Proceedings of the 1996 IEEE International Conference on Control Applications Dearborn, MI September 15-18,1996
WA05 10:50
A ROBUST VARIABLE STRUCTURE CONTROLLER FOR MIXED CONTINUOUS / BANG-BANG SYSTEMS John Valasek Department of Mechanical and Aeronautical Engineering Westem Michigan University, Kalamazoo, Michigan 49008-5065
[email protected] ABSTRACT
separate the piecewise constant part from the continuous part and conduct separate design phases [5]. A systematic design methodology based upon and the sampled-data reguliitor (SDR) of [6] was developed in [2]. The basic concept is to directly design feedback control laws where both types of effectors are accounted for simultaneously. Later work demonstrated the methodology for a generic second-order system [7], and a generic X-29A aircraft fitted with forebody VFC nozzles 181. This paper extends these results by quantifying stability, sensitivity, and robustness properties.
A systematic controller design procedure is developed for the class of linear time-invariant systems which use both continuous and bang-bang control effectors. The sampled-data regulator and its associated weighting matrices and cost function are extended to this class of system, and are used to modulate the bang-bang effector to achieve desired performance levels. Stability, sensitivity, and robustness are quantified, and the controller synthesis methodology is illustrated for a generic second-order plant with one continuous effector and one bang-bang effector. Results indicate that the resulting controllers are stable, not overly sensitive to the designer selected bang-bang weighting, and possesses good robustness with respect to parametric structured uncertainties. The methodology provides the designer with insight required to systematically tune closed-loop performance.
UNIFIED CONTROLLER DESIGN CONCEPT The methodology in [2] considers the complete system to be controlled by two distinct, independent control systems. At every sample interval this variable structure controller uses either a) only the continuous effector and no bang-bang effector; or b) the continuous effector and the bang-bang effector in either the (+) or (-) state. Ideally, the controller should use ithe best features of each effector combination above in a particular situation to achieve a desired performance. This can be accomplished on-line by evaluating a cost function specified by the designer. Since the two types of control effector are not separated out as in previous methods, the controller for both parts of the complete system can now be designed in a unified fashion, i.e., in a single design phase where both controls are accounted for simultaneously. The SDR uses the following discrete cost function for a sampled continuous linear time invariant (LTI)system as the number of samples N approaches infinity:
INTRODUCTION Real world control problems with challenging performance requirements can require dynamic systems to employ a combination of continuous and bang-bang control effectors. An example of this important class of system is the X-29A aircraft fitted with Vortex Flow Control (VFC) [l]. Figure 1 shows a block diagram for a generic control system of this type.
J=
1 N-'
, A
, A
(X,Q X, +U,R U,
+ 2 X;M
U,)
(1)
k- I
where X, E Rn, U, E 'W"' , R E Rmxmis positive definite, and Q E Etnxnand M E R""" are positive semi-definite. Equation (1) is minimized to get constant optimal gains, and the feedback loop is closed with the standard SDR control law
Figure 1 Generic Control System with Continuous and Bang-Bang Control Effectors [2] These systems have distinct continuous and piecewise constant elements and properties. Compared to controller design methodologies for systems with only continuous effectors [3] or only bang-bang effectors [4], existing synthesis techniques for the present class of system
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9
which accounts for the continuous effectors only. The closed loop simulation equation for the sampled-data system is
V(X)
=
SUB-INTERVAL TRAJECTORY EVALUATION At the beginning of each sample interval, the system is simulated on-line forward in time, over a subinterval of time equal to the sample interval, using each of the three candidate controller structures (Figure 2).
-
bang-bang * w e
Figure 2 Sub-Interval Evaluation and Control Selection Scheme [7] The candidate structure which generates the lowest cost in terms of state deviation and control activity, using (l), is selected and used during the upcoming sample interval. The operation of the continuous effector is unaffected. The on-line simulation can use either a full set of nonlinear dynamic equations or a reduced-order linear approximation. A benefit of using SDR gains is that the closed-loop system is stable for a range of initial conditions provided stabilizing gains are used, and the adaptive nature of the controller provides a certain degree of stability robustness with respect to real parameter variations. STABILITY ANALYSIS For the present class of systems with both contmuous and piecewise constant (bang-bang) elements complex d y n m c behaviors are possible, so the stability and range of operatlng conditions are quantfied using analytical and graphcal techniques based upon the second or direct method of Lyapunov [9] The scaler Lyapunov funcuon selected is >
V(X) = X P X
(4)
XPX
+
(5)
With U, = 1 , substituting the time-invariant closed-loop state equation X = {A,
where h is the integration stepsize and U,, contains only bang-bang effectors. With the controller for the continuous effectors determined, the controller for the bang-bang effectors is designed.
x PX
9
-
B K }X
+
B,,U,
(6)
yields: ,
>
V(X) = X [A, P + P A L I ] X + XPB,,+B,,
PX
(7)
For global asymptotic stability, V(X) must be negative definite for all X # 0. Therefore, the entire RHS of equation (7) must be negative definite. The term inside the brackets
which is a quadratic term already satisfies this requirement, and therefore a unique minimum exists [IO]. The presence of the second and third terms which are due to the bangbang effector mean that a unique non-zero solution for all X does not exist. The solution IS instead dependent upon the relative magnitudes and signs of the second and third terms because they are not quadratic. Evaluating the conditions for V(X) to be negative definite in terms of the possible relative magnitudes and signs of each parameter in (7), an inequality is obtained: , , IX [A, P + P A c , l X I > IX PB,, +Bh, PXI (9) Greater insight can be achieved in the scaler case, which reduces (9) to: (IO)
From this result the domain of X for which V(X) negative definite is the set
IS
This set represents the particular contour for which V(X) = 0, the stubiZity boundav. The stability boundary divides the phase-plane into regions of stable and unstable operation. States X, Y whch lie outside the closed curve of the stability boundary are those for which V(X) is negative and trajectories are stable. States X, Y which lie withm the stability boundary are those for which trajectories are unstable. A generic second-order system is used to graphically illustrate the properties of the stability boundary in the phase-plane. Starting with the closed-loop system matrix for the second-order system where subscripts refer to indices of vectorlmatrix elements.
where P is a given symmeuic positive definite matrix. Taking the derivative of 14) ivith respect to time gives
680
Ad =
(4,-B11 K I I ) (4,- BZl Kll )
( 4 2 -Bll
Kl2)
('422 - Bz, KIZ)
1
Equation (17) is the cllosed-form solution of the stability contour defined by V(X) = 0 in terms of the system parameters A, B, B b b , (2, and K. Equation (17) is plotted using a symmetric positive definite matrix P obtained numerically using PC MATLAB. Figure 3 displays the stability boundaries of a generic second-order LTI system for values of Bbbequal to B and ten times greater than B.
(12)
and introducing a symmetric positive definite matrix P, and a new matrix W1 = A',,P + PA,, , equation set (13) is generated :
WIZ2
=2P,,(A,,
-B2,K12)
+2PI2(AI2-B,,K1J
Figure 3 Stability Boundaries for Various B b D ZJsing Equation (17)
Turning to the second term of (7)
[
PBbb
'11
bb 11
'12
'21
bb 11
P22B bbZl
bb21
and third term of (7),
(Bbb
')I1
=
PllBbbll
(Bbb
')lZ
=
PIZBbbll
1
With the bang-bang (effector active, regardless of the magnitude of B b b to B, the area adjacent to the origin is a region of instability. The interior region of instability grows in proportion to the magnitude of the bang-bang control effector. However, examination of equation (17) shows that the instability region is proportional not only to . B,, , but also to the ratio of Bbbto Asymptotic staibility of the system therefore hinges on turning off the bang-bang effector before the trajectory penetrates the stability boundary, or the system will not be asymptotically stable. To quantify this aspect of the controller performance ithe stability boundary is first plotted in the phase-plane. Irrajectories of the system from a sweep Of initial condirtions are overlaid On the Same plot to check for penetration of the stability boundary. Note that the trajectories are plotted only while the bang-bang effector is active. The case of B,,, equal to B is presented in Figure 4. None of the initial conditions investigated led to system instability, and no limit cycles or nonzero equilibriums are observed, because the clontroller does not exhibit the classic "indecision at the switching boundary" prevalent in dualmode systems. All trajectories which begin with zero initial rate turn off the bang-bang effector at exactly the same time, while those with nonzero initial rates exhibit some variation. The system exhibited global asymptotic stability, but it must be noted that global asymptotic stability is possible chiefly because this particular system is linear autonomous, i.e., no constraints on the controls and time-invariant, when the bang-bang effector is not active. The results of tlgs analysis do not directly apply to time-varying systems, or those which have constrained controls.
(14)
+PZIBbbZl
(I5) +
P2ZBbb21
new variables W2 = PB, , W3 = Bb,'P, and second-order system State vector x = [x y]' are defined. Substituting equations (10-15) into (7) results in:
V(X)
Wl,,X2
=
+ Wl,,Y
+ w3,,
x
+
l+
Wl,,XY + Wl,,XY w 2 , , x + W2,,Y
(16)
+ W3,,Y
Setting V(X) = 0 and completing the square,
w1,, [ x +W1,,[ Y
W21, +W3,l 1 2 +
2W1Il +
W%,+ W3,z
(17)
2w122
- (W2,, + W 3 , , ) ' - (W&, + W 3 , , ) ' 4W111
=o
4W1 22
68 1
performance. The uncertainties investigated here are physically motivated by practical considerations rather than purely mathematically motivated. The location of the stability boundary for systems with continuous and bang-bang control effectors moves in the phase-plane according to variations in system parameters. The location and movement are quantified using the concepts of stability center and the stability radius. The stability radius is defined here as the distance from the center of the unstable region (stability center), to the stability boundary. The relation for the stability center is derived by noting that equation (17) for the stability boundary is an ellipse (or a circle if W l , , = W12J centered at Figure 4 Effect of Initial Conditions on Asymptotic Stability, Bbb= B, T = 0.5 sec Based on the results above the system remains asymptotically stable only if the bang-bang effector is inactive. This means that at some point in the trajectory the bang-bang effector must be turned off so that the continuous effector can drive the states to the origin.
The coordinates for the center of the stability region can be expressed in terms of the closed-loop system (A, B, K) and matrix P by substituting for the components of the W1, W2, and W3 matrices. Since P is symmetric, the coordinates of the center reduce to equations (22) and (23):
SENSITIVITY ANALYSIS Sensitivity of the controller selection sequence with respect to the weighting matrices is determined by taking partial derivatives of the sub-interval cost function ,A
, A
J, = X,QX,+U,RUk+2X;MU, The bang-bang effector is an element of U, symmetric R and Q the partial derivatives are:
(18)
.
With
A
a Jk = 2 ( Q X ,
a
(19)
xk
a Jk
a
+MU,)
The stability radius quantifies the robustness of the closed-loop system by measuring how much the stability boundary and therefore the instability region increases or decreases with respect to system parameter variations. Since the origin is always a point on the stability region, an expression for the stability radius is obtained
-
rs = J X ;
A
+
Y:
2(RU, + M ' X k )
k '
The state of the bang-bang effector is weighted by the matrices M in (19) and R in (20). The designer selected bang-bang weighting element (in the continuous R matrix) is expressed only in 8, and since II Q II 2 II R II 2 II M II is usually satisfied, the sub-interval cost function (18) and therefore the selection criteria is sensitive to this weight. Practical experience demonstrates that (18) tends to be extremely sensitive to even small changes in the weighting element on the bang-bang effector. This result is not unexpected considering the limit cycles from indecision at the switching boundary common to dual-mode systems. ROBUSTNESS PROPERTIES The robustness analysis is concerned with structured uncertainties, and the effect of uncertain physical parameters in known model structures on system
with X, and Y, given by equations (22) and (23) respectively. To study the effects of model uncertainties on robustness, the second-order LTI system used in the stability analysis is used to show the effect of bounded parameter variations on the stability boundary. The case of B,, ten times greater than B is presented. A 215% variation is applied individually to the state and control parameters, including the bang-bang effector. No variations are applied to the first row of either the A or B matrix and K is held fixed. The resulting stability boundary is then plotted and compared to the nominal stability boundary. Figure 5 shows that parameter variations have a marked effect on the stability center as Y, is shifted upward, but not much effect on stability radius. The most destabilizing change is a 15% increase in B,, , resulting in a 13% increase in the nominal stability radius.
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control effectors. Stability, sensitivity, and robustness were quantified, and the controller synthesis methodology was illustrated for a generic second-order plant with one continuous effector arid one bang-bang effector. Based upon the results of this research it is concluded that:
1 nominal system
1) The proposed controller design methodology for systems with both continuous and bang-bang control effectors is systematic and permits the designer to directly tradeoff usage of the continuous effectors, bang-bang effectors, and closed-loop system response.
>~~ .IO
*
0
20 lo
;
1
I 20
2) The control effectiveness of the bang-bang effector has a significant effect on system stability. Large values of this parameter are shown to reduce stability, while small values improve stability.
Figure 5 Robustness of Stability Boundary with Respect to Parameter Variations, B, = 10*B, T = 0.5 sec
3) The controller is stmsitive with respect to the designer selected weighting value on the bang-bang effector. This sensitivity is useful, not detrimental, for controller synthesis.
When simulated the worst case results in a 6% increase in cost compared to the nominal case. The least destabilizing change is a 15% decrease in B,, which actually shrinks the stability radius by 15%, thereby improving system stability when the bang-bang effector is operating.
4) Variations of +1549 in state and control parameters do not cause instability in the second-order system studied when controlled by thle nominal feedback gains. In the worst case these variations result in a 3% increase in cost when the bang-bang effector is more effective, and a 6% increase in cost for equally effective bang-bang and continuous effectors.
CONTROLLER DESIGN ILLUSTRATION The example is the classic regulatoF problem. For consistency the same SDR weighting matrices and cost function are used to both design gains and evaluate trajectories. The generic LTI second-order system is:
REFERENCES [ 11 Walchli, Lawrence A., et al, "High Angle-Of-Attack Control Enhancement On A Forward Swept Wing Aircraft," AIAA-92-4427-CP.
with continuous effector U, and bang-bang effector of ten times effectiveness U, . The open-loop system has A,,, = .12 2 2.28j , on = 2.28 radsec, and 4 = 0.054. The elements of the weighting matrices are held fixed at QI1= 0.91; Qz2= 0.33; RI= 3.7. The only parameter which is varied to design the controllers is the weight on the bangbang effector R2 . To compare successive designs a weight of R, = 1.0 on the bang-bang effector generates Design 1, a nominal system performance with only the continuous effector operating (Figure 6). The total discrete cost is 5.66. Successively reducing R2 generates designs which use progressively more bang-bang effector. An R, value of 0.05 generates Design 2 (Figure 7) with improved performance and a discrete cost of 1.68, a reduction of 70%.
[2] Valasek, John, "1Jnified Design of Controllers for Systems with Continuous and Bang-Bang Control Effectors," PhD Dissertation, Department of Aerospace Engineering, University of Kansas, Lawrence, KS, April 1995. [3] Ogata, Katsuhiko, Modern Control Engineering, Prentice-Hall, Inc, Englewood Cliffs, NJ, 1970. [4] Tsypkin, Ya. Z., R!elay Control Systems, Cambridge University Press, Cambridge, England, 1984.
[5] Naumov, Boris Nikolaevich, Philosophy Of Nonlinear Control Systems, Mir Publishers, Moscow, 1990, p. 91. [6] Dorato, Peter, et al, "Optimal Linear Regulators: The
SUMMARY AND CONCLUSIONS
Discrete-Time Case," IEEE Transactions on Automatic
Control, Volume AC-16, Number 6, Dec 1971, pp. 613-
A unified systematic design procedure, based upon the SDR and incorporating on-line future projection and evaluation of states, was demonstrated for controller synthesis of linear time-invariant plants which are controlled by a combination of continuous and bang-bang
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[7] Valasek, John, and Downing, D.R., "A Unified Controller Design Methodology for Systems with Continuous and Bang-Bang Control Effectors," ACC95AIAA-060. [8] Valasek, John, and Downing, D.R., "A Closed-Loop Forebody Vortex Flow Controller for a Generic X-29A Aircraft," AIAA-95-3248-CP. [9] La Salle, Joseph, and Lefschetz, Solomon, Stability by Liapunov's Direct Method with Applications, Academic Press Inc., New York, New York, 1961. I
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I
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[101 Ackermann, Jurgen, Bartlett, Andrew, Kaesbauer, Dieter, Sienel, Wolfgang, and Steinhauser, Reinhold, Robust Control, Systems with Uncertain Physical Parameters, Springer-Verlag, London, 1993, pg. 178.
Figure 6 Design 1 Performance, Cost = 5.66
CI
k
U
60
W U
C
e f -g
3 U
30
o -30 -60
c5
B
U
W
b
P
F
2
10
5
o -5 -10
0
2
I
i
I
1
I
I
I
I
I
I
I
I
I
I
0
5
10
15
20
25
30
1 0
-1 -2
Figure 7 Design 2 Performance, Cost = 1.68 684
