《自动化仪表与过程控制》课程学习资料：Mathematical Control Theory.pdf

Chapter 1 Introduction 1.1 What Is Mathematical Control Theory Mathematical control theory is the area of application-oriented mathematics that deals with the basic principles underlying the analysis and design of control systems. To control an object means to influence its behavior so as to achieve a desired goal. In order to implement this influence, engineers build devices that incorporate various mathematical techniques. These devices range from Watt steam engine governor, designed during the English Industrial Revolution, to the sophisticated microprocessor controllers found in consumer items-such as CD players and automobiles- or in industrial robots and airplane autopilots The study of these devices and their interaction with the object being con- trolled is the subject of this book. While on the one hand one wants to un- derstand the fundamental limitations that mathematics imposes on what is achievable, irrespective of the precise technology being used, it is also true that technology may well influence the type of question to be asked and the choice of mathematical model. An example of this is the use of difference rather than differential equations when one is interested in digital control Roughly speaking, there have been two main lines of work in control the- ry, which sometimes have seemed to proceed in very different directions but which are in fact complementary. One of these is based on the idea that a good model of the object to be controlled is available and that one wants to some- how optimize its behavior. For instance, physical principles and engineering specifications can be -and are- used in order to calculate that trajectory of a spacecraft which minimizes total travel time or fuel consumption. The tech- niques here are closely related to the classical calculus of variations and to other areas of optimization theory; the end result is typically a preprogrammed fight plan. The other main line of work is that based on the constraints imposed by uncertainty about the model or about the environment in which the object operates. The central tool here is the use of feedback in order to correct for deviations from the desired behavior For instance, various feedback control

Chapter 1 Introduction 1.1 What Is Mathematical Control Theory? Mathematical control theory is the area of application-oriented mathematics that deals with the basic principles underlying the analysis and design of control systems. To control an object means to influence its behavior so as to achieve a desired goal. In order to implement this influence, engineers build devices that incorporate various mathematical techniques. These devices range from Watt’s steam engine governor, designed during the English Industrial Revolution, to the sophisticated microprocessor controllers found in consumer items —such as CD players and automobiles— or in industrial robots and airplane autopilots. The study of these devices and their interaction with the object being controlled is the subject of this book. While on the one hand one wants to understand the fundamental limitations that mathematics imposes on what is achievable, irrespective of the precise technology being used, it is also true that technology may well influence the type of question to be asked and the choice of mathematical model. An example of this is the use of difference rather than differential equations when one is interested in digital control. Roughly speaking, there have been two main lines of work in control theory, which sometimes have seemed to proceed in very different directions but which are in fact complementary. One of these is based on the idea that a good model of the object to be controlled is available and that one wants to somehow optimize its behavior. For instance, physical principles and engineering specifications can be —and are— used in order to calculate that trajectory of a spacecraft which minimizes total travel time or fuel consumption. The techniques here are closely related to the classical calculus of variations and to other areas of optimization theory; the end result is typically a preprogrammed flight plan. The other main line of work is that based on the constraints imposed by uncertainty about the model or about the environment in which the object operates. The central tool here is the use of feedback in order to correct for deviations from the desired behavior. For instance, various feedback control 1

2 1. Introduction systems are used during actual space flight in order to compensate for errors from the precomputed trajectory. Mathematically, stability theory, dynamical systems, and especially the theory of functions of a complex variable, have had a strong influence on this approach. It is widely recognized today that these two broad lines of work deal just with different aspects of the same problems, and we do not make an artificial distinction between them in this book. Later on we shall give an axiomatic definition of what we mean by a “system” or “machine.” Its role will be somewhat analogous to that played in mathematics by the definition of “function” as a set of ordered pairs: not itself the object of study, but a necessary foundation upon which the entire theoretical development will rest. In this Chapter, however, we dispense with precise definitions and will use a very simple physical example in order to give an intuitive presentation of some of the goals, terminology, and methodology of control theory. The discussion here will be informal and not rigorous, but the reader is encouraged to follow it in detail, since the ideas to be given underlie everything else in the book. Without them, many problems may look artificial. Later, we often refer back to this Chapter for motivation. 1.2 Proportional-Derivative Control One of the simplest problems in robotics is that of controlling the position of a single-link rotational joint using a motor placed at the pivot. Mathematically, this is just a pendulum to which one can apply a torque as an external force (see Figure 1.1). mg u mg sin θ θ Figure 1.1: Pendulum. We assume that friction is negligible, that all of the mass is concentrated at the end, and that the rod has unit length. From Newton’s law for rotating objects, there results, in terms of the variable θ that describes the counterclockwise angle with respect to the vertical, the second-order nonlinear differential equation m¨θ(t) + mg sin θ(t) = u(t), (1.1)

1.2. Proportional-Derivative Control 3 where m is the mass, g the acceleration due to gravity, and u(t) the value of the external torque at time t (counterclockwise being positive). We call u(·) the input or control function. To avoid having to keep track of constants, let us assume that units of time and distance have been chosen so that m = g = 1. The vertical stationary position (θ = π, ˙ θ = 0) is an equilibrium when no control is being applied (u ≡ 0), but a small deviation from this will result in an unstable motion. Let us assume that our objective is to apply torques as needed to correct for such deviations. For small θ − π, sin θ = −(θ − π) + o(θ − π). Here we use the standard “little-o” notation: o(x) stands for some function g(x) for which limx→0 g(x) x = 0 . Since only small deviations are of interest, we drop the nonlinear part represented by the term o(θ−π). Thus, with ϕ := θ−π as a new variable, we replace equation (1.1) by the linear differential equation ϕ¨(t) − ϕ(t) = u(t) (1.2) as our object of study. (See Figure 1.2.) Later we will analyze the effect of the ignored nonlinearity. u φ Figure 1.2: Inverted pendulum. Our objective then is to bring ϕ and ˙ϕ to zero, for any small nonzero initial values ϕ(0), ϕ˙(0) in equation (1.2), and preferably to do so as fast as possible, with few oscillations, and without ever letting the angle and velocity become too large. Although this is a highly simplified system, this kind of “servo” problem illustrates what is done in engineering practice. One typically wants to achieve a desired value for certain variables, such as the correct idling speed in an automobile’s electronic ignition system or the position of the read/write head in a disk drive controller. A naive first attempt at solving this control problem would be as follows: If we are to the left of the vertical, that is, if ϕ = θ −π > 0, then we wish to move to the right, and therefore, we apply a negative torque. If instead we are to

the right, we apply a positive, that is to say counterclockwise, torque. In other words, we apply proportional feedback (t) (1.3) where a is positive real number, the feedback gain ze the resulting closed-loop equation obtained when the value of the control given by(1.3)is substituted into the open-loop original equation (1.2) p(t-p(t)+oo(t)=0 If a>1, the solutions of this differential equation are all oscillatory, since the roots of the associated characteristic equation are purely imaginary z =tiVa-1. If instead a< 1, then all of the solutions except for those with 中(0)=-p(0)1-a diverge to too. Finally, if a= 1, then each set of initial values with p(0)=0 is an equilibrium point of the closed-loop system. Therefore, in none of the cases is the system guaranteed to approach the desired configuration We have seen that proportional control does not work. We proved this for the linearized model, and an exercise below will show it directly for the origin nonlinear equation(1. 1). Intuitively, the problem can be understood as follows Take first the case a 1. For any initial condition for which p(O) is small but positive and p(0)=0, there results from equation(1.4)that p(0)>0 Therefore, also and hence yp increase, and the pendulum moves away, rather than toward, the vertical position. When a>l the problem is more subtle The torque is being applied in the correct direction to counteract the natural stability of the pendulum, but this feedback helps build too much inertia In particular, when already close to (0)=0 but moving at a relatively large speed, the controller(1.3)keeps pushing toward the vertical, and overshoot and eventual oscillation result The obvious solution is to keep a> l but to modify the proportional feed- back(1.3)through the addition of a term that acts as a brake, penalizing ve- locities. In other words, one needs to add damping to the system. We arrive then at a PD, or proportional-derivative feedback law, u(t)=-ap(t)-B(t with a> I and B>0. In practice, implementing such a controller involves measurement of both the angular position and the velocity. If only the former is easily available, then one must estimate the velocity as part of the control gorithm; this will lead later to the idea of observers, which are techniques for

4 1. Introduction the right, we apply a positive, that is to say counterclockwise, torque. In other words, we apply proportional feedback u(t) = −αϕ(t), (1.3) where α is some positive real number, the feedback gain. Let us analyze the resulting closed-loop equation obtained when the value of the control given by (1.3) is substituted into the open-loop original equation (1.2), that is ϕ¨(t) − ϕ(t) + αϕ(t)=0 . (1.4) If α > 1, the solutions of this differential equation are all oscillatory, since the roots of the associated characteristic equation z2 + α − 1 = 0 (1.5) are purely imaginary, z = ±i √α − 1. If instead α < 1, then all of the solutions except for those with ϕ˙(0) = −ϕ(0)√1 − α diverge to ±∞. Finally, if α = 1, then each set of initial values with ˙ϕ(0) = 0 is an equilibrium point of the closed-loop system. Therefore, in none of the cases is the system guaranteed to approach the desired configuration. We have seen that proportional control does not work. We proved this for the linearized model, and an exercise below will show it directly for the original nonlinear equation (1.1). Intuitively, the problem can be understood as follows. Take first the case α < 1. For any initial condition for which ϕ(0) is small but positive and ˙ϕ(0) = 0, there results from equation (1.4) that ¨ϕ(0) > 0. Therefore, also ˙ϕ and hence ϕ increase, and the pendulum moves away, rather than toward, the vertical position. When α > 1 the problem is more subtle: The torque is being applied in the correct direction to counteract the natural instability of the pendulum, but this feedback helps build too much inertia. In particular, when already close to ϕ(0) = 0 but moving at a relatively large speed, the controller (1.3) keeps pushing toward the vertical, and overshoot and eventual oscillation result. The obvious solution is to keep α > 1 but to modify the proportional feedback (1.3) through the addition of a term that acts as a brake, penalizing velocities. In other words, one needs to add damping to the system. We arrive then at a PD, or proportional-derivative feedback law, u(t) = −αϕ(t) − βϕ˙(t), (1.6) with α > 1 and β > 0. In practice, implementing such a controller involves measurement of both the angular position and the velocity. If only the former is easily available, then one must estimate the velocity as part of the control algorithm; this will lead later to the idea of observers, which are techniques for

1.2. Proportional-Derivative Control reliably performing such an estimation. We assume here that y can indeed be easured. Consider then the resulting closed-loop system p(t)+B(t)+(a-1)y(t)=0 The roots of its associated characteristic equation z2+Bz+a-1=0 2 both of which have negative real parts. Thus all the solutions of(1. 2)converge to zero. The system has been stabilized under feedback. This convergence may be oscillatory, but if we design the controller in such a way that in addition t the above conditions on a and B it is true that then all of the solutions are combinations of decaying exponentials and no os- cillation results We conclude from the above discussion that through a suitable choice of the gains a and B it is possible to attain the desired behavior, at least for the linearized model. That this same design will still work for the original nonlinear model, and, hence, assuming that this model was accurate, for a real pendulum is due to what is perhaps the most important fact in control theory -and for that matter in much of mathematics- namely that first-order approximations re sufficient to characterize local behavior. Informally, we have the following linearization principle Designs based on linearizations work locally for the original system The term "local"refers to the fact that satisfactory behavior only can be ex- pected for those initial conditions that are close to the point about which the linearization was made. Of course, as with any "principle, this is not a theorem. It can only become so when precise meanings are assigned to the various terms and proper technical assumptions are made. Indeed, we will invest some effort this text to isolate cases where this principle may be made rigorous. One of these cases will be that of stabilization, and the theorem there will imply that if we can stabilize the linearized system (1.2) for a certain choice of parameters a, B in the law(1.6), then the same control law does bring initial conditions of (1.1)that start close to 0=T, 6=0 to the vertical equilibrium Basically because of the linearization principle, a great deal of the literature in control theory deals exclusively with linear systems. From an engineering t of view, local solutions to control problems are often enough; when they not, ad hoc methods sometimes may be used in order to "patch"together h local solutions, a procedure called gain scheduling. Sometimes, one may

1.2. Proportional-Derivative Control 5 reliably performing such an estimation. We assume here that ˙ϕ can indeed be measured. Consider then the resulting closed-loop system, ϕ¨(t) + βϕ˙(t)+(α − 1)ϕ(t)=0 . (1.7) The roots of its associated characteristic equation z2 + βz + α − 1 = 0 (1.8) are −β ± pβ2 − 4(α − 1) 2 , both of which have negative real parts. Thus all the solutions of (1.2) converge to zero. The system has been stabilized under feedback. This convergence may be oscillatory, but if we design the controller in such a way that in addition to the above conditions on α and β it is true that β2 > 4(α − 1), (1.9) then all of the solutions are combinations of decaying exponentials and no oscillation results. We conclude from the above discussion that through a suitable choice of the gains α and β it is possible to attain the desired behavior, at least for the linearized model. That this same design will still work for the original nonlinear model, and, hence, assuming that this model was accurate, for a real pendulum, is due to what is perhaps the most important fact in control theory —and for that matter in much of mathematics— namely that first-order approximations are sufficient to characterize local behavior. Informally, we have the following linearization principle: Designs based on linearizations work locally for the original system The term “local” refers to the fact that satisfactory behavior only can be expected for those initial conditions that are close to the point about which the linearization was made. Of course, as with any “principle,” this is not a theorem. It can only become so when precise meanings are assigned to the various terms and proper technical assumptions are made. Indeed, we will invest some effort in this text to isolate cases where this principle may be made rigorous. One of these cases will be that of stabilization, and the theorem there will imply that if we can stabilize the linearized system (1.2) for a certain choice of parameters α,β in the law (1.6), then the same control law does bring initial conditions of (1.1) that start close to θ = π, ˙ θ = 0 to the vertical equilibrium. Basically because of the linearization principle, a great deal of the literature in control theory deals exclusively with linear systems. From an engineering point of view, local solutions to control problems are often enough; when they are not, ad hoc methods sometimes may be used in order to “patch” together such local solutions, a procedure called gain scheduling. Sometimes, one may