Multi-input multi-output control system and methods of making thereof

Abstract

The invention provides a multi-input multi-output (MIMO) control system comprising a controller adapted for receiving an input set of at least two control input parameters and a set of at least two control output parameters, said control system arranged for effecting a modified deadbeat control, in which said modified deadbeat control comprises a robust deadbeat control for an n-th order, linear time invariant (LTI) system based upon a series of cascade proportional integrating-differentiating (PID) controls, each PID control comprising a system transfer function having a nominator and a denominator, wherein for the nominator a constant gain (K) is selected for each PID control. The invention further provides a method for controlling a continuous process using this control system.

Claims

1. A method for controlling a continuous process, said method comprising: providing a control system comprising a controller adapted for receiving an input set of at least two control input parameters and producing a set of at least two control output parameters in response to the at least two control input parameters, said control system arranged for effecting a modified deadbeat control, in which said modified deadbeat control comprises a robust deadbeat control for an n-th order, linear time invariant (LTI) system based upon a series of cascade proportional integrating-differentiating (PID) controls, each PID control comprising a system transfer function having a nominator and a denominator, wherein for the nominator a constant gain (K) is selected for each PID control; providing a chemical processing assembly running a chemical process, said chemical process assembly comprising a series of actuators for setting a series of process conditions of said chemical process; determining during said chemical process a series of process output parameters and providing these output parameters to the control system; said control system determining a set of process input parameters resulting from said process output parameters; said control system providing said input parameters to said actuators during said chemical process for setting said process conditions.

2. The control system of claim 1, wherein the denominator is set functionally equal for each PID control.

3. The control system of claim 1, wherein for each PID control, the gain (K) is set functionally equal.

4. The control system of claim 1, wherein at least one PID control is provided for each input.

5. The control system of claim 1, wherein said control system comprises a continuous time base.

6. The control system of claim 1, wherein the system transfer functions of said PID controls are defined as q(s)/p(s), wherein q(s) and p(s) are polynomials, wherein q(s) is selected as a constant gain K, and p(s) is a polynomial with deadbeat parameters.

7. The control system of claim 1, wherein said denominator is a polynomial.

8. The control system of claim 7, wherein said denominator is a Hurwitz polynomial.

9. The control system of claim 1, wherein said control system comprises a disturbance d(t) which is a function of time, and a size of said disturbance is less or equal to a maximum disturbance dmax.

10. The control system of claim 1, wherein the same PID is used in each loop, minimizing the decoupling effect.

11. The method of claim 1, wherein said chemical process is a continuous polyolefin polymerization reaction.

12. The method of claim 1, wherein actuators are selected from the group consisting of a material feeder, a cooler, a heater, and one or more pressure valves.

13. The method of claim 1, wherein said process output parameters are selected from the group consisting of temperature, pressure, flow rate, viscosity, molecular weight, one or more UV, VIS or IR spectral values, branching index, and a combination thereof.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) Embodiments of the invention will now be described, by way of example only, with reference to the accompanying schematic drawings in which corresponding reference symbols indicate corresponding parts, and in which:

(2) FIG. 1 schematically depicts an example of a deadbeat controller;

(3) FIG. 2 shows an example of a four-loop control system, which are desired pitch, desired airspeed, pitch rate, trim angle of attack;

(4) FIG. 3 shows the response of a small UAV used for research purposes to the set point of 20 degrees as desired pitch angle;

(5) FIG. 4 shows the response for velocity with a set point of 15 m/s;

(6) FIG. 5 shows the Pitch angle tracking;

(7) FIG. 6 shows the response of the UAV, using two method, order reduction and zero gain replacement, FIG. 3 is for order reduction, FIG. 6 is for zero gain replacement to the set point of 20 degrees as desired pitch angle, and

(8) FIG. 7 shows the response for velocity with a set point of 15 m/s

(9) The drawings are not necessarily on scale

DESCRIPTION OF PREFERRED EMBODIMENTS

(10) In many MIMO systems, the control can be defined using a set of Proportional Integrating Differentiating loops, or PID's. In such a system, A system transfer function can be defined:

(11) $G\left(s\right)=\frac{?a_i{s}^i}{?b_i{s}^i}$

(12) In any system transfer function, there thus is a nominator and denominator. The difference between the order of the denominator (od) and the order of the nominator (on) is usually from 0 to 2 (n). In FIG. 1, an example of a Deadbeat control system is shown. After studying the deadbeat system, it was found that deadbeat equation is only applicable if od-on?3. A technique which is called System order reduction may be applied on the nominator of the system in order to increase the order difference (n) from 2 to 3. Afterwards, in addition or alternatively, the idea of replacing the whole nominator with a gain, a constant K, came to me. It was also applied successfully.

(13) In a MIMO system which has multiple inputs and multiple outputs, various system loops can be combined. It was noticed that in these systems, the different system loops have the same denominator.

(14) So, in an embodiment, if a system order reduction was applied, the nominator order can be reduced to zero. Thus, system order reduction can make the difference between the nominator and the denominator more than two. Since the order of nominator becomes zero and taking in consideration that all system transfer functions have the same denominator, it follows that the modified transfer function becomes the same for all system loops.

(15) In another or further elaborated embodiment, It was found that if the nominator can be replaced with a gain and in particular if that gain is the same for all system loops, taking in consideration that it also has the same denominator, then it follows that the modified transfer function becomes the same for all system loops.

(16) Therefore, with either one of these embodiments or a combination thereof, it is possible to solve all the system loops at once instead for solving for each loop individually (like in the classic PID and deadbeat controller). The controller can then be applied on the original transfer function, which means that there will be no effect on the system behaviour. For a four-loop system (see FIG. 2, for instance), originally 12 gains needed to be tuned (no. loops*no. gains in each loop=4*3=12). While after the modification suggested, only one gain (and none for some systems) needs to be tune. This means that the number of gains that need to be tuned is reduced with more than 90%! Applying this controller will mean a huge saving and amazing system performance.

(17) An example of several approaches to the current control system will be demonstrated, applied to UAV autopilot tuning. It is a system of a small UAV used for research purposes. The system state space representation has four loops associated with it, which are: Air speed, speed in z direction, angle of attack, pitch angle.

(18) Usually, there are nine system loops to be tune for a complete autonomous flight of a UAV. To tune those classical nine system loops with the usual three gains for each PID controller (referred to as KP, KI, and KD), a vast amount of man power and time is needed. Applying a deadbeat controller (known as such, see Jay Dawes et al., Design of Deadbeat Robust Systems, Glasgow, UK, pp 1597-1598, 1994), the three gain parameters of each PID controller were reduced to only 1 parameter. This means that for the usual 9 system loops, we now have to tune only 9 gains instead of the original 9?3=27 gains. This means that the number of required tuning parameters was decreased by 66.7%.

(19) It will be demonstrated that using the current control system, the number of tuning parameters can be reduced even further. Since the number of loops may be reduced as well, the number of tuning parameters can eventually be reduced to 1?1=1 parameter only. This means that the number of tuning parameters cab be reduced by 96.5%.

(20) Thus, a control system can be implemented using modified PID controllers using a technique which we will refer to as system order reduction, and/or a technique we will refer to as zero gain replacement. It is possible to combine these techniques.

(21) In this example, first we will demonstrate the control system using system order reduction.

(22) Any aerial system is represented by three motions, which are pitch, roll, and yaw. Pitch and roll are in longitudinal direction, and roll is in lateral direction. In longitudinal direction, there are two forces and one moment. The forces are an X-force and a Z-force. Moment M is a pitching moment. The forces and moment are represented as follow:
{dot over (X)}=Ax+Bu

(23) With

(24) $\left(\begin{array}{c}\stackrel{.}{u}\\ \stackrel{.}{?}\\ \stackrel{.}{q}\\ \stackrel{.}{?}\end{array}\right)=\left(\begin{array}{cccc}{X}_u& X_{?}& 0& g\\ Z_u& Z_{?}& u_0& 0\\ M_u& M_{?}& M_q& 0\\ 0& 0& 1& 0\end{array}\right)\left(\begin{array}{c}u\\ ?\\ q\\ ?\end{array}\right)+\left(\begin{array}{cc}{X}_{{?}_e}& X_{{?}_t}\\ Z_{{?}_e}& Z_{{?}_t}\\ M_{{?}_e}& M_{{?}_t}\\ 0& 0\end{array}\right)\left(\begin{array}{c}{?}_{{?}_e}\\ {?}_{{?}_t}\end{array}\right)$

(25) Suppose a system with the following state space:

(26) $A=\left[\begin{array}{cccc}-\mathrm{.0163}& 9.5179& 0& -9.81\\ -\mathrm{.1362}& -\mathrm{.0854}& 1& 0\\ 0& -2.2604& -2.341& 0\\ 0& 0& 1& 0\end{array}\right]\mathrm{and}$$B=\left[\begin{array}{c}0\\ -5.2708\\ -51.4385\\ 0\end{array}\right]$

(27) The transfer functions for velocity u, G.sub.?.sub.e.sup.?, and for pitch angle ?, G.sub.?.sub.e.sup.u, are as follow:

(28) $\begin{array}{cc}{G}_{{?}_e}^{?}=\frac{?}{{?}_e}=\frac{58.7s^2+5.528s+75.866}{s^4+2.82s^3+4.13s^2+3.544s+3.45}& \left(1\right)\\ G_{{?}_e}^u=\frac{u}{{?}_e}=\frac{5.27s^2+101.8s+81.07}{s^4+2.82s^3+4.13s^2+3.544s+3.45}& \left(2\right)\end{array}$

(29) Next, system order reduction (see R. Prasad et al., New Computing technique for order reduction of linear time invariant systems using stability equation method, Journal of the institution of Engineers IE(I) Journal EL, Vol. 86, September 2005, pp 133-135) is applied on both (1) and (2):

(30) $\begin{array}{cc}{G}_{{?}_e}^u=\frac{u}{{?}_e}=\frac{31}{s^3+1.05s^2+1.26125s+1.2264}& \left(3\right)\\ G_{{?}_e}^{?}=\frac{?}{{?}_e}=\frac{27}{s^3+1.05s^2+1.26125s+1.2264}& \left(4\right)\end{array}$

(31) Now, to verify the solution, the UAV transfer function of ? will be solved using deadbeat equation. Solving for ? we get
D.sub.cD.sub.s=s(s.sup.3+1.05s.sup.2+1.26s+1.226)(5)
D.sub.cN.sub.sH.sub.2=s(27)K.sub.b(6)
NcNsH.sub.1=K.sub.3(s.sup.2+Xs+Y)(27)(1+K.sub.1s)(7)

(32) Now take K.sub.3=1 (to be tuned later) as a starting value. If we further choose the desired settling time to be 2 seconds in the deadbeat equation, we can calculate the value of ?.sub.n, and the values of deadbeat parameters ?, ?, ? (found from deadbeat table below) are as follow:

(33) $?=2.2;?=3.5;?=2.8$${?}_n=\frac{4.81}{\mathrm{.8}\left(2\right)}=3.00625$

(34) When we substitute this into the general Deadbeat equation, the characteristic equation of the deadbeat transfer function becomes:
G.sub.db=s.sup.4+6.6138s.sup.3+31.6314s.sup.2+76.0735s+81.6778(8)

(35) Applying deadbeat equation (G.sub.db), i.e., making the parameters of each s-power equal, we find:
1.05+27*k1=6.6138(9)
1.26+27k1X+27=31.631(10)
1.226+27(K.sub.b+X+k1Y)=76.0735(11)
27Y=81.6771(12)

(36) Solving the above set equations (9)-(12), we find:
K.sub.1=0.214, K.sub.b=1.42, X=0.7857, Y=3.1414

(37) After solving for the unknowns in the way shown above, K.sub.3 was tuned and the final value was found to be 1. Next, the same PID solution is applied on both pitch angle (?) and speed (u).

(38) FIG. 3 shows the response of the UAV to the set point of 20 degrees as desired pitch angle, while FIG. 4 shows the response for velocity with a set point of 15 m/s. FIG. 5 shows the Pitch angle tracking.

(39) Remarks: 1There is no overshooting 2Settling time is as desired

(40) System Robustness

(41) System robustness can be tested by applying some changes on the system parameters. If the system remained stable with the same performance, then the system is robust against disturbance. FIGS. 3, 4 and 5 shows that the system remains stable after applying 100% overestimation and 50% underestimation to the original system.

(42) System Optimality

(43) System optimality can be shown in the solution since the gain K.sub.3 needs to be tuned to reach optimum performance.

(44) Controller Applied to MIMO Systems

(45) Since the UAV system is MIMO system, the controller was successfully applied to the outputs ? and u.

(46) Comparing with Other Similar Work:

(47) The results were showing an oscillation and a settling time of 5 s as well as no reduction in the number of neither tuning parameters nor the control loops. In the current example, no oscillation is occurring, settling time is 2 s, a total reduction in the number of tuning parameters and loops of 96.5%.

(48) Improvement:

(49) The solution was applied on one PID instead of two PIDs. Which means that one PID controller was used to control both the pitch angle (?) and speed (u). This improvement will minimize the decoupling effect of PIDs (the reason why PID controllers doesn't match the performance of other advanced controllers). Which will definitely enhance the system performance and decrease the cost as well as simplifying the control loop. By decreasing the number of tuning parameters, less time and man power as well as less experience is needed to tune the system.

(50) Implementation:

(51) The idea can be implemented on any system in a control loop. By following the steps below.

(52) Again, assume a system with the state space illustrated above, with A and B as defined above (See K. Turkoglu, U. Ozdemir, M. Nikbay, E. Jafarov, PID parameter optimization of a UAV longitudinal Flight control system, World Academy of Science, Engineering and Technology 45, 2008.), the transfer function for velocity and pitch angle are (again) as follow:

(53) $\begin{array}{cc}{G}_{{?}_e}^{?}=\frac{?}{{?}_e}=\frac{58.7s^2+5.528s+75.866}{s^4+2.82s^3+4.13s^2+3.544s+3.45}& \left(13\right)\\ G_{{?}_e}^u=\frac{u}{{?}_e}=\frac{5.27s^2+101.8s+81.07}{s^4+2.82s^3+4.13s^2+3.544s+3.45}& \left(14\right)\end{array}$

(54) We first apply Zeros-gain replacement. In this method, all zeros (s-?)(s-?) . . . are replaced with one single gain K as follow:

(55) $\begin{array}{cc}{G}_{{?}_e}^{?}=\frac{?}{{?}_e}=\frac{K_{?}}{s^3+1.05s^2+1.26125s+1.2264}& \left(15\right)\\ G_{{?}_e}^u=\frac{u}{{?}_e}=\frac{K_u}{s^3+1.05s^2+1.26125s+1.2264}& \left(16\right)\end{array}$

(56) In (13), (14) the order of denominator is 4, the difference between the denominator and the nominator is 4, using deadbeat:
H.sub.1=1+k.sub.1s+k.sub.2s.sup.2(17)

(57) Solving for the denominator of (15) or (16), we get:
D.sub.cD.sub.s=s(s.sup.4+2.82s.sup.3+4.13s.sup.2+3.544s+3.45)(18)
D.sub.cN.sub.sH.sub.2=s(K)K.sub.b(19)
NcNsH.sub.1=K.sub.3(S.sup.2+Xs+Y)(K)(1+K.sub.1s+K.sub.2s.sup.2)(20) Take K.sub.3=1 (to be tuned later)

(58) If we select the desired settling time as 2 seconds, we get the following deadbeat parameters (see deadbeat table below, from J. Dawes, L. Ng, R. Dorf, and C. Tam, Design of deadbeat robust systems, Glasgow, UK, pp1597-1598, 1994)

(59) TABLE-US-00001 Order n.sub.p A ? ? ? ? T.sub.r90.sub.n T.sub.S.sub.n 2.sup.nd 1.82 3.47 4.82 3.sup.rd 1.90 2.20 3.48 4.04 4.sup.th 2.20 3.50 2.80 4.16 4.81 5.sup.th 2.70 4.90 5.40 3.40 4.84 5.43 6.sup.th 3.15 6.50 8.70 7.55 4.05 5.49 6.04
?.sub.1=2.7; ?.sub.2=4.95; ?.sub.3=5.4; ?.sub.4=3.4

(60) Where K denotes K.sub.? or K.sub.u

(61) 0${?}_n=\frac{5.43}{\mathrm{.8}\left(2\right)}=3.3938$

(62) The deadbeat characteristic equation becomes as follow
G.sub.db=s.sup.4+9.1631s.sup.3+56.4359s.sup.2+211.0733s+451.0226(21)

(63) By setting the parameters of each power of s of equation (21) and of equations 18)-(20) equal, the following system of nonlinear equations should be solved to get all the gains K, X and Y:
(K*K.sub.2+2.82)=9.1631(22)
(K*K.sub.1+K*K.sub.2*X+4.125)=56.4359(23)
(K+K*K.sub.1*X+K*K.sub.2*Y+3.544)=211.0733(24)
(K*K.sub.b+K*X+K*K.sub.1*Y+3.45)=451.0226(25)
K*Y=450.1935(26)

(64) The solution of (22)-(26) with selection of K=30 (after optimizing for K, the best value was found to be 30) yields
K.sub.1=1.296, K.sub.b=?6.645, K.sub.2=0.2114, X=2.118, Y=15.01

(65) After solving for the unknowns, K.sub.3 was tuned and the final value was found to be 1. Afterwards, the same PID solution is applied on both pitch angle (?) and speed (u) FIG. 6 shows the response of the UAV to the set point of 20 degrees as desired pitch angle, while FIG. 7 shows the response for velocity with a set point of 15 m/s

(66) Remarks 1There is no overshooting 2Settling time is as desired

(67) System Robustness

(68) System robustness can be tested by applying some changes on the system parameters. If the system remained stable with the same performance, then the system is robust against disturbance. FIGS. 6 and 7 show that the system remains stable after applying 100% overestimation and 50% underestimation to the original system.

(69) System Optimality

(70) System optimality can be shown in the solution since the gain K.sub.3 needs to be tuned to reach optimum performance.

(71) Controller Applied to MIMO Systems

(72) Since the UAV system is a MIMO system, the controller was successfully applied to the outputs ? and u.

(73) Comparing with Other Similar Work:

(74) In the current example, no oscillation is occurring, settling time is 2 s, a total reduction in the number of tuning parameters and loops of 96.5%.

(75) Improvement:

(76) The solution was applied on one PID instead of two PIDs. Which means that here, one PID controller was used to control both the pitch angle (?) and speed (u). This improvement will minimize the decoupling effect of PIDs. This will definitely enhance the system performance and decrease the cost as well as simplifying the control loop.

(77) It will also be clear that the above description and drawings are included to illustrate some embodiments of the invention, and not to limit the scope of protection. Starting from this disclosure, many more embodiments will be evident to a skilled person. These embodiments are within the scope of protection and the essence of this invention and are obvious combinations of prior art techniques and the disclosure of this patent.