International Conference on Computer Aided Systems Theory

Computer Aided Systems Theory – EUROCAST 2015 pp 25-32

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

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9520)

## 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

Consider a linear system of the third order representing the triple integrator
\begin{aligned} \dot{\mathbf {x}}= & {} \mathbf {Ax}+\mathbf {b}u \nonumber \\ y= & {} \mathbf {c}^t \mathbf {x} \end{aligned}
(1)
where
\begin{aligned} \mathbf {A}=\left( \begin{array}{ccc} 0 &{} \,\,1 &{} \,\,0 \\ 0 &{} \,\,0 &{} \,\,1 \\ 0 &{} \,\,0 &{} \,\,0 \end{array} \right) \!,\,\,\, \mathbf {b}=\left( \begin{array}{c} 0 \\ 0 \\ 1 \end{array} \right) \!,\,\,\, \mathbf {c}^t=\left( 1\,\, 0\,\, 0\right) \end{aligned}
(2)
Further take into account that the control signal u is constrained
\begin{aligned} U_ {1} \le u \le U_{2} \end{aligned}
(3)
Then the aim is to design such a time sub-optimal controller that will bring the system from an initial state $$\mathbf {x}=\left( x\,\, y\,\, z\right) ^t$$ to the desired state $$\mathbf {x}_w$$ in the minimum time $$t_{min}$$ under an additional condition that limits the changes between two opposite values $$U_1$$, $$U_2$$ of the control signal 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
\begin{aligned} \frac{\mathrm {d}\rho _i}{\mathrm {d}t}=\alpha _i \rho _i ,\,\, \alpha _i \in \text {R}^-,\,\, i=1,2,3 \end{aligned}
(4)
From this equation the value of the time sub-optimal controller can be calculated. To simplify the problem it is possible to set the desired state $$\mathbf {x}_w$$ to be equal to the origin of the phase space due to a suitable coordinate transformation. Further simplifications will be necessary in order to derive an explicit solution for the control value.

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

The design of the time-suboptimal control is based on a nonlinear dynamics decomposition [3]. This is done in the phase space and it enables to express the state of the system $$\mathbf {x}$$ as a sum of subsystem states $$\mathbf {x}_i, i=1,...,n$$
\begin{aligned} \mathbf {x}=\sum _{i=1}^n \mathbf {x}_i(q_i,t_i),\,\,\,\mathbf {x}_i(q_i,t_i)=e^{-\mathbf {A}t_i}\mathbf {v}_iq_i+\int _0^{-t_i}e^{\mathbf {A}\tau }\mathbf {b}\text {d}\tau q_i \end{aligned}
(5)
where $$\mathbf {A}$$ is the system matrix corresponding to the triple integrator (2), $$\mathbf {b}$$ is the input vector (2), $$\mathbf {v}_i=\left( 1/\alpha _i^n,...,1/\alpha _i\right) ^t$$ represent eigenvectors (with different eigenvalues $$\alpha _i$$), $$U_j, j=1,2$$ are control limits (3), $$q_i$$ and $$t_i$$ are parameters of the decomposition that have to be solved from this system of nonlinear algebraic equations under following conditions
\begin{aligned} \begin{array}{l} \text {if} \,\,\, q_i \in \left[ U_1-\sum \nolimits _{k=1}^{i-1} q_k,\,\, U_2-\sum \nolimits _{k=1}^{i-1} q_k \right] \, \,\, \text {then } \,t_i=0 \\ \text {else} \,\,\, q_i= U_j-\sum \nolimits _{k=1}^{i-1} q_k \,\,\, \text { and } \,\,\, 0<t_i \le t_{i-1} \end{array} \end{aligned}
(6)
for $$i=1,\ldots ,n\,,\,j=1,2,\,t_0=\infty$$. Then the control law can be computed as
\begin{aligned} u=\sum _{i=1}^n q_i \end{aligned}
(7)
The nonlinear decomposition (5) fulfills the aim of the control expressed by the condition (4) when the current state vector is decomposed to the individual subsystem state vectors and each of them is either time optimally controlled by one of the limit values of $$q_i$$ or it decreases the distance $$\rho _i$$ from the phase space origin.
In the case of the triple integrator it is valid $$n=3$$. Suppose we have three different eigenvalues with ordering $$\alpha _3 < \alpha _2 < \alpha _1 <0$$. After substitution of (2) into (5) one gets the system of three algebraic equations (8) that is necessary to be solved under the condition (6) in order to evaluate the parameters $$q_1$$, $$q_2$$ and $$q_3$$ that according to (7) determine the resulting control law u.
\begin{aligned} \mathbf {x}= \left( \begin{array}{c} \sum \nolimits _{i=1}^{3} \left( q_i \left( \frac{1}{\alpha _i^3}-\frac{t_i}{\alpha _i^2}+\frac{1}{2} \frac{t_i^2}{\alpha _i} \right) -\frac{1}{6}q_it_i^3 \right) \\ \sum \nolimits _{i=1}^{3} \left( q_i \left( \frac{1}{\alpha _i^2}-\frac{t_i}{\alpha _i}\right) +\frac{1}{2}q_it_i^2 \right) \\ \sum \nolimits _{i=1}^{3} \left( \frac{q_i}{\alpha _i}-q_i t_i\right) \end{array} \right) \end{aligned}
(8)
Generally, it is not possible to solve this system analytically because of the resulting polynomial of the sixth order. Therefore it is helpful to use graphical interpretation in the phase space. (8) spans the whole phase space but if we consider $$t_3=0$$ then only a part of the phase space is given by it. This part defines the set of states corresponding to the proportional control called proportional band (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.

By application of another limit condition, i.e. $$q_3=0$$, the above mentioned 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 yz. 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.
As an example we describe a procedure of computation of the time sub-optimal control value for one of the regions of 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
\begin{aligned} \left( \begin{array}{c} y\\ z \end{array} \right) = \left( \begin{array}{c} U_j \left( \frac{1}{\alpha _1^2}-\frac{t_1}{\alpha _1}\right) +\frac{1}{2}U_j t_1^2 + \frac{q_2}{\alpha _2^2}\\ \frac{U_j}{\alpha _1}-U_j t_1+\frac{q_2}{\alpha _2} \end{array} \right) \end{aligned}
(9)
After substitution of the calculated parameters $$q_2$$ and $$t_1$$ into the (8) (when $$q_3=0$$) we can get from the subsystem of (8) for the coordinate 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

The simplified time sub-optimal algorithm is applied to the control of the level in the third tank of the hydraulic system (Fig. 3) described by
\begin{aligned} \dot{h}_1= & {} \frac{1}{A_1} q_1 - c_{12} \sqrt{h_1 - h_2}\nonumber \\ \dot{h}_2= & {} c_{12} \sqrt{h_1 - h_2} - c_{23} \sqrt{h_2 - h_3} \nonumber \\ \dot{h}_3= & {} c_{23} \sqrt{h_2 - h_3} - c_3 \sqrt{ h_3} \nonumber \\ y= & {} h_3 \end{aligned}
(10)
where the control action $$q_1$$ represents the inflow in the first tank and $$h_1$$, $$h_2$$, $$h_3$$ are the levels in corresponding tanks. The following parameters have been identified from the real system: the cross-section of the first tank $${A_1=1\cdot 10^{-3}m^2}$$ and the coefficients of corresponding valves $$c_{12}=1.48\cdot 10^{-2}m^{\frac{1}{2}}s^{-1}$$, $$c_{23}=1.52\cdot 10^{-2}m^{\frac{1}{2}}s^{-1}$$, $$c_3=6\cdot 10^{-3}m^{\frac{1}{2}}s^{-1}$$. The control value $$q_1$$ is constrained: $${Q_{min}=0\,m^3s^{-1}}$$ and $${Q_{max}=1.562\cdot 10^{-5}m^3s^{-1}}$$. The aim is to control the height of the level in the third tank.

### 5.2 Exact Linearization Method

Using the exact linearization method [7] the nonlinear system (10) can be expressed in the form of a triple integrator and previously derived control can be applied after taken into account the change of control limits caused by linearization feedback. After denoting the Lie derivative of a scalar function y along a vector field $$\mathbf {f}$$ by $$L_fy$$ the desired exact linearization feedback can be expressed as
\begin{aligned} q_1=\frac{u-L^3_fy}{L_gL^2_fy} ,\,\,\, \mathbf {f}= \left( \begin{array}{c} - c_{12} \sqrt{h_1 - h_2} \\ c_{12} \sqrt{h_1 - h_2} - c_{23} \sqrt{h_2 - h_3}\\ c_{23} \sqrt{h_2 - h_3} - c_3 \sqrt{ h_3} \end{array} \right) \!\!,\, \mathbf {g}= \left( \begin{array}{c} \frac{1}{A_1} \\ 0\\ 0 \end{array} \right) \end{aligned}
(11)
and new control constraints are
\begin{aligned} U_1= & {} L_gL^2_fy \,Q_{min} + L^3_fy \nonumber \\ U_2= & {} L_gL^2_fy \,Q_{max} + L^3_fy \end{aligned}
(12)

### 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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