# Time Sub-Optimal Control of Triple Integrator Applied to Real Three-Tank Hydraulic System

## Abstract

Control of nonlinear systems with input constraints is an interesting topic of control theory that must be solve adequately because of the presence of constraints in almost each real system. There are many authors and several techniques trying to solve this problem (anti-windup structures, positive invariant sets, variable structure systems, global, semi-global and local stabilization of systems with constraints, optimization problems solved by linear matrix inequalities).

This paper shows a different approach that originates in the time optimal control and is improved by decreasing the sensitivity to uncertain model parameters that is balanced by sub-optimality. The design of constrained controller is carried out on the triple integrator system and this is later applied to the nonlinear three-tank hydraulic system after its exact linearization.

### Keywords

Input constraints Time sub-optimal control Nonlinear systems Exact linearization Computer algebra system## 1 Introduction

The optimal control belongs to the basic study field of the control engineering. Its serious development started in 50-ties of the previous century and in 60-ties it was further worked out by many famous scientists (see [1] and references therein). The aim of the optimal control is to minimize or maximize given criteria under existing constraints. For instance, the time optimal control minimizes the time necessary to reach a desired state when the control value or the states values are limited. Using Pontryagin’s Maximum Principle [8] many optimality problems have been theoretically solved. The disadvantage of the optimal control is given by its high sensitivity to model uncertainties, parametric variations, disturbances and noise always presented in real systems. This was also the reason why later the optimal control has been suppressed by pole assignment control.

The idea of the pole assignment control consists in the introduction of a desired dynamics into control circuits. The choice of poles can slow responses and so the sensitivity can be decreased to the level acceptable also in real systems. The problem of the pole assignment control in real applications is determined by the linear character of this method that has been not designed to respect nonlinear elements in a control circuit such as constraints of inputs or states. In order to keep the stability and quality of control the resulting responses are often over-damped.

There exist several methods that are able to cope with input and state constraints. A practical solution is to extend the linear control circuit by an anti-windup structure that will keep the circuit in desired states. Another possibility is to construct positive invariant sets where the states and control are within the specified interval [4]. Also variable structure systems represent the method that is able to respect constraints. Other methods solve global, semi-global and local stabilization of systems with constraints and many optimization problems can be solved by linear matrix inequalities [2, 5].

This paper combines the qualities of the time optimal control with the decreased sensitivity of the “slow” pole assignment control that results in the design of a time sub-optimal controller that is fast enough but it respects given constraints. Similar design methods have been developed for lower order systems in [6]. In [9] the sub-optimal controller has been applied to the triple integrator and in [3] a simplified version of it has been applied to the simulated hydraulic system. The main contribution of this paper consists in application of the previously designed sub-optimal controller to the real system that shows the sophisticated theory can be successfully applied in practice.

The paper is organized in six chapters. After introduction the problem is stated in the second chapter. The third chapter offers analysis in the phase space after applying nonlinear decomposition. The fourth chapter shows possible explicit solutions for the time sub-optimal control algorithm. Its application to the three-level hydraulic system can be found in the fifth chapter and the paper is finished by short conclusions.

## 2 Problem Statement

*u*is constrained

*u*(3). It is required that these changes will correspond to an exponential behavior that could be expressed by the exponential decrease of the distance between the current state \(\mathbf {x}\) and a corresponding part of a switching plane in the phase space. If the distance will be expressed by a scalar function \(\rho _i: \mathbf {R}^3 \rightarrow \text {R}\), the condition of the exponential decrease can be mathematically formulated by the differential equation

After evaluating the sub-optimal controller for the triple integrator it is our goal to apply it for the real hydraulic system. In order to use the control law originally derived for the linear system (triple integrator) a linearization technique must be applied first. This will influence also control limits \(U_1\), \(U_2\) that should be transformed. Exact linearization method uses Lie algebra formalism to convert a nonlinear system to a linear one [7].

## 3 Nonlinear Decomposition

*u*.

*PB*) (Fig. 1). If we were able to localize the actual state within

*PB*the control action

*u*could be easily calculated on the proportional base. This has been used in [9] where the localization has been made with the help of the computer algebra system Maple.

## 4 Time Sub-Optimal Solution

In order to get an analytical solution it is necessary to accept some simplifications in the nonlinear decomposition. For this reason we will change the third subsystem of the decomposition (5) to be \(\mathbf {x}_3= (1 \,\, 0 \,\, 0)^t q_3\). This will enable a projection along the axis *x* and a reduction of the order of equations by one.

*PB*reduces to a two dimensional object in the phase space called reference braking surface

*RBS*. According to values of the parameters \(q_1\), \(q_2\), \(t_1\) and \(t_2\), the

*RBS*consists of several regions depicted in the Fig. 2. If the region of

*RBS*corresponding to the actual state is known the control law

*u*can be evaluated by application of (4) to the distance between the actual state and the

*RBS*region. This procedure has been used in [3] where the

*RBS*has been projected along the axis

*x*into the plane

*y*,

*z*. Then the actual state could be localized into the corresponding

*RBS*region by solving quadratic equations. This has enabled to solve the designed time sub-optimal algorithm analytically.

*RBS*denoted by

*TQ*. Consider the parameters of

*RBS*are \(q_1=U_j\), \(q_2 \in [0\,,\, U_{3-j}-U_j]\), \(t_1>0\) and \(t_2=0\). The region of

*RBS*corresponding to these parameters is depicted in the Fig. 2 by green color. After substitution of these parameters into (8) we can determine the unknown parameters \(q_2\) and \(t_1\) from the subsystem of (8) for coordinates

*y*and

*z*by solving a system with one quadratic equation and one linear equation

*x*the equation representing the region

*TQ*of

*RBS*in the form \(x=f(y,z);\, f(y,z) : \mathbf {R}^2 \rightarrow \text {R}\). Then the distance between the actual state (with the coordinate

*x*) and the corresponding segment of

*RBS*measured along the axis

*x*can be expressed as \(\rho _3=x-f(y,z)\). Finally, the resulting control value

*u*can be evaluated from (4) when \(i=3\).

A similar procedure is possible to derive for each segment of *RBS*. The localization of the actual state \(\mathbf {x}\) to the corresponding region of *RBS* is performed in the \(y,z\,\)-plane where for the triple integrator only the equations of second order are necessary to be solved. The complete description of the time sub-optimal controller design can be found in [3].

## 5 Application to the Real Three-Tank Hydraulic System

### 5.1 Model of the Hydraulic System

### 5.2 Exact Linearization Method

*y*along a vector field \(\mathbf {f}\) by \(L_fy\) the desired exact linearization feedback can be expressed as

### 5.3 Control of the Real System

The real three-tank system has been controlled using the designed time sub-optimal controller and the exact linearization method. The desired height of the third level has been \(0.07\,m\) and the controller parameters have been \(\alpha _1=-0.1\), \(\alpha _2=-0.15\), \(\alpha _3=-0.1\). The output and control responses are shown in the Fig. 4. From the control response one can notice that it does not include the third pulse as it could be expected according to the time optimal control theory. It has been caused by non-modeled dynamics and resulting over-damped sub-optimal control law. Moreover, there is a small overshoot. In practice, to avoid model mismatch, noisy measurements and disturbances it is necessary to extend the whole control structure by the nonlinear disturbance observer that will also influence the transformation of the original control limits.

The designed controller is focused to improve the dynamics of time responses and this has been successfully carried out. Even derived for the triple integrator the sub-optimal controller works well also for nonlinear systems. It respects the constraints of the control signal as it can be seen from the control responses and switching to the opposite control limit is smooth.

## 6 Conclusions

This paper summarizes the design of the time sub-optimal controller based on the nonlinear decomposition and its application to the real hydraulic system. Although the resulting expressions for the control law are sophisticated using todays computer technology it has been no problem to implement the developed control strategies to the real system. Here the simplified nonlinear decomposition has played an important role as it allowed to derive the analytical solution. Using the exact linearization method has been also significant. Due to it the time sub-optimal controller originally designed for the triple integrator system could have been used also for the nonlinear hydraulic system.

## Notes

### Acknowledgments

This work has been partially supported by the grants VEGA 1/0937/14 and APVV-0343-12.

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