ON FORCING SLIDING REGIMES IN AUTOMATIC CONTROL SYSTEMS

E. A. Barbashin, E. I. Gerashchenko

1. It is known that by means of a stepwise change in the parameters of a control system one can cause the system to enter a sliding regime that is robust with respect to changes in the system parameters within wide limits [1—5, 7]. In [2] it was shown that violation of the sliding conditions within certain limits does not reduce the quality of regulation.

In [8] a method was given for choosing sliding parameters that are optimal with respect to the degree of stability. The use of optimal sliding parameters makes it possible to accelerate the course of the process in the sliding regime. However, in this case the equations of motion in the sliding regime remain linear and, consequently, the sliding process is far from best in the sense, for example, of speed of response and disturbance rejection.

On the other hand, in [3] it is shown that by introducing a discontinuous sliding surface one can also accelerate the course of the sliding process, eliminate oscillatory behavior and overshoot.

It seems natural to consider such a sliding surface that the sliding regime would, with time, pass into a sliding regime on a surface of lower dimension as compared with the dimension of the basic surface. In other words, the problem is posed of organizing the exit of a sliding regime into a sliding regime of higher order. Lowering the dimension of the manifold on which sliding occurs should, evidently, lead to a substantial improvement in the quality of the transient process.

A detailed analysis of the problem posed leads to the conclusion that a sliding regime of higher order can exist only in the case where the initial sliding regime is described by differential equations with discontinuous characteristics. But it follows from this that the basic \((n - 1)\)-dimensional sliding surface must be discontinuous. The presence of a discontinuous sliding surface, however, causes the nonideality of a sliding regime of higher order. Thus, by introducing a suitably chosen sliding surface, we effect the transition of the system into an ideal sliding regime, which in turn passes, beginning from a certain instant of time, into a nonideal sliding regime with a corresponding reduction in dimension.

In the present paper, using the example of a third-order system, a detailed justification is given for the propositions stated above.

2. Consider the third-order equation

\[ \dddot{x}+a\ddot{x}+b\dot{x}+cx=-nKx, \tag{2.1} \]

equivalent to the system

\[ \dot{x}=y,\quad \dot{y}=z,\quad \dot{z}=-cx-by-az-nKx . \tag{2.2} \]

Let the parameters \(a, b, c, K\) be arbitrary constants, with the coefficient \(K\) positive.

Fig. 1.

We shall define the quantity \(n\) by the formula

\[ n=\operatorname{sign}\,[xA\operatorname{sign}x(y+Dx)+By+z]\,x, \tag{2.3} \]

where \(A, B, D\) are positive constants.

Thus, the quantity \(n\) can change sign on one of the planes \(x=0\), \(T(x,y)=y+Dx=0\),

\[ R_1(x,y,z)=Ax+By+z=0, \tag{2.4} \]

\[ R_2(x,y,z)=-Ax+By+z=0. \]

The switching surface is composed of the plane \(x=0\), the surface \(S\), which consists of parts \(S_1, S_3\) of the plane \(R_1=0\), parts \(S_2\) and \(S_4\) of the plane \(R_2=0\), and parts \(T_1\) and \(T_2\) of the plane \(T=y+Dx=0\) (Fig. 1). Points of the surface \(S\) for which \(y+Dx\ne 0\) obviously satisfy the equation

\[ R(x,y,z)=xA\operatorname{sign}x(y+Dx)+By+z=0. \tag{2.5} \]

The surface \(S\) and the plane \(x=0\) divide the phase space into four regions \(G_1, G_2, G_3, G_4\), defined respectively by the inequalities

\[ x>0,\quad R>0, \]

\[ x<0,\quad R>0, \]

\[ x<0,\quad R<0, \]

\[ x>0,\quad R<0. \]

Obviously, we have \(n=1\) in the regions \(G_1\) and \(G_3\), and \(n=-1\) in the regions \(G_2\) and \(G_4\).

On the surface \(S\), near the line \(N_1ON_2\), along which the planes \(R_1=0\), \(R_2=0\), and \(x=0\) intersect, the sectors \(L_1OL_2\) and \(L_3OL_4\) are distinguished,

in which the trajectories of system (2.2) “pierce” the surface \(S\). In [2] it is shown that, by choosing a sufficiently large \(K\), the angles of these sectors can be made arbitrarily small. At all remaining points the surface \(S\) is a sliding surface. The plane \(x=0\) is a “pierceable” plane.

Theorem. If the parameters \(A, B, D\) are positive, \(K>K_0\), where \(K_0\) is a sufficiently large number, then all solutions of system (2.2) have the property
\[ \lim x(t)=\lim y(t)=\lim z(t)=0 \quad \text{as } t\to\infty . \]

The greatest interest is presented by the behavior of the trajectories in the case when the inequalities
\[ B^2-4A<0,\quad D<0.5\bigl(B+\sqrt{B^2+4A}\bigr). \tag{2.6} \]
are satisfied.

This case, which is the most difficult, will be considered in detail below. The typical behavior of the trajectories when inequalities (2.6) are satisfied is described as follows. An arbitrary point \(M\) of phase space either reaches the surface \(S\) after a finite interval of time, or after an infinite interval of time—directly at the origin. If the point \(M\) reaches the part \(S_1\) or \(S_3\) of the surface \(S\), then subsequently it will move, by virtue of the system
\[ \dot{x}=y,\quad \dot{y}=-Ax-By \tag{2.7} \]
along a spiral-like curve until it reaches the plane \(T=0\). Having passed through the plane \(T=0\), the point \(M\) must pass from the part \(S_3\) to the part \(S_2\), and from the part \(S_1\) to the part \(S_4\). On the parts \(S_2\) and \(S_4\) of the plane \(R_2=0\) there acts the system
\[ \dot{x}=y,\quad \dot{y}=Ax-By, \tag{2.8} \]
and if \(K_0\) is sufficiently large, then the point \(M\) will go along a curve of hyperbolic type again toward the plane \(T=0\). However, after passing through this plane, the point again enters the region of attraction of the plane \(R_1=0\). Thus, passing from one side of the plane \(T=0\) to the other and oscillating between the planes \(R_1=0\) and \(R_2=0\), the point \(M\) will arrive at the origin. That is, we obtain a nonideal sliding mode of motion of the point \(M\) over the surface \(S\). In the plane \(T=0\) there exists the straight line
\[ y+Dx=0,\quad z+Dy=0, \tag{2.9} \]
which in a certain sense directs the motion of the representative point toward the origin.

In some cases, when inequalities (2.6) are satisfied, the motion of the representative point in the final stage of motion may proceed differently. Having reached the surface \(S_3\) up to the plane \(T=0\), the point \(M\) will leave it, but may fail to reach the part \(S_2\), and instead turn around the line (2.9) back to \(S_3\). In this case the point \(M\), rapidly rotating around the line (2.9), will asymptotically approach the origin.

Below a detailed justification of the assertions just stated is given.

1. We proceed to the proof of the theorem.

Let, for definiteness, the point \(M\) at the initial moment of time lie in the domain \(G_1\). We first show that, as \(t\) increases, the point \(M\) reaches the surface \(S\).

From the proof of the theorem in [4] it follows that, as time increases, the point \(M\) either reaches the plane \(R_1=0\), and consequently the

part \(S_1\) or \(S_4\) of the surface \(S\), or onto the plane \(x=0\). In the latter case the point \(M\) will enter the region \(G_2\). The region \(G_2\) is cut by the integral plane of system (2.2) for \(n=-1\) into two parts. The equation of this plane has the form

\[ (\gamma^2+\delta^2)x-2\gamma y+z=0, \tag{3.1} \]

where \(\gamma\) and \(\delta\) are the real and imaginary parts of the roots of the equation

\[ \lambda^3+a\lambda^2+b\lambda+c-K=0. \tag{3.2} \]

For sufficiently large \(K\), equation (3.2) has a pair of complex roots \(\gamma\pm i\delta\) \((\gamma<0,\ \delta>0)\) and a real positive root \(\nu\). Obviously, as \(K\) increases, the quantities \(|\gamma|,\delta,\nu\) also increase without bound. If the point \(M\) lies above the integral plane (3.1), then, as \(t\) increases, it will reach the plane \(x=0\) (for \(y>0\)) or will fall onto the plane \(R_2=0\), i.e., onto the parts \(S_2\) or \(S_3\) of the surface \(S\) [4].

If, however, the point \(M(x_0,y_0,z_0)\) lies below the plane (3.1), then it is not difficult to show that \(M\) cannot leave the region \(G_2\) through the plane \(x=0\) [4].

On the other hand, it is easy to see that along the trajectory of the point \(M\) we have

\[ R_2(x,y,z)=c e^{\nu t}(\nu^2+B\nu-A)+\varphi(t), \tag{3.3} \]

where \(\varphi(t)\to 0\) as \(t\to\infty\) and

\[ c=\frac{(\gamma^2+\delta^2)x_0-2\gamma y_0+z_0}{(\nu-\gamma)^2+\delta^2}. \]

Since the quantity \(\nu^2+B\nu-A\) is positive for sufficiently large \(K\), the quantity \(R_2(x,y,z)\) must necessarily pass from the region of positive values \((R_2(x_0,y_0,z_0)>0)\) into the region of negative values. Thus, the point \(M\) will either fall onto the part \(S_3\) or \(S_2\) of the surface \(S\), or will reach the plane \(x=0\) with \(y\ge 0\). But in the latter case it was proved in [5] that the point \(M\), having entered the region \(G_1\), must necessarily fall onto the plane \(R_1=0\), i.e., either onto the parts \(S_1,S_4\) of the surface \(S\), or onto the part \(T_1\) of the plane \(T=0\).

Thus, any point of the phase space necessarily reaches the surface \(S\).

1. Let us next consider the motion of the point \(M\) on the surface \(S\), assuming that the inequalities (2.6) are satisfied.

Since \(B^2-4A<0\), the point \(0\) for system (2.7) is a singular point of the “focus” type, and for system (2.8) it is a singular point of the “saddle” type. From the second inequality (2.6) it follows that one of the separatrices of system (2.8) lies on the parts \(S_2\) and \(S_4\) of the surface \(S\).

Thus, the character of the motion of the point on the surface \(S\) is easily established up to the time when the point reaches the plane \(T=y+Dx=0\) or the sectors of sewing \(L_1OL_2,\ L_3OL_4\). But in [2] it was shown that in the latter case the point \(M\), having broken away from the surface \(S\), again falls onto it on the other side of the sector, and the magnitude of the deviation of the point \(M\) from \(S\), as \(K\) increases, becomes arbitrarily small.

It remains to consider the behavior of points that have fallen onto the plane \(T=0\), i.e., onto one of the lines \(OQ_i,\ i=1,2,3,4\) (see Fig. 1). For definiteness, suppose that the point \(M\) lies on the line \(OQ_1\).

It is not difficult to see that, when the conditions (2.6) are satisfied, the trace of the plane

\[ \frac{dT}{dt}=0 \]

on the plane \(T=0\) lies in the sector formed by the traces.

of the planes \(R_1=0\) and \(R_2=0\). Therefore, along the trajectory of motion of the point \(M\), the quantity \(\dfrac{dT}{dt}\) changes sign, and the motion is essentially curvilinear.

Let us pass to new phase coordinates and slow time \(\tau\):

\[ X=x,\quad Y=y,\quad Z=\mu z,\quad t=\mu\tau,\quad \mu=K^{-1/2}. \tag{4.1} \]

For sufficiently large values of \(K\), the quantity \(\mu\) is a small parameter. In the new coordinates, system (2.2) takes the form

\[ \frac{dX}{d\tau}=\mu Y,\quad \frac{dY}{d\tau}=Z,\quad \frac{dZ}{d\tau} \]
\[ =-\mu(aZ+b\mu Y+c\mu X)-nX. \tag{4.2} \]

Fig. 2.

We shall seek the solutions of system (4.2) in the form:

\[ X(\tau)=X^0(\tau)+\mu X^{(1)}(\tau)+\ldots, \]

\[ Y(\tau)=Y^0(\tau)+\mu Y^{(1)}(\tau)+\ldots, \]

\[ Z(\tau)=Z^0(\tau)+\mu Z^{(1)}(\tau)+\ldots, \]

where \(X^0(\tau)\), \(Y^0(\tau)\), \(Z^0(\tau)\) are the solution of system (4.2) for \(\mu=0\).

Obviously, the trajectory of the generating system lies in the plane \(X=X(0)=\mathrm{const}\) and consists of parts of two parabolas (Fig. 2)

\[ Y=Y_0-Z_0^2/2X_0+Z^2/2X_0\quad \text{for } R(DX+Y)\geqslant 0 \tag{4.3} \]

and

\[ Y=Y_0+Z_0^2/2X_0-Z^2/2X_0\quad \text{for } R(DX+Y)<0. \tag{4.4} \]

In order to determine the trajectory of the point \(M\), lying on the line \(OQ_1\), it is necessary in equations (4.3) and (4.4) to substitute, instead of \(X_0, Y_0, Z_0\), the coordinates of the point \(M\), that is, to put \(Y_0=-DX_0,\; Z_0=-\mu(A-BD)X_0\).

From the symmetry of the parabolas (4.3) and (4.4) with respect to the axes \(Z=0,\; X=X_0\), it follows that the motion of the point \(M\) along the trajectory of the generating system will be periodic, with a period of the same order of smallness as \(\mu\).

Substituting the coordinates of the point \(M\) into equations (4.3) and (4.4) and regarding \(X_0\) as variable, we obtain the equation of a certain cone

\[ F(X,Y,Z)= \begin{cases} Y+D_1X-Z^2/2X & \text{for } DX+Y\geqslant 0,\\ Y+D_2X+Z^2/2X & \text{for } DX+Y<0, \end{cases} \]

where \(D_1=D+0.5\mu^2(BD-A)^2,\; D_2=D-0.5\mu^2(BD-A)^2\).

Computing the derivative of the function \(F(X,Y,Z)\) along the trajectory of system (4.2), we find

\[ \frac{dF}{d\tau}= \begin{cases} \mu D_1Y+o(\mu) & \text{for } T\geqslant 0,\\ \mu D_2Y+o(\mu) & \text{for } T<0, \end{cases} \]

where by \(o(\mu)\) is denoted a quantity of higher order of smallness in comparison with \(\mu\).

For sufficiently small \(\mu\) we have \(D_2>0\); the quantity \(Y\) in the domain under consideration is also positive; consequently,

\[ \frac{dF}{d\tau}>0. \]

Thus, by virtue of equations (4.2), the point moves in the direction of the gradient to the surface \(F=0\). It is easy to see that \(\operatorname{grad} F\) is directed toward increasing \(Y\), and consequently the representative point enters the cone \(F=0\) to the left of the plane \(T=0\) (Fig. 2) and leaves the cone to the right of the plane \(T=0\).

The quantity \(Z^0(\tau)\) has the same order of smallness as \(\mu\), while the quantity \(Z(\tau)-Z^0(\tau)\) has the same order of smallness as \(\mu^2\). If the distance between the points \(P\) and \(N\) (the point \(P\) is the point of intersection of the trajectory of the generating system with the plane \(T=0\); the point \(N\) lies on the line \(OQ_2\)) has the same order of smallness as \(\mu\), then the representative point will fall onto the plane \(T=0\) without falling onto the plane \(R_2=0\). If, however, \(PN=o(\mu)\), then the representative point may fall onto the plane \(R_2=0\) and along it return to the plane \(T=0\).

The quantity \(PN\) is equal to \(2\mu BDX_0\); consequently, the second case can occur only when the quantity \(B\cdot D\) is comparable with \(\mu\). It is not difficult to see that if \(B\cdot D\) has an order of smallness lower than \(\mu\), then the trajectory of the representative point either is contracted to the line (2.9) or passes along the surface of a certain cone close to the cone \(F(X,Y,Z)=0\).

If, on the other hand, the quantity \(BD\) is comparable with \(\mu\), then the representative point from the line \(OQ_1\) may fall onto the plane \(R_2=0\), reach along the plane \(R_2=0\) the line \(OQ_2\), then fall onto the plane \(R_1\), and again return to the line \(OQ_1\).

The cases considered differ little from one another. It is important only that in all cases the trajectory of the point \(M\), for sufficiently small \(\mu\), lies entirely in the half-space \(XY<0\). As a consequence of this,

\[ \lim x(t)=\lim y(t)=\lim z(t)=0 \quad \text{as } t\to\infty . \]

Let us note that the time of one revolution around the line (2.9) is a quantity comparable with \(\mu^2\); this gives grounds for separating the motion of the representative point into a “rapid” rotation around the line (2.9) and a “slow” motion along the line (2.9) toward the origin.

1. Let us briefly describe the remaining cases, when at least one of the inequalities (2.6) is not satisfied. If the inequality

\[ B^2-4A \geq 0, \tag{5.1} \]

is satisfied, then the origin for system (2.7) will be a singular point of the “node” type. If, alongside (5.1), the inequality

\[ D<0.5\left(B+\sqrt{B^2+4A}\right), \]

is satisfied, then some of the points \(S_1\) and \(S_3\) will enter the origin along the straight line

\[ y=-0.5\left(B-\sqrt{B^2-4A}\right)x,\quad Ax+By+z=0, \tag{5.2} \]

while the remaining points will again fall onto the plane \(y+Dx=0\) and subsequently execute a sliding motion toward the point \(O\) of exactly the same type as we considered earlier.

If, alongside (5.1), the condition

\[ D>0.5\left(B+\sqrt{B^2+4A}\right), \tag{5.3} \]

is satisfied,

then all points of the surface \(S\) will complete their motion toward the origin along the straight line (5.2). If, along with (5.3), the inequality \(B^{2}-4A<0\) is satisfied, then the “second sliding” will not occur. Let, as before, the point \(M\) lie on the ray \(OQ_{1}\). It is easy to see that the condition (5.3) is equivalent to the fact that the trace of the plane

\[ \frac{dT}{dt}=Dy+z=0 \]

on the plane \(T=0\) lies outside the sector formed by the traces of the planes \(R_{1}=0\) and \(R_{2}=0\). In the region \(T>0,\ R>0,\ x<0\) an unstable system acts \((n=-1)\). The quantity \(\frac{dT}{dt}\) along the trajectory of motion does not change sign. Therefore, for \(K\) sufficiently large, the motion is close to motion along the straight line with directing vector \(F_{1}(M)\)—the velocity vector at the point \(M\).

Obviously, we have

\[ F_{1x}=-Dx_{0},\quad F_{1y}=(BD-A)x_{0}, \]

\[ F_{1z}=(K-c+bD-BD+A)x_{0}. \tag{5.4} \]

Let us find the position \(M'\) of the point \(M\) on the part of the surface \(S_{2}\). Under the assumptions made, the point \(M'\) is determined from the following system of equations:

\[ R_{2}(M')=0, \]

\[ OM'=OM+\Delta t\,F_{1}(M), \tag{5.5} \]

where \(\Delta t\) is the time of reaching the plane \(R_{2}=0\), and \(OM'\) and \(OM\) are vectors drawn to the points \(M'\) and \(M\) from the point \(O\). Solving system (5.5), we find

\[ \Delta t=2A/K+o\left(\frac{1}{K}\right). \tag{5.6} \]

From relation (5.6) it follows that \(\Delta t\) is comparable with the quantity \(A/K\). Let us find the equation of the line \(OQ_{1}'\)—the image of the line \(OQ_{1}\). Computing the coordinates of the point \(M'\) and taking into account that \(K\) is sufficiently large, we obtain the equation of the line \(OQ_{1}'\):

\[ R_{2}(x,y,z)=0,\quad D_{1}x+y=0, \tag{5.7} \]

where \(D_{1}=D\left(1+[2A(BD-A)+2AD^{2}]K^{-1}D^{-1}\right)\).

We note that if the projection of the point \(M\) on the plane \((x,y)\) moved by virtue of system (2.7), then during the time \(\Delta t\) it would arrive at the line \(D_{1}x+y=0\) (up to small terms \(o(K^{-1})\)). This fact suggests the following interpretation of the motion on the surface \(S\): the surface \(R=0\) is continuous, \(R=R_{1}(x,y,z)\), and the sign of \(A\) changes not on the plane \(T=0\), but on the plane \(T'=D_{1}x+y=0\).

If in this case we neglect an error of the same order as \(K^{-2}\), then the motion of a point on the surface \(S\) can be described by the system

\[ \dot{x}=y,\quad \dot{y}=-By-An_{1}x,\quad n_{1}=\operatorname{sign}x(y+D_{1}x). \tag{5.8} \]

The function \(V=n_{1}Ax^{2}+y^{2}\) will be a positive definite function having a discontinuity on the line \(y+D_{1}x=0\). Taking the derivative of \(V\) by virtue of system (5.8), we obtain \(\dot{v}=-2By^{2}\) at points of continuity of \(V\); at the points of discontinuity, the function \(V\), as is easy to see, has a negative increment equal to \(-2Ax^{2}\). Taking into account that the discontinuities of the function \(V\) along the

trajectories are discontinuities of the first kind, we easily convince ourselves of the possibility of applying Theorem 4 of the paper [6]. From this theorem there follows the asymptotic stability of the zero solution of system (5.8).

It is easy to show that in the present case, instead of the discontinuous function \(V\), one may take the positive definite continuous function \(V_1(x,y)=Ax^2+y^2\). Its derivative by virtue of system (5.8) is equal to

\[ \frac{dV_1}{dt} = -2By^2+2Axy(1-n_1) = \begin{cases} -2By^2, & \text{for } (D_1x+y)x \ge 0,\\ -2By^2-4A|xy|, & \text{for } (D_1x+y)x < 0. \end{cases} \]

Consequently, the function \(V_1(x,y)\) satisfies the conditions of the theorem cited above [6].

However, the validity of our theorem in the present case follows from a direct qualitative investigation of the behavior of trajectories on the surface \(S\). Let us note that in this case the representative point moves toward the point \(O\) along a spiral-like curve deviating only slightly from \(S\) in the sectors \(L_1OL_2\), \(L_3OL_4\).

Finally, let us note that the theorem proved could have been treated as a theorem on the asymptotic stability of the zero equilibrium position. For the sake of mathematical rigor, in this case one would have to show that for any \(\varepsilon>0\) one can indicate a positive number \(\delta\) such that from \(x_0^2+y_0^2+z_0^2<\delta^2\) it would follow, for \(t>0\), that \(x^2(t)+y^2(t)+z^2(t)<\varepsilon^2\), where \(x(t), y(t), z(t)\) is the solution of system (2.2) determined by the initial conditions \(x(0)=x_0,\ y(0)=y_0,\ z(0)=z_0\). However, we omit the proof of this fact, noting only that this proof could have been carried out in exactly the same way as was done in the paper [7].

References

1. S. V. Emel’yanov, V. A. Taran. Izvestiya AN SSSR. OTN. Energetika i avtomatika, No. 3, 1962, pp. 183–188.

2. E. A. Barbashin, V. A. Tabueva, R. M. Eydinov. Avtomatika i telemekhanika, 24, No. 7, 1963, pp. 882–889.

3. V. M. Badsov, E. A. Barbashin. Izvestiya AN SSSR. Tekhnicheskaya kibernetika, No. 2, 1964, pp. 121–128.

4. E. A. Barbashin, V. A. Tabueva. Avtomatika i telemekhanika, 23, No. 10, 1962, pp. 1290–1297.

5. E. A. Barbashin, V. A. Tabueva. Avtomatika i telemekhanika, 24, No. 5, 1963, pp. 608–614.

6. E. A. Barbashin, N. N. Krasovskii. DAN SSSR, 86, No. 3, 1952.

7. E. A. Barbashin, V. A. Tabueva. PMM, 28, issue 3, 1964, pp. 523–528.

8. E. I. Gerashchenko. On the degree of stability of nonlinear systems in a sliding regime. Izvestiya AN SSSR. Tekhnicheskaya kibernetika, No. 2, 1964, pp. 114–120.

Received by the editors
September 5, 1964

Sverdlovsk Branch
of the V. A. Steklov Mathematical Institute
of the Academy of Sciences of the USSR