Design of Control Systems with Respect to Constrained Actuators

This paper presents a method for the design of control systems such that actuator performance limits are not exceeded. The maximum energy delivery concept and root locus analysis methods were used to find the gains for a pseudo-derivative feedback controller for a second order system with zero or first order numerator dynamics. The method has been implemented in a computer program which determines the gains and simulates response characteristics.

In practice, control system design often involves expensive trial and error testing in order to design a controller which satisfies a certain set of criteria. Control system performance is ultimately dependent upon the physical limitations of the controlled system, a concept overlooked by most academic approaches to control system design.
This paper presents a method to design control systems considering the physical constraints of the final control elements (actuator) of the system to be controlled. The method has been implemented in the form of a personal computer program which determines the gains for a Pseudo Derivative Feedback (PDF) controller for an adjustable configuration physical system. The system must be modeled as a transfer function with an order of two in the denominator and an order not exceeding one in the numerator. (The gains for the first order denominator configuration can be solved for analytically by direct substitution into equations derived by Phelan (Phelan [1], pp. 152-155).

Motivation
Few authors are concerned with the design of controllers for systems with actuator limits. Of the authors who do consider this limitation, few provide a complete solution in a form useful for a modern control system designer. Phelan describes the actuator limit concept as the single most important point in control system design 1 (Phelan p.11). The general method presented by Phelan for solving such problems provides an analytical solution for a first order control system and a suggested trial and error procedure for second order systems. Other constraints on the configuration of the problem examined by Phelan include only an inertia term in the plant transfer function and no numerator dynamics (no derivatives in the transfer function numerator). The motivation of this investigation was therefore to expand the complexity of the problems to which Phelan's methods (or variations of) could be applied. As a consequence of the incorporation of the methods in a personal computer program, a tool has been developed which is convenient for a control designer to use.

Terminology
The following terms and definitions relate to the block diagram shown in Figure 1-1.
Control system: Any system that controls a supply of energy.
Feedback control system: A control system which uses measurement of the output or controlled variable to help adjust the supply of energy in the system.
Controller: The portion of the control system which encompasses the adjustable parameters which influence how the system responds.
Fixed Elements: The portion of the control system which is not adjustable. The two subsets of the fixed elements are: Actuator: Accepts a low power level command from the controller and converts it to a high power level. Command (Reference) Input, R: The signal or action that is requested of the control system. Often the most severe cqmmand input that can be requested of a system is a step input, which is an instantaneous change from no energy to some maximum value.
Output, C: The desired signal or action as a res~lt of the control process.
Error Signal, E: The difference between the command signal and the output signal.
Controller Signal, V: The signal following the controller in the block diagram.
Actuator Signal, T: The signal following the actuator in the block diagram.
3 1. 4 Summary of the PID family of control laws The most traditional class of controllers are the proportional (P), integral (I) and derivative (D) controllers and various combinations thereof (Phelan p.70 In addition to these desirable characteristics of PID controllers, there are some undesirable ones. Foremost is that the controller which uses some combination of PID (which is more likely than any one action by itself) ·may simultaneously modify conflicting signals, resulting in possibly un-predicted controller performance. Systems designed with the PID family of controllers should therefore never operate on more than one signal in the forward path of the controller. This concept is sometimes referred to as the principle of one master (Phelan p. 150). 4 2.

DESIGN BASED UPON CONSTRAINED ACTUATORS
Phelan makes the following statement about control system design: "Two kinds of automatic control systems -academic and real exist, and they have almost nothing in common." (Phelan p.11) While many aspects of control system design can be understood using basic controller theory, these methods will only be accurate if the system responds linearly. Unfortunately, linearity is not guaranteed unless the physical limitations of the real actuator are taken into account in the determination of controller gains.

Non-linearity in Control Systems
The equations of motion describing the dynamics of every real controlled system are non-linear. Since the mathematical analysis of non-linear systems is much more complicated than linear systems, it is advantageous to simplify the equations of motion so that they are linear. Fortunately, the fundamental idea behind a feedback control system -the comparing of the actual output to the desired output, makes real (vs. academic or theoretical) control systems inherently very tolerant of most nonlinearities, provided they are designed properly.

Non-linearity produced by Actuator Saturation
There are many types of controllers, each of which can provide a wide variety of response characteristics to the signal upon which it operates, the error signal. Most academic lessons in control system design discuss these control methods The problem is that while the actuator is the "muscle" of the control system, it is also the weakest link. The actuator is part of the fixed elements of the control system and therefore is not easily adjustable. Examples of this inflexibility include: 1.
A D.C. motor has a limited torque which it can produce -either deliberately so as to prevent damage to the motors components or accidentally, such as due to improper selection of an amplifier.

2.
A valve cannot be more than fully open or fully closed in a liquid level controller or pressure control system. If a control system operates over a wide range of conditions, it is possible that the output of the controller, V, may request more energy from the actuator than it is capable of delivering. When this maximum value is exceeded, the feedback loop is effectively broken because while the control signal is requesting more energy, the actuator will produce only what it is limited to. When the actuator is at its limit, it is said to be saturated. Some other types of non-linearities include dead-zone, bang-bafig, hysteresis and mechanical backlash (Towill,[3] p. 411). In some of these cases, the non-linearity is actually deliberately produced to improve system performance (Towill p. 415).
A common result of this saturation is called reset-windup. This occurs in controllers which use the integral of the error signal to control the process. The value of the control signal for the integral control algorithm is: The value of V is dependent both upon the magnitude of the error signal and the length of time the error exists. For a step input, the integral term increases rapidly until the actuator saturates and the response overshoots its desired level. The saturation would occur even sooner if proportional control of the error signal were also used (Pl control) because the error is at its maximum value just after time zero. After the response overshoots the set-point value, and the error changes sign, it takes some time before the error is large enough to cancel out the overshoot. Consequently, the actuator signal can not pull away from its saturation limit and the system behaves nonlinearly. The result is that any controller with integral action may have significant overshoot and a longer response time than it would have if the actuator signal did not saturate.
The neglect of the finite energy delivering capability of actuators is the primary reason 7 academic control systems are so different from real ones. Many manufacturers of control equipment use academic methods on real-world systems. As a result their equipment falls short of expected performance which would then require a set of tuning procedures to bring the performance in line. .
There are several methods that have been developed which consider actuator saturation and its effect on overall system performance. A controller which uses Anti-Windup (Astrom [4], p.12 ) has an extra feedback path which measures the actuator signal as a means to prevent saturation. More recently, a numerical method was developed which determines linear controller designs based upon convex optimization techniques (Boyd, et al [5]). The maximum energy delivery concept was developed by Phelan, and is described further in Section 2.3.

The General Method
Actuator saturation and non-linear response can be prevented by simply designing the controller (that is select the control gains for the control scheme) such that the control signal never requires the actuator to saturate. This will require the designer to know three types of information about the system to be controlled: 1. The coefficients of the parameters of the fixed elements of the system. For a second order actuator/plant pair, this would be inertia, damping and restoring terms. 8 2.
The actuator saturation limit.

3.
A maximum operating condition, such as the maximum speed at which a D.C.
motor is expected to operate.
The crucial information is item 2, and is also the most difficult to obtain. The difficulty is that so little emphasis has been placed on actuator limits in the past that data is rarely available on this parameter. The designer may be required to derive this limit from some maximum operating condition of the system. The T characteristics of an ideal actuator with a finite limit on the output, T, v is shown in Figure 2 in the region such that  The minimum and maximum values of T represent the limits on the actuator and may or may not be equal in magnitude. The values of T mm and T max are entirely dependent upon the system requirements and hardware limitations.
1 . . · ~1 .,_I =t= I ···.____I -f As an example, consider PI control of the first order plant shown in Figure 2-2, where m, and ~are the inputs and outputs of the piece-wide linear actuator function respectively. When the actuator function is operating in the linear region, (2)(3) At time t=O, the integral of the error signal is zero, and the value of the error signal is at its maximum -the magnitude of the step input rmaX' This gives K =~.max The standard formula for damping ratio for this second order equation is: Assuming the most desirable response characteristics will be achieved when the system is critically damped (~=l) ~can be determined as such: (Phelan,).

2.4.l Linearity
The differential equation which describes the fixed elements of the control system is assumed to be linear. The transfer functions describing the control system in its entirety are also assumed to be linear, provided that the actuator is prevented from saturating.

Absence of Disturbance Terms
Sometimes random forces and/or deviations (non-linearities) in the parameters of the plant transfer function create a random input preceding the plant in Figure I. For the purposes of this study, disturbances were neglected because their maximum magnitude is difficult to predict, and thus an estimate can not be made on whether they will produce actuator saturation or not. It is assumed that disturbances are second-order effects that don't cause actuator saturation.

Step Functions
The most severe, and therefore most useful, type of reference input is a step function (Phelan p.96). A step change in command input represents an instantaneous, noncontinuous change. No real-world system can respond as such for this would require 11 an infinite amount of energy at time 0. For problems modeled with transfer functions having numerator dynamics, a pure step input is unrealistic. Therefore, a replacement function is used to represent the step function: where M is the magnitude of the step and z is a constant dependent upon the smallest time constant ('tmuJ of the control system. The constant is arbitrarily chosen such that (2-10)

Method applied to a second order fixed element system
The block diagram in Figure 2-3 shows pseudo-derivative-feedback (PDF) control for a second order system with numerator dynamics. PDF control was developed by Phelan (Dec. 1970) as a solution to the problems associated with the principle of one master. In an effort to avoid the undesirable effects of differentiating the error signal, the output of the control system is fed back into the forward path of the loop following an integral, I action, control block. The overall effect of this configuration would be the same as if the outpu·t signal were differentiated and fed back preceding the integral block. I action is chosen over P action in the forward path because it is often unrealistic to expect instantaneous response to a step input as is the case for P action and because I action gives zero steady-state error (Palm p. 417). Note that for the second order plant in Figure 2 and potentiometer/tachometer gains). They do not affect the dynamics of the system, only the magnitude of the PDF gains, and therefore will be neglected in the following derivations. The overall system tr an sf er function for the system in Figure 2 (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12) 13 (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13) (2-14) (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15) Two different methods are presented to determine the gains for satisfactory response of the above control system with respect to a constrained actuator signal. The methods cliff er because of the presence or absence of the first order numerator term (a).

Method with zero order numerator dynamics
The characteristic equation is a third order differential equation, therefore standard formulas for damping ratio and time constant for a second order characteristic equation are of no use in determining the gains which will accomplish the goal. The fact that the system in Figure 2-2 is a multiple loop system will be useful however. The inner loop transfer function is: The characteristic equation is a second order differential equation from which the following equation for damping ratio is found ( where IL refers to inner-loop): One would expect that the optimum values of K2 and K1 would be found when the jnner loop damping ratio is 1 because critically damped systems often have desirable characteristics (fast, smooth response curves). However because ~(s) would never be 50 severe as a step function because of its position following the controller in the block diagram, the value of ~IL can be less than unity. Studies by Phelan and Ulsoy [6] have shown that the optimum value of the inner loop damping ratio for smooth fast response is 0.7 (Phelan,. Through simulations, Phelan determined that the best relationship of K1 to vmax and rmax came out to be From Equation (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17): v: Phelan states that there is no simple way to determine the gain~. analytically. He suggests a trial and error procedure of starting with a low value of ~ and gradually increasing it while providing step changes in the reference input equal to the maximum value expected. At each trial the value of Ki is increased until either the actuator saturates or the output response overshoots.

2.S.2 Method with first order numerator dynamics 15
If the method used by Phelan were to be applied to a fixed element system with first order numerator dynamics, the first step again would be to find the gains K2 and K1 to provide an inner-loop damping ratio of 0.707, where (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20) If Ki were selected as it was in the zero order case, then the solution for K2 would be in the form of a quadratic equation. The difficulties in determining the proper value of K 2 (which may be complex conjugates) make this method more difficult to analyze, that is, a solution might not exist which provide a real value for Ki· The alternative method used to solve this problem uses the root-locus method to find the gains for satisfactory performance, without causing the actuator signal to saturate.
The characteristic equation written in root locus form with ~incorporated into the root locus variable K is: where  There is one zero for this configuration of the characteristic equation. It is s=-l.. a  (2-24) (2-25)  There are also three poles, one of which is at the origin. The other two poles can be placed anywhere by appropriate selection of the gains K 2 and K 1 • By observing the The plot which is constructed using the root-locus plotting guides (Schwarzenback [7] P· !60) shows the locus breaking away at some point between 0 and the position of the poles, and approaching infinity along an asymptote perpendicular to the real axis ( Figure 2-4). The fastest, smoothest response (before adjustment for actuator saturation) will occur at the breakaway point s.,.. The solution for the breakaway point is found by solving the root-locus equation for K and differentiating to find the local minimum. The result is the cubic equation: where  (2-29)  The breakaway point should be the only real root betWeen 0 and the pole position.
The root-locus variable K at the breakaway point is  The gain values K 2 and K 1 selected to place the poles near the zero, and the value of Ki determined by (above) do not guarantee that actuator saturation will not occur.
Therefore, it is necessary to simulate the actuator response, and adjust the values 18 accordingly. The method used to adjust the gains is to incrementally place the real, repeated poles closer to the origin in the root-locus plot, there by slowing the system down. until the actuator does not saturate. A graphical representation of the method is shown in Figure 2-5. This method is easily coded as a computer program algorithm.

2.6.l Saturation Limit Parameter Selection
In practice, the non-linearity that produces saturation can occur anywhere within the fixed elements portion of the control system (Towill p. 411). Likewise it is not practical to design a computer method that analyzes the dynamics of a single type of problem. It is therefore necessary to select a saturation limit that is outside of the fixed elements of the system. The only choice for this parameter must then be the control signal, V.

Response Calculations
The control signal, V, in Figure 2-2 can be represented as: The overall system response, c(t), is found using the fourth order Runge Kutta method for solving third order differential equations.    The maximum operating chacteristics of the actuator must be determined before reducing the fixed elements of the motor into a single transfer function. Note that among the specifications is a maximum pulse current (which is important to avoid demagnetization of the motor's components).

Is+ c
To simplify the problem, it is advantageous to neglect the effects of armature inductance temporarily so that Substitution of the maximum pulse current for i(t) and the armature resistance gives v 1 (t)=(l.55) (24)=37.2Vo1ts The control signal, V(s) is represented as  Vmax=37. 2+ (4. 29) (6. 0) =63 Vol ts The actuator limits for this problem then are +/-63 Volts for a command step input of 6 KRPM. Reducing the fixed elements in Figure 3-1 to a single transfer function gives the general form for a fixed element system with zero order numerator dynamics, where a =0 I= LI c=Rl+Lc (3)(4) For the purposes of this example, the feedback and amplifier gains, KA and K 8 , will be assumed to be unity, for their precensce do not affect the dynamics of the problem as discussed in section 2.5.

Derivation of Problem with First Order Numerator Dynamics.
The liquid level system shown in Figure 3-2 consists of two coupled tanks, each of which has an outflow pipe with known diameters and lengths. The fluid resistance due to laminar pipe flow is given by the Hagen-Poiseuille formula R= 128µ£ npD 4 26 (3)(4)(5) Based upon the block diagram shown in Fig. 3-3, the transfer function relating the volume flow rate, q, to the height of the liquid in the first tank is Is 2 +cs+k (3)(4)(5)(6) where B = g(R 1 + Rz) (3)(4)(5)(6)(7)  In order to select reasonable values for an actuator limit and a corresponding input command height for the first tank, it is necessary to examine the steady state value of the control signal in terms of the command request. The transfer function relating the control signal, V(s), to the command height H1R(s) is:

V(s) = Ki(Is 2 +cs+k)
HlR(s) T 3 s 3 +T 2 s 2 +T 1 s+T 0 (3)(4)(5)(6)(7)(8) Applying the final value theorem (Palm, to the above transfer function and substituting the value for T0 from eqn. the following relationship results: (3)(4)(5)(6)(7)(8)(9) For a maximum step command in liquid level (of the first tank) that would ever be expected, say 2 meters, a corresponding steady state value of V (which in this case is flow rate) can be found required to maintain the height of 2 meters. The flow rate becomes: v < 2 > < 96 · 04 ) =O. 005158m 3 /sec ss= 37 238 The above flow rate is used as the upper actuator limit in the tank problem. A lower actuator limit of 0 is chosen to represent the flow rate when the input valve is completely closed. A reasonable value for the command liquid height in the first tank must be between 0 and 2 meters. A command request of I meter is used. Operating the Computer Program

.3.1 System Requirements and Startup
While the computer method runs adequately on an 80286 type personal computer, speed performance is superior with an 80386 or 486 processor with at least 400K of free randam access memory. A mouse is recommended.
The program can be run from the floppy drive or copied to a hard drive -about 300K of disk space is required. The program is started by typing PDF at the DOS prompt.

Entering System Parameters.
To enter the fixed element characteristics and the desired actuator performance limits,

3.J.3 Calculating Gains and Response.
30 To calculate the gains and view the control signal response, select Go from the menu bar and then the Go sub-menu choice. After a few seconds a line graph is displayed with the control signal response. The PDF gains and control signal maximum and minimum values are displayed in data boxes on the right side of the display. Choose the System Response sub-menu from the Go menu-bar selection to view the overall system response with the gains that have been determined in the previous action.

Saving and Retrieving Specifications and Responses.
The system specifications can be saved to a disk file so that they can be conveniently recalled for another time. Select File from the menu-bar and the Save Specs submenu choice. Type a file name with no extension ( a .PDF will be added). Press ESCAPE not ENTER when done. To recall a saved file select File from the menu-bar and the Retrieve Specs submenu choice. A box is displayed with all the PDF data files in the current directory. Select a file to retrieve by clicking the mouse pointer on the desired file.
To save a control signal or overall system response to a spreadsheet importable file, display the desired line plot (with Go) and choose Save Response from the File menu-bar selection. Type a file name (with or without an extention) for the destination file. Press ESCAPE not ENTER when done.

31
Select Quit from the menu-bar to leave the program and return to DOS. A box is displayed verifying the action. Press the space bar to toggle the Yes/No field in the bOX and click Quit on the menu-bar again to complete the action.

.1 Results for the zero order numerator dynamic problem example.
The results obtained from the computer method for the DC Motor example outlined in  which represent a faster, smoother responding system.
The determination of the control signal limit for this problem involved the selection of some physical limit embedded within the fixed elements of the system and deriving from that a corresponding limit. The physical parameter selected was a maximum   There may be a more appropriate parameter to use to derive the limit (such as the maximum torque a motor can generate) but such data was not available from the manufacturer's specifications.

First Order Numerator Dynamics Problem
The open-loop response for the tank example is shown in Figure 5

General Comments on Computational Error
In addition to the specific results for each of the examples above, there are several important notes pertaining to the computation process. First, for each type of problem, there is a requirement to divide the valid range for the PDF gain ~ into discrete segments for a computer iteration method. Specifically, the algorithm initially chooses an increment size that is 10% of the valid range for stability, and iterates with this value until a non-saturating solution is found. The increment size is then reduced by 39 factor of 10 and a more precise solution is found. An unavoidable result of this segmentation process is that it is nearly impossible to find the absolute optimum gains that will provide a control signal response that does not saturate and have the smallest time constant and smoothest response possible.
Also, there are potential round-off or truncation errors inherent in the Runge Kutta numerical method. The magnitude of any such errors would be far less than is required to cause computational mistakes when comparing the calculated control signal to the limit specified by the user or in generating visual differences in the graphic plots. 40 6.

CONCLUSIONS AND RECOMMENDATIONS
The purpose of this study was to investigate the design of feedback control systems within the limitations of the finite energy delivery capability of the system's physical elements. The development of the computer algorithms to accomplish the investigation indicated that not only is it possible to design control systems in this manner, but that there are many possible combinations of design methods which provide acceptable perfonnance within the saturation prevention constraint.
While the configuration of the problem examined by the computer program is only capable of examining two general classes of physical systems, it does approximate the dynamics of a wide variety of potential systems to be controlled. Some possible areas for further study of this type of problem include: Investigate the effects of different input functions, such as a ramp function, to see if the same or similar methods of solution can be developed.
Apply different methods to prevent actuator saturation such as the antireset windup method.
Expand complexity and flexibility of the physical parameters to be controlled by increasing the order of the dynamics of the fixed elements.