Technical report from Automatic Control at Linköpings universitet
Parametric Controller Strategies
Daniel Petersson
Division of Automatic Control
E-mail: petersson@isy.liu.se
29th April 2015
Report no.: LiTH-ISY-R-3084
Address:
Department of Electrical Engineering Linköpings universitet
SE-581 83 Linköping, Sweden
WWW: http://www.control.isy.liu.se
AUTOMATIC CONTROL REGLERTEKNIK LINKÖPINGS UNIVERSITET
Technical reports from the Automatic Control group in Linköping are available from http://www.control.isy.liu.se/publications.
Abstract
This report describes the modeling of a sounding rocket using a linear time varying model. Different control strategies are investigated to control the time varying model.
Keywords: Linear Parameter Varying Systems; Controller Synthesis; Linear
Parametric Controller Strategies
Daniel Petersson
2015-04-29
1
Introduction
As in many engineering areas today, there is a strive for making lighter and more fuel-efficient constructions. This most often results in constructions that are weaker and more advanced controller strategies are needed to keep the same performance.
The system studied in this report does not only include a flexible structure, the dynamics also varies in time. For systems where the dynamics varies over time the most common controller is a gain-scheduled controller. This type of controllers are synthesized using linear time-invariant theory for different values of relevant parameters that describe the system changing over time. Then these controllers are interpolated over the parameters to get a continuous controller for all parameter values and scheduled on these parameters. For this type of controller to work well it is assumed that the parameters in the model varies slowly with time (see [7]). However, for a sounding rocket this is not the case, where parameters can vary fast, e.g., throttle and mass. This calls for different synthesis methods that can guarantee stability and performance for the whole time period even when the parameters change fast.
In this report a linear time-varying model of a sounding rocket have been derived and new controller strategies have been investigated to be used on this model to be able to guarantee stability and performance for the whole time interval.
2
Model
To model the rocket, Lagrangian mechanics is used. The Lagrangian,L = T −V ,
is defined as the difference between the kinematic energy,T , and the potential
energy,V , of the system.
The dynamic equations, in term of the coordinates used to express the kinetic and potential energy, are obtained as
fi = d dt ∂L ∂ ˙qi −∂L ∂qi (1) where qi are the generalized coordinates, ˙qi are the corresponding generalized
2.1
The flexible model
Using (1), the flexible modes of the model can be described as
L = T − V = 1 2 X i µiξ˙2− 1 2 X i µiωi2ξ2
where ξi is the flexible mode coordinate,ωi the frequency of the flexible mode
andµi its generalized mass. The physical coordinates of the deflectionz in the
pointx are given by
z(x) =X
i
ψi(x)ξi
where ψ(x) defines the deflection of the i:th mode at the point x. A physical
force, F , (or moment) acting at the point x is translated into the components
of the generalized forces,fi =F ψi(x). The dynamic equations for the flexible
system become fi= d dt µiξ˙ + µiω 2 iξi or fi=µi ξ + 2ζ¨ iωiξ + ω˙ 2iξi
where we also have added a damping term with a relative damping of ζi and
assumed that the generalized mass,µi, is time-invariant.
With these equations it is also possible to describe the rigid body modes, translational and rotational, with zero modal frequency,ωi = 0.
2.2
Jet damping
Modeling the rocket we also need to consider jet damping. Jet damping is a phenomena caused by the exhaust from the launcher’s engines. The launcher loses mass throughout the burn of the rocket, hence the derivative of the angular momentum of the mode, µiξi, need to be modified in order to compensate for
the loss caused by the leaving mass. The generalized mass can be expressed as
µi= Z ψi2(x)dx which leads to fi=µi ξ¨i+ 2ζiωiξ˙i+ωi2ξi + ˙µi− ψi2(xNozzle) ˙m ˙ ξi
2.3
Nozzle dynamics
Thrust vector control, TVC, is the concept of controlling a launcher or rocket by means of swiveled or gimbaled nozzles. The gimbal of the nozzle is controlled by a servo system, which inherently has dynamic features and which interacts with the dynamics of the launcher including its flexible modes. To include the TVC dynamics in the dynamic model of the launcher, the dynamics of the TVC and the launcher are integrated and treated as one flexible model.
2.4
Aerodynamics
The aerodynamics is modeled as lateral aerodynamics forces acting on the launcher. The drag is not considered. The aerodynamic forces depend on the local angle of attack
α(x) = w vx +X i ψi0(x)ξi− 1 vx ψiξ˙i ,
wherew is the lateral wind velocity and vx the launcher’s longitudinal velocity.
The local aerodynamic forces can then be computed as
F (x) = qDYNsREF∂CN(x, y)
∂α α(x).
2.5
The complete model
Using the sections above we can now put together the complete model. The ltv state-space model includes the following states:
• Lateral velocity, ˙z
• Pitch rate, ˙θ, and pitch angle, θ
• Flexible modal rates, ˙ξi, and deflections,ξi
and the model becomes ˙
x = (A0+ AAERO+ ATVC+ AJET)x + BTVCu + BWINDw (2a)
y = C0x (2b)
or
˙
x = (A0+ AAERO+ AJET)x + BCANu + BWINDw (3a)
y = C0x, (3b)
where the different matrices are defined below. Note that, as said in Section 2.1, we can regard both the lateral velocity, ˙z = ˙ξz, and attitude angle,θ = ξθ,
as modes with zero frequency and ψz(x) = 1, ψ0z(x) = 0, µz =m, ψθ(x) = x,
ψ0
θ(x) = x and µθ=Iyy. This makes the notation more compact and uniform.
For the modal part we have
A0= 0 0 0 0 0 0 0 . . . 0 0 0 0 0 0 0 . . . 0 1 0 0 0 0 0 . . . 0 0 0 −2ζ1ω1 −ω1 0 0 . . . 0 0 0 1 0 0 0 . . . 0 0 0 0 0 −2ζ1ω1 −ω1 . . . 0 0 0 0 0 1 0 . . . .. . ... ... ... ... ... ... . ..
The aerodynamic part is described as AAERO = 0 ? ? ? ? ? ? . . . 0 ? ? ? ? ? ? . . . 0 0 0 0 0 0 0 . . . 0 ? ? ? ? ? ? . . . 0 0 0 0 0 0 0 . . . 0 ? ? ? ? ? ? . . . 0 0 0 0 0 0 0 . . . .. . ... ... ... ... ... ... . .. , BCAN= ? ? 0 ? 0 ? 0 .. . , BWIND = ? ? 0 ? 0 ? 0 .. . where
AAERO,i,j=qDYNsREF
µi X x∈Body ψi(x) X y∈Body ∂CN(x, y) ∂α −v1 xψj(y) ψ 0 j(y) 0 0 ,
BCAN=qDYNsREF
µi X x∈Body ψi(x) ∂CN(x) ∂δ , BWIND=qDYNsREF
vxµi X x∈Body ψi(x) X y∈Body ∂CN(x, y) ∂α .
Now letψ0(TVC) denote the local relative angular deflection of the TVC. The TVC interaction becomes
ATVC= 0 0 ? 0 ? 0 ? . . . 0 0 ? 0 ? 0 ? . . . 0 0 0 0 0 0 0 . . . 0 0 ? 0 ? 0 ? . . . 0 0 0 0 0 0 0 . . . 0 0 ? 0 ? 0 ? . . . 0 0 0 0 0 0 0 . . . .. . ... ... ... ... ... ... . .. , BTVC=T 1 m x(TVC) Iyy 0 ψ1(TVC) µ1 0 ψ2(TVC) µ2 0 .. . where ATVC,i,j = T µi ψi(TVC)ψj0(TVC) and the jet damping effects are described by
AJ ET = 0 0 0 0 0 0 0 . . . 0 ˙mx2(Nozzle)− ˙Iyy Iyy 0 0 0 0 0 . . . 0 0 0 0 0 0 0 . . . 0 0 0 0 0 0 0 . . . 0 0 0 0 0 0 0 . . . 0 0 0 0 0 0 0 . . . 0 0 0 0 0 0 0 . . . .. . ... ... ... ... ... ... . .. .
G
W1 W2
Figure 1: System with filters
C0=0 0 1 0 0 0 0 . . . .
Remember that these models, (2) and (3), are ltv models. We have omitted the time dependences in the system matrices above but most parameters in the models are time varying.
2.6
Numerical model
Given numerical values for the parameters, e.g., mass and throttle, for a suffi-cient amount of different time instances. Then the parameters can be interpo-lated to be able to get a continuous time ltv model, using the matrices above. These can then be used for both synthesis, analysis and simulation.
3
Controller Strategies
Today the control synthesis is made by synthesizing control laws in a few time instances and then these control laws are interpolated to get one control law covering the whole time interval. This way of doing the control law is called gain-scheduling (see, e.g., [8] or [6]), and is one of the most common ways of doing control synthesis. However, the dynamics for a sounding rocket varies fast during the launch and this is not included in the synthesis of the control law. Since the control laws are only defined for “frozen” time instances and does not include the transitions between different time instances. The goal with the control synthesis/analysis methods in this report is to be able to consider also these effects and to be able to synthesize controllers that we can guarantee are stable for whole time intervals.
There are three main ways to tackle this problem. The first one is to synthe-size controllers using the theory for linear time-varying systems. This approach is both theoretically and computationally hard and has not been investigated further in this report. The second approach is to use linear parameter-varying systems and synthesis for such systems (see, e.g., [3] or [2]). We will go more into detail about these methods in a section below. The third approach is to use a modification of the existing method of gain-scheduling, but to validate the controller using linear time-varying methods. This will also be discussed in a section below.
When synthesizing the controllers two additional systems are involved. These are a body bending filter,W1, where the cut-off frequencies of the filter depends on the resonance frequencies of the first body bending mode, and an integrator,
W2. Thus we have the structure as in Figure 1.
3.1
LPV approach
Here we have investigated methods using lpv techniques to synthesize con-trollers for the ltv system. The method investigated in this family of methods
Gperf K w z y u Figure 2: Feedback G W2−1 w1 W2 z1 W1 w2 W1−1 z2 K Figure 3: H∞ synthesis
is the method described in [2] and also used in [5]. For this method the authors have been given an early release of an implementation of the LPVTools toolbox for Matlab (see [4]). One benefit with linear parameter-varying methods, com-pared to regular gain-scheduling, is that it is possible to incorporate information about rate limitations of the parameters.
To be able to use this method we need to reformulate the problem in the form as in Figure 2. Doing this we get the structure in Figure 3 and the system
Gperf can be written as z1 z2 u = I W2GW1 W2G 0 0 W1−1 W2−1 GW1 G w1 w2 y = I 0 0 0 0 W1−1 W2−1 0 0 + W2 0 I G0 W1 I .
Define the system matrices inGperf as
Gperf:= A B1 B2 C1 C2 D11 D12 D21 D22
the linear matrix inequalities (LMIs) that need to be solved in the method investigated here are of the form
Y(p) ˆAT(p) + ˆA(p)Y(p) −P i± vi∂Y∂pi − γB2(p)BT 2(p) Y(p)CT11(p) B1(p) C11(p)Y(p) −γI 0 BT 1(p) 0 −γI ≺ 0,
˜ AT(p)X(p) + X(p) ˜A(p) +P i± vi∂X∂p i − γCT 2(p)C2(p) X(p)B11(p) CT 1(p) BT 11(p)X(p) −γI 0 C1(p) 0 −γI ≺ 0, X(p) I I Y(p) 0,
where X and Y are the variables to be solved for,vi are the rate limits for the
different parameters and B1 = [B11, B12], CT
1 = [CT11, CT12]T, ˜A = A − B12C2
and ˆA = A − B2C12. These LMIs are then sampled in a sufficiently dense grid of the parameters,p, and solved. Hence, the number of gridpoints that have to
be used grows exponentially with the number of parameters which is a drawback of this method.
In our case, with the model derived above, we have parameters such as throttle, mass and mach, which makes the number of gridpoints grow large. However, all these parameters are not independent of each other, they all depend on time and in reality we only have one parameter, time, which evolve in a linear and predictive manner, ˙t = 1. Now, using the method described in [2] for
our derived LTV model with only time as a parameter have unfortunately not yielded any succesfull results. The reason behind the failure of using this method lies probably in the fact that the only parameter present is time, which have ˙t = 1. Unfortunately, this method requires that the parameters rate limitations are on the form, | ˙p| ≤ ν, and in the case of using only time as a parameter,
as mentioned above, we have that ˙t = 1 and it is not trivial how to make
this condition hold without introducing new states and additional conservatism. Instead we will pursue the method presented in the next section.
3.2
Gain-Scheduling with LTV analysis
In this section we investigate if it is possible to use gain-scheduling and then use LTV analysis methods to validate the interpolated controller over the whole time interval. To be able to use this technique, we first need to modify the old gain-scheduling technique. First we need to be able to sample the model more dense than before, which can be done by using the LTV-model derived in Section 2, where we can get a model for an arbitrary time instance. Secondly we need to make sure that the computed controllers are represented in a coherent state basis. This can be solved by checking if there exists an appropriate similarity transformation between two consecutive, in time, controllers. Given two differ-ent represdiffer-entations of the same systems, G1 =
A1 B1 C1 D1 = A2 B2 C2 D2 , then the we have the relationship
TA1= A2T, TB1= B2,
C1= C2T, D1= D2,
which can be written as T 0 0 I A1 B1 C1 D1 T −A2 B2 C2 D2 T 0 0 I = 0. (4)
Hence, given two consecutive controllers, we can solve (4) to get T to see if it is possible to get a coherent state basis and in that case transform the last controller to the same basis as the first controller. This can be hard to check in the general case of a high order controller. However, for lower order controllers, which is preferable, it is easier. In practice, using the models in this project, it is possible as long as the controllers are of low order (n ≤ 4) and the models are
sampled sufficiently closed (∼ 0.025s apart). Loop shaping is used to compute
the controllers for each of the time instances. To be able to compute low order (or structured, e.g., PID) controllers using loop shaping a modified version of the algorithm has been used. This modification uses the function hinfstruct in Matlab (see [1]).
Now we are able to compute a number of controllers for different time in-stances all in a coherent state basis, and we need to validate that they guarantee performance and stability of the derived LTV-model. To do this we will try to use a time-varying Lyapunov function. To show have this is done, first assume that the system is described as
˙
x = Ax + Bw, (5)
z = Cx + Dw. (6)
The H∞-gain fromw to z is defined as J = xT(T )P Tx(T ) + γ −1/2z 2 2− γ 1/2w 2 2− x T(0)P 0x(0), where the terms xT(T )P
Tx(T ) and xT(0)P0x(0) are the final cost caused by
a non-zero final state and the initial energy in the system available at t = 0,
respectively. Now define V (x) = xTPx to be a positive definite, time-varying Lyapunov function, where P 0 and P(T ) = PT. Assume now that
˙
V + γ−1zTz − γwTw ≤ 0. (7) The assumption (7) can be rewritten as
x w T PA + ATP + ˙P PB BTP −γI +γ−1C T DT C D x w ≤ 0
since ˙V = xTP (Ax + Bw) + (Ax + Bw)T
Px + xTPx. This should hold for all˙ possiblex and w, from which we can infer that
PA + ATP + ˙P PB BTP −γI +γ−1C T DT C D 0. Using Schur complement this is rewritten as
PA + ATP + ˙P PB CT BTP −γI DT C D −γI 0.
G W2 w1 z1 W1 w2 z2 K
Figure 4: Loop shaping
This is the bounded real lemma (see, e.g., [9]) for a linear time-varying system. We can rewrite this again as
PA + ATP + ˙P +PB CT γI −DT −D γI −1 BTP C 0. (8)
Using (8), but with an equality instead, which is also valid, we get ˙ P = − PA + ATP +PB CT γI −D T −D γI −1 BTP C ! (9)
from which we get an equation such that we can integrate P backwards in time. If we are able to integrate P backwards in time, from PT, using (9), with P 0
for all times. Then we can guarantee performance and stability for the linear time-varying system during the whole trajectory.
To be able to use this we need the system in (5) to be the closed loop system including the controller, see Fig. 4. Define the controller as
K := ˆ A Bˆ ˆ C Dˆ ,
andGperf can be written as, including both the system and the filters, as
Gperf= I 0 0 0 0 I I 0 0 + 0 I I W1GW20 I I , with Gperf= A B1 B2 C1 C2 D11 D12 D21 D22 .
Then the closed system becomes
Gcl= Acl Bcl Ccl Dcl = A 0 0 0 B1 0 C1 0 D11 + 0 B2 I 0 0 D12 ˆ A Bˆ ˆ C Dˆ 0 I C2 0 0 D21 .
This system, Gcl, can now be used in (9) to verify the controller for the whole
2 4 6 8 10 12 14 16 18 1.4 1.6 1.8 2 Time [s] γ
γfor the two different controllers
Low order Full order 2 4 6 8 10 12 14 16 18 0 1 2 ·10−3 Time [s] Relativ e γ
Quotient between the two γ’s
Figure 5: Performance measure for a low order controller and a full order con-troller are plotted in the plot on top and below the quotient of the performance measures
In Figure 5 we see, in the upper plot, the performance measure for a full order controller and a controller of order 3 and in the lower plot we see the quotient between them. We see that there are almost no loss in using a 3rd order controller compared to a full order one. This low order controller has also been verified to guarantee performance and stability of the LTV-model over the whole trajectory.
A 4th order controller was also computed and verified. However, as we could see in Figure 5, the increase of performance that can be gained is small. This is also verified in Figure 6, where the Hankel singular values of the 4th order controller have been plotted.
4
Conclusions
An LTV-model of a sounding rocket has been derived. The model includes bending modes of the rocket and interpolation of parameter values have been used to get a continuous model over the whole time period. Additionally, two filters are included in the model, one integrator and a body bending filter where the cut-off frequency of the filter depends on the resonance frequency of the first bending mode.
2 4 6 8 10 12 14 16 18 0 0.2 0.4 0.6 Time [s] Hank el sv
Hankel singular values for the controller
Figure 6: Hankel singular values for the 4th order controller
Evaluation of synthesis methods shows that the LPV method studied is not well suited for LTV systems, probably due to the fact that the only param-eter used is time and ˙t = 1. However, the modified gain-scheduling method,
described in Section 3.2 have shown to work on the derived LTV-model and a verified LTV controller have been computed. All computations described in this report have been done in Matlab with files that are available with the report.
References
[1] P. Apkarian and D. Noll. Nonsmooth H∞ synthesis. Automatic Control,
IEEE Transactions on, 51(1):71–86, Jan 2006.
[2] Jeffrey M. Barker and Gary J. Balas. Comparing linear parameter-varying gina-scheduled control techniques for active flutter suppression. Journal of
Guidance, Control and Dynamics, 23:948–955, 2000.
[3] D. Farret, G. Duc, and J.P. Harcaut. Multirate LPV synthesis: a loop-shaping approach for missile control. In American Control Conference, 2002.
Proceedings of the 2002, volume 5, pages 4092–4097 vol.5, 2002.
[4] Arnar Hjartarson, Peter Seiler, and Gary J Balas. LPV aeroservoelastic control using the LPVTools toolbox. In AIAA Atmospheric Flight Mechanics
Conference, 2013.
[5] Arnar Hjartarson, Peter Seiler, and Gary J Balas. LPV analysis of a gain scheduled control for an aeroelastic aircraft. In American Control Conference
(ACC), 2014, pages 3778–3783. IEEE, 2014.
[6] We. Leithead. Survey of gain-scheduling analysis design. International
Jour-nal of Control, 73:1001–1025, 1999.
[7] Jeff S. Shamma and Michael Athans. Gain scheduling: potential hazards and possible remedies. IEEE Control Systems Magazine, 12(3):101–107, 1992. [8] J.S. Shamma and M. Athans. Guaranteed properties for nonlinear gain
scheduled control systems. In Decision and Control, 1988., Proceedings of
[9] Kemin Zhou, John C. Doyle, and Keith Glover. Robust and optimal control. Prentice-Hall, Inc., 1996.
Avdelning, Institution
Division, Department
Division of Automatic Control Department of Electrical Engineering
Datum Date 2015-04-29 Språk Language Svenska/Swedish Engelska/English Rapporttyp Report category Licentiatavhandling Examensarbete C-uppsats D-uppsats Övrig rapport
URL för elektronisk version
http://www.control.isy.liu.se
ISBN
—
ISRN
—
Serietitel och serienummer
Title of series, numbering ISSN1400-3902
LiTH-ISY-R-3084
Titel
Title Parametric Controller Strategies
Författare
Author Daniel Petersson
Sammanfattning
Abstract
This report describes the modeling of a sounding rocket using a linear time varying model. Different control strategies are investigated to control the time varying model.
