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Some Aspects on Pole-placement Design
C0 I' \I' i ~ hl © I L\C 11 Ih T rie n nial Wor ld COllgress. T allill Il. FSlo lli ;1. l 'SS R. I ~ 1~ Hl
SOME ASPECTS ON POLE-PLACEMENT DESIGN B. Wittenmark and K.
J.
Astrom
Departmmt of Automatic CO lltrol, Lund Institute of T echll olog::;. B ox 11 8. 5 -221 00 Lwui. 5wedfll
Abstract. Pole-placement design for SISO systems are discussed in input-output formulation. It is shown how the different design parameters influence the performance of the closed loop system. For instance, the influences of sampling period, anti-aliasing filter, and desired closed loop performance are treated in detail. Rules of thumb for the choice of the different parameters are given. Keywords:
1.
Pole-placement; controller design; sampled data systems
Introduction
Section 2 is a brief formulation of pole-placement design. The closed loop poles are given physical interpretation and grouped into different categories. This helps in selecting them. The notion of design variables is also introduced. These are the variables, which are at the disposal of the control designer. The control systems design problem is discussed in Section 3. Key issues like response to command signals, load disturbances, and measurement noise are investigated. Further robustness to process variations are discussed. In Section 4 we discuss how the criteria in Section 3 can be met by the pole-placement methods and we provide tools for investigating those aspects of the design problem that is not directly covered in the pole-placement design. Rules of thumb are given for the choice of the design parameters. Examples that illustrate the procedure are given in Section 5 and the paper ends with conclusions.
In the pole-placement design method the dynamics of the process and the disturbances are specified with linear dynamical models. The specifications are given in terms of the desired closed loop poles. There are simple analytical procedures to determine if the problem can be solved and if so to give a solution. Pole-placement control is simple. It is easy to understand and teach. It relates well to classical ideas like root locus and frequency domain methods. There are both state space and polynomial versions of the method. Hence it is a very good vehicle for bridging the gap between the ideas of internal (state space) and external (input-output) model approaches to linear systems. The method is almost identical for continuous and discrete time systems. Many other design methods can also be related to pole-placement. Typical examples are minimum variance, LQG, predictive control etc. It thus appears that the pole-placement method is an almost ideal design method. It has, however, some severe drawbacks:
•
It is difficult to relate engineering specifications to requirements on closed loop poles.
•
All closed loop poles have to be specified. This is often too stringent. Bad choices of some poles may result in poor control
2.
Pole-placement Design
The pole-placement design method for SISO sampled data systems is thoroughly discussed, for instance, in Astrom and Wittenmark (1984). Let the process to be controlled be described by, see Figure 1, B(q)
y=x+e= A(q)(u+v)+e
The controller consists of a feedback from the output of the process and a feedforward from the command signal. This results in a two-degree-of-freedom controller. The purpose of this paper is to suggest some refinements and design rules that make the poleplacement method a truly useful engineering design method. The paper describes the pole-placement design for sampled data single-input-single-output systems .
Figure 1. tion.
2 11
Process and controller configura-
(1)
where u is the control variable, y the measured output, v load disturbance, and e measurement noise. A and B are polynomials in the forward shift operator q. A and B are assumed to be relatively prime. Further, it is assumed that A is monic. The pole-excess d deg A - deg B is the time delay of the process. Let the desired response from the reference signal Yr to the output be described by
Equation (7) has a unique solution if and only if A and B- are relatively prime. Furthermore, it follows from (5) that B- must divide B ... , i.e. B ... B- B!,., and that (8)
=
=
Pole-placement design is obtained by using the controller (3), where the polynomials R, S, and Tare given by (6), (7), and (8). To get a causal controller, the model (2) must have the same or higher pole excess as the process (1). Further it is necessary that the observer polynomial is of sufficiently high order to guarantee that the Diophantine equation has a solution.
(2) Furthermore, let Aa(q) be a predetermined stable polynomial specified by the designer. Aa is called the observer polynomial. The reason is discussed in the end of this section. A general linear regulator can be described by
R(q)u
= T(q)u
c -
S(q)y
Disturbances with Known Dynamics The pole-placement design can take into account disturbances with known dynamics. Let the disturbance v be generated from the dynamical system
(3)
Elimination of u between (1) and (3) gives, when e = 0 and v = 0,
y
=
BT AR+ BS
U
where w is a pulse, a set of widely spread pulses, or white noise. For example, a step disturbance is generated by Ad(q) = q - 1
c
Introduce the polynomial Ac, that may be the same as A... or an approximation of A.... We want the closed loop system from U c to y be governed by Ac. The signal U c is generated from
From the system model (1) and the regulator (3) we get
BT AR+ BS
BR
y
=
This gives the following condition for exact model following design
u
= AR+ BS U c -
BT AR+BS
AT
U
c
AR
+ Ad(AR+ BS) W + AR+ BS e
(4)
BS Ad(AR+ BS)
W -
AS AR+ BS e (9)
The closed-loop characteristic polynomial thus contains the disturbance dynamics as a factor. This polynomial is typically unstable. It follows from (9) that in order to maintain a finite output in case of these disturbances Ad must divide R. This would make y finite, but the controlled input u may be infinite to compensate for an infinite disturbance. The polynomial R must thus be of the form
(5)
where the denominator AR + BS is the closed-loop characteristic polynomial. The denominator in (5) is usually of higher order than Ac. This implies that some poles and zeros are canceled. The polynomial B is facto red as
(10) This implies that a model for the disturbance dynamics is built into the regulator. The idea is called the internal model principle. See Francis and Wonham (1976). Notice that R will contain an integrator if the disturbance has step character. Using (10), the design equation becomes
where B+ is a monic polynomial whose zeros are stable and so well damped that they can be canceled by the regulator. B+ = 1 implies that there is no cancellation of any zeros. Since B+ is canceled, it must also factor the closed-loop characteristic polynomial. The characteristic polynomial of the closed loop system will also contain the observer dynamics. The closed loop characteristic polynomial is thus given by
(11) The controller is now obtained from (8), (10), and
(11). State Space Discussion
which is called the Diophantine equation. It follows that B+ divides R. Hence
State feedback and observer design can be interpreted in the polynomial formulation given above. In the state space formulation the closed loop characteristic polynomial from the reference value to the output is determined by the state feedback controller.
(6) (7)
212
This corresponds to Ae in the polynomial method. The observer dynamics is in the state space formulation not controllable from the reference input. This thus corresponds to the Canceled dynamics specified by Aa in the polynomial design.
The response is thus in essence governed by the plant zeros that are not Canceled and the polynomial Ae . The poles of Am can not be chosen significantly faster than the zeros of B-. The response to command signals does not depend on Aa. The load disturbance response does depend on Ae and Aa . It is possible to make a total separation of responses to command signals and load disturbances by using the twodegree-of-freedom structure shown in Figure 1. The disturbance dynamics is determined by AaAe and the reference value dynamics is shaped through the feedforward given by (4)
Design Parameters The polynomial formulation of the pole-placement problem contains several design parameters. The designer must specify: •
Closed loop dynamics Ae
•
Canceled zeros B+
• • •
Observer polynomial Aa
Sensitivity to Disturbances
Disturbance rejection polynomial Ad
The transfer functions relating the plant output to measurement errors and load disturbances is given by
Sampling period h
• Anti-aliasing filter Go.o. These parameters are discussed in Section 4. 3.
BT BR BS v x= AR+BS Ue + AR+BS - AR+BSe (13) Bm BRIAd B-S = TcUe + AaAe V - AaAe e
Key Design Issues
Control systems can be quite complicated because design is a compromise between many different factors. The following issues must typically be considered: • Command signal following
•
Load disturbances
•
Measurement noise
• •
Model uncertainty
•
State constraint
•
Controller complexity
The properties of the transfer functions can be assessed qualitatively from the location of the closed loop poles. The cancellation of disturbances with known dynamics is clearly seen from (13). For a more accurate assessment we recommend that to plot the Bode diagrams of the transfer functions .
Sensitivity to Model Errors
Actuator saturation
The sensitivity to unmodeled dynamics are well judged from the loop gain. In the sampled case this is given by
We will now discuss how these factors are related to the design parameters given in Section 2. In a good design it is often necessary to grasp all factors . There are few design methods that consider all these factors. With pole-placement design the controller complexity is given directly by the model of the process and the disturbance . Responses to command signals, load disturbances and measurement noise are easy to assess. The effects of model uncertainty have to be investigated separately either by frequency response methods or by simulation. Actuator saturation is typically dealt with by extensions of classical anti-wind up methods. See Astrom and Wittenmark (1984). State constraints are normally dealt with by simulation. The separation of the closed loop poles into control poles (zeros of Ae) and observer poles (zeros of Aa) makes it easy to separate the response to command signals from the response to load disturbances and measurement noise. In the standard case we choose Aa to give appropriate response to load disturbances and measurement noise.
where Hp and He are the pulse transfer functions of the process and the controller respectively. It is well known that the closed loop system is stable under plant variations t:.Hp provided that
It:.;:p I < 11 ~ L 1
A plot of the transfer function (1 + L) / L thus reveals the frequency ranges where high model precision is required. The relation is easy to apply when a design has been done. The right-hand side of (14) can be easily calculated and does not depend on the true transfer function. The conditions on the model precision can thus be expressed in terms of frequency domain conditions.
4.
Command Signal Response With standard pole-placement control the closed loop response is given by
Y
B - B'",.
B- B'",.
= --;r;;-ue = ~Yr
(14)
Choice of Design Parameters
In this section we will discuss the choice of design parameters that the control designer must do. The discussion results in practical rules of thumb that can be used.
(12)
21 3
Closed Loop Dynamics Ac
Sampled Data Systems
The polynomial Ac determines the closed loop behavior from the command signal U c , but in the standard case it also influences the responses from the load disturbance and the measurement noise, see (13). The faster dynamics that are required the larger control signals are demanded. See (9) . It is useful, also for sampled data systems, to specify the desired closed loop poles in continuous time form. Natural frequency and damping can then be used as design variables. A pair of control poles can be specified and the rest of the poles can be made a factor 1.5-2 faster than these poles. On some occasions there are open loop poles that should not be changed, but kept in the closed loop system. One example is fast open loop poles. For discrete time systems it can also be sensible not to change the open loop poles introduced by the antialiasing filter. The pole-placement design method has the drawback that all the poles have to be specified. There are methods, which instead place the poles within a specified region in the complex plane. See, for instance, Furuta and Kim (1987) and Wittenmark et al. (1987).
For sampled systems there are some additional considerations. The sampling period becomes a design parameter together with the anti-aliasing filter. They influence the response to measurement noise, load disturbances and the sensitivity to unmodeled dynamics. They have, however, little influence on the response to command signals. With a sampled system there will always be a delay in responding to load disturbances. A long sampling interval in connection with a properly anti-aliasing filter will make the system less sensitive to load disturbances .
Sampling Period The sampling period should be chosen in relation to the closed loop bandwidth, which also includes the observer polynomial. A rule of thumb is to choose the sampling period h (in seconds) according to wh:::= 0.2 - 0.6
where w is the natural frequency (in rad/s) of the corresponding continuous time control poles . This rule of thumb gives about 4-10 samples per rise time of the closed loop system or 15-45 samples per period. It is also necessary to take the load disturbances into account when selecting the sampling period. A load disturbance occurring just after a sampling time will act on the system a full sampling period before it can be detected. It can thus be necessary to choose the sampling time according to the lower limit in (15) . Too short a sampling period may introduce numerical difficulties in the calculation of the controller parameters and in the implementation of the controller.
Canceled Zeros The poles of the plant can be changed through feedback, but the zeros can be changed only through cancellation (B+), and addition of the desired zeros
(B:") . Sampled data systems frequently have zeros on the negative real axis ("sampling zeros") and outside the unit circle, see Astrom et al. (1984) . These zeros should not be Canceled even if they are inside the unit circle, since they will introduce ringing in the control signal. In the design it is therefore good practice not to change the open loop zeros but to keep them in the desired model.
Anti-aliasing Filter Most analog sensors have filters to attenuate measurement noise in the signals. The filters are seldom chosen for a particular control application or for sampling the signals. It is very important in design of sampled data systems to incorporate an anti-aliasing filter . The filter should ideally remove all frequencies in the signals over the N yquist frequency 7r / h rad/s. The high frequencies will otherwise be folded into low frequencies (aliasing) and will confuse the controller. High order anti-aliasing filters are obtained by cascading filters of the form
Observer Polynomial In the standard case the the observer polynomial is used to shape the responses for load disturbances and measurement noise, see (13). The control designer can choose Aa quite freely. As a rule of thumb the dynamics of Ao can be 1.5- 2 times faster than the desired dynamics Ac. Slow time constants in Ao will decrease the influence of measurement noise, but the reaction will be slow after a load disturbance. The influence of the measurement noise will increase when Ao is made faster. There is thus a compromise to be resolved depending on the nature of the disturbances acting on the system. When the properties of the disturbances are known, optimal filtering theory can be used to determine the optimal observer. For instance, if the system is described by
Ay
(15)
G .... (s)
=
w2
(S/WB)
2
+ 2(w (S/WB) + w 2
(16)
where WB is the desired bandwidth of the filter . The parameters W and ( for different filters are given, for instance, in Astrom and Wittenmark (1984). Unless a quite high sampling rate is chosen it is necessary to include the anti-aliasing filter in the design of the controller. The necessary filter will otherwise deteriorate the performance of the closed loop system. The filter must be taken into account in the design of the regulator if the desired crossover
= Bu+ Ce
where e is white noise, then the optimal choice is Aa C.
=
21 4
Table 1. Approximate time delay Td of Bessel filters of different orders.
Output
Input
3
Order 2
1.3
4
2.1 2.7
6
Ol~--'---~---r--~
o
40
30
10
0
20
30
40
Figure 2 . Simulation of the nominal design for the harmonic oscillator when W 1.5, Wob. = 3, ( (ob. 0 .7 and h 0 .2, with integrator in the regulator.
frequency is larger than about wB/10, where WB is the bandwidth of the filter. The Bessel filter can, however, be approximated with a time delay, since the filter has linear phase for low frequencies . Table 1 shows the delay for different orders of the filter . This implies that the sampled data model including the anti-aliasing filter can be assumed to contain an additional time delay compared to the process.
5.
20
10
=
=
=
=
= =
The nominal parameter values are chosen as ( 0.7, 1.5, (ob. 0.7, Wob. 3 and h 0.2. These specifications imply that significant damping is introduced and that the response speed is increased compared with the open loop system. Figure 2 shows the output and the input when the reference signal is a step at t 0, a step disturbance at the input at t 15 and discrete time white discrete time measurement noise with standard deviation 0.01 at t = 30.
W
Examples
=
=
=
The discussion of pole-placement design based on polynomial methods will be examplefied in this section. The influence of the different design parameters will be illustrated.
=
=
Changing Observer Poles Figure 3 shows the response when the observer poles are changed to Wob. = 4. The load disturbance is eliminated faster with a faster observer dynamics but the noise sensitivity also increases.
Nominal Design Let the process be the harmonic oscillator with the transfer function 2
G(5)-~ -
52
Wo
+ W~
=1
Changing the Sampling Period The sampling period in the nominal design was chosen such that
The sampled pulse transfer function is
H(q)
= (1 -
(3)(q + 1) q2 - 2{3q + 1
= B(q)
(3
A(q)
= cos(woh)
Wob.h
which is according to the upper limit of the rule of thumb. Figure 4 shows the responses when the sampling period is changed to h 0.1 and 1. The controller will respond faster after load disturbances, when the sampling interval is decreased. A too long sampling interval will increase the deviation after the load disturbance. Also the aliasing effect is seen when there is measurement noise since no antialiasing filter is used.
The desired response is characterized by the continuous time characteristic equation 52
+ 2(W5 + w
2
=
=0 =
The sampled data form of this polynomial is Ac (q) Am(q). Since the pulse transfer function has a zero at -1 no zero cancellation is allowed. This implies that B+ 1 and Bm BB . It is specified that the regulator should have integral action. This implies that the observer polynomial should be of second order.
=
=
= 0.6
=
Influence of Anti-aliasing Filter The influence of the measurement noise in Figure 2 indicates that an anti-aliasing filter should be used to eliminate the effect of the disturbance . The filter
This polynomial is transferred to sampled data form
Ao(q) .
Output
The Diophantine equation is
(l- 2{3q+ l)R(q)+ (1- (3)(q+ l)S(q)
= Ao(q)A",(q)
where Rand S are of second order. The integrator is introduced by enforcing that q - 1 is a factor of R. The polynomial T is given by
T( ) q
Input
3
I (
Ol+----r--~----~--~
o
10
20
30
40
0
10
20
30
Figure 3. Response of the pole placement design for the harmonic oscillator for the observer 4. dynamics Wob.
= A",(l)Ao(q)
=
B(l)
21 5
40
Output
,..., 1
10
Input
3
10
0 0
10
20
30
40
0
10
20
30
4
-
2 ........ .....
"
......
,./
' Integrator
40 '.
10° Output
,
Input
3
o inte grator
o
10
20
30
40
0
10
20
30
Figure 6. Bode diagram for the loop transfer function L in the nominal case when the controller is designed with and without integral action.
40
Figure 4. Response of the pole placement design for the harmonic oscillator for sampling intervals a) h = 0 .1 b) h = 1.
Robustness will, however, also influence the closed loop response unless the bandwidth of the filter is an order of magnitude higher than the desired bandwidth of the closed loop system. The closed loop system will be unstable when the nominal controller is used on the system with a Bessel anti-aliasing filter, with WB 27r. The filter introduces too much phase lag at the cross over frequency. The filter dynamics should thus be considered when designing the controller. The Bessel filter can be well approximated by a delay. This simplifies the design and reduces the order of the controller. A sixth order Bessel filter 2.7 /WB. The filter is approximated as a delay T bandwidth is chosen as WB 27r, which gives T = 0.43. To incorporate the delay it is necessary to increase the order of the regulator such that deg R deg S deg T 5. Figure 5 shows the response with anti-aliasing filter when the design is made by approximating the filter by a delay. The filter will significantly reduce the influence of the measurement noise. The filter will, however, also increase the deviation due to the load disturbance. The filter can be disregarded provided the filter bandwidth is more than ten times the desired closed loop bandwidth. This implies , however, also that the ~ampling frequency must be increased by an order of magnitude.
The consequences of added unmodeled dynamics is illustrated next. Figure 6 shows the Bode diagram for the loop transfer function L HpHc when the pole placement controller is designed for the nominal case. For comparison the loop transfer function is shown for the case when no integral action is required in the controller. The enforced integral action implies that less model precision is needed for all frequencies.
=
=
6.
= =
=
=
The paper has given rules of thumb for selecting the design parameters in pole-placement design. With these rules it is quite easy to make a proper design of the closed loop system. The design must, however, be tested through analysis and simulations.
=
Output
7.
ASTRClM, K .J . and B . WITTENMARK (1984) : Computer Controlled Systems, Prentice Hall Inc, Englewood Cliffs, N.J. FRANCIS, B.A . and W .M. WONHAM (1976) : "The internal model principle of control theory," Automatica, 12, 457-465.
2
IV
FURUTA, K. and S .B . KIM (1987) : "Pole assignment in a specified disk," IEEE Trans. Aut. Control, AC-32, 423- 427.
o O~---r---.----~--~
o
10
20
30
40
0
References
ASTRClM, K.J., P . HAGANDER, and J. STERNBY (1984): "Zeros of sampled systems," Automatica, 20 , 31- 38.
Input
3
Summary
10
20
30
40
WITTENMARK, B., R .J . EVANs, and Y .C. SOH (1987): "Constrained pole-placement using transformation and LQ-design," Automatica, 23, 467-469.
Figure 5. Response of pole-placement design when using anti-aliasing filter. The filter is approximated with a delay in the design. 2 16
SOME ASPECTS ON POLE-PLACEMENT DESIGN B. Wittenmark and K.
J.
Astrom
Departmmt of Automatic CO lltrol, Lund Institute of T echll olog::;. B ox 11 8. 5 -221 00 Lwui. 5wedfll
Abstract. Pole-placement design for SISO systems are discussed in input-output formulation. It is shown how the different design parameters influence the performance of the closed loop system. For instance, the influences of sampling period, anti-aliasing filter, and desired closed loop performance are treated in detail. Rules of thumb for the choice of the different parameters are given. Keywords:
1.
Pole-placement; controller design; sampled data systems
Introduction
Section 2 is a brief formulation of pole-placement design. The closed loop poles are given physical interpretation and grouped into different categories. This helps in selecting them. The notion of design variables is also introduced. These are the variables, which are at the disposal of the control designer. The control systems design problem is discussed in Section 3. Key issues like response to command signals, load disturbances, and measurement noise are investigated. Further robustness to process variations are discussed. In Section 4 we discuss how the criteria in Section 3 can be met by the pole-placement methods and we provide tools for investigating those aspects of the design problem that is not directly covered in the pole-placement design. Rules of thumb are given for the choice of the design parameters. Examples that illustrate the procedure are given in Section 5 and the paper ends with conclusions.
In the pole-placement design method the dynamics of the process and the disturbances are specified with linear dynamical models. The specifications are given in terms of the desired closed loop poles. There are simple analytical procedures to determine if the problem can be solved and if so to give a solution. Pole-placement control is simple. It is easy to understand and teach. It relates well to classical ideas like root locus and frequency domain methods. There are both state space and polynomial versions of the method. Hence it is a very good vehicle for bridging the gap between the ideas of internal (state space) and external (input-output) model approaches to linear systems. The method is almost identical for continuous and discrete time systems. Many other design methods can also be related to pole-placement. Typical examples are minimum variance, LQG, predictive control etc. It thus appears that the pole-placement method is an almost ideal design method. It has, however, some severe drawbacks:
•
It is difficult to relate engineering specifications to requirements on closed loop poles.
•
All closed loop poles have to be specified. This is often too stringent. Bad choices of some poles may result in poor control
2.
Pole-placement Design
The pole-placement design method for SISO sampled data systems is thoroughly discussed, for instance, in Astrom and Wittenmark (1984). Let the process to be controlled be described by, see Figure 1, B(q)
y=x+e= A(q)(u+v)+e
The controller consists of a feedback from the output of the process and a feedforward from the command signal. This results in a two-degree-of-freedom controller. The purpose of this paper is to suggest some refinements and design rules that make the poleplacement method a truly useful engineering design method. The paper describes the pole-placement design for sampled data single-input-single-output systems .
Figure 1. tion.
2 11
Process and controller configura-
(1)
where u is the control variable, y the measured output, v load disturbance, and e measurement noise. A and B are polynomials in the forward shift operator q. A and B are assumed to be relatively prime. Further, it is assumed that A is monic. The pole-excess d deg A - deg B is the time delay of the process. Let the desired response from the reference signal Yr to the output be described by
Equation (7) has a unique solution if and only if A and B- are relatively prime. Furthermore, it follows from (5) that B- must divide B ... , i.e. B ... B- B!,., and that (8)
=
=
Pole-placement design is obtained by using the controller (3), where the polynomials R, S, and Tare given by (6), (7), and (8). To get a causal controller, the model (2) must have the same or higher pole excess as the process (1). Further it is necessary that the observer polynomial is of sufficiently high order to guarantee that the Diophantine equation has a solution.
(2) Furthermore, let Aa(q) be a predetermined stable polynomial specified by the designer. Aa is called the observer polynomial. The reason is discussed in the end of this section. A general linear regulator can be described by
R(q)u
= T(q)u
c -
S(q)y
Disturbances with Known Dynamics The pole-placement design can take into account disturbances with known dynamics. Let the disturbance v be generated from the dynamical system
(3)
Elimination of u between (1) and (3) gives, when e = 0 and v = 0,
y
=
BT AR+ BS
U
where w is a pulse, a set of widely spread pulses, or white noise. For example, a step disturbance is generated by Ad(q) = q - 1
c
Introduce the polynomial Ac, that may be the same as A... or an approximation of A.... We want the closed loop system from U c to y be governed by Ac. The signal U c is generated from
From the system model (1) and the regulator (3) we get
BT AR+ BS
BR
y
=
This gives the following condition for exact model following design
u
= AR+ BS U c -
BT AR+BS
AT
U
c
AR
+ Ad(AR+ BS) W + AR+ BS e
(4)
BS Ad(AR+ BS)
W -
AS AR+ BS e (9)
The closed-loop characteristic polynomial thus contains the disturbance dynamics as a factor. This polynomial is typically unstable. It follows from (9) that in order to maintain a finite output in case of these disturbances Ad must divide R. This would make y finite, but the controlled input u may be infinite to compensate for an infinite disturbance. The polynomial R must thus be of the form
(5)
where the denominator AR + BS is the closed-loop characteristic polynomial. The denominator in (5) is usually of higher order than Ac. This implies that some poles and zeros are canceled. The polynomial B is facto red as
(10) This implies that a model for the disturbance dynamics is built into the regulator. The idea is called the internal model principle. See Francis and Wonham (1976). Notice that R will contain an integrator if the disturbance has step character. Using (10), the design equation becomes
where B+ is a monic polynomial whose zeros are stable and so well damped that they can be canceled by the regulator. B+ = 1 implies that there is no cancellation of any zeros. Since B+ is canceled, it must also factor the closed-loop characteristic polynomial. The characteristic polynomial of the closed loop system will also contain the observer dynamics. The closed loop characteristic polynomial is thus given by
(11) The controller is now obtained from (8), (10), and
(11). State Space Discussion
which is called the Diophantine equation. It follows that B+ divides R. Hence
State feedback and observer design can be interpreted in the polynomial formulation given above. In the state space formulation the closed loop characteristic polynomial from the reference value to the output is determined by the state feedback controller.
(6) (7)
212
This corresponds to Ae in the polynomial method. The observer dynamics is in the state space formulation not controllable from the reference input. This thus corresponds to the Canceled dynamics specified by Aa in the polynomial design.
The response is thus in essence governed by the plant zeros that are not Canceled and the polynomial Ae . The poles of Am can not be chosen significantly faster than the zeros of B-. The response to command signals does not depend on Aa. The load disturbance response does depend on Ae and Aa . It is possible to make a total separation of responses to command signals and load disturbances by using the twodegree-of-freedom structure shown in Figure 1. The disturbance dynamics is determined by AaAe and the reference value dynamics is shaped through the feedforward given by (4)
Design Parameters The polynomial formulation of the pole-placement problem contains several design parameters. The designer must specify: •
Closed loop dynamics Ae
•
Canceled zeros B+
• • •
Observer polynomial Aa
Sensitivity to Disturbances
Disturbance rejection polynomial Ad
The transfer functions relating the plant output to measurement errors and load disturbances is given by
Sampling period h
• Anti-aliasing filter Go.o. These parameters are discussed in Section 4. 3.
BT BR BS v x= AR+BS Ue + AR+BS - AR+BSe (13) Bm BRIAd B-S = TcUe + AaAe V - AaAe e
Key Design Issues
Control systems can be quite complicated because design is a compromise between many different factors. The following issues must typically be considered: • Command signal following
•
Load disturbances
•
Measurement noise
• •
Model uncertainty
•
State constraint
•
Controller complexity
The properties of the transfer functions can be assessed qualitatively from the location of the closed loop poles. The cancellation of disturbances with known dynamics is clearly seen from (13). For a more accurate assessment we recommend that to plot the Bode diagrams of the transfer functions .
Sensitivity to Model Errors
Actuator saturation
The sensitivity to unmodeled dynamics are well judged from the loop gain. In the sampled case this is given by
We will now discuss how these factors are related to the design parameters given in Section 2. In a good design it is often necessary to grasp all factors . There are few design methods that consider all these factors. With pole-placement design the controller complexity is given directly by the model of the process and the disturbance . Responses to command signals, load disturbances and measurement noise are easy to assess. The effects of model uncertainty have to be investigated separately either by frequency response methods or by simulation. Actuator saturation is typically dealt with by extensions of classical anti-wind up methods. See Astrom and Wittenmark (1984). State constraints are normally dealt with by simulation. The separation of the closed loop poles into control poles (zeros of Ae) and observer poles (zeros of Aa) makes it easy to separate the response to command signals from the response to load disturbances and measurement noise. In the standard case we choose Aa to give appropriate response to load disturbances and measurement noise.
where Hp and He are the pulse transfer functions of the process and the controller respectively. It is well known that the closed loop system is stable under plant variations t:.Hp provided that
It:.;:p I < 11 ~ L 1
A plot of the transfer function (1 + L) / L thus reveals the frequency ranges where high model precision is required. The relation is easy to apply when a design has been done. The right-hand side of (14) can be easily calculated and does not depend on the true transfer function. The conditions on the model precision can thus be expressed in terms of frequency domain conditions.
4.
Command Signal Response With standard pole-placement control the closed loop response is given by
Y
B - B'",.
B- B'",.
= --;r;;-ue = ~Yr
(14)
Choice of Design Parameters
In this section we will discuss the choice of design parameters that the control designer must do. The discussion results in practical rules of thumb that can be used.
(12)
21 3
Closed Loop Dynamics Ac
Sampled Data Systems
The polynomial Ac determines the closed loop behavior from the command signal U c , but in the standard case it also influences the responses from the load disturbance and the measurement noise, see (13). The faster dynamics that are required the larger control signals are demanded. See (9) . It is useful, also for sampled data systems, to specify the desired closed loop poles in continuous time form. Natural frequency and damping can then be used as design variables. A pair of control poles can be specified and the rest of the poles can be made a factor 1.5-2 faster than these poles. On some occasions there are open loop poles that should not be changed, but kept in the closed loop system. One example is fast open loop poles. For discrete time systems it can also be sensible not to change the open loop poles introduced by the antialiasing filter. The pole-placement design method has the drawback that all the poles have to be specified. There are methods, which instead place the poles within a specified region in the complex plane. See, for instance, Furuta and Kim (1987) and Wittenmark et al. (1987).
For sampled systems there are some additional considerations. The sampling period becomes a design parameter together with the anti-aliasing filter. They influence the response to measurement noise, load disturbances and the sensitivity to unmodeled dynamics. They have, however, little influence on the response to command signals. With a sampled system there will always be a delay in responding to load disturbances. A long sampling interval in connection with a properly anti-aliasing filter will make the system less sensitive to load disturbances .
Sampling Period The sampling period should be chosen in relation to the closed loop bandwidth, which also includes the observer polynomial. A rule of thumb is to choose the sampling period h (in seconds) according to wh:::= 0.2 - 0.6
where w is the natural frequency (in rad/s) of the corresponding continuous time control poles . This rule of thumb gives about 4-10 samples per rise time of the closed loop system or 15-45 samples per period. It is also necessary to take the load disturbances into account when selecting the sampling period. A load disturbance occurring just after a sampling time will act on the system a full sampling period before it can be detected. It can thus be necessary to choose the sampling time according to the lower limit in (15) . Too short a sampling period may introduce numerical difficulties in the calculation of the controller parameters and in the implementation of the controller.
Canceled Zeros The poles of the plant can be changed through feedback, but the zeros can be changed only through cancellation (B+), and addition of the desired zeros
(B:") . Sampled data systems frequently have zeros on the negative real axis ("sampling zeros") and outside the unit circle, see Astrom et al. (1984) . These zeros should not be Canceled even if they are inside the unit circle, since they will introduce ringing in the control signal. In the design it is therefore good practice not to change the open loop zeros but to keep them in the desired model.
Anti-aliasing Filter Most analog sensors have filters to attenuate measurement noise in the signals. The filters are seldom chosen for a particular control application or for sampling the signals. It is very important in design of sampled data systems to incorporate an anti-aliasing filter . The filter should ideally remove all frequencies in the signals over the N yquist frequency 7r / h rad/s. The high frequencies will otherwise be folded into low frequencies (aliasing) and will confuse the controller. High order anti-aliasing filters are obtained by cascading filters of the form
Observer Polynomial In the standard case the the observer polynomial is used to shape the responses for load disturbances and measurement noise, see (13). The control designer can choose Aa quite freely. As a rule of thumb the dynamics of Ao can be 1.5- 2 times faster than the desired dynamics Ac. Slow time constants in Ao will decrease the influence of measurement noise, but the reaction will be slow after a load disturbance. The influence of the measurement noise will increase when Ao is made faster. There is thus a compromise to be resolved depending on the nature of the disturbances acting on the system. When the properties of the disturbances are known, optimal filtering theory can be used to determine the optimal observer. For instance, if the system is described by
Ay
(15)
G .... (s)
=
w2
(S/WB)
2
+ 2(w (S/WB) + w 2
(16)
where WB is the desired bandwidth of the filter . The parameters W and ( for different filters are given, for instance, in Astrom and Wittenmark (1984). Unless a quite high sampling rate is chosen it is necessary to include the anti-aliasing filter in the design of the controller. The necessary filter will otherwise deteriorate the performance of the closed loop system. The filter must be taken into account in the design of the regulator if the desired crossover
= Bu+ Ce
where e is white noise, then the optimal choice is Aa C.
=
21 4
Table 1. Approximate time delay Td of Bessel filters of different orders.
Output
Input
3
Order 2
1.3
4
2.1 2.7
6
Ol~--'---~---r--~
o
40
30
10
0
20
30
40
Figure 2 . Simulation of the nominal design for the harmonic oscillator when W 1.5, Wob. = 3, ( (ob. 0 .7 and h 0 .2, with integrator in the regulator.
frequency is larger than about wB/10, where WB is the bandwidth of the filter. The Bessel filter can, however, be approximated with a time delay, since the filter has linear phase for low frequencies . Table 1 shows the delay for different orders of the filter . This implies that the sampled data model including the anti-aliasing filter can be assumed to contain an additional time delay compared to the process.
5.
20
10
=
=
=
=
= =
The nominal parameter values are chosen as ( 0.7, 1.5, (ob. 0.7, Wob. 3 and h 0.2. These specifications imply that significant damping is introduced and that the response speed is increased compared with the open loop system. Figure 2 shows the output and the input when the reference signal is a step at t 0, a step disturbance at the input at t 15 and discrete time white discrete time measurement noise with standard deviation 0.01 at t = 30.
W
Examples
=
=
=
The discussion of pole-placement design based on polynomial methods will be examplefied in this section. The influence of the different design parameters will be illustrated.
=
=
Changing Observer Poles Figure 3 shows the response when the observer poles are changed to Wob. = 4. The load disturbance is eliminated faster with a faster observer dynamics but the noise sensitivity also increases.
Nominal Design Let the process be the harmonic oscillator with the transfer function 2
G(5)-~ -
52
Wo
+ W~
=1
Changing the Sampling Period The sampling period in the nominal design was chosen such that
The sampled pulse transfer function is
H(q)
= (1 -
(3)(q + 1) q2 - 2{3q + 1
= B(q)
(3
A(q)
= cos(woh)
Wob.h
which is according to the upper limit of the rule of thumb. Figure 4 shows the responses when the sampling period is changed to h 0.1 and 1. The controller will respond faster after load disturbances, when the sampling interval is decreased. A too long sampling interval will increase the deviation after the load disturbance. Also the aliasing effect is seen when there is measurement noise since no antialiasing filter is used.
The desired response is characterized by the continuous time characteristic equation 52
+ 2(W5 + w
2
=
=0 =
The sampled data form of this polynomial is Ac (q) Am(q). Since the pulse transfer function has a zero at -1 no zero cancellation is allowed. This implies that B+ 1 and Bm BB . It is specified that the regulator should have integral action. This implies that the observer polynomial should be of second order.
=
=
= 0.6
=
Influence of Anti-aliasing Filter The influence of the measurement noise in Figure 2 indicates that an anti-aliasing filter should be used to eliminate the effect of the disturbance . The filter
This polynomial is transferred to sampled data form
Ao(q) .
Output
The Diophantine equation is
(l- 2{3q+ l)R(q)+ (1- (3)(q+ l)S(q)
= Ao(q)A",(q)
where Rand S are of second order. The integrator is introduced by enforcing that q - 1 is a factor of R. The polynomial T is given by
T( ) q
Input
3
I (
Ol+----r--~----~--~
o
10
20
30
40
0
10
20
30
Figure 3. Response of the pole placement design for the harmonic oscillator for the observer 4. dynamics Wob.
= A",(l)Ao(q)
=
B(l)
21 5
40
Output
,..., 1
10
Input
3
10
0 0
10
20
30
40
0
10
20
30
4
-
2 ........ .....
"
......
,./
' Integrator
40 '.
10° Output
,
Input
3
o inte grator
o
10
20
30
40
0
10
20
30
Figure 6. Bode diagram for the loop transfer function L in the nominal case when the controller is designed with and without integral action.
40
Figure 4. Response of the pole placement design for the harmonic oscillator for sampling intervals a) h = 0 .1 b) h = 1.
Robustness will, however, also influence the closed loop response unless the bandwidth of the filter is an order of magnitude higher than the desired bandwidth of the closed loop system. The closed loop system will be unstable when the nominal controller is used on the system with a Bessel anti-aliasing filter, with WB 27r. The filter introduces too much phase lag at the cross over frequency. The filter dynamics should thus be considered when designing the controller. The Bessel filter can be well approximated by a delay. This simplifies the design and reduces the order of the controller. A sixth order Bessel filter 2.7 /WB. The filter is approximated as a delay T bandwidth is chosen as WB 27r, which gives T = 0.43. To incorporate the delay it is necessary to increase the order of the regulator such that deg R deg S deg T 5. Figure 5 shows the response with anti-aliasing filter when the design is made by approximating the filter by a delay. The filter will significantly reduce the influence of the measurement noise. The filter will, however, also increase the deviation due to the load disturbance. The filter can be disregarded provided the filter bandwidth is more than ten times the desired closed loop bandwidth. This implies , however, also that the ~ampling frequency must be increased by an order of magnitude.
The consequences of added unmodeled dynamics is illustrated next. Figure 6 shows the Bode diagram for the loop transfer function L HpHc when the pole placement controller is designed for the nominal case. For comparison the loop transfer function is shown for the case when no integral action is required in the controller. The enforced integral action implies that less model precision is needed for all frequencies.
=
=
6.
= =
=
=
The paper has given rules of thumb for selecting the design parameters in pole-placement design. With these rules it is quite easy to make a proper design of the closed loop system. The design must, however, be tested through analysis and simulations.
=
Output
7.
ASTRClM, K .J . and B . WITTENMARK (1984) : Computer Controlled Systems, Prentice Hall Inc, Englewood Cliffs, N.J. FRANCIS, B.A . and W .M. WONHAM (1976) : "The internal model principle of control theory," Automatica, 12, 457-465.
2
IV
FURUTA, K. and S .B . KIM (1987) : "Pole assignment in a specified disk," IEEE Trans. Aut. Control, AC-32, 423- 427.
o O~---r---.----~--~
o
10
20
30
40
0
References
ASTRClM, K.J., P . HAGANDER, and J. STERNBY (1984): "Zeros of sampled systems," Automatica, 20 , 31- 38.
Input
3
Summary
10
20
30
40
WITTENMARK, B., R .J . EVANs, and Y .C. SOH (1987): "Constrained pole-placement using transformation and LQ-design," Automatica, 23, 467-469.
Figure 5. Response of pole-placement design when using anti-aliasing filter. The filter is approximated with a delay in the design. 2 16