SYNTHESIS OF TIME-OPTIMAL CONTROL FOR A DOUBLE-INTEGRATOR-TYPE PLANT UNDER CONDITIONS OF INCOMPLETE OBSERVABILITY

V. D. TSYDENOV, A. M. MALYSHENKO

(Presented by the scientific-technical seminar of the Department of Automation and Telemechanics)

Below we solve the problem of finding a time-optimal control for a plant of the double-integrator type, described by the equation

\[ I\ddot{\alpha}(t)=M_y(t)-M_b(t) \tag{1} \]

under the following conditions:

1. The plant is incompletely observable.
   It is possible to obtain information about the coordinate \(\alpha(t)\) only indirectly, through the coordinate \(g(t)\), related to it by the equation

\[ T_1\dot{g}(t)+g(t)=k\alpha(t). \tag{2} \]

1. The actuator consists of a nonlinear element (NE) and an inertial element, whose equation has the form

\[ T_2\dot{M}_y(t)+M_y(t)=F(u). \tag{3} \]

1. There is a constraint on the control

\[ |F(u)|\leq 1. \tag{4} \]

In equations (1), (2), and (3), \(I\), \(k\), \(T_1\), and \(T_2\) are constants; \(F(u)\) is the regulating action; \(M_b(t)\) is an external action.

It is required to find a time-optimal control satisfying constraint (4) and transferring the system (Fig. 1) from an arbitrary initial state to the terminal state \((\alpha=0,\ \dot{\alpha}=0)\) in minimum time. To solve the posed control problem, we represent the linear part of the automatic-control system in the form

\[ \dddot{g}(t)+\frac{T_1+T_2}{T_1T_2}\ddot{g}(t)+\frac{1}{T_1T_2}\dot{g}(t) = \frac{k}{IT_1T_2}\,[F(u)-Q(t)], \tag{5} \]

where

\[ Q(t)=T_2\dot{M}_b(t)+M_b(t). \]

\(Q(t)\) is assumed to be an admissible action [3]. It turned out that, for the problem described by the fourth-order differential equation (5), it is not possible to obtain the equation of the switching hypersurface \(\{V_1\}\) [1] in explicit form. Therefore we proceed as follows.

Let us define the differential equation of the system (Fig. 1) with respect to \(\alpha(t)\), determining the latter from equation (2). Then we have

\[ \dddot{x}(t)+\frac{1}{T_2}\ddot{x}(t)+\frac{1}{IT_2}[F(u)-Q(t)]. \tag{6} \]

To clarify the qualitative aspect of the problem posed, set

\[ Q(t)\equiv 0. \]

Fig. 1. Structural diagram of the automatic control system: NE—nonlinear element; CD—control device

Taking the latter into account, we have

\[ \dddot{\alpha}(t)+\frac{1}{T_2}\ddot{\alpha}(t)=\frac{1}{IT_2}F(u), \tag{7} \]

where

\[ \alpha(t)=\frac{T_1\dot{g}(t)+g(t)}{\kappa}. \]

Thus, information about \(\alpha(t)\) is obtained through the coordinate \(g(t)\), which is measured directly.

Introduce the variables \(y_1(t)\), \(y_2(t)\), and \(y_3(t)\), putting

\[ y_1(t)=\alpha(t);\quad y_2(t)=\dot{\alpha}(t);\quad y_3(t)=\ddot{\alpha}(t). \tag{8} \]

These variables satisfy the vector differential equation

\[ \begin{bmatrix} \dot{y}_1(t)\\ \dot{y}_2(t)\\ \dot{y}_3(t) \end{bmatrix} = \begin{bmatrix} 0&1&0\\ 0&0&1\\ 0&0&-\dfrac{1}{T_2} \end{bmatrix} \begin{bmatrix} y_1(t)\\ y_2(t)\\ y_3(t) \end{bmatrix} + \begin{bmatrix} 0\\ 0\\ \dfrac{1}{IT_2}F(u) \end{bmatrix}, \]

which can be written more compactly in the form

\[ Y(t)=AY(t)+\lambda(t). \tag{9} \]

The eigenvalues of the \(3\times 3\) matrix \(A\) are

\[ 0,\ 0,\ -\frac{1}{T_2}. \]

Since two of these eigenvalues are identical, by means of a similarity transformation this matrix can be written in Jordan canonical form [1, 2]. Denoting the Jordan canonical form of \(A\) by \(G(A)\), we write

\[ G(A)= \begin{bmatrix} 0 & 1 & 0\\ 0 & 0 & 0\\ 0 & 0 & -\dfrac{1}{T_2} \end{bmatrix}. \tag{10} \]

Since there exists a nonsingular matrix

\[ P= \begin{bmatrix} 1 & 0 & T_2^2\\ 0 & 1 & -T_2\\ 0 & 0 & 1 \end{bmatrix} \tag{11} \]

and its inverse \(P^{-1}\), the relation

\[ G(A)=P^{-1}AP \tag{12} \]

holds. Define the column vector \(Z(t)\) by setting

\[ Z(t)=P^{-1}Y(t). \tag{13} \]

\(Z(t)\) satisfies the differential equation

\[ \dot Z(t)=G(A)Z(t)+P^{-1}\Lambda(t), \]

i.e., the system of equations

\[ \dot Z_1(t)=Z_2(t)-\frac{T_2}{I}F(u); \]

\[ \dot Z_2(t)=\frac{1}{I}F(u); \tag{14} \]

\[ \dot Z_3(t)=-\frac{1}{T^2}Z_3(t)+\frac{1}{IT_2}F(u). \]

To simplify the expressions, as phase coordinates \(x_1(t)\), \(x_2(t)\), and \(x_3(t)\) we take

\[ x_1(t)=\frac{I}{T_2^2}Z_1(t); \]

\[ x_2(t)=\frac{I}{T_2}Z_2(t); \tag{15} \]

\[ x_3(t)=IZ_3(t). \]

These variables satisfy the system of differential equations

\[ \dot x_1(t)=\frac{1}{T_2}x_2(t)-\frac{1}{T_2}F(u); \]

\[ \dot x_2(t)=\frac{1}{T_2}F(u); \tag{16} \]

\[ \dot x_3(t)=-\frac{1}{T_2}x_3(t)+\frac{1}{T_2}F(u). \]

Let us note that system (16) is normal and the optimal control exists and is unique [1].

Applying L. S. Pontryagin’s maximum principle [4] to the system of equations (16), we conclude that the time-optimal control is relay-type and has the form

\[ F(u)=\operatorname{sign}\left[-\Psi_1(0)+\Psi_2(0)-\frac{1}{T_2}\Psi_1(0)\cdot t+\Psi_3(0)\cdot e^{-\frac{t}{T_2}}\right]= \]

\[ =\operatorname{sign}(-\Psi_1+\Psi_2+\Psi_3). \tag{17} \]

Hence we conclude that the function \((-\psi_1+\psi_2+\psi_3)\) can have no more than two zeros, regardless of the values \(\psi_1(0)\), \(\psi_2(0)\), \(\psi_3(0)\), which are the initial values of the quantities \(\psi_i(t)\) \((i=1,2,3)\). Consequently, six control sequences may be optimal:

\[ \{+1\};\quad \{-1\};\quad \{+1,-1\};\quad \{-1,+1\};\quad \{+1,-1,+1\};\quad \{-1,+1,-1\}. \]

Next let us solve equation (16), assuming \(F(u)=\Delta=\pm 1\). Then, eliminating time, we obtain the following equations:

\[ \begin{gathered} x_1=x_1(0)-[x_2-x_2(0)]+\frac{1}{2}\Delta\bigl(x_2^2-x_2^2(0)\bigr),\\[6pt] x_3=\bigl[x_3(0)-\Delta \exp\{-\Delta[x_2-x_2(0)]\}\bigr]+\Delta. \end{gathered} \tag{18} \]

These two equations determine a trajectory in the three-dimensional state space \((x_1,x_2,x_3)\), issuing from the initial point \((x_1(0),x_2(0),x_3(0))\) under the action of the control

\[ F(u)=\Delta. \]

Note that the trajectories do not depend on the magnitude \(T_2\). This circumstance is the reason for choosing \(x_i\) as phase coordinates. Figure 2 shows the curves corresponding to the first equation of (18); solid curves correspond to \(\Delta=+1\), dashed curves to \(\Delta=-1\). Figure 3 shows the curves corresponding to the second equation of (18).

Fig. 2. Projections of phase trajectories; solid curves—for control equal to \(+1\); dashed curves—for control equal to \(-1\)

Fig. 3

Now denote by \(\{V_2\}\) the set of states that are transferred to the origin by means of the control

\[ F(u)=\Delta^*. \]

A set \((x_1,x_2,x_3)\) belonging to \(\{V_2\}\) will be denoted by \(x_i\), \(i=1,2,3\). The set \(\{V_2\}\) may be regarded as the sum of two sets \(\{V_2^+\}\) and \(\{V_2^-\}\). \(\{V_2^+\}\) is the set of states that can be transferred to the origin by the control sequence \(\{+1\}\); analogously, \(\{V_2^-\}\) by the control sequence \(\{-1\}\). The set \(\{V_2^+\}\) is defined as

\[ \{V_2^+\}=\left\{(x_{12},x_{22},x_{32}):\ x_{12}=\frac{1}{2}x_{22}^{2}-x_{22},\ x_{32}=1-\exp(-x_{22}),\right. \tag{19} \]

\[ \left. x_{22}<0\right\}. \]

And the set \(\{V_2^{-}\}\) is defined as

\[ \{V_2^{-}\}=\{(x_{12},x_{22},x_{32}): x_{12}=-\frac{1}{2}x_{22}^{2}-x_{22},\ x_{32}=-1+ \]
\[ +\exp(x_{22}),\ x_{22}>0\}. \tag{20} \]

Thus, we arrive at the conclusion that

\[ \Delta^*=-\operatorname{sign}\{x_{22}\}. \tag{21} \]

So that \(\{V_2\}\) is defined as

\[ \{V_2\}=\{(x_{12},x_{22},x_{32}): x_{12}=\frac{1}{2}\Delta^*x_{22}^{2}-x_{22},\ x_{32}=\Delta^*-\Delta^*\exp(-\Delta^*x_{22})\}. \tag{22} \]

From the preceding definitions it follows that \(\{V_2\}\) is a curve in phase space such that any state \((x_1,x_2,x_3)\in\{V_2\}\) can be brought to the origin by means of the control

\[ F(u)=\Delta^*=-\operatorname{sign}\{x_{22}\}. \tag{23} \]

The projections of \(\{V_2\}\) onto the planes \(x_1x_2\) and \(x_2x_3\) are shown in Figs. 4 and 5, respectively.

Fig. 4

Fig. 5

Denote by \(\{V_1\}\) the set of states that can be brought to the origin by the control sequence \((-\Delta^*,\Delta^*)\). We denote a state belonging to \(\{V_1\}\) by \(x_{i1}\), \(i=1,2,3\). The state \((x_{11},x_{21},x_{31})\) satisfies the equations

\[ x_{12}=x_{11}-(x_{22}-x_{21})-\frac{1}{2}\Delta^*(x_{22}^{2}-x_{21}^{2}), \tag{24} \]

\[ x_{32}=(x_{31}+\Delta^*)\exp\{\Delta^*(x_{22}-x_{21})\}-\Delta^*, \tag{25} \]

where

\[ (x_{12},x_{22},x_{32})\in\{V_2\}. \]

From (22) and (24) it follows that

\[ x_{22}=-\Delta^*\sqrt{\frac{x_{21}^{2}}{2}+\Delta^*(x_{11}+x_{21})}. \tag{26} \]

From (22) and (25) we find that

\[ x_{31}=-\Delta^*+\Delta^*\exp\{-\Delta^*(x_{22}-x_{21})\}[2-\exp\{-\Delta^*x_{22}\}]. \]

Substituting (26) into the last expression, we obtain the relation

\[ x_{31}=-\Delta^*+\Delta^*\exp\left\{\Delta^*x_{21}+\sqrt{\frac{x_{21}^{2}}{2}+\Delta^*(x_{11}+x_{21})}\right\}\times \]
\[ \times\left[2-\exp\left\{\sqrt{\frac{x_{21}^{2}}{2}+\Delta^*(x_{11}+x_{21})}\right\}\right]. \tag{27} \]

It remains to determine \(\Delta^*\) as a function of \(x_{11}, x_{21}, x_{31}\). From Fig. 6 it follows that

\[ \Delta^*=\operatorname{sign}\left\{x_{11}+\frac{1}{2}\operatorname{sign}(x_{21})x_{21}^{2}+x_{21}\right\}. \tag{28} \]

Thus, we have found an expression for \(\Delta^*\) in terms of \(x_{11}\) and \(x_{21}\).

Since the variables \(x_{11}, x_{21}\), and \(x_{31}\) must be real quantities, using (28) we obtain the condition

\[ \frac{x_{21}^{2}}{2}+\operatorname{sign}\left\{x_{11}+\frac{1}{2}\operatorname{sign}(x_{21})x_{21}^{2}+x_{21}\right\}(x_{11}+x_{31})\geqslant 0. \tag{29} \]

We know that \(\{V_1\}\) divides the phase space into two parts. We shall use this property in developing a control strategy for time-optimal systems.

By the method of exclusion one can show that if \((x_1,x_2,x_3)\in\{V_2\}\), then \(F(u)=\Delta^*\) is the optimal control. Note that on the line \(\{V_2\}\), \(\Delta^*\) is determined by the equation

\[ \Delta^*=-\operatorname{sign}\{x_{22}\}. \]

Consider the set of states that can be transferred to the origin by the control sequence \(\{-1,+1\}\). Obviously, switching from the control \(F(u)=-1\) to \(F(u)=+1\) must occur on the set \(\{V_2^+\}\). If the state \((x_1,x_2,x_3)\) can be transferred to the origin by the control sequence \(\{-1,+1\}\), then there exists a state \((x_{12},x_{22},x_{32})\in\{V_2^+\}\) such that \((x_1,x_2,x_3)\) lies on the trajectory obtained under the action of the control \(F(u)=-1\) and ending at the point \((x_{12},x_{22},x_{32})\in\{V_2^+\}\). Next consider the state \(X=(x_1,x_2,x_3)\) on the surface \(\{V_1\}\). For definiteness, suppose that for this particular state \(\Delta^*=+1\). Then the optimal control sequence will be \(\{-1,+1\}\). Indeed, the control sequence \(\{-1,+1\}\) transfers \(X\) to the origin by definition.

Fig. 6

Further, by the method of exclusion one can show that \(\{-1,+1\}\) is the optimal control sequence if the switching from \(F(u)=-1\) to \(F(u)=+1\) occurs on the curve \(\{V_2\}\).

Thus, if \((x_1,x_2,x_3)\in\{V_2\}\), then \(F(u)=-\Delta^*\) is the time-optimal control. Further, if the state \((x_1,x_2,x_3)\) is located above \(\{V_1\}\), then \(F(u)=-1\) is the optimal control.

If \((x_1,x_2,x_3)\) is located below \(\{V_1\}\), then the optimal control is \(F(u)=+1\). The last two assertions are also proved by the method of exclusion.

Choose an arbitrary state \((x_1,x_2,x_3)\) and draw through this point a straight line parallel to \(x_3\). This line intersects the surface \(\{V_1\}\) at the point \((x_{11},x_{21},x_{31})\), where \(x_{11}=x_1,\ x_{21}=x_2\). The coordinate \(x_{31}\) is determined by equations (27) and (28).

If \(x_3 - x_{31} > 0\), we shall say that the point \((x_1, x_2, x_3)\) is located above the surface \(\{V_1\}\), and if \(x_3 - x_{31} < 0\), we shall say that the point \((x_1, x_2, x_3)\) is located below \(\{V_1\}\). If \(x_3 - x_{31} = 0\), then the point \((x_1, x_2, x_3)\) belongs to \(\{V_1\}\).

Thus,

\[ m(t)=x_3-x_{31} \tag{30} \]

controls the relay.

The time-optimal control sequence for each of the states of the phase space is determined by the method of elimination and, in accordance with what was set forth above, is unique.

REFERENCES

1. M. Athans, P. Falb. Optimal Control. Moscow, Mashgiz, 1968.
2. F. R. Gantmacher. The Theory of Matrices. Moscow, Fizmatgiz, 1967.
3. A. A. Pavlov. Synthesis of Relay Systems Optimal in Speed. Moscow, Fizmatgiz, 1966.
4. L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, E. F. Mishchenko. The Mathematical Theory of Optimal Processes. Moscow, Fizmatgiz, 1969.