# Automatic control theory

The science dealing with methods for the determination of laws for controlling systems that can be realized by automatic devices. Historically, such methods were first applied to processes which were mainly technical in nature . Thus, an aircraft in flight is a system the control laws of which ensure that it remains on the required trajectory. The laws are realized by means of a system of transducers (measuring devices) and actuators, which is known as the automatic pilot. This development was due to three reasons: many control systems had been identified by classical science (to identify a control system means to write down its mathematical model, e.g. relationships such as (1) and (2) below); long before the development of automatic control theory, thanks to the knowledge of the fundamental laws of nature, there was a well-developed mathematical apparatus of differential equations and especially an apparatus for the theory of steady motion ; engineers had discovered the idea of a feedback law (see below) and found means for its realization.

The simplest control systems are described by an ordinary (vector) differential equation

$$\tag{1 } \dot{x} = f ( x, u , t )$$

and an inequality

$$\tag{2 } N ( x, u , t ) \geq 0,$$

where $x \{ x _ {1} \dots x _ {n} \}$ is the state vector of the system, $u \{ u _ {1} \dots u _ {r} \}$ is the vector of controls which can be suitably chosen, and $t$ is time. Equation (1) is the mathematical representation of the laws governing the control system, while the inequality (2) establishes its domain of definition.

Let $U$ be some given class of functions $u(t)$( e.g. piecewise-continuous functions) whose numerical values satisfy (2). Any function $u(t) \in U$ will be called a permissible control. Equation (1) will be called a mathematical model of the control system if:

1) A domain $N (x, u , t) \geq 0$ in which the function $f(x, u , t)$ is defined has been specified;

2) A time interval ${\mathcal T} = [ t _ {i} , t _ {f} ]$( or $[t _ {i} , t _ {f} )$, if $t _ {f} = \infty$) during which the motion $x(t)$ is observed, has been specified;

3) A class of permissible controls has been specified;

4) The domain $N \geq 0$ and the function $f(x, u , t)$ are such that equation (1) has a unique solution defined for any $t \in {\mathcal T}$, $x _ {0} \in N$, whatever the permissible control $u(t)$. Furthermore, $f(x, u , t)$ in (1) is always assumed to be smooth with respect to all arguments.

Let $x _ {i} = x(t _ {i} )$ be an initial and let $x _ {f} = x(t _ {f} )$ be a (desired) final state of the control system. The state $x _ {f}$ is known as the target of the control. Automatic control theory must solve two major problems: the problem of programming, i.e. of finding those controls $u(t)$ permitting the target to be reached from $x _ {i}$; and the determination of the feedback laws (see below). Both problems are solved under the assumption of complete controllability (1).

The system (1) is called completely controllable if, for any $x _ {i} , x _ {f} \in N$, there is at least one permissible control $u(t)$ and one interval ${\mathcal T}$ for which the control target is attainable. If this condition is not met, one says that the object is incompletely controllable. This gives rise to a preliminary problem: Given the mathematical model (1), find the criteria of controllability. At the time of writing (1977) only insignificant progress has been made towards its solution. If equation (1) is linear

$$\tag{3 } \dot{x} = Ax + Bu,$$

where $A, B$ are stationary matrices, the criterion of complete controllability is formulated as follows: For (3) to be completely controllable it is necessary and sufficient that the rank of the matrix

$$\tag{4 } Q = \| B AB \dots A ^ {n-1 } B \|$$

be $n$. The matrix (4) is known as the controllability matrix.

If $A, B$ are known differentiable functions of $t$, the controllability matrix is given by

$$\tag{5 } Q = \| L _ {1} ( t ) \dots L _ {n} ( t ) \| ,$$

where

$$L _ {1} ( t ) = B ( t ),\ L _ {k} ( t ) = A ( t ) L _ {k-1 } - dL _ \frac{k-1 }{dt} ,\ k = 2 \dots n.$$

The following theorem applies to this case: For (3) to be completely controllable it is sufficient if at at least one point $t ^ {*} \in {\mathcal T}$ the rank of the matrix (5) equals $n$. Criteria of controllability for non-linear systems are unknown (up to 1977).

The first principal task of automatic control theory is to select the permissible controls that ensure that the target $x _ {f}$ is attained. There are two methods of solving this problem. In the first method, the chief designer of the system (1) arbitrarily determines a certain type of motion for which the target $x _ {f}$ is attainable and selects a suitable control. This solution of the programming problem is in fact used in many instances. In the second method a permissible control minimizing a given cost of controls is sought. The mathematical formulation of the problem is then as follows. The data are: the mathematical model of the controlled system (1) and (2); the boundary conditions for the vector $x$, which will be symbolically written as

$$\tag{6 } ( i, f ) = 0;$$

a smooth function $G(x, t)$; and the cost of the controls used

$$\tag{7 } \Delta G = \left . G ( x, t ) \right | _ {i} ^ {f} .$$

The programming problem is to find, among the permissible controls, a control satisfying conditions (6) and yielding the minimum value of the functional (7). Necessary conditions for a minimum for this non-classical variational problem are given by the L.S. Pontryagin "maximum principle"  (cf. Pontryagin maximum principle). An auxiliary vector $\psi \{ \psi _ {1} \dots \psi _ {n} \}$ and the auxiliary scalar function

$$\tag{8 } H ( \psi , x, u , t ) = \psi \cdot f ( x, u , t )$$

are introduced. The function $H$ makes it possible to write equation (1) and an equation for the vector $\psi$ in the following form:

$$\tag{9 } \dot{x} = \frac{\partial H }{\partial \psi } ,\ \dot \psi = - \frac{\partial H }{\partial x } .$$

Equation (9) is linear and homogeneous with respect to $\psi$ and has a unique continuous solution, which is defined for any initial condition $\psi (t _ {i} )$ and $t \in {\mathcal T}$. The vector $\psi$ will be called a non-zero vector if at least one of its components does not vanish for $t \in {\mathcal T}$. The following theorem is true: For the curve $x ^ {o} , u ^ {o}$ to constitute a strong minimum of the functional (7) it is necessary that a non-zero continuous vector $\psi$, as defined by equation (9), exists at which the function $H( \psi , x, u , t)$ has a (pointwise) maximum with respect to $u$, and that the transversality condition

$$\left [ \delta G - H \delta t + \sum \psi _ \alpha \delta x _ {a} \right ] _ {i} ^ {f} = 0$$

is met. Let $x ^ {o} ( t, x _ {i} , x _ {f} ) , u _ {o} ( t, x _ {i} , x _ {f} )$ be solutions of the corresponding problem. It has then been shown that for stationary systems the function $H( \psi ^ {o} , x ^ {o} , u ^ {o} )$ satisfies the condition

$$\tag{10 } H ( \psi ^ {o} , x ^ {o} , u ^ {o} ) = C ,$$

where $C$ is a constant, so that (10) is a first integral. The solution $u ^ {o} , x ^ {o}$ is known as a program control.

Let $u ^ {o} , x ^ {o}$ be a (not necessarily optimal) program control. It was found that the knowledge of only one program control is insufficient to attain the target. This is because the program $u ^ {o} , x ^ {o}$ is usually unstable with respect to, however small, changes in the problem, in particular to the most important changes, those in the initial and final values $(i, f)$ or, in other words, the problem is ill-posed. However, this ill-posedness is such that it can be corrected by means of automatic stabilization, based solely on the "feedback principle" . Hence the second main task of control: the determination of feedback laws.

Let $y$ be the vector of disturbed motion of the system and let $\xi$ be the vector describing the additional deflection of the control device intended to quench the disturbed motion. To realize the deflection $\xi$ a suitable control source must be provided for in advance. The disturbed motion is described by the equation:

$$\tag{11 } \dot{y} = Ay + B \xi + \phi ( y , \xi , t ) + f ^ {o} ( t ) .$$

where $A$ and $B$ are known matrices determined by the motion of $x ^ {o} , u ^ {o}$, and are assumed to be known functions of the time; $\phi$ is the non-linear part of the development of the function $f(x, u , t)$; $f ^ {o} (t)$ is the constantly acting force of perturbation, which originates either from an inaccurate determination of the programmed motion or from additional forces which were neglected in constructing the model (1). Equation (11) is defined in a neighbourhood $\| y \| \leq \overline{H}\;$, where $\overline{H}\;$ is usually quite small, but in certain cases may be any finite positive number or even $\infty$.

It should be noted that, in general, the fact that the system (1) is completely controllable does not mean that the system (11) is completely controllable as well.

One says that (11) is observable along the coordinates $y _ {1} \dots y _ {r}$, $r \leq n$, if one has at his disposal a set of measuring instruments that continuously determines the coordinates at any moment of time $t \in {\mathcal T}$. The significance of this definition can be illustrated by considering the longitudinal motion of an aircraft. Even though aviation is more than 50 years old, an instrument that would measure the disturbance of the attack angle of the aircraft wing or the altitude of its flight near the ground has not yet been invented. The totality of measured coordinates is called the field of regulation and is denoted by $P ( y _ {1} \dots y _ {r} )$, $r \leq n$.

Consider the totality of permissible controls $\xi$, determined over the field $P$:

$$\tag{12 } \xi = \xi ( y _ {1} \dots y _ {r} , t , p ),\ r \leq n,$$

where $p$ is a vector or a matrix parameter. One says that the control (12) represents a feedback law if the closure operation (i.e. substitution of (12) into (11)) yields the system

$$\tag{13 } \dot{y} = Ay + B \xi ( y _ {1} \dots y _ {r} , t , p ) +$$

$$+ \phi ( y , \xi ( y _ {1} \dots y _ {r} , t , p ), t ),$$

such that its undisturbed motion $y = 0$ is asymptotically stable (cf. Asymptotically-stable solution). The system (13) is said to be asymptotically stable if its undisturbed motion $y = 0$ is asymptotically stable.

There are two classes of problems which may be formulated in the context of the closed system (13): The class of analytic and of synthetic problems.

Consider a permissible control (12) given up to the selection of the parameter $p$, e.g.:

$$\xi = \left \| \begin{array}{ccc} p _ {11} &\dots &p _ {1r} \\ \dots &\dots &\dots \\ p _ {r1} &\dots &p _ {rr} \\ \end{array} \right \| \ \left \| \begin{array}{c} {y _ {1} } \\ . \\ . \\ {y _ {r} } \\ \end{array} \ \right \| .$$

The analytic problem: To determine the domain $S$ of values of the parameter $p$ for which the closed system (13) is asymptotically stable. This domain is constructed by methods developed in the theory of stability of motion (cf. Stability theory), which is extensively employed in the theory of automatic control. In particular, one may mention the methods of frequency analysis; methods based on the first approximation to Lyapunov stability theory (the theorems of Hurwitz, Routh, etc.), on the direct Lyapunov method of constructing $v$- functions, on the Lyapunov–Poincaré theory of constructing periodic solutions, on the method of harmonic balance, on the B.V. Bulgakov method, on the A.A. Andronov method, and on the theory of point transformations of surfaces . The last group of methods makes it possible not only to construct domains $S$ in the space $P$, but also to analyze the parameters of the stable periodic solutions of equation (13) which describe the auto-oscillatory motion of the system (13). All these methods are widely employed in the practice of automatic control, and are studied in the framework of various specifications in schools of higher learning .

If $S$ is non-empty, a control (12) is called a feedback law or regulation law. Its realization, which is effected using a system of measuring instruments, amplifiers, converters and executing mechanisms, is known as a regulator.

Another problem of considerable practical importance, which is closely connected with the analytic problem, is how to construct the boundary of the domain of attraction , . Consider the system (13) in which $p \in S$. The set of values $y(t _ {i} ) = y _ {0}$ containing the point $y = 0$ for which the closed system (13) retains the property of asymptotic stability, is known as the domain of attraction of the trivial solution $y = 0$. The problem is to determine the boundary of the domain of attraction for a given closed system (13) and a point $p \in S$.

Modern scientific literature does not contain effective methods of constructing the boundary of the domain of attraction, except in rare cases in which it is possible to construct unstable periodic solutions of the closed system. However, there are certain methods which allow one to construct the boundary of a set of values of $y _ {0}$ totally contained in the domain of attraction. These methods are based in most cases on the evaluation of a domain in phase space in which the Lyapunov function satisfies the condition $v \geq 0$, $\dot{v} \leq 0$.

Any solution $y(t, y _ {0} , p)$ of the closed system (13) represents a so-called transition process. In most cases of practical importance the mere solution of the stability problem is not enough. The development of a project involves supplementary conditions of considerable practical importance, which require the transition process to have certain additional features. The nature of these requirements and the list of these features are closely connected with the physical nature of the controlled object. In analytic problems it may often be possible, by a suitable choice of the parameter $p$, to preserve the desired properties of the transition process, e.g. the pre-set regulation time $t ^ {*}$. The problem of choosing the parameter $p$ is known as the problem of quality of regulation , and methods for solving this problem are connected with some construction of estimates for solutions $y(t, y _ {0} , p)$: either by actual integration of equation (13) or by experimental evaluation of such solutions with the aid of an analogue or digital computer.

The analytic problems of transition processes have many other formulations in all cases in which $f ^ {o} (t)$ is a random function — in servomechanisms for example, , . Other formulations are concerned with the possibility of a random alteration of the matrices $A$ and $B$ or even of the function $\phi$, . This gave rise to the development of methods for studying random processes, methods of adaptation and learning machines . Transition processes in systems with delay mechanisms and with distributed parameters (see , ) and with a variable structure (see ) have also been studied.

The synthesis problem: Given equation (11), a field of regulation $P (y _ {1} \dots y _ {r} )$, $r \leq n,$ and a set $\xi (y _ {1} \dots y _ {r} , t)$ of permissible controls, to find the whole set $M$ of feedback laws . One of the most important variant of this problem is the problem of the structure of minimal fields. A field $P ( y _ {1} \dots r _ {r} )$, $r \leq n,$ is called a minimal field if it contains at least one feedback law and if the dimension $r$ of the field is minimal. The problem is to determine the structure $P ( y _ {\alpha _ {1} } \dots y _ {\alpha _ {r} } )$ of all minimal fields for a given equation (11) and a set of permissible controls. The following example illustrates the nature of the problem:

$$\dot{z} = Az + Bu,$$

$$A = \left \| \begin{array}{rrrr} 0 & m & 0 & 0 \\ -m & 0 & 0 & n \\ 0 & 0 & 0 & k \\ 0 &-n & k & 0 \\ \end{array} \right \| ,\ B = \left \| \begin{array}{rr} 0 & 0 \\ 1 & 0 \\ 0 & 0 \\ 0 & 1 \\ \end{array} \right \| ,\ u = \left \| \begin{array}{r} u _ {2} \\ u _ {4} \\ \end{array} \right \| ,$$

where $m, n, k$ are given numbers. The permissible controls are the set of piecewise-continuous functions $u _ {2} , u _ {4}$ that take their values from the domain

$$| u _ {2} | \leq \overline{u}\; _ {2} ,\ | u _ {4} | \leq \overline{u}\; _ {4} .$$

The minimal fields in this problem are either the field $P ( z _ {2} )$ or the field $P ( z _ {4} )$. The dimension of each field is one and cannot be reduced .

So far (1977) only one method is known for the synthesis of feedback laws; it is based on Lyapunov functions . A relevant theorem is that of Barbashin–Krasovskii , , : In order for the undisturbed motion $y= 0$ of the closed system

$$\tag{14 } \dot{y} = \phi ( y )$$

to be asymptotically stable, it is sufficient for a positive-definite function $v(y)$ to exist, such that by equation (14) its complete derivative is a function $w(y)$ which is semi-definite negative, and such that on the manifold $w(y) = 0$ no complete trajectory of the system (14), except for $y = 0$, lies. The problem of finding out about the existence and the structure of the minimal fields is of major practical importance, since these fields determine the possible requirements of the chief designer concerning the minimum weight, complexity and cost price of the control system and its maximum reliability. The problem is also of scientific and practical interest in connection with infinite-dimensional systems as encountered in technology, biology, medicine, economics and sociology.

In designing control systems it is unfortunately impracticable to restrict the work to solving problems of synthesis of feedback laws. In most cases the requirements of the chief designer are aimed at securing certain important specific properties of the transition process in the closed system. The importance of such requirements is demonstrated by the importance of monitoring an atomic reactor. If the transition process takes more than 5 seconds or if the maximum value of some of its coordinates exceeds a certain value, an atomic explosion follows. This gives rise to new problems of synthesis of regulation laws, based on the set $M$. Below the formulation of one such problem is given. Consider two spheres $\| y _ {0} \| = R$, $y _ {0} = y _ {i}$, $\| y (t _ {1} ) \| = \epsilon$; $R \gg \epsilon$ are given numerical values. Now consider the set $M$ of all feedback laws. The closure by means of any of them yields the equation:

$$\tag{15 } \dot{y} = Ay + B \xi ( y _ {1} \dots y _ {r} , t ) +$$

$$\phi ( y , \xi ( y _ {1} \dots y _ {r} , t ) , t ) .$$

Consider the entire set of solutions $y (t, y _ {0} )$ of equation (15) which start on the sphere $R$ and call them $R$- solutions. Since the system is asymptotically stable for any $y _ {0}$ on the sphere, there exists a moment of time $t _ {1}$ during which the conditions

$$\| y ( t _ {1} , y _ {0} ) \| = \epsilon ,\ \ \| y ( t , y _ {0} ) \| < \epsilon ,$$

are valid for any $t > t _ {1}$.

Let

$$t ^ {*} = \sup _ {y _ {0} } t _ {1} .$$

It is clear from the definition of $t _ {1}$ that $t ^ {*}$ exists. The interval $t ^ {*} - t _ {i}$ is called the regulation time (the time of damping of the transition process) in the closed system (15) if any arbitrary $R$- solution starts at the $\epsilon$- sphere if $t _ {1} \leq t ^ {*}$, but remains inside it if $t > t _ {1}$. It is clear that the regulation time is a functional of the form $t ^ {*} = t ^ {*} (R, \epsilon , \xi )$. Let $T$ be a given number. There arises the problem of synthesis of fast-acting regulators: Given a set $M$ of feedback laws, one has to isolate its subset $M _ {1}$ on which the regulation time in a closed system satisfies the condition

$$t ^ {*} - t _ {i} \leq T .$$

One can formulate in a similar manner the problems of synthesis of the sets $M _ {2} \dots M _ {k}$ of feedback laws, which satisfy the other $k - 1$ requirements of the chief designer.

The principal synthesis problem of satisfying all the requirements of the chief designer is solvable if the sets $M _ {1} \dots M _ {k}$ have a non-empty intersection .

The synthesis problem has been solved in greatest detail for the case in which the field $P$ has maximal dimension, $r = n$, while the cost index of the system is characterized by the functional

$$\tag{16 } J = \int\limits _ { 0 } ^ \infty w ( y , \xi , t ) dt ,$$

where $w (y, \xi , t)$ is a positive-definite function of $y, \xi$. The problem is then known as the problem of analytic construction of optimum regulators  and is in fact thus formulated. The data include equation (11), a class of permissible controls $\xi (y, t)$ defined over the field $P(y)$ of maximal dimension, and the functional (16). One is required to find a control $\xi = \xi (y, t)$ for which the functional (16) assumes its minimum value. This problem is solved by the following theorem: If equation (11) is such that it is possible to find an upper semi-continuous positive-definite function $v ^ {o} (y, t)$ and a function $\xi ^ {o} (y, t)$ such that the equality

$$\tag{17 } \frac{\partial v ^ {o} }{\partial t } + \frac{\partial v ^ {o} }{\partial y } ( Ay + B \xi ^ {o} + \phi ( y , \xi ^ {o} , t ) ) = 0$$

is true, and the inequality

$$\frac{\partial v ^ {o} }{\partial t } + \frac{\partial v ^ {o} }{\partial y } ( Ay + B \xi + \phi ( y , \xi , t )) \geq 0$$

is also true for all permissible $\xi$, then the function $\xi ^ {o} (y, t)$ is a solution to the problem. Morever, the equality

$$v ^ {o} ( t _ {0} , y _ {0} ) = \mathop{\rm min} _ \xi \int\limits _ { 0 } ^ \infty w ( y , \xi , t ) dt$$

is true. The function $v ^ {o} (y, t)$ is known as the Lyapunov optimum function . It is a solution of the partial differential equation (17), of Hamilton–Jacobi type, satisfying the condition $v(y ( \infty ), \infty ) = 0$. Methods for the effective solution of such a problem have been developed for the case in which the functions $w$ and $\phi$ can be expanded in a convergent power series in $y , \xi$ with coefficients which are bounded continuous functions of $t$. Of fundamental importance is the solvability of the problem of linear approximation to equation (11) and the optimization with respect to the integral of only second-order terms contained in the development of $w$. This problem is solvable if the condition of complete controllability is satisfied .

How to Cite This Entry:
Automatic control theory. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Automatic_control_theory&oldid=45522
This article was adapted from an original article by A.M. Letov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article