- Email: [email protected]
Study of the control-equilibrium of control systems
Journal of Systems Engineering and Electronics Vol. 19, No. 4, 2008, pp.775–778
Study of the control-equilibrium of control systems Liu Qiaoge, Fu Mengyin & Sun Changsheng Dept. of Automatic Control, School of Information Science and Technology, Beijing Inst. of Technology, Beijing 100081, P. R. China (Received December 2, 2006)
Abstract: Not so much had been talked about equilibrium in control area. On the basis of the phenomenon of balance, the concept of control-equilibrium and control-equilibrium of a control system is proposed. According to this theory, a perfect control method should not only guarantee stability of the system, but also ensure the control-equilibrium of the system. To achieve the control-equilibrium, feed-forward control is required.
Keywords: balance, control-equilibrium-state, control-equilibrium, feed-forward control.
1. Introduction The purpose of a control system is to make the changing process of the controlled states as close to the desired one. But there always exist some problems in the transition process such as oscillation, overshoot, and time delay. To overcome these problems, both control laws and control structures should be chosen properly. A balance is a type of weighting devices, with identical weighing pans hung at the either end. The left pan holds an unknown weight while the effective weight in the right pan is increased with the known amounts until the beam is level and motionless. Once the weight in the right pan is decreased, the beam will lose level immediately. So there must be enough weight in the right pan to maintain the “balance” of the balance. On the basis of this phenomenon, the concept of control-equilibrium is proposed and the equilibrium theory for control systems is attempted to build. And in this aspect, the validity of the feed-forward control and limitation of the feedback control is analyzed.
2. Control-equilibrium 2.1
About equilibrium
There are descriptions about equilibrium in some areas, such as in mechanics, thermotics, control area, etc.
In mechanics, if the composite external force in some direction is zero, the object is balanced and it will be either motionless or be in the uniform motion state in this direction. In control area, equilibrium-state is defined in Ref. [1] as follows. Definition 1 The plant is x˙ = f (x, t) and x(t) is an equilibrium-state, If x˙ = 0. That is x(t) = xe (constant vector). 2.2
Control objective
Each control problem has its own task and design methods that maybe different from the others, but there is something alike in control objective. According to the previously published study[2] , all controlled plants have inherent variables to be controlled and it is most important to make these variables approach to the expected value as close as possible, which is the same goal for all control systems. 2.3
Control-equilibrium
On the basis of the concept of equilibrium in mechanics and its combination with the equilibrium-state, we give the concept of control-equilibrium of a control system. First, control-equilibrium-state is defined on the basis of the common control objective talked above. Definition 2 Suppose the difference of a variable between its actual value x(t) and its target value x∗ (t)
776
Liu Qiaoge, Fu Mengyin & Sun Changsheng
is e(t) = x∗ (t)−x(t). If at t0 , the difference is equal to zero and is at equilibrium-state, that is e(t) = 0 and e˙ = 0, then x(t) is at control-equilibrium-state at t0 . When x(t) is at control-equilibrium-state, e(t) is at zero equilibrium-state. From Definition 2, the control-equilibrium of a control system is defined as in Definition 3. Definition 3 For a controlled plant x˙ = f (x, u, t) with the control u(t), if the state vector x(t) is always at control-equilibrium-state from t0 to t1 , then the control system is in control-equilibrium in [t0 , t1 ]. When a control system is in control-equilibrium, the error e(t) of state x(t) between its target value and its actual value is zero every time. That is, in the time slice of [t0 , t1 ], both e(t) and its time derivative corresponding to each order are equal to zero. So the controlled variables have reached the desired values. 2.4 2.4.1
Some notes Staring point
The concept of control-equilibrium comes from the common control objective. For such goal, the control system must have an anticipant response to some control. When the system is at control-equilibrium, the goal is reached. So control-equilibrium can be considered as a reflection of the control action. 2.4.2 Systematicness The control system is composed of a controller and a controlled plant. Control-equilibrium is achieved by the controller’s action and is reflected by the plant’s activity. So control-equilibrium is of systematicness. 2.4.3 Nature According to Definitions 2 and 3, if a control system is at control-equilibrium in some time slice, then the control error between target value and actual value of the controlled variable is at zero equilibrium-state. So the nature of control-equilibrium is that the control error is zero and keeps zero, speaking tersely as “Be and Keep”. “Be” means that the error equals zero at an instant, and “Keep” means that the error equals zero for some time. 2.4.4 Timeliness Control-equilibrium describes the behavior of the sys-
tem for some time, such as [t0 , t1 ]. Suppose the control system lose the control-equilibrium from the instant t2 (t2 > t1 ) for some reason. By the controller operation, the control system may attain the controlequilibrium again starting from the instant t3 (t3 > t2 ). So the desirable control effect is to make the control system in control-equilibrium for all the time, keeping the control error at zero. 2.4.5 Relationship between control-equilibrium and other equilibrium In mechanics, the equilibrium of an object represents the state of zero composite external force. For a control system, the input signal of the whole system can be thought as the external force, including instructions and disturbances. When the composite action of the external force is equal to zero, the control system will be in control-equilibrium. That is, the system responds to the instruction signal properly and restrains the disturbance signal efficiently. Unlike the original equilibrium-state, the controlequilibrium-state is defined not only on the basis of the variable itself, but also the difference between the actual and target value of a variable. If a variable is at control-equilibrium-state, its error is at zero equilibrium-state. So control-equilibrium-state is a relative concept. To make a control system in controlequilibrium, the control error should be kept at zero equilibrium-state. 2.4.6 Control-equilibrium and stability Stability has a progressive character, and its purpose is to make the control error toward zero in infinity. The control-equilibrium emphasizes instantaneous property, and its purpose is to let the control error be and keep at zero for finite time. Both stability and control-equilibrium are highly significant for designing a control system.
3. How to reach control-equilibrium Similar to the previously published study[3] , the reason behind the intelligence of human beings is the utilization of their natural learning ability, but such ability is provided by the physiology structure. So for control systems, not only a complex control law but
Study of the control-equilibrium of control systems
777
also a proper control structure is important. This article mainly discusses which control structure is suitable for control-equilibrium.
forward control structure, the controller is excited by the external signal directly. The importance of such structure shows two aspects.
The main control structure includes closed-loop based on feedback control and open-loop based on feed-forward control. Each of them has some advantages and disadvantages. The next session gives a comparison from the aspect of control-equilibrium.
First, the external signals contain abundant information. The task of a control system is just to respond to the external signals properly. The feed-forward controller is stimulated by the external signals, while the feedback control is commonly based on the error, which can be viewed as a mask to the extern signal.
3.1
Disadvantage of feedback control
In engineering area, the most popular feedback control method is PID. Proportion control is based on the error of the system, but it can’t make the control system in controlequilibrium. Considering the balance as an example, take the weights and the balance as the control system, the balance as the controlled plant, the measured mass as the external input to the control system, and the weights as the control action. The purpose of the system is to make the balance at level. When the balance is at level, the system error is zero. According to the proportion control, once the error is zero, the control action is zero, which means the weights in the right pan are removed. But at the same time, the balance will lose level immediately. So the proportion control can make the system error at zero only for a moment, not for long time. As for the integral control, it is always used in the forward channel to make the steady error at zero. But the integral part adds an open-loop pole at the origin and has 90˚lag in phase that deteriorate the stability of the system. Also the integral action makes the system to respond to necessary overshoots, which is illustrated clearly in [4] by the Theorem 1.3.2. So the integral control can’t benefit control-equilibrium too. 3.2
Advantage of feed-forward control
Consider the balance as an example again. To measure the unknown mass and make the balance level, the right pan must have weights all the time. That means the weight should be added or removed according to the input, not with the error. Similarly for a control system, to make the system in control-equilibrium, the controller should make decision according to the external signal, not with the system error. In the feed-
Second, the external signal represents the direction of development. As in Ref. [5], the control action is needed for the system to work properly and also to ensure the development system in the desired direction. To let the balance level, there must have the same quality in two pans. To maintain the balance level, the right pans should always have these weights. This is similar to the description of control-equilibrium , that is ‘Be and Keep’. In feed-forward control, the external signal stimulates the system directly, which provides the necessary condition for the controlled plant to act toward the desired direction. Thus, to achieve the control-equilibrium of a control system, feed-forward control is necessary.
4. Theory of control-equilibrium for control systems On the basis of the above analysis, a summary is given below and is considered as the basis of equilibrium theory for control systems (1) Control-equilibrium is derived from the control purpose and can reveal the effect of the control action. (2) The nature of control-equilibrium of a system is that the control error is at zero and keeps zero. That is “Be and Keep”. (3) Feed-forward control is an efficient method to ensure control-equilibrium, which provides necessary conditions for the proper working of a system and its desirable development .
5. Conclusions Without restricting certain control problems, controlequilibrium was studied with the control purpose. Except the concept of equilibrium-state, not much has been talked about the equilibrium in control area. So this article proposed the equilibrium theory for control
778
Liu Qiaoge, Fu Mengyin & Sun Changsheng
problems.
of the Department of Automatic Control, School of In-
References
formation Science and Technology, Beijing Institute of Technology, China. Her major research field is motion
[1] Zhang Z F, Sun C S. Linear control system.
Beijing :
Beijing Institute of Technology Press, 2001: 135−142. [2] Shigeyuki Hosoe. System and control. Bai Y L. Ohmsha. Ltd and Science Press, 2001: 2−51. [3] Wiener N. The human use of human beings: Cyberneties and society. Chen Bu. Commerce Press, 1978: 7−56. [4] Seron Maria M, Braslavsky Julio H, Goodwin Graham C. Fundamental limitations in filtering and control. London: Springer, 1997: 1−84. [5] Ya Lerner A. Fundamentals of cybernetics. Liu D Y. Beijing: Science Press, 1980: 80−190.
Liu Qiaoge was born in 1978. She is the Ph. D.
control. E-mail: [email protected] Fu Mengyin was born in 1964. Now he is a professor of the Department of Automatic Control, School of Information Science and Technology, Beijing Institute of Technology, China. His major research fields include intelligent vehicle control and motion control. Sun Changsheng was born in 1944. Now he is a professor of the Department of Automatic Control, School of Information Science and Technology, Beijing Institute of Technology, China. His major research fields are pattern recognition and intelligent control.
Study of the control-equilibrium of control systems Liu Qiaoge, Fu Mengyin & Sun Changsheng Dept. of Automatic Control, School of Information Science and Technology, Beijing Inst. of Technology, Beijing 100081, P. R. China (Received December 2, 2006)
Abstract: Not so much had been talked about equilibrium in control area. On the basis of the phenomenon of balance, the concept of control-equilibrium and control-equilibrium of a control system is proposed. According to this theory, a perfect control method should not only guarantee stability of the system, but also ensure the control-equilibrium of the system. To achieve the control-equilibrium, feed-forward control is required.
Keywords: balance, control-equilibrium-state, control-equilibrium, feed-forward control.
1. Introduction The purpose of a control system is to make the changing process of the controlled states as close to the desired one. But there always exist some problems in the transition process such as oscillation, overshoot, and time delay. To overcome these problems, both control laws and control structures should be chosen properly. A balance is a type of weighting devices, with identical weighing pans hung at the either end. The left pan holds an unknown weight while the effective weight in the right pan is increased with the known amounts until the beam is level and motionless. Once the weight in the right pan is decreased, the beam will lose level immediately. So there must be enough weight in the right pan to maintain the “balance” of the balance. On the basis of this phenomenon, the concept of control-equilibrium is proposed and the equilibrium theory for control systems is attempted to build. And in this aspect, the validity of the feed-forward control and limitation of the feedback control is analyzed.
2. Control-equilibrium 2.1
About equilibrium
There are descriptions about equilibrium in some areas, such as in mechanics, thermotics, control area, etc.
In mechanics, if the composite external force in some direction is zero, the object is balanced and it will be either motionless or be in the uniform motion state in this direction. In control area, equilibrium-state is defined in Ref. [1] as follows. Definition 1 The plant is x˙ = f (x, t) and x(t) is an equilibrium-state, If x˙ = 0. That is x(t) = xe (constant vector). 2.2
Control objective
Each control problem has its own task and design methods that maybe different from the others, but there is something alike in control objective. According to the previously published study[2] , all controlled plants have inherent variables to be controlled and it is most important to make these variables approach to the expected value as close as possible, which is the same goal for all control systems. 2.3
Control-equilibrium
On the basis of the concept of equilibrium in mechanics and its combination with the equilibrium-state, we give the concept of control-equilibrium of a control system. First, control-equilibrium-state is defined on the basis of the common control objective talked above. Definition 2 Suppose the difference of a variable between its actual value x(t) and its target value x∗ (t)
776
Liu Qiaoge, Fu Mengyin & Sun Changsheng
is e(t) = x∗ (t)−x(t). If at t0 , the difference is equal to zero and is at equilibrium-state, that is e(t) = 0 and e˙ = 0, then x(t) is at control-equilibrium-state at t0 . When x(t) is at control-equilibrium-state, e(t) is at zero equilibrium-state. From Definition 2, the control-equilibrium of a control system is defined as in Definition 3. Definition 3 For a controlled plant x˙ = f (x, u, t) with the control u(t), if the state vector x(t) is always at control-equilibrium-state from t0 to t1 , then the control system is in control-equilibrium in [t0 , t1 ]. When a control system is in control-equilibrium, the error e(t) of state x(t) between its target value and its actual value is zero every time. That is, in the time slice of [t0 , t1 ], both e(t) and its time derivative corresponding to each order are equal to zero. So the controlled variables have reached the desired values. 2.4 2.4.1
Some notes Staring point
The concept of control-equilibrium comes from the common control objective. For such goal, the control system must have an anticipant response to some control. When the system is at control-equilibrium, the goal is reached. So control-equilibrium can be considered as a reflection of the control action. 2.4.2 Systematicness The control system is composed of a controller and a controlled plant. Control-equilibrium is achieved by the controller’s action and is reflected by the plant’s activity. So control-equilibrium is of systematicness. 2.4.3 Nature According to Definitions 2 and 3, if a control system is at control-equilibrium in some time slice, then the control error between target value and actual value of the controlled variable is at zero equilibrium-state. So the nature of control-equilibrium is that the control error is zero and keeps zero, speaking tersely as “Be and Keep”. “Be” means that the error equals zero at an instant, and “Keep” means that the error equals zero for some time. 2.4.4 Timeliness Control-equilibrium describes the behavior of the sys-
tem for some time, such as [t0 , t1 ]. Suppose the control system lose the control-equilibrium from the instant t2 (t2 > t1 ) for some reason. By the controller operation, the control system may attain the controlequilibrium again starting from the instant t3 (t3 > t2 ). So the desirable control effect is to make the control system in control-equilibrium for all the time, keeping the control error at zero. 2.4.5 Relationship between control-equilibrium and other equilibrium In mechanics, the equilibrium of an object represents the state of zero composite external force. For a control system, the input signal of the whole system can be thought as the external force, including instructions and disturbances. When the composite action of the external force is equal to zero, the control system will be in control-equilibrium. That is, the system responds to the instruction signal properly and restrains the disturbance signal efficiently. Unlike the original equilibrium-state, the controlequilibrium-state is defined not only on the basis of the variable itself, but also the difference between the actual and target value of a variable. If a variable is at control-equilibrium-state, its error is at zero equilibrium-state. So control-equilibrium-state is a relative concept. To make a control system in controlequilibrium, the control error should be kept at zero equilibrium-state. 2.4.6 Control-equilibrium and stability Stability has a progressive character, and its purpose is to make the control error toward zero in infinity. The control-equilibrium emphasizes instantaneous property, and its purpose is to let the control error be and keep at zero for finite time. Both stability and control-equilibrium are highly significant for designing a control system.
3. How to reach control-equilibrium Similar to the previously published study[3] , the reason behind the intelligence of human beings is the utilization of their natural learning ability, but such ability is provided by the physiology structure. So for control systems, not only a complex control law but
Study of the control-equilibrium of control systems
777
also a proper control structure is important. This article mainly discusses which control structure is suitable for control-equilibrium.
forward control structure, the controller is excited by the external signal directly. The importance of such structure shows two aspects.
The main control structure includes closed-loop based on feedback control and open-loop based on feed-forward control. Each of them has some advantages and disadvantages. The next session gives a comparison from the aspect of control-equilibrium.
First, the external signals contain abundant information. The task of a control system is just to respond to the external signals properly. The feed-forward controller is stimulated by the external signals, while the feedback control is commonly based on the error, which can be viewed as a mask to the extern signal.
3.1
Disadvantage of feedback control
In engineering area, the most popular feedback control method is PID. Proportion control is based on the error of the system, but it can’t make the control system in controlequilibrium. Considering the balance as an example, take the weights and the balance as the control system, the balance as the controlled plant, the measured mass as the external input to the control system, and the weights as the control action. The purpose of the system is to make the balance at level. When the balance is at level, the system error is zero. According to the proportion control, once the error is zero, the control action is zero, which means the weights in the right pan are removed. But at the same time, the balance will lose level immediately. So the proportion control can make the system error at zero only for a moment, not for long time. As for the integral control, it is always used in the forward channel to make the steady error at zero. But the integral part adds an open-loop pole at the origin and has 90˚lag in phase that deteriorate the stability of the system. Also the integral action makes the system to respond to necessary overshoots, which is illustrated clearly in [4] by the Theorem 1.3.2. So the integral control can’t benefit control-equilibrium too. 3.2
Advantage of feed-forward control
Consider the balance as an example again. To measure the unknown mass and make the balance level, the right pan must have weights all the time. That means the weight should be added or removed according to the input, not with the error. Similarly for a control system, to make the system in control-equilibrium, the controller should make decision according to the external signal, not with the system error. In the feed-
Second, the external signal represents the direction of development. As in Ref. [5], the control action is needed for the system to work properly and also to ensure the development system in the desired direction. To let the balance level, there must have the same quality in two pans. To maintain the balance level, the right pans should always have these weights. This is similar to the description of control-equilibrium , that is ‘Be and Keep’. In feed-forward control, the external signal stimulates the system directly, which provides the necessary condition for the controlled plant to act toward the desired direction. Thus, to achieve the control-equilibrium of a control system, feed-forward control is necessary.
4. Theory of control-equilibrium for control systems On the basis of the above analysis, a summary is given below and is considered as the basis of equilibrium theory for control systems (1) Control-equilibrium is derived from the control purpose and can reveal the effect of the control action. (2) The nature of control-equilibrium of a system is that the control error is at zero and keeps zero. That is “Be and Keep”. (3) Feed-forward control is an efficient method to ensure control-equilibrium, which provides necessary conditions for the proper working of a system and its desirable development .
5. Conclusions Without restricting certain control problems, controlequilibrium was studied with the control purpose. Except the concept of equilibrium-state, not much has been talked about the equilibrium in control area. So this article proposed the equilibrium theory for control
778
Liu Qiaoge, Fu Mengyin & Sun Changsheng
problems.
of the Department of Automatic Control, School of In-
References
formation Science and Technology, Beijing Institute of Technology, China. Her major research field is motion
[1] Zhang Z F, Sun C S. Linear control system.
Beijing :
Beijing Institute of Technology Press, 2001: 135−142. [2] Shigeyuki Hosoe. System and control. Bai Y L. Ohmsha. Ltd and Science Press, 2001: 2−51. [3] Wiener N. The human use of human beings: Cyberneties and society. Chen Bu. Commerce Press, 1978: 7−56. [4] Seron Maria M, Braslavsky Julio H, Goodwin Graham C. Fundamental limitations in filtering and control. London: Springer, 1997: 1−84. [5] Ya Lerner A. Fundamentals of cybernetics. Liu D Y. Beijing: Science Press, 1980: 80−190.
Liu Qiaoge was born in 1978. She is the Ph. D.
control. E-mail: [email protected] Fu Mengyin was born in 1964. Now he is a professor of the Department of Automatic Control, School of Information Science and Technology, Beijing Institute of Technology, China. His major research fields include intelligent vehicle control and motion control. Sun Changsheng was born in 1944. Now he is a professor of the Department of Automatic Control, School of Information Science and Technology, Beijing Institute of Technology, China. His major research fields are pattern recognition and intelligent control.