Synthesis of a robust stochastic regulator for the model of electrohydraulic brake system

1.1. Technical description of a control object

According to the technical description of a SEHB object [1]-[3] (Fig. 1), it includes a friction-type brake using the friction power as an energy source for generating a new braking power by transferring the electrohydraulic energy. A dual-action hydraulic cylinder, referred to as a base cylinder here, connects the break support with the railroad car structure, fixing the hydrocylinder piston between two columns of the brake liquid. In the case of braking, the friction force acts on the base cylinder piston, causing a pressure change in its cavities. The liquid pressure in the high hydraulic level does not practically change due to an expansion tank used in the system, whose volume is much larger than the total volume of the hydraulic circuit line and the base cylinder cavity; a spool valve connects the hydraulic lines with the hydraulic motor cavities. In the case where the direct connection of hydraulic lines is active (increasing braking power), the liquid is pumped into the piston cavity of the hydraulic motor, while its rod end is connected to the low-pressure line.

Fig. 1Schematic representation of a SEHB

Note that the selected ratio between the total and the annulus area of the differential cylinder piston gives rise to a self-excited process of the brake power increase. If the reverse connection of hydraulic lines is active (decreasing braking power), then the piston cavity is connected with the low-pressure line and the liquid from the supply pressure line is delivered to the rod end.

A detailed description of the features of the functioning of the considered SEHB control object is considered in publications [1]-[3], here we will be interested only in the formal description of the behavior of its mathematical model in the state space.

1.2. Mathematical apparatus of the synergetic approach for the design of nonlinear regulators

The ADAR method, implemented in what follows, includes the physical aspects of the controlled object via special invariants whose description is a mathematical manifold with an attractive property; the synthesized regulator is robust and satisfies the least action principle.

Let us formulate the following requirements [4]-[7] to the initial description and the necessary a priori information for applying the procedure of synthesizing an ADAR-control:

а) initial mathematical description of the control object is given by a system of ordinary differential equations or a system of difference equations for continuous and discrete control problems, respectively;

b) availability of a globally stable target system for initial model, which satisfies the target analytically set invariant $\mathbf{\psi }\left(\mathbf{x}\left(t\right)\right)=0$, $t\to \infty$, where $\mathbf{\psi }\left(\mathbf{x}\right)\in R^m$, is the continuously differentiated function, $\mathbf{x}\in R^n$, $n\ge m$;

c) boundedness of all solutions of the initial system;

d) stabilizability of all object’s trajectories in the neighborhood of $\mathbf{\psi }\left(\mathbf{x}\right)=0$;

e) implementation of the control system synthesis in the state space in the absence of any restrictions on the controls linearly included into the right-hand parts of the object’s descriptions.

f) ADAR-control ${\mathbf{u}}^A\left(\mathbf{x}\left(t\right)\right)\in R^m$ provides a solution to variational problems ${\mathrm{\Phi }}_C\left(\mathbf{\psi }\right)$ or ${\mathrm{\Phi }}_D\left(\mathbf{\psi }\right)$ for continuous and discrete description of the object, respectively:

where the parameters $T_i$, ${\omega }_i$, $i=\overline{1,m}$ are the settings of the regulators.

3.1. Basic provisions of the NAD-algorithm

It is well known that the use of the ADAR-synthesis guarantees robust properties to the designed regulator, ensuring a suppression of the external and parametric disturbances. Given this, the disturbances can be modeled as piecewise-constant functions in the differential equations of the object, which include controls, and the system of equation is supplemented with the integrator equations [7], [9].

In this case, the algorithm for synthesizing the required control stabilizing the target variable ${\psi }^{*}\left(t\right)=0$, $t\to \infty$ will be implemented by the following steps.

1. Phase space extension or transformation of the initial description $S_{NAD}:=S$ into a closed system, for which a classical ADAR-synthesis is already possible. We obtain a system of the form

where $\eta$ is the designed regulator parameter.

2. Euler-Lagrange equation $T_1{\dot{\psi }}_1\left(t\right)+{\psi }_1\left(t\right)=0$ for the functional ${\mathrm{\Phi }}_1\left({\psi }_1\right)$ only, where ${\psi }_1=x_3-{\phi }_1\left(x_2,x_4,z\right)$ would ensure the following form of the regulator structure:

3. Decomposition of (2) on the set of states $x_3={\phi }_1\left(x_2,x_4,z\right)$, $t\to \infty$ would yield a system of equations:

4. Euler-Lagrange equation $T_2{\dot{\psi }}_2\left(t\right)+{\psi }_2\left(t\right)=0$ for the functional ${\mathrm{\Phi }}_2\left({\psi }_2\right)$ only, where ${\psi }_2=x_4-{\phi }_2\left(x_2,z\right)=0$, $t\to \infty$ would ensure the following form of the regulator ${\phi }_1\left(x_2,x_4,z\right)$:

5. Decomposition of Eq. (4) on the set of states $x_4={\phi }_2\left(x_2,z\right)=0$, $t\to \infty$ would yield a system of equations:

6. Solution of the variational problem ${\mathrm{\Phi }}_3\left({\psi }_3\right)$ with a target invariant of a special form ${\psi }_3={\psi }^{*}+\kappa z=x_2-x_2^{*}+\kappa z=0$, $t\to \infty$, $\kappa >0$ ($\kappa$ – one more system parameter) is provided by the Euler-Lagrange equation $T_3{\dot{\psi }}_3\left(t\right)+{\psi }_3\left(t\right)=0$, from which, including Eqs. (6), find an expression for variable ${\phi }_2\left(x_2,z\right)$:

Having determined the partial derivatives from (5), (7) and combining together Eq. (3), we obtain the final system with the regulator $u$ (see Appendix A2).

Statement 1. Control $u=u_{NAD}$, if any, ensures an asymptotic stability for the controlled object Eq. (1) in the neighborhood of ${\psi }^{*}\left(t\right)=0$, $t\to \infty .$

Proof of Statement 1 is constructive and immediate from the NAD-algorithm of control system synthesis for Eq. (1) (see, e.g., [7]).

4.1. Problem statement of stochastic discrete NAS-control

Represent the description of object Eq. (1) in accordance with the Euler scheme given by

where $\xi \left(k\right)$ is the random bounded function and parameter $\delta$ is the discretization value of the continuous description Eqs. (1).

As above, we will denote the original description as $S_{NAS}=\left(\mathbf{x};u+\tilde{\xi }\right)$, $\mathbf{x}={\left(x_1,x_2,x_3,x_4\right)}^T$, $\tilde{\xi }\left(k\right):=\xi \left(k+1\right)+c\xi \left(k\right)$, $k=\mathrm{0,1},2...$

The problem of stochastic control will be formulated as follows: it is required to design a control system for a stochastic object Eqs. (8), providing:

1) an asymptotically stable on average achievement of the target $\mathbf{E}\left\{{\psi }^{*}\left(k\right)\right\}=\mathbf{E}\left\{x_2\left(k\right)-x_2^{*}\right\}=0$, $k\to \infty$, where $\mathbf{E}\left\{Y\right\}$ is the sign of a mathematical expectation of a random value of $Y$;

2) a minimum to the quality functional $\mathbf{E}\left\{\mathrm{\Phi }\left({\psi }^{*}\right)\right\}$;

3) a minimality of dispersion of the output macrovariable $\mathbf{D}\left\{{\psi }^{*}\right\}\to \mathrm{m}\mathrm{i}\mathrm{n}$ and of all intermediate macrovariables appearing in the implementation of the ADAR-synthesis hierarchical subproblems (subgoals) of the control design).

As far as random noise in the description Eqs. (8) is concerned, assume that ${\left\{\xi \left(k\right)\right\}}_{k\ge 0}$ is a sequence of independent evenly distributed quantities with the properties of $\mathrm{{\rm E}}\left\{\xi \left(k\right)\right\}=0$, $\mathbf{D}\left\{\xi \left(k\right)\right\}={\sigma }^2$; $0<c<1$ is a certain constant interpreted as a noise damping coefficient; $k=\mathrm{0,1},...$.

Restrictions on the choice of command variables are as follows [10], [11]:

1) control strategies are selected from the class of discrete ADAR-controls;

2) those strategies, for which the value of the control variable $u\left(n\right)$ is a function of the previous states and controls, are taken into consideration: $u\left(k\right)=u\left(\mathbf{x}\left(k\right),\mathbf{x}\left(k-1\right),...,\mathbf{x}\left(0\right);u\left(k-1\right),u\left(k-2\right),...,u\left(0\right)\right)$, $k=\mathrm{1,2},...$; $u\left(0\right)=0.$

The peculiarities of the discrete analog of the ADAR-synthesis are accounted for by the following [5] proposition.

Statement 2. The minimum of the average value of functional $J\left(\psi \right)=\sum _{k=0}^{\infty }\left({\alpha }^2{\left(\psi \left(k\right)\right)}^2+{\left(\mathrm{\Delta }\psi \left(k\right)\right)}^2\right)$ is achieved by solving the discrete analog of the Euler-Lagrange equation $\psi \left(k+1\right)+\omega \psi \left(k\right)=0$, $\left|\omega \right|<1$, for which case the relationship between the parameters $\alpha$, $\omega$ is expressed via the formula $\omega =0.5\left(2+{\alpha }^2-\sqrt{{\left(2+{\alpha }^2\right)}^2-4}\right)$ [5].

4.2. Basic provisions of the NAS-algorithm

The algorithm for designing a stochastic control on a target manifold is represented by the following actions.

Step 1. Derive the control structure $\tilde{u}\left(k\right)$, $k=\mathrm{0,1},2...$ relying on the classical discrete variant of the ADAR method at fixed random noise.

Step 2. Take the operation of conditional mathematical expectation $u\left(k\right)=\mathbf{E}\left\{\left.\tilde{u}\left(k\right)\right|{\mathbf{\xi }}^k\right\}$, where ${\mathbf{\xi }}^k=\left(\xi \left(0\right),\xi \left(1\right),..,\xi \left(k\right)\right)$.

Step 3. Decompose description Eq. (6) taking into account the discrete equations. Substitute the found control $u\left(k\right)$ into the expression ${\psi }_1\left(k+1\right)+{\omega }_1{\psi }_1\left(k\right)=0$. Find the dependence $\xi \left(k\right)=\varphi \left(\mathbf{x}\left(k\right),\psi \left(k\right)\right)$ as a function of the observations for the sake of exclusion of variable $\xi \left(k\right)$ from expression $u\left(k\right)=\mathbf{E}\left\{\left.\tilde{u}\left(k\right)\right|{\mathbf{\xi }}^k\right\}$. The synthesis of a discrete stochastic regulator is completed.

Statement 3. Control $u=u_{NAS}$, if any, ensures an asymptotic stability for the object of control Eq. (6) on average in the neighborhood of $\mathbf{E}\left\{\psi \left(k\right)\right\}=0$, $k\to \infty$ and a minimal dispersion of the output macrovariable $\psi \left(k\right)$.

Let us apply the NAS-algorithm to find the stochastic regulator for Eqs. (8).

1. Derivation of a control system structure. We fix the random functions $\xi \left(k\right)$, $k\in \left\{\mathrm{0,1},...\right\}$ and perform a discrete ADAR-synthesis [4] for the control object Eqs. (8) with control quality functional $J\left({\psi }_1\right)$, ${\psi }_1=x_3-{\phi }_1\left(x_2,x_4\right)$, denoting the control of this step as $\tilde{u}$. Enumerate the steps of the discrete ADAR-synthesis for object Eq. (8).

A subscript $«r»$ in Eq. (9) indicates a substitution of the right-hand sides in Eq. (9) for $x_i\left(k+1\right)$, $i=\mathrm{2,4}$; ${\omega }_1=const$, $\left|{\omega }_1\right|<1$.

Then decompose system Eq. (8) on manifold ${\psi }_1=0$ or $x_3={\phi }_1\left(x_2,x_4\right)$:

Afterwards, for the control object (10) with the control quality functional $J_2\left({\psi }_2\right)$, ${\psi }_2\left(k\right)=x_4\left(k\right)-{\phi }_2\left(x_2\left(k\right)\right)=0$, $k\to \infty$ from the Euler-Lagrange equation obtain:

Decomposition of Eq. (10) on the manifold $x_4={\phi }_2\left(x_2\right)=0$, $k\to \infty$ would yield a system of equations:

Perform a discrete ADAR-synthesis for the control object Eq. (12) with control quality functional $J_3\left({\psi }_3\right)$, ${\psi }_3\left(k\right)={\psi }^{*}\left(k\right)=0$, $k\to \infty$:

The outcome of the first step is the control structure (at fixed noise) as a combination of Eqs. (9), (11) and (13).

2. A control for the object under the conditions of unknown (restricted) noise is sought for in the form of a conditional mathematical expectation $u\left(k\right)=\mathbf{E}\left\{\left.\tilde{u}\left(k\right)\right|{\mathbf{\xi }}^k\right\}$, $k=\mathrm{1,2}...$ considering Eqs. (9) as follows:

3. Determination of an estimate of the random function $\xi \left(k\right)$ as a dependence on observations. Let us use the expression for control Eq. (14) including the initial description Eq. (8) in the left-hand part of the functional expression: ${\psi }_1\left(k\right)+{\omega }_1{\psi }_1\left(k-1\right)=\delta \xi \left(k\right)$. Considering the above relation, obtain control in an explicit form:

The final description of the stochastic control system involves a combination of Eqs. (8), (11), (13) and (15):

Here the values $\delta$, $c$, ${\omega }_1$, ${\omega }_2$, ${\omega }_3$ are the settings of the regulator that provide the required quality of transient processes.

It is important to note that the parameters $\left|{\omega }_i\right|<1$, $i=\mathrm{1,2},3$ are directly proportional to the duration of achieving the target manifold ${\psi }^{*}\left(k\right)=0$, $k\to \infty$ for system Eq. (16).

4.3. Results of numerical simulation of three control algorithms under design and off-design conditions

Numerical modeling (Figs. 2-3) of the designed control systems (see Appendix A2) has been performed in the Matlab/Simulink environment using the following parameter values [3]: $a_1=$15820; $a_2=$ 35319.96; $a_3=$628.3; $a_4=$ 394760.89; $\alpha =$0,609; $p=$500000; $T_1=$10^-3; $T_2=$10^-4; $T_3=$2⋅10^-4; $g=$394760.89; $D_v=$0.7; $\mu =$0.4; $x_2^{*}=$ 0.4, where $D_v$ is the valve damping coefficient; $\mu$ is the friction coefficient; $x_2^{*}$ is the target value of $x_2$.

Fig. 2Behavior of the target coordinate x2t, Pa of the SEHB-objects (1) under different disturbances

a)$\zeta \left(t\right)=1.8\mathrm{s}\mathrm{i}\mathrm{n}\left(t\right)$; $E_{ADAR}=\mathrm{}$3.97×10^-2; $E_{NAD}=$ 1.83×10^-3

b)$\zeta \left(t\right)=$ 30; $E_{ADAR}=$ 1.3×10^-3; $E_{NAD}=$ 9×10^-4

Fig. 2 presents the curves of variation of the target variable according to the ADAR and NAD controls, respectively; they imply that a) the control target is being achieved; b) the transient process exhibits an acceptable quality; c) under conditions of constant disturbances, both algorithms are equally effective. The control energies are designated as $E_{ADAR}$ and $E_{NAD}$, respectively.

Fig. 3Comparison of the quality of transient processes of controllable variable x2t, Pa of the SEHB-object in response to controls uADAR, uNAD and uNAS, respectively

a)$E_{NAD}=$ 1.36×10^-2, $E_{ADAR}=$ 1.39×10^-2, $E_{NAS}=$ 19.3

b) Noise is normal $N\left(0;0.3\right)$

As follows from the above results (Figs. 2-3), the ADAR- and NAD-controls are comparable with respect to the quality of transients and the energy spent under constant disturbances.

If the disturbance s function is harmonic, then the use of the NAD control is clearly preferable. For large values of the “noise-to-signal ratio” indicator, the control efficiency will be in favor of the NAD (or NAS)-control.

4.4. Results of numerical simulation of synergetic control algorithms and previously constructed regulators

It was noted above that one of the earlier constructed regulators for a SEHB object is a PID-regulator and the second regulator is based on the feedback-linearization algorithm [3].

Figs. 4(a), 4(b) and Table 1 suggest promising conclusions in favor of the synergetic approach to the control over such an object, which however, needs more detailed research using an object’s prototype and under the conditions of different practical implementations of the object (motor car, train) [12].

Fig. 4Comparison of the control quality indicators under random noise conditions

a) Transient processes of controllable variable $x_2\left(t\right)$, Pa (×10⁵) of the SEHB-object in response to controls $u_{ADAR}$ and $u_{PID}$, respectively

b) Comparison of the energies spent on control upon reaching the target values in conditions of normal random noises $N\left(0;\sigma \right)$

The data are given for the spool damping parameter $D_V=$ 0.4 and the friction coefficient of the brake pads $\mu =$ 0.35.