Passivity-based Adaptive Inventory Control
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1 Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference Shanghai, P.R. China, December 6-8, 29 ThB.2 Passivity-based Adaptive Inventory Control Keyu Li, Kwong Ho Chan and B. Erik Ydstie Abstract A new adaptive inventory control strategy is developed by applying online adaptation in the framework of passivity-based control. Given the definition of inventory, a feedback-feedforward control structure is derived from the passivity theory. The stability analysis is also given in this paper. This control strategy is demonstrated in simulations and an application on a pressure tank system. I. INTRODUCTION Passivity has been an important part of nonlinear control theory since 97s []. The main passivity theorem states that a strictly passive system with a negative feedback of another passive system is stable [3]. In more general cases, the degree of passivity can be quantified by the passivity indices for both passive and non-passive systems. The shortage of passivity of a system can be compensated by the excess of passivity of a feedback controller, which motivated the passivity-based controller design [3]. Passivity theory hasn t been applied in process control until recent years. The notion of process system is first defined by connecting thermodynamics and passivity [4], [5], [6]. By using macroscopic balance of inventories (such as total mass and energy) to construct passive input-output pairs, an inventory control strategy is further proposed based on the idea that the manipulated variables are chosen so that the selected inventories follow their set points [6], [7], [8]. If there are unknown parameters in the system model, an adaptive controller can be used in the passivity-based inventory control frame [7]. In this article, the passive pair is derived from mass and energy balances and unknown model parameters are adapted online, so that the effect of process dynamic changing and disturbances can be compensated. The application part includes simulation tests on transfer function models and the implementation on a pressure tank unit of a chemical plant based on on-site test data. II. PASSIVE SYSTEMS AND THE PASSIVITY THEOREM Definition (Linear Passive Systems []): A linear timeinvariant system Θ, with n n transfer function matrix T (s), is passive if and only if T (s) is Positive Real (PR), or equivalently, ) T (s) is analytic in Re(s) > 2) T ( jω)t ( jω) for all frequency ω that jω is not a pole of T (s). If there are poles p, p 2,..., p m K. Li is with the Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, PA 523, USA keyuli@andrew.cmu.edu K. H. Chan is with Industrial Learning System Co., Pittsburgh, PA 523, USA khchan@ilsystems.net B. E. Ydstie is with the faculty of Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, PA 523, USA ydstie@cmu.edu of T (s) on the imaginary axis, they are nonrepeated and the residue matrix at the poles lim s pi (s p i )T (s) (i =, 2,..., n) is Hermitian and positive semidefinite. System Θ is said to be strictly passive or Strictly Positive Real (SPR) [2] if ) T (s) is analytic in Re(s) 2) T ( jω)t ( jω) > for ω (, ) Definition 2 (Dissipative systems): System H with supply rate w(t) is said to be dissipative if there exists a nonnegative real function S(x) : X R, called the storage function, such that, for all t t, x X and u U, S(x ) S(x ) t t w(t), () where x = φ (t,t,x,u) is the state at t and R is a set of non-negative real numbers. Definition 3 (Excess/shortage of passivity): Let H : u y. System H is said to be: ) Input feedforward passive (IFP) if it is dissipative with respect to supply rate w(u,y) = y T u νu T u for some ν R, denoted as IFP(ν). 2) Output feedback passive (OFP) if it is dissipative with respect to supply rate w(u,y) = y T u ρy T y for some ρ R, denoted as OFP(ρ). For linear systems, the Passivity Theorem states that a feedback system (as shown in Figure ) comprised of a passive system K(s) and a strictly passive system G(s) is asymptotically stable. r e Fig.. K(s) u G(s) Passivity Theorem. If G(s) in Figure is not passive, then there exists such that G (s) = G(s) is passive. Figure and Figure 2 are equivalent. However, since we can find such that G (s) = G(s) is strictly passive, we can shift the negative feedforward of G(s) to become the positive feedback of K(s) as shown in Figure 3. Hence, with G (s) being rendered strictly passive, we need to make sure the controller K(s) with a positive feedback of, i.e., K (s) = [ K(s)] K(s), is passive to y /9/$ IEEE 4529
2 ThB.2 be estimated on-line. Then (4) becomes r e u K(s) G(s) y = uθ µ(x,d) T θ, θ (8) Fig. 2. Closed-loop modification. Based on the passivity theory and inventory control, the system represented by (8) can be controlled with guaranteed stability by using classical feedback and feedforward control so that r K (s) e K(s) u G(s) Fig. 3. Loop shifting. G (s) guarantee the closed-loop stability based on the Passivity Theorem. In other words, G is IFP(ν) where ν < while K is OFP(ρ) where ρ > and ρ ν >. III. ADAPTIVE INVENTORY CONTROL Consider a finite-dimensional dynamic system S represented by ẋ = f(x)g(x,u,d), x() = x (2) y = h(x) (3) where x, u, d and y are the vectors of state variables, inputs and disturbances respectively. An inventory for system S is defined by an additive continuous (C ) function Z : X R where X is the state space, so that the inventory of a system is equal to the sum of the inventories of its subsystems. Examples include total internal energy U, volume V and mass M (number of mole) of each components. From the continuity, the differential equality for the inventory vector Z can be written as where (x) φ(x,u,d) = = φ(x,u,d) p(x) (4) p(x) = Z(x) g(x,u,d) x (5) Z(x) f(x) x (6) where φ is called the rate of supply, p is called the rate of production [6]. It is often possible to write φ(x,u,d) p(x) = uθ µ(x,d) T θ (7) where µ(x,d) is a vector which depends on observable data; θ is a parameter and θ is a vector of parameters which can y uθ µ(x,d) T θ = C(e) (9) where e = Z Z, and C(e) is any input strictly passive (ISP) operator, which can be PID, optimal controller or MPC since they are all examples of ISP controllers. Note that the parameter θ is unknown, so it has to be estimated using the error information e. If the set point Z is a constant, the feedback-feedforward structure of the controller is given as follows d ˆθ d ˆθ u = C(e) ˆθ µ(x,d)t ˆθ ˆθ () = g ue, ˆθ () = Gµe (2) where g > is a tuning parameter and G > is a diagonal tuning matrix. The control scheme is shown in Fig. 4, and the stability of the adaptive controller is given as follows. Theorem 4: Consider the system described by (4-8) and adaptive controller (-2). If θ and θ are constants, then the adaptive controller is stable in a sense that e L 2. Proof: Define the Lyapunov function of the closed-loop with an ISP controller C(e), which has a storage function V c, as followed where V = 2 e2 2 θ T G θ 2 g θ 2 V c (3) e = Z Z θ = θ ˆθ θ = θ ˆθ G = diag(g,g 2,...,g q ) with g i > for all i =, 2,..., q dv c e T C(e) γe T e γ 453
3 ThB.2 Then, dv = e de θ T G d θ θ g d θ ( ( ) = e ) θ T G d ˆθ ( ) θ g d ˆθ ( = e uθ µ T θ ) θ T G d ˆθ (4) (5) θ g d ˆθ (6) ( ) = e u ˆθ µ T ˆθ µ T θ µ T ˆθ uθ u ˆθ θ T G d ˆθ θ g d ˆθ Define C(e) = Then (7) becomes dv u ˆθ µ T ˆθ = e [ C(e) µ T ( θ ˆθ ) u(θ ˆθ ) ] θ T G d ˆθ θ g d ˆθ = ec(e) eµ T θ θ T G d ˆθ eu θ θ g d ˆθ ( ) = ec(e) θ T µe G d ˆθ ( ) θ ue g d ˆθ If we define the unknown parameter estimation as: (7) (8) (9) (2) d ˆθ = g ue (2) d ˆθ = Gµe (22) Substitute (2) and (22) into (2), we have: Thus dv = e T C(e) γe T e e T e (V() V( )) γ < γ V() Hence, e L 2 and closed-loop with adaptive parameters is stable. The above theorem can be further extended to include systems with delays using the loop-shifting techniques described in Section II. Since a system with delay is non-passive, i.e., IFP(ν) where ν <, then the controller needs to be OFP(ρ) where ρ > such that ρ ν > to guarantee the closed-loop stability. In other words, the controller gain is limited by an upper bound, which depends on how non-passive the system is. Fig. 4. control The feedback-feedforward structure of the adaptive inventory IV. APPLICATIONS A. General Transfer Function Test Consider a system is represented by a transfer function G p (s) = G m p(s)g p(s) (23) where G m p is the system model and G p is the un-modeled dynamics. The system transfer function with disturbances is given by Y(s) = G p (s)u(s) n d i= G di D i (s) (24) where D i is the ith disturbance and G di is the ith disturbance model If we let Z = y, the adaptive inventory control strategy can be applied in the system. A PI controller is used as the feedback part, and the adaptive algorithm introduced in last section is used in the feedforward controller. For demonstration, the simulation result is compared with a classical feedforward controller, which is given by G FFi (s) = (G m p(s)) G di (s) (25) The tested transfer function model is given by G m 5 p(s) = 3s and the un-modeled dynamics is given by G p(s) = 5s (26) (27) 453
4 ThB.2 TABLE I PARAMETERS OF THE CONTROL SYSTEM Parameter Value K.5 τ i 3 G(,) 2 G(2,2) 2 The control system parameters are given in Table I. If we just have measured disturbance which is a generated pulse shown in Fig. 5 The disturbance model is given y2 y x x 4 dv Fig. 6. The Output for Adaptive Controller y and Classic Controller y2.2.8 u x 4 Fig. 5. The Disturbance Input d x 4.2 by { G d (s) = s e 3s t < 2s 2 s e 3s t 2s (28) and after that the gain K d changes from to 2 which causes the disturbance dynamic change. Only PI controller is used at the beginning, and the adaptive feed-forward controller is applied to the process at time 3s, and the simulation results are shown in Fig. 6 and Fig. 7. We can see that the performances for both controller are comparable before the disturbance dynamic changes. However, after the disturbance gain changes, the the adaptive feedforward controller has much less oscillations, since the adaptive controller can adapt parameters online, which compensates the dynamic changes. Fig. 8 shows that the parameters converge to new values after the disturbance gain changes from to 2. B. The Pressure Tank Unit ) The System Model: The pressure tank mixes feed flows for the Ethane/Propane cracker process in a chemical plant. The input flows include mass flows of residue ṁ R, lowpressure feed gas ṁ LP, high-pressure feed gas ṁ HP and the recycling feedback off-gas flow ṁ o f f. The output flow is ṁ f urn to the furnace. The pressure tank system is illustrated in Fig. 9. u x 4 Fig. 7. The Input for Adaptive Controller u and Classic Controller u2.8.6 θ θ x x 4 Fig. 8. The Estimates of Parameters for Adaptive Controller 4532
5 ThB.2 Residue Low Pressure Gas High Pressure Gas Fig. 9. Off-gas Outflow to Furnace The Diagram of the Pressure Tank System Since there is usually no high-pressure feed input, the mass balance can be represented as: dm = ṁ R ṁ o f f ṁ LP ṁ f urn (29) where M is the total mass in the tank. The output is the tank pressure P (PV). During the on-site test, we treat the low pressure flow ṁ LP as the manipulated variable (MV), the off-gas flow ṁ o f f and the residue flow ṁ R as measured disturbances (DVs). The data of outflow to the furnace ṁ f urn is not used in the controller design, so it is treated as unmeasured disturbance. We assume ideal behavior of the gas in the tank so that P = MRT m W V (3) where V and T are the temperature and volume of the tank respectively, and m W is the average molecular weight of the mixture in the tank. Main parameters in this model are given in Table II. 2) The Implementation of Adaptive Feedforward Controller: The current control system on the pressure tank plant is a feedback PI control system. We keep the current PI controller with a gain K c =.5 and an integral time constant τ i = 75 as the feedback part and add the adaptive feedforward part. Now we define: y = P (3) Z = M (32) u = ṁ LP (33) d = (ṁ o f f,ṁ R ) T (34) TABLE II PARAMETERS OF THE SYSTEM MODEL Parameter Value V.5 m 3 T 38 K m W 6 g/mol R.26 3 psi m 3 /(mol K) Then (29) can be rewritten in the form of (8) as where = uθ T µ(x,d) (35) θ = (θ,θ 2,θ 3 ) T (36) µ(x,d) = ( ˆṁ o f f, ˆṁ R,) T (37) where θ = and θ 2 =. The inventory, total mass M, can be estimated from (3) with the measurement of pressure P. Since the set point Z is a constant, we have the feedforward controller u f f = ˆθ T µ(x,d) where only θ 3 is estimated online by t ˆθ 3 = g eds (38) e = M M (39) where the adaptive gain g =.5. 3) Simulation Result: The pressure tank unit with adaptive controller is simulated in SIMULINK and compared with the true data from the on-site test. The process model built in SIMULINK is based on the mass balance (29). The model is verified by comparing the true output data and simulated output with PI controller only. Actual data of input and disturbances is used in the simulation. Two measured disturbances are the off-gas flow, and the flow of residue which are shown in Fig.. Off Gas (Mlbs/hr) Residue (Mlbs/hr) Fig.. The Measured Disturbances in the On-site Test The result of the output pressure is shown in Fig.. From the simulation, we can see that the system model is reasonably good to represent the plant. The adaptive feedback-feedforward controller is then applied to the model with actual disturbances data from on-site test. The output result is shown in Fig. 2 and compared with the actual output data of PI controller. We can see the performance of the adaptive controller is much better than current PI control due to the feedforward component. 4533
6 ThB Simulation True Data 6 PI with feedforward PI w/o feedforward 5 4 y (psi) u (Mlbs/hr) Fig.. The Output of Model and Plant Data with PI control Fig. 3. The Input of Adaptive Feedback-feedforward Control PI with feedforward PI w/o feedforward y (psi) (Mlbs/hr) θ 3 Flow to Furnace Fig. 2. The Output of Adaptive Feedback-feedforward Control Fig. 4. Parameter Estimation of θ 3 and Data of ṁ f urn The input flow rates are also shown and compared in Fig. 3. If we compare (35) and (29), we can see that the true value of θ 3 is actually the flow to the furnace ṁ f urn if the measurements of disturbances ˆṁ o f f and ˆṁ R are perfect. The value of θ 3 and ṁ f urn are compared in Fig. 4, and we can see they are very close. V. CONCLUSION The adaptive control is applied in the passivity-based inventory control scheme. Mass-energy balance dynamic models are used and certain model parameters are adapted on line, so that the dynamic changes and the effects of disturbances can be compensated by the adaptive feedforward controller, which makes it outperform the classic feedforward controller. The control strategy is applied on a chemical pressure tank system, and the plant data shows that much better performance can be obtained by adding the passivitybased adaptive controller as the feedforward component to the existing feedback control system. REFERENCES [] J. C. Willems, Dissipative dynamical-systems, Part I, Archive for Rational Mechanics and Analsis, vol. 45 (5), 972, pp [2] W. Sun, P. P. Khargonekar and D. Shim, Solution to the Positive Real Control Problem for Linear Time-Invariant Systems, IEEE Trans. Autom. Control, vol. 39 (), 994, pp [3] J. Bao and P. L. Lee, Process Control: The Passive Systems Approach, Springer, 27. [4] Alonso, A. A. and Ydstie, B. E., Process Systems, Passivity and the Second Law of Thermodynamics, Computers Chem. Engng, vol. 2, Suppl., (996), pp S9 S24. [5] B. E. Ydstie and A. A. Alonso, Process Systems and Passivity via the Clausius-Planck Inequality, Systems and Control Letters, vol. 3, 997, pp [6] C. A. Farschman, K. P. Viswanath and B. E. Ydstie, Process Systems and Inventory Control, AICHE Journal, vol. 44, 998, pp [7] M. Ruszkowski, V. Garcia-Osorio and B. E. Ydstie, Passivity Based Control of Transport Reaction Systems, AICHE Journal, vol. 5, 25, pp [8] B. E. Ydstie and Y.Jiao, Passivity-Based Control of the Float Glass Process, IEEE Control System Magazine, December, 26, pp
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