Feedback Control Design by Lyapunov’s Direct Method Aníbal M. Blanco, José L. Figueroa and J. Alberto Bandoni Planta Piloto de Ingeniería Química, UNS - CONICET, Camino La Carrindanga, Km. 7 8000 Bahía Blanca, ARGENTINA, fax: +54 291 486 1600, E-mail:
[email protected] The purpose of this work is to introduce a systematic technique for feedback control design. Based on Lyapunov´s stability theory a Non-linear Programming Problem is formulated in order to obtain an optimal closed loop design in some sense. The proposed technique is applied to the feedback control design of a classic stirred tank reactor. 1.INTRODUCTION In this work we deal with the so-called first and second fundamental problems in the theory of automatic control, as introduced by Letov (1961). The first fundamental problem, or stability problem, consists of determining the values of the parameters of the controller, which are required to guarantee stability of a steady state point. The second fundamental problem, or control quality problem, deals with the character of the convergence of the motion in terms of response speed. Both aspects will be considered in the control design formulation, by inclusion of Lyapunov’s stability conditions. Lyapunov´s direct (or second) method, is the most general available tool for assessing stability of non-linear dynamic systems, described by a set of differential equations. It is based on an energetic approach and neither explicit nor numerical solutions of the equations are required. Besides the stability issue, it also admits the consideration of transient response speed in an indirect way. 2. LYAPUNOV´S STABILITY THEORY In the present section, most relevant issues of Lyapunov´s Stability Theory are outlined. See, for example, Vidyasagar (1993) for a complete analysis. 2.1. Lyapunov’s linearization method Consider the free, autonomous system:
dx = f (x ) , dt
f ( 0) = 0
(1)
where x , represents the deviation state vector. We can write f ( x ) = Ax + f1 (x ) , where ⎡∂ f ⎤ A = ⎢ ⎥ , and f1 (x ) is the residual. Then, it can be proved (Vidyasagar, 1993) that 0 is ⎣∂ x ⎦ x =0
an exponentially stable local equilibrium of (1) if all eigenvalues of A have negative real parts ( if A is a Hurwitz matrix). Moreover, the quadratic form V ( x ) = x T Px is a suitable Lyapunov function of the system and V ( x ) = − x T Q x + 2 x T P f 1 (x ) , where: A T P + PA = − Q
(2)
being Q an arbitrarily chosen positive definite symmetric matrix. Lyapunov equation (2) is solved for P, which is also symmetric and positive definite. Moreover, denoting eigenvalues by λi and choosen r such that : f1 (x ) x
<
λmin ( Q ) , ∀ x ∈Br 2 λmax ( P )
Br = {x ∈ R n x < r}
(3)
then V ( x ) < 0 , whenever x ∈Br and x ≠ 0 . Br provides an estimate of the domain of attraction of 0, that is, the region of the state space where asymptotically stable trajectories are generated. B
2.2. Estimation of transients ⎧ − dV ( x ) dt ⎫ Consider parameter η, defined as η = min ⎨ ⎬ , which may be loosely regarded as x ⎩ V (x ) ⎭ the reciprocal of the largest time constant, which is descriptive of the motion over the region of asymptotic stability and therefore it is a figure of merit for the control system. A large value of η indicates that the system returns rapidly to the origin. In particular, for a Lyapunov system (2), it is found (Koppel, 1968) that
η = λmin ( P −1Q)
(4)
These results will be applied in the proposed control system design formulation of the next section.
3. CLOSED LOOP DESIGN PROBLEM FORMULATION The basic general problem of (steady state) design, may be posed as a constrained, non linear, optimization problem (NLP): min Φ( y ) y
s. t.
h( y ) = 0
(5)
g( y ) ≤ 0
{
y ∈Y = y y l ≤ y ≤ y u
}
The proposed approach to tackle the fundamental problems of automatic control theory is also posed as a NLP. To do this, let choose Q = I, as is generally suggested. From (4) it can be seen that η = λmin(P-1) becomes a natural objective function to be maximized in order to achieve a fast as possible transient response. Since λmin(P-1) =1/λmax(P), maximizing λmin(P-1) P
P
P
corresponds to minimize λmax(P), which is also a desirable objective in order to enlarge the estimate of the domain of attraction of the origin, as can be concluded from (3). The (nonlinear) steady state model of the closed loop system and the Lyapunov equation (2), conform the set of equality constraints of (5). Positive definite condition on matrix P is also required. The resultant problem turns to become an eigenvalue optimization, non-linear semi-definite programming problem. Large-scale eigenvalue optimization and linear semi-definite programming problems can be efficiently tackled via interior point methods. Boyd et. al., (1997) describe how control system analysis and synthesis can be performed via linear matrix inequalities. Some of such theory may be extended to nonlinear non-convex semi-definite programming problems (see Lewis and Overton (1996) for a comprehensive review of these subjects). In this work, a different approach is proposed in order to transform the nonlinear semi-definite programming problem into a NLP. 3.1. Objective function The problem of eigenvalue optimization of a symmetric matrix P=[pij], min λmax ( P( y )) , may y
be reformulated in terms of a slack variable z as follows (Ringertz, 1997): min z y, z
s. t.
z ≥ λi ( P( y )) i = 1... n
(6)
Since analytic expressions for the eigenvalues of a matrix are not in general available, problem (6) may be approximated as below: min z y, z
s. t.
u z ≥ λmax ( P( y ))
(7)
u where λmax is an upper bound of the maximum eigenvalue of P. For a real symmetric matrix
⎞ ⎛ .However, this upper bound may be (Jennings, 1977): λmax ( P( y )) ≤ ⎜ p ii ( y ) + ∑ p ij ( y ) ⎟ ⎠ max ⎝ i≠ j very conservative. A further property of eigensystems (Jennings, 1977) establishes that the eigenvalues of a matrix remain unaltered if a row is scaled by f and the corresponding column ⎛ ⎞ ⎛ f ⎞ is scaled by 1/f, then: λmax ( P( y )) ≤ ⎜ p ii ( y ) + ∑ ⎜⎜ i ⎟⎟ p ij ( y ) ⎟ may provide a better bound ⎜ ⎟ f ⎝ ⎠ ≠ i j j ⎝ ⎠ max (closer from above) if we let fi to vary within certain ranges. In view of such property, problem (7) may be rewritten as: min z y, z , f i
s. t.
⎛ ⎞ ⎛ f ⎞ z ≥ ⎜ p ii ( y ) + ∑ ⎜⎜ i ⎟⎟ p ij ( y ) ⎟ , i = 1... n ⎜ ⎟ i≠ j ⎝ f j ⎠ ⎝ ⎠
Problem (8) is a non-smooth NLP because of the absolute value function.
(8)
3.2. Positive definiteness on matrix P A symmetric matrix P (n,n) is positive definite if and only if each principal submatrix Pk (k,k) (1 ≤ k ≤ n) has a strictly positive determinant (Noble and Daniel, 1989). As a result, a set of n additional inequalities of the form: Pk ≤ ε , ε > 0
k = 1... n
(9)
should be added to (8) in order to ensure positive definiteness of matrix P. The above ideas are applied to the controller design of a continuous stirred tank reactor in the following section. 4. APPLICATION EXAMPLE In a couple of papers (Berger and Perlmutter, 1964 (a, b)), the authors analyze both, open loop and feedback controlled stability of a chemical reactor, based on Lyapunov’s direct method. They apply a limited version of Krasovskii’s theorem to study the regions of asymptotic stability, in an analytic way. No closed loop performance issues are considered in their contribution. We applied the methodology of section 3 to a somewhat reacher example provided by Devia and Luyben (1978). The system under analysis is a typical continuous stirred tank reactor in which an homogeneous, exothermic, first order, reaction is taking place. The following three states model describes the dynamics of the system: ⎛ F⎞ dC A ⎛ F ⎞ = ⎜ ⎟ C A, 0 − ⎜ ⎟ C A − C Aαe − E / RT dt ⎝ VR ⎠ ⎝ VR ⎠
(10)
⎛ F⎞ UAH λC Aα − E / RT dT ⎛ F ⎞ − e = ⎜ ⎟ T0 − ⎜ ⎟ T − (T − TJ ) dt ⎝ V R ⎠ ρC p ρV R C p ⎝ VR ⎠
(11)
dTJ ⎛ FJ ⎞ UAH = ⎜ ⎟ (TJ ,0 − TJ ) + (T − TJ ) ρJ CJ V J dt ⎝ V J ⎠
(12)
A classic Proportional - Integral feedback control law is applied, pairing T and FJ as measured and manipulated variables respectively: dξ = Tsp − T dt FJ = K c (Tsp − T ) +
(13) Kc ξ + FJ sp τI
(14)
The analyzed example is open loop unstable since the corresponding jacobian matrix is not Hurwitz. Closed loop stability can be achieved within certain ranges of the control parameters Kc and τI. As commented above, slack variable z becomes the objective function, as posed in (8). Closed loop reactor model (10)-(14) in its steady state version, together with Lyapunov equation (2) (which provides ten single equations) conform the set of equality constraints of (5). Four additional inequalities, (9), and bounds on the control parameter values complete the formulation. The above formulation was implemented in GAMS modeling language (Brooke
et. al., 1996) and standard DNLP option, to cope with the non-smoothness of the model, was used in its solution. Numerical data and optimization results can be found in tables 1-4. The resultant system is closed loop stable since matrix A is Hurwitz for such controller parameters. Moreover, it can be seen that the bound strategy for eigenvalue optimization works satisfactorily since the maximum eigenvalue of matrix P is 44497.56 . F CA0 T0 TJ0
556 0.50 530 530
cu.ft./h mol A/cu. ft. R R
Table 1: Streams data
Cp CJ ρ ρJ α E λ
0.75 1 50 62.3 7.08e10 30000 -30000
Btu/lb. R Btu/lb. R lb/cu. ft. lb/cu. ft. 1/hr Btu/mol Btu/mol
Table 2: Physical data
AH V VJ T TSP TJ CA FJSP ξ
399.7 668 167 600 600 553 0.245 1920.5 0
sq. ft. cu. Ft. cu. Ft. R R R mol A/cu. ft. cu. ft./h
Table 3: Reactor data z Kc τI
44497.57 -500 1.363
Table 4: Optimization Results
5. DISCUSSION Most, widely applied control system design techniques resort to linearization. The proposed approach for feedback control system design, admits a non-linear analysis frame, since it is based on the general non-linear Lyapunov’s stability theory. Moreover, no dynamic simulation (or explicit solution) of the system is required for transient quality assessment, since dynamic response speed and the size of the estimate of the domain of attraction are optimized in an indirect way through the eigenvalue optimization. Classical, realistic Proportional-Integral feedback control was applied, although any other control scheme with a formulation expressed in terms of state variables could be considered. Process / Control System design could be carried out simultaneously within the same methodology. In this case, an economic type objective function should be considered, and a multi-objective optimization problem between cost and closed loop performance would probably arise. Future work will consider these topics.
REFERENCES Berger J. S. and D. D. Perlmutter (1964 a); Chemical Reactor Stability by Liapunov’s Direct Method; AIChE J.; 10 (2) pp. 233-238. Berger J. S. and D. D. Perlmutter (1964 b); The Effect of Feedback Control on Chemical
Reactor Stability; AIChE J.; 10 (2) pp. 238-245. Boyd S., Crusius and A. Hansson (1998); Control Applications of Non-linear Convex Programming; Journal of Process Control, 8 (5-6), pp. 313-324. Brooke A., D. Kendrick and A. Meeraus (1996); GAMS Release 2.25 Devia N. and W. L. Luyben (1978); Hydrocarbon Processing; June 1978; pp. 119-122. Jennings A. (1978); Matrix Computation for Engineers and Scientists; John Wiley & Sons. Koppell L. B. (1968); Introduction to Control Theory with Applications to Process Control; Prentice Hall. Letov A. M. (1961); Stability in Non-linear Control Systems; Princeton University Press. Lewis A. S. and M. L. Overton (1996); Eigenvalue Optimization; Acta Numerica (1996); pp. 149-190; Cambridge University Press. Noble B. and J. W. Daniel (1989); Applied Linear Algebra; Prentice-Hall. Ringertz U. T. (1997); Eigenvalues in Optimum Structural Design, in Proceedings of an IMA Workshop on Large-Scale Optimization (A. R. Conn, L. T. Biegler, T. F. Coleman and F. Santosa, eds.) Part I; pp. 135-149 . Vidyasagar M. (1993); Non-linear Systems Analysis; Prentice-Hall.
