On the Stability of a Control System with a Forced Sliding Regime

E. I. Gerashchenko, L. V. Kiselev

1. Suppose there is a system of three equations:

\[ \dot{x}=y,\quad \dot{y}=z,\quad \dot{z}=-az-by-cx-K|x|\operatorname{sign}\sigma_1, \tag{1} \]

where \(a, b, c\) are arbitrary real constants (parameters of the controlled object), \(K\) is the gain coefficient, a positive quantity,

\[ \sigma_1=z+By+A|x|\operatorname{sign}\sigma_2,\quad \sigma_2=y+Dx, \tag{2} \]

\(A, B, D\) are positive numbers (parameters of the regulator).

The following problem is to be solved. Determine the conditions imposed on the regulator parameters \(K, A, B, D\) under which the zero solution of system (1) is asymptotically stable in the large, independently of the parameters \(a, b, c\). The method of proof and the arguments used for solving this problem are closely connected with the results of [1].

It is known [1] that, for certain relations between the quantities \(A, B, D\) and a sufficiently large coefficient \(K\), in system (1) there arises a quasi-ideal (i.e., ideal up to fast motions) sliding regime of second order. In this case the representative point, having begun its motion from an arbitrary position in space, reaches the surface \(\sigma_1=0\), slides along it to the plane \(\sigma_2=0\), and, “rapidly” rotating about the line

\[ \sigma_2=y+Dx=0,\quad \frac{d\sigma_2}{dt}=z+Dy=0, \tag{3} \]

“slowly” moves along it toward the origin.

The described character of sliding along the line (3) makes it possible to conclude that in the region of discontinuity of the surface \(\sigma_1=0\) there exists a certain cone \(C\), which contains within itself a region of attraction for all points of phase space, and moreover for points lying inside the cone \(C\) the only point of attraction is the origin. The separation of the motions of system (1) into motions toward the cone \(C\) and motions within it, corresponding to sliding of second order, makes it possible to carry out the proof of stability of system (1) in three stages: to prove reaching the surface \(\sigma_1=0\), to establish reaching the region of discontinuity of the surface \(\sigma_1=0\), and to construct the cone \(C\).

1. Let us determine the relation between the parameters \(K, A, B, D\) under which the representative point, moving along a trajectory

of system (1) falls at least once on the switching surface. The switching surface \(\sigma_1=0\) may be regarded as continuous (Fig. 1) and composed of the surface

\[ Ax\operatorname{sign}[(y+Dx)x]+By+z=0,\qquad y+Dx\ne 0 \]

Fig. 1

and the sector \(W\) of the plane \(\sigma_2=0\), situated in the region \(\sigma_1^{+}\sigma_1^{-}\le 0\). Here

\[ \sigma_1^{+}=Ax+By+z,\qquad \sigma_1^{-}=-Ax+By+z. \tag{4} \]

Theorem. If the parameters \(A, B, D\), and \(K\) are positive and satisfy the conditions:

a)

\[ B+\sqrt{B^2+4A}>2D; \tag{5} \]

b) the equation

\[ \lambda^3+a\lambda^2+b\lambda+C+K=0 \tag{6} \]

has no real positive roots, then the quantity \(\sigma_1(t)\), varying along a trajectory of system (1), either vanishes or changes sign.

Proof. Let \(M\) be an arbitrary point of the phase space. Suppose, for definiteness, that \(\sigma_1(M)>0\). Introduce the function \(\varphi(t)=\exp Dt\cdot x(t)\). By virtue of system (1), evidently,

\[ \frac{d\varphi}{dt}=\sigma_2(x,y)\exp Dt. \tag{7} \]

If at the point \(M\) the inequality \(\sigma_2(M)<0\) holds, then, by (7), \(\dfrac{d\varphi}{dt}<0\), and \(\varphi(t)\) is a monotonically decreasing function for all points in the region \(\sigma_2<0\). Three possibilities may occur:

1) the representative point, while in the region \(\sigma_2<0\), falls on the switching surface,

2) the point remains indefinitely long in the region \(\sigma_2<0\),

3) the point enters the region \(\sigma_2\ge 0\) with \(\sigma_1>0\).

The first case obviously satisfies the theorem. In the second case the function \(\varphi(t)\) is monotonically decreasing for all finite values of \(t\). Consequently, there exists a proper or improper limit

\[ \varphi_0=\lim \varphi(t)\quad\text{as }t\to\infty. \]

Obviously, \(\varphi_0 < \varphi(t_0) < +\infty\). If \(\varphi_0 \geqslant 0\), then along the trajectory on the right the inequality holds

\[ \exp Dt \cdot x(t) > \varphi_0 \geqslant 0 \]

and, consequently, for all finite values of \(t\) the quantity \(x(t)\) is positive along the trajectory of system (1). Thus, the motion is described by the linear system

\[ \dot{x}=y,\quad \dot{y}=z,\quad \dot{z}=-az-by-cx-Kx, \tag{8} \]

and the quantity \(x(t)\) is bounded above by the exponential

\[ x(t)<x(t_0)\exp[-D(t-t_0)]. \]

By the condition of the theorem, the characteristic equation of system (8) has no positive real roots. Therefore motion of the described type can occur only on the integral manifold corresponding to the negative real roots of equation (6). Otherwise the trajectories of system (8) either intersect the plane \(\sigma_1=0\), or go out onto the plane \(\sigma_2=0\). However, if the motion occurs on the integral manifold corresponding to the negative real roots of equation (6), then the representative point falls directly into the origin as \(t\to\infty\), i.e. \(x(t),\,y(t),\,z(t)\to0\) monotonically as \(t\to\infty\). If \(\varphi_0<0\), then the motion, beginning from some instant of time, is described by the system of equations

\[ \dot{x}=y,\quad \dot{y}=z,\quad \dot{z}=-az-by-cx+Kx. \tag{9} \]

The trajectories of system (9) located in the region \(\sigma_1>0,\ \sigma_2<0,\ x<0\) intersect the switching surface \(\sigma_1=0\) or go out onto the plane \(\sigma_2=0\) with \(\sigma_1>0\).

Thus, if the representative point remains for an infinitely long time in the region \(\sigma_1>0,\ \sigma_2<0\), then it either falls directly into the origin (with \(x(t),\,y(t),\,z(t)\to0\) monotonically as \(t\to\infty\)), or onto the switching surface, or, finally, leaves the region \(\sigma_2<0\) for the region \(\sigma_2>0\) with \(\sigma_1>0\). In consequence of condition (5) of the theorem, the trace of the plane

\[ \frac{d\sigma_2}{dt}=0 \]

on the plane \(\sigma_2=0\) (the straight line (3)) lies between the traces of the planes \(\sigma_1^+=0,\ \sigma_1^-=0\). Consequently, the representative point, having entered the region \(\sigma_1>0,\ \sigma_2>0\), can leave it only through the switching surface. In the region \(\sigma_2>0\) the inequality

\[ \frac{d\varphi}{dt}>0 \]

is satisfied. Therefore, if the point remains for an infinitely long time in the region \(\sigma_2>0,\ \sigma_1>0\) and does not fall on the switching surface, then the nondecreasing function \(\varphi(t)\) tends to some proper or improper limit \(\varphi_1\) as \(t\to\infty\). Obviously, \(\varphi_1(t)>\varphi(t_0)>-\infty\). As above, there may be two cases:

\[ 1)\ \varphi_1\leqslant0,\qquad 2)\ \varphi_1>0. \]

In the first case, along the trajectory of system (1) the inequality \(x(t)\exp Dt<\varphi_1\leqslant0\) is satisfied. Consequently, \(x(t)<0\), and the motion is described by system (9). The only possible trajectories of the system

(9), along which the quantity \(x(t)\) does not change sign and \(\sigma_1(t)\) does not vanish in a finite time, is motion along the integral manifold corresponding to the real negative roots of the characteristic equation of system (9). But then the quantities \(x(t)\), \(y(t)\), \(z(t)\) tend monotonically to zero. If, however, \(\varphi_1>0\), then, beginning with some instant of time, the representative point passes into the region \(\sigma_1 x>0\), \(\sigma_2>0\), and the motion is described by system (8). In this case, if the quantity \(\sigma_1(t)\) does not change sign at some instant of time, then the only possible motion is motion along the integral manifold corresponding to real negative roots. But then \(x(t)\), \(y(t)\), \(z(t)\to0\) monotonically as \(t\to\infty\).

The case \(\sigma_1(M)<0\) is considered similarly. In this case the region \(\sigma_1<0\), \(\sigma_2<0\), \(x<0\) is a “trap” for the trajectories of system (1): having entered this region, the representative point either reaches the switching surface in a finite time, or, along an integral manifold, monotonically approaches the origin. We note that condition (6) of the theorem is also a necessary condition for the representative point to reach the switching surface. The proof of this fact can be carried out using the idea of [2].

1. As is known [1, 3], the surface \(\sigma_1=0\) is a sliding surface everywhere except for the points of the sector \(W\) and the points of a certain sector \(\Pi\) (the penetration sector), in which the trajectories pass through the sliding surface. The sector \(\Pi\) is located near the plane \(x=0\) and is bounded by the straight lines \(L_1L_4\) and \(L_2L_3\), whose equations have the form

\[ L_1L_4:\quad \begin{cases} \sigma_1^- = 0,\\[4pt] y+x\,\dfrac{K-A(B-a)+c}{A+B(B-a)+b}=0, \end{cases} \]

\[ L_2L_3:\quad \begin{cases} \sigma_1^+ = 0,\\[4pt] y-x\,\dfrac{K+A(B-a)+c}{A-B(B-a)-b}=0. \end{cases} \]

The sectors \(N_2OL_4\) and \(N_1OL_2\) are “pierced” by the trajectories of system (1) in the direction \(\sigma_1>0\), and the sectors \(N_2OL_3\) and \(N_1OL_1\) in the direction \(\sigma_1<0\). The points of the sector \(N_2OL_3\) enter the region \(\sigma_1<0\), \(\sigma_2<0\), \(x\le0\), which, as follows from the proof of the theorem, is a “trap” for the points of the half-space \(\sigma_1<0\).

Thus the representative point, having descended from the sector \(N_2OL_3\), will return again to the part \(S_3\) of the surface \(\sigma_1=0\) outside the penetration sector. Moving further according to the equations of the sliding mode, it necessarily reaches either the origin directly, or the ray \(OQ_1\). Similarly, points of the sector \(N_1OL_2\) reach the part \(S_1\) of the surface \(\sigma_1=0\) outside the sector \(\Pi\) and then reach the ray \(OQ_4\) in a finite or infinite time (the latter case corresponds to the system of equations \(\dot x=y\), \(\dot y=-Ax-By\) having a singular point of the “node” type). Therefore, in what follows it is sufficient to consider only the behavior of the points of the sectors \(N_2OL_4\) and \(N_1OL_1\).

As is known from [1, 3], increasing the coefficient \(K\) makes it possible for points that have descended from the sectors \(N_1OL_1\) and \(N_2OL_4\) to return to the switching surface and subsequently reach the straight lines \(Q_1Q_4\) and \(Q_2Q_3\).

We establish a quantitative estimate of the coefficient \(K\), under which the points of the sectors \(N_2OL_4\) and \(N_1OL_1\), after leaving the surface \(\sigma_1=0\), either fall directly into the origin, or into the parts \(S_3\) and \(S_1\) of the surface \(\sigma_1=0\). The return of the points of the sectors \(N_2OL_4\) and \(N_1OL_1\) to the switching surface means that the surface \(\sigma_1=0\) is not perforated.

We pass to new coordinates and to “slow” time, making the substitution

\[ X=x,\qquad Y=y,\qquad Z=\nu z,\qquad t=\nu\tau,\qquad \nu^{-2}=K. \tag{10} \]

Let, in addition, \(A=A_0\nu^{-1}\). System (1) is transformed into the form

\[ \frac{dX}{d\tau}=\nu Y,\qquad \frac{dY}{d\tau}=Z,\qquad \frac{dZ}{d\tau}=-\nu aZ-b\nu^2Y-c\nu^2X-|X|\operatorname{sign}\sigma_1, \tag{11} \]

\[ \sigma_1=A_0|X|\operatorname{sign}\sigma_2+B\nu Y+Z,\qquad \sigma_2=DX+Y. \]

The line \(L_1L_4\) is written in the form

\[ \sigma_1^{-}=0,\qquad Y+X\nu^{-1}C_1(\nu)=0, \tag{12} \]

and the line \(L_2L_3\) in the form

\[ \sigma_1^{+}=0,\qquad Y-X\nu^{-1}C_2(\nu)=0, \]

where

\[ C_1(\nu)=\frac{1-\nu A_0(B-a)+c\nu^2}{A_0+B(B-a)\nu+b\nu}, \qquad C_2(\nu)=\frac{1+\nu A_0(B-a)+c\nu^2}{A_0-\nu B(B-a)-b\nu}. \]

We show that for sufficiently small \(\nu\) there exists a certain plane \(P_1=0\), passing through the line \(OL_4\) and the line \(X=0,\ \alpha Y+Z\) \((\alpha>0)\), and such that it is perforated by trajectories of system (11) in the direction of the surface \(\sigma_1=0\). Indeed, the equation of the plane \(P_1=0\) may be taken in the form

\[ P_1=X[\alpha\nu^{-1}C_1(\nu)-A_0-BC_1(\nu)]+\alpha Y+Z=0. \tag{13} \]

The quantity \(\dfrac{dP_1}{d\tau}\), computed by virtue of system (11) in the domain \(X>0\), \(\sigma_1>0\), \(\sigma_2<0\) on the plane \(P_1=0\), is equal to

\[ \left.\frac{dP_1}{d\tau}\right|_{P_1=0} = -X[1+c\nu^2+(\alpha-\alpha\nu)E]+[-b\nu^2+(E+a\alpha)\nu-\alpha^2]Y, \]

where

\[ E=\alpha\nu^{-1}C_1(\nu)-A_0-BC_1(\nu). \]

Consequently, if the inequalities

\[ 1+c\nu^2+(\alpha-\alpha\nu)E>0,\qquad E\nu-\alpha^2+a\alpha\nu-b\nu^2>0, \tag{14} \]

are satisfied, then \(\dfrac{dP_1}{d\tau}<0\) in the domain under consideration. Obviously, inequalities (14) are satisfied for sufficiently small \(\nu\) and \(\alpha>0\).

Consider the trihedral angle formed by the planes \(\sigma_1^{-}=0\), \(X=0\), \(P_1=0\) (Fig. 2). The trajectories of system (11) that perforate the sector \(N_2OL_4\), or remain for an infinitely long time in the part of the half-space \(X>0\),

\(Y<0,\ Z>0\), bounded by the surface of the trihedral angle, or intersect the plane \(X=0\) in the sector \(N_2OP_1\) between the lines \(Z=0\) and \(\alpha Y+Z=0\). In the first case \(X(t), Y(t), Z(t)\to 0\) monotonically as \(t\to\infty\).

We shall show that all points of the sector \(N_2OP_1\) fall onto the part \(S_3\) of the sliding surface, i.e., thereby establish that, under certain restrictions on the value of the parameter \(v\), the penetrability of the surface \(\sigma_1=0\) is immaterial.

Let \(P_2=0\) be a plane passing through the line \(X=0,\ \alpha Y+Z=0\), given by the equation \(P_2=\beta X+\alpha Y+Z=0\). The line of intersection of the planes \(\sigma_1^+=0\) and \(P_2=0\) has the form

\[ \sigma_1^+=0,\qquad (\alpha-\beta v)Y+(\beta-A_0)X=0. \tag{15} \]

Fig. 2

The line (15) lies on the part \(S_3\) of the surface \(\sigma_1=0\) if and only if the inequality

\[ \frac{\beta-A_0}{\alpha-Bv}<D. \tag{16} \]

is satisfied.

Computing the derivative of the function \(P_2(X,Y,Z)\) for \(P_2=0\), we find

\[ \left.\frac{dP_2}{d\tau}\right|_{P_2=0} = [1-cv^2-\beta(\alpha-av)]X+ [\beta v-bv^2-\alpha(\alpha-av)]Y. \]

If the relations

\[ \beta=bv+\alpha(\alpha-av)v^{-1},\qquad 1-cv^2-\beta(\alpha-av)>0, \tag{17} \]

are assumed to hold, then in the region \(X\leq 0\)

\[ \frac{dP_2}{d\tau}\leq 0 \]

for \(P_2=0\), and hence the plane \(P_2=0\) will be penetrated by the trajectories of system (11) in the direction of the plane \(\sigma_1^+=0\).

Thus, if conditions (14), (16), (17) are fulfilled, then there exists a quadrilateral surface \(OL_4P_1P_2N_2\), formed by the sector \(N_2OL_4\), the part of the plane \(\sigma_1^+=0\), and parts of the planes \(P_1=0\) and \(P_2=0\) (Fig. 2). This surface is such that a representative point, once inside it, can leave it only through the sector \(N_2OL_3\) or through the line of intersection of the planes \(\sigma_1^+=0,\ P_2=0\). But in doing so the point necessarily falls either into the sector \(W\), or onto the part \(S_3\) of the surface \(\sigma_1=0\) outside the sector \(\Pi\), and, moving thereafter according to the system

\[ \dot x=y,\qquad \dot y=-Ax-By, \]

will arrive either at the line \(OQ_1\), or directly at the origin.

We note that the fulfillment of conditions (14), (16), (17) for \(v=v_0\) is still insufficient for their practical application. It is necessary that they be fulfilled for all \(v\leq v_0\). This requirement is satisfied,

if one sets \(\beta - A_0 = 0\), i.e., if \(\alpha\) is chosen equal to the positive root of the quadratic equation \(\alpha^2 - a \alpha \nu + b \nu^2 - A_0 \nu = 0\). Such a root always exists if the inequality \(A_0 > b\nu\) is satisfied.

Thus, one can write down four inequalities with respect to the parameter \(\nu\), the fulfillment of which means that the switching surface is not penetrated:

\[ \begin{aligned} \text{a)}\;& 1 + c\nu^2 + (\alpha - a\nu)E > 0,\\ \text{b)}\;& E\nu - \alpha^2 + a\alpha\nu - b\nu^2 > 0 \quad \text{or} \quad E > A_0,\\ \text{c)}\;& 1 - c\nu^2 - (\alpha - a\nu)\bigl[b\nu + \alpha(a-a\nu)\nu^{-1}\bigr] > 0,\\ \text{d)}\;& A_0 - b\nu > 0, \end{aligned} \tag{18} \]

where

\[ E = \alpha\nu^{-1} C_1(\nu) - \bigl(A_0 + BC_1(\nu)\bigr), \]

\[ \alpha = 0.5\,a\nu + \sqrt{0.25\,a^2\nu^2 + A_0\nu - b\nu^2}. \]

The conditions (18) are satisfied as \(\nu \to 0\). Consequently, they determine an admissible finite value of \(\nu\).

1. Let us construct a cone \(C\) containing within itself all trajectories of system (11) (or (1)) that pass through the sector \(W\) of the plane \(\sigma_2 = 0\).

In what follows, for simplicity we shall assume \(B=0\) (it is intuitively clear that for \(B>0\) the stability of system (1) can only improve). Consider the cone formed by the surface \(\sigma_1=0\) and the surface \(F=0\), where \(F(X,Y,Z)\) is defined as follows:

\[ F = \begin{cases} F_1 = Z^2 - 2q_1XZ - 2XY - p_1X^2 = 0, & \text{for } \sigma_2 X < 0,\\ F_2 = Z^2 + 2q_2XZ + 2XY - p_2X^2 = 0, & \text{for } \sigma_2 X > 0, \end{cases} \tag{19} \]

where

\[ p_1 = A_0^2 + 2q_1A_0 + 2D,\qquad p_2 = A_0^2 + 2q_2A_0 - 2D, \]

and \(q_1, q_2\) are arbitrary positive numbers (Fig. 3).

Fig. 3

The cone constructed in this way lies entirely in the space \(XY<0\) under the condition \(|q_2 + A_0| < \sqrt{2D}\). We shall show that, for certain values of \(\nu\), \(q_1\), and \(q_2\), the cone constructed by us is a cone \(C\) containing within itself all trajectories of system (11) that pass through the sector \(W\).

From (19) it is evident that the gradient to the surface \(F=0\) is directed toward increasing \(F\). Consequently, in order that the trajectories of system (11) not leave the cone through the surface \(F=0\), it is sufficient that, on the part of the cone surface formed by the surface \(F=0\), the inequality \(\dfrac{dF}{d\tau}\leqslant 0\) hold, and that the part of the surface composed of the planes \(\sigma_1^+=0\) and \(\sigma_1^-=0\) lie outside the sector \(\Pi\). Computing \(\dfrac{dF_1}{d\tau}\) for \(F_1=0\), we obtain

\[ \left.\frac{dF_1}{d\tau}\right|_{F_1=0} = -\bigl[2q_1(1-c\nu^2)-p_1\nu(p_1-bq_1\nu)\bigr]X^2 -2a\nu Z^2 -2\nu(c\nu- \]

\[ -aq_1)XZ -2\nu\bigl[Z^2-2q_1XZ-X^2p_1\bigr]\cdot\frac{1}{4X^2} - \]

\[ -\nu Z\cdot\frac{1}{X}(q_1+b\nu)\bigl[Z^2-2q_1XZ-p_1X^2\bigr]. \]

Since we are interested only in that part of the surface \(F_1=0\) in the region \(\sigma_1^+\sigma_1^-<0\), one may put \(Z=\gamma_1A_0X\) and assume that the parameter \(\gamma_1\) varies from \(-1\) to \(+1\).

Then we obtain

\[ \left.\frac{dF_1}{d\tau}\right|_{F_1=0} = -f_1(\gamma_1,q_1,\nu)X^2, \tag{20} \]

where

\[ f_1(\gamma_1,q_1,\nu) = 2q_1(1-c\nu^2)-p_1\nu(p_1-bq_1\nu) +2a\nu\gamma_1^2A_0^2+ \]

\[ +2\nu(c\nu-aq_1)\gamma_1A_0 +\frac{\nu}{2}\bigl(\gamma_1^2A_0^2-2q_1A_0\gamma_1-p_1\bigr)^2+ \]

\[ +\nu(q_1+b\nu)A_0\gamma_1 \bigl(A_0^2\gamma_1^2-2q_1A_0\gamma_1-p_1\bigr). \]

Similarly, computing \(\dfrac{dF_2}{d\tau}\) for \(F_2=0\), we obtain

\[ \left.\frac{dF_2}{d\tau}\right|_{F_2=0} = -f_2(\gamma_2,q_2,\nu)X^2, \]

where

\[ f_2(\gamma_2,q_2,\nu) = 2q_2(1+c\nu^2) -\frac{\nu}{2}\bigl[p_2-\gamma_2^2A_0^2-2q_2A_0\gamma_2\bigr]^2 +2a\nu\gamma_2^2A_0^2+ \]

\[ +\nu(p_2+q_2b\nu)(p_2-\gamma_2^2A_0^2-2q_2A_0\gamma_2) +2A_0\gamma_2\nu(c\nu+aq_2)+ \]

\[ +\nu(b\nu-q_2)(p_2-2q_2A_0\gamma_2-\gamma_2^2A_0^2)A_0\gamma_2. \tag{21} \]

The part of the cone surface composed of the planes \(\sigma_1^+=0\), \(\sigma_1^-=0\), lies outside the sector \(\Pi\) if the inequality

\[ 2q_1A_0+D\leqslant \frac{C_1(\nu)}{\nu} \]

is satisfied.

Thus, if the inequalities

\[ \begin{aligned} \text{a) }&\min_{\gamma_1} f_1(\gamma_1,q_1,\nu)\geqslant 0 \quad \text{for } -1\leqslant \gamma_1\leqslant 1,\\ \text{b) }&\min_{\gamma_2} f_2(\gamma_2,q_2,\nu)\geqslant 0 \quad \text{for } -1\leqslant \gamma_2\leqslant 1, \end{aligned} \tag{22} \]

c) \(2A_0q_1 + D \leq C_1(\nu)\nu^{-1}\),

d) \(|q_2 + A_0| < \sqrt{2D}\),
\[ \tag{22} \]

then the constructed conical surface contains within itself all trajectories of system (11) (or (1)) passing through the sector \(W\) of the plane \(\sigma_2 = 0\), and lies entirely in the space \(XY < 0\).

Conditions (22) contain the free parameters \(q_1\) and \(q_2\), which can be chosen so as to ensure the maximal estimate for the quantity \(\nu\). Obviously, conditions (22) are satisfied for any \(q_1 > 0\), \(q_2 > 0\), and \(\nu \to 0\). Consequently, they determine a certain finite value \(\nu_0\) such that, for all \(\nu \leq \nu_0\), in a neighborhood of the sector \(W\) there exists a stable quasi-ideal sliding regime of the second order.

From what was set forth in Sections 2, 3, and 4, it in fact follows that the zero solution of system (1) is asymptotically stable when conditions (5), (6), (18), and (22) are fulfilled. These conditions are, generally speaking, only sufficient conditions for stability, although, for example, condition (6) is also necessary.

Let us note that the computational work associated with the choice of the parameters \(A\), \(B\), \(D\), and \(K\) is expediently carried out on digital computers, using methods for solving nonlinear inequalities. Such an approach makes it possible, for given characteristics of the controlled plant, to obtain the stability region in the space of the parameters \(K\), \(A\), \(B\), \(D\).

1. In conclusion we give an example of the synthesis of a control system in the case when the characteristic equation of the uncontrolled system has a triple zero root, i.e. \(a=b=c=0\).

Suppose, for example, that it is required to ensure the duration of the transient process in the system equal to \(3.5\) sec.

It is known [1] that, approximately, a system with a forced sliding regime is described by the equation \(\dot{x} + Dx = 0\). The quantity \(T = 3D^{-1}\) determines approximately the time for eliminating the initial deviation \(x(0)\). To obtain the required duration of the transient process it is sufficient to set \(D = 1\).

Let \(A_0 = 0.5\) (generally speaking, the value \(A_0\) determines the frequency of self-oscillations due to the nonideality of the switching device). Conditions (5), (6) give a restriction on the parameter: \(\nu \leq \nu_1 = 0.5\). The conditions of insignificance of perforability (18) reduce to the inequality:
\[ \nu < \nu_2 = \frac{1}{4A_0^3} = 2. \]

Analyzing the conditions for the existence of double sliding (conditions (22)), we obtain
\[ f_1(\gamma_1,q_1,\nu) = 2q_1 - p_1^2\nu + \frac{\nu}{2} \left[A_0^2\gamma_1^2 - p_1\right] \left[A_0^2\gamma_1^2 - 2q_1\gamma_1A_0 - p_1\right]. \]

Simplifying the last expression and taking into account that \(|\gamma_1| < 1\), we shall have
\[ f_1(\gamma_1,q_1,\nu) > 2q_1 - \frac{\nu}{2} \left[ -2q_1A_0^3 + 2p_1A_0^2 + 2p_1q_1A_0 + p_1^2 \right]. \]

Consequently, condition (22,a) will be fulfilled for all \(\nu\) satisfying the relation
\[ \nu < \frac{4q_1}{p_1^2 + 2p_1q_1A_0 + 2p_1A_0^2 - 2q_1A_0^3}. \]

Maximizing the right-hand side of the last inequality with respect to \(q_1\) and substituting the values \(A_0\) and \(D\), we obtain the condition

\[ v \leqslant v_{3a}=\frac{4q_{10}}{p_{10}^{2}+p_{10}q_{10}+0.5p_{10}-0.25q_{10}}=0.284, \]

where \(p_{10}=4.01,\ q_{10}=1.76\). Analyzing condition (22, б) for the function \(f_2(\gamma_2,q_2,v)\), we shall have the relation

\[ f_2(\gamma_2,q_2,v)>2q_2-\frac{v}{2}\,[A_0^4+2A_0^3q_2-2p_2A_0^2-2p_2q_2A_0-p_2^2]. \]

Considering conditions (22, б) and (22, г) jointly, set \(q_2=0.5\). Then \(p_2=A_0^2-2D+2q_2A_0=-1.25\) and \(f_2(\gamma_2,q_2,v)>0\) for all \(v>0\). Condition (22, в) gives the inequality \(v \leqslant A_0^{-1}(2q_{10}+D)^{-1}=0.44\). Thus, finally we have

\[ v \leqslant \min\{0.5,\ 2,\ 0.284,\ 0.44\}=0.284. \]

Passing to the parameters \(K,\ A,\ B,\ D\), we finally obtain

\[ K>12.7,\qquad A>1.78,\qquad B=0,\qquad D=1. \]

The example given shows that conditions (5), (6), (18), and (22) are sufficiently constructive and make it possible to synthesize a control system with prescribed characteristics of the transient process.

References

1. Barbashin E. A., Gerashchenko E. I. Differential Equations, vol. I, No. 1, 25–32, 1965.
2. Emel’yanov S. V., Taran V. A. Izvestiya AN SSSR, OTN, Energetics and Automation, No. 3, 183–188, 1962.
3. Barbashin E. A., Tabueva V. A., Eidinov R. M. Automation and Remote Control, vol. XXIV, No. 7, 882–891, 1963.

Received by the editors
May 12, 1965

V. A. Steklov Mathematical Institute,
Sverdlovsk Branch