the topology of the bulk 3 , 4 , 5.

Being associated with a global order, these phases are robust against local perturbations and promise important applications in metrology, spintronics and quantum computation see, for example, refs 6 , 7 , 8.

Intense studies 9 followed the early discoveries, and topological insulators have by now been engineered in a variety of physical architectures, such as superconducting 10 , mechanical 11 , optomechanical 12 , photonic 13 , atomic 14 and acoustic platforms Such diverse systems have been exposed to either real or synthetic magnetic fields, and their topological properties have been studied by scattering at the interface between different domains 15 , 16 or imaging edge states 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , Direct detection of topological invariants in the bulk of the system with no need of edges has been reported so far by very few experiments 27 , 28 , Topological insulators are classified in terms of dimensionality and discrete symmetries One-dimensional 1D systems with chiral symmetry are characterized by the Zak phase, that is, the Berry phase accumulated by an eigenstate during its parallel transport through the whole Brillouin zone The Zak phase is closely related to the electric polarization in solids and plays a key role in the modern theory of insulators 32 , Periodically driven Floquet systems are attracting an increasing interest, as these show richer topological features than their static counterparts 17 , 24 , 25 , 26 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , Particularly promising Floquet topological systems are discrete-time quantum walks QWs 16 , 17 , 29 , 45 , 46 , 47 , and recent works have reported the observation of topological invariants 16 , 29 , quantum phase transitions 46 and edge states 17 in these systems.

In its simplest version, a QW is the discrete time evolution of a particle the walker on a 1D lattice At each step, the walker moves to neighbouring sites, with the direction of the shift depending on the state of an internal two-level degree of freedom the coin. Between consecutive steps, a rotation modifies the coin state, univoquely determining the following evolution.

Remarkably, this occurs during the free evolution of the system, in absence of any external force or loss mechanism, with the only requirement that the initial wavefunction is localized. We validate experimentally this finding in a photonic discrete-time QW based on the orbital angular momentum OAM of a light beam. We implement the same QW in a shifted inequivalent timeframe and measure a second Zak phase.

Combining the two windings, we extract the complete set of topological invariants characterizing the system. Finally, we prove the robustness of our detection by adding dynamical disorder. These measurements provide therefore a bulk measurement of the Zak phases and complete topological invariants of a 1D chiral QW. Our proposal may be straightforwardly applied to general driven Floquet systems. Results Zak phase detection in the bulk of a QW In one dimension, discrete-time QWs with chiral symmetry display a quantized Zak phase and have been extensively studied in the past years.

Among these implementations, we focus on the photonic platform proposed in ref. Once the system is prepared in an initial state , its state after t timesteps is given by where the single-step operator is obtained by cascading suitable combinations of quarter-wave plates and q-plates 46 , 50 , In Fig.

A light beam, exiting a single-mode fibre depicted on the left, performs a QW by propagating through a sequence of quarter-wave plates purple disks and q-plates turquoise disks.

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Each datapoint is an average over 10 different measurements error bars are the associated s. Purple and red dots refer, respectively, to different input polarizations, and. The translation operator Q is implemented by a q-plate, a liquid crystal device that yields an effective spin—orbit interaction in the light beam. Further details on the q-plates and on the complete experimental setup are provided in the Methods section and Supplementary Fig.

These two sets of teeth are out of alignment with each other by a tooth width, as shown in Figure 1. Specically, the rotor teeth of the left end-cap are displaced one-half pitch from the teeth on the right end-cap.

N Figure 1. N1 S4 S3. S5 S2. N3 N4 S1. Stator Iron windings end-caps Magnet S. Air gap Stator. The left end-cap is magnetized as a north pole, while the right end-cap is magnetized as a south pole.

The slotted stator is equipped with two or more individual coils. We now discuss the way in which the PM stepper motor can be used to perform precise positioning. Assume that the starting point is as represented in Figure 1. The rotor remains in this position as long as stator coil voltageand hence stator polaritiesremains unchanged.

This is because the net torque acting on the rotor is zero. If we now de-energize coil AA and energize coil BB so that the left pole becomes an S-pole and the right pole assumes an N-polarity, then this results in the new steady-state position shown in Figure 1.

We can proceed by alternatively energizing the two stator coils. The PM stepper motor can thus be controlled by digital pulses applied to its two inputs: Our next objective is to devise a mathematical model of the PM stepper motor. In our dis- cussion we use the following notation. We denote by Rt the number of rotor teeth. The angular displacement between rotor teeth is called the tooth pitch and denoted T p. The angular displace- ment of the rotor as we change the input voltage from one pair of windings to the other is called the step length and denoted Sl.

The number of phasesthat is, the number of pairs of stator windingsis denoted N p. In this example, it takes four input voltage switchings, and hence steps, to advance the rotor one tooth pitch. Following Krause and Wasynczuk [, pp.

The ux leaves the left end-cap through the rotor tooth at the top that is aligned with the stator tooth having the A part of the AA winding. The ux then travels up through the stator tooth in the stator iron.

Next, the ux splits and travels around the circumference of the stator, as shown in Figure 1. The ux returns to the south pole of the rotor through the stator tooth positioned at the bottom of the A part of the AA winding, as shown in Figure 1. We denote the peak ux linkage produced by the rotor permanent magnet in the AA stator winding by nm , where n is the number of turns of the winding. Tp Tooth pitch Sl Step length Rt Number of teeth on rotor Np Number of phases, that is, stator winding pairs Angular displacement of the rotor Angular velocity of the rotor va t Voltage applied to phase a, that is, AA stator winding vb t Voltage applied to phase b, that is, BB stator winding i a t Current in phase a i b t Current in phase b R Stator winding resistance per phase L Stator winding inductance per phase B Viscous friction coefcient I Rotor inertia Kb Back emf constant.

The ux then would travel around the circumference of the stator and return to the rotor through the stator pole on which the B part of the BB winding is wound. Combining the above equations, we obtain a model of a two-phase i.

Using the above notation, we represent the PM stepper motor model in state space format: In our derivation of the model, we use Newtons laws, in a fashion similar to that of Kwakernaak and Sivan [] and Ogata [], where linearized models of the stick balancer were developed.

An alternative derivation, using the method of DAlembert, can be found in Cannon [41, Section A free-body diagram of a cart on which a stick i. That is, the force H is the horizontal reaction force that the cart exerts upon the pendulum, whereas H is the force exerted by the pendulum on the cart. Similar convention applies to the forces V and V.

The x and y are the coordinates of the xed, nonrotating coordinate frame xy. The mass of the cart is M, while the mass of the stick is m. The length of the stick is 2l, and its center of gravity is at its geometric center. The control force applied to the cart is labeled u. We assume that the wheels of the cart do not slip.

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Friedland [91, p. Let x G , yG be the coordinates of the center of gravity of the stick. We are now ready to write down equations modeling the system.

The equation that describes the rotational motion of the stick about its center of gravity is obtained by applying the rotational version of Newtons second law. We next write the equation that describes the horizontal motion of the center of gravity of the stick. Applying Newtons second law along the x axis yields. The equation that describes the vertical motion of the center of gravity of the stick is obtained by applying Newtons second law along the y axis.

We have. Using the expression for I , given by 1. Then, by combining 1. While designing a controller for the stick balancer system, we will use an equivalent represen- tation of 1. Then, we use the obtained model to investigate two interacting populations. The simplest model of the population dynamics of the given species is the one resulting from the hypothesis that the net rate of change of x is proportional to the current value of x.

Let r be the rate of growth of the population. It does not account for the limitation of space, food supply, and so on. Therefore, we modify the unrestricted growth rate r to take into account the fact that the environment of the population can only support a certain number, say K , of the species.

The constant K is called the carrying capacity of the environment or the saturation level. K Using the above model of the growth rate, we obtain the following model of the population growth: Then, we represent 1. Equation 1. According to Boyce and DiPrima [32, p.

For reasons somewhat unclear, Verhulst referred to it as the logistic growth; hence 1. We will use the logistic equation to analyze interacting species. We rst model two interacting species that do not prey upon each other, but compete for the same resourcesfor example, for a common food supply. Let x1 and x2 be the populations, at time t, of the two competing species. We assume that the population of each of the species in the absence of the other is governed by a logistic equation.

When both species are present, each will affect the growth rate of the other. Because x1 and x2 use the same resources, the growth rate of x1 will be lowered by an amount proportional to the size of x2. Similarly, the growth rate of x2 is reduced by a2 x1 , where a2 is a measure of the degree by which species x1 lowers the growth rate of x2. We will now study the situation in which one species, the predator, preys upon the other species, the prey, and there is no competition between the predator and prey for a common resource.

Let x1 and x2 denote the populations of the prey and predator, respectively. We will make the following modeling assumptions: The larger the population of the predator, the more prey is eaten. The larger the population of the prey, the easier hunting is for the predator.

We model the growth rate decrease of the prey by including in 1. The growth rate increase of the predator is modeled by adding the term ex1 x2 in 1. They were published by Lotka in and by Volterra in Notes A landmark paper on mathematical description of linear dynamical systems from a controls point of view is by Kalman [].

The rst chapter of Sontags book [] is a nice, easy to read, and comprehensive introduction to the subject of mathematical control theory. Sontags chapter is easily available on the web. For more examples of dynamical system models from mechani- cal and electrical engineering, the reader is referred to Truxal [], Banks [19], Raven [], Mohler [], and DAzzo and Houpis [57].

Wertz [] has many excellent real-life models from space science. A well-written paper on vehicle modeling and control for broad audience is by Ackermann [2].

In addition to references and , vehicle and tire dynamics mod- eling are investigated in detail by Nalecz [], Nalecz and Bindemann [, ], and Smith and Starkey [, ]. For further analysis of the equations modeling competing species and the predatorprey equations, we recommend Boyce and DiPrima [32, Sections 9. Our derivation of the logistic equation follows that of Sandefur []. The logistic equation as well as equations modeling interacting species can be used for mathematical modeling of tumor dynamics and interactions between tumor and immune system.

A survey of models for tumorimmune system dynamics can be found in the well-edited book by Adam and Bellomo [4]. Mayr writes on page in reference , It is still widely believed that the steam-engine governor is the oldest feedback device, and that James Watt had not only invented but also patented it. While both errors are easily refuted, we are still not able to reconstruct the history of this invention in all desired completeness. Watt did not patent the governor. He did not invent it either. On page of his book [], Mayr adds the following: Theorem 1.

Then, the sum of the potential energy and kinetic energy is constant. Hint Denote by C t a differentiable curve along which a particle of mass m moves.

A toilet system is shown in Figure 1. The control objective is to maintain water in the tank at a constant level. Draw a block diagram of the system. Handle Lift wire. Valve seat Overflow tube Outlet valve. Adapted from reference , p. A pumped storage system can be used to supply water to a turbine producing electric power.

The two storage tanks are connected together through a valve that may be modeled as a linear resistance R1. The resistance of the turbine supply pump is modeled by a linear resistance R2. Suppose that we wish to design a controller that would control the inow rate qi in order to maintain the output ow rate qo at some desired value qd. R2 Let A1 and A2 represent the surface areas in the two tanks.

## A variable structure approach to robust control of VTOL aircraft

Numerical data of a two-tank pumped storage system is given in Table 1. The role of an automotive suspension is to support the vehicle body on the axles. A simple linear model of the suspension assembly associated with a single wheel, called the quarter-car model, is shown in Figure 1.

The vehicle body is represented by the sprung mass, m s , while the tire and axle are represented by the unsprung mass, m u. The suspension consists of the spring, the shock absorber, and the variable force element. The role of the variable force element is to generate a force that compensates for the uneven road, which is the source of the system disturbances.

Assume that the variable force element Fa can instantaneously provide any desired force. The above described model of an automotive suspension is referred to as a linear active suspension. Write down equations modeling the dynamics of the two-degree-of-freedom system depicted in Figure 1. Represent the obtained linear model of active suspension in state-space format. Use data of a commercial vehicle given in Table 1. The data come from Sunwoo and Cheok []. Substitute the vehicle data into the modeling equations.

A design goal of an automotive engineer may be to design a control law to isolate the sprung mass from road disturbances. Assume that we can measure the vertical velocity of the sprung mass. Write the output equation. This model comes from Driels [68, pp.

The system operates as follows. The lead screw moves the table with a workpiece on it under the cutter to perform the desired operation. The table is positioned by a control system. We wish to place the table in a desirable position as rapidly and accurately as possible. This means that while modeling the machine tool drive system we should, in particular, take into account the exibility of the lead screw.

The length of the lead screw implies that its rotation at the gear box will not be the same as its rotation at the table. Thus, the angular twist along the length of the lead screw cannot be neglected. Model the lead screw exibility using a torsional spring as illustrated in Figure 1.

The system input is the motor torque m , and the system output is the rotation of the end of the lead screw labeled l. Denote the motor moment of inertia by Im , and denote by 1 the load torque on the motor gear due to the rest of the gear train.

Write the equation modeling the dynamics of the motor shaft. The work done by the rst gear is equal to that of the second. Workpiece Table Lead screw. Adapted from Gear Driels [68]. Model the load shaft exibility with a torsional spring. The torsional spring coefcient is k. Assume that the moment of inertia of the lead screw is negligible compared to that of the load.

Denote the moment of inertia of the load by Il. The torque due to viscous friction of the load is cl l , where cl is the coefcient of the viscous friction of the load. The angular displacement of the end of the lead screw is l. Write the equation modeling the dynamics of the load. Represent the modeling equations in state-space format. Typical numerical data for a machine tool drive system are shown in Table 1. In your derivation assume small angles.

Vehicle data are given in Table 1. Frictionless surface. Treat the eld inductance L f as a parasitic element and denote it as. To be sure, mathematics can be extended to any branch of knowledge, including economics, provided the concepts are so clearly dened as to permit accurate symbolic representation. That is only another way of saying that in some branches of discourse it is desirable to know what you are talking about. James R. Most of dynamical systems analyzed in this book are modeled by ordinary differential equations.

A main use of a mathematical model is to predict the system transient behavior. Unlike linear systems, where closed-form solutions can be written in terms of the systems eigenvalues and eigenvectors, nding analytical, exact, solutions to nonlinear differential equations can be very difcult or impossible. However, we can approximate the solutions of differential equations with difference equations whose solutions can be obtained easily using a computer. In this chapter we discuss a number of methods for solving differential equations.

The rst class of methods allows us to graphically determine solutions to second-order differential equations. Then, we discuss numerical techniques for solving differential equations.

After that, we present two methods of linear approximation of nonlinear systems. Analysis in the state plane is applicable to linear and nonlinear systems modeled by second- order ordinary differential equations. The state-plane methods are graphical procedures for solving such equations. Using state-plane methods, one can graphically determine the transient response of a second-order dynamical system. We now introduce relevant terms.

Consider a class of second-order systems whose dynamics are modeled by the differential equation.

The plane with coordinates x1 and x2 is labeled as the state plane or phase plane. A solution of 2. As t varies, the RP describes a curve in the state plane called a trajectory. A family of trajectories is called a phase portrait. Example 2. The spring is assumed to be linear; that is, it obeys Hookes law.

Applying Newtons law, we obtain the equation of motion of the mass:. Figure 2. We see that a phase portrait of equation 2. Different circular trajectories, in this plane, correspond to different values of the constant A representing initial conditions.

Therefore, as time increases, the RP moves in a clockwise direction along the trajectory. We assume that the satellite is rigid and is in a frictionless environment. It can rotate about an axis perpendicular to the page as a result of torque applied to the satellite by ring the thrusters. The system input is the applied torque, and the system output is the attitude angle.

The satellites moment of inertia is I. Thruster Figure 2. With this control, the system equations are. In the state plane the above equation represents a family of parabolas open toward the positive x1 axis. Typical system trajectories for this case are shown in Figure 2.

## Proceedings and Books 2013

In the above examples, systems trajectories were constructed analytically. Analytical methods are useful for systems modeled by differential equations that can be easily solved. If the system of differential equations cannot be easily solved analytically, we can use graphical or numerical methods.

We now describe a graphical method for solving second-order differential equations or a system of two rst-order differential equations. In other words, an isocline is a locus a set of points , of the trajectories constant slope.

## Integrated fault estimation and fault‐tolerant control for uncertain Lipschitz nonlinear systems

To obtain equations that describe isoclines, we consider a system of two rst-order differential equations of the form. Thus, we eliminated the independent variable t from the set of the rst-order differential equations given by 2.

In equation 2. Note that m is just the slope of the tangent to the trajectory passing through the point [x1 x2 ]T. The locus of the trajectory constant slope,. The curve that satises the above equation is an isocline corresponding to the trajectories slope m because a trajectory crossing the isocline will have its slope equal to m.

The idea of the method of isoclines is to construct several isoclines in the state plane. This then allows one to construct a eld of local tangents m. Then, the trajectory passing through any given point in the state plane is obtained by drawing a continuous curve following the directions of the eld.

This equation describes a family of straight lines through the origin. In Figure 2. This means that trajectories of the springmass system are perpendicular to the isoclines; that is, the trajectories are circles centered at the origin of the state plane. We will nd values of for which there are isoclines along which the trajectory slope and the isocline slope are equal. Such isoclines, if they exist, are called the asymptotes. Then, the second-order differential equation given by 2.

Hence, the condition for the existence of asymptotes is 1. If yes, then write the equations of the asymptotes. The trajectory slope is. An asymptote is the isocline for which its slope and the trajectory slope are equal. We can use a computer to draw phase portraits interactively. An inspection of the phase portrait of the van der Pol equation reveals the existence of a closed trajectory in the state plane.

This solution is an example of a limit cycle. Denition 2. Note that the differential equation modeling the springmass system does not have a limit cycle because the solutions are not isolated.

Limit cycles can only appear in nonlinear systems; they do not exist in linear systems. Theorem 2. Proof We prove the theorem by contradiction. Equation 2. Therefore, if 2.

The proof of the theorem is complete. Bendixsons theorem gives us a necessary condition for a closed curve to be a limit cycle. It is useful for the purpose of establishing the nonexistence of a limit cycle. In general, it is very difcult to establish the existence of a limit cycle. Sufcient conditions for the existence of a limit cycle are discussed, for example, by Hochstadt [, Chapter 7]. A method for predicting limit cycles, called the describing function method, is presented in Section 2.

We begin our presentation of numerical techniques that allow us to determine approximate solutions to the given system of differential equations with a method based on a Taylor series expansion.

This section is based on the Class Notes of Prof. Haas []. If we stop after the qth term of the Taylor series solution, then the remainder after q terms of a Taylor series gives us an estimate of the error. We can expect the Taylor series solution to give a fairly accurate approximation of the solution of the state-space equations only in a small interval about t0.

We illustrate the method with simple examples. We now illustrate the method of Taylor series when applied to a system of rst-order differential equations.

The solution to the given system is then computed as. The Taylor series method can be programmed on a computer. However, this is not a very efcient method from a numerical point of view. In the following, we discuss other methods for numerical solution of the state equations that are related to the method of Taylor series. We rst present two simple numerical techniques known as the forward and backward Euler methods. We call h the step length. The above is known as the forward Euler algorithm.

We then use the rectangular rule for numerical integration that can be stated as follows. A B Figure 2. The backward Euler method differs from the forward Euler method in the way we approximate the derivative: For this reason, we say that the backward Euler method is an implicit integration algorithm. For further discussion of this issue, we refer to Parker and Chua []. Eulers methods are rather elementary, and thus they are not as accurate as some of the more sophisticated techniques that we are going to discuss next.

Applying the trapezoid rule of integration to 2. If the function f is nonlinear, then we will, in general, not be able. We rst divide the interval [t 0 , t f ] into N equal subintervals using evenly spaced subdivision points to obtain. Table 2. We could use x t1 obtained from the corrector formula to construct a next approximation to x t1.

This general algorithm is referred to as a second-order predictorcorrector method. An n-dimensional system of rst-order differential equations can be worked analogously. We will now show that the predictorcorrector method yields a solution that agrees with the Taylor series solution through terms of degree two.

Comparing the right-hand sides of the last where f t , f x , x, two equations, we see that the solution obtained using the predictorcorrector method agrees with the Taylor series expansion of the true solution through terms of degree two. In the new coordinates the three points become. We found x 1 using the predictorcorrector method in Example 2. The calculations for Runges method are summarized in Table 2. Runges method can, of course, be used for systems of rst-order differential equations.

It is based on the use of the formula. Our calculations for the RungeKutta method are summarized in Table 2. The above problem has been worked by the predictorcorrector, Runges, and the RungeKutta methods.

This problem can be solved exactly by means of the Taylor series. Such algorithms are referred to as single-step algorithms. In general, it may be difcult to decide which type of algorithm is more efcient because the performance of a particular algorithm is problem dependent.

Well-known multistep algorithms are: AdamsBashforth, AdamsMoulton, and Gear. More information about the above-mentioned algorithms can be found in Parker and Chua [, Section 4. The motivation for linearization is that the dynamical behavior of many nonlinear system models can be well approximated within some range of variables by linear system models. Then, we can use well-developed techniques for analysis and synthesis of linear systems to analyze a nonlinear system at hand.

However, the results of analysis of nonlinear systems using their linearized models should be carefully interpreted in order to avoid unacceptably large errors due to the approximation in the process of linearization. R R is continuously differentiable; that is, h C 1. Let x0 be an operating point. Then, we can represent 2. We now illustrate the process of linearization on the simple pendulum. Let g be the gravity con- stant. The tangential force component, mg sin , acting on the mass m returns the pendulum to its equilibrium position.

By Newtons second law we obtain. Thus, for small angular displacements , the force F is proportional to the dis- placements. Tangent plane h x0. I I We linearize the above model of an equilibrium state of the form. Equating the right-hand sides of the above enginegovernor equations to zero yields. Performing needed manipulations, we obtain 0 1 0 f g sin2 x1e b 2g sin x1e.

The method can be used to predict limit cycles in nonlinear systems. The describing function method can be viewed as a harmonic linearization of a nonlinear element. The method provides a linear approximation to the nonlinear element based on the assumption that the input to the nonlinear element is a sinusoid of known, constant amplitude.

The fundamental harmonic of the elements output is compared with the input sinusoid to determine the steady-state amplitude and phase relation. This relation is the describing function for the nonlinear element. The describing func- tion method is based on the Fourier series.

In our review of Fourier series, we use the notion of a scalar product over a function space, which we discuss next. Our presentation of the scalar product in the context of spaces of functions follows that of Lang []. The reader familiar with the Fourier series may go directly to Subsection 2.

Using simple properties of the integral, we can verify that the above scalar product satises the following conditions:.

## Introduction

A set of functions from C [a, b] is said to be mutually orthogonal if each distinct pair of functions in the set is orthogonal. This can be veried by direct integration. So far we discussed only continuous functions. In many applications we have to work with more general functions, specically with piecewise continuous functions. The function f is continuous on each open subinterval ti1 , ti.

The graph of a piecewise continuous function is shown in Figure 2. In further discussion, we assume that two piecewise functions f and g are equal if they have the same values at the points where they are continuous. Thus, the functions shown in Figure 2. Let f V. Hence, each such integral must be equal to 0. The function f is continuous on ti1 , ti.

The proof is complete. The above theorem implies that the scalar product on the space V of piecewise continuous functions is positive denite. It follows from the above denition that if T is a period of f , then 2T is also a period, and so is any integral multiple of T. On the set of points where the series 2. Suppose that the series 2. We use the results of the previous subsection to nd expressions for the coefcients Ak and Bk.

We rst compute Ak coefcients. Multiplying 2. It follows from 2. Using 2. Using the orthogonality relations 2.

Then, coefcients Ak and Bk can be computed using 2. However, we do not know whether this series converges for each value of t and, if so, whether its sum is f t. The answer is given by the theorem that we state after explaining the notation used. Then, f has a Fourier series of the form 2. With this knowledge of the Fourier series, it is now possible to discuss the describing function method for analyzing a class of nonlinear control systems.

In our further discussion, we sometimes use a double box to represent a nonlinear element as illustrated in Figure 2. Consider a feedback system that contains a nonlinear element as shown in Figure 2. The output of the nonlinear element is, in general, not sinusoidal in response to a sinusoidal input.

The describ- ing function method relies on the assumption that only the fundamental harmonic is signicant. This assumption is called the ltering hypothesis. The ltering hypothesis is often valid because many, if not most, control systems are low-pass lters. This results in higher harmonics being more attenuated compared with the fundamental harmonic component.

The stability of the nonlinear closed-loop system depicted in Figure 2.

Case i G s plane. We might have the situations shown in Figure 2. Assuming that the poles of G s are in the left-half complex plane, the closed-loop system is asymptotically stable.

At intersection point 1, any increase in X will cause the operating point to be encircled and will consequently cause X to grow. If there is a decrease in X , then the oscillations will decay. Thus, point 1 represents an unstable limit cycle. At intersection point 2, an increase in X would shift the operating point outside the G j plot resulting in the decaying oscillations.

Point 2 represents a stable limit cycle. In case iii , any oscillation will grow without limit. We rst derive the describing function of the ideal relay. When the input of the ideal relay is greater than zero, its output is M. When the relays input is less than zero, its output is M. The output y t of the relay is a rectangular wave. It is drawn horizontally to the right of the relay characteristic in Figure 2. This rectangular wave is an odd function.

Thus, the Fourier series of the relays output has the form. X X A normalized plot of the describing function of the ideal relay nonlinearity is depicted in Figure 2.

X The sinusoidal transfer function of the linear plant model is. Note that the ltering hypothesis is satised in this example. The higher harmonics are even more attenuated compared with the fundamental harmonic.

Notes The system of differential equations 2. Al-Khafaji and Tooley [6] provide an introductory treatment of numerical solutions of differential equations. For further discussion of solving differential equations using numerical methods and analyses of their computational efciency, the reader is referred to Salvadori and Baron [], Conte and de Boor [53, Chapter 8], or Parker and Chua []. For an in-depth treat- ment of the describing function method, we refer to Graham and McRuer [] and Chapters 6 and 7 of Hsu and Meyer [].

Graham and McRuer [] make an interesting remark about the origins of the describing function method. They note on page 92 of their book: While the notion of representing a nonlinearity by an equivalent linear element is quite old and has been used by many writers, the rst instance of a major systematic exploitation of the technique was probably by N.

Kryloff and N. Bogoliuboff, Introduction to Non-linear Mechanics a free translation of the Russian edition of , by S. See also N. Minorsky, Introduction to Non-linear Mechanics, J.

Edwards, Ann Arbor, Mich. The so-called sinusoidal describing function that we ana- lyzed in the previous section, was developed almost simultaneously in several different countries during and just after World War II.

The rst to introduce the method in the open literature appears to be A. The behavior of numerous dynamical systems is often modeled by a set of linear, rst- order, differential equations. Frequently, however, a linear model is a result of linearization of a nonlinear model.

Linear models seem to dominate the controls literature. Yet nature is nonlinear. All physical systems are nonlinear and have time-varying parameters in some degree [, p. How can one reconcile this paradox? Graham and McRuer [, p. Where the effect of the nonlinearity is very small, or if the parameters vary only slowly with time, linear constant-parameter methods of analysis can be applied to give an approximate answer which is adequate for engineering purposes.

The analysis and synthesis of physical systems, predicated on the study of linear constant-parameter mathematical models, has been, in fact, an outstandingly successful enterprise. A system represented as linear, however, is to some extent a mathematical abstraction that can never be encountered in a real world. Either by design or because of natures ways it is often true that experimental facts do not, or would not, correspond with any prediction of linear constant-parameter theory.

In this case nonlinear or time-varying- parameter theory is essential to the description and understanding of physical phenomena.

Find the time it takes the representative point to move from A to B. Sketch a phase-plane portrait for this system. The above nonlinear system was analyzed in reference , p. Represent the equation of motion derived in a in state- space format.

Then, linearize the model about the obtained equilibrium point. The operation of the cyclotron can be described as follows. Positive ions enter the cyclotron from the ion source, S, and are available to be accelerated. An electric oscillator establishes an accelerating potential difference across the gap of the two D-shaped objects, called dees.

An emerging ion from the ion source, S, nds the dee that is negative. It will accelerate toward this dee and will enter it. Once inside the dee, the ion is screened from electric elds by the metal walls of the dees. The dees are immersed in a magnetic eld so that the ions trajectory bends in a circular path. The radius of the path depends on the ions velocity.

After some time, the ion emerges from the dee on the other side of the ion source.

Dee Figure 2. Thus the ion again faces a negative dee. The ion further accelerates and describes a semicircle of larger radius. This process goes on until the ion reaches the outer edge of one dee, where it is pulled out of the cyclotron by a negatively charged deector plate.

The nominal radius of the ions orbit is r. Show that the element v c 1 v 1 cn v n is orthogonal to v1 , v2 ,. The above relation is called the Bessel inequality. A Derive the describing function of the nonlinear element whose characteristic is shown in Figure 2.

K1 Figure 2. K Figure 2. The describing function of the dead-zone nonlinearity is given in Exercise 2. Derive the describing function of the ideal relay with dead zone whose characteristic is shown in Figure 2. In bear country, wear bells to avoid surprising bruins. Dont approach a bear, but dont run away from one either. Movement may incite a charge, and a bear can outrun you.

If a bear approaches you, talk to it calmly in a low, rm voice. Slowly back away until the bear is out of sight. If you climb a tree, get at least 12 feet up. If a bear senses that its food may be stolen or that its cub is in danger, it may charge. If this happens, drop to the ground and assume a fetal position, protecting your head with your arms, and play dead.

Keep your pack on; it may serve as armor. We begin our discussion of linear system theory with a description of the concept of linearity by Mitchell J. Feigenbaum [, p. Linearity means that the rule that determines what a piece of a system is going to do next is not inuenced by what it is doing now. More precisely, this is intended in a differential or incremental sense: For a linear spring, the increase of its tension is proportional to the increment whereby it is stretched, with the ratio of these increments exactly independent of how much it has already been stretched.

Such a spring can be stretched arbitrarily far, and in particular will never snap or break. Accordingly, no real spring is linear. The real world is nonlinear. However, there are a number of reasons to investigate linear systems. Linear models are often used to approximate nonlinear systems. Many times, this approximation is sufcient for the controller design for the underlying nonlinear system. Knowledge of linear system theory helps to understand the intricate theory of nonlinear systems.

In this chapter we acquaint ourselves with basic concepts from linear systems that will be used in subsequent chapters. The above implies that the rank of U i increases by at least one when i is increased by one, until the maximum value of rank U i is attained.

We say that the system 3. Theorem 3. Proof Using 3. A notion closely related to reachability is the notion of controllability. The notion of control- lability is related to the property of a controlled system being transferable from any given state to the origin 0 of the state space by means of an appropriately chosen control law. Proof Using the solution formula for the system 3. Example 3. Using 3. However, the system 3. See appendix, Section A. In summary, for discrete-time linear systems, reachability implies controllability, and the two notions are equivalent if the matrix A of the given discrete system is nonsingular.

We next discuss the notions of reachability and controllability for continuous-time linear systems. It turns out that in the case of continuous, time-invariant systems the two notions are equiv- alent. Indeed, from the solution formula for the controlled system A. On the other hand, from the solution formula for the controlled sys- tem A. Comparing 3. This is not true in the case of discrete-time systems, where reachability implies controllability, and the two notions are equivalent if the matrix A of the given discrete system is nonsingular.

We present a few of them in this section. The system x t 2. The n rows of the matrix e At B are linearly independent over the set of real numbers for all t [0,. We prove this statement by contraposition. By the CayleyHamilton theorem the matrix A satises its own characteristic equation. We will now prove the implication 2 3; that is, if the controllability matrix is of full rank, then the n rows of e At B are linearly independent over R for all t [0,. We now prove, constructively, the implication 4 1.

Note that to construct the above control law the matrix W t0 , t1 must be invertible. Substituting 3. Thus, we have established equivalence between the four statements of the theorem. The gear inertia and friction are negligible. The motor torque is equal to the torque delivered to the load. Plots of x1 , x2 versus time are shown in Figure 3.

A plot of u versus time is given in Figure 3. Proof We rst prove necessity. It is clear that T B must have the above form. A3 B 1 Comparing the above with 3. A3 A1 B 1 Again, comparing the above with 3. Because the similarity transformation pre- serves the reachability property, the pair A, B is nonreachable as well.

If the eigenvalues of the noncontrollable part of a given pair A, B are all in the open left-hand complex planethat is, the nonreachable part is asymptotically stablethen the pair A, B is called stabilizable. We now give yet another test for reachability. We denote the set of eigenvalues of a matrix A as eig A. Proof We rst prove the necessity part by contraposition. We now prove the sufciency part also by contraposition. By Theorem 3. Suppose now that the system state is not directly accessible.

Instead, we have the output of the system,. However, we still want to know the dynamical behavior of the entire state. Note that once x t0 is known, we can determine x t from knowledge of u t and y t over any nite time interval [t0 , t1 ].

## Detection of Zak phases and topological invariants in a chiral quantum walk of twisted photons

We now show how to constructively determine x t0 , given u t and y t. While doing so we will establish a sufciency condition for the system to be observable. The condition is also necessary for the system observability, as we show next.

Our goal now is to determine x t0. Knowledge of x 0 allows us to reconstruct the entire state x t over the interval [t0 , t1 ]. We conclude that if the matrix V t0 , t1 is invertible, then the system is observable. Knowledge of x 0 will allow us to nd the entire state vector for all t [0, 10].

To evaluate the above expression, we rst use formula A. We will now show that the invertibility of V t0 , t1 is also necessary for observability. We rst note that one can use similar arguments to those in the previous section to show that V t0 , t1 is nonsingular if and only if the n columns of the matrix Ce At are linearly independent for all t [0, over the set of real numbers.

In view of the above result, it follows that by repeating essentially the same arguments used in the previous section we can prove the following theorem:. The pair A, C is observable. The n columns of Ce At are linearly independent for all t [0, over the real numbers. The observability matrix C CA. R pnn. C An1 is of full rank n. We can also prove a result analogous to that of Theorem 3. If the eigenvalues of the nonobservable part of a given pair A, C are all in the open left-hand complex planethat is, the nonobservable part is asymptotically stablethen we say that the pair A, C is detectable.

A notion related to observability is the notion of constructability. One can verify that in the case of continuous, time-invariant systems, observability is equiva- lent to constructability. In the case of discrete-time systems, observability implies constructabil- ity, and the two notions are equivalent if the matrix A of the given discrete system is nonsingular. These forms are obtained by a change of state variables and reveal structural properties of the model.

We will use the above-mentioned forms to construct algorithms for pole placement. The pole placement algorithms will then be used to design state-feedback controllers, as well as state estimators. We begin by introducing the controller companion form, or controller form for short. We select the last row of the inverse of the controllability matrix. Let q 1 be that row.Substituting 1.

For example, To guarantee the sliding condition T E 0. This paper presents a new method for automatically generating numerical invariants for imperative programs. L Figure 3. An interconnection of the system and a controller is called a control system.

Once the transformation is performed, we can solve the pole placement problem for the given set of desired closed-loop poles, symmetric with respect to the real axis.

We rst divide the interval [t 0 , t f ] into N equal subintervals using evenly spaced subdivision points to obtain.

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